DISCOVERING CREDIT CARDHOLDERS’ BEHAVIOR
269
Given a set of r variables about the cardholders A = A
1
, . . . , A
i
, let A
t
= A
i 1
, . . . , A
ir
be the development sample of data for the variables, where i = 1, . . . , n and n is the sample size. We try to determine the coefficients of the variables, denoted
by X = x
1
, . . . , x
r
, boundary b
1
to separate G
1
from G
2
, G
3
, and G
4
, boundary b
3
to separate G
4
from G
1
, G
2
, and G
3
, boundary b
2
to separate G
2
and G
3
. This separation can be represented by:
A
i
X ≤ b
1
, A
i
∈ G
1
b
1
≤ A
i
X ≤ b
2
, A
i
∈ G
2
b
2
≤ A
i
X ≤ b
3
, A
i
∈ G
3
, b
3
≤ A
i
X, A
i
∈ G
4
Similar to two-class and three-class models, we apply two measurements for better separations. Let α
1 i
be the overlapping degree with respect of A
i
within G
1
and G
2
, α
2 i
be
the overlapping degree with respect of A
i
within G
2
and G
3
, and α
3 i
be the overlapping
degree with respect of A
i
, within G
3
and G
4
. Let β
1 i
be the distance from A
i
within G
1
and G
2
to its adjusted boundaries A
i
X = b
1
− α
1 ∗
, and A
i
X = b
1
+ α
1 ∗
, β
2 i
be
the distance from A
i
within G
2
and G
3
to its adjusted boundaries A
i
X = b
2
− α
2 ∗
,
and A
i
X = b
2
+ α
2 ∗
, and β
3 i
be the distance from A
i
within G
3
and G
4
to its adjusted
boundaries A
i
X = b
3
− α
3 i
, and A
i
X = b
3
+ α
3 i
. We want to reach the maximization of β
1 i
, β
2 i
, and β
3 i
and the minimization of α
1 i
α
2 i
and α
3 i
simultaneously. After putting α
1 i
, α
2 i
, α
3 i
, β
1 i
, β
2 i
, and β
3 i
into the above four-class separation, we have: M4
Minimize
i
α
1 i
+ α
2 i
+ α
3 i
and Maximize
i
β
1 i
+ β
2 i
+ β
3 i
Subject to: G
1
: A
i
X = b
1
+ α
1 i
− β
1 i
, A ∈ G
1
, Bankrupt charge-off
G
2
: A
i
X = b
1
+ α
1 i
− β
1 i
, A
i
X = b
2
+ α
2 i
− β
2 i
, A i ∈ G
2, Non-bankrupt charge-off
G
3
: A
i
X = b
2
+ α
2 i
− β
2 i
, A
i
X = b
3
− α
3 i
+ β
3 i
, Ai ∈ G
3, Delinquent
G
4
: A
i
X = b
3
− α
3 i
+ β
3 i
, A
i
∈ G
4
Current b
1
+ α
1 i
≤ b
2
− α
2 i
, b
2
+ α
2 i
≤ b
3
− α
3 i
where A
i
, are given, X, b
1
, b
2
, b
3
are unrestricted, and α
1 i
, α
2 i
, α
3 i
, β
1 i
, β
2 i
, and β
3 i
≥ 0.
The constraints b
1
+ α
1 i
≤ b
2
− α
2 i
and b
2
+ α
2 i
≤ b
3
− α
3 i
guarantee the existence of four groups by enforcing b
1
lower than b
2
, and b
2
lower than b
3
. Then we apply the compromise solution approach Shi, 1989, 2001 to reform the model. We assume the
ideal value of −
i
α
1 i
be α
1 ∗
0, −
i
α
2 i
be α
2 ∗
0, −
i
α
3 i
be α
3 ∗
0, and the ideal value of
i
β
1 i
be β
1 ∗
0,
i
β
2 i
be β
2 ∗
0,
i
β
3 i
be β
3 ∗
0.
270
KOU ET AL.
Then, the four-group model M4 is transformed as: M5
Minimize d
− α
1
+ d
+ α
1
+ d
− β
1
+ d
+ β
1
+ d
− α
2
+ d
+ α
2
+ d
− β
2
+ d
+ β
2
+ d
− α
3
+ d
+ α
3
+ d
− β
3
+ d
+ β
3
Subject to: α
1 ∗
+
i
α
1 i
= d
− α
1
− d
+ α
1
, β
1 ∗
−
i
β
1 i
= d
− β
1
− d
+ β
1
, α
2 ∗
+
i
α
2 i
= d
− α
2
− d
+ α
2
, β
2 ∗
−
i
β
2 i
= d
− β
2
− d
+ β
2
, α
3 ∗
+
i
α
3 i
= d
− α
1
− d
+ α
3
, β
3 ∗
−
i
β
3 i
= d
− β
3
− d
+ β
3
, b
1
+ α
1 i
≤ b
2
− α
2 i
, b
2
+ α
2 i
≤ b
3
− α
3 i
, G
1
: A
i
X = b
1
+ α
1 i
− β
1 i
, A
i
= G
1 Bankrupt charge-off
G
2
: A
i
X = b
1
− α
1 i
+ β
1 i
, A
i
X = b
2
+ α
2 i
− β
2 i
, A
i
∈ G
2, Non-bankrupt charge-off
G
3
: A
i
X = b
2
+ α
2 i
− β
2 i
, A
i
X = b
3
− α
3 i
+ β
3 i
, A
i
∈ G
3, Delinquent
G
4
: A
i
X = b
3
− α
3 i
+ β
3 i
, A
i
∈ G
4
, Current
Where A
i
, are given, b
1
≤ b
2
≤ b
3
, X, b
1
, b
2
, b
3
, are unrestricted, and α
1 i
, α
2 i
, α
3 i
, β
1 i
, β
2 i
and β
3 i
≥ 0.
A SAS-based algorithm of this four-class model is proposed as follows:
Algorithm 1.
Step 1 .
Use ReadCHD.sas to convert both the training and verifying data into SAS data sets.
Step 2 .
Use GroupDef.sas to divide the observations within the training data sets into four groups: G
1
, G
2
, G
3
, and G
4
. Step 3
. Use 4GModel.sas to perform the separation task on the training data. Here,
PROC LP in SAS is called to calculate the M5 model for the best solution of the four- class separation given the values of control parameters α
1 ∗
, β
1 ∗
, α
2 ∗
, β
2 ∗
, α
3 ∗
, β
3 ∗
, b
1
, b
2
, b
3
. Step 4
. Use Score.sas to produce the graphical representations of the training results.
Step 3–4 will not terminate until the best training result is found. Step 5
. Use Predict, sas to mine the four classes from the verifying data set.
5. Empirical study and managerial significance of four-class models