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International Journal of Information Technology & Decision MakingVol. 10, No. 6 (2011) 11611174c World Scientific Publishing Company
DOI: 10.1142/S0219622011004750
A FUZZY LINEAR PROGRAMMING-BASED
CLASSIFICATION METHOD
AIHUA LI
School of Management Science and Engineering
Central University of Finance and Economics
Beijing 100080, P. R. China
YONG SHI
Fictitious Economics and Data Technology Research Centre
Chinese Academy of Science
Beijing 100081, P. R. China
College of Information Science and Technology
University of Nebraska
NE 68182, USA
JING HE and YANCHUN ZHANG
Centre for Applied Informatics
Victoria University, Melbourne City MC
VIC 8001, Australia
Fictitious Economics and Data Technology Research Centre
Chinese Academy of Science
Beijing 100081, P. R. [email protected]@gmail.com
Multiple criteria linear programming and multiple criteria quadratic programming clas-sification models have been applied in some field in financial risk analysis and credit riskcontrol such as credit cardholders behavior analysis. In this paper, a fuzzy linear pro-gramming classification method with soft constraints and criteria was proposed based onthe previous findings from other researchers. In this method, the satisfied result can beobtained through selecting constraint and criteria boundary variable di, respectively. A
general framework of this method is also constructed. Two real-life datasets, one from amajor USA bank and the other from a database of KDD 99, are used to test the accuraterate of the proposed method. And the result shows the feasibility of this method.
Keywords: Classification; data mining; MCLP; fuzzy linear programming; membershipfunction.
Corresponding author.
1161
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1. Introduction
Data mining becomes an important international technology with the development
of database and internet, which can extract nontrivial, implicit, previously unknown
and potential useful patterns, or knowledge from database. Classification is one of
the functions in data mining, which is a kind of supervised learning. There are two
steps in the classification process.1 First, hidden pattern or discriminant function
can be derived from the training set. Second, the pattern or discriminant function
is applied to classify the testing dataset. The training accurate rate and testing
accurate rate are often used to evaluate the model.
The term of classification methods initially employ artificial intelligent (AI),
traditional statistics and machine learning tools, such as decision tree,2 linear dis-
criminant analysis (LDA),
3
support vector machine (SVM),
4
and so on. They havebeen applied in real-life medical, communication, and strategic management prob-
lems. For different datasets with different characters, classification methods show
their different advantages and disadvantages. For example, SVM or neural network
(NN) fits well for the output of some dataset, but it may result in overfit problem
sometimes. LDA shows its advantage when the datasets obey normal distribution,
but not a good choice in other conditions.
Linear programming (LP) classification method was first proposed in 1980s,57
which showed its potential applications. In 1990s, multiple criteria linear program-
ming (MCLP) and multiple criteria quadratic programming (MCQP) classificationmodels were developed,810 which have been successfully used in credit cardholders
behavior analysis1114 and network intrusion detection later.15 He et al.16 proposed
a fuzzy linear programming (FLP) model only with soft criteria, in which a sat-
isfied solution could be solved. In this paper, we proposed a FLP classification
method with soft constraints and criteria based on the previous researchers work.
This paper is presented as follows: Sec. 2 reviews LP, MCLP, and FLP method.
Section 3 proposes FLP with soft constraints and criteria, which means decision
maker can choose the reasonable bound for constraints in deriving a satisfied solu-
tion. Section 4 uses two examples, one from a major USA bank and the other fromthe database of KDD 99,17 to test the accurate rate of the proposed method. Some
remarks are given in Sec. 5.
2. LP, MCLP, and FLP Classification Models
In the LP classification method, the objectives of initial forms can be categorized
as MMD and MSD.6 Here, MMD means maximize the minimum distance of obser-
vations from the critical value. MSD means minimize the sum of the distance of theobservations from the critical value. For example, in the credit cardholder behavior
analysis a basic framework of two-class problems can be presented as.
Given a set of r variables (attributes) about a cardholder a = (a1, a2, . . . , ar),
let Ai = (Ai1, Ai2, . . . , Air) be the development sample of data for the variables,
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A Fuzzy Linear Programming-based Classification Method 1163
where i = 1, 2, . . . , n and n is the sample size. We want to determine the best
coefficients of the variables, denoted by X = (x1, x2, . . . , xr)T, and a boundary
value b (a scalar) to separate two classes: G (Good for nonbankrupt accounts) and
B (Bad for bankrupt accounts), that is as follows:
AiX b, Ai B (Bad),
AiX b, Ai G (Good).
