 Open Access
 Total Downloads : 543
 Authors : Annepu Balakrishna
 Paper ID : IJERTV1IS7343
 Volume & Issue : Volume 01, Issue 07 (September 2012)
 Published (First Online): 26092012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
TakagiSugeno And Interval Type2 Fuzzy Logic For Software Effort Estimation
Software Effort Estimation carries inherent risk and this risk would lead to uncertainty and some of the uncertainty factors are project complexity, project size etc. In order to reduce the uncertainty, fuzzy logic is being used as one of the solutions. In this Chapter interval type2 fuzzy logic is applied for software effort estimation. Two different methodologies have been discussed as two models, to estimate effort by using TakagiSugeno and Interval Type2 fuzzy logic. The Formulas that were used to implement these models including Regression Analysis, TakagiSugeno membership functions, foot print of uncertainty intervals and defuzzification process through weighted average method were outlined along with analysis. The experimentation is done with NASA software data set on the proposed models, and the results are tabulated. The measured efforts of these proposed models are compared with available models from literature and finally the performance analysis is done based on parameters such as MARE, VARE and VAF.
A type2 fuzzy set, denoted as A, is characterized by a type2 Membership Function (MF), ÂµA(x, u), where X and Jx, i.e.
~
A
A {(( x, u), ~ (x, u))  x X , x J x [0,1]}
In which 0ÂµA(x, u)1, if the universes of discourse X and the domain of secondary membership function Jx are continuous, A can be expressed as:
A
x x x jx
A (x, ) / (x, ), jx
[0, 1]Where denotes union over all admissible x and . If the universes of discourse X and
Jx are both discrete, is replaced by , A can also be expressed as:
N M k
A
A (xi ,
j ) / (xi , j )
i 1 j 1
Where denotes union over x and u. In the same way, if X is continuous and Jx is discrete or X is discrete and Jx is continuous, A can be expressed as:
Mu
A
x X j 1
A (xi
j ) / (xi ,
j ) (or)
N
A
i 1 jx
A (xi ,
) / (xi , )
The first restriction that is consistent with the T1 constraint 0 A(x) 1. When uncertainties disappear a type2 membership function is reduced to a type1 membership function, in which case the variable u equals A(x)[11,12,13,14,15]. The second restriction that 0 A(x, u) 1 is consistent with the fact that the amplitude of a
membership function should lie between or equal to 0 and 1. When A(x, u) 1, A is an IT2 FS, it can still be expressed as a special case of general T2 FS as follow:
A
x X J x
1/(x, )
x x
1/ / x,
J x
J x [0,1]
If universes of discourse X and Jx are both discrete, the above equation can be expressed as:
A
~ 1/ / x N
N
1 / xi
m1
1/ 1k
/ x1
….
M N
1/ nk
/ xN
x x J x i 1 J xi x 1 x 1
In the above equation + denotes union.
Uncertainty in the primary memberships of an IT2 FS consists of a bounded region named as Footprint of Uncertainty (FOU). It is the union of all primary memberships, i.e.
FOU
(A)
U Jx
x x
This is a verticalslice representation of FOU, because each of primary membership is a vertical slice. The Upper Membership Function (UMF) and Lower Membership Function (LMF) of A are two T1 MFs that bound the FOU. The UMF is associated
with the upper bound of FOU (A) and is denoted as
A (x) ,
x X and the LMF
is associated with the lower bound of FOU(A) and is denoted as
(x)
A
x X ,i.e.
A (x)
(x)
A
FOU ( A) x X
FOU ( A) x X
For an IT2FS, J x
[ (x),A
A (x)], x
x Therefore, the IT2FS A can be denoted
as, Effort=
n
j
[ (x ),
A
i 1
A (xi )] / xi (in discreate situation) or
Effort =
x
[ (xi),A
x
A (xi ) ]/ x
(in continuous situation).
The following Figure 1 shows the components of Interval Type2 Fuzzy Logic.
Fuzzy Rule Bases
Fuzzy Inference
Fuzzificati on
a, b
Size
Effort
www.ijert.org
Type Reducer
Defuzzification
2
Figure 1: Structure of Interval Type2 Fuzzy Logic
Fuzzification is the process which Translates inputs (real values) to fuzzy values. Inference System applies a fuzzy reasoning mechanism to obtain a fuzzy output. Knowledge Base contains a set of fuzzy rules, it is of the form Ri :if x1 is F1i and . xn is Fni then Y is Gi ,i=1,2m and a membership functions set known as the database. Type Reducer transforms a Fuzzy Set into a Type 1 Fuzzy Set. The defuzzification traduces one output to precise values.
