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Variants of the ε-constraint method for biobjective integer programming problems: application to p-median-cover problems

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MATHEMATICAL methodS OF OPERATIONS RESEARCH 2018年第2期87卷 251-283页

作者： Saez-Aguado, Jesus Camelia Trandafir, Paula Univ Valladolid Dept Stat & Operat Res Paseo Belen 7 E-47011 Valladolid Spain Univ Publ Navarra Dept Stat & Operat Res Avda Tarazona S-N Tudela 31500 Spain

We conduct an in-depth analysis of the epsilon-constraint method (ECM) for finding the exact Pareto front for biobjective integer programming problems. We have found up to six possible different variants of the ECM. We first discuss the complexity of each of these variants and their relationship with other exact methods for solving biobjective integer programming problems. By extending some results of Neumayer and Schweigert (OR Spektrum 16: 267-276, 1994), we develop two variants of the ECM, both including an augmentation term and requiring N + 1 integer programs to be solved, where N is the number of nondominated points. In addition, we present another variant of the ECM, based on the use of elastic constraints and also including an augmentation term. This variant has the same complexity, namely N + 1, which is the minimum reached for any exact method. A comparison of the different variants is carried out on a set of biobjective location problems which we call p-median-cover problems;these include the objectives of the p-median and the maximal covering problems. As computational results show, for this class of problems, the augmented ECM with elastic constraint is the most effective variant for finding the Pareto front in an exact manner.

关键词： Biobjective integer programming epsilon-constraint method p-Median-cover problem

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An exact algebraic ε-constraint method for bi-objective linear integer programming based on test sets

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2020年第2期282卷 453-463页

作者： Isabel Hartillo-Hermoso, Maria Jimenez-Tafur, Haydee Maria Ucha-Enriquez, Jose Univ Seville ETS Ingn Informat Dept Matemat Aplicada 1 Av Reina Mercedes S-N E-41012 Seville Spain Univ Seville Inst Matemat Avda Reina Mercedes S-N E-41012 Seville Spain

A new exact algorithm for bi-objective linear integer problems is presented, based on the classic epsilon-constraint method and algebraic test sets for single-objective linear integer problems. Our method provides the complete Pareto frontier N of non-dominated points and, for this purpose, it considers exactly vertical bar N vertical bar single-objective problems by using reduction with test sets instead of solving with an optimizer. Although we use Grobner bases for the computation of test sets, which may provoke a bottleneck in principle, the computational results are shown to be promising, especially for unbounded knapsack problems, for which any usual branch-and-cut strategy could be much more expensive. Nevertheless, this algorithm can be considered as a potentially faster alternative to IP-based methods when test sets are available. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Multiple objective programming Non-dominated set Pareto set epsilon-constraint method Unbounded Knapsack Problem Algebraic test sets

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Improved ε-constraint method for multiobjective programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2008年第3期138卷 375-396页

作者： Ehrgott, M. Ruzika, S. Univ Auckland Dept Engn Sci Auckland 1 New Zealand Univ Nantes CNRS Lab Informat Nantes Atlantique Nantes France Tech Univ Kaiserslautern Fachbereich Math Kaiserslautern Germany

In this paper, we revisit one of the most important scalarization techniques used in multiobjective programming, the epsilon-constraint method. We summarize the method and point out some weaknesses, namely the lack of easy-to-check conditions for properly efficient solutions and the inflexibility of the constraints. We present two modifications that address these weaknesses by first including slack variables in the formulation and second elasticizing the constraints and including surplus variables. We prove results on (weakly, properly) efficient solutions. The improved epsilon-constraint method that we propose combines both modifications.

关键词： multiobjective programming scalarization epsilon-constraint method properly efficient solutions

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Optimizing tree selection for planning cable yarding operations: a multi-objective modelling approach

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FOREST ECOLOGY AND MANAGEMENT 2025年 578卷

作者： Sforza, Francesco Ziesak, Martin Lingua, Emanuele Starke, Michael Univ Padua Dept Land Environm Agr & Forestry Viale Univ 16 I-35020 Padua Legnaro Italy Bern Univ Appl Sci Sch Agr Forest & Food Sci HAFL Langgasse 85 CH-3052 Bern Switzerland

When planning timber harvesting in mountainous areas, selecting the trees to extract is highly challenging. This is due to the need to consider both the forest management plan as well as feasibility aspects of the logging operation. Additionally, the choice using cable yarding systems in steep and difficult terrain increases the complexity and related costs of the harvesting process. The protective function of mountain forests against natural hazards adds another layer of consideration. Decision-makers must strike a balance between maintaining the protective function of the forest directly after cutting and on a long term, while maximizing the economic efficiency of the harvesting operation. In this study, we have developed an optimization model for tree selection using linear mixed programming. The model relies on high-resolution airborne laser scanning data to create a forest inventory through individual tree detection. Constituting the Pareto frontier, by optimizing two primary objectives, 15 optimal trade-off solutions could be identified when minimizing the harvesting area and maximizing the economic benefit of timber harvesting. Four of the solutions were identified, which maintain the maximum protective effect of the remaining forest, with a cumulative basal area ranging from 33.3 to 33.9 m(2)ha(-1), while significantly reducing the economic loss from 17 % to 79.6 % when emphasizing the economic benefit maximization. Furthermore, the model presented four solutions that keep the protective effect above the minimum threshold (cumulative basal area > 30 m(2)ha(-1)) while also resulting in positive economic benefits from harvesting. This model presents a method that integrates a multi-objective approach to decision making, considering the practical constraints of cable yarding technology in the process of tree selection in mountainous areas.

