This research was partially supported by the Serbian Ministry of Science and Ecology under
project 144007. The authors are grateful to Ivana Ljubić for help in testing and to Vladimir Filipović for useful suggestions and comments.In this paper a genetic algorithm (GA) for the task assignment
problem (TAP) is considered.An integer representation with standard genetic operators is used. Computational results are presented for instances
from the literature, and compared to optimal solutions obtained by the
CPLEX solver. It can be seen that the proposed GA approach reaches
17 of 20 optimal solutions. The GA solutions are obtained in a quite a short
amount of computational time

Kratica, Jozef

Marić, Miroslav

Savić, Aleksandar

Tošić, Dušan

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica J. Computing 2 (2008), 267–276GENETIC ALGORITHM APPROACH FOR SOLVING THETASK ASSIGNMENT PROBLEM*Aleksandar Savic´, Dusˇan Tosˇic´, Miroslav Maric´, Jozef KraticaAbstract. In this paper a genetic algorithm (GA) for the task assignmentproblem (TAP) is considered.An integer representation with standard ge-netic operators is used. Computational results are presented for instancesfrom the literature, and compared to optimal solutions obtained by theCPLEX solver. It can be seen that the proposed GA approach reaches17 of 20 optimal solutions. The GA solutions are obtained in a quite a shortamount of computational time.1. Introduction. In recent time, with the growth of computerization,some problems have arisen traced in earlier times, but nowadays mostly associ-ated with the optimization of CPU time. In earlier times, the task assignmentACM Computing Classification System (1998): G.1.6.Key words: evolutionary approach; genetic algorithms; assignment problems, multiprocessorsystems, combinatorial optimization.*This research was partially supported by the Serbian Ministry of Science and Ecology underproject 144007. The authors are grateful to Ivana Ljubic´ for help in testing and to VladimirFilipovic´ for useful suggestions and comments.268 Aleksandar Savic´, Dusˇan Tosˇic´, Miroslav Maric´, Jozef Kraticawas solved manually by the overseer and the distribution was done manually be-tween working parties on construction and other working sites. The idea was toassign tasks to the working parties minimizing jobs interferences and making thetotal working time as short as possible. Nowadays computers replace workingparties and the task assignment is bounded with minimally spent CPU time withthe additional requirement that communication costs between parties should beminimal. Some similar problems can be seen in [3, 6, 12].In this paper a special Task Assignment Problem (TAP), sometimes calledTAS with non-uniform communication costs, is studied. The problem is presentfrom the beginning of computer era, but only now the new techniques capable ofsolving it are being developed.The problem of task assignment with non-uniform communication costscan be modeled as a nonlinear integer 0-1 minimization problem.In this paper the application based on genetic algorithms for solving TAPis described. Because this problem is fully 0-1 integer, it is in the area of researchin recent developments of genetic algorithms (for example, see [1, 9, 10]). Thereare well many documented applications considering large optimization problems[5, 8, 13, 14], and the ideas from these applications could be applied for solvingTAP.Let us first describe the mathematical formulation of TAP, then in briefexplain GA and finally present experimental results.2. Mathematical formulation. The problem of task assignment withnon-uniform communication costs is related to finding an assignment of N tasksto M processors providing that sum of:• total cost of execution for given tasks,• total cost of all communications between processors, while they executingallocated tasks,is minimal.We will use quadratic integer programming formulation from [2].Let there be N tasks, M processors, and eik be the cost of executing taski on a processor k. Let cijkl be the communication cost between tasks i and jif they are respectively assigned to processors