Research Article A Hybrid Grey Relational Analysis...

Research ArticleA Hybrid Grey Relational Analysis and Nondominated SortingGenetic Algorithm-II for Project Portfolio Selection

Farshad Faezy Razi

Department of Industrial Management Islamic Azad University Semnan Branch Semnan Iran

Correspondence should be addressed to Farshad Faezy Razi farshadfaezygmailcom

Received 27 June 2014 Revised 24 November 2014 Accepted 30 November 2014 Published 24 December 2014

Academic Editor Konstantina Skouri

Copyright copy 2014 Farshad Faezy Razi This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Project selection and formation of an optimal portfolio of selected projects are among the main challenges of project managementFor this purpose several factors and indicators are simultaneously examined considering the terms and conditions of the decisionproblem Obviously both qualitative and quantitative factors may influence the formation of a portfolio of projects In this studythe projects were first ranked using grey relational analysis to form an optimal portfolio of projects and to create an expert systemfor the final project selection Because of the fuzzy nature of the environmental risk of each project the environmental risk waspredicted and analyzed using the fuzzy inference system and failure mode and effect analysis based on fuzzy rules Then the rankand risk of each project were optimized using a two-objective zero-onemathematical programmingmodel considering the practicalconstraints of the decision problem through the nondominated sorting genetic algorithm-II (NSGA-II) A case study was used todiscuss the practical methodology for selecting a portfolio of projects

1 Introduction

Project selection is among important issues in industrialmanagement industrial engineering and governmentalnonprofit and commercial organizations [1]The selection ofthe best portfolio or project to achieve full satisfaction in anorganization has been considered in previous studies [2]Theproject selection process can be defined as follows it is startedby continuous collecting analyzing and judging the availableinformation on the project leading to project selection con-sidering the factors influencing the selection process [3] Theproject portfolio selection is a multicriteria decision problemwhich considersmulticriteria quantitative and qualitative fac-tors simultaneously [4] In the multicriteria decision-makingmodel the solution may already exist and therefore the pur-pose is to select the best solution from the available solutionset This class of decision problems is called multicriteriadecision models On the other hand the solution may beunknown In this case the purpose is to find the optimalPareto solution of the problem in the continuous or discretespace [5] Such decision models are called multiple objectivedecision-making models The multicriteria decision models

are formed based on utility theory and human pressures indealing with the behavior of max finder [6] In 1945 JohnNewman published his famous book Theory of Games andEconomic Behavior and proposed a mathematical theoryfor game theory-based economic and social organizationsThis provided the ground for developing multiple attributedecision-making (MADM) models in the decision theory[7] In general MADM models are designed based on oneof the philosophical approaches of choice rank descriptionsort design and portfolio [8] In this study the choice rankand design approaches were combined to form a portfolioof projects According to this approach the projects wereranked through the grey relational analysis in MADM litera-tureThen the environmental risk of the project was analyzedand predicted by a fuzzy inference system Thereafter a two-objective zero-one mathematical programming model wasdesigned to optimize the risk and rank of each project consid-ering the constraints governing the optimal decision problemand the design philosophy in the multiple objective decision-making literature The Pareto solution of the model wasobtained using the nondominated sorting genetic algorithm-II (NSGA-II) The paper proceeds as follows the literature

Hindawi Publishing CorporationAdvances in Operations ResearchVolume 2014 Article ID 954219 8 pageshttpdxdoiorg1011552014954219

2 Advances in Operations Research

is reviewed in Section 2 Section 3 examines the grey rela-tional analysis method Fuzzy inference system is describedin Section 4 NSGA-II is introduced in Section 5 A newapproach for selecting the portfolio of projects is presentedin Section 6 A case study and conclusions are presented inSections 7 and 8 respectively

2 Literature Review

Zarei et al (2009) developed an expert system for portfolioselection The proposed system analyzed the technical riskand return on investment In this model the preferences areweighted and then the optimal portfolio is clustered throughthe rough set theory [9] Lin and Liu proposed a portfoliooptimizationmodel based onMarkowitz linear programmingmodel for project portfolio selection using the minimumswap size The optimal Pareto solution of the model wasobtained by a genetic algorithm [10] Doerner et al presenteda multiobjective integer programming model for optimalportfolio selection using the ant colony optimization algo-rithm [11]Martınez-Lorente et al considered both qualitativeand quantitative objectives to optimize the portfolio ofprojects In this study the path analysis was used to analyzethe qualitative objectives [12] Bilbao-Terol et al consideredthe social responsibility to select the optimal portfolio In thisapproach enterprises do not invest in activities neglectingethical standards Obviously the portfolio is completedthrough the assets observing the ethical standards For thispurpose a measure called social responsibility attractive-ness was used [13] Eshlaghy and Razi proposed a 119896-meanalgorithm-based grey relational analysis model for projectportfolio selection In this model the projects are firstclustered through the 119896-mean algorithm then each cluster isranked using the grey relational analysis Finally the Paretosolutions of rank and risk are analyzed by the geneticalgorithm [14] In another study Razi et al clustered projectsusing the 119862-mean fuzzy algorithm and then analyzed theclusters using the grey relational analysis In this study theproject risk analysis was carried out through a fuzzy inferencesystem [15] Huang et al designed amodel based on the semi-variance index to invest in a portfolio of real estate assets con-sidering the risk preference to optimize the portfolio of realestate assets In the second stage the Pareto optimal solutionof the model was analyzed using the bee colony algorithmThemodeling approach in this study is based on the salesmannetwork model [16]

