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An epsilon-constraint method for fully fuzzy multiobjective linear programming

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS 2020年第4期35卷 600-624页

作者： Perez-Canedo, Boris Luis Verdegay, Jose Miranda Perez, Ridelio Univ Cienfuegos Dept Math Cienfuegos Cuba Univ Granada Dept Comp Sci & Artificial Intelligence Granada Spain

Linear ranking functions are often used to transform fuzzy multiobjective linear programming (MOLP) problems into crisp ones. The crisp MOLP problems are then solved by using classical methods (eg, weighted sum, epsilon-constraint, etc), or fuzzy ones based on Bellman and Zadeh's decision-making model. In this paper, we show that this transformation does not guarantee Pareto optimal fuzzy solutions for the original fuzzy problems. By using lexicographic ranking (LR) criteria, we propose a fuzzy epsilon-constraint method that yields Pareto optimal fuzzy solutions of fuzzy variable and fully fuzzy MOLP problems, in which all parameters and decision variables take on LR fuzzy numbers. The proposed method is illustrated by means of three numerical examples, including a fully fuzzy multiobjective project crashing problem.

关键词： epsilon-constraint method fully fuzzy multiobjective linear programming lexicographic ranking criterion LR fuzzy number project crashing

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An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2006年第3期169卷 932-942页

作者： Laumanns, M Thiele, L Zitzler, E ETH Zentrum Swiss Fed Inst Technol Inst Operat Res CH-8092 Zurich Switzerland ETH Zentrum Swiss Fed Inst Technol Comp Engn & Networks Lab CH-8092 Zurich Switzerland

This paper discusses methods for generating or approximating the Pareto set of multiobjective optimization problems by solving a sequence of constrained single-objective problems. The necessity of determining the constraint value a priori is shown to be a serious drawback of the original epsilon-constraint method. We therefore propose a new, adaptive scheme to generate appropriate constraint values during the run. A simple example problem is presented, where the running time (measured by the number of constrained single-objective sub-problems to be solved) of the original epsilon-constraint method is exponential in the problem size (number of decision variables), although the size of the Pareto set grows only linearly. We prove that-independent of the problem or the problem size-the time complexity of the new scheme is O(k(m-1)), where k is the number of Pareto-optimal solutions to be found and m the number of objectives. Simulation results for the example problem as well as for different instances of the multiobjective knapsack problem demonstrate the behavior of the method, and links to reference implementations are provided. (c) 2004 Elsevier B.V. All rights reserved.

关键词： multiple objective optimization non-dominated set Pareto set epsilon-constraint method generating methods

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A Multi-Objective Model for Home Energy Management System Self-Scheduling using the epsilon-constraint method 14

A Multi-Objective Model for Home Energy Management System Se...

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14th IEEE International Conference on Compatibility, Power Electronics and Power Engineering (CPE-POWERENG)

作者： Javadi, Mohammad Lotfi, Mohamed Osorio, Gerardo J. Ashraf, Abdelrahman Nezhad, Ali Esmaeel Gough, Matthew Catalao, Joao P. S. Univ Porto FEUP Fac Engn P-4200465 Porto Portugal Inst Syst & Comp Engn Technol & Sci INESC TEC P-4200465 Porto Portugal Ontario Tech Univ ERC Oshawa ON L1G 8C4 Canada Univ Beira Interior C MAST Covilha Portugal German Univ Cairo GUC Cairo Egypt Univ Bologna Dept Elect Elect & Informat Engn Bologna Italy

ISBN: (纸本)9781728142180

Self-scheduling of Home Energy Management Systems (HEMS) is one of the most interesting problems for active end-users to reduce their electricity bills. The electricity bill reduction by adopting Demand Response Programs (DRP) considering the flexibility of the end-users is addressed in this paper. The problem is addressed as a multi-objective optimization problem. The first objective function is the minimization of the daily bill, while the second objective aims to minimize the Discomfort Index (DI) regarding shifting the home appliances plugging-in time. The Time-of-Use (ToU) tariff is adopted in this paper and therefore, the end-users can benefit from shifting their flexible loads from peak hours to the off-peak hours and this reduces their bills, accordingly. In this case, the end-users have to change their energy consumption which imposes a level of discomfort on the end-users. Therefore, a two-stage model is proposed in this paper to deal with the mentioned objective functions. The proposed model is represented as standard mixed-integer linear programming (MILP) and for solving this problem the epsilon-constraint method is adopted in this study. The obtained Pareto front from the epsilon-constraint multi-objective framework is fed to the fuzzy satisfying method for final plan selection. These results show that by providing the Pareto set of optimal solutions to the user, they are more informed and can make decisions that better suit their preferences.

