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Journal ArticleDOI

Stability of Solutions for Parametric Inverse Nonlinear Cost Transportation Problem

14 Nov 2020-Vol. 8, Iss: 11, pp 2027
TL;DR: This paper investigates the solution for an inverse of a parametric nonlinear transportation problem, in which, for a certain values of the parameters, the cost of the unit transportation in the basic problem are adapted as little as possible so that the specific feasible alternative become an optimal solution.
Abstract: This paper investigates the solution for an inverse of a parametric nonlinear transportation problem, in which, for a certain values of the parameters, the cost of the unit transportation in the basic problem are adapted as little as possible so that the specific feasible alternative become an optimal solution. In addition, a solution stability set of these parameters was investigated to keep the new optimal solution (feasible one) is unchanged. The idea of this study based on using a tuning parameters λ∈Rm in the function of the objective and input parameters υ∈Rl in the set of constraint. The inverse parametric nonlinear cost transportation problem P(λ,υ), where the tuning parameters λ∈Rm in the objective function are tuned (adapted) as less as possible so that the specific feasible solution x∘ has been became the optimal ones for a certain values of υ∈Rl, then, a solution stability set of the parameters was investigated to keep the new optimal solution x∘ unchanged. The proposed method consists of three phases. Firstly, based on the optimality conditions, the parameter λ∈Rm are tuned as less as possible so that the initial feasible solution x∘ has been became new optimal solution. Secondly, using input parameters υ∈Rl resulting problem is reformulated in parametric form P(υ). Finally, based on the stability notions, the availability domain of the input parameters was detected to keep its optimal solution unchanged. Finally, to clarify the effectiveness of the proposed algorithm not only for the inverse transportation problems but also, for the nonlinear programming problems; numerical examples treating the inverse nonlinear programming problem and the inverse transportation problem of minimizing the nonlinear cost functions are presented.
References
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Journal ArticleDOI
01 Sep 2001
TL;DR: In inverse optimization problems defined as follows, it is proved that if the problemP is a linear programming problem, then its inverse problem under theL1 as well asL8 norm is also alinear programming problem and inverse versions ofP under the L1 andL8 norms are also polynomially solvable.
Abstract: In this paper, we study inverse optimization problems defined as follows. LetS denote the set of feasible solutions of an optimization problemP, letc be a specified cost vector, andx0 be a given feasible solution. The solutionx0 may or may not be an optimal solution ofP with respect to the cost vectorc. The inverse optimization problem is to perturb the cost vectorc tod so thatx0 is an optimal solution ofP with respect tod and ||d- c || p is minimum, where ||d- c || p is some selectedLp norm. In this paper, we consider the inverse linear programming problem underL1 norm (where ||d- c || p= ? j?Jw j|d j-c j|, withJ denoting the index set of variablesx jandw jdenoting the weight of the variablej) and underL8 norm (where||d- c || p= max j?J{w j|d j-c j|} ). We prove the following results: (i) If the problemP is a linear programming problem, then its inverse problem under theL1 as well asL8 norm is also a linear programming problem. (ii) If the problemP is a shortest path, assignment or minimum cut problem, then its inverse problem under theL1 norm and unit weights can be solved by solving a problem of the same kind. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iii) If the problemP is a minimum cost flow problem, then its inverse problem under theL1 norm and unit weights reduces to solving a unit-capacity minimum cost flow problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iv) If the problemP is a minimum cost flow problem, then its inverse problem under theL8 norm and unit weights reduces to solving a minimum mean cycle problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost-to-time ratio cycle problem. (v) If the problemP is polynomially solvable for linear cost functions, then inverse versions ofP under theL1 andL8 norms are also polynomially solvable.

481 citations

Journal ArticleDOI
TL;DR: An algorithm based on the Goldfarb-Idnani method for convex quadratic programming is proposed and analyzed for one of the instances of the inverse shortest paths problem in a graph.
Abstract: The inverse shortest paths problem in a graph is considered, that is, the problem of recovering the arc costs given some information about the shortest paths in the graph The problem is first motivated by some practical examples arising from applications An algorithm based on the Goldfarb-Idnani method for convex quadratic programming is then proposed and analyzed for one of the instances of the problem Preliminary numerical results are reported

297 citations

Journal ArticleDOI
TL;DR: In this paper, a method for solving general inverse LP problems including upper and lower bound constraints is suggested, based on the optimality conditions for LP problems, and when applied to inverse minimum cost flow problem or inverse assignment problem, they are able to obtain strongly polynomial algorithms.

142 citations

Journal ArticleDOI
Lizhi Wang1
TL;DR: Cutting plane algorithms for the inverse mixed integer linear programming problem (InvMILP) are presented, which is to minimally perturb the objective function of a mixedinteger linear program in order to make a given feasible solution optimal.

76 citations

Journal ArticleDOI
TL;DR: In this article, Zhang et al. considered the problem of adjusting the cost coefficients of a given LP problem as less as possible so that a known feasible solution becomes the optimal one.

75 citations