k and l. Let us denote with 0–1integer variable xik which has value 1 if task i is assigned to processor k.Genetic algorithm approach for solving the task assignment problem 269Now we can formulate the quadratic integer programming model for TAPas follows:(1) minN∑i=1M∑k=1eikxik +N−1∑i=1N∑j=i+1M∑k=1M∑l=1cijklxikxjlsubject to(2)M∑k=1xik = 1, i = 1, . . . , N(3) xik ∈ {0, 1} i = 1, . . . , N, k = 1, . . . ,MAs can be seen, the objective function is non-linear and this optimizationproblem cannot be easily solved by using an exact algorithm. The constraints(2) are natural and reflect the fact that any task can be executed only on oneprocessor. Now we will describe the properties of the genetic algorithm and someof its parts for solving the presented problem.3. Proposed GA method. The GA algorithms are stochastic me-thods for searching and finding best solutions. They are motivated by processesin natural world trying to emulate them. Similar to nature, GA works with in-dividuals constituting a population. Each individual represents some solution tothe problem. As in nature, the individuals better suited for survival are at thesame time individuals that are better attain the optimal solution. These indi-viduals are favored for passing into the next generation. The passing on of goodqualities is accomplished by using genetic operators of crossover and mutation.The decision what individual has better quality for passing into next generationis obtained by evaluation of the fitness function. This process of betterment ofindividuals in the population is iteratively continued until an optimum is achievedor some other stopping criterion is applied. The detailed description of GAs isout of this paper’s scope and can be found in [11].The encoding of individuals in our case is integer. The genetic code hasa length of N , where every gene corresponds to one particular task. This means270 Aleksandar Savic´, Dusˇan Tosˇic´, Miroslav Maric´, Jozef Kraticathat every gene represents the ordinal numeral of the processor on which the taskis executed. For example, if the ith gene has value k this means that xik = 1 andxip = 0 for every p, p 6= k.The very important parts of proposed GA are: Fitness Function, Se-lection, Crossover and Mutation. In the following paragraphs, their use andimportance to the whole GA application will be explained.The function find that decides the best suitability of an individual topass into the next generation is the fitness function. The values of this functionare computed by scaling objective values objind of all individuals into the interval[0,1], so that the best suitable individual indmax gets value 1 and the worst indmingets 0. Explicitly, find =objindmin − objindobjindmin − objindmax. Now the individuals are arrangedin a non-increasing order by their best fitness: f1 ≥ f2 ≥ · · · ≥ fNpop, where Npopis number of individuals in population.All elite individuals (their number being Nelite) are automatically passedinto the next generation. All non-elite individuals (their number being Nnnel =Npop − Nelite) are subject to genetic operators. In this way the computationaltime is reduced because the objective function of elite individuals is same in thenext generation and need to be calculated only once, in the first generation.Individuals with the same genetic code in the population must be avoided,so their fitness is set to 0 on all occurrences, except the first one. Also, thenumber of individuals with the same objective function, but different geneticcode, must be limited by some constant Nrv. To avoid this problem, the fitnessof all individuals with the same value of objective function, but different geneticmaterial, will be set to 0 except for the first Nrv of them.Selection operators are applied on all non-elite individuals, so they choosewhich of these individuals will have offspring in the next generation. This is donethrough tournaments. From the whole population an a priori set number of indi-viduals is chosen to participate in the tournament. The number of participants iscalled tournament size. The choosing of individuals is done randomly. The winnerof