Research Article Sequence and Direction Planning ... - Hindawi
Transcript of Research Article Sequence and Direction Planning ... - Hindawi
Research ArticleSequence and Direction Planning ofMultiobjective Attack in Virtual Navigation Based onVariable Granularity Optimization Method
Chengwei Ruan Lei Yu Zhongliang Zhou and Jinfu Wang
Aeronautics and Astronautics Engineering College Air Force Engineering University Xirsquoan 710038 China
Correspondence should be addressed to Chengwei Ruan rcwrff126com
Received 22 April 2015 Accepted 10 September 2015
Academic Editor Konstantinos Karamanos
Copyright copy 2015 Chengwei Ruan et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In multiobjective air attacking tasks it is essential to find the optimal combat preplanning for the attacking flightThis paper solvesthe planning problem by decomposing it into two subproblems attacking sequence planning and attacking direction planningAccording to this decomposition we propose the VGHPSO (Variable Granularity Hybrid Particle Swarm Optimization) methodVGHPSO employs the Particle SwarmOptimization ametaheuristic global optimizationmethod to solve the planning problem inmultiple granularities including optimizing the high-level attacking sequence and optimizing the low-level attacking directions forengagements Furthermore VGHPSO utilizes infeasible individuals in the swarm in order to enhance the capability of searchingin the boundary of the feasible solution space Simulation results show that the proposed model is feasible in the combat and theVGHPSO method is efficient to complete the preplanning process
1 Introduction
The traditional multi-target-attack can be presented as aunique attack style that a single flight fighter ormultiple flightfighters launch and guideweapons towardsmultiple targets atthe same timeWith the development ofmany advanced tech-nologies the air combat scenario becomes more and morecomplex the needed condition to attack different targets atthe same time is hard to satisfy and then the sequential attackstyle is born to make the air combat more reasonable
The multi-target-attack in this paper is defined as anattack process that the multiple targets which are located indistributed spaces are attacked in sequence Here the attacksequence is of a great importance according to different rela-tive situations brought by different targets and the reasonableattack sequence may make the task easier and more effectiveto complete The other important point in this paper is aboutthe transform time during thementionedmulti-target-attackIt is not avoidable to research on how to enter the attack areaof next target after completing the last attack mission Theconcept of attack direction is given to describe the transformstyle duringmission process and a proper attack direction can
shorten the total time of executing the attack task and keepour flight fighter safer
Current researches on multi-target-attack problemmainly focused on the algorithms of calculating the targetthreat [1 2] methods of target assignment [3ndash5] tacticaldecision [6 7] and so on Only few references are concernedabout the transform time during the attack task and thecommon researches on sequential attack pay much attentionto the optimization algorithms [8] the systematic researchesfrom a broader tactical view still need to be focused on
In this study a new sequential attack method basedon the classic multi-target-attack problem is put forwardwhich can make the abstract route planning problem moreconcrete and the whole problem has been transformed into amultiobjective optimization problembased on virtual naviga-tion planning The classic weapon-target-assignment (WTA)problem [9] has been changed to a PSO (Particle SwarmOptimization) problem [10] in solution space The classiccomplicated route planning problem has been decomposedinto a hierarchical problem which pay more attentions to theoptimizing of attack sequence and attack direction In theprocess of optimization the attack sequence and the attack
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 807925 10 pageshttpdxdoiorg1011552015807925
2 Mathematical Problems in Engineering
direction are optimized with relative setting granularities anda surviving policy of noninferior solutions is given to makesure that the best solutions in the Pareto boundary [11ndash14] willnot be neglected
The paper is organized as follows Section 1 is introduc-tion Section 2 presents the basic sequential attack modeland its simplified method Section 3 describes the modifiedswarm optimization algorithm with a new surviving policySection 4 exhibits the simulation results and Section 5 isconclusion
2 Background-Basic Scenario
21 The Analysis of Target Threatening Area In moderncombat scenario the flight fighter may have many advancedsensor and weapon systems which can bring much morethreats [15] And in this paper we define the targetrsquos threatas the threat of targetrsquos comprehensive ability and the threatof targetrsquos destroy ability
Figure 1 describes the traditional 1 versus 1 air combatsituation 119860 is our flight fighter and 119879 is the target V
119886and
V119905are the velocities 119877 is the relative distance between 119860 and
119879 119902 is the targetrsquos approaching angle and 120593 is the targetrsquosazimuth angle In this paper we suppose that the attitudeangle exposed to the targetrsquos radar is equal to the targetrsquosazimuth angle on condition that 119860 and 119879 are in the samehorizontal plane
In air combat the greatest threat to our flight fightercomes from the targetrsquos detection threat and the targetrsquosweapon threat Thus we need to choose an appropriateapproaching method to avoid the threatening area in orderto increase our task success rate So it is necessary to analyzethe attack area and the exposure area
Attack Area We define the attack area in this content as aspecial air-to-air missile attack area around the target andonly in this area our flight fighter can get a probabilitywithin agiven threshold to destroy the target after launching a propermissile The destroy probability may decrease rapidly to zerowhen launching a missile outside this area [16]The real-timecalculating method to get the attack area is hard to satisfythe needs of the modern combat so we apply the data fittingmethod to acquire the approximate targetrsquos attack area [17]Suppose our member carries the middle-range-missile thenthe far and near boundary of the attack area around the targetcan be calculated by the following formulas
119863max = 119891 (1198671198810 119881119879 119879 119881119903 119903 120578119910 120596 119902119879 1199050)
119863min = 119891 (1198671198810 119881119879 119879 119881119903 119903 120578119910 120596 119902119879 1199050) (1)
where119863max119863min are far and near boundary of the attack areaaround the target119867 is attack height119881
0 is missile launching
velocity119881119879 is target velocity
119879 is target accelerated velocity
119881119903 is target angular velocity
119903 is target accelerated angular
velocity 119899119910 ismaneuver overload120596 ismaximummissile off-
boresight launch angle 119902119879 is target approaching angle and 119905
0
is missile working time
R
q
T
A
a
t
120593
Figure 1 The relative situation of friend and foe
Then we can get the whole attack area around the targetin polar coordinate
119885119860= (119902119879 119889 ℎ) | 119902
119879isin [minus120587 120587] ℎ
isin [119867min 119867max] 119889 (119902119879 ℎ)
isin [119863min (119902119879 ℎ) 119863max (119902119879 ℎ)]
(2)
where 119902119879is the targetrsquos approaching angle 119889 is the distance
of the target and ℎ is the relative height to the target where[119867min 119867max]means the attack range in height of missile
In this study we suppose that our member has the sameattack area with the targets and both of them have nearly thesame comprehensive combat capability If there was a greatgap between these two comprehensive combat capabilitiesthen we can describe the combat as an asymmetric combatand the model is not efficient According to current modelwe need to find a proper route planning project to get a betterattack situationwhich can be useful to avoid the targetrsquos attackand to make our member safer
Exposure Area It means an area we exposed to targetrsquos radarThe scanning range of airborne radar is about plusmn60∘ Then wecan also get the exposure area as follows
119885119864= (119902119879 119889 ℎ) | 119902
119879isin [minus120579119877 120579119877] ℎ
isin [119867min 119867max] 119889 (119902119879 ℎ) isin [0 119877119877 (119902119879 ℎ)] (3)
where 120579119877is the scanning angle of targetrsquos radar 119877
119877is the
maximum detection distance and [119867min 119867max] means thedetection range in height of radar
22The Basic Description of the Model Sequential attack [18]problem can be regarded as a multiobjective optimizationproblem and once the attack sequence and the attack direc-tion are assured the whole attack logic is born In order toanalyze the multi-target-attack process in detail we focuson the situation with single target for the first step Beforebringing in the model we should give some definitions aboutthe relative problems
Definition (i) Minimum safety distance this distance can beregarded as the maximum attack distance of the enemyrsquosWhen entering this area ourmembermay get the probabilityof being attacked by the enemies (ii) Attack preparing areathis area is defined between the minimum safety distanceboundary and the far attack area boundary where our flight
Mathematical Problems in Engineering 3
Far attack areaboundary
Minimum safetydistance
Attack preparing area
Threatening areaThreatening area
LAttack direction assuringTactical maneuvering
Retreat routing
2nd circle of tacticalnavigation points
1st circle of tacticalnavigation points
Target TTarget T
L
S1
S2 S3 TT 120579
120579
120578
l1
l2l3
l4
Figure 2 The simplified model of sequential attack
fighter should complete a series of actions to complete itsattack task
From the left side of Figure 2 we can see thewhole processof single-target-attack
119871 is the virtual navigation route across the attack prepar-ing area (an approximate area in this paper) and through thewhole cross process the flight fighter may deal with severalkey stages they are listed as follows
(a) Tactical Maneuvering Stage This stage may decide whichtargets to attack and which routes to choose to enter theminimum safety distance This stage may last till the missilelaunches
(b) Attack Direction Assuring StageThis stage is used to avoidthe threaten area of enemies and find a more effective way toincrease the success rate
(c) Retreat Routing Stage This stage is very important to offeran input for the next attack process and it is also useful to findan safe retreat route to get away from the last threaten area
The three stages can describe thewhole attack process andin order to build amathematical model of the different stageswe simplified the model as the right side of Figure 2
The attack preparing area is described by two circles oftactical navigation points The inner circle is the boundaryof far attack area and the outer circle is the boundaryof minimum safety distance The missile can be launchedtowards the target when the far attack area boundary isreached so we center the target and take sample points inthe far attack area boundary every 120578 angle then we canget the 1st circle tactical navigation points We define the1st circle tactical points of target 119905 as 119877119875119905
1= 119875119905
1 119875
119905
1198731
In order to enter the 1st circle tactical points we cannotavoid the minimum safety distance boundary which can bedecided by the threatening degree of the target as well as therelative combat situations So we also center the target andtake sample points in the minimum safety distance boundaryevery 120578 (120578 = 2120587119899 (119899 = 1 2 )) angle then we canget the 2nd circle tactical points We define the 2nd circletactical points of target 119905 as 119877119875119905
2= 119876119905
1 119876
119905
1198732
Then theproblem can be simplified as a route optimization problemwhich can be described as how to find a best route through
the two circles with shortest time and least threats Then thetransformation between two targets can be completed
If target 119894 would be attacked before target 119895 then thepartial order exists and we make it as 119876119894
1≻ 119876119895
1 The solution
space of virtual navigation can be described as
119875119878 = 1198751
1 119875
1
1198731
1198761
1 119876
1
1198732
119875119879
1 119875
119879
1198731
119876119879
1
119876119879
1198732
(4)
From the right side of Figure 2 we can see that 119871 meansthe attack route through two circles of navigation points and 120579is the angle between the velocity of target and the attack routeand it can describe the attack direction towards the target
From the analysis above we can conclude the problem intwo steps
Step 1 Choose four tactical points from the two circles anddecide the attack sequence according to every single targetin different space We describe it as 119876119905
119894rarr 119875
119905
119898rarr 119875
119905
119899rarr
119876119905
119900(119898 = 119899)
Step 2 Connect the four tactical points in sequence and anew virtual navigation route is born We can describe it as atuple
119871 = ⟨119873 119878 119879119892 119875119903 119864⟩ (5)
where 119873 means the number of navigation route fragmentswhich connect the two kinds of navigation points in differentcircles 119878 is the initial position of our member and it is alsothe first navigation point in the attack route 119879119892 is a set ofserial numbers of targets which records the attack sequence119875119903 contains all the chosen tactical points and their orders 119864ends the navigation
3 Modeling and Algorithm Optimizing
31 Analysis of Sequential Attack and the Attack DirectionThe single-target-attack model is given in Section 2 andthe multi-target-attack model is set up in this chapter FromFigure 2 we can get the general view of sequential attack inmultiple targetsrsquo situation
4 Mathematical Problems in Engineering
New target
Targets area
T1T2
T3
T4
Sequential attack routingManeuver direction routing
Figure 3 The sketch map of multiobjective sequential attack
Sequence attack connection graph
Attack directionconnection graph
Out-tactical point
Feasible solutionTargetTactical point
In-tactical point
Figure 4 Mission decomposition in variable granularities
In Figure 3 the blue line means the attack sequence andthe red linemeans the attack directionWe can easily find thatonce the tactical points are decided the whole attack planincluding attack sequence and direction is acquired Froma tactical view of the combat situation we can solve thisproblem in three different hierarchies
In Figure 4 the problem has been divided into threeparts which describe the feasible region sequential attack
and attack direction respectively The second layer canbe described as a directed graph 119892
119878= 119892(119879 119864
119904) 119879 =
1198791 1198792 1198793 119879
119899 where 119879 is a set of targets and 119864
119904is a set
of connection arcs 119890(119879119894 119879119895) isin 119864
119904means there is a partial
order between 119879119894and 119879
119895 and 119879
119894will be attacked first The
third layer can also be described as a connected directedgraph 119879
119894= 119892119894(119875 119864119901) where 119875 is a set of tactical navigation
points and 119864119901is a set of connection arcs Then the whole
Mathematical Problems in Engineering 5
problem can be rebuilt as a variable granularity directedgraph 119892
119872= 119892(119892
119894(119875 119864119901) 119864) 119864 means the connection arcs
between different granularities
32 Modeling
321 Targetsrsquo Reformation In current air combat scenariothe targetsrsquo flight fighters always appear in formation to exe-cute associated tactical task Thus we can divide these targetsinto different groups according to their unique characters inorder to simplify the complex combat situation And first thegroup should satisfy the following formula
119897avg =119897avg
119871avg (6)
where 119897avg is the mean distance between two targets in thesame group and 119871avg is the mean distance between twodifferent groups The minimum value of 119897avg can be regardedas the important index to choose the proper projects
We give the minimum distance 119871min between groups andmaximum distance 119897max between targets in the same group indifferent projects according to the combat needs When 119897 ge119897max then the target does not belong to the group When 119897 lt119871min we can decide that the target is in the same groupThuswe let 119871min = 119897max = 1198970 = const in order to use a commonmethod for completing the targetsrsquo reformation
The concrete method to reform the target is given asfollows
(a) Set a tactical targetrsquos group as
119866119905= 1198901 1198902 119890
119895 119890119895+1 (7)
1198901 1198902 119890
119895are members in the group and 119890
119895+1is the
119895 + 1member(b) Set the attribute of the member as
119890119894= 119890 (119909
119894 119910119894 119911119894 V119894 120595119894 119896119894) (8)
119909119894 119910119894 119911119894is the space coordinate of the target in
geographic coordinate system V119894is the velocity in
geographic coordinate system 120595119894is the flight direc-
tion angle in geographic coordinate system 119896119894is the
targetrsquos types(c) Set the constraints to reform the target
st 10038161003816100381610038161003816119909119895+1(119905) minus 119909
119894(119905)10038161003816100381610038161003816lt Δ119909
10038161003816100381610038161003816119910119895+1(119905) minus 119910
119894(119905)10038161003816100381610038161003816lt Δ119910
10038161003816100381610038161003816119911119895+1(119905) minus 119911
119894(119905)10038161003816100381610038161003816lt Δ119911
10038161003816100381610038161003816V119895+1(119905) minus V
119894(119905)10038161003816100381610038161003816lt ΔV
10038161003816100381610038161003816120595119895+1(119905) minus 120595
119894(119905)10038161003816100381610038161003816lt Δ120595
119895 + 1 le 119872max 119894 isin (1 2 119895)
(9)
Δ119909 Δ119910 Δ119911 is the threshold to reform the targets intargetsrsquo location ΔV is the velocity threshold of targetΔ120595 is the flight direction angle threshold of target119872max is themaximumnumber of the group Differentconstraints may bring different consequence so it alldepends on the combat situation
322 Attack Sequence Optimization In air combat scenariothe advantages or disadvantages towards enemies can bejudged by four conditions such as aspect relative distancevelocity and height [19] We can get the attack sequence bythe formula below
119891 = 120593 [120583 (120579) 120583 (119877) 120583 (119881) 120583 (119867) 119882]
119882 = 1199081 1199082 1199083 1199084
4
sum
119894=1
119908119894= 1
(10)
where 120583(120579) 120583(119877) 120583(119881) 120583(119867) are linguistic variables in whichgrade values are between 0 and 1 the closer the value ofmembership grade is to 1 the more advantages it is to attackthe target 119882 is the weight variable set which contains fourrelative weight variables in order to stress the importance ofthe linguistic variablesThen the fitness function can be givenas follows
119865 = max119899
sum
119894
119891119894 (11)
According to different targets we can get a sequence of119891119894(119894 = 1 2 119879) and then the total advantages can be
confirmed
323 Attack Direction Optimization In the model we men-tioned above we can easily find that the attack directioncan only be confirmed by connecting two tactical navigationpoints around the target So once the best tactical points arechosen the optimal attack direction is given automatically
33 The Evaluating Indexes of the Optimization The evalu-ation function is needed to evaluate the attack route andin this situation the distance between two different tacticalpoints is far enough with the support of the ability of long-middle-range missile Thus we set up four main indexesto evaluate the whole attack route which contains the totaldistance of attack route the exposure rate the constraints ofattack direction and the constraints when turning aroundFigure 5 describes the detailed definition of each variable
331 Total Distance of Attack Route Consider
119863119871=
119873
sum
119894=1
119897119894 (12)
where 119897119894is the distance of each navigation fragment
6 Mathematical Problems in Engineering
L1
L2
li
120579iT
ei
120583jj+1
Figure 5 Definition of each variable
Initialization Fitnesscalculating
Particleupgrading
Best individualcrossing crossing
Best groupvariation Complete EndYes
NoParticle
Figure 6 Calculating steps
332 Attack Route under Threat Consider
119864119871=
119873
sum
119894=1
119890119894 (13)
where 119890119894is the distance under threat (fire threat) of each
navigation fragment the distance under threat is in relation-ship with the targetrsquos attack area and it can be calculated byanalysis of the combat situation of enemies
333 Attack Direction Constraints The attack directionshould satisfy the maximum missile off-boresight launchangle so we give the constraint as
120579119894= 119891 (119875
119894
119868 119876119894
119868 119879119894) le 120579max (14)
where 120579119894describes the attack direction towards target 119868 and
120579max is the maximummissile off-boresight launch angle
334 Constraints When Turning Around Consider
120583119895119895+1
= infin if 119897119895+ 119897119895+1lt 119903min
120583119895119895+1
ge 120583min else
119895 = 1 2 119873 minus 1
(15)
where 120583119895119895+1
is the angle between route 119895 and route 119895 + 1119897119895means the distance of route 119895 119903min is the minimum turn
radius and 120583min is the minimum turn angleThese four indexes wementioned above canmake contri-
butions to building the fitness function as follows
119865 (119871) = 1199081
119863119871
119863119872
+ 1199082
119864119871
119864119872
+ 1198921(119871) + 119892
2(119871) (16)
where 119865(119871) is the fitness function and the smaller 119865(119871)gets the better the result will be If 119865(119871) rarr infin then thevalue would be meaningless 119863
119871is the total distance of the
attack route 119864119871is the distance under threat 119863
119872and 119864
119872are
normalized coefficients 1199081and 119908
2are weight coefficients
and 1198921(119871) and 119892
2(119871) are correction terms to correct the
conditions when the target approaching angle cannot satisfythe need of maximummissile off-boresight launch angle andthe turning angle is too small to support our action
34 Calculating Process According to the problem simpli-fication above we have changed the complex model intoa typical TSP model thus we use hybrid Particle SwarmOptimization method to solve this problem The detailedsteps can be acquired in Figure 6
341 Encoding Method From Figure 7 we can see that thelength of the gene code is 4119879 + 2 each target may get 4 codeswhich can be described by decimalism and each code rangesfrom 1 to the maximum number of tactical points For betterinitializing the values wemake the two codes in themiddle ofthe gene unequal and all the code will get a random but notrepeated number from 1 to the maximum number of tacticalpoints
342 Route Modifying The two-circle-encode style hasforced all the routes to cross through the tactical points so thesituations of route crossing and repeating exist According tothe rule in Figure 8 we can get the new attack route
