Decision Support Systems 48 (2010) 323–330

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Decision Support Systems

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Combinatorial reverse auction based on revelation of Lagrangian multipliers

Fu-Shiung HsiehDepartment of Computer Science and Information Engineering, Chaoyang University of Technology, Taiwan

E-mail address: fshsieh@cyut.edu.tw.

0167-9236/$ – see front matter © 2009 Elsevier B.V. Aldoi:10.1016/j.dss.2009.08.009

a b s t r a c t

a r t i c l e i n f o

Article history:Received 8 January 2009Received in revised form 3 August 2009Accepted 30 August 2009Available online 13 September 2009

Keywords:Auctione-CommerceInteger programmingOptimizationBid

Recently, researchers have proposed decision support tool for generating suggestions for bids incombinatorial reverse auction based on disclosure of bids. An interesting issue is to design an effectivemechanism to guide the bidders to collectively minimize the overall cost without explicitly disclosing thebids. We consider a winner determination problem for combinatorial reverse auction and study how tosupport the bidders' decisions without explicitly disclosing the bids of others. We propose an informationrevelation scheme for a buyer to guide the sellers to generate potential winning bids to minimize the overallcost. The main results include: (1) a problem formulation for the combinatorial reverse auction problem;(2) a solution methodology based on Lagrangian relaxation; (3) a scheme to guide the sellers to generatepotential winning bids for the bidders in multi-round combinatorial reverse auctions based on revelation ofLagrangian multipliers; (4) a heuristic algorithm for finding a near-optimal feasible solution and (5) resultsand analysis of our solution algorithms.

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© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Auctions are popular, distributed and autonomy preserving waysof allocating items or tasks among multiple agents to maximizerevenue or minimize cost. In economics, different types of auctionshave been proposed and extensively studied, including EnglishAuction (open ascending price auction), Dutch auction (opendescending price auction), sealed first-price auction, etc. Single itemauctions are by far the most common auction format, but they are notalways efficient. Combinatorial auctions [4,16,22,26] enable severalbidders to bid on different combination of goods according to personalpreferences. Allowing bids for bundles of items is the foundation ofcombinatorial auctions. Bidders can select multiple items at one timeand offer those items a price. It enables bidders to decide combina-tions of auction according to personal preferences of bidders.Combinatorial auctions are beneficial if complementarities existbetween the items to be auctioned. Well-known examples are theauctioning of Federal Communications Commission's radio spectrumlicenses, the sales of airport time slots, and allocation of deliveryroutes. Applying combinatorial auctions in corporations' procurementprocesses can lead to significant savings [17,18]. Reverse auction is abusiness auction model that can be applied to corporations'procurement. Combinatorial reverse auction enables buyers tosimultaneously purchase multiple goods with the lowest pricesfrom the sellers.

Combinatorial auctions have attracted considerable attention inthe existing literature. An excellent survey on combinatorial auctions

can be found in [3,19]. Combinatorial auctions have been notoriouslydifficult to solve from a computational point of view [21] due to theexponential growth of the number of combinations [25]. Thecombinatorial auction problem can be modeled as a set packingproblems (SPP) [2,5,10,24] Sandholm et al. mentioned that deter-mining the winners so as to maximize revenue in combinatorialauction is NP-complete [22,23]. Many algorithms have been devel-oped for combinatorial auction problems. Exact algorithms have beendeveloped for the SPP problem, including iterative deepening A*search [23] and the direct application of available CPLEX IP solver [2].Gonen and Lehmann proposed branch and bound heuristics forfinding optimal solutions for multi-unit combinatorial auctions [7].Jones and Koehler studied combinatorial auctions using rule-basedbids [13]. In [8]; [11,12] the authors proposed a Lagrangian heuristicand a Lagrangian relaxation approach for combinatorial reverseauction problems, respectively. Although combinatorial reverseauctions have been extensively studied, most studies focus on thedevelopment of efficient solution algorithms to determine winners.From the viewpoint of a bidder, how to find a potential winning bidthat maximizes the profit is a key issue. In practice, each seller has noidea about the bids placed by other sellers at the beginning of acombinatorial reverse auction. Therefore, each seller is “blind” increating his bids. The bid placed by a seller is usually a profitmaximizing bid that does not take into account the bids placed byother sellers. However, a profit maximizing bid may not be a winningbid. An interesting question is to design an information revelationmechanism to assist a seller to find profit maximizing winning bidswhile minimizing the cost of the buyer.

Recently, decision support in auctions and combinatorial auctionshas received significant attention. Adomavicius and Gupta presented

Fig. 1. Combinatorial reverse auction.

