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F-effort: a fuzzified model of software effortestimationK. K. Aggarwal a , Yogesh Singh b & Jitender Kumar Chhabra ca G.G.S. Indraprastha University , Delhi , Indiab School of Information Technology , G.G.S. Indraprastha University , Delhi , Indiac Department of Computer Engineering , National Institute of Technology , Kurukshetra ,IndiaPublished online: 03 Jun 2013.

To cite this article: K. K. Aggarwal , Yogesh Singh & Jitender Kumar Chhabra (2004) F-effort: a fuzzified modelof software effort estimation, Journal of Discrete Mathematical Sciences and Cryptography, 7:3, 387-400, DOI:10.1080/09720529.2004.10698016

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F-effort : a fuzzified model of software effort estimation

K. K. Aggarwal∗

G.G.S. Indraprastha UniversityDelhi

India

Yogesh Singh†

School of Information TechnologyG.G.S. Indraprastha UniversityDelhiIndia

Jitender Kumar Chhabra‡

Department of Computer EngineeringNational Institute of TechnologyKurukshetraIndia

Abstract

Estimation of effort and resources for a software project is required for planningpurposes in very early stages of development. The traditionally used models to estimateeffort need very accurate inputs and result in a very precise value, but it leads to a numberof problems such as over commitment, which have not been overcome by these models. Thereason for these problems lies with the imprecision and vagueness present in various stagesof this estimation. In this paper, we have proposed a new model named as F -Effort model,which is a generalization of COCOMO model. The size of the software is estimated as a fuzzyset and the imprecision of complexity is managed with help of fuzzy logic. The F -Effortmodel has got two phases. First phase uses the complexity of the software to estimate twoinput parameters, and second phase uses these input parameters along with size to estimatethe effort. The model is able to generate the effort estimation either as a crisp value alongwith degree of possible variation or as a fuzzy set.

∗E-mail: aggarwal krishan@hotmail.com†E-mail: ys66@rediffmail.com‡E-mail: jitenderchhabra@rediffmail.com

—————————————————–Journal of Discrete Mathematical Sciences & CryptographyVol. 7 (2004), No. 3, pp. 387–400c© Taru Publications

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388 K. K. AGGARWAL, Y. SINGH AND J. K. CHHABRA

Introduction

In order to effectively develop software in an increasinglycompetitive and complex environment, many organizations are makingmore and more use of software metrics as part of their productmanagement process [1]. These metrics can then be used as variablesin models for predicting or estimating some aspects of the developmentprocess or product that are of interest [2]. The most common applicationof software metrics is to develop models that predict the effort (oftenmeasured in person-days) needed to be spent for development of thesoftware. These estimates are needed before development is initiated [3].Cost estimation is an important activity for managers, developers, andusers of the software [4, 5]. It is believed that better understanding ofeconomy of software development would reduce the current difficultiesof software production resulting in cost overruns or even projectcancellations [6, 7]. In order to properly estimate the cost of software, lotsof different cost estimation models have been proposed in literature [6,8-13]. All of these models are based on measuring certain size or functionrelated attributes of the software and relating these measurements to thecost or effort necessary for its development.

Software metrics are used to predict the efforts needed to developthe software. The effort estimation is based on mainly three parameters-system size, system complexity, and developer characteristics as shown inFigure 1 [2].

Figure 1Software effort estimation model

All of the traditional cost estimation models are usually developedusing data that contains some measures of system size and complexity,and developer ability in order to predict the effort required for variousstages of the development life cycle. These effort estimation models areusually developed using linear regression analysis on available historicaldata for sufficiently similar projects, although there has been increasing

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F -EFFORT 389

use of other techniques like regression trees [14], neural networks [15],and case based reasoning [16] also. But there are some difficulties with allof these models, which are described below.

Difficulties with traditional effort estimation models

1 Providing Exact Value of Inputs

A major problem that exists with such models is the difficulty projectmanagers face in specifying the exact values for the parameters usedas inputs to the models. Since for many parameters the actual valueis never known with certainty until the project is completed. Accuratespecification of the parameters, using such models therefore demands alevel of accuracy in prediction from project managers that is rarely possibleearly in the project life cycle, which is the crucial time of planning [2].

