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Note on "A new equivalent transformation for interval inequality constraints of interval linear programming"

FUZZY OPTIMIZATION AND DECISION MAKING 2019年第2期18卷 259-262页

作者： Li, Dechao Shi, Jiaheng Huang, Hongying Zhejiang Ocean Univ Sch Math Phys & Informat Sci Zhoushan 316000 Peoples R China Key Lab Oceanog Big Data Min & Applicat Zhejiang Zhoushan 316022 Peoples R China

This note provides a counterexample to demonstrate some flaws in a recent paper by Chen et al. It pinpoints the logic error in the proof of Theorem 5.2 and discusses some remedial works.

关键词： interval linear programming Modeling the interval uncertainty Uniform distribution Stochastic linear programming

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The optimal solution set of the interval linear programming problems

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OPTIMIZATION LETTERS 2013年第8期7卷 1893-1911页

作者： Allahdadi, M. Nehi, H. Mishmast Univ Sistan & Baluchestan Fac Math Zahedan Iran

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 a parts per thousand currency sign lambda (j) , mu (ij) , mu (i) a parts per thousand currency sign 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, mu (ij) = mu (i) = lambda (j) = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of lambda (j) , mu (ij) and mu (i) from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.

关键词： Convex combination problem interval linear programming interval equations system Solution set

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Applying Hybrid interval linear programming and Genetic Algorithm to Coordinate Distance and Directional Over-current Relays

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ELECTRIC POWER COMPONENTS AND SYSTEMS 2016年第17期44卷 1935-1946页

作者： Damchi, Yaser Sadeh, Javad Mashhadi, Habib Rajabi Ferdowsi Univ Mashhad Dept Elect Engn Fac Engn Vakil Abad AveBahonar St Mashhad *** Iran

interval linear programming (ILP) is a powerful tool for modeling problems with uncertain and bounded parameters. In real power systems, planned/unplanned events lead to changes in network topology. The changes cause altered distribution and magnitude of fault currents. Since protective relays are usually coordinated based on the fault currents of the main topology, mal-operation of the relays in such situations is likely. By covering the uncertainties, this paper aims to achieve the robust coordination of distance and directional overcurrent relays (D&DOCRs), as the main protections of transmission/subtransmission systems. Generally, relay coordination is a complicated nonlinear optimization problem. The complexity increases drastically by increasing coordination constraints, due to considering uncertainties. In this study, the problem is modeled as an ILP problem. The salient advantage of using ILP is that the number of constraints remains the same as in the main network topology, which greatly improves solving performance. Finally, to overcome the nonlinearity of the problem, a hybrid genetic algorithm and ILP (HGA/ILP), as a new optimization algorithm, is proposed. The proposed approach is applied to the IEEE 14-bus test system, and the results show that ILP is a useful tool to obtain robust settings of D&DOCRs in a power system.

关键词： interval linear programming uncertainty relay coordination change in network topology directional over-current relay distance relay genetic algorithm

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Weak optimal inverse problems of interval linear programming based on KKT conditions

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Applied Mathematics(A Journal of Chinese Universities) 2021年第3期36卷 462-474页

作者： LIU Xiao JIANG Tao LI Hao-hao School of Statistics and Mathematics Zhejiang Gongshang UniversityHangzhou 310018China Hangzhou College of Commerce Zhejiang Gongshang UniversityHangzhou 310018China School of Data Sciences Zhejiang University of Finance and EconomicsHangzhou 310018China.

In this paper,weak optimal inverse problems of interval linear programming(IvLP)are studied based on KKT ***,the problem is precisely ***,by adjusting the minimum change of the current cost coefficient,a given weak solution can become ***,an equivalent characterization of weak optimal inverse IvLP problems is ***,the problem is simplified without adjusting the cost coefficient of null variable.

关键词： interval linear programming inverse problems KKT conditions weak optimal solution

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Some new results on rough interval linear programming problems and their application to scheduling and fixed-charge transportation problems

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RAIRO-OPERATIONS RESEARCH 2024年第5期58卷 3697-3714页

作者： Allahdadi, Mehdi Rivaz, Sanaz Univ Sistan & Baluchestan Math Fac Zahedan Iran Babol Noshirvani Univ Technol Fac Basic Sci Dept Math Babol Iran

This paper focuses on linear programming problems in a rough interval environment. By introducing four linear programming problems, an attempt is being made to propose some results on optimal value of a linear programming problem with rough interval parameters. To obtain optimal solutions of a linear programming problem with rough interval data, constraints of the four proposed linear problems are applied. In this regard, firstly, the largest and the smallest feasible spaces for a linear constraint set with rough interval coefficients and parameters are introduced. Then, a rough interval for optimal value of such problems is obtained. Further, an upper approximation interval and a lower approximation interval as the optimal solutions of linear programming problems with rough interval parameters are achieved. Moreover, two solution concepts, surely and possibly solutions, are defined. Some numerical examples demonstrate the validity of the results. In particular, a scheduling problem and a fixed-charge transportation problem (FCTP) under rough interval uncertainty are investigated.

