An Exact ε-constraint Method for Bi-objective Combinatorial Optimization Problems - Application to the Traveling Salesman Problem with Profits
TL;DR: It is shown in this paper that the Pareto front of bi-objective problems can be efficiently generated with the [epsilon]-constraint method and heuristics based on information gathered from previous subproblems that significantly speed up the method are described.
About: This article is published in European Journal of Operational Research.The article was published on 2009-04-01. It has received 334 citations till now. The article focuses on the topics: 2-opt & Bottleneck traveling salesman problem.
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TL;DR: The OP is formally described and many relevant variants are presented and all published exact solution approaches and (meta) heuristics are discussed and compared.
882 citations
Cites background from "An Exact ε-constraint Method for Bi..."
...The OP can be formulated as a TSP With Profits (TSPWP) (Feillet et al., 2005a; Bérubé et al., 2009) or as a special case of the Resource Constrained TSP (RCTSP) (Pekny and Miller, 1990)....
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TL;DR: Survey models, algorithmic approaches and methodologies concerning tourist trip design problems, focusing on problem models that best capture a multitude of realistic POIs attributes and user constraints are examined.
Abstract: The tourist trip design problem (TTDP) refers to a route-planning problem for tourists interested in visiting multiple points of interest (POIs). TTDP solvers derive daily tourist tours, i.e., ordered visits to POIs, which respect tourist constraints and POIs attributes. The main objective of the problem discussed is to select POIs that match tourist preferences, thereby maximizing tourist satisfaction, while taking into account a multitude of parameters and constraints (e.g., distances among POIs, visiting time required for each POI, POIs visiting days/hours, entrance fees, weather conditions) and respecting the time available for sightseeing on a daily basis. The aim of this work is to survey models, algorithmic approaches and methodologies concerning tourist trip design problems. Recent approaches are examined, focusing on problem models that best capture a multitude of realistic POIs attributes and user constraints; further, several interesting TTDP variants are investigated. Open issues and promising prospects in tourist trip planning research are also discussed.
243 citations
Cites methods from "An Exact ε-constraint Method for Bi..."
...In [20] the first exact Pareto fronts (sets of non-dominated solutions) have been given for TSPP instances obtained from classical TSP instances, available in the TSPLIB [105]....
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TL;DR: To solve the single-objective constrained optimization problem, an exact solution method as well as a ''math-heuristic'' technique building on a MILP formulation with a heuristically generated constraint pool are proposed.
218 citations
TL;DR: An epsilon-constraint method is proposed and proved that it generates the exact Pareto front and can be applied to any three-objective optimization problem provided that the problem involves at least two integer and conflicting objectives.
199 citations
Cites methods from "An Exact ε-constraint Method for Bi..."
...To this end, we consider two performance measures, used in Bérubé et al. (2009) and originally proposed by Cryzak and Jaszkiewicz (1998), as follows: ∑{ } Where denotes the exact Pareto front, { ( ) } and denotes the value of the kth objective ( in our case)....
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...Recently, Bérubé et al. (2009) consider a bi-objective traveling salesman problem with profits and propose an -constraint method to generate the exact Pareto front for the problem....
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TL;DR: This work considers the three objectives and was motivated by the challenge of supporting decision makers when managing a real case study of a recyclable waste collection system, where strategic decisions on the number and location of depots, vehicles and containers were taken beforehand.
Abstract: The present work aims to support tactical and operational planning decisions of reverse logistics systems while considering economic, environmental and social objectives. In the literature, when addressing such systems economic aspects have been often used, while environmental concerns have emerged only recently. The social component is the one less studied and rarely the combination of the three concerns has been analyzed. This work considers the three objectives and was motivated by the challenge of supporting decision makers when managing a real case study of a recyclable waste collection system, where strategic decisions on the number and location of depots, vehicles and containers were taken beforehand. Tactical and operational decisions are studied involving the establishment of service areas for each depot and the definition and scheduling of collection routes for each vehicle. Such decisions should represent a compromise solution between the three objectives translating a sustainable reverse logistics plan. The problem is modeled as a multi-objective, multi-depot periodic vehicle routing problem with inter-depot routes. A mathematical formulation and a solution approach are proposed. An approximation to the Pareto front is obtained for the case study and the trade-offs between the objectives are discussed. A balanced solution is proposed.
177 citations
References
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Book•
26 Sep 2011
TL;DR: This paper is concerned with the development of methods for dealing with the role of symbols in the interpretation of semantics.
Abstract: Preface. Acknowledgements. Notation and Symbols. Part I: Terminology and Theory. 1. Introduction. 2. Concepts. 3. Theoretical Background. Part II: Methods. 1. Introduction. 2. No-Preference Methods. 3. A Posteriori Methods. 4. A Priori Methods. 5. Interactive Methods. Part III: Related Issues. 1. Comparing Methods. 2. Software. 3. Graphical Illustration. 4. Future Directions. 5. Epilogue. References. Index.
