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

3.2 Capacitated Facilities with Customer Preference
In order to incorporate capacity limitation, let q represent the capacity of facility j
jk
with scale level k. Without loss of generality, we assume that 0 ≤ q ≤ q ≤⋯ ≤ q for all
j,|K|
j,2
j,1
facility j. Recall that it is always possible that all built facilities are not attractive enough
for a customer (e.g., are all too far), and the customer may choose to stay at home without
visiting any facility. To model this, we add a virtual location, location 0, into the set J' and
define J={0}∪ J'. Location 0 has no construction cost (i.e., f for all ), infinite capacity (i.e.,
0,k
f =0 is infinite for all k ∈ K), and zero preference level for all customers in all activity
0,k
sessions (i.e., p = 0 for all i ∈ I, t ∈ T).
i,0,t
We then modify x so that x ∈ [0,1] presents the proportion of customer i going
ijt
ijt
to facility j in activity session t. Note that now x must be fractional instead of binary
ijt
because now facilities are capacitated, and it is possible for customers at the same location
to make different decisions.
We now need to add constraints to ensure that a customer cannot go to a facility-
session pair if there is another more preferred pair that is still available. To do this, we
define three auxiliary variables w , z , and x . A binary variable w is 1 if facility j is still
ijt
jt
i
jt
cap
t
i

hat

acity
i
n

out
either
s
s
a
f
implie
o
ctivity
acility
f
i

se
ssion

available in equilibrium in activity session t with respect to the capacity constraint. The in is

̅ ��� ̅ ��� is 1 implies that either facility is out of capacity in activity session n
1

variable z represents customer i’s preference of the most preferred facility-session pair
i
equilibrium or that facility-session pair is the most preferred out of all available ones.
equilibrium or that fa cility-session pair is the most preferred out of all available ones.
among all available ones. The binary variable x is 1 if at least one customer in location
ijt
i go to facility j in activity session t. In effect, x is 1 implies that either facility j is out
w
f
c
w
e

d
m
o
i
r
e
,
n
p
s
s
o
o
p

t
a
e
n

o
a

i
n
i
t
s

r
h
t

c
e
h
a
g
i
t
W
e
su

t
t
d
customer
o
e
i

l
h
both

With such modifications, we propose a new model that integrates both customer
a
m
ijt
of capacity in activity session t in equilibrium or that facility-session pair is the most
and facility capacity as
and facility capacity as
preferred out of all available ones.
,
(8)
max � � � �
max � � � � � � ��� , (8)
���
� �� � �
� � � � � �� � �
s.t. (2) – (4),
s.t. (2) – (4),
� � �� �� �� � � � � � ��� ≥0 ∀ ∀ , , (9)

(9)
,
� � � � � ≥0

,

���
�
�
� � � � � �
� �
� � �� �� �� � � � � � ��� ≤ ∀ ∀ , , (1 (10)

)
0

� � � � � ≤

,
,
��
�
���
��
�
� � � � � �
� �

)
1
� ≥ ≥ � � ∀ ∀ , , , (1 (11)
,
,

,

���
��� ��
�
10
)
2

��� ��� ��� � �1 � �≥ ∀ ∀ , , , (1 (12)
,
̅ ̅

̅
,

,

�≥

�
1
�
̅
�
���
���
�
���
�

)
̅ ��� ��� ≥ ∀ , , , (1 (13)
3
,
,
̅ ≥
∀

,
���
���

��� ��� ≥ 0 ∀ , , , (1 (14)
)
4

,
0

,
≥
,
∀

�� �� { 0 , 1} ∀ , , (1 (15)
{0, 1} ∀ , ,
)
5

̅ ��� ̅ ��� {0, 1} ∀ , , , (1 (16)
6
)
1}
,

∀

,

{0,
,

,
)
7
�� �� {0, 1} ∀ , , (1 (17)
1}

,
∀

{0,

where is a positive and sufficiently large constant.
wher e i s a positiv e and sufficien tly l arge cons tan t .
The objective function (8) of the model is to maximize the total number of served
The objective function (8 ) of t he m od el i s to m aximize the to ta l n u m b e r o f s e r v e d
.

s
r
ot
co
e
f
a
c
c
e

i
t
y

l
cu customers. Note that the customers served by facility 0 are not counted since facility
i
t
o
t
u
m
n
e
s
i
n
d
s

n

f
a
e
cu
y

i
t
y
h
c
i
l
b

se
o
s
e
r
m
d
s

t
r
v
e
h
ar
t
t

a

o
N
e
e
t

t

0
epresents
(
2)
custo
th

stay
e
r
a
a
ing
at
i

ers
e
hom

–
r
o
same

C
m
s
r
e
ns

4)

s
.
nt
t
a
(
0 0 represents customers staying at home. Constraints (2) – (4) are same as the
uncapacitated ones. C onstraint ( 9) e nsures t hat the customers c an only go to t he l ocation
uncapacitated ones. Constraint (9) ensures that the customers can only go to the location
where a facility is built. Furthermore, the number of customers from all locations cannot
where a fac i lity i s bu ilt. F urther mor e, the nu mber o f cus t ome r s fro m a l l loc ations canno t
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