the tournament is the individual with highest value of the objective function.The number of tournaments is equal to the number of non-elite individuals Nnnel,so that exactly Nnnel parents can be chosen for crossover. The same individualfrom the current generation can participate in more than one tournament. In thestandard tournament selection, the tournament size is integer and this choice canreduce the efficiency of the algorithm.Because of this, for selection an improved tournament selection opera-Genetic algorithm approach for solving the task assignment problem 271tor, Fine-grained tournament selection – FGTS [4, 5] is implemented. Here, thetournament size is a real parameter Ftour, representing the preferable averagetournament size. In this procedure, there are two types of tournaments. One isheld k1 times, with tournament size bFtourc, and the other type is held k2 times,with tournament size dFtoure. From here Ftour ≈k1 · bFtourc + k2 · dFtoureNnnel.To get satisfactory results of GA, a good ratio between the number of eliteand the number of non-elite individuals is necessary. For example, for Npop = 150an adequate proportion is Nelite = 100 and Nnnel = 50. The corresponding k1and k2 parameters for deciding the tournament size are 30 and 20, respectively.Typically, for Npop = 150 an adequate maximum of individuals with the samefitness (with the same value of objective function), is Nrv = 40.As can be seen in [4, 5, 13], FGTS performs best with value of Ftour seton 5.4. Leaning on experience presented in the cited works, the same value isused in this paper.In the crossover operator, non-elite individuals are randomly paired inbNnnel/2c pairs for exchange of genes with the intention to produce offspringwith potentially better suitability.The applying of the crossover operator on cho-sen pair of parents produces two offsprings. In this paper a standard one-pointcrossover operator is used. This operator exchanges all genes between geneticcodes of parents to produce offspring.The probability of applying the crossoveroperator is 85%. This means that approximately 85% of the pairs of individualswill exchange genes.In genetic algorithm the simple mutation operator with frozen genes isused. This operator changes a randomly selected gene in the genetic code of anindividual with some mutation rate. For the improvement of GA, a modificationis included dealing with so-called frozen genes. Sometimes it happens that allindividuals in the population have the same gene in a certain position. Thesegenes are called frozen. With the frozen genes a problem could arise because theyreduce the search space and increase the possibility of premature convergence.Selection and crossover operators cannot change the frozen genes, because allindividuals in the population have them in the same position. The basic mutationrate is 0.4/N .The mentioned improvement of GA consists in increasing the mutationrate only on frozen genes. In each generation it is determined in advance whichgenes are frozen. Then, the mutation rate for these genes is increased. In theproposed GA the increase is 2.5 times greater (1./N) than the basic mutation272 Aleksandar Savic´, Dusˇan Tosˇic´, Miroslav Maric´, Jozef Kraticarate on unfrozen genes.The initialization of population is done in a random way. So, the pop-ulation in the first generation is the most heterogeneous and diversified geneticpool.The performance of the proposed GA is improved by using a cachingtechnique. The main idea behind this technique is to avoid the evaluation ofthe objective function for individuals with the same genetic code. The valuesof individuals for which the objective function was already computed are storedby the least recently used (LRU) caching technique into the hash-queue datastructure. Because of this, whenever an individual with the same genetic codeis generated, the value of its objective function is not computed, but is found incache memory. This procedure can result in significant time saving. The numberof calculated values of the objective function in this implementation is limited to5000. If the cache memory is full, then we