Abstract. 1. Introduction

Transcription

1 Amercan Journal of Operatons Research,,, do:.436/aor..49 Publshed Onlne December ( 49 A Fuzzy Approach for Component Selecton amongst Dfferent Versons of Alternatves for a Fault Tolerant Modular Software System under Recovery Block Scheme Incorporatng Buld-or-Buy Strategy Abstract P. C. Jha, Rtu Arora, U. Dnesh Kumar 3 Department of Operatonal Research, Unversty of Delh, Delh, Inda Maharaa Agrasen Insttute of Technology, GGSIP Unversty, Delh, Inda 3 Indan Insttute of Management, Bangalore, Inda E-mal: hapc@yahoo.com, arora_rtu@yahoo.co.n, dneshk@mb.ernet.n Receved September, ; revsed October, ; accepted November 8, Software proects generally have to deal w producng and managng large and comple software products. As e functonalty of computer operatons become more essental and yet more crtcal, ere s a great need for e development of modular software system. Component-Based Software Engneerng concerned w composng, selectng and desgnng components to satsfy a set of requrements whle mnmzng cost and mamzng relablty of e software system. Ths paper dscusses e fuzzy approach for component selecton usng Buld-or-Buy strategy n desgnng a software structure. We ntroduce a framework at helps developers to decde wheer to buy or buld components. In case a commercal off-e-shelf (COTS) component s selected en dfferent versons are avalable for each alternatve of a module and only one verson wll be selected. If a component s an n-house bult component, en e alternatve of a module s selected. Numercal llustratons are provded to demonstrate e model developed. Keywords: Modular Software, Software Relablty, Software Components (COTS and In-House), Fault Tolerance & Fuzzy Optmzaton. Introducton Computer software s very mportant n today s world. In partcular, scence and technology demand hgh qualty software for makng mprovements and breakroughs. The software development companes are contnuously developng/modfyng/updatng er software accordng to e changng needs and requrements. The concept of software relablty and ts measurement s recevng much attenton n e software development communty. Software relablty s one of e mportant parameters of software qualty and system dependablty. Software relablty engneerng balances customer needs n e maor qualty characterstcs of relablty, avalablty, delvery tme and cost more effectvely. The relablty of software can be controlled durng e development lfe cycle rough e applcaton of relablty mprovement technques. Two of e best-known fault tolerant soft- ware desgn meods are N-verson programmng and Recovery block scheme. The basc mechansm of bo e schemes s to provde redundant software to tolerate software falures. Software whose falure can have bad effects afterwards can be made fault tolerant rough redundancy at module level []. When e desgn of software archtecture reaches a good level of maturty, software engneers have to take a decson on e selecton of software components. Non functonal aspects play a sgnfcant role n determnng software qualty. Gven e fact at lack of proper handlng of non functonal aspects of a software applcaton has led to a seres of software falures [], nonfunctonal attrbutes h as relablty securty and performance should be consdered durng e component selecton phase of software development. Ths paper dscusses a framework at helps developers to decde wheer to buy or buld components of software archtecture on e

2 5 bass of cost and non functonal factors. Whle developng software, components can be bo bought as commercal off-e shelf (COTS) products, and probably adapted to work n e software system, or ey can be developed n-house. Ths decson s known as buld-orbuy decson. Ths decson affects e overall cost and relablty of e system. Most of e current software systems nclude one or more COTS products. COTS are peces of software at can be reused by software proects to buld new systems. Benefts of COTS based development nclude sgnfcant reducton n e development cost, tme and mprovement n e dependablty requrement. The components, whch are not avalable n e market or cannot be purchased economcally, can be developed wn e organzaton and are known as nhouse bult components. Reference [3] dscussed ssues related to relablty of systems, caused by ntegratng COTS components. The optmal selecton