Fundação para a Ciência e a Tecnologia (FCT

Azad, Md. Abul Kalam

Fernandes, Edite Manuela da G. P.

Rocha, Ana Maria A. C.

Universidade do Minho: RepositoriUM

Solving 0–1 Quadratic Knapsack Problems with aPopulation-based Artificial Fish Swarm AlgorithmMd. Abul Kalam Azad1, Ana Maria A.C. Rocha1,2 andEdite M.G.P. Fernandes11 Algoritmi R&D Centre, University of Minho, 4710-057 Braga, Portugalakazad@dps.uminho.pt; emgpf@dps.uminho.pt2 Department of Production and Systems, University of Minho, 4710-057 Braga,Portugal arocha@dps.uminho.ptExtended Abstract1 IntroductionThe 0–1 Quadratic Knapsack Problem (QKP) consists in maximizing a quadraticobjective function subject to a linear capacity constraint. This problem has beenintroduced by Gallo et al. [2] and may be expressed as follows:maximize f(x) ≡n∑i=1pixi +n−1∑i=1n∑j=i+1pijxixjsubject ton∑i=1aixi ≤ bxi ∈ {0, 1}, i = 1, 2, . . . , n,(1)where the coefficients pi, ai(i = 1, 2, . . . , n) and pij(i = 1, 2, . . . , n − 1, j =i + 1, . . . , n) are positive integers and b is an integer such that max{ai : i =1, 2, . . . , n} ≤ b <∑ni=1 ai. Here pi is a profit achieved if item i is selected andpij is a profit achieved if both items i and j (j > i) are selected. The goal is to finda subset of n items that yields maximum profit f without exceeding capacity b.The QKP arises in a variety of real world applications including finance, VLSIdesign, compiler construction, telecommunication, flexible manufacturing systems,locations, hydrological studies. Classical graph and hypergraph partitioning prob-lems can also be formulated as the QKP. Several deterministic [1, 2] as well asstochastic solution methods [4, 8] have been proposed to solve (1).Recently, a population-based artificial fish swarm algorithm that simulates thebehavior of the fish swarm inside water was proposed [3, 6]. Applying to the opti-mization problem, generally a ‘fish’ represents an individual point in a population.Fishes desire to stay close to the swarm, to protect themselves from predators andto look for food, and to avoid collisions within the group. In this paper, we proposea binary version of the artificial fish swarm algorithm for solving (1).12 A Binary Artificial Fish Swarm AlgorithmHere we will present the proposed binary version of the artificial fish swarm algo-rithm for solving (1) and simply denote it by bAFSA.InitializationThe binary artificial fish swarm algorithm uses a population of N individual points(the fish swarm) xi, i = 1, 2, . . . , N to identify promising regions looking for aoptimal solution. In bAFSA, N individual points represented by binary 0/1 stringof length n are randomly initialized. An example is shown in Figure 1.xi,1 = 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0Figure 1: Individual representation in bAFSAHowever, the randomly initialized point xi may not be feasible since the prob-lem (1) has a constraint. The widely used approach to deal with constrained opti-mization problem is based on penalty functions. The performance of penalty-typemethod is not always satisfactory due to the choice of an appropriate penalty pa-rameter, hence alternative techniques have been proposed. In bAFSA, the decodingalgorithm proposed by Sakawa and Kato [7] is used in order to make xi feasible.The visualAfter initializing N feasible individual points, the crucial issue of bAFSA is thevisual of each individual point xi that helps us to create a corresponding trial pointyi. This represents a closed neighborhood of xi with a radius equal to a positivequantity ν. A point is to be considered inside the visual of xi if the distance ofthat point to xi is within ν. The Hamming distance between two points is used todefine the visual. Here we take ν = δ × n, where δ ∈ (0, 1) and n (items) can bethe maximum Hamming distance between two points.Let npi be the number of points inside the visual of point xi. Depending onthe relative positions of the points in the population, three possible scenarios mayoccur: (i) the visual is empty if npi = 0; (ii) the visual is not crowded if npi/N ≤θ; and (iii) the visual is crowded, otherwise. Here θ ∈ (0, 1) is the crowdingparameter.Fish BehaviorDepending on the crowding scenario of the visual, the point xi performs differ-ent behavior. In bAFSA, the fish (point) behavior that create the trial points areoutlined as follows.Chasing behavior: If the visual is not crowded, the point xi performs the chasingbehavior. This behavior is related with a movement towards the point xbest insidethe