Accepted Manuscript. A Stratified Framework for Handling Conditional Preferences: an Extension of the Analytic Hierarchy Process

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1 Accepted Manuscrpt A Stratfed Framework for Handlng Condtonal Preferences: an Extenson of the Analytc Herarchy Process Ivana Ognanovć, Dragan Gaševć, Ebrahm Bagher PII: S () DOI: Reference: ESWA 8078 To appear n: Expert Systems wth Applcatons Please cte ths artcle as: Ognanovć, I., Gaševć, D., Bagher, E., A Stratfed Framework for Handlng Condtonal Preferences: an Extenson of the Analytc Herarchy Process, Expert Systems wth Applcatons (0), do: dx.do.org/0.06/.eswa Ths s a PDF fle of an unedted manuscrpt that has been accepted for publcaton. As a servce to our customers we are provdng ths early verson of the manuscrpt. The manuscrpt wll undergo copyedtng, typesettng, and revew of the resultng proof before t s publshed n ts fnal form. Please note that durng the producton process errors may be dscovered whch could affect the content, and all legal dsclamers that apply to the ournal pertan.

2 A Stratfed Framework for Handlng Condtonal Preferences: an Extenson of the Analytc Herarchy Process Ivana Ognanovć, Dragan Gaševć, Ebrahm Bagher Unversty Medterranean, Athabasca Unversty Vaka Durovca bb, 8000 Podgorca, Montenegro phone/fax numbers: ABSTRACT Representng and reasonng over dfferent forms of preferences s of crucal mportance to many dfferent felds, especally where numercal comparsons need to be made between crtcal optons. Focusng on the well-known Analytcal Herarchcal Process (AHP) method, we propose a two-layered framework for addressng dfferent knds of condtonal preferences whch nclude partal nformaton over preferences and preferences of a lexcographc knd. The proposed formal two-layered framework, called CS-AHP, provdes the means for representng and reasonng over condtonal preferences. The framework can also effectvely order decson outcomes based on condtonal preferences n a way that s consstent wth well-formed preferences. Fnally, the framework provdes an estmaton of the potental number of volatons and nconsstences wthn the preferences. We provde and report extensve performance analyss for the proposed framework from three dfferent perspectves, namely tme-complexty, smulated decson makng scenaros, and handlng cyclc and partally defned preferences. Keywords: condtonal preferences, comparatve preferences, AHP method, S-AHP method, lexcographc order, well-formed preferences. INTRODUCTION Dfferent researchers have been nterested n developng tools and technques for elctng, formalzng and nterpretng stakeholders prortes over the exstng optons such as the par-wse comparson method, prorty groups, networks for decson-makng and cumulatve ratngs (Brafman & Domshlak, 00; Berander & Andrews, 006). The representaton of preferences and ts processng has been studed n many felds such as economcs, especally n proect and rsk management, decson theory, socal choce theory, wth further developments and applcatons n areas such as operatonal research, databases, securty analyss, and artfcal ntellgence. The modelng of user preferences s a great challenge, as t s dffcult to express human opnon n a way that can be easly processed by computers (Yu, Yu, Zhou, & Nakamu, 00). As ntroduced later n ths paper, there s a varety of formalsms and methods for addressng dfferent preference structures wth scales of nput and output nformaton wth dfferent semantcs. Most of the current approaches collect ndependent preferences, under the mutual preference ndependence (MPI) hypothess (Keeny & Raffa, 993), whch means that a user s preference for an opton s ndependent of the other optons (Yu, Yu, Zhou, & Nakamu, 00). However, the MPI hypothess s not always true n practce (Strlng, Frost, Nokleby, & Luo, 007). People often express condtonal preferences they can state ther preference for a partcular opton only when the state of another opton s determned (Yu, Yu, Zhou, & Nakamu, 00). In fact, condtonal preferences appear to be more natural to the human way of thnkng (Yu, Yu, Zhou, & Nakamu, 00).

3 The best-known framework for addressng condtonal preferences s ntroduced by CP-nets and TCP-nets (Boutler, Brafman, Domshlak, Hoos, & Poole, 004; Brafman & Domshlak, 00). They are powerful methods for representng and reasonng over dfferent sorts of qualtatvely defned condtonal preferences. Accordng to Behr et al, qualtatve reasonng s helpful (not completely necessary) but certanly not suffcent for successful performance on quanttatve proportonal reasonng problems (Behr, Guershon, Thomas, & Rchard, 99). Thus, research problem whch rased was related to provdng smlarly clear representatons or semantcs for quanttatve comparsons (McGeache & Doyle, 0), but ths has not been fully explored yet (Boutler, Bacchus, & Brafman, 00; McGeache & Doyle, 0; Mukhtar, Belaïd, & Bernard, 009). On the other hand, tradtonal elctaton methods are typcally developed based on parwse comparsons and they provde quanttatve measurements over uncondtonal preferences. Analytcal Herarchcal Process (AHP) proposed by Saaty (Saaty, 980) s a wdely adopted multcrtera decson makng method to make complex decsons. AHP enables decson makng partes to deal wth both tangble and ntangble optons and montor the degree of consstency n udgments (Roper-Low, 990). As a well-accepted method, AHP has extensvely been used n many mportant decson makng domans such as forecastng, total qualty management, busness process re-engneerng, qualty functon deployment, and the balanced scorecard ust to name a few (Büyüközkan, Çfç, & Güleryüz, 0; Chen & Wang, 00; Forman & Gass, 00; Roper-Low, 990). It mposes the other drecton for resolvng the problem of provdng the method of quanttatve prortzaton: keepng the basc characterstcs of AHP prortzaton method (Ishzaka & Labb, 0), analyze how t can be extended for addressng dfferent knds of condtonal preferences. Although AHP s smple to perform, t suffers from several problems, such as the quadratc number of comparsons, and nablty to compare conceptually dssmlar optons. To resolve these ssues, n our prevous work, we proposed the Stratfed AHP (S-AHP) (Bagher, Asad, Gasevc, & Soltan, 00), whch tames the number of requred comparsons between the avalable optons through the employment of a stratfed two-layered approach. In ths paper, we analyze and extend the S-AHP algorthm for handlng dfferent sorts of condtonal preferences whch nclude partally complete and ncomplete defntons wth the possblty of nducng cycles. Also, a partcular form of strong preference whch s defned usng lexcographc order (Stomenovc, 99) s analyzed as a specal case of preferences, whch s naturally to be expected n dfferent felds and real-lfe problems. Accordngly, our research obectve s defned as: the formulaton of dfferent forms of condtonal preferences usng a twolayered structural model that handles condtonal preferences and partal nformaton over preferences. Our framework, called Condtonal Stratfed AHP (CS-AHP), provdes the followng maor benefts to the process of prortzaton and decson makng:. It presents a framework for representng dfferent forms of preferences over two-layered herarchcal structures;. It proposes an extenson of the well-known AHP method that enables ts use for smultaneously capturng and handlng both condtonal and uncondtonal preferences, whch mght also nclude preferences about lexcographc order, over the two-layered structure; 3. It s able to recognze the avalable volatons of well-formedness rules n the defned condtonal and uncondtonal preferences and to effectvely handle them through a resoluton process.

4 In the rest of the paper, we frst ntroduce our Stratfed Analytc Herarchcal Process (S-AHP) n Sect.. Throughout the paper, we employ a wdely used benchmark case study (Sect. ) from the area of software servce selecton, but t s clear that our proposed work n ths paper s general enough to be appled to any prortzaton process and doman. In the Sect. 3, we show how S-AHP can be extended to address our research obectve wthn the context of condtonal preferences. Sect. 4 ntroduces domnant relatve mportance relatons, whch defne lexcographc order over two-layered herarchcal structures. Fnally, Sec. 5 formalzes the CS-AHP algorthm as a stratfed process for handlng dfferent knds of preferences over the structure ntroduced n the prevous sectons. Sec. 6 presents a tool support developed for the CS-AHP practcal use n the confguraton process of software product lnes. The performance analyses from three dfferent perspectves, namely tme-complexty, weakness under the cycles and partal defntons of preferences, and smulaton analyses, are descrbed n Sect.7. The crtcal revew of methods and frameworks from related work s presented n Sect. 8 before the paper s concluded.. BACKGROUND In ths secton, we formally descrbe the Stratfed Analytc Herarchy Process (S-AHP) by ntroducng a two-layered structural model for representng preferences. Frst, we ntroduce the relatve mportance relaton, denoted as, for the formalzaton of stakeholders preferences. Also, ths secton ntroduces a runnng example, whch practcally explans the steps of the S-AHP algorthm. Throughout the paper, the same example wll be extended for ntroducng the concepts n our approach.. Stratfed Analytc Herarchy Process S-AHP s a prortzaton technque that takes the preferences, busness goals and hgh-level obectves of a gven group of stakeholders nto account n order to fnd the relatve prorty and mportance of the avalable optons (Bagher, Asad, Gasevc, & Soltan, 00). In other words, S-AHP helps the stakeholders fnd the most sutable set of optons for ther target applcaton by creatng a prortzaton over all of the avalable optons based on ther preferences and obectves. Smply stated, S-AHP takes a layered approach to the prortzaton of the avalable optons. To do ths, hgh level obectves and goals of the stakeholders are specfed and are referred to as concerns. Concerns are mportant decson makng crtera for the stakeholders. For nstance, the concerns of the stakeholders of a software desgn process can nclude mplementaton costs, development tme, securty, and sales. Once concerns are dentfed, the optons that need to be prortzed are nterrelated wth the concerns. For nstance, n the software desgn context, one of the optons to be evaluated s mplementaton by usng COTS. For the stakeholders, ths opton entals nsgnfcant mplementaton costs, quck development tme and low securty. However, the other opton s n-house development, whch entals hgh mplementaton costs, tme consumng development and hgh securty. The formaton of ths nterrelatonshp between the concerns and the avalable optons allows S-AHP to create a prortzaton over the avalable optons by valung optons that are related to more mportant concerns hgher. In our smple example, f securty s more mportant for the stakeholders, n-house development wll receve a hgher m- In our orgnal work on S-AHP, we expermented wth the use of S-AHP for confguraton of feature models, whch are commonly used for modelng varablty n software product lnes (SPLs). 3

5 portance and prorty; lkewse, f lower mplementaton costs are more essental, COTS-based development wll be more attractve. Formally sad, the nput to the S-AHP process s a trple (O, C, QT) where O s the set of avalable optons, each of whch s annotated wth concerns and a set of relevant qualfer tags. Qualfer tags are dfferent possble enumeratons for each concern. For nstance, the qualfer tags for the securty concern can be secure, open, and vulnerable or for the mplementaton cost, they can be cheap and expensve; C s the set of defned concerns; and QT s the set of qualfer tags for the concerns. Concerns and ther qualfer tags have a herarchcal structure (whch s the basc characterstc of AHP (Saaty, 980)) wth two-layers. We ntroduce the followng defntons for specfyng the relaton of relatve mportance n the two-layered structure of concerns and qualfer tags: Defnton (Relatve mportance). Relatve mportance between concerns (or qualfer tags) a and b s: a α b ff concern (or qualfer tag) a s more mportant than concern (or qualfer tag) b wth coeffcent α, α>0. Basc characterstcs of ths relaton are:. Reflexvty: a a. α - α / α -Symmetry : a b b a α β Here, transtvty does not hold, whch means that f a b b c, we cannot make any conclusons about the relaton between concerns (or qualfer tags) a and c. Tradtonally, the values, 3, 5, 7, and 9 are used to represent the degree of mportance of dfferent optons over each other n AHP (Saaty, 980). They show equalty (or ndfference), slght value, strong value, very strong and extreme value, respectvely. The equalty a b would show that concern (tag) a s equally mportant as concern (tag) b or that the stakeholder s ndfferent about the relatve mportance between concerns (qualfer tags) a and b. Based on the relatve mportance relatons, matrces for calculatng relatve rankngs are created n two dfferent levels: the level of concerns and the level of qualfer tags. They can be formally defned as follows. Defnton (Matrx of relatve mportance between concerns). Let us suppose that C = { c,..., c n } s the set of concerns. A matrx of relatve mportance between concerns accordng to relaton α α s defned as Rn n ( C) = { R [, ] = α, n, c c }. Defnton 3 (Matrx of relatve mportance for concern c). Let us suppose that concern c s annotated wth a set of qualfer tags QT = qt,..., qt }. A matrx of relatve mportance for concern c { QT accordng to relaton α α s defned as follows: R QT QT ( c, QT ) = { R [, ] = α, QT, qt qt }. As the relaton α s reflexve and α - -symmetrc, matrces R (C) and R ( c, QT ) satsfy the followng condtons: R ( C)[, ] =,.e., R ( c, QT )[, ] = and,.e., ; that R ( C)[, ] = R ( c, QT )[, ] = R ( C)[, ] R ( c, QT )[, ] s, these matrces are unquely determned wth { R ( C)[, ], < },.e., { R ( c, QT)[, ], < }. Accordngly, for cases when we have a set of three concerns { c, c, c3}, ts matrx of relatve mportance s α α3 α3 unquely determned wth { c c, c c3, c c3}. The unque determnaton of the matrx for one concern s defned smlarly. The matrxes ntroduced by Defntons and 3 are flled wth values of relatve mportance between ether each par of concerns or each par of qualfer tags of one concern. They are defned 4

