Geoscience Reference
In-Depth Information
them may have to travel to more distant and more congested facilities than the ones
available in their immediate neighborhood. On the other hand, since the objective
function combines the costs borne by the decision-maker (facility costs
SFC
) with
those borne by the customers (travel cost
STC
and waiting cost
SWC
), the interests
of both parties should be “balanced” in the solution. Customer demand is assumed
to be inelastic, with
j
D
ma
j
for all j
2
J. Since customer utility has no effect
in this model, there is no need to define it. We note that x
ij
are usually assumed
to be binary in NR models (though it is easy to construct examples showing that
higher objective values may be possible with fractional assignments). This is due
to the concern that enforcing fractional demand allocations is likely impractical in
most contexts. Thus, in NR models only the “minimal” constraints (
17.3
)-(
17.5
)
need to be imposed on demand allocations: the decision-maker is free to choose any
allocation that satisfies these constraints.
The other three model types assume some form of customer reaction in the form
of utility-maximizing behavior. The description of the utility mechanism is provided
next.
17.3.1
Customer Utility Functions
Recall that
u
j
is the utility derived by customer j
2
J from the service provided
by the facilities. Note that there are two costs borne by the customer: travel and
waiting. Suppose a customer experiences travel distance d (as before we assume
that distances have been redefined to represent travel costs) and expected system
waiting time
w
. Let the utility U.d;
w
/ be a non-increasing function of d and
w
.To
relate
u
j
to U.d;
w
/ we assume that the total utility derived by customer j is only
affected by the facilities this customer actually visits, letting
u
j
D
X
i2I
U.d.i;j/;W
i
/x
ij
;
(17.26)
Note that this definition remains valid even when the single-sourcing assumption
is relaxed. In this case, x
ij
represents the proportion of time facility i is used by
customer j and
u
j
can be interpreted as the resulting
expected
utility. Observe also
that if a customer does not receive service from any facility, x
ij
D
0 for all i
2
I
and
u
j
D
0.
Perhaps the most natural specification for the utility function U.d;
w
/ is the linear
form
U
L
.d;
w
/
D
.
d
d
C
w
w
/;
(17.27)
where
d
;
w
>0are the relative weights on travel distance and waiting time,
respectively. When
w
D
1, the parameter
d
can be interpreted as the average
travel speed, so that
d
d is the average travel time, and the right-hand side of (
17.27
)
