An artificial chemical reaction optimization algorithm for. multiple-choice; knapsack problem.

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1 An artificial chemical reaction optimization algorithm for multiple-choice knapsack problem Tung Khac Truong 1,2, Kenli Li 1, Yuming Xu 1, Aijia Ouyang 1, and Xiaoyong Tang 1 1 College of Information Science and Engineering, Hunan University, National Supercomputing Center in Changsha, , China. 2 Faculty of Information Technology, Industrial University of Hochiminh city, Hochiminh, Vietnam. Abstract Multiple-choice knapsack problem (MCKP) is a well-known NP-hard problem and it has a lot of applications in the real-world and theory. In this study, the Artificial chemical reaction optimization algorithm () that uses integer string code is developed to solve MCKP. Four specific reaction operators are designed to implicate local and global search. A new penalty function that aims to force the algorithm search in both infeasible and feasible search space is suggested. The experiment on MCKP test set demonstrates that is superior to. Keywords: Artificial chemical reaction optimization; combinatorial; multiple-choice; knapsack problem. 1. Introduction Given m classes N i = {1,... n i }, i = {1,..., m} of items to pack in some knapsack of capacity W. Each item j N i has a cost c ij and a size w ij, and the problem is to choose one item from each class such that the total cost is minimized without having the total size to exceed W. The multiple-choice knapsack problem (MCKP) may thus be formulated as: minimize subject to n i j=1 c ij x ij (1) w ij x ij W, (2) x ij = 1, i {1, 2,..., m}, (3) x ij {0, 1}, i {1,..., m}, j N i. (4) All coefficients c ij, w ij, and W are positive numbers, and the classes N 1,..., N m are mutually disjoint. MCKP is known as an NP-hard (Non-deterministic Polynomial-time hard) problem [1]. The problem has a large range of applications: Capital Budgeting [2], Menu Planning [3], transportation programming [4], nonlinear knapsack problems [2], sales resource allocation [3], design of information systems [5], etc. The MCKP also appear by Lagrange relaxation of several integer programming problems [6]. Since MCKP is an NP-hard problem. The exactly algorithms have complexity time in exponential functions. The heuristic algorithm has an advantage in finding approximate optimal in polynomial time. One of the well-known heuristic algorithm is [7]. Although is pioneer in solving MCKP, yet it has a drawback that it get stack in local optima. Recently, Artificial chemical reaction optimization algorithm has been proposed in [8]. The is mimic from chemical reaction process. The is suggested with two encoding types: real code and binary. It is successful in multiple-sequence alignment, data mining, global numerical optimization, and others [8], [9]. In this study, the that uses integer string code is developed to solve MCKP. In the proposed algorithm, four specific reaction operators are designed to implicate local and global search. A new penalty function that aim to force the algorithm search in both infeasible and feasible search space is suggested. The experiment on MCKP test set demonstrates that superior to. The rest of the paper is organized in sections: Section 3 briefly gives the original framework of, and. Section 4 explains the modification of the original to adapt it to the MCKP problem. We survey the behavior of and compare the simulated results of the with in Section 5. We conclude this paper and suggest potential future work in Section Genetic algorithm In [7], a is proposed to solve MCKP. A chromosome is presented as m genes corresponding to m classes of items. The ith gene takes an integer number from a mutually exclusive set n i, where N i = {1, 2,..., n i }, for i = {1, 2,..., m}. Thus the position of a gene is used to represent class that object belong to, and the value of the gene is used to represent the item selected from the class. For crossover and mutation, uniform crossover and random perturbation are used for genetic operations. For selection, roulette wheel selection is used, it also used elitist method. If the best chromosome of preceding population is not selected to new population, the elitist will replace a randomly chromosome in new population by the best one. For evaluation, A penalty method is proposed to a