6 uncondtonally, and, n the rest of the paper, any set of relatve mportance between ether each par of concerns or each par of qualfer tags would be consdered as stakeholders uncondtonal preferences. The followng stages can be performed consecutvely to produce a vald prortzaton over the avalable optons usng S-AHP: The concern rankng step. Ths stage compares the set of concerns (C) usng standard AHP to determne ther ranks. Thus, frst the relatve mportance of each concern wth respect to the others s defned by the stakeholders and the matrx R (C) s created. Based on the standard AHP n algorthm, set { r,..., r n }, 0 r, r = of relatve ranks for each concern s calculated. The = concerns wth the hghest ranks are then used n the opton rankng step and the others are fltered out. In order to keep one of the basc characterstcs of AHP (Saaty, 980), that s, the sum of ranks at each level must be equal to one, we choose the recprocal dvson, whch keeps the prevous relatve order of concerns. If the set of concerns C = { c,..., c n } s reduced to m n concerns, ther recalculated ' ' ranks { r,..., } are computed as follows: where Δr = r m r r,..., n}\{,..., } { m ' k m = r r r k + Δ =, k {,..., m} s the sum of ranks of less mportant concerns whch are fltered out. The optons rankng step. In order to fnd the actual rank and relatve mportance of the avalable optons, the local ranks of the qualfer tags of the most mportant concerns are computed by performng AHP. For each concern c k, k {,..., m}, the matrx R ( c, QT ) s used for rankng ts k k k k qualfer tags. Ther local ranks are hence r,..., r. The fnal rankng for the qualfer tags s obtaned by multplcaton of ts local ranks by the global rank of an approprate concern, whch k k gves r ' r,..., r ' r, k {,..., m} k k QT k. Afterwards, the rank of each opton s determned based on the rank of the qualfer tags assgned to the concerns attached to each of the optons. If an opton s not related to a concern, ts rank s consdered to be zero. The goal of ths stage s to assgn hgher ranks to the optons whch are related to more mportant concerns from the stakeholders vewpont. Rankngs of optons are calculated by applyng a predefned functon f (.e., mnmum, maxmum, or mean) on the qualfer tag value ranks for each opton. The predefned functon s a functon that s used to select a rank for an opton when an opton s related to more than one concern. Let us denote wth qt the th qualfer tag of the concern c. The fnal rank for an opton annotated k k ' ' k wth the set of qualfer tags { qt,..., qt } s defned as r( qt,..., qt ) = f ( r r,..., r r ). () k. Runnng example We wll use the problem of servce selecton n the doman of Servce-orented Computng as the runnng example. Servce-orented Computng argues the development of servces as selfdescrbng, platform-agnostc computatonal elements that can be composed nto dstrbuted ap- QT k k k k () 5

7 plcatons (Papazoglou, 003). Often, several companes provde dfferent servces wth the same functonalty, but wth dfferent non-functonal propertes, such as dfferent securty levels and costs. In such contexts, makng the best decson about the selecton of servces that accurately satsfes the users needs s a necessary task. The stakeholders' prortes between dfferent nonfunctonal propertes should serve as the bass for the selecton of the most approprate servce from the avalable set. As an example, let us consder payment gateways (e.g., VerSgn, PayPal, and Cyber Source), whch are actually servces used for provdng electronc payments n onlne stores. Let us also consder securty, customer ease, nternatonal sale and cost as non-functonal propertes (concerns). For our approach, each servce contans ts related qualfer tags that descrbe the servces n terms of the concerns (as shown n Fgure ). For example, the PayPal servce contans qualfer tags (C, D, G) meanng that, PayPal has low Securty, hgh Customer ease and hgh Internatonal sale. In the frst stage of S-AHP, concerns are ranked based on relatve mportance between each par of concerns. Let us suppose that a stakeholder defnes relatve mportance between concerns as / 3 follows Sec Cost, Sec ISale, Sec CustEase, Cost ISale, CostCustEase, CustEase ISale. Based on Saaty s standard AHP, ranks for concerns Securty, Cost, Internatonal sale and Customer ease are 0.54, 0.4, 0.4, and 0.08, respectvely. Ths shows that the stakeholders have chosen Internatonal sale and Customer ease as less mportant concerns. Therefore, the S-AHP approach proposes that gven the fact that these concerns are less mportant that they should be removed and the sum of ther ranks should be dstrbuted to Securty and Cost accordng to ther ranks. To follow our example, the value of the summarzed ranks of Internatonal sale and Customer ease s =0.. Ths value s now dstrbuted to Cost and Securty and hence the new values are /( )*0.=0.69, and /( )*0.=0.3, respectvely. In the next stage, the avalable optons are ranked based on the ranks of the qualfer tags of each concern. Analogous to the frst stage, AHP s used for calculatng local ranks of qualfer tags of each concern. Let us suppose that the calculated local ranks for hgh, medum and low Securty are 0.6, 0.3 and 0., respectvely; for hgh, medum and low Cost are 0., 0.3, 0.5, respectvely. Global ranks of qualfer tags are obtaned as multplcaton of ther local ranks and ranks of correspondng concerns. The VerSgn servce s annotated wth hgh Securty, low Internatonal sale and low Cost, and ts fnal rank s an average sum of the global ranks of the annotated qualfer tags. As the Internatonal sale concern s recognzed as a less mportant concern, the average sum conssts only of the remanng two concerns: (0.69* *0.5)/=0.85. On the other sde, any qualfer tag of concern Cost s not assgned to PayPal servce, and thus, the fnal rank of PayPal s: (0.69*0+0.3*0.)/= CONDITIONAL PREFRENCES IN THE TWO-LAYERED STRUCTURAL MODEL In ths secton, the relaton of relatve mportance s extended wth elements of Propostonal Logc for representng condtonally defned preferences. Also, the S-AHP algorthm s analyzed for addressng condtonally defned preferences and the noton of well-formedness rules s ntroduced. 6

8 n = 3. Condtonal preferences Before ntroducng the formal concept of condtonalty, let us ntroduce a smple example for explanng the nature of condtonal preferences about relatve mportance of concerns and ther qualfer tags. Example. In the runnng example, let us consder a stakeholder who defnes relatve mportance between concerns n the followng manner: Uncondtonal relatve mportance: Cost ISale 3 (3) Condtonal relatve mportance: n case of a hgh prorty of Internatonal Sale then Sec CustEase, otherwse SecCustEase, 3 Cost 3 Sec. (4) 5 In the second level,.e., the level of qualfer tags, the stakeholder mght condtonally defne the mportance between the qualfer tags of the Internatonal sale, concern as follows: In case of a hgh prorty gven to Securty, low values of Internatonal sale are strongly unacceptable s comparson to hgh and medum values (.e., ISale.Hgh 5 ISale.Low, ISale.Medum ISale.Low); 5 otherwse, low values are slghtly less acceptable than ether medum or hgh values (.e.,isale.hgh ISale.Medum, ISale.HghISale.Low, 3 ISale.MedumISale.Low). 3 (5) It s mportant to menton that the relatve mportance between the qualfer tags of one concern mght not be defned wth only one condtonal statement,.e., the stakeholders mght defne more condtonal statements about relatve mportance between the qualfer tags of Internatonal sale as follows. In case of a low prorty assgned to Cost, there s a slght preference of medum Internatonal sale over hgh Internatonal sale and strong preference of medum over low values of Internatonal sale (.e. ISale.Medum 3 ISale.Hgh, ISale.MedumISale.Low); 5 otherwse, hgh values of Internatonal sale have a slght and strong prorty n comparson to medum and low values, respectvely (.e. ISale.Hgh ISale.Medum, 3 ISale.Hgh ISale.Low). 5 (6) We can see that the relatve mportance between two concerns mght depend on the qualfer tags of other concerns. Also, relatve mportance between qualfer tags of one concern mght depend on the qualfer tags of other concerns. We consder that any other way of dependency s not to be naturally expected n the two-layered structure of concerns and qualfer tags. In order to formally defne the presented condtonal preferences, we ntroduce Defntons 4 and 5. Defnton 4 (Condtonal preference between concerns). Let us suppose that C = { c,..., c n } s a set of concerns and QT = QT s the set of ther qualfer tags. Condtonal preference between concerns { c,.., c }, l {,.., n}, l {,..., k} accordng to relaton α s defned as a 5- k ( Ψ, C, QT, R ( C), R ( C) ) tuple where:. Ψ s a propostonal logc formula over a set of qualfer tags QT, QT QT/ QT as pro- postonal varables connected wth,, and representng logcal operators conuncton, dsuncton and negaton, respectvely;. QT QT s a set of qualfer tag values for concern c ; R (C) 3. s the relatve mportance matrx for concerns f Ψ s true; R (C) 4. s the relatve mportance matrx for concerns f Ψ s false. Whether Ψ s true or not determnes whch matrx of relatve mportance would be appled. Ths can be represented by pseudo-logcal statements n the followng format: Ψ : R ( C) R ( C). 7 l=, k l

9 Informally nterpreted, relatve mportant matrx R (C) represents the then part of a preference and matrx R (C) represents the else part of the preference. For example, condtonal preference between three gven concerns { c, c, c3}, can be specfed α α3 α3 β β3 β3 as Ψ : c c, c c3, c c3 c c, c c3, c c.that s, f Ψ s satsfed, the matrx of relatve mportance s defned by { c c, c c3, c c3}, otherwse by { c c, c c3, c c3}. Restrcton on the con- 3 α α3 α 3 β β3 β 3 dton of Ψ for the preference between concerns { c,.., c k }, whch cannot contan qualfer tags of any concern from the set { c,.., c k }, s a part of Defnton 4. Ths means that concerns cannot depend on ther own qualfer tags. The formal presentaton of the preference (4) accordng to Defnton 4 s: ISale.Hgh: SecCus- tease SecCustEase, 3 Cost 3 Sec. 5 Defnton 5 (Condtonal preference for concern c ). Let us suppose that C s a set of concerns and QT = s ther approprate set of concern qualfer tags. Condtonal preference for n = QT QT QT \ QT α concern c C accordng to relaton s defned as a 5-tuple ( Ψ, c, QT, R ( c, QT ), R ( c, QT ) ) where:. Ψ s a propostonal logc formula over set of qualfer tags QT as propostonal varables connected wth,, and representng logcal operators conuncton, dsuncton and negaton, respectvely;. c s the concern of nterest; 3. QT QT s the set of qualfer tag values for concern c ; 4. R ( c, QT ) s the relatve mportance matrx for concern c f the condton holds; 5. R ( c, QT ) s the relatve mportance matrx of concern c f the condton does not hold. Whether Ψ s satsfed or not determnes whch matrx of relatve mportance wll be appled. Ths can be defned by pseudo-logcal statements n the form Ψ : R ( c, QT ) R ( c, QT ). For example, condtonal preference for a concern wth only three tag values { qt, qt, qt3}, can be α α3 α3 β β3 β3 specfed as Ψ : qt qt, qt qt3, qt qt3 qt qt, qt qt3, qt qt. Ths means that f condton 3 α α, α 3 3 Ψ s satsfed, the matrx of relatve mportance s determned by { qt qt, qt q t3, qt qt 3} β β3 β 3 otherwse, by { qt qt, qt qt 3, qt qt 3}. Restrctons on condton Ψ for relatve mportance between the qualfer tags of concern c, whch cannot contan qualfer tags of concern c, s a part of Defnton 5. Ths means that qualfer tags cannot depend on themselves. Furthermore, stuatons when optons are not related to concern(s) from condton Ψ are analyzed and dscussed later n the Secton 5. The formal presentaton of the preference (5) s: Sec.Hgh: ISale.Hgh 5 ISale.Low, ISale.Medum 5 ISale.Low ISale.Hgh ISale.Medum, ISale.Hgh 3 ISale.Low, ISale.Medum 3 ISale.Low. 3. Addressng condtonal and uncondtonal preferences In the prevous example, none of the preferences completely defnes the relatve mportance between each par of concerns,.e., each preference only reflects the relatve mportance between 8