visual that has the best objective function value. The point xi performs thechasing behavior if f(xbest) > f(xi). In chasing, the uniform crossover betweenxi and xbest is performed to create the trial point yi.Swarming behavior: This behavior of the point xi is related with a movementtowards the central point xc inside the visual if it is not crowded and f(xbest) ≤2f(xi). In bAFSA, an individual point is represented by binary string of 0/1 bits,so the central point xc inside the visual is calculated according to [5]. Then iff(xc) > f(xi), the point xi performs the swarming behavior. In swarming, theuniform crossover between xi and xc is performed to create the trial point yi.Searching behavior: When the visual is crowded, or it is not crowded but thepoint xi did not perform the chasing or the swarming behavior, the point performsthe searching behavior. Here, a point xrand inside the visual is randomly selectedand the point xi moves towards it if the condition f(xrand) > f(xi) holds. Insearching, the uniform crossover between xi and xrand is also performed to createthe trial point yi.Random behavior: When the visual is empty, or the other fish behavior are notperformed, the point xi performs the random behavior. This behavior is relatedwith a random movement for a better region, and the trial point yi is created byrandomly setting binary 0/1 bits of length n.SelectionAfter creating the N trial points yi,t+1, i = 1, 2, . . . , N , the decoding algorithmis performed to make them feasible. In order to decide whether or not they shouldbecome members of the population in the next iteration t+ 1, the trial point yi,t+1is compared to the current point xi,t using the following greedy criterion:xi,t+1 ={yi,t+1 if f(yi,t+1) ≥ f(xi,t)xi,t otherwise , i = 1, 2, . . . , N. (2)Termination ConditionLet Tmax be the maximum number of iterations. If fmax is the maximum objectivefunction value attained at t and if fopt is the known optimal value, then bAFSAterminates if (t > Tmax or (|fmax − fopt|) ≤ ǫ), for a small positive number ǫ.The bAFSAThe algorithm of the herein proposed binary version of the artificial fish swarmalgorithm for solving (1) is outlined.Step 1: Set parameter values.Step 2: Set t = 1. Randomly initialize xi,1, i = 1, 2, . . . , N .Step 3: Perform decoding and evaluate f . Identify xmax and fmax.Step 4: If termination condition is met, stop.Step 5: For all xi,t,Define visual and identify crowding scenario;Perform fish behavior to create trial point yi,t+1;Perform decoding to make the trial point feasible.Step 6: Perform selection according to (2) to create new current points.Step 7: Evaluate f and identify xmax and fmax.Step 8: Set t = t+ 1 and go to Step 4.33 Preliminary ResultsWe code bAFSA in C and compile with Microsoft Visual Studio 9.0 compiler in aPC having 2.5 GHz Intel Core 2 Duo processor and 4 GB RAM. We set N = 100,δ = 0.5, θ = 0.8 and ǫ = 10−4. We also set Tmax = 1000 for n ≤ 50, otherwiseTmax = 2000. Firstly, 11 QKP are considered to test the performance criteria of theproposed bAFSA. The data were generated randomly according to [1]. The profitcoefficients of objective function were generated with density, d = 1.00. The den-sity means the percentage of non-zeros in the profit coefficients. We solved theseproblems using MINLP1 and compared the results with our solutions. The resultsobtained by MINLP and bAFSA are shown in Table 1. Thirty independent runsTable 1: Results of 11 test problems obtained by MINLP and bAFSAProb. nMINLP bAFSAsolver successful runs among 30 runsfopt FE fmax AITsr AFEsr ATsr Nsr favg AIT AFE AT1 5 282 82 282 1 100 0.00 30 282.00 1 100 0.002 10 1224 162 1224 1 114 0.00 30 1224.00 1 114 0.003 20 5272 5793 5272 32 3212 0.07 30 5272.00 32 3212 0.074 30 1629 46 1629 6 577 0.02 30 1629.00 6 577 0.025 40 31083 3759 31083 47 4688 0.23 29 31082.00 79 7865 0.356 50 58588 2319 58588 40 4004 0.24 30 58588.00 40 4004 0.247 60 66286 8964 66286 550 54997 4.08 11 65706.40 1468 146852 10.748 70 91144 3184 91144 36 3554 0.34 26 91035.73 297 29750 2.699 80 85783 1408 85783 67 6661 0.80 20 85730.77 711 71116 7.9710 90 183475 20298 183475 102 10218 1.16 24 183419.93 482 48179 5.1011 100 99759 258400 99759 149 14902 2.05 11 99406.20 1321 132143 15.96were carried out for each problem using bAFSA. In a run if the algorithm finds theoptimal solution (or near optimal with a tolerance) of a test problem, then the runis considered to be a successful run. In the table,‘fopt’ and ‘FE’, the number offunction evaluations are