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International Journal of Software Engineering and Its Applications

Vol. 7, No. 2, March, 2013

7

A Fuzzy Logic Based Software Cost Estimation Model

Ziauddin

1, Shahid Kamal

2, Shafiullah khan

3 and Jamal Abdul Nasir

4

1COMSATS Institute of Information Technology, Vehari, Pakistan

2FSKSM , Universiti Teknologi Malaysia

3Kohat University of Science & Technology, Kohat, Pakistan

4ICIT, Gomal University, D.I.Khan, Pakistan

[email protected],

[email protected],

[email protected],

[email protected]

Abstract

Software cost estimation is a challenging and onerous task. Estimation by analogy is one

of the expedient techniques in software effort estimation field. However, the methodology

utilized for the estimation of software effort by analogy is not able to handle the categorical

data in an explicit and precise manner. Early software estimation models are based on

regression analysis or mathematical derivations. Todays models are based on simulation,

neural network, genetic algorithm, soft computing, fuzzy logic modelling etc. This paper aims

to utilize a fuzzy logic model to improve the accuracy of software effort estimation. In this

approach fuzzy logic is used to fuzzify input parameters of COCOMO II model and the result

is defuzzified to get the resultant Effort. Triangular fuzzy numbers are used to represent the

linguistic terms in COCOMO II model. The results of this model are compared with

COCOMO II and Alaa Sheta Model. The proposed model yields better results in terms of

MMRE, PRED(n) and VAF.

Keywords: Fuzzy logic, Triangular Fuzzy Number, Membership function and Fuzziness,

Software Effort Estimation

1. Introduction

Estimating the work-effort and the schedule required to develop and/or maintain a software

system is one of the most critical activities in managing software projects. The task is known

as Software Cost Estimation [27]. During the development process cost and time estimates

are useful for the initial rough validation and monitoring of the projects progress; after completion, these estimates may be useful for project productivity assessment for example.

There are different techniques used in software cost estimation:

Algorithm model, also called parametric model, is designed to provide some mathematical

equations to provide software estimation. LOC-based models are algorithm models such as [2,

6, 7, 8]. Ali Idri and Laila Kjjri [9] proposed the use of fuzzy sets in the COCOMO-81

models [8]. Musilek P. and others [18] proposed f-COCOMO model, using fuzzy sets. The

methodology of fuzzy sets giving rise to f-COCOMO [18] is sufficiently general to be applied

to other models of software cost estimation such as function point method [14]. W. Pedrycz

and others [19] found that the concept of information granularity and fuzzy sets, in particular,

plays an important role in making software cost estimation models more user friendly. Harish

Mittal and Pradeep Bhatia [17] used triangular fuzzy numbers for fuzzy logic sizing. Lima,

O.S.J. and Others [13] proposed the use of concepts and properties from fuzzy set theory to


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extend function point analysis to Fuzzy function point analysis, using trapezoid shaped fuzzy

numbers for the linguistic variables of function point analysis complexity matrixes.

In this paper we propose triangular fuzzy numbers to represent the linguistic variables. The

results can be optimised for the given application by varying fuzziness of the triangular fuzzy

numbers. To apply fuzzy logic first fuzzification is done using triangular fuzzy number,

Fuzzy output is then evaluated and estimation is done by defuzzification technique given in

this paper.

1.1. Fuzzy Logic

In 1965, Lofti Zadeh formally developed multi-value set theory, and introduced the term

fuzzy logic [5]. Fuzzy Logic (FL) starts with the concept of fuzzy set theory. It is a theory of

classes with un-sharp boundaries, and considered as an extension of the classical set theory.

The membership of an element x of a classical set A, as subset of the universe X, is defined as follows:

A (x) = 1 if x A A(x) = 0 if x A

A system based on FL has a direct relationship with fuzzy concepts (such as fuzzy sets,

linguistic variables, etc.) and fuzzy logic. The popular fuzzy logic systems can be categorised

into three types: Pure fuzzy logic systems, Takagi and Sugenos fuzzy system, and fuzzy logic system with fuzzifier and defuzzifier . Since most of the engineering applications

produce crisp data as input and expects crisp data as output, the last type is the most widely

used type of fuzzy logic systems. Fuzzy logic system with fuzzifier and defuzzifier, first,

proposed by Mamdani and it has been successfully applied to a variety of industrial processes

and consumer products [16]. The main three steps of applying fuzzy logic to a model are:

Step 1: Fuzzification: It converts a crisp input to a fuzzy set

Step 2: Fuzzy Rule-Based System: Fuzzy logic systems use fuzzy IF-THEN rules.

