A novel software interval type 2 fuzzy effort estimation model using s-fuzzy

International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 3, May – June (2013), © IAEME
477
A NOVEL SOFTWARE INTERVAL TYPE - 2 FUZZY EFFORT
ESTIMATION MODEL USING S-FUZZY CONTROLLER WITH MEAN
AND STANDARD DEVIATION
Mr. Peram Subba Rao1
, Dr.K.Venkata Rao2
, Dr.P.Suresh Varma3
1
Astt.Prof. in School of Computing, department of Information Technology, Vignan
University,
Vadlamudi, Guntur (Dt), Andhra Pradesh, India.
2
Prof. and HOD Department of CSE, Vignan’s Institute of Information Technology,
Gajuwaka, Visakhaptnam (Dt), Andhra Pradesh, India.
3
Dean CDC & Professor, Adikavi Nannaya University,
Rajahmundry, W. Godavari (Dt), Andhra Pradesh, India.
ABSTRACT
Now-a-days software cost estimation has prominent role in software engineering
practice. Cost estimation of software projects is one of the most desired capabilities in
software development organization. It helps the customer to make successful investments and
also assist software project manager in upcoming appropriate plans for projects besides
making reasonable decision during project execution. In order to fulfill the requirements of
cost estimation time effective software is required. By keeping in new of all these
requirements, my paper introduces a novel model of interval type-2 fuzzy logic estimation
effort in software development. This paper touches upon MATLAB which is tuning
parameters for various cost estimation models. Published software projects data model
performance is used along with comparison of my novel model and existing ubiquitous
models.
Keywords: Cost estimation, fuzzy logic, Interval type-2, membership function, regression
analysis.
INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING
& TECHNOLOGY (IJCET)
ISSN 0976 – 6367(Print)
ISSN 0976 – 6375(Online)
Volume 4, Issue 3, May-June (2013), pp. 477-490
© IAEME: www.iaeme.com/ijcet.asp
Journal Impact Factor (2013): 6.1302 (Calculated by GISI)
www.jifactor.com
IJCET
© I A E M E

478
1. INTRODUCTION
Software cost estimation has vital role in software engineering practice. In the last
three decades, many software estimation models and methods have been proposed and used,
such as COCOMO, SLIM, SEER-SEM, Price-S, FP, Delphi, Halsted Equation, Bailey-Basili,
Doty, and Anish Mittal Model and etc COCOMO [2, 4, 5, 13-18]. It determines the success
or failure of contract negotiations and project execution. Cost estimation’s deliverables are
effort, schedule, and staff requirements. These are valuable pieces of information for project
development and implementation. They are used by key inputs for project bidding and
proposals, budget and staff allocation, project planning, progress monitoring and control.
If the estimates are not reasonable and unreliable, there is possibility of project
failure. For example, it is evident in CompTIA survey of 1,000 IT respondents in the year
2007. Identifying these failures which lead to failure of IT projects. This failure is only due to
unrealistic resource estimation.
So, recognizing the importance of software estimation, the software engineer’s
community has put tremendous effort to develop models especially to help estimators
generate accurate cost for software projects.
Software engineering cost (and schedule) models and estimation techniques are used
for other purposes. Which includes budgeting, tradeoff and risk analysis, project planning &
control and software improvement investment analysis. Significant research on software cost
modeling began with an extensive 1965 SDC study of the 104 attributes of 169 software
projects. This led to some useful partial models in the late 1960s and early 1970s. The late
1970s produced a flowering of more robust models such as SLIM, Checkpoint, PRICE-S,
SEER and COCOMO [2, 4, 5, 13-18]. Although most of these researchers started working on
developing models of cost estimation at the same time, all of them have faced the same
problem as such as if software grow in size and importance it also grows in complexity,
makes it very difficult to predict accurate cost of software development. These dynamic fields
of software estimation sustain the interests of researchers who have been succeeded in setting
the stepping-stones of software engineering cost models.
As every coin has two sides, this field of software engineering cost models has its
own pitfalls. Due to fast changing nature of software development made it very difficult to
develop parametric models that yield high accuracy for software development in all domains.
Software development costs continue to increase and practitioners may express their concern
and inability to predict accurate cost. One of the most important objectives of the software
engineering community is the development of useful models that constructively explains the
development life-cycle. Besides predicting the cost for developing a software product. Many
software estimation models have evolved in the last two decades based on the pioneering
efforts of the researchers.
A recent survey shows that the software projects overrun the cost estimation, with
several other problems are found in this area like unrealistic over-optimum, complexity, and
overlooked tasks [1, 15]. So, to overcome all these problems in software development, new
models approached which researchers had showed attention in 1990's. They are artificial
neural networks, fuzzy logic models and genetic algorithms [13-14]. Among these models, it
is proved that fuzzy logic is the best powerful linguistic representation with exact inputs and
outputs. It is also based on model building with logic concepts introduced by Lofti A. Zadeh
[3].

