procurement auctions

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Combinatorial and Quantity-DiscountProcurement Auctions Benefit Mars,Incorporated and Its Suppliers

Gail H ohn er John Rich Ed NgMars, Incorporated, Winnersh Triangle, United Kingdom

Grant ReidMars, Incorporated, McLean, Virginia

An drew J. Davenpo rt Jayant R. Kalagnanam Ho Soo Lee Chae AnIBM T.J. Watson Research Center, PO Box 218, Yorktown Heights, New York 10598

[email protected] [email protected] [email protected]@us.ibm.com [email protected]

Simple auctions neglect the complex business co nstraints required by strategic sourcing. TheMars-IBM team created a procurement auction Web site (www.numberltraders.com) that en-ables buyers to incorporate complex bid structures (such as bundled all-or-nothing bids andquantity-discou nted bids) and business constraints into strategic-sourcing auctions. Outcom esin such auctions must lead to win-win solutions to sustain long-term relationships betweenprocurer and suppliers. These factors are as important or more imp ortant than price. The Marsprocu rem ent auction W eb site sup port s several alternatives to simple auctions that help matchits needs as procurer and the capabilities of suppliers by incorporating optimal bid selectionsubject to constraints based on b usiness rules in a dynam ic environm ent. The ability to considergeographic, volume, and quality factors helps both parties. Feedback from participant sup-pliers has highligh ted the benefits of time efficiency, transp aren cy, and fairness. Altho ugh theyreflect just one side of the benefits ledger, the monetary benefits to Mars (a $14 billion com-pany) and to its suppliers are significant.(Industries: agriculture, food. Games/group decisions: bidding, auctions.)

M ars is one of the world's largest privatelyowned businesses ($14 billion in sales). Its phi-losophy, culture, and style set it apart from other or-ganizations. From its earliest beginnings making but-tercream candies sold door to door. Mars operationshave grown to include global businesses in food, petcare, drinks vending, and electronic automated pay-ment systems. Today the company produces m any topbrands of confectionery, pet food, and rice, includingMars, M&M's, Snickers, Whiskas, Pedigree, an dUncle Ben's. To survive, prosper, and grow in itscompetitive markets, it must develop steady brandloyalty by providing consistently high quality to the

consumer. To prosper. Mars must seek the best valuefor its money in obtaining its inputs. The food and pet-food industries run at much lower margins than manymanufacturing or service industries, making the con-trol of input costs vitally important. Mars needs tomain tain a base of long-term and reliable suppliers andat the same time to design procurement mechanismsto obtain the best value for its money without ad-versely affecting its long-term relationships with sup-pliers. This is the central focus of procurem ent at Mars.Mars bases its world wid e busin ess in over 100 coun-tries on five continents on five core principles: quality,efficiency, responsibility, mutuality, and freedom. It

0092-2102/03/3301 /0023$05.00 INTERFACES, 2003 INFORMS


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HOHNER, RICH, NG, REID, DAVENPORT, KALAGNANAM, LEE, AND ANMars, Incorporated

calls its employees associates and spread s responsibil-ity out into the organization. In procurement, its fiveprinciples guide the firm's m anagem ent of supplier re-lationships and its negotiation protocols.Mars ' Business Environment andSupplier RelationshipsMars relies on a small number of suppliers for eachmaterial it procures. Small supply pools arise by ne-cessity as well as by design. For example, many agri-cultural inpu ts are available from a limited nu mb er oforigins, a limited number of brokers, or under tariffregimes that limit available supplies. Integrated sup-ply pools with highly valued suppliers reinforce a cor-porate culture based on m utually shared benefits. Buy-ers responsible for developing new sources andcontract conditions (including price) maintain relation-ships w ith sup pliers. Single buyers are responsible forlarge portfolios often covering two or three productgroups for a large region (for example, all of Europe)or nearly all inputs required for a specific productionsite. A buyer may be responsible for up to 50relationships.

Mars and its supphers agree on contract conditionsand prices in a num ber of wa ys. Mars has no one-size-fits-all solution. Mars contracts with private busi-nesses, brokers, traded agricultural markets, monop-olies, cartels, and governments. It uses differentbuying techniques to handle a huge number of pur-chasing situations. Negotiation and sealed-bid tender-ing are the most common. Procurement processesbased on many parallel one-to-one negotiations or onsingle sealed-bid tend ers have a num ber of fu ndamen -tal problems:(1) One-to-one negotiations prevent the procure-ment division from fully leveraging the competitiveenvironment across suppliers to negotiate prices. Sin-gle sealed-bid tenders are even more static than one-to-one nego tiations.(2) Status quo methods give the firm no effectiveway of taking adva ntag e of synergies or economies ofscale that suppUers may have for parts of the totalbusiness.(3) In negotiations, buyers spend disproportionateamounts of time determining prices and quantities.

