1

Auction Based Mechanisms for ElectronicProcurement

T. S. Chandrashekar, Y. Narahari, Charles H. Rosa, Devadatta Kulkarni, Jeffrey D. Tew, and Pankaj Dayama

Abstract— This article reviews recent research and currentart in the area of auction based mechanisms for electronicprocurement. These mechanisms are becoming increasinglyrelevant in modern day e-procurement systems since they enablea promising way of automating negotiations with suppliersand achieving the ideal goals of procurement efficiency,cost minimization, and agent based deployment. The surveydelineates different representative scenarios in e-procurementwhere auctions can be deployed and describes the conceptualand mathematical aspects of different categories of procurementauctions. We discuss three categories: (1) multi-unit auctionsfor a single homogeneous type of item; (2) combinatorialprocurement auctions where the buyer seeks to procure abundle of multiple items and the suppliers bid for subsetsof the bundle; and (3) multi-attribute auctions where theprocurement decisions transcend cost considerations alone, totake into account lead times, logistics costs, and other importantattributes. In all three cases, the winner determination problemand the determination of payments turn out to be interestingand challenging combinatorial optimization problems. In ourreview, we present mathematical formulation of procurementscenarios under each category, bring out the challenge involvedin solving the problems, and indicate active research topics. Wealso present a case study of electronic procurement at GeneralMotors.

A Note to Practitioners— Since the burst of the dot.com bub-ble, many procurement professionals and purchasing managershave begun to question the ability of the Internet to redefineprocurement processes within their firms. In this article we setout to show that this would be a misplaced sense of deja vubecause the Internet along with a milieu of decision technologiesbased on Game Theory and Optimization is proving to be asignificant tool in the hands of procurement professionals. Sansall the hype, the dot.com phenomena has left behind useful ideasincluding that of e-platforms for on-line auctions. Building uponthis core conceptual construct, familiar to most procurementprofessionals, we first illustrate the successful implementationof sophisticated auction models by pioneering firms, based onoptimization technologies, that meet the requirements of complexbusiness-to-business procurement. We then discuss the excitingfield of research this has opened up with a vast potential forimmediate and gainful applications. We review the existing state-of-the-art in this field, track its recent developments and classifythe models available for different procurement scenarios. Wealso provide pointers to areas that require further fundamentalas well as applied research which calls for the attention of notjust academic researchers but also practicing professionals.

T.S. Chandrashekar is a doctoral student and Y. Narahari is a Professorat the Department of Computer Science and Automation, Indian Instituteof Science, Bangalore, INDIA and a senior member of IEEE. Email:�

chandra � � hari � @csa.iisc.ernet.inCharles.H. Rosa and Devadatta Kulkarni are members of the research staff

and Jeffrew D Tew is a senior member of the Research Staff at the GeneralMotors Research and Development Center, Warren, MI, USA. Pankaj Dayamais a member of research staff at the GM India Science Lab, Bangalore, India.Email:

�charles.rosa � � datta.kulkarni � � jeffrey.tew � � pankaj.dayama � @gm.com

Index Terms— E-Procurement, negotiations, auctions, multi-attribute auctions, combinatorial auctions, volume discount auc-tions, game theory, mechanism design, NP-hard problems.

ACRONYMS

PR Purchase RequestICT Information and Communication TechnologiesRFQ Request for QuoteLP Linear ProgramIP Integer ProgramMIP Mixed Integer ProgramMILP Mixed Integer Linear ProgramGP Goal ProgrammingCA Combinatorial AuctionGVA Generalized Vickrey AuctionVCG Vickrey-Clarke-Groves (mechanism)NP Non-Deterministic Polynomial TimeFPTAS Fully Polynomial Time Approximation SchemeMAUT Multi-Attribute Utility TheoryMCDA Multi-Criteria Decision AnalysisIRDA Iterative Reverse Dutch Auction

I. INTRODUCTION

THE purchasing function and the associated procurementprocesses in organizations, large and small, have tradi-

tionally been an important area of operations affecting busi-ness performance. In many organizations, procurement costsconstitute a major part of the total costs. For example, about 60percent of the cost of a car is attributed to procurement costs.Recent trends in the business environment suggest that theimportance of procurement is being both reinforced throughthe emergence of global supply chains and amplified by thegrowing incidence of outsourcing in many industrial sectors.Simultaneously, the procurement function itself has undergonemuch transformation. In one large global firm that the au-thors have worked with, decentralized, factored purchasingprocesses have given way to uniform, centralized purchasingpractices, with worldwide purchasing decisions being coordi-nated by a single centralized organization. These changes can,in part, be attributed to the influence that information andcommunication technologies (ICT’s) have had in reshapingprocurement processes both within and between organizations.

The procurement process itself may be hierarchically de-composed and a first level decomposition yields four distinctstages: supplier search and analysis, supplier selection stage,automated transactional stage, and supply chain planning andcontrol. While many aspects of procurement, in each of thefour stages, have benefited from the application of ICT’s anddecision technologies, negotiations, which form a crucial part

2

of the supplier selection stage have so far relied on humanbased processes with little technological support. With therecent advances in ICTs, including the rapid growth of theInternet, this has changed. This vector of change is beingfurther influenced by the application of decision sciences,computational models, and software systems.

Till recently, Negotiation Theory has been a topic of re-search within the economic and social science community. Itprovides a framework that is used to reach a decision throughconsensus whenever a person, organization, or any other entitycannot achieve its goals unilaterally [1], [2]. The increasedcomputing and networking power enabled through the Internethas provided an opportunity to automate negotiations thathappen at the supplier selection stage. The design of suchautomated negotiations requires a computer science and infor-mation systems perspective to feed back into the negotiationmodels and procedures.

These interdisciplinary studies have contributed to threeclosely related approaches to automated negotiation systems:(1) the development of negotiation support systems; (2) soft-ware agents for negotiation, and (3) economic mechanismdesign and on-line auction platforms. Each of these approachesaddresses the requirements of a wide variety of negotia-tion scenarios which is captured in [3] as integrative anddistributive negotiations. For descriptions of the first twoapproaches we refer the reader to [4], [5], [6]. Here our focusis on economic mechanism design and auctions for electronicprocurement.

A. Auctions and Electronic Procurement

Auctions, as a sub field of mechanism design, is con-cerned with the design of the rules of interaction, usingthe tools of game theory and mechanism design [7], [8],for economic transactions that will, in principle, yield somedesired outcome. In the context of negotiations for pro-curement we require rules governing: (1) bidding for con-tracts, (2) the issues and attributes that will be consideredto determine winner(s) of the contract, (3) determination ofwinning suppliers, and (4) the payments that will be made.English auctions, Dutch auctions, and sealed bid contractsare well understood, widely used economic mechanisms inthe procurement context. Since the rules of interaction inthese auctions are well laid out, they have been a nat-ural target for automation, and as a result have formedthe core conceptual constructs on which on-line auctions,like those seen in www.ebay.com, www.onsale.com,www.freemarkets.com, etc., have been based. In orderto address the software requirements of such on-line auctionsfirms such as Ariba, Emptoris, Frictionless Commerce, etc.have appeared. These firms have now expanded their softwareproduct portfolio to include more general e-procurement tools.See http://www.research.ibm.com/absolute/ for more details onthese software vendors.

Also, many large corporations have now either used orare in the process of using automated auction-based negoti-ation methods for their procurement operations. For instance,General Electric has adopted on-line auctions for many of

its procurement operations, procuring more than $6 billionworth of goods and services in on-line auctions in 2000 [9],which led to the Internet Week magazine awarding it thetitle ”E-Business of the Year 2000”. Many more firms, forexample, GlaxoSmithKline [10], Metro [11], Volkswagen [12],and Bechtel [13] have already started using auctions for asignificant share of their procurement operations (up to 30percent in many of them). A study conducted by the Centerfor Advanced Purchasing Studies [14] shows that more thana third of the firms interviewed by them are procuring goodsin excess of $ 100 million through electronic procurementauctions. However, in many cases, the only information thatcan be gleaned about these implementations is from articlesappearing in the popular press. Detailed case studies, apartfrom a few exceptions [15], [16], [17], [18], [19], [20], [21],[22] have been hard to come by possibly because procurementis considered to be a key source of strategic advantage andhence organizations have been unwilling to put the details ofimplementation into the public domain. Our concern here iswith studying auctions for automated negotiations in complexB2B industrial procurement scenarios since they promise afaster emergence of high quality agreements.

B. Contributions of the PaperMotivated by the key role auctions have come to play

in automating negotiations in electronic procurement, weprovide, in this paper, a review of the issues, mathematicalformulations, and the current state of practice in this area. Thispaper differs from other survey articles that have appeared onrelated topics in the following ways.� This paper is not a survey on auctions in general. There

are widely popular books (for example, by Milgrom [23]and Vijay Krishna [24]) and surveys on auctions (forexample, [25], [26], [27], [28], [29]) which deal withauctions in a comprehensive way.� This paper is not a survey on combinatorial auctions(currently an active area of research). Exclusive surveyson combinatorial auctions include the articles by de Vriesand Vohra [30], [31], Pekec and Rothkopf [32], and Nara-hari and Pankaj Dayama [33]. Cramton, Ausubel, andSteinberg [34] have recently brought out a comprehensiveedited volume containing expository and survey articleson varied aspects of combinatorial auctions. This paper isalso not a survey on dynamic pricing. There are excellentsurveys on dynamic pricing [35], [36]. As is well known,auctions represent a particular mechanism for dynamicpricing.� The focus of this paper is on auction based mechanismswhich can be used as part of an e-procurement processto improve the efficiency of the process. It is thereforecloser in its content to the papers by Elmaghraby andKeskinocak [16], Elmaghraby [37], Bichler, Davenport,Hohner, and Kalagnanam [22], and Bichler et al [36].While the papers [16], [37] deal with combinatorialauctions for procurement together with a description ofa case study, the papers [22], [36] deal with volumediscount auctions and multi-attribute auctions. These pa-pers thus specifically deal with certain special categories

3

of procurement auctions. Our paper supplements andcomplements these above papers by taking a bird’s eyeview of almost all categories of procurement auctions.The main goal of this paper, therefore, is to provide anumbrella view of the entire area in a self-sufficient way.This is evident from the structure of the rest of the paperwhich is presented below.

C. Outline of the Paper

The rest of this paper is organized into seven sections.Section II: Procurement Auctions - Scenarios and Case

Summaries: In this section, we describe a range of indus-trial procurement scenarios followed by summaries of twoindustrial cases at MARS Inc. [17] and Home Depot [16],[15], where auction based procurement mechanisms have beenimplemented.

Section III: Issues in Electronic Auctions: In this section,we provide a conceptual discussion of the key technicalissues involved in the design of auction based mechanisms forelectronic procurement. This section is structured as follows:[a] Types of Auctions, [b] Valuation Issues [c] Design ofAuctions [d] Properties Desired from an Auction [e] Relevanceof Game Theoretic and Incentive Issues [f] ComputationalComplexity Issues [g] Design Space for Auction Mechanisms[h] Summary. This subsection is intended to present thebasic intuition behind auctions, game theoretic modeling, andeconomic issues.

Sections IV, V, and VI: In these three sections, we discussthree major categories of procurement auctions:� Procurement of single unit or multiple units of a single

item based on a single attribute (Section IV)� Procurement of single unit or multiple units of multipleitems based on a single attribute (Section V)� Procurement of multiple units of a single item based onmultiple attributes (Section VI)

In each subsection, we begin by discussing the main issuesinvolved followed by a detailed discussion of two or threemechanisms from the literature. We conclude each section witha summarizing discussion on the state-of-the-art and providea tabular listing of important current papers relevant to thetopic.

Section VII: A Case Study from General Motors: In thissection, we discuss a real world case study of electronic pro-curement at General Motors. The last four authors of this paperwere part of a team that developed an auction/optimizationmodel for this procurement scenario.

Section VIII: Summary and Future Work: We summarizethe contents of this paper and we conclude with a discussionof some issues that need to be addressed with the current genreof models and directions for future work.

II. PROCUREMENT AUCTIONS: SCENARIOS AND CASESUMMARIES

In this section we first describe a few practical procurementscenarios, based on the authors’ experience in working withseveral major procurement organizations. The scenarios pro-gressively illustrate the range of complexity in procurement

situations: from buying the proverbial pin (very simple) to aplane (very complex). This is followed by summaries of twocase studies, drawn from recent publications, to demonstratethe applicability of automated negotiations in practical busi-ness situations and to motivate the subsequent discussion ofauction based negotiation mechanisms.

A. Procurement Scenarios

1) Scenario A: Buying a Cutting Tool: A Purchase Request(PR) to buy a specific cutting tool is generated by a userdepartment, such as production. This request is communi-cated to a central buying department called Purchasing. Thebuyer responsible for this category of items acknowledges therequest. If the item is in one of the standard price lists ofvendors already supplying to the organization and a blanketorder already exists then a spot buy order is sent to the supplier.Otherwise a request for quote (RFQ) is prepared by the buyerand sent to a selection of suppliers who respond with quotes.These quotes are analyzed and a sourcing recommendationis made based on the prices that are quoted. This is a verysimple buying situation where the decision to buy a singleitem is made on the basis of a single attribute - price per unit.

Now, consider the same situation as above except that mul-tiple units of the item are to be bought. The suppliers are nowable to exploit economies of scale and hence provide bids withvolume discounts thereby introducing one level of complexityin the buyer’s supplier selection or bid selection problem alsosometimes called the winner determination problem.

In the scenario above, very often it is observed that thesame tool is required by many plants across a geographicalregion or across all plants worldwide. In such cases the buyercreates a blanket order which can be used by all the plants. Theblanket order specifies a single price and the delivery point isthe nearest pickup location operated by third party logisticsproviders on behalf of the manufacturer. Clearly in this casethe sourcing decision involves a greater degree of complexitysince the total cost of procurement - the per unit cost of theitem and the shipment cost, is to be optimized by the buyer.

Fig. 1. Taxonomy of procurement scenarios

4

2) Scenario B: Buying a Family of Cutting Tools: Hereis an extension to the previous scenario. Purchase requestsmay be raised by user departments across the organization formany types of cutting tools. These requests are processed by asingle buyer responsible for the procurement of cutting tools.The requests are aggregated and in some cases bundled and anRFQ is generated. The suppliers respond with bids for bundleswhich provide volume discounts as well as point prices. Thebuyer has to make a decision of allocating the orders so as tominimize his total cost of procurement as well as restrict thenumber of suppliers who receive the orders so as to reducemanagement overhead. The decision problem in this case canbe challenging, and in some cases if the number of suppliersis large then the computations involved can overwhelm thebuyer.

3) Scenario C: Buying a Machine Tool: While it may bepossible to buy cutting tools off-the-shelf by negotiating alongonly one dimension - price, buying a machine tool for aspecific machining requirement depends on many attributes.These may include at least the following: quality of machinetool, lead time to manufacture and deliver, availability ofspares, maintenance spares to be held in inventory, etc. It maybe possible to attach a monetary value to some or all of theattributes that influence the purchase decision. In some cases,additional features and options could be purchased as add-on components to the basic machine tool from some or all ofthe suppliers. Grappling with such multi-attribute procurementdecisions is the raison d’etre of purchasing departments.Figure 1 provides an idea of the range and complexity ofdifferent procurement scenarios.

B. Auction Based Procurement Mechanisms: Case Summaries

Compaq Computer Corp, General Dynamics, Dutch Rail-way, General Electric, Sears Logistics, and Staples Incorpo-rated are a few organizations that are pioneering users ofautomated auction technology for procurement [38], [9], [18].However, as indicated earlier, details of these implementationsare unavailable in the public domain. Two exceptions includeMARS Inc [17] and Home Depot [15], [16]. These case studiesdiscuss the emerging state of practice in procurement and alsocapture the issues that arise in the trenches in a fairly detailedmanner.

1) Combinatorial and Supply Curve Auctions at MARS Inc:Procurement at MARS Inc, a leading global manufacturerof confectioneries, exhibits characteristics which are not allunique to the situation. The supply pool is often small foreach category of materials and contracts are executed withmany types of suppliers including private businesses, tradedagricultural markets, monopolies, cartels, and governments.Negotiation and tendering are the most extensively used con-tracting methods and the structure of typical bids in theseprocurement scenarios include volume discounts and all-or-nothing bids.

The team at MARS realized that several reasons wereresponsible for the inefficiency of the procurement process.Firstly, competitive positions could not be fully leveragedfor price negotiations, because synergies in supply conditions

were not being fully exploited in the item by item negotia-tion process. Secondly, disproportionate amount of time wasspent determining quantities and prices. Automated auctions,generally seen as mechanisms that make negotiations efficientand promote market competition, were sought to eliminate thelimitations in the manual procurement process. But, simpleauctions suffer from certain drawbacks too: firstly, they aretoo reliant on price for market making, making them a bruteforce way of managing relationships and secondly, they areinappropriate when the firm wants to control business volumesor the number of suppliers. Volume discount and combina-torial auction models provide a way of capturing businessrequirements such as minimizing the number of winningsuppliers, limiting the exposure in business volume with asingle supplier, expressing complementarities, etc. But heretoo, identifying the cost minimizing bid set subject to businessconstraints could turn out to be a NP-hard optimization prob-lem. The Mars team working with researchers from the IBMdesigned an auction mechanism that would meet the businessrequirements and also be computationally tractable.

