Further Analysis of an Outcome Set-Based Algorithm for Multiple-Objective Linear Programming

Various difficulties have been encountered in using decision set-based vector maximization methods to solve a multiple-objective linear programming problem (MOLP). Motivated by these difficulties, Benson recently developed a finite, outer-approximation algorithm for generating the set of all efficient extreme points in the outcome set, rather than in the decision set, of problem (MOLP). In this article, we show that the Benson algorithm also generates the set of all weakly efficient points in the outcome set of problem (MOLP). As a result, the usefulness of the algorithm as a decision aid in multiple objective linear programming is further enhanced.     

Determining maximal efficient faces in multiobjective linear programming problem

Finding an efficient or weakly efficient solution in a multiobjective linear programming (MOLP) problem is not a difficult task. The difficulty lies in finding all these solutions and representing their structures. Since there are many convenient approaches that obtain all of the (weakly) efficient extreme points and (weakly) efficient extreme rays in an MOLP, this paper develops an algorithm which effectively finds all of the (weakly) efficient maximal faces in an MOLP using all of the (weakly) efficient extreme points and extreme rays. The proposed algorithm avoids the degeneration problem, which is the major problem of the most of previous algorithms and gives an explicit structure for maximal efficient (weak efficient) faces. Consequently, it gives a convenient representation of efficient (weak efficient) set using maximal efficient (weak efficient) faces. The proposed algorithm is based on two facts. Firstly, the efficiency and weak efficiency property of a face is determined using a relative interior point of it. Secondly, the relative interior point is achieved using some affine independent points. Indeed, the affine independent property enable us to obtain an efficient relative interior point rapidly.     

Hybrid Approach for Solving Multiple-Objective Linear Programs in Outcome Space

Various difficulties arise in using decision set-based vector maximization methods to solve a multiple-objective linear programming problem (MOLP). Motivated by these difficulties, some researchers in recent years have begun to develop tools for analyzing and solving problem (MOLP) in outcome space, rather than in decision space. In this article, we present and validate a new hybrid vector maximization approach for solving problem (MOLP) in outcome space. The approach systematically integrates a simplicial partitioning technique into an outer approximation procedure to yield an algorithm that generates the set of all efficient extreme points in the outcome set of problem (MOLP) in a finite number of iterations. Some key potential practical and computational advantages of the approach are indicated.     

An Outer Approximation Algorithm for Generating All Efficient Extreme Points in the Outcome Set of a Multiple Objective Linear Programming Problem

Various difficulties have been encountered in using decision set-based vector maximization methods to solve a multiple objective linear programming problem (MOLP). Motivated by these difficulties, some researchers in recent years have suggested that outcome set-based approaches should instead be developed and used to solve problem (MOLP). In this article, we present a finite algorithm, called the Outer Approximation Algorithm, for generating the set of all efficient extreme points in the outcome set of problem (MOLP). To our knowledge, the Outer Approximation Algorithm is the first algorithm capable of generating this set. As a by-product, the algorithm also generates the weakly efficient outcome set of problem (MOLP). Because it works in the outcome set rather than in the decision set of problem (MOLP), the Outer Approximation Algorithm has several advantages over decision set-based algorithms. It is also relatively easy to implement. Preliminary computational results for a set of randomly-generated problems are reported. These results tangibly demonstrate the usefulness of using the outcome set approach of the Outer Approximation Algorithm instead of a decision set-based approach. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Semidefinite Programming approach for solving Multiobjective Linear Programming

Several algorithms are available in the literature for finding the entire set of Pareto-optimal solutions of Multiobjective Linear Programmes (MOLPs). However, all of them are based on active-set methods (simplex-like approaches). We present a different method, based on a transformation of any MOLP into a unique lifted Semidefinite Program (SDP), the solutions of which encode the entire set of Pareto-optimal extreme point solutions of any MOLP. This SDP problem can be solved, among other algorithms, by interior point methods; thus unlike an active set-method, our method provides a new approach to find the set of Pareto-optimal solutions of MOLP.     

Constructing efficient solutions structure of multiobjective linear programming

It is not a difficult task to find a weak Pareto or Pareto solution in a multiobjective linear programming (MOLP) problem. The difficulty lies in finding all these solutions and representing their structure. This paper develops an algorithm for solving this problem. We investigate the solutions and their relationships in the objective space. The algorithm determines finite number of weights, each of which corresponds to a weighted sum problems. By solving these problems, we further obtain all weak Pareto and Pareto solutions of the MOLP and their structure in the constraint space. The algorithm avoids the degeneration problem, which is a major hurdle of previous works, and presents an easy and clear solution structure.     

