a hybrid discrete grey wolf optimizer to solve weapon target assignment problems · 2019. 7....

Research ArticleA Hybrid Discrete Grey Wolf Optimizer to SolveWeapon Target Assignment Problems

JunWang 1 Pengcheng Luo 2 Xinwu Hu 1 and Xiaonan Zhang 1

1College of Systems Engineering National University of Defense Technology Changsha 410073 China2Allsim Technology Inc Changsha 410073 China

Correspondence should be addressed to JunWang fisher wangnudteducn

Received 17 June 2018 Revised 3 October 2018 Accepted 29 October 2018 Published 7 November 2018

Academic Editor Pasquale Candito

Copyright copy 2018 JunWang et alThis is an open access article distributed under theCreative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We propose a hybrid discrete greywolf optimizer (HDGWO) in this paper to solve theweapon target assignment (WTA)problem akind of nonlinear integer programming problems Tomake the original grey wolf optimizer (GWO) which was only developed forproblems with a continuous solution space available in the context we first modify it by adopting a decimal integer encodingmethod to represent solutions (wolves) and presenting a modular position update method to update solutions in the discretesolution space By this means we acquire a discrete grey wolf optimizer (DGWO) and then through combining it with a localsearch algorithm (LSA) we obtain the HDGWO Moreover we also introduce specific domain knowledge into both the encodingmethod and the local search algorithm to compress the feasible solution space Finally we examine the feasibility of the HDGWOand the scalability of the HDGWO respectively by adopting it to solve a benchmark case and ten large-scale WTA problems Allof the running results are compared with those of a discrete particle swarm optimization (DPSO) a genetic algorithm with greedyeugenics (GAWGE) and an adaptive immune genetic algorithm (AIGA)Thedetailed analysis proves the feasibility of theHDGWOin solving the benchmark case and demonstrates its scalability in solving large-scale WTA problems

1 Introduction

As a classic military operational problem the purpose ofweapon target assignment (WTA) is to find an optimal or asatisfactory assignment solution which determines the targetattacked by each weapon so as to maximize the total damageexpectancy of hostile targets or minimize the loss expectancyof own-force assets Since first proposed in the 1950s tosupport operation planning command officers training andweapon selection and acquisition [1] the WTA problem hasattracted the attention of relevant researchers for decadesFrom the mathematical point of view it is essentially an NP-complete problem [2] with nonlinear objective functionsdiscrete decision variables and multiple constraints Theseintricate features indicate that seeking out the optimal solu-tion for small-scale WTA problems is relatively realistic butbecomes impracticable for large-scale ones This means insuch circumstances searching for a satisfactory or suboptimalsolution may be more efficient

In the WTA study the process of solving WTA prob-lems can generally be divided into two phases establishinga WTA model and finding an optimal or a suboptimalsolution for the model through an appropriate algorithmIn modeling a variety of factors need to be consideredsuch as the type and quantity of equipment involved thecombat capability of each equipment the characteristics ofthe battlefield and the focus of the commander In additionfor the convenience of modeling certain assumptions maybe made Conventional WTA models include up to threetypes of entities platforms weapons and targets Howeverconsidering that weapons are increasingly dependent onthe information provided by sensors in modern warfareBogdanowicz et al [3 4] believed that sensors should beconsidered in WTAmodels and established the sensor-WTA(S-WTA)model for the first timeThey simplified the S-WTAproblem through the sensortargetweapon decompositionto the following scenario each sensor-weapon pair can beassigned to at most one target and each target can be engaged

HindawiDiscrete Dynamics in Nature and SocietyVolume 2018 Article ID 4674920 17 pageshttpsdoiorg10115520184674920

2 Discrete Dynamics in Nature and Society

by at most one sensor-weapon pair Furthermore throughthe sensorweapontarget augmentation they translated thederived problem into a symmetric optimization problemwithan input consisting of the same numbers of sensors weaponsand targets along with the benefit matrix and presentedthe Swt-opt algorithm derived from the auction algorithmto optimally assign sensors and weapons to targets Sincethen the S-WTA problem has been successively studied bysome scholars Li et al [5 6] proposed an improved Swt-opt algorithm to solve the same problem by combining theconsensus algorithm so that the shortcoming of the Swt-opt algorithm ie highly depending on perfect networktopologies can be overcome Later they put forward adecentralized cooperative auction algorithm for a similar S-WTA problemwhere the number of targets that can be struckvaried in different operational phases [7] Chen et al [8]proposed a particle swarm optimization based on geneticoperators to solve the S-WTAproblemwhere each sensor canguide only one weapon once and each target can be engagedbymultiple weapons onceMu et al [9] proposed amultiscalequantum harmonic oscillator algorithm to solve the S-WTAproblem in intelligent minefields considering the probabilityof detection and killing Xin et al [10] constructed an efficientmarginal-return-based heuristic to solve the S-WTAproblemconsidering the situation where multiple sensorsweaponscan be assigned to one target but each sensor can detect onlyone target at the same time and each weapon can shoot onlyone target simultaneously The proposed heuristic exploitedthe marginal return of each sensor-weapon-target triplet anddynamically updated the threat value of all targets It reliedonly on simple lookup operations to choose each assignmenttriplet thus resulting in very low computational complexity

Both the conventional WTA model and the S-WTAmodel can be classified as follows For ease of presentationthe two models will not be distinguished both called theWTA model A WTA model is called to be dynamic [10ndash12]if the operational process is time related This indicates thereare always several operational stages or new situations usuallyarise during operation Otherwise the model is static [3ndash6 8 9 13ndash16] If two or more objectives such as maximizingthe damage expectancy of hostile targets and minimizing theconsumption of own ammunition and mission completiontime are considered the WTA model is multiobjective [1718] If there is only one objective it is single-objective [3ndash11 19] Depending on the type of combat scenario theWTA model may be asset-based [11] or target-based [17] Indefensive missions the asset-based model is often establishedto minimize the loss expectancy of own-force assets while inoffensive missions the target-based model is always adoptedto maximize the total damage expectancy of hostile targetsIn addition to modeling another research aspect for WTAis to develop algorithms to solve the problem optimally inthe least possible time These algorithms can be generallydivided into two categories exact algorithms and heuristicalgorithms Exact algorithms are developed according to thespecific mathematical properties of WTA problems and cangradually eliminate the nonlinearity of the problem throughtransformation decomposition and other processing means[16] By this way the model is translated to a linear one

and then classical operations research methods can applysuch as the dynamic programming method [12] the branchand bound method [20 21] the branch and price method[22] the mixed integer linear program (MILP) algorithm[17] and the Lagrange relaxationmethod [14]Thesemethodshave demonstrated the feasibility and effectiveness on solvingstatic and dynamic WTA problems but may become difficultto apply when a large number of weapons and targets areinvolved [23]

With the rapid development of heuristic algorithmsmorecomplex WTA problems are able to be well solved Mostresearch to date on solving WTA problems by heuristicalgorithms either constructs some specific search rules basedon the properties of the problem to achieve solutions rapidlyor introduces some local searchmechanisms into the originalalgorithms to improve the solution quality These algorithmsincluding auction algorithms [3ndash6 15] improved geneticalgorithms [18 24ndash26] clonal selection algorithms [27 28]particle swarm algorithms [8 13 29] tabu search algorithms[30] rule-based constructive heuristic algorithms [10 31]and other intelligent optimization algorithms have shownevident advantages over traditional methods in terms ofcomputation time and solution accuracy and however stillsuffer from some drawbacks such as easily falling into pre-mature convergence and local optimum [32] Furthermoreconsidering the variability of the battlefield environmentdecisions always need to be made immediately that is WTAproblems need to be resolved in a very short time Thereforewe propose in this paper a hybrid discrete greywolf optimizer(HDGWO) which is an integration of a discrete grey wolfoptimizer (DGWO) with a local search algorithm (LSA)Moreover in order to enhance the efficiency of the algorithmwe introduce the specific domain knowledge into the encodemethods and the LSA

The rest of this paper is organized as follows The math-ematical model of a typical WTA problem for an offensivemission is formulated in Section 2 Section 3 is a briefintroduction to the original grey wolf optimizer (GWO)presented in [33] and Section 4 is a detailed introductionto the proposed HDGWO In Section 5 we first analyze thefeasibility of HDGWO in solving small-scale WTA problemsby comparing itwith the discrete particle swarmoptimization(DPSO) [13] the genetic algorithm with greedy eugenics(GAWGE) [24] and the adaptive immune genetic algorithm(AIGA) [32] and then investigate the scalability of HDGWOby solving ten different scale WTA problems Finally wemake a conclusion and look ahead to the future research inSection 6

2 Problem Formulation

In general a typical WTA problem for an offensive missioncan be formulated as the following nonlinear integer pro-gramming model [13 32]

max 119891 (119909)= 119873sum119895=1

V119895 [1 minus 119872prod119894=1

(1 minus 119901119894119895 sdot 119890119894119895)119909119894119895]

Discrete Dynamics in Nature and Society 3

Table 1 Symbol declaration

Symbols ExplanationsM The number of weapon platforms that can launchmissiles (eg fighters)N The number of targets119865119894 119879119895 The ith weapon platform and the jth target 119894 = 1 2 sdot sdot sdot119872 119895 = 1 2 sdot sdot sdot 119873119902119894 The number of air-to-groundmissiles available for 119865119894 119894 = 1 2 sdot sdot sdot119872V119895 The value of 119879119895 119895 = 1 2 sdot sdot sdot119873119901119894119895 The kill probability of missiles on 119865119894 against 119879119895 119894 = 1 2 sdot sdot sdot119872 119895 = 1 2 sdot sdot sdot119873119909119894119895 The decision variable indicating the number of missiles launched by 119865119894 against 119879119895 119894 = 1 2 sdot sdot sdot119872 119895 = 1 2 sdot sdot sdot119873

119890119894119895 The engagement constraint factor indicating whether 119865119894 is able to attack 119879119895 When one attack is feasible 119890119894119895 = 1 otherwise119890119894119895 = 0 119894 = 1 2 sdot sdot sdot119872 119895 = 1 2 sdot sdot sdot119873119903119895119866119895(119909) The penalty factor and the penalty function 119866119895(119909) = min0 119892119895(119909) 119892119895(119909) = sum119872119894=1 119909119894119895 minus 1 119895 = 1 2 sdot sdot sdot 119873119873119901 The population size of the proposed HDGWO

997888rarr119878 (119905) A solution in the tth iteration 997888rarr119878 (119905) = [997888rarr119904 1(119905) 997888rarr119904 2(119905) sdot sdot sdot 997888rarr119904 119872(119905)] where 997888rarr119904 119894(119905) (119894 = 1 2 sdot sdot sdot119872) is the assignment scheme of 119865119894The first three best solutions are denoted by 997888rarr119878 120572(119905) 997888rarr119878 120573(119905) and 997888rarr119878 120575(119905) in turn119886 The control parameter determining which rule will be adopted to update solutions119879max The maximum iteration number120578 The elitist retention proportionΔ 119901 The perturbation value used to modify the value of decision variables in local search120582 The selection probability determining whether a solution is to be selected for local search

+ 119873sum119895=1

119903119895 sdot 119866119895 (119909) (1)

subject to

119873sum119895=1

119909119894119895 le 119902119894 119894 = 1 2 sdot sdot sdot 119872 (2)

119872sum119894=1

119909119894119895 ge 1 119895 = 1 2 sdot sdot sdot 119873 (3)

119909119894119895 isin 119873119909119894119895 ge 0

119894 = 1 2 sdot sdot sdot 119872 119895 = 1 2 sdot sdot sdot 119873(4)

119909119894119895 sdot (1 minus 119890119894119895) = 0119894 = 1 2 sdot sdot sdot 119872 119895 = 1 2 sdot sdot sdot 119873 (5)

The symbols in the model are explained in Table 1 For claritywe also list the symbols employed in the following in thistable

In the above model the objective is to maximize thedamage expectancy of all targets Thus we construct theabove objective function (1) which consists of two partsconnected by a plus sign The part before the plus signrepresents the total expectancy of the eliminated value fromall targets and the latter part indicates the penalty for thesolution that violates the constraint set (3) If a solutionviolates the constraint set (3) then the value of the latter

part will become negative which reduces the objective valueThus the solution will be very likely to be discarded inthe next iteration The constraint set (2) limits the numberof missiles launched by each weapon platform to no morethan the available number The constraint set (4) means thatdecision variables are nonnegative integers The constraintset (5) reflects the feasibility of engagement between weaponplatforms and targets When 119865119894 is unable to attack 119879119895 ie119890119894119895 = 0 it should not launch any missiles to 119879119895 ie 119909119894119895 =03 Grey Wolf Optimizer (GWO)

The grey wolf optimizer (GWO) is a swarm intelligencealgorithm first proposed in 2014 by Seyedali Mirjalili et alandmimics the leadership hierarchy and hunting mechanismof grey wolves in nature [33] There are four types of wolvesfrom top to bottom denoted by 120572 120573 120575 and 120596 respectivelyin the leadership hierarchy The former three types of wolves120572 120573 and 120575 guide the hunting process which is followed bythe 120596 wolves The GWO algorithm employs two operators toachieve optimization (1) encircling prey and (2) hunting asintroduced below [33]

31 Encircling Prey Greywolveswill first encircle prey duringthe hunting process The encircling behavior is modeled byformulas (6)-(7) as follows

997888rarr119863 = 100381610038161003816100381610038161003816997888rarr119862 sdot 997888rarr119883119901 (119905) minus 997888rarr119883 (119905)100381610038161003816100381610038161003816 (6)

997888rarr119883 (119905 + 1) = 997888rarr119883 (119905) minus 997888rarr119860 sdot 997888rarr119863 (7)


a1

C1

a3

C3

Dalpℎa

Ddelta

Moe

R

Dbeta

a2

C2

or any other wolvesEstimated positions ofthe prey

Figure 1 The position update process in GWO [33]

where t indicates the current iteration 997888rarr119883119901(119905) and 997888rarr119883(119905) arerespectively the position vector of the prey and the positionvector of a grey wolf in the current iteration 997888rarr119883(119905 + 1) is theposition vector of the grey wolf in the next iteration and 997888rarr119860and 997888rarr119862 are coefficient vectors

