A Fuzzy Optimization Framework for COTS Products Selection of Modular Software Systems

Transcription

1 Internatonal Journal of Fuy Systes, Vol. 5, No., June 0 9 A Fuy Optaton Fraework for COTS Products Selecton of Modular Software Systes Pankaj Gupta, Hoang Pha, Mukesh Kuar Mehlawat, and Shlp Vera Abstract In ths paper, we duss a decson-akng stuaton under uncertanty related to software creaton through Coercal-Off-The-Shelf (COTS) based odules. We propose a b-objectve fuy optaton odel of the COTS selecton proble. The proposed optaton odel sultaneously axe the weghted qualty and ne the total cost of a odular software syste subject to any ltatons ncludng axu threshold on delvery te of the software and ncopatblty aong COTS products. The coeffcents of both the objectve functons and the delvery te constrants are charactered by fuy restrctons wth trangular possblty dstrbutons. Analytcal herarchy process technque s used to assgn weghts to the varous odules accordng to ther access frequences and also usng preferences of the software developer regardng techncal specfcatons of the software syste. Usng possblty theory, an auxlary lnear ultple objectve 0- prograng odel paraetered by possblty (feasblty) level β s obtaned correspondng to the fuy COTS selecton odel. The resultant odel s converted nto an equvalent crsp 0- lnear prograng odel usng ax-n approach eployng lnear ebershp functons for all the fuy sets and we solved the equvalent odel usng an nteractve approach that provdes fuy solutons correspondng to dfferent possblty levels β. A real-world case study of ERP software syste s ncluded for nuercal llustratons to deal wth any stuatons of the precse data n the fuy COTS selecton odel. Keywords: Possblty theory, COTS selecton, Multple Correspondng Author: Pankaj Gupta s wth the Departent of Operatonal Research, Unversty of Delh, Delh, Inda. E-al: pgupta@or.du.ac.n Hoang Pha s wth the Departent of Industral and Systes Engneerng, Rutgers Unversty, Pataway, NJ, USA. E-al: hopha@rc.rutgers.edu Mukesh Kuar Mehlawat, and Shlp Vera are wth the Departent of Operatonal Research, Unversty of Delh, Delh, Inda. E-al: ukesh0980@yahoo.co; verashlp@yahoo.co Manurpt receved 6 June 0; revsed 4 March 0; accepted June 0. crtera optaton, Fuy optaton, Modular software syste.. Introducton Today's need for rapd software developent has generated a great nterest n eployng Coercal-Off- The-Shelf (COTS) software products as a way of anagng cost, developng te and effort []. In order to reale the benefts of COTS products, t s portant that the rght products are selected for varous software systes. The selecton of best COTS products has thus becoe the key to the qualty and cost of the odular software systes based on COTS. Selectng an approprate COTS product s often a non-trval task and s consdered as a ultple crtera decson akng process. Wth an abundance of COTS products to choose fro, the COTS selecton proble becoes how to systeatcally evaluate, rank and select a COTS product that best eets the software project requreents. Several COTS selecton ethods [8, 0,,, 6, 0, 7] have been proposed n lterature. However, t ay be noted that there s no sngle ethod whch s accepted as a standard COTS selecton ethod. A detaled lst of the COTS selecton ethods has been provded by Mohaed et al. [7]. Alternatvely, optaton technques have been used n the COTS selecton process to acheve the dfferent attrbutes of qualty along wth the objectve of nng the cost or keepng cost to a specfed budgetary level. Beran and Ashraf [4] dussed optaton odels for relablty of odular software systes. Ch et al. [7] presented a software relablty optaton odel. Cortellessa et al. [9] developed an optaton odel that supports buld-or-buy decsons n selectng software coponents based on cost-relablty trade-off. Jung and Cho [0] ntroduced two optaton odels for the COTS selecton n the developent of odular software systes consderng cost-relablty trade-off. Kwong et al. [4] presented an optaton odel for deternng the optal selecton of software coponents for coponent-based software syste developent. Neubauer and Stuer [8] presented a two-phase decson support approach based on ultobjectve optaton for the COTS selecton. 0 TFSA

2 9 Internatonal Journal of Fuy Systes, Vol. 5, No., June 0 Tang et al. [5] presented an optaton odel for software coponent selecton under ultple applcatons developent. Zacharah and Ratthall [9] used goal-prograng approach n a ult-crtera optaton odel for the COTS selecton of odular software systes. Zahed and Ashraf [4] dussed software relablty allocaton usng optaton approach based on structure, utlty, prce and cost. All the above entoned optaton odels are based on the assupton that n the software developent process, the decson aker has coplete nforaton. It ay be noted that snce software developent s not an exact ence, there are often plenty of ndefnte and uncertan factors n the estaton of odel paraeters. Hence, the varous odel paraeters are often precse or the process of estaton of these nput paraeters s subjected to uncertanty n a non-stochastc sense. Thus, t s desrable to ncorporate uncertanty n the atheatcal odel tself. Correspondng to the varous possble enaros of uncertanty n the odel paraeters, we ust explore dfferent possble outcoes and then select the best outcoe n a gven decson akng stuaton. The optaton odels n the software developent ay beneft greatly fro the fuy set theory [40] n ters of ntegratng quanttatve and qualtatve nforaton, subjectve preferences of the decson aker and knowledge of the software experts. The crsp optaton odels of the COTS selecton ay be extended to precse/uncertan/fuy COTS selecton odel n order to provde satsfactory solutons to the decson aker n dfferent enaros of nput data uncertanty. Gupta et al. [] forulated b-objectve fuy optaton odel for the COTS selecton usng nonlnear S-shaped functons derbng vague aspraton levels of the decson aker n respect of the weghted qualty and cost. Gupta et al. [] proposed a ebershp functon approach for cost-relablty tradeoff of COTS selecton n fuy envronent. Recently, Gupta et al. [] developed an nteractve fuy approach usng lnear ebershp functons for the COTS selecton proble consderng ultple objectves. Shen et al. [] presented a fuy optaton odel subject to a paraetered budgetary level constrant for selectng the best COTS product consderng weghted qualty as an objectve functon. Jha et al. [9] presented a fuy approach for optal selecton of COTS coponents for odular software syste under consensus recovery block hee ncorporatng executon te. In ths paper, we propose a b-objectve fuy optaton fraework for the COTS selecton under the followng assuptons:. Modular prograng s used for software developent.. Module versons are developed ndependently. It ay be noted that because of assupton () the proposed COTS selecton odel s applcable only to those software systes that are developed usng COTS based odules whch are ndependently generated and tested. The proposed research can be consdered as a generalaton and extenson of the optaton odels proposed by Jung and Cho [0] n ters of provdng a systeatc fraework for COTS selecton that facltates software developent process of a odular software. The proposed b-objectve COTS selecton odel sultaneously axe the weghted qualty and ne the total cost of a odular software syste subject to any ltatons ncludng axu threshold on the delvery te of the software and ncopatblty aong COTS products. We use precse coeffcents charactered by trangular possblty dstrbutons n both the objectve functons and the delvery te constrants. The possblty theory proposed by Zadeh [4] s used to develop an nteractve approach yeldng fuy solutons correspondng to dfferent possblty levels β. The key dfferences between our study and the relevant studes fro lterature are as follows: () Unlke the use of crsp odel paraeters for COTS selecton n extant lterature [,, 0, ], we use precse odel paraeters. In real-world probles, varous odel paraeters, naely, cost, qualty, and delvery te cannot be deterned exactly because these ay be affected by varous ndefnte and uncertan factors whch cannot be easured precsely. Also, ost of the paraeters ebedded n COTS selecton proble are frequently fuy n nature because of ncopleteness and/or unavalablty of requred data over the plannng horon and can be just obtaned subjectvely. Thus, the use of precse odel paraeters provde the decson aker great flexblty n dealng wth dfferent possble enaros of nput data uncertanty; () the proposed approach can easly solve the precse objectve functons and obtan the whole possblstc dstrbuton of the satsfyng objectve value. Furtherore, t can solve rather general precse optaton probles through the nvolveent wth the decson aker thereby capturng hs preferences n the soluton process. Three crtcal values (the ost possble, the pessstc, and the optstc values) are only necessary for each precse objectve functon; () by akng lttle effort t s really possble to obtan a lot of nforaton about the soluton obtaned; and (v) the proposed approach solves COTS selecton proble wth precse paraeters ore effcently snce t preserves orgnal lnear for for the fuy goals. The proposed ethod can be easly pleented usng coercally avalable

