Research ArticleThe Contribution of Particle Swarm Optimization toThree-Dimensional Slope Stability Analysis

Roohollah Kalatehjari1 Ahmad Safuan A Rashid1 Nazri Ali1 and Mohsen Hajihassani2

1 Department of Geotechnics and Transportation Universiti Teknologi Malaysia 81310 Skudai Johor Malaysia2 Construction Research Alliance Universiti Teknologi Malaysia 81310 Skudai Johor Malaysia

Correspondence should be addressed to Ahmad Safuan A Rashid ahmadsafuanutmgmailcom

Received 21 February 2014 Accepted 8 April 2014 Published 1 June 2014

Academic Editors V Bhatnagar and Y Zhang

Copyright copy 2014 Roohollah Kalatehjari et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Over the last few years particle swarm optimization (PSO) has been extensively applied in various geotechnical engineeringincluding slope stability analysis However this contribution was limited to two-dimensional (2D) slope stability analysis Thispaper applied PSO in three-dimensional (3D) slope stability problem to determine the critical slip surface (CSS) of soil slopes Adetailed description of adopted PSOwas presented to provide a good basis for more contribution of this technique to the field of 3Dslope stability problems A general rotating ellipsoid shape was introduced as the specific particle for 3D slope stability analysis Adetailed sensitivity analysis was designed and performed to find the optimum values of parameters of PSO Example problems wereused to evaluate the applicability of PSO in determining the CSS of 3D slopes The first example presented a comparison betweenthe results of PSO and PLAXI-3D finite element software and the second example compared the ability of PSO to determine theCSS of 3D slopes with other optimization methods from the literature The results demonstrated the efficiency and effectiveness ofPSO in determining the CSS of 3D soil slopes

1 Introduction

Slope stability analysis is a major concern in projects relatedto man-made or natural slopes Several techniques areapplied to analyze the stability state of a slope of whichlimit equilibriummethod (LEM) is themost popular [1]Thismethod undertakes the static behavior of the slope at theverge of failure and develops equilibriums of the soil body instatic condition Consequently no stress-strain relationshipis considered and corresponding deformation within the soilbody is not studied [2] As a result the shape of each potentialslip surface which defines the lower boundary of sliding bodyhas to be assumed A numerical ratio as factor of safety (FOS)is used to determine the critical slip surface (CSS) as the leaststable slip surface among all potentials FOS compares theavailable shear strength of the soil with the existing shearstress (mobilized shear strength) on the assumed slip surfaceas follows [3]

FOS =

119878

119879

(1)

where 119878 is mobilized shear strength force (kN) and 119879 isavailable shear strength of the soil (kN) The mobilized shearstrength force is defined as

119878 = (

[119888

1015840+ (120590119899minus 119906119908) tan1206011015840]

FOS) (2)

where FOS is factor of safety 119878 is mobilized shear strengthforce (kN) 1198881015840 is cohesion of the soil in terms of effectivestress (kNm2) 1206011015840 is angle of internal friction of soil in termsof effective stress (kNm2) and 120590

119899is normal stress on the

slip surface (kNm2) and 119906119908is pore water pressure on the

slip surface (kNm2) In general the following principles arerequired to analyze the stability of a slope within LEM [2]

(1) A kinematically admissible slip surface is assumed todefine the mechanism of failure

(2) Two static principles as the assumption of plasticbehavior for soil mass and validity of Mohr-coulombfailure criterion are employed to determine the shear-ing strength along the assumed slip surface

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 973093 12 pageshttpdxdoiorg1011552014973093

2 The Scientific World Journal

(3) Equation of FOS is developed for the assumed slipsurface by dividing the available shear strength at thesurface by the required shear resistance to bring theequilibrium into limiting condition

(4) An iterative process is used to find the satisfying valueof FOS

(5) By using the steps above a search technique isemployed to find the CSS among all assumed slipsurfaces

Although all the LEMs havemutual principles they differin utilizing static equilibrium assumptions and simplifica-tions They can be considered as two-dimensional (2D) andthree-dimensional (3D) methods 2D methods simplify thegeometry of slopes by transforming the problem into anassumed 2D form Consequently some internal and externalforces are simplified or ignored in this process Such simplifi-cations in 2Dmethods may result in different outcomes formthe results of 3D methods Although the assumptions of 3Dmethods are mostly derived from the related 2D basics somenew definitions are only available in 3D methods due to plusone dimension that 3D methods have Ability to consider 3Dshapes of slip surface asymmetric and complex slopes slidingdirection and intercolumn forces are some of the privileges of3Dmethods However 3Dmethods might consider simplifyor ignore any of these aspects

Determining the CSS despite of utilized 2D or 3Dmethod needs a massive search among possible slip sur-face Searching problem is usually defined as optimizationproblem in engineering This problem is framed to findappropriate solution among the candidates by minimizing ormaximizing an objective function If more than one solutionexists among candidates of a problem it turns to globaloptimization Global optimization methods try to find theglobal solution while avoiding local solutions

Particle swarm optimization (PSO) was initially intro-duced by Kennedy and Eberhart [4] as a global optimizationtechnique PSO simulates the birds flock activities when theyrandomly search for food in their path Since PSO has beenreleased its successful application in various engineeringproblems has begun The popularity of PSO is mainly dueto its comprehensible performance as well as its simpleoperation [5] Many researchers applied PSO to solve theirproblems in the fields of structural [6ndash8] environmental [9ndash11] hydrological [12 13] and geotechnical [14ndash16] engineer-ing

Cheng et al [17] tried to determine the CSS of sevenslopes by using PSO as one of the first applications of PSO inslope stability analysis and came to the conclusion that PSOproduces appropriate and reasonable results Furthermore ina comparison with pattern method they [17] reported thatPSO is capable of finding the global minimum FOS and itsrelated CSS in different slopes Ever since PSO has beenused progressively as an effective technique to deal with theproblem of determining the CSS to name a few Cheng etal [17] Cheng et al [18] Zhao et al [19] Tian et al [20]Li et al [21] Kalatehjari et al [22 23] and W Chen and PChen [24] However the contribution of PSO was limited to2D slope stability problem In fact only a few researchers

published their results in determining the CSS in 3D slopestability problems and none of them applied PSO [25ndash30]

Based on the successful performance of PSO in 2D slopestability analysis as well as other problems of geotechnicalengineering it is believed that it can contribute well todetermining the CSS of 3D slopes This paper applies PSO in3D slope stability problem to determine theCSS of soil slopesA detailed description of adopted PSO is presented to providea good basis for more contribution of this technique to thefield of 3D slope stability problems

2 Overview of Particle Swarm Optimization

Kennedy and Eberhart [4] initialized PSO by simulatingthe behavior of a birds swarm with defined instructions forindividual behaviors as well as intercommunications Theseinstructions help in decision making process of individualswhich is based on the following items [4]

(i) experience of individual as its best results so far(ii) outlay of experience of swarmas the best result among

all individuals

Swarm intelligence as the ability of each individual touse the experience of others guides the swarm toward itsoptimumgoalThree principals of the swarmbehavior in PSOwere similar to what described by Reynolds [31]

(i) Individuals are collision-proof(ii) Individuals travel toward swarm objective(iii) Individuals travel to the center of swarm

The standard flowchart of PSO is shown in Figure 1 Thisprocess starts by randomly generating a certain number ofindividuals namely particles where each represents a possi-ble solution for the problem [4 17]The structure of a particlemay contain three sections that separately record its currentposition best position so far and velocity respectively ascoordinates of current position coordinates of best positionso far and velocity vectors in a D-dimensional space whereD starts from one [32] Consequently a 3 times D-dimensionalparticle is fitting for a particle in D-dimensional space

PSO reaches its goal if meets the termination criteriaThese criteria are set to guarantee the ending of iterativesearch process Appropriate termination criteria are neces-sary to accomplish a successful search by avoiding prematureor late convergence [33] The commonly used terminationcriteria are set as follows

(i) reaching a maximum number of iterations(ii) finding a satisfactory solution(iii) achieving a constant fitness for a certain number of

iterations

The closeness of each particle to the best possible solutionis defined by the objective function which is aimed to beminimized or maximized by PSO A fitness function relatedto the objective function is usually set to calculate fitnessvalue of each particle by assessing its current position The

The Scientific World Journal 3

Start

InitializePSO parameters

Generate first swarm

Evaluate the fitnessof all particles

Record personal bestfitness of all particles

Find global best particle

Swarm metthe termination

criteria

End

No

Yes

Update the positionof particles

Update the velocityof particles

Figure 1 Standard flowchart of PSO

velocity of particles is determined by (3) based on theirbest position and global best position in the swarm Tocontinue the search (4) updates the position of all particlesbased on their current position and the obtained velocityThrough an iterative process the improvement of fitness ofparticles continues until PSO meets the termination criteriaThe global solution is then achieved by the current positionof the best particle in the last iteration

V119899(119894)

= V119899(119894minus1)

+ 119906 (0 1205991) (119887119901

119899(119894)minus 119909119899(119894))

+ 119906 (0 1205992) (119887119892

119899(119894)minus 119909119899(119894))

(3)

119909119899(119894+1)

= 119909119899(119894)

+ V119899(119894) (4)

where V119899(119894minus1)

and V119899(119894)

are respectively the velocity of 119899thparticle in past and current iterations 119906(0 120599

1) and 119906(0 120599

2)

are the vectors of random numbers of 119899th particle uniformlydistributed respectively in [0 120599

1] and [0 120599

2] 119887119901119899(119894)

is the bestposition of 119899th particle so far 119887119892

119899(119894)is the position of the

best particle of the swarm so far and 119909119899(119894minus1)

and 119909119899(119894)

are thepositions of 119899th particle respectively in the current and thenext iterations

Initial cognitive and social parts are three componentsof velocity equation The values of 120599

1and 1205992in this equation

control the exploration and exploitation behaviours of theswarm While equal values of 2 are commonly used forthese parameters in early search greater values of 120599

1and

1205992 respectively provide faster convergence to the solution

and enhance discovering the searching space The velocityof particles may increase surprisingly by adjusting these

parameters so a limiting bound of velocity as [minusVmax Vmax]is attached to PSO as constriction coefficients [34] Shi andEberhart [35] modified the original equation of velocity toreduce the role of constriction coefficient and introduced (5)by introducing 120596 as the inertia weight of particles Later onClerc and Kennedy [36] demonstrated that inertia weightsof greater than one may cause converge problems in PSOand proposed (6) by introducing 120585 as the constant multiplierin (7) This modification prevents the swarm to explodeguarantees the mature converge and almost eliminates theneed of constriction coefficient Principally interior param-eters (inertia weight and velocity coefficient) and exteriorparameters (swarm size and topology) of PSO should becarefully adjusted to provide the best results

V119899(119894)

= 120596V119899(119894minus1)

+ 119906 (0 1205991) (119887119901

119899(119894)minus 119909119899(119894))

+ 119906 (0 1205992) (119887119892

119899(119894)minus 119909119899(119894))

(5)

V119899(119894)

= 120585 [V119899(119894minus1)

+ 119906 (0 1205991) (119887119901

119899(119894)minus 119909119899(119894))

+ 119906 (0 1205992) (119887119892

119899(119894)minus 119909119899(119894))]

(6)

120585 =

2

(1205991+ 1205991) minus 2 +

radic(1205991+ 1205991)

2

minus 4 (1205991+ 1205991)

(1205991+ 1205991) gt 4

(7)

Topology in the method of intercommunication betweenparticles controls the convergence of a swarm Topology isdivided into static and dynamic categories In static topolo-gies the number of connected neighbors to a particle isconstant throughout the optimization process However thisnumber increases by the progress of optimization process indynamic topologies to enhance the searching abilities [37]

The original PSO aided a conical static topology basedon intercommunication of all particles with the global bestparticle However Eberhart and Kennedy [38] proposedanother static topology by introducing intercommunicationbetween individuals and local best particles In this modeleach particle was connected to 119870 number of its neighbors inthe swarm array The main advantage of this method was theability of subconvergence in different regions of the searchspace Although the convergence of this method was slowerthan the conical method it was able to better escape fromlocal optima For each problem the appropriate topology canbe defined by performing sensitivity analysis on convergenceand execution time of PSO Figure 2 illustrates conical andlocal (119870 = 2) topologies for randomly generated 100 particlesin a 2D search space

The size of swarm is defined as the number of its particlesWhile a small swarm may fail to converge over a globalsolution a large swarm may have late convergence The sizeof swarm commonly varies from 20 to 50 but the optimumnumber is usually determined through sensitivity analysis onthe convergence parameter of the swarm [36]

4 The Scientific World Journal

0

1

2

3

4

5

6

7

8

9

10

0 2 4 5 6 7 8 10

ParticlesLocal connections

1 3 9

Y-a

xis

X-axis

(a)

ParticlesGlobal connectionsGlobal best

0

1

2

3

4

5

6

7

8

9

10

0 2 4 5 6 7 8 101 3 9

Y-a

xis

X-axis

(b)

Figure 2 (a) Local and (b) global topologies in 2D search space

3 Application of PSO in SlopeStability Analysis

PSO is mainly applied in stability analysis of soil slopeswithin the framework of LEM [1] This analysis involves twoconsequent steps that is calculating FOS of candidate slipsurfaces and determining the CSS among all candidates [39]PSO commonly contributes to the second step to determinethe shape and location of the CSS which are generallyunknown in soil slopes [40]

PSO can be applied in both 2D and 3D slope stabilityanalyses [32] In 2D analysis it can be employed to determinethe shape of the CSS in a predefined 2D section of the slopeDifferent shapes are possible for slip surfaces in 2D analysissuch as circular ellipse spiral and polygonal or arbitrarysurfaces [22 41 42] In contrast 3D slip surfaces such asspherical ellipsoidal and Nonuniform Rational B-Splines(NURBS) are commonly assumed in 3D analysis [23 43 4344]

Figure 3 shows flowchart of PSO to determine the CSSin slope stability analysis The optimization procedure isstarted by setting initial parameters of PSO Then a certainnumber of particles (119873) is generated in a random patternover the search space Since the improvement of swarm hasjust begun personal bests of all particles in initial swarmare identical to the particles themselves Based on the samereason the velocity of all initial particles is set to zero Aftersetting up the initial values the first particle is arranged asits corresponding slip surface This surface is qualified if it

can satisfy the conditions of the problem Otherwise it isdisqualified A predefined minimum fitness value is given todisqualify slip surfaces This value represents a predefinedmaximum FOS For a qualified surface FOS is calculated andthe corresponding fitness values of particle are assigned bythe fitness function of PSO This process is repeated for allparticles of the swarm

Current positions of particles that improved their fitnessvalues are recorded to update their personal bests whileprevious personal bests are used for other particles Theglobal best particle is defined by the greatest fitness value inthe current swarm Through an iterative process subsequentswarms are generated by updating velocities and positionsof former particles The optimization process is terminatedby meeting the termination criteria Eventually the globalbest particle of the last swarm represents the CSS of theslope

31 Coding of the Particles The structure of particles followedthe standard PSO particles involving three sections as currentposition previous best position and the velocity A rotatingellipsoid was selected as the general 3D shape of slip surfacesThis ellipsoid can rotate on119909-119910plane (0 le 120579

119909119910le 120587) to provide

various slip surfaces (Figure 4)In order to achieve the equation of rotated ellipsoid the

parametric equation of general ellipsoid was transferred intonew axes by the rotation angle 120579

119909119910 The result presents the

rotated ellipsoid in (8) It should be noted that this ellipsoid

The Scientific World Journal 5

Start

Generate N particles by random

Set Pbest = particlesSet velocities = 0

Convert the first particleto slip surface

Is the No

YesCalculate 3D FOSof the slip surface

No

Calculate fitness

Was

No

Yes

Find Pbests and gbest

Are the

YesEnd

Con

vert

the n

ext

part

icle

to sl

ip

Calculate and updatenew positions of all particles

Calculate and updatevelocity of all particles

Initialize PSO parametersSwarm size = N

qualifiedslip surface

of the particle

particlethat the last

termination criteria met

surfa

ce

Set t

he fi

tnes

sas

min

imum

Figure 3 Flowchart of PSO to determine the CSS in slope stabilityanalysis

can be easily transformed to spherical and cylindrical slip sur-face by respectively setting equal three and two semiradiuses

(cos 120579119909119910(119909 minus 119883

119888) minus sin 120579

119909119910(119910 minus 119884

119888))

2

119877

2

119909

minus

(cos 120579119909119910(119910 minus 119884

119888) + sin 120579

119909119910(119909 minus 119883

119888))

2

119877

2

119910

+

(119911 minus 119885119888)

2

119877

2

119911

= 1

(8)

where 120579119909119910

is rotation angle of the ellipsoid in 119909-119910 plane 119883119888

119884119888 and 119885

119888are coordinates of center of ellipsoid in 119909- 119910- and

119911-directions 119877119909 119877119910 and 119877

119911are semiradiuses of ellipsoid in

119909- 119910- and 119911-directions and 119909 119910 and 119911 are coordinates of anarbitrary point on the surface of ellipsoid

Based on the parameters of the rotating ellipsoid Figure 5shows schematic structure of a PSO particle The section ofcurrent position records the coordinates of center of ellipsoidits semiradiuses and its rotation angle Considering bestposition and velocity sections PSO has seven-dimensionalsearch space and twenty-one-cell particles

Rx

Ry

120579xy

X

Y

120579xy = 0

120579xy = 1205874

120579xy = 1205872

120579xy = 31205874

120579xy = 120587

Figure 4 Projection of rotating ellipsoid on 119909-119910 plane

Xc Yc Zc Rx Ry Rz 120579xy

Xcp

Y cp

Z cp

R xp

R yp

R zp

120579 p

Vxc

Vyc

Vzc

Vry

Vrx

Vrz

V120579

Current position

Best

posit

ionVelocity

Figure 5 Schematic structures of PSO particles

32 Fitness Function The quality of particles can be cal-culated by the fitness function This function is related tothe objective function and provides quantitative tracking ofimprovement of particles Consequently it makes it possibleto compare and rank particles in the swarm wheremaximumfitness shows the best particle andminimum fitness identifiesthe worst particle of the swarm PSO attempts to increase themaximum fitness of swarms during its iterations Since theobjective function of 3D slope stability analysis is equation ofFOS PSO attempts to find the CSS with the minimum FOSby maximizing the fitness value in

Fitness119899(119894)

=

1

FOS (119909119899(119894))

(9)

where Fitness119899(119894)

is fitness value of the 119899th particle in 119894thiteration and FOS(119909

119899(119894)) is FOS of 3D slip surface described

by 119909119899(119894)

33 Sensitivity Analysis on PSO Parameters The best per-formance of PSO is guaranteed by initializing appropriateparameters for it A sensitivity analysis can help to do soThe optimum values PSO parameters in 3D slope stabilityanalysis were defined by designing and performing severalindependent tests on swarm size coefficients of velocity andinertia weight of the swarm In addition the convergence

6 The Scientific World Journal

Table 1 Properties of slope in sensitivity analysis

Parameters120574 (kNm3) 119888

1015840 (kNm2) 120601

1015840 (∘)Layer 1 1978 12 17Layer 2 1764 245 20

Table 2 Results of sensitivity tests on swarm size

Test number1 2 3 4 5 6 7

Swarm size 5 15 25 35 45 55 65Total CPU time (s) 365 625 549 1726 2792 2241 10101

behavior of PSO as the average fitness of swarms wasobserved during the tests A 3D soil slope was designedwith complex geometry layers of soil and piezometric line(Figure 6) It should be noted that coding of the study wasdone by the authors in MATLAB software (Licensed byUniversiti TeknologiMalaysia)The overall shape of the slopeshows two imbalanced hills with steep sides makes it difficultto find the CSS for conventional slope stability analysesThis specific shape was selected to verify the effectiveness ofPSO in complex 3D slopes The properties of soil layers aredescribed in Table 1

The size of the swarm is defined based on the conditionof search space dimension of particles andor other specifi-cations of the problem The most common population sizesare 20 to 50 [25] However Clerc and Kennedy [36] proposeda relationship to determine the optimum value of swarm sizeas follows

119873119904= 10 + [radic119863

119904] (10)

where 119873119904is swarm size 119863

119904is dimension of the particles

and [ ] is calculator of integer part Since the dimensionof particles in the present problem is seven the proposedoptimum swarm size by this equation is 12 Consideringthe most common range of the swarm size and the resultof equation an interval of swarm sizes was prepared forsensitivity test It should be noted that the first swarm ofall tests was produced by the same random pattern and themaximum iteration number was set to 100 for all tests Table 2shows the results of testsThe phrase ldquoCPU timerdquo in this tablemeans the exact amount of time that CPU spent on each testCPU time was used to produce fair comparisons since somefactors including the operating system and available memorycan affect the overall duration of the tests

Figure 7 illustrates the convergence behavior of PSOin corresponding swarm sizes of Table 2 Three differentconvergence behaviors as good late and failure can describethese trends Swarm sizes 5 15 25 35 and 55 providedgood convergence over maximum iterations while swarmsizes 45 and 65 respectively delivered delay and failurein convergence Among all the tests the best convergencewas obtained by swarm size of 35 that provided the bestconvergence with the highest average fitness

Layer 1 interfaceWater tableLayer 2 interface

5

45

4

35

3

25

2

15

1

05

05

4 32 1

0

Z-a

xis

X-axisY-axis

54

321

0

Figure 6 Generated slope model for sensitivity analysis

1

09

08

07

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 1Test 2Test 3Test 4

Test 5Test 6Test 7

Figure 7 Convergence behavior of PSO with different swarm sizes

The next tests were performed to find the optimumvaluesof coefficients 120599

1and 120599

2of velocity equation Based on the

original coefficients of Kennedy and Eberhart [4] and themodified coefficients of Clerc and Kennedy [36] a seriesof combinations were established as shown in Table 3 Alltests were performed by the same initial swarm with the sizeof 35 (previously obtained as optimum) and the maximumiterations of 100

The results can be presented in two separated groupsincluding unequal and equal coefficients Figure 8 illustratethe results of the tests The first group failed to converge overthe maximum iteration period but the second group showeddifferent performances Overall the best convergence and thegreatest average fitness belonged to equal coefficients of 175that makes it the optimum coefficient of velocity equation

The Scientific World Journal 7

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 1Test 2Test 3

Test 4Test 5Test 6

(a)

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 7Test 8Test 9

Test 10Test 11Test 12

(b)

Figure 8 Results of sensitivity tests on (a) unequal and (b) equal coefficients

Table 3 Combinations of velocity equation coefficients in differenttests

Test number Relationship 1205991

1205992

1205991+ 1205992

1 1205991= 025120599

20800 3200 4

2 1205991= 050120599

21333 2667 4

3 1205991= 075120599

21714 2286 4

4 1205992= 025120599

13200 0800 4

5 1205992= 050120599

12667 1333 4

6 1205992= 075120599

12286 1714 4

7 1205991= 1205992

2500 2500 58 120599

1= 1205992

2000 2000 49 120599

1= 1205992

1750 1750 3510 120599

1= 1205992

1500 1500 311 120599

1= 1205992

1000 1000 212 120599

1= 1205992

0500 0500 1

The last sensitivity tests were performed to find theoptimum inertia weight (120596) of velocity equation The sameinitial swarm with size of 35 and equal coefficients of velocityequation as 175 (previously defined as optimum values) wereapplied for all the tests Five tests with inertia weights of 0025 05 075 and 1 respectively were performed based onthe proposed values of Shi and Eberhart [35] and Clerc andKennedy [36] Figure 9 shows the convergence behavior ofPSO in the tests The results showed successful convergencein all tests except test 3 It should be noted that test 5 wasidentical with the original PSO where no inertia weightwas present in velocity equation Immature convergencehas occurred for tests 1 and 2 Although fast convergenceappears an advantage at first it is a sign of trapping aswarm in local solutions Test 3 failed to converge test 4 hadinstable convergence and test 5 failed to improve its averagefitness over themaximum iterations Consequently it was notpossible to introduce an optimum inertia weight to guarantee

0 10 20 30 40 50 60 70 80 90 100

Iteration

Test 1Test 2Test 3

Test 4Test 5

10908070605040302010

Aver

age fi

tnes

s

Figure 9 Results of sensitivity tests on inertia weight

the convergence of PSO and improvement of average fitnessover the maximum iteration number simultaneously

A dynamic inertia weight was utilized by the presentstudy to overcome the convergence problem of PSOThe pro-posed strategy started with the most anticonvergence inertiaweight (05) continued with the normal convergence inertiaweight (075) and ended upwith themost stable convergence(025) The switching levels of inertia weights were defined asone-third and two-thirds of maximum iterations Figure 10shows the results of sensitivity tests on dynamic inertiaweightwith different maximum iterations from 50 to 300 by stepsof 50 All tests performed well to converge and improve theaverage fitness over their maximum iterations so dynamicinertia weight was adopted for PSO

4 Example Problems

Two example problems were analyzed to verify the perfor-mance of PSO in determining the CSS The properties of theslope materials are shown in Table 4 Example problem 1 wasperformed to verify the performance of PSO in determining

8 The Scientific World Journal

06

05

04

03

02

01

00 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Aver

age fi

tnes

s

Iteration

Dynamic test 1Dynamic test 2Dynamic test 3

Dynamic test 4Dynamic test 5Dynamic test 6

Figure 10 Results of sensitivity tests on dynamic inertia weight

Table 4 Properties of slopes in example problems

Properties 1198881015840 (kNm2) 120601

1015840 (degree) 120574 (kNm3) 120592 119864 (kNm2)Problem 1 15 20 17 03 1119864 + 6

Problem 2 10 10 18 mdash mdash

the CSS in comparison with PLAXIS-3D finite element soft-ware (License byUniversiti TeknologiMalaysia) Alkasawnehet al [44] applied different search techniques to determinethe CSS in 2D slope stability analysis Figure 11 illustrates thegeometry of the slope A 3D model was developed based onthis 2D section inwhich the third dimensionwas extended by100 meters Figure 12 shows the generated 3D models of theslope by the present study and PLAXIS-3D In both methodscylindrical slip surface was employed to determine the CSS ofthe slope

PSO improved average and best fitness of the swarmas shown in Figure 13 Figure 14 shows the minimum FOSversus iterations PSO provided continuous reduction of FOSto find the CSS The present study obtained FOS of 178versus theminimumFOS of 177 of the PLAXIS-3D Figure 15shows the CSS obtained by the present study and the resultof PLAXIS-3D The present study and PLAXIS-3D obtainedsimilar FOS for the CSS with a small difference of 03 Thisresult demonstrates the ability of PSO to determine the CSSwith the minimum FOS in 3D slope stability analysis

Example problem 2 was performed to verify the abilityof PSO to determine the CSS with general ellipsoid shape ina comparison with previous studies from the literature Thisexample was initially analyzed by Yamagami et al [45] andwas reanalyzed by Yamagami and Jiang [25] Yamagami et al[45] used random generation of surfaces to determine theCSS of this slope while Yamagami and Jiang [25] employed acombination of dynamic programming and random numbergeneration to do so It should be noted that the same equationof FOS as previous studies was used to make fair comparisonof the results The example involved a homogeneous slopewith gradient of 2 1 subjected to a square load of 50 kPa onthe top The uniform load was applied on a square surface

30

20

10

00 20 40 60 80

Z(m

)

Y (m)

Figure 11 Geometry of 2D section of example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axis

Y-axis

(a)Z-a

xis

X-axisY-axis

(b)

Figure 12 Generated 3D models of example problem 1 by (a) thepresent study and (b) PLAXIS-3D

with 8 meters sides at the top center of the slope Figure 16illustrates the geometry of example problem 1

Figure 17 shows the generated 3D model by the presentstudy for example problem 2 Figure 18 plots the process ofPSO to improve fitness of the swarmThe trend of average fit-ness value of the swarm experienced some instability duringthe process The main reason of this behavior is the presenceof disqualified slip surfaces in the swarm that dramaticallydecreases the average fitness valueThese surfaces were rarelypresented in the previous example due to adopted simplercylindrical shape compared with more complicated ellipsoidshape in this example The trend of minimum FOS versusiterations is shown in Figure 19 Continuous decrement ofFOS by PSO leads to determining the CSS of the slope

In spite of similar equation of FOS the present studyfound the CSS with a smaller FOS than other methods whichis the best result so far The minimum FOS obtained by PSOwas 095 compared with 114 and 103 of random generationof surfaces [45] and DP with RNG [25] respectively Thisresult demonstrates the ability of PSO to accurately determinethe ellipsoid CSS in 3D slope stability analysis Figure 20

The Scientific World Journal 9

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80

Iteration

Fitn

ess v

alue

Average fitnessMaximum fitness

Figure 13 Fitness values versus iterations in example problem 1

215

21

205

2

195

19

1850 10 20 30 40 50 60 70 80

Iteration

Min

imum

fact

or o

f saf

ety

Figure 14 Minimum FOS versus iterations in example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axisY-axis

(a)

Z-a

xis

X-axis

Y-axis

(b)

Figure 15 (a) The CSS obtained by the present study and (b) exag-gerated displacement vectors of PLAXIS-3D in example problem 1

40

24

20

9

4

0

0 4 14 22 25

x(m

)z

(m)

y (m)

0 4 14 22 25

y (m)

12

(b)

(a)

Figure 16 (a) Half-plan view and (b) central cross-section of slopein example problem 2

10864200

510

1520

2530 35

40 05

1015

2025

Z-a

xis

X-axis Y-axis

Figure 17 Generated 3D models of example problem 2 by thepresent study

illustrates the CSS obtained by the present study in exampleproblem 2

5 Conclusion

Determining the critical slip surface of a soil slope is atraditional problem in geotechnical engineering which is stillchallenging for researchers This problem needs a massivesearching process Although classical searching methodswork for relatively simple problems they are surrounded bylocal minima Moreover their processes become particularlyslow by increasing the number of possible solutions Toeliminate these limitations PSO has been applied in slopestability analysis based on its successful results in advancedengineering problems However this contribution was lim-ited to 2D slope stability analysis This paper applied PSO in

10 The Scientific World Journal

12

1

08

06

04

02

00 10 20 30 40 50 60 70 80 90 100

Fitn

ess v

alue

Iteration

Average fitnessMaximum fitness

Figure 18 Fitness values versus iterations in example problem 2

3

25

35

2

15

1

05

00 10 20 30 40 50 60 70 80 90 100

Min

imum

fac

tor o

f saf

ety

Iteration

Figure 19 Minimum FOS versus iterations in example problem 2

10

8

6

4

2

0051015

2025

303540 0 5 10 15 20 25

Z-a

xis

X-axis

Y-axis

Figure 20 The CSS obtained by the present study in exampleproblem 2

3D slope stability problem to determine theCSS of soil slopesA detailed description of adopted PSO was presented toprovide a good basis for more contribution of this techniqueto the field of 3D slope stability problems

The application of PSO in slope stability analysis wasdescribed by presenting a general flowchart A general rotat-ing ellipsoid shape was introduced as the specific particle for3D slope stability analysis In order to find the optimum val-ues of parameters of PSO a sensitivity analysis was designedand performed The related codes were prepared by theauthors in MATLAB A 3D model with complex geometry

soil layers and piezometric line was used in the analysisThis analysis included three steps to find the optimum swarmsize coefficients and inertia weight of the velocity equationrespectively Moreover the performance of PSO to convergeover a global optimum solution was verified during thetests Based on the obtained values of parameters PSO wasprepared for 3D slope stability analysis

The applicability of PSO in determining the CSS of 3Dslopes was evaluated by analyzing two example problemsThe first example presented a comparison between the resultsof PSO and PLAXI-3D finite element software The secondexample compared the ability of PSO to determine the CSSof 3D slopes with other optimization methods from theliterature Both of the example problems demonstrated theefficiency and effectiveness of PSO in determining the CSS of3D soil slopes Based on the results it is believed that PSO ishighly capable of contributing to the field of 3D slope stabilityanalysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The presented study was undertaken with support fromresearch grant The authors are pleased to acknowledgethe Ministry of Education of Malaysia for the PrototypeResearch Grant Scheme (RJ13000078224L620) and Univer-siti Teknologi Malaysia for providing the research funds

References

[1] R Kalatehjari and N Ali ldquoA review of three-dimensional slopestability analyses based on limit equilibriummethodrdquo ElectronicJournal of Geotechnical Engineering vol 18 pp 119ndash134 2013

[2] D G Fredlund ldquoAnalytical methods for slope stability analysisrdquoin Proceedings of the 4th International Symposium on LandslidesState-of-the-Art pp 229ndash250 Toronto Canada September1984

[3] A W Bishop ldquoThe use of the slip circle in the stability analysisof earth sloperdquo Geotechnique vol 5 no 1 pp 7ndash17 1955