Fuzzy Inference Engine: Once all crisp input values are fuzzified into their respective

linguistic values, the inference engine accesses the fuzzy rule base to derive linguistic values

for the intermediate and the output linguistic variables. Step 3: Defuzzification: It converts

fuzzy output into crisp output.

Use of fuzzy sets in logical expression is known as fuzzy logic. A fuzzy set is

characterized by a membership function, which associates with each point in the fuzzy set a

real number in the interval (0,1), called degree or grade of membership [3, 4, 20, 21]. The

membership function may be triangular, trapezoidal, parabolic etc. Fuzzy numbers are special

convex and normal fuzzy sets, usually with single modal value, representing uncertain

quantitative information.

A fuzzy number is a quantity whose value is imprecise, rather than exact as in the case of

ordinary single valued numbers [4, 9, 20, 21, 23]. Any fuzzy number can be thought of as a

function, called membership function, whose domain is specified, usually the set of real

numbers, and whose range is the span of positive numbers in the closed interval (0, 1). Each

numerical value of the domain is assigned a specific value and 0 represents the smallest

possible value of the membership function, while the largest possible value is 1. In many

respects fuzzy numbers depict the physical world more realistically than single valued

numbers. The curve in Figure 1a is a triangular fuzzy number, the curve in Figure 1b is a

trapezoidal fuzzy number, and the curve in Figure1c is bell shaped fuzzy number.


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Figure 1. Fuzzy Membership Functions

A triangular fuzzy number (TFN) is described by a triplet ( ,m, ), where m is the modal

value, and are the right and left boundary respectively. Fuzziness of a TFN (, m,) is defined as:

Fuzziness of TFN (F) =

, 0 < F< 1

The higher the value of fuzziness, the more fuzzy is TFN.

1.2. COCOMO II

The COCOMO I [7] model is a regression-based software cost estimation model, which

was developed by Boehm in 1981 and thought to be the most cited and the most plausible

model among all traditional cost estimation models. The COCOMO I was a stable model on

that time. One of the problems with the use of COCOMO I today is that it does not match the

development environment of the late 1990s. Therefore, in 1997, Boehm developed the COCOMO II to solve most of the COCOMO I problems.

Figure 2 shows the process of software schedule, cost, and manpower estimation in the

COCOMO II. The COCOMO II includes several software attributes such as: 17 Effort

Multipliers (EMs), 5 Scale Factors (SFs), Software Size (SS), and Effort estimation that are

used in the Post Architecture Model of the COCOMO II.

Figure 2. COCOMO II Estimation Process

The formula for the process is given by:

ffort (P )=A SizeB 0.01 S j

j=1

i

17

i=1

( )

Personnel = Effort/Schedule

Where A=2.94; B=0.91; C=3.67; D=0.28

More details about the Model can be found in [7].

EMs (17)

SF (5)

SS (1)

Effort

Schedule

Personne

l

COCOMO II

Effort Estimation

Model


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2. Proposed Model

Our model is established based on the COCOMO II and Fuzzy Logic. The COCOMO II

includes a set of input software attributes: 17 Effort Multipliers (EMs), 5 Scale Factors (SFs),

one Size in KDLOC (SZ) and one output, Effort. The architecture of the Model is shown in

Figure 3.

Figure 3. Fuzzy Logic Model for Software Effort Estimation

2.1. Inputs

There are three set of inputs

Size in KLOC 17 Effort Multipliers

5 Scale Factors All these inputs are provided as crisp data. These inputs are directed to fuzzification module.

2.2. Fuzzification

This module is sub divided into three sub modules. SZ-Fuzzifications module converts the

Size input to the fuzzy variable, SF-Fuzzification module converts Scale Factors input to

fuzzy variables whereas EM- Fuzzification module converts the Effort Multiplier input to

Fuzzy variable.

The Terms Very Low, Low, Nominal, High, Very High and extra High have been defined

for each variable. We defined a fuzzy set for each linguistic value for Triangular Membership

Function.

A Triangular Membership Function (TMF) is a three-point function [8], defined by

minimum (), aximum () and modal (m) values, that is, T (, m, ), where ( m ).

