Vol. 23, No. 1, January 2014, pp. 138–159 ISSN 1059-1478|EISSN 1937-5956|14|2301|0138

DOI 10.1111/poms.12029 © 2013 Production and Operations Management Society

Dedicated Transportation Subnetworks: Design, Analysis, and Insights Tharanga Rajapakshe Warrington College of Business Administration, University of Florida, Gainesville, Florida 32611,, USA, [email protected]

Milind Dawande Naveen Jindal School of Management, University of Texas at Dallas, Richardson, Texas 75080-3021,, USA, [email protected]

Srinagesh Gavirneni Johnson Graduate School of Management, Cornell University, Ithaca, New York 14853,, USA, [email protected]

Chelliah Sriskandarajah Mays Business School, Texas A&M University, College Station, Texas 77843,, USA, [email protected]

P. Rao Panchalavarapu Schneider Logistics Inc., 3101 South Packerland Drive, Green Bay, Wisconsin 54313-6187,, USA, [email protected]

dedicated subnetwork (DSN) refers to a subset of lanes, with associated loads, in a shipper’s transportation network, for which resources—trucks, drivers, and other equipment—are exclusively assigned to accomplish shipping requirements. The resources assigned to a DSN are not shared with the rest of the shipper’s network. Thus, a DSN is an autonomously operated subnetwork and, hence, can be subcontracted. We address a novel problem of extracting a DSN for outsourcing to one or more subcontractors, with the objective of maximizing the shipper’s savings. In their pure form, the defining conditions of a DSN are often too restrictive to enable the extraction of a sizable subnetwork. We consider two notions—deadheading and lane-sharing—that aid in improving the size of the DSN. We show that all the optimization problems involved are both strongly NP-hard and APX-hard, and demonstrate several polynomially solvable special cases arising from topological properties of the network and parametric relationships. Next, we develop a network-flow-based heuristic that provides near-optimal solutions to practical instances in reasonable time. Finally, using a test bed based on data obtained from a national 3PL company, we demonstrate the substantial monetary impact of subcontracting a DSN and offer useful managerial insights.

A

Key words: transportation network; dedicated subnetwork; deadheading; lane-sharing; heuristics History: Received: April 2012; Accepted: October 2012 by Michael Pinedo, after 1 revision.

freight transportation. Although railroads maintained their dominance until the early 1950s, improvements in productivity following the beginning of the “interstate" era resulted in a rapid growth in the trucking industry. Customers preferred trucking over railroads due to its flexibility and ability to customize services. Nevertheless, the trucking industry remained highly regulated until the early 1980s, with only a handful of national carriers. Thus, despite significant improvements in productivity, trucking companies priced their services considerably higher relative to railroads (Cambridge Systematics and Reebie Associates 2009). The deregulation of the trucking industry in the United States through the “Motor Carriers Act" of 1980 was a significant event that eliminated entry barriers and promoted the competitive pricing of

1. Introduction Dedicated networks operate semi-autonomously within the Schneider One-way truckload network. Using both third-party and internally developed optimization software, Engineering and Research sizes and designs these sub-networks to minimize operating costs, maximize synergy with the broader network, and reduce the impact of freight surges and flow imbalance. from The Schneider Enterprise, Ted Gifford, OR/MS Today, December 2007. Since the mid-1930s, the trucking industry in the United States has competed with the railroads for 138

Rajapakshe, Dawande, Gavirneni, Sriskandarajah, and Panchalavarapu: Dedicated Transportation Subnetworks Production and Operations Management 23(1), pp. 138–159, © 2013 Production and Operations Management Society

services. The existing carriers expanded their services, small trucking companies entered the growing market, and companies were allowed to publish their rates publicly. Consequently, competitive pressures forced companies to be efficient and productive to achieve profitability, resulting in comparatively low and flexible pricing along with customized service arrangements (Supply Chain Digest 2006). According to the American Transportation Research Institute (ATRI 2008), the trucking industry currently accounts for over 80% of revenue and 69% of tonnage of the freight-carrier market. A significant characteristic of the trucking industry is the high percentage of minor players (i.e., local 3PL providers) as compared with major operators (i.e., national 3PL providers). As reported by the ATRI, the number of interstate carriers grew from less than 20,000 in 1980 to more than 290,000 in 2006. In addition, there are about 504,000 private fleets and approximately 235,000 other interstate motor carriers registered with the US Department of Transportation. Although this is an overwhelmingly large number, about 97% of these operate with less than 20 trucks and about 89% with fewer than six trucks (ATRI 2008). It, therefore, comes as no surprise that the trucking industry is fiercely competitive and operates on thin profit margins. In the United States, more than 75% of the Fortune 500 companies use 3PLs to satisfy their logistics requirements (Armstrong Associates 2009a), while the global Fortune 500 3PL market has grown consistently over the recent past and was valued around US$199.7 billion in 2008 (Armstrong Associates, 2009b). In the United States, the trucking industry alone was responsible for about US$680 billion of the total estimated transportation cost of US $872 billion in FY 2008 (Burnson 2009). The large number of minor operators—and the resulting competitive pressures—leaves vulnerable those major operators with significantly owned assets (Baumann 2006). This is especially true during tough economic times (Field 2009). AAI predicts FY 2009 to be “the first recorded negative year in 3PL gross revenue growth" since 1996. According to Tom Sanderson, CEO of Transplace (a national 3PL provider), the 3PL business that is struggling somewhat is the owned fleets (Berman 2008). On the positive side, however, this structure of the industry presents an interesting business opportunity for the national trucking companies, namely, that of collaboration with the local players (Ozener 2008). In other words, national companies could subcontract a portion of their demand to local companies. Such a collaboration will not only benefit the local companies by reducing their market risk but also provide room for the national 3PL providers to shed their assets and concentrate more on investing resources in exploring new business opportunities.

