A Fuzzy Approach for Component Selection amongst Different Versions of Alternatives for a Fault Tolerant Modular Software System under Recovery Block Scheme Incorporating Build-or-Buy Strategy

doi:10.4236/ajor.2011.14029

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American Journal of Operations Research, 2011, 1, 249-258

doi:10.4236/ajor.2011.14029 Published Online December 2011 (http://www.SciRP.org/journal/ajor)

249

A Fuzzy Approach for Component Selection amongst

Different Versions of Alternatives for a Fault Tolerant

Modular Software System under Recovery Block Scheme

Incorporating Build-or-Buy Strategy

P. C. Jha1, Ritu Arora2, U. Dinesh Kumar3

1Department of Operational Research, University of Delhi, Delhi, India

2Maharaja Agrasen Institute of Technology, GGSIP University, Delhi, India

3Indian Institute of Management, Bangalore, India

E-mail: jhapc@yahoo.com, arora_ritu21@yahoo.co.in, dineshk@iimb.ernet.in

Received September 20, 2011; revised October 22, 2011; accepted November 8, 2011

Abstract

Software projects generally have to deal with producing and managing large and complex software products.

As the functionality of computer operations become more essential and yet more critical, there is a great

need for the development of modular software system. Component-Based Software Engineering concerned

with composing, selecting and designing components to satisfy a set of requirements while minimizing cost

and maximizing reliability of the software system. This paper discusses the fuzzy approach for component

selection using “Build-or-Buy” strategy in designing a software structure. We introduce a framework that

helps developers to decide whether to buy or build components. In case a commercial off-the-shelf (COTS)

component is selected then different versions are available for each alternative of a module and only one

version will be selected. If a component is an in-house built component, then the alternative of a module is

selected. Numerical illustrations are provided to demonstrate the model developed.

Keywords: Modular Software, Software Reliability, Software Components (COTS and In-House), Fault

Tolerance & Fuzzy Optimization

1. Introduction

Computer software is very important in today’s world. In

particular, science and technology demand high quality

software for making improvements and breakthroughs.

The software development companies are continuously

developing/modifying/updating their software according

to the changing needs and requirements. The concept of

“software reliability” and its measurement is receiving

much attention in the software development community.

Software reliability is one of the important parameters of

software quality and system dependability. Software re-

liability engineering balances customer needs in the ma-

jor quality characteristics of reliability, availability, de-

livery time and cost more effectively. The reliability of

software can be controlled during the development life

cycle through the application of reliability improvement

techniques. Two of the best-known fault tolerant soft-

ware design methods are N-version programming and

Recovery block scheme. The basic mechanism of both

the schemes is to provide redundant software to tolerate

software failures. Software whose failure can have bad

effects afterwards can be made fault tolerant through

redundancy at module level [1].

When the design of software architecture reaches a

good level of maturity, software engineers have to take a

decision on the selection of software components. Non

functional aspects play a significant role in determining

software quality. Given the fact that lack of proper han-

dling of non functional aspects of a software application

has led to a series of software failures [2], nonfunctional

attributes such as reliability security and performance

should be considered during the component selection

phase of software development. This paper discusses a

framework that helps developers to decide whether to

buy or build components of software architecture on the

P. C. JHA ET AL.

250

basis of cost and non functional factors. While develop-

ing software, components can be both bought as com-

mercial off-the shelf (COTS) products, and probably

adapted to work in the software system, or they can be

developed in-house. This decision is known as “build-or-

buy decision”. This decision affects the overall cost and

reliability of the system. Most of the current software

systems include one or more COTS products. COTS are

pieces of software that can be reused by software pro-

jects to build new systems. Benefits of COTS based de-

velopment include significant reduction in the develop-

ment cost, time and improvement in the dependability

requirement. The components, which are not available in

the market or cannot be purchased economically, can be

developed within the organization and are known as in-

house built components. Reference [3] discussed issues

related to reliability of systems, caused by integrating

COTS components. The optimal selection is achieved

through weighted maximization of system quality subject

to budget as a constraint in which an upper limit is

placed over the constraint.

