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.     

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.     

An achievement rate approach to linear programming problems with an interval objective function

In this paper, we focus on a treatment of a linear programming problem with an interval objective function. From the viewpoint of the achievement rate, a new solution concept, the maximin achievement rate solution, is proposed. Nice properties of this solution are shown: a maximin achievement rate solution is necessarily optimal when a necessarily optimal solution exists, and if not, then it is still a possibly optimal solution. An algorithm for a maximin achievement rate solution is proposed based on a relaxation procedure together with a simplex method. A numerical example is given to demonstrate the proposed solution algorithm.     

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.     

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.     

On duality in multiple objective linear programming

In this paper we present two approaches to duality in multiple objective linear programming. The first approach is based on a duality relation between maximal elements of a set and minimal elements of its complement. It offers a general duality scheme which unifies a number of known dual constructions and improves several existing duality relations. The second approach utilizes polarity between a convex polyhedral set and the epigraph of its support function. It leads to a parametric dual problem and yields strong duality relations, including those of geometric duality.     

Indefinite cubic programming with standard errors in objective function

Summary This paper concerns with the problem of indefinite cubic programming in which the objective function is the product of two factors, one of which is quadratic and contains the terms with standard errors, the other being a linear factor and the constraints being linear. It has been shown that the solution of such a programming problem can be obtained by solving a convex programming problem. The problem has been extended to the case where both the factors in the objective function are quadratic factors containing terms with standard errors. Some already known results have also been deduced as particular cases of the problem discussed.

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Zusammenfassung Die Arbeit behandelt ein indefinites kubisches Programmierungsproblem unter linearen Nebenbedingungen. Hierbei ist die Zielfunktion das Produkt eines quadratischen und eines linearen Faktors. Es wird gezeigt, daß das Problem auf ein konvexes Programm zurückgeführt werden kann. Ferner wird der Fall betrachtet, daß beide Faktoren der Zielfunktion quadratische Ausdrücke mit Fehlertermen sind. Als Sonderfälle der betrachteten Probleme werden einige bekannte Resultate abgeleitet.

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Vorgel. v.:H. P. Künzi.     

Multiple objective linear programming for environmental farm planning

We present a multiple objective linear programming model developed to consider a wide range of farming situations, which allows optimisation of profit or environmental outcome(s) or both. The modelling considers the problem of planning a farming system within a world where environmental considerations are increasing. The objective is to identify the best cropping and machinery options which are both profitable and result in improvements to the environment, depending upon the farm situation of market prices, potential crop yields, soil and weather characteristics. In particular, the model uses a flexible approach to choosing the machinery, timing of operations, crop rotations and levels of inputs. We show for a UK scenario, that large reductions in environmental impact can be achieved for reductions in farm profit which are insignificant relative to the annual variation due to yields and prices.     

Fuzzy linear programming with vague objective coefficients in an uncertain environment

In this paper, a new fuzzy linear programming (FLP)-based methodology using a specific membership function named modified logistic membership function is proposed. The modified logistic membership function is first formulated and its flexibility established by an analytical approach. This membership function is tested for its useful performance through an illustrative example by employing FLP. The developed methodology of FLP has provided confidence in applying to real-life industrial production planning problem. This approach of solving industrial production planning problem can provide feedback to the decision maker, implementer and analyst. In such cases, this approach can be called interactive FLP. There is a possibility to design the self-organizing of the fuzzy system for the product mix selection problem in order to find a satisfactory solution. The decision maker, analyst and implementer can incorporate their knowledge and experience to obtain the best outcome.     

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.     

An algorithm for the multiple objective integer linear programming problem

A technique is presented for solving the multiple objective integer linear programming problem. The technique can be used to identify some or all efficient solutions. While the technique is applicable with any integer programming algorithm, it is well suited for implementation using integer postoptimality techniques. Such an implementation, based on Balas' Additive algorithm, is described for problems with zero-one variables.     

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.     

A WIN-WIN approach to multiple objective linear programming problems

Interactive decision making arose as a means to overcome the uncertainty concerning the decision maker's (DM) value function. So far the search is confined to nondominated alternatives, which assumes that a win–lose strategy is adopted. The purpose of this paper is to suggest a complementary interactive algorithm that uses an interior point method to solve multiple objective linear programming problems. As the algorithm proceeds, the DM has access to intermediate solutions. The sequence of intermediate solutions has a very interesting characteristic: all of the criteria are improved, that is, a solution Open image in new window , that follows another solution Open image in new window , has the values of all objectives greater than those of Open image in new window . This WIN-WIN feature allows the DM to reach a nondominated solution without making any trade-off among the objective functions. However, there is no impediment in proceeding with traditional multiobjective methods.     

The range of the efficient frontier in multiple objective linear programming

Because a rational decision maker should only select an efficient alternative in multiple criterion decision problems, the efficient frontier defined as the set of all efficient alternatives has become a central solution concept in multiple objective linear programming. Normally this set reduces the set of available alternatives of the underlying problem. There are several methods, mainly based on the simplex method, for computing the efficient frontier. This paper presents a quite different approach which uses a nonlinear parametric program, solved by Wolfe's algorithm, to determine the range of the efficient frontier.     

Bilevel programming with discrete lower level problems

In this article, we investigate bilevel programming problems with discrete lower level and continuous upper level problems. We will analyse the structure of these problems and discuss both the optimistic and the pessimistic solution approach. Since neither the optimistic nor the pessimistic solution functions are in general lower semicontinuous, we introduce weak solution function. By using these functions we are able to discuss optimality conditions for local and global optimality.     

Bilevel programming and the separation problem

In recent years, branch-and-cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs (MILPs). Methods for automatically generating inequalities valid for the convex hull of solutions to such MILPs are a critical element of branch-and-cut. This paper examines the nature of the so-called separation problem, which is that of generating a valid inequality violated by a given real vector, usually arising as the solution to a relaxation of the original problem. We show that the problem of generating a maximally violated valid inequality often has a natural interpretation as a bilevel program. In some cases, this bilevel program can be easily reformulated as a simple single-level mathematical program, yielding a standard mathematical programming formulation for the separation problem. In other cases, no such polynomial-size single-level reformulation exists unless the polynomial hierarchy collapses to its first level (an event considered extremely unlikely in computational complexity theory). We illustrate our insights by considering the separation problem for two well-known classes of valid inequalities.     

Multiple objective linear programming models with interval coefficients – an illustrated overview

In most real-world situations, the coefficients of decision support models are not exactly known. In this context, it is convenient to consider an extension of traditional mathematical programming models incorporating their intrinsic uncertainty, without assuming the exactness of the model coefficients. Interval programming is one of the tools to tackle uncertainty in mathematical programming models. Moreover, most real-world problems inherently impose the need to consider multiple, conflicting and incommensurate objective functions. This paper provides an illustrated overview of the state of the art of Interval Programming in the context of multiple objective linear programming models.