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multiple objective linear programming models with interval coefficients - an illustrated overview

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2007年第3期181卷 1434-1463页

作者： Oliveira, Carla Henggeler Antunes, Carlos INESC Coimbra Coimbra Portugal Univ Coimbra Dept Elect & Comp Engn P-3030290 Coimbra Portugal

In most real-world situations, the coefficients of decision support models are not exactly known. In this context, it is convenient to consider an extension of traditional mathematical programming models incorporating their intrinsic uncertainty, without assuming the exactness of the model coefficients. Interval programming is one of the tools to tackle uncertainty in mathematical programming models. Moreover, most real-world problems inherently impose the need to consider multiple, conflicting and incommensurate objective functions. This paper provides an illustrated overview of the state of the art of Interval programming in the context of multiple objective linear programming models. (C) 2006 Elsevier B.V. All rights reserved.

关键词： multiple objective linear programming interval programming satisficing approach optimizing approach

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A multiple-objective linear-programming MODEL FOR POWER-GENERATION EXPANSION PLANNING

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INTERNATIONAL JOURNAL OF ENERGY RESEARCH 1995年第5期19卷 419-432页

作者： CLIMACO, J ANTUNES, CH MARTINS, AG ALMEIDA, AT Faculty of Economics University of Coimbra 3000 Coimbra Portugal INESC—Run Antero de Quental 199 3000 Coimbra Portugal Department of Electrical Engineering—FCT University of Coimbra 3000 Coimbra Portugal

The planning of new units for electrical power generation is a problem which involves different and conflicting aspects. Besides cost, security issues and environmental concerns must be explicitly incorporated into the models. In this way mathematical models become more realistic, and they enhance the decision maker's comprehension of the complex and conflicting nature of the distinct aspects of the problem. A multiple objective linear programming model for power generation expansion planning is presented. The model considers three objective functions (net present cost of the expansion plans, reliability of the supply system, and environmental impacts) and three categories of constraints (load requirements, operational restrictions and budget). Three generating technologies are considered for power system expansion: oil, nuclear and coal.

关键词： POWER GENERATION PLANNING multiple objective linear programming MATHEMATICAL MODELING DECISION SUPPORT SYSTEMS

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An algorithm based on facial decomposition for finding the efficient set in multiple objective linear programming

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OPERATIONS RESEARCH LETTERS 1996年第2期19卷 87-94页

作者： Sayin, S Bilkent University Management Department Faculty of Business Administration 06533 Bilkent Ankara Turkey

We propose a method for finding the efficient set of a multiple objective linear program based on the well-known facial decomposition of the efficient set. The method incorporates a simple linear programming test that identifies efficient faces while employing a top-down search strategy which avoids enumeration of efficient extreme points and locates the maximally efficient faces of the feasible region. We suggest that discrete representations of the efficient faces could be obtained and presented to the Decision Maker. Results of computational experiments are reported.

关键词： multiple objective linear programming vector maximization efficient faces efficient set

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Structural comparison of data envelopment analysis and multiple objective linear programming

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MANAGEMENT SCIENCE 1998年第7期44卷 962-970页

作者： Joro, T Korhonen, P Wallenius, J Helsinki Sch Econ & Business Adm Helsinki 00100 Finland

The concept of efficiency as it applies to Decision Making Units (DMUs), solutions, alternatives plays an important role both in Data Envelopment Analysis (DEA) and multiple objective linear programming (MOLP). Despite this and other apparent similarities, DEA and MOLP research has developed separately. We show that structurally the DEA formulation to identify efficient units is quite similar to the MOLP model based on the reference point or the reference direction approach to generate efficient solutions. DEA and MOLP should not be seen as substitutes, but rather as complements. We show that they cross-fertilize each other. MOLP provides interesting extensions to DEA and DEA provides new areas of application to MOLP.

关键词： data envelopment analysis multiple objective linear programming efficiency performance measurement

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A bilinear algorithm for optimizing a linear function over the efficient set of a multiple objective linear programming problem

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JOURNAL OF GLOBAL OPTIMIZATION 2005年第1期31卷 1-16页

作者： Jorge, JM Univ La Laguna Dept Estadist Invest Operat & Computac Tenerife 38271 Spain

The problem Q of optimizing a linear function over the efficient set of a multiple objective linear program serves several useful purposes in multiple criteria decision making. However, Q is in itself a difficult global optimization problem, whose local optima, frequently large in number, need not be globally optimal. Indeed, this is due to the fact that the feasible region of Q is, in general, a nonconvex set. In this paper we present a monotonically increasing algorithm that finds an exact, globally-optimal solution for Q. Our approach does not require any hypothesis on the boundedness of neither the efficient set E-P nor the optimal objective value. The proposed algorithm relies on a simplified disjoint bilinear program that can be solved through the use of well-known specifically designed methods within nonconvex optimization. The algorithm has been implemented in C and preliminary numerical results are reported.

