research article a novel fused optimization algorithm of...

Research ArticleA Novel Fused Optimization Algorithm of Genetic Algorithmand Ant Colony Optimization

FuTao Zhao1 Zhong Yao1 Jing Luan1 and Xin Song2

1School of Economics and Management Beihang University Beijing 100191 China2School of Computer Science and Engineering Beihang University Beijing 100191 China

Correspondence should be addressed to Zhong Yao iszhyaobuaaeducn

Received 31 May 2016 Accepted 31 July 2016

Academic Editor Anna Vila

Copyright copy 2016 FuTao Zhao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A novel fused algorithm that delivers the benefits of both genetic algorithms (GAs) and ant colony optimization (ACO) is proposedto solve the supplier selection problem The proposed method combines the evolutionary effect of GAs and the cooperative effectof ACO A GA with a great global converging rate aims to produce an initial optimum for allocating initial pheromones of ACOAn ACO with great parallelism and effective feedback is then served to obtain the optimal solution In this paper the approach hasbeen applied to the supplier selection problem By conducting a numerical experiment parameters of ACO are optimized using atraditional method and another hybrid algorithm of a GA and ACO and the results of the supplier selection problem demonstratethe quality and efficiency improvement of the novel fusedmethodwith optimal parameters verifying its feasibility and effectivenessAdopting a fused algorithm of a GA and ACO to solve the supplier selection problem is an innovative solution that presents a clearmethodological contribution to optimization algorithm research and can serve as a practical approach and management referencefor various companies

1 Introduction

Inspired by Darwinrsquos evolution theory and Mendelrsquos hereditytheory Holland first proposed genetic algorithms (GAs) in1975 [1] A GA is a biotic general-purpose search optimiza-tion strategy designed to imitate the evolutionary processes ofnatural selection in biotic populationsThe decision variablesof the problem must be coded as chromosomes and geneticoperations of copying crossover and mutation are employedas the simulated gene pool changes over time in responseto environmental pressures that enable optimal solutionsto survive to the next generation All GA optimizationprocesses are based on these conceptions of chromosomesand biotic populations GAs conform to the genetic andevolutionary principle of ldquosurvival of the fittestrdquo GAs provideself-organization self-adaption and useful global searchability As a global optimization method GAs can handleall types of objective functions and constraints without themathematical limitations that plague numerous approachesto optimization problems therefore GAs have been widelyutilized in various applications However GAs do not have

a practicable feedback mechanism so a large number ofredundant iterations are produced when solutions are withina certain scope resulting in low efficiency [2] AdditionallyGAs require long search times for Big Data problems [3]

Ant colony optimization (ACO) is a class of simulativeevolutionary algorithms mimicking the foraging behavior ofants in nature first proposed by Dorigo and Gambardella [4]and has successfully solved complicated optimization prob-lems such as the traveling salesman problem (TSP) quadraticassignment problem and job shop scheduling problem Real-world ants apply stigmergy to their foraging process whenan ant forages it marks the path that it has chosen byreleasing pheromones as it walks When an ant encountersa fork with no detectable pheromones that ant will randomlychoose one path but when an ant encounters forking pathsmarked with pheromones the antrsquos decision is not entirelyrandom the decision is influenced by the accumulation ofpheromones on the paths Regardless of which route the antchooses the pheromone that the ant releases will influencethe decisions of other ants The probability for an ant tochoose a path depends on the number of ants that previously

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 2167413 10 pageshttpdxdoiorg10115520162167413

2 Mathematical Problems in Engineering

chose that path Therefore in the absence of volatilization apheromone trail on a popular path will accumulate rapidlyand help attract evergrowing numbers of ants to follow thatpath (positive feedback) [5] Through this natural stigmer-gic process without any prior knowledge real-world antcolonies establish optimal foraging paths through exchangesof information between individuals and mutual cooperationAs a swarm intelligence optimization algorithm ACO offersthe advantages of parallel computation self-learning andeffective information feedback However during the initialstages of ACO searches little or no information is availabletherefore ACO searches often converge slowly

The concept of integrating a GA and ACO was firstproposed byAbbattista et al [6] for exploiting the cooperativeeffect of ACO with the evolutionary effect of a GA Theyintegrated these search methods by using GAs to evolveoptimal parameter values for ACO Following their initialpublication numerous efforts have hybridizedGAs andACOand now developed fusing approaches can be roughly dividedinto four categories Approaches in the first category suchas the work of Acan et al [7 8] apply a GA to selectACO parameter values that optimize the performance ofant populations Methods in the second category generateinitial pheromone distributions through GAs which aresubsequently optimized with ACO [9ndash13] Methods in thethird category add genetic operations to ACO to diversifysolutions [14ndash16] Approaches in the fourth category combinethe initialization from the second category and the diversifi-cation from the third category [17] Approaches of the secondcategory fuse the GA and ACO to take advantage of theGArsquos rapid convergence and ACOrsquos parallelism and effectivefeedback this fusion inspires the present study The presentstudy improves the search for an appropriate fusing time thusenhancing the performance of the GA and ACO Through anumerical experiment of the supplier selection problem wedemonstrated the feasibility and efficiency of this novel fusedalgorithm

The supplier selection problem is an essential topic insupply chain management because the selection of propersupply partners can substantially improve a firmrsquos competitiveadvantages and can further influence the quality and pricesof the final products offered to customers [5] In the contextof previous research the supplier selection problem can bedescribed as amultigoal combinatorial optimization problemwith the objectives of achieving production targets andmaximum profits by selecting appropriate suppliers for eachmaterial under the condition of limited resources Becausethe supplier selection problem is a typical combinatorialoptimization problem we consider that solving it with ourfused GA and ACO algorithm could be noteworthy andshed new light on contemporary problems encountered inresearch on supplier selection

The remainder of this paper is organized as followsSection 2 reviews the literature regarding the developmentof the GA ACO and the fused GA and ACO algorithmsSection 3 describes the key points of our fused algorithm andelaborates on the modifications of fusing time the GA andACO Section 4 explains the design of a numerical experi-ment on the supplier selection problem and the optimization

of parameters the performance of the new fused algorithmis evaluated In the final section we offer the conclusions anddirections for future research

2 Literature Review

21 Genetic Algorithms and Ant Colony Optimization GAsand ACO are popular classes of intelligent heuristic algo-rithms which have been broadly applied in the field of opti-mization The initial versions were immediately improvedupon and improvements have continued to advance the per-formance of GAs andACO ForGAs substantial ameliorativework has been performed to optimize search performancewith features such as improved selectionmechanism strategyadaptive mutation probability [18] GA operators [19] andelitism selection mechanisms To improve the global searchcapability and convergence performance of GAs Wang etal [20] proposed four types of improved GAs namelyhierarchic GAs simulated annealing GAs simulated anneal-ing hierarchic GAs and adaptable GAs these methods canovercome the defects of traditional GAs by combining GAswith simulated annealing algorithms and modifying variouscoding methods In terms of ACO the main features of itsimprovement involve mechanisms to intensify the searchinvolving high-quality solutions and preserve a sufficientsearch space [21] Niu et al [22] stated that as a typical greedyheuristic algorithm ACO tends to become trapped in localoptimaThey proposed a method for guiding the search awayfrom local optima by adding a perturbation into the originalprobability Moreover a coefficient representing the effects ofan average pheromone trail updated pheromones to reducethe effects of the parameter Q

GA and ACO are promising because they can substan-tially increase the possibility of determining high-qualitysolutions for some complex combinatorial optimizationproblems such as the supplier selection problem To man-age an integrated multi-item supplier selection problem formaximizing the annual income of an entire supply chainAliabadi et al [23] presented a two-level GA (2LGA) modelbased on two types of variablesmdashbinary variables and realvariablesmdashof which the first layer was used for selectingsuppliers and the following layer was used for orderingthem Simic et al [24] proposed a GA performance valueconstraint model that used a grading variable for assessingthe performance of suppliers Yang et al [25] applied a GA toa stochastic-demand multiproduct supplier selection modelwith constraints of service level and budget where the highestvalue of the average expected profit and the lowest valueof the standard deviation were achieved through differentcombinations of crossover and mutation rates In referenceto the attribute-based ant colony system (AACS) Tsai et al[5] reported an examination of the critical factors the criteriafactors and weights were incorporated in the pheromoneupdate rule and the AACS was used to obtain the optimalsupplier according to a quantitative decision policy