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[5] Y Zhang P Agarwal V Bhatnagar S Balochian and J YanldquoSwarm intelligence and its applicationsrdquo The Scientific WorldJournal vol 2013 Article ID 528069 3 pages 2013

[6] G Poitras G Lefrancois and G Cormier ldquoOptimization ofsteel floor systems using particle swarm optimizationrdquo Journalof Constructional Steel Research vol 67 no 8 pp 1225ndash12312011

[7] S Gholizadeh and E Salajegheh ldquoOptimal design of structuressubjected to time history loading by swarm intelligence and anadvancedmetamodelrdquoComputerMethods in AppliedMechanicsand Engineering vol 198 no 37ndash40 pp 2936ndash2949 2009

[8] S S Gilan H B Jovein and A A Ramezanianpour ldquoHybridsupport vector regressionmdashparticle swarm optimization for

The Scientific World Journal 11

prediction of compressive strength and RCPT of concretescontaining metakaolinrdquo Construction and Building Materialsvol 34 pp 321ndash329 2012

[9] L Yunkai T Yingjie O Zhiyun et al ldquoAnalysis of soil ero-sion characteristics in small watersheds with particle swarmoptimization support vector machine and artificial neuronalnetworksrdquoEnvironmental Earth Sciences vol 60 no 7 pp 1559ndash1568 2010

[10] S Wan ldquoEntropy-based particle swarm optimization withclustering analysis on landslide susceptibility mappingrdquo Envi-ronmental Earth Sciences vol 68 no 5 pp 1349ndash1366 2013

[11] M R Nikoo R Kerachian A Karimi A A Azadnia andK Jafarzadegan ldquoOptimal water and waste load allocationin reservoir-river systems a case studyrdquo Environmental EarthSciences vol 71 no 9 pp 4127ndash4142 2014

[12] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for hourly rainfall-runoff modeling in Bedup Basin Malaysiardquo International Jour-nal of Civil amp Environmental Engineering vol 9 no 10 pp 9ndash182010

[13] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for modeling runoffrdquoInternational Journal of Environmental Science and Technologyvol 7 no 1 pp 67ndash78 2010

[14] J S Yazdi F Kalantary and H S Yazdi ldquoCalibration ofsoil model parameters using particle swarm optimizationrdquoInternational Journal of Geomechanics vol 12 no 3 pp 229ndash238 2012

[15] T V Bharat P V Sivapullaiah and M M Allam ldquoSwarmintelligence-based solver for parameter estimation of laboratorythrough-diffusion transport of contaminantsrdquo Computers andGeotechnics vol 36 no 6 pp 984ndash992 2009

[16] B R Chen and X T Feng ldquoCSV-PSO and its application ingeotechnical engineeringrdquo in Swarm Intelligence Focus on AntandParticle SwarmOptimization F T S Chan andMKTiwariEds chapter 15 pp 263ndash288 2007

[17] Y M Cheng L Li and S C Chi ldquoPerformance studies on sixheuristic global optimization methods in the location of criticalslip surfacerdquoComputers and Geotechnics vol 34 no 6 pp 462ndash484 2007

[18] Y M Cheng L Li Y J Sun and S K Au ldquoA coupled particleswarm and harmony search optimization algorithm for difficultgeotechnical problemsrdquo Structural and Multidisciplinary Opti-mization vol 45 no 4 pp 489ndash501 2012

[19] H B Zhao Z S Zou and Z L Ru ldquoChaotic particle swarmoptimization for non-circular critical slip surface identificationin slope stability analysisrdquo in Proceedings of the InternationalYoung Scholarsrsquo Symposium on Rock Mechanics Boundaries ofRock Mechanics Recent Advances and Challenges for the 21stCentury pp 585ndash588 Beijing China May 2008

[20] D Tian L Xu and S Wang ldquoThe application of particle swarmoptimization on the search of critical slip surfacerdquo in Proceed-ings of the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[21] L Li G Yu X Chu and S Lu ldquoThe harmony search algo-rithm in combination with particle swarm optimization andits application in the slope stability analysisrdquo in Proceedings ofthe International Conference on Computational Intelligence andSecurity (CIS rsquo09) vol 2 pp 133ndash136 Beijing China December2009

[22] R Kalatehjari N Ali M Kholghifard and M HajihassanildquoThe effects of method of generating circular slip surfaceson determining the critical slip surface by particle swarmoptimizationrdquo Arabian Journal of Geosciences vol 7 no 4 pp1529ndash1539 2014

[23] R Kalatehjari N Ali M Hajihassani and M K Fard ldquoTheapplication of particle swarm optimization in slope stabilityanalysis of homogeneous soil slopesrdquo International Review onModelling and Simulations vol 5 no 1 pp 458ndash465 2012

[24] W Chen and P Chen ldquoPSOslope a stand-alone windowsapplication for graphic analysis of slope stabilityrdquo in Advancesin Swarm Optimization Y Tan Y Shi Y Chai and G WangEds vol 6728 of Lecture Notes in Computer Science pp 56ndash632011

[25] T Yamagami and J C Jiang ldquoA search for the critical slipsurface in three-dimensional slope stability analysisrdquo Soils andFoundations vol 37 no 3 pp 1ndash16 1997

[26] J C Jiang T Yamagami and R Baker ldquoThree-dimensionalslope stability analysis based on nonlinear failure enveloperdquoChinese Journal of Rock Mechanics and Engineering vol 22 no6 pp 1017ndash1023 2003

[27] X J E Mowen ldquoA simple Monte Carlo method for locating thethree-dimensional critical slip surface of a sloperdquoActaGeologicaSinica vol 78 no 6 pp 1258ndash1266 2004

[28] X J E Mowen W Zengfu L Xiangyu and X Bo ldquoThree-dimensional critical slip surface locating and slope stabilityassessment for lava lobe of Unzen Volcanordquo Journal of RockMechanics and Geotechnical Engineering vol 3 no 1 pp 82ndash892011

[29] Y M Cheng H T Liu W B Wei and S K Au ldquoLocationof critical three-dimensional non-spherical failure surface byNURBS functions and ellipsoid with applications to highwayslopesrdquo Computers and Geotechnics vol 32 no 6 pp 387ndash3992005

[30] M Toufigh A Ahangarasr and A Ouria ldquoUsing non-linearprogramming techniques in determination of the most prob-able slip surface in 3D slopesrdquo World Academy of ScienceEngineering and Technology vol 17 pp 30ndash35 2006

[31] C W Reynolds ldquoFlocks herbs and schools a distributedbehavorial modelrdquo Computer Graphics vol 21 no 4 pp 25ndash341987

[32] R Kalatehjari An improvised three-dimensional slope stabilityanalysis based on limit equilibrium method by using particleswarm optimization [PhD dissertation] Faculty of Civil Engi-neering Universiti Teknologi Malaysia 2013

[33] A P Engelbrecht Computational intelligence an introductionJohn Wiley amp Sons 2007

[34] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tion an overviewrdquo Swarm Intelligence vol 1 no 1 pp 33ndash572007

[35] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 Piscataway NJ USAMay 1998

[36] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[37] R Mendes P Cortez M Rocha and J Neves ldquoParticle swarmsfor feedforward neural network trainingrdquo in Proceedings of theInternational Joint Conference on Neural Networks (IJCNN rsquo02)pp 1895ndash1899 Honolulu Hawaii USA May 2002

12 The Scientific World Journal

[38] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science pp 39ndash43 IEEENagoya Japan October 1995

[39] R Baker ldquoDetermination of the critical slip surface in slopestability computationsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 4 pp 333ndash3591980

[40] H P J Bolton G Heymann and A A Groenwold ldquoGlobalsearch for critical failure surface in slope stability analysisrdquoEngineering Optimization vol 35 no 1 pp 51ndash65 2003

[41] Y Chen X Wei and Y Li ldquoLocating non-circular critical slipsurfaces by particle swarm optimization algorithmrdquo ChineseJournal of Rock Mechanics and Engineering vol 25 no 7 pp1443ndash1449 2006

[42] L Li S-C Chi R-M Zheng G Lin and X-S Chu ldquoMixedsearch algorithmof non-circular critical slip surface of soil sloprdquoChina Journal of Highway and Transport vol 20 no 6 pp 1ndash62007

[43] L Li S-C Chi andY-M Zheng ldquoThree-dimensional slope sta-bility analysis based on ellipsoidal sliding body and simplifiedJANBU methodrdquo Rock and Soil Mechanics vol 29 no 9 pp2439ndash2445 2008

[44] W Alkasawneh A I H Malkawi J H Nusairat and NAlbataineh ldquoA comparative study of various commerciallyavailable programs in slope stability analysisrdquo Computers andGeotechnics vol 35 no 3 pp 428ndash435 2008

[45] T Yamagami K Kojima and M Taki ldquoA search for the three-dimensional critical slip surface with random generation ofsurfacerdquo in Proceedings of the 36th Symposium on Slope StabilityAnalyses and Stabilizing Construction Methods pp 27ndash34 1991

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

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Applied Computational Intelligence and Soft Computing

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Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 The Scientific World Journal

(3) Equation of FOS is developed for the assumed slipsurface by dividing the available shear strength at thesurface by the required shear resistance to bring theequilibrium into limiting condition

(4) An iterative process is used to find the satisfying valueof FOS

(5) By using the steps above a search technique isemployed to find the CSS among all assumed slipsurfaces

Although all the LEMs havemutual principles they differin utilizing static equilibrium assumptions and simplifica-tions They can be considered as two-dimensional (2D) andthree-dimensional (3D) methods 2D methods simplify thegeometry of slopes by transforming the problem into anassumed 2D form Consequently some internal and externalforces are simplified or ignored in this process Such simplifi-cations in 2Dmethods may result in different outcomes formthe results of 3D methods Although the assumptions of 3Dmethods are mostly derived from the related 2D basics somenew definitions are only available in 3D methods due to plusone dimension that 3D methods have Ability to consider 3Dshapes of slip surface asymmetric and complex slopes slidingdirection and intercolumn forces are some of the privileges of3Dmethods However 3Dmethods might consider simplifyor ignore any of these aspects

Determining the CSS despite of utilized 2D or 3Dmethod needs a massive search among possible slip sur-face Searching problem is usually defined as optimizationproblem in engineering This problem is framed to findappropriate solution among the candidates by minimizing ormaximizing an objective function If more than one solutionexists among candidates of a problem it turns to globaloptimization Global optimization methods try to find theglobal solution while avoiding local solutions

Particle swarm optimization (PSO) was initially intro-duced by Kennedy and Eberhart [4] as a global optimizationtechnique PSO simulates the birds flock activities when theyrandomly search for food in their path Since PSO has beenreleased its successful application in various engineeringproblems has begun The popularity of PSO is mainly dueto its comprehensible performance as well as its simpleoperation [5] Many researchers applied PSO to solve theirproblems in the fields of structural [6ndash8] environmental [9ndash11] hydrological [12 13] and geotechnical [14ndash16] engineer-ing

Cheng et al [17] tried to determine the CSS of sevenslopes by using PSO as one of the first applications of PSO inslope stability analysis and came to the conclusion that PSOproduces appropriate and reasonable results Furthermore ina comparison with pattern method they [17] reported thatPSO is capable of finding the global minimum FOS and itsrelated CSS in different slopes Ever since PSO has beenused progressively as an effective technique to deal with theproblem of determining the CSS to name a few Cheng etal [17] Cheng et al [18] Zhao et al [19] Tian et al [20]Li et al [21] Kalatehjari et al [22 23] and W Chen and PChen [24] However the contribution of PSO was limited to2D slope stability problem In fact only a few researchers

published their results in determining the CSS in 3D slopestability problems and none of them applied PSO [25ndash30]

Based on the successful performance of PSO in 2D slopestability analysis as well as other problems of geotechnicalengineering it is believed that it can contribute well todetermining the CSS of 3D slopes This paper applies PSO in3D slope stability problem to determine theCSS of soil slopesA detailed description of adopted PSO is presented to providea good basis for more contribution of this technique to thefield of 3D slope stability problems

2 Overview of Particle Swarm Optimization

Kennedy and Eberhart [4] initialized PSO by simulatingthe behavior of a birds swarm with defined instructions forindividual behaviors as well as intercommunications Theseinstructions help in decision making process of individualswhich is based on the following items [4]

(i) experience of individual as its best results so far(ii) outlay of experience of swarmas the best result among

all individuals

Swarm intelligence as the ability of each individual touse the experience of others guides the swarm toward itsoptimumgoalThree principals of the swarmbehavior in PSOwere similar to what described by Reynolds [31]

(i) Individuals are collision-proof(ii) Individuals travel toward swarm objective(iii) Individuals travel to the center of swarm

The standard flowchart of PSO is shown in Figure 1 Thisprocess starts by randomly generating a certain number ofindividuals namely particles where each represents a possi-ble solution for the problem [4 17]The structure of a particlemay contain three sections that separately record its currentposition best position so far and velocity respectively ascoordinates of current position coordinates of best positionso far and velocity vectors in a D-dimensional space whereD starts from one [32] Consequently a 3 times D-dimensionalparticle is fitting for a particle in D-dimensional space

PSO reaches its goal if meets the termination criteriaThese criteria are set to guarantee the ending of iterativesearch process Appropriate termination criteria are neces-sary to accomplish a successful search by avoiding prematureor late convergence [33] The commonly used terminationcriteria are set as follows

(i) reaching a maximum number of iterations(ii) finding a satisfactory solution(iii) achieving a constant fitness for a certain number of

iterations

The closeness of each particle to the best possible solutionis defined by the objective function which is aimed to beminimized or maximized by PSO A fitness function relatedto the objective function is usually set to calculate fitnessvalue of each particle by assessing its current position The

The Scientific World Journal 3

Start

InitializePSO parameters

Generate first swarm

Evaluate the fitnessof all particles

Record personal bestfitness of all particles

Find global best particle

Swarm metthe termination

criteria

End

No

Yes

Update the positionof particles

Update the velocityof particles

Figure 1 Standard flowchart of PSO

velocity of particles is determined by (3) based on theirbest position and global best position in the swarm Tocontinue the search (4) updates the position of all particlesbased on their current position and the obtained velocityThrough an iterative process the improvement of fitness ofparticles continues until PSO meets the termination criteriaThe global solution is then achieved by the current positionof the best particle in the last iteration

V119899(119894)

= V119899(119894minus1)

+ 119906 (0 1205991) (119887119901

119899(119894)minus 119909119899(119894))

+ 119906 (0 1205992) (119887119892

119899(119894)minus 119909119899(119894))

(3)

119909119899(119894+1)

= 119909119899(119894)

+ V119899(119894) (4)

where V119899(119894minus1)

and V119899(119894)

are respectively the velocity of 119899thparticle in past and current iterations 119906(0 120599

1) and 119906(0 120599

2)

are the vectors of random numbers of 119899th particle uniformlydistributed respectively in [0 120599

1] and [0 120599

2] 119887119901119899(119894)

is the bestposition of 119899th particle so far 119887119892

119899(119894)is the position of the

best particle of the swarm so far and 119909119899(119894minus1)

and 119909119899(119894)

are thepositions of 119899th particle respectively in the current and thenext iterations

Initial cognitive and social parts are three componentsof velocity equation The values of 120599

1and 1205992in this equation

control the exploration and exploitation behaviours of theswarm While equal values of 2 are commonly used forthese parameters in early search greater values of 120599

1and

1205992 respectively provide faster convergence to the solution

and enhance discovering the searching space The velocityof particles may increase surprisingly by adjusting these

parameters so a limiting bound of velocity as [minusVmax Vmax]is attached to PSO as constriction coefficients [34] Shi andEberhart [35] modified the original equation of velocity toreduce the role of constriction coefficient and introduced (5)by introducing 120596 as the inertia weight of particles Later onClerc and Kennedy [36] demonstrated that inertia weightsof greater than one may cause converge problems in PSOand proposed (6) by introducing 120585 as the constant multiplierin (7) This modification prevents the swarm to explodeguarantees the mature converge and almost eliminates theneed of constriction coefficient Principally interior param-eters (inertia weight and velocity coefficient) and exteriorparameters (swarm size and topology) of PSO should becarefully adjusted to provide the best results

V119899(119894)

= 120596V119899(119894minus1)

+ 119906 (0 1205991) (119887119901

119899(119894)minus 119909119899(119894))

+ 119906 (0 1205992) (119887119892

119899(119894)minus 119909119899(119894))

(5)

V119899(119894)

= 120585 [V119899(119894minus1)

+ 119906 (0 1205991) (119887119901

119899(119894)minus 119909119899(119894))

+ 119906 (0 1205992) (119887119892

119899(119894)minus 119909119899(119894))]

(6)

120585 =

2

(1205991+ 1205991) minus 2 +

radic(1205991+ 1205991)

2

minus 4 (1205991+ 1205991)

(1205991+ 1205991) gt 4

(7)

Topology in the method of intercommunication betweenparticles controls the convergence of a swarm Topology isdivided into static and dynamic categories In static topolo-gies the number of connected neighbors to a particle isconstant throughout the optimization process However thisnumber increases by the progress of optimization process indynamic topologies to enhance the searching abilities [37]

The original PSO aided a conical static topology basedon intercommunication of all particles with the global bestparticle However Eberhart and Kennedy [38] proposedanother static topology by introducing intercommunicationbetween individuals and local best particles In this modeleach particle was connected to 119870 number of its neighbors inthe swarm array The main advantage of this method was theability of subconvergence in different regions of the searchspace Although the convergence of this method was slowerthan the conical method it was able to better escape fromlocal optima For each problem the appropriate topology canbe defined by performing sensitivity analysis on convergenceand execution time of PSO Figure 2 illustrates conical andlocal (119870 = 2) topologies for randomly generated 100 particlesin a 2D search space

The size of swarm is defined as the number of its particlesWhile a small swarm may fail to converge over a globalsolution a large swarm may have late convergence The sizeof swarm commonly varies from 20 to 50 but the optimumnumber is usually determined through sensitivity analysis onthe convergence parameter of the swarm [36]

4 The Scientific World Journal

0

1

2

3

4

5

6

7

8

9

10

0 2 4 5 6 7 8 10

ParticlesLocal connections

1 3 9

Y-a

xis

X-axis

(a)

ParticlesGlobal connectionsGlobal best

0

1

2

3

4

5

6

7

8

9

10

0 2 4 5 6 7 8 101 3 9

Y-a

xis

X-axis

(b)

Figure 2 (a) Local and (b) global topologies in 2D search space

3 Application of PSO in SlopeStability Analysis

PSO is mainly applied in stability analysis of soil slopeswithin the framework of LEM [1] This analysis involves twoconsequent steps that is calculating FOS of candidate slipsurfaces and determining the CSS among all candidates [39]PSO commonly contributes to the second step to determinethe shape and location of the CSS which are generallyunknown in soil slopes [40]

PSO can be applied in both 2D and 3D slope stabilityanalyses [32] In 2D analysis it can be employed to determinethe shape of the CSS in a predefined 2D section of the slopeDifferent shapes are possible for slip surfaces in 2D analysissuch as circular ellipse spiral and polygonal or arbitrarysurfaces [22 41 42] In contrast 3D slip surfaces such asspherical ellipsoidal and Nonuniform Rational B-Splines(NURBS) are commonly assumed in 3D analysis [23 43 4344]

Figure 3 shows flowchart of PSO to determine the CSSin slope stability analysis The optimization procedure isstarted by setting initial parameters of PSO Then a certainnumber of particles (119873) is generated in a random patternover the search space Since the improvement of swarm hasjust begun personal bests of all particles in initial swarmare identical to the particles themselves Based on the samereason the velocity of all initial particles is set to zero Aftersetting up the initial values the first particle is arranged asits corresponding slip surface This surface is qualified if it

can satisfy the conditions of the problem Otherwise it isdisqualified A predefined minimum fitness value is given todisqualify slip surfaces This value represents a predefinedmaximum FOS For a qualified surface FOS is calculated andthe corresponding fitness values of particle are assigned bythe fitness function of PSO This process is repeated for allparticles of the swarm

Current positions of particles that improved their fitnessvalues are recorded to update their personal bests whileprevious personal bests are used for other particles Theglobal best particle is defined by the greatest fitness value inthe current swarm Through an iterative process subsequentswarms are generated by updating velocities and positionsof former particles The optimization process is terminatedby meeting the termination criteria Eventually the globalbest particle of the last swarm represents the CSS of theslope

31 Coding of the Particles The structure of particles followedthe standard PSO particles involving three sections as currentposition previous best position and the velocity A rotatingellipsoid was selected as the general 3D shape of slip surfacesThis ellipsoid can rotate on119909-119910plane (0 le 120579

119909119910le 120587) to provide

various slip surfaces (Figure 4)In order to achieve the equation of rotated ellipsoid the

parametric equation of general ellipsoid was transferred intonew axes by the rotation angle 120579

119909119910 The result presents the

rotated ellipsoid in (8) It should be noted that this ellipsoid

The Scientific World Journal 5

Start

Generate N particles by random

Set Pbest = particlesSet velocities = 0

Convert the first particleto slip surface

Is the No

YesCalculate 3D FOSof the slip surface

No

Calculate fitness

Was

No

Yes

Find Pbests and gbest

Are the

YesEnd

Con

vert

the n

ext

part

icle

to sl

ip

Calculate and updatenew positions of all particles

Calculate and updatevelocity of all particles

Initialize PSO parametersSwarm size = N

qualifiedslip surface

of the particle

particlethat the last

termination criteria met

surfa

ce

Set t

he fi

tnes

sas

min

imum

Figure 3 Flowchart of PSO to determine the CSS in slope stabilityanalysis

can be easily transformed to spherical and cylindrical slip sur-face by respectively setting equal three and two semiradiuses

(cos 120579119909119910(119909 minus 119883

119888) minus sin 120579

119909119910(119910 minus 119884

119888))

2

119877

2

119909

minus

(cos 120579119909119910(119910 minus 119884

119888) + sin 120579

119909119910(119909 minus 119883

119888))

2

119877

2

119910

+

(119911 minus 119885119888)

2

119877

2

119911

= 1

(8)

where 120579119909119910

is rotation angle of the ellipsoid in 119909-119910 plane 119883119888

119884119888 and 119885

119888are coordinates of center of ellipsoid in 119909- 119910- and

119911-directions 119877119909 119877119910 and 119877

119911are semiradiuses of ellipsoid in

119909- 119910- and 119911-directions and 119909 119910 and 119911 are coordinates of anarbitrary point on the surface of ellipsoid

Based on the parameters of the rotating ellipsoid Figure 5shows schematic structure of a PSO particle The section ofcurrent position records the coordinates of center of ellipsoidits semiradiuses and its rotation angle Considering bestposition and velocity sections PSO has seven-dimensionalsearch space and twenty-one-cell particles

Rx

Ry

120579xy

X

Y

120579xy = 0

120579xy = 1205874

120579xy = 1205872

120579xy = 31205874

120579xy = 120587

Figure 4 Projection of rotating ellipsoid on 119909-119910 plane

Xc Yc Zc Rx Ry Rz 120579xy

Xcp

Y cp

Z cp

R xp

R yp

R zp

120579 p

Vxc

Vyc

Vzc

Vry

Vrx

Vrz

V120579

Current position

Best

posit

ionVelocity

Figure 5 Schematic structures of PSO particles

32 Fitness Function The quality of particles can be cal-culated by the fitness function This function is related tothe objective function and provides quantitative tracking ofimprovement of particles Consequently it makes it possibleto compare and rank particles in the swarm wheremaximumfitness shows the best particle andminimum fitness identifiesthe worst particle of the swarm PSO attempts to increase themaximum fitness of swarms during its iterations Since theobjective function of 3D slope stability analysis is equation ofFOS PSO attempts to find the CSS with the minimum FOSby maximizing the fitness value in

Fitness119899(119894)

=

1

FOS (119909119899(119894))

(9)

where Fitness119899(119894)

is fitness value of the 119899th particle in 119894thiteration and FOS(119909

119899(119894)) is FOS of 3D slip surface described

by 119909119899(119894)

33 Sensitivity Analysis on PSO Parameters The best per-formance of PSO is guaranteed by initializing appropriateparameters for it A sensitivity analysis can help to do soThe optimum values PSO parameters in 3D slope stabilityanalysis were defined by designing and performing severalindependent tests on swarm size coefficients of velocity andinertia weight of the swarm In addition the convergence

6 The Scientific World Journal

Table 1 Properties of slope in sensitivity analysis

Parameters120574 (kNm3) 119888

1015840 (kNm2) 120601

1015840 (∘)Layer 1 1978 12 17Layer 2 1764 245 20

Table 2 Results of sensitivity tests on swarm size

Test number1 2 3 4 5 6 7

Swarm size 5 15 25 35 45 55 65Total CPU time (s) 365 625 549 1726 2792 2241 10101

behavior of PSO as the average fitness of swarms wasobserved during the tests A 3D soil slope was designedwith complex geometry layers of soil and piezometric line(Figure 6) It should be noted that coding of the study wasdone by the authors in MATLAB software (Licensed byUniversiti TeknologiMalaysia)The overall shape of the slopeshows two imbalanced hills with steep sides makes it difficultto find the CSS for conventional slope stability analysesThis specific shape was selected to verify the effectiveness ofPSO in complex 3D slopes The properties of soil layers aredescribed in Table 1

The size of the swarm is defined based on the conditionof search space dimension of particles andor other specifi-cations of the problem The most common population sizesare 20 to 50 [25] However Clerc and Kennedy [36] proposeda relationship to determine the optimum value of swarm sizeas follows

119873119904= 10 + [radic119863

119904] (10)

where 119873119904is swarm size 119863

119904is dimension of the particles

and [ ] is calculator of integer part Since the dimensionof particles in the present problem is seven the proposedoptimum swarm size by this equation is 12 Consideringthe most common range of the swarm size and the resultof equation an interval of swarm sizes was prepared forsensitivity test It should be noted that the first swarm ofall tests was produced by the same random pattern and themaximum iteration number was set to 100 for all tests Table 2shows the results of testsThe phrase ldquoCPU timerdquo in this tablemeans the exact amount of time that CPU spent on each testCPU time was used to produce fair comparisons since somefactors including the operating system and available memorycan affect the overall duration of the tests

Figure 7 illustrates the convergence behavior of PSOin corresponding swarm sizes of Table 2 Three differentconvergence behaviors as good late and failure can describethese trends Swarm sizes 5 15 25 35 and 55 providedgood convergence over maximum iterations while swarmsizes 45 and 65 respectively delivered delay and failurein convergence Among all the tests the best convergencewas obtained by swarm size of 35 that provided the bestconvergence with the highest average fitness

Layer 1 interfaceWater tableLayer 2 interface

5

45

4

35

3

25

2

15

1

05

05

4 32 1

0

Z-a

xis

X-axisY-axis

54

321

0

Figure 6 Generated slope model for sensitivity analysis

1

09

08

07

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 1Test 2Test 3Test 4

Test 5Test 6Test 7

Figure 7 Convergence behavior of PSO with different swarm sizes

The next tests were performed to find the optimumvaluesof coefficients 120599

1and 120599

2of velocity equation Based on the

original coefficients of Kennedy and Eberhart [4] and themodified coefficients of Clerc and Kennedy [36] a seriesof combinations were established as shown in Table 3 Alltests were performed by the same initial swarm with the sizeof 35 (previously obtained as optimum) and the maximumiterations of 100

The results can be presented in two separated groupsincluding unequal and equal coefficients Figure 8 illustratethe results of the tests The first group failed to converge overthe maximum iteration period but the second group showeddifferent performances Overall the best convergence and thegreatest average fitness belonged to equal coefficients of 175that makes it the optimum coefficient of velocity equation

The Scientific World Journal 7

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 1Test 2Test 3

Test 4Test 5Test 6

(a)

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 7Test 8Test 9

Test 10Test 11Test 12

(b)

Figure 8 Results of sensitivity tests on (a) unequal and (b) equal coefficients

Table 3 Combinations of velocity equation coefficients in differenttests

Test number Relationship 1205991

1205992

1205991+ 1205992

1 1205991= 025120599

20800 3200 4

2 1205991= 050120599

21333 2667 4

3 1205991= 075120599

21714 2286 4

4 1205992= 025120599

13200 0800 4

5 1205992= 050120599

12667 1333 4

6 1205992= 075120599

12286 1714 4

7 1205991= 1205992

2500 2500 58 120599

1= 1205992

2000 2000 49 120599

1= 1205992

1750 1750 3510 120599

1= 1205992

1500 1500 311 120599

1= 1205992

1000 1000 212 120599

1= 1205992

0500 0500 1

The last sensitivity tests were performed to find theoptimum inertia weight (120596) of velocity equation The sameinitial swarm with size of 35 and equal coefficients of velocityequation as 175 (previously defined as optimum values) wereapplied for all the tests Five tests with inertia weights of 0025 05 075 and 1 respectively were performed based onthe proposed values of Shi and Eberhart [35] and Clerc andKennedy [36] Figure 9 shows the convergence behavior ofPSO in the tests The results showed successful convergencein all tests except test 3 It should be noted that test 5 wasidentical with the original PSO where no inertia weightwas present in velocity equation Immature convergencehas occurred for tests 1 and 2 Although fast convergenceappears an advantage at first it is a sign of trapping aswarm in local solutions Test 3 failed to converge test 4 hadinstable convergence and test 5 failed to improve its averagefitness over themaximum iterations Consequently it was notpossible to introduce an optimum inertia weight to guarantee

0 10 20 30 40 50 60 70 80 90 100

Iteration

Test 1Test 2Test 3

Test 4Test 5

10908070605040302010

Aver

age fi

tnes

s

Figure 9 Results of sensitivity tests on inertia weight

the convergence of PSO and improvement of average fitnessover the maximum iteration number simultaneously

A dynamic inertia weight was utilized by the presentstudy to overcome the convergence problem of PSOThe pro-posed strategy started with the most anticonvergence inertiaweight (05) continued with the normal convergence inertiaweight (075) and ended upwith themost stable convergence(025) The switching levels of inertia weights were defined asone-third and two-thirds of maximum iterations Figure 10shows the results of sensitivity tests on dynamic inertiaweightwith different maximum iterations from 50 to 300 by stepsof 50 All tests performed well to converge and improve theaverage fitness over their maximum iterations so dynamicinertia weight was adopted for PSO

4 Example Problems

Two example problems were analyzed to verify the perfor-mance of PSO in determining the CSS The properties of theslope materials are shown in Table 4 Example problem 1 wasperformed to verify the performance of PSO in determining

8 The Scientific World Journal

06

05

04

03

02

01

00 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Aver

age fi

tnes

s

Iteration

Dynamic test 1Dynamic test 2Dynamic test 3

Dynamic test 4Dynamic test 5Dynamic test 6

Figure 10 Results of sensitivity tests on dynamic inertia weight

Table 4 Properties of slopes in example problems

Properties 1198881015840 (kNm2) 120601

1015840 (degree) 120574 (kNm3) 120592 119864 (kNm2)Problem 1 15 20 17 03 1119864 + 6

Problem 2 10 10 18 mdash mdash

the CSS in comparison with PLAXIS-3D finite element soft-ware (License byUniversiti TeknologiMalaysia) Alkasawnehet al [44] applied different search techniques to determinethe CSS in 2D slope stability analysis Figure 11 illustrates thegeometry of the slope A 3D model was developed based onthis 2D section inwhich the third dimensionwas extended by100 meters Figure 12 shows the generated 3D models of theslope by the present study and PLAXIS-3D In both methodscylindrical slip surface was employed to determine the CSS ofthe slope

PSO improved average and best fitness of the swarmas shown in Figure 13 Figure 14 shows the minimum FOSversus iterations PSO provided continuous reduction of FOSto find the CSS The present study obtained FOS of 178versus theminimumFOS of 177 of the PLAXIS-3D Figure 15shows the CSS obtained by the present study and the resultof PLAXIS-3D The present study and PLAXIS-3D obtainedsimilar FOS for the CSS with a small difference of 03 Thisresult demonstrates the ability of PSO to determine the CSSwith the minimum FOS in 3D slope stability analysis

Example problem 2 was performed to verify the abilityof PSO to determine the CSS with general ellipsoid shape ina comparison with previous studies from the literature Thisexample was initially analyzed by Yamagami et al [45] andwas reanalyzed by Yamagami and Jiang [25] Yamagami et al[45] used random generation of surfaces to determine theCSS of this slope while Yamagami and Jiang [25] employed acombination of dynamic programming and random numbergeneration to do so It should be noted that the same equationof FOS as previous studies was used to make fair comparisonof the results The example involved a homogeneous slopewith gradient of 2 1 subjected to a square load of 50 kPa onthe top The uniform load was applied on a square surface

30

20

10

00 20 40 60 80

Z(m

)

Y (m)

Figure 11 Geometry of 2D section of example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axis

