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GOAL programming-PROBLEMS WITH interval-COEFFICIENTS AND TARGET intervalS

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1991年第3期52卷 345-360页

作者： INUIGUCHI, M KUME, Y UNIV OSAKA PREFECTURE COLL ENGNDEPT IND ENGN804 MOZU UMEMACHI 4 CHOSAKAIOSAKA 591JAPAN

In conventional goal programming, the coefficients of objective functions and constraints, and target values are determined as crisp values. However, it is not frequent that the coefficients and the target values are known precisely. In such cases, the coefficients and target values should be represented by intervals reflecting the imprecision. This paper treats goal programming problems in which coefficients and target values are given by intervals. It is shown that four formulations of the problems can be considered. The properties of the four formulated problems are investigated. An example is given to demonstrate the differences between the four formulations.

关键词： interval programming GOAL programming OPTIMIZATION

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A fuzzy dependent-chance interval multi-objective stochastic expected value programming approach for irrigation water resources management under uncertainty

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DESALINATION AND WATER TREATMENT 2021年 212卷 17-30页

作者： Wang, Youzhi Li, Zhong Liu, Liu Guo, Ping China Agr Univ Ctr Agr Water Res China Beijing 100083 Peoples R China McMaster Univ Dept Civil Engn Hamilton ON L8S 4L8 Canada Minist Agr & Rural Affairs Wuwei Expt Stn Efficient Water Use Agr Wuwei 733000 Peoples R China

In this study, a fuzzy dependent-chance interval multi-objective stochastic expected value programming model is developed for irrigation water resources management under uncertainties. It incorporates fuzzy dependent-chance programming, stochastic expected value programming, interval programming into multi-objective programming. Compared with conventional programming methods, it can quantify the relationship between the expected values of stochastic variables and the fuzzy goals of expected values set by decision-makers through the satisfactory degrees, and trade-off the relationship amid multiple satisfactory degrees selected as objective functions. Besides, it can cope with uncertainties expressed as interval numbers, fuzzy numbers, and stochastic variables. Moreover, the fairness of water allocation constraints formulated by the GINI coefficient can achieve the interactions between fair water allocation and satisfactory degrees. The model is applied to a real case study of irrigation water resources management of different water types (i.e., surface water and groundwater) under different water flow levels (high, medium, and low flow levels) in the midstream region of the Heihe River basin, northwest China. The results reveal that: (1) maximum water demands of wheat and economic crop are satisfied while that of corn is not met under three flow levels;(2) the expected economic benefit and water shortages of crops have positive relationships with water allocation while the expected canal water loss has a negative relationship with water allocation;(3) the bigger expected economic benefit results in the higher satisfactory degree of the expected economic benefit while the lower expected water shortage and canal water loss lead to higher satisfactory degrees of expected water shortage and canal water loss. It shows that the developed model can overcome the disadvantages of the single-objective programming of putting attention to the satisfactory degree of a kind of ex

关键词： Irrigation water allocation interval programming Stochastic expected value programming Fuzzy dependent-chance programming Multi-objective programming Uncertainty

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MINIMAX REGRET SOLUTION TO LINEAR-programming PROBLEMS WITH AN interval OBJECTIVE FUNCTION

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1995年第3期86卷 526-536页

作者： INUIGUCHI, M SAKAWA, M HIROSHIMA UNIV FAC ENGNDEPT IND & SYST ENGN4-1 KAGAMIYAMA 1 CHOMEHIGASHIHIROSHIMA 724JAPAN

This paper deals with a linear programming problem with interval objective function coefficients. A new treatment of an interval objective function is presented by introducing the minimax regret criterion as used in decision theory. The properties of minimax regret solution and the relations with possibly and necessarily optimal solutions are investigated. Next, the minimax regret criterion is applied to the final determination of the solution when a reference solution set is given. A method of solution by a relaxation procedure is proposed. The solution is obtained by repetitional use of the simplex method. The minimax regret solution is obtained by the proposed solution method when the reference solution set is the set of possibly optimal solutions. In order to illustrate the proposed solution method, a numerical example is given.

关键词： interval programming LINEAR programming OPTIMIZATION

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A Goal programming Procedure for Solving interval Valued Multiobjective Fractional programming Problems

A Goal Programming Procedure for Solving Interval Valued Mul...

