Multi-Objective Programming and Goal Programming

This book contains a collection of some of the most important and interesting papers that were presented at the first international conference in Multi-objective programming and Goal programming theories and applications (MOPGP94). It is aimed to present some of the latest theoretical and algorithmic developments in the field as well as presenting a very useful selection of application papers.

This paper formulates a dynamic compromise model where criteria are state variables evolving in time. Motion equations are determined under a few widely accepted assumptions in decision theory. It is noteworthy to point out that the equilibrium is reached at the L_∞ bound of the compromise set. Applications in several fields (economics, sociology, ecology, etc.) seem to be large. We highlight specially two applications: (a) the choice of consumer goods aggregates and (b) the Pigovian-Coasian reparation of a negative externality. Finally, some properties and tentative extensions are indicated.

Water shortage in Jordan is becoming acute, with limited available water and financial resources. With the objective of achieving sustainable development, the Jordan Government is faced with the need to make the best use of these resources. To balance current needs and demands, taking into account the possibility of limited investment in the future, the current Jordanian investment programme (1992–1998) contains various national objectives and constraints to be met. The requirement for this study arises from the need for reliable methodology to select and rank water development projects according to different objectives (criteria). Multicriteria analysis is a realistic approach to this complex problem. In particular, the PROMETHEE method is well adapted to. Its flexibility enables the Decision-Maker to express his preferences more precisely. In this paper the problem is formulated, criteria are presented and quantified, and a partial and complete ranking of projects is obtained by applying the PROMETHEE method to rank a sample of 13 water development projects. The results are discussed and the computational results are given as a step towards the final objective which is a complete ranking of all water sector projects in Jordan taking into account the full range of different constraints, (ie, financial (initial cost, operation and maintenance) and regional development).

Consider a set of decision-making units (DMUs, e.g. branches, departments, firms) which employ a variety of resources to produce multiple outputs. The units being compared may be in the public sector e.g. health or education, so that the outputs and inputs need not be measured in monetary terms. We present two methods for evaluating the relative efficiency with which each individual resource input is utilised at each organizational unit. We therefore end up with a set of scores for each DMU i.e. a profile rather than a single overall score. Unlike conventional DEA (data envelopment analysis) a large data set is not required and the discriminatory power is shown to be higher.

An interactive method, NIMBUS, for nondifferentiable multiobjective optimization problems is introduced. We assume that every objective function is to be minimized The idea of NIMBUS is that the decision maker can easily indicate what kind of improvements are desired and what kind of impairments are tolerable at the point considered.

At each iteration, the decision maker is asked to classify the objectives into up to five different classes: those to be improved, those to be improved down to some aspiration level, those to be accepted as they are, those to be impaired till some upper bound and those allowed to change freely. The aspiration levels and the upper bounds are asked from the decision maker. The decision maker can also attach weighting coefficients to the objective functions. According to this classification, a new (possibly multiobjective) optimization problem is formed.

An MPB (Multiobjective Proximal Bundle) method is employed to solve the new problem. The MPB method is a generalization of the Kiwiel’s proximal bundle approach for nondifferentiable single objective optimization into the multiobjective case. The multiple objectives are treated individually without employing any scalarization. The method is capable of handling several nonconvex Lipschitz continuous objective functions subject to nonlinear (possibly nondifferentiable) constraints.

Although this algorithm is designed to solve problems with two variables, our aim in the future is to extend these results to the general case, even if this implies the impossibility to obtain a graphic representation of the efficient set.

This paper presents a sequential approach for the problem of getting an optimal solution to the multiobjective network flow problem with preemptive priorities. This approach considers a priority every time, while maintaining through the sequence the network structure. This allows to use in each iteration of the process any efficient network-based algorithm for the single objective minimum cost flow problem.

