Multiple Objective Linear Programming Research Articles

On a fuzzy set approach to solving multiple objective linear fractional programming problem

In 1984, Luhandjula used a linguistic variable approach in order to present a procedure for solving multiple objective linear fractional programming problem (MOLFPP). In 1992, Dutta et al. (Fuzzy Sets and Systems 52 (1) (1992) 39–45) modified the linguistic approach of Luhandjula such as to obtain efficient solution to problem MOLFPP. The aim of this paper is to point out certain shortcomings in the work of Dutta et al. and give the correct proof of theorem which validates the obtaining of the efficient solutions. We notice that the method presented there as a general one does only work efficiently if certain hypotheses (restrictive enough and hardly verified) are satisfied. The example considered by Dutta et al. is again used to illustrate the approach.

A weight space-based approach to fuzzy multiple-objective linear programming

In this paper, the effects of uncertainty on multiple-objective linear programming models are studied using the concepts of fuzzy set theory. The proposed interactive decision support system is based on the interactive exploration of the weight space. The comparative analysis of indifference regions on the various weight spaces (which vary according to intervals of values of the satisfaction degree of objective functions and constraints) enables to study the stability and evolution of the basis that correspond to the calculated efficient solutions with changes of some model parameters.

A weight set decomposition algorithm for finding all efficient extreme points in the outcome set of a multiple objective linear program

Various computational difficulties arise in using decision set-based vector maximization methods to solve multiple objective linear programming problems. As a result, several researchers have begun to explore the possibility of solving these problems by examining subsets of their outcome sets, rather than of their decision sets. In this article, we present and validate a basic weight set decomposition approach for generating the set of all efficient extreme points in the outcome set of a multiple objective linear program. Based upon this approach, we then develop an algorithm, called the Weight Set Decomposition Algorithm, for generating this set. A sample problem is solved using this algorithm, and the main potential computational and practical advantages of the algorithm are indicated.

ON THE PROJECTION OF THE EFFICIENT SET AND POTENTIAL APPLICATIONS*

We study the efficient set X E for a multiple objective linear program by using its projection into the linear space L spanned by the independent criteria. We show that in the orthogonally complementary space of L, the efficient points form a polyhedron, while in L an efficiency-equivalent polyhedron for the projection P(X E ) of X E can be constructed by algorithms of outer and inner approximation types. These algorithms can be also used for generating all extreme points of P(X E ). Application to optimization over the efficient set for a multiple objective linear program is considered.

Inverse radiation therapy planning — a multiple objective optimization approach

For some decades, radiation therapy has been proved successful in cancer treatment. It is the major task of clinical radiation treatment planning to realize, on the one hand, a high level dose of radiation in the cancer tissue in order to obtain maximum tumor control. On the other hand, it is obvious that it is absolutely necessary to keep the unavoidable radiation in the tissue outside the tumor, particularly in organs at risk, as low as possible. No doubt, these two objectives of treatment planning—high level dose in the tumor, low radiation outside the tumor—have a basically contradictory nature. Thus, there is need to compromise between overdosing the organs at risk and underdosing the target volume. Differing from the currently used time consuming interactive approach between dosimetrists and physicians, we consider the radiation therapy planning problem as a multiple objective linear programming problem and build a data base of relatively few efficient solutions representing the set of Pareto solutions. This data base can be easily scanned by physicians looking for an adequate treatment plan with the aid of an appropriate online tool. The paper includes a report on first numerical experiments and a list of further research topics which is hoped to stimulate interest of the OR community in a subject, where OR methods can make a difference for many individual's lives.

An Input-output Model for Decision Support in Energy-economy Planning - a Multiobjective Interactive Approach

In this paper an energy-economy planning model is developed, aimed at providing decision support to planners and decision makers in the study of the interactions between the energy system and the whole economy. For this purpose a multiple objective linear programming (MOLP) model is developed by using the coefficients of an input-output table, where the energy sectors as well as the relevant economic activity sectors are disaggregated. The results provided by the MOLP model with a set of data representative of the Portuguese situation are analyzed using the STEM interactive method.

On the Use of Value Efficiency Analysis and Some Further Developments

The purpose of this paper is to discuss the use of Value Efficiency Analysis (VEA) in efficiency evaluation when preference information is taken into account. Value efficiency analysis is an approach, which applies the ideas developed for Multiple Objective Linear Programming (MOLP) to Data Envelopment Analysis (DEA). Preference information is given through the desirable structure of input- and output-values. The same values can be used for all units under evaluation or the values can be specific for each unit. A decision-maker can specify the input- and output-values subjectively without any support or (s)he can use a multiple criteria support system to assist him/her to find those values on the efficient frontier. The underlying assumption is that the most preferred values maximize the decision-maker's implicitly known value function in a production possibility set or a subset. The purpose of value efficiency analysis is to estimate a need to increase outputs and/or decrease inputs for reaching the indifference contour of the value function at the optimum. In this paper, we briefly review the main ideas in value efficiency analysis and discuss practical aspects related to the use of value efficiency analysis. We also consider some extensions.