Y-axis

(a)Z-a

xis

X-axisY-axis

(b)

Figure 12 Generated 3D models of example problem 1 by (a) thepresent study and (b) PLAXIS-3D

with 8 meters sides at the top center of the slope Figure 16illustrates the geometry of example problem 1

Figure 17 shows the generated 3D model by the presentstudy for example problem 2 Figure 18 plots the process ofPSO to improve fitness of the swarmThe trend of average fit-ness value of the swarm experienced some instability duringthe process The main reason of this behavior is the presenceof disqualified slip surfaces in the swarm that dramaticallydecreases the average fitness valueThese surfaces were rarelypresented in the previous example due to adopted simplercylindrical shape compared with more complicated ellipsoidshape in this example The trend of minimum FOS versusiterations is shown in Figure 19 Continuous decrement ofFOS by PSO leads to determining the CSS of the slope

In spite of similar equation of FOS the present studyfound the CSS with a smaller FOS than other methods whichis the best result so far The minimum FOS obtained by PSOwas 095 compared with 114 and 103 of random generationof surfaces [45] and DP with RNG [25] respectively Thisresult demonstrates the ability of PSO to accurately determinethe ellipsoid CSS in 3D slope stability analysis Figure 20

The Scientific World Journal 9

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80

Iteration

Fitn

ess v

alue

Average fitnessMaximum fitness

Figure 13 Fitness values versus iterations in example problem 1

215

21

205

2

195

19

1850 10 20 30 40 50 60 70 80

Iteration

Min

imum

fact

or o

f saf

ety

Figure 14 Minimum FOS versus iterations in example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axisY-axis

(a)

Z-a

xis

X-axis

Y-axis

(b)

Figure 15 (a) The CSS obtained by the present study and (b) exag-gerated displacement vectors of PLAXIS-3D in example problem 1

40

24

20

9

4

0

0 4 14 22 25

x(m

)z

(m)

y (m)

0 4 14 22 25

y (m)

12

(b)

(a)

Figure 16 (a) Half-plan view and (b) central cross-section of slopein example problem 2

10864200

510

1520

2530 35

40 05

1015

2025

Z-a

xis

X-axis Y-axis

Figure 17 Generated 3D models of example problem 2 by thepresent study

illustrates the CSS obtained by the present study in exampleproblem 2

5 Conclusion

Determining the critical slip surface of a soil slope is atraditional problem in geotechnical engineering which is stillchallenging for researchers This problem needs a massivesearching process Although classical searching methodswork for relatively simple problems they are surrounded bylocal minima Moreover their processes become particularlyslow by increasing the number of possible solutions Toeliminate these limitations PSO has been applied in slopestability analysis based on its successful results in advancedengineering problems However this contribution was lim-ited to 2D slope stability analysis This paper applied PSO in

10 The Scientific World Journal

12

1

08

06

04

02

00 10 20 30 40 50 60 70 80 90 100

Fitn

ess v

alue

Iteration

Average fitnessMaximum fitness

Figure 18 Fitness values versus iterations in example problem 2

3

25

35

2

15

1

05

00 10 20 30 40 50 60 70 80 90 100

Min

imum

fac

tor o

f saf

ety

Iteration

Figure 19 Minimum FOS versus iterations in example problem 2

10

8

6

4

2

0051015

2025

303540 0 5 10 15 20 25

Z-a

xis

X-axis

Y-axis

Figure 20 The CSS obtained by the present study in exampleproblem 2

3D slope stability problem to determine theCSS of soil slopesA detailed description of adopted PSO was presented toprovide a good basis for more contribution of this techniqueto the field of 3D slope stability problems

The application of PSO in slope stability analysis wasdescribed by presenting a general flowchart A general rotat-ing ellipsoid shape was introduced as the specific particle for3D slope stability analysis In order to find the optimum val-ues of parameters of PSO a sensitivity analysis was designedand performed The related codes were prepared by theauthors in MATLAB A 3D model with complex geometry

soil layers and piezometric line was used in the analysisThis analysis included three steps to find the optimum swarmsize coefficients and inertia weight of the velocity equationrespectively Moreover the performance of PSO to convergeover a global optimum solution was verified during thetests Based on the obtained values of parameters PSO wasprepared for 3D slope stability analysis

The applicability of PSO in determining the CSS of 3Dslopes was evaluated by analyzing two example problemsThe first example presented a comparison between the resultsof PSO and PLAXI-3D finite element software The secondexample compared the ability of PSO to determine the CSSof 3D slopes with other optimization methods from theliterature Both of the example problems demonstrated theefficiency and effectiveness of PSO in determining the CSS of3D soil slopes Based on the results it is believed that PSO ishighly capable of contributing to the field of 3D slope stabilityanalysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The presented study was undertaken with support fromresearch grant The authors are pleased to acknowledgethe Ministry of Education of Malaysia for the PrototypeResearch Grant Scheme (RJ13000078224L620) and Univer-siti Teknologi Malaysia for providing the research funds

References

[1] R Kalatehjari and N Ali ldquoA review of three-dimensional slopestability analyses based on limit equilibriummethodrdquo ElectronicJournal of Geotechnical Engineering vol 18 pp 119ndash134 2013

[2] D G Fredlund ldquoAnalytical methods for slope stability analysisrdquoin Proceedings of the 4th International Symposium on LandslidesState-of-the-Art pp 229ndash250 Toronto Canada September1984

[3] A W Bishop ldquoThe use of the slip circle in the stability analysisof earth sloperdquo Geotechnique vol 5 no 1 pp 7ndash17 1955

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[5] Y Zhang P Agarwal V Bhatnagar S Balochian and J YanldquoSwarm intelligence and its applicationsrdquo The Scientific WorldJournal vol 2013 Article ID 528069 3 pages 2013

[6] G Poitras G Lefrancois and G Cormier ldquoOptimization ofsteel floor systems using particle swarm optimizationrdquo Journalof Constructional Steel Research vol 67 no 8 pp 1225ndash12312011

[7] S Gholizadeh and E Salajegheh ldquoOptimal design of structuressubjected to time history loading by swarm intelligence and anadvancedmetamodelrdquoComputerMethods in AppliedMechanicsand Engineering vol 198 no 37ndash40 pp 2936ndash2949 2009

[8] S S Gilan H B Jovein and A A Ramezanianpour ldquoHybridsupport vector regressionmdashparticle swarm optimization for

The Scientific World Journal 11

prediction of compressive strength and RCPT of concretescontaining metakaolinrdquo Construction and Building Materialsvol 34 pp 321ndash329 2012

[9] L Yunkai T Yingjie O Zhiyun et al ldquoAnalysis of soil ero-sion characteristics in small watersheds with particle swarmoptimization support vector machine and artificial neuronalnetworksrdquoEnvironmental Earth Sciences vol 60 no 7 pp 1559ndash1568 2010

[10] S Wan ldquoEntropy-based particle swarm optimization withclustering analysis on landslide susceptibility mappingrdquo Envi-ronmental Earth Sciences vol 68 no 5 pp 1349ndash1366 2013

[11] M R Nikoo R Kerachian A Karimi A A Azadnia andK Jafarzadegan ldquoOptimal water and waste load allocationin reservoir-river systems a case studyrdquo Environmental EarthSciences vol 71 no 9 pp 4127ndash4142 2014

[12] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for hourly rainfall-runoff modeling in Bedup Basin Malaysiardquo International Jour-nal of Civil amp Environmental Engineering vol 9 no 10 pp 9ndash182010

[13] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for modeling runoffrdquoInternational Journal of Environmental Science and Technologyvol 7 no 1 pp 67ndash78 2010

[14] J S Yazdi F Kalantary and H S Yazdi ldquoCalibration ofsoil model parameters using particle swarm optimizationrdquoInternational Journal of Geomechanics vol 12 no 3 pp 229ndash238 2012

[15] T V Bharat P V Sivapullaiah and M M Allam ldquoSwarmintelligence-based solver for parameter estimation of laboratorythrough-diffusion transport of contaminantsrdquo Computers andGeotechnics vol 36 no 6 pp 984ndash992 2009

[16] B R Chen and X T Feng ldquoCSV-PSO and its application ingeotechnical engineeringrdquo in Swarm Intelligence Focus on AntandParticle SwarmOptimization F T S Chan andMKTiwariEds chapter 15 pp 263ndash288 2007

[17] Y M Cheng L Li and S C Chi ldquoPerformance studies on sixheuristic global optimization methods in the location of criticalslip surfacerdquoComputers and Geotechnics vol 34 no 6 pp 462ndash484 2007

[18] Y M Cheng L Li Y J Sun and S K Au ldquoA coupled particleswarm and harmony search optimization algorithm for difficultgeotechnical problemsrdquo Structural and Multidisciplinary Opti-mization vol 45 no 4 pp 489ndash501 2012

[19] H B Zhao Z S Zou and Z L Ru ldquoChaotic particle swarmoptimization for non-circular critical slip surface identificationin slope stability analysisrdquo in Proceedings of the InternationalYoung Scholarsrsquo Symposium on Rock Mechanics Boundaries ofRock Mechanics Recent Advances and Challenges for the 21stCentury pp 585ndash588 Beijing China May 2008

[20] D Tian L Xu and S Wang ldquoThe application of particle swarmoptimization on the search of critical slip surfacerdquo in Proceed-ings of the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[21] L Li G Yu X Chu and S Lu ldquoThe harmony search algo-rithm in combination with particle swarm optimization andits application in the slope stability analysisrdquo in Proceedings ofthe International Conference on Computational Intelligence andSecurity (CIS rsquo09) vol 2 pp 133ndash136 Beijing China December2009

[22] R Kalatehjari N Ali M Kholghifard and M HajihassanildquoThe effects of method of generating circular slip surfaceson determining the critical slip surface by particle swarmoptimizationrdquo Arabian Journal of Geosciences vol 7 no 4 pp1529ndash1539 2014

[23] R Kalatehjari N Ali M Hajihassani and M K Fard ldquoTheapplication of particle swarm optimization in slope stabilityanalysis of homogeneous soil slopesrdquo International Review onModelling and Simulations vol 5 no 1 pp 458ndash465 2012

[24] W Chen and P Chen ldquoPSOslope a stand-alone windowsapplication for graphic analysis of slope stabilityrdquo in Advancesin Swarm Optimization Y Tan Y Shi Y Chai and G WangEds vol 6728 of Lecture Notes in Computer Science pp 56ndash632011

[25] T Yamagami and J C Jiang ldquoA search for the critical slipsurface in three-dimensional slope stability analysisrdquo Soils andFoundations vol 37 no 3 pp 1ndash16 1997

[26] J C Jiang T Yamagami and R Baker ldquoThree-dimensionalslope stability analysis based on nonlinear failure enveloperdquoChinese Journal of Rock Mechanics and Engineering vol 22 no6 pp 1017ndash1023 2003

[27] X J E Mowen ldquoA simple Monte Carlo method for locating thethree-dimensional critical slip surface of a sloperdquoActaGeologicaSinica vol 78 no 6 pp 1258ndash1266 2004

[28] X J E Mowen W Zengfu L Xiangyu and X Bo ldquoThree-dimensional critical slip surface locating and slope stabilityassessment for lava lobe of Unzen Volcanordquo Journal of RockMechanics and Geotechnical Engineering vol 3 no 1 pp 82ndash892011

[29] Y M Cheng H T Liu W B Wei and S K Au ldquoLocationof critical three-dimensional non-spherical failure surface byNURBS functions and ellipsoid with applications to highwayslopesrdquo Computers and Geotechnics vol 32 no 6 pp 387ndash3992005

[30] M Toufigh A Ahangarasr and A Ouria ldquoUsing non-linearprogramming techniques in determination of the most prob-able slip surface in 3D slopesrdquo World Academy of ScienceEngineering and Technology vol 17 pp 30ndash35 2006

[31] C W Reynolds ldquoFlocks herbs and schools a distributedbehavorial modelrdquo Computer Graphics vol 21 no 4 pp 25ndash341987

[32] R Kalatehjari An improvised three-dimensional slope stabilityanalysis based on limit equilibrium method by using particleswarm optimization [PhD dissertation] Faculty of Civil Engi-neering Universiti Teknologi Malaysia 2013

[33] A P Engelbrecht Computational intelligence an introductionJohn Wiley amp Sons 2007

[34] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tion an overviewrdquo Swarm Intelligence vol 1 no 1 pp 33ndash572007

[35] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 Piscataway NJ USAMay 1998

[36] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[37] R Mendes P Cortez M Rocha and J Neves ldquoParticle swarmsfor feedforward neural network trainingrdquo in Proceedings of theInternational Joint Conference on Neural Networks (IJCNN rsquo02)pp 1895ndash1899 Honolulu Hawaii USA May 2002

12 The Scientific World Journal

[38] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science pp 39ndash43 IEEENagoya Japan October 1995

[39] R Baker ldquoDetermination of the critical slip surface in slopestability computationsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 4 pp 333ndash3591980

[40] H P J Bolton G Heymann and A A Groenwold ldquoGlobalsearch for critical failure surface in slope stability analysisrdquoEngineering Optimization vol 35 no 1 pp 51ndash65 2003

[41] Y Chen X Wei and Y Li ldquoLocating non-circular critical slipsurfaces by particle swarm optimization algorithmrdquo ChineseJournal of Rock Mechanics and Engineering vol 25 no 7 pp1443ndash1449 2006

[42] L Li S-C Chi R-M Zheng G Lin and X-S Chu ldquoMixedsearch algorithmof non-circular critical slip surface of soil sloprdquoChina Journal of Highway and Transport vol 20 no 6 pp 1ndash62007

[43] L Li S-C Chi andY-M Zheng ldquoThree-dimensional slope sta-bility analysis based on ellipsoidal sliding body and simplifiedJANBU methodrdquo Rock and Soil Mechanics vol 29 no 9 pp2439ndash2445 2008

[44] W Alkasawneh A I H Malkawi J H Nusairat and NAlbataineh ldquoA comparative study of various commerciallyavailable programs in slope stability analysisrdquo Computers andGeotechnics vol 35 no 3 pp 428ndash435 2008

[45] T Yamagami K Kojima and M Taki ldquoA search for the three-dimensional critical slip surface with random generation ofsurfacerdquo in Proceedings of the 36th Symposium on Slope StabilityAnalyses and Stabilizing Construction Methods pp 27ndash34 1991

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

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Applied Computational Intelligence and Soft Computing

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Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

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Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

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International Journal of

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ArtificialNeural Systems

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RoboticsJournal of

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Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 3

Start

InitializePSO parameters

Generate first swarm

Evaluate the fitnessof all particles

Record personal bestfitness of all particles

Find global best particle

Swarm metthe termination

criteria

End

No

Yes

Update the positionof particles

Update the velocityof particles

Figure 1 Standard flowchart of PSO

velocity of particles is determined by (3) based on theirbest position and global best position in the swarm Tocontinue the search (4) updates the position of all particlesbased on their current position and the obtained velocityThrough an iterative process the improvement of fitness ofparticles continues until PSO meets the termination criteriaThe global solution is then achieved by the current positionof the best particle in the last iteration

V119899(119894)

= V119899(119894minus1)

+ 119906 (0 1205991) (119887119901

119899(119894)minus 119909119899(119894))

+ 119906 (0 1205992) (119887119892

119899(119894)minus 119909119899(119894))

(3)

119909119899(119894+1)

= 119909119899(119894)

+ V119899(119894) (4)

where V119899(119894minus1)

and V119899(119894)

are respectively the velocity of 119899thparticle in past and current iterations 119906(0 120599

1) and 119906(0 120599

2)

are the vectors of random numbers of 119899th particle uniformlydistributed respectively in [0 120599

1] and [0 120599

2] 119887119901119899(119894)

is the bestposition of 119899th particle so far 119887119892

119899(119894)is the position of the

best particle of the swarm so far and 119909119899(119894minus1)

and 119909119899(119894)

are thepositions of 119899th particle respectively in the current and thenext iterations

Initial cognitive and social parts are three componentsof velocity equation The values of 120599

1and 1205992in this equation

control the exploration and exploitation behaviours of theswarm While equal values of 2 are commonly used forthese parameters in early search greater values of 120599

1and

1205992 respectively provide faster convergence to the solution

and enhance discovering the searching space The velocityof particles may increase surprisingly by adjusting these

parameters so a limiting bound of velocity as [minusVmax Vmax]is attached to PSO as constriction coefficients [34] Shi andEberhart [35] modified the original equation of velocity toreduce the role of constriction coefficient and introduced (5)by introducing 120596 as the inertia weight of particles Later onClerc and Kennedy [36] demonstrated that inertia weightsof greater than one may cause converge problems in PSOand proposed (6) by introducing 120585 as the constant multiplierin (7) This modification prevents the swarm to explodeguarantees the mature converge and almost eliminates theneed of constriction coefficient Principally interior param-eters (inertia weight and velocity coefficient) and exteriorparameters (swarm size and topology) of PSO should becarefully adjusted to provide the best results

V119899(119894)

= 120596V119899(119894minus1)

+ 119906 (0 1205991) (119887119901

119899(119894)minus 119909119899(119894))

+ 119906 (0 1205992) (119887119892

119899(119894)minus 119909119899(119894))

(5)

V119899(119894)

= 120585 [V119899(119894minus1)

+ 119906 (0 1205991) (119887119901

119899(119894)minus 119909119899(119894))

+ 119906 (0 1205992) (119887119892

119899(119894)minus 119909119899(119894))]

(6)

120585 =

2

(1205991+ 1205991) minus 2 +

radic(1205991+ 1205991)

2

minus 4 (1205991+ 1205991)

(1205991+ 1205991) gt 4

(7)

Topology in the method of intercommunication betweenparticles controls the convergence of a swarm Topology isdivided into static and dynamic categories In static topolo-gies the number of connected neighbors to a particle isconstant throughout the optimization process However thisnumber increases by the progress of optimization process indynamic topologies to enhance the searching abilities [37]

The original PSO aided a conical static topology basedon intercommunication of all particles with the global bestparticle However Eberhart and Kennedy [38] proposedanother static topology by introducing intercommunicationbetween individuals and local best particles In this modeleach particle was connected to 119870 number of its neighbors inthe swarm array The main advantage of this method was theability of subconvergence in different regions of the searchspace Although the convergence of this method was slowerthan the conical method it was able to better escape fromlocal optima For each problem the appropriate topology canbe defined by performing sensitivity analysis on convergenceand execution time of PSO Figure 2 illustrates conical andlocal (119870 = 2) topologies for randomly generated 100 particlesin a 2D search space

The size of swarm is defined as the number of its particlesWhile a small swarm may fail to converge over a globalsolution a large swarm may have late convergence The sizeof swarm commonly varies from 20 to 50 but the optimumnumber is usually determined through sensitivity analysis onthe convergence parameter of the swarm [36]

4 The Scientific World Journal

0

1

2

3

4

5

6

7

8

9

10

0 2 4 5 6 7 8 10

ParticlesLocal connections

1 3 9

Y-a

xis

X-axis

(a)

ParticlesGlobal connectionsGlobal best

0

1

2

3

4

5

6

7

8

9

10

0 2 4 5 6 7 8 101 3 9

Y-a

xis

X-axis

(b)

Figure 2 (a) Local and (b) global topologies in 2D search space

3 Application of PSO in SlopeStability Analysis

PSO is mainly applied in stability analysis of soil slopeswithin the framework of LEM [1] This analysis involves twoconsequent steps that is calculating FOS of candidate slipsurfaces and determining the CSS among all candidates [39]PSO commonly contributes to the second step to determinethe shape and location of the CSS which are generallyunknown in soil slopes [40]

PSO can be applied in both 2D and 3D slope stabilityanalyses [32] In 2D analysis it can be employed to determinethe shape of the CSS in a predefined 2D section of the slopeDifferent shapes are possible for slip surfaces in 2D analysissuch as circular ellipse spiral and polygonal or arbitrarysurfaces [22 41 42] In contrast 3D slip surfaces such asspherical ellipsoidal and Nonuniform Rational B-Splines(NURBS) are commonly assumed in 3D analysis [23 43 4344]

Figure 3 shows flowchart of PSO to determine the CSSin slope stability analysis The optimization procedure isstarted by setting initial parameters of PSO Then a certainnumber of particles (119873) is generated in a random patternover the search space Since the improvement of swarm hasjust begun personal bests of all particles in initial swarmare identical to the particles themselves Based on the samereason the velocity of all initial particles is set to zero Aftersetting up the initial values the first particle is arranged asits corresponding slip surface This surface is qualified if it

can satisfy the conditions of the problem Otherwise it isdisqualified A predefined minimum fitness value is given todisqualify slip surfaces This value represents a predefinedmaximum FOS For a qualified surface FOS is calculated andthe corresponding fitness values of particle are assigned bythe fitness function of PSO This process is repeated for allparticles of the swarm

Current positions of particles that improved their fitnessvalues are recorded to update their personal bests whileprevious personal bests are used for other particles Theglobal best particle is defined by the greatest fitness value inthe current swarm Through an iterative process subsequentswarms are generated by updating velocities and positionsof former particles The optimization process is terminatedby meeting the termination criteria Eventually the globalbest particle of the last swarm represents the CSS of theslope

31 Coding of the Particles The structure of particles followedthe standard PSO particles involving three sections as currentposition previous best position and the velocity A rotatingellipsoid was selected as the general 3D shape of slip surfacesThis ellipsoid can rotate on119909-119910plane (0 le 120579

119909119910le 120587) to provide

various slip surfaces (Figure 4)In order to achieve the equation of rotated ellipsoid the

parametric equation of general ellipsoid was transferred intonew axes by the rotation angle 120579

119909119910 The result presents the

rotated ellipsoid in (8) It should be noted that this ellipsoid

The Scientific World Journal 5

Start

Generate N particles by random

Set Pbest = particlesSet velocities = 0

Convert the first particleto slip surface

Is the No

YesCalculate 3D FOSof the slip surface

No

Calculate fitness

Was

No

Yes

Find Pbests and gbest

Are the

YesEnd

Con

vert

the n

ext

part

icle

to sl

ip

Calculate and updatenew positions of all particles

Calculate and updatevelocity of all particles

Initialize PSO parametersSwarm size = N

qualifiedslip surface

of the particle

particlethat the last

termination criteria met

surfa

ce

Set t

he fi

tnes

sas

min

imum

Figure 3 Flowchart of PSO to determine the CSS in slope stabilityanalysis

can be easily transformed to spherical and cylindrical slip sur-face by respectively setting equal three and two semiradiuses

(cos 120579119909119910(119909 minus 119883

119888) minus sin 120579

119909119910(119910 minus 119884

119888))

2

119877

2

119909

minus

(cos 120579119909119910(119910 minus 119884

119888) + sin 120579

119909119910(119909 minus 119883

119888))

2

119877

2

119910

+

(119911 minus 119885119888)

2

119877

2

119911

= 1

(8)

where 120579119909119910

is rotation angle of the ellipsoid in 119909-119910 plane 119883119888

119884119888 and 119885

119888are coordinates of center of ellipsoid in 119909- 119910- and

119911-directions 119877119909 119877119910 and 119877

119911are semiradiuses of ellipsoid in

119909- 119910- and 119911-directions and 119909 119910 and 119911 are coordinates of anarbitrary point on the surface of ellipsoid

Based on the parameters of the rotating ellipsoid Figure 5shows schematic structure of a PSO particle The section ofcurrent position records the coordinates of center of ellipsoidits semiradiuses and its rotation angle Considering bestposition and velocity sections PSO has seven-dimensionalsearch space and twenty-one-cell particles

Rx

Ry

120579xy

X

Y

120579xy = 0

120579xy = 1205874

120579xy = 1205872

120579xy = 31205874

120579xy = 120587

Figure 4 Projection of rotating ellipsoid on 119909-119910 plane

Xc Yc Zc Rx Ry Rz 120579xy

Xcp

Y cp

Z cp

R xp

R yp

R zp

120579 p

Vxc

Vyc

Vzc

Vry

Vrx

Vrz

V120579

Current position

Best

posit

ionVelocity

Figure 5 Schematic structures of PSO particles

32 Fitness Function The quality of particles can be cal-culated by the fitness function This function is related tothe objective function and provides quantitative tracking ofimprovement of particles Consequently it makes it possibleto compare and rank particles in the swarm wheremaximumfitness shows the best particle andminimum fitness identifiesthe worst particle of the swarm PSO attempts to increase themaximum fitness of swarms during its iterations Since theobjective function of 3D slope stability analysis is equation ofFOS PSO attempts to find the CSS with the minimum FOSby maximizing the fitness value in

Fitness119899(119894)

=

1

FOS (119909119899(119894))

(9)

where Fitness119899(119894)

is fitness value of the 119899th particle in 119894thiteration and FOS(119909

119899(119894)) is FOS of 3D slip surface described

by 119909119899(119894)

33 Sensitivity Analysis on PSO Parameters The best per-formance of PSO is guaranteed by initializing appropriateparameters for it A sensitivity analysis can help to do soThe optimum values PSO parameters in 3D slope stabilityanalysis were defined by designing and performing severalindependent tests on swarm size coefficients of velocity andinertia weight of the swarm In addition the convergence

6 The Scientific World Journal

Table 1 Properties of slope in sensitivity analysis

Parameters120574 (kNm3) 119888

1015840 (kNm2) 120601

1015840 (∘)Layer 1 1978 12 17Layer 2 1764 245 20

Table 2 Results of sensitivity tests on swarm size

Test number1 2 3 4 5 6 7

Swarm size 5 15 25 35 45 55 65Total CPU time (s) 365 625 549 1726 2792 2241 10101

behavior of PSO as the average fitness of swarms wasobserved during the tests A 3D soil slope was designedwith complex geometry layers of soil and piezometric line(Figure 6) It should be noted that coding of the study wasdone by the authors in MATLAB software (Licensed byUniversiti TeknologiMalaysia)The overall shape of the slopeshows two imbalanced hills with steep sides makes it difficultto find the CSS for conventional slope stability analysesThis specific shape was selected to verify the effectiveness ofPSO in complex 3D slopes The properties of soil layers aredescribed in Table 1

The size of the swarm is defined based on the conditionof search space dimension of particles andor other specifi-cations of the problem The most common population sizesare 20 to 50 [25] However Clerc and Kennedy [36] proposeda relationship to determine the optimum value of swarm sizeas follows

119873119904= 10 + [radic119863

119904] (10)

where 119873119904is swarm size 119863

119904is dimension of the particles

and [ ] is calculator of integer part Since the dimensionof particles in the present problem is seven the proposedoptimum swarm size by this equation is 12 Consideringthe most common range of the swarm size and the resultof equation an interval of swarm sizes was prepared forsensitivity test It should be noted that the first swarm ofall tests was produced by the same random pattern and themaximum iteration number was set to 100 for all tests Table 2shows the results of testsThe phrase ldquoCPU timerdquo in this tablemeans the exact amount of time that CPU spent on each testCPU time was used to produce fair comparisons since somefactors including the operating system and available memorycan affect the overall duration of the tests

Figure 7 illustrates the convergence behavior of PSOin corresponding swarm sizes of Table 2 Three differentconvergence behaviors as good late and failure can describethese trends Swarm sizes 5 15 25 35 and 55 providedgood convergence over maximum iterations while swarmsizes 45 and 65 respectively delivered delay and failurein convergence Among all the tests the best convergencewas obtained by swarm size of 35 that provided the bestconvergence with the highest average fitness

Layer 1 interfaceWater tableLayer 2 interface

5

45

4

35

3

25

2

15

1

05

05

4 32 1

0

Z-a

xis

X-axisY-axis

54

321

0

Figure 6 Generated slope model for sensitivity analysis

1

09

08

07

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 1Test 2Test 3Test 4

Test 5Test 6Test 7

Figure 7 Convergence behavior of PSO with different swarm sizes

The next tests were performed to find the optimumvaluesof coefficients 120599

1and 120599

2of velocity equation Based on the

original coefficients of Kennedy and Eberhart [4] and themodified coefficients of Clerc and Kennedy [36] a seriesof combinations were established as shown in Table 3 Alltests were performed by the same initial swarm with the sizeof 35 (previously obtained as optimum) and the maximumiterations of 100

The results can be presented in two separated groupsincluding unequal and equal coefficients Figure 8 illustratethe results of the tests The first group failed to converge overthe maximum iteration period but the second group showeddifferent performances Overall the best convergence and thegreatest average fitness belonged to equal coefficients of 175that makes it the optimum coefficient of velocity equation

The Scientific World Journal 7

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 1Test 2Test 3

Test 4Test 5Test 6

(a)

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 7Test 8Test 9

Test 10Test 11Test 12

(b)

Figure 8 Results of sensitivity tests on (a) unequal and (b) equal coefficients

Table 3 Combinations of velocity equation coefficients in differenttests

Test number Relationship 1205991

1205992

1205991+ 1205992

1 1205991= 025120599

20800 3200 4

2 1205991= 050120599

21333 2667 4

3 1205991= 075120599

21714 2286 4

4 1205992= 025120599

13200 0800 4

5 1205992= 050120599

12667 1333 4

6 1205992= 075120599

12286 1714 4

7 1205991= 1205992

2500 2500 58 120599

1= 1205992

2000 2000 49 120599

1= 1205992

1750 1750 3510 120599

1= 1205992

1500 1500 311 120599

1= 1205992

1000 1000 212 120599

1= 1205992

0500 0500 1

The last sensitivity tests were performed to find theoptimum inertia weight (120596) of velocity equation The sameinitial swarm with size of 35 and equal coefficients of velocityequation as 175 (previously defined as optimum values) wereapplied for all the tests Five tests with inertia weights of 0025 05 075 and 1 respectively were performed based onthe proposed values of Shi and Eberhart [35] and Clerc andKennedy [36] Figure 9 shows the convergence behavior ofPSO in the tests The results showed successful convergencein all tests except test 3 It should be noted that test 5 wasidentical with the original PSO where no inertia weightwas present in velocity equation Immature convergencehas occurred for tests 1 and 2 Although fast convergenceappears an advantage at first it is a sign of trapping aswarm in local solutions Test 3 failed to converge test 4 hadinstable convergence and test 5 failed to improve its averagefitness over themaximum iterations Consequently it was notpossible to introduce an optimum inertia weight to guarantee

0 10 20 30 40 50 60 70 80 90 100

Iteration

Test 1Test 2Test 3

Test 4Test 5

10908070605040302010

Aver

age fi

tnes

s

Figure 9 Results of sensitivity tests on inertia weight

the convergence of PSO and improvement of average fitnessover the maximum iteration number simultaneously

A dynamic inertia weight was utilized by the presentstudy to overcome the convergence problem of PSOThe pro-posed strategy started with the most anticonvergence inertiaweight (05) continued with the normal convergence inertiaweight (075) and ended upwith themost stable convergence(025) The switching levels of inertia weights were defined asone-third and two-thirds of maximum iterations Figure 10shows the results of sensitivity tests on dynamic inertiaweightwith different maximum iterations from 50 to 300 by stepsof 50 All tests performed well to converge and improve theaverage fitness over their maximum iterations so dynamicinertia weight was adopted for PSO

4 Example Problems

Two example problems were analyzed to verify the perfor-mance of PSO in determining the CSS The properties of theslope materials are shown in Table 4 Example problem 1 wasperformed to verify the performance of PSO in determining

8 The Scientific World Journal

06

05

04

03

02

01

00 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Aver

age fi

tnes

s

Iteration

Dynamic test 1Dynamic test 2Dynamic test 3

Dynamic test 4Dynamic test 5Dynamic test 6

Figure 10 Results of sensitivity tests on dynamic inertia weight

Table 4 Properties of slopes in example problems

Properties 1198881015840 (kNm2) 120601

1015840 (degree) 120574 (kNm3) 120592 119864 (kNm2)Problem 1 15 20 17 03 1119864 + 6

Problem 2 10 10 18 mdash mdash

the CSS in comparison with PLAXIS-3D finite element soft-ware (License byUniversiti TeknologiMalaysia) Alkasawnehet al [44] applied different search techniques to determinethe CSS in 2D slope stability analysis Figure 11 illustrates thegeometry of the slope A 3D model was developed based onthis 2D section inwhich the third dimensionwas extended by100 meters Figure 12 shows the generated 3D models of theslope by the present study and PLAXIS-3D In both methodscylindrical slip surface was employed to determine the CSS ofthe slope

PSO improved average and best fitness of the swarmas shown in Figure 13 Figure 14 shows the minimum FOSversus iterations PSO provided continuous reduction of FOSto find the CSS The present study obtained FOS of 178versus theminimumFOS of 177 of the PLAXIS-3D Figure 15shows the CSS obtained by the present study and the resultof PLAXIS-3D The present study and PLAXIS-3D obtainedsimilar FOS for the CSS with a small difference of 03 Thisresult demonstrates the ability of PSO to determine the CSSwith the minimum FOS in 3D slope stability analysis