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Industry experts believe that such a collaboration is indeed feasible: national 3PL providers cater to a large customer base and can, therefore, easily persuade the local trucking companies to work hand-in-hand to reap the benefits from the highly demanding and rapidly expanding market (Caplice and Sheffi 2003, Ergun et al. 2007). In light of these observations, the main goal of this paper is to study the problem of identifying a subnetwork in the network of a national 3PL provider (referred to henceforth as the shipper’s network) that is appropriate for such a collaboration. The identified subnetwork could then be operated independently— and, hence, referred to as a dedicated subnetwork—by one or more local trucking companies (Panchalavarapu 2010). We model the problem from the perspective of the national 3PL provider and measure the total cost saving that could be derived by subcontracting the identified subnetwork. Next, we develop the notion of a dedicated subnetwork more formally. 1.1. Dedicated Subnetworks A typical shipper’s network is defined by the set of origin–destination pairs, referred to as lanes, and the number of truck-loads (hereafter referred to as loads) that need to be shipped on each lane. The number of loads to be shipped in a lane over a certain time period (we discuss the planning horizon relevant to our problem in section 2) is typically referred to as the volume in that lane. We assume that the volume in a lane represents point-to-point traffic and ignore flow-routing decisions. These shipments can be made either by using the shipper’s own resources, that is, an owned fleet, or by using third-party resources, for example, small trucking companies who sell their services on a pre-determined set of lanes (Song and Regan 2003) or a common carrier such as UPS or JB Hunt (Caplice and Sheffi 2003; Sheffi 1990). Another novel alternative, motivated by the substantial presence of small 3PL companies, is to subcontract a part of the shipping network as a dedicated subnetwork (DSN). A DSN refers to a subset of lanes with associated loads in a shipper’s network for which a fleet of resources—trucks, drivers, and other equipment—is exclusively assigned to the subnetwork to carry out all its shipping requirements (Panchalavarapu 2010). The resources assigned to the DSN are not shared for carrying out shipping in the rest of the network. Hence, a DSN is an autonomously operated subnetwork, which could be subcontracted to a third party. We emphasize that the primary motivation behind identifying and subcontracting a DSN is to create an incentive for both the shipper and the subcontractor. On the one hand, subcontracting a DSN enables the shipper to shed its physical assets and expect dedicated service from the subcontractor. On the other

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hand, the requirement of flow balance in the DSN facilitates the efficient utilization of the subcontractor’s assets. Once the DSN is subcontracted, the remainder of the shipper’s network could also be either subcontracted or handled through lane-by-lane auctioning (Sheffi 2004; Song and Regan 2003). This approach has been proposed by practitioners: “[The dedicated subnetwork approach] identifies lanes that are suitable for dedicated contract carriage and those that are suitable for one-way moves. … After identifying the opportunities for dedicated moves, it is important to evaluate the implications for the remaining lanes, which will be served by other means.”(Panchalavarapu 2010). We assume that subcontracting a DSN does not impact the shipper’s operating cost of the remaining network (i.e., the lanes that are not included in the DSN). We discuss this assumption in detail in section 5 and provide a justification for our approach based on interactions with practitioners. The resources assigned to a DSN exclusively cater to the shipping requirements of that subnetwork. This exclusivity restriction necessitates the conservation of flow at each node in the DSN over the planning horizon: the total inflow of loads into a node equals the total outflow of loads from that node. This, in turn, implies that the flow requirements within a DSN can be decomposed into a collection of cyclic routes (see section 2). From a practical point of view, this is an important requirement. Since a subcontractor typically positions its fleet in diverse geographic locations, trucks are required to return to their respective bases after completing the assigned shipments (Chen and Vairaktarakis 2005, Golden et al. 2008, Li and Simchi-Levi 1990). In other words, trucks report to their corresponding home bases on the completion of the assigned deliveries. In an effort to maximize value-added activities, it is preferable to have a positive load on each lane of a vehicle route. Hence, it is natural to view each vehicle route in the subnetwork as a cycle. If the subnetwork does not satisfy flow conservation, then the subcontractor may have to incur additional costs due to a variety of consequences— empty travel of trucks, complex vehicle routing, compensation for the low quality of life of the truck drivers, higher driver turnover, etc.—to facilitate the routing of vehicles in the network. These non-valueadded activities, in turn, impact the pricing of the subcontractor’s services (Ozener and Ergun 2008). By identifying and subcontracting a sizable subnetwork that satisfies flow conservation, the shipper can (a) obtain a competitive price, (b) significantly reduce its owned fleet without diluting its customer base, and (c) focus attention on other non-asset-based businesses (e.g., logistics consulting) or on expanding its services to new markets (Baumann 2006).

Despite the above-mentioned benefits, several practical challenges can be expected in a real-world implementation of this idea.

•

•

While the requirement of conservation of flow at each node of a DSN is well-motivated, it may prevent the identification of a sizable subnetwork. For example, if the shipper’s network is sparse or the volume imbalance at the nodes (i.e., the difference between the total inflow and outflow of loads at the nodes) is high, then imposing flow conservation may result in no non-trivial DSN. In cases where non-trivial DSNs exist, such subnetworks may be too small (in terms of the total volume they address) to be of significance to the shipper and/or the subcontractor. To overcome this difficulty, in section 2, we use two notions— deadheading and lane-sharing—that aid in improving the size of the DSN. Deadheading has been used as an effective resource-repositioning tool in numerous studies; a discussion is provided later in section 2.1. Informally, deadheading allows for the introduction of dummy shipping volumes into the network (Dejax and Crainic 1987) while lane-sharing allows the shipping volume on a lane to be shared by both the shipper and the subcontractor (Caplice and Sheffi 2003). A DSN identified by the shipper may be infeasible for the subcontractor due to the latter’s inability to handle the required volume. Thus, it is natural for the problem of identifying a DSN to incorporate the capacity restrictions of the subcontractor. We consider two different capacities: 1.

2.

Lane capacities: For efficient tactical (day-today) operations of the DSN, the subcontractor often needs to maintain several depots. The routing of trucks has to consider the resources at each depot while ensuring an appropriate repositioning of these resources. Excessive loads on lanes during a certain time window might result in infeasibility. Therefore, as a macro restriction, the subcontractor may impose upper bounds on the possible loads on each lane. Node (origin/destination) capacities: Once a truck arrives at a destination city, it typically requires maintenance before getting assigned to another trip. The maintenance capacity at a city imposes an upper bound on the total inflow to (or outflow from) a city.