3 P. Gupta et al.: A Fuy Optaton Fraework for COTS Products Selecton of Modular Software Systes 9 software such as LINDO and other related atheatcal prograng software. The paper s organed as follows. Secton presents the b-objectve fuy optaton odel of the COTS software selecton proble. In Secton, we develop the nteractve approach and present the soluton procedure. Secton 4 presents nuercal llustratons of a real-world case study nspred fro coponent based software developent to test the effectveness of the proposed approach. Secton 5, presents coparson and benefts of the proposed nteractve possblstc prograng approach and other exstng fuy prograng approaches. Fnally, we furnsh our concludng rearks n Secton 6.. COTS Selecton Forulaton We consder the desgn of a software syste that s developed usng COTS based odules and s requred to perfor one or ore progras as specfed by the user. It s assued that a specfc functon of each progra can call upon a seres of odules. A falure occurs n the software syste f a odule fals to carry out a desgnated operaton. We assue that several alternatve COTS products are avalable for each odule. The odules are developed ndependently usng the best COTS product aong the avalable alternatves. The dagraatc depcton of such a software syste s gven n Fgure that conssts of progras, k odules and several COTS alternatves, for exaple,,,, n are avalable to pleent odule and so on. The herarchy shown n Fgure lnks the user s vew (progra level) wth the software developer s vew (odule level). Note that each odule n the above software syste has dfferent levels of portance that depend on access frequency. The odule whch s called ore frequently by a progra ay be relatvely ore portant than the other odule(s) whose frequency of callng wthn a syste s less. Users often recogne the qualty of a software syste based on the qualty of the odules. Hence, purchasng expensve hgh-qualty COTS products ay be justfed by the frequent use of the odule. Usng the above shown herarchy, we obtan weght paraeters for the odules accordng to ther access frequences and also usng software developer s nputs regardng techncal specfcatons of the software syste. The use of access frequency n deternng the portance of odules s justfed fro the work of Anderson [] on software selecton. In ths paper, we use the Analytc Herarchy Process (AHP) technque [] to assgn weghts to the odules. One of the ost portant advantages of the AHP s that t use par-wse coparsons based on the decson aker s vew of the syste. Besdes, the AHP calculates the nconsstency ndex whch s the rato of the decson aker s nconsstency n par-wse coparsons. The AHP technque s used n any applcatons [,, 8, 0,, 4] to evaluate the qualty of odular software systes. Progra Software Syste Progra Progra Progra Module Module Module Module 4 Module k COTS alternatves Fgure. The herarchy of a odular software syste. There are any factors and ssues that ust be taken nto consderaton durng selecton of the COTS products. Any generc stage-based ethodology for selecton of COTS products broadly conssts of the followng stages [7]: ) Deternng the need for purchasng the product and prelnary nvestgaton of the avalablty of COTS products that ght be sutable canddates, ncludng hgh level nvestgaton of software features and capabltes provded by vendor. ) Short lstng of alternatve COTS products. ) Elnatng those alternatve products that do not have requred features or do not work wth the exstng hardware, operatng syste and database anageent software or network. 4) Usng an evaluaton technque to evaluate reanng products and obtan a ore or overall rankng of the. 5) Purchasng and pleentng ost approprate COTS products. In ths paper, our an focus s on stages 4) and 5) of COTS selecton process. We use qualty crtera for evaluaton of the alternatve COTS products usng ISO/IEC 96-I [6] qualty standard. Further, cost/qualty trade-off s used for selectng best-ft COTS products for odular software systes usng fuy optaton odel. A. COTS Evaluaton Usng Qualty Crtera COTS products are avalable wth the bult-n functonalty and hence the end user has no or lttle

4 94 Internatonal Journal of Fuy Systes, Vol. 5, No., June 0 control on ther functonalty. Thus, the non-functonal requreents whch defne the general qualty of the software syste have receved ore attenton fro researchers n the recent past. We use qualty crtera derved fro the qualty standard ISO/IEC 96-I [6] to access qualty of the COTS products. Currently, ISO/IEC 96-I standard s wdely used by researchers and practtoners to evaluate software qualty. ISO/IEC 96-I qualty odel s defned by eans of general software crtera whch are further represented by subcrtera whch n turn are decoposed nto attrbutes, yeldng to a ultlevel herarchy. At the botto of herarchy appear the easurable software alternatves, whose ores are coputed by usng soe etrc, for exaple, parwse coparsons whch are ale ndependent. Level : GOAL Level : CRITERIA Level : SUBCRITERIA Level 4: ALTERNATIVES Qualty of COTS products Functonalty Relablty Usablty Mantanablty Portablty Effcency Sutablty Accuracy Interoperablty Coplance Maturty Fault tolerance Recoverablty Understandblty Learnablty Operablty Changeablty Operablty Stablty Testablty Adaptablty Installablty Replaceablty Conforance Te behavor Resource behavor COTS product COTS product COTS product Fgure. The herarchy of qualty crtera for COTS evaluaton. Standards for the software qualty odels defnes software qualty crtera as coposed of sx external crtera of nterest, naely, functonalty, relablty, usablty, antanablty, portablty and effcency. Functonalty of the COTS product s nothng but the ablty of the COTS product to perfor accordng to the specfc needs of the organaton. Relablty s ablty of the product to run consstently wthout crashng under specfc condtons. It s used to easure the relablty level to whch the syste satsfes the functonal requreents of the organaton. Usablty s understandablty of the COTS products as well as easness to learn and operate t under certan specfc condtons. Mantanablty s the ablty of the product to be odfed. Modfcatons can nclude correctons, proveents or adaptatons of the software to adjust to changes n the envronent, n ters of functonal requreents and specfcatons. Portablty s the ablty of the product to be transferred fro one envronent to another. Effcency s ablty of the COTS product to provde approprate perforance, relatve to the aount of resources used under certan condtons. Note that, each of these crtera ay be decoposed nto subcrtera. The decson herarchy used for qualty level estaton of COTS products s depcted n Fgure. The qualty level deterned usng parwse coparson etrc provdes values rangng fro 0 to where refers to a very hgh qualty level. B. COTS Selecton Usng Fuy Decson Theory Today s need for rapd software developent has generated a great nterest n eployng COTS software products as a way of anagng cost, developng te and effort. To reale the benefts of COTS products, t s portant that the rght products are selected for a software syste. The selecton of best-ft COTS products has now becoe the key to the qualty and cost of the odular software systes based on COTS. Decson akng n COTS selecton requres consderng conflctng objectves as well as dfferent syste constrants. We study a b-objectve fuy COTS selecton proble that has two objectves, naely, axaton of the qualty of the software syste and naton of the purchasng cost. The task of COTS selecton s sad to be accoplshed optally f the selected COTS products eet the realstc constrants, naely, delvery te, contngent decson and one and only one COTS product for each odule, provdng axu qualty for the software syste requrng nu purchasng cost. We forulate the b-objectve fuy optaton odel of COTS selecton by consderng the followng challenges assocated wth the proble. ) Need for herarchcal decson-akng ) Many slar COTS products ) Two COTS selecton objectves 4) Selecton of one COTS product for each odule 5) Delvery te restrcton on the software 6) Incopatblty ssues aong COTS products Operatonally, forulatng a COTS selecton proble requres an estate of qualty, cost, and delvery te for the varous alternatve COTS n the odules. In realworld probles, these estates cannot be deterned exactly because cost, qualty, and delvery te are affected by varous ndefnte and uncertan factors whch cannot be easured precsely. Also, the decson akers assessent about these estates ay be based on ncoplete knowledge about the COTS product tself

