Hybridization of tabu search with feasible and infeasible local searches for the quadratic multiple knapsack problem
Introduction
The quadratic multiple knapsack problem (QMKP) [12] is an extension of two well-known combinatorial optimization problems, namely the multiple knapsack problem (MKP) and the quadratic knapsack problem (QKP). In QMKP, we are given a set of n objects and a set of K knapsacks . Each object has a positive linear profit pi and a positive weight wi, each pair of objects (i and j) has a positive quadratic profit pij, and each knapsack has a capacity Ck. The profit achieved by assigning objects to knapsacks is the sum of the linear profit of the selected objects and the quadratic profit obtained by pairs of objects belonging to the same knapsack. QMKP is concerned with allocating each object to at most one knapsack in such a way that the total weight of the objects placed in each knapsack does not exceed its capacity and the overall profit of the objects contained in the knapsacks is maximized. Formally, let xik be the decision variable such that if object i is assigned to knapsack k and otherwise. Then QMKP can be formulated as the following integer program [12]:In the above formulation, the objective function (1) is to maximize the total profit of the objects contained in the knapsacks, the constraint (2) stipulates that the total weight of the objects in each knapsack k does not exceed its capacity Ck, and the constraint (3) ensures that no object is allocated to more than one knapsack.
Evidently QMKP is NP-hard as it reduces to the NP-hard 0/1 Knapsack Problem [19] when all the quadratic values pij are set to zero and the number of knapsacks sets to one. In addition to its theoretical challenge as an intractable combinatorial problem, QMKP has many important practical applications in different contexts, including the assignment of workmen to different tasks when their ability to cooperate may affect the results [20], the determination of the locations of the earth stations for communication satellites with a budget constraint, and the determination of the sites of railway stations, freight handling terminals, and airports [19].
Hiley and Julstrom [12] first considered QMKP. Since then, considerable efforts have been made by researchers to address the problem. While effective exact methods have been devised to solve the related knapsack problem, MKP, and QKP [3], [4], [18], to the best of our knowledge, we are unaware of any exact method proposed to solve QMKP in the literature. Instead, heuristics are often employed to find approximate solutions for QMKP in reasonable computing time. The representative heuristics include greedy constructive heuristics, local search methods, and population-based evolutionary algorithms.
For instance, Hiley and Julstrom [12] proposed three methods for QMKP, namely a greedy heuristic, a stochastic hill-climber, and a genetic algorithm, in which the crossover operator preserves the assignments of objects to knapsacks that are common to both selected parents. Singh and Baghel [20] presented a steady-state grouping genetic algorithm to address QMKP. The crossover employed by the genetic algorithm iteratively selects one of the two parents, transmits the knapsack with the largest profit value from the selected parent to the child offspring, and updates the remaining knapsacks associated with both the parents by removing all the objects belonging to the selected knapsack from both the parents. In [19], the authors addressed QMKP with another genetic algorithm using a specialized crossover that interchanges the object assignments between two randomly selected parents. In [21], an artificial bee colony algorithm combined with local search (SS-ABC) was introduced for QMKP. More recently, García-Martínez et al. [7] applied the tabu-enhanced iterated greedy method (TIG-QMKP) to tackle QMKP. This method alternates between constructive and tabu-enhanced destructive phases linked by an improvement process based on exchanging objects from different knapsacks. Later, García-Martínez et al. [5] successfully applied the strategic oscillation method (SO-QMKP) to solve QMKP. This approach iteratively applies an oscillation method that permits the critical boundary to be crossed and uses two specific steps (a divergent step and a convergent step) to generate a new promising solution, a descent-based improvement method to refine the new solution, and an acceptance criterion that decides which configuration becomes the new current solution. Most recently, Chen and Hao [2] presented the iterated responsive threshold search (IRTS), a very effective heuristic approach, for QMKP. This approach alternates between a threshold-based exploration phase and a descent-based improvement phase, and triggers a dedicated perturbation when the search is judged to be stagnating.
For QMKP, the knapsack׳s capacity is imposed as a strict constraint. As for strictly constrained problems, a natural issue is how to effectively explore the search space so as to enhance the performance of the algorithm [8], [16]. Indeed, accounting for problem constraints in the definition of the search space often restricts the search process too much and can lead to mediocre solutions [8]. In this paper we propose a highly effective tabu search (TS) method that incorporates a hybridization scheme combining both feasible and infeasible local searches to effectively explore the search space of QMKP. The proposed TS method embraces several features that contribute to its effectiveness: (1) a multi-neighborhood local search procedure that intensively explores the feasible search space through a joint use of four complementary neighborhoods. (2) An oscillation-based local search procedure that permits searches through the infeasible search space to bring more freedom to the search. (3) An effective hybridization scheme that follows the general idea presented in [22], and combines both feasible and infeasible local search methods in the TS algorithm for solving QMKP.
In order to evaluate the performance of our proposed TS algorithm, we conduct experimental tests on a set of well-known QMKP instances. The computational results indicate that our TS algorithm yields highly impressive outcomes for these instances by finding improved best known solutions for ten instances and matching the best known results for 49 instances. Only for one instance does the TS algorithm fail to reach the previous best known result. Furthermore, we analyze the effect of the hybridization of feasible and infeasible local search methods in the TS paradigm and establish its key role in boosting the performance of the proposed TS algorithm.
The remainder of this paper is organized as follows: in Section 2 we present in detail our proposed TS algorithm for solving QMKP. In Section 3 we discuss the computational results and comparisons with state-of-the-art approaches on a set of benchmark instances, followed by analyses of the important components of the TS algorithm in Section 4. We conclude the paper and suggest topics for future research in Section 5.
Section snippets
Main framework
Algorithm 1 The framework of the tabu search algorithm for QMKP. Require: A QMKP instance Ensure: The best solution Sbest found and its objective value fbest 1. Begin 2. /⁎ Construct a feasible initial solution, Section 2.3 ⁎/ 3. /⁎ Sbest records the best solution found so far ⁎/ 4. /⁎ fbest records the best objective value reached so far ⁎/ 5. While stop condition is not met do 6. /⁎the feasible local search phase ⁎/ 7. /⁎
Computational experiments
We conduct all the experiments on a computer with an Intel Xeon E5440 processor (2.83 GHz CPU and 2GB RAM). We code the proposed TS algorithm in C++ and compiled the source code by the gcc compiler without using any optimization option, although using the compiler׳s optimization options, e.g., -O3 flag, slightly improves the speed of the algorithm.
Analysis of the main components of TS
In this section we investigate two important components of the proposed TS algorithm that contribute to its effectiveness, namely the hybridization scheme and the neighborhood combination method.
Conclusion
One key issue in designing heuristics to solve strictly constrained combinatorial optimization problems is how to effectively explore the search space so as to strengthen the search ability of the algorithm. In this paper we present a tabu search algorithm that integrates both feasible and infeasible local searches for QMKP. By taking advantage of both the global character of infeasible local search and the more intensive focus typically provided by feasible local search, the proposed tabu
Acknowledgment
We thank the anonymous referees for their helpful comments on an earlier version of our paper. This work was supported in part by the National Natural Science Foundation Program of China under grant numbers 71131004 and 71401059.
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