10 concerns that depend on the specfed condton, leavng the others to be defned wth new preferences or to reman undefned. More precsely, preference (3) defnes relatve mportance only between concerns Cost and ISale whle together wth preference (4), relatve mportance between concerns ISale and Sec stll remans undefned. Furthermore, relatve mportance for concern Securty s defned wth two dfferent condtonal preferences, (5) and (6). For all other concerns, preferences are specfed ether condtonally or uncondtonally and only onng them together, the whole nformaton about mportance for the level of concerns can be defned. Formally, stakeholders preferences, consst of parts of matrces ntroduced by Defntons -5, R = { } { } { } {( )} { } Ψ R ( C) C Ρ, R ( c, QT ),, C, QT, R ( C) R C Ψ c n QT R c QT R c QT C,...,, ( ),,,, (, ), (, ),,..., n C Ρ C where P C s a set of all subsets of the set of concerns C. The goal s to fll the whole matrx for the level of concerns (Defnton ) and n matrxes for the level of qualfer tags (Defnton 3: n matrces of dmenson QT QT where QT s the number of qualfer tags for the th concern). The calculaton of local and global ranks s enabled only wth completely flled matrces for both levels. Let us contnue wth Example and show how the matrces should be flled based on the specfed preferences. Example (Contnued). In ths example, there are four concerns for rankng. As prevously stated, preference (3) uncondtonally defnes the mportance between concerns Cost and Internatonal sale; preference (4) defnes n the then part of the mportance between Securty and Customer ease and n the else part between two pars of concerns: Securty and Customer ease, and Cost and Securty. Now, we can see two thngs: frst, we cannot make a decson whch part of the preference (then or else) to use; and second, any part that we take, the matrx for the level of concerns wll not be completely flled (sx postons n the matrx should be flled, but we have at most two or three values dependng on the part of the preference that we choose). The decson about the satsfed part of a condtonal preference cannot be made wthout concrete optons whch should be ranked based on the ranks of the approprate qualfer tags Let us consder two dfferent avalable optons n ths model: VerSgn and Cyber Source from the Secton.. For ether opton, local and global ranks for the concerns and qualfer tags should be calculated separately and based on them, the more approprate opton should be chosen. Let us consder the frst opton. Low Internatonal sale, whch annotates ths opton, makes the else part of condtonal preference (4) satsfed. By onng preference (4) wth uncondtonal preference (3), we have: Cost ISale, Sec CustEase, Cost Sec.Values between other pars of concerns are not specfed. By default, we consder that undefned relatve mportance represents the stakeholders ndfference and accordng to the comments that followed Defnton, ndfference s equal to one. So, for the level of concerns, we have a matrx, whch s defned by: Cost ISale, Sec CustEase, Cost Sec, Cost CustEase, ISale CustEase, ISale Sec. Now, Saaty s standard AHP gves rank values 0.5, 0.9, 0.6 and 0.40 for concerns Securty, Customer ease, Internatonal sale and Cost, respectvely. (7) Ths shows that the frst stage of prortzng concerns s completed. The next stage covers prortzaton of the qualfer tags of each concern. Analogous to the level of concerns, the satsfed parts of preferences (5) and (6) should be chosen, oned together and sets of ranks calculated wth the standard AHP. In the case of the frst avalable opton, the then parts of both preference 9

11 (5) and (6) are satsfed. When oned, they gve the followng preference: ISale. Medum ISale.Hgh, ISale.Hgh ISale.Low, 5 ISale.MedumISale.Low. 5 So, local ranks for qualfer tags of 3 the Internatonal sale concern are: 0.30, 0.60 and 0.0 for hgh, medum and low values, respectvely. Fnally, ther global ranks are: ISale. Hgh : 0.6 * 0.30 = 0.05, ISale. Medum : 0.6 * 0.60 = 0.0, ISale. Low : 0.6 * 0.0 = Now, let us consder the next opton, Cyber Source. The same as n the case of the frst opton, we have the followng values for concerns Securty, Customer ease, Internatonal sale, and Cost: 0.5, 0.9, 0.6 and 0.40, respectvely. For the level of qualfer tags, the else part of preference (5) s satsfed whch gves (ISale.Hgh ISale.Medum, ISale.HghISale.Low, 3 ISale.Medum ISale.Low), and also the else part of preference (6) s satsfed, whch gves (ISale.Hgh 3 IS- 3 ale.medum, ISale.Hgh 5 ISale.Low). However, these two sets of preferences cannot be oned together because they do not unquely defne the relatve mportance between the ISale.Hgh and ISale.Low qualfer tags. Ths makes the calculaton of local ranks for qualfer tags of the Internatonal sale concern mpossble. Stuatons lke the one above should be recognzed as non-well-formed-preferences of stakeholders. In order to detect such non-well-formed preferences, we ntroduce the followng defnton; well-formedness and ts volatons are dscussed n Secton 5. Defnton 6 (Well-formed set of preferences). A set of preferences s well-formed f t satsfes the followng condtons:. Each preference s defned n the form of Defntons -5;. Each value n the matrx for the concern level s undefned or s unquely defned by uncondtonal preferences and under the satsfed part (then or else) of condtonal preferences; 3. Each value n the matrx for each concern n the qualfer tags level s undefned or s unquely defned by uncondtonal preferences and under the satsfed part (then or else) of condtonal preferences. If any of the specfed condtons s not satsfed, the set of preferences s not well-formed. Fnally, as we dscussed n the prevous example, a set of avalable optons mght be ranked based on the preferences defned on an approprate set of concerns and ther qualfer tags f the set of preferences s well-formed. Ths statement s formally specfed and defned n Secton DOMINANT RELATIVE IMPORTANCE In ths secton, we ntroduce the defnton of domnant relatve mportance, whch corresponds to the defnton of the well-known lexcographc order (Stomenovc, 99). Also, we explan how t can be combned wth condtonal and uncondtonal preferences ntroduced n the prevous secton and how t can be addressed by the CS-AHP algorthm. 4. Domnant relatve mportance for two layered structure Lexcographc order s a specal way of sortng, well-known n the lterature (Wlson, 0).It s a generalzaton of the way the alphabetcal order of words s based on the alphabetcal order of letters. Lexcographc orders can be vewed as beng composed of a set of partcular knds of strong preference statements where the choce of values of a varable domnates the assgnments to a set of other (less mportant) varables (Wlson, 0). Ths specal form of preference has applcatons n many dfferent felds and real-lfe problems. In our llustratve example, concernng the selecton of the most approprate servce for elec- 0

12 tronc payments n onlne stores, let us suppose that another stakeholder has very lmted budget, but she needs hgh or medum securty. It means that avalable servces wth low cost should be the hghest ranked regardless of ther other characterstcs. (8) Dfference between them should be made accordng to medum or hgh Securty. (9) Also, t s natural to allow the stakeholder to nclude condtonalty n preferences about relatve mportance between Customer ease and Internatonal sale. We consder that t s not natural to be expected any condtonalty n specfyng domnant mportance of low Cost and later, medum or hgh Securty. Domnant relatve mportance n two-layered structure s ntroduced by the followng defnton. Defnton 7 (Domnant relatve mportance of concern c ). Let us suppose that C = { c,..., c n } s a set of concerns and QT = QT s the set of ther qualfer tags. Concern c has domnant n = D relatve mportance to the others (denoted as ( c ) ) f the rankng over the set of optons s defned as follows: k An opton o, whch s annotated wth the set of qualfer tags{ qt,..., qt }, k m, has hgher l rank than an opton o annotated wth the set of qualfer tags{ qt,..., qt }, l m, ff one of the l followng condtons s satsfed: c) a qualfer tag from the set QT annotates only the opton o ; c) no qualfer tag from the set QT annotates any of optons o and o and an opton o s hgher ranked based on qualfer tags of other concerns attached to t; c3) when both optons are annotated wth qualfer tags from the set QT, the followng two cases are dstngushed: c3.a) both optons are annotated wth the same qualfer tag from the set QT and an opton o s hgher ranked based on qualfer tags of other concerns attached to t; c3.b) opton o s annotated wth the qualfer tag wth hgher rank from the set QT. In all other cases opton o s hgher or equally ranked to the opton o. We can see that ntutve explanaton (8) from the prevous example s formalzed wth (c) and explanaton (9) wth (c3), whle (c) s related to the comparson of optons that are not related to the concern wth domnant relatve mportance (.e. comparson wth no domnant relatve mportance). To exemplfy, let us consder the followng fve optons: o ={Cost.Low, CustEase.Low, ISale.Hgh}, o ={Cost.Hgh, CustEase.Low, ISale.Hgh}, o 3 ={Cost.Hgh, CustEase.Hgh, ISale.Hgh}, o 4 ={CustEase.Low, ISale.Hgh}, o 5 ={CustEase.Hgh, ISale.Hgh}. Also, let us keep the domnant relatve mportance of concern Cost wth the hghest prorty for low values and lower prortes for medum and hgh values, respectvely. Furthermore, let us suppose that combnaton of qualfer tags {CustEase.Hgh, ISale.Hgh} s ranked hgher n comparson to {CustEase.Low, ISale.Hgh}. (0) Now, let us prortze the mentoned fve optons. Accordng to (c), the leadng three optons o, o, o 3 are ranked hgher than last two optons o 4 and o 5. Also, accordng to (c) and (0), opton o 5 s hgher ranked then o 4. The (c3) gves rankng between o, o, o 3 as follows: (c3.a) wth (0) gves hgher rankng to o 3 than to o ; and (c3.b) gves hgher rankng to o n comparson to o, o 3. Fnally, the orderng between avalable optons from the hghest ranked to the lowest s gven wth: o, o 3, o, o 5, o 4. Based on the defnton of domnant relatve mportance of one concern, we can defne domnant relatve mportance of k-tuple of concerns as follows. k

