An Integrated Combinatorial Auction Mechanism for Truckload Transportation Procurement Roy H. Kwon*, Chi-Guhn Lee, Zhong Ma Mechanical and Industrial Engineering, the University of Toronto, Toronto, Ontario, CANADA, M5S 3G8 * Corresponding Author:
[email protected]
We consider an integrated multi-round combinatorial auction mechanism for truckload transportation procurement. Shippers allow bids on packages of lanes and solve a winner determination problem to assign lanes to carriers. Carriers employ optimization models to discover profitable lanes to bid for at a round. Price information derived from the result of a winner determination solution is used by bidders in identifying profitable lanes thereby integrating the shipper and bidder optimizations. We illustrate the benefits of the integrated multi-round mechanism.
1. Introduction Procurement of transportation services is an important outsourcing activity as in house logistics capacity is often inadequate to handle the volume of shipping required for a firm. In particular, truckload (TL) procurement occupies a large segment of the overall outsourcing of transportation services by shippers. Truckload transportation involves the movement of freight via dedicated vehicles (trucks) as opposed to movement of freight that is partitioned for delivery to different shippers. TL transportation is a significant portion of the annual billion dollar freight transportation industry in North America.
The use of auctions (e.g. competitive request for proposals (RFP) process) for truckload procurement has been prevalent as the medium of obtaining carrier services for shippers. Typically, single lane contracts are bid for by carriers where a single lane represents a commitment to move a specified volume from an origin to a destination. However, cost structures for carriers exhibit economies of scope (Caplice 1996, Caplice and Sheffi 2003). That is, carriers would like to obtain a set of contracts that collectively represents traveling as few empty kilometers or miles as possible where an empty distance is the situation where a vehicle is traveling without any load. Empty travel occurs because vehicles need to reposition or travel to another destination for pick up of more freight. For example, empty travel for a truck would occur when servicing a lane contract that specifies movement of freight from origin A to destination B does not include movement of a load from B to wherever the vehicle must go next. If the vehicle would return to city A then another lane contract that involves a load from B to A together with the first contract would exhibit economies of scope. Since no reward is given for empty travel this entails undesirable costs for carriers. A carrier that wishes to obtain a set of lanes that exhibit economies of scope may not be assured of winning such a complete set under auction formats that allow bids on single lanes only. Consequently, carriers may not bid as aggressively for lanes in fear of obtaining an incomplete set of lanes. This phenomenon is known as the exposure problem (Kwasnica et al 2005). One remedy to the exposure problem is to use a combinatorial auction format. Combinatorial auctions are simultaneous multiple item auction formats that allow a single bid for a set of distinct items. In the TL procurement context carriers would place single bids on sets of distinct lanes. Allowing single bids on sets of distinct lanes allow carriers to express synergies that exist for certain lanes. The synergy of interest to carriers would be the degree to which a set of lanes exhibit economies of scope. It is argued that allowing
carriers to form cost efficient lanes for bidding will also result in the reduction of costs for shippers (Caplice and Sheffi 2003). Hence shippers should have incentive to allow combinatorial bidding. Combinatorial auctions (CA) for truckload procurement has been used successfully to enable carriers to achieve economies of scope in the contract procurement process. Elmaghraby and Keskinocak (2003) describe the successful use of a single round combinatorial auction at Home Depot Inc. for TL procurement. In Ledyard et al (2002) the use of a multiple round combinatorial auction for Sears Inc. is documented where savings of millions of dollars are routinely realized compared to costs incurred using former procurement methods. However, the use of a combinatorial auction format for truckload procurement is still not without major challenges. These challenges must be effectively dealt with in order to facilitate the widespread use of combinatorial auctions for truckload procurement. Caplice and Sheffi (2003) give a thorough review of important practical issues related to the execution of combinatorial auctions for TL procurement. In addition, there are important practical design issues and challenges that pertain to the use of combinatorial auctions in general see (Pekec and Rothkopf 2003). In particular, there is the requirement by the auctioneer to solve an NP-hard integer program in order to determine the bidders that are to receive lanes. This process is referred to as winner determination (deVries and Vohra 2003, Rothkopf et al 1998). In addition, there is the complexity associated with the number of possible packages (sets of lanes) a carrier (bidder) has to evaluate. A carrier would ideally consider a set of lanes that would have economies of scope and that give revenue that would exceed the costs of servicing the contracts in a package. Also, it would be common for carriers to have existing commitments to other shippers and so a candidate set of lanes would have a good fit with other existing
business obligations. Given that the number of packages of lanes is often quite large it would be necessary to solve an optimization problem with the appropriate constraints to enforce business requirements and achieve economies of scope and other related factors to identify profit maximizing packages of lanes. For example, bidders in the Sears TL combinatorial auctions had experienced difficulty in forming packages see (Ledyard et al 2003) and decision support to aid bidders in selecting packages of lanes would have been greatly beneficial to carriers. Carrier models have been developed to aid carriers in finding valuable packages in a combinatorial auction see (Song and Regan 2002, Lee et al 2004). Song and Regan (2002) show carriers that use bidder optimization models in a combinatorial auction benefit more than carriers that use simple bid strategies. These models are integer programming formulations that are variants of vehicle routing models and are NP-hard. Mathematical modeling to aid bidders in determining the appropriate package of items in combinatorial auctions is a promising research area that should enable more practical use of combinatorial auctions. For other examples of bidder optimization modeling see Hoffman et al (2004) for bidder decision support in spectrum (radio wave) combinatorial auctions. Another major design decision is the selection of a single or multi-round combinatorial auction format. Multi-round formats involve several iterations of submission of bids by carriers and tentative allocation decisions (winner determination decisions) by the auctioneer/shipper until some stopping criterion is met. In most cases price feedback is given after each tentative allocation of items. A single round format would involve a one time submission of bids by carriers and then the auctioneer/shipper would determine the final winning allocations. Multiround formats provide information feedback for bidders whereby bidders can adjust bids in response to dynamic price information e.g. find other items to bid on if prices are too high for
currently wanted items. Better allocations can result e.g. shipper reduces more cost and more lanes and carriers are allocated profitable lane contracts compared to a single round auction. However, a multi-round format also allows for more strategic behavior in that information that is available after a round may be used for strategic gaming as well. For example, in multi-round spectrum auctions certain bidders would signal through price information to other bidders to not bid on certain items and retaliation occurred for bidders that did bid on the items see Cramton and Schwartz (2002). In this paper we propose a combinatorial auction design for truckload procurement that integrates the winner determination problem and bidder package optimization modeling. In particular, we consider an integrated optimization-based framework where a multi-round descending CA format is used where bidders solve optimization problems at each round to identify valuable packages. Price information from the result of a winner determination at a round is used by bidders in their respective optimizations as revenue coefficients. This price information comes from a pricing optimization problem that creates approximate lane prices that are consistent with package prices. The lane price information provides valuable feedback to bidders. (See Dunford et al (2004), Kwasnica et al (2005), Kwon et al (2005) for recent examples of multi-round combinatorial auctions that employ item prices.) The rationale for a multi-round auction is that carriers have hard valuation problems (i.e. NP-hard bidder optimization problems) and it would be very difficult for bidders to list all relevant packages at once and price information allows bidders to form alternative packages that are valuable. Parkes (1999) has found that the use multi-rounds auctions result in better allocations when bidders have hard valuation problems than when a single round auction is employed. Furthermore, lanes are not strategic elements compared to a case where the items for sale possess substantial strategic value
e.g. spectrum licenses. Thus, our carrier optimization model does not exhibit strategic behavior but, follows a myopic optimization bidding strategy. We assume that issues such as carrier selection have been dealt with and that bidders are viable candidates to perform transportation service that can meet non-price requirements of a shipper. Our combinatorial auction design is the first to use bidder optimization in the context of a multi-round format for truckload procurement where bidder optimization and winner determination are integrated through the use of lanes prices. We give computational results on several classes of problem instances and observe that larger percentage of a shipper’s network gets serviced, more carriers are allocated, and shippers experience lower cost in procurement compared to formats that do not integrate winner determination and bidder optimization.
