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On the projection of the efficient set and potential applications

OPTIMIZATION 2002年第2期51卷 401-421页

作者： Kim, NTB Muu, L Hanoi Inst Math Hanoi 10000 Vietnam

We study the efficient set X-E for a multiple objective linear program by using its projection into the linear space L spanned by the independent criteria. We show that in the orthogonally complementary space of L, the efficient points form a polyhedron, while in L an efficiency-equivalent polyhedron for the projection P(X-E) of X-E can be constructed by algorithms of outer and inner approximation types. These algorithms can be also used for generating all extreme points of P(X-E). Application to optimization over the efficient set for a multiple objective linear program is considered.

关键词： multiple objective linear programming efficient set dimension reduction projection outer and inner approximation optimization over the efficient set

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GOAL programming SENSITIVITY ANALYSIS USING INTERVAL PENALTY WEIGHTS

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MATHEMATICAL programming 1979年第1期17卷 16-31页

作者： STEUER, RE Princeton University Princeton New Jersey USA

This paper presents an application of the vector-maximum research [4–8] to the sensitivity analysis of goal programming problems as several of the criterion function penalty weights are simultaneously and independently varied. A generalized goal programming capability is presented and a six-stage analytic procedure is described. The problem is generalized in the sense that the regular goal programming penalty weights can be expanded to intervals if desired. The solution procedure is new in that it depends upon an algorithm for the vector-maximum problem, “criterion cone” contraction procedures, and “filtering” techniques. Together they are able to generate and process all extreme points on the portion of the surface of the goal programming “augmented” feasible region corresponding to the interval penalty weights specified. In effect, the procedure and adapted algorithm of this paper delivers to goal programming an operational power of sensitivity analysis not previously available to users. A numerical example is provided in order to illustrate the computerized application of the total goal programming procedure outlined.

关键词： Goal programming Sensitivity Analysis Vector-Maximum Algorithms multiple objective linear programming multiple Criteria Decision Making

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Bounding MOLP objective functions: effect on efficient set size

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JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 1998年第5期49卷 549-557页

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

In multiple objective linear programming (MOLP) problems the extraction of all the efficient extreme points becomes problematic as the size of the problem increases. One of the suggested actions, in order to keep the size of the efficient set to manageable limits, is the use of bounds on the values of the objective functions by the decision maker. The unacceptable efficient solutions are screened out from further investigation and the size of the efficient set is reduced. Although the bounding of the objective functions is widely used in practice, the effect of this action on the size of the efficient set has not been investigated. Ln this paper, we study the effect of individual and simultaneous bounding of the objective functions on the number of the generated efficient points. In order to estimate the underlying relationships, a computational experiment is designed in which randomly generated multiple objective linear programming problems of various sizes are systematically examined.

关键词： efficient set multiple objective linear programming

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Standard sensitivity analysis and additive tolerance approach in MOLP

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ANNALS OF OPERATIONS RESEARCH 2010年第1期181卷 219-232页

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

We consider sensitivity analysis of the objective function coefficients in multiple objective linear programming (MOLP). We focus on the properties of the parameters set for which a given extreme solution is efficient. Moreover, we compare two approaches: the standard sensitivity analysis (changing only one coefficient) and the additive tolerance approach (changing all coefficients). We find the connections between these two approaches by giving a theorem describing the upper bound on the maximal additive tolerance.

关键词： multiple objective linear programming Sensitivity analysis

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The natural ordering cone of a polyhedral convex set-valued objective mapping

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OPTIMIZATION 2024年第12期73卷 3653-3665页

作者： Loehne, Andreas Friedrich Schiller Univ Jena Fac Math & Comp Sci Jena Germany

For a given polyhedral convex set-valued mapping we define a polyhedral convex cone which we call the natural ordering cone. We show that the solution behaviour of a polyhedral convex set optimization problem can be characterized by this cone. Under appropriate assumptions, the natural ordering cone is proven to be the smallest ordering cone which makes a polyhedral convex set optimization problem solvable.

关键词： Set optimization vector linear programming multiple objective linear programming

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An inverse optimization model for imprecise data envelopment analysis

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OPTIMIZATION 2015年第11期64卷 2441-2454页

作者： Hadi-Vencheh, A. Hatami-Marbini, A. Beigi, Z. Ghelej Gholami, K. Islamic Azad Univ Isfahan Khorasgan Branch Dept Math Esfahan Iran Catholic Univ Louvain Louvain Sch Management CORE Louvain La Neuve Belgium Islamic Azad Univ Mobarakeh Branch Dept Math Mobarakeh Iran Islamic Azad Univ Bushehr Branch Dept Sci Bushehr Iran

Inverse data envelopment analysis (InDEA) is a well-known approach for short-term forecasting of a given decision-making unit (DMU). The conventional InDEA models use the production possibility set (PPS) that is composed of an evaluated DMU with current inputs and outputs. In this paper, we replace the fluctuated DMU with a modified DMU involving renewal inputs and outputs in the PPS since the DMU with current data cannot be allowed to establish the new PPS. Besides, the classical DEA models such as InDEA are assumed to consider perfect knowledge of the input and output values but in numerous situations, this assumption may not be realistic. The observed values of the data in these situations can sometimes be defined as interval numbers instead of crisp numbers. Here, we extend the InDEA model to interval data for evaluating the relative efficiency of DMUs. The proposed models determine the lower and upper bounds of the inputs of a given DMU separately when its interval outputs are changed in the performance analysis process. We aim to remain the current interval efficiency of a considered DMU and the interval efficiencies of the remaining DMUs fixed or even improve compared with the current interval efficiencies.

