P. C. Chu and J. E. Beasley. A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristics, 4:63--86, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Using a suitable heuristic, the new GA approach outperforms previous hybrid GAs with traditional encodings on nearly all test problems. 2. Prior approaches Several researchers have developed successful GAs for the KP and the more di#cult multi constraint KP, including Chu [3] Chu and Beasley [4], Hinterding [9] Khuri et al. 14] Olsen [17] and Raidl [19] Falkenauer [6] presented a hybrid GA for the BPP. In [20] Raidl and Kodydek observed that these GA approaches can be divided into two categories according to the solution encoding techniques: Some algorithms use direct encoding ....

Chu P. C., Beasley J. E.: A Genetic Algorithm for the Multidimensional Knapsack Problem, working paper at The Management School, Imperial College of Science, London, 1997.

....with different maximum execution times and different number of processors. The column header instance refers to the instance name, n and m are respectively the number of variables (items) and the number of restrictions; best known cost column gives the best known cost in the literature up to now [4] for the instance and best cost found indicates the best cost found by our parallel implementation. More results can be found at [2] and www.lsi.upc.es #mjblesa TSExperiments knapsack.html. 1 http: mscmga.ms.ic.ac.uk info.html 4. Conclusions and Future Work In this paper we have presented two ....

P. Chu and J. Beasley. A Genetic Algorithm for the multidimensional knapsack problem. working paper, 1997.

....done with di erent maximum execution times and di erent number of processors. In this table, instance referes to the instance name, n and m are respectively the number of variables (items) and the number of restrictions; best known cost column gives the best known cost in the literature up to now [3] for the instance obtained by other speci c implementations and best cost found indicates the best cost found by our parallel implementation, p indicates the number of processors (this table corresponds to the execution with p = 4 processors. best best average iters. total instance n m cost ....

P.C. Chu and J.E. Beasley. A Genetic Algorithm for the Multidimensional Knapsack Problem, 1997. Working Paper

....is done with di erent maximum execution times and di erent number of processors. In the table, instance refers to the instance name, n and m are respectively the number of variables (items) and the number of restrictions; best known cost column gives the best known cost in the literature up to now [CB97] for the instance obtained by other speci c implementations and best cost obtained indicates the best cost found by our parallel implementation, p indicates the number of processors. best best cost iters. time instance n m cost cost average dev. average) total known obtained obtained (s) ....

P.C. Chu and J.E. Beasley. A Genetic Algorithm for the Multidimensional Knapsack Problem. working paper, 1997.

.... include surrogate and composite relaxations [Osorio et al. 2000] More recently, several algorithms based on metaheuristics have been developed, including simulated annealing [Drexl, 1988] tabu search [Glover and Kochenberger, 1996; Hanafi and Frville, 1998] and genetic algorithms [Chu and Beasley, 1998] . The metaheuristic approach has allowed to obtain very competitive results on large instances compared with other methods (n = 500 and m = 30) Most of the above heuristics use the so called pseudo utility criterion for selecting the objects to be added into a solution. For the single ....

....computers like PII350, PIII500, ULTRASPARC5 and 30) For all the instances solved below, we run TS MPK with 10 random seeds (0. 9) of the standard srand( C function. We have first tested our approach on the 56 classical problems used in [Aboudi and Jrnsten, 1994; Balas and Martin, 1980; Chu and Beasley, 1998; Dammeyer and Vo, 1993; Drexl, 1988; Frville and Plateau, 1986; 1993; Glover and Kochenberger, 1996; Shih, 1979; Toyoda, 1975] The size of these problems varies from n=6 to 105 items and from m=2 to 30 constraints. These instances are easy to solve for stateof the art algorithms. Indeed, our ....

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P.C. Chu and J.E. Beasley. A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristic, 4:63--86, 1998.

....the multidimensional knapsack problem (MKP) in which only the knapsack constraints are present, has received wide attention in the literature, to our knowledge, no literature exists speci cally devoted to MDMKP. A rich survey on the MKP can be found, e.g. in the paper of Chu and Beasley (1998) [7]. Hereafter we brie y recall some of the main contributions to the solution of MKP, being its structure embedded in that of the proposed problem. The decision version of the well known knapsack problem (m = 1) is weakly NP complete and e ective approximation algorithms for the optimization ....

