IBM Research Report. Winner Determination in Multi-Attribute Auctions

Transcription

1 RC22478 (W ) June 6, 2002 Computer Scence IBM Research Report Wnner Determnaton n Mult-Attrbute Auctons Martn Bchler, Jayant R. Kalagnanam IBM Research Dvson Thomas J. Watson Research Center P.O. Box 218 Yorktown Heghts, NY Research Dvson Almaden - Austn - Bejng - Delh - Hafa - Inda - T. J. Watson - Tokyo - Zurch LIMITED DISTRIBUTION NOTICE: Ths report has been submtted for publcaton outsde of IBM and wll probably be copyrghted f accepted for publcaton. It has been ssued as a Research Report for early dssemnaton of ts contents. In vew of the transfer of copyrght to the outsde publsher, ts dstrbuton outsde of IBM pror to publcaton should be lmted to peer communcatons and specfc requests. After outsde publcaton, requests should be flled only by reprnts or legally obtaned copes of the artcle (e.g., payment of royaltes). Copes may be requested from IBM T. J. Watson Research Center, P. O. Box 218, Yorktown Heghts, NY USA (emal: reports@us.bm.com). Some reports are avalable on the nternet at

2 Martn Bchler and Jayant Kalagnanam IBM T. J. Watson Research Center P.O. Box 218, Yorktown Heghts, NY {bchler, ABSTRACT The theory of procurement auctons tradtonally assumes that the offered quantty and qualty s fxed pror to source selecton. Mult-attrbute reverse auctons allow negotaton over prce and qualtatve attrbutes such as color, weght, or delvery tme. They promse hgher market effcency through a more effectve nformaton exchange of buyer s preferences and suppler s offerngs. Ths paper focuses on a number of wnner determnaton problems n mult-attrbute auctons. Prevous work assumes that multattrbute bds are descrbed as attrbute value pars and that the entre demand s purchased from a sngle suppler. Our contrbuton s twofold: Frst, we wll analyze the wnner determnaton problem n case of multple sourcng. Second, we wll extend the concept of mult-attrbute auctons to allow for confgurable offers. Confgurable offers enable supplers to specfy multple values and prce markups for each attrbute. In addton, supplers can defne confguraton and dscount rules n form of propostonal logc statements. These extensons provde supplers wth more flexblty n the specfcaton of ther bds and allow for an effcent nformaton exchange among market partcpants. We wll present MIP formulatons for the resultng allocaton problems and an mplementaton.

3 1 INTRODUCTION Procurement auctons usually requre the bd to specfy several characterstcs of the contract to be fulflled. Prevous models of procurement auctons have generally assumed that the qualtatve attrbutes are fxed pror to compettve source selecton - hence bddng competton s restrcted to the prce dmenson (see [1] or [2]). Whle such an approach may be approprate for auctons of homogeneous goods, most procurement ncludes heterogeneous offerngs of supplers. Tradtonally, these types of negotatons are resolved through blateral barganng or sealed-bd tenders, where a buyer asks for bds n unstructured or sem-structured format and then the buyer selects one or more of these bds manually. A tenderng procedure allows the sale to be determned by a varety of attrbutes nvolvng not only prce but qualty, lead tme, contract terms, and suppler reputaton. Recently, mult-attrbute reverse auctons have become a popular means of automatng ths process further. The negotable attrbutes are defned n advance, and supplers can compete ether n an open-cry or sealed-bd fashon on multple attrbutes. Ths process allows more degrees of freedom for supplers n formulatng ther bds, whle at the same tme t leverages the compettve forces of an aucton to drve the negotaton to an equlbrum. Expected gans of mult-attrbute auctons are ncreased speed of the negotaton, hgher market transparency, as well as hgher degrees of allocatve effcency. Although the lterature n ths feld s farly young, a number of procurement departments and software vendors have embraced the dea 1. Companes such as ebrevate or PurchasePro have mplemented what s also called the total cost approach. Here buyers specfy monetary values (dscounts and/or mark-ups) for attrbute values, n order to be able to compare dfferent offerngs. Another approach uses decson analyss technques [3] to assgn weghts and ndvdual value functons to the relevant attrbutes, and calculate a value score. Bdders can then compete on ths value score by mprovng one or more of the attrbutes. Ths approach s used by software vendors such as Clarus, IBM/DgtalUnon, Moa, Menerva, and Perfect. TIScover, an Austran destnaton management system, s usng ths type of mult-attrbute aucton to match toursts and hotelers on an accommodaton market [4]. Most of these software packages have been developed durng the past three years. The exstng game-theoretc lterature [5-7] typcally assumes quas-lnearty of buyers scorng functons as well as supplers cost functons to analyze the strategc ssues as well as effce ncy of mult-attrbute auctons. Ths generc format covers a broad varety of functonal forms ncludng lnear addtve functons, whch have found ts way nto most commercal packages as a technque for modelng multattrbute scorng functons. Two fundamental assumptons underle the treatment of mult-attrbute bds n the lterature and n commercal software packages: 1. The bds are pont bds and are specfed as attrbute value pars for each of the attrbutes, and 2. The mult-attrbute auctons usually assume that the contracts are awarded to a sngle bd we call ths the sole sourng assumpton. The restrcton to such smple mult-attrbute bds n prevous approaches s due to techncal dffcultes of specfyng more complex offerngs. The focus on sole sourcng s based on the argument that multattrbute auctons are manly used for contracts wth hgh asset-specfcty, such as Department of Defense (DoD) contracts, whch were also the focus of ntal economc analyss of mult-attrbute auctons [5]. Wth the wde-spread use of mult-attrbute auctons n part because of readly avalable software to support such formats, companes are usng mult-attrbute auctons also for the procurement of large 1 Often also called mutvarate RFQ or multdmensonal aucton. IBM T. J. Watson Research Center, NY, USA 2

