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Analysis of the fuzzy machine repair problem with spares using parametric nonlinear programming

JOURNAL OF INTERDISCIPLINARY MATHEMATICS 2022年第1期25卷 63-77页

作者： Margaret, H. Merlyn Thirunavukarasu, P. SRM Inst Sci & Technol Coll Engn & Technol Dept Math Kanchipuram 603203 Tamil Nadu India Periyar EVR Coll Autonomous PG & Res Dept Math Tiruchirappalli 620023 Tamil Nadu India

Machine repair models find various applications in real-life problems. We carry out a systematic analysis of the classical machine repair problem with N machines and c servers, along with S spares in a fuzzy environment. The machine repair problem is essentially a queueing system with finite calling population. We use trapezoidal membership to model fuzziness in the parameters of the system, namely the service rate and the breakdown (arrival) rate. We propose an approach that uses parametric nonlinear programs to determine the alpha cuts of relevant performance measures of the system. This allows us to construct their membership maps. The problem of determining system characteristics is effectively reduced to an optimization problem. Finally, we present a numerical example to establish the validity of the approach.

关键词： Machine repair problem parametric nonlinear programming Fuzzy sets

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Sequential quadratic programming methods for parametric nonlinear optimization

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2014年第3期59卷 475-509页

作者： Kungurtsev, Vyacheslav Diehl, Moritz Katholieke Univ Leuven Dept Comp Sci & Optimizat Engn Ctr OPTEC B-3001 Leuven Heverlee Belgium Czech Tech Univ Fac Elect Engn Agent Technol Ctr Dept Comp Sci CR-16635 Prague Czech Republic Univ Freiburg Dept Microsyst Engn IMTEK D-79110 Freiburg Germany Katholieke Univ Leuven Elect Engn Dept ESAT STADIUS B-3001 Leuven Heverlee Belgium Katholieke Univ Leuven Optimizat Engn Ctr OPTEC B-3001 Leuven Heverlee Belgium

Sequential quadratic programming (SQP) methods are known to be efficient for solving a series of related nonlinear optimization problems because of desirable hot and warm start properties-a solution for one problem is a good estimate of the solution of the next. However, standard SQP solvers contain elements to enforce global convergence that can interfere with the potential to take advantage of these theoretical local properties in full. We present two new predictor-corrector procedures for solving a nonlinear program given a sufficiently accurate estimate of the solution of a similar problem. The procedures attempt to trace a homotopy path between solutions of the two problems, staying within the local domain of convergence for the series of problems generated. We provide theoretical convergence and tracking results, as well as some numerical results demonstrating the robustness and performance of the methods.

关键词： parametric nonlinear programming nonlinear programming nonlinear constraints Sequential quadratic programming SQP methods Stabilized SQP Regularized methods Model predictive control

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ADJOINT-BASED PREDICTOR-CORRECTOR SEQUENTIAL CONVEX programming FOR parametric nonlinear OPTIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 2012年第4期22卷 1258-1284页

作者： Quoc Tran Dinh Savorgnan, Carlo Diehl, Moritz Katholieke Univ Leuven Dept Elect Engn ESAT SCD B-3001 Heverlee Belgium Katholieke Univ Leuven Optimizat Engn Ctr OPTEC B-3001 Heverlee Belgium Vietnam Natl Univ Dept Math Mech Informat Hanoi Vietnam

This paper proposes an algorithmic framework for solving parametric optimization problems which we call adjoint-based predictor-corrector sequential convex programming. After presenting the algorithm, we prove a contraction estimate that guarantees the tracking performance of the algorithm. Two variants of this algorithm are investigated. The first can be used to treat online parametric nonlinear programming problems when the exact Jacobian matrix is available, while the second variant is used to solve nonlinear programming problems. The local convergence of these variants is proved. An application to a large-scale benchmark problem that originates from nonlinear model predictive control of a hydro power plant is implemented to examine the performance of the algorithms.

关键词： predictor-corrector path-following sequential convex programming adjoint-based optimization parametric nonlinear programming online optimization

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parametric nonlinear programming for analyzing fuzzy queues with finite capacity

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2004年第2期157卷 429-438页

作者： Chen, SP Natl Chung Cheng Univ Dept Business Adm Chiayi 621 Taiwan

This paper proposes a procedure for constructing the membership functions of the performance measures in finite-capacity queueing systems with the arrival rate and service rate being fuzzy numbers. The basic idea is to transform a fuzzy queue with finite capacity to a family of conventional crisp queues with finite capacity by applying the alpha-cut approach. A pair of parametric nonlinear programs is formulated to describe the family of crisp finite-capacity queues, via which the membership functions of the performance measures are derived. To demonstrate the validity of the proposed procedure, two fuzzy queues often encountered in real life serve as examples. Since the performance measures are expressed by membership functions rather than by crisp values, more information is provided for designing queueing systems in real life. By extending to fuzzy environment, the finite-capacity queueing models would have wider applications. (C) 2003 Elsevier B.V. All rights reserved.

关键词： fuzzy sets queueing theory finite capacity membership functions parametric nonlinear programming

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parametric programming approach for batch arrival queues with vacation policies and fuzzy parameters

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APPLIED MATHEMATICS AND COMPUTATION 2006年第1期180卷 217-232页

作者： Ke, Jau-Chuan Huang, Hsin-I Lin, Chuen-Horng Natl Taiwan Inst Technol Dept Stat Taichung 404 Taiwan Natl Taiwan Inst Technol Dept Informat Management Taichung 404 Taiwan

This work constructs the membership functions of the system characteristics of a batch arrival queuing model with vacation policies and fuzzy batch arrival, customer service and server vacation rates. The a-cut approach is used to transform a fuzzy queue into a family of conventional crisp queues in this context. By means of the membership functions of the system characteristics, a set of parametric nonlinear programs is developed to describe the family of crisp queues with a vacationing server. Two numerical examples are solved successfully to illustrate the validity of the proposed approach. Because the system characteristics are expressed and governed by the membership functions, more information is provided for use by management. By extending this model to the fuzzy environment, fuzzy queues with a vacationing server are represented more accurately and the analytic results are more useful for system designers and practitioners. (c) 2006 Published by Elsevier Inc.