Our aim is to analyze the problem of obtaining an optimal DSN when either deadheading or lane-sharing

Rajapakshe, Dawande, Gavirneni, Sriskandarajah, and Panchalavarapu: Dedicated Transportation Subnetworks Production and Operations Management 23(1), pp. 138–159, © 2013 Production and Operations Management Society

or both are allowed. We also examine the relative trade-offs between these two notions and provide transportation managers with guidelines—based on topological properties of the shipper’s network and the relative values of the parameters—to help identify an appropriate DSN.

2. Identifying Dedicated Subnetworks: Models and Assumptions We start by introducing the parameters of our models and justifying the primary assumptions. Then, we formally introduce the basic problem (Problem P) of identifying a DSN in a shipper’s network. Subsequently, in sections 2.1 and 2.2, we introduce the notions of deadheading and lane-sharing and the corresponding two Problems PD and PL , respectively, to enable the extraction of a better DSN. Our most general problem is Problem PDL , where both deadheading and lane-sharing are allowed. The shipper’s network is represented by a directed graph G(V,A), where the nodes in V correspond to origins and/or destinations of loads and A denotes the set of lanes. We assume that the volume from an origin to a destination is an integral number of truck loads. The volume and mileage on each lane (i,j) ∈ A are denoted, respectively, by non-negative integers D(i,j) and M(i,j). The volume D(i,j) is assumed to be the point-to-point traffic between nodes i ∈ V and j ∈ V, and flow routing is ignored. Let the operating cost per load-mile (currently incurred by the shipper) be c0i;j ; ði; jÞ 2 A and purchasing cost of dedicated service per load-mile be csi;j ; ði; jÞ 2 A. As discussed above, let U a ði; jÞ (resp., Uin ) be the upper bound on the subcontractor’s capacity of the arc (i,j) 2 A (resp., node i 2 V). We now justify our assumption of deterministic shipping volumes. We envision D(i,j) to represent the exogenouslyspecified total expected volume (in truck loads) on lane (i,j) over a time period of 2–3 weeks. In practice, it is reasonable to assume that the shipper will have fairly accurate estimates of these numbers. Recall that our models are developed from the perspective of a shipper (a national 3PL provider) that caters to a substantial clientele, usually under long-term contracts. Thus, the volume on each lane is typically generated by several customers. Although the shipping requirements of an individual customer on each lane may exhibit some variance, the variance of the aggregate load over a large number of customers over a reasonably long period of time (e.g., over 2–3 weeks) is negligible. Furthermore, the shipper not only has a good knowledge of the expected volume handled on behalf of each customer, but is also equipped with a substantial amount of historical data to enable the computation of the total volume on each lane over such a time

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period. Assuming a normal business environment, the aggregate demand on a lane over any window of 2–3 weeks is typically expected to remain stable. Therefore, once obtained, the structure of the optimal DSN can be expected to remain the same over a longer horizon (e.g., a year). There is another reason for preferring a window of 2–3 weeks. The purpose of our analysis is to identify a subnetwork which, as discussed earlier, decomposes into a set of cyclic routes. In practice, it is important to ensure that the resources assigned to a cyclic route return to their home base on the route within a reasonable time period (Powell et al. 2002). If aggregate demand over 2–3 weeks is used to identify the subnetwork, then it follows that (in steady state) the resources assigned to any cyclic route in the subnetwork will be typically away from their home base for up to 2–3 weeks, a duration which is common in practice (United States Department of Labor 2010). While the precise day-to-day routing of these resources is not a concern addressed in this study, we do want to ensure that the identified subnetwork lends itself to practically feasible cyclic routes (Bramel et al. 1993; Laporte et al. 2000; Lee and Qi 2009; Li et al. 1992). We are now ready to formally introduce the basic problem of identifying a DSN in the shipper’s network. Problem P: Extracting a DSN (without deadheading and lane-sharing) Instance: A directed network G(V,A), non-negative integers M(i,j), D(i,j), and U a ði; jÞ for each arc (i,j) 2 A, and non-negative integers Uin for each node i 2 V. The operating cost per load-mile is c0i;j ; ði; jÞ 2 A and purchasing cost of dedicated service per loadmile is csi;j ; ði; jÞ 2 A. Solution: A subnetwork SG ðN; EÞ of G(V,A) that satisfies (i) flow conservation at each node in SG ðN; EÞ, that is, for each node i ∈ N, we have P P j:ðj;iÞ2E Dðj; iÞ ¼ j:ði;jÞ2E Dði; jÞ, and (ii) arc and node capacities, that is, Dði; jÞ U a ði; jÞ; 8ði; jÞ 2 E and P n j:ðj;iÞ2E Dðj; iÞ Ui ; 8i 2 N. An example is illustrated in Figure 1. P 0 s Objective function: Maximize ði;jÞ2E ðci;j ci;j Þ Dði; jÞMði; jÞ, the total saving in operating cost due to subcontracting. As mentioned earlier in section 1.1, the primary requirement for a DSN is that there be conservation of flow at each node of the subnetwork. This immediately implies—via the flow decomposition theorem (see, e.g., Ahuja et al. 1993)—that the DSN can be decomposed into a union of cyclic routes. There are two main difficulties with the definition (of a DSN) above that might prevent us from extracting a DSN that offers a substantial cost saving. First, as defined, if an arc (i,j) 2 E, then the entire demand D(i,j) on this arc is required to be assigned to the DSN. In other

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Figure 1 Extracting a Dedicated Subnetwork (DSN) from the Shipper’s Network: (a) Shipper’s Network and (b) A DSN without Deadheading and Lane-Sharing, where Uin = 250; i = 1; 2; . . .; 9, and U a (i; j) = 250; ∀(i; j); i 6= j; i = 1; 2; . . .; 9; and j = 1,2,…,9

D(i , j)