which they could do far more efficiently usiauctions.(4) The end-po int of the negotiation process is somwhat arbitrary (based on how much time the buyinvests to obtain future savings). The lack of transpency in such negotiations can cause disgruntled supliers. For example, a supplier may discover after ngotiations that the buyer has contracted with a pricwould have been willing to better. The buyer may haaccepted an offer without checking all suppliers in pool.Electronic auctions have emerged as a popumechanism for conducting negotiations. Procureme

auctions take the form of reverse auctions w ith a sinbuyer and a set of precertified suppliers negotiatiwithin the context of a private exchange. Mars wantto harness the efficiencies of auction mechanisms procurem ent w hile capturing som e of the complexitof the bid structure s it allowed and the constraints aring from its strategies for maintaining supplrelationships.Auctions are generally thought to promote marcompetition and make the purchasing process efficibut have tw o disadva ntages: (1) they are too reliant

price, making them a brute-force way of managing lationships, and (2) they are inappropriate when tfirm wants to control business volumes or the numbof suppliers. For any auction system to gain wide aceptance, it would have to address these tdisadvantages.

Problem DefinitionThe goal of this project was to enable Mars buyworldwide to run auctions whose design compmented the overall business strategy. A typical buyhas a wide range of materials to procure. Some of theare very straightforward, simple purchases with a sgle contract with a sole source (for example, for offisupplies). Our goal was not to service this type of puchase. Many auction suppliers in the market handsuch purc hases. Our objective was to supp ort the sttegic pu rchases. Strategic pu rchases are typically chacterized by (1) small and fairly static sup ply poo ls, long-term relationships, and (3) significant business tegration. The contracts in strategic purcha ses typica

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HOHNER, RICH, NG, REID, DAVENPORT, KALAGNANAM, LEE, AND ANMars , Incorporated

are of high value, are renewed quarterly or annually,and require the use of special price-negotiationschemes that incorporate appropriate business prac-tices. Typically, bids in these settings have the follow-ing properties:(1) The transaction volume is large, and the sup-pliers provide volum e discou nts that they specify as acurve w ith a quantity rang e associated w ith each pricelevel (for ex am ple, $1,000 per unit up to 100 units , $750per unit over 100 units), and(2) Often the suppliers make all-or-nothing bids ona set of items, offering a special discounted price on abundle (for example, $150 for 30 units of item A and

It takes a long time to change the waypeople think!20 units of item B) and will not sell the items partiallyor separately.Two auction mechanisms incorporate these bidstructures and a set of business requirements centralto strategic procurement:(1) Supply-curve auctions are specifically tailored toindustries in which volumetric discounts are common,for exam ple, bulk chemicals and agricultural com mod-ities. In a supply -curve au ction, suppliers specify thesebids as curves with a quantity range associated witheach price level (for exa mp le, $1,000 per uni t u p to 100units and $750 per unit over TOO units). Such auctionsmay deal with one product or many.

(2) Combinatorial auctions are mu ltilot auctions thatallow bids for combined lots with prices that are con-ditional on winning the entire lot. Combinatorial auc-tions are ideal for situations in which synergies existbetween lots (for example, specifically located freightroutes). Suppliers provide all-or-nothing bids on a setof items, offering a special discounted price on a bun-dle. Bids in this auction can be overlapping (for ex-am ple, bid 1 for A, bid 2 for A + B, bid 3 for A -h B+ D). These auctio ns are also useful for situations inwhich the firm wants to buy small volumes of similarbut not identical materials at once. The combinationsallow it to aggregate business to a level sufficient tointerest suppliers.

Business RulesAfter receiving bids in such auctions, a Mars' buyermust identify the set of bids that minimizes total pro-curement cost subject to the following business rules:(1) The num ber of winn ing su ppliers m ust be at leasta minimum number to avoid Mars depending tooheavily on just a few suppliers.(2) The number of winning suppliers must be atmost a maximum numb er to avoid the adm inistrativeoverhead of managing a large number of suppliers.(3) The maximum amount procured from each sup-plier must be bounded to limit exposure to a singlesupplier.

The minimum amount to be procured from eachsupplier must be bounded to avoid receiving eco-nomically inefficient orders (for example, less than afull truck load), and if two or more sets of bids couldwin, the buyer must pick the set that arrived first. (Thisis imperative if the sup pliers are to see the solution asfair.) Although it might seem computationally expe-dient, approximate solutions are unacceptable becausethe difference between an approximate solution andthe real solution can change how much and exactlywhich business a single supplier receives. For example,a supplier allocated business in the optimal solutionmight get nothing in an approximate solution. Suchoccurrences, if made public, would destroy the credi-bility of an auction mechanism. Identifying the cost-minimizing bid set subject to these business rules is ahard optimization problem; a typical problem with 15items and 10 suppliers is usually too complex to do byhand and is beyond most buyers' programmingability.Previous WorkResearchers have studied procurement problems fromboth the s eller's and bu yer's points of view (Stanley etal . 1954, Kim and Huang 1988, Lee and Rosenblatt1986, Na rsim han and Stoynoff 1986). The use of linearprogramming for evaluating bids is almost as old asthe theory of linear programming itself (Stanley et al.1954). The inventory-control and lot-sizing literatureincludes studies of various discounting schemes, suchas unit discounts (Silverson and Peterson 1979), incre-mental quantity discounts and carload quantity dis-counts Oucker and Rosenblatt 1985, Lee and Rosenblatt