obtained with MINLP solver. The performance criteria ofbAFSA are: (i) for the successful runs – ‘fmax’; the average number of iterations,‘AITsr’; the average number of function evaluations, ‘AFEsr’; the average com-putational time (in seconds), ‘ATsr’ and the number of successful runs, ‘Nsr’; (ii)among the 30 runs – the average of best objective function values obtained, favg;‘AIT’, ‘AFE’ and ‘AT’ bear the same previously defined meanings but among the 30runs. We remark that ‘AFE’ was computed based on the entire population.Secondly, 20 benchmark QKP test problems2 are considered. The results ob-tained by MINLP and bAFSA are shown in Table 2.We may conclude from the results in Tables 1 and 2, that the herein proposedbAFSA is capable of solving 0–1 quadratic knapsack problems although some im-provement in efficiency is still required.1available at http://neos-server.org/neos/2available at http://cedric.cnam.fr/∼soutif/QKP/4Table 2: Results of 20 test problems obtained by MINLP and bAFSAProb.MINLP bAFSAn solver successful runs among 30 runs(d) fopt FE fmax AITsr AFEsr ATsr Nsr favg AIT AFE AT1 100 18558 8364 18558 142 14183 2.51 6 18485.60 1628 162852 27.412 (0.25) 56525 706 56525 111 11062 1.47 18 56407.63 866 86645 10.143 3752 1713 3752 101 10050 1.30 6 3688.43 1620 162025 18.964 50382 61 50382 183 18301 2.47 12 50153.10 1273 127332 15.095 61494 5130 61494 39 3912 0.57 26 61466.40 301 30059 4.546 100 83742 7062 83742 137 13716 1.90 22 83643.43 634 63397 7.737 (0.50) 104856 10071 104856 139 13851 1.88 2 104604.03 1876 187608 22.238 34006 17786 34006 146 14564 2.05 11 33942.83 1320 132018 15.749 105996 157 105996 59 5937 0.91 27 105859.50 253 25345 3.8610 56464 7478 56464 78 7815 1.35 15 56447.57 1039 103918 16.3711 100 189137 7 189137 5 490 0.07 30 189137.00 5 490 0.0712 (0.75) 95074 173495 95074 206 20643 2.81 5 94984.87 1701 170124 20.2613 62098 3447 62098 84 8432 1.56 19 62042.80 787 78681 12.9314 72245 301913 72245 104 10426 1.85 8 72116.53 1494 149461 24.3015 27616 14023 27616 103 10288 1.39 16 27456.83 988 98829 11.8816 100 81978 38198 81978 124 12391 1.75 11 81835.10 1312 131221 15.8217 (1.00) 190424 37144 190424 121 12133 1.90 6 188728.63 1624 162442 24.8118 225434 1590 225434 210 20968 2.66 6 223298.73 1642 164209 19.4019 230076 1370 230076 247 24659 3.10 19 229244.37 890 88958 10.7420 74358 13569 74358 143 14255 2.01 22 74256.80 638 63791 7.714 ConclusionIn this paper, a binary version of the artificial fish swarm algorithm for solving 0–1quadratic knapsack problem has been presented. In this method a point is repre-sented by a binary string of 0/1 bits. The visual of a point is defined using theHamming distance. Depending on the number of points inside the visual, a pointcan perform either chasing, swarming, searching or random behavior. Crossoverand mutation are implemented to create trial points. In order to make points feasi-ble a decoding algorithm is also implemented. A greedy selection criterion is usedto decide whether or not the trial points should be included in the population of thenext iteration.A set of 0–1 QKP were considered to test the performance of bAFSA. It hasbeen shown that the proposed method is capable of solving those problems. Futuredevelopment will focus on improving the algorithm efficiency and the comparisonof the proposed method with the other solution methods available in literature.Acknowledgments. The first author acknowledges Cieˆncia 2007 of FCT – Fundac¸a˜o paraa Cieˆncia e a Tecnologia (Foundation for Science and Technology), Portugal for the finan-cial support under fellowship grant: C2007-UMINHO-ALGORITMI-04. The second andthird authors acknowledge FEDER COMPETE – Programa Operacional Fatores de Com-petitividade (Operational Programme Thematic Factors of Competitiveness) and FCT forthe financial support under project grant: FCOMP-01-0124-FEDER-022674.5References[1] A. Billionnet, ´Eric Soutif, An exact method based on Lagrangian decom-position for the 0–1 quadratic knapsack problem, Eur. J. Oper. Res., 157,pp. 565–575 (2004).[2] G. Gallo, P. L. Hammer, B. Simeone, Quadratic knapsack problems, Math.Program. Study, 12, pp. 132–149 (1980)[3] M. Jiang, Y. Wang, S. Pfletschinger, M. A. Lagunas, D. Yuan, Optimalmultiuser detection with artificial fish swarm algorithm, in: D.-S. Huang,L. Heutte, M. 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Solving 0–1 quadratic knapsack problems with a population-based artificial fish swarm algorithm

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