A Lagrangian approach to the winner determination problem in iterative combinatorial reverse auctions

Decision Support

Accepted Manuscript

A Lagrangian approach to the winner determination problem in iterative combinatorial reverse auctions Bahareh Mansouri, Elkafi Hassini PII: DOI: Reference:

S0377-2217(15)00073-9 10.1016/j.ejor.2015.01.053 EOR 12763

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

21 May 2014 23 December 2014 26 January 2015

Please cite this article as: Bahareh Mansouri, Elkafi Hassini, A Lagrangian approach to the winner determination problem in iterative combinatorial reverse auctions, European Journal of Operational Research (2015), doi: 10.1016/j.ejor.2015.01.053

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Highlights • Investigate a winner determination problem of a multi-unit combinatorial auction. • We develop an efficient Lagrangian-based solution approach.

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• We establish the equivalency between Lagrangian and linear programming relaxation

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• We carry extensive computations using an adaptation of the combinatorial auction test suite.

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Abstract Combinatorial auctions allow allocation of bundles of items to the bidders who value them the most. The N P -hardness of the winner determination problem (WDP) has imposed serious computational challenges when designing efficient solution algorithms. This paper analytically studies the Lagrangian

relaxation of WDP and expounds a novel technique for efficiently solving the relaxation problem.

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Moreover, we introduce a heuristic algorithm that adjusts any infeasibilities from the Lagrangian optimal solution to reach an optimal or a near optimal solution. Extensive numerical experiments illustrate the class of problems on which application of this technique provides near optimal solutions in much less time, as little as a fraction of a thousand, as compared to the CPLEX solver.

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Keywords: reverse multi-unit combinatorial auction, Lagrangian relaxation, iterative auctions

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A Lagrangian approach to the winner determination

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problem in iterative combinatorial reverse auctions Bahareh Mansouria and Elkafi Hassinib a School

of Computational Science and Engineering, McMaster University, Hamilton, Canada, [email protected]

b DeGroote

School of Business, McMaster University, Hamilton, Canada, [email protected]

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February 10, 2015

Introduction

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One important question asked in market places is whom to assign how many units of what items and at what price. Auctions have emerged to answer this question by allocating items to those

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who value them the most. Combinatorial auctions allow bidders to place bids on any bundle of items. This enables them to express complex valuations on the packages of items and report their preferences more precisely. Combinatorial auctions lead to greater auction revenue as well as

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market economic efficiency by allocating items to those who value them the most [8]. However, the winner determination problem (WDP) in these auctions is difficult to solve in practice as

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it is N P -hard [10]. One major stream of research focuses on applying a Lagrangian relaxation followed by heuristic solution algorithms to solve WDP to (sub)optimality. Guo et al. [3] model the

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combinatorial auction problem as a set packing and apply the Lagrangian relaxation method. Song et al. [12] show application of Lagrangian relaxation on business-to-business (B2B) procurement combinatorial auctions. Kameshwaran et al. [5] demonstrate the design of progressive auctions using a Lagrangian relaxation. Hsieh [4] adopts a subgradient method to solve the Lagrangian relaxation problem in a multi-unit multi-item reverse WDP. Mansouri and Hassini [7] propose a model for the bidders’ pricing and lot sizing problem in an iterative combinatorial procurement auction based on the revelation of Lagrangian multipliers.

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A major advantage of the Lagrangian multipliers is that they act as proxies of the item prices. Revealing the multipliers to bidders in an iterative auction format can help improve the auction efficiency [9]. On the down side, the computational effort to solve the Lagrangian relaxation problem depends on the initialization of the Lagrangian multipliers. In a decomposition method, this involves several iterations between the sub and master problems before the convergence

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is obtained. In a subgradient method the convergence rate is dependent on the selection of a Lagrangian step size as well. An improper choice of the step sizes can even cause divergence of the algorithm.

Our main contribution in this paper is that we show that solving the Lagrangian relaxation is equivalent to solving a linear programming relaxation for a wide class of WDPs. This result allows us to efficiently initialize the Lagrangian multipliers. Furthermore, we show that we can find a

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closed form solution for the Lagrangian subproblem. Thus, we are able to solve the Lagrangian relaxation in an efficient way, without recourse to traditional decomposition or subgradient methods. In case the Lagrangian optimal solution is infeasible, we develop an efficient heuristic for adjusting the solution to a (near) optimal one. In addition, we extend the Combinatorial Auction Test Suite (CATS) [6] to generate data for our multi-item iterative combinatorial auction.

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To investigate the efficiency of our proposed heuristic procedure, we carried out extensive numerical computations by solving a total of 7,500 problem instances. On average, the heuristic

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algorithm is able to solve the problems to (near-)optimality in about four thousandth of the time CPLEX 12 takes with an optimality gap of 10%. Our heuristic performs even better for problems

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where suppliers’ bids account for a small proportion of total demand. Such shorter performance time will decrease the time in-between the rounds of an iterative auction. This will in turn shorten

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the duration of the whole auction while providing informative and timely feedback to the bidders. In the remainder of this paper define the reverse WDP and its related Lagrangian relaxations. A heuristic algorithm is introduced to correct for infeasibility from the Lagrangian relaxation.

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The result to this approach is compared against CPLEX 12. We note that there are no other algorithms or test problems to benchmark against in the context of our specific problem (iterative combinatorial auctions with feedback through Lagrange multipliers). We use an adapted version of the general test suite CATS in our computational experiments. Finally we report on our extensive numerical experiments and conclude with some future research directions.

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Problem Formulation

Assume M = {1, 2, .., m} is the set of items and N = {1, ..., n} is the set of bidders who compete to purchase them. Each bidder j ∈ N submits a set of package bids S ⊆ M each of which contains

a subset of items selected by bidder j and a price value PjS also known as price offer or simply price he assigns to it. The bidders’ prices on bundles are non-linear, thus for an arbitrary bidder

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j ∈ N and sets S, S1 , S2 ⊆ M with S = S1 ∪ S2 , we have p jS 6= p jS1 + p jS2 , where p jS is the price for

bundle S offered by bidder j. The prices are non-anonymous meaning that they allow discriminatory pricing so that p jS 6= p j0 S for bidder j 6= j0 . Let qijS be a non-negative integer that represents the

quantity of item i ∈ M that bidder j ∈ N requests for bundle S ⊆ M and di be the auctioneer’s demands of item i with 0 ≤ qijS ≤ di .