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1986). More recently, Katz et al. (1994) studiedbusiness-volume discounts (across several commodi-ties). The focus of all this work was to build decisionmodels for selecting bids. The discounting m odels wedeveloped are similar to those of Sadrian and Yoon(1994) but differ in three ways: (1) the discounts arebased on quantities for each item, (2) the total-allocation constraint on each supplier binds the allo-cation across different items into a single large opti-mization problem, and (3) a minimum number ofsuppliers is specified that further constrains the choiceof winners. Another qualitatively new practice we in-troduced is the use of bun dled bids in a reverse auctionacross different items, which leads to the set-coveringproblem. Combinatorial auctions have been proposedfor forward auctions, such as the FCC spectrum rightsauctions (Rothkopf et al. 1998, Park and Rothkopf2001). Traditionally, buyers have used sequential ne-gotiations or auctions when procuring multiple items.For suppliers with cost complementarities, this couldlead to exposure problems. Suppliers who could pro-vide an overall low cost for multiple items might re-frain from bidding aggressively because their coststructures are contingent on winning subsequent auc-tions. Their reluctance to bid aggressively could leadto inefficient outcomes.However, the more fundamental difference we in-troduced is the iterative negotiation process for boththe bundled and quantity-discounted bids. Gallien andWein (2001) discussed iterative approaches for pro-curement. However, they treated bid evaluation as alinear-programming problem (as did Stanley et al.1954), which allows fractional allocation of a de-mand ed lot to a supplier. We were restricted to integralallocations and hence needed to solve an integer-

programming problem. This led to a reverse-procurement auction that introduces an entirely newdynamic (competition among suppliers, transparency,and efficiency) into the negotiation process. The suc-cess of this approach makes it a forerunner of wide-spread change in procurement practice.None of the commercial products available today isable to provide iterative optimization and conform toall of Mars' business-rule constraints in a dynamic en-vironment. Even those with more sophisticated opti-mization offerings do not support all of Mars' business

rules, provide quick solutions, and prioritize bids acording to the time received.Mars ' Design for ProcurementAuctionsComplex bid structures necessitated an iterative aution design, largely because completely specifying tcost structure using bundled bids or supply-c urve bican result in an exponen tially large bid set. An itera tformat also (1) induces competition among supplie(2) allows suppliers to correct their bids using infmation learned during the process, and (3) elicits bincrementally so that the suppliers do n ot have to spify all of their preferences. W e wante d to pro vid e a fnegotiation process that did not squeeze the supplito a point of unprofitability, an d that would help Malong-term relationships with them. The iterative mecanisms had to terminate at a win-win outcome for tbuyer and the suppliers. The iterative auction desiused by M ars is illustrated in Figure 1.

(Buyer starts Aauction JPossible bid stfuduras:(1) single teen(2) bundted a lt-or..nottiiog(3) supply curve

Buyers use o ptimlzarion tosotve winner determinationI proMam (ot auction

(uyer closes ^aoctton JF i g u r e 1 : T h i s i s th e p r o c e s s r io w r o r a n i t e r a t i v e , m u i ti ro u n t t p r o c u r e ma u c t io n . B o t li t h e a u c t io n s M a r s r u n s h a v e t e r m i n a l i o n c r it e r ia lo sw h e n t h e y r e c e i v e n o n e w b i d s t o r a s p e c i fi e d t i m e .

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HOHNER, RICH, NG, REID, DAVENPORT, KALAGNAN AM, LEE, AND ANMars, Incorporated

Bid SubmissionThe company's existing procurement practice was topost a request for quo te for multiple items, giving thedemand for each item (Table 1). Mars also specifiesdifferent types of responses (or bids). Most negotia-tions are based on a simple-bid structure where thesupplier prov ides a unit price for each item along w iththe minimum and m aximum quantity it is able to pro-vide. The supply-curve bids give the unit price as afunction of quantity. With bundled bids, in contrast,suppliers provide only a single price for a bundledoffer.

Bid EvaluationIdeally, every time M ars received a new bid, it wou ldevaluate the bids to identify the provisional winners.However, the complexity of identifying the set of win-ning bids (often referred to as winner determination)depends on the bid structure. For two classes of bidstructures (all-or-nothing bids and supply-curve bids),the problem of determinin g the winners is NP-hard. Inaddition, the introduction of business rules furthercomplicates these problems, and the integer-programming formulations for solving the winner-determination problems must incorporate the businessrules as side constraints. (We provide integer-programming formulations for determinating the win-ner for each bid type in the Appendix.) As a result, itis difficult to solve the winner-determination problem

H em 1

Item 2

I t e m NPrice

PriceQty, D,PriceQty, Di.