5 P. Gupta et al.: A Fuy Optaton Fraework for COTS Products Selecton of Modular Software Systes 95 and other aspects (e.g. vendors credentals, etc.), whch ay affect the decson of selectng COTS products. Therefore, a decson based on a crsp odel s not the best decson. Under such crcustances, the ssue of COTS selecton becoes one of a choce fro a fuy set of subjectve preferences. The fuy optaton odels can be classfed on the bass of two concepts: () the lts of funess that the decson aker sets- the decson aker s aspratons- wth respect to the objectve and/or constrants, and () the abguty n the coeffcents of the objectve functons and/or constrants (.e., uncertanty n data). The uncertanty n data can be classfed nto two groups: (a) randoness, that coes fro the rando nature of paraeters and usually stochastc prograng approaches are used to odel ths knd of uncertanty; (b) epstec uncertanty that deals wth ll-known paraeters and usually possblstc prograng approaches are used to handle the epstec uncertanty [4]. The research work proposed n ths paper deals wth precson n the paraeters of the objectve functons and constrants nstead of dealng wth vagueness n the aspraton levels of the decson aker studed n our earler works on fuy COTS selecton [,, ]. Accordng to the above-entoned derpton, and because we are confrontng precse paraeters n the COTS selecton proble, possblstc prograng approach s used to handle the uncertan paraeters n the proposed odel. Each ll-known paraeter s assued to follow a possblty dstrbuton n our approach. The possblty dstrbuton represents the possblty degree of occurrence of possble values for each uncertan paraeter and s ostly deterned based on avalable data as well as knowledge of purchasng anagers. B.. Assuptons and Notaton The proposed b-objectve fuy optaton odel for COTS selecton decson s based on the followng assuptons. ) Both the objectve functons,.e. qualty & cost and the delvery te constrants are fuy. ) The pattern of trangular possblty dstrbuton s adopted to represent the precse coeffcents n the objectve functons and constrants. ) Both the objectve functons and the constrants are lnear. 4) The lnear ebershp functons are specfed for all the fuy sets nvolved n the proble. 5) The ax-n approach s used to aggregate all the fuy sets. Assupton relates to the funess of the objectve functons and delvery te constrants n the real-world COTS selecton proble. Assupton relates to the splcty and flexblty of fuy arthetc operatons. Trangular possblty dstrbutons are utled to represent the precse coeffcents n the objectve functons and constrants n order to enhance the coputatonal effcency and facltate data acquston [5]. Assuptons and 4 ensure that the lnearty and proportonalty propertes are techncally satsfed as n a standard lnear prograng for. Assupton 5 s used to convert the b-objectve fuy COTS selecton proble nto an equvalent crsp sngle-objectve lnear prograng proble. The followng notaton are used: Indces : ndex of odules (,,..., k), j : ndex of COTS alternatves n the -th odule ( j,,..., n ), Paraeters k : the nuber of odules n the gven software syste, n : the nuber of COTS alternatves n the -th odule,,,..., k, w : the weght of the -th odule calculated usng AHP,,,..., k, q : the precse qualty level of the j -th COTS alternatve n the -th odule, c : the precse cost of the j -th COTS alternatve n the -th odule, d : the precse delvery te of the j -th COTS alternatve n the -th odule, Decson Varables x : the bnary varable ndcatng whether the j -th COTS alternatve of the -th odule s chosen or not,.e., f the j -th COTS alternatve of the -th x odule s chosen, 0, otherwse. B.. Objectve Functons We consder two portant and conflctng objectves n our COTS selecton proble: the weghted qualty ( ) and the total cost of purchasng ( ). Several other studes [,, 0, ] also consdered slar objectves but not wth precse coeffcents. The use of precse coeffcents provdes the decson aker great flexblty n dealng wth dfferent possble enaros of nput data uncertanty. Weghted Qualty Slar to the work of Jung and Cho [0], we consder the overall qualty as a sngle entty. The overall qualty of dfferent possble alternatves of each

6 96 Internatonal Journal of Fuy Systes, Vol. 5, No., June 0 odule s coputed by cobnng ndvdual easures of qualty lsted n Secton.A. The ndvdual easures of qualty ay conflct each other and coparsons ay have to be reached and hence the overall qualty easure s an optu balance of factors rather than an deal way of representng qualty. Snce the calculatons for qualty levels usng qualty standard odel nvolves subjectve preferences of the decson aker and qualtatve nforaton, we consder the as precse nubers. The objectve of axng the weghted qualty of the COTS selecton proble s expressed as: ax wq x j Cost In ths paper, the cost crteron s based on procureent and adaptaton costs of COTS products. The procureent cost contans lcensng arrangeent cost, product & technology cost and consultng cost. The objectve of nng the total cost of the COTS selecton proble s expressed as: n cx j B.. Model Constrants Delvery Te Constrants The delvery te of each COTS alternatve and also axu threshold T on the delvery te of the software syste s consdered uncertan. The delvery te T of the -th odule s expressed as: n T d x,,,..., k. j Therefore, the delvery te constrant on the software syste s obtaned as: ax ( T ) T,,..., k whch can be decoposed n the set of constrants T T, T T,..., T. k T Alternatvely, the delvery te constrants ay also be reforulated as: n dx T,,,..., k. j One and only one COTS product s selected for each odule n j x,,,..., k. Contngent Decson Constrants In the developent of a odular software syste, soetes a COTS product for one odule s ncopatble wth the COTS alternatves for other odules, due to probles such as pleentaton technology, nterfaces and lcensng. Ths ncopatblty can be denoted by xrs xut, that s, f the r -th odule chooses the s -th COTS alternatve, then u - th odule ust choose the t -th COTS alternatve. Ths condton s called the contngent decson constrant [0]. Suppose there are two contngent decsons n the odel, such as, the COTS alternatve for the odule r s only copatble wth the COTS products t and t for the odule u, that s, ether xut f x rs or xut f xrs. These constrants can be represented as ether xrs xut or xrs xut. Snce the presence of the etheror constrant akes the optaton proble nonlnear, t can be lneared by ntroducng a bnary varable y as follows: 0, f the -th constrant s actve, y, f the -th constrant s nactve. Thus, the constrants correspondng to contngent decsons n the COTS selecton proble can be restated as follows: x x My, h,,...,. rs uth h h y h, yh 0,, h,,...,. The above converson guarantees that only one out of the contngent decson constrants for any COTS product between two odules s actve f M s large (larger than the coeffcents used n the correspondng constrants). Constrants on Varables x 0,,,,..., k, j,,..., n. B.4. The Decson Model The fuy optaton odel of the COTS software selecton s forulated as follows: (BOFOP) ax wq x j subject to n j n cx j dx T,,,..., k, () n x,,,..., k, () j x x My, h,,...,, () rs uth h

7 P. Gupta et al.: A Fuy Optaton Fraework for COTS Products Selecton of Modular Software Systes 97 yh, (4) h y 0,, h,,...,, (5) h x 0,,,,..., k, j,,..., n. (6). Possblstc Prograng Approach In real-world decson akng probles, unavalablty or ncopleteness of data s an portant challenge that poses a hgh degree of uncertanty; hence, n such cases the use of probablty dstrbuton s not desrable. Further, despte the avalablty of hstorcal data, the behavour of odel paraeters does not necessarly coply wth ther hstorcal pattern n future. To deal wth ths challenge n the concerned COTS selecton proble, the uncertan odel paraeters are represented by fuy nubers charactered by ther possblty dstrbuton. The portance of the possblty theory les n the fact that uch of the nforaton on whch huan decsons are based s possblstc rather than probablstc n nature [4]. It also provdes an effectve ethod for proceedng wth nherent abguous phenoena n deternng precse envronent coeffcents and related paraeters [5, 5, 5, 6]. Recallng BOFOP odel, we convert t nto an equvalent auxlary crsp ultobjectve 0- prograng odel usng the possblstc approach. A. Model the Iprecse Data wth Trangular Possblty Dstrbuton In ths study, the pattern of trangular possblty dstrbuton s adopted to represent each fuy paraeter wth a vew to prove the coputatonal effcency and flexblty of fuy arthetc operatons. In practcal stuatons, the decson aker can construct the trangular possblty dstrbuton of c ( p,, o c c c ), see Fgure, based on the three pronent data as follows:. The ost possble value ( c ) that defntely belongs to the set of avalable values (possblty degree = f noraled). o. The optstc value ( c ) that has a very low lkelhood of belongng to the set of avalable values (possblty degree = 0 f noraled). p. The pessstc value ( c ) that has a very low lkelhood of belongng to the set of avalable values (possblty degree = 0 f noraled). B. An Auxlary Mult-Objectve 0- Prograng Model Gven the precse coeffcents n the objectve functons and constrant () of BOFOP odel, n general, one cannot guarantee an optal soluton to the proble. There are several approaches for obtanng coprose solutons n the lterature. We apply an extended verson of a well-known approach proposed by La and Hwang [5] to transfor the BOFOP odel nto an auxlary crsp lnear ultple objectve 0- prograng odel. To do so, we apply approprate strateges for convertng the fuy objectve functons and fuy constrants of BOFOP odel nto the equvalent crsp fors. Fgure. The trangular possblty dstrbuton of c. B.. Strategy for Treatng the Iprecse Objectve Functons We frst consder the weghted qualty objectve functon ( Z ). Geoetrcally, ths precse objectve functon s fully defned by three corner ponts (, ), p o (, 0), and (, 0) and t can be axed by pushng the three corner ponts of trangular possblty dstrbuton to the rght. Snce, the vertcal coordnates of the corner ponts are fxed at ether or 0, the three horontal coordnates are the only consderatons. Usng La and Hwang [5], the strategy adopted here for treatng the precse weghted qualty objectve functon s to sultaneously axe the ost possble value of weghted qualty, ne the rsk p of obtanng lower qualty levels ( )(regon I of the possblty dstrbuton n Fgure 4) and axe the o possblty of obtanng hgher qualty levels ( ) (regon II of the possblty dstrbuton n Fgure 4). 0 (I) (II) Fgure 4. The strategy to solve fuy qualty objectve.