4 amounts of less specfc goods (.e. MRO procurement) where multple sourcng becomes mportant. For example, n a recent procurement aucton run by nternal procurement at IBM for a large quantty of chars for one of ther offce buldngs multple sourcng was entertaned provded some condtons across the supply pool was satsfed. In ths paper we examne two extensons to the current formats used for mult-attrbute auctons. Frst, we wll analyze the wnner determnaton problem n case of multple sourcng. In ths settng we examne the mpact of several busness rules that need to be mposed on the wnner determnaton problem n order to obtan an acceptable supply from multple supplers. Second, we wll extend mult-attrbute auctons to allow for confgurablty n bds. In contrast to tradtonal mult-attrbute offers, confgurable offers enable supplers to specfy multple values and prce markups for each attrbute. The faclty of provdng confgurable offers ntroduces a problem of nformatonal complext y snce the prce functon (over dscrete attrbutes) now needs to be specfed over an exponental number of combnatons. In ths paper, we restrct our attenton to a specal case where the prce dependence on a attrbute s specfed as a markup over a base prce thereby restrctng the prce functon to an addtve form. Ths appears to be suffcent for many real world settngs such as PC, logstcs etc. In addton, we allow supplers to specfy constrants that restrct the set of avalable conf guratons or alternately allow the specfcaton of dscounts based on levels of multple attrbutes. In general, n practce we encounter only a small number of such hgher order terms and hence they can be managed qute effectvely. Moreover, such confguraton rules and dscounts can be adequately represented usng propostonal logc. The advantage of such a restrcted representaton for confgurable bds s that propostonal logc can also be represented by lnear nequaltes whch can then be added to the wnner determnaton problems. The extenson of these mult-attrbute allocaton problems has been motvated by our work on a large-scale procurement marketplace for the retal ndustry and our experence wth nternal procurement auctons. In the next secton we wll provde a bref ntroducton nto the lterature of mult-attrbute auctons. In secton 2 we wll descrbe the standard bd evaluaton technque n mult-attrbute auctons, namely the addtve scorng functon and elaborate on the ssue of preference elctaton n mult-attrbute aucton. In secton 4 we wll analyze mult-attrbute wnner determnaton n the presence of multple sourcng. Secton 5 wll ntroduce the concept of confgurable offers and formulate the assocated allocaton problems. Fnally, n secton 6 we wll summarze the artcle and provde an overvew of a Java object framework mplementng a varety of wnner determnaton algorthms for electronc markets. The framework s currently used n a commercal mplementaton and can be used as a standard solver component n electronc procurement applcatons. 2 REVIEW OF THE LITERATURE Only a small but steadly growng number of academc papers have consdered mult-attrbute auctons so far. A thorough analyss of the desgn of mult-attrbute auctons has been provded by Che [5]. He derved a two-dmensonal verson of the revenue equvalence theorem [8]. Che also desgns an optmal scorng rule based on the assumpton that the buyer knows the probablty dstrbuton of the suppler s cost parameter. Branco s analyss s based on Che s ndependent cost model and derves an optmal aucton mechansm for the case when the bddng frms costs are correlated [6]. Bchler [9] and Bchler and Kaukal [10] show some frst Internet-based mplementatons of the concept and dscuss MAUT as an algorthm for bd evaluaton n sngle -sourcng, mult-attrbute reverse auctons. A varety of dfferent multple ssue aucton algorthms are suggested by Tech, Wallenus and Wallenus [11]. Mult-attrbute Englsh auctons have also been analyzed n the context of servce allocaton amongst artfcal agents [13]. IBM T. J. Watson Research Center, NY, USA 3

