Geoscience Reference
In-Depth Information
In this model, d
j!
represents the demand of node j
2
J under scenario !
2
ǝ,
and a
ij
!
represents the distance (or travel time) between nodes i
2
J and j
2
J
under scenario !
2
ǝ. The minmax objective arises from the combination of (
8.6
)
and (
8.7
).
The solution provided by the previous model tends to be overly conservative. It
reflects a complete aversion of the decision maker towards risk. In fact, by planning
for the worst case scenario (the maximum weighted distance occurring across all
scenarios), the decision maker may be planning for a scenario which turns out to
be very unlikely. A better compromise can be achieved by considering the minmax
regret
1
criterion, in which the decision maker chooses the decision that minimizes
the maximum regret across all scenarios. The corresponding model is obtained by
replacing (
8.7
) with
X
X
d
j!
a
ij
!
x
ij
!
v
!
v; !
2
ǝ;
(8.13)
i2J
j2J
where v
!
is the optimal value of problem (
8.1
)-(
8.5
) solved for scenario !
2
ǝ.
Serra and Marianov (
1998
) consider the above minmax regret model after scaling
the demands. In particular, for each scenario, they divide each demand by the total
demand under that scenario. The authors also note the well-known fact that when
the optimal objective function differs significantly across the different scenarios, the
relative regret is a more appropriate robustness measure (see, for instance, Kouvelis
and Yu
1997
). In this case, (
8.13
) should be replaced by
P
i
2
J
P
j
2
J
d
j!
a
ij
!
x
ij
!
v
!
v
!
v; !
2
ǝ:
(8.14)
For this problem, the same authors propose a heuristic approach.
A different problem is addressed by Serra et al. (
1996
). They consider a firm that
wishes to locate p facilities in a competitive environment. The goal is to maximize
the minimum market captured in a region where competitors are already operating.
The criterion considered corresponds to the “maximization” version of the minmax
“cost” criterion discussed above. Uncertainty is assumed for the demand and for the
location of the competitors. Again, a heuristic approach is proposed for tackling the
problem.
If the allocation of customers to facilities is also an
ex ante
decision, the models
above can be easily adapted. In this case, the scenario index should be removed
from the allocation variables, i.e., the allocation variables become those introduced
in model (
8.1
)-(
8.5
). Furthermore, the location variables y
i
are no longer necessary,
as variables x
ii
(i
2
J) can play their role.
1
In each scenario, the regret of a solution is the difference between the cost of the solution if the
scenario occurs and the optimal cost that can be achieved under that scenario (see Kouvelis and Yu
1997
for further details).