324 F.-S. Hsieh / Decision Support Systems 48 (2010) 323–330

several metrics that bidders can use to evaluate the current auctionsituation and the potential of each bid being among the winners [1].The weakness of solving the winner determination problem (WDP) ofcombinatorial auctions in one shot is the black box nature of integerprogramming. Another stream of literature in combinatorial auctionsis on ascending auctions, in which bidders, instead of bidding oneprice for the bundles, engage in multiple rounds of auctions ofdifferent bundles. Adomavicius and Gupta [1], among other propo-nents of the approach, argued that one advantage over the seal-bidone-shot format is that bidders can get pricing feedback from theprocess. Our approach may open up the black box slightly to thebidders by offering the values of Lagrangian multipliers to them.

Kwasnica et al. provided the bidders with a vector of prices (onefor each commodity) that new bids must beat in order to be accepted[14]. Gallien and Wein presented a system and the underlying theoryfor an optimization-based multi-item auction mechanism that relieson the solution of a linear program for minimizing the buyer's costunder the suppliers' known capacity constraints [6]. They assistsuppliers in finding a winning bid price. The underlying assumption isthat the suppliers are willing to disclose their cost functions to asupposedly neutral third party auction organizer. Hohner et al.provided feedback to nonwinning bidders regarding clearing prices,at which supply for each item equals demand [9]. Leskelä et al. [15]proposed a decision support tool for generating suggestions for bidsthat would be among the current winners of the auction. Their resultsindicate that the quantity support tool is useful as it decreases thetotal cost to the buyer and improves the efficiency of the auction. Thesupport tool of Leskelä et al. requires the information of the bidsplaced by other bidders. To apply the support tool of Leskelä et al.requires disclosing the bids of other bidders. An interesting question ishow to develop an effective method to support bidders' decisionwithout disclosing the bids while minimizing buyer's total cost. Thegoal of this paper is to design an effective mechanism to guide thebidders to collectively minimize the overall cost in combinatorialreverse auction. We consider a winner determination problem forcombinatorial reverse auction inwhich a buyer wants to acquire itemsfrom a set of sellers and each seller can provide a set of items. Wepropose an information revelation scheme for a buyer to guide thebidders to generate potential winning bids to minimize the overallcost without disclosing the bids to all the bidders.

One way to reduce the computational complexity in solving thewinner determination problem (WDP) for combinatorial reverseauction is to set up a fictitious market to determine an allocation andprices in a decentralized way to adapt to dynamic environmentswhere bidders and items may change from time to time. In this paper,we apply Lagrangian relaxation technique to develop a solutionalgorithm for WDP. We propose a multi-round combinatorial reverseauction algorithm to efficiently find a solution based on revelation ofLagrangian multipliers for each type of item at the end of each round.In our multi-round combinatorial reverse auction algorithm, Lagrang-ian relaxation is applied to obtain a solution at the end of each round.To take advantage of Lagrangian multipliers to efficiently guide thebidders in the bidding processes, the Lagrangian multipliers arerevealed to all the bidders. We propose a combinatorial reverseauction information revelation scheme based on Lagrangian multi-pliers. A seller may generate a new bid according to the most recentlyrevealed Lagrangian multipliers.

Lagrangian relaxation provides a systematic approach to deter-mine an allocation and prices based on the introduction of Lagrangianmultipliers, which set prices for each item to be purchased by thebuyer. If two or more sellers compete for the same item, the price willbe adjusted. This saves bidders from specifying their bids for everypossible combination and the buyer from having to process each bidfunction. Based on the price for the individual items, bidders submitbids. The bundle associated with a bid is tentatively assigned to thatbidder only if the price of the bid is the lowest. Based on the iterative

price adjustment mechanism, a solution will be obtained. It should beemphasized that Lagrangian relaxation is not guaranteed to find theoptimal solution to the underlying problem. Furthermore, it is notguaranteed to produce a feasible solution by applying Lagrangianrelaxation technique. In case the resulting solution is not feasible, aheuristic algorithmmust be applied to adjust the infeasible solution toa feasible one. We develop a heuristic algorithm for finding a near-optimal, feasible solution based on the solution of the relaxedproblem. We also demonstrate the advantage of multi-roundcombinatorial reverse auction algorithm (with revelation of Lagrang-ian multipliers) by comparing it with the single-round combinatorialreverse auction algorithm (without information revelation). Ourresults indicate that significant improvement in costs can be achievedby applying our method at the price of more but acceptable CPU time.

In summary, themain results presented in this paper include: (1) aproblem formulation for the combinatorial reverse auction problem;(2) a solution methodology based on Lagrangian relaxation; (3) aprice information revelation scheme to facilitate auction based on aneconomic interpretation of Lagrangian multipliers and (4) results andanalysis of our solution algorithms.

The remainder of this paper is organized as follows. In Section 2,we present the problem formulation. In Section 3, we propose thesolution algorithms for the dual problem and give an economicinterpretation for our solution approach. In Section 4, we propose acombinatorial reverse auction information revelation scheme basedon Lagrangian multipliers. In Section 5, we concentrate on theheuristic algorithm for finding a feasible solution. Finally, wedemonstrate the effectiveness of the proposed algorithms byanalyzing the results of many numerical examples. We concludethis paper in Section 7.

