Geoscience Reference
In-Depth Information
to ensure adequate service given the arrival rate . For many systems and many
specifications of ser
vic
e level constraints it has been shown that this amount grows
proportionately to
p
,i.e.
p
N
./
C
Q
(17.22)
for some constant Q (see, e.g., the discussion in Castillo et al.
2009
). The derivations
in Whitt (
1992
) suggest that, under many conditions, a good interpretation for Q is
provided by
2Q
p
2
C
1P.W > 0/;
where is the coefficient of variation of arrivals. Thus,
p
2Q=
p
2
C
1 is approx-
imately equal to the probability of waiting, a natural service level measure. To
summarize, when the probability of waiting is used as the service-level measure,
the constraint
p
P.W
i
>0/
Ǜ
0
;i
2
I
holds if
i
N
.
i
/
i
C
"r
2
C
1
2
#
p
i
;i
2
I:
Ǜ
0
(17.23)
Similar expressions can be derived with for service level measures where the
threshold for waiting time is set above 0.
As noted earlier, incidence of long waits can be controlled through service level
constraints and/or explicit waiting cost terms in the objective function. While, in
principle, both can be used in the same SLCIS model, it is far more common
to use one or the other. In models where only service level constraints are used,
these constraints will be tight in an optimal solution (since capacity is costly). If,
in addition, the demand is assumed to be inelastic,
i
is a linear function of the
decision variables x
ij
. In this case a significant simplification is achieved by using
the previous expression: setting the SC as an equality, we can eliminate decision
variables
i
from the model, replacing them with the right-hand side of (
17.23
).
17.2.3.3
Facility Costs
We assume that the decision to open a facility at i
2
I incurs two types of costs:
the
fixed cost FC
i
, which depends on the characteristics of the location i,andthe
variable cost VC
.
i
/, which depends on the amount of capacity
i
allocated to
the facility. The function
VC
./ is assumed to be non-decreasing and non-negative
with
VC
.0/
D
0; concavity of
VC
./ is a frequently made assumption, reflecting
