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Optimal Allocation of Capacitated Facilities Considering Time-Dependent User Preference for User Number
Maximization

After the third iteration, there is no other location can build more facilities at, so
both GSAMFE and GSAMF end iterations. Therefore, for GSAMFE algorithm, the final
construction plan y is provided and it will build facility 1, facility 2 and facility 3, all with
scale level 1. To get the exact objective value of our outcome, we solve maximum flow

with such y. Finally, our GSAMFE algorithm ends with the estimated flow value z'(y) is 25
and the actual objective value z(y) is 22. Note that in this example, if we use our GSAMF
algorithm, the final construction plan y will build facility 1 with scale level 1, facility 2

with scale level 2 and facility 3 with scale level 1. It gives objective value z(y)=24.
If we solve this example with mixed integer program through solver, we can find that
the optimal solution is 24 by constructing facility 1 with scale level 1, facility 2 with scale
level 2 and facility 3 with scale level 1. In this example, we find that GSAMFE sometimes
may overestimate and choose a solution worse than GSAMF.

5. Numerical Study

5.1 Experiment Setting
To present the experimental results of the algorithm, we adopt seven factors to
analyze the performance under different circumstances. The first factor is the size of the
problem, which three sizes—small, medium and large—are considered. We set m=10,
n=20 as the small size, m=30, n=60 as the medium size and m=100, n=200 as the large

size for the problem. The second factor is the number of scale levels which each facility
has. We consider two scenarios: one is each facility has one scale level and the other is
each facility has three scale levels. The third factor is the numbers of activity sessions
which is under two scenarios: problems with only one activity session and with three

activity sessions.
The fourth and fifth factors are location preference and time preference which affect
the customer’s time-dependent preference over facilities. In our experiment, the customer
and facility locations are mapped onto a 2-dimentional coordinate. There are two types
of customers’ location distributions in our setting: uniformly distribution and clustering

distribution. In the former type, the locations of customers are randomly separated in the
map; in the latter type, customers tend to locate near several clusters. The parameter d
ij
represents the Euclidian distance between customer i and facility j, and we set customer

24