13 Defnton 8 (Domnant relatve mportance of k-tuple of concerns ( c,..., c k ) ). Let us suppose that C = { c,..., c n } s a set of concerns and n QT = QT s the set of ther qualfer tags. The k- tuple of concerns ( c,..., c k ) has domnant relatve mportance (denoted as ( c,..., c k ) ) f concern c has domnant relatve mportance to the others, and each concern c, {,..., k} has domnant relatve mportance to concerns C \ { c,..., c }. We can see that n the example from the begnnng of ths secton, the par of concerns (Cost, Securty) has domnant relatve mportance. Addtonally, the low Cost s more mportant than medum and hgh, and medum and hgh values of Securty are more mportant than the low value. It s necessary to notce that, by ntroducng domnant relatve mportance of cost, wth the most preferable low value, t s not mportant and meanngful to determne whether low cost s more or much more mportant than medum and hgh values. That nformaton would not nfluence the fnal rankng of optons, because domnant relatve mportance ensures that any servce wth low cost would be hgher ranked than any other servce wth medum or hgh cost. In that sense, stakeholders should only defne qualtatve order between qualfer tags of each concern wth domnant relatve mportance. In the rest of the paper, we assume that the set of qualfer tags of each concern wth domnant relatve mportance s ordered ncreasngly. By r D ( qt ), we denote the ndex of the th qualfer tag n ncreasngly ordered array of qualfer tags of the concern c. So, n the runnng example of the prevous paragraph, we have: r ( Cost. Hgh) =, r ( Cost. Medum) =, r ( Cost. Low) = 3. () D D D By ntroducng the defnton of relatve mportance of the whole set of concerns C, we defne lexcographc order over the two-layered structure of concerns and qualfer tags. For the sake of smplcty, let us assume that an arranged set of concerns { c,..., c n } has the domnant relatve mportance, and qualfer tags of each concern are ordered ncreasngly,.e. qt s the qualfer tag of concern c wth the lowest rank. The fnal rankng over all possble optons over the set of n = concerns C and qualfer tags QT = QT, wth domnant relatve mportance of the whole set C s llustratvely represented n Fgure. The lowest ranked opton s annotated wth the lowest ranked qualfer tag of the concern wth the least domnant relatve mportance. On the other hand, the hghest ranked opton s annotated wth the hghest ranked qualfer tags of each concern. In order to address ths knd of preferences, we ntroduce aggregaton functon F. Based on ths functon, we defne rankng r whch addresses condtonally and uncondtonally defned preferences wth domnant relatve mportance of a subset of concerns. In the followng subsecton, t s also proven that n case of absence of preferences about domnant relatve mportance, r s reduced to the rankng ntroduced n Sectons and Addressng domnant relatve mportance Frst, we ntroduce a well-known example of postonal numeral systems, developed between the st and 5th centures by Indan mathematcans (Glaser, 98), whch served as the nspraton = D

14 for the soluton of the problem of addressng domnant relatve mportance. Postonal numeral system s a system for representaton of numbers by an ordered set of numerals symbols (called dgts) n whch the value of a numeral symbol depends on ts poston. For each poston, a unque symbol or a lmted set of symbols s used. The value of a symbol s gven by the weght of ts poston expressed n the bases (or radces) of the system. The resultant value of each symbol s gven by the value assgned to ts poston (e.g., by a product of the bases) and modfed (e.g., multpled) by the value of the symbol. The total value of the represented number n a postonal number s then sum of the values assgned to the symbols of all postons (Glaser, 98). The orderng between numbers mght be nterpreted as domnant mportance of hgher postons and the value of the number a n... aa0 n base-b postonal system s obtaned n as an b ab + a0, where a { 0,..., b }, {0,..., n}. In the two-layered structure of concerns and qualfer tags, we found an analogy to the postons and dgts n concerns wth domnant relatve mportance, and ther qualfer tags, respectvely. Also, by analogy to the mxed-base postonal system (Glaser, 98), we defne the k-dmenson array of bases to k-tuple of concerns wth the domnant relatve mportance, as follows: Let us suppose that C = { c,..., c n } s a set of concerns and n QT = QT s the set of ther qualfer tags. To the k-tuple of concerns c,..., ) whch has the domnant relatve mportance, we k ( c k assgn bases defned wth: ( QT ),..., ( QT + )( QT + ), QT +, ) ( + k = fnally, aggregated functon for an opton, whch s annotated wth the set of qualfer s tags qt,..., qt }, s defned by: { s s l s F( qt,..., qt ) = bl r D qt + r qt qt qt ( ) ({,..., } \ { k where bl ( QT + ), s k l = l s = k,..., qt, l = k = () < l k = l+ where r()and r D ()are prevously defned, wth () and (), respectvely. k k }), s m, Based on aggregated functon F, we can defne the fnal rank of the opton annotated wth the set s of qualfer tags,..., qt } k s s qt wth r( qt,..., qt ) = F( qt,..., qt ) ( QT + ) { s s For ntroducng the specfed formulas n () and (3), we should prove that: ) Aggregaton functon defned by () defnes the rankng over optons based on rules defned n Defntons 7 and 8 about domnant relatve mportance; ) Rankng defned wth (3) defnes rank values n the nterval [0, ] ; 3) In case of the lack of concerns wth domnant relatve mportance, formula (3) s reduced to formula (). Lemma. In case of domnant relatve mportance of k-tuple of concerns c,..., ) and arbtrary combnaton of qualfer tags{ qt,..., qt s s }, s m, an nequalty s = ( c k k (3) and, 3

15 k l = u u D l s b r ( qt ) + r({ qt,..., qt } \ { qt,..., qt l holds for each u k. s k k }) b u k, where b = ( + ) 0 QT = Proposton. Aggregaton functon defned by (3) defnes rankng over optons based on the set of condtonal and uncondtonal preferences wth relatons of relatve and domnant relatve mportance accordng to Defntons 7 and 8. Detaled proofs of Lemma and Proposton are gven n Appendx A. As a drect consequence of Lemma (case when u=) and the fact that n case of no domnant k l relatve mportance, then sum bl rd ( qt ) equals 0, we formulate the followng proposton. l = Proposton. For each opton o annotated wth the set of qualfer tags followng s satsfed: k s. 0 F( qt,..., qt ) ( QT + ) s = l, ndependently on s; { qt,..., qt s s }, s m, the s s. In case of no domnant relatve mportance then holds: r ( qt,..., qt ) = r( qt,..., qt ). As a concluson of the propostons, we have that rankng r defned n (3) addresses condtonally and uncondtonally defned preferences wth domnant relatve mportance of a subset of concerns n the two-layered structure of concerns and qualfer tags. Now, let us extend our example from the begnnng of ths secton to show how the ntroduced aggregated functon works. Example.Let us suppose that the stakeholder from Secton 4. addtonally defnes the followng preferences: for the level of concerns, 5 Sec. Hgh: ISale CustEase ISaleCustEase ; and, 3 for the level of qualfer tags, CustEase. Hgh CustEaseLow., CustEaseMedum. CustEase. Low 3 3 and ISale. Hgh ISale. Low, ISale. Hgh ISale. Medum. Based on qualtatve ranks for qualfer tags of concerns wth domnant relatve mportance, ntroduced n (), we have that rd ( Cost. Hgh) =, rd ( Cost. Medum) =, rd ( Cost. Low) = 3. Also, the stakeholder defnes that she needs hgh or medum securty, whch means that Sec.Hgh and Sec.Medum are wth the same relatve mportance n decson makng. So, ther ranks are obtaned as follows: rd ( Sec. Hgh) =, rd ( Sec. Medum) =, rd ( Sec. Low) =. Consequently, the bases assgned to the par of concerns wth domnant relatve mportance (Cost, Securty) are (3,). On the other hand, based on the standard AHP method, global ranks for concerns CustEase and ISale are, n case of Sec.Hgh, 0.67 and 0.833, respectvely; otherwse ranks for both concerns are equal to 0.5. Local ranks for qualfer tags of concern CustEase are: 0.43, 0.43, 0.4, respectvely for hgh, medum and low values. Local ranks for hgh, medum and low ISale are 0.6, 0., 0., respectvely. s 3 s 4

16 Fnally, VerSgn payment servce s annotated wth hgh Securty, low Internatonal sale and low Cost, and ts rank s obtaned, accordng to (3) wth r( Cost. Low, Sec. Hgh, ISale. Low) = ( ) (9 + + ) = CS-AHP FOR ADDRESING CONDITIONAL PREFERENCES WITH DOMINANT RELATIVE IMPORTANCE Ths secton explans how the S-AHP algorthm mght be extended for addressng the problem of rankng avalable optons based on condtonally and uncondtonally defned preferences, whch mght nclude domnant relatve mportance. The overall process of CS-AHP s smlar to the orgnal S-AHP prevously ntroduced n Secton. However, some stages have been changed and new stages have been added. So, CS-AHP conssts of the followng stages: Preferences defnton. At ths stage, a stakeholder defnes preferences on both levels concerns and qualfer tags. The number of preferences about each par of concerns and qualfer tags s not lmted. The stakeholder s allowed to defne more condtonal preferences about relatve mportance of the same par of concerns or qualfer tags, or even leave them to be undefned. In case of domnant relatve mportance, relatve ranks should be defned accordng to (). The frst condton from Defnton 6 s checked at ths stage,.e., the stakeholder s not allowed to specfy preferences about relatve mportance between concerns (qualfer tags) that depend on the qualfer tag(s) of any of those concerns. For the next step of CS-AHP, we assume that the set of avalable optons s specfed and annotated wth at most one qualfer tag per concern. Concerns separaton. Based on defned preferences, a subset of concerns wth domnant relatve mportance s sngled out. The followng steps ams at rankng the qualfer tags of the remanng concerns. It s appled n the followng stages for each opton separately. Determnaton of the set of preferences from the level of concerns. For each opton, condtons n each condtonal preference are checked. If the condton s satsfed, the then part of the preference s chosen, otherwse ts else part s selected. It s mportant to menton that not all optons are annotated wth qualfer tags of each concern, so, stuatons where a condton cannot be checked for a certan opton are possble. In that case, we consder the logcal value of the whole condton to be false,.e., the opton does not satsfy the condton. Determnaton the ranks for the level of concerns. Based on the uncondtonal preferences and satsfed parts of the condtonal ones, the matrx for the level of concerns s flled. In case that t cannot be unquely flled, the set of preferences s declared not to be well-formed and the stakeholder s allowed to make a decson about multple values for fllng the matrx. Then, the ranks are calculated and less mportant concerns are fltered out. Ther ranks should be proportonally shared wth others as descrbed n Secton. 5

17 Determnaton of the set of preferences from the level of qualfer tags. Smlarly to the level of concerns, each condtonal preference between the qualfer tags s checked and stuatons when condtons cannot be checked should be addressed n the same way. If the stakeholders have fltered out a concern n the prevous step, t s a sgn that t s of less nterest n the rankng process and condtons whch nclude ts qualfer tags should be modfed. We modfy the preferences, so that they cannot depend on qualfer tags that are of less nterest: If we denote by qt any qualfer tag of the less mportant concerns, each condtonal preference should be transformed teratvely usng the followng rules: Rule : Condtonal preferences n the forms Ψ qt : R ( c, QT ) R ( c, QT ) or Ψ qt : R ( c, QT ) R ( c, QT ) should be transformed nto Ψ : R ( c, QT ) R ( c, QT ) ; Rule : Condtonal preferences n the forms Ψ qt: R ( c, QT ) R ( c, QT ) or Ψ qt: R ( c, QT ) R ( c, QT ) should to be transformed nto Ψ : R ( c, QT ) R ( c, QT ) ; Rule 3: Condtonal preferences n the forms qt : R ( c, QT ) or qt : R ( c, QT ) should be consdered as napproprate ones for that opton and would not be consdered; Rule 4: Condtonal preferences n the forms qt : R ( c, QT ) R ( c, QT ) or qt : R ( c, QT ) R ( c, QT ) should be consdered as not beng well-formed; thus, the process termnates. Rules and should be repeated untl ether of the condtons defned n Rule 3 or 4aresatsfed or a preference, whch does not contan any qualfer tags of the fltered concerns, s obtaned. Determnaton of the ranks for the level of qualfer tags. Based on the uncondtonal preferences and satsfed parts of the condtonal ones, the matrces for the most sgnfcant concerns are flled. In case that any of them cannot be unquely flled, the set of preferences s declared as not beng well-formed and the stakeholder s agan allowed to make a decson about ts resoluton. Then, the local ranks for sets of qualfer tags of each concern are calculated and based on the prevously calculated ranks for the set of concerns, global ranks are delvered. Fnal opton rankng. Fnal opton rankng s obtaned accordng to (3). 6. TOOL SUPPORT One of the man contrbutons of our CS-AHP process s a wde range of ts practcal applcatons n dfferent felds and domans. For the purpose of practcal use of the CS-AHP, we expermented wth the confguraton process of feature models, whch are commonly used for modelng varablty and confguraton management n software product lnes (SPLs). We have created a prototype, a proof of concept, by extendng the Feature Model Plug-n (fmp) (Czarneck & Km, 005) wth the functonalty requred for managng the two-layered structure of concerns and qualfer tags and for supportng feature confguraton enabled by CS-AHP ntroduced n (Bagher, Asad, Gasevc, & Soltan, 00). fmp s a wdely used plug-n for feature modelng and confguraton. Generally, software product lne engneerng conssts of two development lfecycles doman engneerng (development of a famly of products for a partcular doman) and applcaton eng- 6

