A stochastic programming winner determination model for truckload procurement under shipment uncertainty

Transportation Research Part E 46 (2010) 49–60

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

A stochastic programming winner determination model for truckload procurement under shipment uncertainty Zhong Ma, Roy H. Kwon *, Chi-Guhn Lee Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario, Canada, M5S 3G8

a r t i c l e

i n f o

Article history: Received 3 July 2008 Received in revised form 17 December 2008 Accepted 24 February 2009

Keywords: Truckload procurement Combinatorial auctions Stochastic programming Shipment uncertainty

a b s t r a c t We propose a two-stage stochastic integer programming model for the winner determination problem (WDP) in combinatorial auctions to hedge the shipper’s risk under shipment uncertainty. The shipper allows bids on combinations of lanes and solves the WDP to determine which carriers are to be awarded lanes. In addition, many other important comprehensive business side constraints are included in the model. We demonstrate the value of the stochastic solution over one obtained by a deterministic model based on using average shipment volumes. Computational results are given that indicate that moderately sized realistic instances can be solved by commercial branch and bound solvers in reasonable time. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Procurement of truckload (TL) transportation services is an important outsourcing activity for a shipper when the inhouse logistics capacity is insufficient to transport the freight in dedicated movements from origins to destinations. One important means for allocating the shipments to carriers in the TL setting is for the shipper to use an auction. The basic item for sale is a lane which is defined as a contract between the shipper and a carrier that requires a carrier to move a dedicated shipment of a particular (estimated) volume from an origin to a destination. Typically, auctions in practice involve the sale of single lanes only. This often poses risks for bidders since there might be several lanes that that might be of interest and every lane in this set must be won separately. Obtaining an incomplete set of items may render the set much less valuable or even worthless. This phenomenon in simultaneous single item auctions is called the exposure problem see Bykowsky et al. (1995). Consequently, bidders are often less aggressive in bidding due to the fear of obtaining an incomplete set of lanes and as a result economic efficiency of outcomes may suffer. As an alternative, combinatorial auctions have been widely suggested for TL transportation procurement to lessen the exposure of bidders (carriers) which consequently could lead to more efficient allocations. A combinatorial auction is a simultaneous multiple item auction format that allows bidders to place a single bid on a set of distinct items to express synergies that exist for certain items see Parkes (1999). The synergies in lanes arise from combinations of lanes that enable carriers to reduce empty repositioning costs. For example, if a lane A requires the movement of freight from city i to j and there was no freight movement required from j to i, then a truck may have to return to city i without any load thereby incurring empty repositioning costs. Depending on the carrier’s cost of repositioning, it may not be worthwhile for a carrier to accept lane A. A better situation for a carrier would involve obtaining another lane B in addition to lane A that would require transporting a shipment from j to i. The combination of lanes A and B may be more profitable. So carrier may wish to service a package consisting of A and B or nothing at * Corresponding author. Fax: +1 416 978 3274. E-mail address: [email protected] (R.H. Kwon). 1366-5545/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2009.02.002

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Z. Ma et al. / Transportation Research Part E 46 (2010) 49–60