For an interval type2 fuzzy system (ITF2S)
Jf1
Jf2
PL1
2
NL1
2
PL2 ,
NL2 ,
PR1
2
NR1
2
PR2 P (xi ),
NR2 N (xi ),
P (xi
N (xi
are the firing intervals for the membership functions positive and negative
respectively,
PL1,
PL2
are left hand side uncertainty region,
PR1,
PR2
are
right hand side uncertainty region.
The following section deals the two methodologies that have been used on the proposed models in order to estimate effort.
In this model the mean of FOU `s as a firing interval in interval type2, is considered to estimate the cost (effort) of the software.
Step2: The variable size is then fuzzified by two input fuzzy sets named Positive and Negative respective. The mean of the sizes (L) is input for determining the fuzzy memberships. The representation shown in Figure3.2, the membership value P(xi) and N(xi) is either 0 or 1 when xi is outside the interval [L,L]. This process is known as Fuzzification.
N(xi) P(xi)
1
0.5
L 0 L
Figure 2: Universe of Discourse
In this model TakagiSugeno Fuzzy Controller is considered, for determining the memberships, and Interval Type2 logic and fuzzy operator for determining the firing intervals.
Figure3.3: Membership Functions of Fuzzy sets in the Sizes Space

The section analyses the proposed models.
Regression Analysis: By using power regression we calculate [www.xuru.com] a, b parameters Y= axb, Where x is the variable along the xaxis. The function is based on linear regression with both axis are scaled logarithmically.
Membership Functions for Model1, Model2
P ( kloc)
0
kloc L
2L
1
kloc L
L kloc L kloc L
(1.1)
and
1
kloc L
N ( kloc)
kloc L
2L
0
L kloc L kloc L
(1.2)
The value of L affects the control performance and we take the mean of the input
(size) as the L value.
The mathematical definitions of the two Positive fuzzy sets were identical for the input variable size is equation 1.3 and 1.4.
and
P1 (
kloc)
0
kloc L1
2L1
1
0
kloc L1
L1 kloc L1 kloc L1
kloc L2
(1.3)
P 2 (
kloc)
kloc L2
2L2
1
L2 kloc L2
kloc L2
(1.4)
The value of L affects the control performance and we taken (mean+stddev), (mean stddev) of the input (size) as the L1 and L2 value.
The triangular MF in specified by three parameters ( , m, ) as Figure :
m
Figure 3: Triangular Member Function
The parameters ( , m, ) (with < m < ) determine the x coordinates of the three corners of the underlying triangular MF.
Fuzziness of TFN ( , m ) is defined as:
m = model value = Left Boundary = is right boundary
Fuzziness of TFN (F) = , 0 < F< 1
2m
The Higher the value of fuzziness, the more fuzzy is TFN. The value of fuzziness to be taken depends upon the confidence of the estimator. A confident estimator can take smaller values of F. Let (m, 0) divides internally, the base of the triangle in ration K : 1 where K in the real positive number.
So that m =
K
K 1
As per the above definitions, F =
So
2m
1 2KF * m
and
1 2F * m
K 1 K 1
If we consider F = 0.1 and K = 1 then
m* 1
m* 1
2 x 0.1 x 1
2
2 x 0.1 x 1
2
0.9m
m 1
0.1
1.1m
Footprint of Uncertainty in Universe Of Discourse: MODEL1:
After applying fuzzification process on the size by using the positive and negative member functions, the Footprint of Uncertainty (FOU) in universe of discourse with uncertainty regions will be the one as shaded in the following figure 3.5.
Figure 4: Footprint of Uncertainty and Prediction Intervals for Methodology1
PL1,
PL2
are left hand side uncertainty region,
PR1,
PR2
are right hand
side uncertainty region.
After applying fuzzification process on the size by using two positive member functions, the Footprint of Uncertainty (FOU) in universe of discourse with uncertainty regions will be the one as shaded in the following figure 3.6.
Figure 5: Footprint of Uncertainty and Prediction Intervals for Methodology2
P1L1,
P2L2
are left hand side uncertainty region,
P1R1,
P2R2
are right hand
side uncertainty region.