Theproject needs to be accessed again aftermodifying thevirtual navigation If the accessing result was not better thanthe old one we would keep the old policy
Mathematical Problems in Engineering 7
4T
1 2 T
P1I Q
1I Q
1O P
1O P
2I Q
2I Q
2O P
2O P
119879I Q
119879I Q
119879O P
119879O
middot middot middot
Figure 7 The encoding method
Deleted routfragment
Planning virtualnavigation
Modifying virtualnavigation
Figure 8 The display of route modifying method
343 Result Space Expanding Policy When calculating theresult may not get out of the local optimization so after eachiteration we get the global optimal value
bestRout
= (119875V1
119868119876
V1
119868119876
V1
119874119875V1
119874119875V2
119868119876
V2
119868119876
V2
119874119875V2
119874sdot sdot sdot 119875
V119895
119868119876
V119895
119868119876
V119895
119874119875V119895
119874)
(17)
Then we can deal with the existing result as follows andthe description of the space expanding policy can be seen inFigure 9
(a) Rescue the abandoned route when the followingcondition is satisfied1003816100381610038161003816gbest minus fitness
119894
1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)
where 119892119887119890119904119905 is the global optimal value in currenttime and fitness
119894is the abandoned value in the current
circulation when the given condition is satisfied werescue the relative 119877119900119906119905
119894(119894 = 1 2 3 119899) and this
route will take part in the next circulation
(b) Get 119899 random points around each best tactical navi-gation point within the distance 119903
119894= 1205791198771198942 (119894 = 1 2)
(119877119894defines the radius of each tactical area)
(c) Expand the result space with choosing points and geta new result space Ωnew
(d) Calculating 119891 = 119892119887119890119904119905119905 119905 is the simulation time If119891 lt 120575 then we save the current result
344 Calculating Steps See Algorithm 1
4 Simulation
41 Hardware Condition CPU is P4 28GHz and RAM is15 G Matlab 2010
42 The Tactical Scenario Our flight fighter is flying towardsthe enemiesrsquo formation and the initial locations of enemiesand our member are given in Table 1
The relative situation can be seen in Figure 10
43 Simulation Result We suppose the detecting area ofenemies is stable at the moment and we need to find the bestattack sequence and attack direction at this moment We set3000 loops to optimize the final result and the result can beseen in Figures 11 and 12
From Figure 11 our member choose the proper directionto avoid the threaten areas in order to complete the task ina relative short way The attack sequence can be describedas 1198791rarr 1198793rarr 1198792rarr 1198794 and the attack direction (the
direction of enemy velocity is the initial angle 0∘) can bedecided by the angles given in Table 2
Figure 12 shows the updating process when choosing thebest solution and after 600 more loops we get the stablevalue the best route we choose can be shown as follows
bestRout = (11987531198681198763
11986811987623
11987411987522
11987411987510
11986811987615
11986811987617
11987411987518
11987411987510
11986811987615
11986811987617
11987411987518
1198741198752
1198681198765
11986811987619
11987411987517
1198741198759
1198681198764
11986811987621
11987411987523
119874) (19)
In Figure 12 three different methods are compared witheach other The blue line shows the result when using tradi-tional PSOmethod without dividing them in different granu-larities then it has costmore time to get a stable result becauseof much more formations of the available tactical points Thered line means the VGHPSO method we mentioned above
it has cost less time to get a stable value but it is easy to gettrapped in the local optimal value Thus we use VGHPSOlowastmethodwhichmeans the VGHPSOmethodwith result spaceexpanding policy We can see the result reflected in cyan line
In the former chapters we can know that the numbers oftactical points are depended on the sampling angle Thus we
8 Mathematical Problems in Engineering
Table 1 Initial locations
Start point 1198791
1198792
1198793
1198794
End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)
Table 2 Attack direction display
Attack sequence 1198791
1198793
1198792
1198794
Approach angle (∘) 10728 16076 7543 13306Attack angle (∘) 80 12793 2914 17293Retreat angle (∘) 16673 13673 2051 16051
PjO
QjO
r1
r2
n1 n2
n2
n3
bestRout
Figure 9 Result space expanding policy
400 600 800 1000 1200 1400 1600
700
800
900
1000
1100
1200
1300
1400
1500
T1 T2
T3
T4
y(k
m)
x (km)
Figure 10 The initial locations sketch map
do some simulation with different sampling angles and theresults can be seen in Figure 13
From left side of Figure 13 the blue parts mean thetotal distances in different sampling angles and the red partsdescribe the distances under enemiesrsquo threats Then we canget the ratio as shown in Table 3
400 600 800 1000 1200 1400 1600600
700
800
900
1000
1100
1200
1300
1400
1500
1600
T1T2
T3
T4
y(k
m)
x (km)
Figure 11 The optimal route planning project
0 500 1000 1500 2000 2500 30001800
1850
1900
1950
2000
2050
2100
2150
2200
2250
x (times)
VGHPSOPSO
y(k
m)
VGHPSOlowast
Figure 12 Updating curves in different algorithms
FromTable 3 we can see thatwhen choosing the samplingangle as 15∘ we can get a minimum comprehensive indexwhich indicates that we may get a more proper resultefficiently with less threats and shorter times
The usage of the results in this paper can only beregarded as the equal description of attack sequence andattack direction and the tactical points in virtual navigationare not the real navigation points but of great importance tosimplify the engineering problems
Mathematical Problems in Engineering 9
Input (1) Targetsrsquo attributes including the targetsrsquo number119873 the coordinate 119879(119909119879 119910119879 119911119879) the velocity V
119879and the attack area
119882 (2) Our flight fightersrsquo attributes including the number of our fighters119872 the angle 120579 to decide the number of tacticalnavigation points the coordinate 119865(119909
119865 119910119865 119911119865) and the results space Ω (3) A result space expanding policy 120588lowast to optimize
the resultSuppose Condition (1) The targets stay flying in a stable velocity and an unchanged direction (2) The tactical pointsrsquoformation should obey the constraints 120585sequence directionOutput (1) The global optimal fitness result 119892119887119890119904119905 (2) Best tactical points formation 119887119890119904119905119877119900119906119905Initialize randomΩ set the initial stateBeginFor each iteration 119894 = 1 119899 119899 is the iteration number
Cross(119865) rarr Ωnew executing the cross process towards the setting codesIf 119865119894(119894 = 1 2 120587120579) (119865
119894isin Ωnew) do not obey the constraints 120585sequence direction
While (119865119894obey the constraints 120585sequence direction)
Random 119865119894 make sure that all the elements inΩnew obey the constraints
endUpdate (Ωnew)
endCalculating (119891119887119890119904119905
119894)
If 119891119887119890119904119905119894lt min119891119887119890119904119905
1 119891119887119890119904119905
2 119891119887119890119904119905
119894minus1 Then
119892119887119890119904119905119894= 119891119887119890119904119905
119894
Variate(119865) rarr Ω1015840
new executing the variation process towards the setting codesCycling the process as Cross methodExecuting (120588lowast) the saving policy is used to expand the result space to optimize the final result
EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)
End
Algorithm 1
5 15 30 45 60 72 900
500
1000
1500
2000
2500
Dist
ance
(km
)
10 20 30 40 50 60 70 800
10
20
30
40
50
60
70
80
90
Running time (s)Sampling angle (∘)
Sam
plin
g an
gle(
∘ )
Total distanceThreatened distance
Figure 13 The comprehensive values in different sampling angles
10 Mathematical Problems in Engineering
Table 3 Relative calculating results display
Sampling angle (∘) 5 15 30 45 60 72 90Threats ratio 0013 0017 0044 0090 0129 0153 0257Running times (s) 78 45 35 26 24 19 13Comprehensive index 1014 0765 154 234 3096 2907 3341
5 Conclusion
The multiobjective sequence attack and attack directionplanning methods are given in the paper We put forwarda simplified model to create the tactical points in virtualnavigation and the complex air-to-air combat planningproblemhas been transformed into a variable granularity TSPproblem then theVGHPSOmethod is used to solve the givenproblem In the end the result of the simulation shows theefficiency of the method and it also indicates that the modelis available
The model and method we mentioned in the paper areavailable only in static combat planning which belongs toprecombat planning problem But in real combat scenariowe need a dynamic data management system to optimize thereal-time policiesThen we need to pay much attention to theprediction of the future situation according to the historicaldata [20] Thus we may get a more proper and more accurateresult
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Kang H Park S Noh et al ldquoAutonomously decidingcountermeasures against threats in electronic warfare settingsrdquoin Proceedings of the International Conference on ComplexIntelligent and Software Intensive Systems (CISIS rsquo09) pp 177ndash184 IEEE Fukuoka Japan March 2009
[2] H Guo Y Lv P Wang P Ren and L Zhu ldquoTarget threatassessment of air combat based on intervals and SVRrdquo FireControl amp Command Control vol 39 no 8 pp 17ndash21 2014
[3] A Salman I Ahmad and S Al-Madani ldquoParticle swarmoptimization for task assignment problemrdquoMicroprocessors andMicrosystems vol 26 no 8 pp 363ndash371 2002
[4] P Y R Ma E Y S Lee and M Tsuchiya ldquoA task allocationmodel for distributed computing systemsrdquo IEEE Transactionson Computers vol 31 no 1 pp 41ndash47 1982
[5] D Xinghua F Yangwang X Bingsong and et al ldquoTask alloca-tion in cooperative air combat based on multi-agent coalitionrdquoJournal of BeijingUniversity of Aeronautics andAstronautics vol40 no 9 pp 1268ndash1275 2014
[6] X-G Gao Y-H Wang and B Li ldquoDecision-making methodfor the optimal coordination of fire and electronic warfarebased on integrative effectivenessrdquo Systems Engineering andElectronics vol 29 no 12 pp 2081ndash2084 2007
[7] H Guo H Xu D Liu and D Wang ldquoAir combat decision-making for cooperative multiple target attack based on adaptive
hybrid particle swarm algorithmrdquo Journal of Air Force Engineer-ing University vol 11 no 2 pp 16ndash19 2010
[8] K Horie and B A Conway ldquoOptimal fighter pursuit-evasionmaneuvers found via two-sided optimizationrdquo Journal of Guid-ance Control and Dynamics vol 29 no 1 pp 105ndash112 2006
[9] S Matlin ldquoA review of the literature on missile-allocationproblemrdquo Operations Research vol 18 no 2 pp 334ndash373 1970
[10] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on Neural Net-works pp 1942ndash1948 Perth Australia November-December1995
[11] W A Crossley ldquoOptimization for aerospace conceptual designthrough the use of genetic algorithmsrdquo in Proceedings of the1st NASADoD Workshop on Evolvable Hardware pp 200ndash207Pasadena Calif USA
[12] M Camara J Ortega and F de Toro ldquoA single front geneticalgorithm for parallel multi-objective optimization in dynamicenvironmentsrdquo Neurocomputing vol 72 no 16ndash18 pp 3570ndash3579 2009
[13] C-K Goh and K C Tan Evolutionary Multi-objective Opti-mization in Uncertain Environments Issues and AlgorithmsSpringer Berlin Germany 2009
[14] Y Yu and Z-H Zhou ldquoOn the usefulness of infeasible solutionsin evolutionary search a theoretical studyrdquo in Proceedings of theIEEECongress on Evolutionary Computation (CEC rsquo08) pp 835ndash840 IEEE Hong Kong June 2008
[15] J S McGrew J P How B Williams and N Roy ldquoAir-combatstrategy using approximate dynamic programmingrdquo Journal ofGuidance Control and Dynamics vol 33 no 5 pp 1641ndash16542010
[16] Z ZhouGeneral Aviation Fire ControlTheory NationalDefenseIndustry Press Beijing China 2008
[17] L Xiao and J Huang ldquoStudy on calculation for air-to-airmissile attack area based on improved RBF neural networkrdquo inProceedings of the International Conference on Computer Scienceand Artificial Intelligence Advances in Artificial Intelligence pp631ndash637 Sydney Australia December 2012
[18] H Zhang and D Zhou ldquoRoute planning method in air-to-ground attackrdquo Electronics Optics amp Control vol 1 no 1 pp 37ndash42 1999
[19] T-Y Sun S-J Tsai Y-N Lee S-M Yang and S-H Ting ldquoThestudy on intelligent advanced fighter air combat decision sup-port systemrdquo inProceedings of the IEEE International Conferenceon Information Reuse and Integration (IRI rsquo06) pp 39ndash44 IEEEWaikoloa Village Hawaii USA September 2006
[20] X Hai and Y Lei ldquoA situation forecast method based onintuitionistic fuzzy reasoningrdquo Electronics Optics ampControl vol18 no 4 pp 33ndash37 2011
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
direction are optimized with relative setting granularities anda surviving policy of noninferior solutions is given to makesure that the best solutions in the Pareto boundary [11ndash14] willnot be neglected
The paper is organized as follows Section 1 is introduc-tion Section 2 presents the basic sequential attack modeland its simplified method Section 3 describes the modifiedswarm optimization algorithm with a new surviving policySection 4 exhibits the simulation results and Section 5 isconclusion
2 Background-Basic Scenario
21 The Analysis of Target Threatening Area In moderncombat scenario the flight fighter may have many advancedsensor and weapon systems which can bring much morethreats [15] And in this paper we define the targetrsquos threatas the threat of targetrsquos comprehensive ability and the threatof targetrsquos destroy ability
Figure 1 describes the traditional 1 versus 1 air combatsituation 119860 is our flight fighter and 119879 is the target V
119886and
V119905are the velocities 119877 is the relative distance between 119860 and
119879 119902 is the targetrsquos approaching angle and 120593 is the targetrsquosazimuth angle In this paper we suppose that the attitudeangle exposed to the targetrsquos radar is equal to the targetrsquosazimuth angle on condition that 119860 and 119879 are in the samehorizontal plane
In air combat the greatest threat to our flight fightercomes from the targetrsquos detection threat and the targetrsquosweapon threat Thus we need to choose an appropriateapproaching method to avoid the threatening area in orderto increase our task success rate So it is necessary to analyzethe attack area and the exposure area
Attack Area We define the attack area in this content as aspecial air-to-air missile attack area around the target andonly in this area our flight fighter can get a probabilitywithin agiven threshold to destroy the target after launching a propermissile The destroy probability may decrease rapidly to zerowhen launching a missile outside this area [16]The real-timecalculating method to get the attack area is hard to satisfythe needs of the modern combat so we apply the data fittingmethod to acquire the approximate targetrsquos attack area [17]Suppose our member carries the middle-range-missile thenthe far and near boundary of the attack area around the targetcan be calculated by the following formulas
119863max = 119891 (1198671198810 119881119879 119879 119881119903 119903 120578119910 120596 119902119879 1199050)
119863min = 119891 (1198671198810 119881119879 119879 119881119903 119903 120578119910 120596 119902119879 1199050) (1)
where119863max119863min are far and near boundary of the attack areaaround the target119867 is attack height119881
0 is missile launching
velocity119881119879 is target velocity
119879 is target accelerated velocity
119881119903 is target angular velocity
119903 is target accelerated angular
velocity 119899119910 ismaneuver overload120596 ismaximummissile off-
boresight launch angle 119902119879 is target approaching angle and 119905
0
is missile working time
R
q
T
A
a
t
120593
Figure 1 The relative situation of friend and foe
Then we can get the whole attack area around the targetin polar coordinate
119885119860= (119902119879 119889 ℎ) | 119902
119879isin [minus120587 120587] ℎ
isin [119867min 119867max] 119889 (119902119879 ℎ)
isin [119863min (119902119879 ℎ) 119863max (119902119879 ℎ)]
(2)
where 119902119879is the targetrsquos approaching angle 119889 is the distance
of the target and ℎ is the relative height to the target where[119867min 119867max]means the attack range in height of missile
In this study we suppose that our member has the sameattack area with the targets and both of them have nearly thesame comprehensive combat capability If there was a greatgap between these two comprehensive combat capabilitiesthen we can describe the combat as an asymmetric combatand the model is not efficient According to current modelwe need to find a proper route planning project to get a betterattack situationwhich can be useful to avoid the targetrsquos attackand to make our member safer
Exposure Area It means an area we exposed to targetrsquos radarThe scanning range of airborne radar is about plusmn60∘ Then wecan also get the exposure area as follows
119885119864= (119902119879 119889 ℎ) | 119902
119879isin [minus120579119877 120579119877] ℎ
isin [119867min 119867max] 119889 (119902119879 ℎ) isin [0 119877119877 (119902119879 ℎ)] (3)
where 120579119877is the scanning angle of targetrsquos radar 119877
119877is the
maximum detection distance and [119867min 119867max] means thedetection range in height of radar
22The Basic Description of the Model Sequential attack [18]problem can be regarded as a multiobjective optimizationproblem and once the attack sequence and the attack direc-tion are assured the whole attack logic is born In order toanalyze the multi-target-attack process in detail we focuson the situation with single target for the first step Beforebringing in the model we should give some definitions aboutthe relative problems
Definition (i) Minimum safety distance this distance can beregarded as the maximum attack distance of the enemyrsquosWhen entering this area ourmembermay get the probabilityof being attacked by the enemies (ii) Attack preparing areathis area is defined between the minimum safety distanceboundary and the far attack area boundary where our flight
Mathematical Problems in Engineering 3
Far attack areaboundary
Minimum safetydistance
Attack preparing area
Threatening areaThreatening area
LAttack direction assuringTactical maneuvering
Retreat routing
2nd circle of tacticalnavigation points
1st circle of tacticalnavigation points
Target TTarget T
L
S1
S2 S3 TT 120579
120579
120578
l1
l2l3
l4
Figure 2 The simplified model of sequential attack
fighter should complete a series of actions to complete itsattack task
From the left side of Figure 2 we can see thewhole processof single-target-attack
119871 is the virtual navigation route across the attack prepar-ing area (an approximate area in this paper) and through thewhole cross process the flight fighter may deal with severalkey stages they are listed as follows
(a) Tactical Maneuvering Stage This stage may decide whichtargets to attack and which routes to choose to enter theminimum safety distance This stage may last till the missilelaunches
(b) Attack Direction Assuring StageThis stage is used to avoidthe threaten area of enemies and find a more effective way toincrease the success rate
(c) Retreat Routing Stage This stage is very important to offeran input for the next attack process and it is also useful to findan safe retreat route to get away from the last threaten area
The three stages can describe thewhole attack process andin order to build amathematical model of the different stageswe simplified the model as the right side of Figure 2
The attack preparing area is described by two circles oftactical navigation points The inner circle is the boundaryof far attack area and the outer circle is the boundaryof minimum safety distance The missile can be launchedtowards the target when the far attack area boundary isreached so we center the target and take sample points inthe far attack area boundary every 120578 angle then we canget the 1st circle tactical navigation points We define the1st circle tactical points of target 119905 as 119877119875119905
1= 119875119905
1 119875
119905
1198731
In order to enter the 1st circle tactical points we cannotavoid the minimum safety distance boundary which can bedecided by the threatening degree of the target as well as therelative combat situations So we also center the target andtake sample points in the minimum safety distance boundaryevery 120578 (120578 = 2120587119899 (119899 = 1 2 )) angle then we canget the 2nd circle tactical points We define the 2nd circletactical points of target 119905 as 119877119875119905
2= 119876119905
1 119876
119905
1198732
Then theproblem can be simplified as a route optimization problemwhich can be described as how to find a best route through
the two circles with shortest time and least threats Then thetransformation between two targets can be completed
If target 119894 would be attacked before target 119895 then thepartial order exists and we make it as 119876119894
1≻ 119876119895
1 The solution
space of virtual navigation can be described as
119875119878 = 1198751
1 119875
1
1198731
1198761
1 119876
1
1198732
119875119879
1 119875
119879
1198731
119876119879
1
119876119879
1198732
(4)
From the right side of Figure 2 we can see that 119871 meansthe attack route through two circles of navigation points and 120579is the angle between the velocity of target and the attack routeand it can describe the attack direction towards the target
From the analysis above we can conclude the problem intwo steps
Step 1 Choose four tactical points from the two circles anddecide the attack sequence according to every single targetin different space We describe it as 119876119905
119894rarr 119875
119905
119898rarr 119875
119905
119899rarr
119876119905
119900(119898 = 119899)
Step 2 Connect the four tactical points in sequence and anew virtual navigation route is born We can describe it as atuple
119871 = ⟨119873 119878 119879119892 119875119903 119864⟩ (5)
where 119873 means the number of navigation route fragmentswhich connect the two kinds of navigation points in differentcircles 119878 is the initial position of our member and it is alsothe first navigation point in the attack route 119879119892 is a set ofserial numbers of targets which records the attack sequence119875119903 contains all the chosen tactical points and their orders 119864ends the navigation
3 Modeling and Algorithm Optimizing
31 Analysis of Sequential Attack and the Attack DirectionThe single-target-attack model is given in Section 2 andthe multi-target-attack model is set up in this chapter FromFigure 2 we can get the general view of sequential attack inmultiple targetsrsquo situation
4 Mathematical Problems in Engineering
New target
Targets area
T1T2
T3
T4
Sequential attack routingManeuver direction routing
Figure 3 The sketch map of multiobjective sequential attack
Sequence attack connection graph
Attack directionconnection graph
Out-tactical point
Feasible solutionTargetTactical point
In-tactical point
Figure 4 Mission decomposition in variable granularities
In Figure 3 the blue line means the attack sequence andthe red linemeans the attack directionWe can easily find thatonce the tactical points are decided the whole attack planincluding attack sequence and direction is acquired Froma tactical view of the combat situation we can solve thisproblem in three different hierarchies
