Engineering Optimization

26Optimization of Probable Outcomes and Distribution Characteristics

26.1 Introduction

Normally, optimization seeks to minimize or maximize a deterministic single value. However, many situations deal with probability of events, which are the result of uncertainty or environmental vagaries. The objective is to minimize some statistic related to the process, procedure, or product. The statistic could be variance of a quality metric, quantity of off‐spec material that results from manufacturing variability, an undesired event probability, system reliability, economic risk, etc. These objectives could be to minimize the 95% worst outcome, maximize the 99% minimum outcome, minimize variance, etc.

Sometimes the probability distribution associated with the application can be analytically obtained, providing a deterministic value for the objective, such as variance, or the 99% extreme case. However, more often, the outcome is a stochastic value.

Whether deterministic or stochastic, the topic of this chapter is about optimizing aspects of a probability distribution associated with an objective, which is in contrast to optimizing the nominal value.

I think that examples can best reveal the concept.

Example 1

How many dice rolls are needed to maximize the chance of getting at least 2 fives? (If you rolled 10 times and got 3 fives, then it is a success.) The more rolls, the greater the chance; so the answer is to maximize the chance of getting at least 2 fives take infinite rolls. But ...