In many experiments more than one response is of interest for the experimenter. Furthermore, we sometimes want to find a solution for controllable factors which result in the best possible value for each response. This is the context of multiple response optimization, where we seek a compromise between the responses; however, it is not always possible to find a solution for controllable factors which optimize all of the responses simultaneously. Multiple response optimization has an extensive literature in the context of multiple objective optimization which is beyond the scope of this course. Here, we will discuss the basic steps in this area.

As expected, multiple response analysis starts with building a regression model for each response separately. For instance, in Example 11.2 we can fit three different regression models for each of the responses which are Yield, Viscosity and Molecular Weight based on two controllable factors: Time and Temperature.

One of the traditional methods way to analyze and find the desired operating condition one is **overlaid contour plots**. This method is mainly useful when we have two or maybe three controllable factors but in higher dimensions it loses its efficiency. This method simply consists of overlaying contour plot for each of the responses one over another in the controllable factors space and finding the area which makes the best possible value for each of the responses. Figure 11.16 (Montgomery, 7th Edition) shows the overlaid contour plots for example 11.2 in Time and Temperature space.

The unshaded area is where *yield* > 78.5, 62 < *viscosity* < 68, and* molecular weight* < 3400. This area might be of special interest for the experimenter because they satisfy given conditions on the responses.

Another dominant approach for dealing with multiple response optimization is to form a **constrained optimization problem**. In this approach we treat one of the responses as the objective of a constrained optimization problem and other responses as the constraints where the constraint’s boundary is to be determined by the decision maker (DM). The Design-Expert software package solves this approach using a direct search method.

Another important procedure that we will discuss here, also implemented in Minitab, is the **desirability function** approach. In this approach the value of each response for a given combination of controllable factors is first translated to a number between zero and one known as **individual desirability**. Individual desirability functions are different for different objective types which might be Maximization, Minimization or Target. If the objective type is maximum value, the desirability function is defined as

\(d=\left\{\begin{array}

{cl}

0 & y<L \\ \left(\frac{y-L}{T-L}\right)^r & L\leq y\leq T \\ 1 & y>T

\end{array} \right. \)

When the objective type is a minimum value the, the individual desirability is defines as

\(d=\left\{\begin{array}

{cl}

1 & y<T \\ \left(\frac{U-y}{U-T}\right)^r & T\leq y\leq U \\ 0 & y>U

\end{array} \right. \)

Finally the two-sided desirability function with target-the-best objective type is defined as

\(d=\left\{\begin{array}

{cl}

0 & y<L \\ \left(\frac{y-L}{T-L}\right)^{r_1} & L\leq y\leq T \\ \left(\frac{U-y}{U-T}\right)^{r_2} & T\leq y\leq U \\ 0 & y>U

\end{array} \right. \)

Where the \(r_1\) , \(r_2\) and *r* define the shape of the individual desirability function (Figure 11.17 in the text shows the shape of individual desirability for different values of shape parameter). Individual desirability is then used to calculate the overall desirability using the following formula:

\(D=(d_1 d_2 \ldots d_m)^{1/m}\)

where m is the number of responses. Now, the design variables should be chosen so that the overall desirability will be maximized. Minitab’s `Stat` > `DOE` > `Response Surface` > `Response Optimizer` routine uses the desirability approach to optimize several responses, simultaneously.