The vectors 997888rarr119860 and 997888rarr119862 are calculated as follows997888rarr119860 = 2997888rarr119886 sdot 997888rarr119903 1 minus 997888rarr119886 (8)997888rarr119862 = 2 sdot 997888rarr119903 2 (9)

where elements of 997888rarr119886 are linearly decreased from 2 to 0throughout the iteration process and 997888rarr119903 1 and 997888rarr119903 2 are randomvectors in [0 1]32 Hunting In the GWO algorithm the 120572 120573 and 120575 wolvesare supposed to have more knowledge about the potentiallocation of prey Therefore the first three best solutionsobtained so far (ie the 120572 120573 and 120575 wolves in the currentiteration) are saved and employed to assist the other wolvesin updating their positions The process is mathematicallydescribed by formulas (10)-(12) as follows997888rarr119863120572 = 100381610038161003816100381610038161003816997888rarr1198621 sdot 997888rarr119883120572 minus 997888rarr119883100381610038161003816100381610038161003816 997888rarr119863120573 = 100381610038161003816100381610038161003816997888rarr1198622 sdot 997888rarr119883120573 minus 997888rarr119883100381610038161003816100381610038161003816 997888rarr119863120575 = 100381610038161003816100381610038161003816997888rarr1198623 sdot 997888rarr119883120575 minus 997888rarr119883100381610038161003816100381610038161003816

(10)

997888rarr1198831 = 997888rarr119883120572 minus 997888rarr1198601 sdot 997888rarr119863120572997888rarr1198832 = 997888rarr119883120573 minus 997888rarr1198602 sdot 997888rarr119863120573997888rarr1198833 = 997888rarr119883120575 minus 997888rarr1198603 sdot 997888rarr119863120575

(11)

997888rarr119883 (119905 + 1) = (997888rarr1198831 + 997888rarr1198832 + 997888rarr1198833)3 (12)

where 997888rarr119883120572997888rarr119883120573 and

997888rarr119883120575 are the position vectors of 120572 120573 and120575 respectively 997888rarr1198621 997888rarr1198622 and 997888rarr1198623 are random vectors and 997888rarr119883 isthe position vector of a greywolfThe position update processaccording to 120572 120573 and 120575 in a 2D search space is shown inFigure 1 The 120572 120573 and 120575 wolves first estimate the positionof the prey and then other wolves update their positionsrandomly around the prey [33]

From formulas (6)-(12) we find that vectors 997888rarr119860 and 997888rarr119862 aretwo parameters to control the exploration and exploitationability of the GWO algorithm When |119860| gt 1 the algorithmdevotes half of the iterations to explore the search space when|119860| lt 1 the rest of the iterations are dedicated to exploitation[34] As can be seen in formula (9) elements of 997888rarr119862 are inthe interval [0 2] which provide random weights for prey inorder to stochastically emphasize (119862 gt 1) or deemphasize(119862 lt 1) the effect of prey in determining the distance informula (6) This mechanism gives the GWO algorithm a


Solution

Weapon platforms

Targets

F1 F2 FM

T1 T2 TN T1 T2 TN T1 T2 TN

x11 x12 x1N x21 x22 x2N xM1 xM2 xMN

middot middot middot

middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot


Figure 2 The decimal integer encoding method for position vectors [32]

better random search ability throughout the optimizationprocess which is beneficial to exploration and local optimumavoidance [33]

4 Hybrid Discrete Grey Wolf Optimizer(HDGWO) for WTA

The two operators introduced above are originally developedfor solving continuous optimization problems Thus wecannot directly employ the GWO algorithm to address WTAproblems because the search space is discrete and decisionvariables are nonnegative integers And in the field of WTAresearch the application of GWO has not yet been foundFor the above reasons we first modify the original GWO to adiscrete one by a decimal interencoding method and a novelposition update method and then combine it with a localsearch algorithm (LSA) Thus we propose a hybrid discretegrey wolf optimizer (HDGWO) The details of the HDGWOare described in the following subsections

41 Decimal Integer Encoding Method The encoding methodis an important procedure before applying the proposedHDGWO to a particular problem It constructs a bridgebetween solution space of the considered problem andthe search space of the search process Therefore design-ing an appropriate encoding method is a very essen-tial issue that affects the performance of the algorithm[35]

Considering that all decision variables are decimal non-negative integers (see constraint (4) in Section 2) we adoptthe decimal integer encoding method in [32] to represent theposition vector of a grey wolf (in the following sections wewill use the term ldquosolutionrdquo instead of ldquoposition vectorrdquo) Forthe ith weapon platform 119865119894 we use a vector containing Nelements to represent its assignment scheme where the jthelement corresponds to the number ofmissiles launched by119865119894to119879119895 So forM weapon platforms we haveM similar vectorsThen we connect these vectors head to tail in the order of theassignment scheme of1198651 to the assignment scheme of119865119872 andobtain a solution as shown in Figure 2 [32]

During the construction of an initial solution the con-straint sets (2) (4) and (5) will be handled so the searchspace could be effectively compressedWedo not consider theconstraint set (3) here because on one hand we have takeninto account it when constructing the objective function(1) on the other hand the initialization method wouldbecome very complicated if the constraint set (3) should besatisfied Although we cannot guarantee that all the solutions

generated are feasible the penalty function in the objectivefunction (1) can help to eliminate the solution that violates theconstraint set (3) throughout the iteration process In addi-tion to improve the quality of initial solutions we introducethe specific domain knowledge that available missiles shouldbe first assigned to the target with the largest probability tobe destroyed The pseudocode of the initialization method isshown in Algorithm 1

42 Fitness Function The fitness function is used to evaluatethe goodness of a solution with respect to the original objec-tive function For the convenience of solution comparison weset the fitness function as the same as the objective functionintroduced in Section 2 The fitness value of solution 119878 iscomputed as follows

119891 (119878) = 119873sum119895=1

V119895 [1 minus 119872prod119894=1

(1 minus 119901119894119895 sdot 119890119894119895)119909119894119895(119878)]

+ 119873sum119895=1

119903119895 sdot 119866119895 (119878)(13)

43 Searching for Prey On the basis of the encoding methodthe initialization method and the fitness function we canevaluate the goodness of each solution (wolf) and seek out thefirst three best solutions (the120572120573 and 120575wolves)Then the nextcore procedure is to search for prey In HDGWO taking intoaccount the constraints in Section 2 and the characteristicsof the encoding method we propose a novel position updatemethod namely the modular position update method Inaddition we also adopt the elitist retention mechanismwhich is usually used in the genetic algorithm and presenta local search algorithm (LSA) to avoid local optimumThe details are successively introduced in the subsequentsections

431 Modular Position Update Method The search processof the HDGWO is also guided by the first three best solutions

(wolves) denoted by 997888rarr119878 120572 997888rarr119878 120573 and 997888rarr119878 120575 However the specificposition update mechanism is much different from that ofthe original GWO Here we propose a modular positionupdate method which treats the assignment scheme of eachweapon platform as a module We first define the concept ofa module as the assignment scheme of one weapon platformand then denote a solution in the tth iteration by 997888rarr119878 (119905) =[997888rarr119904 1(119905) 997888rarr119904 2(119905) sdot sdot sdot 997888rarr119904119872(119905)] where 997888rarr119904 119894(119905) (119894 = 1 2 sdot sdot sdot 119872)is the ith module of 997888rarr119878 (119905) (ie the assignment scheme of


Input Combat scenario parameters includingM NQ = [119902119894]1times119872 V = [V119895]1times119873 P = [119901119894119895]119872times119873 and E = [119890119894119895]119872times119873and the population size119873119901

Output The initial solutionsProcedures for 119899119901 = 1 997888rarr 119873119901

Set the initial value of the npth solution to a zero-vector [0]1times119872119873for i=1997888rarrMSort all targets in descending order according to P(i) Denote the index vector by INSet q max= 119902119894for j=1997888rarrNif q maxlt=0 determine if all weapons of 119865119894 have been assignedbreak terminates the execution of the for loop ie for j=1997888rarrN

endifif E(i IN(j))==1 119865119894 can attack 119879IN(119895)

Generate a random integer denoted by n rand between 1 and q maxAssign n rand weapons to 119879IN(119895)Update q max= q max-n rand

endifendfor

endforendfor

Algorithm 1 Pseudo-code of the initialization method

119865119894) Based on this the modular position update method isformulated as follows997888rarr119904 119894 (119905 + 1)

= 997888rarr119904 120572119894 (119905) or 997888rarr119904 120573119894 (119905) or 997888rarr119904 120575119894 (119905) if 119903119886119899119889 le 119886997888rarr119904 119877119894 (119905) otherwise

(14)

where 997888rarr119904 119894(119905 + 1) is the ith module of 997888rarr119878 (119905 + 1) (ie theassignment scheme of 119865119894 in the (t+1)th iteration) 997888rarr119904 120572119894 (119905)997888rarr119904 120573119894 (119905) and 997888rarr119904 120575119894 (119905) are respectively the ith module of 997888rarr119878 120572(119905)997888rarr119878 120573(119905) and 997888rarr119878 120575(119905) 997888rarr119904 119877119894 (119905) is a reassignment scheme of 119865119894 119903119886119899119889is a random number in (0 1) and 119886 is a control parameterThe mechanism of the modular position update method isillustrated in Figure 3 and the pseudocode of the method isdescribed in Algorithm 2

As can be seen from Figure 3 and Algorithm 2 theproposed HDGWO generates 997888rarr119904 119894(119905 + 1) according to twodifferent rules When 119903119886119899119889 le 119886 the roulette wheel selectionmethod is employed to select one module among 997888rarr119904 120572119894 (119905)997888rarr119904 120573119894 (119905) and 997888rarr119904 120575119894 (119905) to be997888rarr119904 119894(119905 +1) according to the fitness valueof 997888rarr119878 120572(119905) 997888rarr119878 120573(119905) and 997888rarr119878 120575(119905) This means that 997888rarr119904 120572119894 (119905) acquiresthe largest probability to be selected followed by 997888rarr119904 120573119894 (119905) andfinally 997888rarr119904 120575119894 (119905) In this case the assignment scheme of 119865119894 in thetth iteration whether 997888rarr119904 120572119894 (119905) 997888rarr119904 120573119894 (119905) or 997888rarr119904 120575119894 (119905) is treated as awhole and thuswill not bemodifiedWhen 119903119886119899119889 gt 119886997888rarr119904 119894(119905+1)is generated through a reassignment method which is similarto the initialization method in Section 41 and is also based onthe constraint sets (2) (4) and (5) Unlike the initializationmethod the reassignment method directly assigns a randomnumber of missiles to each target in the order of 1198791 to 119879119873

Whether 997888rarr119904 119894(119905 + 1) is generated through the roulette wheelselection method or the reassignment method the constraintsets (2) (4) and (5) are always satisfied during the positionupdate process

In the proposed HDGWO the control parameter 119886determines which rule will be adopted Based on the searchmechanism of the original GWO in the continuous solutionspace we set 119886 as follows [35]

119886 = 1 minus 119905119879max(15)

where 119905 and 119879max are the current iteration number and themaximum iteration number respectively It can be seen thatas 119905 increases 119886 linearly decreases from 1 to 0 Thereforethe modular position update method generates 997888rarr119904 119894(119905 + 1)mainly through the roulette wheel selection method in thefirst half of the iterations and thus obliges other solutions

to quickly approach 997888rarr119878 120572(119905) 997888rarr119878 120573(119905) and 997888rarr119878 120575(119905) Then in thesecond half the modular position update method acquiresa larger probability to generate 997888rarr119904 119894(119905 + 1) through the reas-signment method hence giving other solutions an increasingprobability to diverge from the prey In summary throughoutthe process of searching for prey the proposed HDGWOmainly focuses on exploitation in the first half of the iterations(when 05 le 119886 le 1) and the second half of the iterations ismainly devoted to exploring the search space (when 0 le 119886 le05)432 Elitist Retention Mechanism The elitist retentionmechanism is employed to select a certain proportiondenoted by 120578 of the best solutions (wolves) in the currentiteration to be a part of the next iteration directly withouttheir positions updated By this means excellent solutions(wolves) are able to be preserved during the iterations


reassign

roulettewheel

selection

rand le a

rand gt a

rarrS

(t)

rarrS

(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs i

rarrs1(t)rarrs i (t)

rarrsM(t)

x11 x12 x1N xi1 xi2 xiNxM1

xM2 xMN

x11 x12 x1N xi1 xi2 xiN xM1 xM2 xMN




rarrS

(t)

rarrS(t+1)

rarrS(t)

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot middotmiddotmiddot


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot

middotmiddotmiddot middotmiddotmiddot middotmiddotmiddot middotmiddotmiddot middotmiddotmiddot


rarrsM(t + 1)(t + 1)rarrs1(t+1)

Figure 3 The mechanism of the modular position update method

Sbase

Skli

xi1

xi1

xil

xil minusΔp

+Δp

middotmiddotmiddot


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

xik

xik +Δp xiN

xiN

middotmiddotmiddotmiddotmiddotmiddot

middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot


Perturbation (pik ge pil)

Fi

Figure 4 The mechanism of the perturbation operator

and thus the convergence speed of the algorithm getsenhanced

433 Local Search Algorithm (LSA) In the proposedHDGWO we proposed a local search algorithm (LSA) basedon a perturbation operator [36] to exploit the neighborhoodof a solution The mechanism of the perturbation operatoris illustrated in Figure 4 Firstly randomly select a solutionto be modified denoted by 119878119887119886119904119890 Secondly randomly selectone weapon platform 119865119894 from the total M ones of whichthe assignment scheme will be adjusted Thirdly randomlychoose two of its targets 119879119896 and 119879119897 (119896 119897 = 1 2 sdot sdot sdot 119873 119896 = 119897)satisfying 119890119894119896 = 119890119894119897 = 1 and 119909119894119897 119909119894119896 gt 0 Finally performthe perturbation operation If 119901119894119896 ge 119901119894119897 modify 119909119894119897 to119909119894119897 minus Δ119901 and 119909119894119896 to 119909119894119896 + Δ119901 and vice versa Note that Δ119901is called the perturbation value used to adjust the valueof decision variables and is set as 1 in this paper After theperturbation operation we obtain a new solution 119878119896119897119894 basedon 119878119887119886119904119890

Considering that local search is relatively time consum-ing and the search process in HDGWO is guided by the

120572 120573 and 120575 solutions (wolves) we just apply the LSA tothese three solutions with a selection probability 120582 thatdetermines whether a solution is to be selected as the basesolution 119878119887119886119904119890 The pseudocode of the LSA is described inAlgorithm 3

44 Programming Procedures The pseudocode of the pro-posed algorithm is specified in Algorithm 4

5 Simulation Results and Analysis

This section aims to validate HDGWO in solving WTAproblems and includes two parts In the first part weadopt the instance in [32] an asset-based defensive WTAproblem with four weapon platforms and five targets asa benchmark case and then compare the HDGWO withthe discrete particle swarm optimization (DPSO) [13] thegenetic algorithm with greedy eugenics (GAWGE) [24] andthe adaptive immune genetic algorithm (AIGA) [32] bysolving this benchmark case so that the feasibility of theHDGWO to solve WTA problems could be demonstratedTo evaluate the solution quality we also solve the caseby the global solver of Lingo 110 a professional softwarepackage for nonlinear programming problems and use therunning result as a reference In the second part in order toexamine the scalability ofHDGWOwefirst employ the abovefour algorithms to solve ten different scale WTA problemsproduced by a test case generator and then compare theresults of all algorithms from three aspects that is thecomparison of the solution quality the comparison of the