22 FusedAlgorithm of aGAandACO A review of the extantalgorithms introduced to solve combinatorial optimization

Mathematical Problems in Engineering 3

problems shows that intelligence optimization algorithms aregradually prevailing Such algorithms include GAs and ACOwhich are inspired by the behavior or processes present innature Each of these has its own advantages and disadvan-tages thus numerous researchers have considered investiga-tions of multiple methods to be notable and hold promisefor overcoming the defects of individual algorithms as well asachieving complementary advantages The hybridization of aGA and ACO has been applied to solve numerous complexcombinatorial optimization problems such as the capacitatedvehicle routing problem [26] logistics distribution routeoptimization [9] the 0-1 knapsack problem and quality ofservice [10] optimization of cloud database route scheduling[11] the virtual enterprise partner selection problem [12 13]and some NP-complete problems including the satisfactionproblem the tripartite matching problem and the TSP [27]

In the relevant literature the key to hybridizing GAsand ACO is to combine the population diversity and globalsearching ability of GAs with the feedback mechanismand rapid convergence of ACO to maximize accuracy andefficiency In Zhang and Wu [17] the fused algorithm hastwo procedures first it approximates the global maximumby using a GA and it then searches for the optimal solutionby using ACO with GA operators Two fusion ideas wereproposed in Xiao and Tan [14] in some cases a GA isused to search for rough initial pheromone solutions whichinitialize ACO information and ACO subsequently seeksan optimal solution however a GA can be used to addcrossover operators into ACO to prevent stagnation at localoptima thereby enhancing the global searching ability ofACO In Liu [28] a GA was used to optimize the coefficientsof pheromones heuristics and pheromone volatilization inACO thus GAs and ACO were integrated to improve theefficiency of ACO With a different approach to fusing aGA and ACO Li et al [15] added a heuristic factor ofgenetic information into an initial fixed heredity proportionto determine the transition probability of ACO this wasintended to minimize computational effort and increase theconvergence rate during the path search

3 Concept of Fusing a Genetic Algorithm andAnt Colony Optimization

31 The Concept of Fusing a GA and ACO In this paperthe basic concept of the dynamic integration of a GA andACO comes from Yao et al [12 13] and Xiong et al [29]We adopted a GA to generate available solutions and updateinitial pheromone values An ACO implementation searchesuntil the optimum is reached Xiong et al [29] presenteda speed-time curve of a GA and ACO (Figure 1) where119905119886is the optimal fusing time In order to achieve a fusion

time approximately equal to 119905119886 they proposed a dynamic

integration strategy that set a minimum iteration 119866119890min (119905119887

moment) a maximum iteration 119866119890max (119905119888moment) and a

constant 119866119890die for their GA If the evolutionary loop returneda result that was less than a constant for119866119890die generations thehybrid algorithm would terminate the GA loop and initiatethe ACO search

V

Va

t0 tb tatc

Ant colony optimization

Genetic algorithm

Figure 1 Difference of speed-time curve of GA and ACO

13702137041370613708

1371137121371413716

001 0009 0008 0007 0006 0005Fi

tnes

s val

ue

Constant

Figure 2 Variation of optimal fitness value among different con-stants

32 Improvements of the Fused Algorithm In this paperwe improved traditional GAs and ACO to enhance theperformance of a hybrid algorithm

321 Fusing Time of a GA and ACO Based on the concept ofsetting the fusing time of the integrated algorithm reportedin Xiong et al [29] in this paper we define the evolutionaryrate as the variation rate of optimal fitness values between twosuccessive iterations When the evolutionary rate is detectedto be less than a certain constant for three iterations of theloop (119866119890die = 3) the efficiency of the GA is consideredto be low enough to end the GA loop and to engage ACOTo determine the constant we compared the optimal fitnessvalues among different constants ranging from 0005 to 001according to a value distribution of the evolutionary rateFigure 2 shows the average fitness values of 10 iterationsunder different constants where 0009 is clearly the optimalconstant

322 Genetic Algorithm with Self-Adaptive Crossover andMutation Probability For a general GA the crossover prob-ability and mutation probability are constants Althoughthe algorithm may initially show a high convergence rateif it lacks an explicit feedback mechanism its efficiencygradually degenerates In the context ofMa [30] self-adaptivecrossover and mutation probabilities are introduced in ouralgorithm By adjusting crossover and mutation probabilitiesautomatically the enhanced GA successfully avoids redun-dant iterations and low search efficiency in its later stagesThe


self-adaptive crossover and mutation probability functionsare as follows

119875119888

=

1198751198880

119891 le 119891

1198751198881

(

1198751198880

1198751198881

)

((119891maxminus119891)(119891maxminus119891))

119891 gt 119891

119875119898

=

1198751198980

1198911015840

le 119891

1198751198981

(

1198751198980

1198751198981

)


1198911015840

gt 119891

(1)

where 1198751198880and 119875

1198980represent the higher crossover and muta-

tion probabilities 1198751198881

(1198751198881

lt 1198751198880

) and 1198751198981

(1198751198981

lt 1198751198980

) are thelower probabilities 119891 and 119891

1015840 are the lower fitness values ofindividuals and 119891max 119891 are the optimal and average fitnessvalues in the population

323 Updating Mechanism of the Pheromone in Ant ColonyOptimization Pheromone updating is a critical process ofACO Stutzle and Hoos [31] presented Max-Min ACOwhich updates only the pheromones of the optimal solutionafter each iteration This concept simplifies the pheromoneupdating method compared with traditional ACO whichupdates the pheromone levels of all solutionsThepheromoneconstant119876 affects the performance of ACO In general 119876 hasan artificial initial value and cannot be changed as the searchproceeds and thus a general ACO implementation is at riskof stagnating at local optima Therefore a self-adaptive 119876 isintroduced in this paper where119876 is not a constant but variesaccording to a step function Based on this the functions ofpheromone updating are as follows

120591119878119894119895

(119905 + 119899) = (1 minus 120588) 120591119878119894119895

(119905) + 120588Δ120591119878119894119895

(119905) (2)

Δ120591119878119894119895

(119905) =

119898

sum

119896=1

Δ120591119896

119878119894119895

(119905) (3)

Δ120591119896

119878119894119895

(119905) =

119876

119891max 119878119894119895

isin 119891max

0 else(4)

119876 = 1198760

lowast (1 minus119908 lowast 119873

119905

max119873) (5)

where 120588 isin [0 1] is pheromone volatilization coefficientΔ120591119878119894119895

(119905) is pheromone variation of the optimal solution andΔ120591119896

119878119894119895

(119905) is the pheromone left by each ant on the traversednodes of the optimal solution Equation (5) is the stepfunction for 119876 where 119876

0is the initial value of 119876 119908 isin [0 1]

is the adjustment coefficient 119873119905is the current iteration and

max119873 is the maximum number of iterations

33 Algorithm Processes

331 Genetic Algorithm

(1) Initialize the control parameters of the GA includingpopulation sizeN high crossover andmutation prob-abilities 119875

1198880and 119875

1198980 lower crossover and mutation

probabilities1198751198881and1198751198981 the end condition of the GA

namely 119866119890min 119866119890max and 119866119890die and evolutionaryrate 119864119903

(2) Randomly generate initial population 119866(0) in accor-dance with constraints and set the index of genera-tions 119892 as 119892 = 0

(3) Calculate the individual fitness value in 119866(119892) andthe maximal and average fitness value inMaxFit andAvgFit

(4) According to the individual fitness value and roulettechoice strategy set 119875(119894) as the choice probability ofeach individual in 119866(119892)

(5) For (119873119905= 0 119873

119905lt 119873 119873

119905= 119873119905+ 2)