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The auctioneer’s problem of deciding which bidders should supply how many units of what

items and at what price with the objective to minimize the total price of procurement while satisfying the auctioneer’s demand is known as the reverse (or procurement) WDP. When bidders are allowed to win any number of the packages they bid on, WDP is referred to as WDPOR . In a

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procurement setting, the WDPOR is formulated as the following integer programme. min Σ j∈ N ΣS⊆ M p jS x jS Σ j∈ N ΣS3i qijS x jS ≥ di ∀i ∈ M

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s.t.

x jS ∈ {0, 1}

(WDPOR )

∀ j ∈ N ∀S ⊆ M.

The first set of constraints (known as demand constraints) state that at least di units of each

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item demanded has to be provided at the optimal solution. An optimal allocation may over satisfy demand. This is acceptable if there is free disposal or no considerable holding costs. Otherwise,

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these constraints are expressed as equality constraints to satisfy exactly the number of units demanded. Unless otherwise stated, in the sequel we assume free disposals.

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In some applications of combinatorial auctions, it is preferred to implement an XOR bidding language to allow each bidder to win at most one bundle. To enhance the model with this requirement, we need to take into account one more constraint with respect to (WDPOR ) and

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formulate WDPXOR as follows. min Σ j∈ N ΣS⊆ M p jS x jS s.t.

Σ j∈ N ΣS3i qijS x jS ≥ di ∀i ∈ M ΣS⊆ M x jS ≤ 1

(WDPXOR )

∀j ∈ N

x jS ∈ {0, 1}

∀ j ∈ N, ∀S ⊆ M.

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The first and second constraints in (WDPXOR ) are respectively referred to as the demand and supply constraints. WDPXOR has the potential to increase the bidders’ precision when pricing the bundles and can decrease the total price of procurement expected from the auction. Moreover, it does not allow bidders to obtain a set of items as singleton bids without having paid for the

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complementarities.

A Lagrangian Relaxation Approach

Lagrangian relaxation is a technique which approximates a difficult problem by relaxing it to a simpler one. The method removes some of the problem constraints and penalizes their violations

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by adding them to the objective function and associating to them a weight parameter known as the Lagrangian multiplier. In fact, the Lagrangian multiplier imposes a cost on the relaxed constraints’

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violations.

Applying the Lagrangian relaxation on reverse combinatorial auctions provides us with an approximate solution to the problem and a lower bound on the total price of procurement. The

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Lagrangian multipliers corresponding to the first set of constraints, in (WDPOR ) or (WDPXOR ), determine how much it costs to have one more unit of an item and thus can be interpreted as

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the item prices. We make use of this information to give feedback to the bidders about the item prices so that they will have a better sense about each other’s valuations. This is economically

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valuable since it can initiate an iterative auction where bidders are provided with the opportunity to modify their bids several times throughout the auction before finalizing them [9]. Consider the winner determination problem modelled in (WDPOR ). Let X = [ x jS ] j∈ N,S⊆ M be

the matrix of assignments and λ be the non-negative Lagrange multipliers assigned to the first set of constraints. This results in the formulation of the Lagrangian relaxation function as L(X, λ) =

∑ ∑

j∈ N S⊆ M

p jS x jS +

∑ λi ( di − ∑ ∑

i∈ N

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j ∈ N S 3i

qijS x jS ),

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and the Lagrange dual function or simply the dual function as g(λ) = min Σ j∈ N ΣS⊆ M p jS x jS + Σi∈ N λi (di − Σ j∈ N ΣS3i qijS x jS ) s.t.

x jS ∈ {0, 1}

∀ j ∈ N, ∀S ⊆ M.

(1)

The Lagrangian dual function is also referred to as the subproblem. For each value of the Lagrange multiplier, we obtain a lower bound on the problem’s optimal value. The optimization

the dual problem is formulated as max g(λ) λ ≥ 0.

s.t.

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problem that seeks to find this value is the Lagrangian dual problem. For the Lagrange function (1)

(2)

Relaxation of the demand constraint in problem (WDPXOR ) can be done in a similar way

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with the exception that the subproblem (1) is solved in the existence of the additional supply constraints.

3.1

Analysis of the Lagrangian Relaxation Bound

This section compares the Lagrangian and linear relaxations for a multi-unit multi-item reverse

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WDP with and without free disposal in an OR or an XOR bidding environment. Define P as the vector of price offers, X as the solution vector, Q as the matrix of bidders’ quantity offers, and R as

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the matrix on supply constraints.

Matrix Q consists of columns (q jS )m×1 whose i − th element (i = 1, . . . , m) identifies the

quantity of item i that bidders j supplies in bundle S and matrix R consists of columns (r jS )n×1

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whose j − th element is 1 for supplier j and 0 elsewhere.

The following propositions provide some properties of relaxing demand constrains in an

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OR, demand only and both supply and demand constraints in an XOR setting. Note that the Lagrangian multipliers associated with the demand constraints are interpreted as item prices for

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an auction environment. Thus, we are not interested in solely relaxing the supply constraints in the XOR model. Proposition 1. The Lagrangian and linear relaxations of reverse OR and XOR versions of WDP yield equivalent bounds. Proof. According to Theorem 16.10 in [1], the linear and Lagrangian relaxation bounds equal if the Lagrangian subproblem satisfies the integrality property, i.e., LP relaxation finds integral solution for the Lagrangian subproblem for any choice of objective function coefficients. 7

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The linear relaxation of subproblems (3) and (5) minimize the objective function subject to the constraint that each variable is less than or equal to 1. This forms a totally unimodular matrix of coefficients and so the subproblems satisfy the integrality property. Also, from the definition of matrix R, each of it’s columns have at most one +1 and they are 0 elsewhere. Thus the integrality of subproblem (6) can be deduced from the total unimodularity of matrix R.

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From Proposition 1 we can conclude that if WDP has an optimal solution with a finite optimum value then from linear programming duality theory it follows that the corresponding Lagrangian relaxation will also have a finite optimal solution. Thus we can state Corollary 1.

Corollary 1. The Lagrangian relaxations of the OR and XOR reverse WDP problems admit optimal

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multipliers as long as the corresponding WDP admits optimal solutions.

Given the result on existence of multipliers in Corollary 1 we can use them to establish closed form solutions for the Lagrangian subproblems.

