Preemptive Two-stage Goal-Programming Formulation of a Strict Version of the Unbounded Knapsack Problem with Bounded Weights

TL;DR

This paper introduces a novel two-stage goal programming approach for a strict version of the unbounded knapsack problem with bounded weights, enabling efficient solutions and greater modeling flexibility.

Contribution

It formulates a strict unbounded knapsack problem with bounded weights as a two-stage goal programming model, improving solution efficiency and modeling flexibility.

Findings

Equivalent reformulation to original problem.

Allows use of efficient multidimensional knapsack algorithms.

Provides immediate feasible solutions for the feasibility stage.

Abstract

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.