3 Grey Relational Analysis

In 1982 Deng published the first paper on the grey systemtheory entitled ldquoThe Control of the Grey Systemsrdquo and thenthe grey system theory was introduced [17] Briefly the basicidea of grey theory is as follows the overall picture of thesystem is imagined considering the partial or limited infor-mation about a system This methodology deals with uncer-tain incomplete and poor problems As one of the mainfeatures of the grey system theory this theory can providesatisfactory outputs using relatively low information and thehigh variability in the criteria Like the fuzzy theory the grey

theory is an effective mathematical model for solving uncer-tain and ambiguous problems [18] There are many differentsystems in the real world each of them has its own compo-nents and subsystems To recognize a system the relationsbetween the components as well as the structure of thesystem should be identified in addition to understandingthe components If the completely known and unknowninformation of a system is respectively shown by white andblack colors the information onmost systems in nature is notwhite (well known) or black (unknown) but it is a mixtureof both colors that is grey information Such systems arecalled grey systemsThemain characteristic of grey systems isincomplete information The aim of the grey systems theoryand its applications is to create a bridge between the social sci-ences andnatural sciencesGrey colormeans the deficiency ofinformation and uncertainty [19]The grey relational analysisincludes the following steps

Formation of a Grey Relation When the performance mea-surement units for different indicators are different it is likelythat the effects of some parameters are ignored Furthermorewhen some performance indicators have a wide range thismay happen In addition the performance indicators withdifferent objectives or directions may lead to inaccurateresultsThus it is necessary to convert all performance valuesof an alternative to comparative series through a processsimilar to normalization process In grey systems theory thisprocess is called the formation of grey relations In a mul-ticriteria decision-making problem with 119898 alternatives and119899 indexes the 119894th alternative is shown by 119910

119894= (1199101198941 1199101198942

119910119894119895 119910

119894119899) in which 119910

119894119895is the performance value of the index

119895 for the alternative 119894119910119894can be converted into the comparative

series 119909119894= (1199091198941 1199091198942 119909119894119895 119909

119894119899) using one of the following

equations

119883119894(119895)lowast=

119909119894(119895) minusmin119909

119894(119895)

max 119909119894(119895) minusmin119909

119894(119895)

(1)

119883119894(119895)lowast=

max119909119894(119895) minus 119909

119894(119895)

max 119909119894(119895) minusmin119909

119894(119895)

(2)

119883119894119895=

1003816100381610038161003816119909119894 (119895) minus 1199090119887(119895)

1003816100381610038161003816

max 119909119894(119895) minus 119909

0119887(119895)

(3)

Equation (1) is used for ldquothe bigger the betterrdquo indexes while(2) is used for ldquothe smaller the betterrdquo indexes Equation (3) isused for the case where ldquovalues closer to the optimal value of119910lowast

119895are betterrdquo [20]

The Reference Target Series Once the grey relations wereformed using (1) (2) or (3) all performance values arelocated in the range [0 1] In the case where the value of119909119894119895generated by the grey relation creation process is equal

to 1 or closer to 1 than the value of any alternative theperformance of the index 119894 in the alternative 119895 is betterthan other alternatives Thus the alternative for which allperformance values are equal to 1 is the best alternative In thisstudy the reference series is defined as 119909

0= (11990901 11990902 119909

0119895

1199090119899) = (1 1 1 1) Accordingly it searches for

Advances in Operations Research 3

an alternative whose comparative series is closer to this targetseries [21]

Grey Relational Coefficient The grey relational coefficient isused to determine the proximity of 119909

119894119895to 1199090119895 Higher grey

relational coefficient closer 119909119894119895to 1199090119895 The grey relational

coefficient is calculated using (4) where119884(1199090119895 119909119894119895) represents

the gray relational coefficient between 119909119894119895

and 1199090119895 The

coefficient of determination is used to expand or limit thedomain of the grey relational coefficient [22]

119884 (1199090119895 119909119894119895) =

Δmin+120585Δmax

Δ119894119895+120585

Δmax

Δ119894119895= 1199090119895minus 119909119894119895

(4)

Grey Relational Rank Once all grey relational coefficients119884(1199090119895 119909119894119895) were calculated the grey relational rank can be

calculated using

Γ (1199090 119909119894) =

119899

sum

119895=1

119908119895sdot 119884 (119909

0119895 119909119894119895) (5)

Equation (5) represents the grey relational rank Γ(1199090 119909119894)

between 119909119894and 119909

0 In fact (5) shows the correlation between

the reference target series and the comparative series inwhich119908119895is the weight of index 119895 119908

119894is usually dependent on the

judgment of the decision-maker or the structure of problemIn addition sum119899

119895=1119908119895= 1 As mentioned earlier the reference

series shows the best achievable performance of each index inthe comparative seriesTherefore the comparative series withthe highest grey relational rank with the reference series hasthe highest similarity with the reference target series Thusthis is the best choice [23]

4 Fuzzy Inference System

Fuzzy inference system provides a systematic process toconvert a knowledge base to a nonlinear mapping This iswhy the knowledge-based systems (fuzzy systems) are usedin engineering and decision-making applications [24] Mam-dani and Assilian used fuzzy inference systems to control asteam engine and boiler combination using a combination oflinguistic control rules and the experience of human opera-tors [25] A fuzzy system has the following components

(i) a fuzzifier to convert the numerical values of thevariables into a fuzzy set

(ii) a fuzzy rules base as a set of ldquoif thenrdquo rules(iii) a fuzzy inference engine to convert inputs to outputs

through a series of actions(iv) a defuzzifier to convert the fuzzy output into a crisp

number [26]

In this study the fuzzy inference system described in Figure 1is used to analyze the environmental risk for each project Asshown in Figure 1 the factors constituting the environmental

Fuzzifier Rules base

Application of inputs to obtain the

membership function

Fuzzy operators

Application of reasoning

method toobtain the output of each rule

Integration of output to obtain the ultimate

fuzzy output

Defuzzifier

Figure 1 A fuzzy inference system for environmental risk analysisof each project

0 5 10 150

01

02

03

04

05

06

07

08

09

1

Gen

eral

ized

bell

-sha

ped

mem

bers

hip

func

tion

Severity

Figure 2 Generalized bell-shaped membership function for Sever-ity

risk of each project are analyzed by failure mode and effectanalysis based on three factors S O and D