关键词： fuzzy sets multiple vacation parametric nonlinear programming single vacation

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A mathematical programming approach to fuzzy queues with batch arrivals

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ENGINEERING OPTIMIZATION 2004年第6期36卷 635-644页

作者： Chen, SP Natl Chung Cheng Univ Dept Business Adm Chiayi 621 Taiwan

This paper proposes a mathematical programming approach for constructing the membership functions of the performance measures in batch-arrival queueing systems with constant batch size and the arrival rate and service rate being fuzzy numbers. The basic idea underlying the proposed method is to transform a fuzzy batch-arrival queue to a family of conventional crisp queues with batch arrivals by applying the alpha-cut approach. Then the family of crisp batch-arrival queues is described by formulating a pair of parametric nonlinear programs, through which the membership functions of the performance measures can be derived. A numerical example is solved successfully to demonstrate the validity of the proposed approach. Since the performance measures are completely expressed by membership functions rather than by crisp values, more information is provided for designing queueing systems. The successful extension of batch-arrival queues to fuzzy environments permits queueing models to have wider applications in the real world.

关键词： fuzzy sets queueing theory batch arrivals membership functions parametric nonlinear programming

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Optimal control and real-time optimization of mechanical multi-body systems

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ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK 2003年第10期83卷 705-719页

作者： Gerdts, M Univ Bayreuth Dept Math D-95440 Bayreuth Germany

Numerical methods for the solution and real-time optimization of parametric optimal control problems for mechanical multibody systems are considered. The equations of motion are given as a differential-algebraic equation (DAE) system of index 3 augmented by additional index I constraints. It turns out that the sensitivity analysis of parametric nonlinear programming problems is of central importance for the numerical solution of the optimal control problem by a direct shooting method and its real-time optimization. On the one hand, a sensitivity analysis of the discretized optimal control problem provides a real-time approximation of the optimal solution of a perturbed optimal control problem. On the other hand, a sensitivity analysis of the projection method used for consistent initialization yields consistent initial values of sensitivity matrices needed within the iterative solution process of the direct shooting method. Two examples conclude the article. (C) 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： optimal control real-time optimization parametric nonlinear programming direct shooting mechanical multi-body systems

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Bifurcation problems for some parametric nonlinear programs in Banach spaces

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1996年第6期34卷 1947-1971页

作者： Poore, AB Department of Mathematics Colorado State University Fort Collins CO 80523 United States

Singularities in a class of parametric nonlinear programming problems in Banach spaces are investigated using bifurcation theory. Motivated by the Fritz John first-order necessary conditions and a nonstandard normalization of the multipliers, this problem is first formulated as a system of nonlinear equations. Conditions for this system to be Fredholm are then derived, and singularities are shown to arise from a violation of one or more of the following conditions: strict complementarity, surjectivity of the Frechet derivative of the active constraints, and a second-order condition. A branching analysis is developed for each of these singularities under a second-order nondegeneracy assumption. Examples from the calculus of variations are then used to illustrate these singularities.

关键词： bifurcation singularity parametric nonlinear programming Banach spaces

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DIRECTIONAL-DERIVATIVES OF THE SOLUTION OF A parametric nonlinear PROGRAM

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MATHEMATICAL programming 1995年第2期70卷 159-172页

作者： RALPH, D DEMPE, S TECH UNIV CHEMNITZ DEPT MATH CHEMNITZ GERMANY

Consider a parametric nonlinear optimization problem subject to equality and inequality constraints. Conditions under which a locally optimal solution exists and depends in a continuous way on the parameter are well known. We show, under the additional assumption of constant rank of the active constraint gradients, that the optimal solution is actually piecewise smooth, hence B-differentiable. We show, for the first time to our knowledge, a practical application of quadratic programming to calculate the directional derivative in the case when the optimal multipliers are not unique.

关键词： parametric nonlinear programming DIRECTIONAL DIFFERENTIABILITY B-DERIVATIVE PIECEWISE SMOOTH FUNCTION NONUNIQUE MULTIPLIERS DEGENERACY

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Bifurcations in parametric nonlinear programming

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Annals of Operations Research 1990年第1期27卷 343-369页

作者： Poore, Aubrey B. Department of Mathematics Colorado State University Fort Collins 80523 CO United States

Bifurcation and continuation techniques are introduced as a class of methods for investigating the parametric nonlinear programming problem. Motivated by the Fritz John first-order necessary conditions, the parametric programming problem is first reformulated as a closed system of nonlinear equations which contains all Karush-Kuhn-Tucker and Fritz John points, both feasible and infeasible solutions, and relative minima, maxima, and saddle points. Since changes in the structure of the solution set and critical point type can occur only at singularities, necessary and sufficient conditions for the existence of a singularity are developed in terms of the loss of a complementarity condition, the linear dependence constraint qualification, and the singularity of the Hessian of the Lagrangian on a tangent space. After a brief introduction to elementary bifurcation theory, some simple singularities in this parametric problem are analyzed for both branching and persistence of local minima. Finally, a brief introduction to numerical continuation and bifurcation procedures is given to indicate how these facts can be used in a numerical investigation of the problem. © 1990 J.C. Baltzer A.G. Scientific Publishing Company.

关键词： Bifurcation continuation parametric nonlinear programming singularities stability

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