Input Combat scenario parameters includingM NQ = [119902119894]1times119872 V = [V119895]1times119873 and E = [119890119894119895]119872times119873The first three best solutions in the tth iteration 997888rarr119878 120572(119905) 997888rarr119878 120573(119905) and 997888rarr119878 120575(119905)The control parameter a

Output The assignment scheme of 119865119894 in the (t+1)th iteration 997888rarr119904 119894(119905 + 1)Procedures Generate a random number between (01) denoted by rand

if rand le a the roulette wheel selection method

Calculate the fitness value of 997888rarr119878 120572(119905) 997888rarr119878 120573(119905) and 997888rarr119878 120575(119905) denoted by 119891120572 119891120573 and 119891120575 in turnCalculate119908119897 = 119891119897(119891120572 + 119891120573 + 119891120575) (119897 = 120572 120573 120575)Set cum=0Generate a random number between (01) denoted by r0for l=120572 120573 120575

cum= cum + 119908119897if r0 le cum997888rarr119904 119894(119905 + 1) = 997888rarr119904 119897119894(119905)

breakendif

endforelse the reassignment method

Set q max= 119902119894for j=1997888rarrN

if q maxlt=0 determine if all weapons of 119865119894 have been assignedbreak terminates the execution of the for loop ie for j=1997888rarrN

endifif E(i j))==1 119865119894 can attack 119879119895

Generate a random integer denoted by n rand between 1 and q maxAssign n rand weapons to 119879119895Update q max= q max-n rand

endifendfor

endif

Algorithm 2 Pseudo-code of the modular position update method

Input Combat scenario parameters includingM N V = [V119895]1times119873 P = [119901119894119895]119872times119873 and E = [119890119894119895]119872times119873The first three best solutions in the tth iteration 997888rarr119878 120572(119905) 997888rarr119878 120573(119905) and 997888rarr119878 120575(119905)The selection probability 120582

Output 997888rarr119878 120572(119905) 997888rarr119878 120573(119905) and 997888rarr119878 120575(119905)Procedures for ls=120572 120573 120575

Generate a random number between (01) denoted by randif rand le 120582 execute the LSA

119878119887119886119904119890 =997888rarr119878 119897119904(119905)Generate a new solution 119878119896119897119894 based on the perturbation operatorCalculate the fitness value of 119878119887119886119904119890 and 119878119896119897119894 denoted by 119891119887119886119904119890 and 119891119899119890119908 respectivelyif 119891119887119886119904119890 lt 119891119899119890119908Replace 119878119887119886119904119890 with 119878119896119897119894 119878119887119886119904119890 = 119878119896119897119894

endif

Update 997888rarr119878 119897119904(119905) = 119878119887119886119904119890endif

endfor

Algorithm 3 Pseudo-code of the local search algorithm


Input Combat scenario parameters includingM NQ = [119902119894]1times119872 V = [V119895]1times119873 P = [119901119894119895]119872times119873 E = [119890119894119895]119872times119873Necessary parameters of HDGWO including the population size119873119901 the maximum iteration number 119879max theelitist retention proportion 120578 the penalty factor vector R = [119903119895]1times119873 the perturbation value Δ 119901 and the selectionprobability 120582

Output The optimal or suboptimal assignment schemeProcedures Step 1 Set t =0

Step 2 Initialize the solutions in the first iterationStep 3 Calculate the fitness value of all solutions in the tth iteration by formula (13) and sort solutions in

descending order according to their fitness value denoted by 997888rarr119878 (1)(119905) 997888rarr119878 (2)(119905) 997888rarr119878 (119873119901)(119905)Step 4 Define three solution sets

119878119887119890119904119905(119905) = 997888rarr119878 120572(119905) 997888rarr119878 120573(119905) 997888rarr119878 120575(119905) = 997888rarr119878 (1)(119905) 997888rarr119878 (2)(119905) 997888rarr119878 (3)(119905)119878119890119897119894119904119905(119905) = 997888rarr119878 (119894)(119905) | 119894 = 1 2 sdot sdot sdot 120578119873119901 and119878120596(119905) = 997888rarr119878 (119895)(119905) | 119895 = 120578119873119901 + 1 120578119873119901 + 2 sdot sdot sdot 119873119901

Step 5 Update 119878120596(119905) to 119878120596(119905 + 1) by the modular position update methodStep 6 Perform the local search on 119878119887119890119904119905(119905) and obtain 119878119887119890119904119905(119905 + 1)Step 7 Select the first119873119901 best solutions set 119878119888119906119903119903119890119899119905(119905 + 1) as the current solutions from119878119887119890119904119905(119905 + 1) cup 119878119890119897119894119904119905(119905) cup 119878120596(119905 + 1)Step 8 Update t =t+1 If 119905 lt 119879max return to Step 3 otherwise continueStep 9 Return the solution with the best fitness value in 119878119888119906119903119903119890119899119905(119905 + 1)

Algorithm 4 Pseudo-code of HDGWO

computation time and the comprehensive comparison basedon a modified performance index (MPI) In this part we alsouse the running results of Lingo 110 after 10000 iterations asreferences All algorithms are coded inMatlab R2015a and allexperimental tests are implemented on a computer with IntelCore i7-4790 36 GHz 8 GB RAM Windows 7 operatingsystem

51 Analysis of Feasibility Although the combat scenarioin [32] is a ground-based air defensive one the WTAmodel in math is essentially the same as that introducedin Section 2 The parameters to describe the combatscenario are the number of weapon platforms is M=4the number of targets is N=5 the missile number vec-tor is Q=[4 3 2 5] the value vector of targets is V=[31 5 4 8] the engagement constraint factor matrix is E= [11 0 1 0 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1] and the lethalityprobability matrix is P= [02 06 01 09 07 08 05 0507 03 03 07 08 06 05 07 05 04 02 09] The nec-essary input parameters of the HGDWO the DPSO theGAWGE and the AIGA are shown in Table 2 Note that inTable 2 we only list part of input parameters of the DPSOthe GAWGE and the AIGA the parameters specific to thesethree algorithms (ie the inertia weight of the DPSO thecrossover probability of the GAWGE etc) are consistent withthose in the corresponding references and will not be listed inthis paper

We independently run each algorithm above for 100trials and record the maximum fitness value in each trial asillustrated in Figure 5 Then we solve the benchmark case bythe global solver of Lingo 110 After iterating 37869 times in13s the global solver found the global optimal solution asillustrated in Figure 6The corresponding objective value was20736

Table 2 The necessary input parameters of the HDGWO theDPSO the GAWGE and the AIGA

119873119901 119879max 120578 r Δ 119901 120582HDGWO 100 500 002 [1 1 1 1 1] 1 06DPSO 100 500 - [1 1 1 1 1] - -GAWGE 100 500 - [1 1 1 1 1] - -AIGA 100 500 002 [1 1 1 1 1] - -

Table 3 The reference solution (assignment scheme)

Target1 Target2 Target3 Target4 Target5Weapon Platform1 0 2 0 2 0Weapon Platform2 2 0 1 0 0Weapon Platform3 0 0 2 0 0Weapon Platform4 1 1 0 0 3

In this paper we take the solution and the correspondingobjective value obtained by Lingo 110 as the referencesolution and the reference objective value as shown in Table 3and Figure 6 respectively Then we define the successfultrials of each algorithm as the trials which can obtain thereference objective value as shown in Figure 7 Table 4 givesthree statistical indicators the average value of the maximumfitness value (AVMFV) of each algorithm for 100 trials withits standard deviation the average value of the computationtime (AVCT) of each algorithm for 100 trials with its standarddeviation and the number of the successful trials (NST)

It can be inferred from Figures 5 and 6 that the maximumfitness value is 20736 which is consistent with that in [32] Asreflected in Figure 7 and Table 4 among the four algorithmsthe proposed HDGWO wins the second largest AVMFVand NST with the least AVCT indicating that the proposed


0 20 40 60 80 100Trial NoHDGWO

0 20 40 60 80 100Trial NoDPSO

0 20 40 60 80 100Trial NoGAWGE

0 20 40 60 80 100Trial No

AIGA

20736 20736

2073620736

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074M

axim

um F

itnes

s Val

ue

Figure 5The maximum fitness value in each trial of the HDGWO the DPSO the GAWGE and the AIGA

Figure 6 The running result of the global solver of Lingo 110

Table 4 Three statistical indicators the AVMFV the AVCT and the NST

HDGWO DPSO GAWGE AIGAAVMFVplusmnstd 207356plusmn00018 207360plusmn00000 207356plusmn00018 207345plusmn00032AVCTplusmnstd (s) 23760plusmn00070 199010plusmn00290 348689plusmn01894 36253plusmn00170NST 95 100 95 81

algorithm is able to obtain a relatively good solution inthe least computation time To be specific in terms of thesolution quality the AVMFV of the HDGWO is respectively-0002 0 and 0005 higher than those of the DPSOthe GAWGE and the AIGA Here the negative sign ldquo-rdquomeans the HDGWO is inferior to the DPSO The similarsuperiority and inferiority can also be reflected in the NSTof all algorithms Furthermore regarding the computationtime the AVCT of the HDGWO is about 881 932 and345 less than those of the DPSO the GAWGE and theAIGA respectively Therefore it can be concluded that theHDGWO is narrowly inferior to the DPSO in terms of thesolution quality but has a great advantage over the DPSO in

terms of the computation time Therefore we can concludethat the HDGWO proposed in this paper achieves a bettertrade-off between the solution quality and the computationtime than the DPSO the GAWGE and the AIGA and thus isfeasible to solve small-scale WTA problems

52 Analysis of Scalability In this subsection we analyze thescalability of theHDGWOby employing it to solve large-scaleWTA problems and compare the running results with thoseof the DPSO the GAWGE and the AIGA For this purposewe first develop a test case generator to produce ten cases asintroduced as follows


HDGWO DPSO GAWGE AIGAAlgorithms

Failed TrialsSuccessful Trials

0102030405060708090

100

Tota

l No

of T

rials

Figure 7 The number of successful trials of the HDGWO theDPSO the GAWGE and the AIGA

521 Test Case Generator

(1) Generate the number of air-to-ground missiles avail-able for the ith weapon platform 119865119894 as 119902119894 =119903119886119899119889119894(119902max) 119894 = 1 2 sdot sdot sdot 119872 The function randi(x)where x is a positive integer returns a pseudorandomscalar integer between 1 and x In this paper we set119902max as 8

(2) Generate the value of the jth target 119879119895 as V119895 =119903119886119899119889119894(Vmax) 119895 = 1 2 sdot sdot sdot 119873 Here Vmax is equal to 10(3) Generate the kill probability of missiles on 119865119894 against119879119895 as 119901119894119895 = 119901119897 + (119901119906 minus 119901119897) lowast 119903119886119899119889 119894 = 1 sdot sdot sdot 119872119895 = 1 2 sdot sdot sdot 119873 The function rand returns a single

uniformly distributed randomnumber between 0 and1 and this ensures 119901119894119895 between the lower bound 119901119897 andthe upper bound 119901119906 of the lethality probability We set119901119897 = 01 and 119901119906 = 09

(4) Generate the engagement constraint factor as 119890119894119895 =[sign(119903119886119899119889 minus 119891119890) + 1]2 119894 = 1 sdot sdot sdot 119872 119895 = 1 2 sdot sdot sdot 119873The function sign(119909) returns 1 if x is a positivenumber and -1 otherwise 119891119890 denotes the probabilitythat 119890119894119895 is equal to 0 and is set as 02

Ten test cases denoted by Case 1 (M=20 N=20) Case2 (M=40 N=40) Case 3 (M=60 N=60) Case 4 (M=80N=80) Case 5 (M=100 N=100) Case 6 (M=120 N=120)Case 7 (M=140 N=140) Case 8 (M=160 N=160) Case 9(M=180 N=180) and Case 10 (M=200 N=200) are thengenerated To reduce the computation time and improve theanalysis efficiency for all algorithms the population sizethe maximum iteration number and the elitist retentionproportion are set as 100 500 and 002 respectively whilethe other parameters remain unchanged

522 Comparison of Solution Quality We execute each algo-rithm for 50 trials for each case independently Based on therunning results we then calculate the following indicators asshown in Table 5 for the comparison of the solution qualitythe average value of the maximum fitness value (AVMFV)

of each algorithm for 50 trials with its standard deviationand the task completion ratio (TCR) and the improvementratio of the solution quality (IRSQ) of one algorithm to thatof another algorithm TCR and IRSQ are defined as formula(16) and formula (17) respectively In fact in formula (16)the denominator should be the fitness value of the optimalsolution but it is too difficult to obtain for large-scale WTAproblems Therefore we use the objective bounds obtainedby the B-and-B (branch and bound) solver of Lingo 110 after10000 iterations instead denoted by Lingo ref which willnot affect the ranking of the four algorithms on the TCRindicator The objective bounds for each case are shown inTable 5 In addition it may be noted that in this paper ourpurpose is to compare the HDGWO with the DPSO theGAWGE and the AIGA so we only need to analyze the IRSQof the HDGWO

119879119862119877 = 119860119881119872119865119881119871119894119899119892119900 119903119890119891 sdot 100 (16)

1198681198781198771198761 = 119860119881119872119865119881119867119863119866119882119874 minus 119860119881119872119865119881119863119875119878119874119860119881119872119865119881119863119875119878119874 sdot 1001198681198781198771198762 = 119860119881119872119865119881119867119863119866119882119874 minus 119860119881119872119865119881119866119860119882119866119864119860119881119872119865119881119866119860119882119866119864 sdot 1001198681198781198771198763 = 119860119881119872119865119881119867119863119866119882119874 minus 119860119881119872119865119881119860119868119866119860119860119881119872119865119881119860119868119866119860 sdot 100

(17)

We can see from Table 5 that the proposed HDGWO lagsbehind the DPSO and the GAWGE and thus achieves thethird ranking on both the AVMFV and TCR indicators inall ten cases which indicates that it is inferior to the DPSOand the GAWGE but superior to the AIGA in terms of thesolution quality Then we analyze the disadvantage of theHDGWO relative to the DPSO and the GAWGE thoughISRQ1 and ISRQ2 The AVMFV of the HDGWO was only307 and 581 smaller than those of the DPSO and theGAWGE in the worst case (ie in Case 10 ISRQ1=-307ISRQ2=-581) respectively And in all ten cases the averagevalues of ISRQ1 and ISRQ2 are separately -155 and -376 indicating that the solution quality of the HDGWOis 155 and 376 worse than those of the DPSO and theGAWGE on average respectively Furthermore we find thatthe TCR indicator of HDGWO in all cases exceeds 90 andaverages 9576 which means that the solution obtained bythe HDGWO is of satisfactory quality even in Case 10 whichinvolves 200 weapon platforms and 200 targets Thereforebased on the above analysis a conclusion can be made thatthe proposed HDGWO has a relatively good scalability toprovide a satisfactory solution for large-scale WTAproblemseven though it is a little inferior to the DPSO and theGAWGE