2. Multi-round descending auction framework 2.1 Auction process In this section, we describe the multi-round auction framework in which bidders (carriers) are allowed to place bids on packages of lanes. A package of lanes is a subset of available lanes. The shipper is an entity that wishes to procure truckload transportation services. A carrier is an agent that represents a logistics truckload service provider that wishes to obtain lane contracts from the shipper. A bid is an amount of reward a carrier would accept for servicing a particular package of lanes. The auctioneer is an agent that represents the shipper and determines the assignment of lanes to carriers and computes prices for lanes based on bids received. The shipper determines which items (lanes) to put up for sale and gives the initial ask prices for individual lanes. Carriers (bidders) determine which packages of lanes to bid on by employing normative optimization models to determine the best package to bid for given lane current lane prices. An
important element in the bidder optimization is the utilization of individual lane prices as opposed to using package prices only. The latter will involve an exponential number of variables to deal with package prices explicitly since the number of possible packages is exponential in the number of lanes. The use of single item prices as revenue coefficients would enable a bidder model to compactly represent all possible packages see Lee et al (2005). The bid price is the price that a carrier offers for a package, which, if she wins, is the revenue awarded by the auctioneer to the carrier for servicing the lanes in the package. After receiving bids from carriers, the auctioneer solves an optimization problem called winner determination problem (WD) at each round to decide which carriers will receive lanes in their corresponding package bids. This is an integer program that ensures a lane is awarded to at most one carrier. The auction starts with no carriers allocated, then based on initial ask prices for lanes bidders determine their respective optimal package of lanes and associated bid prices. Then the auctioneer computes a provisional allocation of lanes to carriers by solving a winner determination problem that seeks to minimize costs of procuring transportation services. Individual lane prices are also computed through an approximate dual pricing program and given to the bidders at the start of the next round. Bidders determine new optimal bids based on the new lane prices and submit them if they are not in the current allocation as determined by the auctioneer. Provisional winners maintain the same winning package from the previous round at the same prices offered in the previous round. Losing bidders are allowed to resubmit a package submitted from the previous round but at possibly new bid prices depending on the new current lane ask prices. Or a losing bidder submits a different package for the next round in which case the bid price of that package must be the lower of the lowest bid price made on that package by any bidder from the previous rounds or the sum of the current ask prices of the lanes in the
package minus a fixed bid decrement ε , which will maintain descending prices during the course of the auction. There are two stopping conditions for the auction. The auction stops when carriers submit same packages for two consecutive rounds or all carriers are allocated at a round. The auction can be seen to resemble an iterative descending version of a well known ascending format for combinatorial auctions iBundle by Parkes (1999). A summary of the auction dynamics is given in Figure 1. Auctioneer announces initial ask prices
Carriers generate and submit package bids
Auctioneer solves Winner Determination Problem
Lanes and business allocated
Stopping condition satisfied?
N
Auctioneer update individual lane prices
Y Stop Provisional allocation and new ask prices
Figure 1 General process of multi-round CA
Some other requirements and assumptions are as follows: •
Each carrier submits at most a single package bid at each round. If the bidder optimization results in multiple optimal solutions only one is selected.
•
The auctioneer will not accept any bids with prices more than the auctioneer’s reservation prices, or ask prices. These initial ask prices for lanes are the maximum prices for which the auctioneer is willing to pay for lanes.
2.2 Notation, definitions and assumptions In this section we give notation and definitions that are to be used throughout the paper. We first describe the shipper network that consists of lanes that are for auctioning. We model the shipper network with a set of geographically dispersed nodes (cities) and edges between nodes
that represents the distance from one city to another city. The nodes represent either an origin or destination of a lane for which a shipper would like to obtain truckload transportation service for. The flow on a lane is the volume that must be transported from an origin to a destination of a lane. There is notation also for lanes that may be a part of an obligation for other shippers and thus notation for the flow on such a lane is differentiated from flow on lanes offered by current shipper. Notation and definitions Let V = {1, 2, ..., n }the set of nodes (cities) S = a subset of V
Γ = the set of all lanes Ψ = the set of all submitted bids at the start of a round St = a valid bid received by the auctioneer, or an element ofΨ, where t is a bidder mt = the number of lanes in the bid St , d jk = the shortest distance between node j and node k, f jk = the volume of an existing flow (i.e. flow on a lane that must be serviced for another
shipper) from node j and node k for a given bidder q jk = the volume of a new flow (i.e. flow on a lane offered by current shipper) available for
bidding from node j and node k, p jk = the current ask price for serving the flow volume q jk ,
p Ini jk = the initial ask price for serving flow volume q jk , Li = the total length of tour i, (A tour is a path over the shippers network and carrier’s network
that a vehicle follows as determined by the requirements of servicing lanes. Note: the lanes could be from the current shipper or from other existing commitments)
ω = maximum length allowed for a tour for a given bidder Vt = the valuation of a package St for a bidder obtained from the bidder’s optimization problem, which is the minimum reward that the bidder is willing to serve this package at the lanes prices for a current round. Figure 2 gives some examples of the notation above. 40 1
q56 = 50 with ask price p56
5
80
50
5
65
100 0
2
125
6
6 f 45 = 65
85
3
50
S1 = {q56 , q64 } with bid price (p56 +p64 )
70
70
4
4
Carrier 1 with existing business & bidding for package S1 = {q56 , q64 } with revenue of (p56 +p64 ) if wins.