Proposition 2. Let λ∗ and γ∗ be the optimal values of the Lagrangian multipliers respectively associated subproblem of the relaxation of

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with demand and supply constraints. We can find closed form optimal solution for the Lagrangian

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1. WDPOR with relaxed demand constraints as:   1 p − λ ∗t q < 0 jS jS ∗ x jS =  0 otherwise,

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2. WDPXOR with relaxed demand and supply constraints as:   1 p − λ ∗t q + γ ∗t r < 0 jS jS jS ∗ x jS =  0 otherwise,

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3. WDPXOR with relaxed demand constraints as   1 p − λ∗t q = min { p − λ∗t q } < 0 S jS jS jS jS ∗ x jS =  0 otherwise.

Proof. First let us assume a free disposal state where the Lagrangian multipliers corresponding to demand constraints take on non-negative values. To prove part 1, we consider the Lagrangian

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optimization problem g(λ) = infX L(X, λ)

= infX {PX + λt (d − QX)}

= dt λ + infX (P − λt Q) X   ∑ ( p − λt q ) p − λt q < 0 jS jS j,S jS jS = dt λ +  0 o.w.

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(3)

maxλ g(λ) = g(λ∗ ) provides the optimal solution for (3). Therefore, g(λ) is maximized when x ∗jS is set to one for negative values of p jS − q jS λ∗ and zero otherwise.

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To prove part 2, consider the WDPXOR formulation as min Pt X s.t.

QX ≥ d

(4)

RX ≤ 1

X

binary.

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Assigning non-negative Lagrangian multipliers λ and γ respectively to the demand and supply constraints in WDPXOR leads to the Lagrangian subproblem

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g(λ, γ) = infX L(X, λ, γ)

(5)

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= infX {PX + λt (d − QX) + γt ( RX − 1)}   ∑ ( p − λt q + γt r ) p − λt q + γt r < 0 jS jS jS jS j,S jS jS = dt λ − 1t γ +  0 o.w.

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g(λ∗ , γ∗ ) optimally solves the Lagrangian relaxation problem.

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Finally, for part 3 the Lagrangian subproblem is g(λ) = infX {PX + λt (d − QX)| RX ≤ 1, X

binary}

(6)

= dt λ + infX {(P − λt Q)X| RX ≤ 1, X binary}. t

In order to maximize g(λ) it suffices to consider the minimum value of p jS − λ∗ q jS for each

bidder j ∈ N. If this values is negative, we set the corresponding variable to 1 and otherwise to 0.

This associates at most one bundle to each bidder j and thus g(λ∗ ) provides us with the optimal solution. 9

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Theorem 1. For the reverse WDP, both in the OR and XOR formats, the dual variables associated with the demand (or supply) constraints of the linear relaxation problem correspond to the Lagrangian multipliers associated with the demand (or supply) constraints of the Lagrangian relaxation problem. Proof. We can reformulate the Lagrangian subproblem (5) corresponding to the relaxation of all

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constraints in an XOR bid setting as g(λ, γ) = λt d − γt 1 + Σ j,S min{0, p jS − λt q jS + γt r jS }.

Based on definition of matrix R, γt r jS = γ j . The corresponding Lagrangian dual problem maximizes g(λ, γ) for nonnegative values of λ and γ as

sup g(λ, γ) = max{dt λ − 1t γ + Σ j,S min{0, p jS − λt q jS + γ j }, λ ≥ 0, γ ≥ 0},

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λ,γ

λ,γ

or equivalently

max dt λ − 1t γ + Σ j,S min{0, p jS − λt q jS + γ j } s.t.

λ ≥ 0, γ ≥ 0.

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Linearizing min{0, p jS − λt q jS + γ j } with variable l jS , we obtain max dt λ − 1t γ + Σ jS l jS

l jS + λt q jS − γ j ≤ p jS ∀ j, S

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s.t.

l jS ≤ 0

∀ j, S

λ ≥ 0, γ ≥ 0.

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Let l jS = −t jS . The corresponding dual problem is formulated as min Σ j,S p jS x jS s.t.

Σ j,S q jS x jS ≥ d ΣS x jS ≤ 1

0 ≤ x jS ≤ 1

∀j

∀ j, S.

Multipliers λ and γ which were initially defined as Lagrangian multipliers associated with

demand and supply constraints also serve as dual variables to the corresponding constraints in the linear relaxation problem. The equivalence in the OR setting can be concluded similarly. Corollary 2. For the OR and XOR reverse WDP problems, initializing Lagrangian multipliers at optimal dual values obtained from the linear relaxation solves the Lagrangian optimization problem to optimality. 10

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We note that based on Theorem 1 initializing the Lagrangian multipliers at the optimal dual values is in fact equivalent to initializing them at their optimal values. As stated in Corlollary 2 this initialization solves the Lagrangian subproblem to optimality without recourse to the dual problem.

Lagrangian Relaxation Solution Algorithm

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3.2

In Theorem 1 we recognize the optimal values of the Lagrangian multipliers. By initializing the Lagrangian multipliers at their optimal values, as stated in Corollary 2, solving the Lagrangian relaxation problem shrinks down to solving the Lagrangian subproblem for which Proposition 2 provides a closed form solution. Algorithm 1 summarizes our methodology for efficiently solving

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the Lagrangian relaxation problem. We note that in the sequel we focus solely on the XOR formulation with free disposal where the demand constraint is relaxed. The focus on the demand constraint stems from our interest in getting information on the value of the items after each iteration of the auction. We focus on the XOR formulation as it more harder to solve than the OR model [11].

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Algorithm 1 Lagrangian Relaxation Solution Algorithm Solve the corresponding LP relaxation.

2:

Initialize the Lagrangian multipliers at the LP’s optimal dual values.

3:

Solve the Lagrangian subproblem using the closed form solution provided in Proposition 2.

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1:

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Table 1 illustrates the savings in execution time when solving the Lagrangian relaxation problem using this methodology as compared to the implementation of a regular decomposition

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method. Each pair of (item, bid) is averaged on 13 runs of different problem instances. The average ratio of 1/88.9, from running a total of 520 problems, indicates that our proposed method provides the Lagrangian optimal solution in about %1 of the time of that of classical Lagrangian

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decomposition methods.