2 fitness function to help the algorithm to search optimal solution from both the feasible and infeasible sides of the solution space. Let x k be the kth chromosome in the current population and x k ij be the corresponding decision variables, the fitness function is given as follows. f k (x k ) = c ij x k ij The penalty coefficient p k is calculated as: { 0 if (2) is satisfied p k = ψ 0 ( m ni i=1 i=1 w ijx k ij W ) if otherwise. where ψ 0 is the large positive penalty value. And, the evaluation function is presented as follows: eval(x k ) = 1 f k (x k ) + p k, k = {1, 2,..., popsize} 3. Artificial chemical reaction optimization algorithm The is a heuristic method proposed by Alatas in [8]. It inspired from the chemical reaction process. In the chemical reaction process, the system tends toward the highest entropy and the lowest enthalpy. The chemical reactions possess efficient objects, states, process, and events that can be designed as a computational method. Enthalpy or potential energy for minimization problem and entropy for maximization problem can be utilized as objective functions for the interested problem. At the beginning, a reactantn um number of reactant (solution) is generated. In the iteration phase, when the terminated criteria no met, then according to the condition (randomly variable), one chemical reaction operator is selected to execute. The reactants update work same as reversible reaction that help the objective function to grow forward to optima. In, there are two chemical reaction types namely bimolecular reactions and monomolecular reactions. Bimolecular reactions are synthesis reaction, redox2 reaction, and displacement reaction; this type of reaction requires two reactants participating. Bimolecular reactions are redox1 reaction and decomposition reaction; one reactant is required in this reaction type. The flow chart of which is depicted in Fig. 1 and an outline of the algorithm is given in Algorithm 1. More details about can be found in [8], [9]. 4. Design for MCKP 4.1 Solution Representation An integer string is used to represent a reactant(solution). The y i receives an integer in N i, it represent y i N i is chosen. The string length is m corresponding to a solution Algorithm 1 algorithm Input: Problem-specific information (the objective function f, constraints, and the dimensions of the problem) Assign parameter values to reactantn um. Setting the initial reactants and evaluation. while stop criterion not met do Do chemical reactions. Reactants update. end while Output: The best solution. Fig. 2: Solution presentation in MCKP. The solution presentation is depict in Fig. 2. By defining an indicator variable, y i is as follows: y i = j if x ij = 1, j N i, i = 1, 2,..., m 4.2 Objective and penalty functions Let x be chromosome in the current population and x ij be the corresponding decision variables. In the reaction, the enthalpy is not negative and the enthalpy is decreasing in the reaction process. To adopt to enthalpy, it is set as follows: enthalpy(x) = c ij x ij + g(x) (5) i=1 i=1 where g(x) is penalty function as following: { 0 if (2) is hold g(x) = Ω 0 + ( m ni i=1 i=1 w ijx ij W ) if otherwise. where Ω 0 is a given positive constant. The idea here is that, for violate solution will have a larger enthalpy. It forces the algorithm search both sides of search space that is feasible and infeasible domains. 4.3 Reaction operators In this paper, we proposed integer string code for MCKP. Five specific problem reaction operators are explained as following Synthesis In this operator one reactant will be created from two original reactants. It responds to the diversifications in the algorithm. The synthesis operator in [10] is redesigned for this problem. The pseudocode of the synthesis operator is described in Algorithm 2.

3 Fig. 1: flow chart [8] Algorithm 2 synthesis(x 1, x 2 ) Input: Reactants x 1 and x 2 for i 1 to n do Get t randomly in [0, 1] if (t > 0.5) then x (i) x 1 (i) else x (i) x 2 (i) end if end for Output: x Displacement This operator creates two new reactants from two original reactants. Each position of the two reactants strings are considered for information swapping based on a randomly generated mask similar to the mask used in uniform crossover used in genetic algorithms [8]. At the position where the mask value is 0, the reactants values are exchanged, otherwise the reactant values does not change Redox2 The two-points crossover is commonly used in genetic algorithm is used. This operator responds to intensifications Decomposition Two random points in the reactant string are selected and the values between those points are reversed. This operator responds to diversifications of the algorithm Redox1 This operator is implemented for diversifications. One new reactant (solution) is generated from one original reactant. One position i th is randomly selected from {1,..., m}, and value of y i is replaced by a random number in {1,..., n i }. 4.4 Reactants update This step inspired reversible chemical reactions, chemical equilibrium test is performed. If the newly generated reactants give better function value, the new reactant set is included and the worse reactant is excluded similar to reversible chemical reactions. It helps the reactants move towards optima.