all. In a combinatorial auction, a carrier could submit single bids for several distinct lanes and if a bid were successful then the carrier would obtain the right to serve all lanes within the set (package) submitted, or otherwise there would be no obligation to ship any incomplete set. This would minimize the risks for carriers in obtaining only a subset of lanes that are not worth much, or that would incur a loss in servicing the incomplete set of won lanes due to different repositioning costs. Caplice and Sheffi (2003) indicate the value of combinatorial auctions in transportation procurement, and discuss some lessons from practice. Ledyard et al. (2002) present their experience of using combinatorial auctions for Sears Logistics Services and report Sears has been savings millions of dollars annually on outsourced transportation costs by using their multiround combinatorial auctions mechanism. Elmaghraby and Keskinocak (2003) describe the experience of Home Depot in using a single round combinatorial auction mechanism for procuring TL transportation service to ship freight among its thousands of stores. Lee et al. (2007) and Song and Regan (2002) describe models that aid carriers in determining a set of valuable lanes to bid for in a combinatorial auction. However, there are still challenges in the use and design of combinatorial auctions see Abrache et al. (2007) for a comprehensive discussion of the issues. In particular, the auctioneer has to solve an NP-hard integer program to determine the bidders (carriers) that are to receive lanes. This problem is also known as the winner determination problem (WDP) see Rothkopf et al. (1998). The winner determination problem in its most basic form is equivalent to the weighted set packing problem see Rothkopf et al., 1998. In this setting a bidder submits a subset of items from a given ground set and places a bid amount associated with the subset. The auctioneer then selects which subsets to take to maximize (minimize) revenue (cost) subject to not allocating more than one of each item. In the context of TL procurement, a subset is a set of lanes that a carrier (bidder) submits with an associated offer amount that represents the payment that the carrier wishes to obtain for servicing the set of lanes for the shipper. There has been much recent activity in developing and analyzing both exact and approximate methods for the weighted set packing problem as well as generalizations of the basic WDP problem. See for instance, Rothkopf et al. (1998), Sandholm (2002), Kwon (2005) and Andersson et al. (2000) where the focus is on the WDP with a single unit of each item. LeytonBrown et al. (2000) and Gonen and Lehmann (2000) extend the WDP for multi-unit combinatorial auctions and Sandholm and Suri (2001) study the impact of side constraints on the WDP. See Lehmann et al. (2006) for a recent discussion of the WDP. Caplice and Sheffi (2003) provide several optimization models for assigning carriers to lanes (winner determination) without and with package bids, and discuss how to enrich the basic assignment model by including the business side constraints. Song and Regan (2002) present alternative formulations for bid structuring and evaluation. In addition, there has been important work in designing bidding languages to enable complex bidding expression see Abrache et al. (2003) and Boutilier and Hoos (2001). One important factor not included in winner determination models for TL procurement is shipment volume uncertainty. Most lanes specify volume as an estimated quantity and lanes are awarded before the actual volume requirements are known. Thus, assignments of lanes to carriers may not be optimal after volume uncertainty has resolved. For example, a carrier that wins a set of lanes may be asked to ship volumes on the lanes that are far less than expected and may not receive as much revenue as originally planned. Or volume may be greater than the expected amount such that an assigned carrier may not have sufficient truckload capacity. In this case, the shipper may need to procure additional third-party services at extra cost to meet actual volume demand. In either case, uncertainty can have a detrimental effect and the winner determination should properly account for this possibility. In this paper, we incorporate uncertainty in shipment volume into the winner determination problem for TL procurement. In particular, we formulate a two-stage stochastic integer program (SIP) to hedge the shipper’s risk under uncertainty. Stochastic programs are mathematical programs that explicitly incorporate uncertainty in parameters see Kall and Wallace (1995) and Birge and Louveaux (1997). Uncertainty is represented as scenarios each of which represents a possible value for a parameter that is random. The first-stage decisions correspond to assigning carriers to lanes based on submitted packages before uncertainty in shipment volumes has been resolved. The second-stage decisions correspond to assigning actual volume to the carriers after uncertainty of shipment volume has been realized. We assume a finite number of scenarios for shipment volume on lanes each occurring with a probability according to a discrete distribution. We also show the value of the stochastic solution over deterministic solutions obtained from a related deterministic model that uses the average volume instead of scenarios. Carrier bids are structured to specify a range of volume that is acceptable for a given subset of lanes and an associated revenue amount to be received per unit volume shipped for a volume amount within the specified range. If a carrier is included in the first-stage decisions, and the realized volume is less than the lower volume bound specified in her bid, then the carrier will have the right to move this lower volume but at a benefit of the lower bound volume times the per unit volume revenue. This feature is included to encourage bidders to participate in the auction without fear of winning a set of lanes in which little volume is realized by guaranteeing a winning carrier’s minimum desired volume for her specified set of lanes. If the shipper experiences volume on allocated lanes that is greater than the upper bounds specified by winning carriers, then the shipper must procure additional third-party services that is assumed to be more costly in general than by procuring through the combinatorial auction. The stochastic program aims to find a set of carriers at the first stage so that the expected costs of paying winning carriers and using additional third-party carriers or paying carriers to ship below their minimum volume requirements are minimized subject to meeting actual volume realizations under all specified scenarios.

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Furthermore, shippers will often wish to guarantee that if a carrier is awarded any business, then it has to be for a certain minimum threshold amount, which is typically used to ensure that a carrier wins enough business to support a pool operation see Caplice and Sheffi (2003). These amounts are called ‘‘threshold volumes” and these type of constraints are ‘‘if-then constraints”. In our model, the inclusion of the range of shipment volumes submitted by each carrier completely captures this basic idea: the upper bound of the range matches the maximum number of trucks committed to the auctioneer; and the lower bound of the range matches the threshold volume. In addition, we include other practical side constraints and features for the shipper such as ensuring that the number of winning carriers is within a certain minimum and maximum range, ensuring minimum and maximum override constraints. Also, the objective function is formulated so that incumbent carriers can be favored based on the setting of an appropriate parameter. The main objective of the paper is to demonstrate that stochastic integer programming can be a relevant and useful modeling framework for winner determination in truckload combinatorial auctions where shipment uncertainty is an important factor along with including a comprehensive set of business side constraints. Stochastic integer programming is a very challenging class of problems. However, we find that branch and bound solvers such as CPLEX can handle reasonable sized instances. Very large instances will require specialized algorithmic approaches and we do not pursue these directions in this paper and instead focus on stochastic programming as a viable modeling framework for winner determination in environments where uncertainty is an issue. The rest of the paper is organized as follows. The paper briefly reviews two-stage stochastic programming in Section 2. Section 3 introduces the two-stage stochastic winner determination model. Section 4 presents a deterministic version of the stochastic model with similar constraints and business considerations. Section 5 introduces the benchmarking method used to measure the performance of the stochastic model and demonstrates the value of the stochastic solution. An example is provided in Section 6 to illustrate the stochastic model. The computational results are given in Section 7. Section 8 concludes the paper. 2. Brief review of stochastic programming In this section, we briefly review two-stage stochastic programming with recourse. Stochastic programs are mathematical programs where some problem data are uncertain see Kall and Wallace (1995) and Birge and Louveaux (1997). The classical two-stage stochastic program with fixed recourse is the problem of finding the optimal solution of:

min z ¼ C T x þ En ½min qðxÞT yðxÞ s:t: Ax ¼ b;