S 2 = {q02 , q03 , q32 } with bid price (p02 + p03 + p32 )
Figure 2 Example of some package bid notation 2.3 Winner Determination problem The winner determination (WD) problem determines an allocation of lanes to carriers given a current set of bids and is usually some form of a mixed 0-1 integer program see deVries and Vohra (2003) for an excellent survey on the WD problem. For the transportation procurement problem, the auctioneer's objective is to minimize the total shipping cost, i.e. the sum of bidding prices of all packages in the final allocation. Although more elaborate winner determination problems can be formulated that capture additional business constraints see Caplice and Sheffi (2004) without loss of generality we consider only a basic version whose primary aim is to cover
as much of the shipper’s network as possible at minimum cost. Additional constraints could be added without much difficulty however doing so would not add much to the analysis. For a general overview of the WD problem and variants see deVries and Vohra (2003). For first introduce some notation, ⎧ 1, if lane from j to k with new business is in bid i, Let a ijk = ⎨ ⎩ 0, otherwise.
and pi be the bidding price for the bid i, and pi = ∑∑ a ijk p jk , where p jk is the announced j
k
price for lane j to k; The decision variables are: ⎧ 1, if bid i is in the optimal allocation, ⎩ 0, otherwise.
ψi = ⎨
Then the Winner Determination problem (WD) can be as follows: Min
∑ p ⋅ψ i
i∈Ψ
(1a)
i
s.t.
∑a i∈Ψ
i jk
⋅ψ i ≥ 1,
∀ j, k ∈ Γ
ψ i ∈ {0, 1}, i ∈ Ψ
(1b) (1c)
The constraints (1b) are set covering constraints, which ensure each lane for bid is covered at least once. The constraints (1c) restricts a package of lanes to be allocated in its entirety or not at all. This set covering model normally provides an optimal allocation with much overlap, i.e. the same flow volume on a single lane allocated to more than one carrier in the optimal allocation. To address the infeasibility issue, a dummy bidder is created for each lane in the shipper’s
network in each round of the auction. There could be as many as n * n dummy bidders, where n is the number of nodes. At the beginning of each round, dummy bidder B jk submits a single bid on a single lane from j to k with the bid price at p Ini jk and dummy bidders maintain the same bid between rounds. This is reasonable because the auctioneer’s reservation price for a certain lane associated with certain new business for bid should be fixed given within the auction. The dummy bidders ensure that no lane be sold for more than the auctioneer’s reservation price. We introduce another set of decision variables:
if lane from j to k is allocated to a dummy bidder ⎧ 1, = ⎨ in the provisional allocation at that round, ⎩ 0, otherwise.
Ω jk
Then the Modified Winner Determination problem (MWD) can be modeled as follows: Min
∑ p ⋅ψ + ∑∑ p i
i∈Ψ
i
j
Ini jk
Ω jk
(1a)
k
s.t.
∑a i∈Ψ
i jk
⋅ψ i + Ω jk = 1,
∀ j, k ∈ Γ
ψ i ∈ {0, 1}, i ∈ Ψ ; Ω jk ∈ {0, 1}, ∀j , k .
( π jk )
(1b) (1c)
From the MWD model, there are two major benefits: first, it ensures each lane is allocated to at most one bidder; furthermore, no lane will be sold for more than its reservation price of the auctioneer. We assume that if there are lanes that were not allocated at the end of the auction, the auctioneer will serve unsold lanes (or lanes sold to dummy bidders) by its private fleet or by negotiation-based contractors.
2.4 The carrier's optimal bid generation problem
We present a carrier’s bid generation optimization model see Lee et al (2005) which aims to identify synergies in the available lanes i.e. economies of scope in a carrier’s transportation operations to determine the optimal profit maximizing packages to bid for. The carrier optimization problem represents simultaneous generation and selection of package of lanes and can incorporate any existing commitment that a carrier may have in determining a set of new lanes to obtain from the current shipper. Carriers employ vehicle routing-type models to identify packages of lanes based on the actual routes that a fleet of trucks will follow in practice. The carrier's optimal bid generation model has the following form: Max Total utility
(4a)
s.t. Each lane with existing business covered exactly once
(4b)
Each lane with new business covered at most once
(4c)
Transportation capacity constraint
(4d)
Other operational constraints
(4e)
The utility can be defined as the revenues associated with servicing lanes minus the operation costs, and measured by the value of the bidding price minus the transportation cost. An important feature of the model is in the use of lane prices (rewards) to determine the profit maximizing package. This facilitates compact representation of prices for all relevant packages. See Appendix A and Lee et al. (2005) for more details on the carrier optimization model.
2.5 Price update As mentioned, the bidder optimization utilizes individual lane prices derived from a WD allocation to derive individual lane prices. The strategy to obtain prices for lanes is to formulate
an optimization problem that is analogous to the dual problem for linear programs. Since integer programs do not in general exhibit strong duality an approximate dual problem is required. Approximate dual information is still very useful price information Hoffman et al (2004). The dual problem (DP) of the linear relaxation of the MWD is given by Max
∑∑ j∈Γ k ∈Γ
π jk
(2a)
s.t.