Firing Intervals :
Here the means of FOUs are taken as firing strength.
JPx
PL1
2
PL2 ,
PR1
2
PR2 (x ),
P
i
(x
P
i
J Nx
NL1
2
NL2 ,
NR1
2
NR2 (x ),
N i
N (xi )
Here the means two positive member functions of FOUs are taken as firing strength.
J P1 L1
P1 L2 ,
P1 R1
P1 R2
(x ),
P (x
P1x 2 2 P1 i 1 i
J
P2 x
P2 L1
2
P2 L2 ,
P2 R1
2
P2 R2
P2
(x ),
i
(x )
P
2 i
The uncertainty considered at the left, right hand side interval i.e. fuzzy operator OR (max) is used to determine the firing interval
J P x
max P (x ), P (x
) , max P (x ), P (x )
1 i 1 i
2 i 2 i
The uncertainty considered only at the right hand side interval to determine the firing interval
J P x
(x ),
P
2 i
(x )
P
2 i
The uncertainty considered only at the left hand side interval to determine the firing interval
J P x
P (x ), P (x )
1 i 1 i
Defuzzification:
In these models weights average method, which is of the following form are considered.
N
wi i
C = i 1
N
wi
i 1 (1.5)
where wi is the weighting factor and i is the membership obtained from triangular
member function.
Performance Measures:
Three criterions were considered and they are outlined below

Variance Accounted For (VAF)
% VAF = 1
var (Measured Effort Estimated Effort)
x 100
var (measured effort)
2 ) Mean Absolute Relative Error (MARE)
% MARE = mean
abs (Measured Effort Estimated Effort)
x 100
(measured effort)
3) Variance Absolute Relative Error (VARE)
% VARE = var
(abs (Measured Effort Estimated Effort)
x 100
(measured effort)
The following section describes the experimentation part of the work, and to conduct the study and in order to establish the effectively of the models dataset of 10 projects from NASA software project data [2] were used .


The membership function definitions and the memberships are shown here using equation 1.1 and 1.2; the L value is the mean of the input sizes i.e. 46. By applying power regression analysis (www.xuru.com) for the input sizes and effort the obtained values are : a=2.7 and b=0.8523. The membership functions are defined as follows using equation 3.1 and 3.2.
N ( kloc)
1
kloc L
2L
kloc
46
46
kloc 46
0 kloc 46
By applying Triangular membership function for the above membership functions the left and right boundaries obtained are shown below.
Foot print of uncertainty intervals for the P is [0.4705 to 0.5751] for left hand side i.e. LMF and [0.9 to 1.1] for right hand side i.e. UMF. Foot print of uncertainty intervals for the N is [0.3277 to 0.4005] for left hand side i.e. LMF and [0.4295 to 0.5249] for right hand side i.e. UMF. The means of FOU intervals is taken as firing strength.
JPx = P (x ), P ( x )
= [0.5228, 1]
i i
J Nx = N (x ), N ( x ) = [0.3641, 0.4772]
i i
The type reducer action by using the triangular membership function which is applied to the uncertainty region as a secondary member function and the results obtained are shown in Table 1and Table 2. Defuzzification process is done through weighted average method.
Table 1:Triangular Fuzzy Number of Adjusted Size and Effort Estimation for Positive Membership Function
S.No
Size(m)
= 0.5228m
m
=m
ab
amb
ab
Ep
1
2.1
1.09788
2.1
2.1
2.923672
5.081492
5.081492
4.978739
2
3.1
1.62068
3.1
3.1
4.074635
7.081925
7.081925
6.938721
3
4.2
2.19576
4.2
4.2
5.27833
9.174008
9.174008
8.9885
4
12.5
6.535
12.5
12.5
13.37204
23.24129
23.24129
22.77132
5
46.5
24.3102
46.5
46.5
40.97051
71.20884
71.20884
69.76892
6
54.5
28.4926
54.5
54.5
46.90638
81.52569
81.52569
79.87715
7
67.5
35.289
67.5
67.5
56.28813
97.83165
97.83165
95.85339
8
78.6
41.09208
78.6
78.6
64.08698
111.3865
111.3865
109.1341
9
90.2
47.15656
90.2
90.2
72.06489
125.2525
125.2525
122.7197
10
100.8
52.69824
100.8
100.8
79.22287
137.6934
137.6934
134.9091
Table 2: Triangular Fuzzy Number of Adjusted Size and Effort Estimation for Negative Membership Function
1
S.No
Size(m)
=
0.3641m
M
=
0.4772m
ab
amb
ab
EN
2.1
0.76461
2.1
1.00212
2.147928
5.081492
2.704878
3.810078
2
3.1
1.12871
3.1
1.47932
2.993503
7.081925
3.769708
5.309992
3
4.2
1.52922
4.2
2.00424
3.877819
9.174008
4.883324
6.878626
4
12.5
4.55125
12.5
5.965
9.824005
23.24129
12.37133
17.4262
5
46.5
0
46.5
0
0
71.20884
0
71.20884
6
54.5
0
54.5
0
0
81.52569
0
81.52569
7
67.5
0
67.5
0
0
97.83165
0
97.83165
8
78.6
0
78.6
0
0
111.3865
0
111.3865
9
90.2
0
90.2
0
0
125.2525
0
125.2525
10
100.8
0
100.8
0
0
137.6934
0
137.6934
The following Table 3 shows the Measured Effort, Estimation Effort, Absolute Error and Relative Error.