The Auction Design: The MARS-IBM team devised aniterative auction scheme to meet the business requirements andto eliminate the need for soliciting complex bid structures. Thescheme allows suppliers to correct their bids using informationlearned during the auction process and also induces a senseof competition among them.

Each iteration proceeds in two stages: first bids are collectedand a subset of bids that minimizes the cost of procurementis found. This is used as an input to the second stage whereanother optimization problem with the procurement cost as ahard constraint is solved.

For the combinatorial auction version, the problem ofchoosing the winning combination of bids from the set ofsubmitted bids, is modeled as a set-covering problem with sideconstraints and in the volume discount auction, it is modeledas a variation of the multiple choice knapsack problem. Thesupplier selection mechanism using the MIP framework wasdeployed using IBM’s eCommerce platform - WebsphereCommerce Suite 4.1 as part of MARS Inc’s procurement website. In order to solve this MIP formulation, a bid evaluationengine was developed as an independent module in C++using IBM’s OSL as the LP/IP solver. This engine, usingheuristic approaches based on customized knapsack coverinequalities and column generation techniques, has proveneffective in solving problems with 500 items and up to 5000bids. Approximate solutions to within one percent of theoptimal were obtained within two minutes response time withthese heuristics. In order to ensure a steady rate of progressof the auction in practice, bids were collected every twominutes and an allocation decision was made. This decisionwas communicated back to the bidders to allow them torestructure their bids.

The impact of the auction mechanism on procurement pro-cesses at MARS Inc has been significant with payback beingachieved in less than a year, mostly from better allocationof contracts. The intangible benefits include a fair, faster andmore transparent negotiation process that has been vouchedfor by many of the participating suppliers.

5

2) Combinatorial Auctions for Logistics Services at HomeDepot: Home Depot(HD) is a large home improvement re-tailer in the Americas. Managing its logistics involves coordi-nating over 7000 suppliers, 1000 stores, 37 distribution centersand numerous carriers. A key component of this logistics effortis the transportation of over 40000 stock keeping units (SKU’s)between different entities in the supply chain. Traditionally, thetrucking services were outsourced and the contracting processwas completely manual.

Home Depot would provide truckers with origin destinationzip codes for each pair of locations within its network and theaggregate demand forecasts for the pair. Based on this sparseinformation, carriers would bid for each origin-destinationpair. Home Depot realized the limitations of such a contractingprocess. Firstly, carriers do not have good visibility intoHome Depot’s network; secondly carriers could not bid oncombinations of lanes to exploit potential synergies therebylimiting their ability to bid more aggressively on synergisticlanes; finally the manual process was extremely inefficient. Toachieve better efficiencies and effectiveness in transportationservices, Home Depot partnered with i2 Technologies, to de-velop and deploy an Internet based flexible auction mechanismthat allowed carriers to bid for individual lanes as well ascombinations of lanes.

Design of the New Bidding Mechanism: The new auctionmechanism consisted of three separate stages. The first is apre-qualification stage where HD provides potential bidderswith information on origin destination pairs, the lane detailsand the forecast demand by truck type. In return, carriers arerequired to provide pre-qualification information. Based onthis information HD eliminates some of the carriers. Qualifiedcarriers are then asked to send in their bids for individuallanes and combinations. The auction design was based on asingle shot sealed bid combinatorial auction, in contrast to theiterative auction used by MARS Inc, to prevent carriers fromengaging in damaging price wars which was seen as a potentiallong-term risk in maintaining the quality of service.

In the second stage HD eliminates some of bids that itconsiders are inferior to other bids that have been received.An integer programming formulation based upon the setpartitioning problem is solved in order to identify the winningcombination of bids. In this stage a subset of the lanes thatwere originally on auction are allocated.

In the third stage, HD used information from previouslysubmitted bids to identify and invite a set of potential carrierswho would be likely to submit “acceptable” bids for theunallocated lanes. The optimization engine was used again toaward the remaining lanes based on the bids collected in thesecond round of bidding.

Auction Software and Implementation Impact: The auctionsoftware consisted of three main modules: (1) Shipper bidsupport (SBS) module, (2) Carrier bid response (CBR) moduleand (3) a Bid selection optimization engine. The SBS modulehelps Home Depot analyze their network and decide whichlanes to put out for an auction. The CBR module helps carriersanalyze the demand data provide by Home Depot and createbids that reflect their cost structures and existing infrastructure.The Bid selection engine is an optimization module that solves

an underlying integer programming formulation of the auctionproblem.

While the auction mechanism has proved to be a successfulinnovation, allowing Home Depot to realize better rates, thereactions of the carriers have also been positive because theycan now exploit their complementarities in cost structuresbetter through the combinatorial bidding mechanism. For moredetails on this case, the reader is referred to [16], [15].

III. ISSUES IN ELECTRONIC AUCTIONS

As discussed in the previous section, auctions constitutea major class of economic mechanisms studied in microe-conomics and game theory [24], [23]. Classical mechanismdesign literature has delineated several useful properties formechanisms, that are detailed later, and the challenge inautomating these mechanisms is to retain them even as weensure computational tractability.

In this section, we first briefly discuss schemes for clas-sification of auctions and for valuations of outcomes. Wethen present the properties that are required from an auctionalong with a discussion of key possibility and impossibilityresults that inform their design space. We then introducegame theoretic issues followed by computational complexityissues that arise in automated auctions. We do this to buildthe necessary background for the subsequent discussion onprocurement mechanisms.

A. Types of Auctions

An Auction is a mechanism to allocate a set of goods toa set of bidders on the basis of bids and asks. In a classicalauction, the auctioneer wants to allocate a single item to abuyer from among a group of bidders. There are four basictypes of auctions described in the literature [25], [26], [28]:Open cry or English auction, Dutch auction, first price sealedbid auction, and second price sealed bid auction (also calledthe Vickrey auction). An English auction is an iterative auctionwhere the bidders submit monotonically increasing bids. Thisiterative process continues until a price is reached where thereis just a single bidder who is willing to buy. The item isawarded to this buyer at the final bid price. The Dutch auctionis the exact reverse of the English auction where the priceis monotonically decreased by the auctioneer until there is abuyer who is willing to buy at the currently announced price.In both these auctions one must note that they are iterativein nature and price signals are continuously being fed backto the bidders. The first and second price sealed bid auctionshowever are single round auctions where bidders submit sealedbids. The winner of the contract is the bidder who submitsthe highest bid. The payment that he makes in the First Pricesealed bid auction is the bid price itself. In the second pricesealed bid auction also called the em Vickrey Auction [39], thepayment that he makes is the second highest price, and thismakes it a truthful auction mechanism. We will have moreon this in the sections that follow. Here, we note the revenueequivalence theorem [25], [26] that states that although theseauction mechanisms are vastly different from each other, they

6

yield the same expected revenue for the auctioneer when oneitem is being sold, under the following set of assumptions:

1) The bidders are risk neutral, in the sense that they tryto maximize their expected profits and their bids avoidany situation that might give them negative payoffs.

2) The valuations of bidders follow the independent privatevalues model (see the section on valuation issues below)

3) The bidders are symmetric (that is all of them draw theirvaluations from the same probability distribution)

4) Payments or prices depend only on bids

A detailed review of results in Auction Theory is beyondthe scope of this paper. The interested reader can refer to[25], [26], [28], and [40] for a game theoretic/microeconomicviewpoint. A computational perspective when auctions are tobe automated over the Internet is provided in [41].

Building on these basic types, auctions have evolved rapidlyto include multiple resources, business level constraints, andmore complex market structures. Kalagnanam and Parkes [29]have suggested a framework for classifying auctions based onsix major factors outlined below.

(1) Resources: An auction involves a set of resources overwhich the negotiation is to be conducted. The resource couldbe a single item or multiple items, with a single or multipleunits of each item. Another common consideration is thetype of the item: a standard commodity or multi-attributecommodity. In the case of multi-attribute items, the agentsmight need to specify the non-price attributes and someutility/scoring function to trade-off across these attributes.

(2) Market Structure: An auction provides a mechanism fornegotiation between buyers and sellers. In forward auctions asingle seller sells resources to multiple buyers. In a reverseauctions, a single buyer attempts to source resources frommultiple suppliers, as is common in procurement. Auctionswith multiple buyers and sellers are called double auctions orexchanges, and these are commonly used for trading securitiesand financial instruments and increasingly within the supplychain. In this paper, our interest is in reverse auctions whichare called procurement auctions in this paper.

(3) Preference Structure: The preferences define an agent’sutility for different outcomes. For example, when negotiatingover multiple units agents might indicate a decreasing marginalutility for additional units. An agent’s preference structureis important when negotiation happens over attributes for anitem, for designing scoring rules used to signal information,etc.

(4) Bid Structure: The structure of the bids within theauction defines the flexibility with which agents can expresstheir resource requirements. For a simple single unit, singleitem commodity, the bids required are simple statements ofwillingness to pay/accept. However, for a multi-unit identicalitems setting bids need to specify price and quantity. This in-troduces the possibility for allowing volume discounts, wherea bid defines the price as a function of the quantity. Withmultiple items, bids may specify all-or-nothing, both withprice on a basket of items. In addition, agents might wishto provide several alternative bids but restrict the choice of

bids.(5) Matching Supply to Demand: A key aspect of an auction

is matching supply to demand, also referred to as marketclearing, or winner determination. The main choice here iswhether to use single-sourcing, in which pairs of buyersand sellers are matched, or multi-sourcing in which multiplesuppliers can be matched with a single buyer, or vice-versa.This form of matching influences the complexity of winnerdetermination, and problems range the entire spectrum fromsimple sorting problems to NP-hard optimization problems.

(6) Information Feedback: An auction protocol may be adirect mechanism or an indirect one. In a direct mechanismsuch as a sealed bid auction, agents submit bids withoutreceiving feedback, such as price signals, from the auction.In an indirect mechanism, such as an ascending-price auction,agents can adjust bids in response to information feedbackfrom the auction. Feedback about the state of the auctionis usually characterized by a price signal and a provisionalallocation, and provides sufficient information about the bidsof winning agents to enable an agent to redefine its bids. Incomplex settings, such as multi-item auctions with bundledbids, a direct mechanism can require an exponential number ofbids to specify an agent’s preference information. With indirectmechanisms the focus of the design is to identify how muchpreference information is sufficient to achieve the desiredeconomic properties and how to implement informationally-efficient mechanisms.

In this paper, our interest is in procurement auctions witha specific attention on three broad types: [a] Single item,single attribute procurement auctions (single unit or multi-unit), [b] Multi-item, single attribute procurement auctions,and [c] Single item, multi-attribute procurement auctions.

B. Valuation Issues

Asymmetry of information is a crucial element in any typeof auction [25]. In procurement auctions, it is unreasonableto expect every bidder to possess the same amount of infor-mation. Also, there are bound to be differences among thevaluations of an item or a set of items. With respect to bid-ders’ valuations, two extreme models which are possible are:Independent-private-values model and common-value model[25].

In the independent-private-values-model model, each bidderknows precisely how she values the item. She does not knowthe valuations of other bidders for this item but perceivesany other bidder’s valuation as a draw from a probabilitydistribution. Also, she knows that the other bidders regard herown valuation as being drawn from a probability distribution.Formally, bidder � is associated with a probability distribution���

from which she draws her valuation � � . Bidder � aloneknows about � � and all other bidders only know the distribu-tion

���. The valuations of any pair of bidders are mutually

independent.In the common-value model, the item has a single objective

value but nobody knows its true value. The bidders, havingaccess to different information sources, have different esti-mates of the item’s valuation. If � is the item’s true value

7

(unobserved), each bidder bidder � , has a perceived value � �which is an independent draw from a probability distribution� � � � ��� . All the bidders know the distribution

.

The above models represent two extremes; real-world pro-curement auctions will have some features of both.

C. Design of Auctions

The design of auctions can be viewed as a problem of de-signing a mechanism that implements a social choice function.Designing a mechanism, in turn, can be viewed as a problemof designing a game with incomplete information having anequilibrium in which the required social choice function isimplemented.

Consider a set of agents or players � ���������������������! with agent � having a type set " � ( �#���$� �%���������&� ). Thetype set of an agent represents the set of perceived valuesof an agent (also called private values). For example, in anauction, the type of an agent may refer to the valuation thatthe agent has for different items up for auction. Let " be theCartesian product of all the type sets of all the agents (thatis " is the set of all type profiles of the agents). Let ' bea set of outcomes. An outcome, in the context of auctions,corresponds to an assignment of auction items and paymentsto the bidders. A social choice function is a mapping from "to ' which associates an outcome with every type profile. Inthe context of auctions, a social choice function correspondsto a desirable way of producing outcomes from given typeprofiles. Let ( � denote the action set of agent � , that is ( � isthe set of all actions that are available to an agent in a givensituation. A strategy ) � of an agent � is a mapping from " �to ( � . That is, a strategy maps each type of an agent to aspecific action the agent will choose if it has that type. In anauction, a strategy corresponds to the bid the agent will placebased on its observed type. Let ( be the Cartesian productof all the strategy sets. A mechanism * is basically a tuple (,+-��(/.$��������� (/01�32 �4��� , where 2 is a mapping from ( to ' .That is, 2 �4� maps each strategy profile into an outcome. Agiven mechanism can always be associated with a game withincomplete information, which is called the game induced bythe mechanism. For details, refer to [7], [42].

We say that a mechanism *5� (6+7��(/.$�������8� (/01�32 �4��� imple-ments a social choice function 9 if there is an equilibrium strat-egy profile

)-:+ �4��� );:. �4����������� );:0 � �&� of the game induced by *such that 2 )7:+ =< +���� );:. >< .;������������)-:0 >< 0��&�?�@9 =< +-� < .7��������� < 0��&�for all possible type profiles

>< +-� < .��������8� < 0�� . That is, a mech-anism implements a social choice function 9 � � if there isan equilibrium of the game induced by the mechanism thatyields the same outcomes as 9 �4� for each possible profileof types. Depending on the type of equilibrium we qualifythe implementation. Two common types of implementationsare dominant strategy implementation and Bayesian Nashimplementation, corresponding respectively to weakly domi-nant strategy equilibrium and Bayesian Nash equilibrium. Fordefinitions of these, the reader is referred to [7]. The weaklydominant strategy equilibrium is an extremely robust solutionconcept that ensures that the equilibrium strategy of each agentis best whatever the strategy profiles of the rest of the agents.

The Bayesian Nash equilibrium is a weaker solution conceptthat only guarantees that the equilibrium strategy of each agentis best with respect to the equilibrium strategies of the otheragents.

A direct revelation mechanism corresponding to a socialchoice function 9 �4� is a mechanism of the form *A� "B+-� "C.$�������8� "C01� 9 �4��� . That is, the strategy sets are the typesets itself and the outcome rule 2 �4� is the social choicefunction itself. A social choice function is said to be incen-tive compatible in dominant strategies (or strategy proof ortruthfully implementable in dominant strategies) if the directrevelation mechanism *D� "E+-� "C.$�������8� "C01� 9 �4��� implements9 �4� in a weakly dominant strategy equilibrium where theequilibrium strategy of each agent is to report its true type.Similarly a social choice function is said to be BayesianNash incentive compatible if the direct revelation mechanism*F� " + � " . ����������" 0 ��9 �4��� implements 9 �4� in a BayesianNash equilibrium where the equilibrium strategy of each agentis to report its true type. The revelation principle [7] states thatif a function can be implemented in dominant strategies (orBayesian Nash equilibrium), it can also be truthfully imple-mented in dominant strategies (or Bayesian Nash equilibrium).The revelation principle enables one to focus attention only onincentive compatible mechanisms.

The mechanism design problem is to determine a mecha-nism that implements a ”good” social choice function. Somedesirable properties which are sought from a social choicefunction and hence from the implementing mechanism (andin the present case, from procurement auctions) are describedbelow [7], [42], [32].

D. Properties Desired from an Auction

We now present an intuitive discussion of properties thatan auction designer looks for while designing an auctionmechanism. For a detailed treatment of these, the reader isreferred to [7], [8].

Solution Equilibrium: The solution of a mechanism isin equilibrium, if no agent wishes to change its bid, giventhe information that it has about other agents. Many typesof equilibria can be computed given the assumptions aboutthe preferences of agents (buyers and sellers), rationality,and information availability. They include: Nash equilibrium,Bayesian Nash Equilibrium, weakly dominant strategy equi-librium.

Efficiency: A general criterion for evaluating a mechanismis Pareto efficiency, meaning that no agent could improve itsallocation without making at least one other agent worse off.Another metric of efficiency is allocative efficiency which isachieved when the total utility of all the winners is maximized.When allocative efficiency is achieved, the resources or itemsare allocated to the agents who value them most. These twonotions are closely related to each other; in fact, when theutility functions take a special form (such as quasi-linear form[7]), Pareto efficiency implies allocative efficiency.

Individual Rationality: A mechanism is individually ratio-nal if its allocations do not make any agent worse off than hadthe agent not participated in the mechanism. That is, every

8

agent gains a non-negative utility by being a participant in themechanism.

