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Two-phases method and Branch and Bound procedures to solve the bi-objective Knapsack problem

JOURNAL OF GLOBAL OPTIMIZATION 1998年第2期12卷 139-155页

作者： Visee, M Teghem, J Pirlot, M Ulungu, EL Fac Polytech Mons Lab Math & Operat Res B-7000 Mons Belgium

The classical 0-1 knapsack problem is considered with two objectives. Two methods of the "two-phases" type are developed to generate the set of efficient solutions. In the first phase, the set of supported efficient solutions is determined by optimizing a parameterized single-objective knapsack problem. Two versions are proposed for a second phase, determining the non-supported efficient solutions: both versions are Branch and Bound approaches, but one is "breadth first", while the other is "depth first". Extensive numerical experiments have been realized to compare the results of both methods.

关键词： combinatorial optimization multi-objective programming Knapsack problem

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SOLVING DISCRETE LINEAR FRACTIONAL BILEVEL PROGRAMS WITH multiPLE objectiveS AT THE UPPER LEVEL

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JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION 2023年第4期19卷 2639-2654页

作者： Fali, Fatima Moulai, Mustapha USTHB Dept Operat Res BP 32 El Alia Algiers 16111 Algeria

This paper proposes an exact method to solve an integer linear fractional bilevel problem with multiple objectives at the upper level, designated by IFMOBP. The proposed algorithm generates a set of efficient solutions using a branch and cut algorithm based on a continuous upper level linear fractional problem. Then, the integer optimal solution obtained is tested for optimality of the lower level problem. First, the integer optimal solution of the bilevel problem is sought with a single objective function at each level. After that, an efficient cut is added and new integer solutions are determined. The efficient set is updated each time a candidate bilevel feasible solution non dominated is got and the process ends when there are no unexplored parts of the original domain. The proposed method is based on a dantzig cut to find the next best integer solution of the first objective function of the upper level, an efficient cut to get the set of efficient solutions for the main problem, and the classical branch and bound technique for integer decision variables. After the presentation of the algorithm, a numerical example and computational experiments are provided.

关键词： multi-objective programming bilevel programming integer programming linear fractional programming branch and cut

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Steepest descent methods for multicriteria optimization

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MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2000年第3期51卷 479-494页

作者： Fliege, J Svaiter, BF Univ Dortmund Fachbereich Math D-44221 Dortmund Germany Inst Matemat Pura & Aplicada Jardim Bot BR-22460320 Rio De Janeiro Brazil

We propose a steepest descent method for unconstrained multicriteria optimization and a "feasible descent direction" method for the constrained case. In the unconstrained case, the objective functions are assumed to be continuously differentiable. In the constrained case, objective and constraint functions are assumed to be Lipshitz-continuously differentiable and a constraint qualification is assumed. Under these conditions, it is shown that these methods converge to a point satisfying certain first-order necessary conditions for Pareto optimality. Both methods do not scalarize the original vector optimization problem. Neither ordering information nor weighting factors for the different objective functions are assumed to be known. In the single objective case, we retrieve the Steepest descent method and Zoutendijk's method of feasible directions, respectively.

关键词： multicriteria optimization multi-objective programming vector optimization Pareto points steepest descent

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multi-objective optimization for a closed-loop network design problem using an improved genetic algorithm

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APPLIED MATHEMATICAL MODELLING 2017年 45卷 14-30页

作者： Shi, Jianmai Liu, Zhong Tang, Luohao Xiong, Jian Natl Univ Def Technol Coll Informat Syst & Management Changsha 410073 Hunan Peoples R China

This paper develops a multi-objective Mixed Integer programming model for a closed-loop network design problem. In addition to the overall costs, the model optimizes overall carbon emissions and the responsiveness of the network. An improved genetic algorithm based on the framework of NSGA II is developed to solve the problem and obtain Pareto-optimal solutions. An example with 95 cities in China is presented to illustrate the approach. Through randomly generated examples with different sizes;the computational performance of the proposed algorithm is also compared with former genetic algorithms in the literature employing the weight-sum technique as a fitness evaluation strategy. Computational results indicate that the proposed algorithm can obtain superior Pareto-optimal solutions. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Closed loop supply chain Carbon emission multi-objective programming Facility location Evaluation algorithm

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An improved bi-level programming model for water resources allocation under multiple uncertainties

