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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)
 
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