8 98 Internatonal Journal of Fuy Systes, Vol. 5, No., June 0 Usng q ( p,, o q q q ), the precse qualty level of the j -th COTS alternatve n the -th odule, the fuy weghted qualty objectve functon n BOFOP odel s replaced by the followng three crsp objectve functons to obtan a coprose soluton: ax w q x j o o ax ( ) w ( q q ) x j p p n ( ) w ( q q) x j Slarly, the strategy adopted for treatng the precse total cost objectve functon ( ) s to sultaneously ne the ost possble value of total cost, ne the rsk of obtanng hgher costs p ( )(regon II of the possblty dstrbuton n Fgure 5) and axe the possblty of obtanng o lower costs ( )(regon I of the possblty dstrbuton n Fgure 5). 0 Fgure 5. The strategy to solve fuy cost objectve. Usng c ( o,, p c c c ), the precse cost of the j -th COTS alternatve for the -th odule, the fuy total cost objectve functon n BOFOP odel s replaced by the followng three crsp objectve functons to obtan a coprose soluton: (I) (II) n c x j o 0 j p p j ax ( ) ( c c ) x n ( ) ( c c ) x B.. Strategy for Treatng the Iprecse Constrants To deal wth the fuy constrant () of BOFOP odel that have precse paraeters both n the left-hand sde and rght-hand sde, we use the fuy rankng concept [5, 9, 4]. Accordngly, for a gven nu acceptable possblty value (0 ) specfed by the decson aker, we replace the fuy constrant wth three equvalent auxlary nequalty constrants. Usng p o d ( d, d, d ), the precse delvery te of the j - th COTS alternatve for the -th odule, the followng auxlary crsp delvery te constrants are obtaned as: n ( d ) x T,,,..., k, (7) j n o o ( d ) x T,,,..., k, (8) j n p p ( d ) x T,,,..., k, (9) j where ( d ) d, ( ) P p ( p ) d d d d, ( d ) o o o d ( d d ), and T T, p p ( p ) T T T T, o o o T T ( T T ). It ay be noted that s the nu satsfacton level of the fuy constrants or the nu desred degree of feasblty and ts ebershp functon s elcted on the bass of the preference concept. Thus, the desred degree of feasblty ay be strongly dependent upon the psychologcal value syste. B.. The Mult-Objectve 0- Model The ultple objectve proble s now presented as follows: (MOP) ax w q x j o o ax ( ) w ( q q ) x j p p n ( ) w ( q q) x j subject to Constrants ()-(9) n c x j o 0 j p p j ax ( ) ( c c ) x n ( ) ( c c ) x C. Solvng the MOP Model There are several ethods n the lterature for solvng MOP proble; aong the the fuy prograng approaches are beng ncreasngly appled. The an advantage of fuy approaches s that they are capable to

9 P. Gupta et al.: A Fuy Optaton Fraework for COTS Products Selecton of Modular Software Systes 99 easure the satsfacton degree of each objectve functon explctly whch helps the decson aker to choose a preferred coprose soluton accordng to the satsfacton degree and preference (relatve portance) of each objectve functon. The MOP odel developed above can be converted nto an equvalent crsp 0- lnear prograng odel usng Zerann s approach [44]. For the purpose, we frst deterne the postve deal soluton (PIS) and negatve deal soluton (NIS) for each objectve functon of the MOP odel as follows: PIS ax, NIS n ax( ), n( ) PIS o NIS o n( ), ax( ) PIS p NIS p n, ax PIS NIS ax( ), n( ) PIS o NIS o n( ), ax( ) PIS p NIS p subject to Constrants ()-(9). It ay be noted that deternng the above deal solutons requre solvng any nteger lnear prograng probles whch could be coputatonally dffcult especally n large-sed proble nstances. In order to reduce the coputatonal coplexty, we ay apply the heurstc rules wheren we solve the correspondng nteger lnear prograng proble heurstcally to obtan a satsfactory feasble nteger soluton []. Next, we specfy the lnear ebershp functons for each objectve functon of the MOP odel as follows:, f ; PIS NIS NIS PIS PIS NIS NIS 0, f. PIS, f ; NIS PIS NIS NIS PIS NIS 0, f., f ;, f ; (0) () Note that the lnear ebershp functons of and are slar to ; whle and are slar to and hence are not specfed explctly. The graphcal representaton of lnear ebershp functons defned by (0) and () are shown n Fgure 6. We now transfor the MOP odel nto an equvalent crsp sngle objectve 0- lnear prograng odel usng the ax-n approach proposed by Bellan and Zadeh []: (LP) ax subject to Z, g,; l,,, gl 0, and Constrants ()-(9), where the auxlary varable (0 ) represents the overall degree of satsfacton wth deterned objectve values. The LP forulaton s capable of yeldng coprose solutons for a gven possblty (feasblty) value β based on the decson aker s preferences. Fgure 6. The ebershp functon of the auxlary objectves and. D. Soluton Procedure The soluton procedure for the fuy optaton odel of COTS selecton proble s suared as under. Step : Calculate the weght of each odule usng AHP. Step : Deterne approprate trangular possblty dstrbutons for the precse coeffcents of both the objectve functons and the fuy constrant and forulate the BOFOP odel of COTS selecton. Step : Convert the fuy objectve functons of the BOFOP odel nto the crsp objectve functons of the MOP odel. Step 4: Gven the nu acceptable possblty (feasblty) value,, convert the fuy constrant of the BOFOP odel nto the

10 00 Internatonal Journal of Fuy Systes, Vol. 5, No., June 0 correspondng crsp ones of the MOP odel usng the fuy rankng concept. Step 5: Calculate the PIS and NIS for each objectve functon of the MOP odel and defne the lnear ebershp functons of the auxlary objectve functons. Step 6: Convert the MOP odel nto ts crsp equvalent LP odel. Solve and odfy the LP odel nteractvely. If the decson aker s not satsfed wth the soluton obtaned then the odel ay be adjusted untl a set of satsfactory solutons s obtaned by varyng values of the nu acceptable possblty (feasblty) value. E. Crsp Transforaton of Fuy Objectves To fnd a representatve crsp value for the fuy nuber Z, we defufy Z to a crsp value. Many defufcaton approaches have been proposed n the lterature [6]. The center of gravty (COG) ethod s the ost trval weghted average. We use the COG ethod to defufy the fuy total cost and fuy weghted qualty. Let Z * be the defufed value of the fuy nuber Z *. The COG ethod calculates Z as follows: U 0 L ( ) * d 0 Z () U 0 L ( d ) 0 If ( ) s represented by trangular fuy nuber (,, ) where and are the lower and upper lts of the support of Z and s the odal value, then we obtan * Z () by actual substtuton of the ebershp functon of the trangular fuy nuber. Usng equatons () and (), we can obtan crsp equvalents of the fuy objectve values of the BOFOP odel. 4. Nuercal Illustratons In order to deonstrate the valdty and practcalty of the proposed COTS selecton odel and the soluton ethod, an ndustral case enaro nspred fro a coponent-based software syste developent s presented. A local software syste suppler planned to develop an Enterprse Resource Plannng (ERP) software syste for sall and edu-se enterprses. An ERP syste s an ntegrated applcaton software package coposed by a set of standard functonal requreents such as plannng producton, sales, huan resources, antanng nventores, fnance, provdng custoer servce and trackng orders. Applcaton software packages are defned by a vendor to provde a set of standard functons that can be adapted to the specfc needs of each custoer. COTS ter refers to applcaton software package. In ths paper, we have dentfed three functonal requreents of the ERP syste, naely, Fnance (Progra ), Operatons (Progra ), and Marketng (Progra ). The software developent tea of the copany has defned four odules, naely, accounts ( ), nventory ( ), sales order ( ), and sales prooton ( ). Progra calls 4 odules and (once each); Progra calls odules and (twce each); Progra calls odules (twce), (once), and (twce). Module 4 has three COTS alternatves denoted as,, and. Module has two COTS alternatves denoted as and. Module has three COTS alternatves denoted as,, and. Module 4 has three COTS alternatves denoted as, 4, and 4. The 4 dagraatc depcton of ERP software syste consdered n ths paper s gven n Fgure 7. A. Data Derpton Because of confdentalty as well as the lack of soe exact requred data, we have used fuy paraeters charactered by trangular possblty dstrbutons n accordance wth the past data and ntegratng the judgeent of the decson aker for the nuercal experents. The trangular possblty dstrbutons of the ndvdual qualty level, cost, and delvery te of all the COTS coponents are shown n Table. B. Module Weghts usng AHP The optaton odel used n ths paper for COTS selecton requres systeatc evaluaton of the user s and software developer s preferences about the software n ters of the progras and odules. B.. The Herarchcal Bass of the Syste Prototype Referrng to Fgure 7, we begn the herarchy fro the top wth the user s vew, whch we defne as the user s overall assessent of the qualty of the software syste. The user s assessent s based on the functonalty of several progras of the software syste, whch are represented at the second level of the herarchy. The syste prototype archtecture of ths study conssts of three progras denoted by Progra, Progra, and Progra. The thrd level of the herarchy lsts the four ndependent odules fro software developer s vew of whch the progras are coposed of. These odules ay have subodules but each subodule belongs to only one odule,.e. a any-to-one relatonshp. We