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JOURNAL OF ENVIRONMENTAL MANAGEMENT 2023年第1期345卷 118882页

作者： Ren, Chongfeng Wang, Yashi Yu, Linghui Zhang, Hongbo Xie, Zhishuai Changan Univ Sch Water & Environm Xian Peoples R China Changan Univ Key Lab Ecohydrol & Water Secur Arid & Semiarid Re Minist Water Resources Xian Peoples R China Changan Univ Key Lab subsurface Hydrol & Ecol Effect Arid Reg Minist Educ Xian Peoples R China Changan Univ Sch Water & Environm Yantastr 126 Xian 710054 Shaanxi Peoples R China

Nowadays, with the scarcity of water resources, competition for water resources among different levels and water sectors is growing increasingly fierce. Furthermore, uncertainties are unavoidable in the water resources system. To address the aforementioned issues, a fuzzy max-min decision bi-level multi-objective interval programming model was proposed, which can not only focus on water conflicts at the same level or between different levels, but also pay attention to optimal allocation of water resources under uncertainty. The developed model was then applied to a case study in Wuwei City, Gansu Province, China, which selected fairness of water distribution and agricultural economic benefits as planning objectives. Based on the developed model, different water resources optimal allocation schemes under different representative hydrological years were provided. From the result, as representative hydrological years changed from wet (P = 25%) to dry (P = 75%), agricultural economic benefit and Gini coefficient of agriculture would vary from [35.19, 37.78] x 108 yuan to [31.12, 31.99] x 108 yuan and from [0.468, 0.429] to [0.505, 0.503], which indicates that as available water resources decrease, agricultural economic benefit would decrease and fairness of water distribution would also decrease. And the water distribution fairness of the upper bound water allocation scheme is higher than that of the lower bound water allocation scheme when in the same representative hydrological year. In addition, no matter what representative hydrological year, the results of the established bi-level programming model are always in the middle of the results of the upper and lower level individual objective, which means that the developed bi-level programming model has great advantage to deal with water competing conflict among different levels. Furthermore, based on the results of developed model, the reasonable water resources optimization schemes can be determined by the decision-ma

关键词： Water resources optimal allocation Bi-level programming multi-objective programming multiple uncertainties

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Trade-off between cost and variance for a multi-objective compromise allocation in stratified random sampling

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2017年第6期46卷 2655-2666页

作者： Shad, Muhamamd Yousaf Husain, Ijaz Quaid I Azam Univ Dept Stat Islamabad Pakistan

In the present paper, a multi-objective goal optimization mechanism is developed by trading off between cost and variance. Both are adversaries to each other while allocating a sample size even in stratified sampling design. Discussion section shows how these adversaries put their influence on optimal selection. This is a dual optimization procedure in which variance or mean square error is optimized in the first step and then considering some compromise on variance, cost is optimized. The process is applied to both individual and multi-objective programming models.

关键词： Compromise allocation lexicographic goal programming multi-objective programming multivariate stratified sampling non linear cost function

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A new method for the selection of distribution centre locations

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IMA JOURNAL OF MANAGEMENT MATHEMATICS 2017年第3期28卷 421-436页

作者： Zhou, Lei Zhang, Guoshan Liu, Wanquan Tianjin Univ Sch Elect Engn & Automat Tianjin 300072 Peoples R China Hebei Univ Sci & Technol Sch Econ & Management Shijiazhuang 050018 Peoples R China Curtin Univ Dept Comp Perth WA 6102 Australia

This paper investigates the distribution centre location problem with inaccurate information, and a general model based on a rough feasible region is established. By means of synthesizing the believable degree of the rough feasible region and objective functions, a solution model termed as rough multi-objective synthesis effect (RMOSE) model is developed;this constitutes a series of crisp multi-objective programming models that reflect different decision consciousness for each decision maker. The optimal solutions of the RMOSE model can be obtained by using the genetic algorithm, and it is demonstrated that the solution of the RMOSE model in proper parameters is same as that of existing model with fuzzy model information. So the proposed RMOSE model is actually an extension of a crisp multi-objective programming model. Two cases of experiments for the distribution centre location problems show that the proposed method can be directly applied to real world practices and it is better than existing methods with fuzzy model information.