关键词： interval linear programming rough interval optimal solution optimal value

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Bounds on the worst optimal value in interval linear programming

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SOFT COMPUTING 2019年第21期23卷 11055-11061页

作者： Mohammadi, Mohsen Gentili, Monica Univ Louisville Dept Ind Engn 132 Eastern Pkwy Louisville KY 40292 USA Univ Salerno Dept Math Giovanni Paolo II 132 I-84084 Fisciano SA Italy

One of the basic tools to describe uncertainty in a linear programming model is interval linear programming, where parameters are assumed to vary within a priori known intervals. One of the main topics addressed in this context is determining the optimal value range, that is, the best and the worst of all the optimal values of the objective function among all the realizations of the uncertain parameters. For the equality constraint problems, computing the best optimal value is an easy task, but the worst optimal value calculation is known to be NP-hard. In this study, we propose new methods to determine bounds for the worst optimal value, and we evaluate them on a set of randomly generated instances.

关键词： interval linear programming Worst optimal value Bounds

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Solving methods for interval linear programming problem: a review and an improved method

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OPERATIONAL RESEARCH 2020年第3期20卷 1205-1229页

作者： Mishmast Nehi, H. Ashayerinasab, H. A. Allahdadi, M. Univ Sistan & Baluchestan Fac Math Zahedan Iran

interval linear programming is used for tackling interval uncertainties in real-world systems. An arbitrary point is a feasible point to the interval linear programming model if it lies in the largest feasible region of the interval linear programming model, and it is optimal if it is an optimal solution to a characteristic model. The optimal solution set to the interval linear programming is the union of all solutions that are optimal for a characteristic model. In this paper, we review some existing methods for solving interval linear programming problems. Using these methods the interval linear programming model is transformed into two sub-models. The optimal solutions of these sub-models form the solution space of these solving methods. A part of the solution space of some of these methods may be infeasible. To eliminate the infeasible part of the solution space of above methods, several methods have been proposed. The solution space of these modified methods may contain non-optimal solutions. Two improvement methods have been proposed to remove the non-optimal solutions of the solution space of above modified methods. Finally, we introduce an improved method and its sub-models. The solution space of our method is absolutely both feasible and optimal.

关键词： Feasibility Optimality interval linear programming Stability Optimal solution set

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Duality Gap in interval linear programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2020年第2期184卷 565-580页

作者： Novotna, Jana Hladik, Milan Masarik, Tomas Charles Univ Prague Fac Math & Phys Prague Czech Republic

This paper deals with the problem of linear programming with inexact data represented by real intervals. We introduce the concept of duality gap to interval linear programming. We give characterizations of strongly and weakly zero duality gap in interval linear programming and its special case where the matrix of coefficients is real. We show computational complexity of testing weakly- and strongly zero duality gap for commonly used types of interval linear programming.

关键词： interval analysis linear programming interval linear programming Duality gap Computational complexity

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An Analysis to Treat the Degeneracy of a Basic Feasible Solution in interval linear programming 9th

An Analysis to Treat the Degeneracy of a Basic Feasible Solu...

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9th International Symposium on Integrated Uncertainty in Knowledge Modelling and Decision Making (IUKM)

作者： Gao, Zhenzhong Inuiguchi, Masahiro Osaka Univ Toyonaka Osaka 5608531 Japan

ISBN: (纸本)9783030980184;9783030980177

When coefficients in the objective function cannot be precisely determined, the optimal solution is fluctuated by the realisation of coefficients. Therefore, analysing the stability of an optimal solution becomes essential. Although the robustness analysis of an optimal basic solution has been developed successfully so far, it becomes complex when the solution contains degeneracy. This study is devoted to overcoming the difficulty caused by the degeneracy in a linear programming problem with interval objective coefficients. We focus on the tangent cone of a degenerate basic feasible solution since the belongingness of the objective coefficient vector to its associated normal cone assures the solution's optimality. We decompose the normal cone by its associated tangent cone to a direct union of subspaces. Several propositions related to the proposed approach are given. To demonstrate the significance of the decomposition, we consider the case where the dimension of the subspace is one. We examine the obtained propositions by numerical examples with comparisons to the conventional techniques.

关键词： interval linear programming Degeneracy Polyhedral convex cone Tangent cone Basic space

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A Hybrid GA - interval linear programming Implementation for Microgrid Relay Coordination Considering Different Fault Locations 7

A Hybrid GA - Interval Linear Programming Implementation for...

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7th International Conference on Power Systems (ICPS)

作者： Srinivas, S. T. P. Swarup, K. Shanti IIT Madras Dept Elect Engn Madras Tamil Nadu India

ISBN: (纸本)9781538617892

This paper proposes hybrid Genetic Algorithm (GA) - interval linear programming (ILP) approach to optimal relay coordination problem for microgrids. Relay coordination in microgrids is complex because of varied and bidirectional fault currents because of the existence of Distributed Generation (DG). Overcurrent relays are the feasible and economic choice of protection for meshed distribution systems. The coordinated relay settings must account for various possible fault scenarios in both grid connected and isolated microgrid modes of operation. Inadequate fault current levels from grid connected to the isolated mode are the major cause of protection miscoordination. This paper systematically formulates the relay coordination problem for microgrids as a linear interval optimization problem and introduces a new method of solution using hybrid GA ILP method. The challenge in using GA to find the optimum in less number of iterations and the feasibility of ILP method to include various fault scenarios as "uncertain but bounded" intervals are combined to find the optimal overcurrent relay settings. The results show the effectiveness of proposed method. The programming

关键词： Terms Optimal relay coordination microgrids genetic algorithm interval linear programming directional overcurrent relays nonlinear optimization

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