18 neerng (dervaton of a concrete product by confguraton of the famly developed n doman engneerng). The fnal goal of the confguraton process n applcaton engneerng s the selecton of the most approprate set of features for a specfc applcaton. The target applcaton stakeholders are allowed to defne preferences of dfferent knds for the both the levels concerns and qualfer tags. The CS-AHP process s used for rankng of a set of avalable features based on the defned preferences, n a smlar manner as ntroduced for S-AHP n (Bagher, Asad, Gasevc, & Soltan, 00). In case volatons of the well-formedness rules durng the rankng process, the applcaton stakeholders mght be asked to resolve them. Fnally, the selecton of features s performed based on the rankngs computed by CS-AHP; and features whch have the least ranks are removed durng the teratve steps of feature confguraton process. A screenshot of the tool s dalog for edtng concerns and ther qualfer tags s gven n Fgure 3a, whle the part for specfyng condtonal preferences for the level of concerns s gven n Fgure 3b. As t can be seen, a dalog for specfyng condtonal preferences conssts of three maor parts: the left part represents the logcal tree of condton of condtonal preferences; the mddle part contans the concerns whose mportance s specfed wth matrces defned on the rght; and the rght part represents two matrces of then (True tab) and else part (False tab) of condtonal preference. Smlarly, the tool allows users to effectvely perform the other steps of CS-AHP. 7. ANALYSES OF CS-AHP Ths secton reports on the results of the analyses based on three dfferent aspects. The theoretcal part of our analyses ncludes complexty analyss and analyss of how both cycles n dependences and the prevously recognzed non-well-formedness n preferences affect the whole rankng process. In addton, the smulaton results are taken as a bass for makng conclusons and suggestons for the most relevant usage of the proposed framework n terms of reducton of potental volatons of well-formedness n preferences. 7. Theoretcal analyss The presented extenson of AHP, named CS-AHP, provdes a framework for representng and reasonng over dfferent forms of preferences on a two-layered herarchcal structure of concerns and qualfer tags. For the nput model (O, C, QT, R ), whch has an addtonalr dmenson n addton to the S-AHP trple and the set of entered preferences, our proposed framework provdes as output the ranks over elements of the set of optons O. The stronger fact holds, as follows. Corollary. In a fnte number of steps, for each set of preferences and for each set of avalable optons, the CS-AHP algorthm offers unque ranks over the set of optons or concludes that the set of preferences s not well-formed. Accordng to the algorthm steps, presented n Secton 5, we can see that the whole algorthm s dvded nto two levels: the level of concerns and the level of qualfer tags. For each opton, approprate matrces on both levels should unquely be flled. Otherwse, as soon as the frst preference volates the unqueness, the algorthm stops and generates the notce about non well-formed preferences. Even f the algorthm recognzes a volaton of the well-formedness rules n the set of preferences, t may allow the stakeholder to make a decson about multple values for fllng the matrx. Alternatvely, the stakeholder mght not be ncluded n the process, and nstead, only the 7

19 frst values can be consdered. That s, as all of the preferences are consdered n the order n whch they are entered, f any preference volates the values entered based the prevous, that preference can be annotated as non-well-formed. Then, the value n the matrx wll not be changed and the algorthm contnues. The two mentoned solutons (wth the stakeholders and by consderng only the frst values) allow the algorthm to always generate the rankng over the set of optons. However, the algorthm wll not always be unque n terms of dependences on the stakeholders decsons about non-well-formed preferences or the order of the preferences entry. In case of volaton of the well-formedness rules, the rankng s not unque to the resoluton of the volaton of the well-formedness. Ths means that n case of a non-well formedness n preferences, ts resoluton has drect mplcatons on the fnal rankng over the optons. As each preference s checked separately for each opton, t dsables any cycles and dependency n the processng of preferences. It s also mportant to menton that preferences about domnant relatve mportance cannot make any volaton of the well-formedness condtons, because ther domnance s defned uncondtonally to others (Wlson, 0). In preferental reasonng, dfferent types of queres make sense n such a settng: ()domnance testng query askng for relatve order between two optons, ()orderng query, seekng an orderng of a subset of optons, and ()optmzaton query that look for a preferentally optmal opton (Domshlak, 008). In preferental reasonng, all three queres are n general NP-hard (Goldsmth, Lang, Truszczyńsk, & Wlson, 008). Corollary.For a gven CS-AHP model (O, C, QT, R ), queres about domnant testng and orderng take polynomal tme complexty. On the other hand, optmzaton queres are n the worst case NP-hard. D Let us denote wth RC as the preferences for the level of concerns, R as the preferences about C domnant relatve mportance of approprate concerns, andr QT as the preferences for the level of D D qualfer tags, R = RC RQT, RC = nc, R = n, RQT = n C C QT.As t can been seen n Secton 5, for each opton, each preference s consdered as follows: for each opton separately, the matrx D on the level of concerns and nc n C matrces on the level of qualfer tags are flled and local ranks are calculated. Prevously, condtons n each condtonal preference are checked, whch D D takes O( nr + ( nc nc ) + ( nc nc ) nqt ) operatons. These operatons are the basc cost of parwse methods, and addtonally they are performed for each opton separately, whch takes D D O( no ( nr + ( nc nc ) + ( nc nc ) nqt )) operatons, where n O s the number of avalable optons. In case of domnance testng and orderng queres, the number of avalable optons s fnte, whch gves polynomal complexty. On the other hand, an optmzaton query wth the presented framework mght be addressed only by backtrackng the generaton of all possble combnatons of qualfer tags and ther overall rankng. It obvously takes exponental tme. It s mportant to menton, that, n case of domnant relatve mportance of the whole set of concerns (.e., lexcographc order), an optmzaton query mght be addressed by smple traversal of each concern, and the set of hghest ranked qualfer tags of each concern represents the optmal opton. In ths case, t has lnear complexty. Also, n case of only uncondtonal preferences, an optmzaton query takes polynomal tme for local rankng of qualfer tags of each concern and optmal opton s annotated wth the set of the hghest ranked qualfer tags of each opton. 8

20 In the prevous paragraph we recognzed two cases when the optmal query s addressed wth lnear and polynomal tme complexty, whch leads to concluson that some structures of preferences optmze optmzaton queres - an ssue left for future research. 7. Smulaton Analyss In ths secton, based on conclusons of Corollary, we fnd the characterzaton of each model (O, C, QT,R ) from the perspectve of the use of the CS-AHP algorthm and get some drectons about the structure of the recommended model, whch would sgnfcantly reduce non-wellformedness n preferences. 7.. Descrptve parameters of the model (O, C, QT, R ) Non-well-formed preferences mght be caused ndependently on both levels of the two-layered process, so, the number of non-well-formed preferences on the level of concerns and the number of non-well-formed preferences on the level of qualfer tags mght be consdered for an analyss from the perspectve of the use of the CS-AHP algorthm. For a gven CS-AHP model (O, C, QT, R ), we have argued that the set O of avalable optons s an nstrument for testng f the ntal set of preferences s well-formed, and the number of nonwell-formedness for both the levels are summarzed up to the whole set of avalable optons. In order to make these numbers ndependent of the avalable set of optons, we use ther average values. So, the usage of the CS-AHP algorthm for an approprate model s analyzed on the bass of the average numbers of non-well-formedness n preferences for both the levels separately. On the other hand, descrptve parameters of the whole model (O, C, QT, R ) mght be recognzed as t follows: - The two-layered herarchcal structure s represented wth the set of concerns and ther qualfer tags. So, the number of concerns and the average number of qualfer tags per each concern are consdered as a par of parameters for the characterzaton of the herarchcal structure. Ths characterzaton s unque due to dfferent elements of the set of concerns and the correspondng set of qualfer tags. - On the other hand, the set of preferences mght have dfferent structures whch only depend on the way n whch users defne preferences related to ther opportunty to specfy even ncomplete preferences or more preferences related to the same par of concerns (or qualfer tags). We recognzed that parameters, whch can descrbe the set of preferences wth respect to the users freedom n defnng and wth potental mportance for our approach, are the followng: o the structure of condtonal preferences whch s related to the form of ther specfcaton; that s, f the preferences consst only of then parts or they nclude else parts, too. o As the number of preferences for each model s unlmted and s not related to the structure of concerns and qualfer tags, the number of preferences cannot be consdered as a parameter of the model. On the level of concerns, each preference defnes the relatve mportance between two or more concerns, so that on ths level, we can defne a condtonal par of concerns as a par of concerns whose relatve mportance s defned by at least one condtonal preference. Also, on the level of qualfer tags, each preference defnes relatve mportance between pars of qualfer tags of one concern, so that we can defne a condtonal concern as a concern whose relatve mportance between at least one par of ts qualfer tags s defned 9

21 condtonally. Fnally, the number of condtonal pars of concerns defned on the level of concerns and the number of condtonal concerns defned on the level of qualfer tags are consdered to be the parameters whch descrbe the set of preferences related to the concept of condtonalty. In the followng sectons, we attempt to dentfy potental relatons between the recognzed descrptve parameters on the one hand, and the numbers of non-well-formedness for both the levels, on the other one. Also, we make estmatons of the expected values of the numbers of nonwell-formedness for both the levels separately. 7.. Hypotheses Our man goal, whch s to fnd sgnfcant characterzaton of the numbers of non-well-formed preferences and ther expected values, s further refned nto the followng more specfc and concrete hypotheses: H. In case of dfferent values of one descrptve parameter and fxed values for others, there are a sgnfcantly dfferent number of non-well-formedness for both levels of each model (O, C, QT, R ). H.The average number of non-well-formed preferences for each level per model (O, C, QT, R ) mght be effcently characterzed at least by the number of condtonal pars of concerns (per level of concerns) and the number of condtonal concerns (per level of qualfer tags). Hypothess H provdes nformaton about the mpact of each recognzed parameter separately on the whole model, whle hypothess H dentfes the best subset of parameters for a sgnfcant predcton of the expected number of non-well-formedness for each model. On the other hand, as condtonalty n preferences determnes dfferent dstrbutons of the set of preferences over the set of concerns, t would be nterestng to consder t as an addtonal descrptve parameter. It s hard to expect to consder t as a real descrptve parameter, because of ts dscrete values and nablty for ts estmaton, but ts nfluence on the observed model s nterestng for analyss. In that sense, we defne an addtonal hypothess. H3. Models (O, C, QT, R ) wth the same descrptve parameters have sgnfcantly dfferent number of non-well-formedness among dfferent dstrbutons of condtonal preferences. These hypotheses are tested by a set of statstcal technques to understand the relatonshp between the number of non-well-formed preferences and recognzed parameters of an approprate model. Regresson and correlaton analyses are used to test f and whch of the recognzed parameters can (most) sgnfcantly determne the number of non-well-formedness n the model (H). Furthermore, the ANOVA test for comparng means for multple ndependent populatons s used for testng f a sgnfcant dfference exsts n the number of non-well-formed preferences n cases of changng the value of one of the descrptve parameters (H), and n case of dfferent dstrbutons of condtonal preferences (H3). As the fnal result, we gve the estmatons of expected values of volatons of well-formedness rules for each group of models wth the same descrptve parameters. They are obtaned as a mean value of the observed average numbers of non-well-formedness on a large ndependent sample Expermental Setup As prevously ndcated that the descrptve parameters mght characterze each model under the use of the CS-AHP algorthm, random generatons are used for developng dfferent models wth 0

