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interval linear programming problem with interval valued decision variables

OPSEARCH 2025年 1-28页

作者： Karmakar, Jewel Karmakar, Samiran Mahato, Sanat Kumar Bankura Zilla Saradamani Mahila Mahavidyapith Dept Math Bankura 722101 West Bengal India Bankura Sammilani Coll Dept Math Bankura 722102 West Bengal India Sidho Kanho Birsha Univ Dept Math Purulia 723104 West Bengal India

In this article, we have proposed a new interval-oriented technique for full interval linear programming problems. This kind of problem not only involves interval-valued model parameters but also the decision variables are anticipated as interval-valued variables. Consideration of decision variables in interval form is much more realistic for decision-making in uncertain or inexact environments. At first, the interval variables are mathematically signified in the light of real-life problems. The problems are reduced to non-interval (crisp) Bi-objective linear programming problems which, normally, have uncountable many Pareto optimal solutions. The main goal lying behind this interval-oriented solution methodology is to achieve a Pareto optimal solution that is highly acceptable because of accuracy, elapsed time, and computational effort. Finally, the method is applied to a good number of test problems most of those are cited from the existing literature and the results are compared with the same.

关键词： interval linear programming interval decision variable Multi-objective optimization Compromised solutions

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Optimizing the selection of the secret parameters for public key cryptosystems by using interval linear programming and fully fuzzy linear programming

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ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE 2025年 144卷

作者： Bas, Esra Istanbul Tech Univ Dept Ind Engn TR-34367 Istanbul Turkiye

One of the basic tasks in public key cryptosystems is to determine the secret parameters. This paper proposes multi-objective interval linear programming and multi-objective fully fuzzy linear programming (FFLP) models to determine the optimum interval numbers and triangular fuzzy numbers (TFNs) of the secret parameters in the public key cryptosystems. The public and private (secret) objective function(s) and constraints can be considered such that the optimum interval numbers and optimum TFNs for the secret parameters can be obtained from which the random selection of the secret parameters can be made for each algorithm's execution. Adapting the general model to the Rivest, Shamir, Adleman (RSA) public key cryptosystem is also illustrated. In the interval programming model for the RSA, the linear combination method combines multiple objective functions into a single objective function, and nonlinear constraints are linearized using the Taylor series expansion method. No multiple objective functions are obtained in the FFLP model of the RSA, but the nonlinear constraints are again linearized using the Taylor series expansion method. The numerical application for the RSA public key cryptosystem is made for both models by considering five cases and twenty-four subcases, and the results are analyzed. To the best of our knowledge, there is no study in the literature that considers interval linear programming and/or FFLP to obtain the optimum interval numbers and TFNs for the secret parameters of the public key cryptosystems. The proposed methodology is systematic and generic and can also be adapted for other public key cryptosystems easily.

关键词： Public key cryptosystem Rivest Shamir Adleman public key cryptosystem interval linear programming Fully fuzzy linear programming Multi-objective optimization

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interval linear programming under transformations: optimal solutions and optimal value range

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CENTRAL EUROPEAN JOURNAL OF OPERATIONS RESEARCH 2019年第3期27卷 601-614页

作者： Garajova, Elif Hladik, Milan Rada, Miroslav Charles Univ Prague Fac Math & Phys Dept Appl Math Malostranske Nam 25 Prague 11800 Czech Republic Univ Econ Dept Financial Accounting & Auditing Nam W Churchilla 4 Prague 13067 Czech Republic

interval linear programming provides a tool for solving real-world optimization problems under interval-valued uncertainty. Instead of approximating or estimating crisp input data, the coefficients of an interval program may perturb independently within the given lower and upper bounds. However, contrarily to classical linear programming, an interval program cannot always be converted into a desired form without affecting its properties, due to the so-called dependency problem. In this paper, we discuss the common transformations used in linear programming, such as imposing non-negativity on free variables or splitting equations into inequalities, and their effects on interval programs. Specifically, we examine changes in the set of all optimal solutions, optimal values and the optimal value range. Since some of the considered properties do not holds in the general case, we also study a special class of interval programs, in which uncertainty only affects the objective function and the right-hand-side vector. For this class, we obtain stronger results.

