A Thermodynamical Selection-Based Discrete Differential Evolution for the 0-1 Knapsack Problem
Many problems in business and engineering can be modeled as 0-1 knapsack problems. However, the 0-1 knapsack problem is one of the classical NP-hard problems. Therefore, it is valuable to develop effective and efficient algorithms for solving 0-1 knapsack problems. Aiming at the drawbacks of the selection operator in the traditional differential evolution (DE), we present a novel discrete differential evolution (TDDE) for solving 0-1 knapsack problem. In TDDE, an enhanced selection operator
... ired by the principle of the minimal free energy in thermodynamics is employed, trying to balance the conflict between the selective pressure and the diversity of population to some degree. An experimental study is conducted on twenty 0-1 knapsack test instances. The comparison results show that TDDE can gain competitive performance on the majority of the test instances. Keywords: differential evolution; discrete optimization; thermodynamical selection; 0-1 knapsack problem 0-1 knapsack problems. In the literature , a hybrid quantum inspired harmony search algorithm is proposed by Layeb for solving 0-1 knapsack problems. Lu and Yu  introduced a quantum multi-objective evolutionary algorithm with an adaptive population (APMQEA) to solve multi-objective 0-1 knapsack problems. In the search process, APMQEA fixes the number of Q-bit individuals assigned to each objective solution, while it adaptively updates its population size according to the number of found non-dominated solutions. Deng et al.  introduced a binary encoding differential evolution for solving 0-1 knapsack problems. Truong et al.  presented a chemical reaction optimization with a greedy strategy algorithm (CROG) for solving 0-1 knapsack problems. The experimental results reveal that CROG is a competitive algorithm for solving 0-1 knapsack problems. In the literature  , an amoeboid organism algorithm is proposed by Zhang et al. to solve 0-1 knapsack problems. Among the evolutionary algorithms, differential evolution, proposed by Storn and Price in 1997 , is a simple, yet effective, global optimization algorithm. According to frequently reported theoretical and experimental studies, differential evolution (DE) has demonstrated better performance than many other evolutionary algorithms in terms of both convergence speed and solution quality over several benchmark functions and real-life problems    . Due to its effectiveness, as well as global search capability, DE has attracted many researchers' interests since its introduction. Thus, it has become a research focus of evolutionary computation in recent years [25, 26] . However, DE adopts a one-to-one greedy selection operator between each pair of the target vector and its corresponding trial vector, which exhibits weak selective pressure due to unbiased selection of parents or target vectors [25, 26] . Furthermore, this weakness may lead to inefficient exploitation when relying on differential mutation, especially on the rotated optimization problems  . A straightforward way to alleviate this shortcoming may be to enhance the selection operator to improve the selective pressure. In fact, such mechanism indeed can promote the exploitation capacity of DE. However, high selective pressure may often lead to rapid loss of population diversity and, thus, increase the probability of being trapped in local minima. Therefore, increasing the selective pressure in DE is often in conflict with promoting the population diversity, which indicates that a feasible solution to alleviate this weakness of DE cannot merely improve one of the selective pressures or the population diversity [25, 26] . Nevertheless, to the best of our knowledge, it is one of the most challenging issues of DE to keep a balance between the selective pressure and the diversity of population. Thus, in this paper, we propose a promising discrete DE (TDDE) using an enhanced selection scheme (MFES) inspired by the principle of the minimal free energy in thermodynamics, trying to palliate the conflict between the selective pressure and the population diversity to some degree. Further, the proposed algorithm is applied to the 0-1 knapsack problems. The rest of the paper is organized as follows. The conventional DE and discrete DE are introduced in Section 2. Section 3 presents the improved selection operator of DE inspired by the principle of the minimal free energy in thermodynamics, which is useful for elaborating the implementation of the proposed algorithm, TDDE, in Section 4. Numerical experiments are presented in Section 5 for the comparison and analysis. Finally, Section 6 concludes this paper.