s acheved rough weghted mamzaton of system qualty subect to budget as a constrant n whch an upper lmt s placed over e constrant. Ths paper dscusses e ssues related to relablty of e software systems and cost caused by ntegratng COTS or n-house bult components. Fault tolerance s acheved rough redundancy n Recovery Block model and redundancy results n addtonal cost. We assume at for all e alternatves avalable for a module, cost ncreases f hgher relablty s desred. Several alternatves of a software module may be avalable as COTS, almost equvalent from e functonal vewpont. Purchase of hgh qualty COTS products can be ustfed by e frequent use of e module. Large software systems possess e modular structure to perform a set of functons. Each functon s performed by dfferent modules havng dfferent alternatves for each module. In case, a COTS component s selected en dfferent versons are avalable for each alternatve of a module and only one verson wll be selected. If a component s an n-house bult component, en e alternatve of a module s selected. A schematc representaton of e software system s gven n Fgure. In e estng research related to e software decson, t s assumed at all e parameters of e problem are known precsely. Varous obectves and restrctons are set by e management and cost coeffcents nvolved n e cost functon are determned based on past eperence and e avalable data base. Ths makes t dffcult for e management to provde precse values of e varous cost coeffcents and obectves to be met. Moreover due to changng customer specfcatons, lack of eperence of testng team or novelty, changng testng envronment, complety n e proect nvolved, unknown emergng factors at e start of e proect adds Fgure. Structure of e software. mprecson and ambguty to e above-mentoned defntons. It may also be possble at e management tself does not set precse values n order to provde some tolerance on ese parameters due to compettve consderatons. All s leads to uncertanty (fuzzness) n e problem formulaton. Crsp maematcal programmng approaches provde no h mechansm to quantfy ese uncertantes. Fuzzy optmzaton s a fleble approach at permts more adequate solutons to real problems n e presence of vague nformaton, provdng well-defned mechansms to quantfy e uncertantes drectly. The dea of fuzzy programmng was frst gven by [4] and en developed by [5-7]. A number of researchers ereafter have contrbuted to e development of fuzzy optmzaton technque [8,9]. Today, smlar to e developments n crsp optmzaton, dfferent knds of maematcal models have been proposed and many practcal applcatons have been mplemented by usng e fuzzy set eory. Reference [] formulated fuzzy mult obectve optmzaton models for selectng e optmal COTS software products n e development of modular software system. Recently, Reference [] de- velops a crsp mult-obectve programmng model from e fuzzy basc data. When a feasble soluton to e problem ests, sngle and multple obectve fuzzy opt- mzaton procedure are used to solve e problem. How- ever, t s assumed at a crsp or a constant value of all e parameters s known. However, n practce, t s not possble for a management to obtan a precse value of relablty and cost for a software system. Or ey may decde not to set precse levels due to e market consderatons and are ready to have some tolerance of er obectves. When e precse values of parameter of e problem are not known, e problem becomes a fuzzy optmzaton problem and e soluton so obtaned s a fuzzy appromaton. Ths paper proposes two fuzzy mult-obectve optmzaton models for selectng e best software product for each module. The frst optmzaton model (optmza-

3 5 ton model-i) of s paper s a ont optmzaton problem at mamzes e system relablty w smultaneous mnmzaton of e cost. The second optmzaton model (optmzaton model-ii) consders e ssue of compatblty between dfferent alternatves of modules as t s observed at some COTS components cannot ntegrate w all e alternatves of anoer module. We apply fuzzy optmzaton procedure to solve e problem, when a feasble soluton of e problem ests, but n case we reach e nfeasblty case, en we apply fuzzy goal programmng optmzaton technque to provde a compromsed soluton for e same. The rest of s paper s organzed as follows. Secton conssts of proposed notatons. In Secton 3, we dscuss e assumptons of optmzaton models