This paper discusses the issues related to reliability of

the software systems and cost caused by integrating

COTS or in-house built components. Fault tolerance is

achieved through redundancy in Recovery Block model

and redundancy results in additional cost. We assume

that for all the alternatives available for a module, cost

increases if higher reliability is desired. Several alterna-

tives of a software module may be available as COTS,

almost equivalent from the functional viewpoint. Pur-

chase of high quality COTS products can be justified by

the frequent use of the module. Large software systems

possess the modular structure to perform a set of func-

tions. Each function is performed by different modules

having different alternatives for each module. In case, a

COTS component is selected then different versions are

available for each alternative of a module and only one

version will be selected. If a component is an in-house

built component, then the alternative of a module is se-

lected. A schematic representation of the software sys-

tem is given in Figure 1.

In the existing research related to the software deci-

sion, it is assumed that all the parameters of the problem

are known precisely. Various objectives and restrictions

are set by the management and cost coefficients involved

in the cost function are determined based on past experi-

ence and the available data base. This makes it difficult

for the management to provide precise values of the var-

ious cost coefficients and objectives to be met. More-

over due to changing customer specifications, lack of

experience of testing team or novelty, changing testing

environment, complexity in the project involved, un-

known emerging factors at the start of the project adds

Figure 1. Structure of the software.

imprecision and ambiguity to the above-mentioned defi-

nitions. It may also be possible that the management it-

self does not set precise values in order to provide some

tolerance on these parameters due to competitive consid-

erations. All this leads to uncertainty (fuzziness) in the

problem formulation. Crisp mathematical programming

approaches provide no such mechanism to quantify these

uncertainties. Fuzzy optimization is a flexible approach

that permits more adequate solutions to real problems in

the presence of vague information, providing well-de-

fined mechanisms to quantify the uncertainties directly.

The idea of fuzzy programming was first given by [4]

and then developed by [5-7]. A number of researchers

thereafter have contributed to the development of fuzzy

optimization technique [8,9]. Today, similar to the de-

velopments in crisp optimization, different kinds of ma-

thematical models have been proposed and many practi-

cal applications have been implemented by using the

fuzzy set theory. Reference [10] formulated fuzzy multi

objective optimization models for selecting the optimal

COTS software products in the development of modular

software system. Recently, Reference [11] de- velops a

crisp multi-objective programming model from the fuzzy

basic data. When a feasible solution to the problem exists,

single and multiple objective fuzzy opti- mization pro-

cedure are used to solve the problem. How- ever, it is

assumed that a crisp or a constant value of all the pa-

rameters is known. However, in practice, it is not possi-

ble for a management to obtain a precise value of reli-

ability and cost for a software system. Or they may de-

cide not to set precise levels due to the market consid-

erations and are ready to have some tolerance of their

objectives. When the precise values of parameter of the

problem are not known, the problem becomes a fuzzy

optimization problem and the solution so obtained is a

fuzzy approximation.

This paper proposes two fuzzy multi-objective opti-

mization models for selecting the best software product

for each module. The first optimization model (optimiza-

251

P. C. JHA ET AL.

tion model-I) of this paper is a joint optimization prob-

lem that maximizes the system reliability with simulta-

neous minimization of the cost. The second optimization

model (optimization model-II) considers the issue of

compatibility between different alternatives of modules

as it is observed that some COTS components cannot

integrate with all the alternatives of another module. We

apply fuzzy optimization procedure to solve the problem,

when a feasible solution of the problem exists, but in

case we reach the infeasibility case, then we apply fuzzy

goal programming optimization technique to provide a

compromised solution for the same. The rest of this pa-

per is organized as follows. Section 2 consists of pro-

posed notations. In Section 3, we discuss the assump-

tions of optimization models and we develop a crisp

model for reliability and cost and in Section 4, we de-

scribe Fuzzy Multi-Objective Optimization Model for

software products selection. In Section 5, Fuzzy optimi-

zation technique is discussed to solve the problem with

numerical illustration. In Section 6, we furnish our con-

cluding observations.

2. Notations

R: System quality measure

: Frequency of use, of function l

: Set of modules required for function l

R: Reliability of module i

: Number of functions, the software is required to

perform

n: Number of modules in the software

V: Number of alternatives available for module i

ij : Number of versions available for alternative

of module

tot

ij : Total number of tests performed on the in-house

developed instance (i.e. alternative of module )

ij : Number of successful (i.e. failure free) test per-

formed on the in-house developed instance (i.e. alterna-

tive of module )

j i

1: Probability that next alternative is not invoked

upon failure of the current alternative

t: Probability that the correct result is judged wrong.