In Figure 4 the problem has been divided into threeparts which describe the feasible region sequential attack
and attack direction respectively The second layer canbe described as a directed graph 119892
119878= 119892(119879 119864
119904) 119879 =
1198791 1198792 1198793 119879
119899 where 119879 is a set of targets and 119864
119904is a set
of connection arcs 119890(119879119894 119879119895) isin 119864
119904means there is a partial
order between 119879119894and 119879
119895 and 119879
119894will be attacked first The
third layer can also be described as a connected directedgraph 119879
119894= 119892119894(119875 119864119901) where 119875 is a set of tactical navigation
points and 119864119901is a set of connection arcs Then the whole
Mathematical Problems in Engineering 5
problem can be rebuilt as a variable granularity directedgraph 119892
119872= 119892(119892
119894(119875 119864119901) 119864) 119864 means the connection arcs
between different granularities
32 Modeling
321 Targetsrsquo Reformation In current air combat scenariothe targetsrsquo flight fighters always appear in formation to exe-cute associated tactical task Thus we can divide these targetsinto different groups according to their unique characters inorder to simplify the complex combat situation And first thegroup should satisfy the following formula
119897avg =119897avg
119871avg (6)
where 119897avg is the mean distance between two targets in thesame group and 119871avg is the mean distance between twodifferent groups The minimum value of 119897avg can be regardedas the important index to choose the proper projects
We give the minimum distance 119871min between groups andmaximum distance 119897max between targets in the same group indifferent projects according to the combat needs When 119897 ge119897max then the target does not belong to the group When 119897 lt119871min we can decide that the target is in the same groupThuswe let 119871min = 119897max = 1198970 = const in order to use a commonmethod for completing the targetsrsquo reformation
The concrete method to reform the target is given asfollows
(a) Set a tactical targetrsquos group as
119866119905= 1198901 1198902 119890
119895 119890119895+1 (7)
1198901 1198902 119890
119895are members in the group and 119890
119895+1is the
119895 + 1member(b) Set the attribute of the member as
119890119894= 119890 (119909
119894 119910119894 119911119894 V119894 120595119894 119896119894) (8)
119909119894 119910119894 119911119894is the space coordinate of the target in
geographic coordinate system V119894is the velocity in
geographic coordinate system 120595119894is the flight direc-
tion angle in geographic coordinate system 119896119894is the
targetrsquos types(c) Set the constraints to reform the target
st 10038161003816100381610038161003816119909119895+1(119905) minus 119909
119894(119905)10038161003816100381610038161003816lt Δ119909
10038161003816100381610038161003816119910119895+1(119905) minus 119910
119894(119905)10038161003816100381610038161003816lt Δ119910
10038161003816100381610038161003816119911119895+1(119905) minus 119911
119894(119905)10038161003816100381610038161003816lt Δ119911
10038161003816100381610038161003816V119895+1(119905) minus V
119894(119905)10038161003816100381610038161003816lt ΔV
10038161003816100381610038161003816120595119895+1(119905) minus 120595
119894(119905)10038161003816100381610038161003816lt Δ120595
119895 + 1 le 119872max 119894 isin (1 2 119895)
(9)
Δ119909 Δ119910 Δ119911 is the threshold to reform the targets intargetsrsquo location ΔV is the velocity threshold of targetΔ120595 is the flight direction angle threshold of target119872max is themaximumnumber of the group Differentconstraints may bring different consequence so it alldepends on the combat situation
322 Attack Sequence Optimization In air combat scenariothe advantages or disadvantages towards enemies can bejudged by four conditions such as aspect relative distancevelocity and height [19] We can get the attack sequence bythe formula below
119891 = 120593 [120583 (120579) 120583 (119877) 120583 (119881) 120583 (119867) 119882]
119882 = 1199081 1199082 1199083 1199084
4
sum
119894=1
119908119894= 1
(10)
where 120583(120579) 120583(119877) 120583(119881) 120583(119867) are linguistic variables in whichgrade values are between 0 and 1 the closer the value ofmembership grade is to 1 the more advantages it is to attackthe target 119882 is the weight variable set which contains fourrelative weight variables in order to stress the importance ofthe linguistic variablesThen the fitness function can be givenas follows
119865 = max119899
sum
119894
119891119894 (11)
According to different targets we can get a sequence of119891119894(119894 = 1 2 119879) and then the total advantages can be
confirmed
323 Attack Direction Optimization In the model we men-tioned above we can easily find that the attack directioncan only be confirmed by connecting two tactical navigationpoints around the target So once the best tactical points arechosen the optimal attack direction is given automatically
33 The Evaluating Indexes of the Optimization The evalu-ation function is needed to evaluate the attack route andin this situation the distance between two different tacticalpoints is far enough with the support of the ability of long-middle-range missile Thus we set up four main indexesto evaluate the whole attack route which contains the totaldistance of attack route the exposure rate the constraints ofattack direction and the constraints when turning aroundFigure 5 describes the detailed definition of each variable
331 Total Distance of Attack Route Consider
119863119871=
119873
sum
119894=1
119897119894 (12)
where 119897119894is the distance of each navigation fragment
6 Mathematical Problems in Engineering
L1
L2
li
120579iT
ei
120583jj+1
Figure 5 Definition of each variable
Initialization Fitnesscalculating
Particleupgrading
Best individualcrossing crossing
Best groupvariation Complete EndYes
NoParticle
Figure 6 Calculating steps
332 Attack Route under Threat Consider
119864119871=
119873
sum
119894=1
119890119894 (13)
where 119890119894is the distance under threat (fire threat) of each
navigation fragment the distance under threat is in relation-ship with the targetrsquos attack area and it can be calculated byanalysis of the combat situation of enemies
333 Attack Direction Constraints The attack directionshould satisfy the maximum missile off-boresight launchangle so we give the constraint as
120579119894= 119891 (119875
119894
119868 119876119894
119868 119879119894) le 120579max (14)
where 120579119894describes the attack direction towards target 119868 and
120579max is the maximummissile off-boresight launch angle
334 Constraints When Turning Around Consider
120583119895119895+1
= infin if 119897119895+ 119897119895+1lt 119903min
120583119895119895+1
ge 120583min else
119895 = 1 2 119873 minus 1
(15)
where 120583119895119895+1
is the angle between route 119895 and route 119895 + 1119897119895means the distance of route 119895 119903min is the minimum turn
radius and 120583min is the minimum turn angleThese four indexes wementioned above canmake contri-
butions to building the fitness function as follows
119865 (119871) = 1199081
119863119871
119863119872
+ 1199082
119864119871
119864119872
+ 1198921(119871) + 119892
2(119871) (16)
where 119865(119871) is the fitness function and the smaller 119865(119871)gets the better the result will be If 119865(119871) rarr infin then thevalue would be meaningless 119863
119871is the total distance of the
attack route 119864119871is the distance under threat 119863
119872and 119864
119872are
normalized coefficients 1199081and 119908
2are weight coefficients
and 1198921(119871) and 119892
2(119871) are correction terms to correct the
conditions when the target approaching angle cannot satisfythe need of maximummissile off-boresight launch angle andthe turning angle is too small to support our action
34 Calculating Process According to the problem simpli-fication above we have changed the complex model intoa typical TSP model thus we use hybrid Particle SwarmOptimization method to solve this problem The detailedsteps can be acquired in Figure 6
341 Encoding Method From Figure 7 we can see that thelength of the gene code is 4119879 + 2 each target may get 4 codeswhich can be described by decimalism and each code rangesfrom 1 to the maximum number of tactical points For betterinitializing the values wemake the two codes in themiddle ofthe gene unequal and all the code will get a random but notrepeated number from 1 to the maximum number of tacticalpoints
342 Route Modifying The two-circle-encode style hasforced all the routes to cross through the tactical points so thesituations of route crossing and repeating exist According tothe rule in Figure 8 we can get the new attack route
Theproject needs to be accessed again aftermodifying thevirtual navigation If the accessing result was not better thanthe old one we would keep the old policy
Mathematical Problems in Engineering 7
4T
1 2 T
P1I Q
1I Q
1O P
1O P
2I Q
2I Q
2O P
2O P
119879I Q
119879I Q
119879O P
119879O
middot middot middot
Figure 7 The encoding method
Deleted routfragment
Planning virtualnavigation
Modifying virtualnavigation
Figure 8 The display of route modifying method
343 Result Space Expanding Policy When calculating theresult may not get out of the local optimization so after eachiteration we get the global optimal value
bestRout
= (119875V1
119868119876
V1
119868119876
V1
119874119875V1
119874119875V2
119868119876
V2
119868119876
V2
119874119875V2
119874sdot sdot sdot 119875
V119895
119868119876
V119895
119868119876
V119895
119874119875V119895
119874)
(17)
Then we can deal with the existing result as follows andthe description of the space expanding policy can be seen inFigure 9
(a) Rescue the abandoned route when the followingcondition is satisfied1003816100381610038161003816gbest minus fitness
119894
1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)
where 119892119887119890119904119905 is the global optimal value in currenttime and fitness
119894is the abandoned value in the current
circulation when the given condition is satisfied werescue the relative 119877119900119906119905
119894(119894 = 1 2 3 119899) and this
route will take part in the next circulation
(b) Get 119899 random points around each best tactical navi-gation point within the distance 119903
119894= 1205791198771198942 (119894 = 1 2)
(119877119894defines the radius of each tactical area)
(c) Expand the result space with choosing points and geta new result space Ωnew
(d) Calculating 119891 = 119892119887119890119904119905119905 119905 is the simulation time If119891 lt 120575 then we save the current result
344 Calculating Steps See Algorithm 1
4 Simulation
41 Hardware Condition CPU is P4 28GHz and RAM is15 G Matlab 2010
42 The Tactical Scenario Our flight fighter is flying towardsthe enemiesrsquo formation and the initial locations of enemiesand our member are given in Table 1
The relative situation can be seen in Figure 10
43 Simulation Result We suppose the detecting area ofenemies is stable at the moment and we need to find the bestattack sequence and attack direction at this moment We set3000 loops to optimize the final result and the result can beseen in Figures 11 and 12
From Figure 11 our member choose the proper directionto avoid the threaten areas in order to complete the task ina relative short way The attack sequence can be describedas 1198791rarr 1198793rarr 1198792rarr 1198794 and the attack direction (the
direction of enemy velocity is the initial angle 0∘) can bedecided by the angles given in Table 2
Figure 12 shows the updating process when choosing thebest solution and after 600 more loops we get the stablevalue the best route we choose can be shown as follows
bestRout = (11987531198681198763
11986811987623
11987411987522
11987411987510
11986811987615
11986811987617
11987411987518
11987411987510
11986811987615
11986811987617
11987411987518
1198741198752
1198681198765
11986811987619
11987411987517
1198741198759
1198681198764
11986811987621
11987411987523
119874) (19)
In Figure 12 three different methods are compared witheach other The blue line shows the result when using tradi-tional PSOmethod without dividing them in different granu-larities then it has costmore time to get a stable result becauseof much more formations of the available tactical points Thered line means the VGHPSO method we mentioned above
it has cost less time to get a stable value but it is easy to gettrapped in the local optimal value Thus we use VGHPSOlowastmethodwhichmeans the VGHPSOmethodwith result spaceexpanding policy We can see the result reflected in cyan line
In the former chapters we can know that the numbers oftactical points are depended on the sampling angle Thus we
8 Mathematical Problems in Engineering
Table 1 Initial locations
Start point 1198791
1198792
1198793
1198794
End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)
Table 2 Attack direction display
Attack sequence 1198791
1198793
1198792
1198794
Approach angle (∘) 10728 16076 7543 13306Attack angle (∘) 80 12793 2914 17293Retreat angle (∘) 16673 13673 2051 16051
PjO
QjO
r1
r2
n1 n2
n2
n3
bestRout
Figure 9 Result space expanding policy
400 600 800 1000 1200 1400 1600
700
800
900
1000
1100
1200
1300
1400
1500
T1 T2
T3
T4
y(k
m)
x (km)
Figure 10 The initial locations sketch map
do some simulation with different sampling angles and theresults can be seen in Figure 13
From left side of Figure 13 the blue parts mean thetotal distances in different sampling angles and the red partsdescribe the distances under enemiesrsquo threats Then we canget the ratio as shown in Table 3
400 600 800 1000 1200 1400 1600600
700
800
900
1000
1100
1200
1300
1400
1500
1600
T1T2
T3
T4
y(k
m)
x (km)
Figure 11 The optimal route planning project
0 500 1000 1500 2000 2500 30001800
1850
1900
1950
2000
2050
2100
2150
2200
2250
x (times)
VGHPSOPSO
y(k
m)
VGHPSOlowast
Figure 12 Updating curves in different algorithms
FromTable 3 we can see thatwhen choosing the samplingangle as 15∘ we can get a minimum comprehensive indexwhich indicates that we may get a more proper resultefficiently with less threats and shorter times
The usage of the results in this paper can only beregarded as the equal description of attack sequence andattack direction and the tactical points in virtual navigationare not the real navigation points but of great importance tosimplify the engineering problems
Mathematical Problems in Engineering 9
Input (1) Targetsrsquo attributes including the targetsrsquo number119873 the coordinate 119879(119909119879 119910119879 119911119879) the velocity V
119879and the attack area
119882 (2) Our flight fightersrsquo attributes including the number of our fighters119872 the angle 120579 to decide the number of tacticalnavigation points the coordinate 119865(119909
119865 119910119865 119911119865) and the results space Ω (3) A result space expanding policy 120588lowast to optimize
the resultSuppose Condition (1) The targets stay flying in a stable velocity and an unchanged direction (2) The tactical pointsrsquoformation should obey the constraints 120585sequence directionOutput (1) The global optimal fitness result 119892119887119890119904119905 (2) Best tactical points formation 119887119890119904119905119877119900119906119905Initialize randomΩ set the initial stateBeginFor each iteration 119894 = 1 119899 119899 is the iteration number
Cross(119865) rarr Ωnew executing the cross process towards the setting codesIf 119865119894(119894 = 1 2 120587120579) (119865
119894isin Ωnew) do not obey the constraints 120585sequence direction
While (119865119894obey the constraints 120585sequence direction)
Random 119865119894 make sure that all the elements inΩnew obey the constraints
endUpdate (Ωnew)
endCalculating (119891119887119890119904119905
119894)
If 119891119887119890119904119905119894lt min119891119887119890119904119905
1 119891119887119890119904119905
2 119891119887119890119904119905
119894minus1 Then
119892119887119890119904119905119894= 119891119887119890119904119905
119894
Variate(119865) rarr Ω1015840
new executing the variation process towards the setting codesCycling the process as Cross methodExecuting (120588lowast) the saving policy is used to expand the result space to optimize the final result
EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)
End
Algorithm 1
5 15 30 45 60 72 900
500
1000
1500
2000
2500
Dist
ance
(km
)
10 20 30 40 50 60 70 800
10
20
30
40
50
60
70
80
90
Running time (s)Sampling angle (∘)
Sam
plin
g an
gle(
∘ )
Total distanceThreatened distance
Figure 13 The comprehensive values in different sampling angles
10 Mathematical Problems in Engineering
Table 3 Relative calculating results display
Sampling angle (∘) 5 15 30 45 60 72 90Threats ratio 0013 0017 0044 0090 0129 0153 0257Running times (s) 78 45 35 26 24 19 13Comprehensive index 1014 0765 154 234 3096 2907 3341
5 Conclusion
The multiobjective sequence attack and attack directionplanning methods are given in the paper We put forwarda simplified model to create the tactical points in virtualnavigation and the complex air-to-air combat planningproblemhas been transformed into a variable granularity TSPproblem then theVGHPSOmethod is used to solve the givenproblem In the end the result of the simulation shows theefficiency of the method and it also indicates that the modelis available
The model and method we mentioned in the paper areavailable only in static combat planning which belongs toprecombat planning problem But in real combat scenariowe need a dynamic data management system to optimize thereal-time policiesThen we need to pay much attention to theprediction of the future situation according to the historicaldata [20] Thus we may get a more proper and more accurateresult
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Kang H Park S Noh et al ldquoAutonomously decidingcountermeasures against threats in electronic warfare settingsrdquoin Proceedings of the International Conference on ComplexIntelligent and Software Intensive Systems (CISIS rsquo09) pp 177ndash184 IEEE Fukuoka Japan March 2009
[2] H Guo Y Lv P Wang P Ren and L Zhu ldquoTarget threatassessment of air combat based on intervals and SVRrdquo FireControl amp Command Control vol 39 no 8 pp 17ndash21 2014
[3] A Salman I Ahmad and S Al-Madani ldquoParticle swarmoptimization for task assignment problemrdquoMicroprocessors andMicrosystems vol 26 no 8 pp 363ndash371 2002
[4] P Y R Ma E Y S Lee and M Tsuchiya ldquoA task allocationmodel for distributed computing systemsrdquo IEEE Transactionson Computers vol 31 no 1 pp 41ndash47 1982
[5] D Xinghua F Yangwang X Bingsong and et al ldquoTask alloca-tion in cooperative air combat based on multi-agent coalitionrdquoJournal of BeijingUniversity of Aeronautics andAstronautics vol40 no 9 pp 1268ndash1275 2014
[6] X-G Gao Y-H Wang and B Li ldquoDecision-making methodfor the optimal coordination of fire and electronic warfarebased on integrative effectivenessrdquo Systems Engineering andElectronics vol 29 no 12 pp 2081ndash2084 2007
[7] H Guo H Xu D Liu and D Wang ldquoAir combat decision-making for cooperative multiple target attack based on adaptive
hybrid particle swarm algorithmrdquo Journal of Air Force Engineer-ing University vol 11 no 2 pp 16ndash19 2010
[8] K Horie and B A Conway ldquoOptimal fighter pursuit-evasionmaneuvers found via two-sided optimizationrdquo Journal of Guid-ance Control and Dynamics vol 29 no 1 pp 105ndash112 2006
[9] S Matlin ldquoA review of the literature on missile-allocationproblemrdquo Operations Research vol 18 no 2 pp 334ndash373 1970
[10] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on Neural Net-works pp 1942ndash1948 Perth Australia November-December1995
[11] W A Crossley ldquoOptimization for aerospace conceptual designthrough the use of genetic algorithmsrdquo in Proceedings of the1st NASADoD Workshop on Evolvable Hardware pp 200ndash207Pasadena Calif USA
[12] M Camara J Ortega and F de Toro ldquoA single front geneticalgorithm for parallel multi-objective optimization in dynamicenvironmentsrdquo Neurocomputing vol 72 no 16ndash18 pp 3570ndash3579 2009
[13] C-K Goh and K C Tan Evolutionary Multi-objective Opti-mization in Uncertain Environments Issues and AlgorithmsSpringer Berlin Germany 2009
[14] Y Yu and Z-H Zhou ldquoOn the usefulness of infeasible solutionsin evolutionary search a theoretical studyrdquo in Proceedings of theIEEECongress on Evolutionary Computation (CEC rsquo08) pp 835ndash840 IEEE Hong Kong June 2008
[15] J S McGrew J P How B Williams and N Roy ldquoAir-combatstrategy using approximate dynamic programmingrdquo Journal ofGuidance Control and Dynamics vol 33 no 5 pp 1641ndash16542010
[16] Z ZhouGeneral Aviation Fire ControlTheory NationalDefenseIndustry Press Beijing China 2008
[17] L Xiao and J Huang ldquoStudy on calculation for air-to-airmissile attack area based on improved RBF neural networkrdquo inProceedings of the International Conference on Computer Scienceand Artificial Intelligence Advances in Artificial Intelligence pp631ndash637 Sydney Australia December 2012
[18] H Zhang and D Zhou ldquoRoute planning method in air-to-ground attackrdquo Electronics Optics amp Control vol 1 no 1 pp 37ndash42 1999
[19] T-Y Sun S-J Tsai Y-N Lee S-M Yang and S-H Ting ldquoThestudy on intelligent advanced fighter air combat decision sup-port systemrdquo inProceedings of the IEEE International Conferenceon Information Reuse and Integration (IRI rsquo06) pp 39ndash44 IEEEWaikoloa Village Hawaii USA September 2006
[20] X Hai and Y Lei ldquoA situation forecast method based onintuitionistic fuzzy reasoningrdquo Electronics Optics ampControl vol18 no 4 pp 33ndash37 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Far attack areaboundary
Minimum safetydistance
Attack preparing area
Threatening areaThreatening area
LAttack direction assuringTactical maneuvering
Retreat routing
2nd circle of tacticalnavigation points
1st circle of tacticalnavigation points
Target TTarget T
L
S1
S2 S3 TT 120579
120579
120578
l1
l2l3
l4
Figure 2 The simplified model of sequential attack
fighter should complete a series of actions to complete itsattack task
From the left side of Figure 2 we can see thewhole processof single-target-attack
119871 is the virtual navigation route across the attack prepar-ing area (an approximate area in this paper) and through thewhole cross process the flight fighter may deal with severalkey stages they are listed as follows
(a) Tactical Maneuvering Stage This stage may decide whichtargets to attack and which routes to choose to enter theminimum safety distance This stage may last till the missilelaunches
(b) Attack Direction Assuring StageThis stage is used to avoidthe threaten area of enemies and find a more effective way toincrease the success rate
(c) Retreat Routing Stage This stage is very important to offeran input for the next attack process and it is also useful to findan safe retreat route to get away from the last threaten area
The three stages can describe thewhole attack process andin order to build amathematical model of the different stageswe simplified the model as the right side of Figure 2
The attack preparing area is described by two circles oftactical navigation points The inner circle is the boundaryof far attack area and the outer circle is the boundaryof minimum safety distance The missile can be launchedtowards the target when the far attack area boundary isreached so we center the target and take sample points inthe far attack area boundary every 120578 angle then we canget the 1st circle tactical navigation points We define the1st circle tactical points of target 119905 as 119877119875119905
1= 119875119905
1 119875
