Doe: Robust Design

Expected Profits and Control-by-noise Interactions:

•         RDPM focuses on the design of engineered systems that produce units. These units could be welded parts in a manufacturing line or patients in a hospital.
•         The goal is to maximize the profit from this activity, which can be calculated as a sum of the revenues produced by the parts minus the cost to repair units that are not acceptable for various reasons.
•        To develop a realistic estimate of these profits as a function of the variables that the decision-maker can control, a number of quantities must be defined:

1.      m is the total number of experimental factors.

 

2.      q is the number of quality characteristics relevant to the system being studied.

 

 

3.      x is an m dimensional vector of all experimental inputs which can be divided into two types, control factors, xc, and noise factors, z.

 

4.       μz and σz are mn dimensional vectors containing the expected values, i.e., μz = E[z], and standard deviations, i.e., σz,i = sqrt[E(zi – μz,i)2], respectively.

 

5.      J is a diagonal matrix with the variances of the noise factors under usual operations along the diagonal, i.e., Ji,i = σz,i for i = 1,…,mn. (More generally, it is the variance-covariance matrix of the noise factors.)

 

6.       yest, 0(x_c,z,ε) is assumed to be the number of parts per year.

 

7.      yest, r(x_c,z,ε) is the rth quality characteristic value function.

 

8.       pr(xc) is the fraction of nonconforming units as a function of the control factors for the rth quality characteristic.

 

9.       w0 is defined as the profit made per conforming unit.

 

10.  w_r is the cost of the nonconformity associated with the rth characteristic. These failure costs include “rework” (e.g., cost of fixing the unit) and customer “loss of good will” .

 

11.  σ_total,r(xc) is the “total variation” at a specific combination of control factors, xc, i.e., the standard deviation of the rth response taking into account the variation of the noise factors during normal system operation.

 

12.   S2 is the set of indices associated with responses that are failure probability estimates, and S1 are all other indices.

 

13.  Φ(x,μ,σ) is the “cumulative normal distribution function,” which is the probability that a normally distributed random variable with mean, μ, and standard deviation, σ, is less than x. Values are given by, e.g., or the NORMDIST function in Excel.