3: Probability that an incorrect result is accepted as

correct.

: Event that output of alternative of module

is rejected.

j i

Yi: Event that correct result of alternative of mod-

ule is accepted.

: Reliability of alternative of module j

r: Reliability of alternative for module i

ijk : Cost of version of alternative for module

k j

ijk

r: Reliability of version k of alternative for

module

ijk : Delivery time of version of alternative

for module

dk j

ij : Unitary development cost for alternative of

module

c j

ij : Estimated development time for alternative of

module

t j



: Average time required to perform a test case for

alternative of module i j

πij: Probability that a single execution of software

fails on a test case chosen from a certain input distribu-

tion





















if constraint is active

if constraint is inactive

if the alternative of module is

in-house develoep.

0otherwise

th th

ijk

















if the version of COTS alternative

of the module is chosen

ot h e r w i s e

th th

z: Binary variable taking value 0 or 1

if alternative is present in module

0otherwise

3. Optimization Models

The first optimization model is developed for the fol-

lowing situations, which also holds good for the second

model, but with additional assumptions related to com-

patibility among alternatives of a module.

The following assumptions are common for optimiza-

tion models:

1) Software system consists of a finite number of mod-

ules.

2) Software system is required to perform a known

number of functions. The program written for a function

can call a series of modules . A failure occurs if a

module fails to carry out an intended operation.



n



3) Codes written for integration of modules do not

contain any bug.

4) Several alternatives are available for each module.

Fault tolerant architecture is desired in the modules (it

has to be within the specified budget). Independently

developed alternatives (primarily COTS/In-House com-

ponents) are attached in the modules and work similar to

the recovery block scheme discussed in [12,13].

5) The cost of an alternative is the development cost, if

developed in house; otherwise it is the buying price for

the COTS product.

P. C. JHA ET AL.

252

6) Different In-house alternatives with respect to uni-

tary development cost, estimated development time, av-

erage time and testability of a module are available.

7) Cost and reliability of an in-house component can

be specified by using basic parameters of the develop-

ment process, e.g., a component cost may depend on a

measure of developer skills, or the component reliability

depends on the amount of testing.

8) Different versions with respect to cost, reliability

and delivery time of a module are available.

9) Other than available cost-reliability versions of an

alternative, we assume the existence of virtual versions,

which has a negligible reliability of 0.001, zero cost and

zero delivery time. These components are denoted by

index one in the third subscript of ijk

, and ijk .

for example 1 denotes the reliability of first version of

alternatives for module .

Cijk r

j i

3.1. Model Formulation

Let be a software architecture made of modules,

with a maximum number of i alternatives available

for each module and each COTS alternatives has differ-

ent versions.

3.1.1. Build versus Buy Decision

For each module , if an alternative is bought (i.e. some

) then there is no in-house development (i.e.

) and vice versa.

ijk

x

y

ij ijk



; and 1, 2,,in1, 2,,i

jm

3.1.2. Redundancy Constrai nt

The equation stated below guarantees that redundancy is

allowed for the components.

ijijk ij







ij ij

xz

j



z1, 2,,in

1, 2,,in

ij 1, 2

; and

;

1, 2,,i

jm

1, 2,,i

jm

z, ,in





3.1.3. Probability of Failure Free In-House

Developed Components

The possibility of reducing the probability that the

alternative of module fails by means of a certain

amount of test cases (represented by the variable ).

Reference [14] define the probability of failure on de-

mand of an in-house developed alternative of

module, under the assumption that the on-field users’

operational profile is the same as the one adopted for

testing [15].

tot

Basing on the testability definition, we can assume

that the number

Nth

j of successful (i.e. failure free)

tests performed on alternative of same module.





1π

uc tot

ijij ij

NN ; 1, 2,,in



 and 1, 2,,i

jm

Let A be the event “

N failure-free test cases have

been performed ” and B be the event “ the alternative is

failure free during a single run”. If ij



is the probability

that the in-house developed alternative is failure free

during a single run given that

N test cases have been

successfully performed, from the Bayes theorem we get

the following.