119905
1198731
In order to enter the 1st circle tactical points we cannotavoid the minimum safety distance boundary which can bedecided by the threatening degree of the target as well as therelative combat situations So we also center the target andtake sample points in the minimum safety distance boundaryevery 120578 (120578 = 2120587119899 (119899 = 1 2 )) angle then we canget the 2nd circle tactical points We define the 2nd circletactical points of target 119905 as 119877119875119905
2= 119876119905
1 119876
119905
1198732
Then theproblem can be simplified as a route optimization problemwhich can be described as how to find a best route through
the two circles with shortest time and least threats Then thetransformation between two targets can be completed
If target 119894 would be attacked before target 119895 then thepartial order exists and we make it as 119876119894
1≻ 119876119895
1 The solution
space of virtual navigation can be described as
119875119878 = 1198751
1 119875
1
1198731
1198761
1 119876
1
1198732
119875119879
1 119875
119879
1198731
119876119879
1
119876119879
1198732
(4)
From the right side of Figure 2 we can see that 119871 meansthe attack route through two circles of navigation points and 120579is the angle between the velocity of target and the attack routeand it can describe the attack direction towards the target
From the analysis above we can conclude the problem intwo steps
Step 1 Choose four tactical points from the two circles anddecide the attack sequence according to every single targetin different space We describe it as 119876119905
119894rarr 119875
119905
119898rarr 119875
119905
119899rarr
119876119905
119900(119898 = 119899)
Step 2 Connect the four tactical points in sequence and anew virtual navigation route is born We can describe it as atuple
119871 = ⟨119873 119878 119879119892 119875119903 119864⟩ (5)
where 119873 means the number of navigation route fragmentswhich connect the two kinds of navigation points in differentcircles 119878 is the initial position of our member and it is alsothe first navigation point in the attack route 119879119892 is a set ofserial numbers of targets which records the attack sequence119875119903 contains all the chosen tactical points and their orders 119864ends the navigation
3 Modeling and Algorithm Optimizing
31 Analysis of Sequential Attack and the Attack DirectionThe single-target-attack model is given in Section 2 andthe multi-target-attack model is set up in this chapter FromFigure 2 we can get the general view of sequential attack inmultiple targetsrsquo situation
4 Mathematical Problems in Engineering
New target
Targets area
T1T2
T3
T4
Sequential attack routingManeuver direction routing
Figure 3 The sketch map of multiobjective sequential attack
Sequence attack connection graph
Attack directionconnection graph
Out-tactical point
Feasible solutionTargetTactical point
In-tactical point
Figure 4 Mission decomposition in variable granularities
In Figure 3 the blue line means the attack sequence andthe red linemeans the attack directionWe can easily find thatonce the tactical points are decided the whole attack planincluding attack sequence and direction is acquired Froma tactical view of the combat situation we can solve thisproblem in three different hierarchies
In Figure 4 the problem has been divided into threeparts which describe the feasible region sequential attack
and attack direction respectively The second layer canbe described as a directed graph 119892
119878= 119892(119879 119864
119904) 119879 =
1198791 1198792 1198793 119879
119899 where 119879 is a set of targets and 119864
119904is a set
of connection arcs 119890(119879119894 119879119895) isin 119864
119904means there is a partial
order between 119879119894and 119879
119895 and 119879
119894will be attacked first The
third layer can also be described as a connected directedgraph 119879
119894= 119892119894(119875 119864119901) where 119875 is a set of tactical navigation
points and 119864119901is a set of connection arcs Then the whole
Mathematical Problems in Engineering 5
problem can be rebuilt as a variable granularity directedgraph 119892
119872= 119892(119892
119894(119875 119864119901) 119864) 119864 means the connection arcs
between different granularities
32 Modeling
321 Targetsrsquo Reformation In current air combat scenariothe targetsrsquo flight fighters always appear in formation to exe-cute associated tactical task Thus we can divide these targetsinto different groups according to their unique characters inorder to simplify the complex combat situation And first thegroup should satisfy the following formula
119897avg =119897avg
119871avg (6)
where 119897avg is the mean distance between two targets in thesame group and 119871avg is the mean distance between twodifferent groups The minimum value of 119897avg can be regardedas the important index to choose the proper projects
We give the minimum distance 119871min between groups andmaximum distance 119897max between targets in the same group indifferent projects according to the combat needs When 119897 ge119897max then the target does not belong to the group When 119897 lt119871min we can decide that the target is in the same groupThuswe let 119871min = 119897max = 1198970 = const in order to use a commonmethod for completing the targetsrsquo reformation
The concrete method to reform the target is given asfollows
(a) Set a tactical targetrsquos group as
119866119905= 1198901 1198902 119890
119895 119890119895+1 (7)
1198901 1198902 119890
119895are members in the group and 119890
119895+1is the
119895 + 1member(b) Set the attribute of the member as
119890119894= 119890 (119909
119894 119910119894 119911119894 V119894 120595119894 119896119894) (8)
119909119894 119910119894 119911119894is the space coordinate of the target in
geographic coordinate system V119894is the velocity in
geographic coordinate system 120595119894is the flight direc-
tion angle in geographic coordinate system 119896119894is the
targetrsquos types(c) Set the constraints to reform the target
st 10038161003816100381610038161003816119909119895+1(119905) minus 119909
119894(119905)10038161003816100381610038161003816lt Δ119909
10038161003816100381610038161003816119910119895+1(119905) minus 119910
119894(119905)10038161003816100381610038161003816lt Δ119910
10038161003816100381610038161003816119911119895+1(119905) minus 119911
119894(119905)10038161003816100381610038161003816lt Δ119911
10038161003816100381610038161003816V119895+1(119905) minus V
119894(119905)10038161003816100381610038161003816lt ΔV
10038161003816100381610038161003816120595119895+1(119905) minus 120595
119894(119905)10038161003816100381610038161003816lt Δ120595
119895 + 1 le 119872max 119894 isin (1 2 119895)
(9)
Δ119909 Δ119910 Δ119911 is the threshold to reform the targets intargetsrsquo location ΔV is the velocity threshold of targetΔ120595 is the flight direction angle threshold of target119872max is themaximumnumber of the group Differentconstraints may bring different consequence so it alldepends on the combat situation
322 Attack Sequence Optimization In air combat scenariothe advantages or disadvantages towards enemies can bejudged by four conditions such as aspect relative distancevelocity and height [19] We can get the attack sequence bythe formula below
119891 = 120593 [120583 (120579) 120583 (119877) 120583 (119881) 120583 (119867) 119882]
119882 = 1199081 1199082 1199083 1199084
4
sum
119894=1
119908119894= 1
(10)
where 120583(120579) 120583(119877) 120583(119881) 120583(119867) are linguistic variables in whichgrade values are between 0 and 1 the closer the value ofmembership grade is to 1 the more advantages it is to attackthe target 119882 is the weight variable set which contains fourrelative weight variables in order to stress the importance ofthe linguistic variablesThen the fitness function can be givenas follows
119865 = max119899
sum
119894
119891119894 (11)
According to different targets we can get a sequence of119891119894(119894 = 1 2 119879) and then the total advantages can be
confirmed
323 Attack Direction Optimization In the model we men-tioned above we can easily find that the attack directioncan only be confirmed by connecting two tactical navigationpoints around the target So once the best tactical points arechosen the optimal attack direction is given automatically
33 The Evaluating Indexes of the Optimization The evalu-ation function is needed to evaluate the attack route andin this situation the distance between two different tacticalpoints is far enough with the support of the ability of long-middle-range missile Thus we set up four main indexesto evaluate the whole attack route which contains the totaldistance of attack route the exposure rate the constraints ofattack direction and the constraints when turning aroundFigure 5 describes the detailed definition of each variable
331 Total Distance of Attack Route Consider
119863119871=
119873
sum
119894=1
119897119894 (12)
where 119897119894is the distance of each navigation fragment
6 Mathematical Problems in Engineering
L1
L2
li
120579iT
ei
120583jj+1
Figure 5 Definition of each variable
Initialization Fitnesscalculating
Particleupgrading
Best individualcrossing crossing
Best groupvariation Complete EndYes
NoParticle
Figure 6 Calculating steps
332 Attack Route under Threat Consider
119864119871=
119873
sum
119894=1
119890119894 (13)
where 119890119894is the distance under threat (fire threat) of each
navigation fragment the distance under threat is in relation-ship with the targetrsquos attack area and it can be calculated byanalysis of the combat situation of enemies
333 Attack Direction Constraints The attack directionshould satisfy the maximum missile off-boresight launchangle so we give the constraint as
120579119894= 119891 (119875
119894
119868 119876119894
119868 119879119894) le 120579max (14)
where 120579119894describes the attack direction towards target 119868 and
120579max is the maximummissile off-boresight launch angle
334 Constraints When Turning Around Consider
120583119895119895+1
= infin if 119897119895+ 119897119895+1lt 119903min
120583119895119895+1
ge 120583min else
119895 = 1 2 119873 minus 1
(15)
where 120583119895119895+1
is the angle between route 119895 and route 119895 + 1119897119895means the distance of route 119895 119903min is the minimum turn
radius and 120583min is the minimum turn angleThese four indexes wementioned above canmake contri-
butions to building the fitness function as follows
119865 (119871) = 1199081
119863119871
119863119872
+ 1199082
119864119871
119864119872
+ 1198921(119871) + 119892
2(119871) (16)
where 119865(119871) is the fitness function and the smaller 119865(119871)gets the better the result will be If 119865(119871) rarr infin then thevalue would be meaningless 119863
119871is the total distance of the
attack route 119864119871is the distance under threat 119863
119872and 119864
119872are
normalized coefficients 1199081and 119908
2are weight coefficients
and 1198921(119871) and 119892
2(119871) are correction terms to correct the
conditions when the target approaching angle cannot satisfythe need of maximummissile off-boresight launch angle andthe turning angle is too small to support our action
34 Calculating Process According to the problem simpli-fication above we have changed the complex model intoa typical TSP model thus we use hybrid Particle SwarmOptimization method to solve this problem The detailedsteps can be acquired in Figure 6
341 Encoding Method From Figure 7 we can see that thelength of the gene code is 4119879 + 2 each target may get 4 codeswhich can be described by decimalism and each code rangesfrom 1 to the maximum number of tactical points For betterinitializing the values wemake the two codes in themiddle ofthe gene unequal and all the code will get a random but notrepeated number from 1 to the maximum number of tacticalpoints
342 Route Modifying The two-circle-encode style hasforced all the routes to cross through the tactical points so thesituations of route crossing and repeating exist According tothe rule in Figure 8 we can get the new attack route
Theproject needs to be accessed again aftermodifying thevirtual navigation If the accessing result was not better thanthe old one we would keep the old policy
Mathematical Problems in Engineering 7
4T
1 2 T
P1I Q
1I Q
1O P
1O P
2I Q
2I Q
2O P
2O P
119879I Q
119879I Q
119879O P
119879O
middot middot middot
Figure 7 The encoding method
Deleted routfragment
Planning virtualnavigation
Modifying virtualnavigation
Figure 8 The display of route modifying method
343 Result Space Expanding Policy When calculating theresult may not get out of the local optimization so after eachiteration we get the global optimal value
bestRout
= (119875V1
119868119876
V1
119868119876
V1
119874119875V1
119874119875V2
119868119876
V2
119868119876
V2
119874119875V2
119874sdot sdot sdot 119875
V119895
119868119876
V119895
119868119876
V119895
119874119875V119895
119874)
(17)
Then we can deal with the existing result as follows andthe description of the space expanding policy can be seen inFigure 9
(a) Rescue the abandoned route when the followingcondition is satisfied1003816100381610038161003816gbest minus fitness
119894
1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)
where 119892119887119890119904119905 is the global optimal value in currenttime and fitness
119894is the abandoned value in the current
circulation when the given condition is satisfied werescue the relative 119877119900119906119905
119894(119894 = 1 2 3 119899) and this
route will take part in the next circulation
(b) Get 119899 random points around each best tactical navi-gation point within the distance 119903
119894= 1205791198771198942 (119894 = 1 2)
(119877119894defines the radius of each tactical area)
(c) Expand the result space with choosing points and geta new result space Ωnew
(d) Calculating 119891 = 119892119887119890119904119905119905 119905 is the simulation time If119891 lt 120575 then we save the current result
344 Calculating Steps See Algorithm 1
4 Simulation
41 Hardware Condition CPU is P4 28GHz and RAM is15 G Matlab 2010
42 The Tactical Scenario Our flight fighter is flying towardsthe enemiesrsquo formation and the initial locations of enemiesand our member are given in Table 1
The relative situation can be seen in Figure 10
43 Simulation Result We suppose the detecting area ofenemies is stable at the moment and we need to find the bestattack sequence and attack direction at this moment We set3000 loops to optimize the final result and the result can beseen in Figures 11 and 12
From Figure 11 our member choose the proper directionto avoid the threaten areas in order to complete the task ina relative short way The attack sequence can be describedas 1198791rarr 1198793rarr 1198792rarr 1198794 and the attack direction (the
direction of enemy velocity is the initial angle 0∘) can bedecided by the angles given in Table 2
Figure 12 shows the updating process when choosing thebest solution and after 600 more loops we get the stablevalue the best route we choose can be shown as follows
bestRout = (11987531198681198763
11986811987623
11987411987522
11987411987510
11986811987615
11986811987617
11987411987518
11987411987510
11986811987615
11986811987617
11987411987518
1198741198752
1198681198765
11986811987619
11987411987517
1198741198759
1198681198764
11986811987621
11987411987523
119874) (19)
In Figure 12 three different methods are compared witheach other The blue line shows the result when using tradi-tional PSOmethod without dividing them in different granu-larities then it has costmore time to get a stable result becauseof much more formations of the available tactical points Thered line means the VGHPSO method we mentioned above
it has cost less time to get a stable value but it is easy to gettrapped in the local optimal value Thus we use VGHPSOlowastmethodwhichmeans the VGHPSOmethodwith result spaceexpanding policy We can see the result reflected in cyan line
In the former chapters we can know that the numbers oftactical points are depended on the sampling angle Thus we
8 Mathematical Problems in Engineering
Table 1 Initial locations
Start point 1198791
1198792
1198793
1198794
End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)
Table 2 Attack direction display
Attack sequence 1198791
1198793
1198792
1198794
Approach angle (∘) 10728 16076 7543 13306Attack angle (∘) 80 12793 2914 17293Retreat angle (∘) 16673 13673 2051 16051
PjO
QjO
r1
r2
n1 n2
n2
n3
bestRout
Figure 9 Result space expanding policy
400 600 800 1000 1200 1400 1600
700
800
900
1000
1100
1200
1300
1400
1500
T1 T2
T3
T4
y(k
m)
x (km)
Figure 10 The initial locations sketch map
do some simulation with different sampling angles and theresults can be seen in Figure 13
From left side of Figure 13 the blue parts mean thetotal distances in different sampling angles and the red partsdescribe the distances under enemiesrsquo threats Then we canget the ratio as shown in Table 3
400 600 800 1000 1200 1400 1600600
700
800
900
1000
1100
1200
1300
1400
1500
1600
T1T2
T3
T4
y(k
m)
x (km)
Figure 11 The optimal route planning project
0 500 1000 1500 2000 2500 30001800
1850
1900
1950
2000
2050
2100
2150
2200
2250
x (times)
VGHPSOPSO
y(k
m)
VGHPSOlowast
Figure 12 Updating curves in different algorithms
FromTable 3 we can see thatwhen choosing the samplingangle as 15∘ we can get a minimum comprehensive indexwhich indicates that we may get a more proper resultefficiently with less threats and shorter times
The usage of the results in this paper can only beregarded as the equal description of attack sequence andattack direction and the tactical points in virtual navigationare not the real navigation points but of great importance tosimplify the engineering problems
Mathematical Problems in Engineering 9
Input (1) Targetsrsquo attributes including the targetsrsquo number119873 the coordinate 119879(119909119879 119910119879 119911119879) the velocity V
119879and the attack area
119882 (2) Our flight fightersrsquo attributes including the number of our fighters119872 the angle 120579 to decide the number of tacticalnavigation points the coordinate 119865(119909
119865 119910119865 119911119865) and the results space Ω (3) A result space expanding policy 120588lowast to optimize
the resultSuppose Condition (1) The targets stay flying in a stable velocity and an unchanged direction (2) The tactical pointsrsquoformation should obey the constraints 120585sequence directionOutput (1) The global optimal fitness result 119892119887119890119904119905 (2) Best tactical points formation 119887119890119904119905119877119900119906119905Initialize randomΩ set the initial stateBeginFor each iteration 119894 = 1 119899 119899 is the iteration number
Cross(119865) rarr Ωnew executing the cross process towards the setting codesIf 119865119894(119894 = 1 2 120587120579) (119865
119894isin Ωnew) do not obey the constraints 120585sequence direction
While (119865119894obey the constraints 120585sequence direction)
Random 119865119894 make sure that all the elements inΩnew obey the constraints
endUpdate (Ωnew)
endCalculating (119891119887119890119904119905
119894)
If 119891119887119890119904119905119894lt min119891119887119890119904119905
1 119891119887119890119904119905
2 119891119887119890119904119905
119894minus1 Then
119892119887119890119904119905119894= 119891119887119890119904119905
119894
Variate(119865) rarr Ω1015840
new executing the variation process towards the setting codesCycling the process as Cross methodExecuting (120588lowast) the saving policy is used to expand the result space to optimize the final result
EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)
End
Algorithm 1
5 15 30 45 60 72 900
500
1000
1500
2000
2500
Dist
ance
(km
)
10 20 30 40 50 60 70 800
10
20
30
40
50
60
70
80
90
Running time (s)Sampling angle (∘)
Sam
plin
g an
gle(
∘ )
Total distanceThreatened distance
Figure 13 The comprehensive values in different sampling angles
10 Mathematical Problems in Engineering
Table 3 Relative calculating results display
Sampling angle (∘) 5 15 30 45 60 72 90Threats ratio 0013 0017 0044 0090 0129 0153 0257Running times (s) 78 45 35 26 24 19 13Comprehensive index 1014 0765 154 234 3096 2907 3341
5 Conclusion
The multiobjective sequence attack and attack directionplanning methods are given in the paper We put forwarda simplified model to create the tactical points in virtualnavigation and the complex air-to-air combat planningproblemhas been transformed into a variable granularity TSPproblem then theVGHPSOmethod is used to solve the givenproblem In the end the result of the simulation shows theefficiency of the method and it also indicates that the modelis available
The model and method we mentioned in the paper areavailable only in static combat planning which belongs toprecombat planning problem But in real combat scenariowe need a dynamic data management system to optimize thereal-time policiesThen we need to pay much attention to theprediction of the future situation according to the historicaldata [20] Thus we may get a more proper and more accurateresult
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Kang H Park S Noh et al ldquoAutonomously decidingcountermeasures against threats in electronic warfare settingsrdquoin Proceedings of the International Conference on ComplexIntelligent and Software Intensive Systems (CISIS rsquo09) pp 177ndash184 IEEE Fukuoka Japan March 2009
[2] H Guo Y Lv P Wang P Ren and L Zhu ldquoTarget threatassessment of air combat based on intervals and SVRrdquo FireControl amp Command Control vol 39 no 8 pp 17ndash21 2014
[3] A Salman I Ahmad and S Al-Madani ldquoParticle swarmoptimization for task assignment problemrdquoMicroprocessors andMicrosystems vol 26 no 8 pp 363ndash371 2002
[4] P Y R Ma E Y S Lee and M Tsuchiya ldquoA task allocationmodel for distributed computing systemsrdquo IEEE Transactionson Computers vol 31 no 1 pp 41ndash47 1982
[5] D Xinghua F Yangwang X Bingsong and et al ldquoTask alloca-tion in cooperative air combat based on multi-agent coalitionrdquoJournal of BeijingUniversity of Aeronautics andAstronautics vol40 no 9 pp 1268ndash1275 2014
[6] X-G Gao Y-H Wang and B Li ldquoDecision-making methodfor the optimal coordination of fire and electronic warfarebased on integrative effectivenessrdquo Systems Engineering andElectronics vol 29 no 12 pp 2081ndash2084 2007
[7] H Guo H Xu D Liu and D Wang ldquoAir combat decision-making for cooperative multiple target attack based on adaptive
hybrid particle swarm algorithmrdquo Journal of Air Force Engineer-ing University vol 11 no 2 pp 16ndash19 2010
[8] K Horie and B A Conway ldquoOptimal fighter pursuit-evasionmaneuvers found via two-sided optimizationrdquo Journal of Guid-ance Control and Dynamics vol 29 no 1 pp 105ndash112 2006
[9] S Matlin ldquoA review of the literature on missile-allocationproblemrdquo Operations Research vol 18 no 2 pp 334ndash373 1970
[10] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on Neural Net-works pp 1942ndash1948 Perth Australia November-December1995
[11] W A Crossley ldquoOptimization for aerospace conceptual designthrough the use of genetic algorithmsrdquo in Proceedings of the1st NASADoD Workshop on Evolvable Hardware pp 200ndash207Pasadena Calif USA
[12] M Camara J Ortega and F de Toro ldquoA single front geneticalgorithm for parallel multi-objective optimization in dynamicenvironmentsrdquo Neurocomputing vol 72 no 16ndash18 pp 3570ndash3579 2009
[13] C-K Goh and K C Tan Evolutionary Multi-objective Opti-mization in Uncertain Environments Issues and AlgorithmsSpringer Berlin Germany 2009
