$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;