ABSTRACT
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.