5 One drecton n aucton desgn s concerned wth effcent mechansms. Here, the prmary objectve of the planner s to maxmze allocatve effcency. Mlgrom provdes proofs that allocatve effcency s acheved n a sngle-sourcng mult-attrbute aucton f the auctoneer announces hs true utlty functon as the scorng rule, and conducts a Vckrey aucton based on the resultng scores [7]. Some frst laboratory experments on the effcency of mult-attrbute auctons have been descrbed n [14]. In these experments and the effcency of prce-only and mult-attrbute auctons dd not show sgnfcant dfferences. In addton, the value scores n mult-attrbute auctons were sgnfcantly hgher than n prce-only auctons. In recent paper, Parkes and Kalagnanam [15] provde an teratve mult-attrbute aucton desgn based on a prmal-dual algorthm. They show that ths desgn s ncentve compatble for the sellers and Pareto effcent wth truthful buyers. Whereas effcent aucton desgn s not concerned wth how the surplus n an aucton s dvded among the bdders and the auctoneer, optmal aucton desgn concentrates on auctons, whch maxmze the expected revenue of the seller. In mult-attrbute auctons t s approprate to speak about buyer utlty maxmzaton nstead of revenue-maxmzaton. As already suggested n Che s analyss, the optmal scorng rule n a mult-attrbute reverse aucton may not be dentcal to the buyer s true value functon. Bel and Wen [17] focus on buyer utlty-maxmzaton n mult-attrbute auctons,.e. optmal aucton desgn. The paper suggests an nverse-optmzaton based approach that allows the buyer va several changes n the announced scorng rule, to learn the supplers cost functons and then determne a scorng rule that maxmzes the buyer s utlty wthn an open-ascendng aucton format. In ths paper we focus on wnner determnaton problems n mult-attrbute auctons wth multple sourcng and confgurable offers. We do not make statements on the optmalty of a scorng functon, ncentve compatblty of the payment schema or other mechansm desgn ssues, but take the scorng functon as gven by the buyer and focus on the resultng allocaton problems. The proposed optmzaton models are desgned to maxmze the buyer s utlty. 3 SOLE SOURCING A popular approach to mplement mult-attrbute auctons s based on tradtonal decson analyss technques. Here bdders submt bds as attrbute-value pars, whch are evaluated by a value or scorng functon provded by the buyer. In ths secton we wll brefly descrbe ths mult-attrbute aucton format and dscuss some of the lmtatons of ths approach. We frst restrct our attenton to the sngle sourcng problem, where the entre demand s purchased from a sngle suppler. 3.1 The Standard Addtve Scorng Functon In practcal mplementatons, the elctaton of a buyer s preferences, and consequently the constructon of an approprate scorng functon s of pvotal mportance. A common approach s based on the use of establshed decson analyss technques, such as MAUT [3], SMART [18] or AHP [19]. Although advanced versons of MAUT and AHP can model nteractons among attrbutes, the basc technques use a lnear, weghted value functon, whch assumes preferental ndependence of all attrbutes. An attrbute x s sad to be preferentally ndependent of y f preferences for specfc outcomes of x do not depend on the value of attrbute y [20]. We next ntroduce some termnology and notaton. Consder I bds (or offers) and J attrbutes. Each attrbute j J has an attrbute space of = j. A bd, receved by the buyer, can then be descrbed as a vector x. = (x 1,, x J ) where x j s the level of attrbute j. The prce p s consdered as one of the attrbute values x j. In the case of an addtve scorng functon S(x ) the buyer evaluates each relevant attrbute x j through a IBM T. J. Watson Research Center, NY, USA 4

6 scorng functon S j (x j ). The overall value S(x ) for a bd x s gven by the sum of all ndvdual scorngs of the attrbutes. It s convenent to scale each of the sngle -attrbute utlty functons S j from zero to one. That s, for a bd x and a scorng functon that has weghts w 1... w J, the overall utlty for a bd s gven by S = S( x ) = wj S j( xj) and wj = 1 j J j J (1) The problem a buyer faces s to determne approprate S j functons and w j weghts. An optmal aucton s allocatng the deal to the supplers n a way that maxmzes the utlty for the buyer. The functon max S wth 1 I provdes the utlty score of the wnnng bd. Bds can be collected through open-cry or sealed-bd aucton schemes. 3.2 Preference Elctaton The assessment of approprate weghts w j s key to MAUT and s an mportant aspect of a good scorng functon n the mult-attrbute aucton. Several technques have been proposed n the tradtonal decson analyss lterature to help users assgn reasonable weghts. One approach s called prcng out because t nvolves determnng the value of one objectve n terms of another (e.g. dolla rs). For example, one mght say that 5 days faster delvery tme s worth $400. The dea s to fnd the ndfference pont,.e. determne the margnal rate of substtuton between two attrbutes. Although ths concept seems straghtforward, t can be a dffcult assessment to make. Snce many decson makers feel unable to provde exact weghts, some of the more recent approaches only ask for uncertan estmates. For example, methods from fuzzy decson analyss use fuzzy sets for weghts and ndvdual scorng functons and fuzzy operators for the aggregaton of those fuzzy sets [21]. AHP uses a dfferent approach to weght determnaton. A prncple used n AHP s that comparatve judgments are appled to construct a symmetrc matrx of par-wse comparsons of all combnatons of attrbutes. The method s based on the mathematcal structure of consstent matrces and ther assocated rght-egenvector s ablty to generate true or approxmate weghts. The rght egenvector of the matrx results n the weghts for the dfferent objectves. More recent approaches try to estmate the buyers preferences based on comparsons of alternatves [22, 23]. These technques assume weaker decson makers and do not ask for attrbute-level utlty assessments. The prevous dscusson assumes lnearty of the buyer s value functon. Interacton effects among attrbutes although relevant n many consderatons are often neglected n real-world mplementatons. Preferental dependences mpact the shape of the utlty functon and requre the modelng of non-lnear utlty functons. Two attrbutes may to some extent be substtutes or may complement each another. In certan applcatons, one mght even argue that prce s preferentally dependent on qualtatve attrbutes. For example, havng the choce between a luxury and low-budget cars, the mportance of prce mght depend on the type of the car evaluated. In order to express these nterdependences among two attrbutes, an addtve utlty functon can be extended towards a so-called multlnear expresson [3]. In general, however, the noton of nteracton among attrbutes s one of the most dffcult concepts n mult-crtera decson makng and s certanly an mportant ssue to consder n the desgn of mult-attrbute auctons. 4 MULTIPLE SOURCING Intally mult-attrbute auctons have been analyzed n the context of DoD purchasng actvtes wth hgh asset specfcty, such as weapon systems. Wth ther more wde-spread use companes start usng mult- IBM T. J. Watson Research Center, NY, USA 5