Table3 : Error Calculations
S.No
Size
Measured
Effort
Estimated
Effort(Ep+En/2)
Absolute
Error
Relative
Error
1
2.1
5
4.394408
0.605592
0.121118
2
3.1
7
6.124357
0.875643
0.125092
3
4.2
9
7.933563
1.066437
0.118493
4
12.5
23.9
20.09876
3.801238
0.159048
5
46.5
79
70.48888
8.51112
0.107736
6
54.5
90.8
80.70142
10.09858
0.111218
7
67.5
98.4
96.84252
1.557483
0.015828
8
78.6
98.7
110.2603
11.5603
0.117126
9
90.2
115.8
123.9861
8.1861
0.070692
10
100.8
138.3
136.3013
1.998739
0.014452
Application on Model2:
The membership function definitions and the memberships shown here are obtained using equation 3.3 and 3.4, the L value is the (mean + stddev) of the input sizes for Positive1 is (46+38.28) 84.28, and L value is the (mean stddev) of the input sizes for Positive2 is (4638.28) 7.72. By applying power regression (www.xuru.com) analysis for the input sizes and effort the obtained values of a, b are a=2.7 and b=0.8523
0 kloc 84.28
P
( kloc)
1
kloc L1
2L1
84.28
kloc
84.28
1 kloc 84.28
and
0 kloc
7.72
P
( kloc)
2
kloc L2
2L2
1
7.72
kloc
kloc
7.72
7.72
By applying Triangular membership function for the above membership functions the left and right boundaries obtained are shown in the following Table3.8 and Table3.9 [P1( , m, ) , P2( , m, )]
Foot print of uncertainty intervals for the P1 is [0.5724 to 0.9] and Foot print of
uncertainty intervals for the P2 is [0.6996 to 1.1]. The means of FOU intervals is taken as firing strength.
JPx = P (x ), P ( x ) = [0.7362, 0.8998]
i i
The type reducer action by using the triangular membership function and associated results are shown in Table3.10. Defuzzification process is done through weighted average method.
Table 4: Triangular Fuzzy Number of Adjusted Size and Effort Estimation for Case1 The following Table 5 shows the Measured Effort, Estimated Effort, Absolute Error and Relative Error.
