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Cost efficiency measurement with fuzzy data in DEA

JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2017年第1期32卷 409-420页

作者： Aghayi, Nazila Islamic Azad Univ Dept Math Ardabil Branch Ardebil Iran

Being a nonparametric method, data envelopment analysis (DEA) is confined to measuring the efficiency of a collection of decision making units (DMUs) consuming multiple crisp inputs to produce multiple crisp outputs. Since not all data in the real world have determined values, and input and output values for DMUs are often subject to fluctuation, the concept of fuzziness has been introduced to deal with such imprecise data. This study intends to evaluate the cost efficiency of DMUs in three different scenarios, with one distinct model proposed for each scenario. The main idea is to use the alpha-cut method and the extension principle to convert the fuzzy cost efficiency model into a family of conventional crisp DEA models by obtaining one lower bound and one upper bound for the cost efficiency score of a DMU for any alpha varying between 0 and 1. When the lower and upper bounds are invertible with respect to alpha, the membership function of the fuzzy efficiency of a DMU-which falls within the scope of parametric programming problems-can be obtained by finding the inverse of the lower and upper bounds as well as employing the extension principle. In this case, the value of fuzzy cost efficiency varies between 0 and 1. Otherwise, the cost efficiency scores of DMUs can be specified as intervals, which are actually alpha-cuts of the fuzzy membership function, by collecting results for different alpha values. Furthermore, we demonstrate that Farrell's decomposition also holds for cost efficiency with fuzzy data. In addition, all DMUs are divided into three classes in each scenario, with cost-efficient and cost-inefficient DMUs falling into independent classes. In other words, units which are cost-inefficient in the upper bound for any alpha ranging between 0 and 1 will definitely be cost-inefficient in the lower bound, too, and the units will be cost-efficient in the upper bound if they are cost-efficient in the lower bound for a specific alpha varying between 0 and 1.

关键词： Data envelopment analysis cost efficiency membership function parametric programming alpha-cut fuzzy

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ON THE OPTIMAL VALUE FUNCTION FOR CERTAIN LINEAR-PROGRAMS WITH UNBOUNDED OPTIMAL SOLUTION SETS

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1985年第1期46卷 55-66页

作者： BENSON, HP 1.Department of Management and Administrative Sciences College of Business Administration University of Florida Gainesville Florida

Often, the coefficients of a linear programming problem represent estimates of true values of data or are subject to systematic variations. In such cases, it is useful to perturb the original data and to either compute, estimate, or otherwise describe the values of the functionf which gives the optimal value of the linear program for each perturbation. If the right-hand derivative off at a chosen point exists and is calculated, then the values off in a neighborhood of that point can be estimated. However, if the optimal solution set of either the primal problem or the dual problem is unbounded, then this derivative may not exist. In this note, we show that, frequently, even if the primal problem or the dual problem has an unbounded optimal solution set, the nature of the values off at points near a given point can be investigated. To illustrate the potential utility of our results, their application to two types of problems is also explained.

关键词： Optimal value function linear programming sensitivity analysis parametric programming multiple-objective linear programming

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Convex Lifting: Theory and Control Applications

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2018年第5期63卷 1243-1258页

作者： Ngoc Anh Nguyen Gulan, Martin Olaru, Sorin Rodriguez-Ayerbe, Pedro Univ Paris Sud Cent Supelec Lab Signals & Syst CNRS F-91192 Gif Sur Yvette France Slovak Univ Technol Bratislava Fac Mech Engn Inst Automat Measurement & Appl Informat Bratislava 81243 Slovakia

This paper presents the concept of convex lifting, which will be proven to enable significant implementation benefits for the class of piecewise affine controllers. Accordingly, two different algorithms to construct a convex lifting for a given polyhedral/polytopic partition will be presented. These two algorithms rely on either the vertex or the halfspace representation of the related polyhedra. Also, we introduce an algorithm to refine a polyhedral partition, which does not admit a convex lifting, into a convexly liftable one. Furthermore, two different schemes will be put forward to considerably reduce both the memory footprint and the online evaluation effort, which play a key role in implementation of piecewise affine controllers. Finally, these results will be illustrated via numerical examples and a complexity analysis.