We take each linguistic variables as a triangular Fuzzy numbers, TFN (, m, ), m, m.

The membership function ((x)) for which is defined as:

0, x (x) = x- / m - , x m -x / -m , m x

0, x

Where m is the mean of input sizes.

EMs (17)

SF (5)

SZ (1)

EM-Fuzzification

SF- Fuzzification

SZ- Fuzzification

Inference

Engine

Rule Base

Database

Defuzzification

EFFORT

Fuzzification


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The Fuzzy Set defined for Size is shown in Figure 4a. The Size range has been divided into

small intervals to get a better fuzzy number. Figure 4b represents Fuzzy Set for Scale Factor

RESL whereas Figure 4c represents Fuzzy Set for Effort Multiplier PERS. All the scale

factors and effort multipliers have scaling range from Very Low to Extra High.

Figure 4a. Fuzzy Set for SIZE

Figure 4b. Fuzzy Set for Risk Resolution (RESL)

Figure 4c. Fuzzy Set For Personnel Capability (PERS)

The Fuzzication Process converts the crisp data to linguistic variables which are passed to

Inference Engine.


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2.3. Inference Engine

Inference engine operates on a set of fuzzy rules. The Fuzzy Model rules contain the

linguistic variables related to the project. We have implemented this model using FIS tool in

MATLAB. A sample of the rules is presented below:

if (PREC is VL) then (EFFORT is XH)

if(PREC is LO) then (EFFORT is VH)

if(FLEX is VL) then (EFFORT is XH)

The number of rules that have been used in the proposed model are 281.

2.4. Defuzzification

This module calculates and converts the fuzzy output into crisp data form. The

defuzzification has been performed using Centeroide Method. It has been used to calculate

the Centre of Gravity (COG) area under the curve as

COG = ( ) ( )

3. Experimental Study

In order to carry out our experiments to evaluate the efficiency of our proposed model, we

have chosen 30 projects data, developed in local software house, ranging from Very Small to

Large. We have chosen two estimation methods, COCOMO II and Alaa Sheta [1a] estimation

method.

We have used following MMRE, PRED (L) and VAF as criteria for assessment of these

models.

The Mean Magnitude of Relative Error, MMRE, is probably the most widely used

evaluation criterion for assessing the performance of competing models. One purpose

of MMRE is to assist us to select the best model as it gives us accuracy prediction of the

competing models. MMRE is calculated as average of the Magnitude of Relative Errors

MREs of each estimate. The MRE and MMRE are calculated as:

MRE =

Variance Account For (VAF) is used in the context of statistical models whose main

purpose is the prediction of future outcomes on the basis of other related information. It is the

proportion of variability in a data set that is accounted for by the statistical model. It provides

a measure of how well future outcomes are likely to be predicted by the model. It is

calculated as:

VAF(%) = ( ( )

( ))

Where is the actual ffort and is the Calculated ffort.


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Prediction PRED (L) is the probability of the model having relative Error less than or equal

to L. It is calculated:

PRED (L) = K / N

Where K is the number of Observations where MRE is less than or equal to L and N are total

number of observations.

Table 1 shows estimated efforts of all three models. It also shows MRE of all three models

for every estimate. Table 2 shows comparison of these models. The estimation results are

graphically shown in Figure 5, 6 and 7. Comparison of the model results in Table 2 shows

that the proposed model has better estimation accuracy as compared to other models having

7.512 MMRE with 98.77% likeliness to have the same ratio of MMRE if the model is tested

under different environment with different information. In order to further verify the

prediction we have analysed the results for 25%, 15%, 10% and 8% prediction. The results

show that the proposed model is 63.33% confident to have its average MRE, which shows the

stability of its estimates.

The stability of estimates of the proposed model can also be verified from Figure 5, 6, 7.

The Figure 7 shows that there are minute variations in actual and estimated calculations.

