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Benders decomposition for bi-objective linear programs

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2025年第2期322卷 376-400页

作者： Raith, Andrea Lusby, Richard Yousefkhan, Ali Akbar Sohrabi Univ Auckland Dept Engn Sci & Biomed Engn Auckland New Zealand Danish Tech Univ Operat Res Sect Dept Technol Management & Econ Lyngby Denmark

In this paper, we develop a new decomposition technique for solving bi-objective linear programming problems. The proposed methodology combines the bi-objective simplex algorithm with Benders decomposition and can be used to obtain a complete set of efficient extreme solutions, and the corresponding set of extreme non-dominated points, for a bi-objective linear programme. Using a Benders-like reformulation, the decomposition approach decouples the problem into a bi-objective master problem and a bi-objective subproblem, each of which is solved using the bi-objective parametric simplex algorithm. The master problem provides candidate efficient solutions that the subproblem assesses for feasibility and optimality. As in standard Benders decomposition, optimality and feasibility cuts are generated by the subproblem and guide the master problem solve. This paper discusses bi-objective Benders decomposition from a theoretical perspective, proves the correctness of the proposed reformulation and addresses the need for so-called weighted optimality cuts. Furthermore, we present an algorithm to solve the reformulation and discuss its performance for three types of bi-objective optimisation problems.

关键词： multiple objective programming Benders decomposition Bi-objective optimisation Linear programming Bi-objective parametric simplex algorithm

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New Dynamic programming algorithm for the Multiobjective Minimum Spanning Tree problem

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COMPUTERS & OPERATIONS RESEARCH 2025年 173卷

作者： de las Casas, Pedro Maristany Sedeno-Noda, Antonio Borndoerfer, Ralf Zuse Inst Berlin Network Optimizat Takustr 7 D-14195 Berlin Germany Univ La Laguna Matemat Estadist & Invest Operat San Cristobal De La Lagun 38271 Spain

The Multiobjective Minimum Spanning Tree (MO-MST) problem generalizes the Minimum Spanning Tree problem by weighting the edges of the input graph using vectors instead of scalars. In this paper, we design a new Dynamic programming MO-MST algorithm. Dynamic programming for a MO-MST instance requests solving a One-to-One Multiobjective Shortest Path (MOSP) instance and both instances have equivalent solution sets. The MOSP instance is defined on a so called transition graph. We study the original size of this graph in detail and reduce its size using cost-dependent arc pruning criteria. To solve the MOSP instance on the reduced transition graph, , we design the Implicit Graph Multiobjective Dijkstra Algorithm (IG-MDA), exploiting recent improvements on MOSP algorithms from the literature. All in all, the new IG-MDA outperforms the current state of the art on a big set of instances from the literature. Our code and results are publicly available.

关键词： multiple objective programming Multiobjective Minimum Spanning Trees Dynamic programming

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Wolfe type higher order multiple objective nondifferentiable symmetric dual programming with generalized invex function

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Journal of Mathematical Modelling and Algorithms in Operations Research 2014年第4期13卷 557-577页

作者： Tripathy, Arun Kumar Devi, Gayatri Department of Mathematics Trident Academy of Technology F2/A Chandaka Industrial Estate Infocity Area Bhubaneswar 751024 Odisha India Department of Computer Science and Engineering Ajaya Binaya Institute of Technology Markat Nagar CDA Sector-1 Cuttack 753014 Odisha India

In this paper, a new class of higher order (φ, ρ)-invex function is introduced with an example, in which the sublinearity and convexity assumption on φ with respect to third argument is relaxed. A pair of higher order Wolfe type multiobjective symmetric dual for a class of nondifferentiable multiobjective programming involving square root term is presented and the weak duality, strong duality and converse duality theorems are established with their proofs under higher order (φ, ρ)-invexity and (φ, ρ)-incavity assumption. Self duality theorem is proved for the proposed dual program. These results are used to discuss Wolfe type higher-order symmetric minimax mixed integer dual problems. A numerical example is developed where the results of weak and strong duality theorems can be applied. Discussion on some particular cases shows that our results generalize earlier results in related domain. © Springer Science+Business Media Dordrecht 2014.

关键词： Higher order (φ, ρ)-invexity multiple objective programming Pareto optimal solution Schwartz inequality Square root term

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A decision support approach for sustainability management using multiple criteria decision making and statistical techniques

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SUSTAINABLE FUTURES 2025年 9卷

作者： Ruiz, Ana B. Gonzalez-Gallardo, Sandra Luque, Mariano Fernandez-Rodero, Pedro M. Ruiz, Francisco Univ Malaga Dept Appl Econ Math C Ejido S-N Malaga 29071 Spain Univ Malaga Dept Math Anal Stat & Operat Res & Appl Math Blvr Louis Pasteur Malaga 29010 Spain

In this paper, we derive a decision support tool to establish the policies a country may carry out to direct its performance toward the Sustainable Development Goals of the 2030 Agenda. A panel data structure for a set of indicators belonging to the three sustainability dimensions (economic, social, and environmental) is used for the countries considered to calculate composite indicators for the three dimensions, and then formulate a multiobjective optimization model to research how to improve each country's sustainability situation. The case of Spain is used as a proof of concept.