and we develop a crsp model for relablty and cost and n Secton 4, we descrbe Fuzzy Mult-Obectve Optmzaton Model for software products selecton. In Secton 5, Fuzzy optmzaton technque s dscussed to solve e problem w numercal llustraton. In Secton 6, we furnsh our concludng observatons.. Notatons R : System qualty measure f l : Frequency of use, of functon l s l : Set of modules requred for functon l R : Relablty of module L : Number of functons, e software s requred to perform n : Number of modules n e software m : Number of alternatves avalable for module V : Number of versons avalable for alternatve of module N : Total number of tests performed on e n-house developed nstance (.e. alternatve of module ) N : Number of cessful (.e. falure free) test performed on e n-house developed nstance (.e. alternatve of module ) t : Probablty at net alternatve s not nvoked upon falure of e current alternatve t : Probablty at e correct result s udged wrong. t3 : Probablty at an ncorrect result s accepted as correct. X : Event at output of alternatve of module s reected. Y : Event at correct result of alternatve of module s accepted. s : Relablty of alternatve of module r : Relablty of alternatve for module Ck : Cost of verson k of alternatve for module r : Relablty of verson k of alternatve for k module dk : Delvery tme of verson k of alternatve for module c : Untary development cost for alternatve of module t : Estmated development tme for alternatve of module : Average tme requred to perform a test case for alternatve of module π : Probablty at a sngle eecuton of software fals on a test case chosen from a certan nput dstrbuton, f t constrant s actve y t :, f t constrant s nactve f e alternatve of module s y : n-house develoep. oerwse, f e k verson of COTS alternatve k : of e module s chosen, oerwse z : Bnary varable takng value or f alternatve s present n module oerwse 3. Optmzaton Models The frst optmzaton model s developed for e followng stuatons, whch also holds good for e second model, but w addtonal assumptons related to compatblty among alternatves of a module. The followng assumptons are common for optmzaton models: ) Software system conssts of a fnte number of modules. ) Software system s requred to perform a known number of functons. The program wrtten for a functon can call a seres of modules n. A falure occurs f a module fals to carry out an ntended operaton. 3) Codes wrtten for ntegraton of modules do not contan any bug. 4) Several alternatves are avalable for each module. Fault tolerant archtecture s desred n e modules (t has to be wn e specfed budget). Independently developed alternatves (prmarly COTS/In-House components) are attached n e modules and work smlar to e recovery block scheme dscussed n [,3]. 5) The cost of an alternatve s e development cost, f developed n house; oerwse t s e buyng prce for e COTS product.

4 5 6) Dfferent In-house alternatves w respect to untary development cost, estmated development tme, average tme and testablty of a module are avalable. 7) Cost and relablty of an n-house component can be specfed by usng basc parameters of e development process, e.g., a component cost may depend on a measure of developer sklls, or e component relablty depends on e amount of testng. 8) Dfferent versons w respect to cost, relablty and delvery tme of a module are avalable. 9) Oer an avalable cost-relablty versons of an alternatve, we assume e estence of vrtual versons, whch has a neglgble relablty of., zero cost and zero delvery tme. These components are denoted by nde one n e rd subscrpt of k, C k and r k. for eample r denotes e relablty of frst verson of alternatves for module. 3.. Model Formulaton Let S be a software archtecture made of n modules, w a mamum number of m alternatves avalable for each module and each COTS alternatves has dfferent versons Buld versus Buy Decson For each module, f an alternatve s bought (.e. some k ) en ere s no n-house development (.e. y ) and vce versa. V y = ;,,, n and,,, m k k 3... Redundancy Constrant The equaton stated below guarantees at redundancy s allowed for e components. y V k k z m z ;,,, n and ;,,, n and z ;,,, n,,, m,,, m Probablty of Falure Free In-House Developed Components The possblty of reducng e probablty at e alternatve of module fals by means of a certan amount of test