ð1aÞ ð1bÞ

TðxÞx þ WyðxÞ ¼ hðxÞ

ð1cÞ

x P 0; yðxÞ P 0

ð1dÞ

The sets of decisions can be partitioned into two categories  A set of decisions, x, is taken before uncertainty is resolved. The time period when these decisions are taken is at the start of the first stage and henceforth the decisions are referred to as first-stage decisions or ‘‘here and now” decisions.  A set of decisions, y(x), can be taken after uncertainty is resolved. These decisions will be associated with the random outcomes x and can be considered to be corrective or recourse decisions. These decisions are called the second-stage decisions and occur at the end of the first-stage (or equivalently at the start of the second stage). In this paper, it is assumed that random outcomes are modeled according to a finite probability distribution. The objective function (1a) is composed of two terms the first of which is the immediate cost for first-stage decisions and the second term is the expected cost of recourse decisions. Constraints (1b) pertain to first-stage decisions only and constraints (1c) represent the recourse constraints (i.e., constraints that pertain to corrective decisions (second-stage decisions). (1d) represents non-negativity requirements on all decisions see Kall and Wallace (1995) and Birge and Louveaux (1997). In combinatorial auctions for TL procurements, most lanes for sale specify volume as an estimated quantity, and the auctioneer has to determine the winners (i.e., assign lanes to carriers) before the actual volume shipments are realized or known. Thus, the assignments of lanes to carriers may not be optimal after volume uncertainty has been resolved, in particular,  a carrier that wins a set of lanes may be asked to ship volumes on the lanes that are far less than expected and may not receive as much revenue as originally planned;  or, volume may be far greater than the expected amount such that an assigned carrier may not have sufficient truckload capacity to serve. In either case, uncertainty can have a detrimental effect and the auctioneer needs to make some corrections after the uncertainty is resolved, i.e., the shipments are known. We develop a WDP incorporating shipment volume uncertainty for TL procurement by using a two-stage stochastic programming modeling approach. The first-stage decisions,

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i.e., the vector x, correspond to assigning carriers to lanes based on submitted packages before uncertainty in shipment volumes has been resolved and the second-stage decisions, i.e., y(x) or (y(x, x)), correspond to assigning actual volume to the carriers after uncertainty of shipment volume has been realized. In particular, the vector x is a binary decision vector that indicates which carriers are to be allocated lanes. In the second stage, the auctioneer needs to allocate integral shipment volume y (x) or (y (x, x)) (in number of TLs) to the carriers that were awarded lanes in the first stage. These decisions take place after volume uncertainty is resolved. The aim of the stochastic model is to determine a set of carriers that will be assigned lanes so that they provide the least expected cost of shipping the realized volume. The two-stage decision dynamics is compatible with common practice in TL transportation procurement where the shippers decide the winners in the auction first, and then tender actual shipment volume to the winning carriers during future daily execution of contract see Caplice and Sheffi (2003).

3. The stochastic WDP model In this section, we provide the necessary definitions, notation and assumptions for the two-stage stochastic programming WDP model for TL procurement. In practice, the shipper may need to consider important side business constraints in the TL transportation procurement winner determination problem, for example, constraints that ensure that the number of winning carriers is between a specified minimum and maximum number of carriers in the final allocation, constraints that ensure to award the minimum and maximum total override amount to certain carriers, constraints that limit certain carriers to win at least or at most certain portions of the total business in the network. The auctioneer may also include other business considerations, for instance, the favoring of incumbents, and performance factors such as on-time percentage, claims performance, refusal rate, Electronic Data Interchange (EDI). These side business considerations are common in practice see (Moore et al., 1991; Caplice and Sheffi, 2003). Our proposed two-stage stochastic programming winner determination model will incorporate all of these side constraints and additional business considerations. Notation, definitions and assumptions: 1. The basic unit of interest is called a lane, which is defined as an origin–destination pair and the volume that is to be shipped from the origin to destination. 2. There is a discrete probability distribution function p(s) of the uncertain shipment volume, s is the index of the scenarios for all possible shipments, s = 1, 2, 3,. . ., S. 3. The auctioneer (shipper) allocates at most one carrier to each lane. In practice, this could be modified by assigning more than one carrier to each lane by assigning to carriers a number of loads on each lane. In this paper, we only consider the case of assigning at most a single carrier to each lane for simplification. 4. Each bidder (carrier) is allowed to submit XOR bids. 5. Each bid may include multiple units of the same unit package, and each unit package may include a set of lanes with one truckload of flow movement on each lane in the set. 6. aijk is equal to one if one truckload of flow movement on lane i is in the unit package k from bidder j, otherwise 0. 7. Kj is a set of lane combinations. 8. Ljk is the minimum units of packages k that the bidder j is willing to accept. 9. Ujk is the maximum units of packages k that the bidder j is willing to or able to accept. 10. Nmin and Nmax are the minimum and maximum number of carriers in the final allocation. 11. qj is the override minimum total award (allocation of lanes), or the minimum coverage, for carrier j. To avoid trivial cases, we should assume qj 6 Ujk, for all j, k. 12. Qj is the override maximum total award (allocation of lanes), or the maximum coverage, for carrier j. 13. rjk is the bidding price for one unit package k by bidder j, or, the revenue/award for bidder j to serve one unit package k if bidder j wins. 14. bj is the favoring or penalty factor the auctioneer applies to carrier j based on the carrier’s performance and service quality in the past (1 6 bj 6 1). The auctioneer may also include the favoring of incumbents in this factor. For example, an incumbent carrier with good performance has a negative factor. i 15. ds is the realized flow movement on lane i (in terms of number of truckloads) when the scenario s occurs. 16. ci is the cost for the auctioneer to satisfy one truckload of flow movement on lane i by the in-house fleet or by negotiating with other 3rd party carriers. Since the stochastic programming model will have two stages where the auctioneer will first determine which carriers will win lanes at the start of the first stage and then allocate shipment volume after volume uncertainty has been resolved, we introduce a new bid structure for carriers to give the flexibility of expressing volume requirement ranges it is willing to serve (as opposed to giving a single volume request that holds constant throughout). A bid k submitted by bidder j is expressed by {rjk, Kj, [Ljk, Ujk]}. For example, if a bidder submits one bid to the auctioneer as {$15, {A, B}, [1, 3]} this means the bidder is willing to serve a package, consisting of lanes A and B, at the price of $15 for each truckload of movement through both lanes, and the bidder is also willing to ship at least 1 truckload and at most three truckloads of movement

