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Violation analysis on two-step method for interval linear programming

INFORMATION SCIENCES 2014年 281卷 85-96页

作者： Wang, Xiuquan Huang, Guohe Univ Regina Fac Engn & Appl Sci Regina SK S4S 0A2 Canada

In many real world problems, system parameters or model coefficients may be bounded between lower and upper bounds due to a variety of uncertainties. Over the past decades, intensive research efforts have focused on interval linear programming (ILP) to tackle such uncertainties. As one of the most popular methods for solving ILP problems, Two-Step Method (TSM, proposed by Huang et al. in 1995) allows uncertain information to be directly communicated into the optimization process and resulting solutions such that decision alternatives could be generated through the interpretation of interval solutions. However, part of optimum solution points obtained through TSM may go beyond the decision space in some cases. This phenomenon, referred to as solution violation, may mislead decision makers to unreasonable policies, plans, or strategies which play important roles in the social and economic development. Therefore, this study first investigates and identifies the essential cause for the solution violation of TSM. Following that, an improved solution method (namely, ITSM) is proposed to avoid resulting violation by introducing extra constraints in the solving process. A numeric example is then presented to demonstrate the effectiveness of ITSM in handling solution violation. (C) 2014 Elsevier Inc. All rights reserved.

关键词： interval linear programming Optimization TSM Uncertainty Violation analysis

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Numerical solutions comparison for interval linear programming problems based on coverage and validity rates

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APPLIED MATHEMATICAL MODELLING 2014年第3期38卷 1092-1100页

作者： Lu, H. W. Cao, M. F. Wang, Y. Fan, X. He, L. North China Elect Power Univ Resources & Environm Res Acad Beijing 102206 Peoples R China Chinese Acad Environm Planning Beijing 100012 Peoples R China

In this paper, two-step method (TSM), alternative solution method (SOM-2) and best-worst case (BWC) method are introduced to solve a type of interval linear programming (ILP) problem. To compare the performance of the methods, Monte Carlo simulation is also used to solve the same ILP problem, whose solutions are assumed to be real solutions. In the comparison, two scenarios corresponding with two assumptions of distribution functions are considered: (i) all the input parameters obey normal distribution;(ii) all the input parameters obey uniform distribution. Based on the simulation results, coverage rate (CR) and validity rate (VR) are proposed as new indicators to measure the quality of the numerical solutions obtained from the methods. Results from a numerical case study indicate that the TSM and SOM-2 solutions can cover the majority of valid values (CR > 50%, VR > 50%), compared to the conventional BWC method. In addition, from the point of CR, TSM is more applicable since the solutions of TSM can identify more feasible solutions. However, from the point of VR, SOM-2 is preferred since it can exclude more baseless solutions (this means more feasible solutions are contained in the SOM-2 solutions). In general, TSM would be preferred when only the range of the system objective needs to be determined, while SOM-2 would be much useful in identifying the effective values of the objective. (C) 2013 Published by Elsevier Inc.

关键词： interval linear programming Coverage rate Valid rate Uncertainty Numerical solution Monte Carlo simulation

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Improving the modified interval linear programming method by new techniques

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INFORMATION SCIENCES 2016年 339卷 224-236页

作者： Allandadi, Mehdi Nehi, Hasan Mishmast Ashayerinasab, Hasan Ali Javanmard, Moslem Univ Sistan & Baluchestan Fac Math Zahedan Iran

In this study, we consider interval linear programming (ILP) problems, which are used to deal with uncertainties resulting from the range of admissible values in problem coefficients. In most existing methods for solving ILP problems, a part of the solution region is not feasible. The solution set obtained through the modified ILP (MILP) method is completely feasible (i.e., it does not violate any constraints), but is not completely optimal (i.e., some points of the region are not optimal). In this paper, two new ILP methods and their sub-models are presented. These techniques improve the MILP method, giving a solution region that is not only completely feasible, but also completely optimal. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Feasibility interval linear programming MILP method Optimality Uncertainty

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Strong optimal solutions of interval linear programming