PriceQty, D,,

Simple Bid$/unit

(Low. Hi)$/unit

(Low, Hi)

$/unit(Low, Hi)

Bundied Bid

Q,

Q..

ap

Suppiy Curvep

o

p0

T a b l e 1 : In t h is e x a m p l e r e q u e s t f o r q u o t e a n d b i d t y p e s , th e i t e m s a r es h o w n in r o w s a n d g i v e t h e p r i c e a n d q u a n t i t y d e m a n d a s s o c i a t e d w i t he a c h i t e m . T h e t a b l e s h o w s d i tt e r e n t b id s t r u c tu r e s a s c o l u m n s .

to identify the provisional winners as every new bidcomes in. In designing the system, we compromisedby allowing two minutes for each evaluation, accu-mulating the new bids within this interval. Thewinner-determination engine therefore had to evaluatebids in two minutes. We chose two minutes to allowenough time for computation under most situationsbut not unduly slow the progress of the auction.We developed the bid-evaluation engine as an in-dependent optimization module in C-1-+ and then in-tegrated it into the auction platform. The engine usesIBM's optimization subroutine library (OSL) as theLP/IP Solver. Tlie implementation of this frameworkwas done using IBM's e-commerce platformWeb-sphere Com merce Suite 4 . 1 , which provides the infra-structure support for the Nu mb erlTrad ers Web site.Feedback on Clearing PricesAfter every provisional allocation with an iterativeauction format. Mars gives the nonwinning supplierssome feedback to help them reformulate their bids. Forsimple single-item auctions, the suppliers are usuallyprovided with a clearing price at which supply for eachitem eq uals dem and. H owev er, for the multi-item auc-tions, no clearing prices exist for each item. We woulddevelop clearing prices for bundles of items, but wecould then have to report an exponential number ofprices. How ever, in any iteration only a small nu mb erof bundles (at most as large as the number of items)have a provisional allocation, and only on these bun-dles would we need to report clearing prices. Over thespan of the entire auction, the buyer must keep a listof all the bund les allocated in any ro und . In the worstcase, this list can become large, but in practice we findthat a small set of bundles provides cost complemen-tarities and the bidding centers around this set.

Bid ReformulatingSuppliers use the feedback provided in each round toreformulate their bids. A rational supplier would re-formulate its bids so as to maximize its outcome overall possible bid strategies of the o ther su ppliers. In re-ality, developing such a complete contingent-bid strat-egy is impractical for several reasons; (1) Suppliers donot have complete information about the cost struc-tures of the other bidders and as a result must c om pute

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their plans based on their expectations to incorporatethe informational uncertainty, and (2) the computa-tional burden of developing such a contingent plan isfar greater than that of solving the winner-determination problem and hence would take longer.We find that the suppliers act in a straightforwardfashion, trying to locally optimize their bids based onthe feedback in the current round. An associated issueconcerning supplier behavior is whether they implic-itly collude. The auctions run so far do not indicatecollusion. The bid price seems to move down quickly,and bidder behavior does not seem to have a (percep-tible) pattern (such as sequential bidding). For someitems purchased, cartels openly engage in price fix-ing. However, such markets are not suited forauctions.

In reformulating these bids, suppliers must followthe rules Mars imposes on subsequent rounds, such asminimum price decrement. For items or bundles forwhich Mars has made an allocation, nonwinning sup-pliers must offer a new price that is less than the cur-rent price by a fixed decrement. Such decrements areused to ensure rapid convergence.

Combinator ial AuctionsIt can be advantageous to aggregate demand over sev-eral locations and plants to form larger transactions.Also, suppliers may make gains in efficiency that per-mit them to provide discounted bids on a bundle (forexample, two sizes of identically printed packagingthat the supplier can print without changing printingpress inks). Such a discounted price would apply onlyif the buyer accepted the entire bid.

For example, the firm might send out an RFQ forpackaging in three formats to four suppliers (Table 2).The suppliers could provide bundled all-or-nothingbids for two or three of these items, at a price shownin the "Seller bid price" row. By introducing simpledecision variables (jl, .T2, X3, and x4) for the four bids,we can formulate an optimization problem that can besolved optimally using integer programming to mini-mize total procurement cost while ensuring that thedemand for each item is satisfied. The optimal supplysolution may oversatisfy demand, if there are no hold-ing costs, this might be acceptable or even desirable.

SuppliersItems SI

101

$150Xi

S201t

S125x2

S3111

$300x3

$

1,000 large Mars' brand display boxes800 small Mars* brand display boxes800 small M&Ms brand display boxesSeller bid priceDecisior) variable

Table 2: This example ot a combinatorial auction sbows the bids proviby four sup pliers.

Bid SubmissionWith all-or-nothing bids, the supplier would have provide exponentially many bids (with respect to tnumber of items in the auction) to completely descriits cost structure. For example, in the simple case three items (Figure 2), each supplier could provide= 8 bids over all possible combinations. For 10 itemthis could lead to over 1,000 bids per supplier. Evensuppliers could determine complete sets of combintorial bids, they would probably be unwilling to prvide this information.

iterative schemes permit suppliers to make onthose bids that maximize their profits at the curreprice levels for the items. They do not need to repotheir entire cost structures.

Winner Determinat ionThe computational complexity of the winnedetermination problem for combinatorial auctions even without side constraints, an NP-hard optimiztion problem. Each supplier is usually allowed mothan one bid, and as the number of items increases, tnumber of bids can get quite large. The combinatoriauction can be formulated as a set-covering problewith side constraints (arising from the business ruleThe formulation differs from the conventional formlation in that the side constraints make even determiing the existence of a feasible solution NP-compleWe model such problems as integer-programmiproblems and solve them using OSL. Integeprogramming techniques have proven to be effectiin solving problems with 500 items and up to 5,00bids.