(a) according to 119875(119894) select two individuals of 119866(119892)

as fathers(b) calculate crossover probability 119875

119888and mutation

probability 119875119898

(c) generate random number 119903 = random [0 1](d) if (119903 le 119875

119898) implement a mutation operation on

the two fathers if the fitness value of the newindividual is higher than that of its father insertit into the next generation group 119866(119892 + 1)

(e) if (119875119898

lt 119903 le 119875119898

+ 119875119888) implement a crossover

operation if the fitness value of the memberof the new generation is higher than its fatherinsert it into next generational group 119866(119892 + 1)

(f) otherwise insert the two fathers into the nextgeneration group 119866(119892 + 1)

(6) Calculate and update the individual fitness valueMaxFit AvgFit and 119892 = 119892 + 1

(7) Judge whether 119864119903 has been invariant for119866119890die genera-tions or 119892 gt 119866119890max if either test is true the algorithmenters the ant colony optimization steps if neither testis true proceed to Step (4)

332 Ant Colony Optimization

(1) Set the initial pheromones for the routes of ACOaccording to the results of the GA

(2) Set the 119894119905119890119903 = 1 (119894119905119890119903 is the index of search iterations)randomness coefficient 119908 optimal value of objectivefunction 119891 = 0 initial number of ants 119898 and pathlength n all the ants start from the beginning

(3) Initiate the feasible sets 119886119897119897119900119908119896(the allowable nodes

for ant 119896) and solution sets 119905119886119887119896(the nodes chosen by

ant 119896 for 119869 types of materials)


(4) According to the transition probability119875 ant 119896movesto the next node and adds the selected node into 119905119886119887updating the feasible set 119886119897119897119900119908

119896

(5) After n iterations all ants have traversed n nodes andone round of the search process is complete Calculatethe fitness value 119891

119896for all the solutions marking

the maximum of 119891119896as 119891max and the corresponding

solution as 119905119886119887best(6) Update the pheromone in the optimal path and set

119894119905119890119903 = 119894119905119890119903 + 1(7) Judge whether 119891 lt 119891max and 119894119905119890119903 le 119894119905119890119903max if so set

119891 = 119891max and return all the ants back to their startingnodes then proceed to Step (3) if not test whether119894119905119890119903 gt 119894119905119890119903max if so the search ends resulting in theoptimal known solution 119891 and 119905119886119887best if not returnall the ants back to their starting nodes and proceedto Step (3)

4 Numerical Experiments

41 Instance Description Supplier selection is a multigoalcombinatorial optimization problem it is an appropriateproblem for swarm intelligence optimization algorithmssuch as our proposed fused algorithm of a GA and ACOIn our numerical experiment we defined 119873 types of rawmaterials and components to be purchased we defined 119869

qualified suppliers All suppliers are grouped into 119873 cate-gories according to the raw materials or components theycan provide and the task is to choose one supplier for eachraw material To ensure product quality each raw materialand component part can be offered by exactly one supplierand each supplier can only offer a limited number of materialtypes Quality (119876) cost (119862) delivery capability and flexibility(119879) and innovation and development capability (119863) arecategorized and considered as the evaluation indices for theselection of suppliers The objective of selecting suppliersis to maximize quality delivery capability and flexibilityinnovation and development capability and to minimizecost denoted as max119876 119879 119863 minus119862 With the increase of 119869the supplier selection problem clearly becomes a combi-natorially explosive problem Because effective selection ofsuppliers to meet all requirements is difficult the problemmust be transformed into a single-objective optimizationproblemHere we adopt the technique for order of preferenceby similarity to ideal solution (TOPSIS) a very effectivemethod in multiobjective decision analysis Its core conceptis to compare distances between each evaluation optionand positivenegative ideal solutions and to evaluate theavailable options In terms of TOPSIS for the 119894thmaterial thesynthetic goal of its 119895th supplier 119891

119894119895(119905) can be written as (6)

where 119889+

(119905) and 119889minus

(119905) are the distances between each indexvalue and positivenegative ideal values (119862+ 119876+ 119879+ 119863+) and(119862minus

119876minus

119879minus

119863minus

) are the positive and negative ideal valuesof the four indices respectively 119902

119894119895 119905119894119895 119889119894119895 and 119888

119894119895represent

the four index values of candidate j for 119894th material and119908119894

119902 119908119894

119889 119908119894

119888 and 119908

119894

119905denote the weights of indices 119876 119863 119862

and 119879 for 119894th material Thus the objective function for this

supplier selection problem can be described as (9) where 119872119894

are the numbers of potential suppliers for 119894th material Basedon TOPSIS we converted a multiobjective combinatorialoptimization problem to a single-objective form

119891119894119895

=

119889minus

119894119895

119889+

119894119895+ 119889minus

119894119895

(6)

119889+

119894119895= 119908119894

119888

10038161003816100381610038161003816119888119894119895

minus 119862+10038161003816100381610038161003816

119862+

+ 119862minus

+ 119908119894

119902

10038161003816100381610038161003816119902119894119895

minus 119876+10038161003816100381610038161003816

119876+

+ 119876minus

+ 119908119894

119905

10038161003816100381610038161003816119905119894119895

minus 119879+10038161003816100381610038161003816

119879+

+ 119879minus

+ 119908119894

119889

10038161003816100381610038161003816119889119894119895

minus 119863+10038161003816100381610038161003816

119863+

+ 119863minus

(7)

119889minus

119894119895= 119908119894

119888

10038161003816100381610038161003816119888119894119895

minus 119862minus10038161003816100381610038161003816

119862+

+ 119862minus

+ 119908119894

119902

10038161003816100381610038161003816119902119894119895

minus 119876minus10038161003816100381610038161003816

119876+

+ 119876minus

+ 119908119894

119905

10038161003816100381610038161003816119905119894119895

minus 119879minus10038161003816100381610038161003816

119879+

+ 119879minus

+ 119908119894

119889

10038161003816100381610038161003816119889119894119895

minus 119863minus10038161003816100381610038161003816

119863+

+ 119863minus

(8)

max119891 =

119873

sum

119894=1

119891119894119895

(119905) 119895 = 1 119872119894 (9)

To examine the time and optimization performance of thehybrid algorithm we coded a simulation case based on thesupplier selection problemdescribed previously In our case amiddle-scale automobile enterprise was required to purchase15 types of accessories in a market with 15 qualified suppliersfor each accessory To ensure the efficiency of suppliers wesupposed that each material could be supplied by only onesupplier and that each supplier could offer only one materialThe fitness value of potential suppliers by TOPSIS and partialdata of simulation case are shown in Appendix Table 1and Appendix Table 2 respectively in Supplementary Mate-rial available online at httpdxdoiorg10115520162167413This numerical experiment comprised two parts Parameteroptimization was conducted to improve the efficiency of thenovel fused algorithm Given those optimal parameters theGA ACO and our fused algorithm were applied separatelyto solve this supplier selection problem

42 Parameter Optimization Because of the lack of criteriafor setting parameters in ACO the main objective of param-eter optimization is to adjust ACOparameters to approximateor reach optimal values These parameters include ant num-ber ant Num pheromone coefficient 119886 heuristic coefficient119887 and pheromone volatilization coefficient 119903 Generally ACOparameters are optimized by trials of their feasible values andempirical selection of values that approximate the optimalsolution as shown in Figure 3

The number of ants can greatly affect the search efficiencyFigure 4 shows the performance levels of our hybrid algo-rithm with ant populations of 5 and 10 The maximal fitnessvalues are plotted against the number of iterations in thethird panel we can conclude that the optimizing capacity of10 ants is superior to that of 5 ants Generally within practicallimits when numbers of ants increase the convergence speedincreases however the improvement cannot be extended