关键词： Convex lifting explicit solutions model predictive control (MPC) parametric programming

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Efficient online solution of multi-parametric mixed-integer quadratic problems

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INTERNATIONAL JOURNAL OF CONTROL 2013年第8期86卷 1386-1396页

作者： Almer, Stefan Morari, Manfred Swiss Fed Inst Technol Automat Control Lab Zurich Switzerland

This paper considers model predictive control of a class of piecewise affine systems and introduces a method to reduce the complexity of the optimisation problem which is solved online. The optimisation problem is a mixed-integer quadratic program which is parameterised by the initial state of the system dynamics. The complexity of the problem solved online is reduced by moving part of the computational burden offline. In the offline computations, the structure of the mixed-integer problem is explored to determine a polytopic partition of the set of initial states. Each polytope of the partition is then considered and it is determined which integer combinations can/cannot be optimal, given that the initial state is contained in the current polytope. The result of the offline computations is that each polytope is associated with the set of potentially optimal integer combinations. The online solution of the mixed-integer problem consists of a two-step procedure: first, it is determined which polytope contains the initial state. Then, the mixed-integer problem is solved, taking into account only those integer combinations associated with the current polytope. The solution approach is shown to give a significant complexity reduction when applied to hybrid control of switched-mode power converters.

关键词： model predictive control hybrid system mixed-integer quadratic problem parametric programming switched-mode power converter

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Sensitivity of optimum designs for spatially varying ground motions

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JOURNAL OF STRUCTURAL ENGINEERING-ASCE 2001年第11期127卷 1324-1329页

作者： Ohsaki, M Kyoto Univ Dept Arch & Arch Sys Sakyo Ku Kyoto 6068501 Japan

An optimization method is presented for long-span frames for specified seismic responses considering spatial variation of ground motions. The response is evaluated by using a response spectrum approach, where incoherency effect, wave passage effect, and local amplification effect are considered. Sensitivity coefficients of the optimum objective value with respect to the parameters defining spatial variation of the seismic motions are computed based on the parametric programming method or postoptimal analysis. It is shown that the second-order parametric sensitivity coefficients are easily obtained if the first-order coefficients vanish due to symmetry and antisymmetry conditions of structure and ground motions. In the examples, optimum designs are found for an arch-type frame and the results of postoptimal analysis are verified in comparison with the parametric sensitivity coefficients by the central difference method. It is shown that the spatial variation of ground motions leads to the increase of the optimal structural volume under constraints on member strains.

关键词： .response spectrum second-order spatial variation parametric sensitivity objective value SPATIALLY VARYING wave passage OPTIMUM DESIGNS incoherency parametric programming postoptimal

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Reliability assessment for fuzzy multi-state systems

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INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 2010年第4期41卷 365-379页

作者： Liu, Yu Huang, Hong-Zhong Univ Elect Sci & Technol China Sch Mechatron Engn Chengdu 610054 Sichuan Peoples R China

Fuzzy multi-state system (FMSS) is defined as a multi-state system (MSS) consisting of multi-state elements (MSE) whose performance rates and transition intensities are presented as fuzzy values. Due to the lack, inaccuracy or fluctuation of data, it is oftentimes impossible to evaluate the performance rates and transition intensities of MSE with precise values. This is true especially in continuously degrading elements that are usually simplified to MSE for computation convenience. To overcome these challenges in evaluating the behaviour of MSS, fuzzy theory is employed to facilitate MSS reliability assessment. Given the fuzzy transition intensities and performance rates, the state probabilities of MSE and MSS are also fuzzy values. A fuzzy continuous-time Markov model with finite discrete states is proposed to assess the fuzzy state probability of MSE at any time instant. The universal generating function with fuzzy state probability function and performance rate is applied to evaluate fuzzy state probability of MSS in accordance with the system structure. A modified FMSS availability assessment approach is introduced to compute the system availability under the fuzzy user demand. In order to obtain the membership functions of the indices of interest, parametric programming technique is employed according to Zadeh's extension principle. The effectiveness of the proposed method is illustrated and verified via reliability assessment of a multi-state power generation system.

关键词： fuzzy multi-state system (FMSS) fuzzy multi-state element (FMSE) fuzzy reliability assessment fuzzy Markov model fuzzy universal generating function (FUGF) parametric programming

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Optimization analysis for design and planning of multi-project programs

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1998年第2期107卷 492-506页

作者： Wiley, VD Deckro, RF Jackson, JA USAF Inst Technol Dept Operat Sci Wright Patterson AFB OH 45433 USA Directorate Requirements Off Aerosp Studies Kirtland AFB NM 87117 USA

Program management concerns the long term planning, coordination, and control of major technological, engineering, scientific, and/or developmental activities. In general, programs tend to be exceptionally large, consisting of several parallel or sequential projects or groups of projects. While a large number of modeling efforts have focused at the project level, this paper looks at the application of optimization techniques to the initial design and development of multi-project programs. The classic work breakdown structure (WBS) is used as a framework to provide an aggregate model to investigate the effects of funding levels, resource allocation, and program, project, and component durations. Decomposition, sensitivity analysis, and parametric programming are utilized to provide the decision maker detailed information for establishing program parameters, conditions, and bounds. (C) 1998 Elsevier Science B.V. All rights reserved.