∑∑a j∈Γ k∈Γ
i jk
⋅π jk ≤ pi ,
0 ≤ π jk ≤ PjkIni ,
∀i ∈ Ψ
(2b)
∀j, k ∈ Γ
(2c)
Where π jk is the lane price from j to k. If the optimal solution of the LP relaxation of the WD is integer, the optimal solution to LP is optimal for the WD and the single lane prices can be obtained by the optimal dual values. However, the integrality condition does not always hold which means that single lane dual prices that support an optimal integral solution may not exist. We follow as in Dunford et al (2004) and Kwon et al. (2005) and form a restricted dual problem (RD) by setting the constraints of the dual problem to active (i.e. binding) that correspond to allocated packages and leaving all other constraints of the dual problem as in the original dual: Max
∑ ∑π j∈Γ k∈Γ
jk
-
∑q
i:ψ i = 0
(3a)
i
s.t.
∑∑a
i jk
⋅π jk = pi ,
∀i: ψ i = 1
(3b)
∑∑a
i jk
⋅π jk − qi ≤ pi ,
∀i: ψ i = 0
(3c)
j∈Γ k∈Γ
j∈Γ k ∈Γ
0 ≤ π jk ≤ PjkIni , ∀ j , k ∈ Γ, qi , ∀i ∈ Ψ
(3d)
RD is always feasible since we can set qi arbitrarily large. The qi can be interpreted as a price discrimination term that ensures that the bid price of a losing bid less than the sum of prices of the lanes in that package. This quantity is essential to ensure that bidders do not select packages that would never be allocated see Kwon et al (2005). By solving the RD, we can get the new prices for each lane, and the price discrimination term for each losing bid, which we call as q terms. Then the q terms can be used to instruct the bidders to evaluate their bidding strategies in multi-round auctions. 3. Worked example In this section, we present a worked example of the integrated multi-round CA with carriers' optimal bid generation and restricted dual price updating as described in the previous sections. We also compare the result to the multi-round CA with fixed package bids but without individual lane price updating within the auction process and so carriers do not perform optimization with lane prices but optimize only over package prices over a finite set of a priori determined set of packages of lanes. We call the integrated multi-round CA as “Price updating CA” (PUCA) because the auctioneer needs to update individual lane prices at each round; and call the second multi-round CA as “Fixed package CA” (FPCA) because each bidder keeps the same packages within the whole auction process and prices are generated for only these packages. The FPCA auction is identical to the FPCA expect price updates are for packages only. 3.1 Lanes for bid and bidding rules In this example, there are 4 carriers bidding for 21 lanes with 260 truckloads of new business in a 6-node complete network. We assume for simplicity that transportation costs depend only on the distance but not on the load level. Moreover, the unit transportation cost may vary between
carriers. Each of the four carriers has her own transportation capacity (in terms of truckload), and her whole fleet is stationed at a home node, in which, without loss any generality, we assume all trucks are identical. The distance matrix of the complete network is given by: ⎛0 ⎜ ⎜ 72 ⎜ 64 ⎜ ⎜ 28 ⎜ 27 ⎜⎜ ⎝ 17
72 0 85 98 85 64
64 85 0 76 48 53
28 98 76 0 30 34
27 85 48 30 0 25
17 ⎞ ⎟ 64 ⎟ 53 ⎟ ⎟ 34 ⎟ 25 ⎟ ⎟ 0 ⎟⎠
The new business matrix q jk for bid is given as. ⎛0 ⎜ ⎜ 10 ⎜ 10 ⎜ ⎜ 20 ⎜ 10 ⎜⎜ ⎝0
10 10 30 20 10 ⎞ ⎟ 0 10 0 10 10 ⎟ 10 0 0 10 0 ⎟ ⎟ 10 0 0 10 0 ⎟ 10 0 20 0 0 ⎟ ⎟ 0 10 10 10 0 ⎟⎠
In this new business matrix, q jk means that there is a volume of q jk truckloads to be moved from node j to node k, i.e, q12 = 10 means 10 truckloads of materials need to be moved from node 1 to node 2 (see Figure 4).
2
10 10 10
10 10
10
10 4
10
20
10
10
10
5
10
10
10 1
0
10
10
10
20 20
3
30
20
Figure 4 The network of available lanes for auction. The auctioneer sets the initial ask prices for the new business on each lane see Table 1. These are the maximum prices for which the auctioneer would be willing to outsource the packages of lanes. Node
1
2
3
4
5
6
1
0
560
580
1606
1160
600
2
961
0
1069
0
1635
1000
3
870
956
0
0
878
0
4
1256
1226
0
0
1096
0
5
490
1100
0
1496
0
0
6
0
0
983.05
458
500
0
Table 1 The initial ask prices of shipper lanes.
The minimal bid decrement is ε = 1000 , the length limit of a tour ω = 300 for each carrier .
3.2 The existing business of carriers The existing business of each bidder is shown in Table 2. For example, 0 ⇒ 4 means that a carrier has existing business with volume of 10 truckloads from node to 4
Carrier
Existing business
Capacity
(each lane has a volume of 10)
Home
Unit
depot
traveling fee
1
0 ⇒ 11 , 0 ⇒ 4 , 4 ⇒ 0
20
0
1.05
2
0 ⇒ 2 ,1 ⇒ 0 , 2 ⇒ 1, 3 ⇒ 2 , 4 ⇒ 5
40
2
0.90
3
0 ⇒ 3 ,1 ⇒ 2 , 2 ⇒ 1
20
3
1.0
4
1 ⇒ 0 ,1 ⇒ 2 , 3 ⇒ 1 , 4 ⇒ 0
30
1
1.0
Table 2
Existing business for each bidder
3.3 Results for Price updating CA For the first round, the 4 carriers solve their own optimal bid generation problem based on the initial ask prices and submit the bids as shown in Table 3. Carrier
Optimal bids at the first round
Bid price
1
0 → 51, 1 → 4, 1 → 5, 3 → 0, 3 → 4, 4 → 1, 4 → 3, 5 →3, 5 → 4
9141
2
0 → 3, 0 →5, 1 → 2,1 → 4, 2 → 0, 2 → 1, 2 → 4, 3 → 0, 3 → 1,
14661.05
3 → 4, 4 → 0, 4 → 3, 5 → 2, 5→ 4 3
1 → 4, 2 → 4, 3 → 1, 3 → 4, 4 → 0, 4 → 3
6821
4
0 → 3, 0 → 4, 2 → 4, 3 → 0, 3 → 1, 3 → 4, 4 → 1, 4 → 3
9818
1: Note i→ k denotes the lane from Node 0 to Node 5 in shipper’s network. Table 3
The bids submitted by 4 carriers at the first round under PUCA.