cases (represented by e varable N ). Reference [4] defne e probablty of falure on de mand of an n-house developed alternatve of module, under e assumpton at e on-feld users operatonal profle s e same as e one adopted for testng [5]. Basng on e testablty defnton, we can assume at e number N of cessful (.e. falure free) tests performed on alternatve of same module. N π N ;,,, n and,,, m Let A be e event N falure-free test cases have been performed and B be e event e alternatve s falure free durng a sngle run. If s e probablty at e n-house developed alternatve s falure free durng a sngle run gven at N test cases have been cessfully performed, from e Bayes eorem we get e followng. PA BPB PB A P A B P B P A B P B The followng equaltes come straghtforwardly: P A B P B π erefore, we have P A B π P B π π N π π π N,,, n and,,, m Relablty Equaton of Bo In-House and COTS Components The relablty ( s ) of alternatve of module of e software. s y r ;,,, n and,,, m where r V r ;,,, n and,,, m k k k Delvery Tme Constrant The mamum reshold T has been gven on e delvery tme of e whole system. In case of a COTS components e delvery tme s smply gven by d k, whereas for an n- house developed alternatve e delv ery tme shall be epressed as t N. n m V y t N dk k k T ;

5 Obectve Functon 3... Relablty Obectve Functon Relablty obectve functon mamzes e system qualty (n terms of relablty) rough a weghted functon of module relabltes. Relablty of modules at are nvoked more frequently durng use s gven hgher weghts. Analytc Herarchy Process (AHP) can be effectvely used to calculate ese weghts. l Mamze R X f R L l sl where R s e relablty of module of e system under Recovery Block stated as follows. m z z R z PXk PY ;,,, n k P X t s t 3 PY s t s t 3... Cost Obectve Functon Cost obectve functon mnmzes e overall cost of e system. The sum of e cost of all e modules s selected from buld- or -buy strategy. The n-house development cost of e alternatve of module can be epressed as c t N n m V Mnmze CX c t N y Ck k k 3.3. Optmzaton Model I In e optmzaton model t s assumed at e alternatves of a module are n a Recovery Block. In a Recovery block, more an one alternatve of a program est. For COTS based software, multple alternatves of a module can be purchased from dfferent vendors. Each module works under a recovery block. Upon nvocaton of a module e frst alternatve s eecuted and e result s submtted for acceptance test. If t s reected, e second alternatve s eecuted w e orgnal nputs. The same process contnues rough attached alternatve untl a result s accepted or e whole recovery block (module) fals. Fault tolerance n a recovery block s acheved by ncreasng e number of redundances. Problem (P) Mamze R X f R L () l l sl n m V Mnmze CX c t N y C k k k () subect to N X S and y k m R z P X P Y are bnary varable z z ;,,, n (3) k k 3 P X t s t st PY s t π N ;,,, n and π π π π N,,, n and,,, m,,, m (4) ; (5) s y r ;,,, n and,,, m (6) V y k ;,,, n and,,, m (7) k V y z ;,,, n and,,, m (8) k k z ;,,, n and,,, m (9) m z ;,,, n () y t N d T n m V k k k () where X s a vector of component k and y ;,,, n ;,,, m ; k,, V Optmzaton Model II, As eplaned n e ntroducton, t s observed at some alternatves of a module may not be compatble w alternatves of anoer module [6]. The net optmzaton model II addresses s problem. It s done, ncorporatng addtonal constrants n e optmzaton models. Ths constrant can be represented as, whch gsq hutc means at f alternatve s for module g s chosen, en alternatve ut, t,, z have to be chosen for module h. We also assume at f two alternatves are compatble, en er versons are also compatble. My gsq hut c t q,, Vgs, c,, V hut, s,, mg ()

6 54 y z V (3) t hu t Constrant () and (3) make use of bnary varable y t to choose one par of alternatves from among dfferent alternatve pars of modules. Fnally, model can be re-wrtten as Problem (P) subect to l Mamze R X f R l sl n m MnmzeCX= c t N y X S My gsq hutc L t V k q,, V, c,, V, gs hu s,, m t g, t y z V If more an one alternatve compatble component s to be chosen for redundancy, constrant (3) can be relaed as follows. y hut 4. Fuzzy Mult-Obectve Optmzaton Model for Software Products Prncple to mult-obectve optmzaton s e concept of an effcent soluton, where any mprovement of one obectve can only be acheved at e epense of anoer. The fuzzy