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A Branch-and-Bound Algorithm for Representative Integer Efficient Solutions in multiple objective Network programming Problems

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NETWORKS 2013年第1期62卷 56-71页

作者： Sun, Minghe Univ Texas San Antonio Dept Management Sci & Stat Coll Business San Antonio TX 78249 USA

In many applications of multiple objective network programming (MONP) problems, only integer solutions are acceptable as the final optimal solution. Representative efficient solutions are usually obtained by sampling the efficient set through the solution of augmented weighted Tchebycheff network programs. Because such efficient solutions are usually not integer solutions, a branch-and-bound (BB) algorithm is developed to find integer efficient solutions. The purpose of the BB algorithm is to support interactive procedures by generating representative integer efficient solutions. To be computationally efficient, the algorithm takes advantage of the network structure as much as possible. An algorithm, used in the BB algorithm and performed on the key tree, is developed to construct feasible solutions from infeasible solutions and basic solutions from nonbasic solutions when bounds on branching variables change. The BB algorithm finds basic and nonbasic or supported and unsupported integer efficient solutions as long as they are optimal. Details of the algorithm are presented, an example is provided and computational results are reported. Computational results show that the BB algorithm performs well. Although the BB algorithm is developed for the purpose of generating integer efficient solutions for MONP problems, it can also solve more general integer network flow problems with linear side constraints. (c) 2013 Wiley Periodicals, Inc.

关键词： network flow algorithms multiple objective programming multiple criteria decision making integer programming branch-and-bound network flow problems with side constraints

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Finding Integer Efficient Solutions for multiple objective Network programming Problems

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NETWORKS 2011年第4期57卷 362-375页

作者： Sun, Minghe Univ Texas San Antonio Dept Management Sci & Stat Coll Business San Antonio TX 78249 USA

For many practical multiple objective network programming (MONP) problems, only integer solutions are meaningful and acceptable. Representative efficient solutions are usually generated by solving augmented weighted Tchebycheff network programs (AWTNPs), sub-problems derived from MONP problems. However, efficient solutions generated this way are usually not integer valued. In this study, two algorithms are developed to construct integer efficient solutions starting from fractional efficient solutions. One algorithm finds a single integer efficient solution in the neighborhood of the fractional efficient solution. The other enumerates all integer efficient solutions in the same neighborhood. Theory supporting the proposed algorithms is developed. Two detailed examples are presented to demonstrate the algorithms. Computational results are reported. The best integer efficient solution is very close, if not equal, to the integer optimal solution. The CPU time taken to find integer efficient solutions is negligible, when compared with that taken to solve AWTNPs. (C) 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(4), 362-375 2011

关键词： network flow algorithms multiple objective programming multiple criteria decision making integer programming

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Multi-criteria branch and bound: A vector maximization algorithm for Mixed 0-1 multiple objective Linear programming

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APPLIED MATHEMATICS AND COMPUTATION 2005年第1期171卷 53-71页

作者： Mavrotas, G Diakoulaki, D Natl Tech Univ Athens Lab Ind & Energy Econ Dept Chem Engn Div 2 Athens 15780 Greece

The paper describes the Multi-Criteria Branch and Bound (MCBB) algorithm. a vector maximization algorithm capable of deriving all efficient extreme points, for small and medium-sized Mixed 0-1 multiple objective Linear programming (Mixed 0-1 MOLP). Particular emphasis is given to computational aspects aiming principally at accelerating the solution procedure. For facilitating the decision maker's search toward the most preferred efficient solution, the notion of efficient combinations of the binary variables is further exploited. It is also shown that the MCBB algorithm can be used in single objective problems (Mixed Integer LP problems) in order to determine all alternative optima, as well as in Mixed Integer MOLP problems and Pure 0-1 MOLP problems that frequently arise in practice. A computational experiment is included in the paper in order to illustrate the performance of the algorithm. (c) 2005 Elsevier Inc. All rights reserved.

关键词： multiple objective programming mixed integer programming branch and bound

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Scrap Steel Recycling strategies optimization problem:a multiple objective programming model

Scrap Steel Recycling strategies optimization problem:a mult...

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第三届（2008）中国管理学年会——市场营销分会场

作者： Yang LIU Jiafu TANG Dept.of Systems Engineering Key Laboratory of Integrated Automation of Process Industry Northeastern University (NEU)

Facing the increasing global competition of steel production and the rapid growth of the output of steel products,to use more scrap steel instead of iron ore has been well acknowledged as an available strategy to support longterm development for the iron and steel *** and practitioners have paid more and more attentions on the issues related effective utilizing scrap steel for the reason that the reverse logistics of scrap steel commonly has many specialties distinguishing from other *** paper studies the combinatorial optimization problem integrating the specialties of scrap steel recycling under considerations of economic profit and protection of environment.A model consisting of multiple suppliers under push and pull policy,one dealer and multiple users is presented,which has two important objective functions,*** is the total cost of the whole network for scrap steel recycling,and the other is the total discharge of industrial pollutants of recycling *** is expected that this work could contribute to strategic management of supply chain for industrial manufacturers and other similar scenarios.

