Geoscience Reference
In-Depth Information
We d e fi n e x
ij
to be
demand allocation
decision variables, specifying what portion
of demand from customer node j
2
J is directed to facility i
2
I. We will initially
assume that demand allocations are binary, with the value of 1 if the demand stream
generated by customer node j is directed to facility i,and0 otherwise. The key
underlying assumption is that once the decisions about the number of facilities,
their locations y
i
and the service capacities
i
for i
2
I are made, the demand
allocations x
ij
can be determined; the exact mechanism for determining demand
allocations depends on the underlying assumptions about system dynamics and is
described later. Mathematically, we assume that x
ij
satisfies the following set of
constraints
X
x
ij
1; j
2
J
(17.3)
i2I
x
ij
y
i
;i
2
I; j
2
J
(17.4)
x
ij
2f
0;1
g
;i
2
I;j
2
J
(17.5)
These constraints are quite standard in location models: (
17.3
) ensures that at most
100 % of customer demand from j is allocated to the facilities, (
17.4
)prevents
allocating a customer to an unopened facility, and (
17.5
) enforces the binary
assumption for the allocations.
The integrality of x
ij
reflects the “single sourcing” assumption made in most
SLCIS models, requiring each customer node to be assigned to at most one
facility. An alternative is to allow “multisourcing”, in which case x
ij
is allowed
to be continuous, by replacing (
17.5
) with its linear relaxation. We also note that
constraints (
17.3
)-(
17.5
) represent “minimal” requirements on x
ij
; they are often
supplemented by other constraints describing the mechanisms by which allocation
of customers to facilities is made.
We allow for the possibility that the demand from j is not assigned to any facility,
i.e.,
P
i2I
x
ij
D
0, which we interpret as the case of “
intentionally
”
lost demand
,
i.e. demand that could have been captured but was lost at the system planning stage,
usually due to insufficient overall system capacity. We note that even when x
ij
D
1
some demand from i may be lost due to congestion at facility J - this portion can be
regarded as “unintentionally” lost demand, since the system did attempt to provide
service to customers at i. The amount of lost demand is typically controlled via a
penalty cost or constraints—we will return to these when we discuss specific model
formulations below. For each facility i we define the set N
i
Df
j
2
J
j
x
ij
D
1
g
,
which represents the
service region
of facility i (clearly N
i
D;
when y
i
D
0).
Observe that once
i
and x
ij
are known, the demand rate facing an open facility
i is a Poisson process with rate
i
D
X
j2N
i
j
D
X
j2J
j
x
ij
:
(17.6)
As mentioned earlier, the Poisson property results from the fact that superposition
of Poisson processes is also a Poisson process. Moreover, the demand streams faced