关键词： Home Energy Management Systems epsilon-constraint method Multi-objective Optimization

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Comparison of nongradient methods with hybrid Taguchi-based epsilon constraint method for multiobjective optimization of cylindrical fin heat sink

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INTERNATIONAL JOURNAL OF ADVANCED MANUFACTURING TECHNOLOGY 2012年第9-12期63卷 1081-1094页

作者： Devi, S. Prasanna Manivannan, S. Rao, K. Suryaprakasa Apollo Engn Coll Dept Comp Sci & Engn Chennai Tamil Nadu India Dr MGR Educ & Res Inst Univ Dept Elect & Instrumentat Engn Chennai Tamil Nadu India Anna Univ Dept Ind Engn Chennai 600025 Tamil Nadu India

The primary aim of the paper is to compare the different nongradient methods of multiobjective optimization for optimizing the geometry parameters of a cylindrical fin heat sink. The methods studied for comparison are Taguchi-based grey relational analysis, epsilon (epsilon) constraint method and genetic algorithm. The various responses that have been studied are electromagnetic emitted radiations, thermal resistance and mass of the heat sink. Since the responses are obtained using complex simulation softwares (HFSS-Ansoft for emitted radiations and CFD-Flotherm for thermal resistance), there is no way of calculating the derivates of the objective functions. Hence, the Taguchi design of experiments design is used to derive the linear regression equations for the responses studied, which are then taken as the objective functions to be optimized. A new hybrid method known as Taguchi-based epsilon constraint method has been proposed in this paper for obtaining nondominated Pareto solution set. The results obtained using the proposed method show that the Pareto optimal set is competitive in terms of diversity of the solutions obtained. It is not likely that there exists a solution, which simultaneously minimizes all the objectives using any of the multiobjective techniques implemented. The value path analysis has been done to compare the trade-off among the design alternatives for the chosen multiple objective parameter optimization problem.

关键词： Multiobjective design optimization Heat sink Linear regression modelling Taguchi design of experiments epsilon-constraint method Grey relational analysis Genetic algorithm Taguchi-based epsilon (epsilon) constraint method

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An Extended C&CG Algorithm for Solving Two-Stage Robust Optimization of Economic and Feasible Scheduling

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2025年第2期205卷 1-29页

作者： Chen, Ruibin Bao, Zhejing Lu, Lingxia Yu, Miao Zhejiang Univ Hangzhou Peoples R China

The extensively researched column-and-constraint-generation (C&CG) algorithm, which utilizes the KKT (Karush-Kuhn-Tucker) condition or duality theory to reformulate the subproblem, encounters challenges when solving two-stage robust optimization (TSRO) problems with extreme parameters that could adversely affect the feasibility of the second-stage decision. After the analysis of the original C&CG algorithm, an extended C&CG algorithm with multiple subproblems is proposed to overcome the challenges, which decompose a TSRO model into the master problem and several subproblems searching for the worst-case scenarios. A simple linear case is given to show the shortcoming of the traditional C&CG algorithm and the advantage of the extended C&CG algorithm. Then, a TSRO model for the scheduling optimization of electricity system considering the optimal power flow (OPF) is proposed, in order to explore the effectiveness of the extended C&CG algorithm in handling the general optimization problem while considering the feasibility. Finally, the proposed solving method is validated by case studies.

关键词： Column and constraint generation Two-stage robust optimization Feasibility check epsilon-constraint method

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Bi-Objective Optimization of a Flow Shop Scheduling Problem Under Time-of-Use Tariffs

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IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING 2025年 22卷 7679-7695页

作者： Wang, Xinyue Chu, Feng Ren, Tao Bai, Danyu Univ Paris Saclay Univ Evry Lab IBISC F-91025 Paris France Northeastern Univ Software Coll Shenyang 110819 Peoples R China Dalian Maritime Univ Sch Maritime Econ & Management Dalian 116026 Peoples R China

Time-of-use (ToU) tariffs flexibly offer markedly cheap electricity prices to industrial and residential users during off-peak periods, encouraging them to shift their peak electricity demands in valley periods. Although ToU tariffs play a crucial role in balancing electricity supply and demand, especially in energy-intensive industries, the best trade-off between industrial performance and energy costs has not been well explored. Manufacturing consumes a substantial amount of energy, primarily in the form of electricity, leading to imbalances in power consumption. The flow shop scheduling (FSS) model is one of the most prevalent models in manufacturing. To explore the significant role of ToU tariffs in manufacturing, this study addresses a bi-objective FSS problem under ToU tariffs. The objective is to find the optimal balance between customer satisfaction and total electricity cost. A tight mixed integer programming model is developed to solve this NP-hard problem using business optimizers. On the bases of the problem properties demonstrated in this study, valid inequalities are designed to reduce the solution space of the problem. For small-scale instances, an improved $\varepsilon $ -constraint method is presented to find the Pareto front. For medium and large-scale instances, a two-stage fruit fly optimization (TFFO) algorithm is developed to obtain the near Pareto front. Experimental results demonstrate the efficiency and effectiveness of the proposed model and algorithms. Note to Practitioners-Scheduling for complex systems remains a formidable challenge in manufacturing. Energy cost saving is a major objective for all energy-intensive industries. Effective scheduling is crucial for businesses achieving eco-friendly performance, especially under ToU tariffs. This study aims to provide efficient scheduling model and methods that can guide decision-makers in fostering ecological transitions. The $\varepsilon $ -constraint method can find globally optimal soluti