523 Comparison of Computation Time Another indicatorfor evaluating the performance of an algorithm is related tothe computation time In this paper we calculate the averagevalue of the computation time (AVCT) of each algorithm


Table5Th

reeind

icatorso

fthe

HDGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

Lingoref

14200

21696

32297

39597

62596

67097

79395

9319

897414

1060

98

HDGWO

AVMFV

1415

521285

31296

3794

160294

63512

74625

88025

92050

9891

3(std)

(057)

(365)

(199)

(546)

(640)

(630)

(846)

(747)

(621)

(609)

TCR(

)9968

9810

9690

9582

9632

9466

9399

9445

9449

9323

ISRQ

1(

)-018

-055

-107

-164

-064

-223

-208

-215

-193

-307

ISRQ

2()

-031

-180

-291

-387

-326

-481

-525

-483

-474

-581

ISRQ

3(

)060

287

175

261

298

369

796

325

276

463

DPS

OAV

MFV

1418

121403

31634

38573

60682

6495

776212

8995

693858

102048

(std)

(003)

(039

)(025)

(060)

(110

)(088)

(148)

(170)

(189)

(180)

TCR(

)9986

9865

9795

9742

9694

9681

9600

9652

9635

9618

GAW

GE

AVMFV

1419

921675

32235

39469

62328

6671

878

760

92489

96636

1050

18(std)

(000)

(003)

(005)

(011)

(019

)(027)

(029)

(034)

(056)

(090)

TCR(

)99

99

9990

9981

9968

9957

9944

9920

9924

9920

9898

AIG

AAV

MFV

14071

20691

3075

73697

758551

61254

6912

185255

89576

94537

(std)

(077)

(381)

(180)

(667)

(806)

(701)

(711)

(878)

(724)

(1091)

TCR(

)9909

9537

9523

9338

9354

9129

8706

9148

9195

8910


for 50 trials in all ten cases and the reduction ratio of thecomputation time (RRCT) of one algorithm to that of anotheralgorithm by formula (18) as shown in Table 6 Consideringour purpose in this paper we only calculate the RRCT of theHDGWO

As reflected in Table 6 on the AVCT indicator theHDGWO is ranked second just behind the AIGA in thefirst five cases (from Case 1 to Case 5) while it achieves thefirst ranking in the last five cases (from Case 6 to Case 10)This means that in the aspect of the computation time theHDGWO is inferior to the AIGA in the first five cases butis superior to the AIGA in the last five cases and both theDPSO and the GAWGE in all cases Since the HDGWO isinferior to both the DPSO and the GAWGE in terms of thesolution quality then we mainly investigate the advantageof the HDGWO over the DPSO and the GAWGE in theaspect of the computation time in this subsection Note thata negative RRCT indicates HDGWO is less time consumingBy analyzing RRCT1 and RRCT2 we can easily find that theAVCT of the HDGWO is at least 75 smaller than thoseof the DPSO and the GAWGE and on average it is 8287and 8907 smaller respectively Thus it can be inferredthat on average the HDGWO consumes less than one-fifthof the computation time of the DPSO and one-ninth ofthe computation time of the GAWGE but only reduces thesolution quality by 155 and 376 respectively This meansthat the advantage of the HDGWO over the DPSO and theGAWGE in terms of the computation time is much moresignificant than the advantages of the DPSO and the GAWGEover the HDGWO in the aspect of the solution quality


(18)

524 Comprehensive Comparison Based on MPI In thissubsection we make a comprehensive comparison based onthe modified performance index (MPI) which is derivedfrom the performance index (PI) defined by Bharti [37] ThePI can be used to test the relative performance of differentalgorithms Later Mohan and Deep et al [38ndash41] adopted theindicator to compare the performance of algorithms in theirresearch The PI of an algorithm is calculated as follows

119875119868=

1119899119906119898 119901119899119906119898 119901sum119895=1

(1198961 119878119903119895119879119903119895 + 1198962119872119905119895119860119905119895 + 1198963119872119891119895119860119891119895 ) if 119878119903119895 gt 0

0 if 119878119903119895 = 0(19)

where 119899119906119898 119901 is the total number of problems solved bythe algorithm 119878119903119895 is the number of successful runs of thejth problem 119879119903119895 is the total number of runs of the jthproblem 119872119905119895 is the minimum of the average computationtime used by all algorithms in obtaining the solution for

jth problem 119860119905119895 is the average computation time used byan algorithm in obtaining the solution for the jth problem119872119891119895 is the minimum of the average number of functionevaluations of successful runs used by all algorithms inobtaining the solution for the jth problem 119860119891119895 is the averagenumber of function evaluations of successful runs used bythe algorithms in obtaining the solution for the jth problemand 1198961 1198962 and 1198963 (1198961 + 1198962 + 1198963 = 1 and 0 le 1198961 1198962 1198963 le1) are the weights assigned to the percentage of successthe computation time and the average number of functionevaluations of successful runs respectively

Since it is too difficult to obtain the optimal solutionsfor large-scale WTA problems the number of successfulruns of the jth problem 119878119903119895 is equal to 0 indicating that PIis equal to 0 Therefore the original PI cannot be directlyused to compare the performance of the four algorithmsConsidering that the performance of an algorithm needs tobe comprehensively evaluated in terms of both the solutionquality and the computation time we modify the originalPI from the perspective of multiattribute decision makingwhere the solution quality is a benefit-type attribute and thecomputation time is a cost-type attribute Then we normalizethe values of the two attributes to the [0 1] interval andaggregate the normalized values in the way similar to theformula (19) The modified performance index (MPI) of theith algorithm is calculated as follows

119872119875119868119894 = 1119899119906119898 119901119899119906119898 119901sum119895=1

(1198961 119860119881119872119865119881119895119894 minus 119860119881119872119865119881minus(119860119881119872119865119881+ minus 119860119881119872119865119881minus)+ 1198962 (119860119881119862119879

+ minus 119860119881119862119879119895119894 )(119860119881119862119879+ minus 119860119881119862119879minus)) (20)

where 119860119881119872119865119881+ = max119894119860119881119872119865119881119895119894 119860119881119872119865119881minus =min119894119860119881119872119865119881119895119894 119860119881119862119879+ = max119894119860119881119862119879119895119894 119860119881119862119879minus =min119894119860119881119862119879119895119894 119899119906119898 119901 is the total number of problems solvedby the ith algorithm 119860119881119872119865119881119895119894 is the 119860119881119872119865119881 of the ithalgorithm in solving the jth problem 119860119881119862119879119895119894 is the 119860119881119862119879of the ith algorithm in solving the jth problem and 1198961 and 1198962are the weights of the solution quality and the computationtime satisfying 1198961 + 1198962 = 1 and 0 le 1198961 1198962 le 1 Note that119860119881119872119865119881 and 119860119881119862119879 are the average value of the maximumfitness value and the average value of the computation timerespectively as defined in Sections 522 and 523

We set 1198961 = 119908 and 1198962 = 1minus119908 and then evaluate the MPIsof all algorithms for119908=0 01 02 03 04 05 06 07 08 09and 1 as shown in Figure 8

It can be seen from Figure 8 that as the weight ofthe solution quality 119908 increases both the MPIs of theHDGWO and the AIGA decrease while those of the DPSOand the GAWGE increase These trends are consistent withour previous analysis The larger the weight of the solutionquality 119908 the more obvious the advantages of the DPSOand the GAWGE over the HDGWO and the AIGA in termsof the solution quality Since no algorithms can keep thefirst ranking on the MPI indicator under different 119908 thenwe further calculate the average value and the standard


Table6Th

eAVC

Tindicatorsof

theH

DGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

HDGWO

AVCT(s)

1339

2751

4458

6522

8877

11478

1415

317210

20502

24045

(std)

(002)

(002)

(01)

(019

)(008)

(012

)(030)

(044)

(038)

(045)

RRCT 1

()

-876

4-8630

-8553

-8453

-8328

-8219

-812

3-8035

-793

2-7835

RRCT 2

()

-916

1-914

5-9052

-9015

-890

0-8874

-8837

-874

6-870

6-8636

RRCT 3

()

11849

10817

6690

2725

239

-1336

-2455

-3373

-393

1-4399

DPS

OAV

CT(s)

10838

20089

30811

4214

695310

764

441

75418

87562

9912

6111

070

(std)

(005)

(005)

(007)

(008)

(013

)(015

)(026)

(039

)(040)

(041)

GAW

GE

AVCT(s)

1596

03218

947036

66213

80718

101913

121685

137222

158385

176251

(std)

(007)

(024)

(032

)(039

)(041)

(053

)(059)

(064)

(070)

(082)

AIG

AAV

CT(s)

613

1322

2671

5125

8670

13248

1875

82597

03378

24292

8(std)

(002)

(004)

(006)

(010

)(025)

(055)

(030)

(033

)(037

)(065)


1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)

01 02 03 04 05 06 07 08 09 10Weight of the solution quality (w)

Figure 8 The MPIs of the HDGWO the DPSO the GAWGE andthe AIGA

Table 7 The avg and the std of the MPI for the HDGWO theDPSO the GAWGE and the AIGA

HDGWO DPSO GAWGE AIGAavg 073 055 050 048std 017 010 033 032

deviation of the MPI for each algorithm The average valueof the MPI reflects the overall performance of the algorithmunder different weights and the standard deviation of theMPI measures the stability of the overall performance of thealgorithm as the weight changes The average value and thestandard deviation of the MPI for each algorithm are shownin Table 7

As reflected in Table 7 the average value of the MPI ofthe HDGWO is the largest and the corresponding standarddeviation is the second smallest only larger than that ofthe DPSO This means that from an overall point of viewthe HDGWO has better performance than the other threealgorithms and has the ability to maintain the performanceat a relatively stable level as the weight changes Thereforewhen the weight of the solution quality is unknown theHDGWO should be recommended first among the fouralgorithms But when the weight of the solution quality hasbeen specified the algorithm is recommended according tothe following rules the HDGWO is preferred if 0 le 119908 lt06533 otherwise the GAWGE is preferred

6 Conclusion and Future Research

In this paper we present the hybrid discrete grey wolfoptimizer (HDGWO) to effectively solve WTA problemson the basis of the original grey wolf optimizer (GWO)whichwas developed only for optimizing the problems with acontinuous solution space TomakeGWOavailable we adoptthe decimal integer encoding method to represent solutionsand propose the modular position update method to updatesolutions in the discrete solution space Thus we obtain the

discrete greywolf optimizer (DGWO) and then by combiningit with the local search algorithm (LSA) we acquire theHDGWO To evaluate the feasibility of HDGWO in solvingWTA problems we first employ it to solve the benchmarkcase in [32] and compare the running results with those of theDPSO the GAWGE and the AIGA To analyze the scalabilityof the HDGWO we compare three kinds of indicators thatrespectively measure the solution quality the computationtime and the comprehensive performance of algorithmsbased on the running results of the HDGWO the DPSO theGAWGE and the AIGA in solving ten different scale WTAproblems produced by the test case generatorThe simulationresults prove the feasibility of the HDGWO in solving small-scale WTA problems and demonstrate the scalability of theHDGWO in solving large-scale WTA problems In additionthe rules for selecting the best one among the four algorithmsare also provided based on the comparison of the modifiedperformance index

We only implement the proposed HDGWO for the staticWTA problems in the current work In future research wewill concentrate on applyingHDGWOto solvemore complexWTA problems such as the dynamic problems and considermore constraints in modeling such as the necessary damagedegree to targets dynamic engagement feasibility factors etcFurthermore we will also consider to employ the HDGWOto optimize similar resource assignment problems in otherareas such as industrial production and transportation etc

Data Availability

Thezip data used to support the findings of this study havebeen deposited in the figshare repository ([httpsfigsharecomarticlesData for HDGWO zip6489926])

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Jun Wang and Pengcheng Luo conceived and designedthe algorithm and experiments Jun Wang performed theexperiments Jun Wang Xinwu Hu and Xiaonan Zhanganalyzed the data Jun Wang wrote the paper

Acknowledgments

The authors disclosed receipt of the following financialsupport for the research authorship andor publication ofthis article this work was supported by theMilitary ScientificResearch Project (15010531150XXXX)

References

[1] A S Manne ldquoA target-assignment problemrdquo OperationsResearch vol 6 pp 346ndash351 1958


[2] S P Lloyd and H S Witsenhausen ldquoWeapon allocation isNP-completerdquo in Proceedings of the IEEE Summer SimulationConference Reno USA 1986

[3] Z R Bogdanowicz and N P Coleman ldquoSensor-target andweapon-target pairings based on auction algorithmrdquo in Pro-ceedings of the 11th WSEAS International Conference on AppliedMathematics pp 92ndash96 Dallas TX USA 2007

[4] Z R Bogdanowicz ldquoA new efficient algorithm for optimalassignment of smart weapons to targetsrdquo Computers amp Math-ematics with Applications vol 58 no 4 pp 1965ndash1969 2009

[5] L Zi-fen L Xiang-min D Jin-jin C Jin-zhu and Z Feng-xialdquoSensor-weapon-target assignment based on improved SWT-opt algorithmrdquo in Proceedings of the 2011 IEEE 2nd InternationalConference on Computing Control and Industrial Engineering(CCIE 2011) pp 25ndash28 Wuhan China August 2011

[6] Z F Li X M Li and J J Dai ldquoSwt-Opt Algorithm for SolvingFormation Air to Ground Joint Fire Distribution ProblemrdquoModern Defense Technology vol 40 no 4 pp 113-114 2012

[7] Z F Li XM Li and J Z Chen ldquoDynamic joint fire distributionmethod based on decentralized cooperative auction algorithmrdquoFire Control ampCommand Control vol 37 no 11 pp 49ndash52 2012

[8] H Chen Z Liu Y Sun andY Li ldquoParticle SwarmOptimizationBased on Genetic Operators for Sensor-Weapon-Target Assign-mentrdquo in Proceedings of the 2012 5th International Symposiumon Computational Intelligence and Design (ISCID) pp 170ndash173Hangzhou China October 2012

[9] L Mu X Qu and P Wang ldquoApplication of SensorWeapon-Target Assignment based on Multi-Scale Quantum HarmonicOscillator Algorithmrdquo in Proceedings of the 2017 2nd Interna-tional Conference on Image Vision and Computing (ICIVC) pp1147ndash1151 Chengdu China June 2017