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H O H N E R , RICH, NG, REID, D A V E N P O R T , K A L A G N A N A M , LE E, A N D A NMars. Incorporated

Feedback for Bid ReformulationFor combinatorial auctions, the winning prices are pro-vided by the buyer on bundles of items specified aswinning bids in some round of the auction. In subse-quen t ro und s, the bids suppliers m ake must be at leastsome fixed decrement delta lower than the currentThe payback on Mars' investmentwas less than a year.winning price on the bundle. The advantage of thisdesign is that one can derive an ex tended integral for-mulation for the set-covering problem with side con-straints that allows for dual prices on bundles. Whensuppliers bid to maximize their profits in the currentround, the outcome is that each winning supplier is atan optimum at the given prices, leading to a priceequilibrium.Bid Submission for Volume-Discount AuctionsIn making supply-curve bids, sellers specify the pricesthey charge for various quantities of an item. For in-stance, a supplier may charge $100 per ton for up to30 tons of sugar , but $50 per ton for more than 30 tons.Bids take the form of supply curves, specifying pricesper unit of an item within particular quantity intervals.Such price schedules can also be represented by an in-finite set of all-or-nothing bund led bids across mu ltipleitems.Winner Determination for Volume-Discount AuctionsIn general, when multiple suppliers provide suchvolume-discount bids, the problem of determining thewinning bids and the amount to buy from each sup-plier is a difficult optimization problem that can bemodeled as an integer progra m an d solved using com-mercial integer-programming software. In addition,the various business rules of the buyer can be capturedas side constraints within the mathematical formula-tion. We based o ur solution approach on modeling theproblem as a variation of the multiple-choice k napsack

problem (Martello and Toth 1984) (Append ix). By us-ing customized knapsack covers, we improv ed the per-formance of the integer-program ming solvers for theseproblems for up to 40 suppliers and 30 items. Forlarger problems we developed heuristics based oncolumn-generation-based heuristics that provide ap-proximate solutions to within one percent of optimal(Ladany i et al. 2001).

Feedback for Bid ReformulationIn volume-discount auctions, suppliers state prices forindividual items. In each rou nd. M ars reports the high-est price paid for any lot to the suppliers. This conven-tion is similar to traditional negotiation practice, andsuppliers find this feedback easy to use in reformulat-ing their bids.

The Impact of Business RulesIn procurement settings, buyers must consider thebusiness rules used in sourcing decisions. Mars' busi-ness rules reflect its practices, and buyers use them toconstrain the allocations they can consider.The business rules comp licate the evaluation of bidsbecause they must be incorporated into the integer-programming formulations as side constraints (Ap-pendix). These side constraints affect the cost of pro-curement and often force Mars to accept higher coststhan it would in a constraint-free solution (Figure 2).Figure 2 presents the results of solving the winner-determination problem (using mixed-integer program-ming) for a randomly generated volume-discount auc-tion with 10 suppliers and 20 lots. The size of thesupply pool is varied, between five and 20 suppliers.This is enforced by setting both the minimum andmaximum number of suppliers allowed in the auctionto be the size of the supply pool required. Each pointon the X-axis represents where we solved the winnerdetermination problem for a particular size of supplypool. (With less than five suppliers, the available b idscould not satisfy the entire demand of the buyer.) Theoptimal (lowest-cost) solution for this auction occursfor a supply pool of 13 suppUers. With 13 suppliers,we also see the shortest CPU tim e required to solve thecorresponding winner-determination problem. As we

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14301420

w 1410 -oS 1400g 1390u- 1380E 1370I 1360 -I 1350 -

13401330

700600

^ 5 0 0o8 400mI 3005 200

10005 10 15 20

Number ol Winning Suppliers10 15 20

Number of Winning Suppl iers

Figure 2: Side constraints for the voiume-discount auction, sucii as the exact number ot winning suppiiers. attect(a) the totai procurement cost to Ma rs, and (b) the CPU time required to solve the winner-determination probiem.

vary the size of the supply pool away from the sizerequired to obtain the optimal procurement cost(which here is 13 suppliers), the total procurement costincreases significantly. The constraint on the minimumquantity awarded to each suppher has a significant ef-fect on how much deterioration we observe as we in-crease the size of the required supply pool. In practice,the buyers have often specified that the size of the sup-ply pool must be fixed or perturbed as little as possiblefrom the status quo. This has the effect of limiting thepossible savings that can be achieved using these typesof auctions. Often the size of the status quo supply poolis larger than that required to find the optimal pro-curement cost, which also has a negative impact on theCPU time required to solve the winner-determinationproblem for these auctions (Figure 2).

Desirable Propert ies for the AuctionDesignTwo main properties are desirable for procurementauctions, fairness and optimaiity, and we have incor-porated them into the auction design for Mars.

Auctions are a competitive mechanism for allocatingthe firm's business to suppliers. It is very importantthat the bidders perceive Mars' auctions to be fair. Ifthey did not, bidders might refuse to participate in theauctions. We tried to ensure optimaiity and win-win

30

outcomes in designing the auctions to make sure thwere fair.