remove the least recently used cashmemory block. Detailed information about caching GA can be found in [7].4. Experimental results. All computations were executed on a QuadCore 2.5 GHz PC computer with 4 GB RAM. The genetic algorithm was codedin C. For experimental testings in our implementation the instances described at[2] were used. These instances include different numbers of tasks (N = 10, 15)and different numbers of processors (M = 3, 5). For each pair of task-processor(N,M), there is a set of ten instances.The finishing criterion of GA is the maximal number of generations Ngen= 5000. The algorithm also stops if the best individual or best objective valueremains unchanged through Nrep = 2000 successive generations. Since the resultsof GA are nondeterministic, our GA was run on a single processor 20 times foreach of the instances.Table 1 summarizes the results of the executions of our application onthese instances. In the first column the names of instances are given. The nameof the instance carries information about the number of tasks N , the numberof processors M and the number of generated cases with same N and M . (Forexample, the instance tassnu 10 3 1 is an instance which has N = 10 tasks onM = 3 processors and it is the first case generated for this N and M). The secondand third columns contain optimal solutions and running times of the CPLEXsolver. The best GA values GAsol are given in the following column. The markopt is given if an optimal solution is reached and there is no difference betweenGenetic algorithm approach for solving the task assignment problem 273that solution and the solution obtained by CPLEX.Average running times needed to detect the best GA values are given inthe t column. On average, GA finished after gen generations. The quality ofthe solution in all 20 executions is evaluated as a percentage gap named agap,with respect to the optimal solution solopt, by using standard deviation σ of theTable 1. GA results on TAP instancesCPLEX GAInstance name sol t sol t gen agap σ eval cache(opt) (sec) (sec) (%) (%) (%)tassnu 10 3 1 −719 < 1 opt 0.162 2014 0.000 0.000 12778 87.3tassnu 10 3 2 −790 < 1 opt 0.165 2023 0.000 0.000 14800 85.4tassnu 10 3 3 −624 < 1 opt 0.210 2624 0.641 1.208 28130 78.5tassnu 10 3 4 −734 < 1 opt 0.172 2128 0.000 0.000 18161 83.0tassnu 10 3 5 −871 < 1 opt 0.162 2021 0.000 0.000 15209 85.0tassnu 10 3 6 −677 < 1 opt 0.165 2018 0.214 0.958 14270 85.9tassnu 10 3 7 −613 < 1 opt 0.162 2026 0.000 0.000 12294 87.9tassnu 10 3 8 −495 < 1 opt 0.164 2016 0.000 0.000 11409 88.7tassnu 10 3 9 −750 < 1 opt 0.163 2020 0.000 0.000 14501 85.7tassnu 10 3 10 −486 < 1 opt 0.164 2017 0.021 0.092 20510 79.7tassnu 15 5 1 −1985 51832 opt 0.254 2285 3.131 6.743 65830 42.4tassnu 15 5 2 −1568 129840 −1539 0.328 2900 4.827 1.579 89965 37.9tassnu 15 5 3 −1892 52955 −1856 0.257 2223 9.905 2.110 75729 32.0tassnu 15 5 4 −1806 91146 opt 0.292 2545 1.462 2.754 85165 32.9tassnu 15 5 5 −1881 78795 opt 0.248 2192 2.818 2.967 69018 37.1tassnu 15 5 6 −1950 79872 opt 0.254 2256 5.285 5.333 73777 34.8tassnu 15 5 7 −1893 51547 opt 0.236 2072 4.691 3.892 70383 32.2tassnu 15 5 8 −1733 108092 opt 0.241 2100 2.796 1.056 73421 30.2tassnu 15 5 9 −1798 107982 −1780 0.246 2183 2.406 2.355 72580 33.8tassnu 15 5 10 −1763 109963 opt 0.246 2159 4.169 2.869 71855 33.3274 Aleksandar Savic´, Dusˇan Tosˇic´, Miroslav Maric´, Jozef Kraticaaverage gap. The percentage gap agap is defined as agap =12020∑i=1gapi, wheregapi = 100 ∗GAi − soloptsoloptand GAi represents the GA solution obtained in thei-th running, while σ is the standard deviation of gapi, i = 1, 2, . . . , 20, obtainedby the formula σ =√12020∑i=1(gapi − agap)2. The last two columns are relatedto the caching: eval represents the average number of evaluations, while cachedisplays the savings (in percent) achieved by using caching technique.As can be seen in Table 1, GA reached 17 of 20 optimal solutions andthe running time on all instances is really small. The average execution on thebiggest instance is less than half a second. The CPLEX solver also runs very faston instances with 10 tasks, but on larger instances, with 