[10] B Xin Y P Wang and J Chen ldquoAn EfficientMarginal-Return-Based Constructive Heuristic to Solve the SensorWeaponTargetAssignment Problemrdquo IEEETransactionsOnSystemsManAndCybernetics Systems

[11] H Naeem and A Masood ldquoAn optimal dynamic threat eval-uation and weapon scheduling techniquerdquo Knowledge-BasedSystems vol 23 no 4 pp 337ndash342 2010

[12] D K Ahner and C R Parson ldquoOptimal multi-stage allocationof weapons to targets using adaptive dynamic programmingrdquoOptimization Letters vol 9 no 8 pp 1689ndash1701 2015

[13] Y Zhou X Li Y Zhu and W Wang ldquoA discrete particleswarm optimization algorithm applied in constrained staticweapon-target assignment problemrdquo in Proceedings of the 201612th World Congress on Intelligent Control and Automation(WCICA) pp 3118ndash3123 Guilin China June 2016

[14] M Ni Z Yu FMa and XWu ldquoA Lagrange RelaxationMethodfor Solving Weapon-Target Assignment Problemrdquo Mathemat-ical Problems in Engineering vol 2011 Article ID 873292 10pages 2011

[15] Z R Bogdanowicz A Tolano K Patel and N P ColemanldquoOptimization ofweapon-target pairings based on kill probabil-itiesrdquo IEEE Transactions on Cybernetics vol 43 no 6 pp 1835ndash1844 2013

[16] O Karasakal ldquoAir defense missile-target allocation models fora naval task grouprdquo Computers amp Operations Research vol 35no 2 pp 1759ndash1770 2008

[17] N Dirik S N Hall and J T Moore ldquoMaximizing strikeaircraft planning efficiency for a given class of ground targetsrdquoOptimization Letters vol 9 no 8 pp 1729ndash1748 2015

[18] Y Yan Y Zha L Qin and K Xu ldquoA research on weapon-target assignment based on combat capabilitiesrdquo in Proceedingsof the 2016 IEEE International Conference on Mechatronicsand Automation pp 2403ndash2407 Harbin Heilongjiang ChinaAugust 2016

[19] M A Sahin and K Leblebicioglu ldquoApproximating the optimalmapping for weapon target assignment by fuzzy reasoningrdquoInformation Sciences vol 255 pp 30ndash44 2014

[20] R K Ahuja A Kumar K C Jha and J B Orlin ldquoExactand heuristic algorithms for the weapon-target assignmentproblemrdquo Operations Research vol 55 no 6 pp 1136ndash11462007

[21] Y-H Cha and Y-D Kim ldquoFire scheduling for planned artilleryattack operations under time-dependent destruction probabili-tiesrdquo Omega vol 38 no 5 pp 383ndash392 2010

[22] O Kwon K Lee D Kang and S Park ldquoA branch-and-pricealgorithm for a targeting problemrdquo Naval Research Logistics(NRL) vol 54 no 7 pp 732ndash741 2007

[23] H Cai J Liu Y Chen and H Wang ldquoSurvey of the researchon dynamic weapon-target assignment problemrdquo Journal ofSystems Engineering and Electronics vol 17 no 3 pp 559ndash5652006

[24] Z-J Lee S-F Su and C-Y Lee ldquoEfficiently solving generalweapon-target assignment problem by genetic algorithms withgreedy eugenicsrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 1 pp 113ndash121 2003

[25] J Chen B Xin Z H Peng L H Dou and J Zhang ldquoEvo-lutionary decision-makings for the dynamic weapon-targetassignment problemrdquo Science in China Series F InformationSciences vol 52 no 11 pp 2006ndash2018 2009

[26] Y M Lu W Q Miao and M Li ldquoThe air defense missileoptimum target assignment based on the improved geneticalgorithmrdquo Journal of Theoretical and Applied InformationTechnology vol 48 pp 809ndash816 2013

[27] H Liang and F Kang ldquoAdaptive chaos parallel clonal selectionalgorithm for objective optimization in WTA applicationrdquoOptik - International Journal for Light and Electron Optics vol127 no 6 pp 3459ndash3465 2016

[28] Y Wang W G Zhang and Y Li ldquoAn efficient clonal selectionalgorithm to solve dynamic weapon-target assignment gamemodel in UAV cooperative aerial combatrdquo in Proceedings ofthe 35th Chinese Control Conference (CCC rsquo16) pp 9578ndash9581IEEE Chengdu China July 2016

[29] Y Wang J Li W Huang and T Wen ldquoDynamic weapon targetassignment based on intuitionistic fuzzy entropy of discreteparticle swarmrdquo China Communications vol 14 no 1 pp 169ndash179 2017

[30] B Xin J Chen J Zhang LDou andZ Peng ldquoEfficient decisionmakings for dynamic weapon-target assignment by virtualpermutation and tabu search heuristicsrdquo IEEE Transactionson Systems Man and Cybernetics Part C Applications andReviews vol 40 no 6 pp 649ndash662 2010

[31] B Xin J Chen Z Peng L Dou and J Zhang ldquoAn efficient rule-based constructive heuristic to solve dynamic weapon-targetassignment problemrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 41 no 3 pp 598ndash606 2011

[32] S Yang M Yang S Wang and K Huang ldquoAdaptive immunegenetic algorithm for weapon system portfolio optimization inmilitary big data environmentrdquo Cluster Computing vol 19 no3 pp 1359ndash1372 2016

[33] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014


[34] S Zhang Y Zhou Z Li and W Pan ldquoGrey Wolf optimizer forunmanned combat aerial vehicle path planningrdquo Advances inEngineering Software vol 99 pp 121ndash136 2016

[35] C Lu S Xiao X Li and L Gao ldquoAn effective multi-objectivediscrete grey Wolf optimizer for a real-world scheduling prob-lem in welding productionrdquo Advances in Engineering Softwarevol 99 pp 161ndash176 2016

[36] Y-C Liang and C-Y Chuang ldquoVariable neighborhood searchfor multi-objective resource allocation problemsrdquo Robotics andComputer-Integrated Manufacturing vol 29 no 3 pp 73ndash782013

[37] Bharti Controlled random search technique and their applica-tions [PhD thesis] Department of Mathematics University ofRoorkee Roorkee India 1994

[38] C Mohan and H T Nguyen ldquoA controlled random searchtechnique incorporating the simulated annealing concept forsolving integer and mixed integer global optimization prob-lemsrdquoComputational Optimization andApplications vol 14 no1 pp 103ndash132 1999

[39] K Deep and M Thakur ldquoA new mutation operator for realcoded genetic algorithmsrdquo Applied Mathematics and Computa-tion vol 193 no 1 pp 211ndash230 2007

[40] K Deep and M Thakur ldquoA new crossover operator for realcoded genetic algorithmsrdquo Applied Mathematics and Computa-tion vol 188 no 1 pp 895ndash911 2007

[41] S Gupta and K Deep ldquoA novel Random Walk Grey WolfOptimizerrdquo Swarm and Evolutionary Computation 2018

Hindawiwwwhindawicom Volume 2018

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by at most one sensor-weapon pair Furthermore throughthe sensorweapontarget augmentation they translated thederived problem into a symmetric optimization problemwithan input consisting of the same numbers of sensors weaponsand targets along with the benefit matrix and presentedthe Swt-opt algorithm derived from the auction algorithmto optimally assign sensors and weapons to targets Sincethen the S-WTA problem has been successively studied bysome scholars Li et al [5 6] proposed an improved Swt-opt algorithm to solve the same problem by combining theconsensus algorithm so that the shortcoming of the Swt-opt algorithm ie highly depending on perfect networktopologies can be overcome Later they put forward adecentralized cooperative auction algorithm for a similar S-WTA problemwhere the number of targets that can be struckvaried in different operational phases [7] Chen et al [8]proposed a particle swarm optimization based on geneticoperators to solve the S-WTAproblemwhere each sensor canguide only one weapon once and each target can be engagedbymultiple weapons onceMu et al [9] proposed amultiscalequantum harmonic oscillator algorithm to solve the S-WTAproblem in intelligent minefields considering the probabilityof detection and killing Xin et al [10] constructed an efficientmarginal-return-based heuristic to solve the S-WTAproblemconsidering the situation where multiple sensorsweaponscan be assigned to one target but each sensor can detect onlyone target at the same time and each weapon can shoot onlyone target simultaneously The proposed heuristic exploitedthe marginal return of each sensor-weapon-target triplet anddynamically updated the threat value of all targets It reliedonly on simple lookup operations to choose each assignmenttriplet thus resulting in very low computational complexity

Both the conventional WTA model and the S-WTAmodel can be classified as follows For ease of presentationthe two models will not be distinguished both called theWTA model A WTA model is called to be dynamic [10ndash12]if the operational process is time related This indicates thereare always several operational stages or new situations usuallyarise during operation Otherwise the model is static [3ndash6 8 9 13ndash16] If two or more objectives such as maximizingthe damage expectancy of hostile targets and minimizing theconsumption of own ammunition and mission completiontime are considered the WTA model is multiobjective [1718] If there is only one objective it is single-objective [3ndash11 19] Depending on the type of combat scenario theWTA model may be asset-based [11] or target-based [17] Indefensive missions the asset-based model is often establishedto minimize the loss expectancy of own-force assets while inoffensive missions the target-based model is always adoptedto maximize the total damage expectancy of hostile targetsIn addition to modeling another research aspect for WTAis to develop algorithms to solve the problem optimally inthe least possible time These algorithms can be generallydivided into two categories exact algorithms and heuristicalgorithms Exact algorithms are developed according to thespecific mathematical properties of WTA problems and cangradually eliminate the nonlinearity of the problem throughtransformation decomposition and other processing means[16] By this way the model is translated to a linear one

and then classical operations research methods can applysuch as the dynamic programming method [12] the branchand bound method [20 21] the branch and price method[22] the mixed integer linear program (MILP) algorithm[17] and the Lagrange relaxationmethod [14]Thesemethodshave demonstrated the feasibility and effectiveness on solvingstatic and dynamic WTA problems but may become difficultto apply when a large number of weapons and targets areinvolved [23]

With the rapid development of heuristic algorithmsmorecomplex WTA problems are able to be well solved Mostresearch to date on solving WTA problems by heuristicalgorithms either constructs some specific search rules basedon the properties of the problem to achieve solutions rapidlyor introduces some local searchmechanisms into the originalalgorithms to improve the solution quality These algorithmsincluding auction algorithms [3ndash6 15] improved geneticalgorithms [18 24ndash26] clonal selection algorithms [27 28]particle swarm algorithms [8 13 29] tabu search algorithms[30] rule-based constructive heuristic algorithms [10 31]and other intelligent optimization algorithms have shownevident advantages over traditional methods in terms ofcomputation time and solution accuracy and however stillsuffer from some drawbacks such as easily falling into pre-mature convergence and local optimum [32] Furthermoreconsidering the variability of the battlefield environmentdecisions always need to be made immediately that is WTAproblems need to be resolved in a very short time Thereforewe propose in this paper a hybrid discrete greywolf optimizer(HDGWO) which is an integration of a discrete grey wolfoptimizer (DGWO) with a local search algorithm (LSA)Moreover in order to enhance the efficiency of the algorithmwe introduce the specific domain knowledge into the encodemethods and the LSA

The rest of this paper is organized as follows The math-ematical model of a typical WTA problem for an offensivemission is formulated in Section 2 Section 3 is a briefintroduction to the original grey wolf optimizer (GWO)presented in [33] and Section 4 is a detailed introductionto the proposed HDGWO In Section 5 we first analyze thefeasibility of HDGWO in solving small-scale WTA problemsby comparing itwith the discrete particle swarmoptimization(DPSO) [13] the genetic algorithm with greedy eugenics(GAWGE) [24] and the adaptive immune genetic algorithm(AIGA) [32] and then investigate the scalability of HDGWOby solving ten different scale WTA problems Finally wemake a conclusion and look ahead to the future research inSection 6

2 Problem Formulation

In general a typical WTA problem for an offensive missioncan be formulated as the following nonlinear integer pro-gramming model [13 32]

max 119891 (119909)= 119873sum119895=1

V119895 [1 minus 119872prod119894=1

(1 minus 119901119894119895 sdot 119890119894119895)119909119894119895]






+ 119873sum119895=1

119903119895 sdot 119866119895 (119909) (1)

subject to

119873sum119895=1

119909119894119895 le 119902119894 119894 = 1 2 sdot sdot sdot 119872 (2)

119872sum119894=1

119909119894119895 ge 1 119895 = 1 2 sdot sdot sdot 119873 (3)

119909119894119895 isin 119873119909119894119895 ge 0











a1

C1

a3

C3

Dalpℎa

Ddelta

Moe

R

Dbeta

a2

C2






(10)


(11)






Solution

Weapon platforms

Targets

F1 F2 FM
















119891 (119878) = 119873sum119895=1

V119895 [1 minus 119872prod119894=1

(1 minus 119901119894119895 sdot 119890119894119895)119909119894119895(119878)]

+ 119873sum119895=1

119903119895 sdot 119866119895 (119878)(13)










endifendfor

endforendfor




(14)





119886 = 1 minus 119905119879max(15)




reassign

roulettewheel

selection

rand le a

rand gt a

rarrS

(t)

rarrS

(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs i


rarrsM(t)


xM2 xMN





rarrS

(t)

rarrS(t+1)

rarrS(t)

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot



middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot






Sbase

Skli

xi1

xi1

xil

xil minusΔp

+Δp

middotmiddotmiddot


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

xik

xik +Δp xiN

xiN





Fi















breakendif






endifendfor

endif






endif


endfor





















0 20 40 60 80 100Trial NoDPSO


0 20 40 60 80 100Trial No

AIGA

20736 20736

2073620736

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074M

axim

um F

itnes

s Val

ue











0102030405060708090

100

Tota

l No

of T

rials










119879119862119877 = 119860119881119872119865119881119871119894119899119892119900 119903119890119891 sdot 100 (16)


(17)




Table5Th

reeind

icatorso

fthe

HDGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

Lingoref

14200

21696

32297

39597

62596

67097

79395

9319

897414

1060

98

HDGWO

AVMFV

1415

521285

31296

3794

160294

63512

74625

88025

92050

9891

3(std)

(057)

(365)

(199)

(546)

(640)

(630)

(846)

(747)

(621)

(609)

TCR(

)9968

9810

9690

9582

9632

9466

9399

9445

9449

9323

ISRQ

1(

)-018

-055

-107

-164

-064

-223

-208

-215

-193

-307

ISRQ

2()

-031

-180

-291

-387

-326

-481

-525

-483

-474

-581

ISRQ

3(

)060

287

175

261

298

369

796

325

276

463

DPS

OAV

MFV

1418

121403

31634

38573

60682

6495

776212

8995

693858

102048

(std)

(003)

(039

)(025)