As shown in Figures 2 3 and 4 the fuzzy membershipfunctions of Severity Occurrence and Detection are gener-alized bell-shaped membership function triangular-shapedmembership function and Gaussian curve membershipfunction respectively

It should be noted that the traditional approach of failuremode and effect analysis (FMEA) employs Risk PriorityNumber (RPN) for prioritization of failure modes using (6)RPN is the product of Severity Occurrence and Detection[27]

RPN = Severity timesOccurrence times Detection (6)


0 5 10 150

01

02

03

04

05

06

07

08

09

1

Tria

ngul

ar-s

hape

d m

embe

rshi

p fu

nctio

n

Occurrence

Figure 3 Triangular-shaped membership for Occurrence

0 5 10 150

010203040506070809

1

Gau

ssia

n cu

rve m

embe

rshi

p fu

nctio

n

Detection

Figure 4 Gaussian curve membership function for Detection

The fuzzy output of RPN is presented as a triangular-shapedmembership in Figure 5

5 NSGA-II

Genetic algorithm (GA) is a probabilistic search methodinspired by the natural process of biological evolution GAoperates on a population of potential solutions This algo-rithm is used for NP-hard problems [28]The general NSGA-II algorithm is as follows

(i) population initialization(ii) fitness calculation(iii) sorting the population according to dominant condi-

tions(iv) crowding distance(v) selection once the initial populationwas sorted based

on dominant conditions the crowding distance willbe calculated and the selection of the initial popula-tion is started The selection is done based on the twofollowing elements

0 2 4 6 8 10 12 14 160

010203040506070809

1

RPN

Tria

ngul

ar-s

hape

d m

embe

rshi

p

Figure 5 Triangular-shaped membership for environmental risk

Rejected

P1

Q1

F1

F2

F3

Ps+1

Figure 6 Sorting population using NSGA-II algorithm

population rank the lower-rank populations are se-lected

distance calculation if 119901 and 119902 are two members ofthe same rank a member with the largest crowdingdistance is selected it should be noted that the selec-tion is first done based on the rank and then thecrowding distance

(vi) crossover and mutation operations to produce newoffspring this is done using a binary selection tech-nique

(vii) integration of the initial population and the pop-ulation obtained from the crossover and mutationoperations

To replace the parentswith the bestmembers of the combinedpopulation in the previous stages at the first stage thelower-rank members are replaced with previous parents andthen are sorted according to the crowding distance Thisprocess is summarized in Figure 6 As shown in Figure 6the initial population and population generated by crossoverand mutation operations are categorized based on the rankThen the lower-rank population is deleted In the next stagethe remaining population is sorted according to crowdingdistance Here sorting is done within a front and all stagesare repeated to reach the target generation (or optimality con-ditions) [29] In Figure 6 119875 and 119876 are the initial populationand the population from crossover and mutation operationsrespectively 119865

119894represents the front


Determination of factors influencing the selection process

Ranking the projects through grey relation analysis

Fuzzy analysis of the environmental risk

The design of a two-objective zero-one programming model

Solving the two-objective model by NSGA-II

Figure 7 Project selection by the hybrid algorithmof grey relationalanalysis and nondominated sorting genetic algorithm-II

6 A New Framework forProject Portfolio Selection

This section describes a hybrid algorithm of grey relationalanalysis and the nondominated sorting genetic algorithm-IIThe main stages of the framework presented in this paper aresummarized in Figure 7

As shown in Figure 7 the parameters affecting the selec-tion and formation of a set of projects are first determinedThen the selected projects are ranked by grey relationalanalysis to form an initial portfolio of projects Thereafterthe environmental risk of each project is analyzed throughfuzzy inference system based on failure mode and effectanalysisThen a two-objective zero-one programmingmodelis designed to optimize the risk and rank The limita-tions include budget constraints staffing independence anddependence of projectsThe two-objective model determinesthe optimal Pareto solutions of risk and rank using NSGA-IIalgorithm

7 Case Study

In this section a case study is presented for project portfolioselection using hybrid grey relational analysis and nondom-inated sorting genetic algorithm-II Table 1 shows the inputdata Among the six criteria for twenty projects ldquothe biggerthe betterrdquo criteria include earnings per project (EP) theimpact of the project on the economic prosperity of the region(EPE) the impact of the project on the social boom of theregion (SPE) the number of personnel employed in eachproject (MP) and the years during which the project is usedwithout significant reconstruction costs (UP) The operatingcosts (CP) of each project are a criterion of ldquothe smaller thebetterrdquo type

The ranking results of grey relational analysis are pre-sented in Table 2 and Figure 8

Table 1 The input data for project selection

Project EP EPE SPE MP UP CPPP1 2285 9 4 776 14 450PP2 1561 9 9 884 10 599PP3 1374 8 9 639 15 502PP4 2745 4 8 845 13 382PP5 1648 3 1 532 10 678PP6 1634 5 1 555 15 567PP7 1244 3 4 858 13 565PP8 1199 7 2 504 10 539PP9 2510 3 3 720 10 528PP10 1581 5 7 531 14 460PP11 1043 8 4 649 12 647PP12 2729 2 5 527 15 522PP13 2475 3 7 678 11 402PP14 2021 2 9 838 12 630PP15 2097 2 4 709 13 669PP16 2934 4 10 902 13 408PP17 1148 8 7 923 13 407PP18 1471 7 7 593 13 439PP19 2525 7 1 716 10 495PP20 2719 10 6 764 11 605

0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Alternatives

GRA

Grey relational analysis

Figure 8 Results of grey relational analysis

The fuzzy inference system was used to analyze theenvironmental risk of the project according to the discussionprovided in Section 4 Figure 9 shows the environmental riskfor the first project