关键词： multiple objective decision making multiple objective programming Composite indicators Sustainability assessment

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KKT transformation approach for multi-objective multi-level linear programming problems

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2002年第1期143卷 19-31页

作者： Sinha, S Sinha, SB Indian Inst Technol Dept Math Kharagpur 721302 W Bengal India

The earlier Karush-Kuhn-Tucker (KKT) transformation method has been applied to multi-level decentralized programming problems (ML(D)PPs) when the decision variable set was divided into subsets where each decision maker (DM) of the system controlled only a particular subset but had no control over any decision variables of some other subset. In this paper we give the mathematical formulation and corresponding development of ML(D)PPs by KKT transformation when DMs have absolute control over certain decision variables but some variables may be shared and hence controlled by two or more DMs. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： multiple objective programming multi-level linear programming Lagrange function Karush-Kuhn-Tucker's optimality conditions local optimal solution

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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 multi-objective mixed integer linear programming model for thesis defence scheduling

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2024年第1期312卷 92-116页

作者： Almeida, Joao Santos, Daniel Figueira, Jose Rui Francisco, Alexandre P. Univ Lisbon CEGIST Inst Super Tecn Lisbon Portugal Univ Lisbon INESC ID Inst Super Tecn Lisbon Portugal

In this paper, we address the thesis defence scheduling problem, a critical academic scheduling management process, which has been overshadowed in the literature by its counterparts, course timetabling and exam scheduling. Specifically, we address the single defence assignment type of thesis defence schedul-ing problems, where each committee is assigned to a single defence, scheduled for a specific day, hour and room. We formulate a multi-objective mixed-integer linear programming model, which aims to be applicable to a broader set of cases than other single defence assignment models present in the literature, which have a focus on the characteristics of their universities. For such a purpose, we introduce a dif-ferent decision variable, propose constraint formulations that are not regulation and policy specific, and cover and offer new takes on the more common objectives seen in the literature. We also include new objective functions based on our experience with the problem at our university and by applying knowl-edge from other academic scheduling problems. We also propose a two-stage solution approach. The first stage is employed to find the number of schedulable defences, enabling the optimisation of instances with unschedulable defences. The second stage is an implementation of the augmented & epsilon;-constraint method, which allows for the search of a set of different and non-dominated solutions while skipping redundant iterations. The methodology is tested for case-studies from our university, significantly outperforming the solutions found by human schedulers. A novel instance generator for thesis scheduling problems is presented. Its main benefit is the generation of the availability of committee members and rooms in availability and unavailability blocks, resembling their real-world counterparts. A set of 96 randomly generated instances of varying sizes is solved and analysed regarding their relative computational performance, the number of schedulable de

关键词： Timetabling Education Integer programming multiple objective programming Scheduling

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A multi-objective multi-period stochastic programming model for public debt management

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2010年第1期205卷 205-217页

作者： Balibek, Emre Koksalan, Murat Middle E Tech Univ Dept Ind Engn TR-06531 Ankara Turkey Turkish Treasury TR-06510 Ankara Turkey

While raising debt on behalf of the government. public debt managers need to consider several possibly conflicting objectives and have to find an appropriate combination for government debt taking into account the uncertainty with regard to the future state of the economy. In this paper, we explicitly consider the underlying uncertainties with a complex multi-period stochastic programming model that captures the trade-offs between the objectives. The model is designed to aid the decision makers in formulating the debt issuance strategy. We apply an interactive procedure that guides the issuer to identify good strategies and demonstrate this approach for the public debt management problem of Turkey. (C) 2009 Elsevier B.V. All rights reserved.

关键词： OR in government multiple objective programming Risk analysis Stochastic programming Public debt management

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A computationally efficient algorithm to approximate the pareto front of multi-objective linear fractional programming problem

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RAIRO-OPERATIONS RESEARCH 2019年第4期53卷 1229-1244页

作者： Stanojevic, Bogdana Stanojevic, Milan Serbian Acad Arts & Sci Math Inst Belgrade Serbia Univ Belgrade Fac Org Sci Belgrade Serbia

The main contribution of this paper is the procedure that constructs a good approximation to the non-dominated set of multiple objective linear fractional programming problem using the solutions to certain linear optimization problems. In our approach we propose a way to generate a discrete set of feasible solutions that are further used as starting points in any procedure for deriving efficient solutions. The efficient solutions are mapped into non-dominated points that form a 0th order approximation of the Pareto front. We report the computational results obtained by solving random generated instances, and show that the approximations obtained by running our procedure are better than those obtained by running other procedures suggested in the recent literature. We evaluated the quality of each approximation using classic metrics.

关键词： Fractional programming multiple objective programming efficient solution non-dominated point 0th order approximation

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Modeling and optimization of multi objective non-linear programming problem in intuitionistic fuzzy environment

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APPLIED MATHEMATICAL MODELLING 2015年第16期39卷 4617-4629页

作者： Singh, Sujeet Kumar Yadav, Shiv Prasad Indian Inst Technol Dept Math Roorkee 247667 Uttar Pradesh India

In dealing with real world practical optimization problems, a decision maker usually faces a state of uncertainty as well as hesitation, due to various unpredictable factors. Sometimes it is necessary to optimize, several non-linear and conflicting objectives simultaneously. To deal with the uncertain parameters which arise in such situations, intuitionistic fuzzy numbers are utilized. We formulate a multiobjective non-linear programming problem in intuitionistic fuzzy environment. We propose a linear ranking function and utilize it to convert the intuitionistic fuzzy model into a crisp model. After converting the problem into equivalent crisp problem, we propose a non-linear membership function and develop various approaches for solving it by using different operators and fuzzy programming technique. We apply our methodologies for justification to a numerical problem in manufacturing systems. (C) 2015 Elsevier Inc. All rights reserved.

关键词： multiple objective programming Intuitionistic fuzzy number Accuracy function Nonlinear programming

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