(060)

(110

)(088)

(148)

(170)

(189)

(180)

TCR(

)9986

9865

9795

9742

9694

9681

9600

9652

9635

9618

GAW

GE

AVMFV

1419

921675

32235

39469

62328

6671

878

760

92489

96636

1050

18(std)

(000)

(003)

(005)

(011)

(019

)(027)

(029)

(034)

(056)

(090)

TCR(

)99

99

9990

9981

9968

9957

9944

9920

9924

9920

9898

AIG

AAV

MFV

14071

20691

3075

73697

758551

61254

6912

185255

89576

94537

(std)

(077)

(381)

(180)

(667)

(806)

(701)

(711)

(878)

(724)

(1091)

TCR(

)9909

9537

9523

9338

9354

9129

8706

9148

9195

8910





(18)


119875119868=

1119899119906119898 119901119899119906119898 119901sum119895=1

(1198961 119878119903119895119879119903119895 + 1198962119872119905119895119860119905119895 + 1198963119872119891119895119860119891119895 ) if 119878119903119895 gt 0

0 if 119878119903119895 = 0(19)




119872119875119868119894 = 1119899119906119898 119901119899119906119898 119901sum119895=1







Table6Th

eAVC

Tindicatorsof

theH

DGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

HDGWO

AVCT(s)

1339

2751

4458

6522

8877

11478

1415

317210

20502

24045

(std)

(002)

(002)

(01)

(019

)(008)

(012

)(030)

(044)

(038)

(045)

RRCT 1

()

-876

4-8630

-8553

-8453

-8328

-8219

-812

3-8035

-793

2-7835

RRCT 2

()

-916

1-914

5-9052

-9015

-890

0-8874

-8837

-874

6-870

6-8636

RRCT 3

()

11849

10817

6690

2725

239

-1336

-2455

-3373

-393

1-4399

DPS

OAV

CT(s)

10838

20089

30811

4214

695310

764

441

75418

87562

9912

6111

070

(std)

(005)

(005)

(007)

(008)

(013

)(015

)(026)

(039

)(040)

(041)

GAW

GE

AVCT(s)

1596

03218

947036

66213

80718

101913

121685

137222

158385

176251

(std)

(007)

(024)

(032

)(039

)(041)

(053

)(059)

(064)

(070)

(082)

AIG

AAV

CT(s)

613

1322

2671

5125

8670

13248

1875

82597

03378

24292

8(std)

(002)

(004)

(006)

(010

)(025)

(055)

(030)

(033

)(037

)(065)


1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)











Data Availability






Acknowledgments


References



















































Journal of












Journal of







Volume 2018






Volume 2018













+ 119873sum119895=1

119903119895 sdot 119866119895 (119909) (1)

subject to

119873sum119895=1

119909119894119895 le 119902119894 119894 = 1 2 sdot sdot sdot 119872 (2)

119872sum119894=1

119909119894119895 ge 1 119895 = 1 2 sdot sdot sdot 119873 (3)

119909119894119895 isin 119873119909119894119895 ge 0











a1

C1

a3

C3

Dalpℎa

Ddelta

Moe

R

Dbeta

a2

C2






(10)


(11)






Solution

Weapon platforms

Targets

F1 F2 FM
















119891 (119878) = 119873sum119895=1

V119895 [1 minus 119872prod119894=1

(1 minus 119901119894119895 sdot 119890119894119895)119909119894119895(119878)]

+ 119873sum119895=1

119903119895 sdot 119866119895 (119878)(13)










endifendfor

endforendfor




(14)





119886 = 1 minus 119905119879max(15)




reassign

roulettewheel

selection

rand le a

rand gt a

rarrS

(t)

rarrS

(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs i


rarrsM(t)


xM2 xMN





rarrS

(t)

rarrS(t+1)

rarrS(t)

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot



middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot






Sbase

Skli

xi1

xi1

xil

xil minusΔp

+Δp

middotmiddotmiddot


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

xik

xik +Δp xiN

xiN





Fi















breakendif






endifendfor

endif






endif


endfor





















0 20 40 60 80 100Trial NoDPSO


0 20 40 60 80 100Trial No

AIGA

20736 20736

2073620736

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074M

axim

um F

itnes

s Val

ue











0102030405060708090

100

Tota

l No

of T

rials










119879119862119877 = 119860119881119872119865119881119871119894119899119892119900 119903119890119891 sdot 100 (16)


(17)




Table5Th

reeind

icatorso

fthe

HDGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

Lingoref

14200

21696

32297

39597

62596

67097

79395

9319

897414

1060

98

HDGWO

AVMFV

1415

521285

31296

3794

160294

63512

74625

88025

92050

9891

3(std)

(057)

(365)

(199)

(546)

(640)

(630)

(846)

(747)

(621)

(609)

TCR(

)9968

9810

9690

9582

9632

9466

9399

9445

9449

9323

ISRQ

1(

)-018

-055

-107

-164

-064

-223

-208

-215

-193

-307

ISRQ

2()

-031

-180

-291

-387

-326

-481

-525

-483

-474

-581

ISRQ

3(

)060

287

175

261

298

369

796

325

276

463

DPS

OAV

MFV

1418

121403

31634

38573

60682

6495

776212

8995

693858

102048

(std)

(003)

(039

)(025)

(060)

(110

)(088)

(148)

(170)

(189)

(180)

TCR(

)9986

9865

9795

9742

9694

9681

9600

9652

9635

9618

GAW

GE

AVMFV

1419

921675

32235

39469

62328

6671

878

760

92489

96636

1050

18(std)

(000)

(003)

(005)

(011)

(019

)(027)

(029)

(034)

(056)

(090)

TCR(

)99

99

9990

9981

9968

9957

9944

9920

9924

9920

9898

AIG

AAV

MFV

14071

20691

3075

73697

758551

61254

6912

185255

89576

94537

(std)

(077)

(381)

(180)

(667)

(806)

(701)

(711)

(878)

(724)

(1091)

TCR(

)9909

9537

9523

9338

9354

9129

8706

9148

9195

8910





(18)


119875119868=

1119899119906119898 119901119899119906119898 119901sum119895=1

(1198961 119878119903119895119879119903119895 + 1198962119872119905119895119860119905119895 + 1198963119872119891119895119860119891119895 ) if 119878119903119895 gt 0

0 if 119878119903119895 = 0(19)




119872119875119868119894 = 1119899119906119898 119901119899119906119898 119901sum119895=1







Table6Th

eAVC

Tindicatorsof

theH

DGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

HDGWO

AVCT(s)

1339

2751

4458

6522

8877

11478

1415

317210

20502

24045

(std)

(002)

(002)

(01)

(019

)(008)

(012

)(030)

(044)

(038)

(045)

RRCT 1

()

-876

4-8630

-8553

-8453

-8328

-8219

-812

3-8035

-793

2-7835

RRCT 2

()

-916

1-914

5-9052

-9015

-890

0-8874

-8837

-874

6-870

6-8636

RRCT 3

()

11849

10817

6690

2725

239

-1336

-2455

-3373

-393

1-4399

DPS

OAV

CT(s)

10838

20089

30811

4214

695310

764

441

75418

87562

9912

6111

070

(std)

(005)

(005)

(007)

(008)

(013

)(015

)(026)

(039

)(040)

(041)

GAW

GE

AVCT(s)

1596

03218

947036

66213

80718

101913

121685

137222

158385

176251

(std)

(007)

(024)

(032

)(039

)(041)

(053

)(059)

(064)

(070)

(082)

AIG

AAV

CT(s)

613

1322

2671

5125

8670

13248

1875

82597

03378

24292

8(std)

(002)

(004)

(006)

(010

)(025)

(055)

(030)

(033

)(037

)(065)


1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)











Data Availability






Acknowledgments


References



















































Journal of












Journal of







Volume 2018






Volume 2018









a1

C1

a3

C3

Dalpℎa

Ddelta

Moe

R

Dbeta

a2

C2






(10)


(11)






Solution

Weapon platforms

Targets

F1 F2 FM
















119891 (119878) = 119873sum119895=1

V119895 [1 minus 119872prod119894=1

(1 minus 119901119894119895 sdot 119890119894119895)119909119894119895(119878)]

+ 119873sum119895=1

119903119895 sdot 119866119895 (119878)(13)










endifendfor

endforendfor




(14)





119886 = 1 minus 119905119879max(15)




reassign

roulettewheel

selection

rand le a

rand gt a

rarrS

(t)

rarrS

(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs i


rarrsM(t)


xM2 xMN





rarrS

(t)

rarrS(t+1)

rarrS(t)

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot



middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot






Sbase

Skli

xi1

xi1

xil

xil minusΔp

+Δp

middotmiddotmiddot


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

xik

xik +Δp xiN

xiN





Fi















breakendif






endifendfor

endif






endif


endfor





















0 20 40 60 80 100Trial NoDPSO


0 20 40 60 80 100Trial No

AIGA

20736 20736

2073620736

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074M

axim

um F

itnes

s Val

ue











0102030405060708090

100

Tota

l No

of T

rials










119879119862119877 = 119860119881119872119865119881119871119894119899119892119900 119903119890119891 sdot 100 (16)


(17)




Table5Th

reeind

icatorso

fthe

HDGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

Lingoref

14200

21696

32297

39597

62596

67097

79395

9319

897414

1060

98

HDGWO

AVMFV

1415

521285

31296

3794

160294

63512

74625

88025

92050

9891

3(std)

(057)

(365)

(199)

(546)

(640)

(630)

(846)

(747)

(621)

(609)

TCR(

)9968

9810

9690

9582

9632

9466

9399

9445

9449

9323

ISRQ

1(

)-018

-055

-107

-164

-064

-223

-208

-215

-193

-307

ISRQ

2()

-031

-180

-291

-387

-326

-481

-525

-483

-474

-581

ISRQ

3(

)060

287

175

261

298

369

796

325

276

463

DPS

OAV

MFV

1418

121403

31634

38573

60682

6495

776212

8995

693858

102048

(std)

(003)

(039

)(025)

(060)

(110

)(088)

(148)

(170)

(189)

(180)

TCR(

)9986

9865

9795

9742

9694

9681

9600

9652

9635

9618

GAW

GE

AVMFV

1419

921675

32235

39469

62328

6671

878

760

92489

96636

1050

18(std)

(000)

(003)

(005)

(011)

(019

)(027)

(029)

(034)

(056)

(090)

TCR(

)99

99

9990

9981

9968

9957

9944

9920

9924

9920

9898

AIG

AAV

MFV

14071

20691

3075

73697

758551

61254

6912

185255

89576

94537

(std)

(077)

(381)

(180)

(667)

(806)

(701)

(711)

(878)

(724)

(1091)

TCR(

)9909

9537

9523

9338

9354

9129

8706

9148

9195

8910





(18)


119875119868=

1119899119906119898 119901119899119906119898 119901sum119895=1

(1198961 119878119903119895119879119903119895 + 1198962119872119905119895119860119905119895 + 1198963119872119891119895119860119891119895 ) if 119878119903119895 gt 0

0 if 119878119903119895 = 0(19)




119872119875119868119894 = 1119899119906119898 119901119899119906119898 119901sum119895=1







Table6Th

eAVC

Tindicatorsof

theH

DGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

HDGWO

AVCT(s)

1339

2751

4458

6522

8877

11478

1415

317210

20502

24045

(std)

(002)

(002)

(01)

(019

)(008)

(012

)(030)

(044)

(038)

(045)

RRCT 1

()

-876

4-8630

-8553

-8453

-8328

-8219

-812

3-8035

-793

2-7835

RRCT 2

()

-916

1-914

5-9052

-9015

-890

0-8874

-8837

-874

6-870

6-8636

RRCT 3

()

11849

10817

6690

2725

239

-1336

-2455

-3373

-393

1-4399

DPS

OAV

CT(s)

10838

20089

30811

4214

695310

764

441

75418

87562

9912

6111

070

(std)

(005)

(005)

(007)

(008)

(013

)(015

)(026)

(039

)(040)

(041)

GAW

GE

AVCT(s)

1596

03218

947036

66213

80718

101913

121685

137222

158385

176251

(std)

(007)

(024)

(032

)(039

)(041)

(053

)(059)

(064)

(070)

(082)

AIG

AAV

CT(s)

613

1322

2671

5125

8670

13248

1875

82597

03378

24292

8(std)

(002)

(004)

(006)

(010

)(025)

(055)

(030)

(033

)(037

)(065)


1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)











Data Availability






Acknowledgments


References



















































Journal of












Journal of







Volume 2018






Volume 2018









Solution

Weapon platforms

Targets

F1 F2 FM
















119891 (119878) = 119873sum119895=1

V119895 [1 minus 119872prod119894=1

(1 minus 119901119894119895 sdot 119890119894119895)119909119894119895(119878)]

+ 119873sum119895=1

119903119895 sdot 119866119895 (119878)(13)










endifendfor

endforendfor




(14)





119886 = 1 minus 119905119879max(15)




reassign

roulettewheel

selection

rand le a

rand gt a

rarrS

(t)

rarrS

(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs i


rarrsM(t)


xM2 xMN





rarrS

(t)

rarrS(t+1)

rarrS(t)

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot



middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot






Sbase

Skli

xi1

xi1

xil

xil minusΔp

+Δp

middotmiddotmiddot


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

xik

xik +Δp xiN

xiN





Fi















breakendif






endifendfor

endif






endif


endfor





















0 20 40 60 80 100Trial NoDPSO


0 20 40 60 80 100Trial No

AIGA

20736 20736

2073620736

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074M

axim

um F

itnes

s Val

ue











0102030405060708090

100

Tota

l No

of T

rials










119879119862119877 = 119860119881119872119865119881119871119894119899119892119900 119903119890119891 sdot 100 (16)


(17)




Table5Th

reeind

icatorso

fthe

HDGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

Lingoref

14200

21696

32297

39597

62596

67097

79395

9319

897414

1060

98

HDGWO

AVMFV

1415

521285

31296

3794

160294

63512

74625

88025

92050

9891

3(std)

(057)

(365)

(199)

(546)

(640)

(630)

(846)

(747)

(621)

(609)

TCR(

)9968

9810

9690

9582

9632

9466

9399

9445

9449

9323

ISRQ

1(

)-018

-055

-107

-164

-064

-223

-208

-215

-193

-307

ISRQ

2()

-031

-180

-291

-387

-326

-481

-525

-483

-474

-581

ISRQ

3(

)060

287

175

261

298

369

796

325

276

463

DPS

OAV

MFV

1418

121403

31634

38573

60682

6495

776212

8995

693858

102048

(std)

(003)

(039

)(025)

(060)

(110

)(088)

(148)

(170)

(189)

(180)