11 P. Gupta et al.: A Fuy Optaton Fraework for COTS Products Selecton of Modular Software Systes 0 stop the herarchcal structure at the level of the ndependent odules. Fnance COTS alternatves ERP Software Syste Operatons Marketng Accounts Inventory Sales Order Sales Prooton Fgure 7. The herarchcal structure of the ERP software syste. B.. Decson Maker s Relatve Preference Our goal s to dentfy the relatve portance of each progra at the second level (user s vew) and each odule (software developer s vew) at the thrd level of the herarchy n the assessent of the qualty of the software syste at the frst level. Clearly, one s unable to ask drectly fro the user or software developer alone to express hs preferences for each progra and each odule because hs vew of the software s only partal vew of the syste. Hence, to acheve our goal, we use AHP technque [] for dentfyng the relatve portance of the progras and the odules. In AHP, the eleents of each level of the decson herarchy are rated usng par-wse coparson. After all the eleents have been copared par by par, a pared coparson atrx s fored. The order of the atrx depends on the nuber of eleents at each level. The nuber of such atrces at each level depends on the nuber of eleents at the edate upper level that t lnks to. After developng all the pared coparson atrces, the egenvectors or the relatve weghts representng the degree of the relatve portance aongst the eleents and the axu egenvalue ( ax ) are calculated for each atrx. The ( ax ) value s an portant valdatng paraeter n AHP. It s used as a reference ndex to reen nforaton by calculatng the consstency rato of the estated vector n order to valdate whether the pared coparson atrx provdes a copletely consstent evaluaton. The consstency rato s calculated as per the followng steps:. Calculate the egenvector or the relatve weghts and ( ax ) for each atrx of order n. Copute the consstency ndex (CI) for each atrx of order n as follows: ax n CI ( n ). The consstency rato (CR) s calculated as follows: CI CR RI Table. Input data of the syste prototype. COTS products Qualty Cost Delvery te Module q (0.80, 0.8, 0.90) q (0.78, 0.8, 0.87) q (0.74, 0.78, 0.8) c (5,0,) c (7,9,4) c (5,8,) d (4,5,7) d (, 4,6) d (5, 7,0) Module q (0.8, 0.85, 0.88) q (0.8, 0.88, 0.9) c (,8,) c (,9,) d (5, 7,0) d (, 6,8) Module Module q (0.8, 0.85, 0.90) q (0.77, 0.79, 0.84) q (0.85, 0.90, 0.98) q 4 (0.84, 0.90, 0.95) q 4 (0.8, 0.88, 0.96) q 4 (0.77, 0.8, 0.87) c (7,8,4) c (4,7,) c (6,9,) c 4 (,9,) c 4 (,8,) c 4 (6,8,) d (,, 6) d (, 7,9) d (,5,8) d 4 (,6,8) d 4 (4,8,0) d 4 (, 6,8)

12 0 Internatonal Journal of Fuy Systes, Vol. 5, No., June 0 where RI s a known rando consstency ndex that has been obtaned fro a large nuber of sulaton runs and vares accordng to the order of atrx. If CI s suffcently sall then par-wse coparsons are probably consstent enough to gve useful estates of the weghts. If CI 0.0, then the degree of RI consstency s satsfactory. However, f CI 0.0, RI then serous nconsstences ay exst and hence AHP ay not yeld eanngful results. The evaluaton process should, therefore, be revewed and proved. Also, the nuber of parwse coparsons depends on () nuber of alternatves to be evaluated and () nuber of crtera consdered for evaluaton. If user requreents of the COTS products changes then relatve ore of each alternatve over the other wth respect to each evaluaton crtera also changes. If nuber of alternatves consdered for evaluaton ncreases then nuber of parwse coparsons also ncreases. Therefore n both the cases: () f user needs of the coponents changes, or () nuber of alternatves to be evaluated changes, parwse coparson ust be done agan. An llustratve par-wse coparson atrx of the attrbutes (Progra, Progra, and Progra ) wth respect to the goal,.e. qualty of software syste s shown n Table. As seen n ths table, the preference ores of qualty of software syste wth respect to the functonalty of the progras are: for Progra -0.4, for Progra -0.6 and for Progra Table. Coparson atrx of the progras w.r.t. the qualty. Attrbutes Progra Progra Progra Relatve preferences Progra 0.4 Progra ½ 0.6 Progra 0.8 Consstency rato: < 0. As seen fro Fgure 7, Progra calls upon odules and. The par-wse coparson atrx of the odules and wth respect to Progra s shown n Table. Table. Coparson atrx of the odules w.r.t. the Progra. (attrbute) Alternatves Module Module Relatve preferences Module Module ½ 0. On the sae lnes the par-wse coparson atrx of the odules and wth respect to Progra and odules,, and 4 wth respect to Progra are shown n Tables 4 and 5, respectvely. Table 4. Coparson atrx of the odules w.r.t. the Progra. (attrbute) Alternatves Module Module Relatve preferences Module 0.5 Module 0.5 Table 5. Coparson atrx of the odules w.r.t. the Progra. (attrbute) Alternatves Module Module Module 4 Relatve preferences Module 0.49 Module / / 0.4 Module Consstency rato: 0 < 0. After fndng all the local weghts, the global (fnal) weght of each odule s deterned by followng what n ters of the AHP herarchy ay be regarded as a botto-up process of successve ultplcaton. For exaple, the local weghts of Module n relaton to the attrbutes -Progra and Progra, are ultpled wth the correspondng local weghts of these attrbutes n relaton to ther parent attrbute,.e. qualty of software syste. The AHP weght of Module s then deterned by addng these products,.e. the AHP weght of Module s obtaned as: 0.667* *0.8=0.45. Slarly, the fnal weghts of odules,, and 4 can be calculated. Thus, we obtan the followng AHP weght vector W= (0.45, 0.68, 0.77, 0.40). It ay be noted that n the above weght vector, the odules havng sae access frequency are not at the sae level of portance because we have cobned access frequency wth the nputs of the software developer n par-wse coparsons. Further, t ay be ponted out that when the nuber of progras or odules s very large at a gven level of herarchy, we can use absolute ratng n place of par-wse coparsons to obtan the weghts. However, absolute ratng of eleents s less accurate than par-wse coparsons n elctng the preferences of the decson aker. B.. The Auxlary MOP Model We use fuy nput data of the syste prototype fro Secton 4.A, odule weght vector W= (0.45, 0.68, 0.77, 0.40) fro Secton 4.B. and axu threshold on the delvery te of the software syste as T (,8,). Consder the followng contngent decson constrants n our COTS selecton proble:

