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NTU Management Review Vol. 33 No. 3 Dec. 2023

2
2
f

a
y
C Collectively, we may formulate the decision maker’s problem as
t
l
i
o
r
o
y
,
l
e
v

m
e

w
c
e
t

t
he decision maker’s problem as
l
e
u
m
a
l

max , ,
(1)
max ���� ����
���
� � ��� (1)
� � � �� �
� �� �
� �
s.t. ≤1 ∀ ,

(2)
s.t. � �
�� �� ≤1 ∀ , (2)
�
� � �

� � ≤ ,
(3)
� � ≤
,
�� �� �� (3)
�
�
�
� � � �
� ��

�� ≤ 1 ∀ ,
(4)

,

1
�� ≤

∀
��� (4)
���
� �� �
� � � �
��� ≤ ≤ � ∀ ∀ , , , (5)

(5)
,

�
,

,

�� ���
��
���
���
� � �
�
{0, 1} ∀ , , ,
��� { 0 , 1} ∀ , , , (6)
(6)
���
{0, 1} ∀ ,
�� { 0 , 1} ∀ , (7)
(7)
��
The objective function (1) of the model is to maximize the total number of served
The objective function (1) of the model is to maximize the total number of served
The objective function (1) of the model is to maximize the total number of served
customers. Constraint (2) ensures that the decision maker can only build at most one
customers. Constraint (2) ensures that the decision maker can only build at most one
customers. Constraint (2) ensures that the decision maker can only build at most one
facility with one scale level at each location. Constraint (3) requires that the total
facility with one scale level at each location. Constraint (3) requires that the total
facility with one scale level at each location. Constraint (3) requires that the total
construction cost should not exceed the given budget B. In (4), the total proportion of
customers visiting facilities cannot exceed 1. The fact that each customer chooses her/
construction cost should not exceed the given budget . In (4), the total proportion of
construction cost should not exceed the given budget . In (4), the total proportion of
his most preferred facility-session pair is modeled in (5), where A is set to 1 if p > 0 or
ijt
ijt
customers visiting facilities cannot exceed 1. The fact that each customer chooses
customers visiting facilities cannot exceed 1. The fact that each customer chooses
0 otherwise. Constraint (5) ensures that a customer chooses a facility-session pair only if
her/his possesses a positive utility over it. Note that because the decision maker cares only
her/his most preferred facility-session pair is modeled in (5), where is set to 1 if
her/his most preferred facility-session pair is modeled in (5), where ��� is s e t to 1 if
���
the total number of served customers, whether a customer goes to her/his most preferred
(
5)
se.
> 0 or 0 otherwise. Constraint (5) ensures that a customer chooses a facility-
i
hooses
t
a
otherw

hat
es

c
e
customer
nsur
fa
Constraint
a
or

cility-
> facility-session pair or any other one is not an issue. Thus, there is no need to write this
0
0
�� ���
�
constraint to restrict one to visit her/his most preferred option. Constraints (6) and (7) state
session pair only if her/his possesses a positive utility over it. Note that because the
session pair on ly if h e r /his poss esse s a pos itiv e utility o v er i t . N ot e t ha t be c a us e t he
that x and y are binary. Note that in this formulation, it does not matter whether x is
ijt
jk
ijt
decision maker cares only the total number of served customers, whether a customer
decision maker cares only t he t ot al number of s erved customers, w h et h er a cu s t o m er
fractional or binary.
mo
a
h
h
/
oth
s
i
i
issue.
not
s
Thus,

n
one
er
ny
r
r

e
fe
o

r
t
o
d

f
a
e
s
cility-session
e
re
pair
g goes to her/his most preferred facility-session pair or any other one is not an issue. Thus,

p
s
a
t
or
there is no need t o write this c onstraint t o r estrict one t o vi s it he r/h i s most p refe rre d
there is no need to write this constraint to restrict one to visit her/his most preferred
9

�0, 1�. However,

o
r
2
2 N o t e th at i n this f o r mu lation , it d o es no t matter wh eth e r we s e t ��� ��� {0, 1} or �0, 1� . Ho we ver ,

Note that in this formulation, it does not matter whether we set {0,
1}
���
���
as we do not intend to solve this uncapacitated problem, we leave the setting to be binary to highlight the
as w e do no t inten d to so lve this u n cap a ci tated p r ob lem, w e lea v e th e s etting to b e bi nar y t o hi g h l i g ht t he
fact that will either be 0 or 1 in an optimal solution. will either be 0 or 1 in an optimal solution.
fact that ��� ���
1 3
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