The two-objective programming model for twenty pro-jects is presented as Model (7) This model maximizes therank of each project whileminimizing the environmental riskgiven the constraints of the problem

Max 1199111= minus06799119909

1minus 05698119909

2minus 05215119909

3

+ 083331199094+ 04810119909

5minus 05372119909

6


Table 2 The degree of grey relation for the studied projects

Project 1 2 3 4 5 6 7 8 9 10GRA 06799 05698 05215 08333 04810 05372 05396 04268 06880 05070Project 11 12 13 14 15 16 17 18 19 20GRA 04473 07815 06734 06267 06067 09506 05706 04899 06918 07777

002

0406

081

0

05

1

05

055

06

065

IE

Risk

Figure 9The environmental risk of the first project obtained by thefuzzy inference system

+ 053961199097+ 04268119909

8+ 06880119909

9

+ 0507011990910minus 04473119909

11+ 07815119909

12

minus 0673411990913minus 06267119909

14+ 06067119909

15

minus 0950611990916minus 05706119909

17minus 04899119909

18

+ 0691811990919minus 07777119909

20

Min 1199112= 061119909

1minus 0532119909

2+ 0243119909

3minus 0752119909

4

minus 05391199095minus 0547119909

6+ 0338119909

7

minus 08211199098minus 0673119909

9+ 0924119909

10

+ 043211990911minus 0532119909

12+ 0724119909

13

minus 032711990914+ 0876119909

15+ 0143119909

16

+ 018511990917minus 0442119909

18+ 0646119909

19

+ 017511990920

subject to 031199091+ 02119909

2+ 04119909

3+ 035119909

4+ 022119909

5

+ 0351199096+ 029119909

7+ 024119909

8+ 033119909

9

+ 0311990910+ 03119909

11+ 02119909

12+ 031119909

13

+ 03211990914+ 032119909

15+ 034119909

16+ 038119909

17

+ 02711990918+ 037119909

19+ 025119909

20le 4

7761199091+ 884119909

2+ 639119909

3+ 845119909

4+ 532119909

5

+ 5551199096+ 858119909

7+ 504119909

8+ 720119909

9

Table 3 The Pareto solution combination

1198851

1198852

minus77505 48860minus68753 65700minus72267 55640minus74242 50580minus69004 57360

+ 53111990910+ 649119909

11+ 527119909

12+ 678119909

13

+ 83811990914+ 709119909

15+ 902119909

16+ 923119909

17

+ 59311990918+ 716119909

19+ 764119909

20ge 5000

11990913+ 11990920le 1

11990911+ 11990918le 1

1199094minus 1199098le 0

1199092minus 1199099le 0

119909119895 isin [0 or 1] 119895 = 1 2 3 20

(7)

In Model (7) the first and second objective functions repre-sent the risk and rank of each project respectively The firstand second constraints are related to funding and staffingfor each project respectively The third to sixth constraintsare related to the selection of independent and dependentprojects given the reasonable constraints Figure 10 shows theoptimal Pareto solutions of the risk and rank of Model (7)obtained from the multiobjective genetic algorithm

The algorithm execution time is 22783 seconds Table 3shows the Pareto solution combination

8 Conclusions

The selection of a portfolio from a large number of poten-tial projects can be modeled as a hybrid model includingmetaheuristic algorithms and multicriteria decision-makingtechniques In such circumstances the criteria governing thedecision problem could be a combination of qualitative andquantitative criteria Therefore such decision problems areinherently complex and ambiguous In this study the follow-ing approach was proposed to select a portfolio of projects

(i) The grey relational analysis was used to rank thecandidate projects

(ii) The hybrid failure analysis model and the fuzzy infer-ence system were used to analyze and predict the riskof project portfolio


5

52

54

56

58

6

62

64

66Pareto plot

minus78 minus77 minus76 minus75 minus74 minus73 minus72 minus71 minus7 minus69 minus68

Z1

Z2

Figure 10 Pareto front of rank and risk

(iii) The environmental protection and green projectmanagement were considered in multicriteria projectselection

(iv) According to the literature on the optimal Paretosolutions of risk and rank the metaheuristic NSGA-II algorithm was used to select the optimal Paretocombination

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] D L Hall and A Nauda ldquoAn interactive approach for selectingIRampDprojectsrdquo IEEETransactions on EngineeringManagementvol 37 no 2 pp 126ndash133 1990

[2] J Wang Y Xu and Z Li ldquoResearch on project selection sys-tem of pre-evaluation of engineering design project biddingrdquoInternational Journal of Project Management vol 27 no 6 pp584ndash599 2009

[3] A Lund N Gorden and A Altounian Anaheim Bid UserrsquosGuide Anaheim Technologies Inc Montreal Canada 1989

[4] J F Bard R Balachandra and P E Kaufmann ldquoInteractiveapproach to RampD project selection and terminationrdquo IEEETransactions on EngineeringManagement vol 35 no 3 pp 139ndash146 1988

[5] M Ehrgott Multicriteria Optimization vol 2 Springer NewYork NY USA 2005

[6] G-H Tzeng and J-J Huang Multiple Attribute Decision Mak-ing Methods and Applications CRC Press Boca Raton FlaUSA 2011

[7] S-J Chen and C-L Hwang Fuzzy Multiple Attribute DecisionMaking vol 375 of Lecture Notes in Economics and Mathemati-cal Systems Springer Berlin Germany 1992

[8] A Ishizaka and P Nemery Multi-criteria Decision AnalysisMethods and Software JohnWiley amp Sons New York NY USA2013

[9] H ZareiM F Zarandi andM KarbasianANew Fuzzy DSSESfor Stock Portfolio Selection Using Technical andFundamentalApproaches in Parallel

[10] C-C Lin and Y-T Liu ldquoGenetic algorithms for portfolioselection problems with minimum transaction lotsrdquo EuropeanJournal of Operational Research vol 185 no 1 pp 393ndash4042008

