Maximal and supremal tolerances in multiobjective linear programming

Let a multiobjective linear programming problem and any efficient solution be given. Tolerance analysis aims to compute interval tolerances for (possibly all) objective function coefficients such that the efficient solution remains efficient for any perturbation of the coefficients within the computed intervals. The known methods either yield tolerances that are not the maximal possible ones, or they consider perturbations of weights of the weighted sum scalarization only. We focus directly on perturbations of the objective function coefficients, which makes the approach independent on a scalarization technique used. In this paper, we propose a method for calculating the supremal tolerance (the maximal one need not exist). The main disadvantage of the method is the exponential running time in the worst case. Nevertheless, we show that the problem of determining the maximal/supremal tolerance is NP-hard, so an efficient (polynomial time) procedure is not likely to exist. We illustrate our approach on examples and present an application in transportation problems. Since the maximal tolerance may be small, we extend the notion to individual lower and upper tolerances for each objective function coefficient. An algorithm for computing maximal individual tolerances is proposed.     

Fuzzy goal programming for multiobjective transportation problems

Several fuzzy approaches can be considered for solving multiobjective transportation problem. This paper presents a fuzzy goal programming approach to determine an optimal compromise solution for the multiobjective transportation problem. We assume that each objective function has a fuzzy goal. Also we assign a special type of nonlinear (hyperbolic) membership function to each objective function to describe each fuzzy goal. The approach focuses on minimizing the negative deviation variables from 1 to obtain a compromise solution of the multiobjective transportation problem. We show that the proposed method and the fuzzy programming method are equivalent. In addition, the proposed approach can be applied to solve other multiobjective mathematical programming problems. A numerical example is given to illustrate the efficiency of the proposed approach.     

Multiobjective data envelopment analysis

Data envelopment analysis (DEA) is popularly used to evaluate relative efficiency among public or private firms. Most DEA models are established by individually maximizing each firm's efficiency according to its advantageous expectation by a ratio. Some scholars have pointed out the interesting relationship between the multiobjective linear programming (MOLP) problem and the DEA problem. They also introduced the common weight approach to DEA based on MOLP. This paper proposes a new linear programming problem for computing the efficiency of a decision-making unit (DMU). The proposed model differs from traditional and existing multiobjective DEA models in that its objective function is the difference between inputs and outputs instead of the outputs/inputs ratio. Then an MOLP problem, based on the introduced linear programming problem, is formulated for the computation of common weights for all DMUs. To be precise, the modified Chebychev distance and the ideal point of MOLP are used to generate common weights. The dual problem of this model is also investigated. Finally, this study presents an actual case study analysing R&D efficiency of 10 TFT-LCD companies in Taiwan to illustrate this new approach. Our model demonstrates better performance than the traditional DEA model as well as some of the most important existing multiobjective DEA models.     

军械物资供应系统中的多目标运输问题

建立了军械物资运输问题的模糊多目标线性规划模型，运用一种解模糊函数和一种基于线性隶属函数的模糊规划算法求其调和解。方法简便、有效，可为部队军械物资的运输供应高效化提供科学依据。     

A Solution Concept for Fuzzy Multiobjective Programming Problems Based on Convex Cones

A solution concept for fuzzy multiobjective programming problems based on ordering cones (convex cones) is proposed in this paper. The notions of ordering cones and partial orderings on a vector space are essentially equivalent. Therefore, the optimality notions in a real vector space can be elicited naturally by invoking a concept similar to that of the Pareto-optimal solution in vector optimization problems. We introduce a corresponding multiobjective programming problem and a weighting problem of the original fuzzy multiobjective programming problem using linear functionals so that the optimal solution of its corresponding weighting problem is also the Pareto-optimal solution of the original fuzzy multiobjective programming problem.     

Preference modelling by estimating local utility functions for multiobjective optimization

This paper is intended to design goal programming models for capturing the decision maker's (DM's) preference information and for supporting the search for the best compromise solutions in multiobjective optimization. At first, a linear goal programming model is built to estimate piecewise linear local utility functions based on pairwise comparisons of efficient solutions as well as objectives. The interactive step trade-off method (ISTM) is employed to generate a typical subset of efficient solutions of a multiobjective problem. Another general goal programming model is then constructed to embed the estimated utility functions in the original multiobjective problem for utility optimization using ordinary nonlinear programming algorithms. This technique, consisting of the ISTM method and the newly investigated search process, facilitates the identification and elimination of possible inconsistent information which may exist in the DM's preferences. It also provides various ways to carry out post-optimality analysis to test the robustness of the obtained best solutions. A modified nonlinear multiobjective management problem is taken as example to demonstrate the technique.     