7 attrbute auctons even for less specfc products such as MRO equpment, where qualty plays a role. In these stuatons, buyers need to buy larger quanttes and are wllng to purchase from multple supplers. 4.1 Allocaton to Multple Supplers A computatonally smple case assumes submsson of dvsble bds,.e. that the bdders are wllng to accept partal quanttes of ther bds for the same unt prce. Smlar to secton 2, the auctoneer can use a scorng functon to sort the bds by descendng score per unt. Snce bds can be dvded nto smaller quanttes wnnng bds are the ones where Σ I q D wth q beng the quantty provded n bd and D beng the demand. However, ths s only applcable n a lmted number of real-world scenaros. A more realstc assumpton s ndvsble bds. We consder a computer vendor who wants to buy 2000 hard dsks on a prvate exchange. Suppose we have decreasng average producton costs,.e. supplers encounter economes of scale and the unt prce of a hard dsk s bound to the quantty sold. Agan, the auctoneer tres to satsfy the buyer s quantty at the lowest cost. The ndvsblty assumpton turns the problem nto a computatonally hard problem, whch cannot be solved by sortng of bd scores. To llustrate ths consder the followng smple example wth 4 bds and a demand of 40. Bd No. Quantty Score/Unt Overall Score Table 1: Example wth Indvsble Bds There are several possble solutons sets to satsfy the buyer s demand: {2}, {1, 3}, or {1, 4}. The optmal soluton (overall score = 1030) s provded by bd set {1, 3}. Selectng the optmal set of bds s related to the well known 0-1 knapsack problem, whch s known to be NP-hard, but can be solved effcently n practce by usng dynamc programmng [24]. Snce, n practce, the combnatons hardly ever sum up to exactly the demand specfed by the buyer, we use the followng nteger program (IP) formulaton wth acceptable lower and upper bounds for the demand (D mn and D max ). subject to max D l (q S ) x l mn l l D max l l l q x (3) ql pl xl C (4) l xl 1 l L (5) x l { 0,1} l L, The objectve n ths optmzaton s to maxmze the overall score (2), where S l s the unt score of a bd from suppler l and q l s the quantty of bd, so that the suppled quantty satsfes the lower and upper bound for the demand (3). The overall reservaton prce C s consdered n (4) where p l s the unt prce of (2) (6) IBM T. J. Watson Research Center, NY, USA 6

8 bd from suppler l. Constrant (5) ensures that only one of the bds of a suppler s selected. We ntroduce the bnary decson varables x l to ndcate the bds selected by the buyer (6). In real-world settngs there are several consderatons besdes cost mnmzaton. These consderatons are specfed as a set of constrants that need to be satsfed whle selectng a set of wnnng bds. We dscuss two such rules, whch have shown to be relevant n practcal applcatons. 4.2 Number of Wnnng Bdders An mportant multple sourcng consderaton s the number of wnners. On the one hand, buyers want to make sure that the entre supply s not sourced from too few supplers, snce ths creates a hgh exposure f some of them are not able to delver on ther promse. On the other hand, havng too many supplers creates a hgh overhead cost n terms of managng a large number of suppler relatonshps. These consderatons ntroduce constrants on the mnmum, L mn, and maxmum, L max, number of wnnng supplers n the soluton to the wnner determnaton problem. 0.1 yl xl ε yl (7) L y l y (8) mn l L max l { 0, } l L 1 We ntroduce an ndcator varable y l for each suppler l, whch takes the value 1 f the suppler has any wnnng bds and 0 otherwse. The frst constrant (7) sets y l to 1 f suppler l2l has any wnnng bds. Note that the constant multpler C, the number of a suppler s bds, ensures that the rght hand sde s large enough when more than one bd of suppler l s selected. 4.3 Homogenety of the Purchase A basc problem, whch arses from mult-attrbute auctons n the case of multple sourcng s the heterogenety of goods purchased from dfferent supplers. The cost mnmzng soluton mght be one where the bds of all wnnng supplers have dfferent values n all attrbutes. Whle ths s not necessarly a problem for all attrbutes, t can be mportant n certan applcatons to enforce homogenety of a certan attrbute n the set of wnnng bds. In order to capture such constrants we ntroduce an ndcator varable z that assumes the value 1 f any supplers are chosen wth a bd at level k for attrbute j. T kj s defned as the set of bds at level K for attrbute J. Ths s formalzed as follows: 0.1z x, l T, l z 1 j J k z T z { 0,1} j J k K, j J Notce that ths constrant s for a gven attrbute of nterest (such as color) at some level (say red). For each attrbute, ths formalsm ntroduces as many constrants as there are levels and one extra constrant to enforce homogenety. (9) (10) (11) (12) IBM T. J. Watson Research Center, NY, USA 7