To measure the separation of Good and Bad, we define:
i = the overlapping of two-class boundary for case Ai (external measurement);
= the maximum overlapping of two-class boundary for all cases Ai(i < );
i = the distance of case Ai from its adjusted boundary (internal measurement);
= the minimum distance for all cases Ai from its adjusted boundary (i > ).
A simple version of Freed and Glovers model which seeks MSD can be written as
Minimizei
i,
Subject to:
AiX b + i, Ai B,
AiX b i, Ai G,
(2.1)
where Ai are given, X and b are unrestricted, and i 0.
The alternative of the above model is to find MMD as follows:
Maximizei
i,
Subject to:
AiX b i, Ai B,
AiX b + i, Ai G,
(2.2)
where Ai are given, X and b are unrestricted, and i 0.
A hybrid model7 that combines models (2.1) and (2.2) can be as follows:
Minimizei
i i
i,
Subject to:
AiX = b + i i, Ai B,
AiX = b i + i, Ai G,
(2.3)
where Ai are given, X and b are unrestricted, and i, i 0, respectively.
Shi et al.8 applied the compromise solution of MCLP to minimize the sum of
i and maximize the sum of i simultaneously. A two-criteria LP model is stated
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as follows:
Minimizei
i and Maximizei
i,
Subject to:
AiX = b + i i, Ai B,
AiX = b i + i, Ai G,
(2.4)
where Ai are given, X and b are unrestricted, and i, i 0, respectively.
For this model MCLP, the system explanation and summary are presented in
these papers.1820
In compromise solution approach,21 the best trade-off between ii and ii
is identified for an optimal solution. To explain this, assume the ideal value ofii be > 0 and the ideal value of ii be > 0. Then, ifii > ,
the regret measure is defined as d+ = ii + . Otherwise, it is defined as 0.
If ii < , the regret measure is defined as d = + ii; otherwise, it
is 0. Thus, the relationship of these measures are (i) + ii = d d+ , (ii)
|+ii| = d +d+ , and (iii) d
, d+ 0. Similarly, we derive
ii = d
d+
,
| ii| = d
+ d+
, and d
, d+
0.
An MCLP model for two-class separation is presented as
Minimize d
+ d
+
+ d
+ d
+
Subject to:
+i
i = d
d+ ,
i
i = d
d+
, (2.5)
AiX = b + i i, Ai B,
AiX = b i + i, Ai G,
where Ai, , and are given, X and b are unrestricted, and i, i, d , d+ , d
,
d+ 0.
In a FLP approach with soft criteria,16 membership functions for the criteria
Minimize
i i and Maximize
i i were expressed respectively by
F1(x) =
1, ifi
i y1U
i i y1Ly1U y1L , if y1L 0, respectively.
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A Fuzzy Linear Programming-based Classification Method 1167
For the model (2.2), we similarly define the membership function as follows:
F2(x) =
1, ifi
i y2U,
i i y2L
y2U y2L, if y2L 0,
(3.2)
where Ai are given, X and b are unrestricted, i, d3, d4 > 0, respectively.In order to unify the sign of models in this research, we use the same membership
function F1 in the model of (3.1), F2 in the model of (3.2) instead of them in
model (2.6), so model (2.6) would be changed into the following format:
Maximize ,
Subject to:i i y1U
y1L y1U ,
i i y2L
y2U y2L ,
AiX = b + i i, Ai G,
AiX = b i + i, Ai B,
(3.3)
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where Ai is known, X and b are unrestricted, and i, i, 0, y1L, y1U, y2L, and
y2U are the same in the models of (3.1) and (3.2).
To identify a fuzzy model for the model (2.4), we first relax the model (2.4)s
constraints to inequality constraints. Then, suppose d1 = d2 = d1, d3 = d4 = d2, afuzzy model with the combinations (3.1) and (3.2) for the relaxed (M4) will be
Maximize ,
Subject to:i i y1U
y1L y1U ,
i i y2L
y2U y2L ,
1 +AiX (b + i i)
d1 , Ai B,
1 AiX (b + i i)
d1 , Ai B,
1 +AiX (b i + i)
d2 , Ai G,
1 AiX (b i + i)
d
2
, Ai G,
1 > 0,
(3.4)
where Ai are given, X and b unrestricted, i, i > 0, respectively. d
i > 0, i = 1, 2
are fixed in the computation. The definitions ofy1L, y1U, y2L, and y2U are the same
as those in models (3.1) and (3.2), respectively.