Given a set of bids, the winner-determination engidetermines the cost-minimizing bid set and therechooses the winning suppliers. This optimizatiproblem is hard to solve computationally. Using heristics to solve the problem might yield an approxmate solution that was within less than one percent the optimal-cost solution. However, such an approxmate solution might call for an allocation drasticadifferent from the optimal. A supplier who wouhave an allocation in the optimal solution might nget any allocation in the approximate solution. Supliers who failed to get an allocation because of tapproximate nature of the algorithm would see tauction as unfair. Therefore, the v^^inner-determinatiengine must provide optimal allocations. The integeprogramming solver we developed and tuned for eficiency provides optimal solutions within two miutes response time.

Time StampsIn multiround auctions, the buyer must decide how treat sets of bids made in different rounds for the samitems at the same price. For example, a supplier migcreate a combinatorial auction to purchase some quatities of items A, B, and C. In the first round of t

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auction. Supplier 1 bids $100 for item s A, B, and C, andSupplier 2 bids $30 for item A. The solution to thewinner-determination problem for this round is thatSupplier 1 wins with its bid because it supplies theentire demand. During the second round. Supplier 3enters the auction and bids $70 for items B and C. Thiswinner-determination problem has two potential so-lutions: either Su pplier 1 's bid to sup ply all three itemsor the com bination of bids by Suppliers 2 and 3, whichtogether would supply all three items. In either case,the buyer would pay $100 for its total demand. Froma fairness point of view, these two solutions are notequally desirable. In a multiround auction, new bidsshould be deemed winning bids only if they result inthe buyer paying a lower price for the items in theauction. In simple auctions, the usual rule governingsuch situations is that the earlier of the identicallypriced bids for the same items is preferred. This ruleis straightfo rward to enforce in a simple, single-itemor multi-item forward or reverse auction. In combi-natorial and volume-discount auctions, this rule be-comes harder to enforce, because the number of pos-sible solutions to the winner-determination problemmay be exponential w ith respect to the num ber of bidsplaced in the auction.In this project, we gave each bid a numeric timestamp, representing the time it was accepted in theauction. We then had two objectives for the winner-determination problem for both types of auctions:(1) To determ ine all the sets of bids that wo uld min-imize the buyers' total procurement cost, and(2) From these sets of bids, to select the set thatwould minimize the sum of the time stamps for thebids in this set.To solve the winner-determination problem withthis new objective, we formulate two integer-programming problems. The first problem we solve isthe winner-determ ination problem (for either the com-binatorial or volume-discount auction) without con-sidering time stamps.The second integer-programming problem differsfrom the first in two ways:(1) We add a constraint stating that the demandshould be bought at the minimum cost determined inthe solution to the first problem.(2) The objective for the second problem does not

consider price at all, because the new constraint willensure the minimum price. Instead we seek to mini-mize the sum of the time stamp s of the candida te win-ning bids in solving the winner-determinationproblem.In practice, we find that it takes far longer (four to20 times) to solve the second integer-programmingproblem than to solve the first.

Deployment IssuesIt takes a long time to change the way people think!Coping with large geographical distances, varyingbusiness environments, and different cultures madethe implementation part of this project at least as hardas the technical part. Mars buyers are spread over fivecontinents and more than one hundred countries. Tolaunch this project we had to inform all the buyersabout the Num berlTrad ers.com Web site, what it doesand how to use it, and convince them of the benefitsso they would change the way they ran theirbusinesses.Training Across the Net and Aroundthe WorldTo inform buyers quickly about the basic ideas un-derlying the new system, we developed a half-daycourse on electronic auctio ns. In 2001, we trained over300 buyers worldwide through classes offered inNorth America, Europe, Australia, and Asia. Wetrained another 150 via video conferences and Netmeetings. Four buyers interested in auctions volun-teered to became buyer-experts and took on addi-tional responsibilities.

The training classes were only the beginning of theeducation and support we provided. Buyers requiredpersonal assistance the first time they ran an auction.Gail Hohner and the buyer-experts helped first-timeusers to choose buying opportunities appropriate forauctions, to design the first auction, and to train sup-pHers in the supply pool. They m ade themselves avail-able for technical support durin g the auctions and doc-umented the results so that other Mars buyers couldlearn from the experience.INTERFACES


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Changing the Business ProcessThe Numberl Traders Web site has been available toMars buyers since September 2001. The growth in us-age has been steady. Buyers see the efficiency of theprocess as a prime reason for adopting the system. Asthe number of buyers who have had positive experi-ences using the system increases, word-of-mouth hasbecome an effective internal advertisement.

Still, we should not underestimate the changes buy-ers must make to adopt a new purchasing practice.With bilateral negotiations, buyers maintain all the so-cial aspects of the supply relationship throughout theprocess, They have many occasions for both formaland informal communication with suppliers. Al-though the buyer and supplier meet at other times dur-ing the year, the majority of the face-to-face meetingtime is concentrated around the contract negotiation.When they change to an auction process, buyers mustcontinue to work on integrating suppliers into theirbusinesses and on maintaining informal social contact(because it strengthens the buyer-supplier relationshipand provides a vector for information); however, theymust avoid negotiating price or quantity allocations.Buyers have a very difficult time making this separa-tion. They must not negotiate prices and quantities, thetask on which they formerly spent most of their time.