9 4.4 Computatonal Issues WINNER DETERMINATION IN MULTI-ATTRIBUTE AUCTIONS A dynamc programmng approach can be used to solve the above problems. We found that commercal nteger programmng software usng a branch-and-bound approach was able to solve problems of 200 bds and 20 attrbutes on the order of a few seconds. As expected, the consderaton of addtonal constrants descrbed n secton 4.2 and 4.3 mpacts the runtme of the program. Fgure 1 presents CPU tme results (n seconds) for solvng a randomly generated nstance of the multattrbute aucton. Ths problem had fxed problem sze of 10 attrbutes and 100 bds. We set the mnmum number of wnnng supplers to be equal to the maxmum number of wnnng supplers, whch we vared between 1 and 20. The optmal allocaton for ths problem wthout any sde constrants on the number of wnnng supplers had fve wnnng supplers. For a very large number of wnnng supplers the problem becomes easer to solve, snce t may not be possble to fnd an allocaton whch satsfes the total demand of the buyer. A constrant on only the maxmum number of wnners dd not have a sgnfcant mpact on the runtme. So the major nfluencng factor for the exponental runtme was the constrant on the mnmum number of wnners CPU tme (mnutes) Mn number of wnners Fgure 1: Expermental results for a mult-attrbute aucton, varyng the number of allowed wnnng supplers constrant Fgure 2 shows the CPU tme results for solvng a problem wth a fxed problem sze of 100 bds, 20 attrbutes and an ncreasng number of homogenety constrants. The mpact of addng these constrants on the runtme s close to lnear. Wth low correlaton of the bd data and a hgh number of possble attrbute levels, ths constrant can often not be satsfed and leads to nfeasblty. We recommend enforcng the homogenety constrant on as few attrbutes as possble n an aucton wth mult-attrbute bds. Homogenety constrants are very applcable to the evaluaton of confgurable offers dscussed n the next secton. IBM T. J. Watson Research Center, NY, USA 8

10 CPU tme (mllseconds) Number of homogeneous attrbutes Fgure 2: Expermental results for a mult-attrbute aucton, varyng the number of homogenety constrants 5 ALLOCATION OF CONFIGURABLE OFFERS A basc assumpton n our prevous dscusson has been that bds are descrbed as sets of attrbute-value pars. In practce, however, many offers are specfed as confguratons, where each attrbute can take a number of dfferent attrbute values. Lets assume a PC has only three attrbutes, namely processor speed, hard dsk sze, and prce. A suppler could specfy that there are three processors avalable {850MHz, 950 MHz, and 1GHz}, as well as two szes of hard dsks {10GB and 15GB}. The base confguraton (850MHz, 10GB) prces for $1000. A confguraton wth a 1GHz computer s $100 more, and one wth a 15GB hard dsk costs an addtonal $200. Most servces such as nsurances or transportaton can be consdered as confgurable offers n a smlar way. The ablty to express optons n mult-attrbute auctons s a crucal feature to further automate negotatons on complex goods and servces. Confgurable mult-attrbute offers exhbt combnatoral features. Wth only 10 attrbutes and 5 possble attrbute values for each of them, there are already 5 10 = 9.7 mllon possble confguratons. Clearly, fndng the best confguraton among all these s not an easy task. Enumeratng all possble confguratons s mostly not a vable alternatve, snce a large amount of ndvdual bds would have to be generated and communcated to the buyer. Therefore, supplers often restrct the number of confguratons they offer to only a small selecton. As a result, buyers mght not fnd the best confguraton and choose the offer of another suppler,.e. the stuaton mght lead to neffcent outcomes. It s n the nterest of both, buyers and supplers, to communcate offers n a compact that cover a large space of possble product confguratons. In the followng we wll propose a procedure to descrbe confgurable offers and determne the best ndvdual confguraton based on a buyers scorng functon. 5.1 Functonal Descrpton of Confgurable Offers There are many possble ways how mult-attrbute offers can be made confgurable. In ths approach we allow supplers to specfy the possble values for each attrbute n an offer n a functonal format. In other words, we descrbe the possble confguratons of a confgurable offer as a functon of prce on quantty and qualtatve attrbutes. Assumng addtvty of the attrbutes, the total prce p for a partcular bd/offer can be wrtten as: p = p( q, x ) = q p ( q ) + q f ( v ) b j j j (13) IBM T. J. Watson Research Center, NY, USA 9

11 where q s quantty, p b (q ) s the base prce per tem as a functon of quantty (.e. specfes a volume dscount), and f j (v j ) s a functonal specfc aton for the mpact of partcular attrbute values v j on the prce of a product. The ndvdual functons p b (q ) and f j (v j, ) can n general be nonlnear. Examples would be dscrete functons, whch specfy prce markups for dfferent types of CPUs (850 MHz, 900 MHz, 1GHz), as well as contnuous functons, whch specfy the mpact of decreasng lead tme on prce. Ths functonal form can be sent to the auctoneer n an XML-based nterchange format. For these purposes, we have desgned CPML, an XML schema to descrbe confgurable mult-attrbute offers. 5.2 Allocaton of Confgurable Offers In ths frst analyss we restrct ourselves to a multple sourcng procedure wth dscrete prce markups, where each attrbute value x of bd has an ndvdual markup m. In addton, each offer specfes a qualty q. The descrpton of a confgurable mult-attrbute offer s now formulated as n (14) wth p beng the unt prce of the confgurable offer. p = p( q, x ) = p b + j k m x Assumng that there are no homogenety constrants, we can solve the wnner determnaton problem n a two-step procedure. In the frst step we select the best possble confguraton for each offer based on the buyer s scorng functon. In a second step, the resultng best confguratons are n the form of conventonal mult-attrbute offers and can be allocated as descrbed n secton (2) (9). The frst step can be modeled as a varaton of the multple-choce knapsack problem [25]. We assocate the bnary decson varable x to each attrbute value k of attrbutes j. The objectve maxmzes the weghted score s for each attrbute value wth weght beng w j. The objectve used n (15) denotes the prce attrbute p of a selected confguraton as an extra varable. Note, that we assume an addtve, quaslnear utlty functon wth a lnear (decreasng) scorng functon on prce, s p (p). Constrant (16) specfes that for each attrbute exactly one value must be selected. Constrant (17) restrcts the unt prce to be smaller or equal to the buyer s unt reservaton prce C. In (18) we determne the value of the contnuous varable p. Ths constrant could also be omtted, so that the prce p needs to determned outsde the optmzaton based on the selected attrbute values. max wj s x + w j k subject to x = 1 j J k j k m x + p b m x p = j x k C p b { 0,1} j J k K, p s p ( p) The procedure allows supplers consderably more flexblty n specfyng offers, whle at the same tme, the bds can be ranked and supplers can compete n an open-cry manner. (14) (15) (16) (17) (18) (19) IBM T. J. Watson Research Center, NY, USA 10

