折扣{0-1}背包问题的简化新模型及遗传算法求解

当前折扣{0-1}背包问题（D{0-1}KP）模型将折扣关系作为一个新的个体，导致求解过程必需采取修复法对个体编码进行修复，求解方式较少。针对求解方法单一的问题，通过改变模型中二进制的编码表达方式，提出折扣关系不在个体编码中的表达方法。首先，设定对任意折扣关系，当且仅当所涉及个体编码值同时为1（即其乘积为1）时，折扣关系成立，据此建立简化折扣{0-1}背包问题（SD{0-1}KP）模型；然后，针对SD{0-1}KP模型，基于杰出者保留策略（EGA），结合贪心策略（GRE），提出改进遗传算法——第一遗传算法（FG）；最后，再结合罚函数法，提出求解SD{0-1}KP高精度罚函数法——第二遗传算法（SG）。结果表明，SD{0-1}KP能够完全覆盖D{0-1}KP问题领域，与FirEGA相比，所提出的两类算法在求解速度方面优势明显，且SG算法首次引入罚函数法，有效地丰富了该问题的求解算法。

New simplified model of discounted {0-1} knapsack problem and solution by genetic algorithm

Current Discounted {0-1} Knapsack Problem (D{0-1}KP) model takes the discounted relationship as a new individual, so the repair method must be adopted in the solving process to repair the individual coding, making the model have less solving methods. In order to solve the problem of single solving method, by changing the binary code expression in the model, an expression method with discounted relationship out of individual code was proposed. Firstly, if and only if each involved individual encoding value was one (which means the product was one), the discounted relationship was established. According to this setting, a Simplified Discounted {0-1} Knapsack Problem (SD{0-1}KP) model was established. Then, an improved genetic algorithm-FG (First Gentic algorithm) was proposed based on Elitist Reservation Strategy (EGA) and GREedy strategy (GRE) for SD{0-1}KP model. Finally, combining penalty function method, a high precision penalty function method-SG (Second Genetic algorithm) for SD{0-1}KP was proposed. The results show that the SD{0-1}KP model can fully cover the problem domain of D{0-1}KP. Compared with FirEGA (First Elitist reservation strategy Genetic Algorithm), the two algorithms proposed have obvious advantages in solving speed. And SG algorithm introduces the penalty function method for the first time, which enriches the solving methods of the problem.