The correct formulation of a large multiple objective linear program (MOLP) can be very difficult. The problem formulator faces the usual difficulties posed by single objective linear programs (LPs), such as isolating and analyzing infeasibility among the constraints. In addition, MOLP problems may raise questions about whether certain relationships should be cast as objectives or constraints, and about how the different objectives interact and interfere with each other. Automated or semi-automated assistance is needed to handle problems of practical scale effectively. We demonstrate how techniques recently developed for the analysis of infeasible LPs can be extended to assist in the formulation of MOLPs, both by analyzing infeasible constraint sets and by illuminating the interactions among objectives and constraints.

In this paper we outline a computer system for analyzing and improving the efficiency of aluminum extrusion operations. The system is capable of examining and optimizing all aspects of an extrusion operation. A primitive ancestor of the system was successfully applied at Alcan Vancouver Works, a division of Alcan Aluminum Ltd., where it led, after a capital outlay of approximately $1 2 million for plant modifications, to average annual savings of about $1.0 million. The pay back period for the investment was estimated to have been eight months.

Multiple objective linear programming (MOLP) is almost universally implemented within a simplex algorithm and relies on some of its properties, in particular the use of bases. Alternative linear programming methods such as the projective method need not use bases. Yet, their most common approach to post-optimal analysis has been conservative, e.g. reconstructing a basis in order to use the classical framework. A second, less investigated approach, is to adapt the scope of MOLP to the new methods. The simplex and the new methods are affected differently by degeneracy that causes numerical difficulties both in the data representation and the solution process. Symbolic solvers can mitigate such difficulties. Formulating MOLP as disjunctive optimization related to logic programming, the constraint logic program CLP(ℜ) is selected for its symbolic treatment of algebraic constraints seamlessly embedded in a Prolog syntax.

Decision Support System for Optimal Resource Allocation (DSS ORA) is an interactive mathematical programming system for optimal resource allocation developed to support decisions of investment in capital intensive telecommunications projects. The system strives to maximize corporate goals while respecting financial constraints such as the availability of capital funds, institutional requirements and varied dependencies or synergies among projects. The Analytical Hierarchy Process (AHP) is used for quantification of qualitative managerial judgment in regard to the relative value of projects through a two stage process. An initial resource allocation is found by a linear program, the objective coefficients of which are determined by the AHP. Then, a modified simplex algorithm proposes some rates of funding substitution. Users choose the amount of substitution or can override the substitutions proposed. Thus, users can build gradual confidence in the constraint checking mechanism of DSS ORA and come to rely on its efficiency-seeking capabilities. DSS ORA has been tested by several groups of managers responsible for the management and the implementation of project portfolios with significantly consistent results. The flexibility, user friendliness and quick time response of DSS ORA make it an effective negotiation tool in a group setting.

We present in this paper a new variant of Ekeland’s variational principle for vector valued functions with applications to the study of Pareto ε-efficiency. A new existence result for Pareto efficiency is also presented.

This paper presents an overview of the current state-of-the-art methods for modelling and solution of goal programming(GP) problems. Strategies for integrating these techniques into computerised software and thus specifications for producing an ‘intelligent’ GP solution and analysis package are suggested. Some recent criticisms of GP are detailed, together with how these perceived problems can be alleviated by means of such a package.

The methodologies of network analysis and goal programming combine to form a powerful modeling framework that enlarges the application base that can be solved by the powerful computational procedures of networks. In this paper, we review and discuss goal programming network structures and some applications.

This survey paper reports on the application of goal programming techniques toward the solutions of nonlinear field problems. The proposed approach involves approximating the solution by a set of trial functions containing unknown coefficients. The technique then minimizes in a weighted residual sense the absolute deviations of the system equations and/or the performance function residuals by nonlinear goal programming. Since the approximate solutions will in general not be able to satisfy all conditions, such as the boundary and initial conditions and at the same time minimize the residual errors, our approach involves reformulating the field problem as a preemptive goal programming model via the use of hard and soft constrains (or stiff and weak springs) in linear programming (or mechanics). Examples from fluid dynamics and control theory will be employed to illustrate the methodolgy. In conclusion, the advantages, limitations and other related issues on the goal programming model in solving nonlinear problems will be discussed.

