Development of a method to linearize the quadratic assignment problem

Authors

DOI:

https://doi.org/10.15587/1729-4061.2021.225311

Keywords:

quadratic assignment problem, Koopmans and Beckmann formulation, linear binary form

Abstract

The paper presents a new powerful technique to linearize the quadratic assignment problem. There are so many techniques available in the literature that are used to linearize the quadratic assignment problem. In all these linear formulations, both the number of variables and the linear constraints significantly increase. The quadratic assignment problem (QAP) is a well-known problem whereby a set of facilities are allocated to a set of locations in such a way that the cost is a function of the distance and flow between the facilities. In this problem, the costs are associated with a facility being placed at a certain location. The objective is to minimize the assignment of each facility to a location. There are three main categories of methods for solving the quadratic assignment problem. These categories are heuristics, bounding techniques and exact algorithms. Heuristics quickly give near-optimal solutions to the quadratic assignment problem. The five main types of heuristics are construction methods, limited enumeration methods, improvement methods, simulated annealing techniques and genetic algorithms. For every formulated QAP, a lower bound can be calculated. We have Gilmore-Lawler bounds, eigenvalue related bounds and bounds based on reformulations as bounding techniques. There are four main classes of methods for solving the quadratic assignment problem exactly, which are dynamic programming, cutting plane techniques, branch and bound procedures and hybrids of the last two. The QAP has application in computer backboard wiring, hospital layout, dartboard design, typewriter keyboard design, production process, scheduling, etc. The technique proposed in this paper has the strength that the number of linear constraints increases by only one after the linearization process.

Author Biography

Elias Munapo, North West University

PhD, Professor of Operations Research

Department of Statistics and Operations Research

School of Economics and Decision Sciences

References

  1. Munapo, E. (2012). Reducing the number of new constraints and variables in a linearised quadratic assignment problem. AFRICAN JOURNAL OF AGRICULTURAL RESEEARCH, 7 (21). doi: https://doi.org/10.5897/ajarx11.073
  2. Koopmans, T. C., Beckmann, M. (1957). Assignment Problems and the Location of Economic Activities. Econometrica, 25 (1), 53. doi: https://doi.org/10.2307/1907742
  3. Drezner, Z. (2008). Extensive experiments with hybrid genetic algorithms for the solution of the quadratic assignment problem. Computers & Operations Research, 35 (3), 717–736. doi: https://doi.org/10.1016/j.cor.2006.05.004
  4. Yang, X., Lu, Q., Li, C., Liao, X. (2008). Biological computation of the solution to the quadratic assignment problem. Applied Mathematics and Computation, 200 (1), 369–377. doi: https://doi.org/10.1016/j.amc.2007.11.016
  5. Adams, W., Johnson, T. (1994). Improved linear programming-based lower bounds for the quadratic assignment problem. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 43–75. doi: https://doi.org/10.1090/dimacs/016/02
  6. Ramakrishnan, K. G., Resende, M. G. C., Ramachandran, B., Pekny, J. F. (2002). Tight QAP bounds via linear programming. Series on Applied Mathematics, 297–303. doi: https://doi.org/10.1142/9789812778215_0019
  7. Cela, E. (1998). The quadratic assignment problem: theory and algorithms. In Combinatorial Optimization. Springer, 287. doi: https://doi.org/10.1007/978-1-4757-2787-6
  8. Nagarajan, V., Sviridenko, M. (2009). On the Maximum Quadratic Assignment Problem. Mathematics of Operations Research, 34 (4), 859–868. doi: https://doi.org/10.1287/moor.1090.0418
  9. Abdel-Basset, M., Manogaran, G., Rashad, H., Zaied, A. N. H. (2018). A comprehensive review of quadratic assignment problem: variants, hybrids and applications. Journal of Ambient Intelligence and Humanized Computing. doi: https://doi.org/10.1007/s12652-018-0917-x
  10. Billionnet, A., Elloumi, S. (2001). Best reduction of the quadratic semi-assignment problem. Discrete Applied Mathematics, 109 (3), 197–213. doi: https://doi.org/10.1016/s0166-218x(00)00257-2
  11. Casares, P. A. M., Martin-Delgado, M. A. (2020). A quantum interior-point predictor–corrector algorithm for linear programming. Journal of Physics A: Mathematical and Theoretical, 53 (44), 445305. doi: https://doi.org/10.1088/1751-8121/abb439
  12. Al-Rabeeah, M., Munapo, E., Al-Hasani, A., Kumar, S., Eberhard, A. (2019). Computational Enhancement in the Application of the Branch and Bound Method for Linear Integer Programs and Related Models. International Journal of Mathematical, Engineering and Management Sciences, 4 (5), 1140–1153. doi: https://doi.org/10.33889/ijmems.2019.4.5-090
  13. Djeumou Fomeni, F., Kaparis, K., Letchford, A. N. (2020). A cut-and-branch algorithm for the Quadratic Knapsack Problem. Discrete Optimization, 100579. doi: https://doi.org/10.1016/j.disopt.2020.100579
  14. Taha, H. A. (2017). Operations Research: An Introduction. Pearson Education.
  15. Arbib, C., Pınar, M. Ç., Rossi, F., Tessitore, A. (2020). Codon optimization by 0-1 linear programming. Computers & Operations Research, 119, 104932. doi: https://doi.org/10.1016/j.cor.2020.104932
  16. Munapo, E. (2016). Second Proof That P=NP. EngOpt 2016. Program of the Conference.
  17. Munapo, E. (2021). The Traveling Salesman Problem. Research Advancements in Smart Technology, Optimization, and Renewable Energy, 88–109. doi: https://doi.org/10.4018/978-1-7998-3970-5.ch006
  18. Han, B., Li, Y., Geng, S. (2017). 0–1 Linear programming methods for optimal normal and pseudo parameter reductions of soft sets. Applied Soft Computing, 54, 467–484. doi: https://doi.org/10.1016/j.asoc.2016.08.052
  19. Kodama, A., Nishi, T. (2017). Petri net representation and reachability analysis of 0–1 integer linear programming problems. Information Sciences, 400-401, 157–172. doi: https://doi.org/10.1016/j.ins.2017.03.014
  20. Demiröz, B. E., Altınel, İ. K., Akarun, L. (2019). Rectangle blanket problem: Binary integer linear programming formulation and solution algorithms. European Journal of Operational Research, 277 (1), 62–83. doi: https://doi.org/10.1016/j.ejor.2019.02.004
  21. Bakhshi–Jafarabadi, R., Sadeh, J., Soheili, A. (2019). Global optimum economic designing of grid-connected photovoltaic systems with multiple inverters using binary linear programming. Solar Energy, 183, 842–850. doi: https://doi.org/10.1016/j.solener.2019.03.019

Downloads

Published

2021-04-30

How to Cite

Munapo, E. (2021). Development of a method to linearize the quadratic assignment problem. Eastern-European Journal of Enterprise Technologies, 2(4 (110), 54–61. https://doi.org/10.15587/1729-4061.2021.225311

Issue

Vol. 2 No. 4 (110) (2021): Mathematics and Cybernetics - applied aspects

Section

Mathematics and Cybernetics - applied aspects

License

Copyright (c) 2021 Elias Munapo

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.

A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with TECHNOLOGY CENTER PC, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.