4 4.5 Termination criterion check When the given termination criterion is met, the report the best solution. Otherwise, the chemical reaction process is repeated. 5. Experiment and analysis 5.1 Data test set One type of randomly generated data instances are considered, each instance tested with data-range R = 1000 for different number of classes m and sizes n i : Strongly correlated data instances (SC): In knapsack problem w j is randomly generated in [1, R] and c j = w j For each class i generate n i items (w j, c j ) as for knapsack problem, and order these by increasing weight. The data instance for MCKP is then w ij = j h=1 w h and c ij = j h=1 c h, j = 1, 2,..., n i. Such instances have no dominated items, and form an upper convex set. For each instances, the capacity W is calculated as follows. W = 1 2 m (min j Ni (w ij ) + max j Ni (w ij )) i=1 5.2 Parameter setting s parameters are set as in [7]: P opsize = 20, P c = 0.8, P m = 0.1. For, the reactantnum is set to 20. The terminated criteria is set the same for and that is function evaluations. 5.3 Experiment results All the algorithms were implemented in Matlab R2011b. The test environment is set up on a personal computer with Pentium E6700 CPU at 3.2 GHz CPU, 2G RAM, running on Windows XP. We have considered how the algorithm behaves for different problem sizes, test instances, and data-ranges. We observe the convergence curves of three test instances in strongly correlated test set. The three instances with (m = 10, n = 10), (m = 100, n = 100), and (m = 1000, n = 100) are used in this experiment. Figure 3 shows the evolution of the mean of the best total costs of CRO over 25 runs in the three instances. It indicates the global search ability and the convergence ability of CRO. There are several observations and they are given as follows: In the case (m = 10, n = 10), as depict in a) the convergence curve of is tied with CRO s, but CRO is still better. For instance (m = 100, n = 100), the Fig. 3b) shows that CRO have a much more quick convergence compares with. For larger instance (m = 1000, n = 100), the Fig. 3c show that CRO still have a good convergence, while x (a) m = 10, n = x (b) m = 100, n = (c) m = 1000, n = 100 Fig. 3: Convergence curves of and on the MCKP. The objective function value were averaged over 25 runs. shows a very slow convergence. From the Fig. 3 shows that CRO much more better in convergent rate and solution quality when solving large MCKP. The simulation results are shown in table 1. m and n are number of class and number of items in a class, respectively. time and stddev represent the elapsed time per run in second and the standard deviation, respectively. mean, worst and best are mean cost, worst cost and best cost, respectively. The maximum number of function evaluations , the number of runs 25. Table 1 shows the experimental results of the strongly correlated instances. For all the proposed instances, CRO yields superior results compared with. The series of experimental results demonstrate the superiority and effectiveness of CRO. In comparison with ; CRO can get better results in shorter time. The smaller standard deviation (StdDev) shows that the new algorithm is more robust than

5 Table 1: Simulation results for strongly correlated instances. Instances CRO m n mean best worst stddev time mean best worst stddev time Conclusion The new approach artificial chemical reaction optimization algorithm using encoding integer string is proposed to solve multiple-choice knapsack problem. Five specific problem reaction operators are presented. A new penalty function is suggested to treat the infeasible solutions. The experiment on a large range of data set demonstrates that the proposed method has superior performance when compared with. It has shown a significant potential in solving MCKP. In the future, we will development a parallel version of this algorithm that improved the efficiency. Acknowledgment This paper was partially funded by the Key Program of National Natural Science Foundation of China (Grant No ), and the National Natural Science Foundation of China (Grant Nos , , and ), Key Projects in the National Science & Technology Pillar Program the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (Grant No ), the Ph.D. Programs Foundation of Ministry of Education of China ( ) the Program for New Century Excellent Talents in University (NCET ), and A Project Supported by Scientific Research Fund of Hunan Provincial Education Department (12A062), A Project Supported by the Science and Technology Research Foundation of Hunan Province (Grant No. 2013GK3082). References [1] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. America: W.H. Freeman, [2] R. M. Nauss, The 0Ű1 knapsack problem with multiple choice constraints, European Journal of Operational Research, vol. 2, no. 2, pp , [3] P. Sinha and A. A. Zoltners, The multiple-choice knapsack problem, Operational Research, vol. 27, no. 3, p. 503, [4] T. Zhong and R. Young, Multiple choice knapsack problem: Example of planning choice in transportation, Evaluation and Program Planning, vol. 33, no. 2, pp , [5] P. C. Yue and C. K. Wong, Storage cost considerations in secondary index selection, International Journal of Parallel Programming, vol. 4, pp , [6] M. L. Fisher, The lagrangian relaxation method for solving integer programming problems, Manage. Sci., vol. 50, no. 12 Supplement, pp , Dec [7] M. R. Gen, R. Cheng, M. Sasaki, and Y. Jin, Multiple-choice knapsack problem using genetic algorithms. Maui, HI: Integrated Technology Systems, [8] B. Alatas, Acroa: Artificial chemical reaction optimization algorithm for global optimization, Expert Systems with Applications, vol. 38, no. 10, pp , [9] B. Alatas, A novel chemistry based metaheuristic optimization method for mining of classification rules, Expert Systems with Applications, vol. 39, no. 12, pp , [10] T. K. Truong, K. Li, and Y. Xu, Chemical reaction optimization with greedy strategy for the 0-1 knapsack problem, Appl. Soft Comput., vol. 13, no. 4, pp , Apr