Budget Balance: A mechanism in the context of procure-ment is said to be weakly budget balanced if the sum ofmonetary transfers between the buyer and the sellers is non-negative while it is said to be strongly budget balanced if thissum is zero. In other words, budget balance ensures that themechanism or the auctioneer does not make losses.

Incentive Compatibility: A mechanism is incentive compat-ible if the agents optimize their expected utilities by biddingtheir true valuations of the goods. Depending on the equi-librium achieved by truthful bidding, an incentive compatiblemechanism is qualified as Bayesian Nash incentive compatibleor dominant strategy incentive compatible. In the latter case,the mechanism is said to be strategy proof . If a mechanism isstrategy proof, each agent’s decision depends only on its localinformation and it gains no advantage in expending effort tomodel other agents’ valuations.

Solution Stability: The solution of a mechanism is stable,if there is no subset of agents that could have done better,coming to an agreement outside the mechanism.

Revenue Maximization or Cost Minimization: In an auctionwhere a seller is auctioning a set of items, the seller wouldlike to maximize total revenue earned. On the other hand,in a procurement auction, the buyer would like to procureat minimum cost. Given the difficulty of finding equilibriumstrategies, designing cost minimizing or revenue maximizingauctions is not easy.

Low Transaction Costs: The buyer and sellers would liketo minimize the costs of participating in auctions. Delay inconcluding the auction is also a transaction cost.

Fairness: This influences bidders’ willingness to participatein auctions. Winner determination algorithms, especially thosebased on heuristics, could lead to different sets of winnersat different times. Since there could be multiple optimalsolutions, different sets of winners could be produced bydifferent algorithms. Bidders who lose even though they couldhave won with a different algorithm could end up feelingunfairly treated.

Failure Freeness: Auction implementations should work asintended under all but the most extreme conditions. Trans-parency is also important because (1) it simplifies bidders’understanding of the situation and eases their decision making(2) increases their trust in the auction process by improvingtheir ability to verify that the auction rules have indeed beenfollowed.

E. Relevance of Game Theory and Incentive Issues in Auto-mated Procurement

Following the description of the case studies in Section II.Band the properties desired by an auction as given in the sectionabove, it may at first appear that there is a disconnect betweentheory and practice. In our opinion such an interpretationwould be specious and misleading for the following reasons.� Firstly, procurement professionals are acutely aware of

the information asymmetry that exists in negotiations andas a result the need to bargain hard. The analysis carried

out by procurement professionals before such bargain-ing often implicitly involves game theoretic reasoning.Witness is provided by the plethora of auction formatsthat procurement professionals employ in practice. Aformal game theoretic analysis only helps to place suchreasoning on a firm basis. In the absence of sound gametheoretic analysis, naive auction strategies may actuallyresult in disastrous consequences. For evidence of thiswe point the reader to a discussion of spectrum auctions[27].� Secondly, often it is argued that human agents fail tomeasure up to the strict requirements of the fundamentalassumptions of rationality and intelligence that gametheory makes. Hence the inapplicability of game theoreticanalysis. However if we extrapolate the vector of develop-ment in this area of application and assume that at somepoint in the not too distant future, automated negotiationscarried out by software agents will become a reality thena game theoretic analysis would be eminently applicablebefore such systems are designed and built [43].

In summary, game theoretic considerations are already usedimplicitly in procurement situations. Therefore, sound use ofgame theoretic principles (for example, providing appropriateincentives) in designing automated procurement mechanismswill enhance the efficacy of automated procurement. We nowturn to issues related to the design of appropriate incentives forprocurement mechanisms based on discussions in [25], [32],[44].

Informally, a procurement mechanism describes any processthat takes as inputs the bids of the sellers and determines whichbidders will be allocated the item(s) and how much paymentis received by the winning bidders. A mechanism is incentivecompatible if the mechanism is structured in a way that eachbidder finds it optimal in some sense to report his valuationtruthfully. An incentive compatible mechanism induces truthrevelation by the bidders by designing the payoff structurein a way that it is in the best interests of the bidders to bidtruthfully.

The second price sealed bid auction or the Vickrey auctionfor a single unit of a single item has been shown to beincentive compatible [39]. The generalized Vickrey auctionis an example of an incentive compatible combinatorial auc-tion mechanism [45], [30]. VCG (Vickrey-Clarke-Groves)mechanisms [39], [46], [47], [42] provide a broader class ofincentive compatible mechanisms. These mechanisms inducetruth revelation by paying a surplus amount to each winningbidder, over and above his actual bid. This surplus which iscalled the Vickrey Surplus is actually the extent by which thetotal procurement cost is decreased due to the presence of thisbidder (marginal contribution of the bidder to the total cost ofprocurement).

VCG mechanisms have very attractive properties. For ex-ample, the GVA mechanism, as already stated, is allocativelyefficient, individual rational, weakly budget balanced, andincentive compatible. However these mechanisms are notcommonly used because the payment of Vickrey surplusesor Vickrey discounts make them revenue inefficient from thepoint of view of the buyer. Secondly, the computation of

9

Vickrey surpluses and Vickrey discounts involves solving asmany problems as the number of winning bidders which inmost cases turn out to be NP-hard problems.

F. Computational Complexity Issues in Auction Design

In an economic mechanism where resource allocation isdone based on decentralized information, computations areinvolved at two levels: first, at the agent level and secondly atthe mechanism level. The complexity questions involved arebriefly indicated below. For a more detailed discussion refer[42], [29].

Complexity at the Agent Level:� Strategic Complexity: Should agents model other agentsand solve game theoretic problems to compute an opti-mal strategy? For instance, in a sealed bid procurementcontract scenario, sellers will need to not only take theirvaluation of the contracts into consideration but alsothe bidding behavior of their competitors. This requiressophisticated bidding capability.� Valuation Complexity: How much computation is requiredto provide preference information within a mechanism?For instance, in a multi item procurement scenario wherethe items exhibit cost complementarities, estimating a bidfor every possible permutation of the bundle of items iscostly.

Complexity at the Mechanism Level:� Winner Determination Complexity: How much compu-tation is expected of the mechanism infrastructure tocompute an outcome given the bid information of theagents.� Communication Complexity: How much communicationis required between agents and the mechanism to computean outcome. For instance, in an English auction, whereindividual valuations are revealed progressively in aniterative manner, the communication costs could be highif the auction were conducted in a distributed mannerover space and/or time.

G. Design Space for Auction Mechanisms

Ideally, one would like to put in place a mechanism thathas all the properties required in an auction mechanism whilesimultaneously achieving computational tractability. Unfortu-nately this may not always be possible. A set of results inmechanism design theory dealing with what combinations ofproperties are possible and what are not possible to be achievedby an economic mechanism such as an auction provide thedesign points around which mechanisms need to be designed.We indicate below some of the important results germane tothe discussion on procurement mechanisms.

Impossibility Results:� Hurwicz [48] showed that it is impossible to achieveallocative efficiency, weak budget balance, and individ-ual rationality in a Bayesian Nash incentive compatiblemechanism.� According to Arrow [49], allocative efficiency and strongbudget balance cannot be achieved simultaneously in adominant strategy equilibrium.

� Myerson and Satterthwaite [50] showed that no exchange(that is with multiple sellers and multiple buyers) canbe efficient, budget balanced, and individual rationalat the same time; this holds with or without incentivecompatibility.

Possibility Results:� It was shown by Groves [47] and Clarke [46] thatallocatively efficient and strategy proof mechanisms arepossible if the utility functions are quasi-linear (that is,of the form utility equals value minus price). Clarkemechanisms are a special class of Groves mechanisms.The generalized Vickrey auction (GVA) [43] is a combi-natorial auction version of Clarke’s mechanisms while theVickrey auction [39] (second price sealed bid auction of asingle indivisible item) is a special case of GVA for non-combinatorial auctions. In fact, GVA satisfies four prop-erties simultaneously: allocative efficiency, individual ra-tionality, weak budget balance, and strategy proofness.All the mechanisms above are also commonly referred toas VCG (Vickrey-Clarke-Groves) mechanisms.� It was shown by Arrow [49] and d’Aspremont andGerard-Verat [51] that under quasi-linear preferences,it is possible to have a mechanism (which is calledthe dAGVA mechanism) that is efficient, Bayesian Nashincentive compatible, strongly budget balanced, and indi-vidually rational.� It was shown by Myerson [52] that revenue maximization,individual rationality, and incentive compatibility can beachieved simultaneously.

For more details on these results, we refer the reader to [42],[7], [53], [24], [29].

The above discussion gives an idea of how the design spacefor auctions is constrained. Another key factor that constrainsthe design space further is the computational complexity (dis-cussed in Section III.F) at the agent level and at the mechanismlevel. The central problem of a designer of procurementauctions faced with a given procurement situation is to comeup with the best possible mechanism that is contained in thisconstrained design space.

H. Summary

In this section, we have discussed important issues inauctions: valuation issues, design of auction mechanisms,properties desired from a procurement auction, relevance ofgame theoretic and incentive issues in electronic procurement,computational complexity issues, and design space for auctionmechanisms. These issues are extremely relevant for electronicprocurement auctions. We will be referring to a few importantproperties (allocative efficiency, cost minimization, and incen-tive compatibility) while discussing procurement mechanismsin subsequent sections. We will also touch upon complexityissues while discussing different procurement mechanisms.

IV. SINGLE ITEM, MULTI-UNIT, SINGLE ATTRIBUTEPROCUREMENT MECHANISMS

In this section, we discuss models related to the procurementof a single item with price as the only consideration for

10

decision making under a variety of different business contexts.Specifically, we review: (a) auction mechanisms for a singleunit of an item, (b) auctions for multiple units of a singleitem with volume discount bids, and (c) auction mechanismsfor procuring multiple units of a single item for multiplemanufacturing points taking logistics costs into account.

A. Procurement Auctions for Single Unit of an Item

In many procurement scenarios for a single unit of anitem, the English auction or a sealed bid tender process isgenerally used to decide the winner of the contract. In thelater case the winning rule is generally a first price rule. Thesecond price auction is rarely used because bidders feel thatthe information may be used to unfair advantage of the buyer.However, in Internet based buying, where anonymity of buyersand sellers and non disclosure of prices may be assured, thesecond price auction may yet be gainfully employed. While,we have seen from the revenue equivalence theorem, that theexpected revenues from all the basic auctions may be the same,there are significantly different implications when the auctionsare to be automated. We summarize the differences below.

1) Computational Implications of the four Basic AuctionMechanisms: As indicated in the previous section, complexitycan be analyzed at the level of the agent - strategic and valua-tion complexities, and the mechanism - winner determinationand communication complexities.

Strategic Complexity: Clearly for an agent participating ina Vickrey auction, which is incentive compatible, the strategiccomplexity is reduced to just bidding one’s true valuationwithout any consideration for the strategies that would befollowed by other agents. However, agents in the English,Dutch, and first price sealed bid auctions require to base theirbids not only on their private valuations but also conditionthem on the valuations of their competitors. This requires themto process additional information - the number of competitors,the probability distributions of their valuations, etc. whichincreases their computational overhead. The English auctionhas certain benefits relative to the Dutch and First PriceSealed auctions. In an English auction, if all bidders bid up totheir true valuations, no single bidder can gain by unilaterallydeviating from this truthful strategy.

Communication Complexity: The communication overhead,and hence the processing overhead, is clearly much higher inmulti round mechanisms like the English and Dutch auctionsas compared to single shot mechanisms such as the Vickreyand first price sealed bid mechanisms.

B. Volume Discount Auctions

In a procurement context when a single buyer and multiplesellers who wish to exploit scale economies are present, avolume discount auction is appropriate. Here suppliers providebids as a function of the quantity that is being purchased[17], [54]. The winner determination problem for this type ofauction mechanism is to select a set of winning bids, wherefor each bid we select a price and quantity so that the totaldemand of the buyer is satisfied at minimum cost.

TABLE INOTATION FOR VOLUME DISCOUNT AUCTIONS

Gquantity of itemHnumber of suppliersIindex for the suppliers (

IKJML�N�O3O&O3N H)P6Q

supply curve (bid) from supplierIR Q

number of price-quantity pairs in bidP QS

index for price-quantity pairs,S JTL�N3O&O3O3N RUQV�QXW

unit price the supplierI

charges if the number of unitsbought from this supplier is withintheS�Y[Z

interval \ G QXW&] ^ _�` NaG QXW&] Z�bdc Z�ef QXW decision variable that takes value 1 if the buyerbuys a quantity in the range \ G QXW&] ^ _�` NaG QXW&] Z�bgc�Z eh QXW a continuous variable that specifies the exact numberof units bought

The winner determination here is fairly straightforward ifthere are no business constraints such as maximum/minimumnumber of units that are to be purchased from a supplier,minimum/maximum number of winning suppliers, etc. Thecomputational problem is to simply find the supplier whooffers the best price and buy the entire quantity from thisseller. However, procurement problems in practice rarely arewithout business constraints. See [55] for a comprehensivetabulation of business constraints that are observed in practice.With inclusion of business constraints, the winner determina-tion problem becomes a mixed integer programming problem(MIP) [17], [56]. In the following subsection, we first presentthe mathematical formulation of a volume discount auctionproblem adapted from [54], [56] and then discuss the compu-tational issues that arise in determining the winning bids.

1) Mathematical formulation of volume discount auctions:The volume discount procurement auction is represented inthe following way [54], [56]. Table I provides the notation.� The buyer needs to procure a quantity i of an item.� The buyer identifies a list of potential suppliers jk����������8��l who can bid in the auction.� Each supplier responds with a bid composed of a supply

curve. A supply curve from supplier j given by abid mon consists of a list of p#n price quantity pairs,� =q n + ��r iEn +�s tvu�w � iEn +�s x �vy x$z �8�������8� {q n�|~}���r iEn8|C} s tvu&w �i�n8|~} s x �gy x7z . Each price quantity pair{q n�����r iEn&� s tvu�w � iEn&� s x �vy x7z � specifies the price

q n&�that the supplier charges per unit of the item if thenumber of units bought from this supplier is withinthe interval r i�n&� s tvu&w � iEn&� s x �vy x$z . It is assumed that thequantity intervals for the supply curve are all pairwisedisjoint. Also, note that different unit prices are usedfor different ranges within the overall quantity i n&�(that is, if a quantity spans multiple intervals, the unitprice for different spanned intervals will be taken as thedesignated unit prices in the intervals).

The MIP formulation is as follows:� A decision variable �/n&� is associated with each price-quantity pair

=q n&� ��r i n&� s t�u�w���i n&� s x �vy x z � for each bid m n .This is a ����� variable taking on the value 1 if webuy some number of units within the price range and0 otherwise.

11

� A continuous variable � n�� is associated with each price-quantity pair, which specifies the exact number of unitsof the item that are purchased from the bid � n within thisprice-quantity pair.

The formulation is (see Table I):

�U�����n�� +

|C}��&� + � n&�

q n&�K���n8� +

|C}��&� + � n��;�Kn&� (1)

subject to:

� n�� � i n&� s x �gy x���i n&� s tvu�w��3� n&�E� ���/j����$��������l5�X�����$�������&p n(2)

|~}��&� + �1n&� � ����jU���$���������&l5� (3)

��n8� +

|C}��&� + �-n&� � �1n&�;iEn&� s tvu&w ����i (4)

� n&�E¡¢�£�¤���$ ¥��jU�¦�$�������8��l5�3�%�����$����������p�n§�� n&� ������jU�¦�$���������&l5�3�%�����$�������8��p n

The coefficient �Kn�� is a constant and computed a priori as:

� n��~��8¨ +�t � +

q n t iEn t[s x �gy x �©iEn t[s tvu&w � (5)

In the formulation provided by [54], it is assumed that �types of items are being bought, which is a more generalizedproblem. However even with a single item, the MIP formula-tion is NP-hard. Additional side constraints such as a limit onthe number of winning suppliers and quantity constraints at thelevel of the supplier increase the complexity of the decisionproblem.

2) Computational Issues: The model is developed to ad-dress the winner determination problem with the implicitassumption that agents participating in the auction have abidding strategy in place, perhaps with the use of gametheoretic analysis. This being the context, we restrict ourattention to computational issues that arise at the mechanismlevel and neglect analysis of any computational issues at theagent (bidder) level. The MIP formulation in the previoussection is a variation of the multiple choice knapsack problemwhich is � q -Hard. Although Knapsack Problems, from atheoretical point of view, are almost intractable as they belongto the family of � q -hard problems, several of the problemsmay be solved to optimality in fractions of a second [57] andcan be considered as the simplest among � q -Hard problemsin combinatorial optimization. This is because the knapsackproblem exhibits special structural properties which makes iteasy to solve. Dynamic programming techniques, branch-and-bound algorithms and polynomial time approximation schemeshave been proposed to solve knapsack problems. For a detailedexposition of several of these methods refer to [57]. Kamesh-waran [58] shows that multi-unit procurement with volumediscount bids leads to piecewise linear knapsack problems and

proposes a variety of exact and heuristic techniques to solvethis class of problems.

C. Single Item, Multi-Unit Procurement to Minimize TotalCost of Procurement (Supply Chain Auction)

In the previous subsections, we examined auction mecha-nisms in automating supplier selection situations which co-incide with a fairly operational view of procurement. Whenthe supplier selection process within a supply chain setting isviewed, at a higher level of granularity, through a strategiclens, a procurement manager is concerned with a larger set ofissues: What is the impact of a sourcing decision on the totalcost of procurement? How does one structure the sourcing poolso that the total cost of procurement is minimized? A partiallist of important cost components that contribute to the totalcost of procurement could include: price per unit, logisticscost, inventory holding costs, and lead time costs. Classicalauction literature, however, has focused on price and ignoredthe impact of other costs on the sourcing decision.