2 Crisp output leading to over commitment

The traditional models always result in crisp value(s) of the output(s)and this often leads to overconfidence in both the accuracy and precisionof the results. For example, using the COCOMO/COCOMO II model,the effort may get estimated as 25 person-months (or even worse as25.3241person-months), there is a risk that this value becomes sacred.This can lead to development time being wasted in the event of anoverestimate, and requirements remaining unfulfilled or the project goingover schedule to an even greater extent where the effort is underestimated.

A fuzzy – logic based approach to effort estimation.

We have suggested a solution here to, at least partially, overcome thepreviously mentioned problems by using fuzzy logic variables for variousinput parameters. In general it is considered that project managers canfairly readily specify independent variables in software metrics modelsusing linguistic labels, such as a large level of project complexity andbigger size of the software, in the early stages of estimation. In thispaper, we have proposed an extension to the COCOMO basic modelby incorporating the concept of fuzziness into measurement of size andvarious other parameters of this model. The reason for such an approachstems from the vagueness present in many of the input parameters ofcost estimation process : size, development modes, and many otherattributes are matter of guessing rather than exact measurements [6].We have chosen to extend the COCOMO model [10] because of three

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390 K. K. AGGARWAL, Y. SINGH AND J. K. CHHABRA

reasons. First, COCOMO model is one of the most frequently referredmodels in cost estimation [3, 17]. Second, COCOMO is based on log-linearformula considered the most plausible for software cost modelling [18].Third, the basic version of COCOMO is simple, and therefore suitable forillustration of use of fuzzy logic in cost estimation. It is believed that suchtype of fuzzy models provide considerable benefits in terms of reducingcommitment, making full use of imprecise knowledge. Some researcherssay to the extent that not only is fuzzy logic useful for effort prediction,but that is essential in order to improve quality of traditional estimationmodels [19].

Introduction to COCOMO model

Barry Boehm [10] introduced a hierarchy of software estimationmodels bearing the name COCOMO, for COnstructive COst MOdel. TheCOCOMO models are defined for three classes of software projects. UsingBoehm’s terminology, these are – Organic mode, semi-detached mode, andembedded mode. The basic equation of the effort/cost estimation of BasicCOCOMO model is [10] :

E = a(KLOC)b (1)

where

E = effort applied in person months

KLOC = estimated kilo lines of code

The values of a and b are [10] :

Software Project a b

Organic 2.4 1.05Semidetached 3.0 1.12Embedded 3.6 1.20

Fuzzification of effort estimation : F-effort model

It has been already proved that use of fuzzy logic in effort estimationgives better results than other traditional models [2]. We have used fuzzylogic to modify the basic COCOMO model to overcome some of thedifficulties of effort estimation models and proposed a new model named

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F -EFFORT 391

as F-Effort model. The changes to the basic COCOMO model along withtheir advantages over the model are as follows.

1 Providing input parameters as fuzzy sets

The input parameters to basic COCOMO model are a, b and size. aand b are two values which are used to decide the software’s complexityand KLOC is used to decide the software size. The complexity of thesoftware is usually difficult to quantify and is only approximated to someestimate. Thus the value of a and b is computed using fuzzy input ofthe complexity. Similarly size of the software is impossible to be specifiedaccurately in the initial phase of the software. So the size of the softwareto be developed is also used as a fuzzy set. The size is not mentioned as aparticular value, but it varies from a lowest value to a highest value withvarying degree of membership. This fuzzification becomes very useful inproviding some tolerance to the variation of size, which is very likely atthe completion of the software [2, 20].

2 Use of fuzzy output to reduce commitment

Particularly at the very early stages of a software developmentproject, estimating to within one person-day or person-month is simplynot realistic. Instead, a fuzzy system may be used to transform linguisticlabels, or numerical values, indicating system size and complexity, etc intoan equally imprecise (but adequate for its purpose) fuzzy set indicatinga continuous range of the predicted effort with some value(s) havingthe highest membership degree. While this approach may well beimprecise, this is justified and should ensure that personnel associatedwith the project do not attach unwarranted accuracy to the figuresproduced. As the project progresses and a greater degree of certaintyis established in relation to the scope of the project (and also as datastarts to become available), then more precise indicators of effort maybe formulated, either through more and smaller membership functions orallowing for numerical defuzzification. There is an inherent trade-off in thedevelopment of effort estimation models ; is it better to be approximatelycorrect most of the time or precisely inaccurate all of the time? [2]

Working of F -effort model.