2. Combinatorial reverse auction problem formulation

In this paper, we first formulate the combinatorial reverse auctionproblem as an integer programming problem. We then developsolution algorithms based on Lagrangian relaxation. Fig. 1 illustratesan application scenario in which Buyer requests to purchase at least abundle of items 2A, 3B, 2C and 1D from the market. There are threebidders, Seller 1, Seller 2 and Seller 3 who place bids in the system.Suppose Seller 1 places two bids: (1A, 2B, p11) and (1C, 1D, p12),where p11 and p12 denote the prices of the bids. Seller 2 places two

325F.-S. Hsieh / Decision Support Systems 48 (2010) 323–330

bids: (1B, 2C, p21) and (2C, 1D, p22). Seller 3 places two bids: (1C, 1D,p31) and (1A,1B, p32). We assume that all the bids entered theauction are recorded. A bid is said to be active if it is in the solution.We assume that there is only one bid active for all the bids placed bythe same bidder. For this example, the solution for this combinatorialreverse auction problem is Seller1: (1A, 2B, p11), Seller 2: (2C, 1D,p22) and Seller 3: (1A,1B, p32).

Consider a buyer who requests a set of items to be purchased. Let Kdenote the number of items requested. Let dk denote the desired unitsof the k-th item, where k∈{1, 2, 3, ... K}. In a combinatorial reverseauction, there are many bidders. Let I denote the number of bidders inthe combinatorial reverse auction. Each i∈{1, 2, 3, ..., I} represents abidder. To model the combinatorial reverse auction problem, the bidmust be represented mathematically. We use a vector bij=(qij1, qij2,qij3, …, qijK, pij) to represent the j-th bid submitted by bidder i, whereqijk is a nonnegative integer that denotes the quantity of the k-th itemsand pij is a real positive number that denotes the price of the bundle.As the quantity of the k-th items cannot exceed the quantity dk, itfollows that the constraint 0≤qijk≤dk must be satisfied. The j-th bidbij is actually an offer to deliver qijk units of items for each k∈ {1, 2, 3, ...,K} a total price of pij. Let ni denote the number of bids placed by bidderi∈ {1, 2, 3, ..., I}. Let J denote themaximumnumber of bids that a biddercan place in each round of combinatorial reverse auction. That is,J = max

i∈f1;2;:::;Igni. To formulate the problem, we use the variable xij to

indicate that the j-th bid placed by bidder i is active (xij=1) or inactive(xij=0). The winner determination problem can be formulated as aninteger programming problem as follows.

Winner determination problem (WDP):

min ∑1

i=1∑ni

j = 1xijpij

s:t:∑1

i=1∑ni

j = 1xijqijk ≥ dk∀k = 1;2;…;K ð2� 1Þ

∑nij = 1xij ≤ 1∀i = 1;…; I ð2� 2Þ

xij∈f0;1g: ð2� 3Þ

Condition (2-1) in WDP assumes “free disposal” as the totalquantity offered by the winners must be greater than or equal to thedesired quantity of the buyer. If there are more quantities providedthan needed, we can dispose of the surplus with no additional cost.One way to reduce the computational burden in solving theWDP is toadopt Lagrangian relaxation approach to set up a fictitious market todetermine an allocation and prices in a decentralized way to adapt todynamic environments where bidders and items may change fromtime to time. The buyer announces which sets of items and sets pricesfor them. If two or more agents compete for the same item, the buyeradjusts the price vector. This saves bidders from specifying their bidsfor every possible combination and the buyer from having to processeach bid function. The bundle associated with the bid is tentativelyassigned to that bidder only if the price of the bid is the lowest.

3. Solution algorithms for dual problem

The basic idea of Lagrangian relaxation is to relax some of theconstraints of the original problem by moving them to the objectivefunction with a penalty term. That is, infeasible solutions to theoriginal problem are allowed, but they are penalized in the objectivefunction in proportion to the amount of infeasibility. The constraintsthat are chosen to be relaxed are selected so that the optimizationproblem over the remaining set of constraints is in some sense easy. InWDP, we observe that the coupling among different operations iscaused by the minimal requirement constraints (2-1). Let λ denotes

the vector with λ(k) representing the Lagrangian multiplier for thek-th items. We define

LðλÞ = min∑I

i=1∑ni

j = 1xijPij + ∑K

k=1λðkÞ dk−∑

I

i=1∑ni

j = 1xijqijk

!

s:t:∑nij = 1xij≤1∀i = 1; :::; I

xij∈f0;1g:

For a given Lagrangian multiplier λ, the relaxation of constraints(2-1) decomposes the original problem into a number of bidders'subproblems (BS). These subproblems can be solved independently.That is, the Lagrangian relaxation results in subproblemswith a highlydecentralized decision making structure. Interactions among subpro-blems are reflected through Lanrange multipliers, which are deter-mined by solving the following dual problem.