Example problem 2 was performed to verify the abilityof PSO to determine the CSS with general ellipsoid shape ina comparison with previous studies from the literature Thisexample was initially analyzed by Yamagami et al [45] andwas reanalyzed by Yamagami and Jiang [25] Yamagami et al[45] used random generation of surfaces to determine theCSS of this slope while Yamagami and Jiang [25] employed acombination of dynamic programming and random numbergeneration to do so It should be noted that the same equationof FOS as previous studies was used to make fair comparisonof the results The example involved a homogeneous slopewith gradient of 2 1 subjected to a square load of 50 kPa onthe top The uniform load was applied on a square surface

30

20

10

00 20 40 60 80

Z(m

)

Y (m)

Figure 11 Geometry of 2D section of example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axis

Y-axis

(a)Z-a

xis

X-axisY-axis

(b)

Figure 12 Generated 3D models of example problem 1 by (a) thepresent study and (b) PLAXIS-3D

with 8 meters sides at the top center of the slope Figure 16illustrates the geometry of example problem 1

Figure 17 shows the generated 3D model by the presentstudy for example problem 2 Figure 18 plots the process ofPSO to improve fitness of the swarmThe trend of average fit-ness value of the swarm experienced some instability duringthe process The main reason of this behavior is the presenceof disqualified slip surfaces in the swarm that dramaticallydecreases the average fitness valueThese surfaces were rarelypresented in the previous example due to adopted simplercylindrical shape compared with more complicated ellipsoidshape in this example The trend of minimum FOS versusiterations is shown in Figure 19 Continuous decrement ofFOS by PSO leads to determining the CSS of the slope

In spite of similar equation of FOS the present studyfound the CSS with a smaller FOS than other methods whichis the best result so far The minimum FOS obtained by PSOwas 095 compared with 114 and 103 of random generationof surfaces [45] and DP with RNG [25] respectively Thisresult demonstrates the ability of PSO to accurately determinethe ellipsoid CSS in 3D slope stability analysis Figure 20

The Scientific World Journal 9

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80

Iteration

Fitn

ess v

alue

Average fitnessMaximum fitness

Figure 13 Fitness values versus iterations in example problem 1

215

21

205

2

195

19

1850 10 20 30 40 50 60 70 80

Iteration

Min

imum

fact

or o

f saf

ety

Figure 14 Minimum FOS versus iterations in example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axisY-axis

(a)

Z-a

xis

X-axis

Y-axis

(b)

Figure 15 (a) The CSS obtained by the present study and (b) exag-gerated displacement vectors of PLAXIS-3D in example problem 1

40

24

20

9

4

0

0 4 14 22 25

x(m

)z

(m)

y (m)

0 4 14 22 25

y (m)

12

(b)

(a)

Figure 16 (a) Half-plan view and (b) central cross-section of slopein example problem 2

10864200

510

1520

2530 35

40 05

1015

2025

Z-a

xis

X-axis Y-axis

Figure 17 Generated 3D models of example problem 2 by thepresent study

illustrates the CSS obtained by the present study in exampleproblem 2

5 Conclusion

Determining the critical slip surface of a soil slope is atraditional problem in geotechnical engineering which is stillchallenging for researchers This problem needs a massivesearching process Although classical searching methodswork for relatively simple problems they are surrounded bylocal minima Moreover their processes become particularlyslow by increasing the number of possible solutions Toeliminate these limitations PSO has been applied in slopestability analysis based on its successful results in advancedengineering problems However this contribution was lim-ited to 2D slope stability analysis This paper applied PSO in

10 The Scientific World Journal

12

1

08

06

04

02

00 10 20 30 40 50 60 70 80 90 100

Fitn

ess v

alue

Iteration

Average fitnessMaximum fitness

Figure 18 Fitness values versus iterations in example problem 2

3

25

35

2

15

1

05

00 10 20 30 40 50 60 70 80 90 100

Min

imum

fac

tor o

f saf

ety

Iteration

Figure 19 Minimum FOS versus iterations in example problem 2

10

8

6

4

2

0051015

2025

303540 0 5 10 15 20 25

Z-a

xis

X-axis

Y-axis

Figure 20 The CSS obtained by the present study in exampleproblem 2

3D slope stability problem to determine theCSS of soil slopesA detailed description of adopted PSO was presented toprovide a good basis for more contribution of this techniqueto the field of 3D slope stability problems

The application of PSO in slope stability analysis wasdescribed by presenting a general flowchart A general rotat-ing ellipsoid shape was introduced as the specific particle for3D slope stability analysis In order to find the optimum val-ues of parameters of PSO a sensitivity analysis was designedand performed The related codes were prepared by theauthors in MATLAB A 3D model with complex geometry

soil layers and piezometric line was used in the analysisThis analysis included three steps to find the optimum swarmsize coefficients and inertia weight of the velocity equationrespectively Moreover the performance of PSO to convergeover a global optimum solution was verified during thetests Based on the obtained values of parameters PSO wasprepared for 3D slope stability analysis

The applicability of PSO in determining the CSS of 3Dslopes was evaluated by analyzing two example problemsThe first example presented a comparison between the resultsof PSO and PLAXI-3D finite element software The secondexample compared the ability of PSO to determine the CSSof 3D slopes with other optimization methods from theliterature Both of the example problems demonstrated theefficiency and effectiveness of PSO in determining the CSS of3D soil slopes Based on the results it is believed that PSO ishighly capable of contributing to the field of 3D slope stabilityanalysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The presented study was undertaken with support fromresearch grant The authors are pleased to acknowledgethe Ministry of Education of Malaysia for the PrototypeResearch Grant Scheme (RJ13000078224L620) and Univer-siti Teknologi Malaysia for providing the research funds

References

[1] R Kalatehjari and N Ali ldquoA review of three-dimensional slopestability analyses based on limit equilibriummethodrdquo ElectronicJournal of Geotechnical Engineering vol 18 pp 119ndash134 2013

[2] D G Fredlund ldquoAnalytical methods for slope stability analysisrdquoin Proceedings of the 4th International Symposium on LandslidesState-of-the-Art pp 229ndash250 Toronto Canada September1984

[3] A W Bishop ldquoThe use of the slip circle in the stability analysisof earth sloperdquo Geotechnique vol 5 no 1 pp 7ndash17 1955

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[5] Y Zhang P Agarwal V Bhatnagar S Balochian and J YanldquoSwarm intelligence and its applicationsrdquo The Scientific WorldJournal vol 2013 Article ID 528069 3 pages 2013

[6] G Poitras G Lefrancois and G Cormier ldquoOptimization ofsteel floor systems using particle swarm optimizationrdquo Journalof Constructional Steel Research vol 67 no 8 pp 1225ndash12312011

[7] S Gholizadeh and E Salajegheh ldquoOptimal design of structuressubjected to time history loading by swarm intelligence and anadvancedmetamodelrdquoComputerMethods in AppliedMechanicsand Engineering vol 198 no 37ndash40 pp 2936ndash2949 2009

[8] S S Gilan H B Jovein and A A Ramezanianpour ldquoHybridsupport vector regressionmdashparticle swarm optimization for

The Scientific World Journal 11

prediction of compressive strength and RCPT of concretescontaining metakaolinrdquo Construction and Building Materialsvol 34 pp 321ndash329 2012

[9] L Yunkai T Yingjie O Zhiyun et al ldquoAnalysis of soil ero-sion characteristics in small watersheds with particle swarmoptimization support vector machine and artificial neuronalnetworksrdquoEnvironmental Earth Sciences vol 60 no 7 pp 1559ndash1568 2010

[10] S Wan ldquoEntropy-based particle swarm optimization withclustering analysis on landslide susceptibility mappingrdquo Envi-ronmental Earth Sciences vol 68 no 5 pp 1349ndash1366 2013

[11] M R Nikoo R Kerachian A Karimi A A Azadnia andK Jafarzadegan ldquoOptimal water and waste load allocationin reservoir-river systems a case studyrdquo Environmental EarthSciences vol 71 no 9 pp 4127ndash4142 2014

[12] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for hourly rainfall-runoff modeling in Bedup Basin Malaysiardquo International Jour-nal of Civil amp Environmental Engineering vol 9 no 10 pp 9ndash182010

[13] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for modeling runoffrdquoInternational Journal of Environmental Science and Technologyvol 7 no 1 pp 67ndash78 2010

[14] J S Yazdi F Kalantary and H S Yazdi ldquoCalibration ofsoil model parameters using particle swarm optimizationrdquoInternational Journal of Geomechanics vol 12 no 3 pp 229ndash238 2012

[15] T V Bharat P V Sivapullaiah and M M Allam ldquoSwarmintelligence-based solver for parameter estimation of laboratorythrough-diffusion transport of contaminantsrdquo Computers andGeotechnics vol 36 no 6 pp 984ndash992 2009

[16] B R Chen and X T Feng ldquoCSV-PSO and its application ingeotechnical engineeringrdquo in Swarm Intelligence Focus on AntandParticle SwarmOptimization F T S Chan andMKTiwariEds chapter 15 pp 263ndash288 2007

[17] Y M Cheng L Li and S C Chi ldquoPerformance studies on sixheuristic global optimization methods in the location of criticalslip surfacerdquoComputers and Geotechnics vol 34 no 6 pp 462ndash484 2007

[18] Y M Cheng L Li Y J Sun and S K Au ldquoA coupled particleswarm and harmony search optimization algorithm for difficultgeotechnical problemsrdquo Structural and Multidisciplinary Opti-mization vol 45 no 4 pp 489ndash501 2012

[19] H B Zhao Z S Zou and Z L Ru ldquoChaotic particle swarmoptimization for non-circular critical slip surface identificationin slope stability analysisrdquo in Proceedings of the InternationalYoung Scholarsrsquo Symposium on Rock Mechanics Boundaries ofRock Mechanics Recent Advances and Challenges for the 21stCentury pp 585ndash588 Beijing China May 2008

[20] D Tian L Xu and S Wang ldquoThe application of particle swarmoptimization on the search of critical slip surfacerdquo in Proceed-ings of the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[21] L Li G Yu X Chu and S Lu ldquoThe harmony search algo-rithm in combination with particle swarm optimization andits application in the slope stability analysisrdquo in Proceedings ofthe International Conference on Computational Intelligence andSecurity (CIS rsquo09) vol 2 pp 133ndash136 Beijing China December2009

[22] R Kalatehjari N Ali M Kholghifard and M HajihassanildquoThe effects of method of generating circular slip surfaceson determining the critical slip surface by particle swarmoptimizationrdquo Arabian Journal of Geosciences vol 7 no 4 pp1529ndash1539 2014

[23] R Kalatehjari N Ali M Hajihassani and M K Fard ldquoTheapplication of particle swarm optimization in slope stabilityanalysis of homogeneous soil slopesrdquo International Review onModelling and Simulations vol 5 no 1 pp 458ndash465 2012

[24] W Chen and P Chen ldquoPSOslope a stand-alone windowsapplication for graphic analysis of slope stabilityrdquo in Advancesin Swarm Optimization Y Tan Y Shi Y Chai and G WangEds vol 6728 of Lecture Notes in Computer Science pp 56ndash632011

[25] T Yamagami and J C Jiang ldquoA search for the critical slipsurface in three-dimensional slope stability analysisrdquo Soils andFoundations vol 37 no 3 pp 1ndash16 1997

[26] J C Jiang T Yamagami and R Baker ldquoThree-dimensionalslope stability analysis based on nonlinear failure enveloperdquoChinese Journal of Rock Mechanics and Engineering vol 22 no6 pp 1017ndash1023 2003

[27] X J E Mowen ldquoA simple Monte Carlo method for locating thethree-dimensional critical slip surface of a sloperdquoActaGeologicaSinica vol 78 no 6 pp 1258ndash1266 2004

[28] X J E Mowen W Zengfu L Xiangyu and X Bo ldquoThree-dimensional critical slip surface locating and slope stabilityassessment for lava lobe of Unzen Volcanordquo Journal of RockMechanics and Geotechnical Engineering vol 3 no 1 pp 82ndash892011

[29] Y M Cheng H T Liu W B Wei and S K Au ldquoLocationof critical three-dimensional non-spherical failure surface byNURBS functions and ellipsoid with applications to highwayslopesrdquo Computers and Geotechnics vol 32 no 6 pp 387ndash3992005

[30] M Toufigh A Ahangarasr and A Ouria ldquoUsing non-linearprogramming techniques in determination of the most prob-able slip surface in 3D slopesrdquo World Academy of ScienceEngineering and Technology vol 17 pp 30ndash35 2006

[31] C W Reynolds ldquoFlocks herbs and schools a distributedbehavorial modelrdquo Computer Graphics vol 21 no 4 pp 25ndash341987

[32] R Kalatehjari An improvised three-dimensional slope stabilityanalysis based on limit equilibrium method by using particleswarm optimization [PhD dissertation] Faculty of Civil Engi-neering Universiti Teknologi Malaysia 2013

[33] A P Engelbrecht Computational intelligence an introductionJohn Wiley amp Sons 2007

[34] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tion an overviewrdquo Swarm Intelligence vol 1 no 1 pp 33ndash572007

[35] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 Piscataway NJ USAMay 1998

[36] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[37] R Mendes P Cortez M Rocha and J Neves ldquoParticle swarmsfor feedforward neural network trainingrdquo in Proceedings of theInternational Joint Conference on Neural Networks (IJCNN rsquo02)pp 1895ndash1899 Honolulu Hawaii USA May 2002

12 The Scientific World Journal

[38] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science pp 39ndash43 IEEENagoya Japan October 1995

[39] R Baker ldquoDetermination of the critical slip surface in slopestability computationsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 4 pp 333ndash3591980

[40] H P J Bolton G Heymann and A A Groenwold ldquoGlobalsearch for critical failure surface in slope stability analysisrdquoEngineering Optimization vol 35 no 1 pp 51ndash65 2003

[41] Y Chen X Wei and Y Li ldquoLocating non-circular critical slipsurfaces by particle swarm optimization algorithmrdquo ChineseJournal of Rock Mechanics and Engineering vol 25 no 7 pp1443ndash1449 2006

[42] L Li S-C Chi R-M Zheng G Lin and X-S Chu ldquoMixedsearch algorithmof non-circular critical slip surface of soil sloprdquoChina Journal of Highway and Transport vol 20 no 6 pp 1ndash62007

[43] L Li S-C Chi andY-M Zheng ldquoThree-dimensional slope sta-bility analysis based on ellipsoidal sliding body and simplifiedJANBU methodrdquo Rock and Soil Mechanics vol 29 no 9 pp2439ndash2445 2008

[44] W Alkasawneh A I H Malkawi J H Nusairat and NAlbataineh ldquoA comparative study of various commerciallyavailable programs in slope stability analysisrdquo Computers andGeotechnics vol 35 no 3 pp 428ndash435 2008

[45] T Yamagami K Kojima and M Taki ldquoA search for the three-dimensional critical slip surface with random generation ofsurfacerdquo in Proceedings of the 36th Symposium on Slope StabilityAnalyses and Stabilizing Construction Methods pp 27ndash34 1991

Submit your manuscripts athttpwwwhindawicom

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Human-ComputerInteraction

Advances in

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4 The Scientific World Journal

0

1

2

3

4

5

6

7

8

9

10

0 2 4 5 6 7 8 10

ParticlesLocal connections

1 3 9

Y-a

xis

X-axis

(a)

ParticlesGlobal connectionsGlobal best

0

1

2

3

4

5

6

7

8

9

10

0 2 4 5 6 7 8 101 3 9

Y-a

xis

X-axis

(b)

Figure 2 (a) Local and (b) global topologies in 2D search space

3 Application of PSO in SlopeStability Analysis

PSO is mainly applied in stability analysis of soil slopeswithin the framework of LEM [1] This analysis involves twoconsequent steps that is calculating FOS of candidate slipsurfaces and determining the CSS among all candidates [39]PSO commonly contributes to the second step to determinethe shape and location of the CSS which are generallyunknown in soil slopes [40]

PSO can be applied in both 2D and 3D slope stabilityanalyses [32] In 2D analysis it can be employed to determinethe shape of the CSS in a predefined 2D section of the slopeDifferent shapes are possible for slip surfaces in 2D analysissuch as circular ellipse spiral and polygonal or arbitrarysurfaces [22 41 42] In contrast 3D slip surfaces such asspherical ellipsoidal and Nonuniform Rational B-Splines(NURBS) are commonly assumed in 3D analysis [23 43 4344]

Figure 3 shows flowchart of PSO to determine the CSSin slope stability analysis The optimization procedure isstarted by setting initial parameters of PSO Then a certainnumber of particles (119873) is generated in a random patternover the search space Since the improvement of swarm hasjust begun personal bests of all particles in initial swarmare identical to the particles themselves Based on the samereason the velocity of all initial particles is set to zero Aftersetting up the initial values the first particle is arranged asits corresponding slip surface This surface is qualified if it

can satisfy the conditions of the problem Otherwise it isdisqualified A predefined minimum fitness value is given todisqualify slip surfaces This value represents a predefinedmaximum FOS For a qualified surface FOS is calculated andthe corresponding fitness values of particle are assigned bythe fitness function of PSO This process is repeated for allparticles of the swarm

Current positions of particles that improved their fitnessvalues are recorded to update their personal bests whileprevious personal bests are used for other particles Theglobal best particle is defined by the greatest fitness value inthe current swarm Through an iterative process subsequentswarms are generated by updating velocities and positionsof former particles The optimization process is terminatedby meeting the termination criteria Eventually the globalbest particle of the last swarm represents the CSS of theslope

31 Coding of the Particles The structure of particles followedthe standard PSO particles involving three sections as currentposition previous best position and the velocity A rotatingellipsoid was selected as the general 3D shape of slip surfacesThis ellipsoid can rotate on119909-119910plane (0 le 120579

119909119910le 120587) to provide

various slip surfaces (Figure 4)In order to achieve the equation of rotated ellipsoid the

parametric equation of general ellipsoid was transferred intonew axes by the rotation angle 120579

119909119910 The result presents the

rotated ellipsoid in (8) It should be noted that this ellipsoid

The Scientific World Journal 5

Start

Generate N particles by random

Set Pbest = particlesSet velocities = 0

Convert the first particleto slip surface

Is the No

YesCalculate 3D FOSof the slip surface

No

Calculate fitness

Was

No

Yes

Find Pbests and gbest

Are the

YesEnd

Con

vert

the n

ext

part

icle

to sl

ip

Calculate and updatenew positions of all particles

Calculate and updatevelocity of all particles

Initialize PSO parametersSwarm size = N

qualifiedslip surface

of the particle

particlethat the last

termination criteria met

surfa

ce

Set t

he fi

tnes

sas

min

imum

Figure 3 Flowchart of PSO to determine the CSS in slope stabilityanalysis

can be easily transformed to spherical and cylindrical slip sur-face by respectively setting equal three and two semiradiuses

(cos 120579119909119910(119909 minus 119883

119888) minus sin 120579

119909119910(119910 minus 119884

119888))

2

119877

2

119909

minus

(cos 120579119909119910(119910 minus 119884

119888) + sin 120579

119909119910(119909 minus 119883

119888))

2

119877

2

119910

+

(119911 minus 119885119888)

2

119877

2

119911

= 1

(8)

where 120579119909119910

is rotation angle of the ellipsoid in 119909-119910 plane 119883119888

119884119888 and 119885

119888are coordinates of center of ellipsoid in 119909- 119910- and

119911-directions 119877119909 119877119910 and 119877

119911are semiradiuses of ellipsoid in

119909- 119910- and 119911-directions and 119909 119910 and 119911 are coordinates of anarbitrary point on the surface of ellipsoid

Based on the parameters of the rotating ellipsoid Figure 5shows schematic structure of a PSO particle The section ofcurrent position records the coordinates of center of ellipsoidits semiradiuses and its rotation angle Considering bestposition and velocity sections PSO has seven-dimensionalsearch space and twenty-one-cell particles

Rx

Ry

120579xy

X

Y

120579xy = 0

120579xy = 1205874

120579xy = 1205872

120579xy = 31205874

120579xy = 120587

Figure 4 Projection of rotating ellipsoid on 119909-119910 plane

Xc Yc Zc Rx Ry Rz 120579xy

Xcp

Y cp

Z cp

R xp

R yp

R zp

120579 p

Vxc

Vyc

Vzc

Vry

Vrx

Vrz

V120579

Current position

Best

posit

ionVelocity

Figure 5 Schematic structures of PSO particles

32 Fitness Function The quality of particles can be cal-culated by the fitness function This function is related tothe objective function and provides quantitative tracking ofimprovement of particles Consequently it makes it possibleto compare and rank particles in the swarm wheremaximumfitness shows the best particle andminimum fitness identifiesthe worst particle of the swarm PSO attempts to increase themaximum fitness of swarms during its iterations Since theobjective function of 3D slope stability analysis is equation ofFOS PSO attempts to find the CSS with the minimum FOSby maximizing the fitness value in

Fitness119899(119894)

=

1

FOS (119909119899(119894))

(9)

where Fitness119899(119894)

is fitness value of the 119899th particle in 119894thiteration and FOS(119909

119899(119894)) is FOS of 3D slip surface described

by 119909119899(119894)

33 Sensitivity Analysis on PSO Parameters The best per-formance of PSO is guaranteed by initializing appropriateparameters for it A sensitivity analysis can help to do soThe optimum values PSO parameters in 3D slope stabilityanalysis were defined by designing and performing severalindependent tests on swarm size coefficients of velocity andinertia weight of the swarm In addition the convergence

6 The Scientific World Journal

Table 1 Properties of slope in sensitivity analysis

Parameters120574 (kNm3) 119888

1015840 (kNm2) 120601

1015840 (∘)Layer 1 1978 12 17Layer 2 1764 245 20

Table 2 Results of sensitivity tests on swarm size

Test number1 2 3 4 5 6 7

Swarm size 5 15 25 35 45 55 65Total CPU time (s) 365 625 549 1726 2792 2241 10101

behavior of PSO as the average fitness of swarms wasobserved during the tests A 3D soil slope was designedwith complex geometry layers of soil and piezometric line(Figure 6) It should be noted that coding of the study wasdone by the authors in MATLAB software (Licensed byUniversiti TeknologiMalaysia)The overall shape of the slopeshows two imbalanced hills with steep sides makes it difficultto find the CSS for conventional slope stability analysesThis specific shape was selected to verify the effectiveness ofPSO in complex 3D slopes The properties of soil layers aredescribed in Table 1

The size of the swarm is defined based on the conditionof search space dimension of particles andor other specifi-cations of the problem The most common population sizesare 20 to 50 [25] However Clerc and Kennedy [36] proposeda relationship to determine the optimum value of swarm sizeas follows

119873119904= 10 + [radic119863

119904] (10)

where 119873119904is swarm size 119863

119904is dimension of the particles

and [ ] is calculator of integer part Since the dimensionof particles in the present problem is seven the proposedoptimum swarm size by this equation is 12 Consideringthe most common range of the swarm size and the resultof equation an interval of swarm sizes was prepared forsensitivity test It should be noted that the first swarm ofall tests was produced by the same random pattern and themaximum iteration number was set to 100 for all tests Table 2shows the results of testsThe phrase ldquoCPU timerdquo in this tablemeans the exact amount of time that CPU spent on each testCPU time was used to produce fair comparisons since somefactors including the operating system and available memorycan affect the overall duration of the tests

Figure 7 illustrates the convergence behavior of PSOin corresponding swarm sizes of Table 2 Three differentconvergence behaviors as good late and failure can describethese trends Swarm sizes 5 15 25 35 and 55 providedgood convergence over maximum iterations while swarmsizes 45 and 65 respectively delivered delay and failurein convergence Among all the tests the best convergencewas obtained by swarm size of 35 that provided the bestconvergence with the highest average fitness

Layer 1 interfaceWater tableLayer 2 interface

5

45

4

35

3

25

2

15

1

05

05

4 32 1

0

Z-a

xis

X-axisY-axis

54

321

0

Figure 6 Generated slope model for sensitivity analysis

1

09

08

07

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 1Test 2Test 3Test 4

Test 5Test 6Test 7

Figure 7 Convergence behavior of PSO with different swarm sizes

The next tests were performed to find the optimumvaluesof coefficients 120599

1and 120599

2of velocity equation Based on the

original coefficients of Kennedy and Eberhart [4] and themodified coefficients of Clerc and Kennedy [36] a seriesof combinations were established as shown in Table 3 Alltests were performed by the same initial swarm with the sizeof 35 (previously obtained as optimum) and the maximumiterations of 100

The results can be presented in two separated groupsincluding unequal and equal coefficients Figure 8 illustratethe results of the tests The first group failed to converge overthe maximum iteration period but the second group showeddifferent performances Overall the best convergence and thegreatest average fitness belonged to equal coefficients of 175that makes it the optimum coefficient of velocity equation

The Scientific World Journal 7

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 1Test 2Test 3

Test 4Test 5Test 6

(a)

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 7Test 8Test 9

Test 10Test 11Test 12

(b)

Figure 8 Results of sensitivity tests on (a) unequal and (b) equal coefficients

Table 3 Combinations of velocity equation coefficients in differenttests

Test number Relationship 1205991

1205992

1205991+ 1205992

1 1205991= 025120599

20800 3200 4

2 1205991= 050120599

21333 2667 4

3 1205991= 075120599

21714 2286 4

4 1205992= 025120599

13200 0800 4

5 1205992= 050120599

12667 1333 4

6 1205992= 075120599

12286 1714 4

7 1205991= 1205992

2500 2500 58 120599

1= 1205992

2000 2000 49 120599

1= 1205992

1750 1750 3510 120599

1= 1205992

1500 1500 311 120599

1= 1205992

1000 1000 212 120599

1= 1205992

0500 0500 1

The last sensitivity tests were performed to find theoptimum inertia weight (120596) of velocity equation The sameinitial swarm with size of 35 and equal coefficients of velocityequation as 175 (previously defined as optimum values) wereapplied for all the tests Five tests with inertia weights of 0025 05 075 and 1 respectively were performed based onthe proposed values of Shi and Eberhart [35] and Clerc andKennedy [36] Figure 9 shows the convergence behavior ofPSO in the tests The results showed successful convergencein all tests except test 3 It should be noted that test 5 wasidentical with the original PSO where no inertia weightwas present in velocity equation Immature convergencehas occurred for tests 1 and 2 Although fast convergenceappears an advantage at first it is a sign of trapping aswarm in local solutions Test 3 failed to converge test 4 hadinstable convergence and test 5 failed to improve its averagefitness over themaximum iterations Consequently it was notpossible to introduce an optimum inertia weight to guarantee

0 10 20 30 40 50 60 70 80 90 100

Iteration

Test 1Test 2Test 3

Test 4Test 5

10908070605040302010

Aver

age fi

tnes

s

Figure 9 Results of sensitivity tests on inertia weight

the convergence of PSO and improvement of average fitnessover the maximum iteration number simultaneously

A dynamic inertia weight was utilized by the presentstudy to overcome the convergence problem of PSOThe pro-posed strategy started with the most anticonvergence inertiaweight (05) continued with the normal convergence inertiaweight (075) and ended upwith themost stable convergence(025) The switching levels of inertia weights were defined asone-third and two-thirds of maximum iterations Figure 10shows the results of sensitivity tests on dynamic inertiaweightwith different maximum iterations from 50 to 300 by stepsof 50 All tests performed well to converge and improve theaverage fitness over their maximum iterations so dynamicinertia weight was adopted for PSO

4 Example Problems

Two example problems were analyzed to verify the perfor-mance of PSO in determining the CSS The properties of theslope materials are shown in Table 4 Example problem 1 wasperformed to verify the performance of PSO in determining

8 The Scientific World Journal

06

05

04

03

02

01

00 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Aver

age fi

tnes

s

Iteration

Dynamic test 1Dynamic test 2Dynamic test 3

Dynamic test 4Dynamic test 5Dynamic test 6

Figure 10 Results of sensitivity tests on dynamic inertia weight

Table 4 Properties of slopes in example problems

Properties 1198881015840 (kNm2) 120601

1015840 (degree) 120574 (kNm3) 120592 119864 (kNm2)Problem 1 15 20 17 03 1119864 + 6

Problem 2 10 10 18 mdash mdash

the CSS in comparison with PLAXIS-3D finite element soft-ware (License byUniversiti TeknologiMalaysia) Alkasawnehet al [44] applied different search techniques to determinethe CSS in 2D slope stability analysis Figure 11 illustrates thegeometry of the slope A 3D model was developed based onthis 2D section inwhich the third dimensionwas extended by100 meters Figure 12 shows the generated 3D models of theslope by the present study and PLAXIS-3D In both methodscylindrical slip surface was employed to determine the CSS ofthe slope

PSO improved average and best fitness of the swarmas shown in Figure 13 Figure 14 shows the minimum FOSversus iterations PSO provided continuous reduction of FOSto find the CSS The present study obtained FOS of 178versus theminimumFOS of 177 of the PLAXIS-3D Figure 15shows the CSS obtained by the present study and the resultof PLAXIS-3D The present study and PLAXIS-3D obtainedsimilar FOS for the CSS with a small difference of 03 Thisresult demonstrates the ability of PSO to determine the CSSwith the minimum FOS in 3D slope stability analysis

Example problem 2 was performed to verify the abilityof PSO to determine the CSS with general ellipsoid shape ina comparison with previous studies from the literature Thisexample was initially analyzed by Yamagami et al [45] andwas reanalyzed by Yamagami and Jiang [25] Yamagami et al[45] used random generation of surfaces to determine theCSS of this slope while Yamagami and Jiang [25] employed acombination of dynamic programming and random numbergeneration to do so It should be noted that the same equationof FOS as previous studies was used to make fair comparisonof the results The example involved a homogeneous slopewith gradient of 2 1 subjected to a square load of 50 kPa onthe top The uniform load was applied on a square surface

30

20

10

00 20 40 60 80

Z(m

)

Y (m)

Figure 11 Geometry of 2D section of example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axis

Y-axis

(a)Z-a

xis

X-axisY-axis

(b)

Figure 12 Generated 3D models of example problem 1 by (a) thepresent study and (b) PLAXIS-3D

with 8 meters sides at the top center of the slope Figure 16illustrates the geometry of example problem 1

Figure 17 shows the generated 3D model by the presentstudy for example problem 2 Figure 18 plots the process ofPSO to improve fitness of the swarmThe trend of average fit-ness value of the swarm experienced some instability duringthe process The main reason of this behavior is the presenceof disqualified slip surfaces in the swarm that dramaticallydecreases the average fitness valueThese surfaces were rarelypresented in the previous example due to adopted simplercylindrical shape compared with more complicated ellipsoidshape in this example The trend of minimum FOS versusiterations is shown in Figure 19 Continuous decrement ofFOS by PSO leads to determining the CSS of the slope

In spite of similar equation of FOS the present studyfound the CSS with a smaller FOS than other methods whichis the best result so far The minimum FOS obtained by PSOwas 095 compared with 114 and 103 of random generationof surfaces [45] and DP with RNG [25] respectively Thisresult demonstrates the ability of PSO to accurately determinethe ellipsoid CSS in 3D slope stability analysis Figure 20

The Scientific World Journal 9

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80

Iteration

Fitn

ess v

alue

Average fitnessMaximum fitness

Figure 13 Fitness values versus iterations in example problem 1

215

21

205

2

195

19

1850 10 20 30 40 50 60 70 80

Iteration

Min

imum

fact

or o

f saf

ety

Figure 14 Minimum FOS versus iterations in example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axisY-axis

(a)

Z-a

xis

X-axis

Y-axis

(b)

Figure 15 (a) The CSS obtained by the present study and (b) exag-gerated displacement vectors of PLAXIS-3D in example problem 1

40

24

20

9

4

0

0 4 14 22 25

x(m

)z

(m)

y (m)

0 4 14 22 25

y (m)

12

(b)

(a)

Figure 16 (a) Half-plan view and (b) central cross-section of slopein example problem 2

10864200

510

1520

2530 35

40 05

1015

2025

Z-a

xis

X-axis Y-axis

Figure 17 Generated 3D models of example problem 2 by thepresent study

illustrates the CSS obtained by the present study in exampleproblem 2

5 Conclusion

Determining the critical slip surface of a soil slope is atraditional problem in geotechnical engineering which is stillchallenging for researchers This problem needs a massivesearching process Although classical searching methodswork for relatively simple problems they are surrounded bylocal minima Moreover their processes become particularlyslow by increasing the number of possible solutions Toeliminate these limitations PSO has been applied in slopestability analysis based on its successful results in advancedengineering problems However this contribution was lim-ited to 2D slope stability analysis This paper applied PSO in

