Geoscience Reference
In-Depth Information
i
D
P
j2J
j
x
ij
). Note that the probability distribution of
i
is quite involved as it
depends on the actual values of x
ij
(j
2
J). Denote by
P
x
.
i
D
s/ the probability
that
i
is equal to s (s
D
0;:::;
z
i
).
Albareda-Sambola et al. (
2011
), consider two possible outsourcing actions. We
use one of them to illustrate the difficulties that may arise in formulating a determin-
istic equivalent. In particular, we consider the so-called customer outsourcing. In
this case, when the number of customers allocated to some facility i
2
I requesting
the service (demand customers) exceeds q
i
,
i
q
i
customers have to be served
directly from an external source. A FIFO policy is assumed for deciding which
customers to serve from the facility and which ones to outsource. The cost for
supplying each outsourced costumer is denoted by g
i
and depends on the facility
to which the customer was originally assigned. Denote by
P
i
.s/ the conditional
probability of serving a demand customer assigned to facility i
2
I given that the
total number of demand customers assigned to facility i
2
I is s (i.e.,
i
D
s). We
have
D
(
1 if s
q
i
q
i
=s otherwise
P
i
.s/
D
min
f
q
i
;s
g
s
(8.62)
Due to the fact that the expected value is additive, the recourse function can be
written as the sum of the expected service cost plus the expected outsourcing cost.
These terms can be computed as follows:
z
i
E
.service cost/
D
X
i2I
X
sD0
P
x
.
i
D
s/
E
.Service cost
j
i
D
s/
2
4
P
x
.
i
D
s/
X
3
D
X
i2I
X
z
i
5
;
j2J
P
.
j
D
1
j
i
D
s/
P
i
.s/c
ij
x
ij
sD0
(8.63)
E
.Outsourcing cost/
D
X
i2I
X
sD0
P
x
.
i
D
s/
z
i
E
.outsourcing cost
j
i
D
s/
0
1
D
X
i2I
X
sDq
i
C1
P
x
.
i
D
s/.s
q
i
/
z
i
@
A
:
g
i
(8.64)
A close look into the above expressions reveals that even for tiny instances, a
deterministic equivalent formulated from these expressions becomes intractable. In
fact, the number of scenarios is huge even for a small number of customers (note
that a scenario is defined not only by the set of customers which request the service
but also by the order of the requests when calling for service). Nevertheless, for the
homogeneous case, i.e., p
j
D
p, j
2
J, it is possible to go farther and derive a
tractable deterministic equivalent, as we show next.