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on all lanes in this package, say, to serve either {A, B} at price $15, or {2A, 2B} at price $30, or {3A, 3B} at price $45 respectively. It is noted that a particular volume level for a carrier for a certain package of lanes is the volume for all lanes in the package. It is reasonable and practical to assume that each carrier is willing to submit, or release the range from the minimum to the maximum volume the carrier is willing to serve on a specified set of lanes. For example, in auctions held by Reynolds Metals Company, each carrier submitted the maximum numbers of trucks committed to Reynolds and to different locations, which are decided by the carrier’s transportation capacity and the existing commitments to other shippers see Moore et al. (1991). We assume that each carrier has determined the economies of scope for a set of lanes and as part of this evaluation can determine the range of volume to ship over the lanes that would profitable. This does not preclude that different lanes can have different volumes shipped for the shipper, but rather that the carrier associates a single capacity level (in terms of TLs) with a package of lanes that could cover the realized volumes of the lanes in the package. If a lane has a realized shipment volume greater than the capacity for a winning carrier for that lane, the shipper must procure additional for-hire services to cover the remaining shipment volume for that lane. Without loss of generality, we assume it is more expensive for the auctioneer to satisfy any demands by the in-house fleet or by negotiating with other for-hire carriers. Thus, it is implicit that the auctioneer prefers to satisfy demands by procuring TL transportation services from the bidders. Furthermore, we assume there are unlimited capacities from other for-hire carriers. Decision variables: 1. ysjk is the integer number of truckloads on all lanes in packages k the auctioneer allocates to bidder j when the scenario s occurs. 2. wsi is the integer number of truckloads on lane i that the auctioneer has to satisfy by her in-house fleet or by negotiating with other 3rd party carriers who did not participate in the auction when the scenario s occurs. The mathematical model for the stochastic winner determination problem (sWDP) is given by:

ðsWDPÞ

X

min

XX X pðsÞ  ð1 þ bj Þ  r jk  ysjk þ ci  wsi

s

XX

s:t:

j

j

! ð2aÞ

i

k

aijk  ysjk þ wsi P dsi ; 8i; s

ð2bÞ

k

Ljk  xjk 6 ysjk ;

8j; k; s

U jk  xjk P ysjk ; 8j; k; s X xjk 6 1; 8i

ð2cÞ ð2dÞ ð2eÞ

k

XX j

k

j

k

j

k

XX XX X

aijk  xjk 6 1; 8i

ð2fÞ

xjk P Nmin ;

ð2gÞ

xjk 6 Nmax ;

ð2hÞ

ysjk P qj ;

8j; s

ð2iÞ

ysjk 6 Q j ;

8j; s

ð2jÞ

k

X k

xjk 2 f0; 1g;

ysjk ;

wsi P 0 and integer; 8i; j; k; s

ð2kÞ

The objective function is to minimize the total expected purchase cost for the auctioneer, including the favoring of incumbents and performance factors, bj. The constraints (2b) ensure that the demands can always be satisfied under each scenario. The constraints (2c) and (2d) make sure that the quantity allocated to each bidder falls in the interval between the bidder’s minimum and maximum supply quantities that the bidder is willing to serve. The constraints (2c) are unique and important, which guarantee each winner wins at least her minimum desired volume no matter which scenario finally realizes. The guarantee can keep each winner always operationally profitable, and there are two purposes: first, it can attract more carriers to participate in this type of auctions; second, it can be viewed as a certain compensation for requesting carriers to reveal more information about their own capacities and profitable business volume levels in this type of auctions. The constraints (2e) are associated with the XOR bids since at most only one bid can be in the final allocation. The constraints (2f) guarantee that at most one winner can be allocated to each lane. The constraints (2g) and (2h) make sure the total number of winning carriers within the minimum and maximum limit for the entire network. The constraints (2i) and (2j) specify the minimum volume and the maximum of shipments that a carrier must be allocated. In this model, xjk is the first-stage decision; ysjk and wsi are the second-stage decision. We should also emphasize that although there is no deterministic cost CT x associated with the first-stage decision in this model, i.e., CT x = 0, the first-stage