3.3

Lagrangian Optimal Solution Adjustment

The Lagrangian relaxation provides an integer solution in step 3 of Algorithm 1 which may not be feasible for the primal problem. In order to fix this, we developed a heuristic algorithm, named as Aggregate heuristic, which consists of several subheuristics and improvement procedures. Starting from the Lagrangian optimal, we observe satisfiability of demand constraints at the 11

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Average ratio of execution time of Lagrangian decomposition to Algorithm 1

(4,20)

31.02

(4,30)

61.88

(5,20)

51.46

(5,30)

65.81

(5,40)

80.35

(5,50)

68.02

(5,100)

59.35

(6,20)

33.87

(6,30)

61.42

(6,40)

83.34

(6,50)

46.17

(6,60)

74.30

(6,70)

85.35

(6,100)

68.56

(6,150)

72.72

(6,200)

81.17

(6,250)

98.92

(7,100)

69.07

(7,150)

99.51

(7,200)

93.15

(7,250)

104.09

(7,300)

123.88

(7,350)

135.70

(8,100)

88.80

(8,200)

59.53

(8,300)

159.96

(8,400)

87.28

(8,500)

98.67

(9,50)

122.55

(9,70)

112.96

(9,100)

93.83

(9,200) (9,250)

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86.98 90.24

108.43 106.14

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(9,300)

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(9,150)

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(#items, #bids)

(9,350)

134.22

(9,400)

112.55

(9,450)

165.41

Average

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Table 1: Execution time comparison of Algorithm 1 and Lagrangian decomposition method

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current solution. Violation of a demand constraint at the current solution indicates insufficient assignment of quantities of the corresponding item. Each subheuristic initially fixes the current solution X at the Lagrangian optimal solution X ∗

LR

and then systematically selects a constraint

and a variable to set to 1. Setting additional variables to 1 continues until the feasibility of all constraints is achieved. Algorithm 2 describes the generic structure of how each subheuristic

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procedure functions for a multi-unit multi-item auction with XOR bidding language as formulated in model (WDPXOR ).

More specifically, in this Algorithm parameter Shi identifies possible shortages corresponding to each of the item (constraint). Set I is defined as the set of all unsatisfied constraints and set J as the set of all bidders who are not currently winning any of their packages. With the existence of an unsatisfied demand, Aggregate heuristic proceeds to run subheuristics each consisting of a

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constraint selection rule (as described in 3.3.1) and a variable selection rule (as described in 3.3.2) to set a new variable to 1. The shortage is calculated again for, not all the constraints, but only for those which were previously recognized with positive shortage.

Note that firstly the algorithm continues the search for as long as there exists bidders who are eligible to win packages. If all bidders are assigned an item and the shortage is not yet

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fully satisfied, then the algorithm exits the loop and returns infeasibility. At this point either the auctioneer decides to supply the remaining shortage from spot market or he induces another

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auction inviting more participants.

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Secondly, with a non empty set of suppliers we do not necessarily rule out the previously chosen constraint i from the set I since the shortage may not be fully resolved. After setting a

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variable to 1, we may satisfy none, some, or all of the unsatisfied constraints. For this reason, we update the set I every time a new variable is selected. Set J is also updated to remove the new winner. This guarantees the feasibility of supply constraints in the optimal solution. As soon as a

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bidder receives a package he is prevented from winning anymore. 3.3.1

Constraint Selection Rules

The constraint selection rules include selecting constraint i ∈ I with Rule 1. the largest shortage i = argmaxi∈ I {Shi |Shi = di − Σ j∈ N ΣS3i qijS X jS }, 13

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Algorithm 2 Subheuristics’ Procedure Structure

3: 4: 5: 6: 7: 8: 9: 10: 11:

LR

Shi ← di − Σ j∈ N ΣS3i qijS x jS I ← {i ∈ M|Shi ≥ 0} J ← { j ∈ N | x jS = 0

while I 6= ∅ do

if J = ∅ then exit loop ‘Infeasible problem Instance’ else

Select constraint i ∈ I based on the Constraint Selection Rules

Select variables j ∈ J, S ⊆ M based on the Variable Selection Rules x jS ← 1

update Shi

13:

I = I \{i |Shi < 0}

∀i ∈ I

J = J \{ j| j selected in line 7}

end if

end while

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for

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15:

∀i ∈ M

∀S}

12:

14:

for

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2:

X ← X∗

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1:

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Rule 2. the minimum slackness value Σ j∈ N ΣS3i qijS i = argmini∈ I { }, di

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Rule 3. the maximum slackness value Σ j∈ N ΣS3i qijS i = argmaxi∈ I { }, di

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Rule 4. the costliest shortage i = argmaxi∈ I {λi .Shi |Shi = di − Σ j∈ N ΣS3i qijS X jS }. Rule 1 looks for the item which has the largest shortage. Rules 2 and 3 search for the items

with maximum or minimum slackness values, where slackness values are equal to the ratio of total item offered quantities to its demand. In other words, we are interested in identifying the items which receive the largest or the smallest total quantity offers with respect to their demand.

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Finally, rule 4 would supply the items whose shortage is the costliest for the auctioneer with respect to the Lagrangian multipliers. Note that these rules are only applied on the set of violated constraints. 3.3.2

Variable Selection Rules

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For each constraint selection rule we perform the following 4 variable selection rules to form a sub-heuristic procedure. Note that each constraint corresponds to an item and each variable of this constraint indicates a package containing this item. Once an item is selected, from all the unallocated packages containing this item, we choose the one that provides the largest value according to one of the following variable selection rules.

Rule 3.

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∑ λi∗ .min(qijS , Shi ) i∈ I

∑ i∈ I

∑ i∈ I

p jS min(qijS , Shi ) Shi p jS λi∗ .min(qijS , Shi ) Shi . p jS

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Rule 4.

p jS

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Rule 2.

i∈ I

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Rule 1.

∑ min(qijS , Shi )

The above ratios are used to provide us with a proxy for the value of an option based on the

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shortage and pricing information. For each rule we calculate its value for all items with unsatisfied demand and pick the variable that corresponds to the largest value. We use min(qijS , Shi ) so that

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we do not value bundles with quantities exceeding the shortage. 3.3.3

Improvement Procedure

Each subheuristic is followed by an improvement procedure to systematically swap a selected

variable with a non-selected one. Since the combination of constraint selection rule 1 and variable selection rule 1 provides the best results in our experiments we use this combination to switch variables.

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To eliminate a variable already selected, the improvement procedure selects a constraint based on the constraint selection rule 1, and a variable with the lowest value of variable selection rule 1. To set a new variable to 1, constraint selection rule 1 and variable selection rule 1 are used once again. Aggregate Heuristic Algorithm

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3.3.4

Let c, v be alternatively the indices on the constraint and variable selection rules. We define each subheuristic as Hc,v and the improvement procedure applied on it as I Hc,v . Considering all combinations of our constraint and variable selection rules each followed by an improvement procedure, provides 32 computationally cheap subroutines. The Aggregate heuristic algorithm

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executes all subroutines and extracts the solution corresponding to the minimum objective value as the optimal solution.