关键词： interval linear programming Optimal set Optimal value range Transformations

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interval linear programming model for long-term planning of vehicle recycling in the Republic of Serbia under uncertainty

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WASTE MANAGEMENT & RESEARCH 2015年第2期33卷 114-129页

作者： Simic, Vladimir Dimitrijevic, Branka Univ Belgrade Fac Transport & Traff Engn Belgrade 11000 Serbia

An interval linear programming approach is used to formulate and comprehensively test a model for optimal long-term planning of vehicle recycling in the Republic of Serbia. The proposed model is applied to a numerical case study: a 4-year planning horizon (2013-2016) is considered, three legislative cases and three scrap metal price trends are analysed, availability of final destinations for sorted waste flows is explored. Potential and applicability of the developed model are fully illustrated. Detailed insights on profitability and eco-efficiency of the projected contemporary equipped vehicle recycling factory are presented. The influences of the ordinance on the management of end-of-life vehicles in the Republic of Serbia on the vehicle hulks procuring, sorting generated material fractions, sorted waste allocation and sorted metals allocation decisions are thoroughly examined. The validity of the waste management strategy for the period 2010-2019 is tested. The formulated model can create optimal plans for procuring vehicle hulks, sorting generated material fractions, allocating sorted waste flows and allocating sorted metals. Obtained results are valuable for supporting the construction and/or modernisation process of a vehicle recycling system in the Republic of Serbia.

关键词： End-of-life vehicle recycling interval linear programming Republic of Serbia Serbian ELV Ordinance uncertainty

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A novel interval linear programming based on probabilistic dominance

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FUZZY SETS AND SYSTEMS 2022年 434卷 20-47页

作者： Qiu, Zhiping Xia, Haijun Beihang Univ Sch Aeronaut Sci & Engn Inst Solid Mech Beijing 100083 Peoples R China

This study investigates a novel interval linear programming based on probabilistic dominance. Firstly, the definition of interval linear programming is briefly reviewed. Then a new interval linear programming model is presented based on probabilistic domi-nance. The probabilistic dominance index treats the intervals as uniformly distributed variables and the interval inequality relation is further defined by probability. To deal with non-linearity in probabilistic dominance index, sequential quadratic programming is used to solve the problem and the performance measure approach is proposed to overcome the convergence difficulties. The deter-mination and sensitivity analysis of the target performance measure are discussed to assess the sequential quadratic programming algorithm. Meanwhile, the extension of the proposed method to fuzzy interval linear programming is discussed. Furthermore, the proposed method is applied to the design of the plane truss structure with interval parameters. Finally, the effectiveness and ratio-nality of the developed method are demonstrated by two mathematical examples and one interval parametric plane truss structure optimization example.(c) 2021 Elsevier B.V. All rights reserved.

关键词： interval linear programming Probabilistic dominance Sequential quadratic programming algorithm Performance measure approach interval parametric plane truss structure optimization

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Necessary and sufficient conditions of some strong optimal solutions to the interval linear programming

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linear ALGEBRA AND ITS APPLICATIONS 2013年第10期439卷 3241-3255页

作者： Li, Wei Luo, Jiajia Deng, Chongyang Hangzhou Dianzi Univ Inst Operat Res & Cybernet Hangzhou 310018 Peoples R China

This paper considers optimal solutions of general interval linear programming problems. Necessary and sufficient conditions of (A, b)-strong and (A, b, c)-strong optimal solutions to the interval linear programming with inequality constraints are proposed. The features of the proposed methods are illustrated by some examples. (C) 2013 Elsevier Inc. All rights reserved.

关键词： interval linear programming Strong feasible solutions Strong optimal solutions Tangent cone

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Solving interval linear programming problems with equality constraints using extended interval enclosure solutions

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SOFT COMPUTING 2019年第16期23卷 7439-7449页

作者： Tabar, Maryam Mohaghegh Keyanpour, Mohammad Lodwick, Weldon A. Univ Guilan Fac Sci Dept Math POB 41335-1914 Rasht Iran Univ Colorado Dept Math Campus Box 170POB 173364 Denver CO 80217 USA

This paper focuses on solving systems of interval linear equations and interval linear programming in a computationally efficient way. Since the computational complexity of most interval enclosure numerical methods is often prohibitive, a procedure to obtain a relaxation of the interval enclosure solution that is computationally tractable is proposed. We show that our approach unifies the four standard interval solutions-the weak, strong, control and tolerance solutions. The interval linear system methods require n2n linear solutions. However, in the case of linear programming problems, we show that this requires just two optimization problem of the size of the problem itself. Numerical examples illustrate our results.