AGGREGATE PRODUCTION PLANNING WITH MULTIPLE FUZZY GOALS

ABSTRACT The purpose of this paper is to develop a multiple fuzzy objective linear programming model (MFOLP) for APP decision problem. This model contains multiple objective functions, more flexible decision procedure, and more extensible decision information. Because this model can also measures decision-maker's decision satisfaction degree level, so it will be more fit practice requirement than other APP decision model.

An interactive interior point algorithm for multiobjective linear programming problems

We propose an interactive interior point method for finding the best compromise solution to a multiple objective linear programming problem. We construct a sequence X 0, X 1,…, X k ,… of smaller and smaller polytopes which shrink towards the compromise solution. During the kth iteration, we move from the center of polytope X k−1 to the center of polytope X k by performing a local optimization which consists of maximizing a linear function over an ellipsoid.

Using objective values to start multiple objective linear programming algorithms

We introduce in this paper a new starting mechanism for multiple-objective linear programming (MOLP) algorithms. This makes it possible to start an algorithm from any solution in objective space. The original problem is first augmented in such a way that a given starting solution is feasible. The augmentation is explicitly or implicitly controlled by one parameter during the search process, which verifies the feasibility (efficiency) of the final solution. This starting mechanism can be applied either to traditional algorithms, which search the exterior of the constraint polytope, or to algorithms moving through the interior of the constraints. We provide recommendations on the suitability of an algorithm for the various locations of a starting point in objective space. Numerical considerations illustrate these ideas.

A COMPROMISE MODEL FOR SOLVING FUZZY MULTIPLE OBJECTIVE LINEAR PROGRAMMING PROBLEMS

ABSTRACT Two-phase approach had been proposed to generate an efficient solution for the multiple objective linear programming problems [MOLP]. In this research, we shall show a revised two-phase approach to the case of the fuzzy multiple objectives linear programming problems [FMOLP]. This revised model can improve the optimal decision obtained from min operator. Moreover, a compromise model embedded two-phase approach and average operator will be proposed to yield a fuzzy-efficient solution between non-compensatory and fully compensatory. More precisely, one compromise index to the membership function can be adjusted by decision-maker to reveal the way of change on degree of satisfaction for each objective and fuzzy constraint. One numerical example is employed to illustrate our assertions.

AN INTERIOR POINT ALGORITHM FOR SOLVING LINEAR OPTIMIZATION OVER THE EFFICIENT SET PROBLEMS

ABSTRACT The efficient set of a multiple objective linear programming problem is usually nonconvex, hence linear optimization over the efficient set problem is classified as a nonlinear optimization problem. This paper presents a modified interior point algorithm for solving linear optimization over the efficient set problems. Using computational experiments, we show that the modified algorithm provides an effective and accurate approach for solving the linear optimization over the efficient set problem.

A comparative analysis of DEA as a discrete alternative multiple criteria decision tool

The application of Data Envelopment Analysis (DEA) as a discrete alternative multiple criteria decision making (MCDM) tool has been gaining more attention in the literature. In this literature, DEA has been applied as an MCDM tool and compared analytically to other MCDM models and their structures, especially those that are based on multiple objective linear programming approaches. In this paper, we provide an empirical evaluation of various DEA ranking approaches and MCDM techniques, which include outranking and multiattribute utility techniques, using case study information. A solid waste management system case study that has been published by Hokkanen and Salminen (1997a) provide the data and results for this comparison. The results of this study show that incorporating decision maker value judgments into the DEA approaches provided comparable results to those found by the `traditional' MCDM approaches. The DEA approaches seem to be advantageous to users by requiring less information from decision makers and analysts, and providing ranked alternative valuations that may be useful for some decision makers.

Optimization over the efficient set of a parametric multiple objective linear programming problem

In this paper we consider the problem ( P) for optimizing a function over the efficient set of a multiple objective linear programming (MOLP) problem with parameters in the right hand side vector. Three solution algorithms in a most general case of problem ( P) and improvements of them in some special cases are presented. In the case when the objective function of problem ( P) is linear, it can be solved based on | T 2| linear programming problems with mixed one-zero integer variables if the parametric set is finite and based on | T 2| linear programming problems if the right hand side vector of the MOLP problem is a linear function of the parameters and the parametric set is a polyhedron, where | T 2| is, in general, the number of maximal efficient faces of the MOLP problem corresponding to a value of the parameters. A numerical example is given to illustrate the working of the algorithms.