Finding DEA-efficient hyperplanes using MOLP efficient faces

This paper suggests a method for finding efficient hyperplanes with variable returns to scale the technology in data envelopment analysis (DEA) by using the multiple objective linear programming (MOLP) structure. By presenting an MOLP problem for finding the gradient of efficient hyperplanes, We characterize the efficient faces. Thus, without finding the extreme efficient points of the MOLP problem and only by identifying the efficient faces of the MOLP problem, we characterize the efficient hyperplanes which make up the DEA efficient frontier. Finally, we provide an algorithm for finding the efficient supporting hyperplanes and efficient defining hyperplanes, which uses only one linear programming problem.     

An exact method for computing the nadir values in multiple objective linear programming

In this paper we propose a new method to determine the exact nadir (minimum) criterion values over the efficient set in multiple objective linear programming (MOLP). The basic idea of the method is to determine, for each criterion, the region of the weight space associated with the efficient solutions that have a value in that criterion below the minimum already known (by default, the minimum in the payoff table). If this region is empty, the nadir value has been found. Otherwise, a new efficient solution is computed using a weight vector picked from the delimited region and a new iteration is performed. The method is able to find the nadir values in MOLP problems with any number of objective functions, although the computational effort increases significantly with the number of objectives. Computational experiments are described and discussed, comparing two slightly different versions of the method.     

Range sets for weak efficiency in multiobjective linear programming and a parametric polytopes intersection problem

ABSTRACT

The aim of this paper is to obtain the range set for a given multiobjective linear programming problem and a weakly efficient solution. The range set is the set of all values of a parameter such that a given weakly efficient solution remains efficient when the objective coefficients vary in a given direction. The problem was originally formulated by Benson in 1985 and left to be solved. We formulate an algorithm for determining the range set, based on some hard optimization problems. Due to toughness of these optimization problems, we propose also lower and upper bound approximation techniques. In the second part, we focus on topological properties of the range set. In particular, we prove that a range set is formed by a finite union of intervals and we propose upper bounds on the number of intervals. Our approach to tackle the range set problem is via the intersection problem of parametric polytopes. Thus, our results have much wider area of applicability since the intersection (and separability) problem of convex polyhedra is important in many fields of optimization.     

A finite,nonadjacent extreme-point search algorithm for optimization over the efficient set

The problem (P) of optimizing a linear function over the efficient set of a multiple-objective linear program serves many useful purposes in multiple-criteria decision making. Mathematically, problem (P) can be classified as a global optimization problem. Such problems are much more difficult to solve than convex programming problems. In this paper, a nonadjacent extreme-point search algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact extreme-point optimal solution for the problem after a finite number of iterations. It can be implemented using only linear programming methods. Convergence of the algorithm is proven, and a discussion is included of its main advantages and disadvantages.The author owes thanks to two anonymous referees for their helpful comments.     

Random problem generation and the computation of efficient extreme points in multiple objective linear programming

This paper looks at the task of computing efficient extreme points in multiple objective linear programming. Vector maximization software is reviewed and the ADBASE solver for computing all efficient extreme points of a multiple objective linear program is described. To create MOLP test problems, models for random problem generation are discussed. In the computational part of the paper, the numbers of efficient extreme points possessed by MOLPs (including multiple objective transportation problems) of different sizes are reported. In addition, the way the utility values of the efficient extreme points might be distributed over the efficient set for different types of utility functions is investigated. Not surprisingly, results show that it should be easier to find good near-optimal solutions with linear utility functions than with, for instance, Tchebycheff types of utility functions.Dedicated to Professor George B. Dantzig on the occasion of his eightieth birthday.     

PROMISE: A DSS for multiple objective stochastic linear programming problems

Most of the multiple objective linear programming (MOLP) methods which have been proposed in the last fifteen years suppose deterministic contexts, but because many real problems imply uncertainty, some methods have been recently developed to deal with MOLP problems in stochastic contexts. In order to help the decision maker (DM) who is placed before such stochastic MOLP problems, we have built a Decision Support System called PROMISE. On the one hand, our DSS enables the DM to identify many current stochastic contexts: risky situations and also situations of partial uncertainty. On the other hand, according to the nature of the uncertainty, our DSS enables the DM to choose the most appropriate interactive stochastic MOLP method among the available methods, if such a method exists, and to solve his problem via the chosen method.     

Group testing with two defectives

Recently the problem of determining the minimax number of group tests for finding two defectives separately contained in two disjoint sets has been completely solved. However, the closely related problem of determining the minimax number of groups tests for finding two defectives contained in one set remains open. This is a surprisingly difficult combinatorial problem with very little known. In this paper we give a partial solution to this problem.     