(a)

words, it is not possible to assign a fraction of the demand D(i,j) to the DSN. Second, if the shipper’s network G(V,A) is a sparse network, then it may not have any cyclic routes. In this case, there is no nontrivial DSN. Furthermore, even if there are cyclic routes, there may not be enough of them to provide a DSN that carries a substantial amount of the shipper’s volume. Motivated by these shortcomings, we now discuss two notions—deadheading and lane-sharing— that allow us to guarantee feasibility and enable the identification of better DSNs. 2.1. Deadheading In order to achieve the conservation of flow at each node in a DSN, we have the option of introducing dummy shipping volumes into the network. These correspond to situations in which trucks go empty from one node to another, an operation commonly referred to as deadheading (Dejax and Crainic 1987). For example, backhauling, the empty return trip of a truck after fulfilling a delivery, could be viewed as a special case of deadheading. The concept of deadheading is more general in that empty trucks can be assigned to travel on any lane, irrespective of whether or not it currently has a positive volume. Clearly, deadheading is a nonvalue-added activity for the shipper, since no real loads get transported. However, it has been widely used as a repositioning tool (see, e.g., Chen et al. 2009, Ergun et al. 2007, Ozener 2008, Ozener and Ergun 2008). Apart from being significantly different from these studies in its context, our work also considers (wherever appropriate) the impact of

(b)

lane-sharing on the solution. For example, Chen et al. (2009) consider an auction mechanism where the shipper is interested in providing carriers with a set of lanes that minimizes the total cost associated with truck repositioning. In this study, while the winner determination problem incorporates several practically relevant constraints such as bounding the total volume assigned to each carrier, the number of carriers selected, and the maximum number of carriers assigned to each lane, the shipper incurs no cost for sharing the volume on a lane among several carriers. While the detailed routing of trucks within the DSN is out of the scope of our study, the work of Ergun et al. (2007), Ozener (2008), and Ozener and Ergun (2008) focuses on analyzing the operational/strategic issues involved in routing. Since the resources assigned to a DSN are to be used exclusively for its shipping requirements, the subcontractor should naturally be compensated for deadheading. Therefore, we assume that the subcontractor charges a per load mile price cdi;j for deadheading; clearly, cdi;j \ csi;j since cdi;j (resp., csi;j ) involves empty (loaded) travel for the subcontractor. Obviously, maintaining too high a deadheading traffic is an inefficient use of resources in the DSN and results in the shipper incurring a higher cost per load-mile for the real shipping volume transported. On the other hand, if we restrict deadheading to be too low, then we may not be able to identify a sufficiently large DSN (Figure 1). We refer to the problem of identifying a DSN when deadheading is allowed as Problem PD . To avoid repetition, we formally define this problem as a special case of Problem PDL below.

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2.2. Lane-Sharing Under lane-sharing, both the shipper and the subcontractor are allowed to share the shipping volume on a lane (Caplice and Sheffi, 2003). In other words, a portion, say X(i,j) < D(i,j), is handled by the subcontractor and the remainder (i.e., D(i,j) X(i,j) > 0) is handled by the shipper. Clearly, lane-sharing is a relaxation of the condition that the volume on a lane be handled entirely by the shipper or entirely by the subcontractor. Like deadheading, lane-sharing also has the ability of enabling flow conservation over a larger DSN (as compared with Problem P). To understand the negative consequences of lanesharing, it is instructive to examine the dispatching process on a lane. The management of loads on each lane—meeting customer requirements, checking resource availability, assigning a load to the corresponding transporter, routing of trucks—is typically handled by one or more dispatchers (Brown and Graves 1981, Spieck et al. 1995). Each dispatcher manages a pre-assigned set of lanes which allows her to gain expertise over these lanes. From the shipper’s viewpoint, subcontracting the entire volume on a lane eliminates the need for a dispatcher on that lane. However, consider a situation when a lane is shared, for example, 40% of the volume is handled by the shipper and the remaining 60% by the subcontractor. Here, it is important to note that the entire demand on the lane does not occur simultaneously, but is instead generated over a period of time. Therefore, when a customer demand arises on a shared lane, the shipper needs to evaluate a host of issues—availability of own shipper’s resources compared with those of the subcontractor, the amount of loads subcontracted so far on this lane, the pros and cons of subcontracting now versus later, etc.—to decide whether or not the load is to be assigned to the subcontractor. Note that these decisions have to be taken regardless of the total volume on the shared lane and regardless of the split in volume between the shipper and subcontractor. We, therefore, model the fixed cost (incurred by the shipper) corresponding to the additional resources expended in making these decisions to manage a shared lane as cl . In addition to the increase in the complexity of the dispatching process discussed above, having multiple carriers serve the same lane also necessitates other costs. In Caplice and Sheffi (2003), the authors identify three different components—the real-time tendering system handling cross-company moves, shippers coming to agreeable terms for payment and cost sharing, and the capability of a freight payment and audit system to handle cross-company moves—that result in additional costs required for the effective collaboration of multiple shippers on the same lane

in freight transportation. The fixed cost cl in our model is also an attempt to capture these costs. Under lane-sharing the problem of identifying a DSN is referred to as Problem PL . Below, we formally define this problem as a special case of Problem PDL . Deadheading and lane-sharing offer two fundamentally different approaches for improving the DSN obtained by solving Problem P. It may be beneficial to use both these approaches to further improve the DSN. Also, if a lane is not entirely subcontracted, deadheading and lane-sharing could substitute each other. In other words, (i) lane-sharing could be eliminated by introducing deadheading on the shared lane and (ii) deadheading on a lane (i,j), where D(i,j) > 0 and X(i,j) = 0, could be eliminated by introducing lane-sharing. We refer to the problem in which both deadheading and lane-sharing are allowed as Problem PDL . If both deadheading and lane-sharing are allowed, the problem of identifying a DSN (Problem PDL ) is defined as follows. Problem PDL : Design of a DSN with deadheading and lane-sharing Instance: A directed network G(V,A) and non-negative integers M(i,j), D(i,j), and U a ði; jÞ for each arc (i,j) 2 A, and non-negative integers Uin for each node i 2 V. The operating cost per load-mile is c0i;j ; ði; jÞ 2 A, purchasing cost of dedicated service per load-mile is csi;j ; ði; jÞ 2 A, deadheading cost per load-mile cdi;j ; ði; jÞ 2 A, and lane-sharing cost cl per lane-shared. Solution: A subnetwork SG ðN; EÞ of G(V,A) that satisfies (i) flow conservation at each node in SG ðN; EÞ, that is, for each node i 2 N, we have X X ðXðj; iÞ þ Yðj; iÞÞ ¼ ðXði; jÞ þ Yði; jÞÞ j:ðj;iÞ2E

j:ði;jÞ2E

and (ii) arc and node capacities, that is, Xði; jÞ P U a ði; jÞ; 8ði; jÞ 2 E and j:ðj;iÞ2E ðXðj; iÞ þ Yðj; iÞÞ Uin ; 8i 2 N, where X(i,j) is the amount of volume subcontracted on lane (i,j) and Y(i,j) is the amount of deadheading on lane (i,j). An example is illustrated in figure 2. Objective Function: Maximize X X ðc0i;j csi;j ÞXði; jÞMði; jÞ cdi;j Yði; jÞMði; jÞ ði;jÞ2E

cl

X

ði;jÞ2E

Zði; jÞ ;