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linear ALGEBRA AND ITS APPLICATIONS 2013年第8期439卷 2479-2493页

作者： Luo, Jiajia Li, Wei Hangzhou Dianzi Univ Sch Sci Hangzhou 310018 Zhejiang Peoples R China

This paper considers optimal solutions of general interval linear programming problems. The new concepts of optimal solutions are introduced in a unified framework. Some existed optimal solution concepts of interval linear program such as weak and strong optimal solutions are special cases in this framework. Necessary and sufficient conditions for checking optimality are developed. Also, 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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Stochastic-based resource expansion planning for a grid-connected microgrid using interval linear programming

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ENERGY 2016年第0期113卷 776-787页

作者： Boloukat, Mohammad Hadi Shaban Foroud, Asghar Akbari Semnan Univ Fac Elect & Comp Engn Semnan Iran

This paper represents a stochastic approach for long-term optimal resource expansion planning of a grid connected microgrid (MG) containing different technologies as intermittent renewable energy resources, energy storage systems and thermal resources. Maximizing profit and reliability, along with minimizing investment and operation costs, are major objectives which have been considered in this model. Also, the impacts of intermittency and uncertainty in renewable energy resources were investigated. The interval linear programming (ILP) was applied for modelling inherent stochastic nature of the renewable energy resources. ILP presents some superiority in modelling of uncertainties in MG planning. The problem was formulated as a mixed-integer linear programming. It has been demonstrated previously that the benders decomposition (BD) served as an effective tool for solving such problems. BD divides the original problem into a master (investment) problem and operation and reliability subproblems. In this paper a multiperiod MG planning is presented, considering life time, maximum penetration limit of each technology, interest rate, capital recovery factor and investment fund. Real-time energy exchange with the utility is covered, with a consideration of variable tariffs at different load blocks. The presented approach can help MG planners to adopt best decision under various uncertainty levels based on their budgetary policies. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Benders decomposition Intermittent renewable energy resources interval linear programming Microgrid resource expansion planning Multiobjective planning

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

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EXPERT SYSTEMS WITH APPLICATIONS 2018年 93卷 39-49页

作者： Ashayerinasab, Hasan Ali Nehi, Hasan Mishmast Allandadi, Mehdi Univ Sistan & Baluchestan Math Fac Zahedan Iran

Based on the binding constraint indices of the optimal solution to the linear programming (LP) model, a feasible system of linear equations can be formed. Because an interval linear programming (ILP) model is the union of numerous LP models, an interval linear equations system (ILES) can be formed, which is the union of these conventional systems. Hence, a new algorithm is introduced in which an arbitrary characteristic model of the ILP model is chosen and solved. The set of indices of its binding constraints is then obtained. This set is used to form and solve an ILES using the enclosure method. If all the components of the interval solutions to this system are strictly non-negative, the optimal solution set (OSS) of the ILP model is determined as the subscription of the zone created by reversing the signs of the binding constraints of the worst model and the binding constraints of the best model. The solutions to several problems obtained by the new algorithm and a Monte Carlo simulation are compared. The proposed algorithm is applicable to large-scale problems. To this end, an ILP model with 270 constraints and 270 variables is solved. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Optimal solution set interval linear programming interval linear equations system Monte Carlo simulation

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Convex contractive interval linear programming for resources and environmental systems management

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STOCHASTIC ENVIRONMENTAL RESEARCH AND RISK ASSESSMENT 2017年第1期31卷 205-224页

作者： Cheng, Guanhui Huang, Guohe Dong, Cong Univ Regina Fac Engn & Appl Sci Regina SK S4S 0A2 Canada Univ Regina Inst Energy Environm & Sustainabil Res Regina SK S4S 0A2 Canada

