parametric analysis in linear fractional programming is significantly more complicated in case of an unbounded feasible region. We propose procedures which are based on a modified version of Martos' algorithm or a...
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parametric analysis in linear fractional programming is significantly more complicated in case of an unbounded feasible region. We propose procedures which are based on a modified version of Martos' algorithm or a modification of Charnes-Cooper's algorithm, applying each to problems where either the objective function or the right-hand side is parametrized.
Jenkins developed an heuristic algorithm for performing the right-hand-side parametric analysis of a Mixed Integer Linear programming (MILP) problem with an approach completely different from all previous work in the ...
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Jenkins developed an heuristic algorithm for performing the right-hand-side parametric analysis of a Mixed Integer Linear programming (MILP) problem with an approach completely different from all previous work in the area: his method can use any software capable of solving MILP problems and able to perform parametric linear programming (PLP). There is no proof of completeness by using Jenkins's algorithm. In this paper we present results that allow us to verify the completeness of the parametrical analysis. Like Jenkins's algorithm our procedure is theoretically independent of the solution method to solve a point-value problem. (C) 1998 Elsevier Science B.V.
This paper presents a method for finding the optimal solutions to an integer program (IP) as the coefficients of the objective function are varied continuously over a wide range. The method requires identifying an opt...
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This paper presents a method for finding the optimal solutions to an integer program (IP) as the coefficients of the objective function are varied continuously over a wide range. The method requires identifying an optimal solution to the IP ata number of point-values of the objective coefficients. These point-value IPs are solved using knapsack facets or Gomory cutting plane. If there are n different optimal solutions in the range of coefficient variation, then optimal solutions must be idenfified at 2n - 1 different coefficient values. By using a facet or cutting plane algorithm and employing post-optimal analysis and re-optimization, the computational effort required may be much less than would be needed to solve 2 n - 1 IPs independently. [ABSTRACT FROM AUTHOR]
Two algorithms for the general case of parametric mixed-integer linear programs (MILPs) are proposed. parametric MILPs are considered in which a single parameter call simultaneously influence the objective function, t...
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Two algorithms for the general case of parametric mixed-integer linear programs (MILPs) are proposed. parametric MILPs are considered in which a single parameter call simultaneously influence the objective function, the right-hand side and the matrix. The first algorithm is based oil branch-and-bound oil the integer variables, solving a parametric linear program (LP) at each node. The second algorithm is based oil the optimality range of a qualitatively invariant solution, decomposing the parametric optimization problem into a series of regular MILPs, parametric LPs and regular mixed-integer nonlinear programs (MINLPs). The number of subproblems required for a particular instance is equal to the number of critical regions. For the parametric LPs an improvement of the well-known rational simplex algorithm is presented, that requires less consecutive operations oil rational functions. Also, an alternative based on predictor-corrector continuation is proposed. Numerical results for a test set are discussed. (C) 2008 Elsevier B.V. All rights reserved.
Two examples of parametric cost programming problems—one in network programming and one in NP-hard 0-1 programming—are given; in each case, the number of breakpoints in the optimal cost curve is exponential in the s...
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Two examples of parametric cost programming problems—one in network programming and one in NP-hard 0-1 programming—are given; in each case, the number of breakpoints in the optimal cost curve is exponential in the square root of the number of variables in the problem.
In this paper, we present algorithms for solving families of nonlinear integer programming problems in which the problems are related by having identical objective coefficients and constraint matrix coefficients. We c...
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In this paper, we present algorithms for solving families of nonlinear integer programming problems in which the problems are related by having identical objective coefficients and constraint matrix coefficients. We consider two types of right-hand sides which have the forms b(l) and b(i) + theta-i(d)i where {b(l)}l = 1,...,L is a given set of vectors, b(i) + theta-i(d)i is a parametric function and the parameter theta-i varies from zero to one. The approach consists primarily of solving the most relaxed problem using branch and search method and then finding the optimal solutions of the proposed parametric programming problems. The application of this methodology to a parametric chance-constrained problem is illustrated with applications in system reliability optimization problems.
Zimmermann's fuzzy approach to the compromise solution concept in the multi-objective linear programming problem is considered. It is shown that given a class of membership functions of fuzzy goals assigned to obj...
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Zimmermann's fuzzy approach to the compromise solution concept in the multi-objective linear programming problem is considered. It is shown that given a class of membership functions of fuzzy goals assigned to objective functions in the problem, wider than primarily proposed by Zimmermann, the use of classical linear programming methods to solve and analyse the problem is also possible. As a result of application of the method based on the parametric programming technique, one can obtain a fuzzy solution of the problem. This solution is a certain fuzzy subset of the set of weakly efficient solutions of the problem. The solution which belongs to this set to the highest degree is a maximizing one (in Bellman and Zadeh's terminology). It may also be obtained by use of the usual non-parametric simplex method.
This study intends to propose an alternative approach to the multi-parametric sensitivity analysis of a linear programming problem by the concept of a maximum volume in the tolerance region. For the purposes of manage...
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This study intends to propose an alternative approach to the multi-parametric sensitivity analysis of a linear programming problem by the concept of a maximum volume in the tolerance region. For the purposes of management and control, the study first proposes the necessary and sufficient conditions of the focal parameters. Then, those nonfocal parameters that have unlimited variations can be identified and thus because of their low sensitivity in practice and simplicity in analysis, they will be deleted. For those focal parameters, their different levels of sensitivity are investigated and conceived by the proposed Maximum Volume Region so that they can be handled with the greatest flexibility, simultaneously and independently. This Maximum Volume Region is bounded by a symmetrically rectangular parallelepiped. It can be characterized by a maximization problem and solved by the existing technique - Dynamic programming. Besides, an extension to the Extended Maximum Volume Region for the case of semi-bounded region is considered. Theoretical proofs are provided with numerical illustrations. The result has been compared to Wendell's approach.
We propose a global optimisation approach for the solution of various classes of bilevel programming problems (BLPP) based on recently developed parametric programming algorithms. We first describe how we can recast a...
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We propose a global optimisation approach for the solution of various classes of bilevel programming problems (BLPP) based on recently developed parametric programming algorithms. We first describe how we can recast and solve the inner (follower's) problem of the bilevel formulation as a multi-parametric programming problem, with parameters being the (unknown) variables of the outer (leader's) problem. By inserting the obtained rational reaction sets in the upper level problem the overall problem is transformed into a set of independent quadratic, linear or mixed integer linear programming problems, which can be solved to global optimality. In particular, we solve bilevel quadratic and bilevel mixed integer linear problems, with or without right-hand-side uncertainty. A number of examples are presented to illustrate the steps and details of the proposed global optimisation strategy.
We propose a global optimisation approach for the solution of various classes of bilevel programming problems (BLPP) based on recently developed parametric programming algorithms. We first describe how we can recast a...
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We propose a global optimisation approach for the solution of various classes of bilevel programming problems (BLPP) based on recently developed parametric programming algorithms. We first describe how we can recast and solve the inner (follower's) problem of the bilevel formulation as a multi-parametric programming problem, with parameters being the (unknown) variables of the outer (leader's) problem. By inserting the obtained rational reaction sets in the upper level problem the overall problem is transformed into a set of independent quadratic, linear or mixed integer linear programming problems, which can be solved to global optimality. In particular, we solve bilevel quadratic and bilevel mixed integer linear problems, with or without right-hand-side uncertainty. A number of examples are presented to illustrate the steps and details of the proposed global optimisation strategy.
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