maxλ≥0

LðλÞ, where

LðλÞ = min∑K

k=1λðkÞdk + ∑

I

i=1∑ni

j = 1xij Pij−∑K

k=1λðkÞqijk

!

s:t:∑nij = 1xij≤1∀i = 1; :::; I

xij∈f0;1g

= ∑K

k=1λðkÞdk + ∑

I

i=1LiðλÞ; with

LiðλÞ = min∑nij = 1xij Pij−∑

K

k=1λðkÞqijk

!

s:t:∑nij = 1xij≤1

xij∈f0;1g:

Li(λ) defines a bidder's subproblem (BS). Our methodology forfinding a near-optimal solution of WDP is developed based on theresult of Lagrangian relaxation and decomposition. It consists of threeparts: (1) an algorithm for solving subproblems, (2) a subgradientmethod for solving the dual problem and (3) a heuristic algorithm forfinding a near-optimal feasible solution. In this section, we focus onpart (1) and part (2). Part (3) will be detailed in Section 5.

(1) An algorithm for solving subproblems

Given λ, the optimal solution to BS subproblem Li(λ) can be solvedas follows.

Let j� = arg minj∈f1;2;:::;niÞ

Pij−∑K

k=1λðkÞqijk

!. The optimal solution to

Li(λ) is as follows.

xij =

0 ∀j∈f1;2; :::;nig n f j�g

1 if Pij�−∑K

k=1λðkÞqij�k < 0

0 if Pij�−∑K

k=1λðkÞqij�k≥0:

8>>>>>><>>>>>>:

Since evaluating Li(λ) for each λ is a snap, if we can find a fast waytodetermine theλ that solves max

λ≥0LðλÞwewouldhavea fast procedure

to find a solution. Although the resulting solution (values of the xvariables) need not be feasible, it could be adjusted to a feasible solutionwithout a great increase in objective function value. Finding λ thatsolves max

λ≥0LðλÞ can be accomplished using the subgradient algorithm.

(2) A subgradient method for solving the dual problem maxλ≥0

LðλÞ

326 F.-S. Hsieh / Decision Support Systems 48 (2010) 323–330

Let xl be the optimal solution to the subproblems for givenLagrangian multipliers λl of iteration l. We define the subgradient ofL(λ) as

glðkÞ = dk−∑I

i=1∑ni

j = 1xlij qijk; where k∈f1;2;…;Kg:

The subgradient method proposed by Polyak [20] is adopted toupdate λ as follows

λl + 1ðkÞ = λlðkÞ + αlglðkÞ if λlðkÞ + αlλlðkÞ≥0;0 otherwise:

�

where αl = c L−LðλÞΣkðglðkÞÞ2, 0≤c≤2 and L is an estimate of the optimal dual

cost. The iteration step terminates if α l is smaller than a threshold.Polyak proved that this method has a linear convergence rate. Thestructure of our algorithms is depicted in Fig. 2.

Iterative application of the algorithms in (1) and (2) may convergeto an optimal dual solution (x*, λ*). It should be emphasized thatLagrangian relaxation is not guaranteed to find the optimal solutionto the underlying problem. Rather, it finds an optimal solution to arelaxation of it. While Lagrangian relaxation will yield the optimalobjective function value for the linear relaxation of the underlyinginteger program, it is not guaranteed toproduce a feasible solution. Thusthe solution generated may not satisfy the complementary slacknessconditions. In case the solution is not feasible, we must develop aheuristic algorithm to find a feasible solution. Development of aheuristic algorithm to find a feasible solution is detailed in Section 5.

4. Revelation of Lagrangian multipliers

Decomposition of the original problem into BS subproblemsprovides us a different viewpoint of the original problem. Decisionmaking of the original problem is composed of those of BSs. BS can beregarded as the decision making problem to acquire the requiredresources and benefits from utilizing the acquired resources. Thedecision processes of each entity are just like the ones made by sellersin a real business environment. That is, a seller is willing to sell a goodonly when its utility is at least equal to its market price. The buyer willraise the market price in case of resource shortage. Such processesoccur in the decision making of the BS. Lagrangian multipliers canoften be given the economic interpretation as marginal costs for usingthe items when they are used to relax demand constraints. In ourrelaxation procedure above, the Lagrangian multipliers λ(k) are usedto relax the demand constraints of item k. Lagrangian multipliers λ(k)can be interpreted as the marginal benefit of using an additional unit

of item k. Suppose the constraint dk−∑I

i=1∑ni

j = 1xijqijk≤0 is violated.