关键词： distribution centre location multi-objective programming rough set genetic algorithm synthesis effect

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Optimal resource allocation in urban transportation networks considering capacity reliability and connectivity reliability: a multi-objective approach

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INTERNATIONAL JOURNAL OF CIVIL ENGINEERING 2013年第1期11卷 33-42页

作者： Mohaymany, A. Shariat Babaei, M. Iran Univ Sci & Technol Fac Civil Engn Tehran Iran

Since the 1990:9, network reliability has been considered as a new index for evaluating transportation networks under uncertainty. A large number of studies have been revealed in the literature in this field, which are mostly dedicated to developing relevant measures that can be utilized for the evaluation of vulnerable networks under different sources of uncertainty, such as daily traffic flow fluctuations, natural disasters, weather conditions, and so fourth. This paper addresses the resource allocation problem in vulnerable transportation networks, in which multiple performance reliability measures should be met at their desired levels, while the overall cost of upgrading links 'performances should be minimized simultaneously. For this purpose, a new approach has been considered to formulate the two well-known performance measures, connectivity and capacity reliability, along with their application in a bi-objective nonlinear mixed integer goal programming model. In order to take into account the uncertain conditions of supply, links' capacities have been assumed to be random variables and follow normal distribution functions. A computationally efficient method has been developed that allows calculating the network-wise performance indices simply by means of a set of functions of links' 'performance reliabilities. Using this approach, as the performance reliability of links are themselves functions of the random links' capacities, they can be simply calculated through numerical integration. To achieve desirable levels for both connectivity reliability and capacity reliability (as network-wise performance reliability measures) two distinct objectives have been considered. One of the objectives seeks to maximize each of the measures regardless of what is happening to the other objective function which minimizes the budget. Since optimization models with two conflicting objectives cannot be solved directly, the well-known goal attainment multi-objective decision-mak

关键词： Network design Resource allocation multi-objective programming Urban transportation networks Connectivity reliability Capacity reliability

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Pareto optimal solutions of a cost-time trade-off bulk transportation problem

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2008年第1期188卷 85-100页

作者： Prakash, Satya Kumar, Pranav Prasad, B. V. N. S. Gupta, Anuj Amity Sch Engn & Technol Dept Math New Delhi India Birla Inst Technol & Sci Dept Math Pilani Rajasthan India Birla Inst Technol & Sci Dept Comp Sci Pilani Rajasthan India Univ Cambridge Dept Chem Engn Cambridge CB2 3RA England

A cost-time trade-off bulk transportation problem with the objectives to minimize the total cost and duration of bulk transportation without according priorities to them is considered. The entire requirement of each destination is to be met from one source only;however a source can supply to any number of destinations subject to the availability of the commodity at it. Two new algorithms are provided to obtain the set of Pareto optimal solutions of this problem. This work extends and generalizes the work related to single-objective and prioritized two-objective bulk transportation problems done in the past while providing flexibility in decision making. (C) 2007 Elsevier B.V. All rights reserved.

关键词： optimization multi-objective programming Pareto optimal transportation bulk transportation

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A stochastic lot-sizing model with multi-supplier and quantity discounts

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INTERNATIONAL JOURNAL OF PRODUCTION RESEARCH 2013年第1期51卷 245-263页

作者： Kang, He-Yau Lee, Amy H. I. Chung Hua Univ Dept Technol Management Hsinchu Taiwan Natl Chin Yi Univ Technol Dept Ind Engn & Management Taipei Taiwan

This research considers a stochastic lot-sizing problem with multi-supplier and quantity discounts. The objectives are to minimise total costs, where the costs include ordering cost, holding cost, purchase cost and shortage cost, and to maximise service level of the system. In this paper, we first formulate the stochastic lot-sizing problem as a multi-objective programming (MOP) model. We then transform the model into a mixed integer programming (MIP) model. Finally, an efficient heuristic dynamic programming (HDP) model is constructed for solving large-scale stochastic lot-sizing problems. An illustrative example with two cases for a touch panel manufacturer is used to illustrate the practicality of these models, and a sensitivity analysis is applied to understand the impact of the changes in parameters to the outcomes. The results demonstrate that the proposed two models are effective and accurate tools for determining the replenishment of touch panels from multiple suppliers for multi-periods.

关键词： stochastic lot-sizing mixed integer programming heuristic dynamic programming multi-objective programming

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