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Optimizing a Binary Integer Program by Identifying Its Optimal Core Problem - A New Optimization Concept Applied to the Multidimensional Knapsack Problem

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Modelling, Computation and Optimization in Information Systems and Management Sciences (MCO 2021)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 363))

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Abstract

The core concept for solving a binary integer program (BIP) is about dividing the BIP’s variables into core and adjunct ones. We fix the adjunct variables to either 0 or 1; consequently, we reduce the problem to the core variables only, forming a core problem (CP). An optimal solution to a CP is not optimal to the original BIP unless adjunct variables are fixed to their optimal values. Consequently, an optimal CP is a CP whose associated adjunct variables are fixed to their optimal values. This paper presents a new optimization concept that solves a BIP by searching for its optimal CP. We use a hybrid algorithm of local search and linear programming to move from a CP to a better one until we find the optimal CP. We use our algorithm to solve 180 multidimensional knapsack (MKP) instances to validate this new optimization concept. Results show that it is a promising approach to investigate because we were able to find the optimal solutions of 149 instances, of which some had 500 variables, by solving several CPs having 30 variables only.

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Author information

Authors and Affiliations

  1. Industrial Engineering and Engineering Management Department, University of Sharjah, PO Box 27272, Sharjah, United Arab Emirates

    Sameh Al-Shihabi

  2. Industrial Engineering Department, The University of Jordan, Amman, Jordan

    Sameh Al-Shihabi

Authors
  1. Sameh Al-Shihabi
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Correspondence to Sameh Al-Shihabi .

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Editors and Affiliations

  1. Computer science and Applications Department, LGIPM, University of Lorraine, Metz Cedex, France

    Hoai An Le Thi

  2. Laboratory of Mathematics, National Institute for Applied Sciences - Rouen, Saint-Etienne-du-Rouvray Cedex, France

    Tao Pham Dinh

  3. Computer science and Applications Department, LGIPM, University of Lorraine, Metz Cedex, France

    Hoai Minh Le

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Al-Shihabi, S. (2022). Optimizing a Binary Integer Program by Identifying Its Optimal Core Problem - A New Optimization Concept Applied to the Multidimensional Knapsack Problem. In: Le Thi, H.A., Pham Dinh, T., Le, H.M. (eds) Modelling, Computation and Optimization in Information Systems and Management Sciences. MCO 2021. Lecture Notes in Networks and Systems, vol 363. Springer, Cham. https://doi.org/10.1007/978-3-030-92666-3_3

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