An Estimation of Distribution Algorithm for solving the Knapsack problem

Transcription

1 Vol.4,No.5, 214 Published olie: May 25, 214 DOI: /jscse.v4.5.1 A Estimatio of Distributio Algorithm for solvig the Kapsack problem 1 Ricardo Pérez, 2 S. Jös, 3 Arturo Herádez, 4 Carlos A. Ochoa *1, PICYTCIATEC A.C., Leó City, México 2, CONACYT, México City, México 3 CIMAT A.C., Guaajuato City, México 4 UACJ, Ciudad Juárez City, México 1 rperez.picyt@ciatec.mx, 2 jos_sachez@hotmail.com, 3 artha@cimat.mx, 4 alberto.ochoa@uacj.mx Abstract. The kapsack problem, a NPhard problem, has bee solved by differet ways durig may years. However, its combiatorial ature is still iterestig for may academics. I this paper, a Estimatio of Distributio Algorithm is applied for solvig the Kapsack problem. for simplicity is called. It cotais a probabilistic model of type chai for samplig ew offsprigs to solve the problem. I additio, we use a ad a to compare the performace of the algorithm. Accordig to the experimets, the geetic algorithm ad the provide good solutios, but the performace from this evolutioary algorithm was able to give better results. Keywords: Estimatio of Distributio Algorithm, Kapsack problem, geetic algorithm, greedy algorithm. * Correspodig Author: Ricardo Pérez Rodríguez, Posgrado Iteristitucioal e Ciecia y Tecología PICYT, CIATEC, A.C., México, rperez.picyt@ciatec.mx Tel: Itroductio Evolutioary ad metaheuristic algorithms have bee extesively used as search ad optimizatio tools durig this decade i several domais from sciece to egieerig, ad others. May demadig applicatios that ivolve the solutio of optimizatio problems of high complexity, a lot of these belogig to a special class of problems called NPhard have bee solved by various methods [1]. Evolutioary ad metaheuristic algorithms are ow cosidered amog the best tools to fid good solutios with a reasoable ivestmet of resources. Estimatio of Distributio Algorithms (EDAs), itroduced by Mühlebei ad Paaβ [2] have bee used satisfactorily to solve complex combiatorial optimizatio problems. Che et al [3], Liu et al [4] ad Pa ad Ruiz [5] ca be cosulted. Disadvatages of EDAs such as loss of diversity ad isufficiet use of locatio iformatio of solutios have bee tackled successfully by icorporatig other methods such as s (GAs) durig the evolutioary process. Che et al [6] use this approach. Several works have bee doe i order to capture the problem structure with more precisio. Advaced probabilistic models to solve combiatorial problems through EDAs have bee proposed attemptig to itegrate higher order iteractios to ehace the solutio quality. Wag et al [7] ad Che et al [8] have cotributed o it. The major procedure of a EDA is listed as follows. 78

2 Vol.4,No.5, 214 Published olie: May 25, 214 DOI: /jscse.v4.5.1 Step 1. Set the geeratio idex g =. Iitialize a iitial populatio S() of size M. Step 2. Select a subset D from S(g) of size N, where N M. Step 3. Establish a probabilistic model P which somehow describes the distributio characteristics of D. Step 4. Geerate a set K of ew idividuals by samplig P. Step 5. Select the best idividuals from K S(g) ad assig them to the ext geeratio S(g+1). Step 6. Let g = g+1. If g<gn, where GN is the maximum umber of geeratios retur step 2. Otherwise, output the best solutio i S(g). The Kapsack problem is a classical combiatorial problem [9][1]. It ca be described as follows: Imagie takig a trip to which you ca oly carry a backpack that, logically, has a limited capacity. Give a set of items, each with a weight ad a value, determie the umber of each item to iclude i a bag so that the total weight is less tha a give limit ad the total value is as large as possible, this problem ca be cosiderate as NPeasy problem but some studies show that the Kapsack problem is a NPhard problem [11]. I the preset paper we itroduce the Estimatio of Distributio Algorithm for solvig the Kapsack problem called for simplicity. The experimets were made o four types of istaces from ucorrelated to subsetsum. All these istaces probe the algorithm varyig the parameters of the profits ad the weight. It was ecessary to have a compariso poit for the ad was used the [9], this is a determiistic algorithm who gives a approximate result for the Kapsack problem. I additio, a or GA was used i order to compare the performace amog them. 2. The Kapsack problem The Kapsack problem [1] is the typical combiatorial problem that has bee studied sice may years ago ad was proved that it is a NPhard problem [12]. The basic problem is the 1 Kapsack problem or Biary Kapsack problem ad it has a search space of 2 1 possible solutios. The Kapsack Problem ca be described as follows: there are objects, each of this objects have a profit ad weight, ad eeds to select those whose sum of their beefits is maximized subject to the sum of the weight of the same objects should ot exceed a amout determied. It ca be formulated mathematically by umberig each of its objects or items from 1 to ad itroducig it to a vector of biary variables = 1,2,3,...,, where each variable represeted here will take the value 1 or depedig o whether it is selected or ot. The solutio to the Kapsack problem is select a subset of objects from the biary vector, solutio vector, that satisfies the costrait o the equatio (2) ad the same time maximize the objective fuctio o the equatio (1). z = j =1 p j x j (1) j =1 w j x j c x j = if the j object is selected otherwise (2) where z represets the profit j represets the jth object x j idicates whether the j object is part of the solutio p j is the jth object profit w j is the jth object weight 79