关键词： program/project management scheduling planning decomposition sensitivity analysis parametric programming

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Global sensitivity analysis via a statistical tolerance approach

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2022年第1期296卷 44-59页

作者： Curry, Stewart Lee, Ilbin Ma, Simin Serban, Nicoleta Georgia Inst Technol H Milton Stewart Sch Ind & Syst Engn 755 Ferst Dr NW Atlanta GA 30332 USA Univ Alberta Alberta Sch Business 2-29B Business Bldg Edmonton AB T6G 2R6 Canada

Sensitivity analysis and multiparametric programming in optimization modeling study variations of optimal value and solutions in the presence of uncertain input parameters. In this paper, we consider simultaneous variations in the inputs of the objective and constraint (jointly called the RIM parameters ), where the uncertainty is represented as a multivariate probability distribution. We introduce a tolerance approach based on principal component analysis, which obtains a tolerance region that is suited to the given distribution and can be considered a confidence set for the random input parameters. Since a tolerance region may contain parameters with different optimal bases, we extend the tolerance approach to the case where multiple optimal bases cover the tolerance region, by studying theoretical properties of critical regions (defined as the set of input parameters having the same optimal basis). We also propose a computational algorithm to find critical regions covering a given tolerance region in the RIM parameter space. Our theoretical results on geometric properties of critical regions contribute to the existing theory of parametric programming with an emphasis on the case where RIM parameters vary jointly, and provide deeper geometric understanding of critical regions. We evaluate the proposed framework using a series of experiments for sensitivity analysis, for model predictive control of an inventory management problem, and for large optimization problem instances. (c) 2021 Elsevier B.V. All rights reserved.

关键词： Robustness and sensitivity analysis Linear programming Sensitivity analysis parametric programming Tolerance sensitivity

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The principle and application of cutting-point virtual tool radius compensation for ellipsoidal outer contour finishing using a ball end mill

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INTERNATIONAL JOURNAL OF ADVANCED MANUFACTURING TECHNOLOGY 2014年第9-12期71卷 1527-1537页

作者： Wang, Zhaoqin Wang, Xiaorong Lanzhou Jiaotong Univ Sch Railway Technol Lanzhou 730070 Peoples R China Lanzhou Jiaotong Univ Sch Mech Engn Lanzhou 730070 Peoples R China

A novel cutter radius compensation method named cutting-point virtual tool radius compensation (CPVTRC) is presented in the finishing of ellipsoidal outer contour. CPVTRC, which corresponds to a virtual tool established at cutting point, is realized through parametric programming which is powerful in solving special manufacturing problems and exhibits great industrial application prospect. The parametric program for ellipsoidal outer contour finishing is given based on the algorithm of CPVTRC. The corresponding amount of the virtual tool radius per layer is calculated via the cutting-point coordinates and the elliptical outer offset curve parametric equation of the current layer. And it is inputted into corresponding registers online using programmable data input command G10L12, and then called through cutter compensation command G41/G42 before interpolation of the ellipse in the current layer. Consequently, the ellipse per layer of ellipsoid is approached via tiny line segments using CPVTRC method. The results of simulation based on VERICUT software and experiment based on a 3-axis vertical computer numerical control (CNC) machine equipped with FANUC 0i-MC exhibit that CPVTRC is highly effective and correct. Furthermore, CPVTRC can extend its application to the CNC parametric programming and machining of swept surfaces.

关键词： parametric programming CAM Ellipsoid Finishing

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A PRIMAL-DUAL EXTERIOR POINT METHOD FOR NONLINEAR OPTIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 2010年第6期20卷 3335-3363页

作者： Yamashita, Hiroshi Tanabe, Takahito Math Syst Inc Shinjuku Ku Tokyo Japan

In this paper, primal-dual methods for general nonconvex nonlinear optimization problems are considered. The proposed methods are exterior point type methods that permit primal variables to violate inequality constraints during the iterations. The methods are based on the exact penalty type transformation of inequality constraints and use a smooth approximation of the problem to form primal-dual iteration based on Newton's method as in usual primal-dual interior point methods. Global convergence and local superlinear/quadratic convergence of the proposed methods are proved. For global convergence, methods using line searches and trust region type searches are proposed. The trust region type method is tested with CUTEr problems and is shown to have similar efficiency to the primal-dual interior point method code IPOPT. It is also shown that the methods can be warm started easily, unlike interior point methods, and that the methods can be efficiently used in parametric programming problems.

关键词： primal-dual method exterior point method warm start parametric programming

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