Because there are 21 lanes, the auctioneer needs to submit 21 dummy bids for each lane, whose bid price is the same as the initial ask price for that lane. And the provisional result for the first round is: Carrier 2 wins a package of 14 lanes, and all other 7 lanes are won by those dummy bidders. Then, the auctioneer solves the RD problem and announces the new ask prices for each lane, and then the auction continues. The auction stops at the 13th round due to no new bids submitted at that round; and the final allocation is recorded in Table 4. The final total cost for the auctioneer is 9267.1. Among total 21 lanes with new business and 260 truckloads of new business for bid, there are 16 lanes and 210 truckloads won by 2 carriers, 5 lanes and 50 truckloads of business volume for bid left for the auctioneer (see Figure 5). Winner
Final allocation 0 → 5, 1 → 2, 1 → 5, 2 → 0, 2 → 1, 4 → 0, 5 →2, 5 → 4
Carrier 2 Carrier 3 Dummy Bidders
Price 4186.1
0 → 3, 0 → 4, 2 → 4, 3 → 0, 3 → 1, 3 → 4, 4 → 1, 4 → 3
887
0 → 1, 0 → 2, 1 → 4, 5 → 3
4194
Table 4
The final allocation under Price updating CA
Existing business for carriers
2
10
10 10
10
10
10
New business for bid 10
10 4
10 5
10
2
10 10
4
10 10 20
10 5
10 0
10
10
1 10
Carrier 2 allocation under PUCA
Carrier 3 allocation under PUCA
Figure 5 Network diagram of final allocation under PUCA
3.4 Results for Fixed package CA (FPCA) For the first round, the 4 carriers solve their own optimal bid generation problem based on the initial ask prices (the price of a package at the start would be the sum of initial ask prices of lanes in carrier’s candidate packages only for the first round or alternatively the shipper could announce a reservation price for each package) and submit the bids as follows see Table 5. Carrier 1
2
Optimal bids at the first round Bid 1
0 → 5, 1 → 4, 1 → 5, 3 → 0, 3 → 4, 4 → 1, 4 → 3, 5 →3, 5 → 4
Bid 2
0 → 5, 1 → 4, 2 → 4, 3 → 0, 3 → 4, 4 → 0, 4 → 3, 5 → 2
Bid 3
0 → 2, 0 → 5, 1 → 4, 2 → 4, 3 → 0, 3 → 4, 4 → 0, 4 → 3, 5 → 3
Bid 1
0 → 3, 0 →5, 1 → 2,1 → 4, 2 → 0, 2 → 1, 2 → 4, 3 → 0, 3 → 1, 3 → 4, 4 → 0, 4 → 3, 5 → 2, 5→ 4
10
10
0 3
1
20
Bid price 9141 8434.05 8489 14661.05
3
Bid 2
0 → 2, 0 → 4, 0 → 5, 1 → 4, 1 → 5, 2 → 0, 2 → 1, 2 → 4, 3 → 0,
13962.05
4 → 0, 4 → 1, 4 → 3, 5 → 2, 5 → 3, 5 → 4 Bid 3
0 → 3, 0 → 5,1 → 2, 1 → 4, 2 → 0, 2 → 1, 2 → 4, 3 → 0, 3 → 1,
13678
3 → 4, 4 → 0, 4 → 3, 5→ 4 3
4
Bid 1
1 → 4, 2 → 4, 3 → 1, 3 → 4, 4 → 0, 4 → 3
6821
Bid 2
1 → 5, 2 → 4, 3 → 4, 4 → 0, 4 → 1, 5 → 3
5022
Bid 3
0 → 5, 1 → 4, 2 → 0, 3 → 1, 4 → 0, 5 → 3
5279
Bid 1
0 → 3, 0 → 4, 2 → 4, 3 → 0, 3 → 1, 3 → 4, 4 → 1, 4 → 3
9818
Bid 2
0 → 3, 0 → 5, 1 → 4, 2 → 4, 3 → 0, 3 → 1, 3 → 4, 4 → 1, 4 → 3,
11393
5→4 Bid 3
Table 5
0 → 3, 1 → 4, 2 → 4, 3 → 0, 3 → 1, 3 → 4, 4 → 1, 4 → 3
10293
The bids submitted by 4 carriers at the first round of Fixed package CA
Similarly, the auctioneer submits 21 dummy bids for each lane for bid, whose bid price is the same as the initial ask price for that lane. And the provisional result for the first round is: Carrier 2 wins its second package of 15 lanes, and all other 6 lanes are won by those dummy bidders. The auction stops at the 11th round due to no new bids submitted for two consecutive rounds; and the final allocation is recorded as in Table 6. The final total cost for the auctioneer is 15480.05. Among total 21 lanes with new business and 260 truckloads of new business for bid, there are 15 lanes and 180 truckloads won by 1 carrier, 6 lanes and 80 truckloads left for the auctioneer (see Figure 6). Winner
Final allocation
Price
0 → 2, 0 → 4, 0 → 5, 1 → 4, 1 → 5, 2 → 0, 2 → 1, 2 → 4, 3 → 0,
Carrier 2
7962.05
4 → 0, 4 → 1, 4 → 3, 5 → 2, 5 → 3, 5 → 4 0 → 1, 0 → 3, 1 → 0, 1 → 2, 3 → 1, 3 → 4
Dummy Bidders
Table 6
7518
The final allocation under FPCA
2
10
10 10
10
10
10
4
10
New business for bid
10 20
10 10
Existing business for carriers
10
10
10
5 10
0
20 20 3
1 10
10
Carrier 2 allocation under FPCA
Figure 6 Final allocation Carrier 2 under FPCA
3.5 Brief result comparison We compare the results of PUCA and FPCA along 5 dimensions: (1) the final cost the shipper is required to pay for the transportation procurement as a percentage of her initial ask prices or her reservation prices for lanes; (2) the percentage of lanes in the shipper’s network won by all carriers in the final allocation; (3) the percentage of total business volume available in shipper’s network allocated to all carriers in final allocation; (4) the number of carriers allocated
a package in the final allocation; and (5) the number of rounds needed to finish the auctions. From Figure 7, it is easy to conclude that Price updating CA works better than Fixed package CA on this example except the number of rounds needed to finish the auction. Price updating CA 80.77%
76.90%
76.92% Fixed package CA
71.43%
68.17%
13 11
45.25%
2
1
Final cost
Percentage of lanes allocated
Percentage of volumes allocated
Number of rounds
Number of winners
Figure 7 Comparison of results.