479
1.1 Membership Functions
Fuzzy numbers are one of the ways to describe data vagueness, obscurity and
imprecision. A fuzzy number is an extension of a regular number in the sense that it does not
refer to one single value but rather to a connected set of possible values, where each possible
value has its own weight between ‘0’ and ‘1’. This weight is called the membership function.
The membership function is increasing towards the mean and decreasing away from it. The
Fuzzy number can be of three types 1) Triangular fuzzy Number 2) Trapezoidal fuzzy
number 3) Bell shaped fuzzy number. Figure 1 shows the three curves [3]
Figure 1: Types of Membership Functions
Fuzziness in a fuzzy set is characterized by its membership functions. A membership
function (MF) is a curve that defines how each point in the input space is mapped to a
membership value (or degree of membership) between 0 and 1. The graphical representations
include different shapes. There are certain restrictions regarding the shapes used. The “shape”
of the membership function is an important criterion that has to be considered. There are
different methods to form membership functions.
Zadeh proposed [3] a series of membership functions that could be classified into two
groups: those made up of straight lines, or “linear,” and Gaussian forms, or “curved.” [3]
Based on this criterion the membership function can be one of the following types. [3]
1.2 S Function
Defined by its lower limit a, it’s upper limit b, and the value m or point of inflection
so that a <m <b. A typical value is: m = (a + b) / 2. Growth is slower when the distance a – b
increases.
… (1)
Figure 2: S Fuzzy Set

480
2. EFFORT ESTIMATION MODELS LITERATURE REVIEW
Within last few decades, to improve the accuracy of cost estimation many software
cost estimation models [2], [4-12] were introduced. It seems to be impractical to expect very
accurate cost estimates of any software because of the inherent obscurity in software
development projects and the complex and dynamic interaction of factors that impact
software development cost use. Still, it is likely that estimates can be improved because
software development cost estimates systematically overoptimistic and very inconsistent. The
primary objective of the software engineers is to develop required models by using software
cost which can be accurately estimated. Estimation models use KDLOC (Thousands of
Delivered Lines of Code) as the primary input. This input is not sufficient for estimating the
accurate cost of products. Several other parameters have to be considered.
COCOMO81 Model
Boehm described COCOMO as a collection of three variants: basic model, intermediate
model, detailed model [12].
Basic model
Effort =a*sizeb
… (2)
Intermediate model
Effort = (a*sizeb
)*EAF … (3)
Detailed model
Effort = (a*sizeb
)*EAF*sum(Wi). … (4)
COCOMO II Model
Boehm and his colleagues have refined and updated COCOMO called as COCOMO II. This
consists of application composition model, early design model, post architecture model.
The Early Design Model
Effort = a*KLOC*EAF … (5)
The Post Architecture Model
Effort=(a*sizeb
)*EAF … (6)
Doty Model
Effort = 5.288(KLOC)
1.047
… (7)
Halsted Equation Model
Effort = 5.2 (KLOC)1.50 …(8)
Bailey-Basili Model
Effort = 5.5 + 0.73 (KLOC)1.16 … (9)
Mittal Model