TCR(

)9986

9865

9795

9742

9694

9681

9600

9652

9635

9618

GAW

GE

AVMFV

1419

921675

32235

39469

62328

6671

878

760

92489

96636

1050

18(std)

(000)

(003)

(005)

(011)

(019

)(027)

(029)

(034)

(056)

(090)

TCR(

)99

99

9990

9981

9968

9957

9944

9920

9924

9920

9898

AIG

AAV

MFV

14071

20691

3075

73697

758551

61254

6912

185255

89576

94537

(std)

(077)

(381)

(180)

(667)

(806)

(701)

(711)

(878)

(724)

(1091)

TCR(

)9909

9537

9523

9338

9354

9129

8706

9148

9195

8910





(18)


119875119868=

1119899119906119898 119901119899119906119898 119901sum119895=1

(1198961 119878119903119895119879119903119895 + 1198962119872119905119895119860119905119895 + 1198963119872119891119895119860119891119895 ) if 119878119903119895 gt 0

0 if 119878119903119895 = 0(19)




119872119875119868119894 = 1119899119906119898 119901119899119906119898 119901sum119895=1







Table6Th

eAVC

Tindicatorsof

theH

DGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

HDGWO

AVCT(s)

1339

2751

4458

6522

8877

11478

1415

317210

20502

24045

(std)

(002)

(002)

(01)

(019

)(008)

(012

)(030)

(044)

(038)

(045)

RRCT 1

()

-876

4-8630

-8553

-8453

-8328

-8219

-812

3-8035

-793

2-7835

RRCT 2

()

-916

1-914

5-9052

-9015

-890

0-8874

-8837

-874

6-870

6-8636

RRCT 3

()

11849

10817

6690

2725

239

-1336

-2455

-3373

-393

1-4399

DPS

OAV

CT(s)

10838

20089

30811

4214

695310

764

441

75418

87562

9912

6111

070

(std)

(005)

(005)

(007)

(008)

(013

)(015

)(026)

(039

)(040)

(041)

GAW

GE

AVCT(s)

1596

03218

947036

66213

80718

101913

121685

137222

158385

176251

(std)

(007)

(024)

(032

)(039

)(041)

(053

)(059)

(064)

(070)

(082)

AIG

AAV

CT(s)

613

1322

2671

5125

8670

13248

1875

82597

03378

24292

8(std)

(002)

(004)

(006)

(010

)(025)

(055)

(030)

(033

)(037

)(065)


1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)











Data Availability






Acknowledgments


References



















































Journal of












Journal of







Volume 2018






Volume 2018














endifendfor

endforendfor




(14)





119886 = 1 minus 119905119879max(15)




reassign

roulettewheel

selection

rand le a

rand gt a

rarrS

(t)

rarrS

(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs i


rarrsM(t)


xM2 xMN





rarrS

(t)

rarrS(t+1)

rarrS(t)

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot



middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot






Sbase

Skli

xi1

xi1

xil

xil minusΔp

+Δp

middotmiddotmiddot


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

xik

xik +Δp xiN

xiN





Fi















breakendif






endifendfor

endif






endif


endfor





















0 20 40 60 80 100Trial NoDPSO


0 20 40 60 80 100Trial No

AIGA

20736 20736

2073620736

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074M

axim

um F

itnes

s Val

ue











0102030405060708090

100

Tota

l No

of T

rials










119879119862119877 = 119860119881119872119865119881119871119894119899119892119900 119903119890119891 sdot 100 (16)


(17)




Table5Th

reeind

icatorso

fthe

HDGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

Lingoref

14200

21696

32297

39597

62596

67097

79395

9319

897414

1060

98

HDGWO

AVMFV

1415

521285

31296

3794

160294

63512

74625

88025

92050

9891

3(std)

(057)

(365)

(199)

(546)

(640)

(630)

(846)

(747)

(621)

(609)

TCR(

)9968

9810

9690

9582

9632

9466

9399

9445

9449

9323

ISRQ

1(

)-018

-055

-107

-164

-064

-223

-208

-215

-193

-307

ISRQ

2()

-031

-180

-291

-387

-326

-481

-525

-483

-474

-581

ISRQ

3(

)060

287

175

261

298

369

796

325

276

463

DPS

OAV

MFV

1418

121403

31634

38573

60682

6495

776212

8995

693858

102048

(std)

(003)

(039

)(025)

(060)

(110

)(088)

(148)

(170)

(189)

(180)

TCR(

)9986

9865

9795

9742

9694

9681

9600

9652

9635

9618

GAW

GE

AVMFV

1419

921675

32235

39469

62328

6671

878

760

92489

96636

1050

18(std)

(000)

(003)

(005)

(011)

(019

)(027)

(029)

(034)

(056)

(090)

TCR(

)99

99

9990

9981

9968

9957

9944

9920

9924

9920

9898

AIG

AAV

MFV

14071

20691

3075

73697

758551

61254

6912

185255

89576

94537

(std)

(077)

(381)

(180)

(667)

(806)

(701)

(711)

(878)

(724)

(1091)

TCR(

)9909

9537

9523

9338

9354

9129

8706

9148

9195

8910





(18)


119875119868=

1119899119906119898 119901119899119906119898 119901sum119895=1

(1198961 119878119903119895119879119903119895 + 1198962119872119905119895119860119905119895 + 1198963119872119891119895119860119891119895 ) if 119878119903119895 gt 0

0 if 119878119903119895 = 0(19)




119872119875119868119894 = 1119899119906119898 119901119899119906119898 119901sum119895=1







Table6Th

eAVC

Tindicatorsof

theH

DGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

HDGWO

AVCT(s)

1339

2751

4458

6522

8877

11478

1415

317210

20502

24045

(std)

(002)

(002)

(01)

(019

)(008)

(012

)(030)

(044)

(038)

(045)

RRCT 1

()

-876

4-8630

-8553

-8453

-8328

-8219

-812

3-8035

-793

2-7835

RRCT 2

()

-916

1-914

5-9052

-9015

-890

0-8874

-8837

-874

6-870

6-8636

RRCT 3

()

11849

10817

6690

2725

239

-1336

-2455

-3373

-393

1-4399

DPS

OAV

CT(s)

10838

20089

30811

4214

695310

764

441

75418

87562

9912

6111

070

(std)

(005)

(005)

(007)

(008)

(013

)(015

)(026)

(039

)(040)

(041)

GAW

GE

AVCT(s)

1596

03218

947036

66213

80718

101913

121685

137222

158385

176251

(std)

(007)

(024)

(032

)(039

)(041)

(053

)(059)

(064)

(070)

(082)

AIG

AAV

CT(s)

613

1322

2671

5125

8670

13248

1875

82597

03378

24292

8(std)

(002)

(004)

(006)

(010

)(025)

(055)

(030)

(033

)(037

)(065)


1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)











Data Availability






Acknowledgments


References



















































Journal of












Journal of







Volume 2018






Volume 2018









reassign

roulettewheel

selection

rand le a

rand gt a

rarrS

(t)

rarrS

(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs 1 (t)

rarrs i (t)

rarrs M(t)

rarrs i


rarrsM(t)


xM2 xMN





rarrS

(t)

rarrS(t+1)

rarrS(t)

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot



middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot






Sbase

Skli

xi1

xi1

xil

xil minusΔp

+Δp

middotmiddotmiddot


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

xik

xik +Δp xiN

xiN





Fi















breakendif






endifendfor

endif






endif


endfor





















0 20 40 60 80 100Trial NoDPSO


0 20 40 60 80 100Trial No

AIGA

20736 20736

2073620736

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074M

axim

um F

itnes

s Val

ue











0102030405060708090

100

Tota

l No

of T

rials










119879119862119877 = 119860119881119872119865119881119871119894119899119892119900 119903119890119891 sdot 100 (16)


(17)




Table5Th

reeind

icatorso

fthe

HDGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

Lingoref

14200

21696

32297

39597

62596

67097

79395

9319

897414

1060

98

HDGWO

AVMFV

1415

521285

31296

3794

160294

63512

74625

88025

92050

9891

3(std)

(057)

(365)

(199)

(546)

(640)

(630)

(846)

(747)

(621)

(609)

TCR(

)9968

9810

9690

9582

9632

9466

9399

9445

9449

9323

ISRQ

1(

)-018

-055

-107

-164

-064

-223

-208

-215

-193

-307

ISRQ

2()

-031

-180

-291

-387

-326

-481

-525

-483

-474

-581

ISRQ

3(

)060

287

175

261

298

369

796

325

276

463

DPS

OAV

MFV

1418

121403

31634

38573

60682

6495

776212

8995

693858

102048

(std)

(003)

(039

)(025)

(060)

(110

)(088)

(148)

(170)

(189)

(180)

TCR(

)9986

9865

9795

9742

9694

9681

9600

9652

9635

9618

GAW

GE

AVMFV

1419

921675

32235

39469

62328

6671

878

760

92489

96636

1050

18(std)

(000)

(003)

(005)

(011)

(019

)(027)

(029)

(034)

(056)

(090)

TCR(

)99

99

9990

9981

9968

9957

9944

9920

9924

9920

9898

AIG

AAV

MFV

14071

20691

3075

73697

758551

61254

6912

185255

89576

94537

(std)

(077)

(381)

(180)

(667)

(806)

(701)

(711)

(878)

(724)

(1091)

TCR(

)9909

9537

9523

9338

9354

9129

8706

9148

9195

8910





(18)


119875119868=

1119899119906119898 119901119899119906119898 119901sum119895=1

(1198961 119878119903119895119879119903119895 + 1198962119872119905119895119860119905119895 + 1198963119872119891119895119860119891119895 ) if 119878119903119895 gt 0

0 if 119878119903119895 = 0(19)




119872119875119868119894 = 1119899119906119898 119901119899119906119898 119901sum119895=1







Table6Th

eAVC

Tindicatorsof

theH

DGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

HDGWO

AVCT(s)

1339

2751

4458

6522

8877

11478

1415

317210

20502

24045

(std)

(002)

(002)

(01)

(019

)(008)

(012

)(030)

(044)

(038)

(045)

RRCT 1

()

-876

4-8630

-8553

-8453

-8328

-8219

-812

3-8035

-793

2-7835

RRCT 2

()

-916

1-914

5-9052

-9015

-890

0-8874

-8837

-874

6-870

6-8636

RRCT 3

()

11849

10817

6690

2725

239

-1336

-2455

-3373

-393

1-4399

DPS

OAV

CT(s)

10838

20089

30811

4214

695310

764

441

75418

87562

9912

6111

070

(std)

(005)

(005)

(007)

(008)

(013

)(015

)(026)

(039

)(040)

(041)

GAW

GE

AVCT(s)

1596

03218

947036

66213

80718

101913

121685

137222

158385

176251

(std)

(007)

(024)

(032

)(039

)(041)

(053

)(059)

(064)

(070)

(082)

AIG

AAV

CT(s)

613

1322

2671

5125

8670

13248

1875

82597

03378

24292

8(std)

(002)

(004)

(006)

(010

)(025)

(055)

(030)

(033

)(037

)(065)


1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)











Data Availability






Acknowledgments


References



















































Journal of












Journal of







Volume 2018






Volume 2018














breakendif






endifendfor

endif






endif


endfor





















0 20 40 60 80 100Trial NoDPSO


0 20 40 60 80 100Trial No

AIGA

20736 20736

2073620736

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074M

axim

um F

itnes

s Val

ue











0102030405060708090

100

Tota

l No

of T

rials










119879119862119877 = 119860119881119872119865119881119871119894119899119892119900 119903119890119891 sdot 100 (16)


(17)




Table5Th

reeind

icatorso

fthe

HDGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

Lingoref

14200

21696

32297

39597

62596

67097

79395

9319

897414

1060

98

HDGWO

AVMFV

1415

521285

31296

3794

160294

63512

74625

88025

92050

9891

3(std)

(057)

(365)

(199)

(546)

(640)

(630)

(846)

(747)

(621)

(609)

TCR(

)9968

9810

9690

9582

9632

9466

9399

9445

9449

9323

ISRQ

1(

)-018

-055

-107

-164

-064

-223

-208

-215

-193

-307

ISRQ

2()

-031

-180

-291

-387

-326

-481

-525

-483

-474

-581

ISRQ

3(

)060

287

175

261

298

369

796

325

276

463

DPS

OAV

MFV

1418

121403

31634

38573

60682

6495

776212

8995

693858

102048

(std)

(003)

(039

)(025)

(060)

(110

)(088)

(148)

(170)

(189)

(180)

TCR(

)9986

9865

9795

9742

9694

9681

9600

9652

9635

9618

GAW

GE

AVMFV

1419

921675

32235

39469

62328

6671

878

760

92489

96636

1050

18(std)

(000)

(003)

(005)

(011)

(019

)(027)

(029)

(034)

(056)

(090)

TCR(

)99

99

9990

9981

9968

9957

9944

9920

9924

9920

9898

AIG

AAV

MFV

14071

20691

3075

73697

758551

61254

6912

185255

89576

94537

(std)

(077)

(381)

(180)

(667)

(806)

(701)

(711)

(878)

(724)

(1091)

TCR(

)9909

9537

9523

9338

9354

9129

8706

9148

9195

8910





(18)


119875119868=

1119899119906119898 119901119899119906119898 119901sum119895=1

(1198961 119878119903119895119879119903119895 + 1198962119872119905119895119860119905119895 + 1198963119872119891119895119860119891119895 ) if 119878119903119895 gt 0

0 if 119878119903119895 = 0(19)




119872119875119868119894 = 1119899119906119898 119901119899119906119898 119901sum119895=1







Table6Th

eAVC

Tindicatorsof

theH

DGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

HDGWO

AVCT(s)

1339

2751

4458

6522

8877

11478

1415

317210

20502

24045

(std)

(002)

(002)

(01)

(019

)(008)

(012

)(030)

(044)

(038)

(045)

RRCT 1

()

-876

4-8630

-8553

-8453

-8328

-8219

-812

3-8035

-793

2-7835

RRCT 2

()

-916

1-914

5-9052

-9015

-890

0-8874

-8837

-874

6-870

6-8636

RRCT 3

()

11849

10817

6690

2725

239

-1336

-2455

-3373

-393

1-4399

DPS

OAV

CT(s)

10838

20089

30811

4214

695310

764

441

75418

87562

9912

6111

070

(std)

(005)

(005)

(007)

(008)

(013

)(015

)(026)

(039

)(040)

(041)

GAW

GE

AVCT(s)

1596

03218

947036

66213

80718

101913

121685

137222

158385

176251

(std)

(007)

(024)

(032

)(039

)(041)

(053

)(059)

(064)

(070)

(082)

AIG

AAV

CT(s)

613

1322

2671

5125

8670

13248

1875

82597

03378

24292

8(std)

(002)

(004)

(006)

(010

)(025)

(055)

(030)

(033

)(037

)(065)


1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)