A first step in incorporating these other cost componentswhile making a strategic sourcing decision is taken by Chen,Janakiraman, Roundy, and Zhang [59]. They provide the de-sign of single item, multi-unit auction mechanisms that achieveoverall supply chain efficiency while taking into accountproduction and transportation costs. The concern is to bringtogether the business requirements within a global supplychain, economic/game theoretic desiderata, and computationalissues in building the decision model. We call this the supplychain auction.

The procurement problem addressed is as follows: A buyerhas requirements, called consumption quantities, for a certaincomponent at a set of geographically diverse locations. It isassumed that the buyer has private valuations of consumptionquantities at the demand locations, which forms the consump-tion vector and that she will act strategically to maximize herutility. Multiple suppliers, each owning a set of productionfacilities, are available to satisfy this demand. Every supplierhas a production cost, which can be described by a convexfunction of the quantities that he produces at his productionlocations. It is also assumed that each supplier is a rational,self interested player who is trying to maximize his payoff(payment received minus the production cost). In addition thebuyer knows the per unit transportation cost along each linkin the supply network and she pays for all shipments alongthe links.

A key requirement in such resource allocation problemsis to ensure that the allocation is efficient, in the sense thatcontracts are allocated to suppliers who provide the best value.To do this the buyer needs to know the production costs ofthe suppliers, truthfully. So, efficient allocation and incentivecompatibility are two of the key desiderata. There are three keymechanisms for multi unit auctions: Pay-as-you-bid, uniform-price, and VCG auctions. From auction literature, it is knownthat VCG based mechanisms are both incentive compatibleand efficient. The auction models presented below for supplychain procurement essentially invoke the VCG structure.

12

TABLE IINOTATION FOR AUCTION T, AUCTION R AND AUCTION S

Hnumber of suppliersI

index for suppliers,IKJTL�N3O&O3O3N Hª

total number of supplier production facilities« index for production facilities, « JML�N&O3O3O�N ªRtotal number of buyer locations¬ index for buyer locations, ¬ JTL�N3O&O3O3N Rª Qset of production facilities owned by supplier kI;­index for supplier that owns production facility n®&¯ consumption at demand center m, ®�¯�°?±f ­ production quantity at production facility n² Q vector of size ³ ª Q ³´ ­ ¯ quantity shipped from the production facility« to demand center ¬µ ­ ¯ cost of transporting unit quantity fromproduction facility « to demand center ¬h Q ¯ ¶ ­7·-¸ } ´ ­ ¯ , total quantity shipped todemand center ¬ by supplier

I¹ Q7º ² Q;» production cost function for supplierI

( ¼B½ ¸ } ½;¾�¼ )¿ Q º ² Q » bidding function from supplierI

( ¼ ½ ¸ } ½ ¾�¼ )

1) Model Formulation: Chen, Janakiraman, Roundy, andZhang [59] present three separate auction mechanisms - Auc-tion T, Auction R, and Auction S. The first two mechanismsincorporate transportation costs explicitly whereas in the thirdmodel, only production costs are considered for the allocationdecision and transportation decisions are made subsequently.This provides a means to compare the two approaches and theimplications for total cost of procurement strategies. Before wepresent the models, the notation used is introduced in TableII.

2) Auction T: In this auction mechanism, the buyer submitsa fixed consumption vector q to the auctioneer. Supplier ksubmits to the auctioneer a bid function ÀCn =Á n7� for supplyingÁ n units, for which he incurs a production cost � n >Á n7� , Á n�¡ÂMÃ Ä } Ã

. The supplier may or may not see the consumptionvector.

The auctioneer decides the quantities awarded to each sup-plier, and the amounts transported from suppliers’ productioncenters to buyer’s demand locations by solving the followingwinner determination problem.

�U�����n8� + À�n

=Á n7� �Ä�0 � +

|�Å � +ÇÆ 0 Å~È 0 Å (6)

subject to:Ä�0 � + È 0 Å �ÊÉ Å �&ËÌ�����������8� pÎÍ (7)

|�Å � + È 0 Å �Ê��01���5���$�������8���ÏÍ (8)

È 0 Å �����&ËÌ�����������8��pÎÍ&�D�¦�$���������&�D� (9)

A Vickrey based payment rule, belonging to the moregeneral truth-inducing VCG family described in [60], is used.The rule, in words, essentially is:

Payment to = Bonus payment on + Bid ofvendor j account of the value vendor j

that the vendor j addsto the system byparticipating in theauction.

That is, if Ð {Ñ � is the optimal value of the objective functionfor a given q; ÒÓ�A� ÑÕÔ?Ñ×ÖÙØ �&Ð =Ñ ��ÚÜÛÝ and werestrict

Ñ ¡©Ò to ensure sufficient supply capacity. If>Á�Þ ��ß Þ �

is an optimal solution, and Ð ¨/n =Ñ � is the optimal value ofthe objective function with the additional constraint that thesupplier k does not participate in the auction (

Á n �à� ), thebuyer will pay supplier j :

á�ân {Ñ �ã�ÝÐ ¨/n =Ñ �ä�TÐ {Ñ � � À�n >Á â n � (10)

While it is true that the payment rule induces rationalsuppliers to bid their true costs, À~n >Á n�� = � n >Á n�� , irrespectiveof other suppliers’ bids, it also results in adverse effects for thebuyer. In situations where the production capacity is tight, andthe production costs are sharply convex, the contribution that asupplier makes to the system can be significantly high resultingin disproportionately high payments even when production isat a low cost. Alternately, as pointed out by the authors, whensuppliers’ capacities are asymmetric, removing a supplier tosolve the optimization problem may result in computing aninfinite value for the contribution that is made, which isan undesirable consequence. To rectify these problems, theauthors propose Auction R.

3) Auction R: In this auction mechanism, the winnerdetermination (optimization) problem solved by the auctioneerretains the essential structure of Auction T except that thepayment rule is modified. This modification results in muchlower payments, compared to Auction T, by the buyer whilesimultaneously ensuring that the sellers bid their true costs. Inessence, incentive compatibility is retained.

The key idea introduced to achieve this result is as follows:Here, the buyer submits a utility function å {Ñ � , which is aproxy for the true consumption utility æ =Ñ � , in place of aconsumption vector as in Auction T. The auctioneer solvesa modified version of the winner determination (optimization)problem using a new payment rule. This rule follows the VCGpayment structure but also incorporates the utility functionå =Ñ � in computing the payments. The utility function es-sentially serves the role of a reservation price function foreach unit of the item that the buyer might acquire. While inmost auctions the payment made to a supplier is based onbids from all suppliers, in Auction R, part of the payment isdetermined by the proxy utility function acting as a reservationprice function. This prevents unusually large payments.

There is however a price to pay for this improvement.In order to submit an optimal utility function åF: {Ñ � , thebuyer needs to know the suppliers’ production cost functions.However this may not hold in reality. So the buyer will haveto expend effort to at least get a probabilistic belief of thecost functions. This causes uncertainty in both payments andconsumption quantities. Numerical examples to illustrate thiseffect are provided in [59].

13

4) Auction S: In order to compare the cost savings, if any,that could be achieved by taking an integrated view of sourcingdecisions, the authors formulate Auction S. Here the buyer’ssourcing decision is based only upon the bids submittedby the sellers, and the transportation costs are determinedsubsequently.

The buyer submits a consumption vector q as in AuctionT. The auctioneer solves two optimization problems, one todetermine allocation and payments and the other to optimizetransportation costs, separately.

Winner Determination Problem:

���v���n�� + À�n

>Á n7� (11)

subject to:

Ä�0 � + È 0 Å �ÊÉ Å �&ËÌ�����������8� pÎÍ (12)

|�Å � + È 0 Å �Ê� 0 ���5���$�������8���ÏÍ (13)

È 0 Å �����&ËÌ�����������8��pÎÍ&�D�¦�$���������&�D� (14)

If Ð1ç {Ñ � is the optimal objective value,Á ç the production

vector, and Ð ¨/nç =Ñ � the optimal objective value without sup-plier j , the buyer’s payment to supplier k is given by the rule:

áKèn =Ñ �é�êÐ ¨/nç =Ñ �!�¢Ð ç {Ñ � � À n >Á çn � (15)

We can observe that this payment rule still retains theVCG structure, and hence the suppliers will submit theirtrue cost functions. Subsequently the transportation costs aredetermined by solving the following optimization problem:

Transportation Problem:

���v� �0 � +

|�Å � + Æ 0 ÅÝë È 0 Å (16)

subject to:

Ä�0 � + È 0 Å �ÊÉ Å �&ËÌ�����������8� pÎÍ (17)

|�Å � + È 0 Å �Ý�

ç0 �&�D�¦�$���������&�ÏÍ (18)

The buyers total outflow, ì ç =Ñ � , is given by ì ç {Ñ �í�¶ �n�� + á çn {Ñ � � ¶ Ä0 � + ¶ |Å � + Æ 0 Åîë Èç0 Å . The numerical exper-

iments clearly show that Auction S minimizes total productioncosts but leads to higher supply chain costs than Auctions Tand R.

5) Computational issues: The crucial contribution of thismodel has been to take an integrated view of the sourcing prob-lem by combining the pricing and the transportation decisions.The mechanism incorporates game theoretic considerations insupply chain formations when products are sourced globallyfor demand points that are distributed geographically. Since themechanism is incentive compatible for the suppliers, the com-plexity of evolving an optimal bid strategy is simply reduced toreporting the actual production costs, thereby eliminating theneed for complex modeling of competitors’ behavior. At themechanism level however,

l � �;� optimization problems needto be solved to decide the allocations and payments. Further,since the mechanism is not incentive compatible for the buyer,the buying agent needs to solve an optimization problemto decide an optimal bidding strategy. Such an optimizationproblem is known to be challenging and hence a numericalapproach may be warranted.

Also, from the analysis presented in this paper, it is clearthat in designing auction mechanisms for complex supplychains, focus on truth revealing auction mechanisms is to beaugmented by pragmatic considerations of limiting the totalpayout of the buyers. This can be achieved by using the ideaof a reservation price suitably expressed as a utility function.

D. Summary and Current Art

In this section we reviewed models proposed to handlethe procurement of both single unit and multiple units of asingle item, with a single attribute - price as the criterion fordecision making. In the first model, we illustrated the gametheoretic considerations that influence the bidding behavior ofagents. In the second model, in the absence of game theoreticrequirements, we pointed out the computational complexity ofthe winner determination problem in a more complex sourcingsituation. The third model brought together the game theoreticas well as computational issues that complicate the sourcingproblem. This was done to illustrate the fact that in the absenceof proper mechanism design, the buyer could end up leavingmoney on the table.

In Section II, we have already presented the case summaryof MARS inc. [17] which shows the successful applicationof single item, multi-unit procurement auction with volumediscount bids from suppliers in a real-world setting. Thecomputational challenges involved in the winner determinationproblem there are described in [56], [54]. Kameshwaran [58],in his recent work, has shown that single item, single attribute,multi-unit procurement with volume discount bids leads topiecewise linear knapsack problems to be solved. In hiswork, Kameshwaran has developed several algorithms (ex-act, heuristic-based, and fully polynomial time approximationschemes) for solving such knapsack problems.

Kothari, Parkes, and Suri [68] consider single-item, singleattribute, multi-unit procurement auctions where the biddersuse marginal-decreasing, piecewise constant functions to bidfor goods. The objective is to minimize cost for the buyer.It is shown that the winner determination problem is a gen-eralization of the classical 0/1 knapsack problem, and henceNP-hard. Computing VCG payments also is addressed. The

14

Scenarios in Single Item, Math.Programming Game Theoretic Case Studies and Other RelevantSingle Attribute Procurement Models Analysis Surveys ReferencesBasic Auction Types [5], [23], [52], [61] [62], [14], [10], [11] [25], [26], [27], [63]

[41], [64] [12], [13], [3], [4] [29], [28], [24][23], [38], [40]

Volume Discount Auction [58], [65], [65] [66], [17], [67], [68] [17]Supply Chain Auction [59], [69]

TABLE IIICLASSIFICATION OF SINGLE ITEM, SINGLE ATTRIBUTE PROCUREMENT AUCTIONS

authors provide a fully polynomial time approximation scheme(FPTAS) for the generalized knapsack problem. This leadsto an FPTAS algorithm for allocation in the auction whichis approximately strategy proof and approximately efficient[68]. It is also shown that VCG payments for the auctions canbe computed in worst-case ï =ðDñvò�ó �,� time, where

ðis the

running time to compute a solution to the allocation problem.Dang and Jennings [67] consider multi-unit auctions where

the bids are piece-wise linear curves. Algorithms are providedfor solving the winner determination problem. In the case ofmulti-unit, single-item auctions, the complexity of the clearingalgorithm is ï � ( � �;� 0 � where � is the number of biddersand ( is an upper bound on the number of segments ofthe piecewise linear pricing functions. The clearing algorithmtherefore has exponential complexity in the number of bids.

To summarize, the two main issues in single item, multi-unit, single attribute auctions are (1) to reduce the complexityof the winner determination problem and (2) to make themechanism strategy proof. Table III provides an appropriatetaxonomy for this type of auctions and gives a listing ofrelevant papers under the various categories. In the nextsection, we review mechanisms that enable the procurementof multiple items with a single attribute (namely price) as thecriterion for decision making.

V. MULTI-ITEM, SINGLE ATTRIBUTE PROCUREMENTMECHANISMS

In the previous sections, we reviewed mechanisms thatsupport single item procurement. In one of the mechanisms,we discussed the inclusion of logistics costs as an additionalelement influencing the procurement decision. Clearly suchsupply chain criteria are of crucial importance to procurementmanagers responsible for large global supply chains. Typicallypurchase requests within such large purchasing organizationsprovide opportunities to exploit complementarities in logisticscosts and often in production costs too. For instance, in oneof the negotiation scenarios depicted in Section II.A, a familyof cutting tools may be procured by a series of sequentialreverse auctions, one for each of the items. This has at leasttwo consequences: Firstly, the price that a supplier may bewilling to offer may depend in complicated ways on whatother items he wins, and since there is uncertainty associatedwith this, the suppliers have no incentive to bid aggressively.Secondly, the suppliers do not have an opportunity to expresstheir unique complementarities in production costs and hencethe buyer cannot exploit the economies of scope associated

with bundling the demands. In such cases it may be beneficialto allow the suppliers to bid on combinations of items ratherthan on single items. Such auctions are called combinatorialauctions.

Simply defined, a combinatorial auction (CA) is a mecha-nism where bidders can submit bids on combinations of items.The winner determination problem is to select a winning setof bids such that each item to be bought is included in at leastone of the selected bids, and the total cost of procurement isminimized. In this section we first present the crucial issuesthat arise when CA’s are applied to procurement settings. Wethen review approaches to modeling multi-item procurementin supply chain settings which take business and capacity levelconstraints into consideration.

A. Issues in Multi-Item Procurement

Important issues in combinatorial auctions are well surveyedin [30], [32], [16], [37], [33]. In the multi-item procurementcase, when suppliers are allowed to respond with combi-natorial bids, the winner determination problem becomes aweighted set covering problem, which is known to be NP-hard.In addition, since the bidders can submit bids on combinationsof items, the representation of the bid is also a crucialimplementation issue. Bidding language issues are discussedin [70], [71].

As opposed to single item procurement scenarios, in multi-item procurement scenarios, the range of issues to be consid-ered is vastly expanded because practical business level issuesand constraints need to be included in the decision model.These could include exclusion constraints (for example, item Acannot be procured from supplier X), aggregation constraints(for example, at least two and at most five suppliers needto be selected for goods of category B), and exposure con-straints (for example, not more than 25% of the procurementvalue should be assigned to any one supplier) [72], [17]. Byincluding these constraints in the decision model, a sensitivityanalysis of side constraints may also prove to be a usefultactical input to procurement planners.

B. A Single Round Combinatorial Procurement Mechanism

As indicated above, the reverse combinatorial auction prob-lem is a set covering problem, which is NP-hard. Additionalside constraints make a fundamental impact on the problem.Even finding a feasible solution when exposure constraints arein place is NP-hard. The basic formulation of the problem andthe additional side constraints are presented below and solution

15

TABLE IVNOTATION FOR COMBINATORIAL PROCUREMENT

ªtotal number of items to be procuredôindex for items,

ô JTL�N3O�O�O�N ªõ bnumber of units of item

ôdemandedH

number of suppliersIindex for suppliers,

IKJML�N�O3O&O3N HRmaximum number of bids allowed for any supplierSindex for a bid,

S JTL�N3O&O3O�N RP6QXWbidS

of supplierIö b QXW 0-1 variable which takes value 1 iff

P QXWwill supply the entire lot corresponding to item

ô÷ QXW price associated with bidP QXWø�Q�] ¯ b ­ minimum quantity that should be allocated to supplier

Iø�Q�] ¯6ù�ú maximum quantity that can be allocated to supplierIf QaW decision variable that takes value 1 iff

P QXWis allocated´ Q indicator variable that takes value 1 iff

supplierI

is allocated any lotû ¯ b ­ minimum number of winners requiredû ¯�ù�ú maximum number of winners allowed

techniques are discussed thereafter. The following formulationis from [54], [17]. Table IV provides the notation.