22 the same parameters. For each generated model, the number of observed non-well-formedness s determned and collected values are used for further analyss. We have suggested that models should at most consst of 0 concerns annotated wth a maxmum of 7 qualfer tags, as those numbers of concerns and qualfer tags are manageable by human users. For each par of concerns, the number of preferences s generated randomly, of whch any number can be condtonally defned wth random defntons of ther condtons. The smlar s done for the level of qualfer tags and preferences for each par of concerns qualfer tags. As ndcated n the comment on Corollary, preferences about domnant relatve mportance do not have any nfluence on the number of volatons of the well-formedness condtons. Consequently, our smulatons nclude only models wth condtonally and uncondtonally defned preferences and the results are later extended to the general case of addtonal preferences about domnant relatve mportance. The number of condtonal pars of concerns on the level of concerns s n the nterval [0, n(n- )/] and each par of them mght depend on a maxmum of n- of other concerns. Also, the total number of condtonal concerns on the level of qualfer tags s n the nterval [0,n], and each of them mght depend on a maxmum of n-of other concerns, where n s the number of concerns n the model. As explaned n Sect. 5, n the stage named Determnaton of the ranks for the levels of concerns, less mportant concerns mght be fltered out. Durng the smulatons, decson about the number of concerns whch are fltered out s randomly generated. Our smulaton ncludes,000 random generatons of models per each possble combnaton of descrptve parameters. Durng the generaton of preferences, ther creaton n the forms ntroduced by Defntons -5 s satsfed. In our prevous study about servce models from Software Product Lne Tools that are publcly avalable and dstrbuted by the software product lne communty (Bagher & Gasevc, 00), we observed that the number of optons n those models was n range of 0 to 9 wth mean value 5.7 (SD=64.4). So, each set of preferences n our smulatons s tested by obtaned mean value (.e.,5) of dfferent optons randomly confgured for an approprate model. For testng hypotheses H and H, motvated by the example ntroduced n Sect.3, the followng three smulatons are done. In the frst smulaton, after each randomly defned then part of a condtonal preference, a decson s randomly made about the creaton of ts else part. In the second smulaton, preferences are created only wth then parts,.e., f the else part were to be defned, t should be done wth a new preference consstng of the negated condton (from the prevous preference) and the then part. The thrd smulaton ncludes the stuaton when each preference always contans the else part. Durng the random generaton, nstead of else parts n preferences, the empty value mght be generated, too. A preference wth empty the else part equals a preference wth only the then part. For testng hypothess H3, random smulaton s done wth three dfferent dstrbutons of condtonal pars of concerns (on the level of concerns) and condtonal concerns (on the level of qualfer tags), over the number of concerns that they depend on. Inverse exponental, normal and completely random dstrbutons have well-known nterpretatons. In our smulatons, the number of concerns, the numbers of condtonal pars of concerns (on the level of concerns) and condtonal concerns (on the level of qualfer tags) are ntegers, so these dstrbutons are generated over ntegers wth nteger values. The random generaton of other parameters s the same as n testng the prevous two hypotheses.

23 7.3 Analyss In ths secton, we present the employed statstcal technques, the purpose of ther use, the obtaned results, and ther nterpretaton Analyss Technques Gven the type of the collected data n the smulatons, we analyzed them wth standard descrptve statstcs (as reported n (Blake, 003) to be a common practce) ncludng mean and standard devaton values. We can consder the parameters representng the number of concerns, the number of condtonal pars of concerns (on the level of concerns) and the number of condtonal concerns (on the level of qualfer tags) as nterval data. The structure of condtonal preferences s the addtonal parameter, whch has three possble values representng only a then part, an optonal else part and a then wth else part. Possble values mght be annotated wth 0, and, but the annotatons do not have a specal numercal meanng; thus, ths parameter s a categorcal type of data. As t would be shown through the results of smulaton, the descrptve parameter whch represents the number of condtonal pars of concerns would be transformed nto a new categorcal parameter. Its categores represent grouped values of the number of condtonal pars of concerns (on the level of concerns). Accordngly, the analyss of dfference among groups, correlaton, and regresson analyss were done by usng parametrc tests (.e., ANOVA, Pearson, multple regresson and multple regresson wth dummy varables, respectvely). For varables whose data were not normally dstrbuted, we used parametrc tests over log-transformed data 3. As only one parameter s categorcal wth three possble values, we decded to splt our data nto three dfferent groups and make a regresson model for each group separately and compare the results. Also, n order to fnd the most approprate regresson model, we use two dfferent models and analyze the results Results As the frst step of our analyss, we hypothesze that a meanngful correlaton exsts between the defned descrptve parameters and the number of volatons of the well-formedness condtons. Table summarzes the results of correlaton studes for both levels. Values for the coeffcent of correlaton below 0.50 generally are consdered unsatsfactory (Blake, 003). Accordngly, our results show that a low correlaton s recognzed wth the average number of qualfer tags (for the level of concerns: 0.0<0.5 and for the level of qualfer tags: 0.9<0.5), so that ths descrptve parameter was not consdered further n our analyss. In order to analyze the nfluence of descrptve parameter whch represents the number of concerns n case of fxed values of other descrptve parameters, we consdered that our data are dvded nto groups accordng to the number of concerns. As the collected data were not normally dstrbuted, a one way ANOVA test was used over log-transformed data to compare the means of the dependent varable (the number of volatons of the non-well-formedness condtons for both the levels) for dfferent groups defned by the number of concerns. The results show a sgnfcant Ths approach wth dummy varables s appled for representng nformaton about group membershp n quanttatve terms wthout mposng unrealstc measurement assumptons on the categorcal varables (Hardy, 993) 3 Ths approach s appled to non-parametrc tests n evdence-based dscplnes such as medcne (Keene, 995). Moreover, ths s consstent wth the fndngs of prevous research n psychologcal measurement (Rasmussen & Dunlap, 99) and the purpose of our approach for addressng users preferences and udgments.

24 dfference n the number of volatons of the well-formedness condtons between dfferent values for the number of concerns: for the level of concerns F(7,460)=69.56, p=0.000, and for the level of qualfer tags F(7,77)=93.38, p=0.000; the mean values per each group for both levels are represented n Table. Also, the Tukey post-hoc test revealed that for each group there s no sgnfcant dfference compared to groups that dffer n the number of concerns by. The smlar consderaton s done for other descrptve parameters. The results show that there s a sgnfcant dfference n the number of volatons of the well-formedness condtons between dfferent structures of preferences: F(,465)= , p=0.000, on the level of concerns, and F(, 777)=50.7, p=0.000 on the level of qualfer tags. The Tukey post-hoc test revealed that there s a sgnfcant dfference between each two groups. The mean values per each group for both levels are represented n Table 3. The results show that there s a sgnfcant dfference n the number of volatons of the wellformedness condtons between dfferent numbers of condtonal pars of concerns on the level of concerns, F(45,4)=58.446, p= In ths case, the Tukey post-hoc test showed that there s no sgnfcant dfference comparng each par of groups. We decded to make supergroups, so that there s no sgnfcant dfference among groups nsde each of them. Accordng to the fact that n case of n concerns, the maxmum number of condtonal pars of concerns s n(n-)/ (prevously dscussed n the Expermental Setup Sect.), we defned supergroups as follows: the frst group contans up to 3 condtonal pars of concerns, the second group contans from 4 to 6 condtonals pars, and others, respectvely, from 7 to 0, from to 5, from 6 to, from to 8, from 9 to 36 and from 37 to 45. The one-way ANOVA test showed that there s no sgnfcant dfference between the mean values of the number of volatons of well-formedness rules among the created supergroups. Also, t showed that there s a sgnfcant dfference n the number of volatons of the well-formedness condtons between dfferent supergroups, F(7,460)=345.93, p=0.000, whle the Tukey post-hoc test showed, n comparson to the case where there s no supergroups, that there s a sgnfcant dfference comparng each par of them. The mean values of volatons of well-formedness for dfferent supergroups are presented n Table 4. On the other hand, results showed that there s a sgnfcant dfference n the number of volatons of the well-formedness condtons between dfferent number of condtonal concerns on the level of qualfer tags, F(9,770)=79.93, p= The Tukey post-hoc test showed, smlar to the prevous analyss of the parameter whch represents the number of concerns, that for each group there s no sgnfcant dfference as compared to groups that dffer n the number of condtonal concerns by. The mean values of volatons of well-formedness for dfferent number of condtonal concerns for the level of qualfer tags are presented n Table 5. Consequently, the frst hypothess s proved n case of the followng descrptve parameters: the number of concerns, the structure of condtonal preferences, the number of condtonal pars of concerns (for the level of concerns) and the number of condtonal concerns (for the level of qualfer tags); and reected for the descrptve parameter whch refers to the number of qualfer tags per each concern. For further analyss we wll use the supergroups as a descrptve parameter representng condtonal preferences on the level of concerns. Dscusson. As shown through testng hypothess H, each of the recognzed descrptve parameters except the number of qualfer tags, makes sgnfcant dfferences n the number of volatons of the well-formedness condtons for fxed values of other parameters. 3

25 Tables 6-8 summarze the results of the regresson analyss for hypothess H. As prevously explaned, we dvded our analyss n three separate parts, accordng to the structures of condtonal preferences. Thus, the tables are organzed as follows: each table corresponds to one of the three dfferent structures of preferences, and, n each of the three table, results for both the levels (the level of concerns and the level of qualfer tags) are presented, separately. Also, durng the testng of hypothess H, we transformed the descrptve parameter, whch represents the number of condtonal pars of concerns, nto a categorcal varable wth eght values. Consequently, n order to use lnear regresson analyss, we defne seven dummy varables, as represented n Fgure 4. Now, two dfferent regresson models are consdered: the frst one, consstng only of the number of dummy varables (on the level of concerns) and condtonal concerns (on the level of qualfer tags) as the parameter wth the hghest correlaton; and the second one, wth an addtonal predctor beng the number of concerns. Dscusson. In all three cases of dfferent structures of condtonal preferences reported n Tables 6-8, both models accurately predct the number of expected volatons of the well-formedness condtons on the both levels, of concerns and qualfer tags. We can see that, the model consstng only of the dummy varables (on the level of concerns) and the number of condtonal concerns (on the level of qualfer tags) can predct above 90% of varablty n the number of volatons of the well-formedness condtons. It means that the descrptve parameter of the set of preferences R mght effcently characterze the model (O, C, QT, R ) under the use of CS-AHP. Thus, the hypothess H s proven. As a value of R measures how much of the varablty n the outcome s accounted for by the predctors, t would be used as a crteron for selectng better model for the both levels, separately. We can see that the model consstng of the number of concerns and the number of dummy varables on the level of concerns, n all three cases of dfferent structures of condtonal preferences, has greater value of R than a model consstng only of the number of dummy varables (Table 6: 0.957=0.957, Table 7: 0.986>0.937, Table 8: 0.963=0.963). The same can be concluded for the level of qualfer tags, where the model consstng of the number of concerns and the number of condtonal concerns s better than one consstng of only the number of condtonal concerns (Table 6: 0.99>0.880, Table 7: 0.9>0.895, Table 8: 0.95>0.93). Ths result re-asserts what we had antcpated, that the key cause for volatons of the well-formedness condtons s n the number of condtonally defned pars of concerns (.e., concerns for the level of qualfer tags), the structure of condtonal preferences, and the number of concerns n the model. From the prevous hypothess, we obtaned that the structure of condtonal preferences sgnfcantly nfluence the number of volatons of the well-formedness condtons and the lowest values are n case of preferences consstng only of then parts. Consequently, we decded to test hypothess H3 only n that case. The results show that there s a sgnfcant dfference n the number of volatons of the well-formedness condtons n dfferent dstrbutons of condtonal preferences over concerns for the level of concerns, F(,465)=44.507, p=0.000; and the level of qualfer tags, F(,777)=40.535, p= The Tukey post-hoc test shows that there s no sgnfcant dfference between completely random smulatons and normal dstrbutons, but there s for the other remanng pars of dstrbutons. The mean values for dfferent dstrbutons are presented n Table 9. 4

