By Tuesday

The goal of a mathematical optimization study is to find the optimal solution to a problem. And, when the problem at hand is simple enough to solve within the available time, we can achieve this goal consistently. However, often we don’t have enough time or computing resources to carry out the number of design evaluations that would be needed to find the optimal solution. In these cases, we have no choice but to relax our goal and to seek the greatest possible design improvement within the available time.

A more direct statement of this would be, “Give me the best design you can find by Tuesday!” Continue reading

One Thing at a Time

Most of us have heard the advice, “Change only one variable at a time to understand how that variable affects your system.” Sometimes this advice is correct, but only in a very local sense. For example, if we want to estimate how sensitive a system is to a change in variable A, then we can hold all other variables constant and change variable A very slightly. The change in the system response divided by the change in variable A is an estimate of the sensitivity derivative at the original design point.

But, the key word in the previous sentence is “point,” because derivatives are defined at a point. If we select a different starting design point, and then repeat the above exercise, we would expect to get a different value for the sensitivity derivative.

Let’s examine this idea further. If we hold variable B constant at value B1, changing variable A will have a certain influence on the system response. But if we hold variable B constant at value B2, the effect of variable A might be very different than before. If so, then the effect of variable A depends on the value of variable B. When this occurs, we say that there is an interaction between variables A and B. We can easily generalize this argument to many variables.  Continue reading

All In

In poker, a player declares “all in” when he decides to bet all of his remaining chips on the cards in his hand. He then waits nervously while the remaining cards are dealt, knowing that he will soon either win big or lose all of his chips (“go bust”). A similar gamble occurs when you apply some optimization approaches based on Design of Experiments (DOE) concepts.  In this case, the actual objective function being minimized is evaluated at a predetermined set of design points. Then, a simple approximation of the objective function is developed by fitting an analytical function to these points. This approximate function is often called a response surface (also a surrogate function). The optimization search is then performed on the response surface, because evaluations of this simpler function are usually much quicker than evaluations of the actual objective function.

However, by defining all of your design evaluations ahead of time (going “all in”), you are risking that the corresponding response surface may not accurately represent the true objective function. If the surface fit is not accurate enough, then searching the response surface may not really give you the optimal design. In fact, it is common for an inaccurate response surface to completely mislead the optimization search, resulting in a very poor solution. So, while an accurate response surface could yield an optimized solution at lower cost than some other optimization approaches, a poorly fit surface may yield no useful results at all (you’ll “go bust”).  Continue reading

The “Multi” in Multidisciplinary Multi means “many” or “multiple.” Multidisciplinary design optimization (MDO) has become popular largely because it allows engineers to optimize over many different disciplines at the same time.

For example, you can use MDO to simultaneously optimize a vehicle body for structural, aerodynamic, thermal and acoustic behaviors. In addition, you can directly include non-performance measures, such as cost and manufacturability, in the optimization statement.  Continue reading