I. INTRODUCTION

II. SET-UNION KNAPSACK PROBLEM

III. MOTH SEARCH ALGORITHM

IV. ENHANCED MS ALGORITHM FOR SUKP

V. COMPUTATIONAL EXPERIMENTS

VI. CONCLUSION

REFERENCES

As an important and novel model with multitudinous practical applications, the set-union knapsack problem (SUKP) is a challenging issue in combinatorial optimization. In this paper, we present an enhanced moth search algorithm (EMS) for solving SUKP, which introduces an enhanced interaction operator (EIO) by integrating differential mutation into the global harmony search and then Lévy flight is replaced by EIO. Comparative experimental results, which were conducted on three types of 30 popular SUKP benchmark instances, demonstrate that EMS algorithm is superior to or competitive with the other state-of-the-art metaheuristic algorithm. In particular, EMS reaches the best-known solutions for the great majority of test instances and improves the best-known solutions for six instances. Two critical ingredients of EIO is investigated to confirm their impact on the performance of EMS. The results show that both components have an important role in improving the performance of EMS.

INTRODUCTION

The classical knapsack problem (KP) [1] is still one of the most challenging problems in combinatorial optimization. Since KP is an NP-hard problem and has many practical applications in reality, new varieties are emerging in recent years. In this paper we consider an extension of KP, namely, the set-union knapsack problem (SUKP) [2], [3], which is a popular binary optimization problem with constraints. Although SUKP was proposed long ago, it has recently attracted more and more researchers to study this issue deeply, because it has been proved that there are many important applications in specific fields, such as public key prototype [4], data stream compression [5], and financial decision making [3]. In addition, SUKP is more complicated and challenging than the classical 0-1 KP. The classical 0-1 KP is characterized by one item with a profit and a weight. Nevertheless, there are a set of items and a set of elements in SUKP, in which each item has a profit and each element has a weight. Particularly, a set of items is required to pack into the knapsack in SUKP. In view of its important application in practice and its theoretical research value, SUKP has attracted much attention in the community. According to the existing literature, the method of solving SUKP problem can be categorized into three groups based on their natures: (1) exact algorithm (2) approximate algorithm, and (3) heuristic approach. Here, we are mainly concerned with the most representative research work. The representative exact approach is dynamic programming (DP) algorithm. SUKP has been first introduced in the literature by Goldschmidt et al. with DP [2].