 



PABPB

PBA PABPB PABPB



 

The following equalities come straightforwardly:







•1

•1π

•π

PAB







therefore, we have



1π

1ππ1π

ij N

ij ijij







 

;

1, 2,,in



 and 1, 2,,i

jm

3.1.4. Reliability Equation of Both In-House and

COTS Components

The reliability (ij

) of alternative of module of

the software.

jth

ijij ijij





; 1,2,,in



 and 1, 2,,i

jm

where

ijijk ijk

rrx



; 1, 2,,in



 and 1, 2,,i

jm

3.1.5. Delivery Time Constraint

The maximum threshold has been given on the de-

livery time of the whole system. In case of a COTS

components the delivery time is simply given by ijk ,

whereas for an in- house developed alternative the deliv-

ery time shall be expressed as .



tot

ijij ij







11 1

ntot

ij ijijijijk ijk

ij k

tN dx



 













253

P. C. JHA ET AL.

3.2. Objective Function

3.2.1. Reliability Objective Function

Reliability objective function maximizes the system

quality (in terms of reliability) through a weighted func-

tion of module reliabilities. Reliability of modules that

are invoked more frequently during use is given higher

weights. Analytic Hierarchy Process (AHP) can be effec-

tively used to calculate these weights.



Maximize

lis

RXf R





where i is the reliability of module of the system

under Recovery Block stated as follows.

R i



iij

iijik ij

RzPXPY















111

ij ij

PXt s 



;

1, 2,,in



2ij











ij ij

PYs t

3.2.2. Cost Objective Function

Cost objective function minimizes the overall cost of the

system. The sum of the cost of all the modules is selected

from “build- or -buy” strategy. The in-house develop-

ment cost of the alternative of module can be

expressed as

j i



tot

ijijij ij

ct N







11 1

Minimize

itot

ij ijijijijijk ijk

ij k

CXc tNyCx



 











3.3. Optimization Model I

In the optimization model it is assumed that the alterna-

tives of a module are in a Recovery Block. In a Recovery

block, more than one alternative of a program exist. For

COTS based software, multiple alternatives of a module

can be purchased from different vendors. Each module

works under a recovery block. Upon invocation of a

module the first alternative is executed and the result is

submitted for acceptance test. If it is rejected, the second

alternative is executed with the original inputs. The same

process continues through attached alternative until a

result is accepted or the whole recovery block (module)

fails. Fault tolerance in a recovery block is achieved by

increasing the number of redundancies.

Problem (P1)



Maximize

lis

RXf R



i

(1)



11 1

Minimize

ntot

ijijijijijijk ijk

ij k

CXctNyCx



 









 (2)

subject to





and are binary variable

ijk ij

XS xy



iij

iijik ij

RzPXPY











; (3) 1, 2,,in













111

ijij ij

PXtstst





 











ij ij

PYst





1π

uc tot

ijij ij

NN; 1,2,,in



 and

(4)

1, 2,,i

jm



1π

;

1ππ1π

1,2,, and 1,2,,

suc

ij N

ij ijij

inj







 

m

(5)

ijij ijij





; 1, 2,,in



 and (6) 1, 2,,i

jm

ij ijk







; 1,2,,in



1, 2,,jm

and (7)



ijijk ij







; 1, 2,,in1, 2,,jm and (8)





ij ij



; 1, 2,,in



 and (9) 1, 2,,i

jm





; (10) 1, 2,,in



11 1

ntot

ijijijijijk ijk

ij k

ytNdx T



 











 (11)



where

is a vector of component ijk

and ;

1, 2in, ,



; 1, 2,,i



; .

1, ij

kV2, ,

3.4. Optimization Model II

As explained in the introduction, it is observed that some

alternatives of a module may not be compatible with

alternatives of another module [16]. The next optimiza-

tion model II addresses this problem. It is done, incorpo-

rating additional constraints in the optimization models.

This constraint can be represented as t

sqhu c

x, which

means that if alternative

for module

is chosen,

then alternative ,

u1, ,tz



 have to be chosen for

module . We also assume that if two alternatives are

compatible, then their versions are also compatible.

sqhu ct

xMy





2, ,



, 2,,t

cV, 1, ,

m (12)

P. C. JHA ET AL.

254





thu

yzV

 (13)

Constraint (12) and (13) make use







subject to

of binary variable

fer

to choose one pair of alternatives from among dif-

ent alternative pairs of modules.