1) Mathematical Formulation: For each item type �í��$�������8��� there is a demand ü � . Each supplier jî¡Dl is allowedup to p bids indexed by � . Associated with each bid mEn&� isa zero-one vector ý � n&� , �B�@�$�������8�&� where ý � n&� �@� if mon&�will supply the entire lot corresponding to item � , and zerootherwise. Associated with each bid m n&� is price þ n�� at whichthe bidder is willing to supply the combination of items in thebid. A mixed integer programming (MIP) formulation can bewritten as follows:

�U�����n�� +

|��&� + þ n&���1n&� (19)

)$� ÿ��n�� +

|��&� + ý

�n&� � n��B� ��� �����$�������8�&� (20)

� n&� ¡M�;�����7 ��jU���$���������&l �%���¦���������8��p

åTn s Å � 0 È n �Ä�� � +

|��&� + ý

�n&� ü� �1n&�8�/j����$�������8��l (21)

�� � +

|��&� + ý

�n�� ü� � n�� � å¢n s Å����7È n �/j���������������l (22)

|��&� + � n��B� È n ��jU���$�������8�&l (23)

( Å � 0 ���n�� + È n�� ( Å���� (24)

È n ¡¢�£�¤���7 �/j����$�������8�&l

å n s Å � 0 and å n s Å���� are the minimum and maximumquantities that can be allocated to any supplier j ; Constraints(21) and (22) restrict the total allocation to any supplier tolie within

å n s Å � 0���å n s Å���� � . È n is an indicator variable thattakes the value 1 if supplier j is allocated any lot. ( Å � 0 and( Å���� are respectively the minimum and maximum number ofwinners required for the allocation and constraint (24) restrictsthe winners to be within that range.

2) Solution Approach and Discussion: Although the deci-sion model is NP-hard, integer programming techniques areknown to be effective in solving problems with 500 items andup to 5000 bids [17]. For the purpose of this paper, the authorsuse IBM’s OSL (Optimization Solutions and Library) to solvethe IP formulation. From the experiments that were conducted,the following conclusions could be derived:� Firstly, varying the aggregation constraint seems to have a

large impact on the computation time. This is because, aswe commented earlier, the problem of finding a feasiblesolution itself is NP-complete.� Secondly, the min-max exposure constraints also havea significant effect on the computational time. In somesense, the exposure constraint could itself be a proxy forthe aggregation constraint and hence the behavior appearssimilar.

C. An Iterative Reverse Dutch Auction for CombinatorialProcurement

In the previous section, we discussed a purely computationalview of the multi-item procurement problem without regard toeconomic desiderata. It is however important for a variety ofreasons to lend credence to economic issues in the modelingand analysis of multi-item procurement. Firstly, we would liketo ensure that procurement contracts are allocated to thosethat value them most. In the absence of economic analysis,the game is open for suppliers to indulge in strategic biddingbehavior in an effort to extract the maximum possible surplusutility from the negotiation. This is especially true when singleshot mechanisms are used. Secondly, even if multi-roundcombinatorial auctions are used where allocation and pricinginformation is disclosed at the end of each round, it is notclear to suppliers as to how they need to reformulate theirbids. The two issues together may prompt wasteful usage ofresources by each supplier in trying to outsmart the buyer andother suppliers. One way to amend the situation is to design anincentive compatible mechanism, that is. a mechanism whichprovides incentives for truthful bidding thereby making it thedominant strategy for all participants in the mechanism.

Generalized Vickrey Auctions (GVA) generalize the singleitem second price auction (Vickrey auction) proposed in [39].This is an incentive compatible mechanism for combinatorialauctions, which can be applied to the procurement context[73]. However, one issue that needs to considered is that theGVA requires an optimal solution to the allocation problemwhich is NP-hard. In the absence of an optimal solution to theallocation problem, the resulting mechanism, whose allocationis obtained through some approximation scheme, is no longerincentive compatible [60]. However, iterative algorithms for

16

combinatorial auction problems have been proposed by [74]and [75] which try to resolve the tension between economicand computational efficiencies. In a similar vein, Biswas andNarahari [73] have proposed an iterative reverse Dutch auctionscheme for combinatorial procurement auctions which wediscuss next.

1) Integer Programming Formulation for the Reverse DutchAuction: The basic idea behind this approach is to formulatethe procurement problem as a weighted set covering problemand use the Dutch auction format to develop an iterativesolution scheme. This approach reduces the computationalcomplexity by breaking up one large GVA into several smallerGVA’s. The iterative algorithm itself is motivated by [76],and involves a two step process: (1) progressively increasingthe average reserve price of the items in each round, and(2) allocating items for which bids satisfy the reserve price.Classical (forward) Dutch auctions are decreasing auctionswhich have been typically used for multi-unit homogeneousitems. In the single unit Dutch auction the auctioneer beginsat a high price and incrementally lowers it until some biddersignals acceptance. Similarly in the multi-unit case the price isincrementally reduced till all the items are sold or the seller’sreserve price is reached.

In the reverse auction for procurement the buyer has aprocurement budget and tries to procure a bundle of itemsat minimal cost not exceeding the procurement budget. Shestarts with a low initial willingness to pay (say equal to zero)and keeps on increasing the willingness to pay until the totalbundle is procured or the budget limit is reached. This totalbudget cannot be divided linearly into budget for each itembecause of the complementarities involved.

The iterative mechanism proposed in [73], [77] consists ofmultiple bidding rounds denoted by ÿä�Ê�¤������������� ( ÿ!� � is theinitial round). The buyer sets å m���� , maximum willingnessto pay for the remaining bundle m�� to be procured in roundÿ . The pricing of items is not linear, therefore the cost of theallocated bundles cannot be divided into price of individualitems. Therefore we compute þ � , the average willingness ofthe buyer to pay for each item in round ÿ .

þ��!� å m�� �� m � �Ê� � ����� m������� (25)

The payment made by the buyer for the subset ( � in iterationÿ is ��: ( � � . The average price paid by the buyer for each itemprocured is � � � ����� ç����à ç�� à . Iterations in which no items areprocured are ignored.

The reserve price of the seller for any bundle ( in iteration ÿis� ( � ���X¨ + . GVA with reserve prices [64] used in each iteration

to solve the allocation and payment problems efficiently. Thebundle procured in round ÿ is denoted by (�� . Therefore theinteger programming formulation of the GVA problem withreserve prices in iteration ÿ becomes

�U: ( � �� �������n�� +

�ç "! � n (ä� È (���j��

TABLE VNOTATION FOR ITERATIVE REVERSE DUTCH AUCTION ALGORITHM#

set of items to be procuredôindex for an item to be procured,

ô%$ #ûa subset of items,

û'&(#Htotal number of suppliersIindex for suppliers,

I�JTL�N�O�O3O3N H)iteration numberP Y bundle remaining to be procured in iteration

)º û Y » set procured in iteration)*,+�º û Y » buying price of the setû Y in iteration

)- Y average buying price of each item in iteration)÷ Y average price set by the auctioneer in iteration)ø©º[P Y » maximum price set by the auctioneer for the bundleP Y in iteration

)- Q º û » valuation of setû

to sellerI´ º û%NaI » indicator variable that takes value 1 iff

bundleû $ #

is allocated to agentI. increment in buyer’s willingness to pay

)�� ÿ�� �ç0/ �

��n�� + È

(!� j����í��� �ä¡21�ç0 %! È (!��j�� � �Ï��jU���$�������8�&l�$n (ä� È (!��j���� � ( � ���X¨ + ��(4351�� ��jU�¦�$���������&l��n�� +

�ç0 %! � n (ä� È (!��j¤� � å m � �5�/(4361È (���j��ä�Ê�������/(7361��3�/j��¦���������8��l (26)

The notation for the Iterative Reverse Dutch Auction(IRDA) algorithm is provided in Table V.

The IRDA Algorithm: As opposed to a naive approach, theIRDA algorithm relies on choosing an increment in each roundsuch that it is based on the size of the bundle to be procured.

1) Suppose the buyer’s initial willingness to pay for theentire bundle m�8 is zero i.e. å m�8-�ä�Ê� . Therefore thewillingness to pay for each item is also zero i.e. þ%8~� � .Since no sellers are likely to be interested to bid at thisprice, therefore

� : ("8-�¢�Ê��8Ê� �2) Increment the average willingness to pay for each item

by 9 to þ/+�� ��8 � 9 . This actually means that thebuyer’s willingness to pay for the bundle mE+ is changedto å mB+8� � � mB+ �0: þ1+ . We assume that the increment9 in every iteration is constant. The reserve price of anybundle ( for the sellers becomes

� ( � � 8 .3) Solve the allocation problem if there are any bids i.e.

for iteration ÿ¢� � solve Eq. 26 and calculate � + . Thisis again a combinatorial optimization problem. But thisis much smaller than the complete problem.

4) Allocate the subsets to the winners. Remove the allo-cated items from the set to be procured and incrementthe average willingness to pay for each item to þ . ���+ � 9 , i.e. the maximum willingness of the buyer to payfor the remaining bundle mo. is å mC.��¦� � m~. �0: þ . .The new reserve price of any bundle ( of items for thesellers is

� ( � �§+ .

17

TABLE VINOTATION FOR PROCUREMENT UNDER CAPACITY CONSTRAINTS

ªnumber of components (items) to be procuredôindex for components,

ô JML�N�O3O&O3N ª® b number of units of componentô

to be procuredHnumber of suppliersIindex for resources (suppliers),

IKJTL�N3O&O3O�N H; Q capacity of supplierIö Q b amount of resourceI

required for componentô)

index that represents auction rounds< Q b º ) » bid submitted in round)

by supplierI

for any quantity between 0 and ® bf Q b º ) » decision variable for bid< Q b º ) »

5) Go to step = and repeat until the buyer can procure theentire bundle or the upper limit i.e. the total procurementbudget is reached . In any iteration ÿ the followingcondition should be satisfied:> ò >@? ñ A � òCB�D �@� � � � >FE D0G%ó � > �Õå m��3� � �X¨ +�

� � 8 � : ( � �This approach to the combinatorial procurement problem,

solves the problem of capturing cost complementarities. How-ever, in typical strategic procurement settings, capacity plan-ning at suppliers is a crucial task meriting detailed attention ifsupply lines are not to be disrupted. This planning of capacityand allocation of contracts can be significantly difficult whenmultiple contracts are to be established simultaneously. Wediscuss this issue in the next subsection.

D. An Iterative Procurement Mechanism for Capacity Con-strained Environments

In some strategic sourcing settings, where suppliers arelooked upon as extensions of the enterprise, it is imperativeto engage in prudent capacity management. Failure to do socould result in production line disruptions and high overtimecosts. So when making multi-item procurement decisions, evenwhen the product attributes do not show complementarities incosts, but share resources it is necessary to factor in capacityplanning at least at the aggregate levels. One recent effortto incorporate this aspect into auction based procurementmechanisms has been due to Gallien and Wein[69].

The procurement scenario that they analyze is as follows(see Table VI for notation): The buyer wants to procure É �units of component �?¡��§�$�������8���© . There are l supplierswilling to supply some or all of these quantities subject tooverall capacity constraints. Overall capacity constraints ofresource (supplier) j is H n and the usage can be describedby a linear model where ý n � is the amount of resource jrequired for component � . An auctioneer acts as an interme-diary between the buyer and the seller thereby decouplingthe information between the two, much like what happens onhttp://www.freemarkets.com/.

1) The Auction Mechanism: The mechanism is designedas a multi round auction where in each round ÿ the supplierj submits a bid � n � ÿ�� to sell any quantity between 0 andÉ � of component � . Further non-reneging and minimum bid

decrement rules are in place. In each round of the auctiona potential allocation

Áä ÿ���� � n � ÿ���� nJILK + sNMNMNM4s �PO s � ILK + sNMNMNM s Ä O isobtained by solving the linear program given below.

���v���n8� +

�� � + �8n

�& ÿ�� � n �� ÿ�� (27)

)�� ÿÄ�� � + ý%n

� � n �� ÿ�� � H�n ��jU���$�������8�&l#Í (28)

��n�� + � n

� �êÉ � � �����$���������&�ÏÍ (29)

�1n � ÿ����Ý� ��jU���$�������8�&l � �6���$���������&�At the end of each round, the potential allocation

Á n ÿ�� isdisplayed to the supplier j which can be used to better thebid in the following round. The supplier can choose to followa myopic best response (MBR) strategy embedded within asoftware based bid suggestion device. To use this feature, thesupplier j is required to submit his actual production costs�$n � for component � to the auctioneer. The next bid Qé:n ÿ � �£�to be submitted by supplier j is such that he maximizes hispotential payoff RCn Q ÿ����,S ¶ 0� � � �8n �� ÿ������$n � �3� n � ÿ�� in roundÿ by carrying out a suitable computation [69].

E. Summary and Current Art

In this section we reviewed mechanisms designed to sup-port multi-item procurement scenarios. In the first part, wediscussed a purely computational approach to the procurementdecision problem to illustrate the computational problems thatarise. In the next section we introduced economic concerns anddiscussed the limitations to achieving both computation andeconomic efficiencies. Finally we discussed the implicationsof capacity constraints at suppliers on decisions made throughcombinatorial auction models. Also, we deliberated on oneapproach to providing the suppliers with a bid suggestion de-vice to help in reformulating bids in a complex combinatorialenvironment.

Currently, combinatorial auctions constitute a very activeresearch area. There are several surveys that have appearedon this topic, for example, see [30], [31], [32], [16], [33],[42]. There is also a recent comprehensive book by Cramton,Shoham, and Steinberg [34]. There have been numerousefforts in solving the winner determination problems arisingin complex combinatorial auctions. The reader is referredto the papers [45], [88], [90], [80], [78], [79], [81], [94],[67]. Another topic of active research is design of truthfulcombinatorial auctions. Please refer to [83], [87], [60], [31] formore details. Design of iterative combinatorial auctions [84],[85], [42], [86], [93], [77], [91], [92] is one more direction inwhich a fair amount of research activity is in progress. Multi-attribute combinatorial auctions, however, have received littleattention due to the intrinsic difficulties involved.

Table VII provides an appropriate taxonomy of combinato-rial procurement auctions and gives a listing of relevant papersunder each category.

18

Scenarios in Math.Programming Game Theoretic Case Studies, Surveys,Multi-Item Procurement Models Analysis References

Combinatorial Auction [78], [79], [80], [81], [45], [22], [16], [68], [82] [17], [15], [37], [18], [19][83], [84], [85], [42], [86] [73], [77], [87], [88], [20], [21], [89], [31]

[90], [91], [92], [67] [32], [33], [34], [22][93], [77], [91], [92]

Volume Discount Auction [66], [65], [58] [66], [65] [17]

TABLE VIICLASSIFICATION OF MULTI-ITEM PROCUREMENT AUCTIONS

In the next section, we review models that address procure-ment scenarios with one more degree of complexity - sourcingdecisions based upon multiple attributes.

VI. SINGLE ITEM, MULTI-ATTRIBUTE PROCUREMENTMECHANISMS

The procurement mechanisms of the two previous sectionsinvolve a single attribute based on price. In practice, thesemechanisms address only a limited band of the negotiationspectrum, whereas sourcing decisions involve multiple criteria- both quantitative and qualitative [95]. Benyoucef, Ding, andXie [55] present an exhaustive, hierarchical list of criteria thatare used in evaluating and selecting suppliers. Incorporatingthese criteria into an automated negotiation tool to supportsourcing decisions has been the holy grail of purchasingprofessionals, industrial engineers, and computer scientists.Many software solution vendors now support multi attributereverse auctions in their e-sourcing solutions. Weighted, multi-parameter, multi-line item request-for-quotes (RFQ’s) andreverse auction capabilities are provided by Ariba, freemar-kets, and Procuri; i2 Technologies goes one step further andclaims to support all these within an optimization modulethat allows business level constraints to be incorporated [38].Since these are commercial products, it is not possible toindependently verify the mechanics of the automated sourcingsolutions. However, we conjecture that they use one or moreof the known approaches to multiple criteria decision analysis(MCDA) such as additive value models, analytic hierarchyprocess (AHP), lexicographic ordering, multi-attribute util-ity theory (MAUT), simple multi-attribute rating technique(SMART), and traditional weight assessment. For a detailedtreatment of these approaches, we refer the reader to [96].

In this section, our discussion focuses on single item,multi-attribute procurement problems to investigate (1) theproblem scenarios addressed, (2) solution approaches, and (3)computational and game theoretic issues.