There are two pieces of difference between the models (3.4) and (2.4). First,
instead of optimal solution, a satisfying solution is obtained based on the member-
ship function from the FLP. Second, with these soft constraints to the model (2.4),
the boundary b can be flexibly moved by the upper bound and the lower bound
with the separated distance di, i = 1, 2, 3, 4 according to the characteristics of thedata.
4. Experimental Studies
There are two datasets used here to test the accuracy rate of the proposed fuzzy
classification method with both soft criteria and constraints. The first dataset came
from a major US bank with 65 attributes which include the credit cardholders over
limit fee, over charge fee, and other information in credit card using history, etc.
There are in total 6000 records in the dataset. Here we compare the proposed FLPwith both soft criteria and constrains with MSD, MMD, and MCLP model.
We select 1400 records with 700 Good (nonbankrupt) and 700 Bad (bankrupt)
randomly from the dataset for training, and the left 4600 are used to test the clas-
sifier accuracy, which is based on the method of cross validation. In the experiment,
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A Fuzzy Linear Programming-based Classification Method 1169
b is given as 0.5 for all models, d1 = d2 = d
1 = 1, d3 = d4 = d
2 = 1.5, respectively
for fuzzy-model (3.4). There are five groups training results in Table 1 and testing
results in Table 2 listed below. In Tables 13, we define
Absolute accurate rate of Good = Sensitivity =t Good
Good,
Absolute accurate rate of Bad = Specificity =t Bad
Bad,
Catch rate = Accuracy = Sensitivity Good
Good + Bad
+ Specificity Bad
Good + Bad
,
where t Good is the number of the Good (Good records that were correctly
classified as much). Good is the number of Good; t Bad is the number of the
Bad (Bad records that were correctly classified as much). Bad is the number
of Bad. In this case, to catch a bad person is more important than to catch a
good cardholder in order to avoid the defaulting.
Tables 1 and 2 show us that model (2.5)-MCLP, fuzzy-model (3.3)-FLP1,
and the proposed model (3.4)-FLP2 are better than (2.1)-MSD and (2.2)-MMD.
Although model (2.2)-(MMD) is the best for Bad catching it cannot be selected
due to its poor Good catching and instability in the experiment. MCLP shows its
trade-off with the balanced Good and Bad accuracy rate. FLP1 works well
for the overall catch rate and a little worse than FLP2 for Bad catching. Thus,
among MSD, MCLP, and FLP if we give importance to catching Bad cardholder
and keeping a satisfied absolute accuracy rate, fuzzy model (3.4) would be a good
choice. Table 3 shows us that the choice of boundary value di in the model (3.4)
affects the result of classification. By adjusting the value of di , we can get the
satisfied result of classification in the training process.
The second dataset came from KDD 99. Here a connection is a sequence of TCP
packets starting and ending between which data flows from a source IP address to
a target IP address under some well-defined protocol. Each connection is labeled
as either normal or an attack, here dos is exactly one specific attack type. In this
task, we select 38 characters needed. There are 1,060,078 records in the dataset we
used in this example; 812,812 Normalrecords and 247,266 Dos records. First,
4000 records was selected randomly from the dataset for training, 2000 of which is
labeled Normal, the other 2000 is labeled Dos. Second, the left records, 810,812
records for Normal and 245,266 records for Dos were used for testing. Tables 4
and 5 show us the training and testing results.In Tables 4 and 5, we use:
Absolute accurate rate of Normal = Sensitivity =t Normal
Normal,
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Table1.
Trainingresultsof1400records.
