Geoscience Reference
In-Depth Information
equilibria when solving realistic-size problem instances for mammography clinics
in Montreal, Canada. The obvious advantage of the single-sourcing approach is that
the system (
17.30
)-(
17.35
) is much easier to solve and can be embedded as part of
constraints in a larger SLCIS model.
The second approach is to use distance-only utilities U
D
.d/ given by (
17.28
).
Since these are independent of waiting times, the existence of customer flow
equilibria is no longer an issue; utility-maximizing behavior by customers merely
implies that once facility locations are specified, each customer travels to the closest
facility, replacing (
17.30
) with
d.i;j/
d.k;j/y
k
C
M.1
x
ij
/; i;k
2
I;j
2
J;
(17.39)
which leads to significant simplifications (obviously, single-sourcing assumption
can be retained here as well).
Yet another alternative to customer flow equilibrium is to use market share
allocation approach, as discussed in Sect.
17.3.2.4
below.
17.3.2.2
DR: Models with Demand-Only Reaction
In this model class, the decision-maker has the control of the demand allocation
vector
, however, the demand
j
D
.
u
j
/ for customer node j
2
J is assumed to
be a function of the utility
u
j
realized by customers at j. Following Brandeau et al.
(
1995
) we assume that
x
j
D
ma
j
h.
u
j
/;
where, as defined earlier,
ma
j
is the maximum possible demand rate at node j and
h.
u
/
2
Œ0;1 is a strictly decreasing, twice differentiable function with h.0/
D
1 and
h.
u
/
!
0 as
u
!
u
mi
j
,where
u
mi
j
is the lower bound on the utility for customers
at j (e.g., if utilities are scaled to be non-negative, then we can set
u
mi
j
D
0). Thus,
h.
u
j
/ can be interpreted as the percentage of the maximum available demand at j
that is “captured” by the system; it is often called the “participation rate”.
Recall that by (
17.26
), the utility
u
j
is a function of the waiting time and travel
distance faced by customers at j. As in the case of
NR
models, we will assume
that x
ij
is binary, motivated by the same considerations as before: when customer
demand allocations are dictated by the decision-maker, rather than by an equilibrium
condition of the previous section, enforcing fractional assignments is typically
unrealistic. Thus, assuming all customers at j will be served (as will be shown
below, this assumption holds automatically in DR models), x
ij
D
1 for exactly one
i
D
i.j/
2
I. Then, we have
j
.d.i.j/;j/;W
i.j/
/
D
max
j
h.U.d.i.j/;j/;W
i.j/
//; j
2
J:
(17.40)