10 The Scientific World Journal

12

1

08

06

04

02

00 10 20 30 40 50 60 70 80 90 100

Fitn

ess v

alue

Iteration

Average fitnessMaximum fitness

Figure 18 Fitness values versus iterations in example problem 2

3

25

35

2

15

1

05

00 10 20 30 40 50 60 70 80 90 100

Min

imum

fac

tor o

f saf

ety

Iteration

Figure 19 Minimum FOS versus iterations in example problem 2

10

8

6

4

2

0051015

2025

303540 0 5 10 15 20 25

Z-a

xis

X-axis

Y-axis

Figure 20 The CSS obtained by the present study in exampleproblem 2

3D slope stability problem to determine theCSS of soil slopesA detailed description of adopted PSO was presented toprovide a good basis for more contribution of this techniqueto the field of 3D slope stability problems

The application of PSO in slope stability analysis wasdescribed by presenting a general flowchart A general rotat-ing ellipsoid shape was introduced as the specific particle for3D slope stability analysis In order to find the optimum val-ues of parameters of PSO a sensitivity analysis was designedand performed The related codes were prepared by theauthors in MATLAB A 3D model with complex geometry

soil layers and piezometric line was used in the analysisThis analysis included three steps to find the optimum swarmsize coefficients and inertia weight of the velocity equationrespectively Moreover the performance of PSO to convergeover a global optimum solution was verified during thetests Based on the obtained values of parameters PSO wasprepared for 3D slope stability analysis

The applicability of PSO in determining the CSS of 3Dslopes was evaluated by analyzing two example problemsThe first example presented a comparison between the resultsof PSO and PLAXI-3D finite element software The secondexample compared the ability of PSO to determine the CSSof 3D slopes with other optimization methods from theliterature Both of the example problems demonstrated theefficiency and effectiveness of PSO in determining the CSS of3D soil slopes Based on the results it is believed that PSO ishighly capable of contributing to the field of 3D slope stabilityanalysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The presented study was undertaken with support fromresearch grant The authors are pleased to acknowledgethe Ministry of Education of Malaysia for the PrototypeResearch Grant Scheme (RJ13000078224L620) and Univer-siti Teknologi Malaysia for providing the research funds

References

[1] R Kalatehjari and N Ali ldquoA review of three-dimensional slopestability analyses based on limit equilibriummethodrdquo ElectronicJournal of Geotechnical Engineering vol 18 pp 119ndash134 2013

[2] D G Fredlund ldquoAnalytical methods for slope stability analysisrdquoin Proceedings of the 4th International Symposium on LandslidesState-of-the-Art pp 229ndash250 Toronto Canada September1984

[3] A W Bishop ldquoThe use of the slip circle in the stability analysisof earth sloperdquo Geotechnique vol 5 no 1 pp 7ndash17 1955

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[5] Y Zhang P Agarwal V Bhatnagar S Balochian and J YanldquoSwarm intelligence and its applicationsrdquo The Scientific WorldJournal vol 2013 Article ID 528069 3 pages 2013

[6] G Poitras G Lefrancois and G Cormier ldquoOptimization ofsteel floor systems using particle swarm optimizationrdquo Journalof Constructional Steel Research vol 67 no 8 pp 1225ndash12312011

[7] S Gholizadeh and E Salajegheh ldquoOptimal design of structuressubjected to time history loading by swarm intelligence and anadvancedmetamodelrdquoComputerMethods in AppliedMechanicsand Engineering vol 198 no 37ndash40 pp 2936ndash2949 2009

[8] S S Gilan H B Jovein and A A Ramezanianpour ldquoHybridsupport vector regressionmdashparticle swarm optimization for

The Scientific World Journal 11

prediction of compressive strength and RCPT of concretescontaining metakaolinrdquo Construction and Building Materialsvol 34 pp 321ndash329 2012

[9] L Yunkai T Yingjie O Zhiyun et al ldquoAnalysis of soil ero-sion characteristics in small watersheds with particle swarmoptimization support vector machine and artificial neuronalnetworksrdquoEnvironmental Earth Sciences vol 60 no 7 pp 1559ndash1568 2010

[10] S Wan ldquoEntropy-based particle swarm optimization withclustering analysis on landslide susceptibility mappingrdquo Envi-ronmental Earth Sciences vol 68 no 5 pp 1349ndash1366 2013

[11] M R Nikoo R Kerachian A Karimi A A Azadnia andK Jafarzadegan ldquoOptimal water and waste load allocationin reservoir-river systems a case studyrdquo Environmental EarthSciences vol 71 no 9 pp 4127ndash4142 2014

[12] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for hourly rainfall-runoff modeling in Bedup Basin Malaysiardquo International Jour-nal of Civil amp Environmental Engineering vol 9 no 10 pp 9ndash182010

[13] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for modeling runoffrdquoInternational Journal of Environmental Science and Technologyvol 7 no 1 pp 67ndash78 2010

[14] J S Yazdi F Kalantary and H S Yazdi ldquoCalibration ofsoil model parameters using particle swarm optimizationrdquoInternational Journal of Geomechanics vol 12 no 3 pp 229ndash238 2012

[15] T V Bharat P V Sivapullaiah and M M Allam ldquoSwarmintelligence-based solver for parameter estimation of laboratorythrough-diffusion transport of contaminantsrdquo Computers andGeotechnics vol 36 no 6 pp 984ndash992 2009

[16] B R Chen and X T Feng ldquoCSV-PSO and its application ingeotechnical engineeringrdquo in Swarm Intelligence Focus on AntandParticle SwarmOptimization F T S Chan andMKTiwariEds chapter 15 pp 263ndash288 2007

[17] Y M Cheng L Li and S C Chi ldquoPerformance studies on sixheuristic global optimization methods in the location of criticalslip surfacerdquoComputers and Geotechnics vol 34 no 6 pp 462ndash484 2007

[18] Y M Cheng L Li Y J Sun and S K Au ldquoA coupled particleswarm and harmony search optimization algorithm for difficultgeotechnical problemsrdquo Structural and Multidisciplinary Opti-mization vol 45 no 4 pp 489ndash501 2012

[19] H B Zhao Z S Zou and Z L Ru ldquoChaotic particle swarmoptimization for non-circular critical slip surface identificationin slope stability analysisrdquo in Proceedings of the InternationalYoung Scholarsrsquo Symposium on Rock Mechanics Boundaries ofRock Mechanics Recent Advances and Challenges for the 21stCentury pp 585ndash588 Beijing China May 2008

[20] D Tian L Xu and S Wang ldquoThe application of particle swarmoptimization on the search of critical slip surfacerdquo in Proceed-ings of the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[21] L Li G Yu X Chu and S Lu ldquoThe harmony search algo-rithm in combination with particle swarm optimization andits application in the slope stability analysisrdquo in Proceedings ofthe International Conference on Computational Intelligence andSecurity (CIS rsquo09) vol 2 pp 133ndash136 Beijing China December2009

[22] R Kalatehjari N Ali M Kholghifard and M HajihassanildquoThe effects of method of generating circular slip surfaceson determining the critical slip surface by particle swarmoptimizationrdquo Arabian Journal of Geosciences vol 7 no 4 pp1529ndash1539 2014

[23] R Kalatehjari N Ali M Hajihassani and M K Fard ldquoTheapplication of particle swarm optimization in slope stabilityanalysis of homogeneous soil slopesrdquo International Review onModelling and Simulations vol 5 no 1 pp 458ndash465 2012

[24] W Chen and P Chen ldquoPSOslope a stand-alone windowsapplication for graphic analysis of slope stabilityrdquo in Advancesin Swarm Optimization Y Tan Y Shi Y Chai and G WangEds vol 6728 of Lecture Notes in Computer Science pp 56ndash632011

[25] T Yamagami and J C Jiang ldquoA search for the critical slipsurface in three-dimensional slope stability analysisrdquo Soils andFoundations vol 37 no 3 pp 1ndash16 1997

[26] J C Jiang T Yamagami and R Baker ldquoThree-dimensionalslope stability analysis based on nonlinear failure enveloperdquoChinese Journal of Rock Mechanics and Engineering vol 22 no6 pp 1017ndash1023 2003

[27] X J E Mowen ldquoA simple Monte Carlo method for locating thethree-dimensional critical slip surface of a sloperdquoActaGeologicaSinica vol 78 no 6 pp 1258ndash1266 2004

[28] X J E Mowen W Zengfu L Xiangyu and X Bo ldquoThree-dimensional critical slip surface locating and slope stabilityassessment for lava lobe of Unzen Volcanordquo Journal of RockMechanics and Geotechnical Engineering vol 3 no 1 pp 82ndash892011

[29] Y M Cheng H T Liu W B Wei and S K Au ldquoLocationof critical three-dimensional non-spherical failure surface byNURBS functions and ellipsoid with applications to highwayslopesrdquo Computers and Geotechnics vol 32 no 6 pp 387ndash3992005

[30] M Toufigh A Ahangarasr and A Ouria ldquoUsing non-linearprogramming techniques in determination of the most prob-able slip surface in 3D slopesrdquo World Academy of ScienceEngineering and Technology vol 17 pp 30ndash35 2006

[31] C W Reynolds ldquoFlocks herbs and schools a distributedbehavorial modelrdquo Computer Graphics vol 21 no 4 pp 25ndash341987

[32] R Kalatehjari An improvised three-dimensional slope stabilityanalysis based on limit equilibrium method by using particleswarm optimization [PhD dissertation] Faculty of Civil Engi-neering Universiti Teknologi Malaysia 2013

[33] A P Engelbrecht Computational intelligence an introductionJohn Wiley amp Sons 2007

[34] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tion an overviewrdquo Swarm Intelligence vol 1 no 1 pp 33ndash572007

[35] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 Piscataway NJ USAMay 1998

[36] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[37] R Mendes P Cortez M Rocha and J Neves ldquoParticle swarmsfor feedforward neural network trainingrdquo in Proceedings of theInternational Joint Conference on Neural Networks (IJCNN rsquo02)pp 1895ndash1899 Honolulu Hawaii USA May 2002

12 The Scientific World Journal

[38] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science pp 39ndash43 IEEENagoya Japan October 1995

[39] R Baker ldquoDetermination of the critical slip surface in slopestability computationsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 4 pp 333ndash3591980

[40] H P J Bolton G Heymann and A A Groenwold ldquoGlobalsearch for critical failure surface in slope stability analysisrdquoEngineering Optimization vol 35 no 1 pp 51ndash65 2003

[41] Y Chen X Wei and Y Li ldquoLocating non-circular critical slipsurfaces by particle swarm optimization algorithmrdquo ChineseJournal of Rock Mechanics and Engineering vol 25 no 7 pp1443ndash1449 2006

[42] L Li S-C Chi R-M Zheng G Lin and X-S Chu ldquoMixedsearch algorithmof non-circular critical slip surface of soil sloprdquoChina Journal of Highway and Transport vol 20 no 6 pp 1ndash62007

[43] L Li S-C Chi andY-M Zheng ldquoThree-dimensional slope sta-bility analysis based on ellipsoidal sliding body and simplifiedJANBU methodrdquo Rock and Soil Mechanics vol 29 no 9 pp2439ndash2445 2008

[44] W Alkasawneh A I H Malkawi J H Nusairat and NAlbataineh ldquoA comparative study of various commerciallyavailable programs in slope stability analysisrdquo Computers andGeotechnics vol 35 no 3 pp 428ndash435 2008

[45] T Yamagami K Kojima and M Taki ldquoA search for the three-dimensional critical slip surface with random generation ofsurfacerdquo in Proceedings of the 36th Symposium on Slope StabilityAnalyses and Stabilizing Construction Methods pp 27ndash34 1991

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

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The Scientific World Journal 5

Start

Generate N particles by random

Set Pbest = particlesSet velocities = 0

Convert the first particleto slip surface

Is the No

YesCalculate 3D FOSof the slip surface

No

Calculate fitness

Was

No

Yes

Find Pbests and gbest

Are the

YesEnd

Con

vert

the n

ext

part

icle

to sl

ip

Calculate and updatenew positions of all particles

Calculate and updatevelocity of all particles

Initialize PSO parametersSwarm size = N

qualifiedslip surface

of the particle

particlethat the last

termination criteria met

surfa

ce

Set t

he fi

tnes

sas

min

imum

Figure 3 Flowchart of PSO to determine the CSS in slope stabilityanalysis

can be easily transformed to spherical and cylindrical slip sur-face by respectively setting equal three and two semiradiuses

(cos 120579119909119910(119909 minus 119883

119888) minus sin 120579

119909119910(119910 minus 119884

119888))

2

119877

2

119909

minus

(cos 120579119909119910(119910 minus 119884

119888) + sin 120579

119909119910(119909 minus 119883

119888))

2

119877

2

119910

+

(119911 minus 119885119888)

2

119877

2

119911

= 1

(8)

where 120579119909119910

is rotation angle of the ellipsoid in 119909-119910 plane 119883119888

119884119888 and 119885

119888are coordinates of center of ellipsoid in 119909- 119910- and

119911-directions 119877119909 119877119910 and 119877

119911are semiradiuses of ellipsoid in

119909- 119910- and 119911-directions and 119909 119910 and 119911 are coordinates of anarbitrary point on the surface of ellipsoid

Based on the parameters of the rotating ellipsoid Figure 5shows schematic structure of a PSO particle The section ofcurrent position records the coordinates of center of ellipsoidits semiradiuses and its rotation angle Considering bestposition and velocity sections PSO has seven-dimensionalsearch space and twenty-one-cell particles

Rx

Ry

120579xy

X

Y

120579xy = 0

120579xy = 1205874

120579xy = 1205872

120579xy = 31205874

120579xy = 120587

Figure 4 Projection of rotating ellipsoid on 119909-119910 plane

Xc Yc Zc Rx Ry Rz 120579xy

Xcp

Y cp

Z cp

R xp

R yp

R zp

120579 p

Vxc

Vyc

Vzc

Vry

Vrx

Vrz

V120579

Current position

Best

posit

ionVelocity

Figure 5 Schematic structures of PSO particles

32 Fitness Function The quality of particles can be cal-culated by the fitness function This function is related tothe objective function and provides quantitative tracking ofimprovement of particles Consequently it makes it possibleto compare and rank particles in the swarm wheremaximumfitness shows the best particle andminimum fitness identifiesthe worst particle of the swarm PSO attempts to increase themaximum fitness of swarms during its iterations Since theobjective function of 3D slope stability analysis is equation ofFOS PSO attempts to find the CSS with the minimum FOSby maximizing the fitness value in

Fitness119899(119894)

=

1

FOS (119909119899(119894))

(9)

where Fitness119899(119894)

is fitness value of the 119899th particle in 119894thiteration and FOS(119909

119899(119894)) is FOS of 3D slip surface described

by 119909119899(119894)

33 Sensitivity Analysis on PSO Parameters The best per-formance of PSO is guaranteed by initializing appropriateparameters for it A sensitivity analysis can help to do soThe optimum values PSO parameters in 3D slope stabilityanalysis were defined by designing and performing severalindependent tests on swarm size coefficients of velocity andinertia weight of the swarm In addition the convergence

6 The Scientific World Journal

Table 1 Properties of slope in sensitivity analysis

Parameters120574 (kNm3) 119888

1015840 (kNm2) 120601

1015840 (∘)Layer 1 1978 12 17Layer 2 1764 245 20

Table 2 Results of sensitivity tests on swarm size

Test number1 2 3 4 5 6 7

Swarm size 5 15 25 35 45 55 65Total CPU time (s) 365 625 549 1726 2792 2241 10101

behavior of PSO as the average fitness of swarms wasobserved during the tests A 3D soil slope was designedwith complex geometry layers of soil and piezometric line(Figure 6) It should be noted that coding of the study wasdone by the authors in MATLAB software (Licensed byUniversiti TeknologiMalaysia)The overall shape of the slopeshows two imbalanced hills with steep sides makes it difficultto find the CSS for conventional slope stability analysesThis specific shape was selected to verify the effectiveness ofPSO in complex 3D slopes The properties of soil layers aredescribed in Table 1

The size of the swarm is defined based on the conditionof search space dimension of particles andor other specifi-cations of the problem The most common population sizesare 20 to 50 [25] However Clerc and Kennedy [36] proposeda relationship to determine the optimum value of swarm sizeas follows

119873119904= 10 + [radic119863

119904] (10)

where 119873119904is swarm size 119863

119904is dimension of the particles

and [ ] is calculator of integer part Since the dimensionof particles in the present problem is seven the proposedoptimum swarm size by this equation is 12 Consideringthe most common range of the swarm size and the resultof equation an interval of swarm sizes was prepared forsensitivity test It should be noted that the first swarm ofall tests was produced by the same random pattern and themaximum iteration number was set to 100 for all tests Table 2shows the results of testsThe phrase ldquoCPU timerdquo in this tablemeans the exact amount of time that CPU spent on each testCPU time was used to produce fair comparisons since somefactors including the operating system and available memorycan affect the overall duration of the tests

Figure 7 illustrates the convergence behavior of PSOin corresponding swarm sizes of Table 2 Three differentconvergence behaviors as good late and failure can describethese trends Swarm sizes 5 15 25 35 and 55 providedgood convergence over maximum iterations while swarmsizes 45 and 65 respectively delivered delay and failurein convergence Among all the tests the best convergencewas obtained by swarm size of 35 that provided the bestconvergence with the highest average fitness

Layer 1 interfaceWater tableLayer 2 interface

5

45

4

35

3

25

2

15

1

05

05

4 32 1

0

Z-a

xis

X-axisY-axis

54

321

0

Figure 6 Generated slope model for sensitivity analysis

1

09

08

07

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 1Test 2Test 3Test 4

Test 5Test 6Test 7

Figure 7 Convergence behavior of PSO with different swarm sizes

The next tests were performed to find the optimumvaluesof coefficients 120599

1and 120599

2of velocity equation Based on the

original coefficients of Kennedy and Eberhart [4] and themodified coefficients of Clerc and Kennedy [36] a seriesof combinations were established as shown in Table 3 Alltests were performed by the same initial swarm with the sizeof 35 (previously obtained as optimum) and the maximumiterations of 100

The results can be presented in two separated groupsincluding unequal and equal coefficients Figure 8 illustratethe results of the tests The first group failed to converge overthe maximum iteration period but the second group showeddifferent performances Overall the best convergence and thegreatest average fitness belonged to equal coefficients of 175that makes it the optimum coefficient of velocity equation

The Scientific World Journal 7

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 1Test 2Test 3

Test 4Test 5Test 6

(a)

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 7Test 8Test 9

Test 10Test 11Test 12

(b)

Figure 8 Results of sensitivity tests on (a) unequal and (b) equal coefficients

Table 3 Combinations of velocity equation coefficients in differenttests

Test number Relationship 1205991

1205992

1205991+ 1205992

1 1205991= 025120599

20800 3200 4

2 1205991= 050120599

21333 2667 4

3 1205991= 075120599

21714 2286 4

4 1205992= 025120599

13200 0800 4

5 1205992= 050120599

12667 1333 4

6 1205992= 075120599

12286 1714 4

7 1205991= 1205992

2500 2500 58 120599

1= 1205992

2000 2000 49 120599

1= 1205992

1750 1750 3510 120599

1= 1205992

1500 1500 311 120599

1= 1205992

1000 1000 212 120599

1= 1205992

0500 0500 1

The last sensitivity tests were performed to find theoptimum inertia weight (120596) of velocity equation The sameinitial swarm with size of 35 and equal coefficients of velocityequation as 175 (previously defined as optimum values) wereapplied for all the tests Five tests with inertia weights of 0025 05 075 and 1 respectively were performed based onthe proposed values of Shi and Eberhart [35] and Clerc andKennedy [36] Figure 9 shows the convergence behavior ofPSO in the tests The results showed successful convergencein all tests except test 3 It should be noted that test 5 wasidentical with the original PSO where no inertia weightwas present in velocity equation Immature convergencehas occurred for tests 1 and 2 Although fast convergenceappears an advantage at first it is a sign of trapping aswarm in local solutions Test 3 failed to converge test 4 hadinstable convergence and test 5 failed to improve its averagefitness over themaximum iterations Consequently it was notpossible to introduce an optimum inertia weight to guarantee

0 10 20 30 40 50 60 70 80 90 100

Iteration

Test 1Test 2Test 3

Test 4Test 5

10908070605040302010

Aver

age fi

tnes

s

Figure 9 Results of sensitivity tests on inertia weight

the convergence of PSO and improvement of average fitnessover the maximum iteration number simultaneously

A dynamic inertia weight was utilized by the presentstudy to overcome the convergence problem of PSOThe pro-posed strategy started with the most anticonvergence inertiaweight (05) continued with the normal convergence inertiaweight (075) and ended upwith themost stable convergence(025) The switching levels of inertia weights were defined asone-third and two-thirds of maximum iterations Figure 10shows the results of sensitivity tests on dynamic inertiaweightwith different maximum iterations from 50 to 300 by stepsof 50 All tests performed well to converge and improve theaverage fitness over their maximum iterations so dynamicinertia weight was adopted for PSO

4 Example Problems

Two example problems were analyzed to verify the perfor-mance of PSO in determining the CSS The properties of theslope materials are shown in Table 4 Example problem 1 wasperformed to verify the performance of PSO in determining

8 The Scientific World Journal

06

05

04

03

02

01

00 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Aver

age fi

tnes

s

Iteration

Dynamic test 1Dynamic test 2Dynamic test 3

Dynamic test 4Dynamic test 5Dynamic test 6

Figure 10 Results of sensitivity tests on dynamic inertia weight

Table 4 Properties of slopes in example problems

Properties 1198881015840 (kNm2) 120601

1015840 (degree) 120574 (kNm3) 120592 119864 (kNm2)Problem 1 15 20 17 03 1119864 + 6

Problem 2 10 10 18 mdash mdash

the CSS in comparison with PLAXIS-3D finite element soft-ware (License byUniversiti TeknologiMalaysia) Alkasawnehet al [44] applied different search techniques to determinethe CSS in 2D slope stability analysis Figure 11 illustrates thegeometry of the slope A 3D model was developed based onthis 2D section inwhich the third dimensionwas extended by100 meters Figure 12 shows the generated 3D models of theslope by the present study and PLAXIS-3D In both methodscylindrical slip surface was employed to determine the CSS ofthe slope

PSO improved average and best fitness of the swarmas shown in Figure 13 Figure 14 shows the minimum FOSversus iterations PSO provided continuous reduction of FOSto find the CSS The present study obtained FOS of 178versus theminimumFOS of 177 of the PLAXIS-3D Figure 15shows the CSS obtained by the present study and the resultof PLAXIS-3D The present study and PLAXIS-3D obtainedsimilar FOS for the CSS with a small difference of 03 Thisresult demonstrates the ability of PSO to determine the CSSwith the minimum FOS in 3D slope stability analysis

Example problem 2 was performed to verify the abilityof PSO to determine the CSS with general ellipsoid shape ina comparison with previous studies from the literature Thisexample was initially analyzed by Yamagami et al [45] andwas reanalyzed by Yamagami and Jiang [25] Yamagami et al[45] used random generation of surfaces to determine theCSS of this slope while Yamagami and Jiang [25] employed acombination of dynamic programming and random numbergeneration to do so It should be noted that the same equationof FOS as previous studies was used to make fair comparisonof the results The example involved a homogeneous slopewith gradient of 2 1 subjected to a square load of 50 kPa onthe top The uniform load was applied on a square surface

30

20

10

00 20 40 60 80

Z(m

)

Y (m)

Figure 11 Geometry of 2D section of example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axis

Y-axis

(a)Z-a

xis

X-axisY-axis

(b)

Figure 12 Generated 3D models of example problem 1 by (a) thepresent study and (b) PLAXIS-3D

with 8 meters sides at the top center of the slope Figure 16illustrates the geometry of example problem 1

Figure 17 shows the generated 3D model by the presentstudy for example problem 2 Figure 18 plots the process ofPSO to improve fitness of the swarmThe trend of average fit-ness value of the swarm experienced some instability duringthe process The main reason of this behavior is the presenceof disqualified slip surfaces in the swarm that dramaticallydecreases the average fitness valueThese surfaces were rarelypresented in the previous example due to adopted simplercylindrical shape compared with more complicated ellipsoidshape in this example The trend of minimum FOS versusiterations is shown in Figure 19 Continuous decrement ofFOS by PSO leads to determining the CSS of the slope

In spite of similar equation of FOS the present studyfound the CSS with a smaller FOS than other methods whichis the best result so far The minimum FOS obtained by PSOwas 095 compared with 114 and 103 of random generationof surfaces [45] and DP with RNG [25] respectively Thisresult demonstrates the ability of PSO to accurately determinethe ellipsoid CSS in 3D slope stability analysis Figure 20

The Scientific World Journal 9

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80

Iteration

Fitn

ess v

alue

Average fitnessMaximum fitness

Figure 13 Fitness values versus iterations in example problem 1

215

21

205

2

195

19

1850 10 20 30 40 50 60 70 80

Iteration

Min

imum

fact

or o

f saf

ety

Figure 14 Minimum FOS versus iterations in example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axisY-axis

(a)

Z-a

xis

X-axis

Y-axis

(b)

Figure 15 (a) The CSS obtained by the present study and (b) exag-gerated displacement vectors of PLAXIS-3D in example problem 1

40

24

20

9

4

0

0 4 14 22 25

x(m

)z

(m)

y (m)

0 4 14 22 25

y (m)

12

(b)

(a)

Figure 16 (a) Half-plan view and (b) central cross-section of slopein example problem 2

10864200

510

1520

2530 35

40 05

1015

2025

Z-a

xis

X-axis Y-axis

Figure 17 Generated 3D models of example problem 2 by thepresent study

illustrates the CSS obtained by the present study in exampleproblem 2

5 Conclusion

Determining the critical slip surface of a soil slope is atraditional problem in geotechnical engineering which is stillchallenging for researchers This problem needs a massivesearching process Although classical searching methodswork for relatively simple problems they are surrounded bylocal minima Moreover their processes become particularlyslow by increasing the number of possible solutions Toeliminate these limitations PSO has been applied in slopestability analysis based on its successful results in advancedengineering problems However this contribution was lim-ited to 2D slope stability analysis This paper applied PSO in

10 The Scientific World Journal

12

1

08

06

04

02

00 10 20 30 40 50 60 70 80 90 100

Fitn

ess v

alue

Iteration

Average fitnessMaximum fitness

Figure 18 Fitness values versus iterations in example problem 2

3

25

35

2

15

1

05

00 10 20 30 40 50 60 70 80 90 100

Min

imum

fac

tor o

f saf

ety

Iteration

Figure 19 Minimum FOS versus iterations in example problem 2

10

8

6

4

2

0051015

2025

303540 0 5 10 15 20 25

Z-a

xis

X-axis

Y-axis

Figure 20 The CSS obtained by the present study in exampleproblem 2

3D slope stability problem to determine theCSS of soil slopesA detailed description of adopted PSO was presented toprovide a good basis for more contribution of this techniqueto the field of 3D slope stability problems

The application of PSO in slope stability analysis wasdescribed by presenting a general flowchart A general rotat-ing ellipsoid shape was introduced as the specific particle for3D slope stability analysis In order to find the optimum val-ues of parameters of PSO a sensitivity analysis was designedand performed The related codes were prepared by theauthors in MATLAB A 3D model with complex geometry

soil layers and piezometric line was used in the analysisThis analysis included three steps to find the optimum swarmsize coefficients and inertia weight of the velocity equationrespectively Moreover the performance of PSO to convergeover a global optimum solution was verified during thetests Based on the obtained values of parameters PSO wasprepared for 3D slope stability analysis

The applicability of PSO in determining the CSS of 3Dslopes was evaluated by analyzing two example problemsThe first example presented a comparison between the resultsof PSO and PLAXI-3D finite element software The secondexample compared the ability of PSO to determine the CSSof 3D slopes with other optimization methods from theliterature Both of the example problems demonstrated theefficiency and effectiveness of PSO in determining the CSS of3D soil slopes Based on the results it is believed that PSO ishighly capable of contributing to the field of 3D slope stabilityanalysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The presented study was undertaken with support fromresearch grant The authors are pleased to acknowledgethe Ministry of Education of Malaysia for the PrototypeResearch Grant Scheme (RJ13000078224L620) and Univer-siti Teknologi Malaysia for providing the research funds

References

[1] R Kalatehjari and N Ali ldquoA review of three-dimensional slopestability analyses based on limit equilibriummethodrdquo ElectronicJournal of Geotechnical Engineering vol 18 pp 119ndash134 2013

[2] D G Fredlund ldquoAnalytical methods for slope stability analysisrdquoin Proceedings of the 4th International Symposium on LandslidesState-of-the-Art pp 229ndash250 Toronto Canada September1984

[3] A W Bishop ldquoThe use of the slip circle in the stability analysisof earth sloperdquo Geotechnique vol 5 no 1 pp 7ndash17 1955

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[5] Y Zhang P Agarwal V Bhatnagar S Balochian and J YanldquoSwarm intelligence and its applicationsrdquo The Scientific WorldJournal vol 2013 Article ID 528069 3 pages 2013

[6] G Poitras G Lefrancois and G Cormier ldquoOptimization ofsteel floor systems using particle swarm optimizationrdquo Journalof Constructional Steel Research vol 67 no 8 pp 1225ndash12312011

[7] S Gholizadeh and E Salajegheh ldquoOptimal design of structuressubjected to time history loading by swarm intelligence and anadvancedmetamodelrdquoComputerMethods in AppliedMechanicsand Engineering vol 198 no 37ndash40 pp 2936ndash2949 2009

[8] S S Gilan H B Jovein and A A Ramezanianpour ldquoHybridsupport vector regressionmdashparticle swarm optimization for

The Scientific World Journal 11

prediction of compressive strength and RCPT of concretescontaining metakaolinrdquo Construction and Building Materialsvol 34 pp 321ndash329 2012

[9] L Yunkai T Yingjie O Zhiyun et al ldquoAnalysis of soil ero-sion characteristics in small watersheds with particle swarmoptimization support vector machine and artificial neuronalnetworksrdquoEnvironmental Earth Sciences vol 60 no 7 pp 1559ndash1568 2010

[10] S Wan ldquoEntropy-based particle swarm optimization withclustering analysis on landslide susceptibility mappingrdquo Envi-ronmental Earth Sciences vol 68 no 5 pp 1349ndash1366 2013

[11] M R Nikoo R Kerachian A Karimi A A Azadnia andK Jafarzadegan ldquoOptimal water and waste load allocationin reservoir-river systems a case studyrdquo Environmental EarthSciences vol 71 no 9 pp 4127ndash4142 2014

[12] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for hourly rainfall-runoff modeling in Bedup Basin Malaysiardquo International Jour-nal of Civil amp Environmental Engineering vol 9 no 10 pp 9ndash182010

[13] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for modeling runoffrdquoInternational Journal of Environmental Science and Technologyvol 7 no 1 pp 67ndash78 2010

[14] J S Yazdi F Kalantary and H S Yazdi ldquoCalibration ofsoil model parameters using particle swarm optimizationrdquoInternational Journal of Geomechanics vol 12 no 3 pp 229ndash238 2012

[15] T V Bharat P V Sivapullaiah and M M Allam ldquoSwarmintelligence-based solver for parameter estimation of laboratorythrough-diffusion transport of contaminantsrdquo Computers andGeotechnics vol 36 no 6 pp 984ndash992 2009

[16] B R Chen and X T Feng ldquoCSV-PSO and its application ingeotechnical engineeringrdquo in Swarm Intelligence Focus on AntandParticle SwarmOptimization F T S Chan andMKTiwariEds chapter 15 pp 263ndash288 2007

[17] Y M Cheng L Li and S C Chi ldquoPerformance studies on sixheuristic global optimization methods in the location of criticalslip surfacerdquoComputers and Geotechnics vol 34 no 6 pp 462ndash484 2007

[18] Y M Cheng L Li Y J Sun and S K Au ldquoA coupled particleswarm and harmony search optimization algorithm for difficultgeotechnical problemsrdquo Structural and Multidisciplinary Opti-mization vol 45 no 4 pp 489ndash501 2012

[19] H B Zhao Z S Zou and Z L Ru ldquoChaotic particle swarmoptimization for non-circular critical slip surface identificationin slope stability analysisrdquo in Proceedings of the InternationalYoung Scholarsrsquo Symposium on Rock Mechanics Boundaries ofRock Mechanics Recent Advances and Challenges for the 21stCentury pp 585ndash588 Beijing China May 2008

[20] D Tian L Xu and S Wang ldquoThe application of particle swarmoptimization on the search of critical slip surfacerdquo in Proceed-ings of the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[21] L Li G Yu X Chu and S Lu ldquoThe harmony search algo-rithm in combination with particle swarm optimization andits application in the slope stability analysisrdquo in Proceedings ofthe International Conference on Computational Intelligence andSecurity (CIS rsquo09) vol 2 pp 133ndash136 Beijing China December2009