That is, dk−∑I

i=1∑ni

j = 1xijqijk > 0. In this case, adding an additional

Fig. 2. Structure of sol

unit of item k reduces the total cost by λ(k). On the other hand, if the

constraint dk−∑I

i=1∑ni

j = 1xijqijk≤0 holds, adding an additional unit of

item increases the total cost by λ(k).To take advantage of the Lagrangianmultipliers to efficiently guide

the bidders in a combinatorial reverse auction, we propose aninformation revelation scheme. In this section, we propose aninformation revelation scheme to guide the bidders without revealingthe details of all the bids submitted. The information revealed in ourscheme is based on Lagrangianmultipliers obtained at the end of eachround of the combinatorial reverse auction. Let λ* denote theLagrangian multipliers obtained at the end of the m-th round.Lagrangian multipliers λ*(k) for each item k of the m-th round arerevealed. By revealing λ*(k) for each item k, each bidder will know themarginal benefit of adding a unit of item k. Thus, each bidder mayplace a new bid based on λ*(k) for each k. To determine whether anew bid should be placed, consider the following BS for bidder i∈ I:

Suppose

LiðλÞ = min∑nij = 1xij Pij−∑

K

k=1λ�ðkÞqijk

!s:t:∑ni

j = 1xij≤1

xij∈f0;1g:

Suppose a new bid bij′=(qij′1, qij′2, qij′3, ..., qij′K, pij′) is placed bybidder i. Then

L′iðλÞ = min ∑nij = 1xij Pij−∑

K

k=1λ�ðkÞqijk

!+ xij′ Pij′−∑

K

k=1λ�ðkÞqij′k

! !

s:t:∑nij = 1xij≤1

xij∈f0;1g:Suppose

Pij′−∑K

k=1λ�ðkÞqij′k≤Pij−∑

K

k=1λ�ðkÞqijk∀j∈f1;2; :::;nig: ð4� 1Þ

In this case, bij′ is the better than all the existing bids placed bybidder i. It will be the bid selected by the algorithm. Therefore, if λ*(k)is revealed for each item k, inequality (4−1) imposes a condition forbidder i to place a new bid that will be selected by the Lagrangianrelaxation solution algorithm.

To be a winner in the next round of combinatorial reverse auction,a biddermay draw up a new bid under the constraint that theminimalprofit must be satisfied. A bidder determines whether he shouldsubmit a new bid bij′ to maximize the profit while satisfyingconstraints (4−1). To achieve this objective, the following optimi-zation problem is formulated, where ci* denotes the minimal profitsbidder i expects to take from the combinatorial reverse auction.

ution algorithms.

327F.-S. Hsieh / Decision Support Systems 48 (2010) 323–330

Profit Maximization for Bidder (PMB)

max Pij′−∑K

k=1cij′kqij′k

s:t:Pij′−∑K

k=1cij′kqij′k≥ci

�

Pij′−∑K

k=1λ�ðkÞqij′k≤Pij−∑

K

k=1λ�ðkÞqijk∀j∈f1;2; :::;nig

Pij′≥0

qij′k≥0 and qij′k∈Zþ∪f0g∀k∈f1;2; :::;Kg:

Aproblemobtained by relaxing the constraints qij′k∈Z+∪{0}∀k∈{1,2, ..., K} in PMB is formulated as follows.

Linear Programming for PMB (LPPMB)

max Pij′−∑K

k=1cij′kqij′k

s:t:Pij′−∑K

k=1cij′kqij′k≥ci

�

Pij′−∑K

k=1λ�ðkÞqij′k≤Pij−∑

K

k=1λ�ðkÞqijk∀j∈f1;2; :::;nig

Pij′≥0

qij′k≥0∀k∈f1;2; :::;Kg:

Bidder i determines whether to submit a new bid by solvingLPPMB. If there exists a solution for LPPMB, a newbidwill be submittedbased on the solution. Otherwise, no new bid will be submitted.

Suppose the combinatorial reverse auction is conductedM rounds.Our information revelation scheme is applied at the end of eachround. Each bidder may submit J bids in each round. Therefore, in theM-th round, a bidder may submit at most MJ bids.

5. A heuristic algorithm for finding a feasible solution

Although revelation of Lagrangian multipliers may improve theefficiency as well as reduce the costs, it is still possible that theoutcome of themulti-round combinatorial reverse auction is still not afeasible solution. In this case, we develop a heuristic algorithm to finda feasible solution based on the solution x* and the Lagrangianmultipliers λ* obtained at the end of the m-th round. The heuristicalgorithm for finding a near-optimal feasible solution x ̅ based on thesolution (x*, λ*) of the relaxed problem is proposed in this section.

The solution obtained by applying the subgradient method maynot be a feasible. If it is not feasible, it could be adjusted to a feasiblesolution without a great increase in objective function value. To adjustthe solution of the dual problem to a feasible one, one must identifythe set of demand constraints violated K0 and the set of bidders I0

that is not a winner in the solution of the dual problem. Then we pickthe bidders from the set I0 according to the rule of minimal cost firstto fulfill the insufficient quantity required by the set of violateddemand constraints K0.

