# Doe: Robust Design

**Example:****Write out a functional form which is a second order polynomial with two control factors and two noise factors and calculates the related c vector and C and D matrices assuming there is only one quality characteristic?**

**Answer**:

**Robust Design Based on Profit Maximization:**

Robust Design based on Profit Maximization (RDPM) methods generally require all of the inputs that response surface methods (RSM) require. These include (1) an “experimental design”, **D***s *and (2) vectors that specify the highs, **H**, and lows, **L**, of each factor. In addition, they require (3) the declaration of which factors **x**c are control and which are noise **z**.

**Algorithm ****Robust Design based on Profit Maximization:**

**Step 1****:**If models of the *pr*(**x**c) for all quality characteristics are available, go to *Step 6*. Otherwise continue.

**Step 2****:**For each quality characteristic for which *pr*(**x**c) is not available, include the associated response index in the set S1 if the response is a quality characteristic. Include the response in the set S2 if the response is the fraction nonconforming with respect to at least one type of nonconformity. Also, identify the specification limits, *LSLk *and *USLk*, for the responses in the set S1.

**Step 3****:**Apply a response surface method (all steps except the last, optimization step) to obtain an empirical model of all quality characteristics including the production rate, *yest*,*r*(**x ***c*,**z**) for *r *= 1,…,*q*.

**Step 4****:**Estimate the expected value, μz,*i*, and standard deviation, σz,*i*, of all the noise

factors relevant under normal system operation for all *i *= 1, …, *mn*.

**Step 5****:**Estimate the failure probabilities as a function of the control factors, *pr*(**x**c) for all quality characteristics, *r *∈*S*1

**Step 6****:**Obtain cost information in the form of revenue per unit, *w*0, and rework and/or scrap costs per defect or nonconformity of type *wr *for *r *= 1,…,*q*.

**Step 7****:**Maximize the profit, Profit(**x **c), in Equation (14.3) as a function of **x**c.