[22] R Kalatehjari N Ali M Kholghifard and M HajihassanildquoThe effects of method of generating circular slip surfaceson determining the critical slip surface by particle swarmoptimizationrdquo Arabian Journal of Geosciences vol 7 no 4 pp1529ndash1539 2014

[23] R Kalatehjari N Ali M Hajihassani and M K Fard ldquoTheapplication of particle swarm optimization in slope stabilityanalysis of homogeneous soil slopesrdquo International Review onModelling and Simulations vol 5 no 1 pp 458ndash465 2012

[24] W Chen and P Chen ldquoPSOslope a stand-alone windowsapplication for graphic analysis of slope stabilityrdquo in Advancesin Swarm Optimization Y Tan Y Shi Y Chai and G WangEds vol 6728 of Lecture Notes in Computer Science pp 56ndash632011

[25] T Yamagami and J C Jiang ldquoA search for the critical slipsurface in three-dimensional slope stability analysisrdquo Soils andFoundations vol 37 no 3 pp 1ndash16 1997

[26] J C Jiang T Yamagami and R Baker ldquoThree-dimensionalslope stability analysis based on nonlinear failure enveloperdquoChinese Journal of Rock Mechanics and Engineering vol 22 no6 pp 1017ndash1023 2003

[27] X J E Mowen ldquoA simple Monte Carlo method for locating thethree-dimensional critical slip surface of a sloperdquoActaGeologicaSinica vol 78 no 6 pp 1258ndash1266 2004

[28] X J E Mowen W Zengfu L Xiangyu and X Bo ldquoThree-dimensional critical slip surface locating and slope stabilityassessment for lava lobe of Unzen Volcanordquo Journal of RockMechanics and Geotechnical Engineering vol 3 no 1 pp 82ndash892011

[29] Y M Cheng H T Liu W B Wei and S K Au ldquoLocationof critical three-dimensional non-spherical failure surface byNURBS functions and ellipsoid with applications to highwayslopesrdquo Computers and Geotechnics vol 32 no 6 pp 387ndash3992005

[30] M Toufigh A Ahangarasr and A Ouria ldquoUsing non-linearprogramming techniques in determination of the most prob-able slip surface in 3D slopesrdquo World Academy of ScienceEngineering and Technology vol 17 pp 30ndash35 2006

[31] C W Reynolds ldquoFlocks herbs and schools a distributedbehavorial modelrdquo Computer Graphics vol 21 no 4 pp 25ndash341987

[32] R Kalatehjari An improvised three-dimensional slope stabilityanalysis based on limit equilibrium method by using particleswarm optimization [PhD dissertation] Faculty of Civil Engi-neering Universiti Teknologi Malaysia 2013

[33] A P Engelbrecht Computational intelligence an introductionJohn Wiley amp Sons 2007

[34] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tion an overviewrdquo Swarm Intelligence vol 1 no 1 pp 33ndash572007

[35] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 Piscataway NJ USAMay 1998

[36] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[37] R Mendes P Cortez M Rocha and J Neves ldquoParticle swarmsfor feedforward neural network trainingrdquo in Proceedings of theInternational Joint Conference on Neural Networks (IJCNN rsquo02)pp 1895ndash1899 Honolulu Hawaii USA May 2002

12 The Scientific World Journal

[38] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science pp 39ndash43 IEEENagoya Japan October 1995

[39] R Baker ldquoDetermination of the critical slip surface in slopestability computationsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 4 pp 333ndash3591980

[40] H P J Bolton G Heymann and A A Groenwold ldquoGlobalsearch for critical failure surface in slope stability analysisrdquoEngineering Optimization vol 35 no 1 pp 51ndash65 2003

[41] Y Chen X Wei and Y Li ldquoLocating non-circular critical slipsurfaces by particle swarm optimization algorithmrdquo ChineseJournal of Rock Mechanics and Engineering vol 25 no 7 pp1443ndash1449 2006

[42] L Li S-C Chi R-M Zheng G Lin and X-S Chu ldquoMixedsearch algorithmof non-circular critical slip surface of soil sloprdquoChina Journal of Highway and Transport vol 20 no 6 pp 1ndash62007

[43] L Li S-C Chi andY-M Zheng ldquoThree-dimensional slope sta-bility analysis based on ellipsoidal sliding body and simplifiedJANBU methodrdquo Rock and Soil Mechanics vol 29 no 9 pp2439ndash2445 2008

[44] W Alkasawneh A I H Malkawi J H Nusairat and NAlbataineh ldquoA comparative study of various commerciallyavailable programs in slope stability analysisrdquo Computers andGeotechnics vol 35 no 3 pp 428ndash435 2008

[45] T Yamagami K Kojima and M Taki ldquoA search for the three-dimensional critical slip surface with random generation ofsurfacerdquo in Proceedings of the 36th Symposium on Slope StabilityAnalyses and Stabilizing Construction Methods pp 27ndash34 1991

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

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Applied Computational Intelligence and Soft Computing

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Artificial Intelligence

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Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 The Scientific World Journal

Table 1 Properties of slope in sensitivity analysis

Parameters120574 (kNm3) 119888

1015840 (kNm2) 120601

1015840 (∘)Layer 1 1978 12 17Layer 2 1764 245 20

Table 2 Results of sensitivity tests on swarm size

Test number1 2 3 4 5 6 7

Swarm size 5 15 25 35 45 55 65Total CPU time (s) 365 625 549 1726 2792 2241 10101

behavior of PSO as the average fitness of swarms wasobserved during the tests A 3D soil slope was designedwith complex geometry layers of soil and piezometric line(Figure 6) It should be noted that coding of the study wasdone by the authors in MATLAB software (Licensed byUniversiti TeknologiMalaysia)The overall shape of the slopeshows two imbalanced hills with steep sides makes it difficultto find the CSS for conventional slope stability analysesThis specific shape was selected to verify the effectiveness ofPSO in complex 3D slopes The properties of soil layers aredescribed in Table 1

The size of the swarm is defined based on the conditionof search space dimension of particles andor other specifi-cations of the problem The most common population sizesare 20 to 50 [25] However Clerc and Kennedy [36] proposeda relationship to determine the optimum value of swarm sizeas follows

119873119904= 10 + [radic119863

119904] (10)

where 119873119904is swarm size 119863

119904is dimension of the particles

and [ ] is calculator of integer part Since the dimensionof particles in the present problem is seven the proposedoptimum swarm size by this equation is 12 Consideringthe most common range of the swarm size and the resultof equation an interval of swarm sizes was prepared forsensitivity test It should be noted that the first swarm ofall tests was produced by the same random pattern and themaximum iteration number was set to 100 for all tests Table 2shows the results of testsThe phrase ldquoCPU timerdquo in this tablemeans the exact amount of time that CPU spent on each testCPU time was used to produce fair comparisons since somefactors including the operating system and available memorycan affect the overall duration of the tests

Figure 7 illustrates the convergence behavior of PSOin corresponding swarm sizes of Table 2 Three differentconvergence behaviors as good late and failure can describethese trends Swarm sizes 5 15 25 35 and 55 providedgood convergence over maximum iterations while swarmsizes 45 and 65 respectively delivered delay and failurein convergence Among all the tests the best convergencewas obtained by swarm size of 35 that provided the bestconvergence with the highest average fitness

Layer 1 interfaceWater tableLayer 2 interface

5

45

4

35

3

25

2

15

1

05

05

4 32 1

0

Z-a

xis

X-axisY-axis

54

321

0

Figure 6 Generated slope model for sensitivity analysis

1

09

08

07

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 1Test 2Test 3Test 4

Test 5Test 6Test 7

Figure 7 Convergence behavior of PSO with different swarm sizes

The next tests were performed to find the optimumvaluesof coefficients 120599

1and 120599

2of velocity equation Based on the

original coefficients of Kennedy and Eberhart [4] and themodified coefficients of Clerc and Kennedy [36] a seriesof combinations were established as shown in Table 3 Alltests were performed by the same initial swarm with the sizeof 35 (previously obtained as optimum) and the maximumiterations of 100

The results can be presented in two separated groupsincluding unequal and equal coefficients Figure 8 illustratethe results of the tests The first group failed to converge overthe maximum iteration period but the second group showeddifferent performances Overall the best convergence and thegreatest average fitness belonged to equal coefficients of 175that makes it the optimum coefficient of velocity equation

The Scientific World Journal 7

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 1Test 2Test 3

Test 4Test 5Test 6

(a)

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 7Test 8Test 9

Test 10Test 11Test 12

(b)

Figure 8 Results of sensitivity tests on (a) unequal and (b) equal coefficients

Table 3 Combinations of velocity equation coefficients in differenttests

Test number Relationship 1205991

1205992

1205991+ 1205992

1 1205991= 025120599

20800 3200 4

2 1205991= 050120599

21333 2667 4

3 1205991= 075120599

21714 2286 4

4 1205992= 025120599

13200 0800 4

5 1205992= 050120599

12667 1333 4

6 1205992= 075120599

12286 1714 4

7 1205991= 1205992

2500 2500 58 120599

1= 1205992

2000 2000 49 120599

1= 1205992

1750 1750 3510 120599

1= 1205992

1500 1500 311 120599

1= 1205992

1000 1000 212 120599

1= 1205992

0500 0500 1

The last sensitivity tests were performed to find theoptimum inertia weight (120596) of velocity equation The sameinitial swarm with size of 35 and equal coefficients of velocityequation as 175 (previously defined as optimum values) wereapplied for all the tests Five tests with inertia weights of 0025 05 075 and 1 respectively were performed based onthe proposed values of Shi and Eberhart [35] and Clerc andKennedy [36] Figure 9 shows the convergence behavior ofPSO in the tests The results showed successful convergencein all tests except test 3 It should be noted that test 5 wasidentical with the original PSO where no inertia weightwas present in velocity equation Immature convergencehas occurred for tests 1 and 2 Although fast convergenceappears an advantage at first it is a sign of trapping aswarm in local solutions Test 3 failed to converge test 4 hadinstable convergence and test 5 failed to improve its averagefitness over themaximum iterations Consequently it was notpossible to introduce an optimum inertia weight to guarantee

0 10 20 30 40 50 60 70 80 90 100

Iteration

Test 1Test 2Test 3

Test 4Test 5

10908070605040302010

Aver

age fi

tnes

s

Figure 9 Results of sensitivity tests on inertia weight

the convergence of PSO and improvement of average fitnessover the maximum iteration number simultaneously

A dynamic inertia weight was utilized by the presentstudy to overcome the convergence problem of PSOThe pro-posed strategy started with the most anticonvergence inertiaweight (05) continued with the normal convergence inertiaweight (075) and ended upwith themost stable convergence(025) The switching levels of inertia weights were defined asone-third and two-thirds of maximum iterations Figure 10shows the results of sensitivity tests on dynamic inertiaweightwith different maximum iterations from 50 to 300 by stepsof 50 All tests performed well to converge and improve theaverage fitness over their maximum iterations so dynamicinertia weight was adopted for PSO

4 Example Problems

Two example problems were analyzed to verify the perfor-mance of PSO in determining the CSS The properties of theslope materials are shown in Table 4 Example problem 1 wasperformed to verify the performance of PSO in determining

8 The Scientific World Journal

06

05

04

03

02

01

00 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Aver

age fi

tnes

s

Iteration

Dynamic test 1Dynamic test 2Dynamic test 3

Dynamic test 4Dynamic test 5Dynamic test 6

Figure 10 Results of sensitivity tests on dynamic inertia weight

Table 4 Properties of slopes in example problems

Properties 1198881015840 (kNm2) 120601

1015840 (degree) 120574 (kNm3) 120592 119864 (kNm2)Problem 1 15 20 17 03 1119864 + 6

Problem 2 10 10 18 mdash mdash

the CSS in comparison with PLAXIS-3D finite element soft-ware (License byUniversiti TeknologiMalaysia) Alkasawnehet al [44] applied different search techniques to determinethe CSS in 2D slope stability analysis Figure 11 illustrates thegeometry of the slope A 3D model was developed based onthis 2D section inwhich the third dimensionwas extended by100 meters Figure 12 shows the generated 3D models of theslope by the present study and PLAXIS-3D In both methodscylindrical slip surface was employed to determine the CSS ofthe slope

PSO improved average and best fitness of the swarmas shown in Figure 13 Figure 14 shows the minimum FOSversus iterations PSO provided continuous reduction of FOSto find the CSS The present study obtained FOS of 178versus theminimumFOS of 177 of the PLAXIS-3D Figure 15shows the CSS obtained by the present study and the resultof PLAXIS-3D The present study and PLAXIS-3D obtainedsimilar FOS for the CSS with a small difference of 03 Thisresult demonstrates the ability of PSO to determine the CSSwith the minimum FOS in 3D slope stability analysis

Example problem 2 was performed to verify the abilityof PSO to determine the CSS with general ellipsoid shape ina comparison with previous studies from the literature Thisexample was initially analyzed by Yamagami et al [45] andwas reanalyzed by Yamagami and Jiang [25] Yamagami et al[45] used random generation of surfaces to determine theCSS of this slope while Yamagami and Jiang [25] employed acombination of dynamic programming and random numbergeneration to do so It should be noted that the same equationof FOS as previous studies was used to make fair comparisonof the results The example involved a homogeneous slopewith gradient of 2 1 subjected to a square load of 50 kPa onthe top The uniform load was applied on a square surface

30

20

10

00 20 40 60 80

Z(m

)

Y (m)

Figure 11 Geometry of 2D section of example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axis

Y-axis

(a)Z-a

xis

X-axisY-axis

(b)

Figure 12 Generated 3D models of example problem 1 by (a) thepresent study and (b) PLAXIS-3D

with 8 meters sides at the top center of the slope Figure 16illustrates the geometry of example problem 1

Figure 17 shows the generated 3D model by the presentstudy for example problem 2 Figure 18 plots the process ofPSO to improve fitness of the swarmThe trend of average fit-ness value of the swarm experienced some instability duringthe process The main reason of this behavior is the presenceof disqualified slip surfaces in the swarm that dramaticallydecreases the average fitness valueThese surfaces were rarelypresented in the previous example due to adopted simplercylindrical shape compared with more complicated ellipsoidshape in this example The trend of minimum FOS versusiterations is shown in Figure 19 Continuous decrement ofFOS by PSO leads to determining the CSS of the slope

In spite of similar equation of FOS the present studyfound the CSS with a smaller FOS than other methods whichis the best result so far The minimum FOS obtained by PSOwas 095 compared with 114 and 103 of random generationof surfaces [45] and DP with RNG [25] respectively Thisresult demonstrates the ability of PSO to accurately determinethe ellipsoid CSS in 3D slope stability analysis Figure 20

The Scientific World Journal 9

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80

Iteration

Fitn

ess v

alue

Average fitnessMaximum fitness

Figure 13 Fitness values versus iterations in example problem 1

215

21

205

2

195

19

1850 10 20 30 40 50 60 70 80

Iteration

Min

imum

fact

or o

f saf

ety

Figure 14 Minimum FOS versus iterations in example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axisY-axis

(a)

Z-a

xis

X-axis

Y-axis

(b)

Figure 15 (a) The CSS obtained by the present study and (b) exag-gerated displacement vectors of PLAXIS-3D in example problem 1

40

24

20

9

4

0

0 4 14 22 25

x(m

)z

(m)

y (m)

0 4 14 22 25

y (m)

12

(b)

(a)

Figure 16 (a) Half-plan view and (b) central cross-section of slopein example problem 2

10864200

510

1520

2530 35

40 05

1015

2025

Z-a

xis

X-axis Y-axis

Figure 17 Generated 3D models of example problem 2 by thepresent study

illustrates the CSS obtained by the present study in exampleproblem 2

5 Conclusion

Determining the critical slip surface of a soil slope is atraditional problem in geotechnical engineering which is stillchallenging for researchers This problem needs a massivesearching process Although classical searching methodswork for relatively simple problems they are surrounded bylocal minima Moreover their processes become particularlyslow by increasing the number of possible solutions Toeliminate these limitations PSO has been applied in slopestability analysis based on its successful results in advancedengineering problems However this contribution was lim-ited to 2D slope stability analysis This paper applied PSO in

10 The Scientific World Journal

12

1

08

06

04

02

00 10 20 30 40 50 60 70 80 90 100

Fitn

ess v

alue

Iteration

Average fitnessMaximum fitness

Figure 18 Fitness values versus iterations in example problem 2

3

25

35

2

15

1

05

00 10 20 30 40 50 60 70 80 90 100

Min

imum

fac

tor o

f saf

ety

Iteration

Figure 19 Minimum FOS versus iterations in example problem 2

10

8

6

4

2

0051015

2025

303540 0 5 10 15 20 25

Z-a

xis

X-axis

Y-axis

Figure 20 The CSS obtained by the present study in exampleproblem 2

3D slope stability problem to determine theCSS of soil slopesA detailed description of adopted PSO was presented toprovide a good basis for more contribution of this techniqueto the field of 3D slope stability problems

The application of PSO in slope stability analysis wasdescribed by presenting a general flowchart A general rotat-ing ellipsoid shape was introduced as the specific particle for3D slope stability analysis In order to find the optimum val-ues of parameters of PSO a sensitivity analysis was designedand performed The related codes were prepared by theauthors in MATLAB A 3D model with complex geometry

soil layers and piezometric line was used in the analysisThis analysis included three steps to find the optimum swarmsize coefficients and inertia weight of the velocity equationrespectively Moreover the performance of PSO to convergeover a global optimum solution was verified during thetests Based on the obtained values of parameters PSO wasprepared for 3D slope stability analysis

The applicability of PSO in determining the CSS of 3Dslopes was evaluated by analyzing two example problemsThe first example presented a comparison between the resultsof PSO and PLAXI-3D finite element software The secondexample compared the ability of PSO to determine the CSSof 3D slopes with other optimization methods from theliterature Both of the example problems demonstrated theefficiency and effectiveness of PSO in determining the CSS of3D soil slopes Based on the results it is believed that PSO ishighly capable of contributing to the field of 3D slope stabilityanalysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The presented study was undertaken with support fromresearch grant The authors are pleased to acknowledgethe Ministry of Education of Malaysia for the PrototypeResearch Grant Scheme (RJ13000078224L620) and Univer-siti Teknologi Malaysia for providing the research funds

References

[1] R Kalatehjari and N Ali ldquoA review of three-dimensional slopestability analyses based on limit equilibriummethodrdquo ElectronicJournal of Geotechnical Engineering vol 18 pp 119ndash134 2013

[2] D G Fredlund ldquoAnalytical methods for slope stability analysisrdquoin Proceedings of the 4th International Symposium on LandslidesState-of-the-Art pp 229ndash250 Toronto Canada September1984

[3] A W Bishop ldquoThe use of the slip circle in the stability analysisof earth sloperdquo Geotechnique vol 5 no 1 pp 7ndash17 1955

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[5] Y Zhang P Agarwal V Bhatnagar S Balochian and J YanldquoSwarm intelligence and its applicationsrdquo The Scientific WorldJournal vol 2013 Article ID 528069 3 pages 2013

[6] G Poitras G Lefrancois and G Cormier ldquoOptimization ofsteel floor systems using particle swarm optimizationrdquo Journalof Constructional Steel Research vol 67 no 8 pp 1225ndash12312011

[7] S Gholizadeh and E Salajegheh ldquoOptimal design of structuressubjected to time history loading by swarm intelligence and anadvancedmetamodelrdquoComputerMethods in AppliedMechanicsand Engineering vol 198 no 37ndash40 pp 2936ndash2949 2009

[8] S S Gilan H B Jovein and A A Ramezanianpour ldquoHybridsupport vector regressionmdashparticle swarm optimization for

The Scientific World Journal 11

prediction of compressive strength and RCPT of concretescontaining metakaolinrdquo Construction and Building Materialsvol 34 pp 321ndash329 2012

[9] L Yunkai T Yingjie O Zhiyun et al ldquoAnalysis of soil ero-sion characteristics in small watersheds with particle swarmoptimization support vector machine and artificial neuronalnetworksrdquoEnvironmental Earth Sciences vol 60 no 7 pp 1559ndash1568 2010

[10] S Wan ldquoEntropy-based particle swarm optimization withclustering analysis on landslide susceptibility mappingrdquo Envi-ronmental Earth Sciences vol 68 no 5 pp 1349ndash1366 2013

[11] M R Nikoo R Kerachian A Karimi A A Azadnia andK Jafarzadegan ldquoOptimal water and waste load allocationin reservoir-river systems a case studyrdquo Environmental EarthSciences vol 71 no 9 pp 4127ndash4142 2014

[12] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for hourly rainfall-runoff modeling in Bedup Basin Malaysiardquo International Jour-nal of Civil amp Environmental Engineering vol 9 no 10 pp 9ndash182010

[13] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for modeling runoffrdquoInternational Journal of Environmental Science and Technologyvol 7 no 1 pp 67ndash78 2010

[14] J S Yazdi F Kalantary and H S Yazdi ldquoCalibration ofsoil model parameters using particle swarm optimizationrdquoInternational Journal of Geomechanics vol 12 no 3 pp 229ndash238 2012

[15] T V Bharat P V Sivapullaiah and M M Allam ldquoSwarmintelligence-based solver for parameter estimation of laboratorythrough-diffusion transport of contaminantsrdquo Computers andGeotechnics vol 36 no 6 pp 984ndash992 2009

[16] B R Chen and X T Feng ldquoCSV-PSO and its application ingeotechnical engineeringrdquo in Swarm Intelligence Focus on AntandParticle SwarmOptimization F T S Chan andMKTiwariEds chapter 15 pp 263ndash288 2007

[17] Y M Cheng L Li and S C Chi ldquoPerformance studies on sixheuristic global optimization methods in the location of criticalslip surfacerdquoComputers and Geotechnics vol 34 no 6 pp 462ndash484 2007

[18] Y M Cheng L Li Y J Sun and S K Au ldquoA coupled particleswarm and harmony search optimization algorithm for difficultgeotechnical problemsrdquo Structural and Multidisciplinary Opti-mization vol 45 no 4 pp 489ndash501 2012

[19] H B Zhao Z S Zou and Z L Ru ldquoChaotic particle swarmoptimization for non-circular critical slip surface identificationin slope stability analysisrdquo in Proceedings of the InternationalYoung Scholarsrsquo Symposium on Rock Mechanics Boundaries ofRock Mechanics Recent Advances and Challenges for the 21stCentury pp 585ndash588 Beijing China May 2008

[20] D Tian L Xu and S Wang ldquoThe application of particle swarmoptimization on the search of critical slip surfacerdquo in Proceed-ings of the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[21] L Li G Yu X Chu and S Lu ldquoThe harmony search algo-rithm in combination with particle swarm optimization andits application in the slope stability analysisrdquo in Proceedings ofthe International Conference on Computational Intelligence andSecurity (CIS rsquo09) vol 2 pp 133ndash136 Beijing China December2009

[22] R Kalatehjari N Ali M Kholghifard and M HajihassanildquoThe effects of method of generating circular slip surfaceson determining the critical slip surface by particle swarmoptimizationrdquo Arabian Journal of Geosciences vol 7 no 4 pp1529ndash1539 2014

[23] R Kalatehjari N Ali M Hajihassani and M K Fard ldquoTheapplication of particle swarm optimization in slope stabilityanalysis of homogeneous soil slopesrdquo International Review onModelling and Simulations vol 5 no 1 pp 458ndash465 2012

[24] W Chen and P Chen ldquoPSOslope a stand-alone windowsapplication for graphic analysis of slope stabilityrdquo in Advancesin Swarm Optimization Y Tan Y Shi Y Chai and G WangEds vol 6728 of Lecture Notes in Computer Science pp 56ndash632011

[25] T Yamagami and J C Jiang ldquoA search for the critical slipsurface in three-dimensional slope stability analysisrdquo Soils andFoundations vol 37 no 3 pp 1ndash16 1997

[26] J C Jiang T Yamagami and R Baker ldquoThree-dimensionalslope stability analysis based on nonlinear failure enveloperdquoChinese Journal of Rock Mechanics and Engineering vol 22 no6 pp 1017ndash1023 2003

[27] X J E Mowen ldquoA simple Monte Carlo method for locating thethree-dimensional critical slip surface of a sloperdquoActaGeologicaSinica vol 78 no 6 pp 1258ndash1266 2004

[28] X J E Mowen W Zengfu L Xiangyu and X Bo ldquoThree-dimensional critical slip surface locating and slope stabilityassessment for lava lobe of Unzen Volcanordquo Journal of RockMechanics and Geotechnical Engineering vol 3 no 1 pp 82ndash892011

[29] Y M Cheng H T Liu W B Wei and S K Au ldquoLocationof critical three-dimensional non-spherical failure surface byNURBS functions and ellipsoid with applications to highwayslopesrdquo Computers and Geotechnics vol 32 no 6 pp 387ndash3992005

[30] M Toufigh A Ahangarasr and A Ouria ldquoUsing non-linearprogramming techniques in determination of the most prob-able slip surface in 3D slopesrdquo World Academy of ScienceEngineering and Technology vol 17 pp 30ndash35 2006

[31] C W Reynolds ldquoFlocks herbs and schools a distributedbehavorial modelrdquo Computer Graphics vol 21 no 4 pp 25ndash341987

[32] R Kalatehjari An improvised three-dimensional slope stabilityanalysis based on limit equilibrium method by using particleswarm optimization [PhD dissertation] Faculty of Civil Engi-neering Universiti Teknologi Malaysia 2013

[33] A P Engelbrecht Computational intelligence an introductionJohn Wiley amp Sons 2007

[34] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tion an overviewrdquo Swarm Intelligence vol 1 no 1 pp 33ndash572007

[35] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 Piscataway NJ USAMay 1998

[36] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[37] R Mendes P Cortez M Rocha and J Neves ldquoParticle swarmsfor feedforward neural network trainingrdquo in Proceedings of theInternational Joint Conference on Neural Networks (IJCNN rsquo02)pp 1895ndash1899 Honolulu Hawaii USA May 2002

12 The Scientific World Journal

[38] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science pp 39ndash43 IEEENagoya Japan October 1995

[39] R Baker ldquoDetermination of the critical slip surface in slopestability computationsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 4 pp 333ndash3591980

[40] H P J Bolton G Heymann and A A Groenwold ldquoGlobalsearch for critical failure surface in slope stability analysisrdquoEngineering Optimization vol 35 no 1 pp 51ndash65 2003

[41] Y Chen X Wei and Y Li ldquoLocating non-circular critical slipsurfaces by particle swarm optimization algorithmrdquo ChineseJournal of Rock Mechanics and Engineering vol 25 no 7 pp1443ndash1449 2006

[42] L Li S-C Chi R-M Zheng G Lin and X-S Chu ldquoMixedsearch algorithmof non-circular critical slip surface of soil sloprdquoChina Journal of Highway and Transport vol 20 no 6 pp 1ndash62007

[43] L Li S-C Chi andY-M Zheng ldquoThree-dimensional slope sta-bility analysis based on ellipsoidal sliding body and simplifiedJANBU methodrdquo Rock and Soil Mechanics vol 29 no 9 pp2439ndash2445 2008

[44] W Alkasawneh A I H Malkawi J H Nusairat and NAlbataineh ldquoA comparative study of various commerciallyavailable programs in slope stability analysisrdquo Computers andGeotechnics vol 35 no 3 pp 428ndash435 2008

[45] T Yamagami K Kojima and M Taki ldquoA search for the three-dimensional critical slip surface with random generation ofsurfacerdquo in Proceedings of the 36th Symposium on Slope StabilityAnalyses and Stabilizing Construction Methods pp 27ndash34 1991

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 7

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 1Test 2Test 3

Test 4Test 5Test 6

(a)

01005

05505045040350302502015

00

10 20 30 40 50 60 70 80 90 100

Aver

age fi

tnes

s

Iteration

Test 7Test 8Test 9

Test 10Test 11Test 12

(b)

Figure 8 Results of sensitivity tests on (a) unequal and (b) equal coefficients

Table 3 Combinations of velocity equation coefficients in differenttests

Test number Relationship 1205991

1205992

1205991+ 1205992

1 1205991= 025120599

20800 3200 4

2 1205991= 050120599

21333 2667 4

3 1205991= 075120599

21714 2286 4

4 1205992= 025120599

13200 0800 4

5 1205992= 050120599

12667 1333 4

6 1205992= 075120599

12286 1714 4

7 1205991= 1205992

2500 2500 58 120599

1= 1205992

2000 2000 49 120599

1= 1205992

1750 1750 3510 120599

1= 1205992

1500 1500 311 120599

1= 1205992

1000 1000 212 120599

1= 1205992

0500 0500 1

The last sensitivity tests were performed to find theoptimum inertia weight (120596) of velocity equation The sameinitial swarm with size of 35 and equal coefficients of velocityequation as 175 (previously defined as optimum values) wereapplied for all the tests Five tests with inertia weights of 0025 05 075 and 1 respectively were performed based onthe proposed values of Shi and Eberhart [35] and Clerc andKennedy [36] Figure 9 shows the convergence behavior ofPSO in the tests The results showed successful convergencein all tests except test 3 It should be noted that test 5 wasidentical with the original PSO where no inertia weightwas present in velocity equation Immature convergencehas occurred for tests 1 and 2 Although fast convergenceappears an advantage at first it is a sign of trapping aswarm in local solutions Test 3 failed to converge test 4 hadinstable convergence and test 5 failed to improve its averagefitness over themaximum iterations Consequently it was notpossible to introduce an optimum inertia weight to guarantee

0 10 20 30 40 50 60 70 80 90 100

Iteration

Test 1Test 2Test 3

Test 4Test 5

10908070605040302010

Aver

age fi

tnes

s

Figure 9 Results of sensitivity tests on inertia weight

the convergence of PSO and improvement of average fitnessover the maximum iteration number simultaneously

A dynamic inertia weight was utilized by the presentstudy to overcome the convergence problem of PSOThe pro-posed strategy started with the most anticonvergence inertiaweight (05) continued with the normal convergence inertiaweight (075) and ended upwith themost stable convergence(025) The switching levels of inertia weights were defined asone-third and two-thirds of maximum iterations Figure 10shows the results of sensitivity tests on dynamic inertiaweightwith different maximum iterations from 50 to 300 by stepsof 50 All tests performed well to converge and improve theaverage fitness over their maximum iterations so dynamicinertia weight was adopted for PSO

4 Example Problems

Two example problems were analyzed to verify the perfor-mance of PSO in determining the CSS The properties of theslope materials are shown in Table 4 Example problem 1 wasperformed to verify the performance of PSO in determining

8 The Scientific World Journal

06

05

04

03

02

01

00 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Aver

age fi

tnes

s

Iteration

Dynamic test 1Dynamic test 2Dynamic test 3

Dynamic test 4Dynamic test 5Dynamic test 6

Figure 10 Results of sensitivity tests on dynamic inertia weight

Table 4 Properties of slopes in example problems

Properties 1198881015840 (kNm2) 120601

1015840 (degree) 120574 (kNm3) 120592 119864 (kNm2)Problem 1 15 20 17 03 1119864 + 6

Problem 2 10 10 18 mdash mdash

the CSS in comparison with PLAXIS-3D finite element soft-ware (License byUniversiti TeknologiMalaysia) Alkasawnehet al [44] applied different search techniques to determinethe CSS in 2D slope stability analysis Figure 11 illustrates thegeometry of the slope A 3D model was developed based onthis 2D section inwhich the third dimensionwas extended by100 meters Figure 12 shows the generated 3D models of theslope by the present study and PLAXIS-3D In both methodscylindrical slip surface was employed to determine the CSS ofthe slope

PSO improved average and best fitness of the swarmas shown in Figure 13 Figure 14 shows the minimum FOSversus iterations PSO provided continuous reduction of FOSto find the CSS The present study obtained FOS of 178versus theminimumFOS of 177 of the PLAXIS-3D Figure 15shows the CSS obtained by the present study and the resultof PLAXIS-3D The present study and PLAXIS-3D obtainedsimilar FOS for the CSS with a small difference of 03 Thisresult demonstrates the ability of PSO to determine the CSSwith the minimum FOS in 3D slope stability analysis

Example problem 2 was performed to verify the abilityof PSO to determine the CSS with general ellipsoid shape ina comparison with previous studies from the literature Thisexample was initially analyzed by Yamagami et al [45] andwas reanalyzed by Yamagami and Jiang [25] Yamagami et al[45] used random generation of surfaces to determine theCSS of this slope while Yamagami and Jiang [25] employed acombination of dynamic programming and random numbergeneration to do so It should be noted that the same equationof FOS as previous studies was used to make fair comparisonof the results The example involved a homogeneous slopewith gradient of 2 1 subjected to a square load of 50 kPa onthe top The uniform load was applied on a square surface