ði;jÞ2E

where Z(i,j) = 1 if lane (i,j) is shared with the subcontractor and Z(i,j) = 0 otherwise. Next, we define two subcases of Problem PDL : Problems PD and PL . Problem PD corresponds to the problem with X(i,j) = D(i,j), ∀(i,j) 2 A. Problem PL corresponds to the problem with Y(i,j) = 0, ∀i,j 2 V.

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Figure 2 Extracting a Dedicated Subnetwork from the Shipper’s Network in Figure 1: (a) With Deadheading, (b) with Lane-Sharing, and (c) with Both Deadheading and Lane-Sharing

D(i , j)

(a)

(b)

(c)

Notes: Deadheading is indicated by broken lines, M(i, j ) = 10, ∀(i,j ), i 6¼ j, i = 1,2,…,9, and j = 1,2,…,9, Uin = 250, i = 1,2,…,9, and Ua(i,j ) = 250, ∀(i, j ), i 6¼ j, i = 1,2,…,9, and j = 1,2,…,9.

To illustrate the advantage of solving Problem PDL , consider the shipper’s network in Figure 1a under the following cost parameters: c0i;j ¼ 1; csi;j ¼ 0:8; cdi;j ¼ 0:2; U a ði; jÞ ¼ 250; 8ði; jÞ 2 A, Uin ¼ 250; 8i 2 V, and cl ¼ 50. Figure 2 illustrates the optimal solutions for Problems PD , PL , and PDL . Without the DSN, the total operating cost over the entire network of Figure 1a for the shipper is 10,250. The savings in cost corresponding to Problems PD , PL , and PDL are, respectively, 900, 720, and 1020. Thus, Problem PDL has the potential to substantially improve savings. Our next goal is to theoretically analyze the four problems: P, PD , PL , and PDL . In particular, we resolve their computational complexities and explore several interesting special cases arising from simplifications in the parametric relationships and the topological properties of the shipper’s network.

3. Problems P, PD , PL , and PDL: Theoretical Results and Heuristics To precisely define the problems discussed in the previous section, we formulate them as mixed-integer programs (MIPs). The formulations are provided in section A.1 of the Online Appendix. In section 3.1, we show that all these problems are strongly NP-hard as well as APX-hard. Furthermore, these problems remain hard for practically common topological structures such as Hub-Spoke and Trees. In section 3.2, we discuss some interesting polynomially solvable special cases that are derived from two sources: (i) simplifications resulting from the relationships between the parameters of the

models (section 3.2.1) and (ii) simplifications in the structural properties of the shipper’s network (section 3.2.2). Using the insights obtained from the theoretical analysis, we develop a heuristic that can provide efficient solutions to practical problem instances within a reasonable time limit. The heuristic is described in section 3.3 The heuristic is later evaluated on a comprehensive test bed and the results are discussed in section 4.3.2. The proofs of all technical results are placed in section A.2 of the Online Appendix. 3.1. Hardness Results for Problems P, PD , PL , and PDL In section 3.1.1 we first show that Problem P is strongly NP-hard on hub-spoke networks. The strong NP-hardness of the four problems on general networks then follows from this result. Furthermore, in Theorem 3, we establish that these four problems are APX-hard as well. 3.1.1. Hardness Results for Hub-Spoke Networks and Trees. Hub-Spoke and Trees are two commonly observed network structures in transportation (Hu 2010, Kim 1998, Kunnumkal and Topaloglu 2010, Pirkul and Schilling 1998). In a hub-spoke network (also referred to as a pool-point network—see Figure 3a), a set of (well-separated) hubs act as depots, collecting and redistributing loads to nearby destinations; the link between a hub and such a nearby destination is referred to as a spoke. Typically, a hub is linked by spokes to several destinations; these destinations can then only be served by the assigned hub. Hence, it is clear that the flow of loads between a

Rajapakshe, Dawande, Gavirneni, Sriskandarajah, and Panchalavarapu: Dedicated Transportation Subnetworks Production and Operations Management 23(1), pp. 138–159, © 2013 Production and Operations Management Society

Figure 3

Special Network Topologies: (a) Hub-Spoke and (b) Tree

Hub

(a)

hub and a corresponding destination point is bidirectional. In practical hub-spoke networks, the volume of transportation between the hubs is typically much more than that between a hub and a spoke (Elhedhli and Hu 2005, Geismar et al. 2011, Pirkul and Schilling 1998). Some other practically common transportation structures such as a star network (introduced later in section 3.2.2) are special cases of a general hub-spoke structure. A hub-spoke topology is capable of offering fast delivery and a high level of customer service (Elhedhli and Hu 2005, Pirkul and Schilling 1998). A tree (or a hierarchical network; see Figure 3b) is a more general acyclic network. For example, a hierarchical network of plants, warehouses, and retailers in which unidirectional transportation occurs from plants to warehouses and from warehouses to retailers, is a tree. Such a structure typically occurs in import distribution channels (Kim 1998). Theorems 1, 2, and Corollary 1 comment on the hardness of our problems for these two special network structures. THEOREM 1. Problem P is strongly NP-hard for hubspoke networks. COROLLARY 1. NP-hard.

Problems PD , PL , and PDL are strongly

THEOREM 2. For tree networks, Problems P and PL are infeasible while Problems PD and PDL are strongly NPhard. Theorem 3 below shows that all four problems are APX-hard as well. Hence, unless P = NP, there is no polynomial time approximation scheme for any of the four problems. THEOREM 3. hard.