The solution (x*, λ*) may result in one type of constraint violationdue to relaxation: assignment of the quantity of items less than thedemand of the items. Our heuristic scheme first checks all the demand

constraints ∑I

i=1∑ni

j = 1xij�qijk≥dk∀k = 1;2; :::;K that have not been

satisfied. Let K0 = fk jk∈f1;2;3;…;Kg;∑I

i=1∑ni

j = 1xij�qijk < dkg. K 0

denotes the set of demand constraints violated. Let I0={i|i∈{1, 2, 3,...., I}, xij*=0}. I0 denotes the set of bidders that is not a winner insolution x*. To make the set of constraints K0 satisfied, we first pick

k∈K0with k = arg mink∈K0

dk−∑I

i=1∑ni

j = 1xij�qijk. The heuristic algorithm

proceeds as follows tomake constraint k satisfied. Select i∈ I0 and j∈{1,2, ..., ni} with j = arg min

j∈f1;2;:::nig;qijk>0pij and set x i̅j=1. After performing

the above operation, we set I0← I0\{i}. If the violation of the k-thconstraint cannot be completely resolved, the same procedure repeats.Eventually, all the constraints will be satisfied.

Heuristic algorithm for finding a feasible solution

Step 0. x ̅←x*.Step 1. Find the set of demand constraints violated.

Find K0 = fk jk∈f1;2;3;…;Kg; ∑I

i=1∑ni

j = 1xij�qijk < dkg:

Step 2. Find the set of bidders that is not a winner in solution x*.

Find I0 = fi j i∈f1;2;3;…; Ig; xij� = 0g:

Step 3. While K0≠Φ.

Select k = arg mink∈K0

dk−∑I

i=1∑ni

j = 1xij�qijk from K0

:

Select i∈I0 and j∈f1;2;…;nigwith j = arg minj∈f1;2;:::nig;qijk>0

pij:

Set xi̅j=1.

I0←I0 n fig:

End whileThe effectiveness of the solution algorithms can be evaluated

based on the duality gap, which is the ratio of the difference betweenprimal and dual objective values divided by the primal objective

value. That is, duality gap is defined by f ðxÞ−Lðλ�Þf ðxÞ .

f ðxÞ = ∑I

i=1∑ni

j = 1xijpij:

6. Experimental results and analysis

Based on the proposed algorithms for combinatorial reverseauction, we conduct several examples to illustrate the effectivenessof our method.

Example 1. Consider a buyer who will purchase a set of five items.The desired units of each type of items are one. Suppose there are twosellers. Suppose each seller only places two bids. For this example, wehave

I = 2; J = 2; K = 5dk = 1ðd1 = 1;d2 = 1; d3 = 1;d4 = 1;d5 = 1Þ:

Suppose the four bids submitted by the two bidders are as follows:

q111 = 1; q112 = 0; q113 = 1; q114 = 0; q115 = 1q121 = 0; q122 = 1; q123 = 0; q124 = 1; q125 = 0q211 = 1; q212 = 0; q213 = 1; q214 = 0; q215 = 1q221 = 0; q222 = 1; q223 = 0; q224 = 1; q225 = 0:

P11 = 100; P12 = 60; P21 = 90; P22 = 40:

Supposewe initialize the Lagrangianmultipliers as follows: λ(1)=10.0, λ(2)=10.0, λ(3)=10.0, λ(4)=10.0, λ(5)=10.0 Our subgra-dient algorithm converges to the following solution:

x11� = 1; x12

� = 0; x21� = 0; x22

� = 1:

Fig. 3. CPU time (in millisecond) respect to I.

328 F.-S. Hsieh / Decision Support Systems 48 (2010) 323–330

As the above solution is a feasible one, the heuristic algorithm needsnot be applied. Therefore, x 1̅1=1, x 1̅2=0, x 2̅1=0, and x 2̅2=1.Indeed, the solution x* is also an optimal solution. The duality gap ofthe solution is as follows:

Duality gap=0%.

Example 2. Consider a buyer who will purchase a set of four items.The desired units of each type of items are

d1 = 2; d2 = 1;d3 = 2;d4 = 1:

Suppose there are three sellers. Suppose each bidder only placestwo bids. For this example, we have

I = 3; J = 2; K = 4;d1 = 2;d2 = 1; d3 = 2;d4 = 1:

Suppose the four bids submitted by the two bidders are as follows:

q111 = 1; q112 = 0; q113 = 1; q114 = 0q121 = 1; q122 = 1; q123 = 0; q124 = 0q211 = 0; q212 = 0; q213 = 1; q214 = 0q221 = 0; q222 = 1; q223 = 0; q224 = 1q311 = 0; q312 = 0; q313 = 1; q314 = 0q321 = 0; q322 = 0; q323 = 0; q324 = 1:

P11 = 70; P12 = 75; P21 = 40; P22 = 80; P31 = 45; P32 = 50:

Suppose we initialize the Lagrangian multipliers as follows.