4. Computational Experiments In this section, we report on additional computational experiments. We compare the results obtained under PUCA and FPCA for all instances. For the carrier’s optimal bid generation problem, we use the same algorithms in Lee et al. (2005). All programs are implemented in C and run on a personal computer with a Pentium 4 3.4GHz CPU. We use CPLEX (version 9.0) to solve all optimization problems occurring in the auctions and sub-problems in the algorithm for the carrier model. 4.1 Configuration of test problems We generate distance matrices for the shipper’s network, matrices for flow volumes on shipper’s network, initial ask price matrices and the limit on the length of tours. At the same time, we also need to generate matrix for existing flows, truck capacity, home node, and unit traveling cost for each carrier.
•
The distance matrices d jk are derived from actual distances between major cities in North American and scaled down appropriately;
•
The existing flow volume matrix f jk for each carrier, and new flow volume matrix q jk are randomly generated according to a uniform distribution in a range of [0, 70]. The ask price matrix PjkIni is proportional to the product of flow volume between and the distance on the lane from j to k, in which, the ratio of PjkIni and the product are uniformly distributed in range of [1, 2];
•
For problem instances with 6, 8, 10, 12, and 15 nodes, the average numbers of lanes with new business and the range of existing business for each carrier are shown in the Table 5;
•
The truck capacity for each carrier is determined randomly and ranged between 50% to 100% of the sum of all existing flow volumes for that carrier;
•
The limit on the total length of feasible tours is the product of the number of working hours in each unit time period and the average traveling speed per hour, where, the number of working hours set at 40 hours/week, the average traveling speed at 70 miles/hour, then scale down to one of ten, say, set the limit on the total length of tours as 280 unit distance. For large examples with nodes more than or equal to 12, the limit on the total length of tours is a little more than the above in order to be long enough to cover all lanes in this network. Examples No. of lanes with Range of No. of lanes (Nodes)
new business
with existing business
6
21-22
1-6
8
35
13-19
10
46
19-26
12
76
6-32
15
126
10-43
Table 5
The data of lanes with new business and existing business
4.2 General results Table 6 presents the main computational results. In the first column “Ex”, the first number in each name reflects the number of nodes while the second of the index represents the number of carriers in the problem instance. For instance, N6C4 means there are 6 nodes and 4 carriers in this problem instance. The second column “Rd” is the number of rounds needed to finish the auction. In the third column: Final Cost = Final cost for final allocation / Total initial ask prices for lanes of the shipper’s network, which is the final cost the shipper should pay for the transportation procurement in percentage of her initial ask prices or her reservation prices for those services, so “Final Cost” can be viewed as the benefits for the shipper due to the auction. “Number of lanes allocated” represents the percentage of lanes won by all carriers in total lanes from the shipper’s network in the final allocation. “Business volume allocated” means the percentage of business volume allocated to all carriers in total in final allocation. “CPU” means the CPU time in seconds to solve the problem instance. No. Ex
1
N6C4-1
Auction
PUCA
Rd
13
Final Cost
45.25%
Num.
of Business
lanes
volume
allocated
Allocated
76.90%
80.77%
CPU(s)
24.44
2
3
4
5
6
7
8
9
10
11
12
N6C4-2
N6C6-1
N6C6-2
N8C4
N8C6
N8C8
N8C10
N8C12
N10C4
N10C6
N10C8
FPCA
11
60.17%
71.43%
76.92%
4.22
PUCA
3
22.98%
81.82%
85.45%
24.97
FPCA
3
76.11%
72.73%
72.73%
12.78
PUCA
2
68.66%
77.27%
81.82%
30.19
FPCA
6
76.11%
72.73%
72.73%
14.98
PUCA
2
71.76%
42.86%
38.46%
7.00
FPCA
3
88.56%
38.10%
30.77%
3.00
PUCA
7
51.50%
82.86%
84.35%
109.05
FPCA
6
63.09%
80.0%
81.74%
38.11
PUCA
4
83.93%
85.71%
85.22%
122.97
FPCA
13
54.81%
80.0%
81.74%
102.73
PUCA
11
17.50%
80.0%
81.74%
735.55
FPCA
15
59.40%
77.14%
79.13%
151.33
PUCA
10
13.63%
80.0%
81.74%
965.78
FPCA
15
59.40%
77.14%
79.13%
196.03
PUCA
9
17.60%
85.71%
89.57%
1183.73
FPCA
13
85.24%
51.43%
42.61%
281.73
PUCA
7
19.22%
84.78%
83.57%
420.08
FPCA
8
67.25%
78.26%
81.43%
140.58
PUCA
6
19.01%
86.96%
90.71%
1248.53
FPCA
7
26.20%
73.91%
71.43%
322.23
PUCA
6
22.63%
86.96%
90.71%
2299.0
FPCA
7
80.35%
71.74%
69.29%
509.20
13
14
15
16
17
18
19
20
N10C10
N10C12
N12C4
N12C6
N12C8
N12C10
N12C12
N15C4
PUCA
6
25.77%
86.96%
90.71%
4123.80
FPCA
7
80.35%
71.74%
69.29%
1213.56
PUCA
4
22.33%
86.96%
90.71%
3258.95
FPCA
7
80.35%
71.74%
69.29%
1230.42
PUCA
5
34.60%
89.47%
87.03%
1369.94
FPCA
5
62.52%
80.26%
76.76%
250.59
PUCA
5
30.15%
86.84%
82.70%
3433.53
FPCA
5
62.52%
80.26%
76.76%
672.28
PUCA
5
27.17%
88.16%
85.41%
3268.20
FPCA
5
62.52%
80.26%
76.76%
8103.56
PUCA
5
33.35%
88.16%
85.41%
5652.52
FPCA
5
62.52%
80.26%
76.76%
8515.66
PUCA
5
38.19%
88.16%
85.41%
9207.91
FPCA
5
62.52%
80.26%
76.76%
9652.94
PUCA
9
16.68%
96.83%
97.16%
15180.63
FPCA
11
56.30%
95.24%
94.89%
1507.81
Table 6
General computational results
We have the following additional observations: •
For Price updating CA, the average final cost is 30.83% of the sum of initial ask prices for all new business, comparing to 66.64% for Fixed package CA (see Figure 8); we should also mention why we measure final cost by the percentage of initial ask
prices. Intuitively, it is more reasonable to measure the final cost by the minimum true valuation of all carriers for new business. However, there are two difficulties. First, it is very hard to determine the true valuation for all possible packages for each carrier, in which, the number of all possible packages is exponential; secondly, it is private information for each carrier. •
There are 82.97% of lanes with new business for bid allocated to carriers on average for all problem instances by Price updating CA, comparing to 74.32% by Fixed package CA, see Figure 9;
•
There are 83.84% of new business flow volume allocated to carriers on average for all problem instances by Price updating CA, comparing to 72.93% of Fixed package CA, see Figure 10;
•
As the number of carriers increases, both of percentage of lanes from shipper’s network allocated to carriers and percentage of business flow volume allocated to carriers trend to increase;
•
Due to a relatively long time period between each round in a combinatorial auction in reality, i.e., one month time interval between two rounds for Sears case (Ledyard et al, 2002), the CPU time to solve a problem instance seems to be less important in this research. We claim that the number of rounds to solve a problem instance is more significant in practice. From Figure 11, we conclude PUCA framework works better for most cases (except for 2 problem instances) than FPCA in terms of the number of rounds to finish auctions.