approach can be used as an effectve tool for quckly obtanng a good compromse soluton. Conventonal optmzaton meods assume at all parameters and goals of an optmzaton model are precsely known. However, for many practcal problems ere are ncompleteness and unrelablty of nput nformaton. Ths has resulted n use of fuzzy mult-obectve optmzaton meod w fuzzy parameters. For nstance, a desgner s requred to mnmze e system cost whle smultaneously mamzng e system relablty. Therefore multple obectve functons become an mportant aspect n e relablty desgn of e engneerng systems. In general relablty optmzaton problem s solved w e assumpton at e coeffcents or cost of components s specfed n a precse way. In real lfe, ere are many dverse stuatons due to uncertanty n udgments, lack of evdence, etc. Sometmes t s not possble to get relevant precse data for e relablty system. Ths type of mprecse data s not always well represented by random varable selected from a probablty z V t hu t C k k dstrbuton. Fuzzy number may represent s data, so fuzzy optmzaton meod w fuzzy parameters s needed for a fuzzy relablty optmzaton model. Therefore, we formulate fuzzy mult-obectve optmzaton model for software products selecton based on vague aspraton levels, e decson maker may decde hs aspraton levels on e bass of past eperence and knowledge possessed by hm. To epress vague aspraton levels of e decson, varous membershp functons have been proposed [6,7]. A fuzzy maematcal programmng problem w non lnear membershp functon results n a non lnear programmng problem. Usually, a lnear membershp functon s employed to avod nonlnearrty. Furer, f membershp functon s nterpreted as e fuzzy utlty of e decson maker, whch descrbes e behavor of ndfference, preference or averson towards uncertanty, a non lnear membershp functon s a better representaton an a lnear membershp functon. 4.. Problem Formulaton Fuzzy mult-obectve optmzaton model for software products selecton based on mamzng e fuzzfer relablty functon and mnmzng e fuzzfer cost functon subect to crsp constrants are stated as follows. Problem (P3) Mamze RX ( ) f R subect to X S Here, we have defned e two obectve functons, e relablty and cost at are consdered to be vague and uncertan (.e. fuzzy n nature) and e constrants are of crsp nature. Cut-roat competton n e estng market, system complety, and ntended fleblty makes t dffcult for e management to precsely defne er goals and constrants. Moreover a slght shft on bounds can provde a more effcent soluton. Hence, we have used fuzzy optmzaton technque (fuzzy maematcal programmng) to solve e fuzzy mult-obectve optmzaton problem. L l l sl Mnmze C(X) = c t N y C 4.. Problem Soluton n m V k k k The followng steps are used to solve e fuzzy maematcal programmng problem. Step. Compute e crsp equvalent of e fuzzy parameters usng a defuzzfcaton functon. Same defuzzfcaton functon s to be used for each of e parameters.

7 55 Here we use e defuzzfcaton functon of e type 3 F A a a a 4 3 where a, a, a are e trangular fuzzy numbers. Step. Incorporate e obectve functon of e fuzzfer m n (ma) as a fuzzy constrant w a restrcton (aspraton) level. The above problem (P3) can be rewrtten as Problem (P4) Fnd X subect to RX fl R R m L l sl n C X c t N y Ck k C k X S Step 3. Defne approprate membershp functons for each fuzzy nequaltes as well as constrant correspondng to e obectve functon. The membershp functon for e fuzzy less an or equal to and greater an or equal to type are gven as R RX R ; V ; R X R X R R X R R R ; RX R where R s e aspraton level and R s e tolerance level to e fuzzy relablty obectve functon constrant. C ; C X C C CX X C C X C ; C C ; C( X) C where C s e restrcton level and C s e tolerance level to e fuzzy budget constrant. Step 4. Etenson prncple s used to dentfy e fuzzy decson, whch results n a crsp maematcal programmng problem gven by Problem (P5) Mamze subect to R X, X, X S C can be solved by e standard crsp maematcal programmng algorms. Step 5. Whle solvng e problem, e obectve of e problem s also treated as a constrant. Each constrant s consdered to be an obectve for e decson maker and e problem can be looked as a fuzzy multple obectve maematcal programmng problem. Furer each obec- tve can have a dfferent level of mportance and can be assgned weghts accordng to er relatve mportance. The