On stability in multiobjective programming — A stochastic approach

In this paper we assume that a deterministic multiobjective programming problem is approximated by surrogate problems based on estimations for the objective functions and the constraints. Making use of a large deviations approach, we investigate the behaviour of the constraint sets, the sets of efficient points and the solution sets if the size of the underlying sample tends to infinity. The results are illustrated by applying them to stochastic programming with chance constraints, where (i) the distribution function of the random variable is estimated by the empirical distribution function, (ii) certain parameters have to be estimated.     

A multiobjective based approach for mathematical programs with linear flexible constraints

In this paper a mathematical problem with linear flexible constraints is considered. In order to solve the problem an approach is proposed based on multiobjective linear programming. Indeed, allowing violations for the constraints, and using multiobjective linear programming to minimize these violations, a subset of solution set which has less violations, namely efficiently feasible set, is obtained. Then, the corresponding objective function is optimized over efficiently feasible set in order to obtain an optimal solution. An application of the proposed approach in pattern classification is introduced.     

Fuzzy goal programming approach to multilevel programming problems

The purpose of this paper is to propose a procedure for solving multilevel programming problems in a large hierarchical decentralized organization through linear fuzzy goal programming approach. Here, the tolerance membership functions for the fuzzily described objectives of all levels as well as the control vectors of the higher level decision makers are defined by determining individual optimal solution of each of the level decision makers. Since the objectives are potentially conflicting in nature, a possible relaxation of the higher level decision is considered for avoiding decision deadlock. Then fuzzy goal programming approach is used for achieving highest degree of each of the membership goals by minimizing negative deviational variables. Sensitivity analysis with variation of tolerance values on decision vectors is performed to present how the solution is sensitive to the change of tolerance values. The efficiency of our concept is ascertained by comparing results with other fuzzy programming approaches.     

物流运输网络多目标最短路问题的模糊满意解

本文对物流运输网络多目标最短路问题进行了研究。提出了一种求解多目标最短路问题的目标集成方法和对集成后目标函数求解的扩展标号法。在将多目标转化为单目标时,综合考虑了每个目标的边缘评价和所有目标的整体评价因素,通过对每个目标的权重分配将决策者的偏好充分体现到决策过程中,采用广义的模糊目标集成算子形成了相应的折衷规划模型。最后,通过实例对本文所提方法进行了说明。     

New approach for nonseparable dynamic programming problems

A general class of nonseparable dynamic problems is studied in a dynamic programming framework by introducingkth-order separability. The solution approach uses multiobjective dynamic programming as a separation strategy forkth-order separable dynamic problems. The theoretical grounding on which the optimal solution of the original nonseparable dynamic problem can be attained by a noninferior solution of the corresponding multiobjective dynamic programming problem is established. The relationship between the overall optimal Lagrangian multipliers and the stage-optimal Lagrangian multipliers and the relationship between the overall weighting vector and the stage weighting vector are explored, providing the basis for identifying the optimal solution of the original nonseparable problem from among the set of noninferior solutions generated by the envelope approach.This work was supported in part by NSF Grant No. CES-86-17984. The authors appreciate the comments from Dr. V. Chankong and the editorial work by Mrs. V. Benade and Dr. S. Hitchcock.     

Multiobjective duality for convex ratios

In this paper we present a duality approach for a multiobjective fractional programming problem. The components of the vector objective function are particular ratios involving the square of a convex function and a positive concave function. Applying the Fenchel-Rockafellar duality theory for a scalar optimization problem associated to the multiobjective primal, a dual problem is derived. This scalar dual problem is formulated in terms of conjugate functions and its structure gives an idea about how to construct a multiobjective dual problem in a natural way. Weak and strong duality assertions are presented.     

A class of multiobjective linear programming models with random rough coefficients

In the present paper, we concentrate on dealing with a class of multiobjective programming problems with random rough coefficients. We first discuss how to turn a constrained model with random rough variables into crisp equivalent models. Then an interactive algorithm which is similar to the interactive fuzzy satisfying method is introduced to obtain the decision maker’s satisfying solution. In addition, the technique of random rough simulation is applied to deal with general random rough objective functions and random rough constraints which are usually hard to convert into their crisp equivalents. Furthermore, combined with the techniques of random rough simulation, a genetic algorithm using the compromise approach is designed for solving a random rough multiobjective programming problem. Finally, illustrative examples are given in order to show the application of the proposed models and algorithms.     