Suppliers have generally reacted favorably to theseauctions. They have commented on some of their ad-vantages. Many suppliers value the transparency ofthe auction process. In negotiations, they had no in-formation on competing suppliers' prices. In auctions,the prices they offer are compared to those of othersuppliers. Also suppliers now control their last bidsrather than having the buyer end the process at an ar-bitrary point. Suppliers value their time. When the sys-tem becomes very efficient (as it does when auctionshave been run many times), suppliers have been veryhappy with the process. Suppliers view these auctionsas more equitable than any other auction mechanismthey have encountered because many suppliers winbusiness, not just one. These auctions also allow sup-pliers to present their unique sales propositions (forexample, discounts reflecting unique synergies). SchutCees of Thermphos commented, "We have partici-pated in auctions that have just been price gouging

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exercises. This is the most equitable design we haever seen."

Procurement auctions have received a lot of prerecently, and spectacular savings have been reportefor reverse English auctions. But where do those saings come from? In many cases, they come frosqueezing suppliers to the point of (or beyond) zeprofit. Consequently, a strong backlash is developinin some industries against using auctions at all. Wfind that our approach to auctions draws our supplipool closer rather than alienating it. No system is pefect, however, and not everybody is happy, but geerally the response has been very positive.

ImpactThe auctions have yielded consistent cost savings. Wdon't mean that suppliers are selling to us below cosThe efficiencies come from matching supplier capabities and the company's needs and thus increasing supliers' margins, part of which are to provide Mars wisavings. We have found that when buyers were willinto change the size of the supplier pool or shift largamounts of business, the auctions yielded greater saings. The payback on Mars' investment was less thaa year.

To determine savings, we compared the buyers' predictions made one or two days before the auction ato the best outcome possible from a negotiation. Suca prediction incorporates the market knowledge date and is the best way we know of making a comparison. Buyers do not make such predictions solelas benchmarks for auctions but for use in Marproduction-planning system. Mars assesses its buyeon the quality of their predictions and their undestanding of the markets they cover. Any systematdistortion would come from buyers' optimism abouwhat they could negotiate. In such cases, we were conservative in estimating savings. Once they are integrated into the business process, auctions take mucless time than negotiations. Mars buyers, as a resulhave more time to ahgn businesses and to seek synergistic value from suppliers.

It usually takes a day or two to design, set up, antrain suppliers for auctions the first time they are runAs we repeat an auction, training times for supplie

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drop from one hour for first-time user s to less than 10minutes for repeat users. Auctions have never takenmore time than negotiations. In the most significantreported time savings, a 40-minute auction replaced aprice-only negotiation process that had lasted over twoweeks and required the buyer tomake nine separateair trips to finalize only the prices and volumes of th econtract.Adoption has been encouraging. Many more buyershave expressed interest in the process, but they mustwait un til the ir next contract cycle (which can be a yearaway). By the end of 2002, we expect to have con-ducted nearly 60 auctions.

The Fu ture of Combinatorial andSupply Curve Auctions at MarsMars has started newbusiness units (for example, Qi/,;,?;J at price P| per unit; it takesvalue 0 otherwise.We associate a continuous variable 2* with eachprice-quantity pair, which specifies the exact numberof units of the lot that is to be purchased from the bidB( within this price-quantity pair.The MIP formulation is then given as follows:mmimize

subject to+

fceK ie N

V( e N, V ; G Ml

V V fr* J. r^r^ , 1 :> O* Vt f= k"Xi X. vi,'; X îi\-c.iilow' >< ' m f - ^ " ^ ^

4 G iO, II V/ GN, V; GMf, Vit G K,4 > 0 ViG N, V; G K.

(a )(b )(c)

(1)The coefficient Cf is a constant and computed a priorias

(2)Constraint (a) specifies, for each price-quantity pairiPîr [QîjMc'


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the range [(^fjow Qj,?,,^?,]). Constraint (b) specifies thatfor each winning bid, we can only buy at a price andquantity that corresponds to a single price-quantitypair. Constraint (c) states that we must determine awinning set of bids such that the total demand of thebuyer for each lot k is satisfied.

We have included all lots that the buyer wishes toprocure in this formulation, even though, as it stands,we could solve the winner-determination problem foreach lot independently. This is done for convenience:in the next section, we introduce side constraints thatinvolve all the lots in the auction.

Side ConstraintsThe constraints on the minimum S^in and maximumSmax nu mb er of win ning sup plier s in the solution tothe winner-determination problem are encoded in theMIP formulation in the following way. We introducean indicator variable y, for each supplier i which takesthe value 1 if the supplier has any winning bids and 0otherwise. The first constraint sets y, to 1 if supplier /has any winn ing bids. Note that the constant m ultiplierensures that the right-hand side is large enough whenifi is one. (The constant m ultiplier K follows from con-straint (c) in the formulation.)