S.No
Size (m)
= 0.7362m
M
=0.8998
m
ab
amb
ab
Effort
1
2.1
1.54602
2.1
1.88958
3.914099
5.081492
4.644189
4.817663
2
3.1
2.28222
3.1
2.78938
5.454964
7.081925
6.472469
6.714233
3
4.2
3.09204
4.2
3.77916
7.066424
9.174008
8.384511
8.697696
4
12.5
9.2025
12.5
11.2475
17.90196
23.24129
21.24119
22.03461
5
46.5
34.2333
46.5
41.8407
54.84972
71.20884
65.08075
67.5117
6
54.5
40.1229
54.5
49.0391
62.79644
81.52569
74.50975
77.2929
7
67.5
49.6935
67.5
60.7365
75.35636
97.83165
89.41245
92.75225
8
78.6
57.86532
78.6
70.72428
85.79716
111.3865
101.8008
105.6033
9
90.2
66.40524
90.2
81.16196
96.47767
125.2525
114.4735
118.7494
10
100.8
74.20896
100.8
90.69984
106.0605
137.6934
125.8438
130.5444
Table 5: Error Calculations for Case1
S.No
Size
Measured
Effort
Estimated
Effort
Absolute
Error
Relative
Error
1
2.1
5
4.81766
0.182337
0.036467
2
3.1
7
6.71423
0.285767
0.040824
3
4.2
9
8.6977
0.302304
0.033589
4
12.5
23.9
22.0346
1.865395
0.07805
5
46.5
79
67.5117
11.4883
0.145422
6
54.5
90.8
77.2929
13.5071
0.148757
7
67.5
98.4
92.7523
5.647746
0.057396
8
78.6
98.7
105.603
6.903304
0.069942
9
90.2
115.8
118.749
2.949393
0.02547
10
100.8
138.3
130.544
7.75559
0.056078
This case deals with uncertainty at the left, right hand side interval i.e. fuzzy operator OR (max) is used here to determined the firing interval
J P x
max P (x ), P (x
) , max P (x ), P (x )
1 i 1 i
2 i 2 i
Foot print of uncertainty intervals for the P1 is [0.5724 to 0.9], Foot print of uncertainty intervals for the P2 is [0.6996 to 1.1].The fuzzy operator max of FOU intervals is taken as firing strength.
JPx = P (x ), P ( x ) = [0.9, 1.1]
i i
The Table 6 shows the Effort estimation using above firing intervals.
Table 6: Triangular Fuzzy Number of Adjusted Size and Effort Estimation for Case2
S.No
Size(m)
= 0.9m
m
=1.1m
ab
amb
ab
Effort
1
2.1
1.89
2.1
2.31
4.645
5.081
5.511
5.265
2
3.1
2.79
3.1
3.41
6.473
7.081
7.681
7.337
3
4.2
3.78
4.2
4.62
8.386
9.174
9.95
9.506
4
12.5
11.25
12.5
13.75
21.245
23.241
25.208
24.082
5
46.5
41.85
46.5
51.15
65.093
71.208
77.234
73.786
6
54.5
49.05
54.5
59.95
74.523
81.525
88.424
84.476
7
67.5
60.75
67.5
74.25
89.429
97.831
106.11
101.373
8
78.6
70.74
78.6
86.46
101.82
111.386
120.812
115.419
9
90.2
81.18
90.2
99.22
114.495
125.252
135.851
129.786
10
100.8
90.72
100.8
110.88
125.867
137.693
149.345
142.678
The Table 7 shows the Measured Effort, Absolute Error, Estimated Effort and Relative Error.
Table 7: Error Calculations for Case2
S.No
Size
Measured Effort
Estimated Effort
Absolute Error
Relative Error
1
2.1
5
5.265
0.265
0.0529
2
3.1
7
7.337
0.337
0.0481
3
4.2
9
9.506
0.506
0.0562
4
12.5
23.9
24.082
0.182
0.0076
5
46.5
79
73.786
5.214
0.066
6
54.5
90.8
84.476
6.324
0.0696
7
67.5
98.4
101.373
2.973
0.0302
8
78.6
98.7
115.419
16.719
0.1693
9
90.2
115.8
129.786
13.986
0.1207
10
100.8
138.3
142.678
4.378
0.0316
In this case the uncertainty considered only at the right hand side interval i.e. Firing interval
J P x
(x ),
P
2 i
(x )
P
2 i
Foot print of uncertainty intervals for the P1 is [0.5724 to 0.9] Foot print of uncertainty intervals for the P2 is [0.6996 to 1.1].The more uncertainty is on the right hand side.
JPx = P (x ), P ( x ) = [0.6996, 1.1]
i i
The Table 8 shows the Effort estimation using above firing intervals.
Table 8: Triangular Fuzzy Number of Adjusted Size and Effort Estimation Case3
S.No
Size(m)
=
0.6996m
M
=1.1m
ab
amb
ab
Effort
1
2.1
1.469
2.1
2.31
3.747
5.081
5.511
5.222
2
3.1
2.168
3.1
3.41
5.221
7.081
7.681
7.278
3
4.2
2.938
4.2
4.62
6.765
9.174
9.95
9.428
4
12.5
8.745
12.5
13.75
17.14
23.241
25.208
23.887
5
46.5
32.531
46.5
51.15
52.516
71.208
77.234
73.187
6
54.5
38.128
54.5
59.95
60.125
81.525
88.424
83.791
7
67.5
47.223
67.5
74.25
72.151
97.831
106.11
100.55
8
78.6
54.988
78.6
86.46
82.147
111.386
120.812
114.482
9
90.2
63.103
90.2
99.22
92.373
125.252
135.851
128.733
10
100.8
70.519
100.8
110.88
101.548
137.693
149.345
141.52
The Table 9 shows the Measured Effort, Absolute Error, Estimated Effort and Relative Error.