....procedures for xing variables at their optimal values in large size instances. Also tabu search heuristics (see e.g. Vo (1993) 28] Glover and Kochenberger (1996) 13] L kketangen and Glover (1998) 17] Hana and Freville (1998) 14] and genetic algorithms (see e. g Chu and Beasley (1998) [7] and Haul and Vo (1998) 15] have been applied to the MKP. In particular, in order to test their genetic algorithm, Chu and Beasley (1998) 7] generated 270 test problems of larger sizes and of more dicult types than the standard problems used by many authors in the past to validate their ....

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P.C. Chu and J.E. Beasley. A Genetic Algorithm for the Multidimensional Knapsack Problem. Journal of Heuristics, 4:63-86, 1998.

....is a dimension and including a bid in the solution bid set corresponds to putting the bid into the knapsack. MDKP has been the subject of several theoretical analyses [6, 9, 12, 15, 29, 38] and experimental investigations involving all manner of search methods, including genetic algorithms[8, 20, 24], TABU search [2, 19] local search [5, 11, 30] and classical complete algorithms such as branch and bound [16, 37] and dynamic programming [41] A good review of previous work is given in [8] A standard set of test problems for the MDKP is available through J. Beasley s ORLIB[3] Files mknap1 ....

.... experimental investigations involving all manner of search methods, including genetic algorithms[8, 20, 24] TABU search [2, 19] local search [5, 11, 30] and classical complete algorithms such as branch and bound [16, 37] and dynamic programming [41] A good review of previous work is given in [8]. A standard set of test problems for the MDKP is available through J. Beasley s ORLIB[3] Files mknap1 [31] and mknap2 [14, 36, 37, 41] contain real world test problems widely used in the literature. The others were generated with the aim of creating more difficult problems[8] Each problem has ....

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P.C. Chu and J.E. Beasley. A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristics, 4:63--86, 1998. 8

....) and x i =0 , then N(s) has exactly Z neighboring configurations. Note that similar neighborhoods based on adding dropping have been used in many heuristic algorithms for MKP [Dammeyer Voss 93, Frville Plateau 94, 97, Glover Kochenberger 96, Lokketangen Glover 98, Hanafi Frville 98, Chu Beasley 98] However, one difference remains that concerns the repair operation after adding an element: constraint repairing here is much simpler since it concerns only binary and ternary logic constraints. 3.3.3. Incremental evaluation of the neighborhood TS uses an aggressive search strategy to exploit ....

P.C. Chu and J.E. Beasley, A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristics 4: 63-86.

.... contributed to the parents fitness (for much larger projects, even more than two cuts may probably be advisable) In contrast, the uniform crossover operator (which yields good results for problems with a different structure such as e.g. the multidimensional knapsack problem, cf. Chu and Beasley [4]) does not seem to be well suited for sequencing problems. A randomized selection strategy seems to be advantageous only if a much larger number of individuals is considered. p mutation crossover selection av. dev. max. dev. optimal CPU sec 0.01 two point ranking 0.64 9.7 77.5 0.54 0.05 ....

Chu, P.C. and J.E. Beasley (1997): A genetic algorithm for the multidimensional knapsack problem. Working paper, The Management School, Imperial College, London.

....m ) and x =0 , then N(s) has exactly Z neighboring configurations. Note that similar neighborhoods based on adding dropping have been used in many heuristic algorithms for MKP [Dammeyer Voss 93, Fr vill Plateau 94, 97, Glover Kochenberger 96, Glover Lokketangen 98, Hanafi Fr ville 98, Chu Beasley 98] However, one difference remains concerning the repair operation after adding an element, here constraint repairing is much simpler since it concerns only binary and ternary logic constraints. 3.3.3. Incremental evaluation of the neighborhood TS uses an aggressive search strategy to exploit its ....

P.C. Chu and J.E. Beasley, A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristics 4: 63-86.

....http: mscmga.ms.ic.ac.uk jeb orlib mknapinfo.html. We tested this approach on data where the number of constraints (m) is 5, the number of variables (n) is 100, and a tightness ratio (ff) is 0.25, 0.50, or 0.75. The tightness ratio for right hand side coefficients is b i = ff P n j=1 r ij [3]. Ten problems for each m n ff combination were tested. The number of generated examples was always one hundred. The results are obtained on the Sun Station Ultra Sparc 2 machine, running at 170 MHz frequency and having 370 Mb of RAM. In addition, three different thresholds (T) for good bad ....