resultng problem can be solved by e weghted mnma approach. The crsp formulaton of e weghted problem s gven as Problem (P6) Mamze subect to X w X w, R C, X S w, w, ww where, represents e degree up to whch e aspraton of e decson maker s met. If e constrants are fuzzy as well as crsp, en n e equvalent crsp maematcal programmng problem, ere wll be no change n e orgnal crsp constrants snce er tolerances are zero ecept for ose constrants whch are fuzzy n nature. The problem (P6) can be solved usng standard maematcal programmng approach. Step 6. On substtutng e values for X R X C e problem becomes Problem (P7) Mamze subect to X w R R R R C X C w C C and X S, w, w, ww Step 7. If a feasble soluton s not obtanable for e problem (P7) en we can use fuzzy goal programmng approach to obtan a compromsed soluton [9]. The meod s dscussed n detal n e numercal llustraton. 5. Illustratve Eamples Consder a software system havng two modules w more an one alternatve for each module. The data sets for COTS and n-house developed components are gven n Table and Table, respectvely. Let L 3, s,, 3, s, 3, s3, f.5, f.3, f3.. It s also assumed at t., t.5 and t.. The assgnment of weg 3 h ts s based on e epert s udgment for e relablty and e cost crtera. Weghts assgned for relablty and cost are.7 and.3 respectvely. 5.. Mnmum and Mamum Level of Relablty and Cost Frstly, e trangular fuzzy relablty and cost values are computed usng fuzzed values of ese parameters and

8 56 Modules 3 Modules 3 Modules 3 Table. Data set for COTS components. Alternatves Verson Cost Relablty Delvery Tme Alternatves Verson Cost Relablty Delvery Tme Alternatves Verson 3 Cost Relablty Delvery Tme Table. Data set for n-house components. Module Alternatves t 3 c en defuzzed usng Helpern s def uzzer. If e aval- able relablty and cost are specfed as TFN (trangular fuzzy number) gven as follows R.93,.95,.97; C 7,75,8. The aspraton level of relablty s R.95 and e restrcton on cost s C 75. The tolerance level for relablty and cost s R.85 and C Fuzzy Goal Program mng Approach On solvng e problem, we found at e problem (P7) s not feasble; hence e management goal cannot be acheved for a feasble value of,. Now we use fuzzy goal programmng technque to obtan a compromsed soluton. The approach s based on e goal programmng technque for solvng crsp goal programmng problem (Mohamed, 997). The mamum value of any membershp functon can be ; mamzaton of, s equvalent to makng t as close to as best as possble. Ths can be acheved by mnmzng e negatve devatonal varables of goal programmng (.e. ) from. The fuzzy goal programmng formulaton for e gven problem (P7) ntroducng e negatve and postve devatonal varables and s gven as Mnmze u subect to R X X C u w ; ;,, X S, ; w, w ; w w ; u 5.3. Optmzaton Model I ; Table 3 presents e soluton for optmzaton model I. The problem s solved usng software package LINGO [7]. The soluton to e model gves e optmal component selecton for e software system along w e correspondng cost and relablty of e overall system. The senstvty analyss to e delvery tme constrant has been performed. It s clearly seen from e table at n case, when e delvery tme was 5 unts en one n-house and oer COTS components were selected whle n all oer cases when e delvery tme ncreases along w n-house components ere wll be a correspondng change n relablty and cost. In case, when e delvery tme was 8 unts, our system relablty and cost also ncreases and n case 3 as compared to case, when delvery tme was unts, system relablty ncreases and cost decreases. Therefore, f e customer s ready to wat en case 3 s an optmal soluton. Redundancy s also present n all e ree cases Optmzaton Model II To llustrate optmzaton model for compatblty, we use prevous results. Case. Delvery tme s assumed to be 5 unts. We assume at second alternatve of frst module s compatble w second and rd alternatve of second module. Followng soluton was obtaned usng LINGO.

