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A criterion space search algorithm for Biobjective Integer Programming: The Balanced Box Method

INFORMS JOURNAL ON COMPUTING 2015年第4期27卷 735-754页

作者： Boland, Natashia Charkhgard, Hadi Savelsbergh, Martin Georgia Inst Technol Sch Ind & Syst Engn Atlanta GA 30332 USA Univ Newcastle Sch Math & Phys Sci Callaghan NSW 2308 Australia

We present a new criterion space search algorithm, the balanced box method, for finding all nondominated points of a biobjective integer program. The method extends the box algorithm, is easy to implement, and converges quickly to the complete set of nondominated points. Because the method maintains, at any point in time, a diverse set of nondominated points, it is ideally suited for fast approximation of the efficient frontier. In addition, we present several enhancements of the well-known. epsilon-constraint, augmented weighted Tchebycheff, and perpendicular search methods. An extensive computational study, using instances from different classes of combinatorial optimization problems, demonstrates the efficacy of the balanced box method.

关键词： biobjective integer programming criterion space search algorithm balanced box method

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Learning to project in a criterion space search algorithm: an application to multi-objective binary linear programming

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OPTIMIZATION LETTERS 2024年第9期18卷 2051-2078页

作者： Sierra-Altamiranda, Alvaro Charkhgard, Hadi Dayarian, Iman Eshragh, Ali Javadi, Sorna Univ S Florida Dept Ind & Management Syst Engn Tampa FL 33620 USA Univ Alabama Culverhouse Coll Business Tuscaloosa AL 35487 USA Johns Hopkins Univ Carey Business Sch Washington DC USA

In this paper, we investigate the possibility of improving the performance of multi-objective optimization solution approaches using machine learning techniques. Specifically, we focus on multi-objective binary linear programs and employ one of the most effective and recently developed criterion space search algorithms, the so-called KSA, during our study. This algorithm computes all nondominated points of a problem with p objectives by searching on a projected criterion space, i.e., a (p-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p-1)$$\end{document}-dimensional criterion apace. We present an effective and fast learning approach to identify on which projected space the KSA should work. We also present several generic features/variables that can be used in machine learning techniques for identifying the best projected space. Finally, we present an effective bi-objective optimization-based heuristic for selecting the subset of the features to overcome the issue of overfitting in learning. Through an extensive computational study over 2000 instances of tri-objective knapsack and assignment problems, we demonstrate that an improvement of up to 18% in time can be achieved by the proposed learning method compared to a random selection of the projected space. To show that the performance of our algorithm is not limited to instances of knapsack and assignment problems with three objective functions, we also report similar performance results when the proposed learning approach is used for solving random binary integer program instances with four objective functions.

关键词： Multiple objective programming Machine learning Binary linear program criterion space search algorithm Learning to project

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A criterion space search algorithm for mixed integer linear maximum multiplicative programs: a multiobjective optimization approach

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INTERNATIONAL TRANSACTIONS IN OPERATIONAL REsearch 2022年第3期29卷 1659-1687页

作者： Saghand, Payman Ghasemi Charkhgard, Hadi Univ S Florida Dept Ind & Management Syst Engn Tampa FL 33620 USA

We study a class of mixed integer optimization problems with linear constraints and a multilinear objective function, the so-called mixed integer linear maximum multiplicative programs (MIL-MMPs). Such a problem can be transformed into a second-order cone program (SOCP) and can be solved effectively by a commercial solver such as CPLEX. However, MIL-MMPs can also be viewed as special cases of the problem of optimization over the set of efficient solutions in multiobjective optimization. Using this observation, we develop a criterion space search algorithm for solving any MIL-MMP. An extensive computational study on around 2000 instances illustrates that the proposed algorithm significantly outperforms not only the CPLEX mixed integer SOCP solver but also a state-of-the-art algorithm that is capable of solving special cases of MIL-MMPs. Moreover, the computational study illustrates that even if we linearize the objective function and solve the linearized problem by CPLEX, the proposed algorithm still performs significantly better.