A. Issues in Multi-Attribute Procurement

Simply defined, multi-attribute procurement refers to thedecision process related to the determination of a contractby considering a variety of attributes involving not just pricebut also aspects such as quality, delivery time, contract terms,warranties, after sales service, etc. In practice, multi-attributeprocurement scenarios come in various hues and shades. Toaid in analysis, we group these problems into three distinctcategories which we believe adequately reflect the issues to be

considered from a computational / automation point of view.The groups and their characteristics are:� Multiple attributes are known a priori and are uncorre-

lated; individual attributes have point values. The suppli-ers provide point bids where each bid has a single priceand each attribute has a single value, either provided bythe supplier or computed by the buyer.� Multiple attributes are known a priori and are uncorre-lated; each attribute can take an individual value from adomain of possible values for the attribute. The suppliersprovide configurable bids which specify multiple valuesand price markups for each attribute.� The multiple attributes are not known a priori and theyare uncorrelated. The suppliers may provide point bidsor configurable bids. This is not unlike many negotiationsituations where the buyer has only a rough idea of herrequirements but relies on suppliers to educate him aboutattributes relevant to the procurement decision.

A further level of complexity is involved when the attributesare correlated in each of the above cases. Also, each of theabove scenarios can occur in combinations.

In the first scenario above, if we assume that there existssome decision rule which, as a function of all the multiple at-tributes, ranks the bids, then the winner determination problemis relatively straightforward. In the latter two cases, however,the options are combinatorial in nature and hence the winnerdetermination problem will have exponential complexity. Thisraises the issue of compactness of information or bid repre-sentation. We now focus on methods to develop the decisionrule for ranking of bids.

Traditional approaches to developing the decision rule haverelied on either a hierarchical elimination process or onweighted average techniques. Since determining the hierar-chy or the choice of weights is not always straightforward,especially in the face of a large number of attributes, manysophisticated approaches, including the use of optimizationtools, have been proposed which we review next.

1) Elimination Methods: In this method, at each level,we eliminate from the bid list, bids that do not satisfy theselection rule. The selection rule may be a conjunctive rule ora lexicographic rule. In either case, a hierarchy of the attributesneeds to be established in the order of their importance, whichis fuzzy for most decision makers. To overcome this handicap,optimization based approaches have been devised. We brieflydescribe these techniques below:

Lexicographic Ordering: Here, on the first level, we selectthe most significant criterion and we compare bids based on

19

this. If a bid satisfies this criterion much better than the othersuppliers, then it is chosen, otherwise the bids are comparedwith respect to the second criterion, and so on.

Satisficing: In this technique we set minimum levels forevery attribute except one, which is the target attribute, suchas price. We select bids which satisfy the minimum levels andchoose the one with the optimal value of the target. It is alsopossible to iterate through the minimum levels.

Analytic Hierarchy Process (AHP): This is an analyticaltool, supported by simple techniques, that enables people toexplicitly rank tangible and intangible factors against oneanother for the purpose of resolving conflict or for settingpriorities. The process involves structuring a problem froma primary objective (e.g. selecting the best bid) to secondarylevels of objectives (e.g. performance objectives, quality needs,etc.). Once these hierarchies have been established, a pairwisecomparison matrix of each element within each level is con-structed.

Goal Programming: This is a way to handle multipleobjectives in what would otherwise be a LP. The basic conceptis to set ”aspiration levels” (targets) for each objective andprioritize them. One would then optimize the highest priorityobjective with respect to the original (”hard”) constraints.Next, a constraint is added saying that the first objectivefunction’s value must be at least as good as what was achievedor the aspiration level, whichever is worse. Now the secondobjective is optimized, turned into a constraint, etc.

2) Weighted Average Methods: The weighted averagemethods essentially rely on the introduction of a virtualcurrency which expresses the overall utility of a bid to thebuyer. The computation of the virtual currency is based uponthe utility function specified by the buyer and bids that achievethe best overall utility are declared winners. It is obviouslyof interest to the buyer to specify the best possible utilityfunction. We briefly describe some of these techniques below:

Traditional weighted average technique: Purchasing pro-fessionals have traditionally relied upon assigning weights toindividual attributes and using a simple additive rule to derivethe virtual currency. This is not dissimilar to approaches inmicro-economic theory for developing utility functions. Thishas been further refined by the swing weighting approachdescribed in [97]. The application of this approach can beseverely restricting when the number of attributes to be con-sidered is large.

Weight Determination Based on Ordinal Ranking of Alter-natives (WORA): This approach recognizes the pitfalls in usingthe weighted average technique and hence relies on linearprogramming techniques to compute the optimal weights foreach of the attributes. We detail this approach in the nextsection.

Inverse Optimization Methods: This technique is an im-provement over WORA, in the sense that it does not rely on asingle central agency (the buyer) to provide inputs to computethe optimal decision rule. Rather it uses a novel and intelligenttechnique based on inverse optimization to gather informationdistributed among various agents (sellers) in the marketplaceto develop the optimal scoring (decision) rule.

In the following sections, we emphasize two different ap-

TABLE VIIINOTATION FOR ORDINAL RANKING OF ALTERNATIVES

PSet of Bids,

P J � PUT N�O3O�O3N PWV �Iindex for supplier bids (

I�J5L�N&O3O3O&N H)« number of attributesô

index for attributes,ô JTL�N�O�O3O�N «G

vector of attributesf b Q level of attributeô

in bidI² Q vector of attributes of bidP�QX b º ² b Q » utility function for attributeô

in bidP Qû Q

score of BidP QY b weight for attribute

ô

proaches that have been proposed to develop multi attributeprocurement mechanisms.

B. Additive Value Model Based on Ordinal Ranking of Alter-natives

A straightforward approach to multi-attribute procurementis to assume that the utilities of each of the attributes areadditive in nature and hence a virtual currency, like the stockmarket index can be developed. Formally, we have a vector iof relevant attributes of a bid. We index the attributes by � andthe set of bids m ��;m + ���������&m � by j . See Table VIII fornotation. A vector

Á n?� � +n ����������� 0 n � is specified, where � � nis the level of the attribute � in bid ��n . In the simple case ofan additive utility function æ >Á n$� , each attribute is evaluatedthrough a utility function æ ��>Á � n � and the overall utility is thesum of all weighted utilities. This produces a virtual currencyto be used in mechanisms of the type proposed for singleattributes. The crucial issue however is the following: How arethe weights or the utility function decided? Naive approachesinclude the elicitation of buyers preferences through the designof smart web forms [1]. More recently techniques based ondecision analysis have been proposed by [97] and [98] whichwe describe next.

The WORA technique has been developed as part of anapplication framework to provide buyer side decision supportfor e-sourcing [97]. It is based on the realization that thebuyer can make deductions about the superiority of bidsthrough simple pairwise comparisons. By making many suchcomparisons, not entirely exhaustive, it is possible to builda larger information set to elicit intelligently the scoringfunction. This is done in a two step process.

In the first step, sample ordinal rankings of bids are pro-vided by the buyer. They are of the form m +[Z m .\Zm�] . These rankings are checked for intransitive preferencesand dominance violations. These sample rankings are thentransformed into constraints to a linear program, which thengenerates estimates for the decision maker’s weights. The LPis formulated as follows:

Let m � �£m + ��m . ����������mP�� be a subset of bids that areranked in the order m +^Z m ._Z m�] Z ����� Z mP� . The score(�n of each bid mBn is computed as:

(�nB�0�� � +

` � æ �& � � n � ��j (30)

20

TABLE IXNOTATION FOR INVERSE OPTIMIZATION TECHNIQUES

Hnumber of suppliersIindex for suppliers (

I�JTL�N3O&O3O3N H)« number of attributesô

index for attributes,ô JTL�N�O�O3O3N «V

number of cost parameters÷ price specified in a bida index for rounds of auction ( a J5L�N&O3O�O�N Vcb L )f b magnitude of non-price attributeôd b Qfe magnitude of parameter ÷

for attributeô

of supplierIg b e magnitude of parameter ÷ affecting attribute

ôof buyerh b eji magnitude of parameter ÷ affecting attributeô

in round a of the auctionk b º f b » unparameterized scoring rule in roundV^b L

where the weights ` � are unknown and � is the number of theattribute, and these satisfy the bid rankings. The LP is:

Maximize �subject to:( + �Ý( .(/.o�Ý( ]

...

("�X¨ + �Ý(l�¶ 0� � + ` � �¦��� ` � ��� for each �

A feasible solution to this LP can be found and results in aset of weights to be used in the scoring rule. The authors alsoprovide some numerical experiments to show the efficacy ofthe technique. These weights are then used within a standardsingle attribute like procurement mechanism with the virtualcurrency, obtained by combining price and all other attributes,substituting for price.

While this technique is an improvement over ad-hoc weightassignment models, it still relies on a single buying agentto indicate an ordinal ranking in order to come up withan optimal scoring function. This approach does nothing toexploit the cost complementarities that production systems ofthe suppliers may exhibit in providing certain attribute levels.In such cases it may be beneficial for the buyer to understandthe nature of the suppliers’ cost functions in terms of the non-price attributes. Understandably, it is not likely to be in theinterest of suppliers to part with this information. In the nextsection we discuss one approach which tries to overcome thisproblem.

C. Additive Value Model with Inverse Optimization Techniques

In some procurement scenarios where the establishment ofa contract depends upon multiple attributes (price, quality,delivery time, features and options, etc), a Request-for-Quote(RFQ) process is preferred. In this process, the buyer an-nounces a scoring rule in terms of the bid price and the variousattributes to be considered. It is not uncommon for the buyerto change the scoring rule to reflect any new information thathas been gleaned during the RFQ/negotiation process. This

idea is formalized in the eRFQ mechanism in [98] which isdesigned to address a procurement scenario with the followingcharacteristics and assumptions (see Table IX for notation):� A single item with multiple attributes is to be procured.

The attributes are indexed by �����$�������8��� ; k indexes thel suppliers; and mE�����������8� q � � indexes the rounds ofthe auction, where

qis the number of cost (and utility)

parameters.� Each bid submitted by a supplier is of the form þ,�&� + ����������� 0 � where þ denotes the price and � � is themagnitude of non-price attributes which are continuous,nonnegative variables.� Supplier k’s cost function ¶ 0� � + H � n � � � <-� n + �������8� <-� n�n�� ,is additive across attributes and H � n is increasing, convexand twice continuously differentiable in � � .� The auctioneer is assumed to know the form of thesuppliers’ cost function but not the actual parametervalues

>< � n +7��������� < � n�n �� ¶ 0� � + � � � � � á � +-�������8� á � n ��� þ , is the true utilityfunction of the buyer and the scoring rule is¶ 0� � + � �� � � �@� � +fo �������8�j� � n o �!��þ , in round pE�¦���������8� q ,which is increasing and concave in � � . Further, to guar-antee existence of solutions to the optimization problem,described later, a set of technical requirements are im-posed upon the cost, scoring and valuation functions.

The eRFQ mechanism is based upon an open ascendingauction format consisting of

q � � rounds. The firstq

rounds enable learning of theq

parameters, through InverseOptimization, and the last round is with an optimized scoringrule. Ahuja and Orlin [99] describe Inverse Optimization inthe following manner: ”A typical optimization problem is aforward problem because it identifies the values of observableparameters (optimal decision variables), given the values ofthe model parameters (cost coefficients, right-hand side vector,and the constraint matrix). An inverse optimization problemconsists of inferring the values of the model parameters (costcoefficients, right-hand side vector, and the constraint matrix),given the values of observable parameters (optimal decisionvariables)”.

In each round of the auction, the buyer announces a scoringrule in response to which suppliers submit bids. Activityrules and transition rules are imposed to move from oneround to the other. In each round the buyer ranks the bidsaccording to the latest scoring rule and announces the rankingswithout revealing the bidders or the actual bids. The winnerdetermination, at the end of

q � � rounds, is based simplyupon an English auction like rule, with the bidder providingthe highest utility at the end of

q � � rounds being offeredthe contract at his bid price.

The key questions that arise are: (1) what is the likelybidding behavior of the suppliers; (2) how does the buyer esti-mate the suppliers’ cost functions; and (3) how is the optimalscoring rule determined after learning the cost functions.

Bidding Behavior of Suppliers: The supplier is assumed tofollow a myopic best response bidding behavior which is inline with the approaches used in [100], and [69]. Here, thesupplier chooses his bid such that he maximizes his currentprofit with the assumption that other suppliers do not change

21

their bids. The bid is chosen by solving the following Non-Linear Optimization problem:

� ?�qr s s�t�sNMNMNM s s�u þ?� 0�� � + H � n � � � < � n +7��������� < � n�n � (31)

subject to

0�� � + �

�& � � �@� � +fo �������8�j� � n o �!�wvD�êl � 9 (32)

In the final round however, the constraint would be:

0�� � + 9

�� � � ���xv�� l � 9 (33)

While the bidding behavior does not include a game theo-retic analysis of competing suppliers, it still requires biddersto be sophisticated enough to solve optimization problems.This is striking a middle ground with respect to the bidders’rationality.

Estimating cost functions of the suppliers: By virtue of theauction design, the auctioneer at the end of

qrounds has for

each attribute � and each supplier j ,q

equations. Theseq

equations obtained by solving the first order conditions for�!�����������8�&� ,y � �� � � �j� � +o ���������@� � n o �y � � �y H � n � � � <-� n + ��������� <-� n�nã�y � � (34)

form a set of simultaneous linear equations. By solving thisset of equations for each supplier j the true cost parametersfor each attribute � can be obtained.

Computing the optimal scoring rule: After learning thesuppliers’ cost functions, the optimal scoring rule for roundq � � is computed by solving the following optimizationproblem:

� ?�qzf{ | 0�� � + �

�& � :� + � á � + ��������� á � nã��� 0�� � + 9

�� � :� + ��}�| 0�� � + 9

� � :� . �!�0�� � + H � . � :� . � < � .�+;��������� < � . n ��} (35)

subject to:

(�no�0�� � + 9

�& � :� �!�0�� � + H � n � :� � <-� n + ��������� <-� n~nã� (36)

( . Ö 9 (37)

( + Ö ( . � 9 (38)

The objective function maximizes the buyer’s utility withthe last three terms of (24) making up the winning suppliersprice; (36) is supplier ) ’s maximum drop-out score; (37) and(38) ensure that the top two suppliers participate in roundq � � .

In the model described above, assumptions about (1) undis-torted bidding behavior by suppliers and (2) knowledge ofthe form of the suppliers’ cost functions may be untenable inpractice. The authors propose several extensions to the basicmodel in order to make it more robust. First, in developingthe scoring rule for the last round, the authors envisage threeproblems: (a) a difficult mathematical programming problemneeds to be solved, (b) the scoring rule may turn out to be toocomplex, and (c) the scoring rule may force the losing supplierto submit bids with negligible values of non-price attributes. Toovercome these problems the authors propose to (1) changingthe method of finding the best competitor, (2) providing thescoring rules in graphical form, and (3) introducing lowerbound constraints on the attribute levels in the final round.Secondly, the bidding behavior of suppliers may not followthe undistortedness assumption either because of strategicintentions of the supplier or the lack of sophistication on theirpart. This would result in an inconsistent set of simultaneouslinear equations. The authors suggest using a weighted averageleast squares procedure to reduce the effects of bid distortion.We however would like to emphasize that this approach doesnot absolve buyer of need to compute true utility.

D. Summary and Current Art

In this section we reviewed mechanisms designed for theprocurement of items with multiple attributes. We brieflyreviewed the approaches to multi-criteria decision making andidentified two broad categories of techniques - eliminationmethods and weighted average methods. The crucial issue inmulti criteria procurement is the assignment of weights to eachof the attributes to facilitate the development of a scoringfunction which captures the buyers’ utility. Two intelligentapproaches - the first relying on a central agency to indicatea pairwise preference among a sample of received bids andthe second based upon estimating the suppliers cost functions,were detailed.

Currently, developing efficient approaches for multi-attribute procurement is an active research area. The readeris referred to [105], [104], [102], [72], [58] for some recentwork. The papers [105], [104], [102], [101] build upon theapproaches described above. Kameshwaran and Narahari [72]have proposed an approach based on goal programming. Goalprogramming (GP) is one of the tools of choice for multicriteria decision analysis [106]. In [72], the authors showthat GP can be used to model procurement scenarios wheresuppliers provide bids with configurable offers. Here the bidsare assumed to be piece-wise linear and the buyer has ahierarchy of goals or aspiration levels which are to be satisfied.The authors propose the use of Weighted GP, LexicographicGP, and Interactive Sequential GP techniques to solve themulti-attribute procurement problem.

There are many issues that remain unresolved in multi-attribute procurement. No single approach seems to workuniformly well and it is intrinsically a challenging problem.Much work remains to be done in all the areas: winnerdetermination algorithms, payment rules, and achieving truthrevelation.

22

Scenarios in Math.Programming Game Theoretic Case Studies, Surveys,Multi-attribute Procurement Models Analysis ReferencesAdditive Value Models [101], [72], [102] [1], [103], [97], [104], [105], [55], [95], [2]

Inverse Optimization Models [98] [99]

TABLE XCLASSIFICATION OF MULTI-ATTRIBUTE PROCUREMENT AUCTIONS

We summarize the discussion on multi-attribute procure-ment auctions by presenting Table X which gives snapshot ofthe possible scenarios and a listing of relevant references.

VII. A CASE STUDY FROM GENERAL MOTORS

Procurement business professionals need to collect largenumbers of bids from prospective suppliers, and, then, de-termine an allocation of awards to bids that minimizes pro-curement costs subject to various constraints on supplierdelivery risk, vendor award count on commodities or subsetsof commodities, and logistics costs. This need is presentduring the initial stage of awarding business to suppliers onnew products, and is also present when primary suppliers areunable to deliver supplies (e.g., in the case of a strike, naturaldisaster, financial default, or other event that causes a workstoppage) to existing products.