1352135413561358

1361362136413661368

1371372

30 50 70 90 110 130 150 170

Fitn

ess v

alue

Coefficient value

(a) Ant number

135

1355

136

1365

137

1375

01 02 03 04 05 06 07 08 09 1

Fitn

ess v

alue

Coefficient value

12913

131132133134135136137138

1 2 3 4 5 6 7 8 9

Fitn

ess v

alue

Coefficient value

(c) Heuristic coefficient b

135713581359

136136113621363136413651366

01 02 03 04 05 06 07 08 09 1

Fitn

ess v

alue

Coefficient value

(d) Pheromone delay coefficient r

(b) Pheromone coefficient a

Figure 3 The influence of parameters

indefinitely Figure 3(a) shows that 110 is a pivotal point Theblue line shows the average optimal values for 10 iterationswith ant populations ranging from 30 to 170 Given 110ants the iterative optimal value and performance of novelalgorithm are optimal

Regarding the pheromone coefficient a which can causethe search to stagnate at local optima the larger its valueis the more influence it exerts on transition probability 119875The orange line in Figure 3(b) shows the influence of thispheromone coefficient on the optimal fitness value and 119886 =

04 performed better The heuristic coefficient 119887 reflects theeffect of the heuristic on algorithm efficiency As the greenline in Figure 3(c) indicates 119887 = 8 is the most suitable valuefor our algorithmThe pheromone volatilization coefficient 119903

determines the degree of pheromone volatilization Specifi-cally the greater 119903 is the more the pheromones are left andthe more easily the algorithm can stagnate If 119903 is excessivelylow the pheromones volatilize too rapidly and the traces ofan optimal path disappear before the ants can reinforce thatpath In Figure 3(d) the red line demonstrates that 119903 = 03 isthe proper value

Traditional parametric optimization involves setting allthe other variables constant and only adjusting one param-eter but this traditional method requires excessive time andcomputational workload By adopting another feasible fusionof aGAandACO Liu [28] used aGA to search for the optimalACOparameter combination theGAwas applied to generatea parameter combination and parameter performance was

evaluated by comparing ACO solutions premised on thoseparameters In this study we also attempted to utilize thefusion of a GA and ACO to optimize parameters Specificallyparameters 119886 119887 and 119903 were coded as chromosomes in theGA Seven-digit codes were used for each parameter eachchromosome had 21 digits in total The parameter combina-tion generated by the GA was converted to decimal numbersaccording to the parameter scope and was applied by ACOfor solving the supplier selection problemThe specific codingand converting scheme is shown in Table 1 and the results aredisplayed in Figure 5The optimal values of pheromone coef-ficient 119886 heuristic coefficient 119887 and pheromone volatilizationcoefficient 119903 were 04 8 and 03 respectively these resultswere equivalent to those of the traditional method but theywere reached after 60 iterationsMoreover we discovered thatthe optimal fitness value obtained from the fused algorithmwas inferior to that of the integrated algorithm of the GA andACO for supplier selection (13238 versus 13729) This mayhave been caused by the influence of parameter uncertaintythis disparity indicates the pivotal role that parameters playin ACO

43 Simulation Results and Analysis To solve the problemdescribed previously we conducted a GA ACO and ourfused algorithmwith the optimal parametersWe used JAVA6to code the algorithm and simulated the numerical exampleon aWindows 7 Ultimate platform Figures 6 and 7 show the


135

13

125

12

115

11

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant 1Ant 2Ant 3

Ant 4Ant 5

Ant 1Ant 2Ant 3Ant 4Ant 5


11

115

12

125

13

135

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

124126128

13132134136138

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant number = 5

Ant number = 10

Figure 4 Performance of hybrid algorithm of GA and ACO with different numbers of ants

123124125126127128129

13

011 022 033 044 055 066 077 088 099

Fitn

ess v

alue

Coefficient value

Pheromone coefficient aHeuristic coefficient bPheromone volatilization coefficient r

Figure 5 Parameter optimization by fused algorithm of GA andACO

Table 1 Binary-coding and converting scheme of parameters

Parameter Binary-coding scheme Decimalization119886 0 1 1 1 0 1 1 05119887 1 1 1 0 0 0 1 9119903 0 0 0 1 0 0 1 01

results and the settings of the initial parameters are shown inAppendix Table 3

02468

10121416

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Supp

lier

Material

123

45The best route

Figure 6 The result of supplier selection

Figure 6 displays the search results (six routes) of theants of the hybrid algorithm where the orange line showsthe optimum It verifies the feasibility of our new hybridalgorithm and the effectiveness of the optimal parametersand demonstrates that the hybrid algorithm can retain thesuperior solution and increase the diversity of solutions

Figure 7 displays comparisons of GA-ACO the GAand ACO during 100 iterations including the variations offitness value and evolutionary rate Figure 7(a1) plots the


10105

11115

12125

13135

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Fitn

ess v

alue

Iteration

GA-ACOGAACO

GA-ACOGAACO

GA-ACOGAACO

1341345

1351355

1361365

1371375

138

25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Fitn

ess v

alue

Iteration

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

Iteration

(a1) Variation of Fitness value of 100 iterations

00005

0010015

0020025

0030035

004

Evol

utio

nary

rate

(a2) Variation of fitness value of 25thndash100th iteration

(b) Variation of evolutionary rate

Figure 7 A comparison of operation process with GA-ACO GAand ACO

fitness variations of GA-ACO the GA and ACO against 100iterations Figure 7(a2) shows the variations of fitness valuesfor GA-ACO the GA and ACO from the 25th to the 100thiteration which are easily observed In detail the GA (orangeline) was stable at 13729 after 86 iterations ACO (red line)required 93 iterations to be stable at 13729 whereas for theintegrated algorithm (blue line) the function value reachedthe optimum 13729 after 66 iterations Figure 7(b) showsthat at the early stage of searching the fused algorithmhas a higher convergence rate than ACO has and at thelater stage the fused algorithm has a faster evolutionary ratethan GA has This demonstrates the main improvement andcontribution of our novel fused algorithm compared with

traditional single algorithms and demonstrates its advantagesof shorter time expenditure and higher efficiency

Details of the comparison of these three algorithms are asfollows

431 Genetic Algorithm As the orange curves in Figures7(a1) and 7(a2) show after 24 iterations the variance offitness values was dramatic from 95679 to 128549 From the25th iteration the convergence rate gradually slowed and thefitness value changed from 13424 to 13545 From the 47thto the 65th iteration the fitness value varied from 13545 to13671 From the 66th to the 76th iteration the evolutionaryrate declined continuously and after 11 iterations the fitnessvalue was 13689 From the 77th to the 85th iteration thesearching process was smooth with a low changing ratio Atthe 86th iteration the algorithm reached its optimal value13729 Until then the searching algorithm had been stableWhen considering the orange curve in Figure 7(b) althoughthe evolutionary rate declined substantially at the initialsearching stage the GA clearly had an excellent convergencerate and high efficiency However from the 25th iteration thealgorithm required excessive time to seek a better solutionthat is as the iterations increased the convergence ratedropped even though it obtained the optimum 13729 afterthe 86th iteration This verifies that at the later stage of theGA its search efficiency was relatively low and redundantiterations occurred frequently

432 Ant Colony Optimization Consider the red curve inFigure 7 at the early stages of the search the overall changeof the ACO fitness value was lower than that of the GA upto the 39th iteration However at the later stages from the66th to the 92nd iteration the solving process of ACO wasrelatively shorter than that of the GA and the convergencerate was faster until the 96th iteration At the 93rd iterationACO reached stability at the optimum 13729This illustratesthat ACO had the capacity to converge quickly to a localoptimum However this also exposes a flaw of ACO namelythat our search stagnated at a local optimum from the 12th tothe 53rd iteration From the red line ACOclearly had a higherinitial value than the GA and the new fused algorithm (ACOhad 13527 the GA had 95679 and our fused algorithm had10555) This is because the GA is a random algorithm andits original populations are generated at random howeverin ACO each transfer of ants is determined by probabilityTherefore ACO is 119873-level decision-making problem andACO can likely obtain a better value than the GA can obtainAdditionally because of the randomness for the GA theorange line fluctuates more frequently as does the fitnessvalue but for ACO the red line is flatter and the fitness valuechanges only a few times