[14] Y Yu and Z-H Zhou ldquoOn the usefulness of infeasible solutionsin evolutionary search a theoretical studyrdquo in Proceedings of theIEEECongress on Evolutionary Computation (CEC rsquo08) pp 835ndash840 IEEE Hong Kong June 2008
[15] J S McGrew J P How B Williams and N Roy ldquoAir-combatstrategy using approximate dynamic programmingrdquo Journal ofGuidance Control and Dynamics vol 33 no 5 pp 1641ndash16542010
[16] Z ZhouGeneral Aviation Fire ControlTheory NationalDefenseIndustry Press Beijing China 2008
[17] L Xiao and J Huang ldquoStudy on calculation for air-to-airmissile attack area based on improved RBF neural networkrdquo inProceedings of the International Conference on Computer Scienceand Artificial Intelligence Advances in Artificial Intelligence pp631ndash637 Sydney Australia December 2012
[18] H Zhang and D Zhou ldquoRoute planning method in air-to-ground attackrdquo Electronics Optics amp Control vol 1 no 1 pp 37ndash42 1999
[19] T-Y Sun S-J Tsai Y-N Lee S-M Yang and S-H Ting ldquoThestudy on intelligent advanced fighter air combat decision sup-port systemrdquo inProceedings of the IEEE International Conferenceon Information Reuse and Integration (IRI rsquo06) pp 39ndash44 IEEEWaikoloa Village Hawaii USA September 2006
[20] X Hai and Y Lei ldquoA situation forecast method based onintuitionistic fuzzy reasoningrdquo Electronics Optics ampControl vol18 no 4 pp 33ndash37 2011
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
New target
Targets area
T1T2
T3
T4
Sequential attack routingManeuver direction routing
Figure 3 The sketch map of multiobjective sequential attack
Sequence attack connection graph
Attack directionconnection graph
Out-tactical point
Feasible solutionTargetTactical point
In-tactical point
Figure 4 Mission decomposition in variable granularities
In Figure 3 the blue line means the attack sequence andthe red linemeans the attack directionWe can easily find thatonce the tactical points are decided the whole attack planincluding attack sequence and direction is acquired Froma tactical view of the combat situation we can solve thisproblem in three different hierarchies
In Figure 4 the problem has been divided into threeparts which describe the feasible region sequential attack
and attack direction respectively The second layer canbe described as a directed graph 119892
119878= 119892(119879 119864
119904) 119879 =
1198791 1198792 1198793 119879
119899 where 119879 is a set of targets and 119864
119904is a set
of connection arcs 119890(119879119894 119879119895) isin 119864
119904means there is a partial
order between 119879119894and 119879
119895 and 119879
119894will be attacked first The
third layer can also be described as a connected directedgraph 119879
119894= 119892119894(119875 119864119901) where 119875 is a set of tactical navigation
points and 119864119901is a set of connection arcs Then the whole
Mathematical Problems in Engineering 5
problem can be rebuilt as a variable granularity directedgraph 119892
119872= 119892(119892
119894(119875 119864119901) 119864) 119864 means the connection arcs
between different granularities
32 Modeling
321 Targetsrsquo Reformation In current air combat scenariothe targetsrsquo flight fighters always appear in formation to exe-cute associated tactical task Thus we can divide these targetsinto different groups according to their unique characters inorder to simplify the complex combat situation And first thegroup should satisfy the following formula
119897avg =119897avg
119871avg (6)
where 119897avg is the mean distance between two targets in thesame group and 119871avg is the mean distance between twodifferent groups The minimum value of 119897avg can be regardedas the important index to choose the proper projects
We give the minimum distance 119871min between groups andmaximum distance 119897max between targets in the same group indifferent projects according to the combat needs When 119897 ge119897max then the target does not belong to the group When 119897 lt119871min we can decide that the target is in the same groupThuswe let 119871min = 119897max = 1198970 = const in order to use a commonmethod for completing the targetsrsquo reformation
The concrete method to reform the target is given asfollows
(a) Set a tactical targetrsquos group as
119866119905= 1198901 1198902 119890
119895 119890119895+1 (7)
1198901 1198902 119890
119895are members in the group and 119890
119895+1is the
119895 + 1member(b) Set the attribute of the member as
119890119894= 119890 (119909
119894 119910119894 119911119894 V119894 120595119894 119896119894) (8)
119909119894 119910119894 119911119894is the space coordinate of the target in
geographic coordinate system V119894is the velocity in
geographic coordinate system 120595119894is the flight direc-
tion angle in geographic coordinate system 119896119894is the
targetrsquos types(c) Set the constraints to reform the target
st 10038161003816100381610038161003816119909119895+1(119905) minus 119909
119894(119905)10038161003816100381610038161003816lt Δ119909
10038161003816100381610038161003816119910119895+1(119905) minus 119910
119894(119905)10038161003816100381610038161003816lt Δ119910
10038161003816100381610038161003816119911119895+1(119905) minus 119911
119894(119905)10038161003816100381610038161003816lt Δ119911
10038161003816100381610038161003816V119895+1(119905) minus V
119894(119905)10038161003816100381610038161003816lt ΔV
10038161003816100381610038161003816120595119895+1(119905) minus 120595
119894(119905)10038161003816100381610038161003816lt Δ120595
119895 + 1 le 119872max 119894 isin (1 2 119895)
(9)
Δ119909 Δ119910 Δ119911 is the threshold to reform the targets intargetsrsquo location ΔV is the velocity threshold of targetΔ120595 is the flight direction angle threshold of target119872max is themaximumnumber of the group Differentconstraints may bring different consequence so it alldepends on the combat situation
322 Attack Sequence Optimization In air combat scenariothe advantages or disadvantages towards enemies can bejudged by four conditions such as aspect relative distancevelocity and height [19] We can get the attack sequence bythe formula below
119891 = 120593 [120583 (120579) 120583 (119877) 120583 (119881) 120583 (119867) 119882]
119882 = 1199081 1199082 1199083 1199084
4
sum
119894=1
119908119894= 1
(10)
where 120583(120579) 120583(119877) 120583(119881) 120583(119867) are linguistic variables in whichgrade values are between 0 and 1 the closer the value ofmembership grade is to 1 the more advantages it is to attackthe target 119882 is the weight variable set which contains fourrelative weight variables in order to stress the importance ofthe linguistic variablesThen the fitness function can be givenas follows
119865 = max119899
sum
119894
119891119894 (11)
According to different targets we can get a sequence of119891119894(119894 = 1 2 119879) and then the total advantages can be
confirmed
323 Attack Direction Optimization In the model we men-tioned above we can easily find that the attack directioncan only be confirmed by connecting two tactical navigationpoints around the target So once the best tactical points arechosen the optimal attack direction is given automatically
33 The Evaluating Indexes of the Optimization The evalu-ation function is needed to evaluate the attack route andin this situation the distance between two different tacticalpoints is far enough with the support of the ability of long-middle-range missile Thus we set up four main indexesto evaluate the whole attack route which contains the totaldistance of attack route the exposure rate the constraints ofattack direction and the constraints when turning aroundFigure 5 describes the detailed definition of each variable
331 Total Distance of Attack Route Consider
119863119871=
119873
sum
119894=1
119897119894 (12)
where 119897119894is the distance of each navigation fragment
6 Mathematical Problems in Engineering
L1
L2
li
120579iT
ei
120583jj+1
Figure 5 Definition of each variable
Initialization Fitnesscalculating
Particleupgrading
Best individualcrossing crossing
Best groupvariation Complete EndYes
NoParticle
Figure 6 Calculating steps
332 Attack Route under Threat Consider
119864119871=
119873
sum
119894=1
119890119894 (13)
where 119890119894is the distance under threat (fire threat) of each
navigation fragment the distance under threat is in relation-ship with the targetrsquos attack area and it can be calculated byanalysis of the combat situation of enemies
333 Attack Direction Constraints The attack directionshould satisfy the maximum missile off-boresight launchangle so we give the constraint as
120579119894= 119891 (119875
119894
119868 119876119894
119868 119879119894) le 120579max (14)
where 120579119894describes the attack direction towards target 119868 and
120579max is the maximummissile off-boresight launch angle
334 Constraints When Turning Around Consider
120583119895119895+1
= infin if 119897119895+ 119897119895+1lt 119903min
120583119895119895+1
ge 120583min else
119895 = 1 2 119873 minus 1
(15)
where 120583119895119895+1
is the angle between route 119895 and route 119895 + 1119897119895means the distance of route 119895 119903min is the minimum turn
radius and 120583min is the minimum turn angleThese four indexes wementioned above canmake contri-
butions to building the fitness function as follows
119865 (119871) = 1199081
119863119871
119863119872
+ 1199082
119864119871
119864119872
+ 1198921(119871) + 119892
2(119871) (16)
where 119865(119871) is the fitness function and the smaller 119865(119871)gets the better the result will be If 119865(119871) rarr infin then thevalue would be meaningless 119863
119871is the total distance of the
attack route 119864119871is the distance under threat 119863
119872and 119864
119872are
normalized coefficients 1199081and 119908
2are weight coefficients
and 1198921(119871) and 119892
2(119871) are correction terms to correct the
conditions when the target approaching angle cannot satisfythe need of maximummissile off-boresight launch angle andthe turning angle is too small to support our action
34 Calculating Process According to the problem simpli-fication above we have changed the complex model intoa typical TSP model thus we use hybrid Particle SwarmOptimization method to solve this problem The detailedsteps can be acquired in Figure 6
341 Encoding Method From Figure 7 we can see that thelength of the gene code is 4119879 + 2 each target may get 4 codeswhich can be described by decimalism and each code rangesfrom 1 to the maximum number of tactical points For betterinitializing the values wemake the two codes in themiddle ofthe gene unequal and all the code will get a random but notrepeated number from 1 to the maximum number of tacticalpoints
342 Route Modifying The two-circle-encode style hasforced all the routes to cross through the tactical points so thesituations of route crossing and repeating exist According tothe rule in Figure 8 we can get the new attack route
Theproject needs to be accessed again aftermodifying thevirtual navigation If the accessing result was not better thanthe old one we would keep the old policy
Mathematical Problems in Engineering 7
4T
1 2 T
P1I Q
1I Q
1O P
1O P
2I Q
2I Q
2O P
2O P
119879I Q
119879I Q
119879O P
119879O
middot middot middot
Figure 7 The encoding method
Deleted routfragment
Planning virtualnavigation
Modifying virtualnavigation
Figure 8 The display of route modifying method
343 Result Space Expanding Policy When calculating theresult may not get out of the local optimization so after eachiteration we get the global optimal value
bestRout
= (119875V1
119868119876
V1
119868119876
V1
119874119875V1
119874119875V2
119868119876
V2
119868119876
V2
119874119875V2
119874sdot sdot sdot 119875
V119895
119868119876
V119895
119868119876
V119895
119874119875V119895
119874)
(17)
Then we can deal with the existing result as follows andthe description of the space expanding policy can be seen inFigure 9
(a) Rescue the abandoned route when the followingcondition is satisfied1003816100381610038161003816gbest minus fitness
119894
1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)
where 119892119887119890119904119905 is the global optimal value in currenttime and fitness
119894is the abandoned value in the current
circulation when the given condition is satisfied werescue the relative 119877119900119906119905
119894(119894 = 1 2 3 119899) and this
route will take part in the next circulation
(b) Get 119899 random points around each best tactical navi-gation point within the distance 119903
119894= 1205791198771198942 (119894 = 1 2)
(119877119894defines the radius of each tactical area)
(c) Expand the result space with choosing points and geta new result space Ωnew
(d) Calculating 119891 = 119892119887119890119904119905119905 119905 is the simulation time If119891 lt 120575 then we save the current result
344 Calculating Steps See Algorithm 1
4 Simulation
41 Hardware Condition CPU is P4 28GHz and RAM is15 G Matlab 2010
42 The Tactical Scenario Our flight fighter is flying towardsthe enemiesrsquo formation and the initial locations of enemiesand our member are given in Table 1
The relative situation can be seen in Figure 10
43 Simulation Result We suppose the detecting area ofenemies is stable at the moment and we need to find the bestattack sequence and attack direction at this moment We set3000 loops to optimize the final result and the result can beseen in Figures 11 and 12
From Figure 11 our member choose the proper directionto avoid the threaten areas in order to complete the task ina relative short way The attack sequence can be describedas 1198791rarr 1198793rarr 1198792rarr 1198794 and the attack direction (the
direction of enemy velocity is the initial angle 0∘) can bedecided by the angles given in Table 2
Figure 12 shows the updating process when choosing thebest solution and after 600 more loops we get the stablevalue the best route we choose can be shown as follows
bestRout = (11987531198681198763
11986811987623
11987411987522
11987411987510
11986811987615
11986811987617
11987411987518
11987411987510
11986811987615
11986811987617
11987411987518
1198741198752
1198681198765
11986811987619
11987411987517
1198741198759
1198681198764
11986811987621
11987411987523
119874) (19)
In Figure 12 three different methods are compared witheach other The blue line shows the result when using tradi-tional PSOmethod without dividing them in different granu-larities then it has costmore time to get a stable result becauseof much more formations of the available tactical points Thered line means the VGHPSO method we mentioned above
it has cost less time to get a stable value but it is easy to gettrapped in the local optimal value Thus we use VGHPSOlowastmethodwhichmeans the VGHPSOmethodwith result spaceexpanding policy We can see the result reflected in cyan line
In the former chapters we can know that the numbers oftactical points are depended on the sampling angle Thus we
8 Mathematical Problems in Engineering
Table 1 Initial locations
Start point 1198791
1198792
1198793
1198794
End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)
Table 2 Attack direction display
Attack sequence 1198791
1198793
1198792
1198794
Approach angle (∘) 10728 16076 7543 13306Attack angle (∘) 80 12793 2914 17293Retreat angle (∘) 16673 13673 2051 16051
PjO
QjO
r1
r2
n1 n2
n2
n3
bestRout
Figure 9 Result space expanding policy
400 600 800 1000 1200 1400 1600
700
800
900
1000
1100
1200
1300
1400
1500
T1 T2
T3
T4
y(k
m)
x (km)
Figure 10 The initial locations sketch map
do some simulation with different sampling angles and theresults can be seen in Figure 13
From left side of Figure 13 the blue parts mean thetotal distances in different sampling angles and the red partsdescribe the distances under enemiesrsquo threats Then we canget the ratio as shown in Table 3
400 600 800 1000 1200 1400 1600600
700
800
900
1000
1100
1200
1300
1400
1500
1600
T1T2
T3
T4
y(k
m)
x (km)
Figure 11 The optimal route planning project
0 500 1000 1500 2000 2500 30001800
1850
1900
1950
2000
2050
2100
2150
2200
2250
x (times)
VGHPSOPSO
y(k
m)
VGHPSOlowast
Figure 12 Updating curves in different algorithms
FromTable 3 we can see thatwhen choosing the samplingangle as 15∘ we can get a minimum comprehensive indexwhich indicates that we may get a more proper resultefficiently with less threats and shorter times
The usage of the results in this paper can only beregarded as the equal description of attack sequence andattack direction and the tactical points in virtual navigationare not the real navigation points but of great importance tosimplify the engineering problems
Mathematical Problems in Engineering 9
Input (1) Targetsrsquo attributes including the targetsrsquo number119873 the coordinate 119879(119909119879 119910119879 119911119879) the velocity V
119879and the attack area
119882 (2) Our flight fightersrsquo attributes including the number of our fighters119872 the angle 120579 to decide the number of tacticalnavigation points the coordinate 119865(119909
119865 119910119865 119911119865) and the results space Ω (3) A result space expanding policy 120588lowast to optimize
the resultSuppose Condition (1) The targets stay flying in a stable velocity and an unchanged direction (2) The tactical pointsrsquoformation should obey the constraints 120585sequence directionOutput (1) The global optimal fitness result 119892119887119890119904119905 (2) Best tactical points formation 119887119890119904119905119877119900119906119905Initialize randomΩ set the initial stateBeginFor each iteration 119894 = 1 119899 119899 is the iteration number
Cross(119865) rarr Ωnew executing the cross process towards the setting codesIf 119865119894(119894 = 1 2 120587120579) (119865
119894isin Ωnew) do not obey the constraints 120585sequence direction
While (119865119894obey the constraints 120585sequence direction)
Random 119865119894 make sure that all the elements inΩnew obey the constraints
endUpdate (Ωnew)
endCalculating (119891119887119890119904119905
119894)
If 119891119887119890119904119905119894lt min119891119887119890119904119905
1 119891119887119890119904119905
2 119891119887119890119904119905
119894minus1 Then
119892119887119890119904119905119894= 119891119887119890119904119905
119894
Variate(119865) rarr Ω1015840
new executing the variation process towards the setting codesCycling the process as Cross methodExecuting (120588lowast) the saving policy is used to expand the result space to optimize the final result
EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)
End
Algorithm 1
5 15 30 45 60 72 900
500
1000
1500
2000
2500
Dist
ance
(km
)
10 20 30 40 50 60 70 800
10
20
30
40
50
60
70
80
90
Running time (s)Sampling angle (∘)
Sam
plin
g an
gle(
∘ )
Total distanceThreatened distance
Figure 13 The comprehensive values in different sampling angles
10 Mathematical Problems in Engineering
Table 3 Relative calculating results display
Sampling angle (∘) 5 15 30 45 60 72 90Threats ratio 0013 0017 0044 0090 0129 0153 0257Running times (s) 78 45 35 26 24 19 13Comprehensive index 1014 0765 154 234 3096 2907 3341
5 Conclusion
The multiobjective sequence attack and attack directionplanning methods are given in the paper We put forwarda simplified model to create the tactical points in virtualnavigation and the complex air-to-air combat planningproblemhas been transformed into a variable granularity TSPproblem then theVGHPSOmethod is used to solve the givenproblem In the end the result of the simulation shows theefficiency of the method and it also indicates that the modelis available
The model and method we mentioned in the paper areavailable only in static combat planning which belongs toprecombat planning problem But in real combat scenariowe need a dynamic data management system to optimize thereal-time policiesThen we need to pay much attention to theprediction of the future situation according to the historicaldata [20] Thus we may get a more proper and more accurateresult
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Kang H Park S Noh et al ldquoAutonomously decidingcountermeasures against threats in electronic warfare settingsrdquoin Proceedings of the International Conference on ComplexIntelligent and Software Intensive Systems (CISIS rsquo09) pp 177ndash184 IEEE Fukuoka Japan March 2009
[2] H Guo Y Lv P Wang P Ren and L Zhu ldquoTarget threatassessment of air combat based on intervals and SVRrdquo FireControl amp Command Control vol 39 no 8 pp 17ndash21 2014
[3] A Salman I Ahmad and S Al-Madani ldquoParticle swarmoptimization for task assignment problemrdquoMicroprocessors andMicrosystems vol 26 no 8 pp 363ndash371 2002
[4] P Y R Ma E Y S Lee and M Tsuchiya ldquoA task allocationmodel for distributed computing systemsrdquo IEEE Transactionson Computers vol 31 no 1 pp 41ndash47 1982
[5] D Xinghua F Yangwang X Bingsong and et al ldquoTask alloca-tion in cooperative air combat based on multi-agent coalitionrdquoJournal of BeijingUniversity of Aeronautics andAstronautics vol40 no 9 pp 1268ndash1275 2014
[6] X-G Gao Y-H Wang and B Li ldquoDecision-making methodfor the optimal coordination of fire and electronic warfarebased on integrative effectivenessrdquo Systems Engineering andElectronics vol 29 no 12 pp 2081ndash2084 2007
[7] H Guo H Xu D Liu and D Wang ldquoAir combat decision-making for cooperative multiple target attack based on adaptive
hybrid particle swarm algorithmrdquo Journal of Air Force Engineer-ing University vol 11 no 2 pp 16ndash19 2010
[8] K Horie and B A Conway ldquoOptimal fighter pursuit-evasionmaneuvers found via two-sided optimizationrdquo Journal of Guid-ance Control and Dynamics vol 29 no 1 pp 105ndash112 2006
[9] S Matlin ldquoA review of the literature on missile-allocationproblemrdquo Operations Research vol 18 no 2 pp 334ndash373 1970
[10] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on Neural Net-works pp 1942ndash1948 Perth Australia November-December1995
[11] W A Crossley ldquoOptimization for aerospace conceptual designthrough the use of genetic algorithmsrdquo in Proceedings of the1st NASADoD Workshop on Evolvable Hardware pp 200ndash207Pasadena Calif USA
[12] M Camara J Ortega and F de Toro ldquoA single front geneticalgorithm for parallel multi-objective optimization in dynamicenvironmentsrdquo Neurocomputing vol 72 no 16ndash18 pp 3570ndash3579 2009
[13] C-K Goh and K C Tan Evolutionary Multi-objective Opti-mization in Uncertain Environments Issues and AlgorithmsSpringer Berlin Germany 2009
[14] Y Yu and Z-H Zhou ldquoOn the usefulness of infeasible solutionsin evolutionary search a theoretical studyrdquo in Proceedings of theIEEECongress on Evolutionary Computation (CEC rsquo08) pp 835ndash840 IEEE Hong Kong June 2008
[15] J S McGrew J P How B Williams and N Roy ldquoAir-combatstrategy using approximate dynamic programmingrdquo Journal ofGuidance Control and Dynamics vol 33 no 5 pp 1641ndash16542010
[16] Z ZhouGeneral Aviation Fire ControlTheory NationalDefenseIndustry Press Beijing China 2008
[17] L Xiao and J Huang ldquoStudy on calculation for air-to-airmissile attack area based on improved RBF neural networkrdquo inProceedings of the International Conference on Computer Scienceand Artificial Intelligence Advances in Artificial Intelligence pp631ndash637 Sydney Australia December 2012
[18] H Zhang and D Zhou ldquoRoute planning method in air-to-ground attackrdquo Electronics Optics amp Control vol 1 no 1 pp 37ndash42 1999
[19] T-Y Sun S-J Tsai Y-N Lee S-M Yang and S-H Ting ldquoThestudy on intelligent advanced fighter air combat decision sup-port systemrdquo inProceedings of the IEEE International Conferenceon Information Reuse and Integration (IRI rsquo06) pp 39ndash44 IEEEWaikoloa Village Hawaii USA September 2006