4

Computational Simulation of Multi-unit Combinatorial Auction

In this section we describe the computational steps taken to simulate a multi-unit auction environ-

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ment. All our computational experiments are performed on an Intel Xeon E5440 @2.83 Ghz 2.83 Ghz (2 processors) machine with memory (RAM) 12.0 Gb running with a 64-bit operating system

Using CATS for Data Generation

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4.1

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on a standard windows server.

To imitate actual bidding behaviour in combinatorial auction we used CATS (Combinatorial

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Auction Test Suite) [6]. Given the number of goods, bids and required distribution type, CATS determines how many items to include in a bid, what items to include, and what price to attach to the whole package. Once all the items are enumerated, CATS starts counting dummy bids. Dummy

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bids are used to determine what packages are received from which bidders, i.e., packages sharing similar dummy goods are received from a single bidder.

4.2

Algorithm Coding

We used General Algebraic Modeling System (GAMS) platform for coding the multi-unit reverse combinatorial auction problem. All required information regarding the bidders submitting bids, the items included in each package, and the price associated with it are extracted from CATS 16

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output file within our GAMS code, after several necessary preprocessing of input data that are explained in the next Section. We use CPLEX 12 for solving our problem instances directly. The execution time and optimal solution is compared against application of linear relaxation, Lagrangian decomposition method, and our proposed solution methodology for solving the Lagrangian relaxation problem and

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eventually our heuristic algorithm for fixing the infeasibility of the Lagrangian optimal solution, if needed .

Despite the benefits of using CATS for generating realistic data, there is no convenient interface between CATS and GAMS. To incorporate this test suite in our methodology, changes need to be applied on the text file generated by CATS before introducing it as an input file for GAMS. In a large scale set of data, it is almost impossible to apply these changes manually. For this reason,

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we generated VBA codes in excel and produced several macros to automate implementing all the required modifications on the text file produced by CATS.

Data generated by CATS represents a single-unit bidding environment. In a multi-unit combinatorial auction we are concerned about bundle prices as well as how many units of each item to include in each package. To modify the pricing and bundling schemes in CATS, we derive

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the supplier quantities from a uniform distribution and define the new price corresponding to each package P as

Size of Problem Instances

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4.3

PCATS price × Total number of units included in P . Total number of items included in P

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PNew price =

We explore generation of 10, 20, and 30 items each with 10, 200, . . . , 1000 number of bids. We fixed

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the upper bound for demand generation at 50, and varied the upper bound for quantities of items included in the packages at 5, 10, 15, . . . , 50 to represent a supply capacity of 10%, 20%, . . . , 100%

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of demand. In order to get more reliable results, for each combination of number of items, bids, and the specified value of the quantities’ upper-bound, we averaged results over 25 problem instances of data generation. This adds up to a generation of 7, 500 problem instances. For more clarification, let m represent the number of items, n the number of bids, qUB the upper-bound considered for the random quantity generation and iter the number of iterations a new instance is generated. Figure 1 illustrates the size of our data generated.

17

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4.4

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Figure 1: Dat Size Diagram              iter = 1                   .        qUB = 5 ..                           iter = 25          n = 100            qUB = 10              .    ..   m = 10                qUB = 50                 n = 200          .    ..              n = 1000          m = 20       m = 30

Adjusting Lagrangian Constraint Satisfaction Ratio

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Define Z IP , ZLR , and ZH respectively as the optimal values of the CPLEX solver, the Lagrangian relaxation, and the Aggregate heuristic and TIP , TLR and TH as their execution times. Let w be the ratio of the number of positive variables (corresponding to distinct winners) in the CPLEX

ED

optimal solution to that of Aggregate heuristic. To have a better understanding of the quality of the objective value that the heuristic produces, we record the heuristic optimality gap as

PT

( ZH − Z IP )/Z IP and the Lagrangian duality Gap as ( Z IP − ZLR )/ZLR . Table 2 displays these gaps

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as LR Gap and H Gap respectively.

In our next set of experiments, we investigate how the quality of the Aggregate heuristic optimality gap relates to the Lagrangian Satisfiability Ratio (LSR). We define LSR as the ratio of the

AC

primal constraints that are satisfied at the Lagrangian optimal solution to the total number of constraints.

We implemented our experiments on 2,500 problem instances generated for 10 items and bids ranging from 100 to 1000 in increments of 100. For each combination of item and bids, we generated random supply quantities covering a maximum of 10%,20%,. . . ,100% of demand. Finally, for each problem combination we generated 25 different data instances so as to have more 18

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significant statistics.

We observed that an increase in the LSR value increases the optimality gap obtained from the Lagrangian relaxation as well as the Aggregate heuristic. In other words, as illustrated in Figure 2, the more constraints the Lagrangian optimal solution satisfies, the larger the optimal Lagrangian

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and heuristic gaps become. A possible explanation of this observation is that when LSR is close to 1 the heuristic spends less effort in improving the Lagrangian solution and we end up with a relatively large gap.

Figure 2: Lagrangian and Aggregate heuristic optimality gaps when reducing satisfiability ratios below the threshold

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0.35 0.3

Gap

0.25 0.2 0.15

0.05 0 0.1

0.2

0.3

0.4

0.5

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0

M

0.1

0.6

0.7

0.8

0.9

Lagrangian Gap Heuristic Gap (No threshold) Heuristic Gap (threshold=0.7)

1

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Lagrangian Satisfiability Ratio (LSR)

For this reason, in our next set of experiments whenever the Lagrangian optimal solution

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satisfies α% or more of the primal constraints, we systematically select and zero out positive variables (from the Lagrangian optimal solution) until the LSR value drops down below our

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threshold value. Algorithm 3 describes this procedure for the Lagrangian optimal solution X ∗ . As illustrated in Figure 2, we observe an improvement in the average heuristic optimality gap

for LSR ≥ α = 0.7. The overall average optimality gap drops down from 11% with no violation to 7% with the implementation of this violation.