Finally, model can be re-written as

Problem (P2)



ximize

lis

RXf R







ntot

ij ijij ijij

CXctNy















Minimize

ijk ijk





S

sqthu c

xMy



2, ,

s, t

hu , qV2, ,cV1, ,

m,



atible co





t

yzV

If more than one alternative compmponent is

to be c

4. Fuzzy Multi-Objective Optimization

ris the concept

sed on

tion model for software

roducts selection based on maximizing the fuzzifier

subject to

hosen for redundancy, constraint (13) can be re-

laxed as follows.



thu

yzV



Model for Software Products

inciple to multi-objective optimization P

of an “efficient solution”, where any improvement of one

objective can only be achieved at the expense of another.

The fuzzy approach can be used as an effective tool for

quickly obtaining a good compromise solution. Conven-

tional optimization methods assume that all parameters

and goals of an optimization model are precisely known.

However, for many practical problems there are incom-

pleteness and unreliability of input information. This has

resulted in use of fuzzy multi-objective optimization

method with fuzzy parameters. For instance, a designer

is required to minimize the system cost while simultane-

ously maximizing the system reliability. Therefore mul-

tiple objective functions become an important aspect in

the reliability design of the engineering systems.

In general reliability optimization problem is solved

with the assumption that the coefficients or cost of com-

ponents is specified in a precise way. In real life, there

are many diverse situations due to uncertainty in judg-

ments, lack of evidence, etc. Sometimes it is not possible

to get relevant precise data for the reliability system.

This type of imprecise data is not always well repre-

sented by random variable selected from a probability

distribution. Fuzzy number may represent this data, so

fuzzy optimization method with fuzzy parameters is

needed for a fuzzy reliability optimization model.

Therefore, we formulate fuzzy multi-objective opti-

mization model for software products selection ba

gue aspiration levels, the decision maker may decide

his aspiration levels on the basis of past experience and

knowledge possessed by him. To express vague aspira-

tion levels of the decision, various membership functions

have been proposed [6,7]. A fuzzy mathematical pro-

gramming problem with non linear membership function

results in a non linear programming problem. Usually, a

linear membership function is employed to avoid non-

linearrity. Further, if membership function is interpreted

as the fuzzy utility of the decision maker, which describes

the behavior of indifference, preference or aversion to-

wards uncertainty, a non linear membership function is a

better representation than a linear membership function.

4.1. Problem Formulation

Fuzzy multi-objective optimiza

reliability function and minimizing the fuzzifier cost

function subject to crisp constraints are stated as follows.

Problem (P3)



ximize ()

lis

RXf R



 Ma



111

Minimize(X) =

ntot

ijijijijijijk ijk

ij k

CctNyC



 









 





Here, we have defined the two objective functions, the

and cost that are considered to be vague and

tion

sed to solve the fuzzy mathe-

atical programming problem.

function. Same defuzzi-

fic

reliability

certain (i.e. fuzzy in nature) and the constraints are of

crisp nature. Cut-throat competition in the existing mar-

ket, system complexity, and intended flexibility makes it

difficult for the management to precisely define their

goals and constraints. Moreover a slight shift on bounds

can provide a more efficient solution. Hence, we have

used fuzzy optimization technique (fuzzy mathematical

programming) to solve the fuzzy multi-objective opti-

mization problem.

4.2. Problem Solu

The following steps are u

mStep 1. Compute the crisp equivalent of the fuzzy pa-

rameters using a defuzzification

ation function is to be used for each of the parameters.

255

P. C. JHA ET AL.

Here we use the defuzzification function of the type



123

24FAaa a

where are the triangular fuzzy numbers.

Step Ie the objective function of the fuzzi-

n (maxn

ncor

porat

fier mi) as a fuzzy constraint with a restrictio

spiration) level. The above problem (P3) can be re-

written as

Problem (P4)

Find

subject to



lis

RXfR R











11 1

tot

ijijijijijijk ijk

ij k

CXc tNyCxC



 









 

n







S

Step 3. Define appropriate membership functions for

each fuzzy inequalities as well constraint correspond-

g to the objective function. The membership function

for the fuzzy less than or equal to and greater than or

equal to type are given as



 



1; RX R

;

RX R

RRXR

RX R

















where is the aspiration level and is the toler-

fuzzy reliability objfunction con-







ance level to the

ective

straint.



  

1; CXC

;

0;( )

CCXC

CX C



















where is the restriction level and is the toler-

he fuzzy budget constraint.