13 P. Gupta et al.: A Fuy Optaton Fraework for COTS Products Selecton of Modular Software Systes 0 x4 x or x4 x (4) Constrant (4) ples that f Module 4 chooses the second COTS product then Module ust choose the frst COTS product or the thrd COTS product. Consderng 0.5, we obtan the followng auxlary MOP odel: ax 0.45(0.8x 0.8x 0.78 x ) 0.68(0.85x 0.88 x) 0.77(0.85x 0.79x 0.90 x) 0.40(0.90x4 0.88x 0.8 x ) 4 4 ax 0.45(0.07x 0.05x 0.04 x) 0.68(0.0x 0.05 x) 0.77(0.05x 0.05x 0.08 x) 0.40(0.05x4 0.08x x4) n 0.45(0.0x 0.04x 0.04 x) 0.68(0.0x 0.06 x) 0.77(0.0x 0.0x 0.05 x) 0.40(0.06x4 0.07x x4) n 0x 9x 8x 8x 9x 8x 7x 9x 9x4 8x4 8x4 ax 5x x x 7x 6x x x x 6x4 6x4 x4 n x 5x x 4x x 6x 4x 4x x4 4x4 x4 subject to x x x, (5) x x, (6) x x x, (7) x4 x4 x4, (8) 4.5x.5x 6x 5.5, (9) 5x 4x 7x 8, (0) 6x 5x 8.5x 0, () 6x 4.5x 5.5, () 7x 6x 8, () 8.5x 7x 0, (4) x 5x 4x 5.5, (5) x 7x 5x 8, (6) 4.5x 8x 6.5x 0, (7) 4.5x4 6x4 4.5x4 5.5, (8) 6x4 8x4 6x4 8, (9) 7x4 9x4 7x4 0, (0) x x 5y, () 4 x x x 5y, (), () 4 y y y, y 0,, (4) 0,,, j. (5) Next, we calculate PIS and NIS for each objectve functon of the MOP odel. The coputatonal results are presented n Table 6. Table 6. PIS and NIS for fuy objectve functons. Objectves PIS NIS B.4. The Equvalent Crsp LP Model The lnear ebershp functon for each objectve functon n the auxlary MOP odel can be defned usng (0) and (). The equvalent LP odel s obtaned usng the ax-n approach to aggregate all the fuy sets. ax subject to 0.45(0.8x 0.8x 0.78 x ) 0.68 (0.85x 0.88 x) 0.77(0.85x 0.79x 0.90 x) 0.40(0.90x4 0.88x4 0.8 x4) , 0.45(0.07x 0.05x 0.04 x) 0.68 (0.0x 0.05 x) 0.77(0.05x 0.05x 0.08 x) 0.40(0.05x4 0.88x x4) , 0.45(0.0x 0.04x 0.04 x) 0.68 (0.0x 0.06 x) 0.77(0.0x 0.0x 0.05 x) 0.40(0.06x4 0.07x x4) , 0x 9x 8x 8x 9x 8x 7x 9x 9x4 8x4 8x4 4 7, 5x x x 7x 6x x x x 6x4 6x4 x4 9, x 5x x 4x x 6x 4x 4x x4 4x4 x4 5 6, 0, and Constrants (5) (5). We use LINDO software [] to solve the above LP odel. The obtaned solutons have the followng trangular possblty dstrbutons:

14 04 Internatonal Journal of Fuy Systes, Vol. 5, No., June 0 (weghted qualty) = ( , 0.846, ), (cost) = (5, 5, 46), and (overall degree of satsfacton) = The solutons varables are x, x, x, and x 4. The selected COTS products for odules,, and 4 are,,, and, respectvely. It ay be noted 4 that the qualty of the software syste s fuy, ts ost lkely value falls at 0.846, and ts value s possble to fall outsde the range of and Slarly, the total cost of the software syste s fuy, but ts ost lkely value falls at 5, and ts value s possble to fall outsde the range of 5 and 46. Thus, the obtaned qualty and cost ranges provde a useful reference for the software developent tea of the ERP software syste. respectvely Fgure 8. The possblty dstrbutons for qualty objectve. B.5. Senstvty Analyss w.r.t. -values If the decson aker s not satsfed wth the obtaned COTS selecton, ore COTS selecton strateges can be generated by varyng value of the nu acceptable possblty (feasblty) paraeter,, n the MOP odel. Table 7 lsts the results of pleentng dfferent - values. The possblty dstrbutons of the obtaned solutons usng dfferent -values for the fuy qualty and cost objectve functons are shown n Fgure 8 and Fgure 9, -value Degree of satsfacton -value Degree of satsfacton Table 7. Results of senstvty analyss on -values. Table 7. Results of senstvty analyss on -values Objectve values Objectve values (0.807, , ) (,5,49) (0.807, , ) (,5,49) ( ,0.846,0.9044) (5,5,46) ( ,0.846,0.9044) (5,5,46) (0.808,0.8808,0.890) (,4,47) (0.808,0.8808,0.890) (,4,47) Fgure 9. The possblty dstrbutons for cost objectve. In ths paper, we have assued that the decson aker has deterned the possblty dstrbuton of each precse coeffcent as trangular fuy nuber. The proposed soluton approach yelds a coprose soluton when copared wth the optal goal value COTS products for odules COTS products for odules

15 P. Gupta et al.: A Fuy Optaton Fraework for COTS Products Selecton of Modular Software Systes 05 obtaned by solvng the sngle objectve lnear prograng proble correspondng to each objectve functon. It ay be noted that all the obtaned solutons are effcent solutons. As shown n Table 7, the change n β -value nfluences the overall degree of satsfacton, ndvdual degree of satsfacton of the auxlary objectve functons, g,; l,,, objectve gl functon values and soluton varables. The -value ncreases fro 0. to 0.5, when the β -value ncreases fro 0. to 0.9. Generally, the acheveent level ay not be large enough to satsfy the decson aker because of ultple objectve nature of the proble and also consderng the fact that we are dealng wth 0- prograng proble. Next, by defufyng the fuy total cost and fuy weghted qualty usng () and (), the crsp values of the two fuy objectves and are provded n Table 8. Table 8. Crsp values of the fuy objectve functons. Objectve Cost Weghted qualty - value Fuy value (,5,49) (5,5,46) (,4,47) (0.807, , ) ( ,0.846,0.9044) (0.808,0.8808,0.890) Crsp value It can be seen fro Table 8 that acceptng deteroraton n one of the objectve functon ay provde proveent n the other objectve functon. In ost cases, t s possble that poor perforance on one crteron ay be copensated by good perforance on other crtera. Thus, correspondng to decson aker s preferences, any dfferent solutons can be reached at dfferent satsfacton degrees. In the proposed ethodology, both the qualty and cost objectves are treated equvalently n the LP odel. However, the decson aker can choose dfferent solutons accordng to hs needs n dfferent stuatons. For exaple, cost (qualty) can be the ost portant objectve for the decson aker n a deterned perod of software developent. Then, the decson aker wll choose the soluton whch satsfes the cost (qualty) objectve functon the ost. However, ths can cause poor perforance of the satsfacton degree of other objectve(s). Therefore, dfferent solutons at dfferent β -values wll enable the decson aker to deterne approprate COTS selecton strateges usng qualty and cost objectves. The proposed soluton ethod provdes a wde range of nforaton and flexblty n the sense that by changng possblty (feasblty) values nteractvely provdes dfferent enaro analyss for precse stuatons. Also, t s possble that the lower (upper) bounds of the objectve functons can be changed usng decson aker s preferences so that new ebershp functons are generated. Ths wll lead to the forulaton of a new crsp LP odel. Hence, possbly a new coprose soluton ay be obtaned. 5. Coparson and Benefts of the Proposed Approach To better justfy the applcablty of the proposed nteractve possblstc approach for COTS selecton, we present the perforance analyss and the benefts of the proposed approach. A. Perforance Analyss We evaluate perforance of the proposed nteractve possblstc prograng approach and the fuy nteractve approach developed by Gupta et al. [] for COTS selecton. To analye and copare perforance of the nteractve approaches, we use a well-known perforance easure, naely, dstance easure. The dstance easure s used for deternng the degree of closeness of each soluton to the correspondng best soluton. In ths regard, we defne the followng faly of dstance functons [4, 8, 4]: p p p dp( x) gl ( x) gl ; g l p and nteger, (6) where gl, g,; l,, ndcates relatve portance of the objectve functons. Snce the satsfacton degree of each objectve s defned as the relatve closeness of the soluton to the best pont or the relatve reoteness to the worst pont, they are used explctly n (6). The power p represents a dstance paraeter and especally p,, and are operatonally portant so that d (the Manhattan dstance) and d (the Eucldean dstance) are the longest and shortest dstances n the geoetrcal sense; and d (the Tchebycheff dstance) s the shortest dstance n the nuercal sense. Generally, when p ncreases, the aount of dstance d p and also the credblty of the dstance functon d p decreases [4, 8, 4]. It ay be noted that based on the defnton of d p, the fuy approach wth nu d p (especally for p = ), would be preferred to the other ethods. To evaluate perforance of both the nteractve approaches, we solve the bobjectve odel BOFOP usng the nput data provded n Table, AHP weght vector W (0.45, 0.68, 0.77, 0.40) and β = 0.5 wth