[20] X Hai and Y Lei ldquoA situation forecast method based onintuitionistic fuzzy reasoningrdquo Electronics Optics ampControl vol18 no 4 pp 33ndash37 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
problem can be rebuilt as a variable granularity directedgraph 119892
119872= 119892(119892
119894(119875 119864119901) 119864) 119864 means the connection arcs
between different granularities
32 Modeling
321 Targetsrsquo Reformation In current air combat scenariothe targetsrsquo flight fighters always appear in formation to exe-cute associated tactical task Thus we can divide these targetsinto different groups according to their unique characters inorder to simplify the complex combat situation And first thegroup should satisfy the following formula
119897avg =119897avg
119871avg (6)
where 119897avg is the mean distance between two targets in thesame group and 119871avg is the mean distance between twodifferent groups The minimum value of 119897avg can be regardedas the important index to choose the proper projects
We give the minimum distance 119871min between groups andmaximum distance 119897max between targets in the same group indifferent projects according to the combat needs When 119897 ge119897max then the target does not belong to the group When 119897 lt119871min we can decide that the target is in the same groupThuswe let 119871min = 119897max = 1198970 = const in order to use a commonmethod for completing the targetsrsquo reformation
The concrete method to reform the target is given asfollows
(a) Set a tactical targetrsquos group as
119866119905= 1198901 1198902 119890
119895 119890119895+1 (7)
1198901 1198902 119890
119895are members in the group and 119890
119895+1is the
119895 + 1member(b) Set the attribute of the member as
119890119894= 119890 (119909
119894 119910119894 119911119894 V119894 120595119894 119896119894) (8)
119909119894 119910119894 119911119894is the space coordinate of the target in
geographic coordinate system V119894is the velocity in
geographic coordinate system 120595119894is the flight direc-
tion angle in geographic coordinate system 119896119894is the
targetrsquos types(c) Set the constraints to reform the target
st 10038161003816100381610038161003816119909119895+1(119905) minus 119909
119894(119905)10038161003816100381610038161003816lt Δ119909
10038161003816100381610038161003816119910119895+1(119905) minus 119910
119894(119905)10038161003816100381610038161003816lt Δ119910
10038161003816100381610038161003816119911119895+1(119905) minus 119911
119894(119905)10038161003816100381610038161003816lt Δ119911
10038161003816100381610038161003816V119895+1(119905) minus V
119894(119905)10038161003816100381610038161003816lt ΔV
10038161003816100381610038161003816120595119895+1(119905) minus 120595
119894(119905)10038161003816100381610038161003816lt Δ120595
119895 + 1 le 119872max 119894 isin (1 2 119895)
(9)
Δ119909 Δ119910 Δ119911 is the threshold to reform the targets intargetsrsquo location ΔV is the velocity threshold of targetΔ120595 is the flight direction angle threshold of target119872max is themaximumnumber of the group Differentconstraints may bring different consequence so it alldepends on the combat situation
322 Attack Sequence Optimization In air combat scenariothe advantages or disadvantages towards enemies can bejudged by four conditions such as aspect relative distancevelocity and height [19] We can get the attack sequence bythe formula below
119891 = 120593 [120583 (120579) 120583 (119877) 120583 (119881) 120583 (119867) 119882]
119882 = 1199081 1199082 1199083 1199084
4
sum
119894=1
119908119894= 1
(10)
where 120583(120579) 120583(119877) 120583(119881) 120583(119867) are linguistic variables in whichgrade values are between 0 and 1 the closer the value ofmembership grade is to 1 the more advantages it is to attackthe target 119882 is the weight variable set which contains fourrelative weight variables in order to stress the importance ofthe linguistic variablesThen the fitness function can be givenas follows
119865 = max119899
sum
119894
119891119894 (11)
According to different targets we can get a sequence of119891119894(119894 = 1 2 119879) and then the total advantages can be
confirmed
323 Attack Direction Optimization In the model we men-tioned above we can easily find that the attack directioncan only be confirmed by connecting two tactical navigationpoints around the target So once the best tactical points arechosen the optimal attack direction is given automatically
33 The Evaluating Indexes of the Optimization The evalu-ation function is needed to evaluate the attack route andin this situation the distance between two different tacticalpoints is far enough with the support of the ability of long-middle-range missile Thus we set up four main indexesto evaluate the whole attack route which contains the totaldistance of attack route the exposure rate the constraints ofattack direction and the constraints when turning aroundFigure 5 describes the detailed definition of each variable
331 Total Distance of Attack Route Consider
119863119871=
119873
sum
119894=1
119897119894 (12)
where 119897119894is the distance of each navigation fragment
6 Mathematical Problems in Engineering
L1
L2
li
120579iT
ei
120583jj+1
Figure 5 Definition of each variable
Initialization Fitnesscalculating
Particleupgrading
Best individualcrossing crossing
Best groupvariation Complete EndYes
NoParticle
Figure 6 Calculating steps
332 Attack Route under Threat Consider
119864119871=
119873
sum
119894=1
119890119894 (13)
where 119890119894is the distance under threat (fire threat) of each
navigation fragment the distance under threat is in relation-ship with the targetrsquos attack area and it can be calculated byanalysis of the combat situation of enemies
333 Attack Direction Constraints The attack directionshould satisfy the maximum missile off-boresight launchangle so we give the constraint as
120579119894= 119891 (119875
119894
119868 119876119894
119868 119879119894) le 120579max (14)
where 120579119894describes the attack direction towards target 119868 and
120579max is the maximummissile off-boresight launch angle
334 Constraints When Turning Around Consider
120583119895119895+1
= infin if 119897119895+ 119897119895+1lt 119903min
120583119895119895+1
ge 120583min else
119895 = 1 2 119873 minus 1
(15)
where 120583119895119895+1
is the angle between route 119895 and route 119895 + 1119897119895means the distance of route 119895 119903min is the minimum turn
radius and 120583min is the minimum turn angleThese four indexes wementioned above canmake contri-
butions to building the fitness function as follows
119865 (119871) = 1199081
119863119871
119863119872
+ 1199082
119864119871
119864119872
+ 1198921(119871) + 119892
2(119871) (16)
where 119865(119871) is the fitness function and the smaller 119865(119871)gets the better the result will be If 119865(119871) rarr infin then thevalue would be meaningless 119863
119871is the total distance of the
attack route 119864119871is the distance under threat 119863
119872and 119864
119872are
normalized coefficients 1199081and 119908
2are weight coefficients
and 1198921(119871) and 119892
2(119871) are correction terms to correct the
conditions when the target approaching angle cannot satisfythe need of maximummissile off-boresight launch angle andthe turning angle is too small to support our action
34 Calculating Process According to the problem simpli-fication above we have changed the complex model intoa typical TSP model thus we use hybrid Particle SwarmOptimization method to solve this problem The detailedsteps can be acquired in Figure 6
341 Encoding Method From Figure 7 we can see that thelength of the gene code is 4119879 + 2 each target may get 4 codeswhich can be described by decimalism and each code rangesfrom 1 to the maximum number of tactical points For betterinitializing the values wemake the two codes in themiddle ofthe gene unequal and all the code will get a random but notrepeated number from 1 to the maximum number of tacticalpoints
342 Route Modifying The two-circle-encode style hasforced all the routes to cross through the tactical points so thesituations of route crossing and repeating exist According tothe rule in Figure 8 we can get the new attack route
Theproject needs to be accessed again aftermodifying thevirtual navigation If the accessing result was not better thanthe old one we would keep the old policy
Mathematical Problems in Engineering 7
4T
1 2 T
P1I Q
1I Q
1O P
1O P
2I Q
2I Q
2O P
2O P
119879I Q
119879I Q
119879O P
119879O
middot middot middot
Figure 7 The encoding method
Deleted routfragment
Planning virtualnavigation
Modifying virtualnavigation
Figure 8 The display of route modifying method
343 Result Space Expanding Policy When calculating theresult may not get out of the local optimization so after eachiteration we get the global optimal value
bestRout
= (119875V1
119868119876
V1
119868119876
V1
119874119875V1
119874119875V2
119868119876
V2
119868119876
V2
119874119875V2
119874sdot sdot sdot 119875
V119895
119868119876
V119895
119868119876
V119895
119874119875V119895
119874)
(17)
Then we can deal with the existing result as follows andthe description of the space expanding policy can be seen inFigure 9
(a) Rescue the abandoned route when the followingcondition is satisfied1003816100381610038161003816gbest minus fitness
119894
1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)
where 119892119887119890119904119905 is the global optimal value in currenttime and fitness
119894is the abandoned value in the current
circulation when the given condition is satisfied werescue the relative 119877119900119906119905
119894(119894 = 1 2 3 119899) and this
route will take part in the next circulation
(b) Get 119899 random points around each best tactical navi-gation point within the distance 119903
119894= 1205791198771198942 (119894 = 1 2)
(119877119894defines the radius of each tactical area)
(c) Expand the result space with choosing points and geta new result space Ωnew
(d) Calculating 119891 = 119892119887119890119904119905119905 119905 is the simulation time If119891 lt 120575 then we save the current result
344 Calculating Steps See Algorithm 1
4 Simulation
41 Hardware Condition CPU is P4 28GHz and RAM is15 G Matlab 2010
42 The Tactical Scenario Our flight fighter is flying towardsthe enemiesrsquo formation and the initial locations of enemiesand our member are given in Table 1
The relative situation can be seen in Figure 10
43 Simulation Result We suppose the detecting area ofenemies is stable at the moment and we need to find the bestattack sequence and attack direction at this moment We set3000 loops to optimize the final result and the result can beseen in Figures 11 and 12
From Figure 11 our member choose the proper directionto avoid the threaten areas in order to complete the task ina relative short way The attack sequence can be describedas 1198791rarr 1198793rarr 1198792rarr 1198794 and the attack direction (the
direction of enemy velocity is the initial angle 0∘) can bedecided by the angles given in Table 2
Figure 12 shows the updating process when choosing thebest solution and after 600 more loops we get the stablevalue the best route we choose can be shown as follows
bestRout = (11987531198681198763
11986811987623
11987411987522
11987411987510
11986811987615
11986811987617
11987411987518
11987411987510
11986811987615
11986811987617
11987411987518
1198741198752
1198681198765
11986811987619
11987411987517
1198741198759
1198681198764
11986811987621
11987411987523
119874) (19)
In Figure 12 three different methods are compared witheach other The blue line shows the result when using tradi-tional PSOmethod without dividing them in different granu-larities then it has costmore time to get a stable result becauseof much more formations of the available tactical points Thered line means the VGHPSO method we mentioned above
it has cost less time to get a stable value but it is easy to gettrapped in the local optimal value Thus we use VGHPSOlowastmethodwhichmeans the VGHPSOmethodwith result spaceexpanding policy We can see the result reflected in cyan line
In the former chapters we can know that the numbers oftactical points are depended on the sampling angle Thus we
8 Mathematical Problems in Engineering
Table 1 Initial locations
Start point 1198791
1198792
1198793
1198794
End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)
Table 2 Attack direction display
Attack sequence 1198791
1198793
1198792
1198794
Approach angle (∘) 10728 16076 7543 13306Attack angle (∘) 80 12793 2914 17293Retreat angle (∘) 16673 13673 2051 16051
PjO
QjO
r1
r2
n1 n2
n2
n3
bestRout
Figure 9 Result space expanding policy
400 600 800 1000 1200 1400 1600
700
800
900
1000
1100
1200
1300
1400
1500
T1 T2
T3
T4
y(k
m)
x (km)
Figure 10 The initial locations sketch map
do some simulation with different sampling angles and theresults can be seen in Figure 13
From left side of Figure 13 the blue parts mean thetotal distances in different sampling angles and the red partsdescribe the distances under enemiesrsquo threats Then we canget the ratio as shown in Table 3
400 600 800 1000 1200 1400 1600600
700
800
900
1000
1100
1200
1300
1400
1500
1600
T1T2
T3
T4
y(k
m)
x (km)
Figure 11 The optimal route planning project
0 500 1000 1500 2000 2500 30001800
1850
1900
1950
2000
2050
2100
2150
2200
2250
x (times)
VGHPSOPSO
y(k
m)
VGHPSOlowast
Figure 12 Updating curves in different algorithms
FromTable 3 we can see thatwhen choosing the samplingangle as 15∘ we can get a minimum comprehensive indexwhich indicates that we may get a more proper resultefficiently with less threats and shorter times
The usage of the results in this paper can only beregarded as the equal description of attack sequence andattack direction and the tactical points in virtual navigationare not the real navigation points but of great importance tosimplify the engineering problems
Mathematical Problems in Engineering 9
Input (1) Targetsrsquo attributes including the targetsrsquo number119873 the coordinate 119879(119909119879 119910119879 119911119879) the velocity V
119879and the attack area
119882 (2) Our flight fightersrsquo attributes including the number of our fighters119872 the angle 120579 to decide the number of tacticalnavigation points the coordinate 119865(119909
119865 119910119865 119911119865) and the results space Ω (3) A result space expanding policy 120588lowast to optimize
the resultSuppose Condition (1) The targets stay flying in a stable velocity and an unchanged direction (2) The tactical pointsrsquoformation should obey the constraints 120585sequence directionOutput (1) The global optimal fitness result 119892119887119890119904119905 (2) Best tactical points formation 119887119890119904119905119877119900119906119905Initialize randomΩ set the initial stateBeginFor each iteration 119894 = 1 119899 119899 is the iteration number
Cross(119865) rarr Ωnew executing the cross process towards the setting codesIf 119865119894(119894 = 1 2 120587120579) (119865
119894isin Ωnew) do not obey the constraints 120585sequence direction
While (119865119894obey the constraints 120585sequence direction)
Random 119865119894 make sure that all the elements inΩnew obey the constraints
endUpdate (Ωnew)
endCalculating (119891119887119890119904119905
119894)
If 119891119887119890119904119905119894lt min119891119887119890119904119905
1 119891119887119890119904119905
2 119891119887119890119904119905
119894minus1 Then
119892119887119890119904119905119894= 119891119887119890119904119905
119894
Variate(119865) rarr Ω1015840
new executing the variation process towards the setting codesCycling the process as Cross methodExecuting (120588lowast) the saving policy is used to expand the result space to optimize the final result
EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)
End
Algorithm 1
5 15 30 45 60 72 900
500
1000
1500
2000
2500
Dist
ance
(km
)
10 20 30 40 50 60 70 800
10
20
30
40
50
60
70
80
90
Running time (s)Sampling angle (∘)
Sam
plin
g an
gle(
∘ )
Total distanceThreatened distance
Figure 13 The comprehensive values in different sampling angles
10 Mathematical Problems in Engineering
Table 3 Relative calculating results display
Sampling angle (∘) 5 15 30 45 60 72 90Threats ratio 0013 0017 0044 0090 0129 0153 0257Running times (s) 78 45 35 26 24 19 13Comprehensive index 1014 0765 154 234 3096 2907 3341
5 Conclusion
The multiobjective sequence attack and attack directionplanning methods are given in the paper We put forwarda simplified model to create the tactical points in virtualnavigation and the complex air-to-air combat planningproblemhas been transformed into a variable granularity TSPproblem then theVGHPSOmethod is used to solve the givenproblem In the end the result of the simulation shows theefficiency of the method and it also indicates that the modelis available
The model and method we mentioned in the paper areavailable only in static combat planning which belongs toprecombat planning problem But in real combat scenariowe need a dynamic data management system to optimize thereal-time policiesThen we need to pay much attention to theprediction of the future situation according to the historicaldata [20] Thus we may get a more proper and more accurateresult
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Kang H Park S Noh et al ldquoAutonomously decidingcountermeasures against threats in electronic warfare settingsrdquoin Proceedings of the International Conference on ComplexIntelligent and Software Intensive Systems (CISIS rsquo09) pp 177ndash184 IEEE Fukuoka Japan March 2009
[2] H Guo Y Lv P Wang P Ren and L Zhu ldquoTarget threatassessment of air combat based on intervals and SVRrdquo FireControl amp Command Control vol 39 no 8 pp 17ndash21 2014
[3] A Salman I Ahmad and S Al-Madani ldquoParticle swarmoptimization for task assignment problemrdquoMicroprocessors andMicrosystems vol 26 no 8 pp 363ndash371 2002
[4] P Y R Ma E Y S Lee and M Tsuchiya ldquoA task allocationmodel for distributed computing systemsrdquo IEEE Transactionson Computers vol 31 no 1 pp 41ndash47 1982
[5] D Xinghua F Yangwang X Bingsong and et al ldquoTask alloca-tion in cooperative air combat based on multi-agent coalitionrdquoJournal of BeijingUniversity of Aeronautics andAstronautics vol40 no 9 pp 1268ndash1275 2014
[6] X-G Gao Y-H Wang and B Li ldquoDecision-making methodfor the optimal coordination of fire and electronic warfarebased on integrative effectivenessrdquo Systems Engineering andElectronics vol 29 no 12 pp 2081ndash2084 2007
[7] H Guo H Xu D Liu and D Wang ldquoAir combat decision-making for cooperative multiple target attack based on adaptive
hybrid particle swarm algorithmrdquo Journal of Air Force Engineer-ing University vol 11 no 2 pp 16ndash19 2010
[8] K Horie and B A Conway ldquoOptimal fighter pursuit-evasionmaneuvers found via two-sided optimizationrdquo Journal of Guid-ance Control and Dynamics vol 29 no 1 pp 105ndash112 2006
[9] S Matlin ldquoA review of the literature on missile-allocationproblemrdquo Operations Research vol 18 no 2 pp 334ndash373 1970
[10] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on Neural Net-works pp 1942ndash1948 Perth Australia November-December1995
[11] W A Crossley ldquoOptimization for aerospace conceptual designthrough the use of genetic algorithmsrdquo in Proceedings of the1st NASADoD Workshop on Evolvable Hardware pp 200ndash207Pasadena Calif USA
[12] M Camara J Ortega and F de Toro ldquoA single front geneticalgorithm for parallel multi-objective optimization in dynamicenvironmentsrdquo Neurocomputing vol 72 no 16ndash18 pp 3570ndash3579 2009
[13] C-K Goh and K C Tan Evolutionary Multi-objective Opti-mization in Uncertain Environments Issues and AlgorithmsSpringer Berlin Germany 2009
[14] Y Yu and Z-H Zhou ldquoOn the usefulness of infeasible solutionsin evolutionary search a theoretical studyrdquo in Proceedings of theIEEECongress on Evolutionary Computation (CEC rsquo08) pp 835ndash840 IEEE Hong Kong June 2008
[15] J S McGrew J P How B Williams and N Roy ldquoAir-combatstrategy using approximate dynamic programmingrdquo Journal ofGuidance Control and Dynamics vol 33 no 5 pp 1641ndash16542010
[16] Z ZhouGeneral Aviation Fire ControlTheory NationalDefenseIndustry Press Beijing China 2008
[17] L Xiao and J Huang ldquoStudy on calculation for air-to-airmissile attack area based on improved RBF neural networkrdquo inProceedings of the International Conference on Computer Scienceand Artificial Intelligence Advances in Artificial Intelligence pp631ndash637 Sydney Australia December 2012
[18] H Zhang and D Zhou ldquoRoute planning method in air-to-ground attackrdquo Electronics Optics amp Control vol 1 no 1 pp 37ndash42 1999
[19] T-Y Sun S-J Tsai Y-N Lee S-M Yang and S-H Ting ldquoThestudy on intelligent advanced fighter air combat decision sup-port systemrdquo inProceedings of the IEEE International Conferenceon Information Reuse and Integration (IRI rsquo06) pp 39ndash44 IEEEWaikoloa Village Hawaii USA September 2006
[20] X Hai and Y Lei ldquoA situation forecast method based onintuitionistic fuzzy reasoningrdquo Electronics Optics ampControl vol18 no 4 pp 33ndash37 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
L1
L2
li
120579iT
ei
120583jj+1
Figure 5 Definition of each variable
Initialization Fitnesscalculating
Particleupgrading
Best individualcrossing crossing
Best groupvariation Complete EndYes
NoParticle
Figure 6 Calculating steps
332 Attack Route under Threat Consider
119864119871=
119873
sum
119894=1
119890119894 (13)
where 119890119894is the distance under threat (fire threat) of each
navigation fragment the distance under threat is in relation-ship with the targetrsquos attack area and it can be calculated byanalysis of the combat situation of enemies
333 Attack Direction Constraints The attack directionshould satisfy the maximum missile off-boresight launchangle so we give the constraint as
120579119894= 119891 (119875
119894
119868 119876119894
119868 119879119894) le 120579max (14)
where 120579119894describes the attack direction towards target 119868 and
120579max is the maximummissile off-boresight launch angle
334 Constraints When Turning Around Consider
120583119895119895+1
= infin if 119897119895+ 119897119895+1lt 119903min
120583119895119895+1
ge 120583min else
119895 = 1 2 119873 minus 1
(15)
where 120583119895119895+1
is the angle between route 119895 and route 119895 + 1119897119895means the distance of route 119895 119903min is the minimum turn
radius and 120583min is the minimum turn angleThese four indexes wementioned above canmake contri-
butions to building the fitness function as follows
119865 (119871) = 1199081
119863119871
119863119872
+ 1199082
119864119871
119864119872
+ 1198921(119871) + 119892
2(119871) (16)
where 119865(119871) is the fitness function and the smaller 119865(119871)gets the better the result will be If 119865(119871) rarr infin then thevalue would be meaningless 119863
119871is the total distance of the
attack route 119864119871is the distance under threat 119863
119872and 119864
119872are
normalized coefficients 1199081and 119908
2are weight coefficients
and 1198921(119871) and 119892
2(119871) are correction terms to correct the
conditions when the target approaching angle cannot satisfythe need of maximummissile off-boresight launch angle andthe turning angle is too small to support our action
34 Calculating Process According to the problem simpli-fication above we have changed the complex model intoa typical TSP model thus we use hybrid Particle SwarmOptimization method to solve this problem The detailedsteps can be acquired in Figure 6
