5. Process Improvement
5.5. Advanced topics
5.5.3. How do you optimize a process?
5.5.3.2. Multiple response case

5.5.3.2.1.

Multiple responses: Path of steepest ascent

Objective: consider and balance the individual paths of maximum improvement

When the responses exhibit adequate linear fit (i.e., the response models are all linear), the objective is to find a direction or path that simultaneously considers the individual paths of maximum improvement and balances them in some way. This case is addressed next.

When there is a mix of linear and higher-order responses, or when all empirical response models are of higher-order, see sections 5.5.3.2.2 and 5.5.3.2.3. The desirability method (section 5.5.3.2.2) can also be used when all response models are linear.

Procedure: Path of Steepest Ascent, Multiple Responses

A weighted priority strategy is described using the path of steepest ascent for each response

The following is a weighted priority strategy using the path of steepest ascent for each response.

1. Compute the gradients g_i (i = 1, 2, . . ., k) of all responses as explained in section 5.5.3.1.1. If one of the responses is clearly of primary interest compared to the others, use only the gradient of this response and follow the procedure of section 5.5.3.1.1. Otherwise, continue with step 2.
2. Determine relative priorities \( \pi_{i} \) for each of the k responses. Then, the weighted gradient for the search direction is given by
\[ g = \frac{\pi_{1}g_{1} +\pi_{2}g_{2} + \cdots + \pi_{k}g_{k}} {\sum_{i=1}^{k}{\pi_{i}}} \]

and the weighted direction is

\( d = \frac{g} {\parallel g \parallel} \)

Weighting factors based on R²

The confidence cone for the direction of maximum improvement explained in section 5.5.3.1.2 can be used to weight down "poor" response models that provide very wide cones and unreliable directions. Since the width of the cone is proportional to (1 - R²), we can use

\[ \pi_{j} \frac{R_{j}^{2}} {\sum_{i=1}^{k}{R_{i}^{2}}} \hspace{.5in} j = 1, 2, \dots , k \]

Single response steepest ascent procedure

Given a weighted direction of maximum improvement, we can follow the single response steepest ascent procedure as in section 5.5.3.1.1 by selecting points with coordinates x^* = ρd_i, i = 1, 2, ..., k. These and related issues are explained more fully in Del Castillo (1996).

Example: Path of Steepest Ascent, Multiple Response Case

An example using the weighted priority method

Suppose the response model:

\( \hat{y}_{1} = 711.0 + 50.9 x_{1} + 154.8 x_{2} \)

with \( R_{1}^{2} = 0.8968 \) represents the average yield of a production process obtained from a replicated factorial experiment in the two controllable factors (in coded units). From the same experiment, a second response model for the process standard deviation of the yield is obtained and given by

\( \hat{y}_{2} = 19.26 + 6.31 x_{1} + 6.28 x_{2} \)

with \( R_{2}^{2} = 0.5977 \). We wish to maximize the mean yield while minimizing the standard deviation of the yield.

Step 1: compute the gradients:

Compute the gradients

We compute the gradients as follows.

\[ \begin{array}{lcl} g_{1}^{'} & = & \left( \frac{50.9}{\sqrt{50.9^{2} + 154.8^{2}}}, \frac{154.8}{\sqrt{50.9^{2} + 154.8^{2}}} \right) \\ & = & (0.3124, 0.9500) \end{array} \]

\[ \begin{array}{lcl} g_{2}^{'} & = & \left( \frac{-6.31}{\sqrt{6.31^{2} + 6.28^{2}}}, \frac{-6.28}{\sqrt{6.31^{2} + 6.28^{2}}} \right) \\ & = & (-0.7088, -0.7054) \end{array} \]

(recall we wish to minimize y₂).

Step 2: find relative priorities:

Find relative priorities

Since there are no clear priorities, we use the quality of fit as the priority:

\( \pi_{1} = \frac{0.8968}{0.8968 + 0.5977} = 0.6 \)

\( \pi_{2} = \frac{0.5977}{0.8968 + 0.5977} = 0.4 \)

Then, the weighted gradient is

g' = (0.6(0.3124) + 0.4(-0.7088), 0.6(0.95) + 0.4(-0.7054)) = (-0.096, 0.2878)

which, after scaling it (by dividing each coordinate by \( \small \sqrt{(-0.096)^{2} + 0.2878^{2}} \)), gives the weighted direction d' = (-.03164, 0.9486).

Therefore, if we want to move ρ = 1 coded units along the path of maximum improvement, we will set x₁ = (1)(-0.3164) = -0.3164, x₂ = (1)(0.9486) = 0.9486 in the next run or experiment.