λð1Þ = 30:0;λð2Þ = 40:0;λð3Þ = 35:0;λð4Þ = 50:0:

Our subgradient algorithm converges to the following solution:

x12� = 1; x22

� = 1; x32� = 1:

As the above solution is not a feasible one, the heuristic algorithmneeds to be applied. Our heuristic algorithm leads to the followingfeasible solution

x11 = 1; x21 = 1; x22 = 1; x32 = 1:

The duality gap of the solution is as follows:Duality gap=2.448%.Despite the duality gap is not zero, the solution x ̄ is an optimal

solution for this example.In addition to Example 1 and Example 2, Table 1 illustrates the

duality gap of several cases based on the problem size (I, J, K).According the results, the duality gaps are within 3%. This means thesolution methodology generates near-optimal solution.

In addition to the two examples above, we also conduct severalexperiments to study the computational efficiency of our proposedalgorithms. These experiments show the growth of CPU time withrespect to I, J and K, respectively.

Fig. 3 shows the CPU time for a number of problems in whichparameters J and K are fixed while parameter I is changed. The

Table 1Duality gap of several cases.

I J K Duality gap

5 10 5 2.8%10 5 10 2.2%30 10 20 2.4%

increase in the CPU time is not significant as parameter I is increased.According to the following equation:

LðλÞ = ∑K

k=1λðkÞdk + ∑

I

i=1LiðλÞ:

This result justifies the fact that the CPU time to compute L(λ) for agiven λ grows approximately linearly with respect to I.

The CPU time for a number of problems in which parameters I and Kare fixed while parameter J is changed is shown in Fig. 4. As J=max ni,it only influences the computation of

LiðλÞ = min∑nij = 1xij Pij−∑

K

k=1λðkÞqijk

!

s:t:∑nij = 1xij≤1

xij∈f0;1g:

Therefore, the CPU time to compute L(λ) for a given λ growsapproximately linearly with J.

We also study the growth of the CPU time with respect toparameter K.

As LðλÞ = ∑K

k=1λðkÞdk + ∑

I

i=1LiðλÞ with

LiðλÞ = min∑nij = 1xij Pij−∑

K

k=1λðkÞqijk

!

s:t:∑nij = 1xij≤1

xij∈f0;1g:

The CPU time to calculate∑K

k=1λðkÞqijkwill increase approximately

proportionally with K. Note that the Lagrangian multiplier is updatedas follows:

λl + 1ðkÞ = λlðkÞ + αlglðkÞ if λlðkÞ + αlλlðkÞ≥0;0 otherwise:

�

Fig. 4. CPU time (in millisecond) respect to J.

Fig. 6. Cost for each round m=1, 2, 3, 4, 5.

329F.-S. Hsieh / Decision Support Systems 48 (2010) 323–330

As the number of Lagrangian multipliers is proportional to K, theCPU time computation involved in updating λ will increase approx-imately proportionally with K2.

Fig. 5 shows the growth of CPU time with respect to K.Fig. 6 shows the reduction on the cost for the combinatorial

reverse auction Example 2 with five rounds and

c1j′1 = 1; c1j′2 = 2; c1j′3 = 1; c1j′4 = 3c2j′1 = 2; c2j′2 = 2; c2j′3 = 1; c2j′4 = 1c3j′1 = 1; c3j′2 = 1; c3j′3 = 3; c3j′4 = 1:

Note that significant cost reduction can be achieved in the secondround and the third round. The cost reduction diminishes at the fourthround and the fifth round. This implies that the cost reduction can beexpected in the first few rounds.

Before concluding this paper, we compare the difference betweenthe method proposed in this paper and the existing literature oncombinatorial auctions. This paper is differentiated from the methodproposed in [23] to find the optimal solution in that we adopt theLagrangian relaxation approach to find approximate solutions. Theadvantage of our Lagrangian relaxation approach is to find approx-imate solutions efficiently. Numerical examples indicate that ouralgorithm generates near-optimal solution within acceptable CPUtime. Our proposed algorithm is different from the one [8] based onLagrangian heuristic in existing literature as our algorithm is based onthe subgradient algorithm to adjust Lagrangian multipliers in multi-round combinatorial reverse auctions. We propose an effectivemechanism to guide the bidders to collectively minimize the overallcost without explicitly disclosing the bids. Our multi-round combi-natorial reverse auction algorithm to guide sellers to generatepotentially winning bids only relies on revelation of Lagrangianmultipliers. It is different from the methods proposed in [6] and [15],which require revelation of all bids to generate suggestions for bids.