3 Vol.4,No.5, 214 Published olie: May 25, 214 DOI: /jscse.v4.5.1 c is the volume or capacity of the kapsack 3. Types of Kapsack problem istaces The Kapsack Problem is affected by the relatioship betwee the profit ad the weight of the objects; these types of istaces are the followig: Ucorrelated: the profits ad weight are distributed uiformly betwee oe ad a maximum T umber. p j 1, T ; w j 1, T (3) Weakly correlated: the weight is distributed uiformly betwee oe ad a maximum T umber ad the profits are distributed uiformly aroud the weight ad a R ratio. w j 1, T ; p j w j R, w j + R (4) Strogly correlated: the weight is uiformly distributed betwee oe ad a maximum T umber; the profits are the weight plus oe K costat. w j 1, T ; p j = w j + K (5) Subsetsum: the profits ad weight have the same value ad are distributed uiformly betwee oe ad a maximum T umber. w j 1, T ; p j = w j (6) 4. The This algorithm gives a ituitive approach cosiderig the profit ad weight of each item; it is kow as the efficiecy which is based o the Equatio (7). The objective is to try to put the items with highest efficiecy ito the Kapsack. It is ecessary sort all the items based o the efficiecy, usig the Equatio (8), before to apply the Greedy algorithm to the problem. e j = p j w j (7) p w p 1 w 1 p w (8) 5. for the kapsack problem Our approach is to use the MIMIC algorithm to build the probabilistic graph model. Itroduced by De Boet et al [13], the MIMIC algorithm uses a chai structured probabilistic model where the probability distributio of all the variables except the head ode is coditioed o the value of the variable precedig them i the chai. It meas a margial uivariate fuctio ad 1 pairs of coditioal desity fuctios to build the probabilistic graph model. Solutio represetatio: ay solutio of the problem metioed should be a specific biary vector that represets the objects i the kapsack. Thus, a solutio ca be expressed by or 1 for each object accordig to sectio 2. Probability model: I this paper, the probability model is desiged as a probability matrix. The elemet p j of the probability matrix represets the probability that the object j be loaded i the kapsack. For all j (j = 1, 2, ), p j is iitialized as 8

4 Vol.4,No.5, 214 Published olie: May 25, 214 p j = j =1 x j = 1 DOI: /jscse.v4.5.1 (9) where represets the umber of elemets. Via samplig accordig to the probability matrix ew promisig idividuals may be geerated. 6. Experimets To test the was used the Geerator of Kapsack Test Istaces [14]; it requires the umber of elemets ad the coefficiets rage to geerate a test istace. We geerate the four types of test istaces described, ad was used the same parameters for each. Each algorithm was ru 1 times for obtaiig their average ad stadard deviatio. The Table 1 shows the parameters used i this research. Table 1. Parameters used for solvig the Kapsack problem Parameters Values Geerator of Kapsack Test Istaces Number of items or elemets Rage of Coefficiets Number of istaces Number of test i series Type of selectio Cross rate Mutatio rate Probabilistic Model Touramet of size 2 8% 1% Structure of type chai 7. Results We show the results obtaied by testig each type istaces with the differet algorithms, i.e., the average, the best ad the worst profit, ad their fitess's stadard deviatio for each algorithm. We also show the algorithms behavior through some graphics. Ucorrelated: the Table 2 depicts the results for all algorithms where the offers the best std. deviatio ad the best average. Table 2. Results for ucorrelated istace same result i ay iteractio, there is o exist std. deviatio The Figure 1 depicts the performace for each algorithm i each iteractio. 81