Data Availability






Acknowledgments


References



















































Journal of












Journal of







Volume 2018






Volume 2018



























0 20 40 60 80 100Trial NoDPSO


0 20 40 60 80 100Trial No

AIGA

20736 20736

2073620736

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074

Max

imum

Fitn

ess V

alue

2072

20725

2073

20735

2074M

axim

um F

itnes

s Val

ue











0102030405060708090

100

Tota

l No

of T

rials










119879119862119877 = 119860119881119872119865119881119871119894119899119892119900 119903119890119891 sdot 100 (16)


(17)




Table5Th

reeind

icatorso

fthe

HDGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

Lingoref

14200

21696

32297

39597

62596

67097

79395

9319

897414

1060

98

HDGWO

AVMFV

1415

521285

31296

3794

160294

63512

74625

88025

92050

9891

3(std)

(057)

(365)

(199)

(546)

(640)

(630)

(846)

(747)

(621)

(609)

TCR(

)9968

9810

9690

9582

9632

9466

9399

9445

9449

9323

ISRQ

1(

)-018

-055

-107

-164

-064

-223

-208

-215

-193

-307

ISRQ

2()

-031

-180

-291

-387

-326

-481

-525

-483

-474

-581

ISRQ

3(

)060

287

175

261

298

369

796

325

276

463

DPS

OAV

MFV

1418

121403

31634

38573

60682

6495

776212

8995

693858

102048

(std)

(003)

(039

)(025)

(060)

(110

)(088)

(148)

(170)

(189)

(180)

TCR(

)9986

9865

9795

9742

9694

9681

9600

9652

9635

9618

GAW

GE

AVMFV

1419

921675

32235

39469

62328

6671

878

760

92489

96636

1050

18(std)

(000)

(003)

(005)

(011)

(019

)(027)

(029)

(034)

(056)

(090)

TCR(

)99

99

9990

9981

9968

9957

9944

9920

9924

9920

9898

AIG

AAV

MFV

14071

20691

3075

73697

758551

61254

6912

185255

89576

94537

(std)

(077)

(381)

(180)

(667)

(806)

(701)

(711)

(878)

(724)

(1091)

TCR(

)9909

9537

9523

9338

9354

9129

8706

9148

9195

8910





(18)


119875119868=

1119899119906119898 119901119899119906119898 119901sum119895=1

(1198961 119878119903119895119879119903119895 + 1198962119872119905119895119860119905119895 + 1198963119872119891119895119860119891119895 ) if 119878119903119895 gt 0

0 if 119878119903119895 = 0(19)




119872119875119868119894 = 1119899119906119898 119901119899119906119898 119901sum119895=1







Table6Th

eAVC

Tindicatorsof

theH

DGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

HDGWO

AVCT(s)

1339

2751

4458

6522

8877

11478

1415

317210

20502

24045

(std)

(002)

(002)

(01)

(019

)(008)

(012

)(030)

(044)

(038)

(045)

RRCT 1

()

-876

4-8630

-8553

-8453

-8328

-8219

-812

3-8035

-793

2-7835

RRCT 2

()

-916

1-914

5-9052

-9015

-890

0-8874

-8837

-874

6-870

6-8636

RRCT 3

()

11849

10817

6690

2725

239

-1336

-2455

-3373

-393

1-4399

DPS

OAV

CT(s)

10838

20089

30811

4214

695310

764

441

75418

87562

9912

6111

070

(std)

(005)

(005)

(007)

(008)

(013

)(015

)(026)

(039

)(040)

(041)

GAW

GE

AVCT(s)

1596

03218

947036

66213

80718

101913

121685

137222

158385

176251

(std)

(007)

(024)

(032

)(039

)(041)

(053

)(059)

(064)

(070)

(082)

AIG

AAV

CT(s)

613

1322

2671

5125

8670

13248

1875

82597

03378

24292

8(std)

(002)

(004)

(006)

(010

)(025)

(055)

(030)

(033

)(037

)(065)


1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)











Data Availability






Acknowledgments


References



















































Journal of












Journal of







Volume 2018






Volume 2018











0102030405060708090

100

Tota

l No

of T

rials










119879119862119877 = 119860119881119872119865119881119871119894119899119892119900 119903119890119891 sdot 100 (16)


(17)




Table5Th

reeind

icatorso

fthe

HDGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

Lingoref

14200

21696

32297

39597

62596

67097

79395

9319

897414

1060

98

HDGWO

AVMFV

1415

521285

31296

3794

160294

63512

74625

88025

92050

9891

3(std)

(057)

(365)

(199)

(546)

(640)

(630)

(846)

(747)

(621)

(609)

TCR(

)9968

9810

9690

9582

9632

9466

9399

9445

9449

9323

ISRQ

1(

)-018

-055

-107

-164

-064

-223

-208

-215

-193

-307

ISRQ

2()

-031

-180

-291

-387

-326

-481

-525

-483

-474

-581

ISRQ

3(

)060

287

175

261

298

369

796

325

276

463

DPS

OAV

MFV

1418

121403

31634

38573

60682

6495

776212

8995

693858

102048

(std)

(003)

(039

)(025)

(060)

(110

)(088)

(148)

(170)

(189)

(180)

TCR(

)9986

9865

9795

9742

9694

9681

9600

9652

9635

9618

GAW

GE

AVMFV

1419

921675

32235

39469

62328

6671

878

760

92489

96636

1050

18(std)

(000)

(003)

(005)

(011)

(019

)(027)

(029)

(034)

(056)

(090)

TCR(

)99

99

9990

9981

9968

9957

9944

9920

9924

9920

9898

AIG

AAV

MFV

14071

20691

3075

73697

758551

61254

6912

185255

89576

94537

(std)

(077)

(381)

(180)

(667)

(806)

(701)

(711)

(878)

(724)

(1091)

TCR(

)9909

9537

9523

9338

9354

9129

8706

9148

9195

8910





(18)


119875119868=

1119899119906119898 119901119899119906119898 119901sum119895=1

(1198961 119878119903119895119879119903119895 + 1198962119872119905119895119860119905119895 + 1198963119872119891119895119860119891119895 ) if 119878119903119895 gt 0

0 if 119878119903119895 = 0(19)




119872119875119868119894 = 1119899119906119898 119901119899119906119898 119901sum119895=1







Table6Th

eAVC

Tindicatorsof

theH

DGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

HDGWO

AVCT(s)

1339

2751

4458

6522

8877

11478

1415

317210

20502

24045

(std)

(002)

(002)

(01)

(019

)(008)

(012

)(030)

(044)

(038)

(045)

RRCT 1

()

-876

4-8630

-8553

-8453

-8328

-8219

-812

3-8035

-793

2-7835

RRCT 2

()

-916

1-914

5-9052

-9015

-890

0-8874

-8837

-874

6-870

6-8636

RRCT 3

()

11849

10817

6690

2725

239

-1336

-2455

-3373

-393

1-4399

DPS

OAV

CT(s)

10838

20089

30811

4214

695310

764

441

75418

87562

9912

6111

070

(std)

(005)

(005)

(007)

(008)

(013

)(015

)(026)

(039

)(040)

(041)

GAW

GE

AVCT(s)

1596

03218

947036

66213

80718

101913

121685

137222

158385

176251

(std)

(007)

(024)

(032

)(039

)(041)

(053

)(059)

(064)

(070)

(082)

AIG

AAV

CT(s)

613

1322

2671

5125

8670

13248

1875

82597

03378

24292

8(std)

(002)

(004)

(006)

(010

)(025)

(055)

(030)

(033

)(037

)(065)


1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)











Data Availability






Acknowledgments


References



















































Journal of












Journal of







Volume 2018






Volume 2018









Table5Th

reeind

icatorso

fthe

HDGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

Lingoref

14200

21696

32297

39597

62596

67097

79395

9319

897414

1060

98

HDGWO

AVMFV

1415

521285

31296

3794

160294

63512

74625

88025

92050

9891

3(std)

(057)

(365)

(199)

(546)

(640)

(630)

(846)

(747)

(621)

(609)

TCR(

)9968

9810

9690

9582

9632

9466

9399

9445

9449

9323

ISRQ

1(

)-018

-055

-107

-164

-064

-223

-208

-215

-193

-307

ISRQ

2()

-031

-180

-291

-387

-326

-481

-525

-483

-474

-581

ISRQ

3(

)060

287

175

261

298

369

796

325

276

463

DPS

OAV

MFV

1418

121403

31634

38573

60682

6495

776212

8995

693858

102048

(std)

(003)

(039

)(025)

(060)

(110

)(088)

(148)

(170)

(189)

(180)

TCR(

)9986

9865

9795

9742

9694

9681

9600

9652

9635

9618

GAW

GE

AVMFV

1419

921675

32235

39469

62328

6671

878

760

92489

96636

1050

18(std)

(000)

(003)

(005)

(011)

(019

)(027)

(029)

(034)

(056)

(090)

TCR(

)99

99

9990

9981

9968

9957

9944

9920

9924

9920

9898

AIG

AAV

MFV

14071

20691

3075

73697

758551

61254

6912

185255

89576

94537

(std)

(077)

(381)

(180)

(667)

(806)

(701)

(711)

(878)

(724)

(1091)

TCR(

)9909

9537

9523

9338

9354

9129

8706

9148

9195

8910





(18)


119875119868=

1119899119906119898 119901119899119906119898 119901sum119895=1

(1198961 119878119903119895119879119903119895 + 1198962119872119905119895119860119905119895 + 1198963119872119891119895119860119891119895 ) if 119878119903119895 gt 0

0 if 119878119903119895 = 0(19)




119872119875119868119894 = 1119899119906119898 119901119899119906119898 119901sum119895=1







Table6Th

eAVC

Tindicatorsof

theH

DGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

HDGWO

AVCT(s)

1339

2751

4458

6522

8877

11478

1415

317210

20502

24045

(std)

(002)

(002)

(01)

(019

)(008)

(012

)(030)

(044)

(038)

(045)

RRCT 1

()

-876

4-8630

-8553

-8453

-8328

-8219

-812

3-8035

-793

2-7835

RRCT 2

()

-916

1-914

5-9052

-9015

-890

0-8874

-8837

-874

6-870

6-8636

RRCT 3

()

11849

10817

6690

2725

239

-1336

-2455

-3373

-393

1-4399

DPS

OAV

CT(s)

10838

20089

30811

4214

695310

764

441

75418

87562

9912

6111

070

(std)

(005)

(005)

(007)

(008)

(013

)(015

)(026)

(039

)(040)

(041)

GAW

GE

AVCT(s)

1596

03218

947036

66213

80718

101913

121685

137222

158385

176251

(std)

(007)

(024)

(032

)(039

)(041)

(053

)(059)

(064)

(070)

(082)

AIG

AAV

CT(s)

613

1322

2671

5125

8670

13248

1875

82597

03378

24292

8(std)

(002)

(004)

(006)

(010

)(025)

(055)

(030)

(033

)(037

)(065)


1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)











Data Availability






Acknowledgments


References



















































Journal of












Journal of







Volume 2018






Volume 2018












(18)


119875119868=

1119899119906119898 119901119899119906119898 119901sum119895=1

(1198961 119878119903119895119879119903119895 + 1198962119872119905119895119860119905119895 + 1198963119872119891119895119860119891119895 ) if 119878119903119895 gt 0

0 if 119878119903119895 = 0(19)




119872119875119868119894 = 1119899119906119898 119901119899119906119898 119901sum119895=1







Table6Th

eAVC

Tindicatorsof

theH

DGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

HDGWO

AVCT(s)

1339

2751

4458

6522

8877

11478

1415

317210

20502

24045

(std)

(002)

(002)

(01)

(019

)(008)

(012

)(030)

(044)

(038)

(045)

RRCT 1

()

-876

4-8630

-8553

-8453

-8328

-8219

-812

3-8035

-793

2-7835

RRCT 2

()

-916

1-914

5-9052

-9015

-890

0-8874

-8837

-874

6-870

6-8636

RRCT 3

()

11849

10817

6690

2725

239

-1336

-2455

-3373

-393

1-4399

DPS

OAV

CT(s)

10838

20089

30811

4214

695310

764

441

75418

87562

9912

6111

070

(std)

(005)

(005)

(007)

(008)

(013

)(015

)(026)

(039

)(040)

(041)

GAW

GE

AVCT(s)

1596

03218

947036

66213

80718

101913

121685

137222

158385

176251

(std)

(007)

(024)

(032

)(039

)(041)

(053

)(059)

(064)

(070)

(082)

AIG

AAV

CT(s)

613

1322

2671

5125

8670

13248

1875

82597

03378

24292

8(std)

(002)

(004)

(006)

(010

)(025)

(055)

(030)

(033

)(037

)(065)


1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)











Data Availability






Acknowledgments


References



















































Journal of












Journal of







Volume 2018






Volume 2018









Table6Th

eAVC

Tindicatorsof

theH

DGWOthe

DPS

Othe

GAW

GEandtheA

IGAfore

achcase

Case

1Ca

se2

Case

3Ca

se4

Case

5Ca

se6

Case

7Ca

se8

Case

9Ca

se10

HDGWO

AVCT(s)

1339

2751

4458

6522

8877

11478

1415

317210

20502

24045

(std)

(002)

(002)

(01)

(019

)(008)

(012

)(030)

(044)

(038)

(045)

RRCT 1

()

-876

4-8630

-8553

-8453

-8328

-8219

-812

3-8035

-793

2-7835

RRCT 2

()

-916

1-914

5-9052

-9015

-890

0-8874

-8837

-874

6-870

6-8636

RRCT 3

()

11849

10817

6690

2725

239

-1336

-2455

-3373

-393

1-4399

DPS

OAV

CT(s)

10838

20089

30811

4214

695310

764

441

75418

87562

9912

6111

070

(std)

(005)

(005)

(007)

(008)

(013

)(015

)(026)

(039

)(040)

(041)

GAW

GE

AVCT(s)

1596

03218

947036

66213

80718

101913

121685

137222

158385

176251

(std)

(007)

(024)

(032

)(039

)(041)

(053

)(059)

(064)

(070)

(082)

AIG

AAV

CT(s)

613

1322

2671

5125

8670

13248

1875

82597

03378

24292

8(std)

(002)

(004)

(006)

(010

)(025)

(055)

(030)

(033

)(037

)(065)


1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)











Data Availability






Acknowledgments


References



















































Journal of












Journal of







Volume 2018






Volume 2018









1

HDGWODPSOGAWGEAIGA

0

02

04

06

08

Mod

ified

Per

form

ance

Inde

x (M

PI)











Data Availability






Acknowledgments


References



















































Journal of












Journal of







Volume 2018






Volume 2018

























































Journal of












Journal of







Volume 2018






Volume 2018








a hybrid discrete grey wolf optimizer to solve weapon target assignment problems · 2019. 7....

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