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decision plays a significant part in determining the expected purchase cost. Also, wsi is an integral decision variable in this model and in the presence of the other integer decisions in the model this variable becomes integer regardless if the integer restriction is imposed or not. Thus, wsi can be relaxed as linear. 4. Deterministic WDP model In this section, we present a deterministic WDP model in order to introduce a benchmark for the comparison with sWDP in the next section. The deterministic WDP is similar to sWDP but uses single volume point estimates which are often just averages of possible volume realizations. The deterministic model does not allow for uncertainty in volume. All notation, definitions and decision variables we use for the deterministic model are the same as those defined in the previous section except the following. Notation and definition: 1. di be the expected volume movement on lane i (in terms of number of truckloads). Decision variables: 1. yjk is the integer number of truckloads on all lanes in packages k the auctioneer allocates to bidder j. 2. wi is the integer number of truckloads on lane i that the auctioneer has to satisfy by her in-house fleet or by negotiating with other 3rd party carriers. The mathematical model for the deterministic winner determination problem (dWDP) is given by:

ðdWDPÞ

XX

min

j

k

j

k

XX

s:t:

ð1 þ bi Þ  r jk  yjk þ

X

ci  wi

ð3aÞ

i

aijk  yjk þ wi P di ; 8i

Ljk  xjk 6 yjk ;

8j; k

U jk  xjk P yjk ; 8j; k X xjk 6 1; 8j

ð3bÞ ð3cÞ ð3dÞ ð3eÞ

k

XX j

k

j

k

j

k

XX XX X

aijk  xjk 6 1; 8i

ð3fÞ

xjk  xjk P Nmin ;

ð3gÞ

xjk  xjk 6 Nmax;

ð3hÞ

yjk P qj ;

8j

ð3iÞ

yjk P Q j ;

8j

ð3jÞ

k

X k

X jk ; zj 2 f0; 1g; yjk ; wi P 0 and integer; 8i; j; k

ð3kÞ

The objective function is to minimize the total expected purchase cost for the auctioneer. The constraints (3b) ensure that the auctioneer’s expected demands can be satisfied. All other constraints are similar to those in the sWDP model. 5. Benchmark problem pWDP and the value of the stochastic solution In this section, we introduce the benchmarking procedure to measure the performance of the sWDP and to show the value of the stochastic solution see Birge (1982). The benchmarking strategy is to form a deterministic version of a stochastic model and use the optimal solution corresponding to first-stage decisions in sWDP. The idea is to compare the robustness of the first-stage decisions generated by the deterministic model versus the optimal first-stage decisions from solving the stochastic model (sWDP) see Birge (1982) for more details. The benchmark model is generated as follows: 1. Solve the dWDP, and get the optimal solution for the deterministic model. 2. Use the optimal values of xd from dWDP and substitute in sWDP, and obtain a new problem which will then include only the second-stage decisions from sWDP as variables. We refer to this problem as Plug-in WDP or pWDP.

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The constraints (2e), (2f), (2g), (2h) will not appear in the pWDP model, and the constraints (2c) and (2d) are replaced by the following constraints (4c) and (4d) respectively. The mathematical model for the plug-in stochastic winner determination problem (pWDP) is given by:

ðpWDPÞ

X

min

XX

pðsÞ 

s

XX

s:t:

j

j

ð1 þ bi Þ  r jk  ysjk þ

X

! ci  wsi

ð4aÞ

i

k

aijk  ysjk þ wsi P dsi ; 8i; s

ð4bÞ

k

Y sjk P Ljk ;

8j ; k; s : xjk ¼ 1;

ð4cÞ

6 U jk ; 8j ; k; s : xjk ¼ 1; X ysjk P qi ; 8j ; s

ð4dÞ

Y sjk

k X

ysjk P Q i ; j; s k ysjk ; wsi P 0 and integer;

ð4eÞ

8

ð4fÞ

8i; j; k; s

ð4gÞ

Next, we show the value of the stochastic solution from sWDP i.e., that the first-stage decisions generated by sWDP dominate the first-stage decisions from pWDP. First, we introduce the following lemma first. Lemma. pWDP is always feasible whenever sWDP is feasible and qj 6 U jk . Proof. We know there is at most one k such that ysjk can be positive for each pair of j and s and must be the one with xjk = 1. Then the constraints (4e) can be rewritten as ysjk P qj , for all j, k, s, although they are not as tight as the constraints (4e). Similarly, the constraints (4f) can be rewritten as ysjk P Q j , for all j, k, s, although they are not as tight as the constraints (4f). Combining two sets of the constraints (4c) and (4e) and the assumption that qj 6 Ujk, for all j, k, it is straightforward to conclude that the constraints (4c), (4e), and (4f) will be satisfied if we set

ysjk ¼ maxfLjk;qj g;

