Artificial Bee Colony Algorithm Merged with Pheromone Communication Mechanism for the 0-1 Multidimensional Knapsack Problem

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 676275, 13 pages http://dx.doi.org/10.1155/2013/676275 Research Article Artificial Bee Colony Algorithm Merged with Pheromone Communication Mechanism for the 0-1 Multidimensional Knapsack Problem Junzhong Ji, Hongkai Wei, Chunnian Liu, and Baocai Yin College of Computer Science and Technology, Beijing University of Technology, Beijing Municipal Key Laboratory of Multimedia and Intelligent Software Technology, Beijing 100124, China Correspondence should be addressed to Junzhong Ji; Received 24 March 2013; Accepted 11 June 2013 Academic Editor: Vishal Bhatnagar Copyright © 2013 Junzhong Ji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Given a set of n objects, the objective of the 0-1 multidimensional knapsack problem (MKP 01) is to find a subset of the object set that maximizes the total profit of the objects in the subset while satisfying m knapsack constraints. In this paper, we have proposed a new artificial bee colony (ABC) algorithm for the MKP 01. The new ABC algorithm introduces a novel communication mechanism among bees, which bases on the updating and diffusion of inductive pheromone produced by bees. In a number of experiments and comparisons, our approach obtains better quality solutions in shorter time than the ABC algorithm without the mechanism. We have also compared the solution performance of our approach against some stochastic approaches recently reported in the literature. Computational results demonstrate the superiority of the new ABC approach over all the other approaches. 1. Introduction Given a set 𝐽 = {𝑜1 , 𝑜2 , . . . , 𝑜𝑛 } of 𝑛 objects and a knapsack with a set 𝐶 = {𝑐1 , 𝑐2 , . . . , 𝑐𝑚 } of 𝑚 dimensions, the 01 multidimensional knapsack problem (MKP 01) seeks a subset of 𝐽 in such a way that the total profit of objects included in the subset is maximized, while 𝑚 resource constraints remain satisfied. More formally, each object 𝑜𝑗 ∈ 𝐽 has profit 𝑝𝑗 and weight 𝑟𝑖𝑗 in dimension 𝑖 (1 ≤ 𝑖 ≤ 𝑚), and each dimension of the knapsack has a capacity 𝑐𝑖 . By introducing binary decision variable 𝑥𝑗 to indicate whether object 𝑜𝑗 is included into the knapsack (𝑥𝑗 = 1) or not (𝑥𝑗 = 0), the MKP 01 can be formulated as 𝑛 Maximize ∑ 𝑝𝑗 𝑥𝑗 𝑗=1 𝑛 Subject to ∑ 𝑟𝑖𝑗 𝑥𝑗 ≤ 𝑐𝑖 , 𝑖 = 1, . . . , 𝑚, 𝑗=1 𝑥𝑗 ∈ {0, 1} , 𝑗 = 1, . . . , 𝑛. (1) The MKP 01 is a well-known NP-Hard problem. There are many practical applications which can be formulated as a MKP 01, for example, the capital budgeting problem, the cargo loading, the processor allocation in distributed systems, and the project selection. Therefore, more and more people recently focus on the research for solving the MKP 01. In general, the solving algorithms can be divided into two kinds: exact and heuristic methods [1]. The exact methods used to employ some typical search techniques, such as Enumeration algorithm [2], Branch and Bound method [3], and Approximate Dynamic programming [4]. These methods can be only applied to some small-scaled MKP 01 because the computation complexity is rather high. Subsequently, many heuristic search methods, including Genetic Algorithm (GA) [5, 6], Evolutionary Algorithm (EA) [7], Particle Swarm Optimization (PSO) [8], Ant Colony Optimization (ACO) [9–12], and Artificial Bee Colony (ABC) [13, 14], were proposed by simulating some natural phenomena. As these populationbased methods are versatile and robust, thus they have been proved to be very effective methods. Thereinto, ABC algorithm is a recently proposed method, which employs the 2 mechanism of combining local and global searches to effectively solve MKPs. However, since the algorithm framework itself is flawed, there is a main bottleneck that the iteration number is too large and the convergence time is too long, which strongly restricts the development of ABC algorithm. In [15], we proposed an artificial bee colony algorithm for the MKP 01, which introduced the pheromone into ABC algorithm and gave the corresponding transition strategy. Though there are still some problems, the preliminary experimental results in [15] are very encouraging, and these results are significant motivations for the present research. This paper conducts a further and thorough investigation along this line. In comparison with our previous work, the main new contributions of this paper include the following. (1) Based on the researches of entomologists, a new algorithm, combining chemical communication way and behavior communication way for solving MKP 01 (ABCPUDMKP), has been developed. First, the paper extends our previous work, analyzes, and explicitly presents the pheromone communication mechanism. Second, the paper designs the constructing method of feasible solutions based on the new mechanism. Third, the paper introduces the special updating and diffusing strategies of the inductive pheromone. An important characteristic of the new algorithm is that the collaborations among bees can be strengthened, and the intelligent foraging behavior of honey swarm can be mimicked more faithfully by means of updating and diffusion of the inductive pheromone. (2) Systematic experiments have been conducted to compare the proposed algorithm with the previous work and some other algorithms proposed recently in respective literatures on many instances of two benchmark data sets. Moreover, the sensitivity to algorithmic parameters and effects of different parameter selections have been experimentally investigated. The rest of this paper is organized as follows. Section 2 provides an introduction to the artificial bee colony algorithm and the basic idea of the ABC for the MKP 01. In Section 3, we describe our new algorithm for the MKP 01. Computational results are presented in Section 4. Finally, we conclude the paper in Section 5. 2. ABC Algorithm and Its Application in the MKP_01 2.1. The Artificial Bee Colony (ABC) Algorithm. The artificial bee colony algorithm is a population-based metaheuristic approach proposed recently; it finds near-optimal solutions to the difficult optimization problems by simulating the intelligent foraging behavior of a honeybee swarm. There are three groups of the foraging bees, employed ones, onlookers, and scouts. All bees that are currently exploiting food sources are called employed bees. These bees bring nectar from different food sources to their hive. Onlooker bees are those bees who are waiting in the hive for the information on food sources to be shared by the employed bees, and scout bees are those bees that are currently searching for new food sources near the hive. By dancing in a common area Mathematical Problems in Engineering of the hive, employed bees share the information on food sources with onlooker bees. The duration of a dance of an em (...truncated)