Engineering Optimization

5Path Analysis

5.1 Introduction

We often have to evaluate events of a situation along a path and use either a summation of events or a rate of change as the objective function. This chapter reveals some examples and the mathematical methods for assessing the OF.

Many people use the term line integral, but to help eliminate confusion over the term line search and to admit that the path for accumulation does not need to be a straight line, I’ll use path integral.

5.2 Path Examples

Following are five examples to reveal the concepts of path analysis:

Example 1

What path should you walk along to get from point A to point B? Figure 5.1 shows two paths: one walking directly and one weaving among the trees. If it has just started lightly raining and getting wet is undesired, then the longer path under the protection of the trees might be best. How does one calculate the amount of rain encountered on each path to determine which is better? If a path was described by a function such as , where x and y represent the ground map coordinates, and the walking speed was where S is the distance along the path, then path analysis provides the time in the path section ΔS at any particular spot ...