On multiple objective programming problems with set functions

The convexity of a subset of a σ-algebra and the convexity of a set function on a convex subset are defined. Related properties are also examined. A Farkas-Minkowski theorem for set functions is then proved. These results are used to characterize properly efficient solutions for multiple objective programming problems with set functions by associated scalar problems.     

Linear complementarity problems and multiple objective programming

An equivalence is demonstrated between solving a linear complementarity problem with general data and finding a certain subset of the efficient points of a multiple objective programming problem. A new multiple objective programming based approach to solving linear complementarity problems is presented. Results on existence, uniqueness and computational complexity are included.     

Fuzzy programming and linear programming with several objective functions

In the recent past numerous models and methods have been suggested to solve the vectormaximum problem. Most of these approaches center their attention on linear programming problems with several objective functions. Apart from these approaches the theory of fuzzy sets has been employed to formulate and solve fuzzy linear programming problems. This paper presents the application of fuzzy linear programming approaches to the linear vectormaximum problem. It shows that solutions obtained by fuzzy linear programming are always efficient solutions. It also shows the consequences of using different ways of combining individual objective functions in order to determine an “optimal” compromise solution.     

Linear programming with a possibilistic objective function

This paper proposes some ways for dealing with a linear program when the coefficients of the objective function are subject to possibilistic imprecision, i.e. they are characterized by fuzzy restrictions. Emphasis is placed upon a passive approach that yields a satisfying solution via an appropriate semi-infinite program, and an active one that allows to reach a solution with a high possibility level of optimality. Extensions to the possibilistic constraints case and to the case of multiple-objective programming problems with possibilistic coefficients are also hinted. We end up with some concluding remarks and indicate axes for further developments.     

Mathematical programming problems with quasi-convex objective functions

This paper generalizes results obtained by Arrow and Enthoven in [1] for functions of ClassC ² on the non-negative orthant.This research was supported by Hydro—Quebec; University of Montreal; Office of Naval Research, Contract N-0014-67-A0112-001; National Science Foundation, Grant GP 25738.     

On nonlinear multiple objective fractional programming involving semilocally type-I univex functions

The aim of this paper is to obtain sufficient optimality conditions for a nonlinear multiple objective fractional programming problem involving semilocally type-I univex and related functions. Furthermore, a general dual is formulated and duality results are proved under the assumptions of generalized semilocally type-I univex and related functions. Our results generalize several known results in the literature.     

Cutting plane method for multiple objective stochastic integer linear programming

This paper grapples with the problem of incorporating integer variables in the constraints of a multiple objective stochastic linear program (MOSLP). After representing uncertain aspirations of the decision maker by converting the original problem into a deterministic multiple objective integer linear program (MOILP), a cutting plane technique may be used to compute all the efficient solutions of the last model leaving the decision maker to choose a solution according to his preferences. A numerical example is also included for illustration.     

Linear programming with multiple fuzzy goals

This paper describes the use of fuzzy set theory in goal programming (GP) problems. In particular, it is demonstrated how fuzzy or imprecise aspirations of the decision maker (DM) can be quantified through the use of piecewise linear and continuous functions. Models are then presented for the use of fuzzy goal programming with preemptive priorities, with Archimedean weights, and with the maximization of the membership function corresponding to the minimum goal. Examples are also provided.     

Linear programming with spheres and hemispheres of objective vectors

Given a linear program with a boundedp-dimensional feasible region let the objective vector range over ans-sphere, that is, ans-dimensional sphere centered at the origin wheres does not exceedp–1. If the feasible region and the sphere are in general position with respect to each other, then the corresponding set of all optimal solutions is a topologicals-sphere. Similar results are developed for unbounded feasible regions and hemispheres of objective vectors.This research is based on work supported in part by the National Science Foundation under Grant DMS-86-03232.     

Problems and methods with multiple objective functions

LetA be a set of feasible alternatives or decisions, and supposen different indices, measures, or objectives are associated with each possible decision ofA. How can a “best” feasible decision be made? What methods can be used or experimented with to reach some decision? The purpose of this paper is to attempt a synthesis of the main approaches to this problem which have been studied to date. Four different classes of approaches are distinguished: (1) aggregation of multiple objective functions into a unique function defining a complete preference order; (2) progressive definition of preference together with exploration of the feasible set; (3) definition of a partial order stronger than the product of then complete orders associated with then objective functions; and (4) maximum reduction of uncertainty and incomparability.     