12 5.3 Wnner Determnaton wth Homogenety Constrants The formulatons n sectons 5.2 work only under the assumpton that there are no homogenety constrants on the buyer s sde. Beng able to formulate homogenety constrants s a very useful feature for the evaluaton of confgurable offers, however, at the expense of complexty of the wnner determnaton. The followng formulas (20) (28) show the overall MIP model for the evaluaton of confgurable mult-attrbute offers consderng multple sourcng and homogenety constrants. max q wj sx + wps p mx + wp( s p pb + d) y j k j k x = y j J, k q m x pb y + C (22) j k D L q y (23) mn D max mn y L max 0.1z (24) x, l T z = 1 j J k x y T z j J { 0,1}, j J k K, { 0, } 1 The objectve maxmzes the overall score of the selected confguratons. As n (15) we assume an addtve and quas-lnear scorng functon wth a lnearly decreasng functon on prce. The varable s agan denotes the score for a partcular qualtatve attrbute value, whereas x s a bnary ndcator varable whch ndcates whether a partcular attrbute value has been chosen. The varable s p descrbes the slope of the lnear scorng functon for prce, whereas d s ts ntercept. In (21) we select exactly one attrbute value for each attrbute n an offer, and ntroduce y as an ndcator varable for a partcular offer. The constrant n (22) specfes a reservaton prce C. (23) restrcts the quantty to match an upper and lower bound (D mn and D max ) specfed by the buyer, and (24) lmts the number of wnners. Fnally, (25) and (26) specfy homogenety constrants. In (25) we ntroduce the ndcator varable z that assumes the value 1 f any supplers are chosen wth a bd at level k for attrbute j. T kj s defned as the set of bds at level K for attrbute J. Compared to formula (2) (6) we have dropped the ndex l, because we assume every bdder to submt only one confgurable offer. (20) (21) (25) (26) (27) (28) IBM T. J. Watson Research Center, NY, USA 11

13 5.4 Treatment of Logcal Confguraton and Dscount Rules It s often essental for supplers to express rules, whch defne constrants on the combnaton of attrbute values, or dscounts and markups based on some combnaton of attrbute values. For nstance, a confguraton rule may nclude compatblty restrctons, sayng attrbute value x 12 cannot be connected to attrbute value x 23, or requrements lke attrbute value x 12 and x 32 needs attrbute value x 23. CPML provdes the possblty to express these rules as logcal mplcatons. For example, the proposton x 23 => x 31 descrbes the confguraton rule that f a certan motherboard, x 23 s selected by the user, then the buyer s restrcted from usng a certan type of CPU, x 31. Logcal mplcaton (=>) allows, that f any other knd of motherboard s selected, the partcular CPU may or may not be chosen. Another type of rules, whch s often found n practce are so called dscount rules. For example, x 12 x 31 p - where x agan descrbe partcular attrbute values and p - descrbes a certan dscount (or markup) enforces a dscount upon selecton of these attrbute values. The dscount s only gven, f and only f x 12 x 31 s true. If x 12 x 31 s false, then no dscount wll be granted. Therefore, we use the equvalence operator ( ) for dscount rules. We use x to denote the logcal as well as the bnary varable n the MIP formulaton. For ease of readng we omt the frst subscrpt for bds n the frst part of ths secton. For the evaluaton of a confgurable offerng, these addtonal rules have to be consdered n the IP formulaton. In order to obtan an equvalent mathematcal representaton for any propostonal logc expresson, one must frst consder basc logcal operators to determne how each can be transformed nto an equvalent representaton n the form of an equaton or nequalty. Raman and Grossman [26] specfy transformatons, whch can then be used to convert general logcal expressons nto an equvalent mathematcal representaton. Some of these transformatons are descrbed n Table 2. (29) (30) Logcal relaton Pure logcal expresson Representaton as lnear nequaltes Logcal OR x 1 x 2 x n x 1 + x 2 + x n õ 1 Logcal AND x 1 x 2 x n x 1 õ 1; x 1 õ 1; ; x n õ 1 Implcaton (=>) x 1 x 2 1- x 1 + x 2 õ 1 Equvalence ( ) ( x 1 x 2) ( x 2 x 1) x 1 - x 2 ï 0; x 2 - x 1 ï 0 Table 2: Representaton of logcal relatons wth lnear nequaltes A common approach to convert a general logcal expresson nto nequaltes s to frst transform t n ts equvalent conjunctve normal form (CNF) representaton. CNF nvolves the applcaton of pure logcal operatons (and, or, not ), and s a conjuncton of clauses. A clause s defned as a set of basc lterals separated by -operators, such as (x 12 x 23 ) ( x 34 x 45) (31) CNF can then be expressed as a set of lnear nequalty constrants, as shown n Table 2. We have chosen ths approach to transform the confguraton and dscount rules n CPML nto approprate constrants n IBM T. J. Watson Research Center, NY, USA 12