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Problems P, PD , PL , and PDL , are APX-

Supply/ Demand Location

(b)

3.2. Polynomially Solvable Special Cases In this section, we identify several polynomially solvable special cases of the four problems: P, PD , PL , and PDL . These special cases are derived from two sources: (i) relationships between the parameters of the models (section 3.2.1) and (ii) characteristics of the topological properties of the shipper’s network (section 3.2.2). In our discussion, we will often use the well-known minimum cost flow problem on a directed network G(V,A). The formulation of the minimum cost flow problem is provided in section A.1 of the Online Appendix. 3.2.1. Simplifications Due to Relationships Between the Parameters of the Model. When a shipper caters to a wide variety of customers from diverse geographic markets, it is possible that the shipper’s network does not exhibit any specific topological structure (Kleywegt et al. 2006). In such a general network, one can exploit the relationships between the various parameters of the model to derive efficient algorithms for the four problems. We start by discussing a basic property (Lemma 1), which is used in our subsequent discussion in this section. LEMMA 1. In any optimal DSN to Problem PDL , on a given lane (i,j), either Y(i,j) = 0 or Xði; jÞ ¼ minfDði; jÞ; U a ði; jÞg. As a consequence of the above result, we conclude that if an optimal solution to PDL has deadheading on lane (i,j), then the total volume that can be subcontracted (without violating the subcontractor’s resource requirement) on this lane is already in the DSN (i.e., Xði; jÞ ¼ minfDði; jÞ; U a ði; jÞg). For Problems P and PL , the subcontracted volumes (i.e., the values of X(i,j)) satisfy flow conservation at each node, while for Problems PD and PDL , the sum of the

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subcontracted and deadheading volumes (i.e., X(i,j) + Y(i,j)) satisfy flow conservation at each node. It is, therefore, clear that any feasible solution to these problems can be decomposed into a union of distinct maximal cyclic flows in the corresponding variables (i.e., X(i,j) for P and PL ; X(i,j) + Y(i,j) for PD and PDL ). Recall from section 2.2 that the lane-sharing cost cl is the fixed cost corresponding to the additional resources expended in managing dispatching decisions on a shared lane. If additional resources are already available or if such resources are cheap to acquire, then the lane-sharing cost will typically be negligible (and can, therefore, be ignored) or be insignificant as compared with the operating and deadheading costs. In such cases, Lemma 2 shows that an optimal or near optimal solution to Problems PL and PDL can be obtained in polynomial time.

In this case, we can obtain a polynomial-time solution to Problems PD , PL , and PDL with an additive guarantee that is a linear function of r.

3.2.2. Simplifications Due to Structural Properties of the Transportation Network. We focus our attention on some widely observed network structures in practice: a star network together with its variants and a hub-spoke network. The special characteristics of these networks allow for several polynomially solvable special cases. In several practical settings, shippers are involved in transporting goods originating from a single point. For instance, consider a shipper responsible only for transporting imported goods from a harbor to several main warehouse locations serving a wide customer base (Horizon Logistics 2009, 2010). In this setting, loads originate from the harbor and the destination points (warehouses) can be considered as pure demand points (Horizon Logistics 2010). Such a network is typically referred to as a star network (Figure 4a). In such a network, since a node is either a pure supply or a pure demand point, deadheading is necessary to ensure the conservation of flow required by a DSN. Furthermore, since each destination point is exclusively connected to the origin, it is reasonable to assume that the distance between any two destination points is significantly larger than the distance between any origin-destination pair (Chen 2010). We, therefore, assume this to be the case. This assumption implies that, in an optimal solution, deadheading can occur only on the origin-destination lanes of the network. Accordingly, we assume that there is no transportation between two destination points of a star network. Two other more-sophisticated variants of the star topology are also observed in practice: (i) a connected-star network, where two or more stars are connected through one or more common destination points (Figure 4b) and (ii) a hybrid-star network, where the destination points of a star act as origin points to other star networks (Figure 4c). In turn, a hybrid-star is a special case of the more-general

LEMMA 2. When the lane-sharing cost cl ¼ 0, Problems PL and PDL can both be solved in polynomial time. More generally, if 0 \ cl e, then a near-optimal solution, with an additive guarantee that is a linear function of e, can be obtained in polynomial time. The following two results address the situation when (i) the demand on each lane of the shipper’s network is (approximately) the same and (ii) the subcontractor has the capacity to handle any amount of volume. Lemma 3 is used for proving the more general statement of Lemma 4. LEMMA 3. When U a ði; jÞ ¼ 1; 8ði; jÞ 2 A; i 6¼ j and Uin ¼ 1; 8i 2 V, and the demand on each lane in the shipper’s network is a constant, that is, D(i,j) = D, ∀(i,j) 2 A, Problems P, PD , PL , and PDL can be solved to optimality in polynomial time. Furthermore, there exists optimal solutions to PL and PDL that do not have lanesharing. LEMMA 4. Let U a ði; jÞ ¼ 1; 8ði; jÞ 2 A; i 6¼ j, Uin ¼ 1; 8i 2 V, and the demands in the shipper’s network D(i,j) 2 [(1 r)D,(1 + r)D], ∀(i,j) 2 A, where 0 < r 1. Figure 4

Special Network Topologies: (a) Star, (b) Connected-Star, and (c) Hybrid-Star

Origin

(a)

Destination

(b)

(c)

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hub-spoke topology (section 3.1.1). Again, we assume that, for these topologies, there is no transportation between two points that are destination points of the origin node of some star network. For these special structures, since deadheading is necessary for a feasible DSN, Problems P and PL are infeasible. We, therefore, comment only on Problems PD and PDL . Theorem 4 comments on the computational complexities of the star, connected-star, and hybrid-star network topologies. THEOREM 4. Problems PD and PDL are weakly NP-hard for star, connected-star, and hybrid-star network topologies. When the node capacity at the origin is unlimited, both the problems can be solved to optimality in polynomial time. Next, consider a hub-spoke network (section 3.1.1) in which there is no hub-to-hub transportation. In this case, the shipper’s network decomposes into multiple independent hub-spoke networks, each with a single hub. Note that these single-hub networks are different from star networks since the transportation between a hub and its destination points is bidirectional (unlike the unidirectional flow in a star network). Recall that each spoke location exclusively communicates with the corresponding hub and there is no transportation between spoke locations. Therefore, it is sufficient to examine the volumes between the hub and a spoke location to decide on the subcontracting of the corresponding lane. As before, we assume that there is no deadheading between spoke locations. The following lemma addresses the complexity of our four problems on such a network. LEMMA 5. Problems P, PD , PL , and PDL are weakly NP-hard on a hub-spoke network, even if there is no hubto-hub transportation. When there is no hub-to-hub transportation and no capacity restriction at the hubs, these problems are all polynomially solvable.