V/ G N,, y , G {0,11

y,- ^Local lot-level constraints state for each supplier /and lot k, the minimum (/frnm ^^^ maximum quantity^im^\ that can be allocated to the supplier of this lottype. For each supplier ;, the buyer provides bounds

on the total allocation, across all lots, to lie betweenW, n,i,, and W, max- If* the winn ing allocation , if su pp lier/ has any allocation at all then it must lie in this range.These constraints can be expressed in the followingway in the formulation:(4 + 4 0

- y,w,,^,, < 0 m34

Note that these global lot-level constraints imply tfirst constraint in Equation 3.Reservation prices, maximum allowable prices punit can be provided by the buyer for each lot. Thconstraint is not encoded in the MIP formulation fthe winner-determination problem. Instead bids, portions of the supply curve in each bid, that fail satisfy this constraint are disregarded by the solvengine.

Mathematical Formulation of theCombinatorial Auction Winner-Determination ProblemWe are given a set of k items, where for each item kK there is a dema nd for Q^ units of the item (calledlot). Each supplier i S N is allowed up to M' bids idexed by ;. We associate with each bid 6,y a zero-onvector a'li, k = 1, ..., K wh ere 4 = 1 if bid B^ wsupply the (entire) lot corresponding to item k, anzero otherwise. Each bid 6,y offers a price pa at whithe bidder is willing to supply the combination oitems in the bid. A mixed-integer-programming fomulation for the reverse combinatorial auction can bwritten as follows:minimize

^^' subject to 2 2ieN / E M '

X,X, G (0, 11 V; G N, V; G M '.

((

The decision variable X/, takes the value 1 if the bid Bis a winning bid in the auction, and 0 otherwise. Constraint (a) states that the total number of units of eacitem in all the winning bids must satisfy the demanthe buyer has for this item.

Side ConstraintsThese constraints can be added to the MIP formulatioas follows:

4 tkeK jeM'

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HOHNER, RICH, NG, REID, DAVENPORT, KALAGNANAM, LEE, AND ANMars, Incorporateii

2 2 . !)tf^ y , V i G N ,

^2ieN

^ S.10, II Vi G N.

(b)keK

(c)(d )(6)

The terms W,,niin and W,,n,a,, define the minimum andmaximum quantity that can be allocated to any sup-plier (. Constraints (a) and (b) restrict the total alloca-tion to any supplier to lie within the range (Wy,,,jn,W,,niax)' Note that y, is an indicator variable that takesthe value 1 if supplier / is allocated any lot. If W, ,! =0, then y, becom es a free varia ble. To fix this, we intro -duce constraint (c), which ensures that y, = 0 if no bidsfrom supplier / are chosen. S^jn ^^^ ^max relate to theminimum and maximum number of winners requiredfor the allocation. Constra int (d) restricts the total nu m -ber of winners to be within the range (Sn j , S^^^).ReferencesBender, I'. S., R. W. Brown, M . H. Isaac, J. F. Shapiro. 1986. Improv-ing purchase productivity with a normative decision supportsystem. Interfaces 15(3) 106-115.Gallien, J., L. Wein. 2002. A smart m arket for industrial procurement

with capacity constraints. Submitted to Management Sci.Jucker, J. V., M. J. Rosenblatt. 1985. Single period inventory mod elswith demand uncertainty and quantity discounts: Behavioralimplications and a new solution procedure. Naval Res. Logis.Quart. 32(4) 537-550.

Katz, P., A. Sadrian, P. Tendrick. 1994. Telephone companies analyzeprice quotations with BellCore's PDSS software. Interfaces 24(1)50-63.

Kim, K. H., H. Hwang. 1988. An incremental discount pricing withmultiple customers and a single price break. Eur. ]. Oper. Res.35(1)71-79.Ladany i, L., ]. Forrest, J. Kalagnanam. 2001, Column generation ap-proach to the multiple knapsack problem with color con-straints. IBM Research Report RC 22013, Yorktown Heights,New York.Lee, H. L., M.J. Rosenblatt. 1986. A generalized quantity discountpricing model to increase supplier's profit. Management Sci.32(9)1177-1188.Martello, S., P. Toth. 1984. Knapsack Problems. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley,

New York.Narasimhan, R., M. K. Stoynoff. 1986. Optimizing aggregate pro-curement decisions. /. Purcliasing Materials Management 22(22-30.Park, S., M. H. Rothkopf. 2001. Auctions with endogenously deter-mined allowable combinations. RUTCOR research report RRR-

3-2001, January. Rutgers Center for Operations Research, Rut-gers University, NJ.Rothkopf, M. H., A. Pekec, R. M. Ha rstad. 1998, C omp utationallymanageable combinatorial auctions. Management Sci. 44 1131-1147.Sadrian, A., Y. S. Yoon. 1994. A procurement decision support sys-tem in business volume discount environm ents. Oper. Res. 42(1)

14-23.Silverson, R., E. A. Peterson. 1979. Decision Systems for Inventory Management an d Production Planning. John Wiley and Sons, NeYork.Stanley, E. D., D. Honig, L. Gainen, 1954. Linear programming in bidevaluation. Naval Res. Logist. Quart. 1 48-54.

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