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


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 higherorder responses, or when all empirical response models are of higherorder, 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.
and the weighted direction is


Weighting factors based on R^{2} 
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  R^{2}),
we can use


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:


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 y_{2}). 

Step 2: find relative priorities:  
Find relative priorities 
Since there are no clear priorities, we use the quality of fit as
the priority:
Therefore, if we want to move ρ = 1 coded units along the path of maximum improvement, we will set x_{1} = (1)(0.3164) = 0.3164, x_{2} = (1)(0.9486) = 0.9486 in the next run or experiment. 