Optimization in Practice with MATLAB® For Engineering Students and Professionals

Overview

Engineering design problems are multiobjective in nature. These problems usually optimize two or more conflicting objectives – simultaneously. An approach to multiobjective problem formulation combines the multiple objectives into a single objective function, also known as the Aggregate Objective Function (AOF). This AOF is solved to obtain one Pareto solution. One of several challenges in the area of multiobjective optimization is to judiciously construct an AOF that satisfactorily models the designer's preferences. This chapter provides a concise presentation of the Physical Programming method, which defines a framework to effectively incorporate the designer's preferences into the AOF (Ref. [1]).

Several methods to solve multiobjective optimization problems have been discussed in Chapter 6, such as the weighted sum method, compromise programming, and goal programming. These weight-based approaches require the designer to specify numerical weights in defining the AOF. This process can be ambiguous. For example, consider the following: (1) How does the designer specify weights in weight-based approaches? (2) Do the weights reflect the designer's preferences accurately? If the designer chooses to increase the importance of a particular objective, by how much should he/she increase the weight? Is 25% adequate? Or is 200% adequate? (3) Does the AOF denote a true mathematical representation of the designer's preferences?

The above questions begin to explain that the problem of determining “good weights” can be difficult and dubious. Because of this ambiguity, the weight selection process is often a computational bottleneck in large scale multiobjective design optimization problems. The above discussion paves the way for a multiobjective problem formulation framework that alleviates these ambiguities: Physical Programming (PP) developed by Messac [2].

Physical Programming systematically develops an AOF that effectively reflects the designer's wishes. This approach eliminates the need for iterative weight setting, which alleviates the above discussed ambiguities. Instead of choosing weights, the designer chooses ranges of desirability for each objective. The PP method formulates the AOF from these ranges of desirability, while yielding interesting and useful properties for the AOF.