Constrained optimization using multiple objective programming

In practical applications of mathematical programming it is frequently observed that the decision maker prefers apparently suboptimal solutions. A natural explanation for this phenomenon is that the applied mathematical model was not sufficiently realistic and did not fully represent all the decision makers criteria and constraints. Since multicriteria optimization approaches are specifically designed to incorporate such complex preference structures, they gain more and more importance in application areas as, for example, engineering design and capital budgeting. The aim of this paper is to analyze optimization problems both from a constrained programming and a multicriteria programming perspective. It is shown that both formulations share important properties, and that many classical solution approaches have correspondences in the respective models. The analysis naturally leads to a discussion of the applicability of some recent approximation techniques for multicriteria programming problems for the approximation of optimal solutions and of Lagrange multipliers in convex constrained programming. Convergence results are proven for convex and nonconvex problems.     

Selecting tolerances in chance-constrained programming: A multiple objective linear programming approach

An importance issue concerning the practical application of chance-constrained programming is the lack of a rational method for choosing risk levels or tolerances on the chance constraints. While there has also been much recent debate on the relationship, equivalence, usefulness, and other characteristics of chance-constrained programming relative to stochastic programming with recourse, this paper focuses on the problem of improving the selection of tolerances within the chance-constrained framework. An approach is presented, based on multiple objective linear programming, which allows the decision maker to be more involved in the tolerance selection process, but does not demand a priori decisions on appropriate tolerances. An example is presented which illustrates the approach.     

On computing objective function values in multiple objective quadratic-linear programming

In this paper, we will consider the computation of objective function values when a nondominated frontier is searched in multiple objective quadratic-linear programming (MOQLP). Reference directions and weighted-sums constitute a methodological basis for the search. This idea leads to a parametric linear complementarity model formulation. A critical task of making a search procedure efficient, is to compute the changes in quadratic and linear objective functions efficiently when a search direction is changed or a basis change is performed. Those changes in objective functions can be computed by a so-called direct or indirect method. The direct method is a straightforward one and based on the use of unit changes in basic decision variables. Instead, the indirect method utilizes some other basic variables of the model. We will introduce the indirect method and make theoretical and empirical comparisons between the methods. Based on the comparisons, we point out that the indirect method is clearly much more efficient than the direct one.     

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.     

Two theorems on multilevel programming problems with dominated objective functions

In this paper, we present two theorems on the structure of a type of multilevel programming problems. The theorems explore relations among a multilevel programming problem, a dynamical programming, and a nonlinear programming problem.     

Optimality and duality in fractional multiple objective programming involving semilocally preinvex and related functions

Necessary and sufficient optimality conditions are obtained for a nonlinear fractional multiple objective programming problem involving η-semidifferentiable functions. Also, a general dual is formulated and duality results are proved using concepts of generalized semilocally preinvex functions.     

Analysis of the objective space in multiple objective linear programming

A multiple objective linear program is defined by a matrix C consisting of the coefficients of the linear objectives and a convex polytope X defined by the linear constraints. An analysis of the objective space Y = C[X] for this problem is presented. A characterization between a face of Y and the corresponding faces of X is obtained. This result gives a necessary and sufficient condition for a face to be efficient. The theory and examples demonstrate the collapsing (simplification) that occurs in mapping X to Y. These results form a basis for a new approach to analyzing multiple objective linear programs.     

Optimality conditions and duality in multiple objective programming involving semilocally convex and related functions

A nonlinear multiple objective programming problem is considered where the functions involved are nondifferentiable. By considering the concept of weak minima, the Fritz John type and Karush-Kuhn- Tucker type necessary optimality conditions and Wolfe and Mond-Weir type duality results are given in terms of the right differentials of the functions. The duality results are stated by using the concepts of generalized semilocally convex functions     

An interactive method of tackling uncertainty in interval multiple objective linear programming

Mathematical programming models for decision support must explicitly take account of the treatment of the uncertainty associated with the model coefficients along with multiple and conflicting objective functions. Interval programming just assumes that information about the variation range of some (or all) of the coefficients is available. In this paper, we propose an interactive approach for multiple objective linear programming problems with interval coefficients that deals with the uncertainty in all the coefficients of the model. The presented procedures provide a global view of the solutions in the best and worst case coefficient scenarios and allow performing the search for new solutions according to the achievement rates of the objective functions regarding both the upper and lower bounds. The main goal is to find solutions associated with the interval objective function values that are closer to their corresponding interval ideal solutions. It is also possible to find solutions with non-dominance relations regarding the achievement rates of the upper and lower bounds of the objective functions considering interval coefficients in the whole model.