# Takagi-Sugeno And Interval Type-2 Fuzzy Logic For Software Effort Estimation

DOI : 10.17577/IJERTV1IS7343

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#### Takagi-Sugeno And Interval Type-2 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 type-2 fuzzy logic is applied for software effort estimation. Two different methodologies have been discussed as two models, to estimate effort by using Takagi-Sugeno and Interval Type-2 fuzzy logic. The Formulas that were used to implement these models including Regression Analysis, Takagi-Sugeno membership functions, foot print of uncertainty intervals and de-fuzzification 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 type-2 fuzzy set, denoted as A, is characterized by a type-2 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 type-2 membership function is reduced to a type-1 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 vertical-slice 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 Type-2 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 Type-2 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 type-2 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 type-2, 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 Takagi-Sugeno Fuzzy Controller is considered, for determining the memberships, and Interval Type-2 logic and fuzzy operator for determining the firing intervals.

Figure3.3: Membership Functions of Fuzzy sets in the Sizes Space

1. 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 x-axis. The function is based on linear regression with both axis are scaled logarithmically.

Membership Functions for Model-1, Model-2

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: MODEL-1:

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 Methodology-1

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 Methodology-2

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

1. 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 .

2. 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 47.1566 90.2 90.2 72.0649 125.252 125.252 122.72 10 100.8 52.6982 100.8 100.8 79.2229 137.693 137.693 134.909

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 Model-2:

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 (46-38.28) 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 Case-1 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 Case-1

 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 6.9033 0.069942 9 90.2 115.8 118.749 2.94939 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 Case-2

 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 Case-2

 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 Case-3

 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 Case-3

 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 Case-4

 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 Case-4

 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
3. One of the objective of the present work is to employ Interval Type-2 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 Baily-Basili, Alaa F. Sheta and Harish.

Comparison with other models:

The Table 12 compares effort estimation of TSFC- Interval Type-2 Models with other available models. The resulting data indicate that the approximation accuracy of the type-2 fuzzy systems methodology which is used in this chapter is comparable with the Bailey-Basili, AlaaF. Sheta, Harish models. The fuzzy systems approach to effort estimation has an advantage over the other models as the Interval Type-2 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 Man-Months of Various Models with Interval Type-2 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 Type-2 Model- I TSFC Model 2 Case-I Case-II Case-III Case-IV 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 type-2 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 Takagi-Sugeno and Interval Type-2 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 Baily-Basili, 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 Type-2 Model1 99.276 9.602 0.228 TSFC Model 2 Case-I 99.1 6.858 0.19 TSFC Model 2 Case-II 98.63 6.522 0.22 TSFC Model 2 Case-III 98.74 5.991 0.225 TSFC Model 2 Case-IV 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

4. CONCLUSION :

In this study we proposed new model structures to estimate the software cost (Effort) estimation. Interval Type-2 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 Type-2 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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