•
The number of carriers allocated under PUCA range from 20 to 70 percent of total number of bidders where the average number of allocated bidders was over 50% whereas under FPCA only one bidder was allocated in most cases.
5. Conclusions Combinatorial auction design for truckload procurement involves challenges for both shippers and carriers. We have presented an integrated multi-round combinatorial auction design where the auctioneer winner determination is integrated with bidder optimization through individual lane prices derived from a current allocation at a round. The results indicate that better allocations can be obtained for both shippers and carriers i.e. shippers reduce costs of procurement of services and have more of their network serviced and carriers are able to identify alternative valuable packages of lanes. The elements in the design that have the greatest impact for producing good allocations are the multi-round nature which allows price feedback to be used by bidders and the use of optimization modeling for carriers to identify valuable packages. The conclusion is that good combinatorial auction design should involve the use of optimization not only for shippers but for carriers as well in an iterative environment.
90.00%
Percentage
80.00% 70.00% 60.00% 50.00%
PUCA
40.00%
FPCA
30.00% 20.00% 10.00%
Example index
1
3
5
7
9
11
13
15
17
19
Figure 8 Comparison of Final Cost
100.00%
Percentage
90.00% 80.00% 70.00% PUCA
60.00%
FPCA
50.00% 40.00% 30.00%
Example index
1
3
5
7
9
11
13
15
17
19
Figure 9 Comparison of Number of lanes for bid finally allocalted
100.00%
Percentage
90.00% 80.00% 70.00% PUCA
60.00%
FPCA
50.00% 40.00% 30.00%
Example index
1
3
5
7
9
11
13
15
17
Figure 10 Comparison of Business volume for bid finally allocalted
19
15
Rounds
12 9
PUCA FPCA
6 3 0
Example index
1
3
5
7
9
11
13
15
17
19
Figure 11 Comparison of Rounds to finish the auction
References
Caplice, C., 1996. An optimization based bidding process: a new framework for shippercarrier relationships, Doctoral dissertation, MIT, Department of Civil Engineering. Caplice, C. and Sheffi, Y., (2003). Optimization-Based Procurement for Transportation Services." Journal of Business Logistics, Volume 24, Number 2 Cramton, P. and Schwartz, J. (2002) Collusive Bidding in FCC Spectrum Auctions, Contributions to Economic Analysis and Policy, Vol. 1, Issue 1. Article 11. de Vries, S. and Vohra, R.,(2003). Combinatorial Auctions: A Survey, INFORMS J. of Computing, Vol. 15, 3, 284-309 Elmaghraby, W. and Keskinocak, P., (2003). "Combinatorial Auctions in Procurement," in The Practice of Supply Chain Management, C. Billington, T. Harrison, H. Lee, J. Neale (editors), Kluwer Academic Publishers Dunford, M., K. Hoffman, D. Menon, R. Sultana, T. Wilson. (2004) Testing Linear Pricing Algorithms for use in Ascending Combinatorial Auctions. Working Paper, Department of Systems Engineering and Operations Research, George Mason University, Farifax, VA.
Hoffman, K., D. Menon, S.A. van den Heever. (2004). A Bidder Aid Tool for DynamicPackage Creation in the FCC Spectrum Auctions. Draft. Department of Systems Engineering and Operations Research, George Mason University, Fairfax, VA. Kwasnica , A., Ledyard, J., Porter, D., and DeMartini, C., (2005). A New and Improved Design For Multi-Object Iterative Auctions, forthcoming Management Science Kwon, R.H., Anandalingam, G., and Ungar, L.H., (2005). Iterative combinatorial auctions with bidder-determined combinations, forthcoming in Management Science Ledyard, J., Olson, M., Porter, D., Swanson, J., and Torma, D., (2002), The first use of a combined value auction for transportation services, Interfaces, Sept-Oct 2002, volume 32 issue 5 pp 4-12 Lee, C.G., Kwon, R.H., and Ma, Z., (2005), Carrier's Optimal Bid Generation Problem in Combinatorial Auctions for Transportation Procurement, forthcoming in Transportation Research E Parkes, D. (1999) iBundle: an efficient ascending price bundle auction, In Proc. ACM Conference on Electronic Commence (EC-99), Denver. Parkes, D. (1999) Optimal Auction Design for Agents with Hard Valuation Problems In the Proc. IJCAI-99 Workshop on Agent Mediated Electronic Commerce, Stockholm, Sweden, pages 206-219, July. Pekec, A. and Rothkopf, M. (2003), Combinatorial Auction Design, Management Science, Vol. 49, No. 11, November, pp.1485-1503. Rothkopf, M., Pekec A., and Harstad, R., (1998). Computationally manageable combinatorial auctions, Management science, Vol.44, No.8, pp.1131-1147
Song, J., and Regan, A., (2002). Combinatorial auctions for trucking service procurement: the carrier perspective, Transportation Research Record, Journal of the Transportation Research Board, 1833, pp. 40-46.