[11] K Doerner W J Gutjahr R F Hartl C Strauss and C Stum-mer ldquoPareto ant colony optimization a metaheuristic approachto multiobjective portfolio selectionrdquo Annals of OperationsResearch vol 131 no 1ndash4 pp 79ndash99 2004

[12] A R Martınez-Lorente F Dewhurst and B G Dale ldquoTotalquality management origins and evolution of the termrdquo TQMMagazine vol 10 no 5 pp 378ndash386 1998

[13] A Bilbao-Terol M Arenas-Parra and V Canal-FernandezldquoSelection of socially responsible portfolios using goal program-ming and fuzzy technologyrdquo Information Sciences vol 189 pp110ndash125 2012

[14] A T Eshlaghy and F F Razi ldquoA hybrid grey-basedK-means andgenetic algorithm for project selectionrdquo International Journal ofBusiness Information Systems vol 19 no 2 2015

[15] A T E F F Razi J Nazemi M Alborzi and A PoorebrahimildquoA hybrid grey based fuzzy C-means and multiple objectivegenetic algorithms for project portfolio selectionrdquo InternationalJournal of Industrial and Systems Engineering In press

[16] D Huang S Zhu F J Fabozzi and M Fukushima ldquoPortfolioselection under distributional uncertainty a relative robustCVaR approachrdquo European Journal of Operational Research vol203 no 1 pp 185ndash194 2010

[17] J L Deng ldquoIntroduction to grey system theoryrdquoThe Journal ofGrey System vol 1 no 1 pp 1ndash24 1989

[18] C-C Yang and B-S Chen ldquoSupplier selection using combinedanalytical hierarchy process and grey relational analysisrdquo Jour-nal of Manufacturing Technology Management vol 17 no 7 pp926ndash941 2006

[19] K-H Chang Y-C Chang and I-T Tsai ldquoEnhancing FMEAassessment by integrating grey relational analysis and thedecision making trial and evaluation laboratory approachrdquoEngineering Failure Analysis vol 31 pp 211ndash224 2013

[20] P Mujumdar and S Karmakar ldquoGrey fuzzy multi-objectiveoptimizationrdquo in Fuzzy Multi-Criteria Decision Making pp453ndash482 Springer 2008

[21] Y Kuo T Yang and G-W Huang ldquoThe use of grey relationalanalysis in solving multiple attribute decision-making prob-lemsrdquoComputers amp Industrial Engineering vol 55 no 1 pp 80ndash93 2008

[22] Z Li D Zhang and Q Gao ldquoA grey method of prioritizingengineering characteristics in QFDrdquo in Proceedings of theChinese Control andDecision Conference (CCDC rsquo09) pp 3443ndash3447 IEEE 2009

[23] P Wang P Meng J-Y Zhai and Z-Q Zhu ldquoA hybrid methodusing experiment design and grey relational analysis for mul-tiple criteria decision making problemsrdquo Knowledge-Based Sys-tems vol 53 pp 100ndash107 2013

[24] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993

[25] E H Mamdani and S Assilian ldquoAn experiment in linguisticsynthesis with a fuzzy logic controllerrdquo International Journal ofMan-Machine Studies vol 7 no 1 pp 1ndash13 1975


[26] N K Kasabov and Q Song ldquoDENFIS dynamic evolvingneural-fuzzy inference systemand its application for time-seriespredictionrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 144ndash154 2002

[27] H-T Liu and Y-L Tsai ldquoA fuzzy risk assessment approachfor occupational hazards in the construction industryrdquo SafetyScience vol 50 no 4 pp 1067ndash1078 2012

[28] E G Bekele and J W Nicklow ldquoMulti-objective automaticcalibration of SWAT using NSGA-IIrdquo Journal of Hydrology vol341 no 3-4 pp 165ndash176 2007

[29] C A C Coello D A van Veldhuizen and G B Lamont Evo-lutionary Algorithms for Solving Multi-Objective Problems vol242 Springer 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


is reviewed in Section 2 Section 3 examines the grey rela-tional analysis method Fuzzy inference system is describedin Section 4 NSGA-II is introduced in Section 5 A newapproach for selecting the portfolio of projects is presentedin Section 6 A case study and conclusions are presented inSections 7 and 8 respectively

2 Literature Review

Zarei et al (2009) developed an expert system for portfolioselection The proposed system analyzed the technical riskand return on investment In this model the preferences areweighted and then the optimal portfolio is clustered throughthe rough set theory [9] Lin and Liu proposed a portfoliooptimizationmodel based onMarkowitz linear programmingmodel for project portfolio selection using the minimumswap size The optimal Pareto solution of the model wasobtained by a genetic algorithm [10] Doerner et al presenteda multiobjective integer programming model for optimalportfolio selection using the ant colony optimization algo-rithm [11]Martınez-Lorente et al considered both qualitativeand quantitative objectives to optimize the portfolio ofprojects In this study the path analysis was used to analyzethe qualitative objectives [12] Bilbao-Terol et al consideredthe social responsibility to select the optimal portfolio In thisapproach enterprises do not invest in activities neglectingethical standards Obviously the portfolio is completedthrough the assets observing the ethical standards For thispurpose a measure called social responsibility attractive-ness was used [13] Eshlaghy and Razi proposed a 119896-meanalgorithm-based grey relational analysis model for projectportfolio selection In this model the projects are firstclustered through the 119896-mean algorithm then each cluster isranked using the grey relational analysis Finally the Paretosolutions of rank and risk are analyzed by the geneticalgorithm [14] In another study Razi et al clustered projectsusing the 119862-mean fuzzy algorithm and then analyzed theclusters using the grey relational analysis In this study theproject risk analysis was carried out through a fuzzy inferencesystem [15] Huang et al designed amodel based on the semi-variance index to invest in a portfolio of real estate assets con-sidering the risk preference to optimize the portfolio of realestate assets In the second stage the Pareto optimal solutionof the model was analyzed using the bee colony algorithmThemodeling approach in this study is based on the salesmannetwork model [16]

