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All Journal International Journal of Industrial Optimization (IJIO)
Yun Lu
Kutztown University
Decision Sciences, Operations Research & Management Engineering Industrial & Manufacturing Engineering

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Using general-purpose integer programming software to generate bounded solutions for the multiple knapsack problem: a guide for or practitioners Emre Shively-Ertas; Yun Lu; Myung Song; Francis Vasko
International Journal of Industrial Optimization Vol. 4, No. 1 (2023)
Publisher : Universitas Ahmad Dahlan

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.12928/ijio.v4i1.6446

Abstract

An NP-Hard combinatorial optimization problem that has significant industrial applications is the Multiple Knapsack Problem. If approximate solution approaches are used to solve the Multiple Knapsack Problem there are no guarantees on solution quality and exact solution approaches can be intricate and challenging to implement.  This article demonstrates the iterative use of general-purpose integer programming software (Gurobi) to generate solutions for test problems that are available in the literature. Using the software package Gurobi on a standard PC, we generate in a relatively straightforward manner solutions to these problems in an average of less than a minute that are guaranteed to be within 0.16% of the optimum.  This algorithm, called the Simple Sequential Increasing Tolerance (SSIT) algorithm, iteratively increases 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. This approach is attractive to management for solving industrial problems because it is both cost and time effective and guarantees the quality of the generated solutions.  Finally, comparing SSIT results for 480 large multiple knapsack problem instances to results using published multiple knapsack problem algorithms demonstrates that SSIT outperforms these specialized algorithms.
The role of mathematical formulation in solving the unbalanced assignment problem Francis J. Vasko; Yun Lu; Myung Soon Song
International Journal of Industrial Optimization Vol. 7 No. 1 (2026)
Publisher : Universitas Ahmad Dahlan

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.12928/ijio.v7i1.12987

Abstract

In a 2019 paper, the authors claim to have developed a modified Hungarian method that performs better than a number of other solution methods for the unbalanced assignment problem (UAP) based on the solution of one UAP instance that has been discussed in the literature. The purpose of this short paper is to demonstrate that the math formulation used in the 2019 paper was not as restrictive as the standard one commonly used in the literature and therefore the comparison is not valid. The commonly used UAP math formulation not only tries to minimize cost, but also tries to level load the jobs onto the machines. The formulation from the 2019 paper allows many jobs to be assigned to a low-cost machine. Hence solutions (not even optimums) to the 2019 formulation can be better than the optimal solution using the standard UAP math formulation. Additionally, it will be shown that the Modified Hungarian method proposed in the 2019 paper does not generate guaranteed optimums to the math formulation used in that paper (let alone the standard UAP formulation). An 8-job and 5-machine assignment problem that appeared in the literature will be used to illustrate the points mentioned above.
Co-Authors Emre Shively-Ertas Francis J. Vasko Francis Vasko Myung Song Myung Soon Song