Polynomial goal programming: A procedure for modeling preference trade‐offs
Authors: Deckro Richard F. , Hebert John E. 
Published: October 2002

In the production/operations management environment it is common for decision makers to be faced with problems involving multiple objectives or goals, which cannot be satisfied simultaneously. The conflicting nature of these objectives results in solutions that involve tradeoffs or compromise. In production planning, for example, we typically wish to meet forecasted demand levels (or maximize customer service) over a given planning horizon, while simultaneously attempting to fully utilize normal capacity and also limit or minimize such factors as employee turnover, overtime, subcontracting, and inventory levels. Goal programming is a technique that is often useful in assisting decision makers find good solutions to problems involving multiple, conflicting goals.

The objective of preemptive goal programming models is to minimize the sum of the weighted deviations from a set of ordered (or prioritized) goals. Current procedures for formulating and solving goal programming models rely heavily on the use of preemptive priorities to reflect the goal preferences of the decision maker. The nature of preemptive priorities is such that the weight associated with the deviational variables on one priority level is infinitely greater than the weight associated with the deviational variables on the next lower priority level. The capability to consider trade‐offs involving the satisfaction of goals that are associated with different priority levels is precluded by this requirement. Clearly, it is more common for a decision maker to be content with a solution near a goal, rather than demand that it be exactly satisfied, particularly if this attitude would result in an increase in overall satisfaction generated by a greater achievement of lower level goals.

This paper develops an approach for a generalized goal programming model and solution technique, which will permit trade‐offs between preference levels, while retaining the flexibility to model intra‐level preferences. The Polynomial Goal Programming (PGP) model accommodates both intra‐level and inter‐level preference trade‐offs via the specification of the objective function as a polynomial expression.

The exponents and coefficients associated with the terms in the objective function are selected (by the decision maker) to reflect the relative importance of satisfying goals as the corresponding deviational variables approach zero. These powers and weights establish the marginal rates of substitution involved in the satisfaction of both inter‐level and intra‐level goals.

A key factor in the acceptance of a new modeling procedure is the ability to conveniently solve the resulting models with little or no modification to proven solution techniques. Polynomial Goal Programming models are particularly amenable to solution by Rosen's gradient projection method, as well as a number of related gradient‐based, non‐linear programming routines.

A comparative example demonstrates the enhanced modeling capability of the Polynomial Goal Programming approach in contrast to the more traditional linear and quadratic preemption models. The example clearly demonstrates the capability of the PGP model to recommend inter‐goal trade‐offs when an increase in total satisfaction will result.

This increase in “accuracy” of the model does not come without cost. Each refinement of the objective function attempts to bring the resulting solution closer to the decision maker's true preference. However, each refinement tends to increase both the complexity of the objective function and the cost of obtaining the corresponding solution. Thus, the use of the Polynomial Goal Programming Procedure is recommended only for those situations in which a cost‐benefit analysis indicates that the increased accuracy of the PGP solution is cost‐effective.