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16th International Conference on Advanced Computers and Communications

作者： Pal, Bijay Baran Sen, Shyamal Univ Kalyani Dept Math Kalyani 741235 W Bengal India BKC Coll Dept Math Kolkata 700108 W Bengal India

ISBN: (纸本)9781424429622

This article presents a goal programming (GP) procedure for solving interval valued multiobjective fractional programming problems (MOFPPs) with interval objective functions in an inexact environment. In the proposed approach, the interval objective functions are first converted into the standard objective goals in the fractional GP formulation by using the interval arithmetic technique. Then, in the decision process, the fractional goals are transformed into the linear goals by linearization approach [31] studied previously. In solution process, the executable GP model of the problem is formulated with the objective to minimize the regret with the view to achieve the goals in their specified ranges and thereby arriving at a most satisfactory solution in the decision making environment. Two numerical examples are solved to illustrate the proposed approach and the model solution of one problem is compared with the solution of a fuzzy programming approach [28] studied previously.

关键词： Fractional programming Fuzzy programming Fuzzy Goal programming interval programming interval Arithmetic

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Evolutionary technique based goal programming approach to chance constrained interval valued bilevel programming problems

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OPSEARCH 2016年第2期53卷 390-408页

作者： Chakraborti, Debjani Narula Inst Technol Dept Math 15 Nilgunj Rd2nd FloorPO Sodepur Kolkata 700110 W Bengal India

The real world multiobjective decision environment involves great complexity and uncertainity. Many decision making problems often need to be modelled as a class of bilevel programming problems with inexact coefficients and chance constraints. To deal with these problems, a genetic algorithm (GA) based goal programming (GP) procedure for solving interval valued bilevel programming (BLP) problems in a large hierarchical decision making and planning organization is proposed. In the model formulation of the problem the chance constraints are converted to their deterministic equivalent using the notion of mean and variance. Further, the individual best and least solutions of the objectives of the decision makers (DMs) located at different hierarchical levels are determined by using GA method. The target intervals for achievement of each of the objectives as well as the target interval of the decision vector controlled by the upper-level DM are defined. Then, using interval arithmetic technique the interval valued objectives and control vectors are transformed into the conventional form of goal by introducing under-and over-deviational variables to each of them. In the solution process, both the aspects of minsum and minmax GP formulations are adopted to minimize the lower bounds of the regret intervals for goal achievement within the specified interval from the optimistic point of view. The potential use of the approach is illustrated by a numerical example

关键词： Bilevel programming Chance constrained programming Genetic algorithm Goal programming interval programming interval valued bilevel programming

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Some results in interval multiobjective linear programming for recognizing different solutions

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OPSEARCH 2015年第1期52卷 75-85页

作者： Rivaz, S. Yaghoobi, M. Shahid Bahonar Univ Kerman Fac Math & Comp Dept Math Kerman Iran

Based on the concepts of efficiency and weak efficiency, different solutions are defined to multiobjective linear programming problems with interval objective functions coefficients. This paper introduces a new solution concept namely necessarily weak efficient. Moreover, we propose some new results for recognizing different kinds of solutions by using some linear and nonlinear programming models. To illustrate the results, a numerical example is given.

关键词： Multiobjective linear programming interval programming Efficiency Weak efficiency

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A Genetic Algorithm Based Goal programming Method for Solving Patrol Manpower Deployment Planning Problems with interval-Valued Resource Goals in Traffic Management System: A Case Study

A Genetic Algorithm Based Goal Programming Method for Solvin...

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1st International Conference on Advanced Computing

作者： Pal, Bijay Baran Chakraborti, Debjani Biswas, Papun Univ Kalyani Dept Math Kalyani 741235 W Bengal India

ISBN: (纸本)9781424447862

This article demonstrates how the genetic algorithm (GA) method can be used to solve interval-valued goal programming (GP) model of patrol manpower allocation problem to various road-segment areas in different shifting times of Metropolitan cities to deter traffic violations and accidents. In the model formulation of the problem, the goals with target intervals are first converted into the standard goals in GP approach by using interval arithmetic technique. Then, the defined goals are transformed into the conventional form of goals by introducing under- and over-deviational variables to each of them to make a reasonable balance of decision in the deployment planning context. In the achievement function of the executable GP model, both the minsum and minmax aspects of GP are addressed to construct the achievement function for minimizing the possible regret towards achieving the goal whiles from the optimistic point of view in the decision making environment. A demonstrative example of the city Kolkata, West Bengal, India is solved and the model solution is compared with the solution of conventional GP approach [1] studied previously.