关键词： mixed integer maximum multiplicative programming multiobjective optimization optimization over the efficient set criterion space search algorithm

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An exact criterion space search method for a bi-objective nursing home location and allocation problem

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MATHEMATICS AND COMPUTERS IN SIMULATION 2023年 206卷 166-180页

作者： Gargari, Masoud Atashpaz Sahraeian, Rashed Shahed Univ Coll Engn Dept Ind Engn Tehran Iran

A health care system will be survived if it is economically affordable and elderly people can receive its services in case of availability. Hence, a network should be designed for home nursing of elderly people considering both economic and well-fare objectives. Finding optimal solutions to both mentioned objectives in health care systems, especially for elderly people, is challenging nowadays. Hence, it is necessary to find near-optimal solutions in action. This paper presents a new criterion space search algorithm for achieving all non-dominated points in a bi-objective nursing home location-allocation problem. Concepts of Box and Balanced Box methods and modified e-constraint are used in designing the proposed algorithm. The main focus of the research is to decrease potential areas that could involve non-dominated points by defining Minimum Feasible Rectangles during the search procedure. The performance of the proposed algorithm is analyzed by using two real data sets taken from the literature in the field of nursing home location-allocation problems. The proposed algorithm can outperform previous recently developed exact methods based on obtained results, especially from a computational time viewpoint.(c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： Nursing home system Bi-objective optimization Location and allocation criterion space search algorithm Minimum Feasible Rectangle

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A New Exact algorithm to Optimize a Linear Function over the Set of Efficient Solutions for Biobjective Mixed Integer Linear Programs

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INFORMS JOURNAL ON COMPUTING 2019年第4期31卷 823-840页

作者： Altamiranda, Alvaro Sierra Charkhgard, Hadi Univ S Florida Dept Ind & Management Syst Engn Tampa FL 33620 USA

We present the first (criterion space search) algorithm for optimizing a linear function over the set of efficient solutions of biobjective mixed integer linear programs. The proposed algorithm is developed based on the triangle splitting method [Boland N, Charkhgard H, Savelsbergh M (2015) A criterion space search algorithm for biobjective mixed integer programming: The triangle splitting method. INFORMS J. Comput. 27(4): 597-618.], which can find a full representation of the nondominated frontier of any biobjective mixed integer linear program. The proposed algorithm is easy to implement and converges quickly to an optimal solution. An extensive computational study shows the efficacy of the algorithm. We numerically show that the proposed algorithm can be used to quickly generate a provably high-quality approximate solution because it maintains a lower and an upper bound on the optimal value of the linear function at any point in time.

关键词： biobjective mixed integer linear programming criterion space search algorithm optimization over efficient frontier triangle splitting method

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Multi-objective optimization based algorithms for solving mixed integer linear minimum multiplicative programs

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COMPUTERS & OPERATIONS REsearch 2021年 128卷 105178-105178页

作者： Mahmoodian, Vahid Charkhgard, Hadi Zhang, Yu Univ S Florida Dept Ind & Management Syst Engn Tampa FL 33620 USA Univ S Florida Dept Civil & Environm Engn Tampa FL 33620 USA

We present two new algorithms for a class of single-objective non-linear optimization problems, the socalled Mixed Integer Linear minimum Multiplicative Programs (MIL-mMPs). This class of optimization problems has a desirable characteristic: a MIL-mMP can be viewed as a special case of the problem of optimization over the efficient set in multi-objective optimization. The proposed algorithms exploit this characteristic and solve any MIL-mMP from the viewpoint of multi-objective optimization. A computational study on 960 instances demonstrates that the proposed algorithms outperform a generic purpose solver, SCIP, by a factor of more than 10 on many instances. We numerically show that selecting the best algorithm among our proposed algorithms highly depends on the class of instances used. Specifically, since one of the proposed algorithms is a decision space search algorithm and the other one is a criterion space search algorithm, one can significantly outperform the other depending on the dimension of decision space and criterion space. Although it is possible to linearize some instances of MIL-mMPs, we show that a commercial solver, CPLEX, struggles to directly solve such linearized instances because linearization introduces additional constraints and binary decision variables. (c) 2020 Elsevier Ltd. All rights reserved.

关键词： Multi-objective optimization Minimum multiplicative programming Optimization over the efficient set Decision space search algorithm criterion space search algorithm

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