## Optimum Process Level

The probabilities in Eqs. (6.2) and (6.3) are represented as shaded areas on the density function

f

(

x

) in Figure 6.1. Now, the expected costs can be written as:

(6.4)

and

(6.5)

Hence, the total expected cost is

(6.6)

In Eq. (6.6), the total expected cost per item is denoted by

TC

(

µ

) to signify that it is a function of the decision variable

µ

. The expression in Eq. (6.6) is the objective function of our optimization prob-

lem. It consists of components

E

(

C

) and

E

(

C

), out of which

E

(

C

) decreases and

E

(

C

) increases as

µ

increases and vice versa (see Figure 6.2).This is based on the assumption that the function

TC

(

µ

) has a unique minimum value corresponding to the optimum process level

µ

. At this stage, the optimum value

E CUSL( ) CUSL probability that X is greater than USL×= CUSL P X USL>[ ]×=

E CLSL( ) CLSL f x( ) xd Xmin

LSL∫=

E CUSL( ) CUSL f x( ) dx USL

TC µ( ) CLSL f x( ) dx CUSL f x( ) dx USL

of

µ

can be obtained by increasing

µ

systematically from a small value and numerically evaluating

TC

(

µ

) at those values of

µ

until it attains its lowest value. This trial-and-error procedure will yield the global optimal solution if there is a unique optimal solution, or just one of the local optimal solutions otherwise.