Compensation and Weights for Trade-offs in Engineering Design: Beyond the Weighted Sum

Multicriteria decision support methods are common in engineering design. These methods typically rely on a summation of weighted attributes to accomplish trade-offs among competing objectives. It has long been known that a weighted sum, when used for multicriteria optimization, may fail to locate all points on a nonconvex Pareto frontier. More recent results from the optimization literature relate the curvature of an objective function to its ability to capture Pareto points, but do not consider the significance of the objective function parameters in choosing one Pareto point over another. A parametrized family of aggregations appropriate for engineering design is shown to model decisions capturing all possible trade-offs, and therefore can direct the solution to any Pareto optimum. This paper gives a mathematical and theoretical interpretation of the parameters of this family of aggregations as defining a degree of compensation among criteria as well as a measure of their relative importance. The inability to reach all Pareto optima is shown to be surmounted by this consideration of degree of compensation as an additional parameter of the decision. Additionally, the direct specification of importance weights is common to many decision methods. The choice of a single point from a Pareto frontier by specifying importance weights alone is shown to depend on the degree of compensation implicit in the aggregation. Thus both the degree of compensation and weights must be considered to capture all potentially acceptable decisions. A simple truss design example is used here to illustrate the concepts.