26 Dscusson. Ths result confrms what we had antcpated accordng to the nterpretaton of each of the three well-known dstrbutons. Normal dstrbuton favors the dependency on the medum number of concerns, whle nverse exponental to the low number of concerns, and consequently, t ncreases the number of volatons of the well-formedness condtons. 7.4 Crtcal Recommendatons The results of the hypothess reveal the nfluence of each of the recognzed parameters on the number of volatons of the-well-formedness condtons for both levels of an approprate model. The analyss of hypothess H shows that they can be wth the hghest sgnfcance characterzed wth the followng parameters: the number of concerns and the number of condtonal pars of concerns (on the level of concern) and the number of condtonal concerns (on the level of qualfer tags). As our smulatons generated,000 dfferent models for each combnaton of parameters and ther well-formedness s tested under 5 randomly generated optons, the obtaned results allow us to estmate expected values of volatons of the well-formedness condtons wth approprate mean values. The results are presented n Tables B. and B. n the Appendx B of the paper. We suggest usng the values from Tables B. and B. for the assessment of the ranks obtaned wth the usage of the CS-AHP algorthm on an approprate model (O, C, QT, R ), n the followng steps:. The average numbers of obtaned non-well formedness n the preferences R for both the levels are compared to correspondng numbers n the tables n Appendx B. The values n Tables B. and B. are determned based on descrptve parameters of the observed model: the number of concerns (decreased by the number of concerns wth domnant relatve mportance) and the number of condtonal pars of concerns (on the level of concern) and the number of condtonal concerns (on the level of qualfer tags);. If both values are equal or less than the values n the tables, t means that the obtaned values of non-well formedness s below the average and can be gnored. The obtaned ranks should be consdered to be approprate for the model; 3. If any of the values s hgher than the value n the table, t means that the number of volatons of non-well-foremdness s above the average. The stakeholder should make a decson about the acceptance of the obtaned ranks because t mght be concluded that the model s below the expectaton accordng to the usage of the CS-AHP algorthm. Ths means that the obtaned ranks mght be consdered too dstant from the ntal set of preferences and all caused by the hgher than expected value of non-well-formed preferences. Based on the results and analyzes of the smulatons, we can make recommendatons about the usage of the proposed algorthm and the best ways of defnng an approprate model and ts related set of preferences. Recommendaton. The least average number of non-well-formed preferences s n cases when condtonal preferences consst of only then-parts,.e., there are a lower number of conflcts n ther satsfacton; Recommendaton.A model wth less concerns and a hgher number of qualfer tags s more sutable than a model wth more concerns, as there s no sgnfcant ncrease of the number of volatons of the non-well-formedness condtons by ncreasng the number of qualfer tags; Recommendaton3.A lower number of dfferent concerns n the condtons s not a guarantee for a lower number of volatons of the well-formedness condtons under the set of avalable optons. It means that volatons of the well-formedness n condtonal preferences are caused only 5

27 by stuatons when condtons from dfferent condtonal preferences are satsfed at the same tme. 8. DISCUSSION AND RELATED WORK 8. Quanttatve Prortzatons and Condtonally defned Preferences In the followng, we dscuss methods and formalsms from dfferent felds that are the most related to quanttatve prortzatons and condtonally defned preferences. An example of a famly of methods based on quanttatve measurements (based on categorzaton of Larchev (Tursk, 008)) s the TOPSIS (Technque for Order Preference by Smlarty to an Ideal Soluton) method. There are many algorthms belongng to ths category, such as SAW (Smple Addtve Weghtng) (Zavadskas, Kaklauskas, Peldschus, & Tursks, 007; MacCrmmon, 968; Zavadskas, Tursks, Deus, & Vtekene, 007), LINMAP (Lnear Programmng Technques for Multdmensonal Analyss of Preference) (Srnvasan & Shocker, 973), CORPAS (Complex Proportonal Assessment) (Zavadskas & Kaklauskas, 996; Zavadskas, Kaklauskas, Peldschus, & Tursks, 007), but ther characterstcs do not dffer much based on the ssues presented n Table0. The basc prncple of the TOPSIS method (Hwang & Yoon, 98) s that the chosen alternatve should have the shortest dstance from the postve deal soluton and the farthest dstance from the negatve deal soluton. The best deal and negatve deal solutons are found based on statcally defned preferences. That s, the best prce s the lowest one, the best qualty s the hghest one and so on. Ths n fact presents the lmtaton for ts use for addressng dfferent sorts of preferences ( Amr, 00). An example of comparatve preference methods based on parwse comparson of alternatves s the ELECTRE famly of methods developed n md-sxtes, whch had a strong mpact on the operatonal research communty (Fguera, Mousseau, & Roy, 005). The ELECTRE evaluaton method allows for handlng qualtatve and quanttatve crtera smultaneously. In contrast to the tradtonal approach, ELECTRE ntroduces the concept of an ndfference threshold, and the preference relatonshps are redefned as follows: preferred, ndfferent, and cannot be compared. The whole famly of methods s developed n order to support heterogenety of scales, nterpretaton of outrankng relaton as a fuzzy relaton and stuatons when relatve mportance coeffcents of crtera are not completely defned. These methods are well-known for decson optmzaton n the case of decsons taken n crcumstances of certanty, and as n the TOPSIS methods, preferences are statcally defned, whch s the man reason why the ELECTRE famly cannot be appled n the doman addressed by ths paper. Also related, the Bubble sort technque has been used to rank order the preference statements. Bubble sort s n essence very smlar to AHP wth the slght dfference that preference comparsons are made to determne whch preference has a hgher prorty, but not to what extent. It s clear that Bubble sort suffers from smlar ssues to those of AHP (e.g., the large number of requred comparsons). There have been proposals attemptng to reduce the number of requred comparsons n comparson-based technques, whch are generally referred to as ncomplete parwse comparson methods (Berander & Andrews, 006). These technques are based on some local and/or global stoppng rule, whch determnes when a further comparson wll not reveal more useful nformaton wth regards to the prortzaton of the optons. Such technques can be benefcal f used along wth technques such as AHP, S-AHP, CS-AHP and Bubble sort. 6

28 Addtonally, Herarchcal Cumulatve Votng (HCV) has been used to prortze preferences where top vote-getter preferences are prortzed hgher than the others (Berander & Jönsson, 006). One of the drawbacks of ths approach s that as the number of preferences (optons) ncreases, t becomes very hard for the stakeholders (voters) to select the best votng tactc, whch would reveal ther preferences about the hghest prorty preferences. In addton, HCV assumes that t s possble to herarchcally dvde the obects of nterest nto dfferent levels, but does not contan any mechansm for dong so (Karlsson, Olsson, & Ryan, 997). If we would lke to extend HCV for both condtonal and uncondtonal stuatons, the frst soluton would be to dvde condtonal and uncondtonal optons n two dfferent groups. Although n such a case t may happen that we have only one uncondtonal opton (or even none) and a large number of condtonal optons whch means that we have a new problem of properly dvdng the optons nto groups. On the contrary, f all of the optons are n one block, there s a problem of compensaton whch s well known for ths method. 8. Condtonal Preferences Numerous studes have specfcally examned condtonal preferences and dfferent sorts of preferences wth condtonalty. Often, they defne the structure as networks and graphs for representng condtonalty n preferences, and the best well-known are CP-nets/TCP-nets. The Condtonal Preference Network (CP-net) (Boutler, Brafman, Domshlak, Hoos, & Poole, 004) s a formalsm for compactly expressng condtonal preferences n multvarate problems. It s a qualtatve graphcal representaton of preferences that reflects condtonal dependence and ndependence of preference statements under a ceters parbus (all else beng equal) nterpretaton. One of nce propertes of the CP-net model s that determnng the optmal outcome s straghtforward and can be done n the lnear tme wth respect to the number of varables by a smple topologcal order of a gven network (Domshlak & Brafman, 00). The stuaton wth domnance testng s not as sharp (Domshlak, Hullermeer, Kac, & Prade, 0). However, there are some studes that address certan topologes of CP-net and that optmze domnance testng query, but n general, t s NP-hard (Goldsmth, Lang, Truszczyńsk, & Wlson, 008). Tradeoffs-enhanced CP-nets (TCP-nets) (Brafman & Domshlak, 00) capture nformaton about condtonal ndependence and nformaton about condtonal relatve mportance. Thus, they provde a rcher framework for representng user preferences, allowng stronger conclusons to be drawn, yet reman commtted to the use of ntutve, qualtatve nformaton as ther source. In case of a condtonally acyclc TCP-net, the formalsm keeps the lnear complexty tme for determnng an optmal outcome. Smlar computatonal analyss for precse complexty of domnance testng n TCP-nets s stll an open theoretcal queston, to the best of our knowledge. Condtonal Outcome Preference Network (COP-network) (Chen, Buffett, & Flemng, 007) s one more example of users preferences presentaton wth drected graphs. Ths formalsm models preferences drectly elcted from a user and then extends to ndcate all preferences that can be nferred as a result. In addton to other methods, ths method develops a utlty functon to predct all known utltes and can be used to determne quckly whether one outcome s preferred over another one. Snce a COP-net contans a node for every possble outcome, run-tme for buldng and traversng the tree s very expensve n the worst case. But, n comparson to other technques, whch only handle preferences specfed over values for a partcular attrbute, COP-net can nfer preferences over outcomes when they are specfed for values across attrbutes as well. 7

29 CP-nets/TCP-nets are general frameworks for addressng condtonalty n preferences, but they have a weak performance for domnant queres and focus on acyclcty, whch s a strong restrcton, lmtng ther potental applcablty (Wlson, 0). A few extensons have recently been proposed n dfferent drectons. Recently an extenson nto a more general cp-theory n (Wlson, 0) ntroduces a new formalsm, whch can be vewed as a smple logc of condtonal preferences. It s shown that gven semantcs can represent a stronger knd of preference statements, whch can be used, for example, to construct a lexcographc order on outcomes, whch s not expressble wthn the formalsms of CP-nets and TCP-nets. Cp-theores consder weaker forms of acyclcty, whch are suffcent condtons for the cp-theory to be consstent. The work on cp-theores presented n (Wlson, 0) dd not addressed the mportant (but very hard, as shown for CP-nets and TCP-nets n (Goldsmth, Lang, Truszczyńsk, & Wlson, 008; Brafman & Domshlak, 00; Boutler, Brafman, Domshlak, Hoos, & Poole, 004) problem of domnance testng. There s another varant of TCP-nets, known as UCP-nets (Boutler, Bacchus, & Brafman, 00) that capture quanttatve preferences and relatve mportance nformaton usng utlty functons. They combne the theory of CP-nets and GAI-nets (Generalzed Addtve Decomposable Utlty Functons) (Gonzales & Perny, 004). By extendng CP-nets wth quanttatve utlty nformaton, the expressve power s enhanced and domnance queres become computatonally effcent. On the other hand, UCP-net shave a lmtaton, as they do not make any assumpton on the knd of nteractons between attrbutes that need to be prortzed. There s a work dfferent to the one proposed n (McGeache & Doyle, 0), where the TCPnets theory s analyzed as a model for representng and reasonng over quanttatve preferences (Mukhtar, Belaïd, & Bernard, 009). Preferences and constrants are specfed qualtatvely and then mapped onto a quanttatve utlty model. The user preferences are evaluated aganst avalable optons by buldng a preference tree contanng optons propertes and user preference values for them. Consequently, domnance query mght be addressed n polynomal tme, but an optmzaton query addtonally ncludes constructon of a preference tree wth all possble combnaton of optons. Addtonally, despte the advantages of TCP-nets, they do not allow complete quantfcaton of relatve mportance over attrbutes. The most related to our work s a recent extenson of CP-nets wth quanttatve trade-off statements (McGeache & Doyle, 0). The extenson s made by concepts from elementary geometry and usage of addtve lnear value functon whch corresponds to users preference relatons (Brafman, Domshlak, & Kogan, 004) and representaton of tradeoff statements as constrants on the partal dervatves of the value functon. They have demonstrated that for each acyclc CP-net an addtve value functon can represent all forms of preferences (Brafman, Domshlak, & Kogan, 004). Unfortunately, ths extenson also suffers of cycles nduced by condtonally defned preferences as CP-nets/TCP-nets. Furthermore, prorty statements specfc for TCP-nets are not completely addressed and that novel representaton rases many new questons for further research, especally about nteracton wth ceters parbus preference statements (McGeache & Doyle, 0). Ths method proposes the transformaton of a set of preferences nto a system of lnear nequaltes, and n general, that system does not have a soluton, whch s an addtonal complexty to that CP-nets extenson approach. 8.3 Semantcs of (Condtonal) Preferences There s one more topc of nterest related to our work: semantcs of stakeholders preferences wth specal emphass on condtonal preferences. Snce condtonal sentences n natural lan- 8