It is likely that the most reliable estimation of system uncertainty in resources and environmental systems management (RESM) is a value range with an unknown distribution. Stochastic programming would be challenged by distortion of the original uncertain information through fabricating an inexistent probabilistic distribution function. Instead, interval linear programming (ILP), i.e. a synthesis of interval-set coefficients and the conventional linear programming, has been employed to identify the desired schemes for a number of RESM problems under interval uncertainty. However, its effectiveness is disabled by constraint violation which may lead to severe penalties on socio-economic or eco-environmental development. To mitigate such a challenge, a convex contractive interval linear programming (CCILP) approach is proposed in this study. It mainly consists of six modules: parameterizing an RESM problem as an ILP model, initializing a hyperrectangle decision space by two linear programming sub-models, revealing causes of constraint violation given a criterion, inferring feasibilities of potential solutions, finalizing a feasible hyperrectangle decision space by another linear programming sub-model, and supporting RESM of various complexities through alternative variants. A simple ILP model for RESM is introduced to demonstrate the procedures of CCILP and verify its advantages over existing ILP methods. The result indicates that CCILP is capable of robustly incorporating interval uncertainties into the optimization process, avoiding heavy computation burdens on complicated sub-models, eliminating occurrence of constraint violation, enabling provision of a hyperrectangle decision space, adapting to diverse system requirements, and increasing reliability of decision support for interval linear RESM problems.

关键词： Resources and environmental systems management interval uncertainty interval linear programming Constraint violation

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Checking weak optimality and strong boundedness in interval linear programming

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SOFT COMPUTING 2019年第9期23卷 2937-2945页

作者： Garajova, Elif Hladik, Milan Charles Univ Prague Fac Math & Phys Dept Appl Math Malostranske Nam 25 Prague 11800 Czech Republic

interval programming provides a mathematical tool for dealing with uncertainty in optimization problems. In this paper, we study two properties of interval linear programs: weak optimality and strong boundedness. The former property refers to the existence of a scenario possessing an optimal solution, or the problem of deciding non-emptiness of the optimal set. We investigate the problem from a complexity-theoretic point of view and prove that checking weak optimality is NP-hard for all types of programs, even if the variables are restricted to a single orthant. The property of strong boundedness implies that each feasible scenario of the program has a bounded objective function. It is co-NP-hard to decide for inequality-constrained interval linear programs. For this class of programs, we derive a sufficient and necessary condition for testing strong boundedness using the orthant decomposition method. We also discuss the open problem of checking strong boundedness of programs described by equations with nonnegative variables.

关键词： interval linear programming Weak optimality Strong boundedness Computational complexity

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On the optimal solution set in interval linear programming

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2019年第1期72卷 269-292页

作者： Garajova, Elif Hladik, Milan Charles Univ Prague Dept Appl Math Fac Math & Phys Malostranske Nam 25 CR-11800 Prague Czech Republic

Determining the set of all optimal solutions of a linear program with interval data is one of the most challenging problems discussed in interval optimization. In this paper, we study the topological and geometric properties of the optimal set and examine sufficient conditions for its closedness, boundedness, connectedness and convexity. We also prove that testing boundedness is co-NP-hard for inequality-constrained problems with free variables. Furthermore, we prove that computing the exact interval hull of the optimal set is NP-hard for linear programs with an interval right-hand-side vector. We then propose a new decomposition method for approximating the optimal solution set based on complementary slackness and show that the method provides the exact description of the optimal set for problems with afixed coefficient matrix. Finally, we conduct computational experiments to compare our method with the existing orthant decomposition method.

关键词： interval linear programming Optimal solution set Decomposition methods Topological properties

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Checking strong optimality of interval linear programming with inequality constraints and nonnegative constraints

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2014年 260卷 180-190页

作者： Luo, Jiajia Li, Wei Wang, Qin Hangzhou Dianzi Univ Sch Sci Hangzhou 310018 Zhejiang Peoples R China China Jiliang Univ Coll Sci Hangzhou 310018 Zhejiang Peoples R China

This paper considers optimal solutions of interval linear programming problems, in a unified framework. Necessary and sufficient conditions for checking (A, b)-strong optimality and (A, b, c)-strong optimality of interval linear programming with inequality constraints are developed. The features of the proposed methods are illustrated by some examples. (C) 2013 Elsevier B.V. All rights reserved.

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

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