4,976 citations
"An Exact ε-constraint Method for Bi..." refers background or methods in this paper
...Theorems 1 and 2 are proved for general multi-objective problems (see [6,23]) and are therefore valid for BOCO problems....
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...Besides weighting sum algorithms, the -constraint method [6,23] is the best known approach for solving MOCO problems, according to [9]....
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...Thereby, the interest for multi-criteria decision making has continually grown during the last decades, as attested by the number of books and surveys on the topic (see [6,10,23,28], among others)....
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TL;DR: In this article, the authors present a survey of the state of the art in multiple criterion decision analysis (MCDA) with an overview of the early history and current state of MCDA.
Abstract: In two volumes, this new edition presents the state of the art in Multiple Criteria Decision Analysis (MCDA). Reflecting the explosive growth in the field seen during the last several years, the editors not only present surveys of the foundations of MCDA, but look as well at many new areas and new applications. Individual chapter authors are among the most prestigious names in MCDA research, and combined their chapters bring the field completely up to date. Part I of the book considers the history and current state of MCDA, with surveys that cover the early history of MCDA and an overview that discusses the “pre-theoretical” assumptions of MCDA. Part II then presents the foundations of MCDA, with individual chapters that provide a very exhaustive review of preference modeling, along with a chapter devoted to the axiomatic basis of the different models that multiple criteria preferences. Part III looks at outranking methods, with three chapters that consider the ELECTRE methods, PROMETHEE methods, and a look at the rich literature of other outranking methods. Part IV, on Multiattribute Utility and Value Theories (MAUT), presents chapters on the fundamentals of this approach, the very well known UTA methods, the Analytic Hierarchy Process (AHP) and its more recent extension, the Analytic Network Process (ANP), as well as a chapter on MACBETH (Measuring Attractiveness by a Categorical Based Evaluation Technique). Part V looks at Non-Classical MCDA Approaches, with chapters on risk and uncertainty in MCDA, the decision rule approach to MCDA, the fuzzy integral approach, the verbal decision methods, and a tentative assessment of the role of fuzzy sets in decision analysis. Part VI, on Multiobjective Optimization, contains chapters on recent developments of vector and set optimization, the state of the art in continuous multiobjective programming, multiobjective combinatorial optimization, fuzzy multicriteria optimization, a review of the field of goal programming, interactive methods for solving multiobjective optimization problems, and relationships between MCDA and evolutionary multiobjective optimization (EMO). Part VII, on Applications, selects some of the most significant areas, including contributions of MCDA in finance, energy planning problems, telecommunication network planning and design, sustainable development, and portfolio analysis. Finally, Part VIII, on MCDM software, presents well known MCDA software packages.
4,055 citations
"An Exact ε-constraint Method for Bi..." refers methods in this paper
...Furthermore, we describe heuristics based on information gathered from previous subproblems that significantly speed up the method....
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TL;DR: This study provides a rigorous analysis of the limitations underlying this type of quality assessment in multiobjective evolutionary algorithms and develops a mathematical framework which allows one to classify and discuss existing techniques.
Abstract: An important issue in multiobjective optimization is the quantitative comparison of the performance of different algorithms. In the case of multiobjective evolutionary algorithms, the outcome is usually an approximation of the Pareto-optimal set, which is denoted as an approximation set, and therefore the question arises of how to evaluate the quality of approximation sets. Most popular are methods that assign each approximation set a vector of real numbers that reflect different aspects of the quality. Sometimes, pairs of approximation sets are also considered. In this study, we provide a rigorous analysis of the limitations underlying this type of quality assessment. To this end, a mathematical framework is developed which allows one to classify and discuss existing techniques.
3,702 citations
"An Exact ε-constraint Method for Bi..." refers background in this paper
...The last column I in both Tables is the -indicator metric reported in [31]....
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TL;DR: This paper contains the description of a traveling salesman problem library (TSPLIB) which is meant to provide researchers with a broad set of test problems from various sources and with various properties.
Abstract: This paper contains the description of a traveling salesman problem library (TSPLIB) which is meant to provide researchers with a broad set of test problems from various sources and with various properties. For every problem a short description is given along with known lower and upper bounds. Several references to computational tests on some of the problems are given. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.
2,397 citations
"An Exact ε-constraint Method for Bi..." refers methods in this paper
...We report the first exact Pareto fronts for TSPP instances obtained from classical VRP and TSP instances available in the TSPLIB [25]....
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...We transformed VRP and TSP instances of the TSPLIB [25] into TSPP instances using the rules provided in [5] and [12]....
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Book•
01 Jan 1983
2,227 citations
"An Exact ε-constraint Method for Bi..." refers background or methods in this paper
...Besides weighting sum algorithms, the ǫ-constraint method [6, 22] is the best known approach for solving MOCO problems, according to [9]....
[...]
...Theorems 1 and 2 are proved for general multi-objective problems (see [6, 22]) and are therefore valid for BOCO problems....
[...]
...Thereby, the interest for multi-criteria decision making has continually grown during the last decades, as attested by the number of books and surveys on the topic (see [6, 10, 22, 27], among others)....
[...]