The proposed F-Effort model uses fuzzy values of the complexity

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392 K. K. AGGARWAL, Y. SINGH AND J. K. CHHABRA

and size and produces the output as a fuzzy set. In order to keep thesoftware managers away from the intricacies of the fuzzy logic, the modelalso generates a crisp value of the effort estimation along with a degree ofpossible variation. The details of the output are explained in the later partof the paper. The model works in two phases. First phase uses the fuzzyinput of the complexity of the software, and generates the crisp values fora and b . The output of the first phase (value of a and b) along with thesize of the software become input to the second phase of the model andgenerate the effort estimation as a fuzzy set. The most likely value of theeffort is also generated along with the possible degree of variation. Theblock diagram of the F-Effort model is shown below in Figure 2.

Figure 2Block diagram of F-effort model

1 Fuzzy-model (Part I)

In the Fuzzy Model (Part-I), the crisp value of a and b is generatedusing fuzzy value of software complexity. The advantage of using FuzzyModel (Part-I) is two fold. One is that the software manager can easilygive the input of complexity as a linguistic variable and second advantageis that the value of a and b can be anything between 2.4-3.6 and 1.05-1.20 respectively instead of just 3 fixed value. The software complexity isinherently continuous in nature and thus considering only three distinctvalues of a and b towards computation of effort is not very realistic.A software may be thought of belonging to more than one categoryof complexity, which obviously cannot be handled with help of crisplogic [21, 22]. So software complexity is estimated as a fuzzy set and 3membership functions – Low, Medium and High are defined as shown inFigure 3.

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F -EFFORT 393

Figure 3Input variable complexity

The outputs of this model are two values - a and b . These twovalues are used in the next phase as input to fuzzy model (Part-II), andare defined as three singleton functions each as shown in Figure 4 and 5respectively. Minimum and maximum value of a and b is used from thebasic COCOMO model, but in-between values are also possible for bothof these depending upon the rules getting fired.

As the outputs of this model are singleton functions, the Sugenoinferencing is applicable in this case [23, 24]. For the inferencing of thefuzzy model (Part-I), six simple rules are sufficient, 3 for computing valueof a and 3 for computing value of b .

1. If complexity is low then a is a-low.

2. If complexity is med then a is a-med.

3. If complexity is high then a is a-high.

4. If complexity is low then b is b-low.

5. If complexity is med then b is b-med.

6. If complexity is high then b is b-high.

Figure 4Output variable ab

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394 K. K. AGGARWAL, Y. SINGH AND J. K. CHHABRA

Figure 5Output Variable bb

Based on these inputs, it is possible that a software project maybelong to more than one category. For example a software may becategorized as of low complexity as well as of med complexity. In thatcase the value of a will be 2.7 and b will be 1.08. This concept is illustratedwith help of Figure 6, which corresponds to the output screen generatedfrom simulation of this model implemented with help of Matlab [25].

2 Fuzzy-model (Part-II)

The second phase of proposed F -Effort model uses the outputs offuzzy model (Part-I) (value a and b) and size (in KLOC) as input. Thesize of the software under consideration is taken as input in the form ofa fuzzy set from the software manager and the effort is computed usingequation (I). As the input to this phase consists of one fuzzy variable, afuzzy model (Part-II) is developed for computing the effort. The size ofthe software is considered as a fuzzy set. The trapezoidal membership

Figure 6Output screen for value a and b of Fuzzy Model (Part-I)

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F-EFFORT 395

function is one of the most frequently used fuzzy set, which is ageneralization of triangular membership function, and the triangularmembership function has been already accepted as an approximationto many different membership functions [26]. A trapezoidal fuzzy set(TFS) K , is described by a quartet {d, e, f , g} as shown in Figure 7. Themembership function for trapezoidal fuzzy set K is defined in form ofequation (2).

Figure 7Trapezoidal fuzzy set

K(x) =

x− de− d

for x ∈ [d, e]

1 for x ∈ [e, f ]g− xg− f

for x ∈ [ f , g]

0 for x 6∈ [d, g] .

(2)

Once the size has been estimated as a fuzzy set the effort is estimated usingequation (1) as

E = a(size)b .

