#### Objective Functions

Most optimization problems, by convention, have one objective. This is at least partially because the mathematics of optimization were developed for single objective problems. Often in optimization the selection of the objective will be obvious: we wish to minimize pressure drop or maximize heat transfer, etc. However, sometimes we end up with a surrogate objective because we can’t quantify or easily compute the real objective.

For example, it is common in structures to minimize weight, with the assumption the minimum weight structure will also be the minimum cost structure. This may not always be true, however. (In particular, if minimum weight is achieved by having every member be a different size, the optimum could be very expensive!) Thus the designer should always keep in mind the assumptions and limitations associated with the objective of the optimization problem.

Often in design problems there are other objectives or constraints for the problem which we can’t include. For example, aesthetics or comfort are objectives which are often difficult to quantify. For some products, it might also be difficult to estimate cost as a function of the design variables.

These other objectives or constraints must be factored in at some point in the design process. The presence of these other considerations means that the optimization problem only partially captures the scope of the design problem. Thus the results of an optimization should be considered as one piece of a larger puzzle. Usually a designer will explore the design space and develop a spectrum of designs which can be considered as final decisions are made.

#### Financial Objective Functions

One popular way to formulate objective functions is to bring everything back to a Net Present Value (NPV). The NPV considers the time value of money and helps distinguish between multiple investment options. In the case of optimization, we typically want to maximize the NPV for our problem.

Worksheet on Financial Objectives

Another popular way to analyze an investment decision is to evaluate the Return on Capital Employed (ROCE). This exercise walks through the investment decision for a $50,000 heat pump that saves $12,000 per year of alternative expenses.