Solving the interval linear programming problem: A new algorithm for a general case

Ashayerinasab, Hasan Ali; Nehi, Hassan Mishmast; Allahdadi, Mehdi

doi:10.1016/j.eswa.2017.10.020

Cited by 29 publications

(24 citation statements)

References 13 publications

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“…3) It can be followed that a IR 1 b and b IR 1 c, and also it is clear that a IR 1 c, which implies that IR 1 is transitive; the same is true for the other four relations. 4) Neither a IR 1 e nor e IR 1 a, holds, which means that a = [2,3] and e = [2.2, 2.4] cannot be compared by IR 1 , the same is true for the other four relations; as a result, all the five ordering relations are not totally ordering on I(R). Particularly, more solutions cannot be distinguished by relation IR 5 since its condition is very strong.…”

Section: Ordering Relations and Possibility Degrees Of Intervalsmentioning

confidence: 99%

“…In numerical optimization, the approaches for solving them can be grouped into three categories: satisficing approaches, optimizing approaches and minimax regret approaches. The satisficing approaches transform an interval programming model into one or more deterministic model(s) by using ordering relations or possibility degrees between intervals, and then solutions of the original model are evaluated according to the values of the transformed deterministic objective function(s) [1,2,6,9,10,21,33]. The optimizing approaches put forward the concepts of weak or strong optimal solutions for interval linear programming [19,26,30,36] or extend the concept of efficiency used in conventional multi-objective programming to interval environment [5,17,34].…”

Section: Introductionmentioning

confidence: 99%

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An ensemble framework for assessing solutions of interval programming problems

Sun

Gong

Zeng

et al. 2018

Information Sciences

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Interval programming is a commonly used technique in real-world situations. Its related theories and methods have been widely researched. There are a variety of approaches for assessing solutions of an interval programming problem due to the particularity of intervals. It is well-known that different assessing approaches may produce different optimal solution(s) for the same interval programming problem, and it is rather difficult to choose from these assessing approaches for users, especially for those who have little knowledge about interval arithmetic, which greatly restricts its extensive applications. In this paper, we develop an ensemble framework for assessing solutions of interval programming problems. At the start, interval dominance rules are defined, and their correlations are described via exclusion, inclusion and equivalence; then, a rule reduction strategy is developed through inspecting the impact of different rules on the sorting of solutions, and a novel ensemble dominance relation for interval programming is proposed to evaluate solutions; furthermore, their complexities are analyzed; finally, the experimental results empirically validate the correctness and effectiveness of the proposed framework.

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Section: Ordering Relations and Possibility Degrees Of Intervalsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An ensemble framework for assessing solutions of interval programming problems

Sun

Gong

Zeng

et al. 2018

Information Sciences

View full text Add to dashboard Cite

show abstract

“…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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“…Interval linear programming (ILP) is used to deal with uncertainties of many real-world problems, where parameters may be specified as lying between lower and upper bounds. There are several methods for solving ILP models, which are divided into two sub-models to obtain the solution set [2,3,4,10,12,14,24,26,27,28].…”

Section: Introductionmentioning

confidence: 99%

An improved three-step method for solving the interval linear programming problems

Allahdadi

Deng

2018

YUGOSLAV J OPERATION

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Feasibility condition, which ensures that the solution space does not violate any constraints, and optimality condition, which guarantees that all points of the solution space are optimal, are very significant conditions for the solution space of interval linear programming (ILP) problems. Among the existing methods for ILP problems, the best-worst cases (BWC) method and two-step method (TSM) do not ensure feasibility condition, while the modified ILP (MILP), robust TSM (RTSM), improved TSM (ITSM), and three-step method (ThSM) guarantee feasibility condition, whose solution spaces may not be completely optimal. We propose an improved ThSM (IThSM) for ILP problems, which ensures both feasibility and optimality conditions, i.e., we introduce an extra step to optimality.

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Solving the interval linear programming problem: A new algorithm for a general case

Cited by 29 publications

References 13 publications

An ensemble framework for assessing solutions of interval programming problems

An ensemble framework for assessing solutions of interval programming problems

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

An improved three-step method for solving the interval linear programming problems

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