Geoscience Reference
In-Depth Information
primarily, with the fact that customer utility is a function of the waiting time W
i
,
which is not directly controlled by the decision-maker, but rather arises as a result
of joint actions of the decision-maker and
all
customers: the former determines
facility locations and capacities
i
, while the latter determine the demand rates
i
.
This gives rise to traffic equilibrium conditions, where the actions of one customer
(adjusting their demand rate
j
and/or demand allocation x
ij
) change the waiting
times at the facilities and thus affect the utilities of all other customers. Thus, not
only is there a bi-level game being played between the decision-maker and the
customers, but there is also a simultaneous non-cooperative game being played
between the customers themselves. Moreover, the response functions in the latter are
rather complicated, which may lead to lack of equilibria (if customers are restricted
to simple strategies), or to multiple equilibria, not to mention serious difficulties
in computing these equilibria. We discuss these issues briefly below, referring the
interested reader to more general references on spatial equilibria like Nagurney
(
1999
).
17.3.2.1
AR: Models with Allocation-Only Reaction
In this type of models, it is assumed that the demand rate of each customer node is
fixed a priori, with
j
D
ma
j
for all j
2
J. However, the customers determine their
demand allocations, i.e., the values of x
ij
variables, in a utility-maximizing fashion.
For concreteness, we will assume the linear specification of the utility function
U
L
.d;
w
/ given by (
17.27
), though much of the discussion extends to alternative
specifications as well.
We first consider the original “single-sourcing” assumption. Since the customer
will allocate all of their demand to a utility-maximizing facility, x
ij
D
1 implies that
U
L
d.i;j/;W
i
U
L
d.k;j/;W
k
for all k
2
I with y
k
D
1;
which, assuming for simplicity that
w
D
d
D
1 in (
17.27
), is equivalent to
d.i;j/
C
W
i
d.k;j/
C
W
k
if y
k
D
1;k
2
I:
Recalling that
i
is given by (
17.6
)andW
i
by (
17.15
), this leads to the following
equilibrium conditions that must be satisfied by allocations x
ij
:
d.i;j/
C
W
i
Œd.k;j/
C
W
k
y
k
C
M.1
x
ij
/; i;k
2
I;j
2
J
(17.30)
W
i
D
.1
C
2
/
i
y
i
i
C
M.1
y
i
/
;
2
i
.
i
i
/
C
i
2
I
(17.31)
i
D
X
j2J
max
j
x
ij
;
j
2
J
(17.32)