Outcome Space Partition of the Weight Set in Multiobjective Linear Programming

Approaches for generating the set of efficient extreme points of the decision set of a multiple-objective linear program (P) that are based upon decompositions of the weight set W0 suffer from one of two special drawbacks. Either the required computations are redundant, or not all of the efficient extreme point set is found. This article shows that the weight set for problem (P) can be decomposed into a partition based upon the outcome set Y of the problem, where the elements of the partition are in one-to-one correspondence with the efficient extreme points of Y. As a result, the drawbacks of the decompositions of W0 based upon the decision set of problem (P) disappear. The article explains also how this new partition offers the potential to construct algorithms for solving large-scale applications of problem (P) in the outcome space, rather than in the decision space.

Optimizing Over the Efficient Set Using a Top-Down Search of Faces

The problem of optimizing a linear function over the efficient set of a multiple objective linear programming problem is studied. The decomposition of the efficient set into efficient faces is used as the basis of a search-based algorithm to solve this problem. The faces of the feasible region are characterized by the set of constraints that hold as equality in that face. The search is conducted over the indices of the constraints in a way that explores faces of possibly higher dimension first. Computational tests are performed to establish the behavior of the algorithm when the objective function is built according to different schemes that have received attention in the literature. The results indicate that different objective function types may lead to varying computation time requirements. In general, computational requirements of the algorithm increase significantly with problem size. A heuristic modification of the algorithm is proposed to solve large problems within reasonable time limits. Tests to measure the quality of the heuristic solutions show that the heuristic approach constitutes a practical alternative for finding good solutions for the problem.

Computing the NTU-Shapley value of NTU-games defined by multiple objective linear programs

In this paper we study a class of cooperative games with non-transferable utility (NTU) derived from multiple objective linear programs (MOLP's). This is done in order to introduce the NTU-Shapley value as a solution to multiple objective linear programming. We present an algorithm for the computation of the set of all NTU-Shapley values for the bicriterion cases, which is based on the simplex algorithm. Furthermore, we give a new optimal recursive procedure for the computation of the Shapley value of TU-games.

A multiple criteria approach to data envelopment analysis

In this paper, we present a Multiple Criteria Data Envelopment Analysis (MCDEA) model which can be used to improve discriminating power of DEA methods and also effectively yield more reasonable input and output weights without a priori information about the weights. In the proposed model, several different efficiency measures, including classical DEA efficiency, are defined under the same constraints. Each measure serves as a criterion to be optimized. Efficiencies are then evaluated under the framework of multiple objective linear programming (MOLP). The method is illustrated through three examples in which data sets are taken from previous research on DEA's discriminating power and weight restriction.

An energy planning approach based on mixed 0–1 Multiple Objective Linear Programming

Multiple Objective Linear Programming (MOLP) models have been widely used in the energy sector for taking into account several conflicting objectives pursued in energy planning. However, continuous variables are not sufficient to accurately represent discrete phenomena encountered in many practical decision situations, such as the power generation expansion problem. This paper presents a new approach based on a mixed 0–1 MOLP model and applied to the Greek electricity generation sector for identifying the number and output of each type of power units needed to satisfy the expected electricity demand in the future. The core of the model is a branch and bound algorithm, which has been properly modified for the multi-objective case and is capable of generating the whole set of efficient solutions. The results provided by this method is the extraction of the efficient combinations of the power generation units, and for each combination the efficient solutions determining electricity production from each unit.

An energy planning approach based on mixed 0–1 Multiple Objective Linear Programming

AbstractMultiple Objective Linear Programming (MOLP) models have been widely used in the energy sector for taking into account several conflicting objectives pursued in energy planning. However, continuous variables are not suffcient to accurately represent discrete phenomena encountered in many practical decision situations, such as the power generation expansion problem. This paper presents a new approach based on a mixed 0–1 MOLP model and applied to the Greek electricity generation sector for identifying the number and output of each type of power units needed to satisfy the expected electricity demand in the future. The core of the model is a branch and bound algorithm, which has been properly modified for the multi‐objective case and is capable of generating the whole set of effcient solutions. The results provided by this method is the extraction of the effcient combinations of the power generation units, and for each combination the effcient solutions determining electricity production from each unit.