DifferentGroups
Model1(MSD)
Model2(MMD)
M
odel5(MCLP)
FuzzyM
odel3(FLP1)
FuzzyMode
l4(FLP2)
AbsoluteAccuracyRateA
bsoluteAccuracyRate
Abso
luteAccuracyRate
Absolute
AccuracyRate
AbsoluteAcc
uracyRate
Good
Bad
CatchRateG
ood
Bad
CatchRate
Good
Bad
CatchRate
GoodB
ad
CatchRate
Good
Bad
CatchRate
Group1
0.6
7
0.6
8
0.6
8
0.0
6
0.9
3
0.5
0
0.74
0.7
4
0.7
4
0.7
7
0.8
0
0.7
8
0.6
2
0.8
2
0.7
2
Group2
0.7
0
0.7
0
0.7
0
0.0
4
0.9
0
0.4
7
0.79
0.7
9
0.7
9
0.7
5
0.7
8
0.7
6
0.6
8
0.8
3
0.7
5
Group3
0.6
9
0.7
0
0.7
0
0.0
4
0.9
3
0.4
9
0.77
0.7
7
0.7
7
0.7
4
0.7
8
0.7
6
0.6
7
0.7
9
0.7
3
Group4
0.6
9
0.7
0
0.6
9
0.0
6
0.9
1
0.4
8
0.76
0.7
5
0.7
6
0.7
5
0.7
9
0.7
7
0.6
2
0.8
4
0.7
3
Group5
0.7
2
0.6
9
0.7
1
0.2
8
0.6
0
0.4
4
0.73
0.7
8
0.7
5
0.2
8
0.6
0
0.4
4
0.5
8
0.8
4
0.7
1
Table2.
Testingresultsof4600records.
DifferentGroups
Model1(MSD)
Model2(MMD)
M
odel5(MCLP)
FuzzyM
odel3(FLP1)
FuzzyMode
l4(FLP2)
AbsoluteAccuracyRateA
bsoluteAccuracyRate
Abso
luteAccuracyRate
Absolute
AccuracyRate
AbsoluteAcc
uracyRate
Good
Bad
CatchRateG
ood
Bad
CatchRate
Good
Bad
CatchRate
GoodB
ad
CatchRate
Good
Bad
CatchRate
Group1
0.7
0
0.7
7
0.7
0
0.0
3
0.9
2
0.0
8
0.75
0.7
4
0.7
5
0.7
2
0.7
8
0.7
3
0.6
1
0.8
4
0.6
2
Group2
0.6
9
0.7
1
0.6
9
0.0
3
0.9
1
0.0
8
0.73
0.7
9
0.7
4
0.7
4
0.7
4
0.7
4
0.6
3
0.7
9
0.6
4
Group3
0.7
2
0.7
3
0.7
2
0.0
3
0.9
0
0.0
8
0.75
0.7
2
0.7
5
0.7
6
0.7
5
0.7
6
0.6
7
0.7
8
0.6
7
Group4
0.6
9
0.7
0
0.6
9
0.0
4
0.8
8
0.0
9
0.77
0.6
8
0.7
6
0.7
4
0.7
0
0.7
4
0.6
5
0.8
2
0.6
6
Group5
0.7
0
0.6
8
0.7
0
0.2
8
0.6
3
0.3
0
0.75
0.7
2
0.7
5
0.7
2
0.6
3
0.7
2
0.6
1
0.7
8
0.6
2
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A Fuzzy Linear Programming-based Classification Method 1171
Table 3. Training and testing result of 1400 records for fuzzy model 4.
Different di Training: Absolute Accuracy Rate Testing Absolute Accuracy Rate
d1=
d2 d3=
d4Good Bad Catch Rate Good Bad Catch Rate
1 3 0.547 0.897 0.722 0.538 0.896 0.5581 2 0.574 0.863 0.719 0.570 0.877 0.5881 1.5 0.619 0.823 0.721 0.607 0.838 0.6201 1 0.673 0.746 0.709 0.670 0.777 0.676
Absolute accurate rate of Dos = Specificity =t Dos
Dos,
Catch rate = Accuracy = Sensitivity NormalNormal + Dos
+ Specificity DosNormal + Dos
,
where t Normal is the number of the Normal (Normal records that were cor-
rectly classified as much). Normal is the number of Normal; t Dos is the number
of the Dos (Dos records that were correctly classified as much). Dos is the num-
ber of Dos. In this experimental study, MMD shows the same character as the
credit cardholder dataset analysis. But the result of comparison is not very clear
from the separate group training and testing result, so we compute the average
value to analyze the classification efficiency. The average value tells that MCLPand FLP2 show better catch rate in testing. MSD works well for Dos catching
and fuzzy model (3.4) FLP2 does a little worse than that.
