James P. Ignizio

Generalized goal programming An overview

Abstract Generalized goal programming is a practical and robust tool for use in multiobjective mathematical programming. In this paper we introduce this approach, describe its underlying philosophy, and indicate the specific subclass of models and methods which serve to comprise the overall approach.

Sequential linear goal programming: Implementation via MPSX

Abstract This paper presents the actual results obtained when a traditional linear programming computer code is used sequentially so as to solve the linear goal programming problem. The approach, denoted as Sequential Linear Goal Programming (SLGP), was implemented by incorporating the IBM linear programming code known as MPSX. The results obtained, the advantages and drawbacks, as well as a comparison with an alternate linear goal programming code, are also described.

Multiobjective mathematical programming via the MULTIPLEX model and algorithm

Abstract The increased interest in the existence and consideration of multiple objectives has made itself evident in the significant growth in the development and implementation of multiobjective mathematical programming. Unfortunately, it is our opinion that this field is now characterized by such a diversity of philosophies, models, approaches and terminology that any unifying theme is obscured. In fact, rather than stressing the (substantial) degree of inherent commonality between multiobjective models and methods, most presentations seem to focus on their real, or imagined, differences. We believe that such treatment can be counterproductive and thus propose, herein, what we hope is a more unified treatment of multiobjective mathematical programming via the use of the multiphase simplex, or Multiplex model and algorithm. While none of the components and concepts of the Multiplex method are, in themselves, new, we do believe that the specific arrangement of these ideas, in the form presented, does serve to clarify the close relationships between the models and (simplex based) algorithms for most forms of multiobjective mathematical programming (and, in turn, their relationship to ‘conventional,’ single objective programming).

Fuzzy multicriteria integer programming via fuzzy generalized networks

Recent advances in network based mathematical programming methods have given rise to practical, economical, and computationally efficient solution methodologies for truly large-scale problems. Today, a wide variety of real world problems have been solved in such a manner even though, on the surface, they do not have an apparent network structure. One type of network, the generalized network, has been successfully applied to solve large-scale zero-one or mixed integer mathematical programming models. However, one drawback to such approaches is their typical focus on but a single objective. The purpose of this paper then is to document the results of an ongoing research effort which combines the generalized network concept with the technique of fuzzy programming. The resultant, hybrid approach has been found to provide a computationally efficient approach to multi-objective zero-one (or integer) programming problems.

The determination of a subset of efficient solutions via goal programming

Abstract A wide variety of models and methods have been proposed to solve the vectormaximum problem. Many of these approaches center their attention on linear programming with several objective functions and seek to obtain the set of efficient (Pareto optimal) solutions. Another approach to the same problem is to rank the objectives according to a priority structure and seek the lexicographic minimum of an ordered function of goal deviations. This latter approach, known as goal programming with preemptive priorities, has, in the literature, usually been treated as a separate topic. In this paper we show that the solution to the linear goal programming problem can be made to always be an efficient solution from which we may conduct a practical investigation of a subset of efficient solutions which form a useful compromise set. While perhaps lacking the elegance of the more esoteric approaches, this technique nonetheless has worked well in practice on actual problems.

An introduction to goal programming with applications in urban systems

Abstract Decisions involving large-scale, complex systems, particularly those in which “society” serves as an ultimate judge of their outcome and effectiveness, also involve multiple, conflicting and noncommensurable goals. Traditional models for the representation and solution of such problems have generally been forced to ignore the multiobjective nature of such problems. As a result, we obtain “optimal” solutions to the simplified models but, since the models do not reflect the actual situation, these solutions can sometimes cause more harm than good. Since large-scale, complex and multiobjective systems are so predominate in urban systems, it is vital that any improved methodology for modeling and solution be at least considered. In this paper we direct our attention to just one of the several new approaches to multiobjective decision analysis; the tool known as goal programming. Considerable interest seems to have been generated in this area recently but the perceptions of goal programming are varied and conflicting and, all too often, erroneous. In this paper an attempt is made to present the reader with a logical structuring of multiobjective optimization and, in particular, to identify goal programmings place and role within this framework. In doing this we hope to dispel a number of myths and misconceptions that have arisen while, at the same time, present an accurate view of the scope and limitations of the methodology. While the paper is primarily tutorial, we will however, also consider the actual and potential implementation of goal programming in problems encountered in the study of urban systems.

Antenna array beam pattern synthesis via goal programming

It is interesting to note that the field of operations research and that of radar both began in World War II and, initially, the success of one was greatly influenced by that of the other. It is a matter of historical record [6] that one of the very first problems investigated by the early operational research groups was that of the siting and implementation of radar in both air battles and U-boat detection. Since World War II, the tools of the OR analyst have been refined and increased while, at the same time, the modern radar system has become vastly larger and more complex. In this paper we address one particular problem associated with the radar system: the synthesis of the beam patterns produced by largescale antenna arrays. The solution of this important problem is achieved by means of the implementation of a relatively new tool of optimization (speci-

An enhanced conversion scheme for lexicographic, multiobjective integer programs

Abstract A number of approaches have been proposed (and several implemented) for the solution of lexicographic, multiobjective programming problems. These approaches may be divided into two classes. The first encompasses the development of algorithms specifically designed to deal directly with the initial model while the second attempts to transform, efficiently, the lexicographic, multiobjective model into an equivalent, single objective programming problem. This second approach would appear particularly attractive since it permits the use of conventional, readily available, mathematical programming software. In this paper we address a particular form of the lexicographic, multiobjective model; specifically one in which all functions are linear and all variables integer. It is then shown how a recently developed scheme for the transformation of this model may be substantially improved. As a result, lexicographic, multiobjective integer linear programs may be easily converted into conventional linear integer programs wherein the magnitude of the objective function coefficients are minimized.

A Generalized Goal Programming Approach to the Minimal Interference, Multicriteria N × 1 Scheduling Problem

Abstract The conventional approach to the modeling and solution of most scheduling problems involves the development of a mathematical model which (1) employs discrete variables (e.g., linear integer programs), and (2) includes only a single objective to be maximized or minimized (e.g., minimization of makespan). Unfortunately, models involving discrete variables are inherently combinatorially explosive (i.e., methods such as branch-and-bound will exhibit computation times which grow exponentially with problem size). Further, scheduling problems encountered in the real world invariably involve multiple conflicting objectives, and thus using a single-objective representation can lead to gross oversimplification. In this paper we address a specific class of scheduling problem encountered in several real-world applications that may be efficiently addressed as a linear multiobjective model having only continuous variables. The model and its solution are compared with those of a highly acclaimed recent approac...

A methodology for multicriteria network partitioning

Abstract Conventional solution techniques are not appropriate for the multiobjective partitioning of large-scale distributed computer networks. The multiple (conflicting) goals and constraints include the assignment of all nodes without violating partition capacity limitations, maximal communication between nodes assigned to the same partition, minimal dispersion of technology classes, minimal diversity of technology classes within partitions, maximal reliability and minimal cost. This problem has been mathematically formulated and a two-stage, heuristic solution method was developed and programmed for solution and evaluation. The modelling and solution methodologies are flexible and may easily be applied to a wide range of network partitioning problems.