433 Fusing Algorithm The blue curve in Figure 7 shows theprocess of the integrated algorithm the first 19 iterations usedthe GA and ACO began from the 20th iterationThe optimalvalue varied quickly from 13495 to 13655 between the 20thand the 28th iterationMost of the solving process was shorterthan those of the GA and ACOThis clearly demonstrates the


merits of the GA namely a high convergence rate at earlysearch stages and also illustrates the advantages of ACOnamely the ability to converge quickly to a local optimumAlthough ACO is often limited by a low improvement ratein its early iterations because of the lack of pheromonesthe proposed method overcame that obstacle Moreover theproposed method efficiently avoided the redundant late stageiterations that are typical of a GA

5 Conclusions and Future Research

In this paper we described a novel fused algorithm thatemploys a GA and ACO for the supplier selection problemIt provides the advantages of a GA and ACO and effectivelyavoids their defects Each part of the fused algorithm isimproved and in the context of Xiong et al [29] the rationalintegration of these two algorithms is carefully observedand designed To test the feasibility and effectiveness of thenew fused algorithm three separate instances of a supplierselection problem were implemented for the GA ACO andour new fused algorithm The results show that our newfused algorithm delivered a better time than the times ofits competitors and the new fused algorithm delivered theoptimal known value as the solution of its objective function

The present study has some limitations The proposedideas deserve to be improved and explored For example thescale of the simulation case applied in this paper is relativelysmall and some large-scale studies should test our fusedalgorithmTherefore further research can focus on verifyingour fused algorithm in terms of other typical combinatorialoptimization problems such as the TSP Additionally theuniversality of our new fused algorithm must be testedand numerous previously unresolved challenges can be fur-ther investigated with our new fused method Furthermoreparameters and their influence on optimization performanceshould be studied in greater detail identifying the optimaltime to cease the GA and engage ACO would be warranted

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work has been supported by the Natural ScienceFoundation of China (Projects nos 71271012 71671011 and71332003)

References

[1] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Oxford UK 1975

[2] P F Peng ldquoImprovement and simulation of ant colony algo-rithm based on genetic generdquo Computer Engineering amp Appli-cations vol 46 no 4 pp 43ndash45 2010

[3] Q Zhu and S Chen ldquoA new ant evolution algorithm to resolveTSP problemrdquo in Proceedings of the 6th International Conference

on Machine Learning and Applications (ICMLA rsquo07) pp 62ndash66Cincinnati Ohio USA December 2007

[4] M Dorigo and L M Gambardella ldquoAnt colonies for thetravelling salesman problemrdquo BioSystems vol 43 no 2 pp 73ndash81 1997

[5] Y L Tsai Y J Yang and C-H Lin ldquoA dynamic decisionapproach for supplier selection using ant colony systemrdquo ExpertSystems with Applications vol 37 no 12 pp 8313ndash8321 2010

[6] F Abbattista N Abbattista and L Caponetti ldquoAn evolutionaryand cooperative agents model for optimizationrdquo in Proceedingsof the IEEE International Conference on Evolutionary Computa-tion pp 668ndash671 IEEE Perth Australia 1995

[7] A Acan ldquoGAACO A GA+ACO hybrid for faster and bettersearch capabilityrdquo in Proceedings of the 3rd International Work-shop on Ant Algorithms vol 2483 of Lecture Notes in ComputerScience pp 300ndash301 ANTS Brussels Belgium 2002

[8] D X Gong andX G Ruan ldquoA hybrid approach of GA andACOfor TSPrdquo in Proceedings of the 5th World Congress on IntelligentControl and Automation pp 2068ndash2072 IEEE HangzhouChina July 2004

[9] S Zhu W Dong and W Liu ldquoLogistics distribution routeoptimization based on genetic ant colony algorithmrdquo Journal ofChemical amp Pharmaceutical Research vol 6 no 6 pp 2264ndash2267 2014

[10] W G Zhang and T Y Lu ldquoThe research of genetic ant colonyalgorithm and its applicationrdquo Procedia Engineering vol 37 no2012 pp 101ndash106 2012

[11] Y H Zhang L Feng and Z Yang ldquoOptimization of clouddatabase route scheduling based on combination of geneticalgorithm and ant colony algorithmrdquo Precedia Engineering vol15 pp 3341ndash3345 2011

[12] Z Yao J Liu and Y-G Wang ldquoFusing genetic algorithm andant colony algorithm to optimize virtual enterprise partnerselection problemrdquo in Proceedings of the IEEE Congress onEvolutionary Computation (CEC rsquo08) pp 3614ndash3620 IEEEHong Kong June 2008

[13] Z Yao R Pan and F Lai ldquoImprovement of the fusing geneticalgorithmand ant colony algorithm in virtual enterprise partnerselection problemrdquo in Proceedings of the World Congress onComputer Science and Information Engineering (CSIE rsquo09) pp242ndash246 Los Angeles Calif USA April 2009

[14] H F Xiao and G Z Tan ldquoStudy improvement of the fusinggenetic algorithm and ant colony algorithm in virtual enterprisepartner selection problem on fusing genetic algorithm into antcolony algorithmrdquo Journal of Chinese Computer System vol 30no 3 pp 512ndash517 2009

[15] X M Li Z Mao and E Qi ldquoResearch on multi-supplier per-formancemeasurement based on genetic ant colony algorithmrdquoin Proceedings of the 1st ACMSIGEVO Summit on Genetic andEvolutionary Computation (GEC rsquo09) pp 867ndash870 2009

[16] S Gao Z Zhang andC Cao ldquoA novel ant colony genetic hybridalgorithmrdquo Journal of Software vol 5 no 11 pp 1179ndash1186 2010

[17] Y D Zhang and L N Wu ldquoA novel genetic ant colonyalgorithmrdquo Journal of Convergence Information Technology vol7 no 1 pp 268ndash274 2012

[18] M Bessedik F B-S Tayeb H Cheurfi and A BlizakldquoAn immunity-based hybrid genetic algorithms for permuta-tion flowshop scheduling problemsrdquo International Journal ofAdvanced Manufacturing Technology vol 85 no 9 pp 2459ndash2469 2016


[19] ZHAhmed ldquoExperimental analysis of crossover andmutationoperators on the quadratic assignment problemrdquo Annals ofOperations Research 2015

[20] X M Wang X Liu and G Liu ldquoPerformance comparisonof several kinds of improved genetic algorithmrdquo Journal ofChemical and Pharmaceutical Research vol 6 no 9 pp 463ndash468 2014

[21] M Lopez-Ibanez T Stutzle and M Dorigo ldquoAnt colonyoptimization a component-wise overviewrdquo IRIDIA-TechnicalReport Series TRIRIDIA2015-006 2015

[22] S H Niu S K Ong and A Y C Nee ldquoAn enhanced antcolony optimiser for multi-attribute partner selection in virtualenterprisesrdquo International Journal of Production Research vol50 no 8 pp 2286ndash2303 2012

[23] D E Aliabadi A Kaazemi and B Pourghannad ldquoA two-levelGA to solve an integrated multi-item supplier selection modelrdquoApplied Mathematics and Computation vol 219 no 14 pp7600ndash7615 2013

[24] D Simic V Svircevic and S Simic ldquoA hybrid evolutionarymodel for supplier assessment and selection in inbound logis-ticsrdquo Journal of Applied Logic vol 13 no 2 pp 138ndash147 2015

[25] P C Yang H M Wee S Pai and Y F Tseng ldquoSolving astochastic demandmulti-product supplier selectionmodel withservice level and budget constraints using genetic algorithmrdquoExpert Systems with Applications vol 38 no 12 pp 14773ndash14777 2011

[26] A Mazidi M Fakhrahmad and M Sadreddini ldquoA meta-heuristic approach to CVRP problem local search optimizationbased on GA and ant colonyrdquo Journal of Advance in ComputerResearch vol 7 no 1 pp 1ndash22 2016

[27] G F Dong W W Guo and K Tickle ldquoSolving the travelingsalesman problem using cooperative genetic ant systemsrdquoExpert Systems with Applications vol 39 no 5 pp 5006ndash50112012