30

20

10

00 20 40 60 80

Z(m

)

Y (m)

Figure 11 Geometry of 2D section of example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axis

Y-axis

(a)Z-a

xis

X-axisY-axis

(b)

Figure 12 Generated 3D models of example problem 1 by (a) thepresent study and (b) PLAXIS-3D

with 8 meters sides at the top center of the slope Figure 16illustrates the geometry of example problem 1

Figure 17 shows the generated 3D model by the presentstudy for example problem 2 Figure 18 plots the process ofPSO to improve fitness of the swarmThe trend of average fit-ness value of the swarm experienced some instability duringthe process The main reason of this behavior is the presenceof disqualified slip surfaces in the swarm that dramaticallydecreases the average fitness valueThese surfaces were rarelypresented in the previous example due to adopted simplercylindrical shape compared with more complicated ellipsoidshape in this example The trend of minimum FOS versusiterations is shown in Figure 19 Continuous decrement ofFOS by PSO leads to determining the CSS of the slope

In spite of similar equation of FOS the present studyfound the CSS with a smaller FOS than other methods whichis the best result so far The minimum FOS obtained by PSOwas 095 compared with 114 and 103 of random generationof surfaces [45] and DP with RNG [25] respectively Thisresult demonstrates the ability of PSO to accurately determinethe ellipsoid CSS in 3D slope stability analysis Figure 20

The Scientific World Journal 9

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80

Iteration

Fitn

ess v

alue

Average fitnessMaximum fitness

Figure 13 Fitness values versus iterations in example problem 1

215

21

205

2

195

19

1850 10 20 30 40 50 60 70 80

Iteration

Min

imum

fact

or o

f saf

ety

Figure 14 Minimum FOS versus iterations in example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axisY-axis

(a)

Z-a

xis

X-axis

Y-axis

(b)

Figure 15 (a) The CSS obtained by the present study and (b) exag-gerated displacement vectors of PLAXIS-3D in example problem 1

40

24

20

9

4

0

0 4 14 22 25

x(m

)z

(m)

y (m)

0 4 14 22 25

y (m)

12

(b)

(a)

Figure 16 (a) Half-plan view and (b) central cross-section of slopein example problem 2

10864200

510

1520

2530 35

40 05

1015

2025

Z-a

xis

X-axis Y-axis

Figure 17 Generated 3D models of example problem 2 by thepresent study

illustrates the CSS obtained by the present study in exampleproblem 2

5 Conclusion

Determining the critical slip surface of a soil slope is atraditional problem in geotechnical engineering which is stillchallenging for researchers This problem needs a massivesearching process Although classical searching methodswork for relatively simple problems they are surrounded bylocal minima Moreover their processes become particularlyslow by increasing the number of possible solutions Toeliminate these limitations PSO has been applied in slopestability analysis based on its successful results in advancedengineering problems However this contribution was lim-ited to 2D slope stability analysis This paper applied PSO in

10 The Scientific World Journal

12

1

08

06

04

02

00 10 20 30 40 50 60 70 80 90 100

Fitn

ess v

alue

Iteration

Average fitnessMaximum fitness

Figure 18 Fitness values versus iterations in example problem 2

3

25

35

2

15

1

05

00 10 20 30 40 50 60 70 80 90 100

Min

imum

fac

tor o

f saf

ety

Iteration

Figure 19 Minimum FOS versus iterations in example problem 2

10

8

6

4

2

0051015

2025

303540 0 5 10 15 20 25

Z-a

xis

X-axis

Y-axis

Figure 20 The CSS obtained by the present study in exampleproblem 2

3D slope stability problem to determine theCSS of soil slopesA detailed description of adopted PSO was presented toprovide a good basis for more contribution of this techniqueto the field of 3D slope stability problems

The application of PSO in slope stability analysis wasdescribed by presenting a general flowchart A general rotat-ing ellipsoid shape was introduced as the specific particle for3D slope stability analysis In order to find the optimum val-ues of parameters of PSO a sensitivity analysis was designedand performed The related codes were prepared by theauthors in MATLAB A 3D model with complex geometry

soil layers and piezometric line was used in the analysisThis analysis included three steps to find the optimum swarmsize coefficients and inertia weight of the velocity equationrespectively Moreover the performance of PSO to convergeover a global optimum solution was verified during thetests Based on the obtained values of parameters PSO wasprepared for 3D slope stability analysis

The applicability of PSO in determining the CSS of 3Dslopes was evaluated by analyzing two example problemsThe first example presented a comparison between the resultsof PSO and PLAXI-3D finite element software The secondexample compared the ability of PSO to determine the CSSof 3D slopes with other optimization methods from theliterature Both of the example problems demonstrated theefficiency and effectiveness of PSO in determining the CSS of3D soil slopes Based on the results it is believed that PSO ishighly capable of contributing to the field of 3D slope stabilityanalysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The presented study was undertaken with support fromresearch grant The authors are pleased to acknowledgethe Ministry of Education of Malaysia for the PrototypeResearch Grant Scheme (RJ13000078224L620) and Univer-siti Teknologi Malaysia for providing the research funds

References

[1] R Kalatehjari and N Ali ldquoA review of three-dimensional slopestability analyses based on limit equilibriummethodrdquo ElectronicJournal of Geotechnical Engineering vol 18 pp 119ndash134 2013

[2] D G Fredlund ldquoAnalytical methods for slope stability analysisrdquoin Proceedings of the 4th International Symposium on LandslidesState-of-the-Art pp 229ndash250 Toronto Canada September1984

[3] A W Bishop ldquoThe use of the slip circle in the stability analysisof earth sloperdquo Geotechnique vol 5 no 1 pp 7ndash17 1955

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[5] Y Zhang P Agarwal V Bhatnagar S Balochian and J YanldquoSwarm intelligence and its applicationsrdquo The Scientific WorldJournal vol 2013 Article ID 528069 3 pages 2013

[6] G Poitras G Lefrancois and G Cormier ldquoOptimization ofsteel floor systems using particle swarm optimizationrdquo Journalof Constructional Steel Research vol 67 no 8 pp 1225ndash12312011

[7] S Gholizadeh and E Salajegheh ldquoOptimal design of structuressubjected to time history loading by swarm intelligence and anadvancedmetamodelrdquoComputerMethods in AppliedMechanicsand Engineering vol 198 no 37ndash40 pp 2936ndash2949 2009

[8] S S Gilan H B Jovein and A A Ramezanianpour ldquoHybridsupport vector regressionmdashparticle swarm optimization for

The Scientific World Journal 11

prediction of compressive strength and RCPT of concretescontaining metakaolinrdquo Construction and Building Materialsvol 34 pp 321ndash329 2012

[9] L Yunkai T Yingjie O Zhiyun et al ldquoAnalysis of soil ero-sion characteristics in small watersheds with particle swarmoptimization support vector machine and artificial neuronalnetworksrdquoEnvironmental Earth Sciences vol 60 no 7 pp 1559ndash1568 2010

[10] S Wan ldquoEntropy-based particle swarm optimization withclustering analysis on landslide susceptibility mappingrdquo Envi-ronmental Earth Sciences vol 68 no 5 pp 1349ndash1366 2013

[11] M R Nikoo R Kerachian A Karimi A A Azadnia andK Jafarzadegan ldquoOptimal water and waste load allocationin reservoir-river systems a case studyrdquo Environmental EarthSciences vol 71 no 9 pp 4127ndash4142 2014

[12] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for hourly rainfall-runoff modeling in Bedup Basin Malaysiardquo International Jour-nal of Civil amp Environmental Engineering vol 9 no 10 pp 9ndash182010

[13] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for modeling runoffrdquoInternational Journal of Environmental Science and Technologyvol 7 no 1 pp 67ndash78 2010

[14] J S Yazdi F Kalantary and H S Yazdi ldquoCalibration ofsoil model parameters using particle swarm optimizationrdquoInternational Journal of Geomechanics vol 12 no 3 pp 229ndash238 2012

[15] T V Bharat P V Sivapullaiah and M M Allam ldquoSwarmintelligence-based solver for parameter estimation of laboratorythrough-diffusion transport of contaminantsrdquo Computers andGeotechnics vol 36 no 6 pp 984ndash992 2009

[16] B R Chen and X T Feng ldquoCSV-PSO and its application ingeotechnical engineeringrdquo in Swarm Intelligence Focus on AntandParticle SwarmOptimization F T S Chan andMKTiwariEds chapter 15 pp 263ndash288 2007

[17] Y M Cheng L Li and S C Chi ldquoPerformance studies on sixheuristic global optimization methods in the location of criticalslip surfacerdquoComputers and Geotechnics vol 34 no 6 pp 462ndash484 2007

[18] Y M Cheng L Li Y J Sun and S K Au ldquoA coupled particleswarm and harmony search optimization algorithm for difficultgeotechnical problemsrdquo Structural and Multidisciplinary Opti-mization vol 45 no 4 pp 489ndash501 2012

[19] H B Zhao Z S Zou and Z L Ru ldquoChaotic particle swarmoptimization for non-circular critical slip surface identificationin slope stability analysisrdquo in Proceedings of the InternationalYoung Scholarsrsquo Symposium on Rock Mechanics Boundaries ofRock Mechanics Recent Advances and Challenges for the 21stCentury pp 585ndash588 Beijing China May 2008

[20] D Tian L Xu and S Wang ldquoThe application of particle swarmoptimization on the search of critical slip surfacerdquo in Proceed-ings of the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[21] L Li G Yu X Chu and S Lu ldquoThe harmony search algo-rithm in combination with particle swarm optimization andits application in the slope stability analysisrdquo in Proceedings ofthe International Conference on Computational Intelligence andSecurity (CIS rsquo09) vol 2 pp 133ndash136 Beijing China December2009

[22] R Kalatehjari N Ali M Kholghifard and M HajihassanildquoThe effects of method of generating circular slip surfaceson determining the critical slip surface by particle swarmoptimizationrdquo Arabian Journal of Geosciences vol 7 no 4 pp1529ndash1539 2014

[23] R Kalatehjari N Ali M Hajihassani and M K Fard ldquoTheapplication of particle swarm optimization in slope stabilityanalysis of homogeneous soil slopesrdquo International Review onModelling and Simulations vol 5 no 1 pp 458ndash465 2012

[24] W Chen and P Chen ldquoPSOslope a stand-alone windowsapplication for graphic analysis of slope stabilityrdquo in Advancesin Swarm Optimization Y Tan Y Shi Y Chai and G WangEds vol 6728 of Lecture Notes in Computer Science pp 56ndash632011

[25] T Yamagami and J C Jiang ldquoA search for the critical slipsurface in three-dimensional slope stability analysisrdquo Soils andFoundations vol 37 no 3 pp 1ndash16 1997

[26] J C Jiang T Yamagami and R Baker ldquoThree-dimensionalslope stability analysis based on nonlinear failure enveloperdquoChinese Journal of Rock Mechanics and Engineering vol 22 no6 pp 1017ndash1023 2003

[27] X J E Mowen ldquoA simple Monte Carlo method for locating thethree-dimensional critical slip surface of a sloperdquoActaGeologicaSinica vol 78 no 6 pp 1258ndash1266 2004

[28] X J E Mowen W Zengfu L Xiangyu and X Bo ldquoThree-dimensional critical slip surface locating and slope stabilityassessment for lava lobe of Unzen Volcanordquo Journal of RockMechanics and Geotechnical Engineering vol 3 no 1 pp 82ndash892011

[29] Y M Cheng H T Liu W B Wei and S K Au ldquoLocationof critical three-dimensional non-spherical failure surface byNURBS functions and ellipsoid with applications to highwayslopesrdquo Computers and Geotechnics vol 32 no 6 pp 387ndash3992005

[30] M Toufigh A Ahangarasr and A Ouria ldquoUsing non-linearprogramming techniques in determination of the most prob-able slip surface in 3D slopesrdquo World Academy of ScienceEngineering and Technology vol 17 pp 30ndash35 2006

[31] C W Reynolds ldquoFlocks herbs and schools a distributedbehavorial modelrdquo Computer Graphics vol 21 no 4 pp 25ndash341987

[32] R Kalatehjari An improvised three-dimensional slope stabilityanalysis based on limit equilibrium method by using particleswarm optimization [PhD dissertation] Faculty of Civil Engi-neering Universiti Teknologi Malaysia 2013

[33] A P Engelbrecht Computational intelligence an introductionJohn Wiley amp Sons 2007

[34] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tion an overviewrdquo Swarm Intelligence vol 1 no 1 pp 33ndash572007

[35] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 Piscataway NJ USAMay 1998

[36] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[37] R Mendes P Cortez M Rocha and J Neves ldquoParticle swarmsfor feedforward neural network trainingrdquo in Proceedings of theInternational Joint Conference on Neural Networks (IJCNN rsquo02)pp 1895ndash1899 Honolulu Hawaii USA May 2002

12 The Scientific World Journal

[38] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science pp 39ndash43 IEEENagoya Japan October 1995

[39] R Baker ldquoDetermination of the critical slip surface in slopestability computationsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 4 pp 333ndash3591980

[40] H P J Bolton G Heymann and A A Groenwold ldquoGlobalsearch for critical failure surface in slope stability analysisrdquoEngineering Optimization vol 35 no 1 pp 51ndash65 2003

[41] Y Chen X Wei and Y Li ldquoLocating non-circular critical slipsurfaces by particle swarm optimization algorithmrdquo ChineseJournal of Rock Mechanics and Engineering vol 25 no 7 pp1443ndash1449 2006

[42] L Li S-C Chi R-M Zheng G Lin and X-S Chu ldquoMixedsearch algorithmof non-circular critical slip surface of soil sloprdquoChina Journal of Highway and Transport vol 20 no 6 pp 1ndash62007

[43] L Li S-C Chi andY-M Zheng ldquoThree-dimensional slope sta-bility analysis based on ellipsoidal sliding body and simplifiedJANBU methodrdquo Rock and Soil Mechanics vol 29 no 9 pp2439ndash2445 2008

[44] W Alkasawneh A I H Malkawi J H Nusairat and NAlbataineh ldquoA comparative study of various commerciallyavailable programs in slope stability analysisrdquo Computers andGeotechnics vol 35 no 3 pp 428ndash435 2008

[45] T Yamagami K Kojima and M Taki ldquoA search for the three-dimensional critical slip surface with random generation ofsurfacerdquo in Proceedings of the 36th Symposium on Slope StabilityAnalyses and Stabilizing Construction Methods pp 27ndash34 1991

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 The Scientific World Journal

06

05

04

03

02

01

00 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Aver

age fi

tnes

s

Iteration

Dynamic test 1Dynamic test 2Dynamic test 3

Dynamic test 4Dynamic test 5Dynamic test 6

Figure 10 Results of sensitivity tests on dynamic inertia weight

Table 4 Properties of slopes in example problems

Properties 1198881015840 (kNm2) 120601

1015840 (degree) 120574 (kNm3) 120592 119864 (kNm2)Problem 1 15 20 17 03 1119864 + 6

Problem 2 10 10 18 mdash mdash

the CSS in comparison with PLAXIS-3D finite element soft-ware (License byUniversiti TeknologiMalaysia) Alkasawnehet al [44] applied different search techniques to determinethe CSS in 2D slope stability analysis Figure 11 illustrates thegeometry of the slope A 3D model was developed based onthis 2D section inwhich the third dimensionwas extended by100 meters Figure 12 shows the generated 3D models of theslope by the present study and PLAXIS-3D In both methodscylindrical slip surface was employed to determine the CSS ofthe slope

PSO improved average and best fitness of the swarmas shown in Figure 13 Figure 14 shows the minimum FOSversus iterations PSO provided continuous reduction of FOSto find the CSS The present study obtained FOS of 178versus theminimumFOS of 177 of the PLAXIS-3D Figure 15shows the CSS obtained by the present study and the resultof PLAXIS-3D The present study and PLAXIS-3D obtainedsimilar FOS for the CSS with a small difference of 03 Thisresult demonstrates the ability of PSO to determine the CSSwith the minimum FOS in 3D slope stability analysis

Example problem 2 was performed to verify the abilityof PSO to determine the CSS with general ellipsoid shape ina comparison with previous studies from the literature Thisexample was initially analyzed by Yamagami et al [45] andwas reanalyzed by Yamagami and Jiang [25] Yamagami et al[45] used random generation of surfaces to determine theCSS of this slope while Yamagami and Jiang [25] employed acombination of dynamic programming and random numbergeneration to do so It should be noted that the same equationof FOS as previous studies was used to make fair comparisonof the results The example involved a homogeneous slopewith gradient of 2 1 subjected to a square load of 50 kPa onthe top The uniform load was applied on a square surface

30

20

10

00 20 40 60 80

Z(m

)

Y (m)

Figure 11 Geometry of 2D section of example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axis

Y-axis

(a)Z-a

xis

X-axisY-axis

(b)

Figure 12 Generated 3D models of example problem 1 by (a) thepresent study and (b) PLAXIS-3D

with 8 meters sides at the top center of the slope Figure 16illustrates the geometry of example problem 1

Figure 17 shows the generated 3D model by the presentstudy for example problem 2 Figure 18 plots the process ofPSO to improve fitness of the swarmThe trend of average fit-ness value of the swarm experienced some instability duringthe process The main reason of this behavior is the presenceof disqualified slip surfaces in the swarm that dramaticallydecreases the average fitness valueThese surfaces were rarelypresented in the previous example due to adopted simplercylindrical shape compared with more complicated ellipsoidshape in this example The trend of minimum FOS versusiterations is shown in Figure 19 Continuous decrement ofFOS by PSO leads to determining the CSS of the slope

In spite of similar equation of FOS the present studyfound the CSS with a smaller FOS than other methods whichis the best result so far The minimum FOS obtained by PSOwas 095 compared with 114 and 103 of random generationof surfaces [45] and DP with RNG [25] respectively Thisresult demonstrates the ability of PSO to accurately determinethe ellipsoid CSS in 3D slope stability analysis Figure 20

The Scientific World Journal 9

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80

Iteration

Fitn

ess v

alue

Average fitnessMaximum fitness

Figure 13 Fitness values versus iterations in example problem 1

215

21

205

2

195

19

1850 10 20 30 40 50 60 70 80

Iteration

Min

imum

fact

or o

f saf

ety

Figure 14 Minimum FOS versus iterations in example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axisY-axis

(a)

Z-a

xis

X-axis

Y-axis

(b)

Figure 15 (a) The CSS obtained by the present study and (b) exag-gerated displacement vectors of PLAXIS-3D in example problem 1

40

24

20

9

4

0

0 4 14 22 25

x(m

)z

(m)

y (m)

0 4 14 22 25

y (m)

12

(b)

(a)

Figure 16 (a) Half-plan view and (b) central cross-section of slopein example problem 2

10864200

510

1520

2530 35

40 05

1015

2025

Z-a

xis

X-axis Y-axis

Figure 17 Generated 3D models of example problem 2 by thepresent study

illustrates the CSS obtained by the present study in exampleproblem 2

5 Conclusion

Determining the critical slip surface of a soil slope is atraditional problem in geotechnical engineering which is stillchallenging for researchers This problem needs a massivesearching process Although classical searching methodswork for relatively simple problems they are surrounded bylocal minima Moreover their processes become particularlyslow by increasing the number of possible solutions Toeliminate these limitations PSO has been applied in slopestability analysis based on its successful results in advancedengineering problems However this contribution was lim-ited to 2D slope stability analysis This paper applied PSO in

10 The Scientific World Journal

12

1

08

06

04

02

00 10 20 30 40 50 60 70 80 90 100

Fitn

ess v

alue

Iteration

Average fitnessMaximum fitness

Figure 18 Fitness values versus iterations in example problem 2

3

25

35

2

15

1

05

00 10 20 30 40 50 60 70 80 90 100

Min

imum

fac

tor o

f saf

ety

Iteration

Figure 19 Minimum FOS versus iterations in example problem 2

10

8

6

4

2

0051015

2025

303540 0 5 10 15 20 25

Z-a

xis

X-axis

Y-axis

Figure 20 The CSS obtained by the present study in exampleproblem 2

3D slope stability problem to determine theCSS of soil slopesA detailed description of adopted PSO was presented toprovide a good basis for more contribution of this techniqueto the field of 3D slope stability problems

The application of PSO in slope stability analysis wasdescribed by presenting a general flowchart A general rotat-ing ellipsoid shape was introduced as the specific particle for3D slope stability analysis In order to find the optimum val-ues of parameters of PSO a sensitivity analysis was designedand performed The related codes were prepared by theauthors in MATLAB A 3D model with complex geometry

soil layers and piezometric line was used in the analysisThis analysis included three steps to find the optimum swarmsize coefficients and inertia weight of the velocity equationrespectively Moreover the performance of PSO to convergeover a global optimum solution was verified during thetests Based on the obtained values of parameters PSO wasprepared for 3D slope stability analysis

The applicability of PSO in determining the CSS of 3Dslopes was evaluated by analyzing two example problemsThe first example presented a comparison between the resultsof PSO and PLAXI-3D finite element software The secondexample compared the ability of PSO to determine the CSSof 3D slopes with other optimization methods from theliterature Both of the example problems demonstrated theefficiency and effectiveness of PSO in determining the CSS of3D soil slopes Based on the results it is believed that PSO ishighly capable of contributing to the field of 3D slope stabilityanalysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The presented study was undertaken with support fromresearch grant The authors are pleased to acknowledgethe Ministry of Education of Malaysia for the PrototypeResearch Grant Scheme (RJ13000078224L620) and Univer-siti Teknologi Malaysia for providing the research funds

References

[1] R Kalatehjari and N Ali ldquoA review of three-dimensional slopestability analyses based on limit equilibriummethodrdquo ElectronicJournal of Geotechnical Engineering vol 18 pp 119ndash134 2013

[2] D G Fredlund ldquoAnalytical methods for slope stability analysisrdquoin Proceedings of the 4th International Symposium on LandslidesState-of-the-Art pp 229ndash250 Toronto Canada September1984

[3] A W Bishop ldquoThe use of the slip circle in the stability analysisof earth sloperdquo Geotechnique vol 5 no 1 pp 7ndash17 1955

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[5] Y Zhang P Agarwal V Bhatnagar S Balochian and J YanldquoSwarm intelligence and its applicationsrdquo The Scientific WorldJournal vol 2013 Article ID 528069 3 pages 2013

[6] G Poitras G Lefrancois and G Cormier ldquoOptimization ofsteel floor systems using particle swarm optimizationrdquo Journalof Constructional Steel Research vol 67 no 8 pp 1225ndash12312011

[7] S Gholizadeh and E Salajegheh ldquoOptimal design of structuressubjected to time history loading by swarm intelligence and anadvancedmetamodelrdquoComputerMethods in AppliedMechanicsand Engineering vol 198 no 37ndash40 pp 2936ndash2949 2009

[8] S S Gilan H B Jovein and A A Ramezanianpour ldquoHybridsupport vector regressionmdashparticle swarm optimization for

The Scientific World Journal 11

prediction of compressive strength and RCPT of concretescontaining metakaolinrdquo Construction and Building Materialsvol 34 pp 321ndash329 2012

[9] L Yunkai T Yingjie O Zhiyun et al ldquoAnalysis of soil ero-sion characteristics in small watersheds with particle swarmoptimization support vector machine and artificial neuronalnetworksrdquoEnvironmental Earth Sciences vol 60 no 7 pp 1559ndash1568 2010

[10] S Wan ldquoEntropy-based particle swarm optimization withclustering analysis on landslide susceptibility mappingrdquo Envi-ronmental Earth Sciences vol 68 no 5 pp 1349ndash1366 2013

[11] M R Nikoo R Kerachian A Karimi A A Azadnia andK Jafarzadegan ldquoOptimal water and waste load allocationin reservoir-river systems a case studyrdquo Environmental EarthSciences vol 71 no 9 pp 4127ndash4142 2014

[12] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for hourly rainfall-runoff modeling in Bedup Basin Malaysiardquo International Jour-nal of Civil amp Environmental Engineering vol 9 no 10 pp 9ndash182010

[13] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for modeling runoffrdquoInternational Journal of Environmental Science and Technologyvol 7 no 1 pp 67ndash78 2010

[14] J S Yazdi F Kalantary and H S Yazdi ldquoCalibration ofsoil model parameters using particle swarm optimizationrdquoInternational Journal of Geomechanics vol 12 no 3 pp 229ndash238 2012

[15] T V Bharat P V Sivapullaiah and M M Allam ldquoSwarmintelligence-based solver for parameter estimation of laboratorythrough-diffusion transport of contaminantsrdquo Computers andGeotechnics vol 36 no 6 pp 984ndash992 2009

[16] B R Chen and X T Feng ldquoCSV-PSO and its application ingeotechnical engineeringrdquo in Swarm Intelligence Focus on AntandParticle SwarmOptimization F T S Chan andMKTiwariEds chapter 15 pp 263ndash288 2007

[17] Y M Cheng L Li and S C Chi ldquoPerformance studies on sixheuristic global optimization methods in the location of criticalslip surfacerdquoComputers and Geotechnics vol 34 no 6 pp 462ndash484 2007

[18] Y M Cheng L Li Y J Sun and S K Au ldquoA coupled particleswarm and harmony search optimization algorithm for difficultgeotechnical problemsrdquo Structural and Multidisciplinary Opti-mization vol 45 no 4 pp 489ndash501 2012

[19] H B Zhao Z S Zou and Z L Ru ldquoChaotic particle swarmoptimization for non-circular critical slip surface identificationin slope stability analysisrdquo in Proceedings of the InternationalYoung Scholarsrsquo Symposium on Rock Mechanics Boundaries ofRock Mechanics Recent Advances and Challenges for the 21stCentury pp 585ndash588 Beijing China May 2008

[20] D Tian L Xu and S Wang ldquoThe application of particle swarmoptimization on the search of critical slip surfacerdquo in Proceed-ings of the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[21] L Li G Yu X Chu and S Lu ldquoThe harmony search algo-rithm in combination with particle swarm optimization andits application in the slope stability analysisrdquo in Proceedings ofthe International Conference on Computational Intelligence andSecurity (CIS rsquo09) vol 2 pp 133ndash136 Beijing China December2009

[22] R Kalatehjari N Ali M Kholghifard and M HajihassanildquoThe effects of method of generating circular slip surfaceson determining the critical slip surface by particle swarmoptimizationrdquo Arabian Journal of Geosciences vol 7 no 4 pp1529ndash1539 2014

[23] R Kalatehjari N Ali M Hajihassani and M K Fard ldquoTheapplication of particle swarm optimization in slope stabilityanalysis of homogeneous soil slopesrdquo International Review onModelling and Simulations vol 5 no 1 pp 458ndash465 2012

[24] W Chen and P Chen ldquoPSOslope a stand-alone windowsapplication for graphic analysis of slope stabilityrdquo in Advancesin Swarm Optimization Y Tan Y Shi Y Chai and G WangEds vol 6728 of Lecture Notes in Computer Science pp 56ndash632011

[25] T Yamagami and J C Jiang ldquoA search for the critical slipsurface in three-dimensional slope stability analysisrdquo Soils andFoundations vol 37 no 3 pp 1ndash16 1997

[26] J C Jiang T Yamagami and R Baker ldquoThree-dimensionalslope stability analysis based on nonlinear failure enveloperdquoChinese Journal of Rock Mechanics and Engineering vol 22 no6 pp 1017ndash1023 2003

[27] X J E Mowen ldquoA simple Monte Carlo method for locating thethree-dimensional critical slip surface of a sloperdquoActaGeologicaSinica vol 78 no 6 pp 1258ndash1266 2004

[28] X J E Mowen W Zengfu L Xiangyu and X Bo ldquoThree-dimensional critical slip surface locating and slope stabilityassessment for lava lobe of Unzen Volcanordquo Journal of RockMechanics and Geotechnical Engineering vol 3 no 1 pp 82ndash892011

[29] Y M Cheng H T Liu W B Wei and S K Au ldquoLocationof critical three-dimensional non-spherical failure surface byNURBS functions and ellipsoid with applications to highwayslopesrdquo Computers and Geotechnics vol 32 no 6 pp 387ndash3992005

[30] M Toufigh A Ahangarasr and A Ouria ldquoUsing non-linearprogramming techniques in determination of the most prob-able slip surface in 3D slopesrdquo World Academy of ScienceEngineering and Technology vol 17 pp 30ndash35 2006

[31] C W Reynolds ldquoFlocks herbs and schools a distributedbehavorial modelrdquo Computer Graphics vol 21 no 4 pp 25ndash341987

[32] R Kalatehjari An improvised three-dimensional slope stabilityanalysis based on limit equilibrium method by using particleswarm optimization [PhD dissertation] Faculty of Civil Engi-neering Universiti Teknologi Malaysia 2013

[33] A P Engelbrecht Computational intelligence an introductionJohn Wiley amp Sons 2007

[34] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tion an overviewrdquo Swarm Intelligence vol 1 no 1 pp 33ndash572007

[35] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 Piscataway NJ USAMay 1998

[36] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[37] R Mendes P Cortez M Rocha and J Neves ldquoParticle swarmsfor feedforward neural network trainingrdquo in Proceedings of theInternational Joint Conference on Neural Networks (IJCNN rsquo02)pp 1895ndash1899 Honolulu Hawaii USA May 2002

12 The Scientific World Journal

[38] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science pp 39ndash43 IEEENagoya Japan October 1995

[39] R Baker ldquoDetermination of the critical slip surface in slopestability computationsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 4 pp 333ndash3591980

[40] H P J Bolton G Heymann and A A Groenwold ldquoGlobalsearch for critical failure surface in slope stability analysisrdquoEngineering Optimization vol 35 no 1 pp 51ndash65 2003

[41] Y Chen X Wei and Y Li ldquoLocating non-circular critical slipsurfaces by particle swarm optimization algorithmrdquo ChineseJournal of Rock Mechanics and Engineering vol 25 no 7 pp1443ndash1449 2006

[42] L Li S-C Chi R-M Zheng G Lin and X-S Chu ldquoMixedsearch algorithmof non-circular critical slip surface of soil sloprdquoChina Journal of Highway and Transport vol 20 no 6 pp 1ndash62007

[43] L Li S-C Chi andY-M Zheng ldquoThree-dimensional slope sta-bility analysis based on ellipsoidal sliding body and simplifiedJANBU methodrdquo Rock and Soil Mechanics vol 29 no 9 pp2439ndash2445 2008

[44] W Alkasawneh A I H Malkawi J H Nusairat and NAlbataineh ldquoA comparative study of various commerciallyavailable programs in slope stability analysisrdquo Computers andGeotechnics vol 35 no 3 pp 428ndash435 2008

[45] T Yamagami K Kojima and M Taki ldquoA search for the three-dimensional critical slip surface with random generation ofsurfacerdquo in Proceedings of the 36th Symposium on Slope StabilityAnalyses and Stabilizing Construction Methods pp 27ndash34 1991

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 9

06

05

04

03

02

01

00 10 20 30 40 50 60 70 80

Iteration

Fitn

ess v

alue

Average fitnessMaximum fitness

Figure 13 Fitness values versus iterations in example problem 1

215

21

205

2

195

19

1850 10 20 30 40 50 60 70 80

Iteration

Min

imum

fact

or o

f saf

ety

Figure 14 Minimum FOS versus iterations in example problem 1

3025201510500

2040

6080

100 0 10 20 30 40 50 60 70 80

Z-a

xis

X-axisY-axis

(a)

Z-a

xis

X-axis

Y-axis

(b)

Figure 15 (a) The CSS obtained by the present study and (b) exag-gerated displacement vectors of PLAXIS-3D in example problem 1

40

24

20

9

4

0

0 4 14 22 25

x(m

)z

(m)

y (m)

0 4 14 22 25

y (m)

12

(b)

(a)

Figure 16 (a) Half-plan view and (b) central cross-section of slopein example problem 2

10864200

510

1520

2530 35

40 05

1015

2025

Z-a

xis

X-axis Y-axis

Figure 17 Generated 3D models of example problem 2 by thepresent study

illustrates the CSS obtained by the present study in exampleproblem 2

5 Conclusion

Determining the critical slip surface of a soil slope is atraditional problem in geotechnical engineering which is stillchallenging for researchers This problem needs a massivesearching process Although classical searching methodswork for relatively simple problems they are surrounded bylocal minima Moreover their processes become particularlyslow by increasing the number of possible solutions Toeliminate these limitations PSO has been applied in slopestability analysis based on its successful results in advancedengineering problems However this contribution was lim-ited to 2D slope stability analysis This paper applied PSO in

