有界权无界背包问题严格型的抢占式两阶段目标规划
Preemptive Two-stage Goal-Programming Formulation of a Strict Version of the Unbounded Knapsack Problem with Bounded Weights
摘要：
有界权重的无界背包问题是传统二进制背包问题的一个变体;关键的变化是放松二进制约束，并允许每个项目的单位权重落在一个范围内。在本文中，我们制定了这个问题的一个变种，我们称之为严格无界背包问题与有界的重量，通过取代的不等式约束的总重量与一个等式。我们表明，这个问题可以分解成两个阶段，先发制人的目标规划问题，第一阶段是一个2维背包问题，第二阶段是一个线性可行性方案（每规范制定）或简单的线性约束程序在一般情况下。这种重新被证明是相当于原来的配方，但允许使用良好的研究，高效的多维背包问题的算法。此外，它将围绕背包中放置什么的建模工作与应该为每个物品类型分配什么单位重量的考虑分开，为建模者提供了更大的灵活性，而不会增加背包配置选择的复杂性。最后，我们表明，对于第二阶段的可行性版本，可以立即得到第一阶段的解决方案的可行解。
英文摘要：
The unbounded knapsack problem with bounded weights is a variant of the well-studied variant of the traditional binary knapsack problem; key changes being the relaxation of the binary constraint and allowing the unit weights of each item to fall within a range. In this paper, we formulate a variant of this problem, which we call the strict unbounded knapsack problem with bounded weights, by replacing the inequality constraint on the total weight with an equality. We show that this problem can be decomposed into a two-stage, pre-emptive goal programming problem, with the first stage being a 2-dimensional knapsack problem and the second being either a linear feasibility program (per canonical formulation) or simply a linearly-constrained program in the general case. This reformulation is shown to be equivalent to the original formulation but allows the use of well-studied, efficient algorithms for multidimensional knapsack problems. In addition, it separates the modeling effort around what to put in the knapsack from considerations around what unit weight one should assign to each item type, providing substantially more flexibility to the modeler without adding complexity to the choice of knapsack configuration. Finally, we show that for the feasibility version of the second stage, one can immediately get a feasible solution to the first stage solution.
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