关键词： interval linear system interval linear programming Constraint intervals Extended interval enclosure solutions

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Life cycle cost optimization of biofuel supply chains under uncertainties based on interval linear programming

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BIORESOURCE TECHNOLOGY 2015年 187卷 6-13页

作者： Ren, Jingzheng Dong, Liang Sun, Lu Goodsite, Michael Evan Tan, Shiyu Dong, Lichun Univ Southern Denmark Dept Technol & Innovat DK-5230 Odense M Denmark Chongqing Univ Sch Chem & Chem Engn Chongqing 400044 Peoples R China Univ Padua Dept Ind Engn CESQA Qual & Environm Res Ctr I-35131 Padua Italy Natl Inst Environm Studies Ctr Social & Environm Syst Res Tsukuba Ibaraki 3058506 Japan Chongqing Univ Minist Educ Key Lab Low Grade Energy Utilizat Technol & Syst Chongqing 400044 Peoples R China

The aim of this work was to develop a model for optimizing the life cycle cost of biofuel supply chain under uncertainties. Multiple agriculture zones, multiple transportation modes for the transport of grain and biofuel, multiple biofuel plants, and multiple market centers were considered in this model, and the price of the resources, the yield of grain and the market demands were regarded as interval numbers instead of constants. An interval linear programming was developed, and a method for solving interval linear programming was presented. An illustrative case was studied by the proposed model, and the results showed that the proposed model is feasible for designing biofuel supply chain under uncertainties. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Life cycle cost Bioethanol Supply chain Uncertainty interval linear programming

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Risk Explicit interval linear programming Model for Uncertainty-Based Nutrient-Reduction Optimization for the Lake Qionghai Watershed

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JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT-ASCE 2011年第1期137卷 83-91页

作者： Liu, Yong Zou, Rui Guo, Huaicheng Peking Univ Minist Educ Key Lab Water & Sediment Sci Coll Environm Sci & Engn Beijing 100871 Peoples R China Tetra Tech Inc Water Resource Ctr Fairfax VA 22030 USA

Water quality management is subject to large uncertainties due to inherent randomness in the natural system and vagueness in the decision-making process. For water quality management optimization models, this means that some model coefficients can be represented by probability distributions, while others can be expressed only by ranges. interval linear programming (ILP) and risk explicit interval linear programming (REILP) models for optimal load reduction at the watershed scale are developed for the management of Lake Qionghai Watershed, China. The optimal solution space of an ILP model is represented using intervals corresponding to the lower and upper bounds of each decision variable. The REILP model extends the ILP model through introducing a risk function and aspiration levels (lambda(pre)) into the model formulation. The REILP model is able to generate practical solutions and trade-offs through solving a series of submodels, minimizing the risk function under different aspiration levels. This is illustrated in the present study by solving 11 submodels corresponding to different aspiration levels. The results show that the ILP model suffers severe limitations in practical decision support, while the REILP model can generate solutions explicitly relating system performance to risk level. Weighing the optimal solutions and corresponding risk factors, decision makers can develop an efficient and practical implementation plan based directly on the REILP solution.

关键词： interval linear programming Risk explicit Load reduction Aspiration level Lake Qionghai Watershed

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

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OPTIMIZATION LETTERS 2019年第4期13卷 875-890页

作者： Rada, Miroslav Hladik, Milan Garajova, Elif Univ Econ Fac Finance & Accounting Dept Financial Accounting & Auditing W Churchills Sq 4 Prague 13067 Czech Republic Charles Univ Prague Fac Math & Phys Dept Appl Math Malostranske Nam 25 Prague 11800 Czech Republic

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. In particular, the proposed algorithm runs in polynomial time for interval linear programs with no equality constraints.

关键词： interval linear programming Weakly optimal solution Weak optimality testing

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