关键词： Linear programming epsilon-constraint method Protective forest Forest management Timber harvest Economic benefit

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A New Approach for Solving Bi-Objective Redundancy Allocation Problem Using DOE, Simulation and ε-constraint method

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INFORMATICA 2017年第1期28卷 79-104页

作者： Keshavarz Ghorabaee, Mehdi Amiri, Maghsoud Turskis, Zenonas Allameh Tabatabai Univ Dept Ind Management Fac Management & Accounting Tehran Iran Vilnius Gediminas Tech Univ Res Inst Smart Bldg Technol Vilnius Lithuania

The redundancy allocation problem (RAP) has been studied for many different system structures, objective functions, and distribution assumptions. In this paper, we present a problem formulation and a solution methodology to maximize the system steady-state availability and minimize the system cost for the repairable series-parallel system designs. In the proposed approach, the components' time-to-failure (TTF) and time-to-repair (TTR) can follow any distribution such as the Gamma, Normal, Weibull, etc. We estimate an approximation of the steady-state availability of each subsystem in the series-parallel system with an individual meta-model. Design of experiment (DOE), simulation and the stepwise regression are used to build these meta-models. Face centred design, which is a type of central composite design is used to design experiments. According to a max-min approach, obtained meta-models are utilized for modelling the problem alongside the cost function of the system. We use the augmented epsilon-constraint method to reformulate the problem and solve the model. An illustrative example which uses the Gamma distribution for TTF and TTR is explained to represent the performance of the proposed approach. The results of the example show that the proposed approach has a good performance to obtain Pareto (near-Pareto) optimal solutions (system configurations).

关键词： redundancy allocation problem bi-objective RAP design of experiment simulation epsilon-constraint method

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A simple augmented ε-constraint method for multi-objective mathematical integer programming problems

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2014年第1期234卷 15-24页

作者： Zhang, Weihua Reimann, Marc Beihang Univ Sch Econ & Management Beijing 100191 Peoples R China Graz Univ Inst Prod & Operat Management A-8010 Graz Austria

A simple augmented epsilon-constraint (SAUGMECON) method is put forward to generate all non-dominated solutions of multi-objective integer programming (MOIP) problems. The SAUGMECON method is a variant of the augmented epsilon-constraint (AUGMECON) method proposed in 2009 and improved in 2013 by Mavrotas et al. However, with the SAUGMECON method, all non-dominated solutions can be found much more efficiently thanks to our innovations to algorithm acceleration. These innovative acceleration mechanisms include: (1) an extension to the acceleration algorithm with early exit and (2) an addition of an acceleration algorithm with bouncing steps. The same numerical example in Lokman and Koksalan (2012) is used to illustrate workings of the method. Then comparisons of computational performance among the method proposed by (Ozlen and Azizoglu (2009), Ozlen et al. (2012), the method developed by Lokman and Koksalan (2012) and the SAUGMECON method are made by solving randomly generated general MOIP problem instances as well as special MOIP problem instances such as the MOKP and MOSP problem instances presented in Table 4 in Lolcman and Koksalan (2012). The experimental results show that the SAUGMECON method performs the best among these methods. More importantly, the advantage of the SAUGMECON method over the method proposed by Lokman and Koksalan (2012) turns out to be increasingly more prominent as the number of objectives increases. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Multi-objective programming epsilon-constraint method AUGMECON method SAUGMECON method

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An improved version of the augmented ε-constraint method (AUGMECON2) for finding the exact pareto set in multi-objective integer programming problems

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APPLIED MATHEMATICS AND COMPUTATION 2013年第18期219卷 9652-9669页

作者： Mavrotas, George Florios, Kostas Natl Tech Univ Athens Sch Chem Engn Lab Ind & Energy Econ Athens 15780 Greece

Generation (or a posteriori) methods in Multi-Objective Mathematical Programming (MOMP) is the most computationally demanding category among the MOMP approaches. Due to the dramatic increase in computational speed and the improvement of Mathematical Programming algorithms the generation methods become all the more attractive among today's decision makers. In the current paper we present the generation method AUGMECON2 which is an improvement of our development, AUGMECON. Although AUGMECON2 is a general purpose method, we will demonstrate that AUGMECON2 is especially suitable for Multi-Objective Integer Programming (MOIP) problems. Specifically, AUGMECON2 is capable of producing the exact Pareto set in MOIP problems by appropriately tuning its running parameters. In this context, we compare the previous and the new version in a series of new and old benchmarks found in the literature. We also compare AUGMECON2's performance in the generation of the exact Pareto sets with established methods and algorithms based on specific MOIP problems (knapsack, set packing) and on published results. Except from other Mathematical Programming methods, AUGMECON2 is found to be competitive also with Multi-Objective Meta-Heuristics (MOMH) in producing adequate approximations of the Pareto set in Multi-Objective Combinatorial Optimization (MOCO) problems. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Multi-objective programming epsilon-constraint method Exact pareto set