15 tasks, the runningtimes are huge (> 50 000 seconds). Although the GA algorithm did not reach theoptimal solution for 3 of 20 instances, this fact is overwhelmed by its speed. Itis obvious that CPLEX cannot handle larger instances with more than 15 tasks,but GA will quickly give solutions of acceptable quality even in these instances.5. Conclusions. The GA metaheuristic for solving TAP is presented.The integer representation of the task assignment was used based on the non-linear integer 0-1 optimization problem. The fine-grained tournament selection,one-point crossover and simple mutation with frozen genes were used. The com-putational performance of GA was additionally improved by caching GA tech-nique. For almost all instances, except three, GA calculates solutions matchingoptimal ones and obtained in a very small running time.Based on the presented results, we can conclude that GA has the potentialof being a useful metaheuristic for solving other similar problems, while TAPcould be considered with additional constraints. Parallelization of the GA andits hybridization with exact methods are most promising directions of futurework.REFERE NCES[1] Djuric´ B., J. Kratica, D. Tosˇic´, V. Filipovic´. Solving the maximallybalanced connected partition problem in graphs by using genetic algorithm.Computing and Informatics, 27, No. 3, 2008, 341–354.Genetic algorithm approach for solving the task assignment problem 275[2] Elloumi S. The task assignment problem, a library of instances.http://cedric.cnam.fr/oc/TAP/TAP.html, 2004.[3] Fidanova S., M. Durchova. Ant Algorithm for Grid Scheduling Prob-lem. Large Scale computing, Lecture Notes in Computer Science, vol. 3743,Springer, Germany, 2006, 405–412.[4] Filipovic´ V. Fine-grained Tournament Selection Operator in Genetic Al-gorithms. Computing and Informatics, 22 (2003), 143–161.[5] Filipovic´ V. Selection and migration operators and Web services in paral-lel evolutionary algorithms. PhD thesis, University of Belgrade, Faculty ofMathematics, 2006 (in Serbian).[6] Gong L., X.H. Sun, E. Waston. Performance Modeling and Predictionof Non-Distributed Network Computing. IEEE Transaction on Computers,51, No 9 (2002), 1041–1055.[7] Kratica J. Improving Performances of the Genetic Algorithm by Caching.Computers and Artificial Intelligence, 18 (1999), 271–283.[8] Kratica J., V. Kovacˇevic´-Vujcˇic´, M. Cˇangalovic´. Computing strongmetric dimension of some special classes of graphs by genetic algorithms.Yugoslav Journal of Operations Research, 18, No. 2 (2008), 143–151.[9] Kovacˇevic´ J. Hybrid Genetic Algorithm For Solving The Low-Autocorrelation Binary Sequence Problem. Yugoslav Journal of OperationsResearch, in press.[10] Maric´ M. An efficient genetic algorithm for solving the multi-level unca-pacitated facility location problem. Computing and Informatics, in press.[11] Mitchell M. Introduction to genetic algorithms. Cambridge, Massa-chusetts, MIT Press, 1999.[12] Schopf M. J. General Architecture for Scheduling on the Grid. Specia issueof JPDC on Grid Computing, 2002,http://www.mcs.anl.gov/~schopf/Pubs/sched.arch.2002.pdf, 2008.[13] Stanimirovic´ Z. Genetic algorithms for solving some NP-hard hub locationproblems. Ph. D. thesis, University of Belgr. Faculty of Mathematics, 2007.276 Aleksandar Savic´, Dusˇan Tosˇic´, Miroslav Maric´, Jozef Kratica[14] Stanimirovic´ Z. A genetic algorithm approach for the capacitated singleallocation p-hub median problem. Computing and Informatics, 27 (2008),in press.Aleksandar Savic´Dusˇan Tosˇic´Miroslav Maric´Faculty of MathematicsUniversity of Belgrade,Studentski trg 16/IV11 000 Belgrade, Serbiae-mail: aleks3rd@eunet.yudtosic@matf.bg.ac.yumaricm@matf.bg.ac.yuJozef KraticaMathematical InstituteSerbian Academy of Sciences and ArtsKneza Mihaila 36/III11000 Belgrade, Serbiae-mail: jkratica@mi.sanu.ac.yuReceived September 25, 2008Final Accepted October 15, 2008

Genetic Algorithm Approach for Solving the Task Assignment Problem

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Genetic Algorithm Approach for Solving the Task Assignment Problem

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