procmean.gms : Optimal Process Mean

**Description**

Find the optimal process mean when the quality characteristic follows a Beta distribution and using a linear quality loss.

**References**

- Chen, C H, and Chou, C Y, Determining the Optimum Process Mean under a Beta Distribution. Journal of the Chinese Institute of Industrial Engineers 18 (3) (2003), 27–32.
- Phillips, M D, and Cho, B R, Determining the Optimum Process Mean under a Beta Distribution. A Nonlinear model for determining the most economic process mean under a beta distribution 7 (2000), 61–74.

**Small Model of Type :** NLP

**Category :** GAMS Model library

**Main file :** procmean.gms

```
$title Optimal Process Mean (PROCMEAN,SEQ=301)
$onText
Find the optimal process mean when the quality characteristic
follows a Beta distribution and using a linear quality loss.
Erwin Kalvelagen, April 2004
Chen, C H, and Chou, C Y, Determining the Optimum Process Mean under a
Beta Distribution. Journal of the Chinese Institute of Industrial
Engineers 18 (3) (2003), 27--32.
Phillips, M D, and Cho, B R, Determining the Optimum Process Mean
under a Beta Distribution. A Nonlinear model for determining the most
economic process mean under a beta distribution 7 (2000), 61--74.
Keywords: nonlinear programming, statistics, process target, quality loss function,
beta distribution, process optimization
$offText
Scalar
a 'minimum value of quality characteristic' / 113 /
b 'maximum value of quality characteristic' / 119 /
alpha 'shape parameter' / 2 /
beta 'shape parameter' / 4 /
T 'target value' / 115 /
k1 'quality loss coefficient when x < T' / 2 /
k2 'quality loss coefficient when x > T' / 3 /;
Scalar g1, g2, g3;
g1 = gamma(alpha + beta)/(gamma(alpha)*gamma(beta));
g2 = gamma(alpha + 1)*gamma(beta)/gamma(alpha + beta + 1);
g3 = g1*g2;
Variable
TC 'total expected cost per unit'
delta 'location parameter'
y 'transformation';
Equation
tcdef 'cost model'
ydef;
tcdef.. tc =e= k1*T*betareg(y,alpha,beta)
- k1*{(delta + a)*betareg(y,alpha,beta)
+(b - a)*betareg(y,alpha + 1,beta)*g3}
+ k2*{(delta + a)*[1 - betareg(y,alpha,beta)]
+(b - a)*[1 - betareg(y,alpha + 1,beta)*g3]}
- k2*T*[1 - betareg(y,alpha,beta)];
ydef.. y =e= (T - delta - a)/(b - a);
y.lo = 0.0001;
y.up = 0.9999;
y.l = 0.5;
Model m / all /;
solve m using nlp minimizing tc;
```