A team of researchers from General Motors Research(which included the last four authors of this paper) recentlyused an auction-optimization approach similar to those de-scribed earlier in this paper to solve this problem. Thisapproach, soon to be deployed as a web application withinGeneral Motors Corporation (GM Corp), allows business usersto determine an optimal allocation of awards to bids usingthe application over the company’s intranet. Because theoptimization approach is math driven, automatic, and can berolled out over the businesses intranet, business users can nowfocus on the business constraints that drive the award process.Business users can dynamically adjust the business constraintsto understand the impact of the stated constraints on the awardallocation. That is, they can understand the sensitivity of theawards to the constraint settings. This process can happen inreal time, thus making the procurement business professionals’much easier, and allowing them to focus on developing therelationship with suppliers rather than spending their timecrunching numbers in spreadsheets in an effort to discover theoptimal award allocation using more primitive manual typeapproaches.

The sourcing corresponds to that of an important rawmaterial for automotive manufacturing at GM. The overallcommodity sourcing process is shown in Figure 2. Within GM,a huge amount of this commodity is sourced every year. Togain maximum cost savings (at a sufficiently high level ofdesired quality), GM uses a centralized demand aggregationand reselling application for the whole supply chain. Thisapplication attempts to combine the individual commodityrequirements of its processors with GM’s direct commodityrequirements to create large orders. These larger orders oftenqualify for significant volume discounts with the commodity

Fig. 2. Overall Business Process

suppliers. GM then resells a portion of the purchased com-modity to its processors to cover their material needs. Someof the processing is performed by external processors, andthe rest is done by GM’s internal processing group. So, theoverall process is very complex and manual approaches fordetermining an allocation of awards to the suppliers requireenormous effort. Moreover, the solution may not be optimal.By using an optimization model similar to those describedearlier in this paper, superior solutions can be obtained withless effort.

A. System Overview

A general commodity sourcing application in the automotiveindustry will look like the one shown in Figure 3. Here,the overall process is the same as shown in Figure 2. Therequirements for the commodities are aggregated within acentralized system (sourcing tool) and an effective bundlingof the commodity is found. The tool looks at the catalogs ofthe approved suppliers and sends RFQs to each of the suppliersfor those bundles that they are capable of supplying. Each ofthese suppliers submits a list of configurable bids to the toolin response to the RFQ. The tool evaluates the bids and feedsthem into the optimization model. The optimization model isalso provided with plant specific constraints and other businessconstraints. The tool outputs a cost effective allocation to thesuppliers.

23

Fig. 3. Commodity Sourcing Application

Fig. 4. Main Elements of a Configurable Bid

The main elements of a configurable bid submitted by asupplier are shown in the tree-like structure shown in Figure4. A configurable bid gives either a base price for a bundle andquantity, or a volume discount price, which is a function ofquantity. The bid consists of various attributes, which can befixed attributes, range attributes, or optional attributes. Fixedattributes are simple attributes with one corresponding value.Range attributes allow suppliers to specify an attribute as a stepfunction, where each step has a fixed impact on price. Optionalattributes can either be included or not. A supplier can alsospecify some logical rules for assigning some discounts tospecific combination of selected attributes.

B. A Model for Minimizing Procurement Spend

The commodity requirements of all the plants are aggre-gated and bids are invited from the suppliers. Each plant hasa set of preferred suppliers for procuring the commodities.Also, suppliers are capable of supplying certain commodities

to one or more particular plants. Each bid submitted hasvariable and fixed cost components. They are submitted formultiple units of multiple heterogeneous commodities destinedfor multiple plants. There are some bounds on the number ofunits of a particular commodity that a supplier can supplyto a plant. Also, there are some restrictions on the numberof suppliers that can be selected for a plant and also onthe number of suppliers that can be awarded business for acommodity. All of these constraints are determined by takinginto consideration the suppliers’ capacities, their disruptionrisk, their overhead costs, etc. The model that the team createdto solve the problem determines an efficient allocation ofawards to bids so that overall procurement cost is minimized,and the various constraints on the number of suppliers fora plant/commodity, bounds on the award to each supplier,plant preferred suppliers, and other constraints are satisfied.The mathematical programming model used combines severalfeatures of the formulations discussed in IV.B and V.B andextends those formulations with business constraints uniqueto GM.

C. Results and Conclusions

Using the above approach, the team was able to determinea cost minimizing potential re-allocation of awards that couldbe used for negotiations with the current suppliers (A and B)regarding the status of expiring contracts. The set of specialtycommodities that the team considered were C1, C2, C3, andC4. The solution obtained by the conventional method (usingcurrent suppliers A and B) gave an Annual Purchase Value(APV) of $ 31.11 M while that obtained using our optimizationapproach gave an APV of $ 30.4M. The results are shown inTable XI and Table XII respectively. These preliminary resultsgave significant savings of about $ 0.71M (2.3% of APV) ina solution time in the single digit seconds range.

Also, the team considered how the optimization modelmight be used to help determine an optimal supply re-allocation strategy in the event of a supplier disruption event.For this, the team considered the reallocation of a portionof the commodity from one of the suppliers to two othersuppliers. Three scenarios were investigated:

1) Reallocating only the highest volume parts2) Reallocating so as to minimize the total cost impact3) Same as Scenario 1 above, but excluding certain parts

for reallocationThe results are shown in Table XII. So, if the number of

reallocations (Scenario 1) is sought to be minimized, the costimpact is high ($ 3.2M). On the other hand, trying to minimizethe total cost impact increases the number of reallocationsto 280. Assuming a switching cost of approximately $10,000for reallocating a part to a new supplier, the cost minimizingreallocation has an impact of $ 2.7M with 146 reallocations.

In conclusion, the team demonstrated that significant costsavings are possible when applying auctions and optimizationto a real-world procurement problem. The savings could be ashigh as 3%. Of course, given that the analysis was limited toa particular commodities group, there is reason to believe thatthis estimate is only a lower bound on what is possible using

24

Mill/Commodity C1 C2 C3 C4 Grand Total

A $107,010 $ 2,863,928 $ 7,069,270 $1,502,302 $ 11, 542, 510B $303,039 $ 1, 739, 721 $ 10, 153, 908 $ 7, 367, 029 $ 19, 563, 697

Grand Total $ 410, 049 $ 4, 603, 649 $ 17, 223, 178 $ 8, 869, 331 $ 31, 106, 207

TABLE XICURRENT COMMODITY ALLOCATION (APV: $ 31.22M)

Mill/Commodity C1 C2 C3 C4 Grand Total

A - - - $5, 117, 764 $ 5, 117, 764B $ 71, 708 $ 1, 629, 282 $ 7,600, 918 $1, 276, 440 $ 10, 578, 348C $ 5, 981 $ 1, 763, 698 $ 6, 970, 686 - $ 8, 739, 365D $ 277, 243 $ 1, 007, 565 $ 775, 921 $ 2, 182, 372 $ 4, 243, 101E $ 27, 945 $ 176, 621 $ 1, 513, 435 - $ 1, 718, 001

Grand Total $ 382, 877 $ 4, 577, 166 $ 16, 860, 960 $ 8, 576, 576 $ 30, 396, 579Savings ($) $ 27, 172 $ 27, 483 $ 362, 218 $ 292, 755 $ 709, 628Savings (%) 6.627 % 0.597% 2.103 % 3.301% 2.2813 %

TABLE XIIOPTIMAL COMMODITY ALLOCATION (APV: $ 30.40M)

Scenario Cost Impact (Increase) No. of Reallocations

1 $ 3.2M 772 $ 2.5 M 2803 $ 3.8M 81

TABLE XIIIREALLOCATION OF MATERIAL

this approach. It was also shown that using this type of mod-eling approach allows business analysts to explore multipledifferent constraints on the award and reallocation process.This leads to a much better understanding of the sensitivityof the optimal sourcing decisions to business constraints thanpossible using a more traditional manual process.

In the following section, we summarize the discussion inthis paper by providing a few concluding remarks and pointersto potential research opportunities.

VIII. CONCLUSIONS AND FUTURE WORK

In this paper, we have surveyed the state-of-the-art inauction-based mechanisms for automating negotiations in elec-tronic procurement. We have discussed these mechanismsunder three categories:� Procurement of single unit or multiple units of a single

item based on a single attribute� Procurement of single unit or multiple units of multipleitems based on a single attribute� Procurement of multiple units of a single item based onmultiple attributes

A fourth category would be: procurement of multiple units ofmultiple items based on multiple attributes. Given that the firstthree categories are areas of active research with unresolvedissues, this fourth category would be even more challenging. Ineach of the three categories of procurement, we discussed sev-eral procurement scenarios, provided problem formulations,and discussed issues of computational complexity related tothe mechanism and the bidding process. We have also provideda description of several case studies, including a detailed onefrom General Motors.

Currently, procurement auctions is an active area of researchwith many interesting problems which merit further study.Here, we provide pointers to a few selected categories:

Winner Determination Problem in Procurement Auctions

Though extensive work has been done on solving the winnerdetermination problem in different types of procurement auc-tions, there is still wide scope for future work here. One of thepromising directions is to come up with efficient approxima-tion algorithms with provable bounds for solving the winner

25

determination problem. There are many efforts in this directionalready, see for example, [58], [68]. The rich body of literatureavailable in combinatorial optimization and approximation andrandomized algorithms will be extremely useful here. Anotherkey opportunity here is in the area of on-line algorithms. Inmany situations, it becomes imperative to quickly evaluate thebids received and allocate quantities/orders to suppliers. Insuch situations, on-line algorithms would be useful. As of thepresent art, there is not much work in this direction and thiswould be a valuable topic for further investigation.

Iterative Procurement Auctions

As already stated in the paper, iterative mechanisms haveseveral advantages over one shot mechanisms. Also, there havebeen several iterative mechanisms developed for procurement[42], [77], [45], [93], [107]. Iterative mechanisms are ap-pealing to the buyer and suppliers because they enable thebidders to apply corrections to their bids in a continuous way.More work is required in ensuring that iterative mechanismssatisfy desirable economic properties also. The use of on-linealgorithms for rapid winner determination in successive roundsof bidding is an important direction for future work.

Design of Incentive Compatible Mechanisms

Inducing truth revelation will continue to be a major is-sue in the design of procurement mechanisms. Designingsuch mechanisms has intrinsic difficulties such as very highcomputational complexity and loss of efficiency. It would beinteresting to study how the use of approximate algorithms forwinner determination and payment computation would affectincentive compatibility. There is already a fair amount of workin this direction [68], [86], [60].

Multi-Attribute Procurement

Multi-attribute procurement is an intrinsically difficult prob-lem, but at the same time an important problem that needsimmediate attention. There are a few results available [95],[104], [107] and there are a few promising approaches such asgoal programming [72], but much more needs to be researchedin this area.

Use of Learning in Procurement Auctions

Procurement auctions provide an ideal platform for useof machine learning techniques in improving the efficiencyof the process. The history of bidding by a supplier is animportant parameter for winner determination. To incorporatehistory into procurement decision making will call for use ofappropriate machine learning techniques such as reinforcementlearning. Learning based models would be useful in iterativeprocurement auctions to help the buyer estimate the costfunctions of the suppliers and in optimally incrementing theprocurement budgets. One such application is discussed in[82]. Another interesting application is discussed in [108]. Webelieve machine learning techniques have powerful applica-tions in procurement auctions.

Procurement Auctions from a Total Supply Chain Perspective

It is important to design procurement mechanisms based ona total cost approach where the total cost captures all aspectsof the entire supply chain. This has been explored in a fewpapers already [59] but a deeper understanding and a moresystematic approach is required here.

Deployment Issues

Practical deployment of procurement auctions will throwup numerous challenges. Several authors have addressed theseissues: security of transactions [109]; collusion of suppli-ers [61]; user interface issues [105]; fairness issues [32];failure freeness and robustness against failures [32], [109],[41]. These issues need immediate attention for successfuladoption of auctions by purchasing departments. Designingsoftware implementation frameworks so as to allow sensitivityanalysis of procurement decisions in complex supply chainenvironments is also an important issue. Use of multi-agentagent technology in automating standard electronic procure-ment problems is one more issue. Using emerging Internettechnology standards such as ebXML in implementing e-procurement solutions is an immediate practical issue.

Procurement Exchanges

Procurement exchanges are those where there are multiplebuyers and multiple suppliers and the exchange facilitatesmatching of buyers with suppliers. All the issues become morecomplex with exchanges because of the presence of multiplebuyers. There is a large body of literature on exchanges; forexample, see [110], [23], [111].

ACKNOWLEDGMENT

The first two authors would like acknowledge the supportof General Motors Research and Development Center, Warren,Michigan, USA for carrying out this research under theIMPROVE research grant.

REFERENCES

[1] M. Bichler, M. Kaukal, and A. Segev, “Multi-attribute auctions forelectronic procurement,” in Proceedings of the First IBM IAC Workshopon Internet Based Negotiation Technologies, Yorktown Heights, NY,USA, 1999.

[2] M. Bichler, G. Kersten, and S. Strecker, “Towards a structured designof electronic negotiations,” Group Decisions and Negotiations, vol. 12,no. 4, pp. 311–335, 2003.

[3] G. E. Kersten, S. J. Noronha, and T. Jeffrey, “Are all e-commercenegotiations auctions?” in Fourth International Conference on theDesign of Cooperative Systems, Sophia-Antipolis, France, May 2000.

[4] C. Beam and A. Segev, “Automated negotiations: A survey of the stateof the art,” Wirtschaftsinformatik, vol. 39, no. 3, pp. 263–268, 1997.

[5] A. Chavez and P. Maes, “Kasbah: An agent marketplace for buying andselling goods,” in First International Conference on the Practical Ap-plication of Intelligent Agents and Multi-Agent Technology (PAAM’96),1996, pp. 75–90.

[6] D. Zeng and K. Sycara, “Bayesian learning in negotiation,” in WorkingNotes for the AAAI Symposium on Adaptation, Co-evolution andLearning in Multiagent Systems, S. Sen, Ed., Stanford University, CA,USA, 1996, pp. 99–104.

[7] A. Mas-Collel, M. D. Whinston, and J. R. Green, MicroeconomicTheory. Oxford University Press, 1995.

[8] R. B. Myerson, Game Theory: Analysis of Conflict. Cambridge,Massachusetts: Harvard University Press, 1997.

26

[9] G. E. Corporation, “Letter to share owners,” GE Annual Report, 2000.[10] S. Beall, C. Carter, and et. al., “The role of reverse auctions in strate-

gic sourcing: Case Study Glaxo Smith Kline,” Center for AdvancedPurchasing Studies, Temple, AZ, USA, Tech. Rep., 2003.

[11] ——, “The role of reverse auctions in strategic sourcing: Case StudyMETRO,” Center for Advanced Purchasing Studies, Temple, AZ, USA,Tech. Rep., 2003.

[12] ——, “The role of reverse auctions in strategic sourcing: Case StudyVolkswagen,” Center for Advanced Purchasing Studies, Temple, AZ,USA, Tech. Rep., 2003.

[13] ——, “The role of reverse auctions in strategic sourcing: Case StudyBechtel,” Center for Advanced Purchasing Studies, Temple, AZ, USA,Tech. Rep., 2003.

[14] ——, “The role of reverse auctions in strategic sourcing,” Center forAdvanced Purchasing Studies, Temple, AZ, USA, Tech. Rep., 2003.

[15] W. Elmaghraby and P. Keskinocak, “Technology for transportationbidding at the home depot,” in Practice of Supply Chain Management:Where Theory and Practice Converge. Kluwer Academic Publishers,2003.

[16] ——, “Combinatorial auctions in procurement,” in Practice of SupplyChain Management: Where Theory and Practice Converge. KluwerAcademic Publishers, 2003.

[17] G. Hohner, J. Rich, E. Ng, G. Reid, A. J. Davenport, J. R. Kalagnanam,S. H. Lee, and C. An, “Combinatorial and quantity discount pro-curement auctions provide benefits to Mars, Incorporated and to itssuppliers,” INTERFACES, vol. 33, no. 1, pp. 23–35, 2003.

[18] J. Ledyard, M. Olson, D. Porter, J. Swanson, and D. Torma, “The firstuse of a combined value auction for transportation services,” Interfaces,vol. 32, no. 5, pp. 4–12, 2002.

[19] C. Caplice and Y. Sheffi, “Combinatorial auctions for truckload trans-portation,” in Combinatorial Auctions, P. Cramton, Y. Shoham, andR. Steinberg, Eds. The MIT Press, Cambridge, Massachusetts, USA,2005.

[20] Y. Sheffi, “Combinatorial auctions in the procurement of transportationservices,” Interfaces, vol. 34, 2004.

[21] R. Epstein, L. Henriquez, J. Catalan, G. Weintraub, and C. Martinez,“A combinatorial auction improves school meals in Chile,” Interfaces,vol. 32, no. 6, pp. 1–14, 2002.

[22] M. Bichler, A. Davenport, G. Hohner, and J. Kalagnanam, “Indus-trial procurement auctions,” in Combinatorial Auctions, P. Cramton,Y. Shoham, and R. Steinberg, Eds. The MIT Press, Cambridge,Massachusetts, USA, 2005.