3 Grey Relational Analysis

In 1982 Deng published the first paper on the grey systemtheory entitled ldquoThe Control of the Grey Systemsrdquo and thenthe grey system theory was introduced [17] Briefly the basicidea of grey theory is as follows the overall picture of thesystem is imagined considering the partial or limited infor-mation about a system This methodology deals with uncer-tain incomplete and poor problems As one of the mainfeatures of the grey system theory this theory can providesatisfactory outputs using relatively low information and thehigh variability in the criteria Like the fuzzy theory the grey

theory is an effective mathematical model for solving uncer-tain and ambiguous problems [18] There are many differentsystems in the real world each of them has its own compo-nents and subsystems To recognize a system the relationsbetween the components as well as the structure of thesystem should be identified in addition to understandingthe components If the completely known and unknowninformation of a system is respectively shown by white andblack colors the information onmost systems in nature is notwhite (well known) or black (unknown) but it is a mixtureof both colors that is grey information Such systems arecalled grey systemsThemain characteristic of grey systems isincomplete information The aim of the grey systems theoryand its applications is to create a bridge between the social sci-ences andnatural sciencesGrey colormeans the deficiency ofinformation and uncertainty [19]The grey relational analysisincludes the following steps

Formation of a Grey Relation When the performance mea-surement units for different indicators are different it is likelythat the effects of some parameters are ignored Furthermorewhen some performance indicators have a wide range thismay happen In addition the performance indicators withdifferent objectives or directions may lead to inaccurateresultsThus it is necessary to convert all performance valuesof an alternative to comparative series through a processsimilar to normalization process In grey systems theory thisprocess is called the formation of grey relations In a mul-ticriteria decision-making problem with 119898 alternatives and119899 indexes the 119894th alternative is shown by 119910

119894= (1199101198941 1199101198942

119910119894119895 119910

119894119899) in which 119910

119894119895is the performance value of the index

119895 for the alternative 119894119910119894can be converted into the comparative

series 119909119894= (1199091198941 1199091198942 119909119894119895 119909

119894119899) using one of the following

equations

119883119894(119895)lowast=

119909119894(119895) minusmin119909

119894(119895)

max 119909119894(119895) minusmin119909

119894(119895)

(1)

119883119894(119895)lowast=

max119909119894(119895) minus 119909

119894(119895)

max 119909119894(119895) minusmin119909

119894(119895)

(2)

119883119894119895=

1003816100381610038161003816119909119894 (119895) minus 1199090119887(119895)

1003816100381610038161003816

max 119909119894(119895) minus 119909

0119887(119895)

(3)

Equation (1) is used for ldquothe bigger the betterrdquo indexes while(2) is used for ldquothe smaller the betterrdquo indexes Equation (3) isused for the case where ldquovalues closer to the optimal value of119910lowast

119895are betterrdquo [20]

The Reference Target Series Once the grey relations wereformed using (1) (2) or (3) all performance values arelocated in the range [0 1] In the case where the value of119909119894119895generated by the grey relation creation process is equal

to 1 or closer to 1 than the value of any alternative theperformance of the index 119894 in the alternative 119895 is betterthan other alternatives Thus the alternative for which allperformance values are equal to 1 is the best alternative In thisstudy the reference series is defined as 119909

0= (11990901 11990902 119909

0119895

1199090119899) = (1 1 1 1) Accordingly it searches for




119894119895to 1199090119895 Higher grey




and 1199090119895 The


119884 (1199090119895 119909119894119895) =

Δmin+120585Δmax

Δ119894119895+120585

Δmax

Δ119894119895= 1199090119895minus 119909119894119895

(4)


calculated using

Γ (1199090 119909119894) =

119899

sum

119895=1

119908119895sdot 119884 (119909

0119895 119909119894119895) (5)


between 119909119894and 119909












number [26]




