The Multiple Knapsack Problem Approached by a Binary Differential Evolution Algorithm with Adaptive Parameters

Abstract

In this paper the well-known 0-1 Multiple Knapsack Problem (MKP) is approached by an adaptive Binary Differential Evolution (aBDE) algorithm. The MKP is a NP-hard optimization problem and the aim is to maximize the total profit subjected to the total weight in each knapsack that must be less than or equal to a given limit. The aBDE self adjusts two parameters, perturbation and mutation rates, using a linear adaptation procedure that changes their probabilities at each generation. Results were obtained using 11 instances of the problem with different degrees of complexity. The results were compared using aBDE, BDE, a standard Genetic Algorithm (GA) and its adaptive version (aGA), and an island-inspired Genetic Algorithm (IGA) and its adaptive version (aIGA). The results show that aBDE obtained better results than the other algorithms. This indicates that the proposed approach is an interesting and a promising strategy to control the parameters and for optimization of complex problems.Index Terms-Adaptive parameter control, binary differential evolution, multiple knapsack problem, evolutionary computation. I. INTRODUCTIONT HE 0-1 Multiple Knapsack Problem (MKP) is a binary NP-hard combinatorial optimization problem that consists in given a set of items and a set of knapsacks, each item with a mass and a value, determine which item to include in which knapsack. The aim is to maximize the total profit subjected to the total weight in each knapsack that must be less than or equal to a given limit.Different variants of the MKP can be easily adapted to real problems, such as, capital budgeting, cargo loading and others [1]. Hence, the optimization of resource allocation is one major concern in several areas of logistics, transportation and production [2]. In this way, the search for efficient methods to achieve such optimization aims to increase profits and reduce the use of raw materials.According to the size of an instance (number of items and number of knapsacks) of the MKP, the search space can become too large to apply exact methods. Hence, a large number of heuristics and metaheuristics have been applied to the MKP. Some examples are the modified binary Manuscript

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“…An elitism routine can also help algorithm convergence to the optimal solution point and make an always upward fitness curve (best individual fitness in the current population). The elitism copies the best individual of the current population to the next algorithm generation [28].…”

“…Binary Differential Evolution (BDE) is a probabilistic and population-based metaheuristic inspired by the canonical Differential Evolution (DE) algorithm [29]. BDE is adapted to handle binary search space problems using a simple modification of the DE/rand/1/bin variant for binary coding, which combines each individual of the current population with another randomly chosen one using the crossover operator [28]. Besides the binary representation, the main modification is the insertion of a bitflip mutation operator inspired by the GA that improves its global search ability, which enables diversity [28].…”

“…BDE is adapted to handle binary search space problems using a simple modification of the DE/rand/1/bin variant for binary coding, which combines each individual of the current population with another randomly chosen one using the crossover operator [28]. Besides the binary representation, the main modification is the insertion of a bitflip mutation operator inspired by the GA that improves its global search ability, which enables diversity [28]. Thus, the BDE consists of applying these simple operators (crossover and bit-flip mutation) in candidate solutions represented as binary strings.…”