Geoscience Reference
In-Depth Information
X
x
ij
1;
j
2
J
(17.33)
i2I
x
ij
y
i
;
i
2
I;j
2
J
(17.34)
x
ij
2f
0;1
g
;
i
2
I;j
2
J
(17.35)
where M is a suitably large constan
t.
We assume that some finite limit can
b
e
imposed on the expected waiting time W
i
at any facility and that M
d.i;j/
C
W
i
for all j and i.
Of course a trivi
al
solution to this system is to have x
ij
D
0 for j
2
J;i
2
I
(which also implies W
i
D
0 for all i
2
I), i.e., to have complete loss of all customer
demand. Clearly, we are interested in non-trivial solutions where at least some
customers choose to obtain service. On the other hand, the system may not have
enough capacity to serve all customers. We therefore make the following definition.
Definition 17.1
A subset of customer nodes J
0
J is
serviceable
if
X
ma
j
X
i2I
i
:
j2J
0
A subset J
0
is
fully served
if
P
i2I
x
ij
D
1 for all j
2
J
0
,i.e.if(
17.33
) holds as
equality for all j
2
J
0
.
This definition simply assures that there is sufficient capacity to serve any service-
able subset. We are interested solutions where at least some serviceable subsets of
J are fully served. Unfortunately, the system (
17.30
)-(
17.35
) may have no such
solutions.
Example 17.1
Consider a network with one customer node j and two facility nodes
0;1 both of which contain facilities, i.e., y
0
D
y
1
D
1. Assume further that
0
D
1
>
ma
j
, and thus J
Df
j
g
is serviceable. Assume d.j;0/
D
d.j;1/. Then, since
W
i
D
0 if x
ij
D
0 and W
i
>0when x
ij
D
1 for i
D
0;1, there is no feasible solution
to the system (
17.30
)-(
17.35
). Indeed, if customers at j select facility i, it creates
non-zero waiting time at that facility, making the other facility a utility-maximizing
choice. Other similar examples of non-existence of equilibria with binary allocation
vectors are easy to construct.
The underlying reason for the phenomena illustrated above is that single-sourcing
strategies create discontinuities (a facility receives either all of customer's demand,
or none of it), while the existence of equilibria typically requires continuity of the
underlying functions. Indeed, intuitively it is clear that in the previous example
equilibrium allocations are achieved if the customers at j visit each facility with
equal frequency. This, of course, requires the relaxation of the single-sourcing
assumption, allowing x
ij
to take on fractional values, which are interpreted as
visit frequencies. In addition to replacing (
17.35
) with its linear relaxation, the
equilibrium-defining inequality (
17.30
) has to be adjusted as follows.