10 The Scientific World Journal

12

1

08

06

04

02

00 10 20 30 40 50 60 70 80 90 100

Fitn

ess v

alue

Iteration

Average fitnessMaximum fitness

Figure 18 Fitness values versus iterations in example problem 2

3

25

35

2

15

1

05

00 10 20 30 40 50 60 70 80 90 100

Min

imum

fac

tor o

f saf

ety

Iteration

Figure 19 Minimum FOS versus iterations in example problem 2

10

8

6

4

2

0051015

2025

303540 0 5 10 15 20 25

Z-a

xis

X-axis

Y-axis

Figure 20 The CSS obtained by the present study in exampleproblem 2

3D slope stability problem to determine theCSS of soil slopesA detailed description of adopted PSO was presented toprovide a good basis for more contribution of this techniqueto the field of 3D slope stability problems

The application of PSO in slope stability analysis wasdescribed by presenting a general flowchart A general rotat-ing ellipsoid shape was introduced as the specific particle for3D slope stability analysis In order to find the optimum val-ues of parameters of PSO a sensitivity analysis was designedand performed The related codes were prepared by theauthors in MATLAB A 3D model with complex geometry

soil layers and piezometric line was used in the analysisThis analysis included three steps to find the optimum swarmsize coefficients and inertia weight of the velocity equationrespectively Moreover the performance of PSO to convergeover a global optimum solution was verified during thetests Based on the obtained values of parameters PSO wasprepared for 3D slope stability analysis

The applicability of PSO in determining the CSS of 3Dslopes was evaluated by analyzing two example problemsThe first example presented a comparison between the resultsof PSO and PLAXI-3D finite element software The secondexample compared the ability of PSO to determine the CSSof 3D slopes with other optimization methods from theliterature Both of the example problems demonstrated theefficiency and effectiveness of PSO in determining the CSS of3D soil slopes Based on the results it is believed that PSO ishighly capable of contributing to the field of 3D slope stabilityanalysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The presented study was undertaken with support fromresearch grant The authors are pleased to acknowledgethe Ministry of Education of Malaysia for the PrototypeResearch Grant Scheme (RJ13000078224L620) and Univer-siti Teknologi Malaysia for providing the research funds

References

[1] R Kalatehjari and N Ali ldquoA review of three-dimensional slopestability analyses based on limit equilibriummethodrdquo ElectronicJournal of Geotechnical Engineering vol 18 pp 119ndash134 2013

[2] D G Fredlund ldquoAnalytical methods for slope stability analysisrdquoin Proceedings of the 4th International Symposium on LandslidesState-of-the-Art pp 229ndash250 Toronto Canada September1984

[3] A W Bishop ldquoThe use of the slip circle in the stability analysisof earth sloperdquo Geotechnique vol 5 no 1 pp 7ndash17 1955

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[5] Y Zhang P Agarwal V Bhatnagar S Balochian and J YanldquoSwarm intelligence and its applicationsrdquo The Scientific WorldJournal vol 2013 Article ID 528069 3 pages 2013

[6] G Poitras G Lefrancois and G Cormier ldquoOptimization ofsteel floor systems using particle swarm optimizationrdquo Journalof Constructional Steel Research vol 67 no 8 pp 1225ndash12312011

[7] S Gholizadeh and E Salajegheh ldquoOptimal design of structuressubjected to time history loading by swarm intelligence and anadvancedmetamodelrdquoComputerMethods in AppliedMechanicsand Engineering vol 198 no 37ndash40 pp 2936ndash2949 2009

[8] S S Gilan H B Jovein and A A Ramezanianpour ldquoHybridsupport vector regressionmdashparticle swarm optimization for

The Scientific World Journal 11

prediction of compressive strength and RCPT of concretescontaining metakaolinrdquo Construction and Building Materialsvol 34 pp 321ndash329 2012

[9] L Yunkai T Yingjie O Zhiyun et al ldquoAnalysis of soil ero-sion characteristics in small watersheds with particle swarmoptimization support vector machine and artificial neuronalnetworksrdquoEnvironmental Earth Sciences vol 60 no 7 pp 1559ndash1568 2010

[10] S Wan ldquoEntropy-based particle swarm optimization withclustering analysis on landslide susceptibility mappingrdquo Envi-ronmental Earth Sciences vol 68 no 5 pp 1349ndash1366 2013

[11] M R Nikoo R Kerachian A Karimi A A Azadnia andK Jafarzadegan ldquoOptimal water and waste load allocationin reservoir-river systems a case studyrdquo Environmental EarthSciences vol 71 no 9 pp 4127ndash4142 2014

[12] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for hourly rainfall-runoff modeling in Bedup Basin Malaysiardquo International Jour-nal of Civil amp Environmental Engineering vol 9 no 10 pp 9ndash182010

[13] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for modeling runoffrdquoInternational Journal of Environmental Science and Technologyvol 7 no 1 pp 67ndash78 2010

[14] J S Yazdi F Kalantary and H S Yazdi ldquoCalibration ofsoil model parameters using particle swarm optimizationrdquoInternational Journal of Geomechanics vol 12 no 3 pp 229ndash238 2012

[15] T V Bharat P V Sivapullaiah and M M Allam ldquoSwarmintelligence-based solver for parameter estimation of laboratorythrough-diffusion transport of contaminantsrdquo Computers andGeotechnics vol 36 no 6 pp 984ndash992 2009

[16] B R Chen and X T Feng ldquoCSV-PSO and its application ingeotechnical engineeringrdquo in Swarm Intelligence Focus on AntandParticle SwarmOptimization F T S Chan andMKTiwariEds chapter 15 pp 263ndash288 2007

[17] Y M Cheng L Li and S C Chi ldquoPerformance studies on sixheuristic global optimization methods in the location of criticalslip surfacerdquoComputers and Geotechnics vol 34 no 6 pp 462ndash484 2007

[18] Y M Cheng L Li Y J Sun and S K Au ldquoA coupled particleswarm and harmony search optimization algorithm for difficultgeotechnical problemsrdquo Structural and Multidisciplinary Opti-mization vol 45 no 4 pp 489ndash501 2012

[19] H B Zhao Z S Zou and Z L Ru ldquoChaotic particle swarmoptimization for non-circular critical slip surface identificationin slope stability analysisrdquo in Proceedings of the InternationalYoung Scholarsrsquo Symposium on Rock Mechanics Boundaries ofRock Mechanics Recent Advances and Challenges for the 21stCentury pp 585ndash588 Beijing China May 2008

[20] D Tian L Xu and S Wang ldquoThe application of particle swarmoptimization on the search of critical slip surfacerdquo in Proceed-ings of the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[21] L Li G Yu X Chu and S Lu ldquoThe harmony search algo-rithm in combination with particle swarm optimization andits application in the slope stability analysisrdquo in Proceedings ofthe International Conference on Computational Intelligence andSecurity (CIS rsquo09) vol 2 pp 133ndash136 Beijing China December2009

[22] R Kalatehjari N Ali M Kholghifard and M HajihassanildquoThe effects of method of generating circular slip surfaceson determining the critical slip surface by particle swarmoptimizationrdquo Arabian Journal of Geosciences vol 7 no 4 pp1529ndash1539 2014

[23] R Kalatehjari N Ali M Hajihassani and M K Fard ldquoTheapplication of particle swarm optimization in slope stabilityanalysis of homogeneous soil slopesrdquo International Review onModelling and Simulations vol 5 no 1 pp 458ndash465 2012

[24] W Chen and P Chen ldquoPSOslope a stand-alone windowsapplication for graphic analysis of slope stabilityrdquo in Advancesin Swarm Optimization Y Tan Y Shi Y Chai and G WangEds vol 6728 of Lecture Notes in Computer Science pp 56ndash632011

[25] T Yamagami and J C Jiang ldquoA search for the critical slipsurface in three-dimensional slope stability analysisrdquo Soils andFoundations vol 37 no 3 pp 1ndash16 1997

[26] J C Jiang T Yamagami and R Baker ldquoThree-dimensionalslope stability analysis based on nonlinear failure enveloperdquoChinese Journal of Rock Mechanics and Engineering vol 22 no6 pp 1017ndash1023 2003

[27] X J E Mowen ldquoA simple Monte Carlo method for locating thethree-dimensional critical slip surface of a sloperdquoActaGeologicaSinica vol 78 no 6 pp 1258ndash1266 2004

[28] X J E Mowen W Zengfu L Xiangyu and X Bo ldquoThree-dimensional critical slip surface locating and slope stabilityassessment for lava lobe of Unzen Volcanordquo Journal of RockMechanics and Geotechnical Engineering vol 3 no 1 pp 82ndash892011

[29] Y M Cheng H T Liu W B Wei and S K Au ldquoLocationof critical three-dimensional non-spherical failure surface byNURBS functions and ellipsoid with applications to highwayslopesrdquo Computers and Geotechnics vol 32 no 6 pp 387ndash3992005

[30] M Toufigh A Ahangarasr and A Ouria ldquoUsing non-linearprogramming techniques in determination of the most prob-able slip surface in 3D slopesrdquo World Academy of ScienceEngineering and Technology vol 17 pp 30ndash35 2006

[31] C W Reynolds ldquoFlocks herbs and schools a distributedbehavorial modelrdquo Computer Graphics vol 21 no 4 pp 25ndash341987

[32] R Kalatehjari An improvised three-dimensional slope stabilityanalysis based on limit equilibrium method by using particleswarm optimization [PhD dissertation] Faculty of Civil Engi-neering Universiti Teknologi Malaysia 2013

[33] A P Engelbrecht Computational intelligence an introductionJohn Wiley amp Sons 2007

[34] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tion an overviewrdquo Swarm Intelligence vol 1 no 1 pp 33ndash572007

[35] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 Piscataway NJ USAMay 1998

[36] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[37] R Mendes P Cortez M Rocha and J Neves ldquoParticle swarmsfor feedforward neural network trainingrdquo in Proceedings of theInternational Joint Conference on Neural Networks (IJCNN rsquo02)pp 1895ndash1899 Honolulu Hawaii USA May 2002

12 The Scientific World Journal

[38] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science pp 39ndash43 IEEENagoya Japan October 1995

[39] R Baker ldquoDetermination of the critical slip surface in slopestability computationsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 4 pp 333ndash3591980

[40] H P J Bolton G Heymann and A A Groenwold ldquoGlobalsearch for critical failure surface in slope stability analysisrdquoEngineering Optimization vol 35 no 1 pp 51ndash65 2003

[41] Y Chen X Wei and Y Li ldquoLocating non-circular critical slipsurfaces by particle swarm optimization algorithmrdquo ChineseJournal of Rock Mechanics and Engineering vol 25 no 7 pp1443ndash1449 2006

[42] L Li S-C Chi R-M Zheng G Lin and X-S Chu ldquoMixedsearch algorithmof non-circular critical slip surface of soil sloprdquoChina Journal of Highway and Transport vol 20 no 6 pp 1ndash62007

[43] L Li S-C Chi andY-M Zheng ldquoThree-dimensional slope sta-bility analysis based on ellipsoidal sliding body and simplifiedJANBU methodrdquo Rock and Soil Mechanics vol 29 no 9 pp2439ndash2445 2008

[44] W Alkasawneh A I H Malkawi J H Nusairat and NAlbataineh ldquoA comparative study of various commerciallyavailable programs in slope stability analysisrdquo Computers andGeotechnics vol 35 no 3 pp 428ndash435 2008

[45] T Yamagami K Kojima and M Taki ldquoA search for the three-dimensional critical slip surface with random generation ofsurfacerdquo in Proceedings of the 36th Symposium on Slope StabilityAnalyses and Stabilizing Construction Methods pp 27ndash34 1991

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 The Scientific World Journal

12

1

08

06

04

02

00 10 20 30 40 50 60 70 80 90 100

Fitn

ess v

alue

Iteration

Average fitnessMaximum fitness

Figure 18 Fitness values versus iterations in example problem 2

3

25

35

2

15

1

05

00 10 20 30 40 50 60 70 80 90 100

Min

imum

fac

tor o

f saf

ety

Iteration

Figure 19 Minimum FOS versus iterations in example problem 2

10

8

6

4

2

0051015

2025

303540 0 5 10 15 20 25

Z-a

xis

X-axis

Y-axis

Figure 20 The CSS obtained by the present study in exampleproblem 2

3D slope stability problem to determine theCSS of soil slopesA detailed description of adopted PSO was presented toprovide a good basis for more contribution of this techniqueto the field of 3D slope stability problems

The application of PSO in slope stability analysis wasdescribed by presenting a general flowchart A general rotat-ing ellipsoid shape was introduced as the specific particle for3D slope stability analysis In order to find the optimum val-ues of parameters of PSO a sensitivity analysis was designedand performed The related codes were prepared by theauthors in MATLAB A 3D model with complex geometry

soil layers and piezometric line was used in the analysisThis analysis included three steps to find the optimum swarmsize coefficients and inertia weight of the velocity equationrespectively Moreover the performance of PSO to convergeover a global optimum solution was verified during thetests Based on the obtained values of parameters PSO wasprepared for 3D slope stability analysis

The applicability of PSO in determining the CSS of 3Dslopes was evaluated by analyzing two example problemsThe first example presented a comparison between the resultsof PSO and PLAXI-3D finite element software The secondexample compared the ability of PSO to determine the CSSof 3D slopes with other optimization methods from theliterature Both of the example problems demonstrated theefficiency and effectiveness of PSO in determining the CSS of3D soil slopes Based on the results it is believed that PSO ishighly capable of contributing to the field of 3D slope stabilityanalysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The presented study was undertaken with support fromresearch grant The authors are pleased to acknowledgethe Ministry of Education of Malaysia for the PrototypeResearch Grant Scheme (RJ13000078224L620) and Univer-siti Teknologi Malaysia for providing the research funds

References

[1] R Kalatehjari and N Ali ldquoA review of three-dimensional slopestability analyses based on limit equilibriummethodrdquo ElectronicJournal of Geotechnical Engineering vol 18 pp 119ndash134 2013

[2] D G Fredlund ldquoAnalytical methods for slope stability analysisrdquoin Proceedings of the 4th International Symposium on LandslidesState-of-the-Art pp 229ndash250 Toronto Canada September1984

[3] A W Bishop ldquoThe use of the slip circle in the stability analysisof earth sloperdquo Geotechnique vol 5 no 1 pp 7ndash17 1955

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[5] Y Zhang P Agarwal V Bhatnagar S Balochian and J YanldquoSwarm intelligence and its applicationsrdquo The Scientific WorldJournal vol 2013 Article ID 528069 3 pages 2013

[6] G Poitras G Lefrancois and G Cormier ldquoOptimization ofsteel floor systems using particle swarm optimizationrdquo Journalof Constructional Steel Research vol 67 no 8 pp 1225ndash12312011

[7] S Gholizadeh and E Salajegheh ldquoOptimal design of structuressubjected to time history loading by swarm intelligence and anadvancedmetamodelrdquoComputerMethods in AppliedMechanicsand Engineering vol 198 no 37ndash40 pp 2936ndash2949 2009

[8] S S Gilan H B Jovein and A A Ramezanianpour ldquoHybridsupport vector regressionmdashparticle swarm optimization for

The Scientific World Journal 11

prediction of compressive strength and RCPT of concretescontaining metakaolinrdquo Construction and Building Materialsvol 34 pp 321ndash329 2012

[9] L Yunkai T Yingjie O Zhiyun et al ldquoAnalysis of soil ero-sion characteristics in small watersheds with particle swarmoptimization support vector machine and artificial neuronalnetworksrdquoEnvironmental Earth Sciences vol 60 no 7 pp 1559ndash1568 2010

[10] S Wan ldquoEntropy-based particle swarm optimization withclustering analysis on landslide susceptibility mappingrdquo Envi-ronmental Earth Sciences vol 68 no 5 pp 1349ndash1366 2013

[11] M R Nikoo R Kerachian A Karimi A A Azadnia andK Jafarzadegan ldquoOptimal water and waste load allocationin reservoir-river systems a case studyrdquo Environmental EarthSciences vol 71 no 9 pp 4127ndash4142 2014

[12] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for hourly rainfall-runoff modeling in Bedup Basin Malaysiardquo International Jour-nal of Civil amp Environmental Engineering vol 9 no 10 pp 9ndash182010

[13] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for modeling runoffrdquoInternational Journal of Environmental Science and Technologyvol 7 no 1 pp 67ndash78 2010

[14] J S Yazdi F Kalantary and H S Yazdi ldquoCalibration ofsoil model parameters using particle swarm optimizationrdquoInternational Journal of Geomechanics vol 12 no 3 pp 229ndash238 2012

[15] T V Bharat P V Sivapullaiah and M M Allam ldquoSwarmintelligence-based solver for parameter estimation of laboratorythrough-diffusion transport of contaminantsrdquo Computers andGeotechnics vol 36 no 6 pp 984ndash992 2009

[16] B R Chen and X T Feng ldquoCSV-PSO and its application ingeotechnical engineeringrdquo in Swarm Intelligence Focus on AntandParticle SwarmOptimization F T S Chan andMKTiwariEds chapter 15 pp 263ndash288 2007

[17] Y M Cheng L Li and S C Chi ldquoPerformance studies on sixheuristic global optimization methods in the location of criticalslip surfacerdquoComputers and Geotechnics vol 34 no 6 pp 462ndash484 2007

[18] Y M Cheng L Li Y J Sun and S K Au ldquoA coupled particleswarm and harmony search optimization algorithm for difficultgeotechnical problemsrdquo Structural and Multidisciplinary Opti-mization vol 45 no 4 pp 489ndash501 2012

[19] H B Zhao Z S Zou and Z L Ru ldquoChaotic particle swarmoptimization for non-circular critical slip surface identificationin slope stability analysisrdquo in Proceedings of the InternationalYoung Scholarsrsquo Symposium on Rock Mechanics Boundaries ofRock Mechanics Recent Advances and Challenges for the 21stCentury pp 585ndash588 Beijing China May 2008

[20] D Tian L Xu and S Wang ldquoThe application of particle swarmoptimization on the search of critical slip surfacerdquo in Proceed-ings of the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[21] L Li G Yu X Chu and S Lu ldquoThe harmony search algo-rithm in combination with particle swarm optimization andits application in the slope stability analysisrdquo in Proceedings ofthe International Conference on Computational Intelligence andSecurity (CIS rsquo09) vol 2 pp 133ndash136 Beijing China December2009

[22] R Kalatehjari N Ali M Kholghifard and M HajihassanildquoThe effects of method of generating circular slip surfaceson determining the critical slip surface by particle swarmoptimizationrdquo Arabian Journal of Geosciences vol 7 no 4 pp1529ndash1539 2014

[23] R Kalatehjari N Ali M Hajihassani and M K Fard ldquoTheapplication of particle swarm optimization in slope stabilityanalysis of homogeneous soil slopesrdquo International Review onModelling and Simulations vol 5 no 1 pp 458ndash465 2012

[24] W Chen and P Chen ldquoPSOslope a stand-alone windowsapplication for graphic analysis of slope stabilityrdquo in Advancesin Swarm Optimization Y Tan Y Shi Y Chai and G WangEds vol 6728 of Lecture Notes in Computer Science pp 56ndash632011

[25] T Yamagami and J C Jiang ldquoA search for the critical slipsurface in three-dimensional slope stability analysisrdquo Soils andFoundations vol 37 no 3 pp 1ndash16 1997

[26] J C Jiang T Yamagami and R Baker ldquoThree-dimensionalslope stability analysis based on nonlinear failure enveloperdquoChinese Journal of Rock Mechanics and Engineering vol 22 no6 pp 1017ndash1023 2003

[27] X J E Mowen ldquoA simple Monte Carlo method for locating thethree-dimensional critical slip surface of a sloperdquoActaGeologicaSinica vol 78 no 6 pp 1258ndash1266 2004

[28] X J E Mowen W Zengfu L Xiangyu and X Bo ldquoThree-dimensional critical slip surface locating and slope stabilityassessment for lava lobe of Unzen Volcanordquo Journal of RockMechanics and Geotechnical Engineering vol 3 no 1 pp 82ndash892011

[29] Y M Cheng H T Liu W B Wei and S K Au ldquoLocationof critical three-dimensional non-spherical failure surface byNURBS functions and ellipsoid with applications to highwayslopesrdquo Computers and Geotechnics vol 32 no 6 pp 387ndash3992005

[30] M Toufigh A Ahangarasr and A Ouria ldquoUsing non-linearprogramming techniques in determination of the most prob-able slip surface in 3D slopesrdquo World Academy of ScienceEngineering and Technology vol 17 pp 30ndash35 2006

[31] C W Reynolds ldquoFlocks herbs and schools a distributedbehavorial modelrdquo Computer Graphics vol 21 no 4 pp 25ndash341987

[32] R Kalatehjari An improvised three-dimensional slope stabilityanalysis based on limit equilibrium method by using particleswarm optimization [PhD dissertation] Faculty of Civil Engi-neering Universiti Teknologi Malaysia 2013

[33] A P Engelbrecht Computational intelligence an introductionJohn Wiley amp Sons 2007

[34] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tion an overviewrdquo Swarm Intelligence vol 1 no 1 pp 33ndash572007

[35] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 Piscataway NJ USAMay 1998

[36] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[37] R Mendes P Cortez M Rocha and J Neves ldquoParticle swarmsfor feedforward neural network trainingrdquo in Proceedings of theInternational Joint Conference on Neural Networks (IJCNN rsquo02)pp 1895ndash1899 Honolulu Hawaii USA May 2002

12 The Scientific World Journal

[38] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science pp 39ndash43 IEEENagoya Japan October 1995

[39] R Baker ldquoDetermination of the critical slip surface in slopestability computationsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 4 pp 333ndash3591980

[40] H P J Bolton G Heymann and A A Groenwold ldquoGlobalsearch for critical failure surface in slope stability analysisrdquoEngineering Optimization vol 35 no 1 pp 51ndash65 2003

[41] Y Chen X Wei and Y Li ldquoLocating non-circular critical slipsurfaces by particle swarm optimization algorithmrdquo ChineseJournal of Rock Mechanics and Engineering vol 25 no 7 pp1443ndash1449 2006

[42] L Li S-C Chi R-M Zheng G Lin and X-S Chu ldquoMixedsearch algorithmof non-circular critical slip surface of soil sloprdquoChina Journal of Highway and Transport vol 20 no 6 pp 1ndash62007

[43] L Li S-C Chi andY-M Zheng ldquoThree-dimensional slope sta-bility analysis based on ellipsoidal sliding body and simplifiedJANBU methodrdquo Rock and Soil Mechanics vol 29 no 9 pp2439ndash2445 2008

[44] W Alkasawneh A I H Malkawi J H Nusairat and NAlbataineh ldquoA comparative study of various commerciallyavailable programs in slope stability analysisrdquo Computers andGeotechnics vol 35 no 3 pp 428ndash435 2008

[45] T Yamagami K Kojima and M Taki ldquoA search for the three-dimensional critical slip surface with random generation ofsurfacerdquo in Proceedings of the 36th Symposium on Slope StabilityAnalyses and Stabilizing Construction Methods pp 27ndash34 1991

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 11

prediction of compressive strength and RCPT of concretescontaining metakaolinrdquo Construction and Building Materialsvol 34 pp 321ndash329 2012

[9] L Yunkai T Yingjie O Zhiyun et al ldquoAnalysis of soil ero-sion characteristics in small watersheds with particle swarmoptimization support vector machine and artificial neuronalnetworksrdquoEnvironmental Earth Sciences vol 60 no 7 pp 1559ndash1568 2010

[10] S Wan ldquoEntropy-based particle swarm optimization withclustering analysis on landslide susceptibility mappingrdquo Envi-ronmental Earth Sciences vol 68 no 5 pp 1349ndash1366 2013

[11] M R Nikoo R Kerachian A Karimi A A Azadnia andK Jafarzadegan ldquoOptimal water and waste load allocationin reservoir-river systems a case studyrdquo Environmental EarthSciences vol 71 no 9 pp 4127ndash4142 2014

[12] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for hourly rainfall-runoff modeling in Bedup Basin Malaysiardquo International Jour-nal of Civil amp Environmental Engineering vol 9 no 10 pp 9ndash182010

[13] K K Kuok S Harun and S M Shamsuddin ldquoParticle swarmoptimization feedforward neural network for modeling runoffrdquoInternational Journal of Environmental Science and Technologyvol 7 no 1 pp 67ndash78 2010

[14] J S Yazdi F Kalantary and H S Yazdi ldquoCalibration ofsoil model parameters using particle swarm optimizationrdquoInternational Journal of Geomechanics vol 12 no 3 pp 229ndash238 2012

[15] T V Bharat P V Sivapullaiah and M M Allam ldquoSwarmintelligence-based solver for parameter estimation of laboratorythrough-diffusion transport of contaminantsrdquo Computers andGeotechnics vol 36 no 6 pp 984ndash992 2009

[16] B R Chen and X T Feng ldquoCSV-PSO and its application ingeotechnical engineeringrdquo in Swarm Intelligence Focus on AntandParticle SwarmOptimization F T S Chan andMKTiwariEds chapter 15 pp 263ndash288 2007

[17] Y M Cheng L Li and S C Chi ldquoPerformance studies on sixheuristic global optimization methods in the location of criticalslip surfacerdquoComputers and Geotechnics vol 34 no 6 pp 462ndash484 2007

[18] Y M Cheng L Li Y J Sun and S K Au ldquoA coupled particleswarm and harmony search optimization algorithm for difficultgeotechnical problemsrdquo Structural and Multidisciplinary Opti-mization vol 45 no 4 pp 489ndash501 2012

[19] H B Zhao Z S Zou and Z L Ru ldquoChaotic particle swarmoptimization for non-circular critical slip surface identificationin slope stability analysisrdquo in Proceedings of the InternationalYoung Scholarsrsquo Symposium on Rock Mechanics Boundaries ofRock Mechanics Recent Advances and Challenges for the 21stCentury pp 585ndash588 Beijing China May 2008

[20] D Tian L Xu and S Wang ldquoThe application of particle swarmoptimization on the search of critical slip surfacerdquo in Proceed-ings of the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[21] L Li G Yu X Chu and S Lu ldquoThe harmony search algo-rithm in combination with particle swarm optimization andits application in the slope stability analysisrdquo in Proceedings ofthe International Conference on Computational Intelligence andSecurity (CIS rsquo09) vol 2 pp 133ndash136 Beijing China December2009

[22] R Kalatehjari N Ali M Kholghifard and M HajihassanildquoThe effects of method of generating circular slip surfaceson determining the critical slip surface by particle swarmoptimizationrdquo Arabian Journal of Geosciences vol 7 no 4 pp1529ndash1539 2014

[23] R Kalatehjari N Ali M Hajihassani and M K Fard ldquoTheapplication of particle swarm optimization in slope stabilityanalysis of homogeneous soil slopesrdquo International Review onModelling and Simulations vol 5 no 1 pp 458ndash465 2012

[24] W Chen and P Chen ldquoPSOslope a stand-alone windowsapplication for graphic analysis of slope stabilityrdquo in Advancesin Swarm Optimization Y Tan Y Shi Y Chai and G WangEds vol 6728 of Lecture Notes in Computer Science pp 56ndash632011

[25] T Yamagami and J C Jiang ldquoA search for the critical slipsurface in three-dimensional slope stability analysisrdquo Soils andFoundations vol 37 no 3 pp 1ndash16 1997

[26] J C Jiang T Yamagami and R Baker ldquoThree-dimensionalslope stability analysis based on nonlinear failure enveloperdquoChinese Journal of Rock Mechanics and Engineering vol 22 no6 pp 1017ndash1023 2003

[27] X J E Mowen ldquoA simple Monte Carlo method for locating thethree-dimensional critical slip surface of a sloperdquoActaGeologicaSinica vol 78 no 6 pp 1258ndash1266 2004

[28] X J E Mowen W Zengfu L Xiangyu and X Bo ldquoThree-dimensional critical slip surface locating and slope stabilityassessment for lava lobe of Unzen Volcanordquo Journal of RockMechanics and Geotechnical Engineering vol 3 no 1 pp 82ndash892011

[29] Y M Cheng H T Liu W B Wei and S K Au ldquoLocationof critical three-dimensional non-spherical failure surface byNURBS functions and ellipsoid with applications to highwayslopesrdquo Computers and Geotechnics vol 32 no 6 pp 387ndash3992005

[30] M Toufigh A Ahangarasr and A Ouria ldquoUsing non-linearprogramming techniques in determination of the most prob-able slip surface in 3D slopesrdquo World Academy of ScienceEngineering and Technology vol 17 pp 30ndash35 2006

[31] C W Reynolds ldquoFlocks herbs and schools a distributedbehavorial modelrdquo Computer Graphics vol 21 no 4 pp 25ndash341987

[32] R Kalatehjari An improvised three-dimensional slope stabilityanalysis based on limit equilibrium method by using particleswarm optimization [PhD dissertation] Faculty of Civil Engi-neering Universiti Teknologi Malaysia 2013

[33] A P Engelbrecht Computational intelligence an introductionJohn Wiley amp Sons 2007

[34] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tion an overviewrdquo Swarm Intelligence vol 1 no 1 pp 33ndash572007

[35] Y Shi and R Eberhart ldquoModified particle swarm optimizerrdquo inProceedings of the IEEE International Conference on Evolution-ary Computation (ICEC rsquo98) pp 69ndash73 Piscataway NJ USAMay 1998

[36] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[37] R Mendes P Cortez M Rocha and J Neves ldquoParticle swarmsfor feedforward neural network trainingrdquo in Proceedings of theInternational Joint Conference on Neural Networks (IJCNN rsquo02)pp 1895ndash1899 Honolulu Hawaii USA May 2002

12 The Scientific World Journal

[38] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science pp 39ndash43 IEEENagoya Japan October 1995

[39] R Baker ldquoDetermination of the critical slip surface in slopestability computationsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 4 pp 333ndash3591980

[40] H P J Bolton G Heymann and A A Groenwold ldquoGlobalsearch for critical failure surface in slope stability analysisrdquoEngineering Optimization vol 35 no 1 pp 51ndash65 2003

[41] Y Chen X Wei and Y Li ldquoLocating non-circular critical slipsurfaces by particle swarm optimization algorithmrdquo ChineseJournal of Rock Mechanics and Engineering vol 25 no 7 pp1443ndash1449 2006

[42] L Li S-C Chi R-M Zheng G Lin and X-S Chu ldquoMixedsearch algorithmof non-circular critical slip surface of soil sloprdquoChina Journal of Highway and Transport vol 20 no 6 pp 1ndash62007

[43] L Li S-C Chi andY-M Zheng ldquoThree-dimensional slope sta-bility analysis based on ellipsoidal sliding body and simplifiedJANBU methodrdquo Rock and Soil Mechanics vol 29 no 9 pp2439ndash2445 2008

[44] W Alkasawneh A I H Malkawi J H Nusairat and NAlbataineh ldquoA comparative study of various commerciallyavailable programs in slope stability analysisrdquo Computers andGeotechnics vol 35 no 3 pp 428ndash435 2008

[45] T Yamagami K Kojima and M Taki ldquoA search for the three-dimensional critical slip surface with random generation ofsurfacerdquo in Proceedings of the 36th Symposium on Slope StabilityAnalyses and Stabilizing Construction Methods pp 27ndash34 1991

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

12 The Scientific World Journal

[38] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micro Machine and Human Science pp 39ndash43 IEEENagoya Japan October 1995

[39] R Baker ldquoDetermination of the critical slip surface in slopestability computationsrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 4 pp 333ndash3591980

[40] H P J Bolton G Heymann and A A Groenwold ldquoGlobalsearch for critical failure surface in slope stability analysisrdquoEngineering Optimization vol 35 no 1 pp 51ndash65 2003

[41] Y Chen X Wei and Y Li ldquoLocating non-circular critical slipsurfaces by particle swarm optimization algorithmrdquo ChineseJournal of Rock Mechanics and Engineering vol 25 no 7 pp1443ndash1449 2006

[42] L Li S-C Chi R-M Zheng G Lin and X-S Chu ldquoMixedsearch algorithmof non-circular critical slip surface of soil sloprdquoChina Journal of Highway and Transport vol 20 no 6 pp 1ndash62007

[43] L Li S-C Chi andY-M Zheng ldquoThree-dimensional slope sta-bility analysis based on ellipsoidal sliding body and simplifiedJANBU methodrdquo Rock and Soil Mechanics vol 29 no 9 pp2439ndash2445 2008

[44] W Alkasawneh A I H Malkawi J H Nusairat and NAlbataineh ldquoA comparative study of various commerciallyavailable programs in slope stability analysisrdquo Computers andGeotechnics vol 35 no 3 pp 428ndash435 2008

[45] T Yamagami K Kojima and M Taki ldquoA search for the three-dimensional critical slip surface with random generation ofsurfacerdquo in Proceedings of the 36th Symposium on Slope StabilityAnalyses and Stabilizing Construction Methods pp 27ndash34 1991

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014