The optimal solution set of the interval linear programming problems

Allahdadi, Mehdi; Nehi, Hassan Mishmast

doi:10.1007/s11590-012-0530-4

Cited by 64 publications

(35 citation statements)

References 6 publications

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“…A survey on results can be found in [6]. Since that time, a partial characterization of the weakly optimal solution set was given in [2] and an inner approximation was considered in [1]. More general concepts of solutions, extending weak and strong solutions, were recently addressed in [10,12,13].…”

Section: Related Workmentioning

confidence: 99%

Testing weak optimality of a given solution in interval linear programming revisited: NP-hardness proof, algorithm and some polynomially-solvable cases

2018

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We address the problem of testing weak optimality of a given solution of a given interval linear program. The problem was recently wrongly stated to be polynomially solvable. We disprove it. We show that the problem is NP-hard in general. We propose a new algorithm for the problem, based on orthant decomposition and solving linear systems. Running time of the algorithm is exponential in the number of equality constraints. Interval linear programs with inequality constraints only can be processed in polynomial time.

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Section: Related Workmentioning

confidence: 99%

Testing weak optimality of a given solution in interval linear programming revisited: NP-hardness proof, algorithm and some polynomially-solvable cases

2018

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“…An interval arithmetic‐based LP approach is considered in Refs. and . The first research on taking into account real‐world soft constraints in the LP problem by using fuzzy sets was suggested in the famous work by Zadeh and Bellman .…”

Section: Introductionmentioning

confidence: 97%

Z-Number-Based Linear Programming

Aliev¹,

Alizadeh

Huseynov

et al. 2015

Int. J. Intell. Syst.

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Linear programming (LP) is the operations research technique frequently used in the fields of science, economics, business, management science, and engineering. Although it is investigated and applied for more than six decades, and LP models with different level of generalization of information about parameters including models with interval, fuzzy, generalized fuzzy, and random numbers are considered, until now there is no approach to account for reliability of information within the framework of LP. Professor L. Zadeh introduced the concept of a Z‐number to describe uncertain information, which is a more generalized notion closely related to reliability. The use of Z‐information is more adequate and intuitively meaningful for formalizing information structure of a decision problem. In this paper, we suggest a study of fully Z‐number based LP (Z‐LP) model to better fit real‐world problems within the framework of LP. We propose the method to solve Z‐LP problems, which utilize differential evolution optimization and Z‐number arithmetic developed by the authors. The suggested model and solution method for Z‐LP are illustrated on the basis of a benchmark LP problem, where we conduct comparative analysis, which shows validity of the approach.

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“…Over the past decades, interval mathematical programming methods were developed to tackle such uncertainties [4][5][6][7][8][9][10][11][12][13]. Many papers studied the problem of computing the range of optimal values of interval linear programming problems, see e.g., [7,10,[14][15][16][17] among others.…”

Section: Introductionmentioning

confidence: 99%

New method for computing the upper bound of optimal value in interval quadratic program

Xia

2015

Journal of Computational and Applied Mathematics

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Please cite this article as: W. Li, M. Xia, H. Li, New method for computing the upper bound of optimal value in interval quadratic program, Journal of Computational and Applied Mathematics (2015), http://dx.Abstract We consider the interval quadratic programming problems. The aim of this paper is to present a new method to compute the upper bound of the optimal values, under weaker conditions. Moreover, we discuss the relations between the new method and previous results. The features of the proposed methods are illustrated by some examples.

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The optimal solution set of the interval linear programming problems

Cited by 64 publications

References 6 publications

Testing weak optimality of a given solution in interval linear programming revisited: NP-hardness proof, algorithm and some polynomially-solvable cases

Testing weak optimality of a given solution in interval linear programming revisited: NP-hardness proof, algorithm and some polynomially-solvable cases

Z-Number-Based Linear Programming

New method for computing the upper bound of optimal value in interval quadratic program

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