3.3. The Flow-Fix-Improve Heuristic In this section, we provide a detailed description of a heuristic, referred to as Flow-Fix-Improve (FFI), for Problem PDL . The efficiency and the robustness of the heuristic is demonstrated later in section 4.3. The corresponding heuristics for Problems PD and PL can be obtained in a similar manner. We first describe the basic intuition behind FFI and then provide a formal procedure. Let I denote an instance of PDL . As shown in Lemma 3, when the shipper’s network has the same demand on each lane, then I can be solved optimally by transforming it to a minimum cost flow problem. Furthermore, this solution does not have any shared

147

lanes. Motivated by this result, our first attempt is to extract a subproblem, say I1 , of I, which can be used to identify a set of lanes on which lane-sharing is absent. This task, however, poses a challenge. When the demands on the lanes in the shipper’s network vary significantly, the solution obtained using the transformation in Lemma 3 may not be a good approximation to PDL since it ignores the impact of lane-sharing cost. By ignoring lane-sharing cost, the minimum cost flow solution is expected to have more unsaturated lanes as compared with the optimal DSN. To overcome this drawback, the first step of the heuristic identifies a subset of lanes with small variation in demand. Note that our ability to extract cycles with no lane-sharing also improves with the density of the subnetwork corresponding to I1 . Thus, there is a need to balance the following two properties: the lanes under I1 should have limited variation in demand and the density of the subnetwork should be sufficiently high. Assuming that the demands in I follow a Gamma distribution, we select (for inclusion in I1 ) the demands that fall within the range [min{0, (k1)har},(k1)h + ar], where a 0 is a constant that decides the range. This assumption is later justified using real-world data in section 4.1. Note that a Gamma distribution is characterized by two parameters: scale h > 0 and shape k > 0; the mean of the distribution is kh and the mode is (k 1)h for k 1. A tentative DSN is obtained by solving a minimum cost flow problem on I1 . Clearly, this solution is also feasible for the original instance I. Due to the manner in which the instance I1 is extracted, it is reasonable to expect that there are only a few lanes shared between the shipper and subcontractor in the minimum cost flow solution corresponding to I1 . The second step of the heuristic creates a new instance, say I2 , using the original instance I and the solution obtained for I1 . To get I2 , we first fix the flow on the lanes with no lane-sharing in the solution to I1 . Then, we include all the other lanes from I with the corresponding node and lane capacities. The generated instance I2 is solved by CPLEX using the MIP formulation provided in the Appendix. Two points concerning the second step deserve mentioning. Since we fix the flow on a set of lanes (identified in the first step), the number of remaining lanes, on which flow is to be decided, in I2 is typically small. Therefore, the MIP can be solved effectively. Also, the inclusion of the lanes on which demands vary significantly and the consideration of the lane-sharing cost (i) improves the size of the DSN and the cost saving and (ii) possibly eliminates unnecessary lane-sharing through efficient utilization of the deadheading. Thus, the second step can be considered as one that improves the solution of the first step. We now formally describe the heuristic.

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The Flow-Fix-Improve Heuristic (FFI) Input: The network G(V,A), along with its parameters cl ; Dði; jÞ; Mði; jÞ; c0i;j ; csi;j ; cdi;j ; U a ði; jÞ; Uin ; the parameters of the Gamma distribution k,h,r, and the range constant a. Step 1: Define G0 ðV 0 ; A0 Þ such that V0 ¼ V; A0 ¼ ði; jÞ : minf0; ðk 1Þh arg Dði; jÞ ðk 1Þh þ ar; ði; jÞ 2 A : Step 2: (flow solution). Solve for the minimum cost flow solution on G0 ðV 0 ; A0 Þ. Let f be the flow vector obtained after solving for the minimum cost flow solution. Step 3: (fixing step). Fix the flows on the arcs in A00 A0 A, where A00 ¼ ði; jÞ : fði; jÞ ¼ Dði; jÞ, Dði; jÞ U a ði; jÞ; ði; jÞ 2 A0 . Step 4: (improvement step). Solve the Problem PDL on G(V,A) with the constraint Xði; jÞ ¼ Dði; jÞ; ði; jÞ 2 A00 , where X(i,j) is the volume subcontracted on lane (i,j). Output: The subcontracted and deadheading volumes X(i,j) and Y(i,j), respectively.

4. Understanding the Trade-off between Deadheading and Lane-Sharing Our aim in this section is to (i) test the performance of the heuristic, (ii) examine the monetary impact of subcontracting a DSN, and (iii) offer managers insights into the influence of density and size of the shipper’s network on the relative extents of deadheading and lane-sharing in an optimal DSN. To focus our investigation on realistic instances, we utilize real-world data obtained from a national 3PL provider. Before proceeding further, we briefly summarize the main issues addressed in our computational study: 1. Performance of the heuristic: In section 4.3.2, we evaluate the efficiency as well as the effectiveness of the FFI heuristic on a comprehensive test bed. 2. Monetary impact: In section 4.3, we demonstrate the attractiveness of the DSN concept by computing the cost saving offered by an optimal DSN as a percentage of the shipper’s total operating cost. 3. Dense vs. sparse networks: The density of the shipper’s network, that is, the fraction of the number of lanes in the network to the total possible number of lanes, is expected to be a fundamental factor in the extent to which deadheading and lane-sharing are utilized. In

section 4.3.3, we analyze DSNs on networks with varying densities to understand the relative behavior of deadheading and lane-sharing. We also study the impact of the subcontractor’s cost effectiveness (relative to the shipper) on the amount of deadheading and lane-sharing in the DSN. 4. Strategic alliances: Fierce competition has forced logistics providers to investigate novel approaches to lower costs. One emerging thought is that two or more providers could engage in a strategic alliance to benefit from the resulting increase in shipping volumes in the combined network (Ergun et al. 2007). In section 4.3.4, we examine the impact of merging the networks of two shippers on the DSN.