Chu, P.C., Beasley, J.E: A Genetic Algorithm for the Multidimensional Knapsack Problem, http://mscmga.ms.ic.ac.uk/jeb/jeb.html, current working paper.

....of multiple knapsack constraints. The multidimensional version of the knapsack problem is NP hard whereas in the case of a single constraint the problem is not strongly NP hard. Reviews of heuristic and exact algorithms for the single constraint knapsack problem can be found in [11] and in [3] for the multidimensional version. Several sets of test instances for the MDKP are available at [1] The set of large test instances discussed here consists of 270 randomly generated instances, 30 problems for each combination NK Gamma N I , with NK 2 f5; 10; 30g knapsack constraints and N I 2 ....

....test instances discussed here consists of 270 randomly generated instances, 30 problems for each combination NK Gamma N I , with NK 2 f5; 10; 30g knapsack constraints and N I 2 f100; 250; 500g variables. Table 1: Aggregated results for large MKNP instances in comparison to results reported in [3]. Column gap lists the average LP gap, opt. indicates the number of instances where optimality was proven during the corresponding run. Problem CP VF VF(10) XP XS(1500s) XP(1500s) MKHEUR GA instance gap gap gap gap gap opt. gap opt. gap gap set ( 5 100 1.26 1.57 ....

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P.C. Chu and J.E. Beasley. A Genetic Algorithm for the Multidimensional Knapsack Problem. http://mscmga.ms.ic.ac.uk/jeb/jeb.html, 1998.

.... heuristic (examining as we do a large number of possible solutions) Note here that the strategy adopted in Algorithm 1, namely to change (if possible) the GA solution into a feasible solution for the original problem, is a strategy that we have used, with success, in our previous GA work [7,13]. We used a population size of 100. Parents were chosen by binary tournament selection which works by forming two pools of individuals, each consisting of two 18 individuals drawn from the population randomly. The individuals with the best fitness, one taken from each of the two tournament ....

P.C. Chu and J.E. Beasley, A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristics 4 (1998) 63-86.

....seek to improve this single solution in some way. Population heuristics, by contrast, explicitly work with a population of solutions and combine them together in some way to generate new solutions. In recent years the author has been working with such heuristics (both in the academic world [4,6,10,11,12] and in the commercial world) and has, through personal experience, come to believe that: There is something about a population heuristic which means that it is often capable of producing better quality solutions than single solution heuristics Note here however, as will be discussed below, ....

....(MKP) is: maximise p j x j (1) subject to r ij x j b i i=1, m (2) x j (0,1) j=1, n (3) where p j , r ij and b i 0. Each of the m constraints in equation (2) is called a knapsack constraint, which is how the problem derives its name. More about this problem can be found in [11]. Throughout this paper we shall use the notation (x j ) to signify variables and (X j ) to signify a set of values for these variables which may, or may not, be feasible. 4 To make the MKP clearer we present a small example with n=7 variables (columns) and m=2 constraints (rows) in Figure 1 ....

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P.C. Chu and J.E. Beasley, A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristics 4 (1998) 63-86.

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P. C. Chu and J. E. Beasley. A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristics, 4:63--86, 1998.

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Chu, P.C., Beasley, J.E.: A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristics 4 (1998) 63--86

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P. C. Chu and J. E. Beasley. A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristics, 4(1):63--86, June 1998.

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P.C. Chu and J. E. Beasley. A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristics, 4:63--86, 1998.

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P.C. Chu and J.E. Beasley. A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristic, 4:63--86, 1998.

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P. C. Chu and J. E. Beasley. A Genetic Algorithm for the Multidimensional Knapsack Problem. Journal of Heuristics, 4(1):63--86, 1998.

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P.C. Chu and J.E. Beasley. A Genetic Algorithm for the Multidimensional Knapsack Problem. Journal of Heuristics, 4, 63-86, 1998.

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Chu PC and Beasley JE (1998). A genetic algorithm for the multidimensional knapsack problem. Journal of Heuristics 4:63-86.

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