481
Fuzzification: u(E) = …(10)
Defuzzification: E= …(11)
Harish model1
… (12)
Harish model2
… (13)
3. PROPOSED METHOD
Introduction on “interval type-2 fuzzy logic” for software cost estimation.
Figure 3: Interval Type-2 Fuzzy Logic
Structure of Type-2 fuzzy logic [20-27] is shown above. Fuzzification is the process
of which it 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 F1
i
and …. xn is Fn
i
then Y is Gi
,i=1,2…m 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.
Interval Type-2 Fuzzy Logic
A type-2 fuzzy set [20-27], denoted as A, is characterized by a type-2 Membership Function
(MF), µA(x, u), where x∈X and u∈Jx, i.e.
]}1,0[,|)),(),,{((
~
~ ⊆∈∀∈∀= xA
JxXxuxuxA µ … (14)

482
In which 0≤µA(x, u)≤1, if the universes of discourse X and the domain of secondary
membership function Jx are continuous, A can be expressed as:
∫∫ ∈∈
⊆=
xjx
xA
xx
jxxA ]1,0[),,(/),( µµµ … (15)
Where ∫ denotes union over all admissible x and µ. If the universes of discourse X and Jx are
both discrete, ∫ is replaced by Σ, A can also be expressed as:
∑∑ ==
=
kM
j
jijiA
N
i
xxA
11
),(/),( µµµ … (16)
If universes of discourse X and Jx are both discrete, (4) can be expressed as:
N
M
x
nk
m
x
k
N
J
N
iJxx
xxxixA
N
xix
//1....//1/1//1
~
1
1
1
1
1
1






++





=





=





= ∑∑∑∑∑∑ ==∈=∈∈
µµµµ
µµ
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) [20-27]. It is the
union of all primary memberships, i.e.
x
xx
JUAFOU
∈
=)(
… (17)
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 )(xAµ , Xx ∈∀ and the LMF is associated with the
lower bound of FOU (A) and is denoted as XxxA
∈∀)(µ , i.e.
XxAFOUx
XxAFOUx
A
A
∈∀=
∈∀=
)()(
)()(
µ
µ
… (18)
xxxx AA
∈∀= )],(),([JIT2FS,anFor x µµ Therefore, the IT2FS A can be denoted as.
orsituationdiscreateinxixx iAjA
)(/)](),([
n
1i
µµ∑= … (19)
Effort = xxxi iA
xx
A
/])(,)([ µµ∫∈
(in continuous situation). … (20)
The Upper Membership Function (UMF) and Lower Membership Function (LMF) of A are
two T1 MF`s that bound the FOU. For an interval type-2 fuzzy system (ITF2S).

483
[ ])(),(
2
21,
2
21
px
J
i
xPi
x
P
PRPRPLPL µµ
µµµµ
=







 ++
=
… (21)
[ ])(),(
2
21,
2
21
Nx
J
i
xNi
x
N
NRNRNLNL µµ
µµµµ
=







 ++
=
…. (22)
Are the firing intervals for positive and negative functions
2
,
1 PLPL
µµ are left hand side
uncertainty region.
2
,
1 PRPR
µµ are right hand side uncertainty region .similarly for
negative membership function.
3.1 Model Description
In this model I consider the mean of FOU `s as a firing interval in interval type-2 [20-
27], to estimate the cost (Effort) of the software.
1. In this first step by regression analysis (power regression) estimate the parameters of
a,b (the amplitude a and the exponent b).
2. In the second step, the variable “size” is fuzzified by two input fuzzy sets named
“Positive” and “Negative” respective. The mean of the size(L) is input for
determining the fuzzy memberships.
Figure 4: Universe Of Discourse
The membership value µP(xi) and µN(xi) is either 0 or 1 when xi is outside the interval [-L,L].
This process is known as fuzzification.
3. Next step is to apply S membership function to reduce the uncertainty (FOU). We
calculate the uncertainty at the left side (LMF) and right side (UMF).This is known as
fuzzy inference.
4. After identifying the LMF and UMF for the both positive and negative membership
functions, we the mean of the LMF and UMF FOU as a firing interval for converting
Type-2 into Type-1 fuzzy sets. This is known as Type Reducer.
5. Finally, In order to convert Fuzzy values into Output (Effort), weighted average
defuzzification (centroid) method is used.