8j; k; s : xjk ¼ 1

ð5aÞ

Due to the assumption that there are unlimited capacities of other 3rd party carriers, or, 0 6 wi 6 +1, for all i, we know the constraints (4b) can be satisfied for any i under any realization of scenario s, i.e., by letting s

wsi ¼ di 

XX j

aijk  ysjk

ð5bÞ

k

It is clear that the combination of wsi and ysjk defined by (5a) and (5b) is a feasible solution for the pWDP. QED. Let EðC p Þ be the optimal total cost for the auctioneer offered by the pWDP, and EðC s Þ be the optimal total cost offered by the sWDP, then we have the following results which gives the value of the stochastic solution. h Theorem 1. EðC p Þ P EðC p Þ holds. Proof. By the lemma pWDP is feasible and thus feasible for sWDP and so the result follows. h

6. Illustration In this section, we illustrate the sWDP model. In this example, there are four bidders, and each bidder submits up to three bids at each round. There are shipments on three lanes for bid, say: A, B and C, and three different shipment scenarios (see Table 1): low, normal, and high with the probabilities as follows (see Table 2): The favoring/penalty factors for carriers are given in Table 3. In this example, Nmin = 1 and Nmax = 3. For simplification, we set qj = 0, Qj = 6, for all j. The details of bids from each bidder are given by Table 4. For example, bidder 1 submits three bids where the first bid is on lanes {A, B}, and the bidder is willing to ship a volume of 2–4 units over lanes {A, B} for a revenue of 13 per unit volume shipped.

Table 1 The demands for scenarios. Lane

Low demand

Normal demand

High demand

A B C

1 2 1

2 3 4

5 6 5

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Z. Ma et al. / Transportation Research Part E 46 (2010) 49–60

Table 2 The probabilities for demand scenarios. Scenario

Low demand

Normal demand

High demand

Probability

0.4

0.5

0.1

Table 3 The favoring/penalty factors. Bidder

1

2

3

4

Favoring/penalty factor

0.0

0.02

0.01

0.0

Table 4 The detail of all bids. Bidder

Bid No.

Package

Range of volume on packages

Unit revenue

j=l

k=1 k=2 k=3 k=1 k=2 k=3 k=1 k=2 k=3 k=1 k=2 k=3

A+B B+C B B A+B A+C C A+C A A+C B B+C

(2, (1, (2, (1, (2, (2, (2, (1, (2, (1, (1, (1,

13.0 20.0 10.0 9.0 14.0 16.0 11.0 14.0 5.0 15.0 8.0 18.0

j=2

j=3

j=4

4) 5) 4) 5) 4) 4) 4) 5) 4) 5) 5) 3)

Table 5 The unit costs for the auctioneer to move shipment by other means. Lane A

Lane B

Lane C

9.0

17.5

12.5

The costs ci for the auctioneer to satisfy one truckload of shipment by her in-house capacity or by negotiating with other 3rd party suppliers are given in Table 5. For the deterministic model, if the realized demands are more than the expected demands, the auctioneer has to purchase more transportation services by holding another auction or negotiating with other 3rd party carriers, or satisfy more demands by her own in-house capacities. It is the most convenient and most likely for the auctioneer to obtain the best results in terms of total costs minimizing when the unallocated bids from all bidders are still valid, or the auctioneer can still procure more transportation services from those bidders within their remaining capacities. Due to the assumption that it is more expensive for the auctioneer to satisfy any shipment demands by her in-house capacity or by negotiating with other 3rd party carriers, the auctioneer procures as many transportation services as possible from the bidders, and then the auctioneer has to move the remaining shipments by her own in-house capacities or negotiating with other 3rd Party carriers when the high demands are realized. We use the deterministic model with the above strategy when we analyze the performance of sWDP in the next section.

Table 6 The allocations and corrections for sWDP and pWDP. Model

1st stage decision

2nd stage correction

Total exp. cost ($)

Low

Normal

High

sWDP

x32 = x42 = 1

y132 ¼ 1; y142 ¼ 2 with cost $29.86

y232 ¼ 2; y242 ¼ 3; w23 ¼ 2 with cost $76.72

y332 ¼ 5; y242 ¼ 5 w32 ¼ 1 with cost $125.0

62.98

pWDP

x33 = x43 = 1

y133 ¼ 2; y143 ¼ 1; w12 ¼ 1 with cost $45.40

y233 ¼ 2; y243 ¼ 3; w23 ¼ 1 with cost $76.4

y33 = 4, y43 = 3 w13 ¼ 1; w32 ¼ 3; w33 ¼ 2 with $160.3

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Fig. 1. Allocation of pWDP and actual shipments under different scenarios.

Fig. 2. Allocation of sWDP and actual shipments under different scenarios.