9 57 Table 3. Soluton for optmzaton model I. Case No. Delvery Tme In-House COTS System Relablty Overall System Cost α Value 5 y 3 = 8 y 4 = y 3 = 3 y 4 = y 3 = = 3 = 3 = = 3 = 3 = 4 = = = 3 = 3 = = 3 = 3 = 3 = = 3 = 3 = = 3 = 3 = 3 = y It s observed at due to e compatblty condton, rd alternatve of second module s chosen as t s compatble w second alternatve of frst module. The system relablty for e above soluton s.86 and cost s 85 unts. The achevement level of e membershp functon s.4. Case. Delvery tme s assumed to be unts. We assume at second alternatve of second module s compatble w second and rd alternatve of frst module. Followng soluton was obtaned usng LINGO. y y It s observed at due to e compatblty condton, rd alternatve of frst module s chosen as t s compatble w second alternatve of second module. The system relablty for e above soluton s.93 and cost s 77 unts. The achevement level of e membershp functon s Conclusons We have presented optmzaton models at supports e decson wheer to buy software components for software archtecture or to buld em n-house. We have formulated b-crtera optmzaton models based on decson varables ndcatng e set of structural components to buy and to buld n order to mnmze e software cost w smultaneous mamzaton of system relablty. The problem s formulated for Recovery Block fault-tolerant software system. It may be apprecated at when dfferent alternatves of e same module are avalable w varatons n e attrbutes of relablty and cost, en t nvolves mult-obectve decson makng envronment at befts more of fuzzy appromaton an determnstc formulaton. Therefore, we have drawn on fuzzy meodology for e estmaton of relablty and cost. Ths developed approach can effectvely deal w e vagueness and subectvty of epert s nformaton. Fuzzy predctons of e trangular fuzzy statstcal data have been defuzzfed usng Hepern s defuzzfer and a crsp mult-obectve model has been developed usng e defuzzfed values. The component selecton problem s formulated as a mult-obectve programmng problem and fuzzy goal programmng technque s used to provde a feasble soluton. 7. Acknowledgements One of e auors, Rtu Arora, gratefully acknowledges e Drector and Management of Maharaa Agrasen Insttute of Technology for er permsson to publsh s research work. 8. References [] F. Bell and P. Jadrzeowcz, An Approach to Relablty Optmzaton Software w Redundancy, IEEE Transac- Engneerng, Vol. 7, No. 3, 99, pp. ton of Software 3-3. [] L. M. Cysneros and J. C. S. Lete, Nonfunctonal Re- Qurements: From Elctaton to Conceptual Models, IEEE Transactons on Software Engneerng, Vol. 3, No. 5, 4, pp do:.9/tse.4. [3] P. K. Kapur, A. K. Bardhan and P. C. Jha, Optmal Relablty Allocaton Problem for a Modular Software System, OPSEARCH, Vol. 4, No., 3, pp [4] R. E. Bellman and L. A. Zadeh, Decson-Makng n a Fuzzy Envronment, Management Scence, Vol. 7, No. 4, 97, pp. B4-B64. do:.87/mnsc.7.4.b4 [5] H. Tanaka, T. Okuda and K. Asa, On Fuzzy Maematcal Programmng, Journal of Cybernetcs, Vol. 3, No. 4, 974, pp do:.8/ [6] H. J. Zmmermann, Descrpton and Optmzaton of Fuzzy Systems, Internatonal Journal of General Systems, Vol., No. 4, 976, pp do:.8/ [7] H. J. Zmmermann, Fuzzy Set Theory and Its Applca-

10 58 [8] H. J. Zmmermann, Fuzzy Set Theory and Its Applca- 99. tons, Academc Publsher, Dordrecht, [9] R. H. Mohamed, The Relatonshp between Goal Programmng and Fuzzy Programmng, Fuzzy Sets and Sys- tems, Vol. 89, No., 997. pp. 5-. do:.6/s65-4(96)-5 [] P. Gupta, M. K. Mehlawat, G. Mttal and S. Verma, A Hybrd Approach for Selectng Optmal COTS Products, Computatonal Scence and Its Applcaton-ICCSA, Sprnger Publcaton, Berln, 9, pp [] B. P. Gladsh, I. Gonzalez; A. B. Terol and M. A. Parra, Plannng a TV Advertsng Campagn: A Crsp Mult obectve Programmng Model from Fuzzy Basc Data, Omega, Vol. 38, No. -,, pp do:.6/.omega [] O. Berman and U. D. Kumar, Optmzaton Models for Relablty of Modular Software System, IEEE Transacton of Software Engneerng, Vol. 9, No., 993, pp. tons, Kluwer Academc Publshers, Boston, [3] U. D. Kumar, Relablty Analyss of Fault Tolerant Recovery Block, OPSEARCH, Vol. 35, No. 4, 998, pp [4] V. Cortellessa, F. Marnell and P. Potena, An Optmzaton Framework for Buld-Or-Buy Decsons n Software Archtecture, Computers and Operatons Research, Vol. 35, No., 8, pp do:.6/.cor.7.. [5] A. Bertolno and L. Strgn, On e Use of Testablty Measures for Dependablty Assessment, IEEE Transactons on Software Engneerng, Vol., No., 996, pp do:.9/3.485 [6] H. W. Jung and B. Cho, Optmzaton Models for Qualty and Cost of Modular Software System, European Journal of Operatons Research, Vol., No. 3, 999, pp do:.6/s377-7(98)69-6 [7] H. Threz OR Software LINGO, European Journal of Operatonal Research, Vol.,, pp