The input variable size being fuzzy in nature, the effort is computed usingfuzzy arithmetic [6, 27, 28]. The determination of the membership functionof the effort is based on the extension principle as

E(y) = supx∈R : y=ab

X

[K(x)] (3)

where K(x) and E(y) denote membership function of the size andmembership function of the effort respectively. The above equation (3)

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396 K. K. AGGARWAL, Y. SINGH AND J. K. CHHABRA

can be rewritten in a different format by eliminating the constraintsand moving them directly into the membership function, and then theequation becomes of the form

E(y) = supx∈R : y=ab

X

[K(x)] = K

((Ya

)1/b)

. (4)

When the trapezoidal fuzzy set is passed as input to the fuzzy model (Part-II), the output effort is obtained by substituting (2) into (5)

E(y) =

( ya

)1/b − d

e− dfor y ∈ [ab

d, abe ]

1 for y ∈ [abe , ab

f ]

g−( y

a

)1/b

g− ffor y ∈ [ad

f , abg]

0 for y 6∈ [abd, ab

g] .

(5)

Using equation 4 and equation 5, the effort estimated can be computed,which will be a fuzzy set, and will give a better estimate of effort needed todevelop the software under consideration. If the software manager is stillinterested in a crisp value, then the fuzzy set is defuzzified using centroidtechnique, and a crisp value is generated, which represents the most likelyestimate of effort [29, 30]. In order to inform the software manager aboutthe possible variation in the effort estimate, standard deviation of thisfuzzy set is also computed, and this value reflects the possible variationin the estimated value, which may occur depending upon the final sizeof the completed software. Thus software manager gets a range of effortestimation instead of just a single value, which is definitely much moreuseful in planning of the software.

The results of F -effort model for some selected software projects havebeen computed and are illustrated below in figure 8 with help of a selectivesoftware project, where software complexity belongs to low as well as medcategory, and size of the software is defined as a trapezoidal set as shownin Figure 8. As complexity belongs to low as well as med, the value of aand b will be 2.7 and 1.08 respectively, as already shown in Figure 6.

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F -EFFORT 397

Figure 8Estimated size and equivalent effort estimated with a = 2.7 ,

b = 1.08

The above shown fuzzy set of effort estimated is an output of theproposed F -Effort model, which will be very useful for the softwaremanager to avoid over commitment. This set helps to have anapproximate correct result than a precise inaccurate result. In order toget a single value the above estimated effort, effort can be defuzzified. Itmay be noted that the plotting of output fuzzy set of effort gives a shape,which is very-very close to a trapezoidal set, and hence defuzzificationusing centroid method is done, as shown below [29-30].

Effort =

∫ 15.36

8.84(mx + c)xdx +

∫ 18.7

15.36xdx +

∫ 25.5

18.7(mx + c)x dx

∫ 15.36

8.84(mx + c)dx +

∫ 18.7

15.36dx +

∫ 25.5

18.7(mx + c)dx

=

∫ 15.36

8.84(0.15x− 1.36)xdx +

∫ 18.7

15.36xdx +

∫ 25.5

18.7(−0.15x + 3.75)xdx

∫ 15.36

8.84(0.15x− 1.36)dx +

∫ 18.7

15.36dx +

∫ 25.5

18.7(−0.15x + 3.75)dx

=

[0.15x3

3− 1.36x2

2

]15.36

8.84+

[x2

2

]18.7

15.36+

[−.015x3

3+

3.75x2

2

]25.5

18.7[0.15x2

2− 1.36x

]15.36

8.84+

[ x1

]18.7

15.36+

[−0.15x3

2+ 3.75x

]25.5

18.7

= 17.03 .

The F -Effort model has been implemented by the authors with help ofMATLAB and C LANGUAGE. The same value of 17.03 is obtained fromthis implementation also for the corresponding specified inputs, whichverifies the results. Similarly the standard deviation is also computed

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398 K. K. AGGARWAL, Y. SINGH AND J. K. CHHABRA

for the output fuzzy set and comes out to be 5.18. This gives a hint tothe software manager that the effort is going to be approximately 17 unitswith a variation of ±5 . It is obvious that if there is more ambiguity (i.e.fuzziness) in the size input, the standard deviation will be more and vice-versa.

Conclusion

The idea of using fuzzy logic for defining input parameters of effortestimation model by defining them as linguistic parameters and formodeling the process has been outlined in this paper. The motivationfor this approach has been the difficulties faced by software managers interms of avoiding premature and costly commitment, using the availableimprecise knowledge. A new model named as F -Effort model has beenpresented in this paper, which is able to handle the vagueness of variousinput parameters, and generates the output in such a way that softwaremanager can have a better software project planning and management.

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Received October, 2002

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