341 Encoding Method From Figure 7 we can see that thelength of the gene code is 4119879 + 2 each target may get 4 codeswhich can be described by decimalism and each code rangesfrom 1 to the maximum number of tactical points For betterinitializing the values wemake the two codes in themiddle ofthe gene unequal and all the code will get a random but notrepeated number from 1 to the maximum number of tacticalpoints
342 Route Modifying The two-circle-encode style hasforced all the routes to cross through the tactical points so thesituations of route crossing and repeating exist According tothe rule in Figure 8 we can get the new attack route
Theproject needs to be accessed again aftermodifying thevirtual navigation If the accessing result was not better thanthe old one we would keep the old policy
Mathematical Problems in Engineering 7
4T
1 2 T
P1I Q
1I Q
1O P
1O P
2I Q
2I Q
2O P
2O P
119879I Q
119879I Q
119879O P
119879O
middot middot middot
Figure 7 The encoding method
Deleted routfragment
Planning virtualnavigation
Modifying virtualnavigation
Figure 8 The display of route modifying method
343 Result Space Expanding Policy When calculating theresult may not get out of the local optimization so after eachiteration we get the global optimal value
bestRout
= (119875V1
119868119876
V1
119868119876
V1
119874119875V1
119874119875V2
119868119876
V2
119868119876
V2
119874119875V2
119874sdot sdot sdot 119875
V119895
119868119876
V119895
119868119876
V119895
119874119875V119895
119874)
(17)
Then we can deal with the existing result as follows andthe description of the space expanding policy can be seen inFigure 9
(a) Rescue the abandoned route when the followingcondition is satisfied1003816100381610038161003816gbest minus fitness
119894
1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)
where 119892119887119890119904119905 is the global optimal value in currenttime and fitness
119894is the abandoned value in the current
circulation when the given condition is satisfied werescue the relative 119877119900119906119905
119894(119894 = 1 2 3 119899) and this
route will take part in the next circulation
(b) Get 119899 random points around each best tactical navi-gation point within the distance 119903
119894= 1205791198771198942 (119894 = 1 2)
(119877119894defines the radius of each tactical area)
(c) Expand the result space with choosing points and geta new result space Ωnew
(d) Calculating 119891 = 119892119887119890119904119905119905 119905 is the simulation time If119891 lt 120575 then we save the current result
344 Calculating Steps See Algorithm 1
4 Simulation
41 Hardware Condition CPU is P4 28GHz and RAM is15 G Matlab 2010
42 The Tactical Scenario Our flight fighter is flying towardsthe enemiesrsquo formation and the initial locations of enemiesand our member are given in Table 1
The relative situation can be seen in Figure 10
43 Simulation Result We suppose the detecting area ofenemies is stable at the moment and we need to find the bestattack sequence and attack direction at this moment We set3000 loops to optimize the final result and the result can beseen in Figures 11 and 12
From Figure 11 our member choose the proper directionto avoid the threaten areas in order to complete the task ina relative short way The attack sequence can be describedas 1198791rarr 1198793rarr 1198792rarr 1198794 and the attack direction (the
direction of enemy velocity is the initial angle 0∘) can bedecided by the angles given in Table 2
Figure 12 shows the updating process when choosing thebest solution and after 600 more loops we get the stablevalue the best route we choose can be shown as follows
bestRout = (11987531198681198763
11986811987623
11987411987522
11987411987510
11986811987615
11986811987617
11987411987518
11987411987510
11986811987615
11986811987617
11987411987518
1198741198752
1198681198765
11986811987619
11987411987517
1198741198759
1198681198764
11986811987621
11987411987523
119874) (19)
In Figure 12 three different methods are compared witheach other The blue line shows the result when using tradi-tional PSOmethod without dividing them in different granu-larities then it has costmore time to get a stable result becauseof much more formations of the available tactical points Thered line means the VGHPSO method we mentioned above
it has cost less time to get a stable value but it is easy to gettrapped in the local optimal value Thus we use VGHPSOlowastmethodwhichmeans the VGHPSOmethodwith result spaceexpanding policy We can see the result reflected in cyan line
In the former chapters we can know that the numbers oftactical points are depended on the sampling angle Thus we
8 Mathematical Problems in Engineering
Table 1 Initial locations
Start point 1198791
1198792
1198793
1198794
End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)
Table 2 Attack direction display
Attack sequence 1198791
1198793
1198792
1198794
Approach angle (∘) 10728 16076 7543 13306Attack angle (∘) 80 12793 2914 17293Retreat angle (∘) 16673 13673 2051 16051
PjO
QjO
r1
r2
n1 n2
n2
n3
bestRout
Figure 9 Result space expanding policy
400 600 800 1000 1200 1400 1600
700
800
900
1000
1100
1200
1300
1400
1500
T1 T2
T3
T4
y(k
m)
x (km)
Figure 10 The initial locations sketch map
do some simulation with different sampling angles and theresults can be seen in Figure 13
From left side of Figure 13 the blue parts mean thetotal distances in different sampling angles and the red partsdescribe the distances under enemiesrsquo threats Then we canget the ratio as shown in Table 3
400 600 800 1000 1200 1400 1600600
700
800
900
1000
1100
1200
1300
1400
1500
1600
T1T2
T3
T4
y(k
m)
x (km)
Figure 11 The optimal route planning project
0 500 1000 1500 2000 2500 30001800
1850
1900
1950
2000
2050
2100
2150
2200
2250
x (times)
VGHPSOPSO
y(k
m)
VGHPSOlowast
Figure 12 Updating curves in different algorithms
FromTable 3 we can see thatwhen choosing the samplingangle as 15∘ we can get a minimum comprehensive indexwhich indicates that we may get a more proper resultefficiently with less threats and shorter times
The usage of the results in this paper can only beregarded as the equal description of attack sequence andattack direction and the tactical points in virtual navigationare not the real navigation points but of great importance tosimplify the engineering problems
Mathematical Problems in Engineering 9
Input (1) Targetsrsquo attributes including the targetsrsquo number119873 the coordinate 119879(119909119879 119910119879 119911119879) the velocity V
119879and the attack area
119882 (2) Our flight fightersrsquo attributes including the number of our fighters119872 the angle 120579 to decide the number of tacticalnavigation points the coordinate 119865(119909
119865 119910119865 119911119865) and the results space Ω (3) A result space expanding policy 120588lowast to optimize
the resultSuppose Condition (1) The targets stay flying in a stable velocity and an unchanged direction (2) The tactical pointsrsquoformation should obey the constraints 120585sequence directionOutput (1) The global optimal fitness result 119892119887119890119904119905 (2) Best tactical points formation 119887119890119904119905119877119900119906119905Initialize randomΩ set the initial stateBeginFor each iteration 119894 = 1 119899 119899 is the iteration number
Cross(119865) rarr Ωnew executing the cross process towards the setting codesIf 119865119894(119894 = 1 2 120587120579) (119865
119894isin Ωnew) do not obey the constraints 120585sequence direction
While (119865119894obey the constraints 120585sequence direction)
Random 119865119894 make sure that all the elements inΩnew obey the constraints
endUpdate (Ωnew)
endCalculating (119891119887119890119904119905
119894)
If 119891119887119890119904119905119894lt min119891119887119890119904119905
1 119891119887119890119904119905
2 119891119887119890119904119905
119894minus1 Then
119892119887119890119904119905119894= 119891119887119890119904119905
119894
Variate(119865) rarr Ω1015840
new executing the variation process towards the setting codesCycling the process as Cross methodExecuting (120588lowast) the saving policy is used to expand the result space to optimize the final result
EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)
End
Algorithm 1
5 15 30 45 60 72 900
500
1000
1500
2000
2500
Dist
ance
(km
)
10 20 30 40 50 60 70 800
10
20
30
40
50
60
70
80
90
Running time (s)Sampling angle (∘)
Sam
plin
g an
gle(
∘ )
Total distanceThreatened distance
Figure 13 The comprehensive values in different sampling angles
10 Mathematical Problems in Engineering
Table 3 Relative calculating results display
Sampling angle (∘) 5 15 30 45 60 72 90Threats ratio 0013 0017 0044 0090 0129 0153 0257Running times (s) 78 45 35 26 24 19 13Comprehensive index 1014 0765 154 234 3096 2907 3341
5 Conclusion
The multiobjective sequence attack and attack directionplanning methods are given in the paper We put forwarda simplified model to create the tactical points in virtualnavigation and the complex air-to-air combat planningproblemhas been transformed into a variable granularity TSPproblem then theVGHPSOmethod is used to solve the givenproblem In the end the result of the simulation shows theefficiency of the method and it also indicates that the modelis available
The model and method we mentioned in the paper areavailable only in static combat planning which belongs toprecombat planning problem But in real combat scenariowe need a dynamic data management system to optimize thereal-time policiesThen we need to pay much attention to theprediction of the future situation according to the historicaldata [20] Thus we may get a more proper and more accurateresult
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Kang H Park S Noh et al ldquoAutonomously decidingcountermeasures against threats in electronic warfare settingsrdquoin Proceedings of the International Conference on ComplexIntelligent and Software Intensive Systems (CISIS rsquo09) pp 177ndash184 IEEE Fukuoka Japan March 2009
[2] H Guo Y Lv P Wang P Ren and L Zhu ldquoTarget threatassessment of air combat based on intervals and SVRrdquo FireControl amp Command Control vol 39 no 8 pp 17ndash21 2014
[3] A Salman I Ahmad and S Al-Madani ldquoParticle swarmoptimization for task assignment problemrdquoMicroprocessors andMicrosystems vol 26 no 8 pp 363ndash371 2002
[4] P Y R Ma E Y S Lee and M Tsuchiya ldquoA task allocationmodel for distributed computing systemsrdquo IEEE Transactionson Computers vol 31 no 1 pp 41ndash47 1982
[5] D Xinghua F Yangwang X Bingsong and et al ldquoTask alloca-tion in cooperative air combat based on multi-agent coalitionrdquoJournal of BeijingUniversity of Aeronautics andAstronautics vol40 no 9 pp 1268ndash1275 2014
[6] X-G Gao Y-H Wang and B Li ldquoDecision-making methodfor the optimal coordination of fire and electronic warfarebased on integrative effectivenessrdquo Systems Engineering andElectronics vol 29 no 12 pp 2081ndash2084 2007
[7] H Guo H Xu D Liu and D Wang ldquoAir combat decision-making for cooperative multiple target attack based on adaptive
hybrid particle swarm algorithmrdquo Journal of Air Force Engineer-ing University vol 11 no 2 pp 16ndash19 2010
[8] K Horie and B A Conway ldquoOptimal fighter pursuit-evasionmaneuvers found via two-sided optimizationrdquo Journal of Guid-ance Control and Dynamics vol 29 no 1 pp 105ndash112 2006
[9] S Matlin ldquoA review of the literature on missile-allocationproblemrdquo Operations Research vol 18 no 2 pp 334ndash373 1970
[10] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on Neural Net-works pp 1942ndash1948 Perth Australia November-December1995
[11] W A Crossley ldquoOptimization for aerospace conceptual designthrough the use of genetic algorithmsrdquo in Proceedings of the1st NASADoD Workshop on Evolvable Hardware pp 200ndash207Pasadena Calif USA
[12] M Camara J Ortega and F de Toro ldquoA single front geneticalgorithm for parallel multi-objective optimization in dynamicenvironmentsrdquo Neurocomputing vol 72 no 16ndash18 pp 3570ndash3579 2009
[13] C-K Goh and K C Tan Evolutionary Multi-objective Opti-mization in Uncertain Environments Issues and AlgorithmsSpringer Berlin Germany 2009
[14] Y Yu and Z-H Zhou ldquoOn the usefulness of infeasible solutionsin evolutionary search a theoretical studyrdquo in Proceedings of theIEEECongress on Evolutionary Computation (CEC rsquo08) pp 835ndash840 IEEE Hong Kong June 2008
[15] J S McGrew J P How B Williams and N Roy ldquoAir-combatstrategy using approximate dynamic programmingrdquo Journal ofGuidance Control and Dynamics vol 33 no 5 pp 1641ndash16542010
[16] Z ZhouGeneral Aviation Fire ControlTheory NationalDefenseIndustry Press Beijing China 2008
[17] L Xiao and J Huang ldquoStudy on calculation for air-to-airmissile attack area based on improved RBF neural networkrdquo inProceedings of the International Conference on Computer Scienceand Artificial Intelligence Advances in Artificial Intelligence pp631ndash637 Sydney Australia December 2012
[18] H Zhang and D Zhou ldquoRoute planning method in air-to-ground attackrdquo Electronics Optics amp Control vol 1 no 1 pp 37ndash42 1999
[19] T-Y Sun S-J Tsai Y-N Lee S-M Yang and S-H Ting ldquoThestudy on intelligent advanced fighter air combat decision sup-port systemrdquo inProceedings of the IEEE International Conferenceon Information Reuse and Integration (IRI rsquo06) pp 39ndash44 IEEEWaikoloa Village Hawaii USA September 2006
[20] X Hai and Y Lei ldquoA situation forecast method based onintuitionistic fuzzy reasoningrdquo Electronics Optics ampControl vol18 no 4 pp 33ndash37 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
4T
1 2 T
P1I Q
1I Q
1O P
1O P
2I Q
2I Q
2O P
2O P
119879I Q
119879I Q
119879O P
119879O
middot middot middot
Figure 7 The encoding method
Deleted routfragment
Planning virtualnavigation
Modifying virtualnavigation
Figure 8 The display of route modifying method
343 Result Space Expanding Policy When calculating theresult may not get out of the local optimization so after eachiteration we get the global optimal value
bestRout
= (119875V1
119868119876
V1
119868119876
V1
119874119875V1
119874119875V2
119868119876
V2
119868119876
V2
119874119875V2
119874sdot sdot sdot 119875
V119895
119868119876
V119895
119868119876
V119895
119874119875V119895
119874)
(17)
Then we can deal with the existing result as follows andthe description of the space expanding policy can be seen inFigure 9
(a) Rescue the abandoned route when the followingcondition is satisfied1003816100381610038161003816gbest minus fitness
119894
1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)
where 119892119887119890119904119905 is the global optimal value in currenttime and fitness
119894is the abandoned value in the current
circulation when the given condition is satisfied werescue the relative 119877119900119906119905
119894(119894 = 1 2 3 119899) and this
route will take part in the next circulation
(b) Get 119899 random points around each best tactical navi-gation point within the distance 119903
119894= 1205791198771198942 (119894 = 1 2)
(119877119894defines the radius of each tactical area)
(c) Expand the result space with choosing points and geta new result space Ωnew
(d) Calculating 119891 = 119892119887119890119904119905119905 119905 is the simulation time If119891 lt 120575 then we save the current result
344 Calculating Steps See Algorithm 1
4 Simulation
41 Hardware Condition CPU is P4 28GHz and RAM is15 G Matlab 2010
42 The Tactical Scenario Our flight fighter is flying towardsthe enemiesrsquo formation and the initial locations of enemiesand our member are given in Table 1
The relative situation can be seen in Figure 10
43 Simulation Result We suppose the detecting area ofenemies is stable at the moment and we need to find the bestattack sequence and attack direction at this moment We set3000 loops to optimize the final result and the result can beseen in Figures 11 and 12
From Figure 11 our member choose the proper directionto avoid the threaten areas in order to complete the task ina relative short way The attack sequence can be describedas 1198791rarr 1198793rarr 1198792rarr 1198794 and the attack direction (the
direction of enemy velocity is the initial angle 0∘) can bedecided by the angles given in Table 2
Figure 12 shows the updating process when choosing thebest solution and after 600 more loops we get the stablevalue the best route we choose can be shown as follows
bestRout = (11987531198681198763
11986811987623
11987411987522
11987411987510
11986811987615
11986811987617
11987411987518
11987411987510
11986811987615
11986811987617
11987411987518
1198741198752
1198681198765
11986811987619
11987411987517
1198741198759
1198681198764
11986811987621
11987411987523
119874) (19)
In Figure 12 three different methods are compared witheach other The blue line shows the result when using tradi-tional PSOmethod without dividing them in different granu-larities then it has costmore time to get a stable result becauseof much more formations of the available tactical points Thered line means the VGHPSO method we mentioned above
it has cost less time to get a stable value but it is easy to gettrapped in the local optimal value Thus we use VGHPSOlowastmethodwhichmeans the VGHPSOmethodwith result spaceexpanding policy We can see the result reflected in cyan line
In the former chapters we can know that the numbers oftactical points are depended on the sampling angle Thus we
8 Mathematical Problems in Engineering
Table 1 Initial locations
Start point 1198791
1198792
1198793
1198794
End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)
Table 2 Attack direction display
Attack sequence 1198791
1198793
1198792
1198794
Approach angle (∘) 10728 16076 7543 13306Attack angle (∘) 80 12793 2914 17293Retreat angle (∘) 16673 13673 2051 16051
PjO
QjO
r1
r2
n1 n2
n2
n3
bestRout
Figure 9 Result space expanding policy
400 600 800 1000 1200 1400 1600
700
800
900
1000
1100
1200
1300
1400
1500
T1 T2
T3
T4
y(k
m)
x (km)
Figure 10 The initial locations sketch map
do some simulation with different sampling angles and theresults can be seen in Figure 13
From left side of Figure 13 the blue parts mean thetotal distances in different sampling angles and the red partsdescribe the distances under enemiesrsquo threats Then we canget the ratio as shown in Table 3
400 600 800 1000 1200 1400 1600600
700
800
900
1000
1100
1200
1300
1400
1500
1600
T1T2
T3
T4
y(k
m)
x (km)
Figure 11 The optimal route planning project
0 500 1000 1500 2000 2500 30001800
1850
1900
1950
2000
2050
2100
2150
2200
2250
x (times)
VGHPSOPSO
y(k
m)
VGHPSOlowast
Figure 12 Updating curves in different algorithms
FromTable 3 we can see thatwhen choosing the samplingangle as 15∘ we can get a minimum comprehensive indexwhich indicates that we may get a more proper resultefficiently with less threats and shorter times
The usage of the results in this paper can only beregarded as the equal description of attack sequence andattack direction and the tactical points in virtual navigationare not the real navigation points but of great importance tosimplify the engineering problems
Mathematical Problems in Engineering 9
Input (1) Targetsrsquo attributes including the targetsrsquo number119873 the coordinate 119879(119909119879 119910119879 119911119879) the velocity V
119879and the attack area
119882 (2) Our flight fightersrsquo attributes including the number of our fighters119872 the angle 120579 to decide the number of tacticalnavigation points the coordinate 119865(119909
119865 119910119865 119911119865) and the results space Ω (3) A result space expanding policy 120588lowast to optimize
the resultSuppose Condition (1) The targets stay flying in a stable velocity and an unchanged direction (2) The tactical pointsrsquoformation should obey the constraints 120585sequence directionOutput (1) The global optimal fitness result 119892119887119890119904119905 (2) Best tactical points formation 119887119890119904119905119877119900119906119905Initialize randomΩ set the initial stateBeginFor each iteration 119894 = 1 119899 119899 is the iteration number
Cross(119865) rarr Ωnew executing the cross process towards the setting codesIf 119865119894(119894 = 1 2 120587120579) (119865
119894isin Ωnew) do not obey the constraints 120585sequence direction
While (119865119894obey the constraints 120585sequence direction)
Random 119865119894 make sure that all the elements inΩnew obey the constraints
endUpdate (Ωnew)
endCalculating (119891119887119890119904119905
119894)
If 119891119887119890119904119905119894lt min119891119887119890119904119905
1 119891119887119890119904119905
2 119891119887119890119904119905
119894minus1 Then
119892119887119890119904119905119894= 119891119887119890119904119905
119894
Variate(119865) rarr Ω1015840
new executing the variation process towards the setting codesCycling the process as Cross methodExecuting (120588lowast) the saving policy is used to expand the result space to optimize the final result
EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)
End
Algorithm 1
5 15 30 45 60 72 900
500
1000
1500
2000
2500
Dist
ance
(km
)
10 20 30 40 50 60 70 800
10
20
30
40
50
60
70
80
90
Running time (s)Sampling angle (∘)
Sam
plin
g an
gle(
∘ )
Total distanceThreatened distance
Figure 13 The comprehensive values in different sampling angles
10 Mathematical Problems in Engineering
Table 3 Relative calculating results display
Sampling angle (∘) 5 15 30 45 60 72 90Threats ratio 0013 0017 0044 0090 0129 0153 0257Running times (s) 78 45 35 26 24 19 13Comprehensive index 1014 0765 154 234 3096 2907 3341
5 Conclusion
The multiobjective sequence attack and attack directionplanning methods are given in the paper We put forwarda simplified model to create the tactical points in virtualnavigation and the complex air-to-air combat planningproblemhas been transformed into a variable granularity TSPproblem then theVGHPSOmethod is used to solve the givenproblem In the end the result of the simulation shows theefficiency of the method and it also indicates that the modelis available
The model and method we mentioned in the paper areavailable only in static combat planning which belongs toprecombat planning problem But in real combat scenariowe need a dynamic data management system to optimize thereal-time policiesThen we need to pay much attention to theprediction of the future situation according to the historicaldata [20] Thus we may get a more proper and more accurateresult
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Kang H Park S Noh et al ldquoAutonomously decidingcountermeasures against threats in electronic warfare settingsrdquoin Proceedings of the International Conference on ComplexIntelligent and Software Intensive Systems (CISIS rsquo09) pp 177ndash184 IEEE Fukuoka Japan March 2009
[2] H Guo Y Lv P Wang P Ren and L Zhu ldquoTarget threatassessment of air combat based on intervals and SVRrdquo FireControl amp Command Control vol 39 no 8 pp 17ndash21 2014
[3] A Salman I Ahmad and S Al-Madani ldquoParticle swarmoptimization for task assignment problemrdquoMicroprocessors andMicrosystems vol 26 no 8 pp 363ndash371 2002
[4] P Y R Ma E Y S Lee and M Tsuchiya ldquoA task allocationmodel for distributed computing systemsrdquo IEEE Transactionson Computers vol 31 no 1 pp 41ndash47 1982
[5] D Xinghua F Yangwang X Bingsong and et al ldquoTask alloca-tion in cooperative air combat based on multi-agent coalitionrdquoJournal of BeijingUniversity of Aeronautics andAstronautics vol40 no 9 pp 1268ndash1275 2014
[6] X-G Gao Y-H Wang and B Li ldquoDecision-making methodfor the optimal coordination of fire and electronic warfarebased on integrative effectivenessrdquo Systems Engineering andElectronics vol 29 no 12 pp 2081ndash2084 2007
[7] H Guo H Xu D Liu and D Wang ldquoAir combat decision-making for cooperative multiple target attack based on adaptive
hybrid particle swarm algorithmrdquo Journal of Air Force Engineer-ing University vol 11 no 2 pp 16ndash19 2010