4. to

mathematical





ance level to t

Step Extension principle is used identify the

fuzzy decision, which results in a crisp

ogramming problem given by

Problem (P5)

Maximize



subject to









,







,

S

can be solved by the standard crisp mathem

gramming algorithms.

onstraint. Each constraint is

atical pro-

Step 5. While solving the problem, the objective of the

problem is also treated as a c

nsidered to be an objective for the decision maker and

the problem can be looked as a fuzzy multiple objective

mathematical programming problem. Further each objec-

tive can have a different level of importance and can be

assigned weights according to their relative importance.

The resulting problem can be solved by the weighted

minmax approach. The crisp formulation of the weighted

problem is given as

Problem (P6)

Maximize



subject to









, C









,

S

where,

ww,0, 12

1ww



represents the gree up to which the aspira-

tion of the decision maker is met. If thnstraints are

e co

fuzzy as well as crisp, then in the equivalent crisp ma-

thematical programming problem, there will be no

change in the original crisp constraints since their toler-

ances are zero except for those constraints which are

fuzzy in nature. The problem (P6) can be solved using

standard mathematical programming approach.

Step 6. On substituting the values for







and







the problem becomes

Problem (P7)

mize Maxi



subject to











wR R





1R RX0











020

1CXCwC C



 







0, 1





,0ww, 12

1ww

Step 7. If a feasible solution is not otainable for the

problem (P7) then we can use fuzzy googramming

aving two modules with

ore than one alternative for each module. The data sets

al pr

proach to obtain a compromised solution [9]. The me-

thod is discussed in detail in the numerical illustration.

5. Illustrative Examples

Consider a software system h

for COTS and in-house developed components are given

in Table 1 and Table 2, respectively.

Let 3L







11, 2, 3s,



21, 3s,





32s,

10.5f



, 20.3f



, 30.2f



. It is also assumed that

t0.01



, 2

t0.05



and

ent of weig

is based on the expert’s

0.01.

tsThe assignmh

judgment for the reliability and the cost criteria. Weights

nimum and Maximum Level of

Reliability and Cost

First liability and cost values are

omputed using fuzzied values of these parameters and

signed for reliability and cost are 0.7 and 0.3 respec-

tively.

5.1. Mi

ly, the triangular fuzzy re

P. C. JHA ET AL.

256

Table 1. Data set for COTS components.

Version 1

Modules Alternatives

Cost Reliability Delivery Time

1 0 0.00 01

2 0 0.001 0

3 0 0.001 0

1 0 0.001 0

2 0 0.001 0

3 0 0.001 0

1 0 0.001 0

2 0 0.001 0

3 0 0.001 0

Version 2

Modules Alternatives

Cost Reliability Delivery Time

1 14 0.90 3

2 12.5 0.86 4

3 17 0.90 2

1 13 0.87 4

2 11 0.91 5

3 18 0.89 2

1 13 0.86 4

2 16 0.85 3

3 16 0.89 3

Version 3

Modules Alternatives

Cost Reliability Delivery Time

1 11 0.88 4

2 18 0.92 2 1

3 15 0.88 3

1 17.5 0.86 2

2 12 0.89 4

3 15 0.86 3

1 14 0.88 3

2 18 0.90 2

3 17 0.87 2

Table 2. Data set n-h omponen

for iouse cts.

dule i Alternatives i

t i





1 8 0.005 5 0.002

2 60.005 4 0.00

3 7 0.005 4 0.002

1 9 0.005 5 0.002

2 5 0.005 2 0.002

3 6 0.005 4 0.002

4 5 0.005 3 0.002

1 6 0.005 4 0.002

2 5 0.005 3 0.002

then defuzzied using Hn’ s f thil-

ble reliability and cost are specified as TFN (triangular

The aspiration level of reliability is and

the restriction on cost is

eilper deuzzier.If e ava

fuzzy number) given as follows



00.93,0.95,0.97R

;



C

.

070,75,80

00.95R

075C



. The tolerance

085C.

level

ming

(P7)

not feasible; hence the management goal cannot be

r reliability and cost is *

00.85R and

5.2. Fuzzy Goal Program Approach

On solving the problem, we found that the problem

achieved for a feasible value of





0,1



. Now we use

fuzzy goal programming technique to obtain a compro-

mised solution. The approach is the goal pro-

gramming technique for solving crisp goal programming

problem (Mohamed, 1997). The maximum value of any

membership function can be 1; maximization of

based on





0,1



 is equivalent to making it as close to 1 as best

as possible. This can be achieved by minimizi

deviational variables of goal programming (i.e.