[28] M J Liu Research on integration and performance of antcolony algorithm and genetic algorithm [PhD thesis] School ofScience China University of Geosciences Beijing China 2013

[29] Z-H Xiong S-K Li and J-H Chen ldquoHardwaresoftware par-titioning based on dynamic combination of genetic algorithmand ant algorithmrdquo Journal of Software vol 16 no 4 pp 503ndash512 2005

[30] Z J Ma ldquoPartner selection of supply chain alliance basedon genetic algorithmrdquo Academic Journal of System EngineeringTheory and Practice (Chinese Journal) vol 9 pp 81ndash84 2003

[31] T Stutzle and H H Hoos ldquoMAX-MIN ant systemrdquo FutureGeneration Computer Systems vol 16 no 8 pp 889ndash914 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


chose that path Therefore in the absence of volatilization apheromone trail on a popular path will accumulate rapidlyand help attract evergrowing numbers of ants to follow thatpath (positive feedback) [5] Through this natural stigmer-gic process without any prior knowledge real-world antcolonies establish optimal foraging paths through exchangesof information between individuals and mutual cooperationAs a swarm intelligence optimization algorithm ACO offersthe advantages of parallel computation self-learning andeffective information feedback However during the initialstages of ACO searches little or no information is availabletherefore ACO searches often converge slowly

The concept of integrating a GA and ACO was firstproposed byAbbattista et al [6] for exploiting the cooperativeeffect of ACO with the evolutionary effect of a GA Theyintegrated these search methods by using GAs to evolveoptimal parameter values for ACO Following their initialpublication numerous efforts have hybridizedGAs andACOand now developed fusing approaches can be roughly dividedinto four categories Approaches in the first category suchas the work of Acan et al [7 8] apply a GA to selectACO parameter values that optimize the performance ofant populations Methods in the second category generateinitial pheromone distributions through GAs which aresubsequently optimized with ACO [9ndash13] Methods in thethird category add genetic operations to ACO to diversifysolutions [14ndash16] Approaches in the fourth category combinethe initialization from the second category and the diversifi-cation from the third category [17] Approaches of the secondcategory fuse the GA and ACO to take advantage of theGArsquos rapid convergence and ACOrsquos parallelism and effectivefeedback this fusion inspires the present study The presentstudy improves the search for an appropriate fusing time thusenhancing the performance of the GA and ACO Through anumerical experiment of the supplier selection problem wedemonstrated the feasibility and efficiency of this novel fusedalgorithm

The supplier selection problem is an essential topic insupply chain management because the selection of propersupply partners can substantially improve a firmrsquos competitiveadvantages and can further influence the quality and pricesof the final products offered to customers [5] In the contextof previous research the supplier selection problem can bedescribed as amultigoal combinatorial optimization problemwith the objectives of achieving production targets andmaximum profits by selecting appropriate suppliers for eachmaterial under the condition of limited resources Becausethe supplier selection problem is a typical combinatorialoptimization problem we consider that solving it with ourfused GA and ACO algorithm could be noteworthy andshed new light on contemporary problems encountered inresearch on supplier selection

The remainder of this paper is organized as followsSection 2 reviews the literature regarding the developmentof the GA ACO and the fused GA and ACO algorithmsSection 3 describes the key points of our fused algorithm andelaborates on the modifications of fusing time the GA andACO Section 4 explains the design of a numerical experi-ment on the supplier selection problem and the optimization

of parameters the performance of the new fused algorithmis evaluated In the final section we offer the conclusions anddirections for future research

2 Literature Review

21 Genetic Algorithms and Ant Colony Optimization GAsand ACO are popular classes of intelligent heuristic algo-rithms which have been broadly applied in the field of opti-mization The initial versions were immediately improvedupon and improvements have continued to advance the per-formance of GAs andACO ForGAs substantial ameliorativework has been performed to optimize search performancewith features such as improved selectionmechanism strategyadaptive mutation probability [18] GA operators [19] andelitism selection mechanisms To improve the global searchcapability and convergence performance of GAs Wang etal [20] proposed four types of improved GAs namelyhierarchic GAs simulated annealing GAs simulated anneal-ing hierarchic GAs and adaptable GAs these methods canovercome the defects of traditional GAs by combining GAswith simulated annealing algorithms and modifying variouscoding methods In terms of ACO the main features of itsimprovement involve mechanisms to intensify the searchinvolving high-quality solutions and preserve a sufficientsearch space [21] Niu et al [22] stated that as a typical greedyheuristic algorithm ACO tends to become trapped in localoptimaThey proposed a method for guiding the search awayfrom local optima by adding a perturbation into the originalprobability Moreover a coefficient representing the effects ofan average pheromone trail updated pheromones to reducethe effects of the parameter Q

GA and ACO are promising because they can substan-tially increase the possibility of determining high-qualitysolutions for some complex combinatorial optimizationproblems such as the supplier selection problem To man-age an integrated multi-item supplier selection problem formaximizing the annual income of an entire supply chainAliabadi et al [23] presented a two-level GA (2LGA) modelbased on two types of variablesmdashbinary variables and realvariablesmdashof which the first layer was used for selectingsuppliers and the following layer was used for orderingthem Simic et al [24] proposed a GA performance valueconstraint model that used a grading variable for assessingthe performance of suppliers Yang et al [25] applied a GA toa stochastic-demand multiproduct supplier selection modelwith constraints of service level and budget where the highestvalue of the average expected profit and the lowest valueof the standard deviation were achieved through differentcombinations of crossover and mutation rates In referenceto the attribute-based ant colony system (AACS) Tsai et al[5] reported an examination of the critical factors the criteriafactors and weights were incorporated in the pheromoneupdate rule and the AACS was used to obtain the optimalsupplier according to a quantitative decision policy

22 FusedAlgorithm of aGAandACO A review of the extantalgorithms introduced to solve combinatorial optimization










V

Va

t0 tb tatc


Genetic algorithm


13702137041370613708

1371137121371413716

001 0009 0008 0007 0006 0005Fi

tnes

s val

ue

Constant







119875119888

=

1198751198880

119891 le 119891

1198751198881

(

1198751198880

1198751198881

)


119891 gt 119891

119875119898

=

1198751198980

1198911015840

le 119891

1198751198981

(

1198751198980

1198751198981

)


1198911015840

gt 119891

(1)

where 1198751198880and 119875



(1198751198881

lt 1198751198880

) and 1198751198981

(1198751198981

lt 1198751198980




120591119878119894119895

(119905 + 119899) = (1 minus 120588) 120591119878119894119895

(119905) + 120588Δ120591119878119894119895

(119905) (2)

Δ120591119878119894119895

(119905) =

119898

sum

119896=1

Δ120591119896

119878119894119895

(119905) (3)

Δ120591119896

119878119894119895

(119905) =

119876

119891max 119878119894119895

isin 119891max

0 else(4)

119876 = 1198760


119905

max119873) (5)



119878119894119895








1198880and 119875







(5) For (119873119905= 0 119873

119905lt 119873 119873

119905= 119873119905+ 2)



119888and mutation





(e) if (119875119898

lt 119903 le 119875119898














119896










119894119895(119905) can be written as (6)

where 119889+

(119905) and 119889minus


119876minus

119879minus

119863minus


119894119895 119905119894119895 119889119894119895 and 119888



119902 119908119894

119889 119908119894

119888 and 119908

119894





119891119894119895

=

119889minus

119894119895

119889+

119894119895+ 119889minus

119894119895

(6)

119889+

119894119895= 119908119894

119888

10038161003816100381610038161003816119888119894119895

minus 119862+10038161003816100381610038161003816

119862+

+ 119862minus

+ 119908119894

119902

10038161003816100381610038161003816119902119894119895

minus 119876+10038161003816100381610038161003816

119876+

+ 119876minus

+ 119908119894

119905

10038161003816100381610038161003816119905119894119895

minus 119879+10038161003816100381610038161003816

119879+

+ 119879minus

+ 119908119894

119889

10038161003816100381610038161003816119889119894119895

minus 119863+10038161003816100381610038161003816

119863+

+ 119863minus

(7)

