Abstract
This paper describes methods to merge two cover inequalities and also simultaneously merge multiple cover inequalities in a multiple knapsack instance. Theoretical results provide conditions under which merged cover inequalities are valid. Polynomial time algorithms are created to find merged cover inequalities. A computational study demonstrates that merged inequalities improve the solution times for benchmark multiple knapsack instances by about 9% on average over CPLEX with default settings.
Highlights
Introduction to Inequality MergingAn integer program (IP) is a common type of optimization problem, defined as maximize cT x subject to Ax ≤ b and x ∈ n + whereA ∈ m×n, b ∈ m, and c ∈ n where m and n are integers both greater than or equal to 1
Notice that following the recommended implementation strategies always improved the solution times. This provides strong evidence that inequality merging is a beneficial technique for multiple knapsack (MK) problems, and the reduction of computational ticks correlates to hours of time savings for large problems
This paper provides the theoretical foundations needed to build merged cover inequalities in MK instances
Summary
An integer program (IP) is a common type of optimization problem, defined as maximize cT x subject to. (2015) On Merging Cover Inequalities for Multiple Knapsack Problems. If the valid inequality separates the linear relaxation solution from the convex hull of the IP, it is called a cutting plane. Their paper creates a single cutting plane by merging two inequalities. Information from two or more cover inequalities in an MK instance may be merged into a single cutting plane. Simultaneous merging of cover inequalities may occur across multiple rows at the same time. The section describes the process of cover inequality merging for MK instances and provides theoretical results and examples. The third section offers the results of a computational study that highlights the computational benefits of employing merged cover inequalities in test MK problems.
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