[23] P. Milgrom, Putting Auction Theory to Work. Cambridge UniversityPress, 2004.

[24] V. Krishna, Auction Theory. Academic Press, 2002.[25] R. McAfee and J. McMillan, “Auctions and bidding,” Journal of

Economic Literature, vol. 25, pp. 699–738, 1987.[26] P. Milgrom, “Auctions and bidding: A primer,” Journal of Economic

Perspectives, vol. 3, no. 3, pp. 3–22, 1989.[27] P. Klemperer, Auctions: Theory and Practice. The Toulouse Lectures

in Economics. Princeton University Press, 2004.[28] E. Wolfstetter, “Auctions: An introduction,” Economic Surveys, vol. 10,

pp. 367–421, 1996.[29] J. Kalagnanam and D. C. Parkes, “Auctions, bidding, and exchange

design,” in Handbook of Quantitative Supply Chain Analysis: Modelingin the E-Business Era, ser. Int. Series in Operations Research andManagement Science, D. Simchi-Levi, D. S. Wu, and M. Shen, Eds.Kluwer Academic Publishers, 2005, ch. 10.

[30] S. de Vries and R. V. Vohra, “Combinatorial auctions: A survey,”INFORMS Journal of Computing, vol. 15, no. 1, pp. 284–309, 2003.

[31] S. de Vries and R. H. Vohra, “Design of combinatorial auctions,” inHandbook of Quantitative Supply Chain Analysis: Modeling in theE-Business Era. International Series in Operations Research andManagement Science, Kluwer Academic Publishers, Norwell, MA,USA, 2005, pp. 247–292.

[32] A. Pekec and M. Rothkopf, “Combinatorial auction design,” Manage-ment Science, vol. 49, pp. 1485–1503, 2003.

[33] Y. Narahari and P. Dayama, “Combinatorial auctions for electronicbusiness,” Sadhana (Special Issue on Electronic Commerce and Elec-tronic Business), vol. 30, pp. 179–211, April 2005.

[34] P. Cramton, Y. Shoham, and R. Steinberg, “Introduction to combinato-rial auctions,” in Combinatorial Auctions, P. Cramton, Y. Shoham, andR. Steinberg, Eds. The MIT Press, Cambridge, Massachusetts, USA,2005.

[35] W. J. Elmaghraby and P. Keskinocak, “Dynamic pricing: Researchoverview, current practices and future directions,” Management Sci-ence, vol. 49, no. 10, pp. 1287– 1309, October 2003.

[36] M. Bichler, R. Lawrence, J. Kalagnanam, H. Lee, K. Katircioglu,G. Lin, A. King, and Y. Lu., “Applications of flexible pricingin business-to-business electronic commerce,” IBM Systems Journal,vol. 41, no. 2, pp. 287–302, 2002.

[37] W. Elmaghraby, “Pricing and auctions in emarketplaces,” in Handbookof Quantitative Supply Chain Analysis: Modeling in the E-BusinessEra. International Series in Operations Research and ManagementScience, Kluwer Academic Publishers, Norwell, MA, USA, 2005, pp.213–246.

[38] T. A. Minahan, H. Francis, and V. Mark, “Making e-sourcing strategic:From tactical technology to core business strategy,” Aberdeen GroupInc, Boston, Massachusetts, Tech. Rep., 2002.

[39] W. Vickrey, “Counter speculation, auctions, and competitive sealedtender,” Journal of Finance, vol. 16, pp. 8–37, 1961.

[40] M. Herschlag and R. Zwick, “Internet auctions-a popular and profes-sional literature review,” Electronic Commerce, vol. 1, no. 2, pp. 161–186, 2000.

[41] K. Manoj and S. I. Feldman, “Internet auctions,” in Proceedings ofThird Usenix Workshop on Electronic Commerce, 1999.

[42] D. Parkes, “Iterative combinatorial auctions: Achieving economic andcomputational efficiency,” Ph.D. dissertation, Department of Computerand Information Science, University of Pennsylvania, May 2001.

[43] H. R. Varian, “Economic mechanism design for computerized agents,”in Proceedings of the USENIX Workshop on Electronic Commerce,1995, minor update, 2000.

[44] J. Laffont and J. Tirole, A Theory of Incentives in Procurement andRegulation. The MIT Press, Cambridge, MA, USA, 1993.

[45] L. Ausubel and P. Milgrom, “Ascending auctions with package bid-ding,” Frontiers of Theoretical Economics, vol. 1, no. 1, 2002.

[46] E. Clarke, “Multi-part pricing of public goods,” Public Choice, vol. 11,pp. 17–23, 1971.

[47] T. Groves, “Incentives in teams,” Econometrica, vol. 41, pp. 617–631,1973.

[48] L. Hurwicz, “On informationally decentralized systems,” in Decisionand Organization: A Volume in Honor of Jacob Marchak. North-Holland, 1972.

[49] K. Arrow, “The property rights doctrine and demand revelation underincomplete information revelation,” in Economics and Human Welfare.Academic Press, New York, 1979.

[50] R. B. Myerson and M. A. Satterthwaite, “Efficient mechanisms forbilateral trading,” Journal of Economic Theory, vol. 28, pp. 265–283,1983.

[51] C. d’Aspremont and L. Gerard-Verat, “Incentives and incompleteinformation,” Journal of Public Economics, vol. 11, pp. 25–45, 1979.

[52] R. B. Myerson, “Optimal auction design,” Mathematics of OperationsResearch, vol. 6, pp. 58–73, 1981.

[53] J. Green and J. Laffont, Incentives in Public Decision Making. North-Holland, Amsterdam, 1979.

[54] A. Davenport and J. Kalagnanam, “Price negotiations for direct pro-curement,” IBM Research, Yorktown Heights, NJ, USA, ResearchReport RC 22078, 2001.

[55] L. Benyoucef, H. Ding, and X. Xie, “Supplier selection problem:Selection criteria and methods,” INRIA, Lorraine, Tech. Rep., 2003.

[56] M. Eso, S. Ghosh, J. Kalagnanam, and L. Ladanyi, “Bid evaluationin procurement auctions with piece-wise linear supply curves,” IBMResearch, Yorktown Heights, NJ, USA, Research Report RC 22219,2001.

[57] D. Pisinger and P. Toth, “Knapsack problems,” in Handbook of Com-binatorial Optimization, D. D.-Z and P. M. Pardolas, Eds. KluwerAcademic Publishers, 1998, pp. 299–428.

[58] S. Kameshwaran, “Algorithms for piecewise linear knapsack problemswith applications in electronic commerce,” Ph.D. dissertation, Depart-ment of Computer Science and Automation, Indian Institute of Science,Bangalore, India, August 2004.

[59] R. R. Chen, G. Janakiraman, R. Robin, and R. Q. Zhang, “Efficientauctions for supply chain procurement,” Johnson Graduate School ofManagement, Cornell University, Ithaca, NY, Tech. Rep., 2002.

[60] N. Nisan and A. Ronen, “Computationally feasible VCG mechanisms,”in Proceedings of the Second ACM Conference on Electronic Com-merce, 2000, pp. 242–252.

[61] P. Bajaria and G. Summers, “Detecting collusion in procurementauctions: A selective survey of recent research,” Stanford University,Department of Economics, Working Paper 01014, Tech. Rep., 2002.

[62] R. Narasimhan, S. Talluri, and et.al., “Evaluating e-procurement so-lutions,” Center for Advanced Purchasing Studies, Temple, AZ, USA,Tech. Rep., 2003.

27

[63] J. Kagel, “Auctions: A survey of experimental research,” in TheHandbook of Experimental Economics. Princeton University Press,Princeton, 1995, pp. 501–587.

[64] L. Ausubel and P. Cramton, “Vickrey auctions with reserve pricing,”Working Paper, University of Maryland, Tech. Rep., 1999.

[65] M. Eso, S. Ghosh, J. R. Kalagnanam, and L. Ladanyi, “Bid evaluationin procurement auctions with piece wise linear supply curves,” IBM,Tech. Rep. RC22219 (W0110-087), 2001.

[66] A. Davenport and J. Kalagnanam, “Price negotiations for procurementof direct inputs,” in Mathematics of the Internet: E-Auction andMarkets, B. Dietrich and R. Vohra, Eds. Springer Verlag, 2001, IMAVolumes in Mathematics and its Applications.

[67] V. Dang and N. Jennings, “Optimal clearing algorithms for multi-unit single-item and multi-unit combinatorial auctions with demand-supply function bidding,” Dept of Electronics and Computer Science,University of Southampton, UK, Tech. Rep., 2003.

[68] A. Kothari, D. Parkes, and S. Suri, “Approximately strategyproof and tractable multi-unit auctions,” in Proceedingsof ACM Conference on Electronic Commerce (EC-03),San Diego CA, June 9-12, 2003. [Online]. Available:citeseer.ist.psu.edu/kothari03approximatelystrategyproof.html

[69] J. Gallien and L. Wein, “A smart market for industrial procurementwith capacity constraints,” Management Science, vol. 51, no. 1, pp.76–91, January 2005.

[70] N. Nisan, “Bidding and allocation in combinatorial auctions,” inProceedings of the Second ACM Conference on Electronic Commerce,2000.

[71] C. Boutlier and H. Hoos, “Bidding languages for combinatorial auc-tions,” in Proceedings of the 17th International Joint Conference onArtificial Intelligence, 2001.

[72] S. Kameshwaran and Y. Narahari, “E-procurement using goal pro-gramming,” in Proceedings of International Conference on ElectronicCommerce and Web Technologies, DEXA 2003 Conferences, Linz,Austria, 2003.

[73] S. Biswas and Y. Narahari, “Iterative reverse dutch auction for elec-tronic procurement,” in Proceedings of the International Conference onElectronic Commerce Research, ICECR-5, Montreal, Canada, 2002.

[74] D. Parkes, “iBundle: An efficient ascending price bundle auction,” inProceedings of ACM Conference on Electronic Commerce (EC-99),2000, pp. 148–157.

[75] M. Wellman, W. Walsh, P. Wurman, and J. MacKie-Mason, “Auction protocols for decentralized scheduling,”University of Michigan, Tech. Rep., 1998. [Online]. Available:citeseer.nj.nec.com/article/wellman98auction.html

[76] V. Chvatal, “A greedy heuristic for the set cover problem,” Mathematicsof Operations Research, vol. 4, pp. 233–235, 1979.

[77] S. Biswas and Y. Narahari, “Iterative Dutch combinatorial auctions,”Annals of Mathematics and Artificial Intelligence, 2005, to appear inthe special issue on the Foundations of Electronic Commerce.

[78] T. Sandholm, “An algorithm for optimal winner determination incombinatorial auctions,” Artificial Intelligence, vol. 135, no. 1, pp. 1–54, 2002.

[79] T. Sandholm and S. Suri, “Bob: Improved winner determination incombinatorial auctions and generalizations,” Artificial Intelligence, vol.145, pp. 33–58, 2003.

[80] R. Gonen and D. Lehmann, “Optimal solutions for multi-unit combina-torial auctions: Branch and bound heuristics,” in Proceedings of ACMConference on Electronic Commerce (EC-00), 2000, pp. 13–20.

[81] E. Zurel and N. Nisan, “An efficient approximate allocation algorithmfor combinatorial auctions,” in Proceedings of ACM Conference onElectronic Commerce (EC-01), 2001, pp. 125–136.

[82] C. V. L. Raju and Y. Narahari, “Use of reinforcement learning initerative bundle auctions for procurement,” in Proceedings of theInternational Conference on Automation, Energy, and InformationTechnology, EAIT-2001, Indian Institute of Technology, Kharagpur,2001.

[83] C. DeMartini, A. Kwasnica, J. Ledyard, and D. Porter, “A new andimproved design for multi-object iterative auctions,” Social ScienceWorking Paper No. 1054, Pennsylvania State University, Tech. Rep.,1998.

[84] D. Parkes, “Iterative combinatorial auctions,” in Combinatorial Auc-tions, P. Cramton, Y. Shoham, and R. Steinberg, Eds. The MIT Press,Cambridge, Massachusetts, USA, 2005.

[85] S. Bikhchandani and J. Ostroy, “From the assignment model to combi-natorial auctions,” in Combinatorial Auctions, P. Cramton, Y. Shoham,and R. Steinberg, Eds. The MIT Press, Cambridge, Massachusetts,USA, 2005.

[86] D. C. Parkes, “An iterative generalized vickrey auction: Strategy-proofness without complete revelation,” in Proceedings of AAAI SpringSymposium on Game Theoretic and Decision Theoretic Agents, 2001.

[87] Y. Bartal, R. Gonen, and N. Nisan, “Incentive compatible multiunitcombinatorial auctions,” in Proceedings of Dagstuhl Seminar on Elec-tronic Market Design, 2002.

[88] M. Rothkopf, A. Pekec, and R. Harstad, “Computationally manageablecombinatorial auctions,” Management Science, vol. 44, pp. 1131–1147,1998.

[89] S. de Vries and R. V. Vohra, “Combinatorial auctions: A survey,”INFORMS Journal on Computing, 2002.

[90] K. Leyton-Brown, Y. Shoham, and M. Tennenboltz, “An algorithmfor multi-unit combinatorial auctions,” in Proceedings of NationalConference on Artificial Intelligence (AAAI-00), 2000.

[91] D. Mishra and D. C. Parkes, “Ascending price Vickrey auctions usingprimal-dual algorithms,” Harvard Univeristy, Tech. Rep., 2004.

[92] ——, “A Vickrey-Dutch clinching auction,” Harvard University, Tech.Rep., 2004.

[93] P. R. Wurman and M. P. Wellman, “Akba: A progressive, anonymous-price combinatorial auction,” in Proceedings of ACM Conference onElectronic Commerce (EC-00), 2000, pp. 21–29.

[94] M. Tennenholtz, “Some tractable combinatorial auctions,” in Proceed-ings of National Conference on Artificial Intelligence (AAAI-00), 2000.

[95] Y. K. Che, “Design competition through multidimensional auctions,”RAND Journal of Economics, vol. 24, pp. 668–679, 1993.

[96] R. Keeney and H. Raiffa, Decisions with multiple objectives: Prefer-ences and value tradeoffs. Wiley, 1976.

[97] M. Bichler, J. Lee, H. S. Lee, and J. Y. Chung, “Absolute: An intelligentdecision making framework for e-sourcing,” in Proceedings of theThird Workshop on Electronic Commerce and Web Based InformationSystems, San Jose, CA, 2001.

[98] D. R. Beil and L. M. Wein, “An inverse optimization based auctionmechanism to support a multi-attribute rfq process,” ManagementScience, vol. 49, no. 11, pp. 1529–1545, November 2003.

[99] R. K. Ahuja and J. B. Orlin, “Inverse optimization,” OperationsResearch, vol. 49, no. 5, pp. 771–783, 2001.

[100] D. C. Parkes and L. H. Ungar, “Iterative combinatorial auctions: Theoryand practice,” in Proceedings of the 17th National Conference onArtificial Intelligence, 2000, pp. 74–81.

[101] M. Bichler and J. Kalagnanam, “Configurable offers and winner de-termination in multi-attribute offers,” European Journal of OperationsResearch (to appear), 2005.

[102] T. W. Sandholm and S. Suri, “Side constraints and non priceattributes in markets,” in In IJCAI-2001 Workshop on DistributedConstraint Reasoning, Seattle, WA, 2001. [Online]. Available:citeseer.ist.psu.edu/sandholm01side.html

[103] M. Bichler and H. Werthner, “Simulation of multidimensional pro-curement auctions,” in Proceedings of the European Simulation Multi-Conference, Gent, Belgium, 2000.

[104] M. Bichler and J. Kalagnanam, “Configurable offers and winner deter-mination in multi-attribute auctions,” European Journal of OperationalResearch, vol. 160, no. 2, 2005, iBM Research Report RC22478.

[105] M. Bichler, J. Kalagnanam, and H. S. Lee, “Reco: Representation andevaluation of configurable offers,” IBM Research Report, RC 22288,Tech. Rep., 2002.

[106] R. E. Steur, Multiple Criteria Optimization: Theory, Computation andApplication. Wiley, 1986.

[107] D. C. Parkes and J. Kalagnanam, “Iterative Multiattribute VickreyAuctions,” Management Science, 2004, special Issue on ElectronicMarkets.

[108] S. Lahaie and D. C. Parkes, “Applying learning algorithms to prefer-ence elicitation in the generalized vickrey auction,” Harvard University,Tech. Rep., 2004.

[109] Y. Sakurai, M. Yokoo, and S. Matsubara, “A limitation of generalizedVickrey auction in electronic commerce: Robustness against false-namebids,” in Proceedings of National Conference on Artificial Intelligence(AAAI-99, vol. 16, 1999, pp. 86–92.

[110] S. Biswas, “Iterative algorithms for combinatorial auctions and ex-changes,” Ph.D. dissertation, Department of Computer Science andAutomation, Indian Institute of Science, Bangalore, India, April 2004.

[111] D. C. Parkes, R. Cavallo, N. Elprin, A. Juda, S. Lahaie, B. Lubin,L. Michael, J. Shneidman, and H. Sultan, “ICE: An iterative combi-natorial exchange,” in Proc. 6th ACM Conf. on Electronic Commerce(EC’05), 2005.