7. Conclusion

Combinatorial auctions enable several bidders to bid on differentcombination of goods according to personal preferences. Bidders canselect multiple items at one time and offer those items a price. Itenables bidders to decide combinations of auction according topersonal preferences and more effectively arranges bid winners.Combinatorial auctions have been notoriously difficult to solve from acomputational point of view due to the exponential growth of thenumber of combinations. Although combinatorial reverse auctionshave attracted a lot of attention recently, most studies focus ondevelopment of efficient solution algorithms to determine winners.From the viewpoint of a buyer, an important issue is to design aneffective mechanism to guide the bidders to collectively minimize theoverall cost. On the other hand, from the viewpoint of a bidder, how tofind a potential winning bid that maximizes the profit is a key issue. Inpractice, each seller has no idea about the bids placed by other sellersat the beginning of a combinatorial reverse auction. Therefore, each

Fig. 5. CPU time (in millisecond) with respect to K.

seller is “blind” in creating his bids. The bid placed by a seller is usuallya profit maximizing bid that does not take into account the bids placedby other sellers. However, a profit maximizing bid may not be awinning bid. An interesting issue is to design an information reve-lation mechanism to assist a seller to find profit maximizing winningbid while minimizing the cost of the buyer.

Motivated by the deficiency of existing results, we consider awinner determination problem for combinatorial reverse auction inwhich a buyer wants to acquire items from a set of sellers and eachseller can provide a set of items. We propose an informationrevelation scheme for a buyer to guide the sellers to generatepotential winning bids to minimize the overall cost ultimately. Thispaper is different from the one presented in [8] as we adopt thesubgradient method to find Lagrangian multipliers. Our multi-roundcombinatorial reverse auction algorithm to guide sellers to generatepotentially winning bids only relies on revelation of Lagrangianmultipliers. It is different from the methods proposed in [6] and [15],which require revelation of all bids to generate suggestions for bids.

We formulate a winner determination optimization problem forcombinatorial auction. The demands of the buyer impose additionalconstraints on determination of the winners. By applying Lagrangianrelaxation technique, the original optimization can be decomposedinto a number of bidders' subproblems. Our methodology consists offive parts: (1) an algorithm for solving bidders' subproblems byexploiting their individual structures; (2) a subgradient method forsolving the non-differentiable dual problem; (3) an informationrevelation scheme to facilitate auction based on an economicinterpretation of Lagrangian multipliers; (4) a heuristic algorithm forfinding a near-optimal, feasible solution and (5) results and analysis ofour solution algorithms. The information revealed is based on theLagrangianmultipliers that are iteratively updated in our algorithm. Abidder may submit a new bid based on the price information revealed.Although the price information revelation scheme may improve theperformance of combinatorial reverse auctions, it does not guarantee afeasible solution. In case the resulting solution is not feasible, aheuristic algorithmmust be applied to adjust the infeasible solution toa feasible one. We develop a heuristic algorithm for finding a near-optimal, feasible solution based on the solution of the relaxedproblem.Numerical results indicate that our proposed algorithms yield near-optimal solutions. We also demonstrate the advantage of multi-roundcombinatorial reverse auction algorithm (with revelation of Lagrang-ian multipliers) by comparing it with the single-round combinatorialreverse auction algorithm (without information revelation). Ourresults indicate significant improvement in costs can be achieved byapplying our method with acceptable CPU time.

Although we concentrate on combinatorial reverse auction, thesolution method proposed in this paper can be applied to regularauctions in which there are multiple buyers bidding for a set of goods

330 F.-S. Hsieh / Decision Support Systems 48 (2010) 323–330

offered by one seller. The WDP of combinatorial auctions is similar tothe winner determination problem (WDP) of combinatorial reverseauctions. By changing the objective function tomaximize the revenue,changing sellers to buyers and changing buyer to seller in the WDP ofcombinatorial reverse auctions, we can formulate the WDP ofcombinatorial auctions. Lagrangian relaxation approach can also beapplied to solve the WDP of combinatorial auctions by following aprocedure similar to the one proposed in this paper.

Acknowledgement

This paper is supported in part by National Science Council ofTaiwan, R.O.C. under grant NSC97-2410-H-324-017-MY3.

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Fu-Shiung Hsieh received his B.S. and M.S. degrees in control engineering from theNational Chiao-Tung University, Taiwan, Republic of China in 1987 and 1989,respectively. He received his Ph.D. degree from the National Taiwan University in1994. He served as a researcher in Industrial Technology Research Institute (ITRI),Taiwan, from 1994 to 1999. He was an Assistant Professor and Associate Professor withThe Overseas Chinese Institute of Technology (OCIT) from 1999 to July 2004 and August2004 to July 2005, respectively. He was the Director of the International ElectronicCommerce Center of OCIT from August 2001 to July 2002. Since August 2005, he hasbeen with Chaoyang University of Technology, where he is currently an AssociateProfessor at the Department of Computer Science and Information Engineering. He hasover fifty publications, including international journal and conference papers. He islisted in Who's Who in the World 2006, 2007, and 2009 and Who's Who in Asia 2007.His research interests are in electronic commerce, workflows, multi-agent systems andmanufacturing systems.