) from 1. The fuzzy goal programming formulation for

the given problem (P7) introducing the negative and pos-

itive deviational variables

ng the

negative



and



is given as

Minimize u subject to





11R1X











  



; 0





; ,0





1, 2j





;





0, 1



;

1ww

,0ww;



; 1u





5.3. Optimization Model I

for optimization model I.

he problem is solved using software package LINGO

l for compatibility, we

se previous results.

ond alternative of first module is

co econd

Table 3 presents the solution

[17]. The solution to the model gives the optimal com-

ponent selection for the software system along with the

corresponding cost and reliability of the overall system.

The sensitivity analysis to the delivery time constraint

has been performed. It is clearly seen from the table that

in case 1, when the delivery time was 15 units then one

in-house and other COTS components were selected

while in all other cases when the delivery time increases

along with in-house components there will be a corre-

sponding change in reliability and cost. In case 2, when

the delivery time was 18 units, our system reliability and

cost also increases and in case 3 as compared to case 2,

when delivery time was 20 units, system reliability in-

creases and cost decreases. Therefore, if the customer is

ready to wait then case 3 is an optimal solution. Redun-

dancy is also present in all the three cases.

5.4. Optimization Model II

To illustrate optimization mode

uCase 1. Delivery time is assumed to be 15 units.

We assume that sec

mpatible with second and third alternative of s

odule. Following solution was obtained using LINGO.

P. C. JHA ET AL.

257

Case No. Delivery Time In-House ty Overall System Cost α Value

Table 3. Solution for optimization model I.

COTS System Reliabili

1 15 y32 = 1

x111 = = 1

x21 0.92 75 0.84

x123 = x132

1 = x223 = x232 = x241 = 1

x311 = 1

2 18 y24 = y32 = 1

111 2

0.93 77.5 0.86

3 20 y24 = y32 = 1

111 2

0.94 76 0.90

x = x123 = x13 = 1

x211 = x223 = x232 = 1

x311 = 1

x = x123 = x13 = 1

x211 = x223 = x232 = 1

x311 = 1

It is observed that due to the comatibility condition,

32 1y

223

x

111123132 1xxx

233241 1xxx 311

211 1

ird alternative of second module is chosen as it is

compatible with second alternative of first module. The

system reliability for the above solution is 0.86 and cost

is 85 units. The achievement level of the membership

function is 0.40



.

Case 2. Delieme is assumed to be 20 units. vry ti

odule

We assume that second alternative of second m

compatible with second and third alternative of first

module. Following solution was obtained using LINGO.

2432 1yy 111123133 1xxx





211 222

xx

231 1x 311 1x

It is observed that due to the compatibility condition,

third alternative of first module is chosen as it is com-

patible with second alternative of second module. The

system reliability for the above solution is 0.93 and cost

is 77 units. The achievement level of the membership

function is 0.90



.

. Conclusions

ptimization models that supports the

and cost. This developed approach can effectively deal

efully acknowledges

nt of Maharaja Agrasen In-

ir permission to publish this

adrzejowicz, “An Approach to Reliability

oftware with Redundancy,” IEEE Transac-

re Engineering, Vol. 17, No. 3, 1991, pp.

310-312.

e have presented oW

decision whether to buy software components for soft-

ware architecture or to build them in-house. We have

formulated bi-criteria optimization models based on de-

cision variables indicating the set of structural compo-

nents “to buy” and “to build” in order to minimize the

software cost with simultaneous maximization of system

reliability. The problem is formulated for Recovery

Block fault-tolerant software system. It may be appreci-

ated that when different alternatives of the same module

are available with variations in the attributes of reliability

and cost, then it involves multi-objective decision mak-

ing environment that befits more of fuzzy approximation

than deterministic formulation. Therefore, we have drawn

on fuzzy methodology for the estimation of reliability

with the vagueness and subjectivity of expert’s informa-

tion. Fuzzy predictions of the triangular fuzzy statistical

data have been defuzzified using Heipern’s defuzzifier

and a crisp multi-objective model has been developed

using the defuzzified values. The component selection

problem is formulated as a multi-objective programming

problem and fuzzy goal programming technique is used

to provide a feasible solution.

7. Acknowledgements

ne of the authors, Ritu Arora, gratO

the Director and Manageme

titute of Technology for thes

research work.

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