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A new multi-objective particle swarm optimization method for solving reliability redundancy allocation problems

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RELIABILITY ENGINEERING & SYSTEM SAFETY 2013年 111卷 58-75页

作者： Khalili-Damghani, Kaveh Abtahi, Amir-Reza Tavana, Madjid Islamic Azad Univ Dept Ind Engn S Tehran Branch Tehran Iran Univ Econ Sci Dept Knowledge Engn & Decis Sci Fac Econ Inst Management Tehran Iran La Salle Univ Lindback Distinguished Chair Informat Syst & Deci Philadelphia PA 19141 USA

In this paper, a new dynamic self-adaptive multi-objective particle swarm optimization (DSAMOPSO) method is proposed to solve binary-state multi-objective reliability redundancy allocation problems (MORAPs). A combination of penalty function and modification strategies is used to handle the constraints in the MORAPs. A dynamic self-adaptive penalty function strategy is utilized to handle the constraints. A heuristic cost-benefit ratio is also supplied to modify the structure of violated swarms. An adaptive survey is conducted using several test problems to illustrate the performance of the proposed DSAMOPSO method. An efficient version of the epsilon-constraint (AUGMECON) method, a modified non-dominated sorting genetic algorithm (NSGA-II) method, and a customized time-variant multi-objective particle swarm optimization (cTV-MOPSO) method are used to generate non-dominated solutions for the test problems. Several properties of the DSAMOPSO method, such as fast-ranking, evolutionary-based operators, elitism, crowding distance, dynamic parameter tuning, and tournament global best selection, improved the best known solutions of the benchmark cases of the MORAP. Moreover, different accuracy and diversity metrics illustrated the relative preference of the DSAMOPSO method over the competing approaches in the literature. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Multi-objective redundancy allocation problem Meta-heuristics Dynamic self-adaptive multi-objective particle swarm optimization epsilon-constraint method NSGA-II

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Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems

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APPLIED MATHEMATICS AND COMPUTATION 2009年第2期213卷 455-465页

作者： Mavrotas, George Natl Tech Univ Athens Lab Ind & Energy Econ Sch Chem Engn Athens 15780 Greece

As indicated by the most widely accepted classification, the Multi-Objective Mathematical Programming (MOMP) methods can be classified as a priori, interactive and a posteriori, according to the decision stage in which the decision maker expresses his/her preferences. Although the a priori methods are the most popular, the interactive and the a posteriori methods convey much more information to the decision maker. Especially, the a posteriori (or generation) methods give the whole picture (i.e. the Pareto set) to the decision maker, before his/her final choice, reinforcing thus, his/her confidence to the final decision. However, the generation methods are the less popular due to their computational effort and the lack of widely available software. The present work is an effort to effectively implement the epsilon-constraint method for producing the Pareto optimal solutions in a MOMP. We propose a novel version of the method (augmented epsilon-constraint method - AUGMECON) that avoids the production of weakly Pareto optimal solutions and accelerates the whole process by avoiding redundant iterations. The method AUGMECON has been implemented in GAMS, a widely used modelling language, and has already been used in some applications. Finally, an interactive approach that is based on AUGMECON and eventually results in the most preferred Pareto optimal solution is also proposed in the paper. (C) 2009 Elsevier Inc. All rights reserved.

关键词： Multi-Objective Programming epsilon-constraint method GAMS

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Multi-objective geometric programming problem with ε-constraint method

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APPLIED MATHEMATICAL MODELLING 2014年第2期38卷 747-758页

作者： Ojha, A. K. Biswal, K. K. Indian Inst Technol Bhubaneswar Sch Basic Sci Bhubaneswar 751013 Orissa India NIIS Inst Business Adm Dept Math Bhubaneswar 752054 Orissa India

In multi-objective geometric programming problem there are more than one objective functions: There is no single optimal solution which simultaneously optimizes all the objective functions. Under these conditions the decision makers always search for the most "preferred" solution, in contrast to the optimal solution. A few mathematical programming methods namely fuzzy programming, goal programming and weighting methods have been applied in the recent past to find the compromise solution. In this paper epsilon-constraint method has been applied to find the non-inferior solution. A brief solution procedure of epsilon-constraint method has been presented to find the non-inferior solution of the multi-objective programming problems. Further, the multi-objective programming problems is solved by the fuzzy programming technique to find the optimal compromise solution. Finally, two numerical examples are solved by both the methods and compared with their obtained solutions. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Geometric programming Vector minimization epsilon-constraint method Fuzzy programming Duality theory Pareto optimal solution

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