Tightening α to 0.6 decreases the optimality gap to 6% while increasing the heuristic execution time. In fact, the average execution time ratio of CPLEX to the Aggregate heuristic decreases from 19.45, for α = 0.7, to 6.32 when α = 0.6. Thus, while decreasing α to 0.6 slightly improves the 19

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Algorithm 3 Satisfiability Violation

3: 4: 5: 6: 7: 8: 9:

Shi ← di − Σ j∈ N ΣS3i qijS x jS I ← {i ∈ M|Shi ≥ 0} J ← { j ∈ N | x jS = 0

Ratio ←

Select constraint i ∈ M − I Select j ∈ J, S ⊆ M

such that

such that

I = I \{ei |Shei < 0}

i = argmaxi {Shi }

jS = argmax jS {

x jS ← 0

11:

14:

∀S}

while Ratio ≥ threshold do

update Shi

13:

∀i ∈ M

card( M )−card( I ) card( I )

10:

12:

for

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2:

X ← X∗

∑i∈ I min(qijS ,Shi ) } PjS

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1:

J = J \{ej|ej selected in line 8}

update Ratio end while

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optimality gap, it yields much larger execution time. For this reason we keep the threshold value

4.5

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α at 0.7 for the rest of our experiments.

Efficiency of the Proposed Algorithm

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In this section, we compare the efficiency of the Aggregate heuristic algorithm against CLPEX 12 on the data set explained in Section 4.3. Table 2 summarizes our results.

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For more clarification on this Table, we illustrate the diagram of the 250 problem instances generated for its first row in Figure 3. This row demonstrates average results for bids with 10 items, 100,. . . ,1000 total number of bids, and provision of maximum of 10% of demand for each

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item in the bundles generated. For each class of items, we include the total average, the standard deviation, and the coefficient of variation values in order to provide better understanding of our original data. These values are respectively denoted as Ave, STDev, and CV in Table 2. The following subsections are devoted to explaining our computational results in more details.

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Figure 3: Data size diagram of the first row of Table 2

of demand

   n = 200      ..   .       n = 1000

Optimality Gap

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4.5.1

qUB = 10%

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m = 10,

       iter = 1         .   n = 100 ..             iter = 25 

Our experiments show that the quality of the heuristic optimality gap improves with the decrease in the maximum percentage of the demand supplied in the submitted bids. To show this pattern more clearly we designed Table 2 to classify problem instances based on the maximum percentage

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of the demand satisfied by submitted bids. Figure 4(a) shows the growth of the average optimality gap (along side the average standard deviation) with the increase in the bids’ supplied quantity for 10, 20, and 30 items. The plot also compares the average optimality gap with its overall expected

ED

value when fixing the number of items. For 10, 20, and 30 items the average optimality gap derived is respectively 8, 10, and 12 percent.

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As observed in Figure 4, the heuristic provides lower gaps on the class of problems in which each bidder offers a low percentage of demand. Specifically, when the bidder offers to supply

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at most 50% of demand, as shown in Table 3, the average heuristic gap for 10, 20, 30 items and submissions of 100 to 1000 bids (in increments of 100) drops down to respectively 6, 7, and 8

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percent.

4.5.2 Execution Time In order to observe how fast Aggregate heuristic solves a problem instance to a solution, we plot the execution time ratio of CPLEX to Aggregate heuristic in Figure 4(b). This figure illustrates that

when bidders supply low percentages of demand, CPLEX 12 takes much more time as compared to Aggregate heuristic. In Figure 5, we have a closer look at this ratio for the class of problems in 21

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Demand %

LR Gap

H Gap

T IP (sec)

T H (sec)

T IP/T H

w

10

0.02

0.03

0.89

0.4

2.54

0.93

20

0.05

0.05

2.03

0.44

4.29

0.9

30

0.1

0.06

1.03

0.44

2.3

0.87

40

0.15

0.07

0.86

0.43

1.86

0.87

50

0.19

0.08

0.54

0.43

1.25

0.85

60

0.24

0.08

0.5

0.44

1.11

0.88

70

0.24

0.08

0.5

0.44

1.11

0.88

80

0.34

0.09

0.36

0.42

0.93

0.89

90

0.41

0.1

0.29

0.42

0.77

0.88

100

0.44

0.11

0.25

0.41

0.71

0.86

Ave

0.22

0.08

0.73

0.43

1.69

STDev

0.07

0.05

1.3

0.22

2.26

CV

0.32

0.62

1.8

0.51

1.34

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10 item

0.88

0.18 0.21

20 items

0.02

0.05

137.02

0.56

321.59

20

0.07

0.06

189.79

0.57

370.28

30

0.12

0.08

76.75

0.55

138.72

40

0.17

0.08

37.79

0.56

77.18

50

0.22

0.09

18.7

0.53

35.25

60

0.27

0.1

8.29

0.52

70

0.27

0.1

8.29

0.52

80

0.38

0.11

3.5

0.52

90

0.43

0.13

2.27

0.5

100

0.49

0.15

1.04

Ave

0.24

0.1

48.34

STDev

0.053

0.06

90.15

CV

0.22

0.66

1.86

0.9

0.89 0.86 0.87 0.87

M

10

18.03

0.85

18.03

0.85 0.85

4.84

0.83

0.5

2.37

0.84

0.53

99.36

0.86

0.28

203.1

0.15

0.52

2.04

PT

ED

7.29

0.18 30 items

0.03

0.06

783.43

0.66

1425.32

0.9

20

0.07

0.08

758.21

0.73

1151.58

0.87

30

0.12

0.08

672.66

0.72

1027.39

0.86

40

0.17

0.09

421.74

0.68

655.54

0.86

50

0.22

0.11

307.56

0.64

490.61

0.85

0.27

0.13

223.21

0.65

353.49

0.86

0.27

0.14

223.21

0.65

353.49

0.86

0.41

0.14

39.12

0.56

68.65

0.86

0.44

0.16

24.01

0.6

39.89

0.89

100

0.51

0.16

11.44

0.59

19.38

0.87

Ave

0.25

0.12

346.46

0.65

558.53

0.87

STDev

0.08

0.07

264.64

0.34

490.21

0.22

CV

0.31

0.63

0.76

0.52

0.88

0.26

Overall Ave

0.24

0.1

131.84

0.54

219.86

0.87

70 80

AC

90

CE

10

60

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Table 2: Efficiency comparison of Aggregate heuristic versus CPLEX 12

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Table 3: Comparison of Aggregate heuristic and CPLEX 12 with supplied quantity of less than half of demand

Demand %

LR Gap

H Gap

T IP (sec)

T H (sec)