membership function

Fuzzy operators




fuzzy output

Defuzzifier


0 5 10 150

01

02

03

04

05

06

07

08

09

1

Gen

eral

ized

bell

-sha

ped

mem

bers

hip

func

tion

Severity







0 5 10 150

01

02

03

04

05

06

07

08

09

1

Tria

ngul

ar-s

hape

d m

embe

rshi

p fu

nctio

n

Occurrence


0 5 10 150

010203040506070809

1

Gau

ssia

n cu

rve m

embe

rshi

p fu

nctio

n

Detection



5 NSGA-II





0 2 4 6 8 10 12 14 160

010203040506070809

1

RPN

Tria

ngul

ar-s

hape

d m

embe

rshi

p


Rejected

P1

Q1

F1

F2

F3

Ps+1


















7 Case Study





0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Alternatives

GRA





Max 1199111= minus06799119909

1minus 05698119909

2minus 05215119909

3

+ 083331199094+ 04810119909

5minus 05372119909

6




002

0406

081

0

05

1

05

055

06

065

IE

Risk


+ 053961199097+ 04268119909

8+ 06880119909

9

+ 0507011990910minus 04473119909

11+ 07815119909

12

minus 0673411990913minus 06267119909

14+ 06067119909

15

minus 0950611990916minus 05706119909

17minus 04899119909

18

+ 0691811990919minus 07777119909

20

Min 1199112= 061119909

1minus 0532119909

2+ 0243119909

3minus 0752119909

4

minus 05391199095minus 0547119909

6+ 0338119909

7

minus 08211199098minus 0673119909

9+ 0924119909

10

+ 043211990911minus 0532119909

12+ 0724119909

13

minus 032711990914+ 0876119909

15+ 0143119909

16

+ 018511990917minus 0442119909

18+ 0646119909

19

+ 017511990920

subject to 031199091+ 02119909

2+ 04119909

3+ 035119909

4+ 022119909

5

+ 0351199096+ 029119909

7+ 024119909

8+ 033119909

9

+ 0311990910+ 03119909

11+ 02119909

12+ 031119909

13

+ 03211990914+ 032119909

15+ 034119909

16+ 038119909

17

+ 02711990918+ 037119909

19+ 025119909

20le 4

7761199091+ 884119909

2+ 639119909

3+ 845119909

4+ 532119909

5

+ 5551199096+ 858119909

7+ 504119909

8+ 720119909

9


1198851

1198852


+ 53111990910+ 649119909

11+ 527119909

12+ 678119909

13

+ 83811990914+ 709119909

15+ 902119909

16+ 923119909

17

+ 59311990918+ 716119909

19+ 764119909

20ge 5000

11990913+ 11990920le 1

11990911+ 11990918le 1

1199094minus 1199098le 0

1199092minus 1199099le 0

119909119895 isin [0 or 1] 119895 = 1 2 3 20

(7)



8 Conclusions





5

52

54

56

58

6

62

64

66Pareto plot


Z1

Z2






References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894119895to 1199090119895 Higher grey




and 1199090119895 The


119884 (1199090119895 119909119894119895) =

Δmin+120585Δmax

Δ119894119895+120585

Δmax

Δ119894119895= 1199090119895minus 119909119894119895

(4)


calculated using

Γ (1199090 119909119894) =

119899

sum

119895=1

119908119895sdot 119884 (119909

0119895 119909119894119895) (5)


between 119909119894and 119909












number [26]




membership function

Fuzzy operators




fuzzy output

Defuzzifier


0 5 10 150

01

02

03

04

05

06

07

08

09

1

Gen

eral

ized

bell

-sha

ped

mem

bers

hip

func

tion

Severity







0 5 10 150

01

02

03

04

05

06

07

08

09

1

Tria

ngul

ar-s

hape

d m

embe

rshi

p fu

nctio

n

Occurrence


0 5 10 150

010203040506070809

1

Gau

ssia

n cu

rve m

embe

rshi

p fu

nctio

n

Detection



5 NSGA-II





0 2 4 6 8 10 12 14 160

010203040506070809

1

RPN

Tria

ngul

ar-s

hape

d m

embe

rshi

p


Rejected

P1

Q1

F1

F2

F3

Ps+1


















7 Case Study





0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Alternatives

GRA





Max 1199111= minus06799119909

1minus 05698119909

2minus 05215119909

3

+ 083331199094+ 04810119909

5minus 05372119909

6




002

0406

081

0

05

1

05

055

06

065

IE

Risk


+ 053961199097+ 04268119909

8+ 06880119909

9

+ 0507011990910minus 04473119909

11+ 07815119909

12

minus 0673411990913minus 06267119909

14+ 06067119909

15

minus 0950611990916minus 05706119909

17minus 04899119909

18

+ 0691811990919minus 07777119909

20

Min 1199112= 061119909

1minus 0532119909

2+ 0243119909

3minus 0752119909

4

minus 05391199095minus 0547119909

6+ 0338119909

7

minus 08211199098minus 0673119909

9+ 0924119909

10

+ 043211990911minus 0532119909

12+ 0724119909

13

minus 032711990914+ 0876119909

15+ 0143119909

16

+ 018511990917minus 0442119909

18+ 0646119909

19

+ 017511990920

subject to 031199091+ 02119909

2+ 04119909

3+ 035119909

4+ 022119909

5

+ 0351199096+ 029119909

7+ 024119909

8+ 033119909

9

+ 0311990910+ 03119909

11+ 02119909

12+ 031119909

13

+ 03211990914+ 032119909

15+ 034119909

16+ 038119909

17

+ 02711990918+ 037119909

19+ 025119909

20le 4

7761199091+ 884119909

2+ 639119909

3+ 845119909

4+ 532119909

5

+ 5551199096+ 858119909

7+ 504119909

8+ 720119909

9


1198851

1198852


+ 53111990910+ 649119909

11+ 527119909

12+ 678119909

13

+ 83811990914+ 709119909

15+ 902119909

16+ 923119909

17

+ 59311990918+ 716119909

19+ 764119909

20ge 5000

11990913+ 11990920le 1

11990911+ 11990918le 1

1199094minus 1199098le 0

1199092minus 1199099le 0

119909119895 isin [0 or 1] 119895 = 1 2 3 20

(7)