16 06 Internatonal Journal of Fuy Systes, Vol. 5, No., June 0 both the nteractve approaches. It ay be noted that to solve the BOFOP odel usng the nteractve approach of Gupta et al. [] we defufy the precse data (trangular fuy nubers) n respect of the fuy total cost and fuy weghted qualty usng (). Note that the data n respect of delvery te s used n precse for (trangular fuy nubers) only. The coputatonal results are suared n Table 9. For the gven llustraton, t s assued that. It ay be noted that the above 6 defned faly of dstance functons needs to be odfed approprately n order to apply on the nteractve approach of Gupta et al. []. Fro Table 9, t s clear that the proposed nteractve possblstc prograng approach provdes an coprose soluton better than the soluton obtaned by the fuy nteractve approach [] for the dstance functons d and d. B. Benefts of the Proposed Approach The anageral plcatons n applyng the proposed ethodology to practcal COTS selecton are as follows. The proposed approach yelds a coprose soluton for a fuy COTS selecton proble. For a gven acceptable possblty (feasblty) level β, the obtaned solutons provdes a range n whch the objectves, that s, the qualty of the software syste and the total cost of the software syste wll appear. To exeplfy, for β = 0.5, the obtaned objectve value of the qualty of the software syste s ( , 0.846, ) and the total cost of the software syste s (5, 5, 46). The trangular dstrbuton range of qualty so obtaned s nterpreted as: the ost lkely value of the qualty s Table 9. Coparsons of solutons , the best value of the qualty s , and the worst value of the qualty s Slarly, the ost lkelest value of the cost s 5, the best value of the cost s 5, and the worst value of the cost s 46. Also, t s possble to have the qualty < and > and cost < 5 and > 46. The solutons presented n Table 7 reveal that the changes n the acceptable possblty (feasblty) β value nfluence both the overall degree of satsfacton and objectve values. If the aspraton levels are properly gven, they ay create a ore coplcated procedure. For ths reason, we use the PIS and NIS values n the soluton approach. The benefts of the proposed soluton approach are suared as under.. The proposed approach effectvely handles vagueness and precson n the goal values usng trangular fuy nubers to odel the precse data. Several dfferent patterns ncludng trangular, trapeod, S-curve, exponental, hyperbolc are used to odel precse data n the exstng lterature. Aong the, the trangular dstrbuton s used ost often to represent precse data owng to the ease t offers n defnng the axu and nu lts of devaton of a fuy nuber fro ts central value; although, certan practcal applcatons ay prefer other patterns.. The proposed approach exhbts greater coputatonal effcency and flexblty by eployng lnear ebershp functons to specfy the fuy goal values. The an advantage of usng a lnear ebershp functon s that t produces a coputatonally tractable functon that closely reflects the real-world structure of the decson Degree of satsfacton Objectve values COTS products for odules Proposed approach Fuy nteractve approach [] d d d d ( ,0.846,0.9044) (5,5,46)

17 P. Gupta et al.: A Fuy Optaton Fraework for COTS Products Selecton of Modular Software Systes 07. proble. A lnear ebershp functon s ost coonly used because t s sple and t s defned by fxng two ponts: the upper and lower levels of acceptablty; however, certan decsonstuatons ay prefer nonlnear patterns. 4. The proposed approach solves COTS selecton proble wth precse coeffcents ore effcently snce t preserves orgnal lnear for for the fuy goals. The proposed ethod can be easly pleented usng coercally avalable software such as LINDO [] and other related atheatcal prograng software. 6. Conclusons In real-world COTS selecton proble, the decson aker ust sultaneously handle conflctng objectves. Moreover, nput data and/or related odel paraeters are often uncertan n non-stochastc sense because ether nforaton s ncoplete or unavalable over the plannng horon. Ths work has developed an nteractve approach for solvng a b-objectve fuy COTS selecton odel consderng precse qualty, cost and delvery te coeffcents wth trangular possblty dstrbutons. The nteractve approach based on possblty theory developed n ths paper provde a systeatc fraework that facltates software developent process of a odular software, enablng the decson aker to nteractvely odfy the possblty (feasblty) levels untl a set of satsfactory solutons s obtaned. A real-world case study for nuercal llustratons has been provded to deonstrate the feasblty of applyng the nteractve approach to the COTS selecton proble. The an ltaton of the approach developed here s the assupton of trangular possblty dstrbutons to represent precse coeffcents n the objectve functons and constrants. In real-world COTS selecton proble, the decson aker ay generate sutable possblty dstrbutons based on subjectve judgents and/or hstorcal data. Thus, t would be nterestng to apply trapeodal, bell-shaped, exponental, hyperbolc or other possblty dstrbuton patterns for representng precse nubers n solvng COTS selecton odel usng the proposed approach. However, t ay be noted that ore general fors of possblty dstrbutons ncreases the coplexty of solvng the fuy optaton proble. The use of AHP n ths paper depends on the nuber of alternatves and the nuber of crtera used for evaluaton. Although, excellent software systes are avalable for pleentng the AHP technque, yet, one ay explore the possblty of usng soe other technques for parwse coparsons whch are coputatonally less te consung and ore relable. The b-objectve fuy COTS selecton odel studed n ths work concerns the assupton that both the qualty and cost objectve functons have sae relatve portance and that the operaton of the COTS product s statstcally ndependent. Thus, the proposed odel can be extended further to ake t better suted to practcal applcatons by consderng dfferent levels of portance for the objectve functons. Also, the sngle applcaton developent task consdered n ths paper can be extended to ultple applcaton tasks ncorporatng the reusablty of COTS products. Acknowledgent We are thankful to the Edtor-n-Chef, Assocate Edtor, and the anonyous referees for ther valuable coents and suggestons to prove presentaton of the paper. The frst author acknowledges the research grant receved under a hee for strengthenng R & D Doctoral Research Prograe of Unversty of Delh, Delh, Inda. References [] Alves and A. Fnkelsten, Investgatng conflcts n COTS decson akng, Internatonal Journal of Software Engneerng and Knowledge Engneerng, vol., pp , 00. [] E. E. Anderson, A heurstc for software evaluaton and selecton, Software-Practce and Experence, vol. 9, pp , 989. [] R. E. Bellan and L. A. Zadeh, Decson akng n a fuy envronent, Manageent Scence, vol. 7, pp. 4-64, 970. [4] O. Beran and N. Ashraf, Optaton odels for relablty of odular software systes, IEEE Trans. on Software Engneerng, vol. 9, pp. 9-, 99. [5] J. J. Buckley, Possblstc lnear prograng wth trangular fuy nubers, Fuy Sets and Systes, vol. 6, pp. 5-8, 998. [6] S. J. Chen, C. L. Hwang, and F. P. Hwang, Fuy Multple Attrbute Decson Makng: Methods and Applcatons, Sprnger, New York, 99. [7] D.-H. Ch, H.-H. Ln, and W. Kuo, Software relablty and redundancy optaton, Proc. of the Annual Relablty and Mantanablty Syposu, IEEE, pp. 4-45, 989. [8] L. Chung, K. Cooper, and S. Courtney, COTS- Aware Requreents Engneerng: The CARE Process, Proc. of the nd Internatonal Workshop on Requreents Engneerng for COTS Coponents (RECOTS 004), Kyoto, Japan, Sep 7, 004.