Table 9: Error Calculations for Case3
S.No
Size
Measured
Effort
Estimated
Effort
Absolute
Error
Relative
Error
1
2.1
5
5.222
0.222
0.0444
2
3.1
7
7.278
0.278
0.0397
3
4.2
9
9.428
0.428
0.0475
4
12.5
23.9
23.887
0.013
0.0005
5
46.5
79
73.187
5.813
0.0735
6
54.5
90.8
83.791
7.009
0.0771
7
67.5
98.4
100.55
2.15
0.0218
8
78.6
98.7
114.482
15.782
0.1598
9
90.2
115.8
128.733
12.933
0.1116
10
100.8
138.3
141.52
3.22
0.0232
The uncertainty in this case is considered only at the left hand side interval i.e. firing interval
J P x
P (x ), P (x )
1 i 1 i
Foot print of uncertainty intervals for the P1 is [0.5724 to 0.9] Foot print of uncertainty intervals for the P2 is [0.6996 to 1.1].The more uncertainty is on the left hand side.
JPx = P (xi ),
P(xi)
= [0.5724, 0.9]
The Table 10 shows the Effort estimation using above firing intervals.
Table 10: Triangular Fuzzy Number of Adjusted Size and Effort Estimation for Case4
S.No
Size(m)
=
0.5724m
M
=0.9m
ab
amb
ab
Effort
1
2.1
1.202
2.1
1.89
3.158
5.081
4.645
4.781
2
3.1
1.774
3.1
2.79
4.4
7.081
6.473
6.663
3
4.2
2.404
4.2
3.78
5.702
9.174
8.386
8.633
4
12.5
7.155
12.5
11.25
14.445
23.241
21.245
21.871
5
46.5
26.616
46.5
41.85
44.26
71.208
65.093
67.012
6
54.5
31.195
54.5
49.05
50.672
81.525
74.523
76.721
7
67.5
38.637
67.5
60.75
60.808
97.831
89.429
92.067
8
78.6
44.99
78.6
70.74
69.233
111.386
101.82
104.823
9
90.2
51.63
90.2
81.18
77.852
125.252
114.495
117.872
10
100.8
57.697
100.8
90.72
85.584
137.693
125.867
129.58
The Table 11 shows the Measured Effort, Absolute Error, Estimated Effort and Relative Error.
Table 11: Error Calculations for Case4
S.No
Size
Measured
Effort
Estimated
Effort
Absolute
Error
Relative
Error
1
2.1
5
4.781
0.219
0.0438
2
3.1
7
6.663
0.337
0.0481
3
4.2
9
8.633
0.367
0.0407
4
12.5
23.9
21.871
2.029
0.0848
5
46.5
79
67.012
11.988
0.1517
6
54.5
90.8
76.721
14.079
0.155
7
67.5
98.4
92.067
6.333
0.0643
8
78.6
98.7
104.823
6.123
0.062
9
90.2
115.8
117.872
2.072
0.0178
10
100.8
138.3
129.58
8.72
0.063

One of the objective of the present work is to employ Interval Type2 fuzzy logic for tuning the effort parameters and test its suitability for software effort estimation. This methodology is then tested using NASA dataset provided by Boehm. The results are then compared with the models in the literature such as BailyBasili, Alaa F. Sheta and Harish.
Comparison with other models:
The Table 12 compares effort estimation of TSFC Interval Type2 Models with other available models. The resulting data indicate that the approximation accuracy of the type2 fuzzy systems methodology which is used in this chapter is comparable with the BaileyBasili, AlaaF. Sheta, Harish models. The fuzzy systems approach to effort estimation has an advantage over the other models as the Interval Type2 fuzzy systems architecture determines the firing intervals for inputs which reduces the factors of uncertainty, and the fuzzy rules be extracted from numerical data, which may easily be analyzed and the implementation is also relatively easy.