A modified method for constructing efficient solutions structure of MOLP

This paper deals with a recently proposed algorithm for obtaining all weak efficient and efficient solutions in a multi objective linear programming (MOLP) problem. The algorithm is based on solving some weighted sum problems, and presents an easy and clear solution structure. We first present an example to show that the algorithm may fail when at least one of these weighted sum problems has not a finite optimal solution. Then, the algorithm is modified to overcome this problem. The modified algorithm determines whether an efficient solution exists for a given MOLP and generates the solution set correctly (if exists) without any change in the complexity.     

Experiments with an interactive paired comparison simplex method for molp problems

In this paper, an interactive paired comparison simplex based method formultiple objective linear programming (MOLP) problems is developed and compared to other interactive MOLP methods. The decision maker (DM)’s utility function is assumed to be unknown, but is an additive function of his known linearized objective functions. A test for ‘utility efficiency’ for MOLP problems is developed to reduce the number of efficient extreme points generated and the number of questions posed to the DM. The notion of ‘strength of preference ’ is developed for the assessment of the DM’s unknown utility function where he can express his preference for a pair of extreme points as ‘strong ’, ‘weak ’, or ‘almost indifferent ’. The problem of ‘inconsistency of the DM’ is formalized and its resolution is discussed. An example of the method and detailed computational results comparing it with other interactive MOLP methods are presented. Several performance measures for comparative evaluations of interactive multiple objective programming methods are also discussed. All rights reserved. This study, or parts thereof, may not be reproduced in any form without written permission of the authors.     

Computational Study of the Relationships Between Feasible and Efficient Sets and an Approximation

The computational difficulty of obtaining the efficient set in multi-objective programming, specially in nonlinear problems, suggest the need of considering an approximation approach to this problem. In this paper, we provide the computational results of the relationships between an approximation to the efficient set and the feasible and efficient sets. Random problem generation is considered for different sizes of the feasible set and we study the implications with respect to the number of objective functions and various kinds of objective functions. Computational experience with this approximation suggests that we obtain a substantial improvement when it increases the number of objective functions.     

An MOLP based procedure for finding efficient units in DEA models

In this paper a multiple objective linear programming (MOLP) problem whose feasible region is the production possibility set with variable returns to scale is proposed. By solving this MOLP problem by multicriterion simplex method, the extreme efficient Pareto points can be obtained. Then the extreme efficient units in data envelopment analysis (DEA) with variable returns to scale, considering the specified theorems and conditions, can be obtained. Therefore, by solving the proposed MOLP problem, the non-dominant units in DEA can be found. Finally, a numerical example is provided.     

Bounding MOLP objective functions: effect on efficient set size

In multiple objective linear programming (MOLP) problems the extraction of all the efficient extreme points becomes problematic as the size of the problem increases. One of the suggested actions, in order to keep the size of the efficient set to manageable limits, is the use of bounds on the values of the objective functions by the decision maker. The unacceptable efficient solutions are screened out from further investigation and the size of the efficient set is reduced. Although the bounding of the objective functions is widely used in practice, the effect of this action on the size of the efficient set has not been investigated. In this paper, we study the effect of individual and simultaneous bounding of the objective functions on the number of the generated efficient points. In order to estimate the underlying relationships, a computational experiment is designed, in which randomly generated multiple objective linear programming problems of various sizes are systematically examined.     

Characterizing Nonemptiness and Compactness of the Solution Set of a Convex Vector Optimization Problem with Cone Constraints and Applications

In this paper, we characterize the nonemptiness and compactness of the set of weakly efficient solutions of a convex vector optimization problem with cone constraints in terms of the level-boundedness of the component functions of the objective on the perturbed sets of the original constraint set. This characterization is then applied to carry out the asymptotic analysis of a class of penalization methods. More specifically, under the assumption of nonemptiness and compactness of the weakly efficient solution set, we prove the existence of a path of weakly efficient solutions to the penalty problem and its convergence to a weakly efficient solution of the original problem. Furthermore, for any efficient point of the original problem, there exists a path of efficient solutions to the penalty problem whose function values (with respect to the objective function of the original problem) converge to this efficient point.     

Iterative algorithms with the regularization for the constrained convex minimization problem and maximal monotone operators

In this paper, we prove a strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the set of solutions of a finite family of variational inclusion problems in Hilbert space. A strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the solution sets of a finite family of zero points of the maximal monotone operator problem in Hilbert space is also obtained. Using our main result, we have some additional results for various types of non-linear problems in Hilbert space.