[8] K Horie and B A Conway ldquoOptimal fighter pursuit-evasionmaneuvers found via two-sided optimizationrdquo Journal of Guid-ance Control and Dynamics vol 29 no 1 pp 105ndash112 2006
[9] S Matlin ldquoA review of the literature on missile-allocationproblemrdquo Operations Research vol 18 no 2 pp 334ndash373 1970
[10] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on Neural Net-works pp 1942ndash1948 Perth Australia November-December1995
[11] W A Crossley ldquoOptimization for aerospace conceptual designthrough the use of genetic algorithmsrdquo in Proceedings of the1st NASADoD Workshop on Evolvable Hardware pp 200ndash207Pasadena Calif USA
[12] M Camara J Ortega and F de Toro ldquoA single front geneticalgorithm for parallel multi-objective optimization in dynamicenvironmentsrdquo Neurocomputing vol 72 no 16ndash18 pp 3570ndash3579 2009
[13] C-K Goh and K C Tan Evolutionary Multi-objective Opti-mization in Uncertain Environments Issues and AlgorithmsSpringer Berlin Germany 2009
[14] Y Yu and Z-H Zhou ldquoOn the usefulness of infeasible solutionsin evolutionary search a theoretical studyrdquo in Proceedings of theIEEECongress on Evolutionary Computation (CEC rsquo08) pp 835ndash840 IEEE Hong Kong June 2008
[15] J S McGrew J P How B Williams and N Roy ldquoAir-combatstrategy using approximate dynamic programmingrdquo Journal ofGuidance Control and Dynamics vol 33 no 5 pp 1641ndash16542010
[16] Z ZhouGeneral Aviation Fire ControlTheory NationalDefenseIndustry Press Beijing China 2008
[17] L Xiao and J Huang ldquoStudy on calculation for air-to-airmissile attack area based on improved RBF neural networkrdquo inProceedings of the International Conference on Computer Scienceand Artificial Intelligence Advances in Artificial Intelligence pp631ndash637 Sydney Australia December 2012
[18] H Zhang and D Zhou ldquoRoute planning method in air-to-ground attackrdquo Electronics Optics amp Control vol 1 no 1 pp 37ndash42 1999
[19] T-Y Sun S-J Tsai Y-N Lee S-M Yang and S-H Ting ldquoThestudy on intelligent advanced fighter air combat decision sup-port systemrdquo inProceedings of the IEEE International Conferenceon Information Reuse and Integration (IRI rsquo06) pp 39ndash44 IEEEWaikoloa Village Hawaii USA September 2006
[20] X Hai and Y Lei ldquoA situation forecast method based onintuitionistic fuzzy reasoningrdquo Electronics Optics ampControl vol18 no 4 pp 33ndash37 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Table 1 Initial locations
Start point 1198791
1198792
1198793
1198794
End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)
Table 2 Attack direction display
Attack sequence 1198791
1198793
1198792
1198794
Approach angle (∘) 10728 16076 7543 13306Attack angle (∘) 80 12793 2914 17293Retreat angle (∘) 16673 13673 2051 16051
PjO
QjO
r1
r2
n1 n2
n2
n3
bestRout
Figure 9 Result space expanding policy
400 600 800 1000 1200 1400 1600
700
800
900
1000
1100
1200
1300
1400
1500
T1 T2
T3
T4
y(k
m)
x (km)
Figure 10 The initial locations sketch map
do some simulation with different sampling angles and theresults can be seen in Figure 13
From left side of Figure 13 the blue parts mean thetotal distances in different sampling angles and the red partsdescribe the distances under enemiesrsquo threats Then we canget the ratio as shown in Table 3
400 600 800 1000 1200 1400 1600600
700
800
900
1000
1100
1200
1300
1400
1500
1600
T1T2
T3
T4
y(k
m)
x (km)
Figure 11 The optimal route planning project
0 500 1000 1500 2000 2500 30001800
1850
1900
1950
2000
2050
2100
2150
2200
2250
x (times)
VGHPSOPSO
y(k
m)
VGHPSOlowast
Figure 12 Updating curves in different algorithms
FromTable 3 we can see thatwhen choosing the samplingangle as 15∘ we can get a minimum comprehensive indexwhich indicates that we may get a more proper resultefficiently with less threats and shorter times
The usage of the results in this paper can only beregarded as the equal description of attack sequence andattack direction and the tactical points in virtual navigationare not the real navigation points but of great importance tosimplify the engineering problems
Mathematical Problems in Engineering 9
Input (1) Targetsrsquo attributes including the targetsrsquo number119873 the coordinate 119879(119909119879 119910119879 119911119879) the velocity V
119879and the attack area
119882 (2) Our flight fightersrsquo attributes including the number of our fighters119872 the angle 120579 to decide the number of tacticalnavigation points the coordinate 119865(119909
119865 119910119865 119911119865) and the results space Ω (3) A result space expanding policy 120588lowast to optimize
the resultSuppose Condition (1) The targets stay flying in a stable velocity and an unchanged direction (2) The tactical pointsrsquoformation should obey the constraints 120585sequence directionOutput (1) The global optimal fitness result 119892119887119890119904119905 (2) Best tactical points formation 119887119890119904119905119877119900119906119905Initialize randomΩ set the initial stateBeginFor each iteration 119894 = 1 119899 119899 is the iteration number
Cross(119865) rarr Ωnew executing the cross process towards the setting codesIf 119865119894(119894 = 1 2 120587120579) (119865
119894isin Ωnew) do not obey the constraints 120585sequence direction
While (119865119894obey the constraints 120585sequence direction)
Random 119865119894 make sure that all the elements inΩnew obey the constraints
endUpdate (Ωnew)
endCalculating (119891119887119890119904119905
119894)
If 119891119887119890119904119905119894lt min119891119887119890119904119905
1 119891119887119890119904119905
2 119891119887119890119904119905
119894minus1 Then
119892119887119890119904119905119894= 119891119887119890119904119905
119894
Variate(119865) rarr Ω1015840
new executing the variation process towards the setting codesCycling the process as Cross methodExecuting (120588lowast) the saving policy is used to expand the result space to optimize the final result
EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)
End
Algorithm 1
5 15 30 45 60 72 900
500
1000
1500
2000
2500
Dist
ance
(km
)
10 20 30 40 50 60 70 800
10
20
30
40
50
60
70
80
90
Running time (s)Sampling angle (∘)
Sam
plin
g an
gle(
∘ )
Total distanceThreatened distance
Figure 13 The comprehensive values in different sampling angles
10 Mathematical Problems in Engineering
Table 3 Relative calculating results display
Sampling angle (∘) 5 15 30 45 60 72 90Threats ratio 0013 0017 0044 0090 0129 0153 0257Running times (s) 78 45 35 26 24 19 13Comprehensive index 1014 0765 154 234 3096 2907 3341
5 Conclusion
The multiobjective sequence attack and attack directionplanning methods are given in the paper We put forwarda simplified model to create the tactical points in virtualnavigation and the complex air-to-air combat planningproblemhas been transformed into a variable granularity TSPproblem then theVGHPSOmethod is used to solve the givenproblem In the end the result of the simulation shows theefficiency of the method and it also indicates that the modelis available
The model and method we mentioned in the paper areavailable only in static combat planning which belongs toprecombat planning problem But in real combat scenariowe need a dynamic data management system to optimize thereal-time policiesThen we need to pay much attention to theprediction of the future situation according to the historicaldata [20] Thus we may get a more proper and more accurateresult
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Kang H Park S Noh et al ldquoAutonomously decidingcountermeasures against threats in electronic warfare settingsrdquoin Proceedings of the International Conference on ComplexIntelligent and Software Intensive Systems (CISIS rsquo09) pp 177ndash184 IEEE Fukuoka Japan March 2009
[2] H Guo Y Lv P Wang P Ren and L Zhu ldquoTarget threatassessment of air combat based on intervals and SVRrdquo FireControl amp Command Control vol 39 no 8 pp 17ndash21 2014
[3] A Salman I Ahmad and S Al-Madani ldquoParticle swarmoptimization for task assignment problemrdquoMicroprocessors andMicrosystems vol 26 no 8 pp 363ndash371 2002
[4] P Y R Ma E Y S Lee and M Tsuchiya ldquoA task allocationmodel for distributed computing systemsrdquo IEEE Transactionson Computers vol 31 no 1 pp 41ndash47 1982
[5] D Xinghua F Yangwang X Bingsong and et al ldquoTask alloca-tion in cooperative air combat based on multi-agent coalitionrdquoJournal of BeijingUniversity of Aeronautics andAstronautics vol40 no 9 pp 1268ndash1275 2014
[6] X-G Gao Y-H Wang and B Li ldquoDecision-making methodfor the optimal coordination of fire and electronic warfarebased on integrative effectivenessrdquo Systems Engineering andElectronics vol 29 no 12 pp 2081ndash2084 2007
[7] H Guo H Xu D Liu and D Wang ldquoAir combat decision-making for cooperative multiple target attack based on adaptive
hybrid particle swarm algorithmrdquo Journal of Air Force Engineer-ing University vol 11 no 2 pp 16ndash19 2010
[8] K Horie and B A Conway ldquoOptimal fighter pursuit-evasionmaneuvers found via two-sided optimizationrdquo Journal of Guid-ance Control and Dynamics vol 29 no 1 pp 105ndash112 2006
[9] S Matlin ldquoA review of the literature on missile-allocationproblemrdquo Operations Research vol 18 no 2 pp 334ndash373 1970
[10] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on Neural Net-works pp 1942ndash1948 Perth Australia November-December1995
[11] W A Crossley ldquoOptimization for aerospace conceptual designthrough the use of genetic algorithmsrdquo in Proceedings of the1st NASADoD Workshop on Evolvable Hardware pp 200ndash207Pasadena Calif USA
[12] M Camara J Ortega and F de Toro ldquoA single front geneticalgorithm for parallel multi-objective optimization in dynamicenvironmentsrdquo Neurocomputing vol 72 no 16ndash18 pp 3570ndash3579 2009
[13] C-K Goh and K C Tan Evolutionary Multi-objective Opti-mization in Uncertain Environments Issues and AlgorithmsSpringer Berlin Germany 2009
[14] Y Yu and Z-H Zhou ldquoOn the usefulness of infeasible solutionsin evolutionary search a theoretical studyrdquo in Proceedings of theIEEECongress on Evolutionary Computation (CEC rsquo08) pp 835ndash840 IEEE Hong Kong June 2008
[15] J S McGrew J P How B Williams and N Roy ldquoAir-combatstrategy using approximate dynamic programmingrdquo Journal ofGuidance Control and Dynamics vol 33 no 5 pp 1641ndash16542010
[16] Z ZhouGeneral Aviation Fire ControlTheory NationalDefenseIndustry Press Beijing China 2008
[17] L Xiao and J Huang ldquoStudy on calculation for air-to-airmissile attack area based on improved RBF neural networkrdquo inProceedings of the International Conference on Computer Scienceand Artificial Intelligence Advances in Artificial Intelligence pp631ndash637 Sydney Australia December 2012
[18] H Zhang and D Zhou ldquoRoute planning method in air-to-ground attackrdquo Electronics Optics amp Control vol 1 no 1 pp 37ndash42 1999
[19] T-Y Sun S-J Tsai Y-N Lee S-M Yang and S-H Ting ldquoThestudy on intelligent advanced fighter air combat decision sup-port systemrdquo inProceedings of the IEEE International Conferenceon Information Reuse and Integration (IRI rsquo06) pp 39ndash44 IEEEWaikoloa Village Hawaii USA September 2006
[20] X Hai and Y Lei ldquoA situation forecast method based onintuitionistic fuzzy reasoningrdquo Electronics Optics ampControl vol18 no 4 pp 33ndash37 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Input (1) Targetsrsquo attributes including the targetsrsquo number119873 the coordinate 119879(119909119879 119910119879 119911119879) the velocity V
119879and the attack area
119882 (2) Our flight fightersrsquo attributes including the number of our fighters119872 the angle 120579 to decide the number of tacticalnavigation points the coordinate 119865(119909
119865 119910119865 119911119865) and the results space Ω (3) A result space expanding policy 120588lowast to optimize
the resultSuppose Condition (1) The targets stay flying in a stable velocity and an unchanged direction (2) The tactical pointsrsquoformation should obey the constraints 120585sequence directionOutput (1) The global optimal fitness result 119892119887119890119904119905 (2) Best tactical points formation 119887119890119904119905119877119900119906119905Initialize randomΩ set the initial stateBeginFor each iteration 119894 = 1 119899 119899 is the iteration number
Cross(119865) rarr Ωnew executing the cross process towards the setting codesIf 119865119894(119894 = 1 2 120587120579) (119865
119894isin Ωnew) do not obey the constraints 120585sequence direction
While (119865119894obey the constraints 120585sequence direction)
Random 119865119894 make sure that all the elements inΩnew obey the constraints
endUpdate (Ωnew)
endCalculating (119891119887119890119904119905
119894)
If 119891119887119890119904119905119894lt min119891119887119890119904119905
1 119891119887119890119904119905
2 119891119887119890119904119905
119894minus1 Then
119892119887119890119904119905119894= 119891119887119890119904119905
119894
Variate(119865) rarr Ω1015840
new executing the variation process towards the setting codesCycling the process as Cross methodExecuting (120588lowast) the saving policy is used to expand the result space to optimize the final result
EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)
End
Algorithm 1
5 15 30 45 60 72 900
500
1000
1500
2000
2500
Dist
ance
(km
)
10 20 30 40 50 60 70 800
10
20
30
40
50
60
70
80
90
Running time (s)Sampling angle (∘)
Sam
plin
g an
gle(
∘ )
Total distanceThreatened distance
Figure 13 The comprehensive values in different sampling angles
10 Mathematical Problems in Engineering
Table 3 Relative calculating results display
Sampling angle (∘) 5 15 30 45 60 72 90Threats ratio 0013 0017 0044 0090 0129 0153 0257Running times (s) 78 45 35 26 24 19 13Comprehensive index 1014 0765 154 234 3096 2907 3341
5 Conclusion
The multiobjective sequence attack and attack directionplanning methods are given in the paper We put forwarda simplified model to create the tactical points in virtualnavigation and the complex air-to-air combat planningproblemhas been transformed into a variable granularity TSPproblem then theVGHPSOmethod is used to solve the givenproblem In the end the result of the simulation shows theefficiency of the method and it also indicates that the modelis available
The model and method we mentioned in the paper areavailable only in static combat planning which belongs toprecombat planning problem But in real combat scenariowe need a dynamic data management system to optimize thereal-time policiesThen we need to pay much attention to theprediction of the future situation according to the historicaldata [20] Thus we may get a more proper and more accurateresult
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Kang H Park S Noh et al ldquoAutonomously decidingcountermeasures against threats in electronic warfare settingsrdquoin Proceedings of the International Conference on ComplexIntelligent and Software Intensive Systems (CISIS rsquo09) pp 177ndash184 IEEE Fukuoka Japan March 2009
[2] H Guo Y Lv P Wang P Ren and L Zhu ldquoTarget threatassessment of air combat based on intervals and SVRrdquo FireControl amp Command Control vol 39 no 8 pp 17ndash21 2014
[3] A Salman I Ahmad and S Al-Madani ldquoParticle swarmoptimization for task assignment problemrdquoMicroprocessors andMicrosystems vol 26 no 8 pp 363ndash371 2002
[4] P Y R Ma E Y S Lee and M Tsuchiya ldquoA task allocationmodel for distributed computing systemsrdquo IEEE Transactionson Computers vol 31 no 1 pp 41ndash47 1982
[5] D Xinghua F Yangwang X Bingsong and et al ldquoTask alloca-tion in cooperative air combat based on multi-agent coalitionrdquoJournal of BeijingUniversity of Aeronautics andAstronautics vol40 no 9 pp 1268ndash1275 2014
[6] X-G Gao Y-H Wang and B Li ldquoDecision-making methodfor the optimal coordination of fire and electronic warfarebased on integrative effectivenessrdquo Systems Engineering andElectronics vol 29 no 12 pp 2081ndash2084 2007
[7] H Guo H Xu D Liu and D Wang ldquoAir combat decision-making for cooperative multiple target attack based on adaptive
hybrid particle swarm algorithmrdquo Journal of Air Force Engineer-ing University vol 11 no 2 pp 16ndash19 2010
[8] K Horie and B A Conway ldquoOptimal fighter pursuit-evasionmaneuvers found via two-sided optimizationrdquo Journal of Guid-ance Control and Dynamics vol 29 no 1 pp 105ndash112 2006
[9] S Matlin ldquoA review of the literature on missile-allocationproblemrdquo Operations Research vol 18 no 2 pp 334ndash373 1970
[10] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on Neural Net-works pp 1942ndash1948 Perth Australia November-December1995
[11] W A Crossley ldquoOptimization for aerospace conceptual designthrough the use of genetic algorithmsrdquo in Proceedings of the1st NASADoD Workshop on Evolvable Hardware pp 200ndash207Pasadena Calif USA
[12] M Camara J Ortega and F de Toro ldquoA single front geneticalgorithm for parallel multi-objective optimization in dynamicenvironmentsrdquo Neurocomputing vol 72 no 16ndash18 pp 3570ndash3579 2009
[13] C-K Goh and K C Tan Evolutionary Multi-objective Opti-mization in Uncertain Environments Issues and AlgorithmsSpringer Berlin Germany 2009
[14] Y Yu and Z-H Zhou ldquoOn the usefulness of infeasible solutionsin evolutionary search a theoretical studyrdquo in Proceedings of theIEEECongress on Evolutionary Computation (CEC rsquo08) pp 835ndash840 IEEE Hong Kong June 2008
[15] J S McGrew J P How B Williams and N Roy ldquoAir-combatstrategy using approximate dynamic programmingrdquo Journal ofGuidance Control and Dynamics vol 33 no 5 pp 1641ndash16542010
[16] Z ZhouGeneral Aviation Fire ControlTheory NationalDefenseIndustry Press Beijing China 2008
[17] L Xiao and J Huang ldquoStudy on calculation for air-to-airmissile attack area based on improved RBF neural networkrdquo inProceedings of the International Conference on Computer Scienceand Artificial Intelligence Advances in Artificial Intelligence pp631ndash637 Sydney Australia December 2012
[18] H Zhang and D Zhou ldquoRoute planning method in air-to-ground attackrdquo Electronics Optics amp Control vol 1 no 1 pp 37ndash42 1999
[19] T-Y Sun S-J Tsai Y-N Lee S-M Yang and S-H Ting ldquoThestudy on intelligent advanced fighter air combat decision sup-port systemrdquo inProceedings of the IEEE International Conferenceon Information Reuse and Integration (IRI rsquo06) pp 39ndash44 IEEEWaikoloa Village Hawaii USA September 2006
[20] X Hai and Y Lei ldquoA situation forecast method based onintuitionistic fuzzy reasoningrdquo Electronics Optics ampControl vol18 no 4 pp 33ndash37 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
Table 3 Relative calculating results display
Sampling angle (∘) 5 15 30 45 60 72 90Threats ratio 0013 0017 0044 0090 0129 0153 0257Running times (s) 78 45 35 26 24 19 13Comprehensive index 1014 0765 154 234 3096 2907 3341
5 Conclusion
The multiobjective sequence attack and attack directionplanning methods are given in the paper We put forwarda simplified model to create the tactical points in virtualnavigation and the complex air-to-air combat planningproblemhas been transformed into a variable granularity TSPproblem then theVGHPSOmethod is used to solve the givenproblem In the end the result of the simulation shows theefficiency of the method and it also indicates that the modelis available
The model and method we mentioned in the paper areavailable only in static combat planning which belongs toprecombat planning problem But in real combat scenariowe need a dynamic data management system to optimize thereal-time policiesThen we need to pay much attention to theprediction of the future situation according to the historicaldata [20] Thus we may get a more proper and more accurateresult
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Kang H Park S Noh et al ldquoAutonomously decidingcountermeasures against threats in electronic warfare settingsrdquoin Proceedings of the International Conference on ComplexIntelligent and Software Intensive Systems (CISIS rsquo09) pp 177ndash184 IEEE Fukuoka Japan March 2009
[2] H Guo Y Lv P Wang P Ren and L Zhu ldquoTarget threatassessment of air combat based on intervals and SVRrdquo FireControl amp Command Control vol 39 no 8 pp 17ndash21 2014
[3] A Salman I Ahmad and S Al-Madani ldquoParticle swarmoptimization for task assignment problemrdquoMicroprocessors andMicrosystems vol 26 no 8 pp 363ndash371 2002
[4] P Y R Ma E Y S Lee and M Tsuchiya ldquoA task allocationmodel for distributed computing systemsrdquo IEEE Transactionson Computers vol 31 no 1 pp 41ndash47 1982
[5] D Xinghua F Yangwang X Bingsong and et al ldquoTask alloca-tion in cooperative air combat based on multi-agent coalitionrdquoJournal of BeijingUniversity of Aeronautics andAstronautics vol40 no 9 pp 1268ndash1275 2014
[6] X-G Gao Y-H Wang and B Li ldquoDecision-making methodfor the optimal coordination of fire and electronic warfarebased on integrative effectivenessrdquo Systems Engineering andElectronics vol 29 no 12 pp 2081ndash2084 2007
[7] H Guo H Xu D Liu and D Wang ldquoAir combat decision-making for cooperative multiple target attack based on adaptive
hybrid particle swarm algorithmrdquo Journal of Air Force Engineer-ing University vol 11 no 2 pp 16ndash19 2010
[8] K Horie and B A Conway ldquoOptimal fighter pursuit-evasionmaneuvers found via two-sided optimizationrdquo Journal of Guid-ance Control and Dynamics vol 29 no 1 pp 105ndash112 2006
[9] S Matlin ldquoA review of the literature on missile-allocationproblemrdquo Operations Research vol 18 no 2 pp 334ndash373 1970
[10] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on Neural Net-works pp 1942ndash1948 Perth Australia November-December1995
[11] W A Crossley ldquoOptimization for aerospace conceptual designthrough the use of genetic algorithmsrdquo in Proceedings of the1st NASADoD Workshop on Evolvable Hardware pp 200ndash207Pasadena Calif USA
[12] M Camara J Ortega and F de Toro ldquoA single front geneticalgorithm for parallel multi-objective optimization in dynamicenvironmentsrdquo Neurocomputing vol 72 no 16ndash18 pp 3570ndash3579 2009
[13] C-K Goh and K C Tan Evolutionary Multi-objective Opti-mization in Uncertain Environments Issues and AlgorithmsSpringer Berlin Germany 2009
[14] Y Yu and Z-H Zhou ldquoOn the usefulness of infeasible solutionsin evolutionary search a theoretical studyrdquo in Proceedings of theIEEECongress on Evolutionary Computation (CEC rsquo08) pp 835ndash840 IEEE Hong Kong June 2008
[15] J S McGrew J P How B Williams and N Roy ldquoAir-combatstrategy using approximate dynamic programmingrdquo Journal ofGuidance Control and Dynamics vol 33 no 5 pp 1641ndash16542010
[16] Z ZhouGeneral Aviation Fire ControlTheory NationalDefenseIndustry Press Beijing China 2008
[17] L Xiao and J Huang ldquoStudy on calculation for air-to-airmissile attack area based on improved RBF neural networkrdquo inProceedings of the International Conference on Computer Scienceand Artificial Intelligence Advances in Artificial Intelligence pp631ndash637 Sydney Australia December 2012
[18] H Zhang and D Zhou ldquoRoute planning method in air-to-ground attackrdquo Electronics Optics amp Control vol 1 no 1 pp 37ndash42 1999
[19] T-Y Sun S-J Tsai Y-N Lee S-M Yang and S-H Ting ldquoThestudy on intelligent advanced fighter air combat decision sup-port systemrdquo inProceedings of the IEEE International Conferenceon Information Reuse and Integration (IRI rsquo06) pp 39ndash44 IEEEWaikoloa Village Hawaii USA September 2006
[20] X Hai and Y Lei ldquoA situation forecast method based onintuitionistic fuzzy reasoningrdquo Electronics Optics ampControl vol18 no 4 pp 33ndash37 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of