14 our IP formulaton descrbed n (20) (28). Formulas (32) to (37) show how the proposton n (32) can be translated nto lnear constrants n our IP formulaton. x 12 x 23 p - ( (x 12 x 23 ) p - ) ( p - (x 12 x 23 )) (33) ( x 12 x 23 p - ) ( p - x 12 ) ( p - x 23 ) (34) x 12 + x 23 - p - ï 1 (35) x 12 - p - õ 0 (36) x 23 - p - õ 0 (37) In (33) the equvalence operator has been transformed nto a proposton wth pure logc operators. Usng DeMorgan s Theorem the negaton operator of the frst term n brackets s moved nwards, so that we get CNF n (34). Fnally, n (35) to (37) CNF s translated nto nequaltes, whch can be added to the nteger programmng formulaton. In addton, we have to ntroduce an addtonal bnary ndcator varable for p - n our model, whch ndcates the dscount f the rule takes effect. Logcal expresson Equvalent lnear nequaltes.x => x rs and ci, j,rcj, kck j, sck r, j! r ( 1 x ) + x 1 R -x => x rs and ci, j,rcj, kck j, sck r, j! r x x 0 R,.x p - and ci, jcj, kck j ( 1 x ) + p 1 rs R x rs p 0 R, -x p - and ci, jcj, kck j x p 0 R p x 0 R, Table 3: Translaton of typcal confguraton and dscount rules (32) The logcal expressons n Table 3 descrbe common forms of confguraton and dscount rules wth only conjunctons or only dsjunctons n the antecedent and one lteral n the consequent. We have used the notaton wth three subscrpts so that the addtonal constrants can be added to the optmzaton formulaton n (20) (28). R s defned as the set of attrbute values n the antecedent of a rule n an offer. Of course, the antecedent and the consequent of these rules can n general be any combnaton of conjunctons and dsjunctons. In other words, wth the relatons gven n Table 2 one can systematcally model an arbtrary propostonal logc expresson as a set of lnear equalty and nequalty constrants. 5.5 Computatonal Issues From a computatonal pont of vew the allocaton of confgurable offers wthout homogenety constrants s consderably easer to solve than the problem wth homogenety constrants descrbed n secton 5.3. Wthout homogenety constrants, the overall wnner determnaton can be splt n several smaller problems (see secton 5.2), n whch the best possble confguraton for each confgurable offer s selected IBM T. J. Watson Research Center, NY, USA 13

15 based on a buyer s scorng functon. In our numercal smulatons, the selecton of the best confguraton for an offer wth four confguraton rules, and ten attrbutes wth four attrbute values each could fnd the best confguraton n the order of mllseconds usng a commercal optmzaton package. The results of these ndvdual selecton problems are then used n the overall wnner determnaton descrbed n equatons (2) (9), the runtme of whch has been analyzed n CPU Tme (mllseconds) No of bdders Fgure 3: Expermental results for the allocaton confgurable offers wth ncreasng numbers of bdders The wnner determnaton problem s consderably harder to solve n the presence of homogenety constrants, because all bds have to be consdered at the same tme. Fgure 3 shows the CPU tmes of a randomly generated problem nstance wth 30 attrbutes, a sngle homogenety constrant on one of the attrbutes, and an ncreasng number of bdders. Fgure 4 nvestgates the mpact of the homogenety constrants on the runtme of the wnner determnaton. The problem sze was constant wth 60 bds and 20 attrbutes and no other sde constrants were set for the experment CPU tme (mllsecondes) Number of homogenety constrants 19 Fgure 4: Expermental results for the allocaton confgurable offers wth ncreasng numbers of homogenety constrants In our future research, we plan to extend the analyss towards confgurable offers, whch allow the specfcaton of volume dscounts. Ths aspect adds an addtonal degree of flexblty to supplers, however, agan at the expense of complexty n the wnner determnaton. IBM T. J. Watson Research Center, NY, USA 14