Appendix A: Carrier’s optimal bid generation problem In this appendix we describe the carrier optimization model. See Lee et al. (2005) for more details. Let yi = integral times of tour i are included in the submitted packages, ⎧ 1, if lane from j to k carrying new business is in the bid, z jk = ⎨ ⎩ 0, otherwise. ⎧ 1, if bid package n is chosen to submit for the next round, bn = ⎨ ⎩ 0, otherwise. ⎧ 1, if lane j to k with business of type α is in the tour i, x ijkα = ⎨ ⎩ 0, otherwise. ⎧ 1, if node j is in the tour i, z ij = ⎨ ⎩ 0, otherwise.
And other there more notations for this subsection: qt = the q term of the bid t, in which, t ∈ Ψ ; T = the capacity or the number of all trucks one carrier has; v = the cost of a truck traveling a unit distance. Then the authors present a quadratic integer model for the carrier's bid generation problem as follows. Max
∑∑ p j
k
jk
⋅ z jk -
v⋅d ∑∑∑∑ α i
j
k
jk
⋅ x ijkα ⋅ y i -
∑q ⋅b t∈Ψ
t
t
(5a)
s.t.
∑y
i
≤T
(5b)
i
∑y ⋅x
i0 jk
= f
∑y ⋅x
i1 jk
= z jk ⋅ q jk ,
i
jk
∀ j, k
,
(5c)
i
i
∀ j, k
(5d)
i
x α = ∑∑ x α , ∑∑ α α i jk
i kj
k
∀i , j
(5e)
k
xα ≤ z , ∑ α
∀i , j , k
(5f)
xα ≤ z , ∑ α
∀i , j , k
(5g)
xα ≥1 ∑∑ α
∀i
(5h)
i jk
i
k
i jk
i
j
i 1k
k
∑ ∑∑d α
jk
∑ ∑∑ α
x ijkα ≥ z li , S ⊂ V ,1 ∈ S , l ∈ V \ S , ∀i
j
⋅ x ijkα ≤ ω
∀i
(5i)
k
j∈S k∈V \ S
(5j)
bt + mt − 1 ≥ ∑ ∑ z jk , ∀ lane j to k ⊆ St , ∀t
(5k)
yi ≥ 0, and integer, ∀i; z li ∈ {0,1}, ∀i, j
(5l)
xαjk ∈ {0,1}, ∀i, j, k , α ; z jk ∈ {0,1}, ∀j, k ; bt ∈ {0,1}, ∀t.
(5m)
j
k
The objective function (5a) maximizes the profit, in which,
∑∑ p j
jk
⋅ z jk is the total revenue
k
due to bidding on the set of lanes (the pjk is the lane price from node k to nodej and can be obtained from an approximate dual pricing program see Dunford et al (2004) or Kwon et al
(2005)),
v⋅d ∑∑∑∑ α i
j
jk
⋅ x ijkα ⋅ y i is the total traveling cost,
k
∑ q ⋅b t∈Ψ
t
t
is the additional cost for
certain high demanding packages if a penalty applies. The constraints (5b) ensure that the number of trucks needed to serve all submitted packages (tours) doesn't exceed the total truck capacity of the carrier, which express the total trucks a carrier owns. The constraints (5c) make sure that the existing flow volume f jk can be and must be covered exactly once in all submitted packages. The constraints (5d) ensure that the new flow volume q jk can be covered at most once in all submitted packages. The constraints (5e) make sure that only pure cycles and cycles with sub-tours are allowed. The constraints (5f) and (5g) enforce that if one lane from node j to node k is included in the optimal solution, then both of node j and node k must be selected, which correspond to maintain the property that each lane has at most single unit of flow volume. The constraints (5h) enforce any tour to include the depot (imply z ij = 1 always true), or one truck must visit the depot at least once in each tour. The constraints (5i) ensure each tour has to satisfy the total length limit. The constraints (5j) eliminate those solutions that consist of subtours for which there does not exist a node j in common (node-disjointed subtours), that means, all connected subtours are allowed in this model. The constraints (5k) define the packages apply penalties. The constraints (5l) and (5m) are for binary variables or integral variables. The authors present a decomposition of the carrier's optimal bid generation model into a master problem and sub-problem for the purpose of algorithmic development. The master problem provides the optimal solution, or the best bidding package(s) in terms of utility. Based on the dual information of the MP, the sub-problem generates feasible tours that can offer the most promising improvement in utility.
Appendix B: Epsilon effect on performance In this section, we report on the effect of the minimum bid decrement, epsilon ε , on two examples: N6C6-1, N6C6-2 to illustrate sensitivity of auction results for different bid decrements. •
Epsilon and the number of rounds. From the results, we observe that the number of rounds roughly monotonically decreases in Epsilon. We also observe that the more rounds the auction proceeds, the lower final cost likely (see Figure 12).
•
Epsilon and the total cost for the final allocation. Intuitively, the smaller Epsilon, the lower cost for the final allocation. However, according to our observation, we can not claim this (also see Figure 12). We guess the reason may be due to the special cost structure and private information for each carrier in auction, which is determined by the fact that every carrier has his or her own existing service commitment with the auctioneer or other shippers. 50 45 40 35 30 25
Final cost (in 0.5 k)
20
Rounds
15 10 5
Epsilon
0 30
60
125
250
500
1000
2000
3750
5250
7500
8000 10000 15000
Figure 12 Epsilon effects on Example N6C6-1
•
Epsilon and the number of carriers in final allocation. Some auctioneers may have interest in the number of winners in final allocation. We use example N6C6-2 to show the effect. When Epsilon changes, the number of carriers in final allocation also varies. For Table 7, we also have one interesting observation for this example: the
number of carriers in the final allocation, the lower the final cost for the final allocation. Epsilon
Round
No. of carriers won
Final Cost
1500
5
2
81.82%
1000
7
2
71.76%
500
12
3
49.71%
300
19
4
49.71%
200
16
2
87.15%
10
27
1
92.22%
Table 7 Epsilon effect for Example N6C6-2