From Table 6, the stochastic model is better (also see Figs. 1 and 2), and the computational results also support the property in Theorem 1. 7. Computational study In this section, we present the computational results in solving instances of sWDP, dWDP, and pWDP. All problem instances were solved to optimality by a commercial optimization software AMPL/CPLEX 9.0 on a personal computer with a Pentium 4 3.4 GHz CPU. We randomly generate all data for each problem instance tested. We recognize that different data may have different distribution, but our primary goal is to show that it is possible to solve the problem instances similar to the realistic large-size instances as reported in the literature e.g. see Caplice and Sheffi (2003). For simplification, we assume all data are uniformly distributed except the probabilities for all different shipment scenarios. We assume that the probabilities are normally distributed for all different shipment scenarios. 1. In generating scenarios, we allow a combination of different possibilities for each lane in the shipper’s network. For instance, suppose there are three lanes, and three possible shipments on each lane, e.g. {H (igh), N (ormal), and L (ower)}. If the shipments on all three lanes are with the same probabilities for each scenario, then the shipper can set three scenarios for the problem instance; otherwise, the shipper may set up to nine scenarios for the problem instance, for example, {H, H, H}, {H, H, N}, {H, H, L}, {H, N, H}, and so forth. 2. We assume that each carrier submits the same number of packages: the maximum of packages the auctioneer allowed for each bidder to submit, which is reasonable for large-size problem instances. 3. We assume that the auctioneer sets an upper limit on the number lanes contained in a bid. This enables the auctioneer to control distribution of allocation so that only a few carriers are winning most of the lanes. 4. We assume that the shipping cost ci for the shipper to move one unit of shipment on lane i is uniformly distributed in the range of [1, 5] if the auctioneer has to move the unit of shipment by herself or other 3rd logistics parties (non-bidders). 5. The possible shipment on a lane is uniformly distributed in the range of [1, 20]. 6. Each bidder’s shipping cost of one unit of shipment on lane i is uniformly distributed in the range of [1.5, 2]  ci. 7. The favoring/penalty factors bj are uniformly distributed in the range of [0.05, 0.05].

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Table 7 Computational results for dWDP, pWDP, and sWDP. Example

60-30-3-10-20 60-30-3-12-50 100-20-3-10-20 100-20-5-10-20 100-20-8-10-20 100-20-10-10-20 100-20-15-10-20 100-20-20-10-20 100-20-40-10-20 150-30-3-10-20 150-30-5-10-20 150-30-8-10-20 150-30-10-10-20 150-30-15-10-20 200-30-3-10-20 200-30-5-10-30 300-10-3-10-30 300-10-S-10-30 400-40-3-10-30 400-40-5-10-30 600-50-3-10-30

dWDP

pWDP

sWDP

Opt (%)

CPU (s)

Opt (%)

CPU (5)

Opt (%)

CPU (s)

119.84 126.09 112.0 112.44 109.43 109.30 107.72 104.48 97.79 116.48 114.46 111.98 109.09 106.12 109.11 113.65 118.63 111.34 114.64 110.80 111.93

9.140 1.313 24.562 14.766 15.937 15.719 4.125 10.11 7.422 8.047 25.297 14.484 17.53 48.812 46.765 17.328 19.0 75.156 148.828 47.360 64.750

107.15 106.44 103.51 102.90 102.75 105.11 101.78 102.04 102.18 101.05 100.56 101.45 101.34 100.52 105.19 100.59 100.80 100.59 100.11 100.80 100.10

0.031 <0.00l <0.001 <0.001 0.015 0.016 0.015 <0.001 0.016 <0.001 0.016 <0.001 0.015 <0.001 <0.00l <0.00l <0.00l <0.001 <0.001 <0.001 <0.001

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

72.531 7.172 331.313 831.094 1530.73 3532.97 1041.36 7804.l2 25383.2 58.719 592.017 2900.95 3660.06 68635.50 449.703 304.672 247.25 7157.97 4047.34 3157.34 7336.66

8. The minimum override volumes qj are uniformly distributed in the range of [0, 5]. 9. The maximum override volumes Qj are uniformly distributed in the range of qj+ [1, 10]. The computational results can be summarized as follows: 1. AMPL/CPLEX 9.0 can optimally solve instances of sWDP (as well as dWDP) with up to 600 lanes, 50 bidders in reasonable time. An ‘‘example” or instance is characterized by five parameters written as (total number of lanes for sale in auction)(total number of bidders)-(number of scenarios)-(number of packages submitted by each bidder)-(upper limit on number of lanes in any package for a bidder) e.g. ‘‘60-30-3-10-20” means an instance of sWDP with 60 lanes for sale, 30 bidders, 3 scenarios, 10 packages per bidder, and up to 20 lanes in any bidder’s package. Computation times for sWDP ranged from 7.172 s to 19 h. dWDP instances took from 1.131 s to 148.828 s. Corresponding instances of pWDP required no more than one second as these are just linear programming problems. The results are summarized in Table 7. The Opt values represent the equivalent percent of the corresponding ratio of the optimal objective function value for each problem instance type over the optimal objective function value of the corresponding sWDP instance. So an OPT value for a dWDP instance is the ratio of the optimal objective function value of the dWDP instance over the optimal objective function value of the corresponding sWDP instance. It is noted that in one case the dWDP instance had a ratio of less than one indicating that the deterministic problem had an objective function value less than the corresponding sWDP instance. This can happen when the scenario probabilities and realizations are such that decisions in the sWDP model induces decisions more heavily in recourse decisions with higher weighted probabilities/costs. In general, depending on the nature of scenarios and their probabilities of occurrence the dWDP cost can be greater than or less than the cost of the associated sWDP instance. 2. The value of the stochastic solution for each instance is given in Table 8 and is simply the difference in percent between the objective values of dWDP and sWDP. The values ranged from 7.15% to 0.40%. 3. We have observed that increasing the number of scenarios causes significant increases in the computational times. This is to be expected as each scenario added essentially increases the size of the problem by dimensions of the corresponding deterministic version. For moderate sized problems it is possible to include up to 40 scenarios and still have an optimal solution in well under an hour. 4. If an auctioneer has a large quantity of shipments on a large-sized shipment network involving many lanes and bidders, the auctioneer should then limit the number of scenarios to a low level e.g. five scenarios. In general, having at least three scenarios that represent average, above average, and below average possibilities would be a reasonable minimal requirement 5. It should be noted that sWDP is a comprehensive model including many additional side constraints in addition to incorporating uncertainty. Some of these side constraints may be unnecessary for a particular auction and so additional speedups can result by omitting the unnecessary constraints. 6. In summary, it is generally found that optimization-based procurement methods such as combinatorial-based auctions are useful in environments where shippers require more than 12 lanes to be serviced and involve more than 20 carriers see Caplice and Sheffi (2003). The results we present indicate that the sWDP instances can be solved optimally with far