30 guage have several nterpretatons (Yu, Yu, Zhou, & Nakamu, 00), a statement of the condtonal preferences may also be understood n dfferent ways by dfferent users. Addressng condtonal preferences wth rgorous mathematcal theores has several lmtatons for the applcaton n real world. Frst, users need to specfy too many statements to draw a complete pcture of preferences n a doman. Second, users should understand the statements n the way they are defned n order to be adopted (Yu, Yu, Zhou, & Nakamu, 00). In (Yu, Yu, Zhou, & Nakamu, 00), the authors have proposed an ontology-based quanttatve model for condtonal preferences wth specal ssues on dfferent nterpretatons of the condtonal preference statements such as the herarchcal relatonshps of the concepts n the ontology (Langley, 995), the connotaton of suffcent and necessary condtons (Brennan, 003), and the bpolar property (Benferhat, Dubos, Kac, & Prade, 006) of preferences n human thnkng. Ther expermental results show that t s possble to nfer all ratngs from a few rules, thus lessenng users workload. The nhertance property, whch clams users preference for an ancestor tem can be nherted by ts descendant tems, s also proved. 8.4 Summary The characterstcs of the mentoned approaches are summarzed n Table 0. For each approach, we have lsted the scale of nput and output data, f t supports condtonalty n preferences or any other specfc knd of preferences, and complexty for dfferent reasonng queres. It s clear that a unque representatonal and reasonng technque whch can effectvely address all dfferent knd of preferences and reasonng queres does not exst. Dependng on the characterstcs of a consderng problem n an approprate feld, the most approprate method should be chosen. In ths paper, we recognze the problem of quanttatve orderng between optons based on dfferent knds of condtonally and uncondtonally defned preferences whch cannot be completely addressed by exstng methods, to the best of our knowledge. We decded to use the wdely accepted and adopted AHP technque and extend t for dfferent knds of preferences. Our use of the two-level herarchy has several explanatons, such as possblty of weak reductons of the number of comparsons (shown n (Bagher, Asad, Gasevc, & Soltan, 00)), suffcent expressveness (accordng to (Yu, Yu, Zhou, & Nakamu, 00)) and analogous to the concept of attrbutes n all developed technques for addressng condtonal preferences. We prevously showed ts effectve usage for confguraton of feature models, whch are commonly used for modelng varablty n software product lnes (SPLs) (Ognanovc, Gasevc, Bagher, & Asad, 0). 9. CONCLUSION AND FUTURE WORK In ths paper, we nvestgated how the well-known AHP technque mght be extended for reasonng over dfferent sorts of condtonal preferences n a two-layered structure. The proposed CS- AHP approach satsfes the preferences of an effectve prortzaton technque based on the challenges and characterstcs that have been ntroduced n (McManus, 004; Sommervlle & Sawyer, 997; Yu & Reff-Marganec, 008): - Stakeholders are able to defne condtonal and uncondtonal preferences about the avalable concerns and ts qualfer tags. Also, they mght defne preferences about domnant relatve mportance as a specal form of preferences. Compared wth the S-AHP method that does not support condtonal preferences, concerns and ther qualfer tags do not have statc fnal ranks because they depend on optons whch mght ether satsfy the condtons defned by stakeholder s preferences or not (Ognanovc, Gasevc, Bagher, & Asad, 0). 9

31 - The actvtes wthn CS-AHP are easy to perform and are based on a smple par-wse comparson method. It can be nexpensvely mplemented n a spreadsheet program such as MS Excel wth addtonal usage of any program for checkng satsfacton of condtons n condtonal defned preferences. - Durng the whole process, n each step, t mght be checked whether an nconsstency exsts n the preferences of the stakeholders, and f so, stakeholders nterventon mght be necessary (Ognanovc, Gasevc, Bagher, & Asad, 0). - We decded to have a local calculaton of ranks, as t dsables any cycles and dependency n the processng of preferences. In comparson to all other technques for addressng condtonal preferences, only CS-AHP does not suffer any knd of cyclcty n preferences. - Non-well-formedness recognzed n ths paper represents a logcal nconsstency n defnng multple condtonal preferences; consequently, such nconsstency s an obstacle for any reasonng technque. CS-AHP enables for checkng non-well-formedness n preferences n each step of prortzaton and n that sense represents a unque framework, whch also gves the estmaton of expected values of the number of non-well-formedness for an approprate model. - Tme-complexty for domnance testng and orderng queres are polynomal whch makes CS- AHP an effectve technque for quanttatve orderng of sets of optons based on dfferent sorts of condtonal and uncondtonal preferences. Possbltes of defnng cycles n dependences, multple and ncomplete preferences represent addtonal benefcal characterstcs of our proposed technque. Also, the extensve usage of AHP n many mportant decson makng domans such as forecastng, total qualty management, busness process re-engneerng, qualty functon deployment, and the balanced scorecard (Forman & Gass, 00; Saaty, 980) can be effectvely extended wth a wde set of condtonal preferences over the two-layered structure. Addtonal two characterstcs of the AHP method, not addressed n ths paper, are: ()group decson makng (Roper-Low, 990; Saaty, 980; Forman & Gass, 00) AHP consders two dfferent approaches: aggregaton of ndvdual udgments (AIJ) and aggregaton of ndvdual prortes (AIP) (Escobar, Aguaron, & Moreno-Jmenez, 004; Aguaron & Moreno-Jmenez, 003); ()analytcal measure to evaluate the nconsstency of the decson maker when elctng the udgments, called Consstency Rato (CR) (Saaty, 003). We beleve that future research needs to be undertaken for CS-AHP n order to: ()nclude preferences from dfferent stakeholders wth dfferent nterests and vews, and ()extend the exstng consstency ndex and nclude the measurement of non-well-formedness ntroduced n ths paper. In the followng we lst some of the maor drectons for future work: - As the man characterstc of the standard AHP algorthm s that a user needs to specfy too many statements to draw a complete pcture of preferences (Hsu & Wang, 0), t s hard to expect to make any optmzaton n that sense. On the other hand, as two alternatves mght be annotated wth the same subset of qualfer tags wth dfferences on the others, the optmzaton mght be n reducton of checkng preferences related to that ntersecton set. Also, another drecton s n pre-recognton of nconsstences n condtonally defned preferences. In a general case, t s a SAT problem, but the research problem s w.r.t. the recognton of stuatons where optmzaton can be done. - The language of condtonal preferences presented n ths paper only allows preferences of a sngle varable (condtonal on other varables); some natural preference statements nvolve preferences over more than one varable, so t would be desrable to consder more general languages (Lang, 004; McGeache & Doyle, 004; Wlson, 0; Mouhoub & Sukpan, 006). 30

32 APPENDIX A Proof (Lemma). Accordng to the defnton of r D () as an ndex of an approprate qualfer tag n ncreasngly ordered array of qualfer tags of a concern wth domnant relatve mportance, the followng nequalty holds: k l = u k ( QT + ) QT + = ( QT + ) k k l s k b + urd ( qt ) r({ qt,..., qt } \{ qt,..., qt }) k l, l s l = u = l + = u where the last equalty s easly shown wth mathematcal nducton. Proof (Proposton ). The proof of ths proposton should be performed n two drectons. In the frst drecton, n order to prove that, f optons o and o satsfy one of the condtons (c)-(c3) of Defnton 7, then t mples that F(o )>F(o ), we use mathematcal nducton per the number of concerns wth domnant relatve mportance. Frst, let us suppose that only concern c has the domnant relatve mportance over the others, k and two arbtrary optons o, annotated wth { qt,..., qt }, k m and o, annotated wth { qt,..., qt l l }, l msatsfy one of condtons (c)-(c3) of Defnton 7. Let us suppose that condton (c) s satsfed, e.g., an opton o s annotated wth the qualfer tag qt of concern c (.e. r ( qt ) ). Then, the followng holds: k D k k l F ({ qt,..., qt }) = r ( qt ) + r({ qt,..., qt }/{ qt }) > r( qt,..., qt ) = F( qt D k k l,..., qt In case of condton (c) where optons o and o are not annotated wth qualfer tag of concern c and opton o s hgher ranked then o accordng to other concerns, t s obvous that the requred nequalty holds. Also, n case of condton (c3.a)of Defnton 7 where optons o and o are annotated wth the same qualfer tag qt of concern c, t s obvous that requred nequalty k l holds ff r({ qt,..., qt } \{ qt }) > r({ qt,..., qt } \{ qt }),.e., ff the opton o k s hgher ranked l then o accordng to the other concerns. Now, let us suppose that condton (c3.b) of Defnton 7 s satsfed,.e., optons o and o are annotated respectvely wth the qualfer tags qt and qt of concern c, where qualfer tag qt s hgher ranked than qt. We have that the followng holds: l l ). 3

33 F( qt = F ( qt,..., qt k k,..., qt ) = r l l ) D ( qt ) + r({ qt,..., qt k k } \ { qt }) r D ( qt ) + r D ( qt ) + r({ qt,..., qt l l } \ { qt. Now, f we assume that asserton holds for each number of concerns whch s less then k, let us D ( k D ( k D ( D ( k prove that t holds n case of k concerns wth domnant relatve mportance c,..., c ). Accordng to Defnton 8, c,..., c ) holds ff c ) and c,..., c ). We have F( qt = b r,..., qt D ( qt k l k ) = bl r l= ) + F({ qt D ( qt l,..., qt l k l ) + r({ qt } \ { qt }),..., qt k l } \ { qt,..., qt If any of condtons (c), (c) or (c3a) of Defnton 7 s satsfed, t s, by analogy to the prevous, easy to prove the requred nequalty. Now, let us consder that optons o and o are annotated respectvely wth the qualfer tags qt and qt of concern c, where qualfer tag qt s hgher ranked than qt. We have that the followng nequaltes hold as drect mplcaton of the proposed Lemma: F( qt b,..., qt k ( rd ( qt ) + ) k ) = b r D ( qt + F({ qt ) + F({ qt,..., qt k k } \ { qt,..., qt k k }) > b r } \ { qt D ( qt }) ) + F({ qt,..., qt l l } \ { qt k k }) }) = F( qt,..., qt In the opposte drecton, n order to prove that nequalty F(o )>F(o ) mples satsfacton of one of the condtons (c)-(c3), smple reducton to the absurd and prevous drecton mght be appled. APPENDIX B Table B.. Expected values (Mean value, Std dev) of the number of volatons of the non-well-formedness condtons for the level of concerns Table B.. Expected values (Mean value, Std dev) of the number of volatons of the non-well-formedness condtons for the level of qualfer tags l l ) }) 3

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38 LIST OF FIGURES Fgure. Electronc payments servces annotated wth qualfer tags. Fgure. Schematc representaton of lexcographcal order over two-layered structure of concerns and qualfer tags. Fgure 3. CS-AHP extenson of the Feature Plug-n: a) Edtng concerns and assocatng wth features; b) Specfyng condtonal preferences on the level of concerns Fgure 4. Dummy codng for the number of condtonal pars of concerns on the level of concerns 37

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