A Simple and Efficient Technique to Generate Bounded Solutions for the Multidimensional Knapsack Problem: a Guide for OR Practitioners

Abstract

The 0-1 Multidimensional Knapsack Problem (MKP) is a NP-Hard problem that has important applications in business and industry. Approximate solution approaches for the MKP in the literature typically provide no guarantee on how close generated solutions are to the optimum. This article demonstrates how general-purpose integer programming software (Gurobi) is iteratively used to generate solutions for the 270 MKP test problems in Beasley’s OR-Library such that, on average, the solutions are guaranteed to be within 0.094% of the optimums and execute in 88 seconds on a standard PC. This methodology, called the simple sequential increasing tolerance (SSIT) matheuristic, uses a sequence of increasing tolerances in Gurobi to generate a solution that is guaranteed to be close to the optimum in a short time. This solution strategy generates bounded solutions in a timely manner without requiring the coding of a problem-specific algorithm. The SSIT results (although guaranteed within 0.094% of the optimums) when compared to known optimums deviated only 0.006% from the optimums—far better than any published results for these 270 MKP test instances.

Using general-purpose integer programming software to generate bounded solutions for the multiple knapsack problem: a guide for or practitioners

“…More benefits of SSIT are detailed in [19]. Successful applications of SSIT to solve several binary integer programs (BIP) have been documented in the literature [14][15][16][17][18][19]. For these applications, SSIT typically generates solutions guaranteed to be within 0.1% of the optimums in about 60 seconds on standard PCs.…”

Simple strategies that generate bounded solutions for the multiple‐choice multi‐dimensional knapsack problem: a guide for OR practitioners

“…Statistical analyses demonstrated that these SSIT results were as good as the best published results obtained from algorithms specifically designed to solve SKCPs. Also, using Gurobi, Lu et al (2021) employed the SSIT methodology to quickly (average of 88 seconds on a standard PC) generate solutions guaranteed to be, on average, within 0.09% of the optimum on 270 MKP instances commonly used in the literature. These results are far better than other published metaheuristic results for the MKP.…”

“…Also, using Gurobi, Lu et al. (2021) employed the SSIT methodology to quickly (average of 88 seconds on a standard PC) generate solutions guaranteed to be, on average, within 0.09% of the optimum on 270 MKP instances commonly used in the literature. These results are far better than other published metaheuristic results for the MKP.…”

A Simple and Effective Methodology for Generating Bounded Solutions for the Set K-Covering and Set Variable K-Covering Problems: A Guide for or Practitioners

“…In this article a methodology first discussed in McNally [10] referred to as the simple sequential increasing tolerance (SSIT) matheuristic is used to quickly solve 135 SKCPs commonly used in the literature and 65 SVKCPs introduced in this article. Lu, et al [11] used SSIT to quickly (average of 88 seconds on a standard PC) generate solutions guaranteed to be close (average within 0.094% of the optimums) on 270 multidimensional knapsack problem (MKP) instances commonly used in the literature. These results are far better than other published metaheuristic results for the MKP.…”