T IP/T H

w

10

0.02

0.03

0.89

0.4

2.54

0.93

20

0.05

0.05

2.03

0.44

4.29

30

0.1

0.06

1.03

0.44

2.3

40

0.15

0.07

0.86

0.43

1.86

50

0.19

0.08

0.54

0.43

1.25

Ave

0.10

0.06

1.07

0.43

2.45

CV

0.68

0.33

0.53

0.04

0.47

0.04

10

0.02

0.05

137.02

0.56

321.59

0.9

20

0.07

0.06

189.79

0.57

370.28

0.89

30

0.12

0.08

76.75

0.55

138.72

0.86

40

0.17

0.08

37.79

0.56

77.18

0.87

50

0.22

0.09

18.7

0.53

35.25

0.87

Ave

0.12

0.07

92.01

0.55

188.60

0.88

CV

0.66

0.23

0.77

0.03

0.79

0.02

0.06

783.43

0.66

1425.32

0.9

0.07

0.08

758.21

0.73

1151.58

0.87

0.12

0.08

672.66

0.72

1027.39

0.86

0.17

0.09

421.74

0.68

655.54

0.86

50

0.22

0.11

307.56

0.64

490.61

0.85

Ave

0.12

0.08

588.72

0.69

950.09

0.87

CV

0.62

0.22

0.36

0.06

0.40

0.02

Overall Ave

0.11

0.07

227.27

0.56

380.38

0.88

20 30

AC

40

0.03

CE

10

ED

PT

30 items

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M

20 items

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10 item

23

0.9

0.87 0.87 0.85 0.88

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Figure 4: Comparison of Lagrangian and Aggregate heuristic when reducing satisfiability ratios beyond the threshold (a) Optimality gap comparison Heuristic Gap

Average

(b) Execution time comparison Time Ratio

STDEV

0 0

20 40 60 80 100 Max demand percentage supplied in each bid

Heuristic Gap

Average

10

5

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Execution time ratio

Gap

0.1

0.05

0

0

50 100 Max demand percentage supplied in each bid Time Ratio

STDEV

STDEV

Average

1500

20 items

0.2 0.15 0.1 0.05 0 0

Average

500

0

0 50 100 Max demand percentage supplied in each bid

50 100 Max demand percentage supplied in each bid Heuristic Gap

1000

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0.25

20 items

Execution Time Ratio

0.3

STDEV

Time Ratio

STDEV

Average

3000

0.3 0.25

M

0.1 0.05 0 0

50

100

2000

1000

0 0

ED

Max demand percentage supplied in each bid

30 items

30 items

0.2 0.15

Execution Time Ratio

Gap

Average

10 items

10 items

0.15

Gap

STDEV

15

0.2

50 100 Max demand percentage supplied in each bid

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which each bidder offers to supply at most 50% of the demand. In this figure, the x axis shows

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the number of bids submitted and the y axis shows the time ratio).

It can be seen that this ratio exceeds 2500 for 30 items and 400 bids and when no bid contains

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more than 10% of the demand (note that we use the term exceed, since each problem instance is characterized by a specific combination of the number of items, the number of bids and the

percentage of the demand offered in the package, is averaged over 25 generations of random problem instances). As observed in Table 3, averaging over all number of bid submissions, this ratio exceeds 950 for 30 items.

24

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Figure 5: CPU Time Ratio of CPLEX to the Aggregate heuristic for quantity offers less than half of the demand T IP / TH (q <=20% d)

T IP / T H (q <= 10% d) 3000

2500

2500

2000

2000

1500

10 items 1500

20 items

10 items 20 items

1000

30 items

30 items

1000

0

0 0

200

400

600

800

1000

0

1200

200

400

600

CR IP T

500

500

800

1000

1200

T IP / TH (q <=40% d)

T IP / TH (q <=30% d) 1400

2000 1800 1600 1400

1200 1000 10 items

800

20 items

600

30 items

400

10 items 20 items 30 items

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1200 1000 800 600 400 200 0

200

0

0

200

400

600

800

1000

0

1200

200

400

600

800

1000

1200

T IP / TH (q <=50% d) 1200 1000 800

10 items

400 200 0

Robustness

200

400

ED

0

4.5.3

20 items

M

600

600

800

30 items

1000

1200

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To show the extent of variability in relation to mean of our data, we use the coefficient of variation (CV) defined as the ratio of the standard deviation σ to the mean µ; CV = µσ . In fact, CV combines

CE

information about the mean and standard deviation of the system. A system associated with large values of CV is considered weakly robust, due to the large variation of standard deviation

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as compared to the mean, conversely, low values of CV indicate strong robustness of the system. Comparison of the CV values for the CPLEX and the Aggregate heuristic execution times assures that the time-wise performance of the Aggregate heuristic is more stable and thus robust compared to CPLEX.

25

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4.5.4

Aggregate Heuristic Optimal Solution

Since the heuristic algorithms for solving procurement WDP provide a feasible solution which is not necessarily optimal, the optimal heuristic value is greater than (or equal to) the objective value of CPLEX. Thus, it is likely that we get more positive solution variables in the heuristic optimal solution. In other words, the heuristic algorithms introduce more winners in the auction than

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CPLEX. Nonetheless, it is essential for auctioneers to know how many more winners they should expect, and whether this additional number of winners is dependent on the problem size or the percentage of demand supplied in each bid.

Figure 6 and Figure 7 illustrate this correspondence with the increase in the maximum percentage of demand offered in the packages received and the increase in the number of bids

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received in the auction, respectively.

Figure 6: Comparison of the number of winners in Aggregate heuristic versus CPLEX 12 with respect to the increase in maximum demand percentage supplied in the bids

0.88 0.86 0.84 0

50

0.9

0.88 0.86 0.84

100

0

50

100

Max demand percentage supplied in each bid

30 items 0.92 0.9 0.88 0.86 0.84 0

50

100

Max demand percentage supplied in each bid

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Max demand percentage supplied in each bid

0.82

Ratio of # winners in CPLEX to Heuristic

0.9

0.92

M

0.92

Ratio of # winners in CPLEX to Heuristic

20 items

0.94

ED

Ratio of # winners in CPLEX to Heuristic

10 items

Figure 7: Comparison of the number of winners in Aggregate heuristic versus CPLEX 12 with respect to the increase

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the number of bids received

0.88 0.87 0.86 0.85

0

500

Number of bids

1000

0.88 0.87 0.86 0.85 0.84 0.83 0.82 0

500

Number of bids

1000

Ratio of # winners in CPLEX to Heuristic

0.89

Ratio of # winners in CPLEX to Heuristic

0.9

30 items

20 items

AC

Ratio of # winners in CPLEX to Heuristic

10 items

0.91

0.9 0.89 0.88 0.87 0.86 0.85 0.84 0.83 0

500

1000

Number of bids

As observed from the figures, ratio w lacks any particular increasing or decreasing pattern. Thus, our first conclusion is that w may increase or decrease irrespective of the increase in the 26