119889minus

119894119895= 119908119894

119888

10038161003816100381610038161003816119888119894119895

minus 119862minus10038161003816100381610038161003816

119862+

+ 119862minus

+ 119908119894

119902

10038161003816100381610038161003816119902119894119895

minus 119876minus10038161003816100381610038161003816

119876+

+ 119876minus

+ 119908119894

119905

10038161003816100381610038161003816119905119894119895

minus 119879minus10038161003816100381610038161003816

119879+

+ 119879minus

+ 119908119894

119889

10038161003816100381610038161003816119889119894119895

minus 119863minus10038161003816100381610038161003816

119863+

+ 119863minus

(8)

max119891 =

119873

sum

119894=1

119891119894119895

(119905) 119895 = 1 119872119894 (9)





1352135413561358

1361362136413661368

1371372

30 50 70 90 110 130 150 170

Fitn

ess v

alue

Coefficient value

(a) Ant number

135

1355

136

1365

137

1375

01 02 03 04 05 06 07 08 09 1

Fitn

ess v

alue

Coefficient value

12913

131132133134135136137138

1 2 3 4 5 6 7 8 9

Fitn

ess v

alue

Coefficient value


135713581359

136136113621363136413651366

01 02 03 04 05 06 07 08 09 1

Fitn

ess v

alue

Coefficient value












135

13

125

12

115

11

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant 1Ant 2Ant 3

Ant 4Ant 5



11

115

12

125

13

135

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

124126128

13132134136138

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant number = 5

Ant number = 10


123124125126127128129

13

011 022 033 044 055 066 077 088 099

Fitn

ess v

alue

Coefficient value






02468

10121416

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Supp

lier

Material

123

45The best route





10105

11115

12125

13135

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Fitn

ess v

alue

Iteration

GA-ACOGAACO

GA-ACOGAACO

GA-ACOGAACO

1341345

1351355

1361365

1371375

138

25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Fitn

ess v

alue

Iteration

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

Iteration


00005

0010015

0020025

0030035

004

Evol

utio

nary

rate















Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















V

Va

t0 tb tatc


Genetic algorithm


13702137041370613708

1371137121371413716

001 0009 0008 0007 0006 0005Fi

tnes

s val

ue

Constant







119875119888

=

1198751198880

119891 le 119891

1198751198881

(

1198751198880

1198751198881

)


119891 gt 119891

119875119898

=

1198751198980

1198911015840

le 119891

1198751198981

(

1198751198980

1198751198981

)


1198911015840

gt 119891

(1)

where 1198751198880and 119875



(1198751198881

lt 1198751198880

) and 1198751198981

(1198751198981

lt 1198751198980




120591119878119894119895

(119905 + 119899) = (1 minus 120588) 120591119878119894119895

(119905) + 120588Δ120591119878119894119895

(119905) (2)

Δ120591119878119894119895

(119905) =

119898

sum

119896=1

Δ120591119896

119878119894119895

(119905) (3)

Δ120591119896

119878119894119895

(119905) =

119876

119891max 119878119894119895

isin 119891max

0 else(4)

119876 = 1198760


119905

max119873) (5)



119878119894119895








1198880and 119875







(5) For (119873119905= 0 119873

119905lt 119873 119873

119905= 119873119905+ 2)



119888and mutation





(e) if (119875119898

lt 119903 le 119875119898














119896










119894119895(119905) can be written as (6)

where 119889+

(119905) and 119889minus


119876minus

119879minus

119863minus


119894119895 119905119894119895 119889119894119895 and 119888



119902 119908119894

119889 119908119894

119888 and 119908

119894





119891119894119895

=

119889minus

119894119895

119889+

119894119895+ 119889minus

119894119895

(6)

119889+

119894119895= 119908119894

119888

10038161003816100381610038161003816119888119894119895

minus 119862+10038161003816100381610038161003816

119862+

+ 119862minus

+ 119908119894

119902

10038161003816100381610038161003816119902119894119895

minus 119876+10038161003816100381610038161003816

119876+

+ 119876minus

+ 119908119894

119905

10038161003816100381610038161003816119905119894119895

minus 119879+10038161003816100381610038161003816

119879+

+ 119879minus

+ 119908119894

119889

10038161003816100381610038161003816119889119894119895

minus 119863+10038161003816100381610038161003816

119863+

+ 119863minus

(7)

119889minus

119894119895= 119908119894

119888

10038161003816100381610038161003816119888119894119895

minus 119862minus10038161003816100381610038161003816

119862+

+ 119862minus

+ 119908119894

119902

10038161003816100381610038161003816119902119894119895

minus 119876minus10038161003816100381610038161003816

119876+

+ 119876minus

+ 119908119894

119905

10038161003816100381610038161003816119905119894119895

minus 119879minus10038161003816100381610038161003816

119879+

+ 119879minus

+ 119908119894

119889

10038161003816100381610038161003816119889119894119895

minus 119863minus10038161003816100381610038161003816

119863+

+ 119863minus

(8)

max119891 =

119873

sum

119894=1

119891119894119895

(119905) 119895 = 1 119872119894 (9)





1352135413561358

1361362136413661368

1371372

30 50 70 90 110 130 150 170

Fitn

ess v

alue

Coefficient value

(a) Ant number

135

1355

136

1365

137

1375

01 02 03 04 05 06 07 08 09 1

Fitn

ess v

alue

Coefficient value

12913

131132133134135136137138

1 2 3 4 5 6 7 8 9

Fitn

ess v

alue

Coefficient value


135713581359

136136113621363136413651366

01 02 03 04 05 06 07 08 09 1

Fitn

ess v

alue

Coefficient value












135

13

125

12

115

11

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant 1Ant 2Ant 3

Ant 4Ant 5



11

115

12

125

13

135

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

124126128

13132134136138

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant number = 5

Ant number = 10


123124125126127128129

13

011 022 033 044 055 066 077 088 099

Fitn

ess v

alue

Coefficient value






02468

10121416

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Supp

lier

Material

123

45The best route





10105

11115

12125

13135

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Fitn

ess v

alue

Iteration

GA-ACOGAACO

GA-ACOGAACO

GA-ACOGAACO

1341345

1351355

1361365

1371375

138

25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Fitn

ess v

alue

Iteration

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

Iteration


00005

0010015

0020025

0030035

004

Evol

utio

nary

rate















Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119875119888

=

1198751198880

119891 le 119891

1198751198881

(

1198751198880

1198751198881

)


119891 gt 119891

119875119898

=

1198751198980

1198911015840

le 119891

1198751198981

(

1198751198980

1198751198981

)


1198911015840

gt 119891

(1)

where 1198751198880and 119875



(1198751198881

lt 1198751198880

) and 1198751198981

(1198751198981

lt 1198751198980




120591119878119894119895

(119905 + 119899) = (1 minus 120588) 120591119878119894119895

(119905) + 120588Δ120591119878119894119895

(119905) (2)

Δ120591119878119894119895

(119905) =

119898

sum

119896=1

Δ120591119896

119878119894119895

(119905) (3)

Δ120591119896

119878119894119895

(119905) =

119876

119891max 119878119894119895

isin 119891max

0 else(4)

119876 = 1198760


119905

max119873) (5)



119878119894119895








1198880and 119875







(5) For (119873119905= 0 119873

119905lt 119873 119873

119905= 119873119905+ 2)



119888and mutation





(e) if (119875119898

lt 119903 le 119875119898














119896










119894119895(119905) can be written as (6)

where 119889+

(119905) and 119889minus


119876minus

119879minus

119863minus


119894119895 119905119894119895 119889119894119895 and 119888



119902 119908119894

119889 119908119894

119888 and 119908

119894





119891119894119895

=

119889minus

119894119895

119889+

119894119895+ 119889minus

119894119895

(6)