关键词： bilinear programming global optimization multiple objective linear programming optimization over efficient sets

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Solving DC programs with a polyhedral component utilizing a multiple objective linear programming solver

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JOURNAL OF GLOBAL OPTIMIZATION 2017年第2期69卷 369-385页

作者： Loehne, Andreas Wagner, Andrea Friedrich Schiller Univ Jena Dept Math D-07737 Jena Germany Vienna Univ Econ & Business Inst Stat & Math A-1020 Vienna Austria

A class of non-convex optimization problems with DC objective function is studied, where DC stands for being representable as the difference of two convex functions g and h. In particular, we deal with the special case where one of the two convex functions g or h is polyhedral. In case g is polyhedral, we show that a solution of the DC program can be obtained from a solution of an associated polyhedral projection problem. In case h is polyhedral, we prove that a solution of the DC program can be obtained by solving a polyhedral projection problem and finitely many convex programs. Since polyhedral projection is equivalent to multiple objective linear programming (MOLP), a MOLP solver (in the second case together with a convex programming solver) can be used to solve instances of DC programs with polyhedral component. Numerical examples are provided, among them an application to locational analysis.

关键词： DC programming Global optimization Polyhedral projection multiple objective linear programming linear vector optimization

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A regression study of the number of efficient extreme points in multiple objective linear programming

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2005年第2期162卷 484-496页

作者： Steuer, RE Piercy, CA Univ Georgia Terry Coll Business Athens GA 30602 USA

In this paper we employ regression analysis to construct relationships for predicting the number of efficient extreme points in MOLPs (multiple objective linear programs) with up to 120,000 efficient extreme points, and the CPU time to compute them. Principal among the factors affecting the number of efficient extreme points and CPU time are the number of objectives, criterion cone size, number of constraints, number of variables, and the nonzero density of the constraint matrix. The regression equations show the degree to which interactions are present among the factors and provide a more formal basis for understanding how the complexity of the efficient set, an indicator of the difficulty involved in solving a multiple criteria problem, increases with problem size. (C) 2003 Elsevier B.V. All rights reserved.

关键词： multiple objective linear programming large MOLPs random problem generator efficient extreme points regression computational experiments

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An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem

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JOURNAL OF GLOBAL OPTIMIZATION 1998年第1期13卷 1-24页

作者： Benson, HP Univ Florida Coll Business Adm Dept Informat & Decis Sci Gainesville FL 32611 USA

Various difficulties have been encountered in using decision set-based vector maximization methods to solve a multiple objective linear programming problem (MOLP). Motivated by these difficulties, some researchers in recent years have suggested that outcome set-based approaches should instead be developed and used to solve problem (MOLP). In this article, we present a finite algorithm, called the Outer Approximation Algorithm, for generating the set of all efficient extreme points in the outcome set of problem (MOLP). To our knowledge, the Outer Approximation Algorithm is the first algorithm capable of generating this set. As a by-product, the algorithm also generates the weakly efficient outcome set of problem (MOLP), Because it works in the outcome set rather than in the decision set of problem (MOLP), the Outer Approximation Algorithm has several advantages over decision set-based algorithms. It is also relatively easy to implement. Preliminary computational results for a set of randomly-generated problems are reported. These results tangibly demonstrate the usefulness of using the outcome set approach of the Outer Approximation Algorithm instead of a decision set-based approach.

关键词： efficient set global optimization multiple objective linear programming outer approximation vector maximization

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SENSITIVITY ANALYSIS OF WEAK EFFICIENCY IN multiple objective linear programming

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ASIA-PACIFIC JOURNAL OF OPERATIONAL RESEARCH 2011年第4期28卷 445-455页

作者： Sitarz, Sebastian Univ Silesia Inst Math PL-40007 Katowice Poland

This paper, studies the sensitivity analysis of weakly efficient extreme solutions in multiple objective linear programming (MOLP). The aim of the paper is to compute the set of the parameters (corresponding to one coefficient) for which a given extreme point is a weakly efficient solution. We also focus on the properties of the parameters set by proving convexity and closeness of this set. Moreover, we compare the results of the sensitivity analysis of efficiency and of weak efficiency in MOLP.

关键词： multiple objective linear programming sensitivity analysis weak efficiency

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Global optimization in practice: An application to interactive multiple objective linear programming

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JOURNAL OF GLOBAL OPTIMIZATION 1998年第4期12卷 353-372页

作者： Benson, HP Lee, D McClure, P Univ Florida Dept Informat & Decis Sci Gainesville FL 32611 USA Korea Ind Technol Assoc Kangnam Gu Seoul 135703 South Korea Becker Groves Inc Ft Pierce FL 34954 USA

A multiple objective linear programming problem (F') involves the simultaneous maximization of two or more conflicting linear objective functions over a nonempty polyhedron X. Many of the most popular methods for solving this type of problem, including many well-known interactive methods, involve searching the efficient set X-E of the problem. Generally however, X-E is a complicated, nonconvex set. As a result, concepts and methods from global optimization may be useful in searching X-E. In this paper, we will explain in theory, and show via an actual application to citrus rootstock selection in Florida, how the potential usefulness of the well-known interactive method STEM for solving problem (P) in this way, can depend crucially upon how accurately certain global optimization problems involving minimizations over X-E are solved. In particular, we will show both in theory and in practice that the choice of whether to use the popular but unreliable "payoff table" approach or to use one of the lesser known, more accurate global optimization methods to solve these problems can determine whether STEM succeeds or fails as a decision aid. Several lessons and conclusions of transferable value derived from this research are also given.

关键词： multiple objective linear programming optimization over the efficient set interactive methods global optimization citrus rootstock selection

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