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problem size, the number of bids, and the percentage of the demand supplied in each bid. Secondly w is always below 1, confirming obtaining a larger number of positive variables in the optimal solution obtained from the Aggregate heuristic. For the Aggregate heuristic algorithm this average ratio remains above 83%. Thus, the auctioneers knows he could expect obtaining a maximum of 17% extra positive variables, i.e. , suppliers, in the auction. This helps the auctioneers

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involve more suppliers in the auction. However, for the auctioneers who do not prefer such an outcome, it is possible to add further restricting constraints in advance in order to limit the total number of winning suppliers in the auction. 4.5.5

Efficiency of subheuristics

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In order to determine which combination of constraints and variable selection rule performs best, we record the total number of times a subheuristic is at minimum. Note that multiple subheuristic can provide equal minimum values at the same time. Therefore, we also recorded the number of times a subheuristic is the sole minimum, meaning that it is performing strictly better than all other procedures. Table 4 summarizes our results.

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Table 4 shows that variable selection rule 1 provides the highest percentages with all different constraint selection rules. However, while H42 provides the min value in over 10% of times, it never produces a strict minimum value. Therefore, it is possible to remove this procedure and

Conclusion

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5

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reduce execution time. Figure 8 visualizes the efficiency comparison of subheuristic procedures.

CE

Combinatorial auction allows several bidders to submit bids on different selections of items based on their personal preferences. From the computational point of view this problem is difficult to solve due to the exponential growth of the number of combinations. Lagrangian relaxation has

AC

been suggested as a promising method for solving the winner determination problem. Revelation of the Lagrangian multipliers guides sellers to adjust their prices on the bundles they require and eventually improves auction results. In this paper, we analytically established the equivalence of Lagrangian and linear relaxation for a multi-item multi-unit reverse WDP. We also show the equivalence of the Lagrangian multipliers and the dual variables of the LP relaxation. Therefore, solving the linear relaxation of WDP

provides fast access to the Lagrangian multipliers as approximates for item prices and a lower 27

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% H is the sole min

13.65

4.24

2

IH11

7.69

2.49

3

H12

9.08

2.35

4

IH12

5.53

2.21

5

H13

10

2.39

6

IH13

6.53

1.77

7

H14

6.64

1.2

8

IH14

4.71

1.32

9

H21

11.13

2.99

10

IH21

7.77

2.43

11

H22

6.51

1.56

12

IH22

4.55

1.41

13

H23

8.27

2.11

14

IH23

5.77

1.55

15

H24

5.79

1.19

16

IH24

3.73

0.99

17

H31

10.51

3.19

18

IH31

7.32

3

19

H32

9.24

3.04

20

IH32

6.03

1.68

21

H33

6.64

2.49

22

IH33

5.21

2.55

23

H34

6.95

1.73

24

IH34

5.11

1.69

25

H41

11.29

0.23

26

IH41

7.23

0.39

27

H42

10.31

0

28

IH42

6.47

0.03

29

H43

9.19

1.48

30

IH43

6.48

1.44

31

H44

5.63

0.73

32

IH44

3.24

0.65

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% H is min

H 11

M

H Name

1

ED

H Number

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Table 4: Efficiency comparison of subheuristics

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bound on the total price of procurement.

Based on this equivalence we propose a procedure which determines the Lagrangian solution

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by solving a single subproblem. This method saves significant amount of time in finding the optimal Lagrange solution as compared to traditional Lagrangian relaxation methods. In order to adjust infeasibility of the Lagrangian optimal solution, we design an Aggregate heuristic algorithm

AC

consisting of 32 computationally efficient heuristic procedures. Our extensive numerical experiments indicate that on the class of problems whose maximum

quantity of items included in each package is less than 50% of demand, the Aggregate heuristic

provides a near optimal solution for respectively 10, 20, and 30 items which is on average 5, 6, and 7 percent off from the optimal in around 1/3, 1/227, and 1/1065 of CPLEX time as we show in Table 5. This solution scheme is most efficient for iterative combinatorial auctions in large marketplaces 28

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15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

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Percentage

Figure 8: Efficiency comparison of subheuristics

H is Min

H is the sole Min

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

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Heuristic Number

Table 5: Comparison of Aggregate heuristic and CPLEX 12 with supplied quantity of less than half of demand

Ave_Gap

Ave_STD_Gap

10 items

0.0525

0.03

20 items

0.0675

0.035

30 items

0.0775

0.045

Ave_Time Ratio

Ave_STD_Time Ratio

2.7475

4.33

226.9425

461.46

1064.9575

757.63

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M

Num_items

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with capacitated suppliers. While application of iterative combinatorial auctions increases the auctioneers’ overall payoff, it requires finding solution efficiently in each round of the auction. Our experiments show that CPLEX 12 behaves poorly in particular when suppliers are only able

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to provide a limited amount of the demand. Application of the Aggregate heuristic scheme can provide a good quality of the solution in relatively much less time.

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There are several ways in which our work can be extended in the future. First, in our numerical

study we focused on the XOR formulation with free disposal where the demand constraint is relaxed. It will be interesting to also consider the effect of relaxing the supply constraint on the quality of the solution. In addition, it is worthwhile investigating environments where free disposal is not applicable. When solution time is not of concern, it is also worthwhile comparing the solution quality obtained from our heuristic to that of other traditional methods. Second, the bidder’s problem of how to package items and what prices to attach to them has not received

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significant attention in the literature, especially when compared to the auctioneer’s WDP. This problem becaomes more significant in mluti-item combinatorial auctions due to the complexity of the bid generation problem. We are aware of handful of studies in this area. Ergun et al. [2] and Triki et al. [13] consider the bid pricing problem in in the context of the bid generation problem for transportation auctions. Hsieh [4] suggests a bidder’s profit maximization model based on the

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Lagrangian multipliers revealed in each round of the auction. Mansouri and Hassini [7] propose closed form solutions to the capacitated and uncapacitated versions of the bidders’ pricing and lot sizing problem and analyze the resulting prices in comparison with prior auction round’s prices. Finally, another possible extension is to consider bidders’ delivery risks such as discrepancy in quantity, quality, and delivery time due to frequent and/or infrequent risks.

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Acknowledgements

Research in this paper has been supported through a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). We are grateful to the reviewers for their

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feedback that has improved the quality of the paper.

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