Table 1. Estimated Efforts and MREs of all Three Models

P.No Size Actual

COCOMO

II

Alaa

Sheta Proposed

MRE

COCOMO

MRE

Alaa

Sheta

MRE

Proposed

1 18 301 214 243 293 28.90365 19.2691 2.657807

2 50 1063 841 931 984 20.88429 12.41769 7.431797

3 40 605 601 453 537 0.661157 25.12397 11.23967

4 22 243 282 181 201 16.04938 25.5144 17.28395

5 13 141 128 95 127 9.219858 32.62411 9.929078

6 12 112 131 91 121 16.96429 18.75 8.035714

7 3 16 15 13 16 6.25 18.75 0

8 11 91 78 65 82 14.28571 28.57143 9.89011

9 42 781 751 634 722 3.841229 18.82202 7.554417

10 16 233 241 201 249 3.433476 13.73391 6.866953

11 21 334 304 289 318 8.982036 13.47305 4.790419

12 52 982 1043 756 890 6.211813 23.01426 9.368635

13 23 305 361 300 281 18.36066 1.639344 7.868852

14 27 421 459 398 401 9.026128 5.463183 4.750594

15 9 45 48 36 41 6.666667 20 8.888889

16 15 189 197 178 192 4.232804 5.820106 1.587302

17 4 17 19 14 17 11.76471 17.64706 0

18 6 22 23 19 22 4.545455 13.63636 0

19 2 8 8 7 8 0 12.5 0


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20 7 34 39 31 29 14.70588 8.823529 14.70588

21 13 84 97 72 79 15.47619 14.28571 5.952381

22 18 274 329 241 268 20.07299 12.0438 2.189781

23 24 403 398 365 379 1.240695 9.42928 5.955335

24 26 371 352 322 362 5.121294 13.20755 2.425876

25 61 1241 1459 871 1102 17.56648 29.81467 11.20064

26 29 677 605 561 638 10.63516 17.13442 5.760709

27 19 291 271 202 263 6.872852 30.58419 9.621993

28 10 110 128 94 119 16.36364 14.54545 8.181818

29 21 239 201 189 213 15.89958 20.9205 10.87866

30 12 145 168 156 189 15.86207 7.586207 30.34483

Table 2. Comparison of the Models

COCOMO

II

Alaa Sheta Proposed

MMRE 11.003% 16.838% 7.512%

VAF 95.86% 93.90% 98.77%

PRED(25) 93.33% 80% 96.33%

PRED(15) 63.33% 53.33% 93.33%

PRED(10) 50% 20% 80%

PRED(8) 40% 10% 63.33%

Figure 5. COCOMO II Vs Actual Calculation of Effort


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Figure 6. Alaa Sheeta Vs Actual Calculation of Effort

Figure 7. Proposed Model Vs actual Calculation of Effort

4. Conclusion

In this paper, we have presented a Software Effort Estimation Model using Fuzzy Logic

Model. Triangular membership function has been used to generate linguistic fussy values.

One can easily develop the same model using trapezoidal or Gauss-Bell membership

functions. Similarly ther is still margin of improvement in fuzzification process. If

fuzziffication process is further optimized, it will certainly result in substantial reduction in

number of fuzzy rules implemented in inference engine.


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Authors

Ziauddin

Dr. Ziauddin is working as Assistant Professor in COMSATS Institute

of Information Technology, Vehari. He has worked in Gomal University

for 22 years. He has published more than 15 research papers in reputed

International journals. His area of expertise is Software Engineering,

Software Process Improvement, Software Reliability Engineering,

Software Development Improvement, Software Quality Assurance and

Requirement Engineering. He is Gold Medalist from Peshawar

University in M.Sc Computer Science. He has Diploma in Computer

Forensics from School of Education, Indiana University USA.

Shahid Kamal

Shahid Kamal Tipu is working as Assistant Professor in Institute of

Computing & Information Technology, Gomal University, D.I.Khan

since 2000. Currently he is doing his Ph.D. in Malaysia. His area of

expertise is Information System Security and Software Quality. He has

published his research in reputed International Journals. He is a

renowned software engineer in local industry.

Shafiullah Khan

Dr. Shafiullah Khan is Assistant professor in Institute of Information

Technology, Kohat University of Science and Technology, Pakistan. He

has done his PhD from the School of Engineering and Design, Brunel

University, UK. His research mainly focuses on wireless broadband

network architecture, security and privacy, security threats and mitigating

techniques.

Jamal Abdul Nasir

Jamal Abdul Nasir is working as Assistant Professor in Institute of

Computing & Information Technology, Gomal University, D.I.Khan

since 1988. Currently he is doing his Ph.D. in CIIT, Islamabad. His area

of expertise is Data Mining and Artificial Intelligence. He has published

his research in reputed International Journals. He is a one of the pioneers

of Information technology in the country.


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