关键词： Flow shop scheduling time-of-use electricity tariffs mixed integer programming fruit fly optimization metaheuristic epsilon-constraint method Flow shop scheduling time-of-use electricity tariffs mixed integer programming fruit fly optimization metaheuristic

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ε-constraint procedures for Pareto front optimization of large size discrete time/cost trade-off problem

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2025年第3期322卷 753-769页

作者： Aminbakhsh, Saman Sonmez, Rifat Atan, Tankut Atilim Univ Civil Engn Ankara Turkiye Middle East Tech Univ Civil Engn Ankara Turkiye Bahcesehir Univ Ind Engn Istanbul Turkiye

The discrete time/cost trade-off problem (DTCTP) optimizes the project duration and/or cost while considering the trade-off between activity durations and their direct costs. The complete and non-dominated time-cost profile over the set of feasible project durations is achieved within the framework of Pareto front problem. Despite the importance of Pareto front optimization in project and portfolio management, exact procedures have achieved very limited success in solving the problem for large size instances. This study develops exact procedures based on combinations of mixed-integer linear programming (MILP), epsilon-constraint method, network and problem reduction techniques, and present new bounding strategies to solve the Pareto problem for large size instances. This study also provides new large size benchmark problem instances aiming to represent the size of real-life projects for the DTCTP. The new instances, therefore, are generated to include up to 990 activities and nine execution modes. Computational experiments reveal that the procedures presented herein can remarkably outperform the state-of-the-art exact methods. The new exact procedures enabled obtaining the optimal Pareto front for instances with serial networks that include more than 200 activities for the first time.

关键词： Discrete time/cost trade-off problem Pareto front Large instances Multi-objective optimization epsilon-constraint method

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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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Optimal operational and carbon neutrality strategies for private hospitals: A multi-objective approach considering patient flow

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TRANSPORTATION RESEARCH PART E-LOGISTICS AND TRANSPORTATION REVIEW 2025年 198卷

作者： Li, Haoran Wu, Yiwei Xin, Baogui Xu, Min Wu, Shining Shandong Univ Sci & Technol Coll Econ & Management Qingdao 266590 Peoples R China Hong Kong Polytech Univ Dept Logist & Maritime Studies Hong Kong 999077 Peoples R China Hong Kong Polytech Univ Dept Ind & Syst Engn Hong Kong 999077 Peoples R China

Hospitals are now the fifth-largest source of global greenhouse gas (GHG) emissions. Despite this significant impact, the critical goal of achieving carbon neutrality is often overlooked in optimization studies on healthcare operations management. To address this gap, this paper proposes a novel multi-objective decision-making framework for private hospitals that balances patient health, environmental impact, and economic performance. This study tackles the intricate challenge of multi-objective optimization that maximizes patient utility and hospital profit while simultaneously minimizing net carbon emissions. An exact algorithm based on the epsilon-constraint method is employed to optimize hospital operational decisions by discretizing patient utility and net carbon emission targets into linear constraints on the non-concave profit objective. This approach guarantees Pareto-optimal solutions, thus enabling hospital decision-makers to select their preferred options among the three criteria: patient utility, environmental sustainability, and hospital profitability, while exploring the resulting trade-offs. Additionally, this study discusses the impacts of hospital operational decision changes on these objectives, particularly under scenarios where patient flow increases waiting time due to limited hospital capacity. To validate the practical significance of the proposed multi-objective optimization model and evaluate the effectiveness of the proposed algorithm, two case studies and sensitivity analyses are conducted. The results demonstrate that this approach effectively supports hospital decision-makers in making informed decisions that balance patient utility, carbon neutrality, and hospital profitability objectives.

关键词： Healthcare operations management Multi-objective optimization epsilon-constraint method Carbon neutrality Patient flow

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Research on Multi-Objective Programming Model of Profits and Carbon Emission Reduction in Manufacturing Industry

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ENERGIES 2025年第6期18卷 1411-1411页

作者： Tsai, Wen-Hsien Wu, Yi-Han Natl Cent Univ Dept Business Adm Taoyuan 320 Taiwan

As the issue of global climate change becomes increasingly severe, governments worldwide have implemented carbon reduction policies, such as carbon taxes and industrial low-carbon transitions, to effectively control total carbon emissions. This study applies a multi-objective programming approach and uses the plastic raw material manufacturing process in the petrochemical industry as an example to explore how companies can balance profit maximization with minimizing production-related carbon emissions. By integrating Activity-Based Costing (ABC) and the Theory of constraints (TOC), this study develops a production decision-making model and employs the epsilon-constraint method to impose carbon emission constraints, analyzing the resulting changes in corporate profitability. The model considers three different policy scenarios: basic carbon tax costs (including the use of renewable energy), continuous incremental progressive carbon tax costs, and discontinuous incremental progressive carbon tax costs. The results indicate that adopting renewable energy effectively reduces carbon emissions during production, while the discontinuous incremental carbon tax model provides better control over emissions. Under different carbon emission constraints, significant variations in optimal profits and production volumes are observed across the models, offering valuable insights for governments and enterprises in formulating carbon reduction strategies.

关键词： multi-objective programming epsilon-constraint method carbon emissions carbon tax activity-based costing (ABC) theory of constraints petrochemical industry

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