484
3.2 Model Analysis: Regression Analysis
By using polynomial regression, [www.xuru.com] a,b parameters Y= a xb
is
calculated. Where x is the variable along the x-axis. The function is based on linear
regression with both axis are scaled logarithmically.
3.3 S membership function(SMF)
We Use S function Fuzzy number S(N) which is defined as Follows:
S(n) = … (23)
Figure 5: Representation of Effort
Where n is size as input. E is effort as output are the parameters of membership
function S(n).
Fuzziness of SFN (α, m, β) in defined as:
m = model value α = Left Boundary, β = is right boundary
Fuzziness of TFN (F) =
2m
αβ −
, 0 < F< 1 … (24)
The Higher the value of fuzziness, the more fuzzy in SFN. 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 why K in the real
positive number.
So that m =
1K
K βα
+
+
… (25)
As per the above definitions. F =
2m
αβ −
… (26)

485
So m*
1K
2KF
1α 





+
−=
and m*
1K
2F
1β 





+
+=
… (27)
3.4 Interval Type-2 TMF (α,m,β) Firing intervals
[ ])(),(
2
21,
2
21
Px
J
i
xPi
x
P
PRPRPLPL µµ
µµµµ
=







 ++
=
[ ])(),(
2
21,
2
21
Nx
J
i
xNi
x
N
NRNRNLNL µµ
µµµµ
=







 ++
=
3.5 Defuzzification
In this model, centroid method (weights average) is considered, which is of the form
C =
∑
∑
=
=
N
i
i
N
i
ii
w
w
1
1
µ
… (28)
4. EXPERIMENTAL RESULTS AND DISCUSSION
For the membership functions the L value is the mean of the input sizes [5, 19] i.e.
207.3385 and stddev() is 186.3325 (L1=393.671, L2= 21.006). By applying polynomial
regression [www.xuru.org] analysis for the input sizes and effort is obtained a=70.737 and
b=0.004.
By applying S membership function for the membership functions the left and right
boundaries are [µP (α, m, β), µN(α, m, β)] measured with α=0.9m and β=1.1m.
Foot print of uncertainty intervals for the µP is [0.501212 to 0.612593] 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 µN is [0.13737 to 0.167896] for left hand side i.e LMF and [0.206148 to 0.251959] for
right hand side i.e UMF. The means of FOU intervals is taken as firing strength.
JPx = [µp (Xi), µp (Xi)] =[0.556903, 1]
JNx= [µN (Xi), µN (Xi)] =[0.152633, 0.229054]
The type reducer action by using the S membership function and defuzzification is done
through centroid method and results are shown in the table II.

486
4.1 Estimated Efforts in Man-Months of Various Models
TABLE I
EFFORTS OF VARIOUS MODELS
TABLE II
EFFORT ESTIMATION
4.2 Following graph comparison of interval type-2 proposed model results with
measured efforts
Figure 6: Measured Effort Vs Proposed Model Estimated Effort

487
Figure 7: Measured Effort Vs Estimated Efforts of various Models
4.3 Performance Measures
The performance of proposed software effort estimation model is assessed by
comparing against various software cost estimation models. The methodology used in
empirical evaluation is described as follows:
• For each model, using MRE we evaluate the impact of estimation accuracy using
(MARE, VARE, StdDev) evaluation criteria.
• Criterion for measurement of software effort estimation model performance. The three
criterions are
1. Mean Absolute Relative Error (MARE)
% MARE =
Mean
2. Variance Absolute Relative Error (VARE)
% VARE =
var
3. Standard Deviation ( StdDev )
% StdDev =

488
4.4 Comparison of various models
TABLE III
PERFORMANCE CALCULATIONS
Figure 8: Mean Absolute Relative Error for above models
Figure 9: Variance Absolute Relative Error for above models

489
Figure 10: Standard Deviation for above models
5. CONCLUSION & FUTURE WORK
Software development life cycle is an important for project managers to estimate the
accuracy and reliability at the early stages of software development. This paper postulates
about the fuzzy software cost estimation model and with other popular software cost
estimation models. It concludes by empirical evaluation is better software effort with
proposed and traditional estimation models by MARE, VARE and Standard Deviation
evaluation criteria. To identify the problem of obscurity and vagueness that are existed in
software effort drivers’ fuzzy logic methods are applied. This proves the fuzzy logic
application is used in software engineering successfully and the work of Interval Type-2
fuzzy sets can be applied to other models of software cost estimation.
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