关键词： Genetic algorithm Goal programming interval arithmetic interval programming Patrol manpower deployment

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Extended Goal programming Approach with interval Data Uncertainty for Resource Allocation in Farm Planning: A Case Study

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48th Annual Convention of Computer Society of India (CSI)

作者： Pal, Bijay Baran Kumar, Mousumi Univ Kalyani Dept Math Kalyani 741235 W Bengal India Alipurduar Coll Dept Math W Bengal 736122 India

ISBN: (纸本)9783319031071

This paper presents an extended version of goal programming (GP) approach for modeling and solving farm planning problems having objectives with interval parameter sets by utilizing farming resources in the planning horizon. In the model formulation of the problem, the defined goals with interval parameters are converted into conventional goals by using interval arithmetic technique in interval programming and introducing under- and over-deviational variables to each of them. In the decision process, extended GP (EGP) approach, i.e. convex combination of both the modelling aspects, minsum GP and minmax GP are addressed in the achievement function for minimizing the possible regret towards goal achievement from the optimistic point of view in the inexact decision making environment. The potential use of the approach is demonstrated via a case example.

关键词： Farm Planning Goal programming interval arithmetic interval programming

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Solution of Nonlinear programming Problem with interval Grey Parameters

Solution of Nonlinear Programming Problem with Interval Grey...

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Proceedings of 2015 IEEE International Conference on Grey Systems and Intelligent Services (GSIS)

作者： Mao, Shu-Hua Zhu, Min Wuhan Univ Technol Coll Sci Wuhan Hubei Peoples R China Wuhan Univ Technol Reliabil Engn Inst Wuhan Hubei Peoples R China

ISBN: (纸本)9781479983759

We considered the nonlinear programming problems with interval grey number in the objective function when the distribution of grey number is known, and when it is unknown, according to historical data and related information of parameters, and combining with statistical knowledge, analyzed the solving method of that programming problem. When the probability distribution was known, we established the interval programming model for different instances whose objective function with grey numbers, and used the distribution information of data and historical information, adopted classic statistical method, Bayesian statistical method and minimized the posterior risk to estimate grey number, then transformed the interval programming model into a general programming model, and this solved the uncertain model.

关键词： interval grey number Bayesian statistical analysis Time series forecasting interval programming

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A Goal programming approach for solving interval valued Multiobjective Fractional programming problems using Genetic Algorithm

A Goal Programming approach for solving Interval valued Mult...

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IEEE Region 10 Colloquium/3rd International Conference on Industrial and Information Systems

作者： Pal, Bijay Baran Gupta, Somsubhra Univ Kalyani Dept Math Kalyani 741235 West Bengal India JIS Coll Engn Dept Informat Technol Kalyani West Bengal India

ISBN: (纸本)9781424428052

In this article, the efficient use of a genetic algorithm (GA) to the goal programming (GP) formulation of interval valued multiobjective fractional programming problems (MOFPPs) is presented. In the proposed approach, first the interval arithmetic technique ill is used to transform the fractional objectives with interval coefficients into the standard form of an interval programming problem with fractional criteria. Then, the redefined problem is converted into the conventional fractional goal objectives by using interval programming approach [2] and then introducing under-and over-deviational variables to each of the objectives. In the model formulation of the problem, both the aspects of GP methodologies, minsum GP and minimax GP [3] are taken into consideration to construct the interval function (achievement function) for accommodation within the ranges of the goal intervals specified in the decision situation where minimization of the regrets (deviations from the goal levels) to the extent possible within the decision environment is considered. In the solution process, instead of using conventional transformation approaches [4, 5, 6] to fractional programming, a GA approach is introduced directly into the GP framework of the proposed problem. In using the proposed GA, based on mechanism of natural selection and natural genetics, the conventional roulette wheel selection scheme and arithmetic crossover are used for achievement of the goal levels in the solution space specified in the decision environment. Here the chromosome representation of a candidate solution in the population of the GA method is encoded in binary form. Again, the interval function defined for the achievement of the fractional goal objectives is considered the fitness function in the reproduction process of the proposed GA. A numerical example is solved to illustrate the proposed approach and the model solution is compared with the solutions of the approaches [6, 7] studied previously.

关键词： genetic algorithm goal programming multiobjective fractional programming fuzzy goal programming interval programming

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