The goal programming model is probably the most known in mathematical programming with multiple objectives. Available in various versions, this model has been applied in much varied fields, It has also been the target of many criticisms among which are those related to the difficulty of determining precisely the goals as well as those concerning the decision-maker’s near absence in this modelling process. In the actual paper, we focus mainly on the explicit integration of the decision-maker in this model, especially when he may not express his goals in a precise way. To this end, we are building up with the help of the decision-maker his functions of satisfaction related to the type of deviation in accordance with each of his imprecise goals.

In conventional Mathematical Goal Programming problems, the coefficients of objetive functions, constraints and target values are expressed as crisp numbers. In real world problems, however, it is no frequent for coefficients and target values to be known precisely. In such case, the parameters of the model should be represented by fuzzy sets for reflecting imprecision.

Our object is the formulation of a fuzzy linear goal programming problem to obtain a reasonable solution under consideration of the ambiguity in the achievement level desired for each attribute and to solve that fuzzy linear goal programming problem. The problem is transformed into a parametric one, and the solution is given in fuzzy terms; a numerical example is provided.

The basic philosophy underlying investor portfolio stems from economic utility theory. Many mathematical models have been applied to portfolio selection, however a major drawback of these methods is that a vast majority of input data is needed which requires a large amount of computation.

The aim of this paper is to investigate the multi-objective approach of Goal Programming(GP) and its application to portfolio evaluation and selection. A two stage GP model is proposed. The first stage predicts the sensitivity of the shares to specific indicators. The second stage of the model selects a portfolio based on the decision maker’s priorities and goals together with the information produced by the first stage.

This paper presents an application of MOPGP to wildlife management. The case study is the development of a game management operational plan for deer in Southern Spain. As there are two species, ecologic and economic conflicts are in conflict. The solution procedures start with a Multiobjective analysis of the efficient set, and is complemented by using Lexicographical Goal Programming to obtain a “satisfactory” operational plan. This research proves that management models can be improved by using multicriteria approaches, especially when a satisfactory combination of methods is employed.

A sequential goal programming (GP) approach is considered for nonlinear optimal control problems. This paper puts emphasis on flight trajectory problems which have no feasible solutions satisfying all constraints. The prioritization of multiple goals and fuzzy objectives are utilized to deal with unsatisfied constraints and multiple performance requirements. In order to apply a conventional simplex algorithm for a nonlinear problem, a linearized problem with respect to a set of discrete control variables is solved sequentially. A rocket ascending problem is shown as numerical examples of a well-defined problem. As a practical ill-defined problem, a takeoff trajectory of a jet airplane through a severe wind called a microburst is investigated.

This paper presents an investigation of equipment replacement policy problem using an approach which solve geometric programming problems with fuzzy parameters and fuzzy goals. Almost all types of equipment are subject to deterioration over time through age and usage and therefore decisions regarding the need for replacement and cost reduction are required. Such an analysis is based on existing functional relationships between costs (overhaul, replacement, and operating costs) and predictor variables (replacement, overhaul, and inspection intervals). Replacement strategies are directly connected with deterioration and its permanent dynamic changes. Deterioration is continuously subjected to influence of many factors that cannot always be predicted by company engineers and experts. As a result the optimisation model is based on uncertain of data as well as target values and model parameter values are usually specified by experts, which implies for subjectivity. The approach adopted here solves a series of multiobjective linear programming problems in order to build fuzzy regression models to represent such responses. The obtained geometric programming problem with fuzzy parameters is completed with fuzzy target values of the cost objective function. Deterministic transformation of the model as well as its computational treatment are considered.

This paper presents a comparison of Linear and Goal Programming methods applied to the downstream oil industry. A realistic, hypothetical model is developed. Results are presented and conclusions are drawn.