16 6 CONCLUSIONS In ths paper we have dscussed a number of wnner determnaton problems n the context of multattrbute auctons. Durng the past few years many advances have been made n the area of computatonal mechansm desgn. Besdes mult-attrbute auctons, many new aucton mechansms such as combnatoral auctons [27-29] or volume dscount auctons [30] have been developed. The wnner determnaton n these auctons s usually a computatonally hard problem. Ths computatonal complexty has been a sgnfcant hurdle for the wdespread use of these advanced aucton models. In an attempt to foster a more wdespread (re-)use of multdmensonal aucton mechansms, we have mplemented the Multdmensonal Aucton Platform (MAP) [31], an object framework wth a generc API to bd evaluaton and allocaton algorthms. MAP conssts of a generc Java API for dfferent allocaton mechansms, an XML schema to defne varous knds of bds and asks, and a database schema to make these bds and asks persstent. Currently t mplements wnner determnaton algorthms for combnatoral and volume-dscount auctons, as well as the mult-attrbute allocaton algorthms descrbed n ths paper. Ths framework enables applcaton programmers to specfy buyer preferences, allocaton rules and suppler offerngs n a declaratve manner, and solve the allocaton problems wthout havng to re-mplement the computatonally complex algorthms. MAP s currently beng used n a large-scale procurement marketplace for the retal ndustry. 7 REFERENCES [1] W. Vckrey, "Counterspeculaton, Auctons, and Compettve Sealed Tenders," Journal of Fnance, pp. 8-37, [2] S. Dasgupta and D. F. Spulber, "Managng procurement auctons," Informaton Economcs and Polcy, vol. 4, pp. 5-29, [3] R. L. Keeny and H. Raffa, Decson Makng wth Multple Objectves: Preferences and Value Tradeoffs. Cambrdge, UK: Cambrdge Unversty Press, [4] M. Bchler, "BdTaker: An Applcaton of Mult-Attrbute Aucton Markets n Toursm," presented at Wrtschaftsnformatk 2002, Augsburg, Germany, [5] Y.-K. Che, "Desgn Competton through Multdmensonal Auctons," RAND Journal of Economcs, vol. 24, pp , [6] F. Branco, "The Desgn of Multdmensonal Auctons," RAND Journal of Economcs, vol. 28, pp , [7] P. Mlgrom, "An Economst's Vson of the B-to-B Marketplace," Perfect.com, Whte Paper October [8] J. G. Rley and J. G. Samuleson, "Optmal auctons," Amercan Economc Revew, vol. 71, pp , [9] M. Bchler, "Decson Analyss - A Crtcal Enabler for Mult-attrbute Auctons," presented at 12th Electronc Commerce Conference, Bled, Slovena, [10] M. Bchler, M. Kaukal, and A. Segev, "Mult-attrbute auctons for electronc procurement," presented at Frst IBM IAC Workshop on Internet Based Negotaton Technologes, Yorktown Heghts, NY, USA, [11] J. Tech, H. Wallenus, and J. Wallenus, "Multple-ssue aucton and market algorthms for the world wde web," Decson Support Systems, vol. 26, pp , IBM T. J. Watson Research Center, NY, USA 15

17 [12] J. E. Tech, H. Wallenus, J. Wallenus, and A. Zatsev, "Desgnng Electronc Auctons: An Internet-Based Hybrd Procedure Combnng Aspects of Negotatons and Auctons," Electronc Commerce Research, vol. 1, pp , [13] N. Vulkan and N. R. Jennngs, "Effcent mechansms for the supply of servces n mult-agent envronments," Decson Support Systems, vol. 28, pp. 5-19, [14] M. Bchler, "An Expermental Analyss of Mult-Attrbute Auctons," Decson Support Systems, vol. 28, [15] D. C. Parkes and J. Kalagnanam, "Multattrbute Reverse Auctons," presented at AAAI, [16] R. McAfee and P. J. McMllan, "Auctons and Bddng," Journal of Economc Lterature, vol. 25, pp , [17] D. R. Bel and L. M. Wen, "An Inverse-Optmzaton-Based Aucton Mechansm To Support a Mult- Attrbute RFQ Process," MIT, Boston, MA, USA, Research Report December [18] W. Edwards, "How to use mu ltattrbute utlty measurement for socal decsonmakng," IEEE Transactons on Systems, Man, and Cybernetcs SMC, vol. 7, pp , [19] T. L. Saaty, The Analytc Herarchy Process. New York, USA: McGraw Hll, [20] D. L. Olson, Decson Ads for Selecton Problems. New York, et al.: Sprnger, [21] R. A. Rbero, "Fuzzy multple attrbut decson makng: A revew and new preference elctaton technques," Fuzzy Sets and Systems, vol. 78, pp , [22] M. Bchler, J. Lee, H. S. Lee, and J.-Y. Chung, "ABSolute: An Intellgent Decson Makng Framework for E-Sourcng," presented at 3rd Internatonal Workshop on Advanced Issues of E-Commerce and Web-Based Informaton Systems, San Jose, CA, [23] V. S. Iyengar, J. Lee, and M. Campbell, "Q-Eval: Evaluatng Multple Attrbute Items Usng Queres," IBM Research, New York, Research Report [24] W. L. Wnston, Operatons Research - Applcatons and Algorthms, 3 ed. Bemont, CA: Duxbury Press, [25] S. Martello and P. Toth, Knapsack Problems. Chchester, New York: John Wley & Sons, [26] R. Raman and I. E. Grossmann, "Relaton between MILP modellng and logcal nference for process synthess," Computers and chemcal engneerng, vol. 15, pp , [27] S. Rassent, V. L. Smth, and R. L. Bulfn, "A Combnatoral Aucton Mechansm for Arport Tme Slot Allocatons," Bell Journal of Economcs, vol. 13, pp , [28] M. H. Rothkopf and A. Pekec, "Computatonally Manageable Combnatoral Auctons," presented at Maryland Aucton Conference, Maryland, USA, [29] T. Sandholm, "Approaches to wnner determnaton n combnatoral auctons," Decson Support Systems, vol. 28, pp , [30] A. Davenport and J. Kalagnanam, "Prce Negotatons for Drect Procurement," IBM T.J. Watson Research, New York, Research Report RC 22078, February [31] M. Bchler, J. Lee, H. S. Lee, and J. Kalagnanam, "Resource Allocaton Algorthms for Electronc Auctons: A Framework Desgn," presented at 3rd Internatonal Conference on Electronc Commerce and Web Technologes (EC-Web), Ax-en-Provence, France, IBM T. J. Watson Research Center, NY, USA 16