关键词： Index Terms – combinatorial optimization multiple objective programming scrap steel recycling the iron and steel industry

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A comparison of two reference point methods in multiple objective mathematical programming

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2003年第1期149卷 17-34页

作者： Buchanan, J Gardiner, L Univ Waikato Dept Management Syst Hamilton New Zealand Calif State Univ Los Angeles Coll Business Dept Accounting & Management Informat Syst Chico CA 95929 USA

When making decisions with multiple criteria, a decision maker often thinks in terms of an aspiration point or levels of achievement for the criteria. In multiple objective mathematical programming, solution methods based on aspiration points can generate nondominated solutions using a variety of scalarizing functions. These reference point solution methods commonly use a scalarizing function that reaches down from the ideal solution, in a direction specified by the aspiration point. Conversely, a similar scalarizing function can push out from the nadir point toward a specified aspiration point. These scalarizing functions are similar in structure but diametrically opposed in their reference points. In this paper we examine how these approaches behave from a technical point of view and conduct an experiment to understand better the human behaviour of users of these approaches. Insights we gain on the evenness of dispersion are relevant when attempting to construct a representation of the nondominated set. Further, the technical characteristics of the two formulations' solutions, combined with behavioural tendencies, allow us to comment on the implications for their use in interactive multiple objective methods. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： multiple objective programming decision analysis

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An exact method for computing the nadir values in multiple objective linear programming

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2009年第2期198卷 637-646页

作者： Alves, Maria Joao Costa, Joao Poulo Univ Coimbra INESC Coimbra Fac Econ P-3004512 Coimbra Portugal

In this paper we propose a new method to determine the exact nadir (minimum) criterion values over the efficient set in multiple objective linear programming (MOLP). The basic idea of the method is to determine, for each criterion, the region of the weight space associated wit It the efficient solutions that have a value in that criterion below the minimum already known (by default, the minimum in the payoff table). If this region is empty, the nadir value has been found. Otherwise, a new efficient solution is computed using a weight vector picked from the delimited region and a new iteration is performed. The method is able to find the nadir Values ill MOLP problems with ally number of objective functions, although the computational effort increases significantly with the number of objectives. Computational experiments are described and discussed, comparing two slightly different versions of the method. (C) 2008 Elsevier B.V. All rights reserved.

关键词： multiple criteria analysis multiple objective programming Nadir point

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Reducing wall-clock time for the computation of all efficient extreme points in multiple objective linear programming

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2019年第2期277卷 653-666页

作者： Piercy, Craig A. Steuer, Ralph E. Univ Georgia Terry Coll Business Athens GA 30602 USA

This paper describes an approach for markedly reducing the time required to obtain all efficient extreme points of a multiple objective linear program (MOLP) with three objectives. The approach is particularly useful when working with such MOLPs possessing large numbers of efficient extreme points. By subdividing the criterion cone into sub-cones, the paper shows how the task of computing all efficient extreme points can be broken down into parts so that the parts can be solved concurrently, thus allowing all efficient extreme points to be computed in much reduced elapsed time. The paper investigates several schemes for conducting this task and reports on a volume of computational experience. (C) 2019 Published by Elsevier B.V.

关键词： multiple objective programming Efficient extreme points Nondominated vertices Distributed processing Criterion cones

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A common formula to compute the efficient sets of a class of multiple objective linear programming problems

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OPTIMIZATION 2015年第10期64卷 2065-2092页

作者： Ta Van Tu Corvinus Univ Budapest Dept Operat Res Budapest Hungary

Finding the efficient set of a multiple objective linear programming (MOLP) problem is difficult and finding the efficient sets of many MOLP problems is still more difficult. In this paper, a common formula to compute the efficient sets of an arbitrary number of the MOLP problems corresponding to different right-hand side vectors is dealt with. We show that all the previously discovered methods and future methods for determining the efficient set of a MOLP problem which are not based on the efficient basic set of this MOLP problem cannot find such a common formula. In addition, our common formula is the union of the least number of descriptor sets for faces of the constraint polyhedrons among all possible common formulae for computing the efficient sets of the above MOLP problems. In order to increase the usefulness of the common formula, we give an efficient method for determining the efficient basic set of a MOLP problem. Some comparisons between our method and the methods for finding all extreme points of a convex polyhedral set in determining the efficient basic set of a MOLP problem are presented. A numerical example is given to illustrate the method.

关键词： efficient basic set multiple objective programming maximal efficient faces 90C90 90C50 90C29

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multiple objective programming with piecewise linear functions

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Journal of Multi-Criteria Decision Analysis 2000年第6期8卷

作者： Stefan Nickel Margaret M. Wiecek Fachbereich Mathematik Universität Kaiserslautern Germany On sabbatical leave from the Department of Mathematical Sciences Clemson University South Carolina USA.

An approach to generating all efficient solutions for multiple objective programs with piecewise linear objective functions and linear constraints is presented. The approach is based on the decomposition of the feasible set into subsets, referred to as cells, so that the original problem reduces to a series of single objective linear programs and feasibility tests over the cells. The concepts of cell-efficiency and complex-efficiency are introduced and their relationship with efficiency is examined. A generic algorithm for finding efficient solutions for bi-objective piecewise linear programs is proposed. Applications in location theory as well as in worst case analysis are highlighted. Copyright © 1999 John Wiley & Sons, Ltd.

关键词： multiple objective programming piecewise linear functions cell-efficiency complex-efficiency

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Efficient solution generation for multiple objective linear programming based on extreme ray generation method

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2005年第1期160卷 242-251页

作者： Ida, M Natl Inst Acad Degrees & Univ Evaluat Fac Univ Evaluat & Res Kodaira Tokyo 1878587 Japan

In this paper we consider solution generation method for multiple objective linear programming problems. The set of efficient or Pareto optimal solutions for the problems can be regarded as global information in multiple objective decision making situation. In the past three decades as solution generation techniques various conventional algorithms based on simplex-like approach with heavy computational burden were developed. Therefore, the development of novel and useful directions in efficient solution generation method have been desired. The purpose of this paper is to develop theoretical results and computational techniques of the efficient solution generation method based on extreme ray generation method that sequentially generates efficient points and rays by adding inequality constraints of the polyhedral feasible region. (C) 2003 Elsevier B.V. All rights reserved.

关键词： multiple objective programming polyhedra efficient solution extreme ray generation

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