关键词： inverse data envelopment analysis decision-making unit multiple objective linear programming interval data 90C05 90C90 90B50

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Hybrid efficiency measurement and target setting based on identifying defining hyperplanes of the PPS with negative data

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OPERATIONAL RESEARCH 2020年第2期20卷 1055-1092页

作者： Ghazi, Amineh Hosseinzadeh Lotfi, Farhad Sanei, Masoud Islamic Azad Univ Tehran Cent Branch Dept Math Tehran Iran Islamic Azad Univ Sci & Res Branch Dept Math Tehran Iran

The traditional data envelopment analysis (DEA) models can be used for efficiency evaluation of decision making units (DMUs) with nonnegative data. However, in the real world there are DMUs which have negative inputs and/or outputs. Consequently, in the literature of DEA various approaches have been offered in order to deal with negative data. The current research proposes an algorithm concerned with a pure mathematical optimization method to measure hybrid efficiency of DMUs in the presence of negative data, and also a pure mathematical procedure for target setting. In doing so, explicit form equations of strong and weak defining hyperplanes of production possibility set (PPS) based on the multiple criteria decision making methodology are obtained. In characterizing these hyperplanes, a new multiple objective linear programming (MOLP) problem is presented whose feasible region of criterion space is similar to the PPS with variable returns to scale technology on nonnegative and negative data. The MOLP problem is solved using the multicriteria simplex method where its process of solving leads to construct all strong and weak defining hyperplanes. Then, using the strong and weak hyperplanes obtained and also without using any DEA optimization model, a hybrid measure of efficiency and a strong efficient target unit for each inefficient DMU with negative data are obtained. Finally, the results are discussed using two examples, where the first one, in detail, explains the proposed methods and algorithm, and the second one briefly shows the contributions are applicable for the real data.

关键词： Data envelopment analysis multiple objective linear programming Negative data Defining hyperplane Economic interpretation Lagrange multiplier

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A STOCHASTIC POSSIBILISTIC programming-MODEL FOR BANK HEDGING DECISION-PROBLEMS

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FUZZY SETS AND SYSTEMS 1993年第3期57卷 351-363页

作者： LAI, YJ HWANG, CL Department of Industrial Engineering Kansas State University Manhattan KS 66506 USA

For a practical bank hedging decision optimization problem, interest rates and price of futures contract may involve both fuzziness and randomness. For subjective nature of satisfaction, maximum desired values of loan demand, deposit supply and ratio of desired loan to deposit are often fuzzy. In this study, we consider and solve a stochastic possibilistic programming model of bank hedging decision problems with the above characters. We first use the expected value to obtain an auxiliary possibilistic linear programming problem which is further resolved by use of beta-level cut. An (crisp) auxiliary bi-objective linear programming model is then proposed and solved by our augmented maximin approach. For illustration purpose. a numerical bank hedging decision problem is solved.

关键词： BALANCE-SHEET OPTIMIZATION BANK HEDGING DECISIONS LINGUISTIC INTEREST RATES UNDER multiple STATES STOCHASTIC POSSIBILISTIC programming POSSIBILISTIC programming multiple objective linear programming AUGMENTED MAXIMIN

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Using aspiration levels in an interactive interior multiobjective linear programming algorithm

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1996年第1期89卷 193-201页

作者： Abel, A Korhonen, P HELSINKI SCH ECON & BUSINESS ADM SF-00100 HELSINKIFINLAND TEL AVIV UNIV DEPT IND ENGNIL-69978 TEL AVIVISRAEL

In this paper, we propose the use of an interior-point linear programming algorithm for multiple objective linear programming (MOLP) problems. At each iteration, a Decision Maker (DM) is asked to specify aspiration levels for the various objectives, and an achievement scalarizing function is applied to project aspiration levels onto the nondominated set. The interior-point algorithm is used to find an interior solution path from a starting solution to a nondominated solution corresponding to the optimum of the achievement scalarizing function. The proposed approach allows the DM to re-specify aspiration levels during the solution process and thus steer the interior solution path toward different areas in objective space. We illustrate the use of the approach with a numerical example.

关键词： multiple objective linear programming aspiration levels scalarizing functions interior point algorithms affine-scaling primal algorithm

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An interactive analytic center trade-off cutting plane algorithm for multiobjective linear programming

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COMPUTERS & INDUSTRIAL ENGINEERING 1999年第3期37卷 649-669页

作者： Trafalis, TB Alkahtani, RM Univ Oklahoma Sch Ind Engn Norman OK 73019 USA King Saud Univ Riyadh 11451 Saudi Arabia

This paper proposes the use of an interior point algorithm for Multiobjective linear programming problems. At each iteration of the algorithm, the decision maker furnishes his precise trade-offs. From these trade-offs, a cut is formed in the objective space. This cut induces a cut in the decision space that defines a half-space of promising points. We compute the analytic center of the restricted feasible region in the decision space and then we calculate the trade-offs of the decision maker at the image of the analytic center in the objective space. Therefore, we obtain a trajectory of analytic centers that converges to the best compromise solution. Since the proposed algorithm moves through the interior of the feasible region, it avoids the combinatorial difficulties of visiting extreme points and is less sensitive to problem size. We illustrate the method through a numerical example and provide computational experience. (C) 2000 Elsevier Science Ltd. All rights reserved.

关键词： multiple objective linear programming interactive methods trade-off cutting plane interior point algorithm method of analytic centers

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