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

that 0≤ f ≤ f ≤ ...≤ f for all facility j. For ease of exposition, we may call the facility
1
j,|K|
j,1
j,2
built on location j as facility j from time to time. I={1,2,3,…,|I|} is the set of customer
locations, where d is the population size at customer location i. For ease of exposition,
i
we may call the customers at location i as customer i, and d as the demand of customer i
i
from time to time. It is assumed that one day is split into several non-overlapping activity
sessions in which one customer chooses at most one session to visit a facility. To represent
the fact, let T={1,2,3,…,|T|} be the set of activity sessions. Customer i has a preference

level over facility j in activity session t, represented by p . We have p i,j1,t >p i,j2,t if customer
ijt
i prefers facility j to facility j in activity session t. For those customers at the same
1
2
location, we assume that they have identical preference for the same facility in the same
activity session. The total budget for building facilities is B.
The decision maker’s decision is to choose locations to build facilities at a certain

scale level. To model this, let y ∈{0,1} be 1 if a facility is built at location j with scale
jk
level k or 0 otherwise. The special case y for any k is always 1 since customers can
0,k
decide to stay at home in any activity sessions. After facilities are built, each customer

either chooses one activity session to visit one facility or stays at home. That decision
is made according to her/his preferences. We assume that customers’ preferences are
exogenous; i.e., the preference over one facility will not be affected by other customers’
decisions or whether other facilities are built or not.
We first model customers’ choice when facilities all have ample capacity, i.e., the

number of customers going to the same facility in the same activity session is unlimited.
In this case, let x ∈{0,1} present whether customers at location i go to facility j in activity
ijt
session t (x =1) or not (x =0). Note that x will not be fractional in equilibrium, i.e., all
ijt
ijt
ijt
customers at the same location may make the same decisions, because the capacity of
facility is infinite.
2
Collectively, we may formulate the decision maker’s problem as

1 If the number of the scale level candidates of location j is less than |K|, set f j,k' to infinite for those k' ∈
K which cannot be chosen for location j.
2 Note that in this formulation, it does not matter whether we set x ijt ∈ {0,1} or x ijt ∈ [0,1]. However, as
we do not intend to solve this uncapacitated problem, we leave the setting to be binary to highlight
the fact that x ijt will either be 0 or 1 in an optimal solution.

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