Solving a multiobjective possibilistic problem through compromise programming

Real decision problems usually consider several objectives that have parameters which are often given by the decision maker in an imprecise way. It is possible to handle these kinds of problems through multiple criteria models in terms of possibility theory.Here we propose a method for solving these kinds of models through a fuzzy compromise programming approach.To formulate a fuzzy compromise programming problem from a possibilistic multiobjective linear programming problem the fuzzy ideal solution concept is introduced. This concept is based on soft preference and indifference relationships and on canonical representation of fuzzy numbers by means of their α-cuts. The accuracy between the ideal solution and the objective values is evaluated handling the fuzzy parameters through their expected intervals and a definition of discrepancy between intervals is introduced in our analysis.     

Solving a multiobjective traveling salesman problem by dynamic programming

In this paper Klötzler's method of multiobjective dynamic programming is applied to the solution of a two-dimensional traveling salesman problem. In this way Bellman's and Held/Karp's dynamic programming approach to one-dimensional traveling salesman problems is extended to the multiobjective case.     

Solving the multiobjective possibilistic linear programming problem

In conventional multiobjective decision making problems, the estimation of the parameters of the model is often a problematic task. Normally they are either given by the decision maker (DM), who has imprecise information and/or expresses his considerations subjectively, or by statistical inference from past data and their stability is doubtful. Therefore, it is reasonable to construct a model reflecting imprecise data or ambiguity in terms of fuzzy sets for which a lot of fuzzy approaches to multiobjective programming have been developed. In this paper we propose a method to solve a multiobjective linear programming problem involving fuzzy parameters (FP-MOLP), whose possibility distributions are given by fuzzy numbers, estimated from the information provided by the DM. As the parameters, intervening in the model, are fuzzy the solutions will be also fuzzy. We propose a new Pareto Optimal Solution concept for fuzzy multiobjective programming problems. It is based on the extension principle and the joint possibility distribution of the fuzzy parameters of the problem. The method relies on α-cuts of the fuzzy solution to generate its possibility distributions. These ideas are illustrated with a numerical example.     

The role of weights and scales in the application of multiobjective decision making

The application of multiobjective solution techniques requires, at some point in the analysis, the specification of the preference structure of the decision maker over the set of objectives. In practice, this step must often take the form of determining a set of weights that attempts to quantify the relative importance of the various competing objectives. This paper looks at how some of the well known techniques make use of these weights in attempting to achieve the best compromise solution. The definition of the weights (i.e., what meaning they carry into the analysis) is also explored. It is found that one must be very careful about applying weights when using these techniques. A different set of weights for the same objectives may apply when implementing two different techniques. In addition, a related factor, the specification of scales or ranges over which the objectives are to be evaluated, is explored. Again, it is found that different techniques use these scales in ways that change the manner in which they should be defined. The techniques reviewed are compromise programming, multiattribute utility theory, ELECTRE and cooperative game theory. Concepts are illustrated through the use of a water resources example problem.     

On the transformation of lexicographic nonlinear multiobjective programs to single objective programs

This paper deals with multiobjective optimization programs in which the objective functions are ordered by their degree of priority. A number of approaches have been proposed (and several implemented) for the solution of lexicographic (preemptive priority) multiobjective optimization programs. These approaches may be divided into two classes. The first encompasses the development of algorithms specifically designed to deal directly with the initial model. Considered only for linear multiobjective programs and multiobjective programs with a finite discrete feasible region, the second one attempts to transform, efficiently, the lexicographic multiobjective model into an equvivalent model, i.e. a single objective programming problem. In this paper, we deal with the second approach for lexicographic nonlinear multiobjective programs.     

Multiobjective management of mining under water hazard by game theory

The management of an aquifer is studied under the assumption that the solution of the multiobjective programming model describing the management problem should satisfy a certain set of axioms. It is shown that a certain class of multiobjective problems may be solved by a game-theoretical concept leading to a single objective quasiconvex programming problem. The method is generalization of Nash's cooperative game theoretical model, and may lean on Zeuther's bargaining process. The methodology is applied to the Transdanubian Karstic region in Hungary where three objectives are present: mining costs, water supply and environmental protection. Results are compared with the solution previously obtained by compromise programming with an l₁-norm. It is found that results obtained by the two methods are comparable.     

求群体多目标决策问题的一种可行方向交互算法

对于群体多目标决策问题,决策者可以各自的关于目标之间的权衡比表达其偏爱信息并进行决策.当个体权衡比具有加性性质时可得群体权衡比.本文以此构造一种求解群体非线性规划问题的交互算法.迭代中基于求解决非线性规划的Topkis-Veinott方法构造可行方向.在一定的条件下,算法收敛于所讨论问题的群体满意解.