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Z. Ma et al. / Transportation Research Part E 46 (2010) 49–60 Table 8 Value of the stochastic solution. Example

Value of the stochastic solution (%)

60-30-3-10-20 60-30-3-12-50 100-20-3-10-20 100-20-5-10-20 100-20-8-10-20 100-20-10-10-20 100-20-20-10-20 100-20-40-10-20 150-30-3-10-20 150-30-5-10-20 150-30-8-10-20 150-30-10-10-20 150-30-15-10-20 200-30-3-10-30 200-30-5-10-30 300-40-3-10-30 300-40-5-10-30 400-40-3-10-30 400-40-5-10-30 600-50-3-10-30

7.15 6.44 3.51 2.90 2.75 5.11 2.04 2.18 1.05 0.56 1.45 1.34 0.52 5.49 0.59 0.80 0.59 0.41 0.80 0.40

more lanes than 12 with up to 50 carriers in the context of many important side constraints. Although, it is clear that for very large instances an off the shelf branch and bound solver would be inadequate. Thus, sWDP could be used effectively with a commercial solver for moderate sized instances that yet are of the size range for which optimization-based procurement would be beneficial. If a very large number of scenarios (in addition to large number of lanes and bidders) are deemed necessary then decomposition methods must be developed e.g. dual decomposition see Caroe and Schultz (1999), or more recent methods based on local branching based acceleration of decomposition methods see Rei et al. (2009). 8. Conclusion and future research We have developed a two-stage stochastic integer programming model for the winner determination problem in combinatorial auctions for TL procurement that incorporates shipment volume uncertainty and that includes many important business side constraints. We have shown the value of the stochastic solution over the deterministic version of the problem. Moderately sized instances of the stochastic winner determination can be solved by commercial solvers in reasonable time. The contribution of the paper to the literature is the incorporation of uncertainty in TL combinatorial auctions through the use of stochastic programming. In particular, to the best of our knowledge this work represents one of the first attempts to model uncertainty in combinatorial auctions and the first to use stochastic programming in combinatorial auctions. Important future research directions include developing decomposition methods for the stochastic model in order to handle largescale instances and of methods to generate problem instances that are reflective of real world TL procurement auctions. Also, in the current model, volume-based discount is not allowed. However, in TL transportation practice, it is common. Including the volume-based discount to the bid structure will lead to a non-linear integer model, i.e., a step function associated with the volume-based discount. This could also be promising extension for future research. References Abrache, J., Bourbeau, B., Crainic, T.G., Gendreau, M., 2003. A new bidding framework for combinatorial E-auctions. Computers and Operations Research 31, 1177–1803. Abrache, J., Crainic, T.G., Gendreau, M., Rekik, M., 2007. Combinatorial auctions. Annals of Operations Research 153, 131–164. Andersson, A., Tenhunen, M., Ygge, F., 2000. Integer programming for combinatorial auction winner determination. In: Proceedings of the Fourth International Conference on Multi-Agent Systems, pp. 39–46. Birge, J.R., 1982. The value of the stochastic solution in stochastic linear programs, with fixed recourse. Mathematical Programming 24, 314–325. Birge, J.R., Louveaux, F., 1997. Introduction to Stochastic Programming. Springer. Boutilier, C., Hoos, H.H., 2001. Bidding languages for combinatorial auctions. In: Proceedings of the 17th International Joint Conference on Artificial Intelligence (IJCAI-01), pp. 1211–1217. Bykowsky, M., Cull, R., Ledyard, J., 1995. Mutually destructive bidding: the FCC auction design problem. Social science working paper 916, California Institute of Technology. Caplice, C., Sheffi, Y., 2003. Optimization-based procurement for transportation services. Journal of Business Logistics 24 (2), 109–128. Caroe, C.C., Schultz, R., 1999. Dual decomposition of stochastic integer programming. Operations Research Letters 24 (1), 37–45. Elmaghraby, W., Keskinocak, P., 2003. Combinatorial auctions in procurement. In: Billington, C., Harrison, T., Lee, H., Neale, J. (Eds.), The Practice of Supply Chain Management. Kwler Academic publishers. Gonen, R., Lehmann, D., 2000. Optimal solutions for multi-unit combinatorial auctions: branch-and-bound heuristics. In: Proceedings of the ACM Conference on Electronic Commerce, EC 2000, Minneapolis, MN, pp. 13–20. Kall, P., Wallace, S.W., 1995. Stochastic Programming. Wiley John & Sons.

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