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where
WC
;
FC
;
VC
are the waiting cost, fixed cost and variable cost parameters
respectively. This function is minimized subject to constraints (
17.53
), (
17.55
)
specified as equality, as well as (
17.54
), (
17.56
)and(
17.57
).
Observe that for any specified values of x
ij
and y
i
, the optimal capacity
i
can
be determined separately for each facility. Indeed, it is not difficult to show that
i
D
i
C
r
WC
VC
i
:
Observe the similarity of this expression to (
17.22
) discussed earlier. It also has the
same interpretation: the optimal capacity at facility i consists of the minimal level
i
, necessary to ensure system stability, and “capacity cushion” which grows with
the square root of
i
and whose size depends on the ratio of waiting and capacity
costs. Substituting the last expression into (
17.60
) and performing some algebraic
manipulations allows us to re-state the objective function as
s
X
mininize Z
D
LJ
X
j
X
d.i;j/
j
x
ij
C
2
p
WC
VC
X
j
X
j
x
ij
C
FC
X
i
y
i
;
2
J
i
2
I
2
J
i
2
I
j
2
J
2
I
subject to constraints (
17.53
), (
17.56
), and (
17.55
) specified as equality; the
variables
i
and
i
are no longer needed.
This is a MIP with a single concave (more specifically, square root) term in
the objective. Several methods are available to obtain exact solutions for models
of this type, which also arise in location-inventory models, competitive location
models and other contexts. One approach, based on Lagrangian Relaxation, is
described in Shen (
2005
); a variant of this is used in Castillo et al. (
2009
). Another
approach, based on piecewise linear approximation of the concave term, is presented
in Aboolian et al. (
2007
).
It should be noted that in view of the discussion preceding (
17.22
), a similar
“trick” for replacing the congestion cost term with a concave form should work for
more general queueing systems as well, at least as an approximation.
The second approach for obtaining exact solutions to balanced-type SLCIS is
based on Elhedhli (
2006
). Once again we start with the model whose objective
function is given by (
17.60
) and assume an M=M=1 queue at each facility. In
addition, it is assumed that processing capacity of a facility must be equal to one
of H
C
1 discrete values, i.e., that
i
2f
0;
1
;
2
;:::;
H
g
for all i
2
I,where
1
<
2
<:::<
H
.
Treating the expected queue length L
i
as a decision variable, we rewrite
(
17.59
)as
H
X
L
i
1
C
L
i
h
z
ih
;i
2
I;
i
D
(17.61)
hD1