8 Conclusions





5

52

54

56

58

6

62

64

66Pareto plot


Z1

Z2






References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 150

01

02

03

04

05

06

07

08

09

1

Tria

ngul

ar-s

hape

d m

embe

rshi

p fu

nctio

n

Occurrence


0 5 10 150

010203040506070809

1

Gau

ssia

n cu

rve m

embe

rshi

p fu

nctio

n

Detection



5 NSGA-II





0 2 4 6 8 10 12 14 160

010203040506070809

1

RPN

Tria

ngul

ar-s

hape

d m

embe

rshi

p


Rejected

P1

Q1

F1

F2

F3

Ps+1


















7 Case Study





0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Alternatives

GRA





Max 1199111= minus06799119909

1minus 05698119909

2minus 05215119909

3

+ 083331199094+ 04810119909

5minus 05372119909

6




002

0406

081

0

05

1

05

055

06

065

IE

Risk


+ 053961199097+ 04268119909

8+ 06880119909

9

+ 0507011990910minus 04473119909

11+ 07815119909

12

minus 0673411990913minus 06267119909

14+ 06067119909

15

minus 0950611990916minus 05706119909

17minus 04899119909

18

+ 0691811990919minus 07777119909

20

Min 1199112= 061119909

1minus 0532119909

2+ 0243119909

3minus 0752119909

4

minus 05391199095minus 0547119909

6+ 0338119909

7

minus 08211199098minus 0673119909

9+ 0924119909

10

+ 043211990911minus 0532119909

12+ 0724119909

13

minus 032711990914+ 0876119909

15+ 0143119909

16

+ 018511990917minus 0442119909

18+ 0646119909

19

+ 017511990920

subject to 031199091+ 02119909

2+ 04119909

3+ 035119909

4+ 022119909

5

+ 0351199096+ 029119909

7+ 024119909

8+ 033119909

9

+ 0311990910+ 03119909

11+ 02119909

12+ 031119909

13

+ 03211990914+ 032119909

15+ 034119909

16+ 038119909

17

+ 02711990918+ 037119909

19+ 025119909

20le 4

7761199091+ 884119909

2+ 639119909

3+ 845119909

4+ 532119909

5

+ 5551199096+ 858119909

7+ 504119909

8+ 720119909

9


1198851

1198852


+ 53111990910+ 649119909

11+ 527119909

12+ 678119909

13

+ 83811990914+ 709119909

15+ 902119909

16+ 923119909

17

+ 59311990918+ 716119909

19+ 764119909

20ge 5000

11990913+ 11990920le 1

11990911+ 11990918le 1

1199094minus 1199098le 0

1199092minus 1199099le 0

119909119895 isin [0 or 1] 119895 = 1 2 3 20

(7)



8 Conclusions





5

52

54

56

58

6

62

64

66Pareto plot


Z1

Z2






References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















7 Case Study





0 5 10 15 20 250

01

02

03

04

05

06

07

08

09

1

Alternatives

GRA





Max 1199111= minus06799119909

1minus 05698119909

2minus 05215119909

3

+ 083331199094+ 04810119909

5minus 05372119909

6




002

0406

081

0

05

1

05

055

06

065

IE

Risk


+ 053961199097+ 04268119909

8+ 06880119909

9

+ 0507011990910minus 04473119909

11+ 07815119909

12

minus 0673411990913minus 06267119909

14+ 06067119909

15

minus 0950611990916minus 05706119909

17minus 04899119909

18

+ 0691811990919minus 07777119909

20

Min 1199112= 061119909

1minus 0532119909

2+ 0243119909

3minus 0752119909

4

minus 05391199095minus 0547119909

6+ 0338119909

7

minus 08211199098minus 0673119909

9+ 0924119909

10

+ 043211990911minus 0532119909

12+ 0724119909

13

minus 032711990914+ 0876119909

15+ 0143119909

16

+ 018511990917minus 0442119909

18+ 0646119909

19

+ 017511990920

subject to 031199091+ 02119909

2+ 04119909

3+ 035119909

4+ 022119909

5

+ 0351199096+ 029119909

7+ 024119909

8+ 033119909

9

+ 0311990910+ 03119909

11+ 02119909

12+ 031119909

13

+ 03211990914+ 032119909

15+ 034119909

16+ 038119909

17

+ 02711990918+ 037119909

19+ 025119909

20le 4

7761199091+ 884119909

2+ 639119909

3+ 845119909

4+ 532119909

5

+ 5551199096+ 858119909

7+ 504119909

8+ 720119909

9


1198851

1198852


+ 53111990910+ 649119909

11+ 527119909

12+ 678119909

13

+ 83811990914+ 709119909

15+ 902119909

16+ 923119909

17

+ 59311990918+ 716119909

19+ 764119909

20ge 5000

11990913+ 11990920le 1

11990911+ 11990918le 1

1199094minus 1199098le 0

1199092minus 1199099le 0

119909119895 isin [0 or 1] 119895 = 1 2 3 20

(7)



8 Conclusions





5

52

54

56

58

6

62

64

66Pareto plot


Z1

Z2






References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












002

0406

081

0

05

1

05

055

06

065

IE

Risk


+ 053961199097+ 04268119909

8+ 06880119909

9

+ 0507011990910minus 04473119909

11+ 07815119909

12

minus 0673411990913minus 06267119909

14+ 06067119909

15

minus 0950611990916minus 05706119909

17minus 04899119909

18

+ 0691811990919minus 07777119909

20

Min 1199112= 061119909

1minus 0532119909

2+ 0243119909

3minus 0752119909

4

minus 05391199095minus 0547119909

6+ 0338119909

7

minus 08211199098minus 0673119909

9+ 0924119909

10

+ 043211990911minus 0532119909

12+ 0724119909

13

minus 032711990914+ 0876119909

15+ 0143119909

16

+ 018511990917minus 0442119909

18+ 0646119909

19

+ 017511990920

subject to 031199091+ 02119909

2+ 04119909

3+ 035119909

4+ 022119909

5

+ 0351199096+ 029119909

7+ 024119909

8+ 033119909

9

+ 0311990910+ 03119909

11+ 02119909

12+ 031119909

13

+ 03211990914+ 032119909

15+ 034119909

16+ 038119909

17

+ 02711990918+ 037119909

19+ 025119909

20le 4

7761199091+ 884119909

2+ 639119909

3+ 845119909

4+ 532119909

5

+ 5551199096+ 858119909

7+ 504119909

8+ 720119909

9


1198851

1198852


+ 53111990910+ 649119909

11+ 527119909

12+ 678119909

13

+ 83811990914+ 709119909

15+ 902119909

16+ 923119909

17

+ 59311990918+ 716119909

19+ 764119909

20ge 5000

11990913+ 11990920le 1

11990911+ 11990918le 1

1199094minus 1199098le 0

1199092minus 1199099le 0

119909119895 isin [0 or 1] 119895 = 1 2 3 20

(7)



8 Conclusions





5

52

54

56

58

6

62

64

66Pareto plot


Z1

Z2






References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5

52

54

56

58

6

62

64

66Pareto plot


Z1

Z2






References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Hybrid Grey Relational Analysis...

Documents

Transcript of Research Article A Hybrid Grey Relational Analysis...