119889+

119894119895= 119908119894

119888

10038161003816100381610038161003816119888119894119895

minus 119862+10038161003816100381610038161003816

119862+

+ 119862minus

+ 119908119894

119902

10038161003816100381610038161003816119902119894119895

minus 119876+10038161003816100381610038161003816

119876+

+ 119876minus

+ 119908119894

119905

10038161003816100381610038161003816119905119894119895

minus 119879+10038161003816100381610038161003816

119879+

+ 119879minus

+ 119908119894

119889

10038161003816100381610038161003816119889119894119895

minus 119863+10038161003816100381610038161003816

119863+

+ 119863minus

(7)

119889minus

119894119895= 119908119894

119888

10038161003816100381610038161003816119888119894119895

minus 119862minus10038161003816100381610038161003816

119862+

+ 119862minus

+ 119908119894

119902

10038161003816100381610038161003816119902119894119895

minus 119876minus10038161003816100381610038161003816

119876+

+ 119876minus

+ 119908119894

119905

10038161003816100381610038161003816119905119894119895

minus 119879minus10038161003816100381610038161003816

119879+

+ 119879minus

+ 119908119894

119889

10038161003816100381610038161003816119889119894119895

minus 119863minus10038161003816100381610038161003816

119863+

+ 119863minus

(8)

max119891 =

119873

sum

119894=1

119891119894119895

(119905) 119895 = 1 119872119894 (9)





1352135413561358

1361362136413661368

1371372

30 50 70 90 110 130 150 170

Fitn

ess v

alue

Coefficient value

(a) Ant number

135

1355

136

1365

137

1375

01 02 03 04 05 06 07 08 09 1

Fitn

ess v

alue

Coefficient value

12913

131132133134135136137138

1 2 3 4 5 6 7 8 9

Fitn

ess v

alue

Coefficient value


135713581359

136136113621363136413651366

01 02 03 04 05 06 07 08 09 1

Fitn

ess v

alue

Coefficient value












135

13

125

12

115

11

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant 1Ant 2Ant 3

Ant 4Ant 5



11

115

12

125

13

135

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

124126128

13132134136138

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant number = 5

Ant number = 10


123124125126127128129

13

011 022 033 044 055 066 077 088 099

Fitn

ess v

alue

Coefficient value






02468

10121416

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Supp

lier

Material

123

45The best route





10105

11115

12125

13135

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Fitn

ess v

alue

Iteration

GA-ACOGAACO

GA-ACOGAACO

GA-ACOGAACO

1341345

1351355

1361365

1371375

138

25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Fitn

ess v

alue

Iteration

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

Iteration


00005

0010015

0020025

0030035

004

Evol

utio

nary

rate















Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896










119894119895(119905) can be written as (6)

where 119889+

(119905) and 119889minus


119876minus

119879minus

119863minus


119894119895 119905119894119895 119889119894119895 and 119888



119902 119908119894

119889 119908119894

119888 and 119908

119894





119891119894119895

=

119889minus

119894119895

119889+

119894119895+ 119889minus

119894119895

(6)

119889+

119894119895= 119908119894

119888

10038161003816100381610038161003816119888119894119895

minus 119862+10038161003816100381610038161003816

119862+

+ 119862minus

+ 119908119894

119902

10038161003816100381610038161003816119902119894119895

minus 119876+10038161003816100381610038161003816

119876+

+ 119876minus

+ 119908119894

119905

10038161003816100381610038161003816119905119894119895

minus 119879+10038161003816100381610038161003816

119879+

+ 119879minus

+ 119908119894

119889

10038161003816100381610038161003816119889119894119895

minus 119863+10038161003816100381610038161003816

119863+

+ 119863minus

(7)

119889minus

119894119895= 119908119894

119888

10038161003816100381610038161003816119888119894119895

minus 119862minus10038161003816100381610038161003816

119862+

+ 119862minus

+ 119908119894

119902

10038161003816100381610038161003816119902119894119895

minus 119876minus10038161003816100381610038161003816

119876+

+ 119876minus

+ 119908119894

119905

10038161003816100381610038161003816119905119894119895

minus 119879minus10038161003816100381610038161003816

119879+

+ 119879minus

+ 119908119894

119889

10038161003816100381610038161003816119889119894119895

minus 119863minus10038161003816100381610038161003816

119863+

+ 119863minus

(8)

max119891 =

119873

sum

119894=1

119891119894119895

(119905) 119895 = 1 119872119894 (9)





1352135413561358

1361362136413661368

1371372

30 50 70 90 110 130 150 170

Fitn

ess v

alue

Coefficient value

(a) Ant number

135

1355

136

1365

137

1375

01 02 03 04 05 06 07 08 09 1

Fitn

ess v

alue

Coefficient value

12913

131132133134135136137138

1 2 3 4 5 6 7 8 9

Fitn

ess v

alue

Coefficient value


135713581359

136136113621363136413651366

01 02 03 04 05 06 07 08 09 1

Fitn

ess v

alue

Coefficient value












135

13

125

12

115

11

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant 1Ant 2Ant 3

Ant 4Ant 5



11

115

12

125

13

135

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

124126128

13132134136138

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant number = 5

Ant number = 10


123124125126127128129

13

011 022 033 044 055 066 077 088 099

Fitn

ess v

alue

Coefficient value






02468

10121416

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Supp

lier

Material

123

45The best route





10105

11115

12125

13135

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Fitn

ess v

alue

Iteration

GA-ACOGAACO

GA-ACOGAACO

GA-ACOGAACO

1341345

1351355

1361365

1371375

138

25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Fitn

ess v

alue

Iteration

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

Iteration


00005

0010015

0020025

0030035

004

Evol

utio

nary

rate















Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1352135413561358

1361362136413661368

1371372

30 50 70 90 110 130 150 170

Fitn

ess v

alue

Coefficient value

(a) Ant number

135

1355

136

1365

137

1375

01 02 03 04 05 06 07 08 09 1

Fitn

ess v

alue

Coefficient value

12913

131132133134135136137138

1 2 3 4 5 6 7 8 9

Fitn

ess v

alue

Coefficient value


135713581359

136136113621363136413651366

01 02 03 04 05 06 07 08 09 1

Fitn

ess v

alue

Coefficient value












135

13

125

12

115

11

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant 1Ant 2Ant 3

Ant 4Ant 5



11

115

12

125

13

135

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

124126128

13132134136138

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant number = 5

Ant number = 10


123124125126127128129

13

011 022 033 044 055 066 077 088 099

Fitn

ess v

alue

Coefficient value






02468

10121416

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Supp

lier

Material

123

45The best route





10105

11115

12125

13135

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Fitn

ess v

alue

Iteration

GA-ACOGAACO

GA-ACOGAACO

GA-ACOGAACO

1341345

1351355

1361365

1371375

138

25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Fitn

ess v

alue

Iteration

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

Iteration


00005

0010015

0020025

0030035

004

Evol

utio

nary

rate















Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










135

13

125

12

115

11

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant 1Ant 2Ant 3

Ant 4Ant 5



11

115

12

125

13

135

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

124126128

13132134136138

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Fitn

ess v

alue

Iteration

Ant number = 5

Ant number = 10


123124125126127128129

13

011 022 033 044 055 066 077 088 099

Fitn

ess v

alue

Coefficient value






02468

10121416

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Supp

lier

Material

123

45The best route





10105

11115

12125

13135

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Fitn

ess v

alue

Iteration

GA-ACOGAACO

GA-ACOGAACO

GA-ACOGAACO

1341345

1351355

1361365

1371375

138

25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Fitn

ess v

alue

Iteration

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

Iteration


00005

0010015

0020025

0030035

004

Evol

utio

nary

rate















Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10105

11115

12125

13135

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Fitn

ess v

alue

Iteration

GA-ACOGAACO

GA-ACOGAACO

GA-ACOGAACO

1341345

1351355

1361365

1371375

138

25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Fitn

ess v

alue

Iteration

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

Iteration


00005

0010015

0020025

0030035

004

Evol

utio

nary

rate















Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a novel fused optimization algorithm of...

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