Table 12: Effort Efforts in ManMonths of Various Models with Interval Type2 Models
S.No
Size
Measured effort
Bailey Basili Estimate
Alaa F. Sheta G.E.model
Estimate
Alaa F. ShetaModel 2 Estimate
Harish model1
Harish model2
Interval Type2 Model
I
TSFC Model 2
CaseI
CaseII
CaseIII
CaseIV
1
2.1
5
7.226
8.44
11271
6.357
4.257
4.394
4.822
5.265
5.222
4.781
2
3.1
7
8.212
11.22
14.457
8.664
7.664
6.124
6.721
7.337
7.278
6.663
3
4.2
9
9.357
14.01
19.976
11.03
13.88
7.933
8.707
9.506
9.428
8.633
4
12.5
23.9
19.16
31.098
31.686
26.252
24.702
20.099
22.06
24.082
23.887
21.871
5
46.5
79
68.243
81.257
85.007
74.602
77.452
70.489
67.591
73.786
73.187
67.012
6
54.5
90.8
80.929
91.257
94.977
84.638
86.938
80.701
77.385
84.476
83.791
76.721
7
67.5
98.4
102.175
106.707
107.254
100.329
97.679
96.842
92.863
101.373
100.55
92.067
8
78.6
98.7
120.848
119.27
118.03
113.237
107.288
110.26
105.73
115.419
114.482
104.823
9
90.2
115.8
140.82
131.898
134.011
126.334
123.134
123.986
118.891
129.786
128.733
117.872
10
100.8
138.3
159.434
143.0604
144.448
138.001
132.601
136.301
130.7
142.678
141.52
129.58
Assessment through Graph Representation of Measured Effort Vs Estimated Effort:
The Figure 6 shows measured effort Vs estimated effort of interval type2 models and one can notice that the estimated efforts are very close to the measured effort.
Figure 6: Measured Effort Vs Estimated Effort of TSFC models
Figure 7: Effort Estimations of Various models Vs TSFC Models
PERFORMANCE ANALYSIS:
Parameters such as VAF, MARE, and VARE are employed to asses as well as to compare the performance of the estimation models. The integration of TakagiSugeno and Interval Type2 fuzzy logic can be powerful tool when tackling the problem of
effort estimation. It can be seen from the resulting data that the Fuzzy logic models for Effort estimation outperform the BailyBasili, Alaa F. Sheta and Harish models. The computed MARE, VARE and VAF for all the models are indicated in Table 13.
Table 13 Summary Results of VAF, MARE and VARE
Model
Variance Accounted For (VAF%)
Mean Absolute Relative Error(MARE%)
Variance Absolute Relative
Error(VARE%)
Bailey Basili Estimate
93.147
17.325
1.21
Alaa F. Sheta G.E.Model Estimate
98.41
26.488
6.079
Alaa F. Sheta Model 2 Estimate
98.929
44.745
23.804
Harish model1
98.5
12.17
80.859
Harish model2
99.15
10.803
2.25
Interval Type2 Model1
99.276
9.602
0.228
TSFC Model 2 CaseI
99.1
6.858
0.19
TSFC Model 2 CaseII
98.63
6.522
0.22
TSFC Model 2 CaseIII
98.74
5.991
0.225
TSFC Model 2 CaseIV
98.98
7.312
0.21
Figure 8: Variance Accounted For Of Various Models Vs TSFC Models
Figure 9: Mean Absolute Relative Error of Various models Vs TSFC Models
Figure 10: Variance Absolute Relative Error of various models Vs TSFC Models

CONCLUSION :
In this study we proposed new model structures to estimate the software cost (Effort) estimation. Interval Type2 fuzzy sets is used for modeling uncertainty and impression to better the effort estimation. Rather than using a single number, the software size can be regarded as a fuzzy set yielding the cost estimate also in the form of a fuzzy set. These proposed models were able to provide good estimation capabilities as per the as per the experimental study taking parameters like VAF, MARE, and VARE. The work of Interval Type2 fuzzy sets can be applied to other models of software cost estimation. However in these models only the size is used as input for estimating the effort. But there are so many Cost Drivers which have to be considered for measuring effort. In fact the main difficulty is to determine which cost driver really capture the reason for differences in estimated effort among the projects. Therefore for large projects of size>100 KDLOC the estimation process requires data to be more accurate, consistent with appropriate cost drivers. It is reasonable to assume that one should specify cost drivers for large projects as they are essential to calibrate the estimation model.

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