5 Beefit Iteratioal Joural of Vol.4,No.5, 214 Published olie: May 25, 214 DOI: /jscse.v Performace o Ucorrelated istace Beefit Greed algorithm Geetic algorithm algorithm Figure 1. Trials for ucorrelated istace Weakly correlated: the Table 3 details the results for all algorithms where the agai offers the best std. deviatio ad the best average. Table 3. Results for weakly correlated istace same result i ay iteractio, there is o exist std. deviatio The Figure 2 details the performace for each algorithm i each iteractio Performace o Weakly correlated istace Greed algorithm Geetic algorithm algorithm Figure 2. Trials for weakly correlated istace Strogly correlated: the Table 4 shows the results for all algorithms where the agai offers the best std. deviatio ad the best average. Table 4. Results for strogly correlated istace 82

6 Beefit Iteratioal Joural of Vol.4,No.5, 214 Published olie: May 25, same result i ay iteractio, there is o exist std. deviatio DOI: /jscse.v The Figure 3 shows the performace for each algorithm i each iteractio. 12 Performace o Strogly correlated istace 1 8 Beefit Greed algorithm Geetic algorithm algorithm Figure 3. Trials for strogly correlated istace Subsetsum: the Table 5 presets the results for all algorithms where the agai offers the best std. deviatio ad the best average. Table 5. Results for subsetsum istace same result i ay iteractio, there is o exist std. deviatio The Figure 4 presets the performace for each algorithm i each iteractio. 12 Performace o Subsetsum istace Greed algorithm Geetic algorithm algorithm Figure 4. Trials for subsetsum istace 83

7 Vol.4,No.5, 214 Published olie: May 25, 214 DOI: /jscse.v Coclusios There are may evolutioary algorithms ad metaheuristics to solve the Kapsack problem; i this work we itroduce the algorithm which is a evolutioary algorithm. The experimets were desiged with the same parameters for the three algorithms to give them the same characteristics i order to be equal betwee them. We ca see i all the graphics the algorithms behavior, ad we ca observe that the Greedy Algorithm was the worst because it always yields the same result for ay iteractio. The ad the yield more cosistet results, its graphs show that these algorithms i the most of the cases give good results. So we ca coclude that the is a alterative to solve the Kapsack problem, because each time that it's ru, it gives a good solutio, ad this solutio always is better that the solutios obtaied by the other algorithms. Overall the results preset a low stadard deviatio. Refereces [1] McDuffSpears W, Usig eural etworks ad geetic algorithms as heuristics for NPcomplete problems, Thesis of Master of Sciece i Computer Sciece, George Maso Uiversity, Virgiia, USA, [2] Mühlebei H, Paaß G, From recombiatio of gees to the estimatio of distributios: I. biary parameters, i Parallel Problem Solvig from Nature PPSN IV, Voigt H, Ebelig W, Recheberg I, Schwefel H, Eds., Berli: Spriger, pp , [3] Che S, Che M, Chag P, Zhag Q, Che Y, Guidelies for developig effective Estimatio of Distributio Algorithms i solvig sigle machie schedulig problems, Expert Systems with Applicatios, vol. 37, pp , 21. [4] Liu H, Gao L, Pa Q, A hybrid particle swarm optimizatio with estimatio of distributio algorithm for solvig permutatio flowshop schedulig problem, Experts Systems with Applicatios, vol. 38, pp , 211. [5] Pa Q, Ruiz R, A estimatio of distributio algorithm for lotstreamig flow shop problems with setup times, Omega, vol. 4, pp , 212. [6] Che S, Chag P, Cheg T, Zhag Q, A Selfguided for permutatio flowshop schedulig problems, Computers ad Operatios Research, vol. 39, pp , 212. [7] Wag L, Wag S, Xu Y, Zhou G, Liu M, A bipopulatio based estimatio of distributio algorithm for the flexible jobshop schedulig problem, Computers ad Idustrial Egieerig, vol. 62, pp , 212. [8] Che Y, Che M, Chag P, Che S, Exteded artificial chromosomes geetic algorithm for permutatio flowshop schedulig problems, Computers ad Idustrial Egieerig, vol. 62, pp , 212. [9] Kellerer H, Pferschy U, Pisiger D, Kapsack Problems, Spriger, Berli, Germay, 24. [1] Silvao M, Toth P, Kapsack Problem, Algorithms, ad Computer Implemetatios, Joh Wiley ad Sos, New York, USA, 199. [11] Garey M, David S, Computers ad Itractibility: A Guide to the Theory of NPCompleteess I, [12] Pisiger D, Where Are The Hard Kapsack problems?, Computers ad Operatios Research, vol. 32, pp , 25. [13] De Boet J, Isbell C, Viola P, MIMIC: Fidig Optima by Estimatio Probability Desities, Advaces i Neural Iformatio Processig Systems, vol. 9, [14] Pisiger D, Core Problems i Kapsack Algorithms, Operatios Research, vol. 32, pp ,