18 08 Internatonal Journal of Fuy Systes, Vol. 5, No., June 0 [9] V. Cortellessa, F. Marnell, and P. Potena, An optaton fraework for buld-or-buy decsons n software archtecture, Coputers & Operatons Research, vol. 5, pp , 008. [0] G. Grau et al., J. P. Carvallo, X. Franch, and C. Quer, DesCOTS: A software syste for selectng COTS coponents, Proc. of the 0th IEEE Eurocro Conference (EUROMICRO'04), 004. [] P. Gupta, M. K. Mehlawat, G. Mttal, and S. Vera, A hybrd approach for selectng optal COTS products, Lecture Notes n Coputer Scence, vol. 559, pp , 009. [] P. Gupta, S. Vera, and M. K. Mehlawat, A ebershp functon approach for cost-relablty trade-off of COTS selecton n fuy envronent, Internatonal Journal of Relablty, Qualty and Safety Engneerng, vol. 8, no. 6, pp , 0. [] P. Gupta, M. K. Mehlawat, and S. Vera, COTS selecton usng fuy nteractve approach, Optaton Letters, vol. 6, pp. 7-89, 0. [4] P. Gupta and M. K. Mehlawat, A new possblstc prograng approach for solvng fuy ultobjectve assgnent proble, IEEE Transactons on Fuy Systes, DOI: 0.09/ TFUZZ , 0. [5] H. M. Hsu and W. P. Wang, Possblstc prograng n producton plannng of assebleto-order envronents, Fuy Sets and Systes, vol. 9, pp , 00. [6] ISO/IEC Standard 96-I, Software Engneerng- Product Qualty, Part : Qualty Model, 00. [7] A. S. Jadhav and R. M. Sonar, Evaluatng and selectng software packages: A revew, Inforaton and Software Technology, vol. 5, pp , 009. [8] A. S. Jadhav and R. M. Sonar, Fraework for evaluaton and selecton of software packages: A hybrd knowledge based syste approach, The Journal of Systes and Software, vol. 84, pp , 0. [9] P. C. Jha, S. Bal, and U. Kuar, A fuy approach for optal selecton of COTS coponents for odular software syste under consensus recovery block hee ncorporatng executon te, Turksh Journal of Fuy Systes, vol., no., pp. 45-6, 0. [0] H.-W. Jung and B. Cho, Optaton odels for qualty and cost of odular software systes, European Journal of Operatonal Research, vol., pp. 6-69, 999. [] A. P. Kanyalkar and G. K. Adl, Aggregate and detaled producton plannng ntegratng procureent and dstrbuton plans n a ult-ste envronent, Internatonal Journal of Producton Reserach, vol. 45, pp , 007. [] J. Konto et al., A COTS selecton ethod and experences of ts use, Twenteth Annual Software Engneerng Workshop, NASA Goddard Space Flght Center, Greenbelt, Maryland, Nov., 995. [] G. Kotonya and J. Hutchnson, Vewponts for specfyng coponent-based systes, Lecture Notes n Coputer Scence, vol. 054, pp. 4-, 004. [4] C. K. Kwong et al., Optaton of software coponents selecton for coponent-based software syste developent, Coputers & Industral Engneerng, vol. 58, pp , 00. [5] Y.-J. La and C.-L. Hwang, A new approach to soe possblstc lnear prograng probles, Fuy Sets and Systes, vol. 49, pp. -, 99. [6] K. R. P. H. Leung and H. K. N. Leung, On the effcency of doan-based COTS product selecton ethod, Inforaton and Software Technology, vol. 44, pp , 00. [7] A. Mohaed, G. Ruhe, and A. Eberlen, COTS selecton: past, present, and future, Proc. of the 4th Annual IEEE Internatonal Conference and Workshops on the Engneerng of Coputer-Based Systes (ECBS'07), 007. [8] T. Neubauer and C. Stuer, Interactve decson support for ultobjectve COTS selecton, Proc. of the 40th Annual Hawa Internatonal Conference on Syste Scences (HICSS'07), 007. [9] J. Raík and J. Ríánek, Inequalty relaton between fuy nubers and ts use n fuy optaton, Fuy Sets and Systes, vol. 6, pp. -8, 985. [0] C. Rolland, Requreent engneerng for COTS based systes, Inforaton and Software Technology, vol. 4, pp , 999. [] T. L. Saaty, Fundaentals of Decson Makng and Prorty Theory wth the AHP, Second Edton, RWS Publcatons, Pttsburgh PA, 000. [] L. Schrage, Optaton Modelng wth LINDO, Duxbury Press, CA, 997. [] X. Shen, Y. Chen, and L. Xng, Fuy optaton odels for qualty and cost of software systes based on COTS, Proc. of the Sxth Internatonal Syposu on Operatons Research and Its Applcatons (ISORA' 06), Xnjang, Chna, Aug. 8-, ORSC & APORC, pp. -8, 006. [4] H. Tanaka, H. Ichhash, and K. Asa, A forulaton of fuy lnear prograng proble based on coparson of fuy nubers, Control and Cybernetcs, vol., pp , 984. [5] J. F. Tang, L. Mu, C. K. Kwong, and X. G. Luo, An optaton odel for software coponent

19 P. Gupta et al.: A Fuy Optaton Fraework for COTS Products Selecton of Modular Software Systes 09 [6] selecton under ultple applcatons developent, European Journal of Operatonal Research, vol., pp. 0-, 0. [7] R.-C. Wang and T.-F. Lang, Applyng possblstc lnear prograng to aggregate producton plannng, Internatonal Journal of Producton Econocs, vol. 98, pp. 8-4, 005. [8] T. Wanyaa and B. H. Far, Towards provdng decson support for COTS selecton, Proc. of the Canadan Conference on Electrcal and Coputer Engneerng, Saskatoon Saskatchewan, Canada (CCECE 005), May -4, 005. [9] P. L. Yu, A class of solutons for group decson probles, Manageent Scence, vol. 9, pp , 97. [40] B. Zacharah and R. N. Ratthall, A ultcrtera optaton odel for qualty of odular software systes, Asa-Pacfc Journal of Operatonal Research, vol. 4, pp , 007. [4] L. A. Zadeh, Fuy sets, Inforaton and Control, vol. 8, pp. 8-5, 965. [4] L. A. Zadeh, Fuy sets as a bass for a theory of possblty, Fuy Sets and Systes, vol., pp. - 8, 978. [4] F. Zahed and N. Ashraf, Software relablty allocaton based on structure, utlty, prce and cost, IEEE Transactons on Software Engneerng, vol. 7, pp , 99. [44] M. Zeleny, Coprose prograng, In Cochrane, J. L. et al. (eds.), Multple crtera decson akng, Unversty of South Carolna Press, Coluba, pp. 6-0, 97. [45] H.-J. Zerann, Fuy prograng and lnear prograng wth several objectve functons, Fuy Sets and Systes, vol., pp , 978. Pankaj Gupta receved hs Ph.D. degree fro Departent of Matheatcs, Unversty of Delh, Delh, Inda n 000. Currently he s assocated wth the Departent of Operatonal Research, Unversty of Delh, Delh, Inda. Hs research nterests nclude optaton: theory and applcatons, fuy optaton and portfolo optaton. He has publshed sgnfcantly n reputed nternatonal journals ncludng European Journal of Operatonal Research, Fuy Sets and Systes, Inforaton Scences, Expert Systes wth Applcatons, Appled Matheatcs and Coputaton, Journal of Global Optaton, Journal of Optaton Theory and Applcatons, Optaton Letters, TOP, Journal of Nonlnear and Convex Analyss, Optaton, Appled Matheatcal Modellng, Knowledge-Based Systes, Internatonal Journal of Fuy Systes, Insurance: Matheatcs and Econocs, IEEE Transactons on Fuy Systes, Internatonal Journal of Inforaton Technology and Decson Makng. He s recpent of any prestgous natonal and nternatonal fellowshps. He has vsted any unverstes abroad for research collaboratons. Hoang Pha receved the M.S. degree n Statstcs fro the Unversty of Illnos at Urbana Chapagn; and the M.S., and Ph.D. degrees n Industral Engneerng fro State Unversty of New York at Buffalo. He s a professor and charan of the Departent of Industral and Systes Engneerng at Rutgers Unversty. He s the author or coauthor of fve books: Software Relablty (Sprnger, 000), Syste Software Relablty (006), Relablty and Optal Mantenance (006), Mantenance for Industral Systes (00), and Software Relablty Assessent wth OR Applcatons (0). He s the edtor of eght books ncludng the Handbook of Relablty Engneerng (00), and Handbook of Engneerng Statstcs (006), aong any others. He s also the edtor of Sprnger Seres n Relablty Engneerng, and has publshed over 0 journal artcles. He s the edtor-n-chef of the Internatonal Journal of Relablty, Qualty and Safety Engneerng; and s an edtoral board eber of several journals. He s an assocate edtor of IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS. Hs recent awards nclude 009 IEEE Relablty Socety Engneer of the Year Award. He s a fellow of the IEEE, and IIE. Mukesh Kuar Mehlawat receved hs Ph.D. degree fro Departent of Operatonal Research, Unversty of Delh, Delh, Inda n 0. Currently he s assocated wth the Departent of Operatonal Research, Unversty of Delh, Delh, Inda. Hs research nterests nclude: optaton theory and applcatons, fuy optaton and fnancal optaton. He has publshed any research papers n reputed nternatonal journals ncludng Fuy Sets and Systes, Inforaton Scences, Expert Systes wth Applcatons, Journal of Global Optaton, Optaton Letters, TOP, Knowledge-Based Systes, Internatonal Journal of Fuy Systes, Insurance: Matheatcs and Econocs, IEEE Transactons on Fuy Systes, Internatonal Journal of Inforaton Technology and Decson Makng. He has receved any teachng and research awards. Shlp Vera s workng as an Assstant Professor n Northern Inda Engneerng College, New Delh, Inda. Presently, she s pursung her Ph.D. degree fro Departent of Operatonal Research, Unversty of Delh, Delh, Inda. Her research nterests nclude fuy optaton and COTS selecton. She has publshed few research papers n reputed nternatonal journals.