In this paper, we just compared the proposed FLP classification method with
MMD, MSD, and MCLP in two real-life datasets. As references, the readers can find
the previous works comparing MCLP and FLP with soft criteria, decision tree, and
neural network in Refs. 9, 10 and 17. Thus, we shall not elaborate the comparison
of this FLP method with other classification methods.
5. Remarks
In this paper, a FLP classification method with both soft criteria and constraints is
proposed based on the previous researchers works. The relationship between this
model and other related models was discussed. Two real-life datasets, one from
the real bank in USA and the other from KDD 99, have been used to evaluate
the accurate rate of classification. The result shows the feasibility of this method.
Moreover, the general framework of FLP for classification have been described for
the first time systemically and evaluated. However, there is some new research workto be considered and continued in the line of research. For example, how does the
value di affect the result of classification? How can we consider ensemble analysis
to improve the selection of the best classifier? We shall report the significant results
of these ongoing projects in the near future.
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Table4.
Trainingresultsof4000records.
DifferentGroup
s
Model1(MSD)
Model2(MMD)
Model5(MCL
P)
FuzzyModel4
(FLP2)
AbsoluteAccuracyRate
AbsoluteAccuracyR
ate
AbsoluteAccuracyRate
AbsoluteAccura
cyRate
Normal
Dos
CatchR
ate
Normal
Dos
CatchRate
Normal
Dos
CatchRate
Normal
Dos
CatchRate
Group1
0.9
89
0.9
97
0.99
3
0.5
08
0.9
19
0.7
13
0.9
98
0.9
93
0.9
95
0.9
89
0.9
97
0.9
93
Group2
0.9
91
0.9
95
0.99
3
0.2
69
0.9
72
0.6
21
0.9
92
0.9
98
0.9
95
0.9
9
0.9
96
0.9
93
Group3
0.9
87
0.9
98
0.99
2
0.2
32
0.9
92
0.6
12
0.9
93
0.9
98
0.9
95
0.9
87
0.9
97
0.9
92
Group4
0.9
89
0.9
97
0.99
3
0.2
63
0.9
82
0.6
22
0.9
94
0.9
97
0.9
95
0.9
90
1.0
00
0.9
95
Average
0.9
89
0.9
97
0.99
3
0.3
18
0.9
66
0.6
42
0.9
94
0.9
97
0.9
95
0.9
89
0.9
98
0.9
93
Table5.
Testingresultsofotherrecords.
DifferentGroup
s
Model1(MSD)
Model2(MMD)
Model5(MCL
P)
FuzzyModel4
(FLP2)
AbsoluteAccuracyRate
AbsoluteAccuracyR
ate
AbsoluteAccuracyRate
AbsoluteAccura
cyRate
Normal
Dos
CatchR
ate
Normal
Dos
CatchRate
Normal
Dos
CatchRate
Normal
Dos
CatchRate
Group1
0.9
18
0.9
90
0.93
5
0.4
99
0.9
16
0.5
95
0.9
75
0.9
83
0.9
77
0.9
53
0.9
88
0.9
61
Group2
0.9
71
0.9
86
0.97
5
0.3
23
0.9
80
0.4
76
0.9
68
0.9
89
0.9
73
0.9
59
0.9
88
0.9
65
Group3
0.9
25
0.9
89
0.94
0
0.2
54
0.9
88
0.4
24
0.9
30
0.9
87
0.9
43
0.9
66
0.9
88
0.9
72
Group4
0.9
14
0.9
89
0.93
1
0.2
90
0.9
82
0.4
51
0.9
63
0.9
85
0.9
68
0.9
54
0.9
88
0.9
62
Average
0.9
32
0.9
89
0.94
5
0.3
42
0.9
67
0.4
87
0.9
59
0.9
86
0.9
65
0.9
58
0.9
88
0.9
65
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A Fuzzy Linear Programming-based Classification Method 1173
Acknowledgments
The authors would like to thank Professor S. Cheng for his patience and encourage-
ments on this work. They also express their thanks to Mr. G. Kou and P. Zhang for
their constructive comments in preparing this paper. This research is partially sup-
ported by the grants (70531040, 70472074 and 70921061) from the National NSFC,
the third 211 construction funding and Program for Innovation Research in CUFE.
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