4.1. A Real-World Instance Data obtained from a national 3PL provider1 helped us in designing a test bed that is based in reality. We first briefly summarize the details of the real-world logistics network of the provider and then compute the cost saving (section 4.3.1) from subcontracting a DSN for this network. The provider serves a diverse set of customers (companies) spread across the continental United States. The total number of distinct customers is about 600, with each customer generating demand on several lanes spread over multiple states. We provide aggregate statistics over the 6-month period from March to August 2007. The actual demand is generated incrementally over this time period. The network has about 12,000 distinct lanes with positive demand, with about 5000 distinct origin/destination points. Figure 5 shows a snapshot of the origin/destination points. The mileages of the lanes vary from 10 miles to about 3000 miles; over all lanes, the average mileage is 668.50 miles. For a broader view, it is convenient to cluster all origin/destination points from the same state to a single node. Thus, this clustered, state-based network has 50 nodes and a density of about 35%. The volumes on the lanes in the clustered network vary from 1 to 3199 truck loads. Figure 6 provides a better understanding of the distribution of demand in the shipper’s network. As illustrated in the figure, the frequencies of the low-demand numbers are significantly higher than the high-demand numbers. We observe the same behavior under other clustering methods based on cities and zip codes. Moreover, this pattern is consistent even when we consider the demands over shorter time windows. We now provide an explanation for the demand behavior in Figure 6. The average traffic on a lane in the clustered network is around 15 truck loads. A careful analysis

Rajapakshe, Dawande, Gavirneni, Sriskandarajah, and Panchalavarapu: Dedicated Transportation Subnetworks Production and Operations Management 23(1), pp. 138–159, © 2013 Production and Operations Management Society

Figure 5

Figure 6

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A Snapshot of the Origin/Destination Points for the Real-World Example

Demand Distribution in the Real-World Network of a National Shipping Company

200

Frequency

150

100

50

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 More

0

Demand (in Truck Loads)

reveals that the states with high to-and-from traffic (e.g., exceeding 200 truck loads) represent some of the main population centers, industrial hubs, and states in which the company has a substantial presence. There is significant inter-state traffic between Indiana, Texas, Illinois, Virginia, Georgia, Missouri, California, Colorado, Pennsylvania, and Ohio. These represent major demographic/industrial hubs, and we observe high demands between these states from transportation of goods between key geographic sites (of the customers), for example, between ports and

warehouse locations, between distribution centers. The shipper has a service hub in Texas, with around 45% of the total traffic in the network either entering or leaving this state. The shipper’s reputation and efficiency in serving such geographical locations (i.e., those where it has significant presence) attracts considerable business. Therefore, we observe high to-and-from traffic associated with such locations. The demand originating from the states mentioned above to those with moderate populations and industrial activities are in the medium range (around

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75–200 truck loads per lane). Some examples include Georgia–Arkansas, Texas–Florida, and Georgia–Florida. A majority of the small customers are geographically dispersed and generate small demand volumes. For example, Massachusetts, Idaho, Maine, Montana, North Dakota, etc. have significantly lower to-andfrom traffic. The above observations indicate that the demand distribution of Figure 6 —where frequencies of lower demands are significantly larger than higher demands—can be expected for a typical national shipping network. This justifies our earlier assumption in section 3.3 that the demand on a lane follows a Gamma distribution. 4.2. Test Beds for Computational Experimentation To investigate the four issues mentioned at the beginning of this section, we now describe four separate test beds. 4.2.1. Test Bed 1: To Test the Performance of Heuristic FFI. The efficiency and the effectiveness of the heuristic is tested by varying (i) the number of nodes, (ii) the density of the shipper’s network, and (iii) the shape parameter k of the Gamma distribution. For the real-world data, about 50% of the demands are less than 60 truckloads. Hence, to anchor our test bed on practice, we set the scale parameter h = 60/k. We assume that fuel cost can vary in the range of USD 3–4 per gallon and a truck can travel about 10 miles per gallon. Therefore, as a base case, the cost parameters c0i;j and csi;j are both generated from U[0.3,0.4]; the ranges for these two costs are varied later for our other experiments. The deadheading cost shared by the shipper for a lane is set at 30% of the operating cost of the subcontractor for the same lane. Based on the structure of the clustered network described in section 4.1, the mileages of the lanes are generated from U[100,1000]. Recall that the lane-sharing cost, cl , represents the fixed cost incurred for the additional resources required to manage dispatching decisions on a shared lane. Based on the information received from the 3PL provider, we assume that one extra dispatcher, together with the required infrastructure, would be enough to manage this additional workload on a shared lane. We assume that the annual salary of a dispatcher is US$60,000 (http://swz.salary.com) and scale it appropriately (to be consistent with the time period of 2–3 weeks for the shipping volumes on the lanes, network density, and the typical workload assigned to a dispatcher per day). The lane capacities are generated from U l þ 2r; l 0:5r and the node capacities are generated pﬃﬃﬃﬃﬃ from U jVjdl þ r; jVjdl r where l ¼ kh; r ¼ kh, and d is the network density. Table 1 summarizes the parametric setting used in this experiment. For each combination of the

Table 1. Parameter Settings to Evaluate the Performance of the Heuristic Parameter

Values

Number of nodes, |V| Network density, d Demand on a lane, D(i,j) Mileage of a lane, M(i,j) 0 Shipper’s operating cost, ci;j Subcontractor’s operating cost, ci;js Lane-sharing cost, cl

100, 200, 300 0.4, 0.6 Γ[k,h],k = 1,2,3 U[100, 1000] U[0.3, 0.4] U[0.3, 0.4] 3600

parameters we generate three instances of PDL , for a total of 3 9 2 9 3 9 3 = 54 instances. 4.2.2. Test Bed 2: To Test the Impact of Density, Size, and Costs on the DSN. To understand the trade-off between deadheading and lane-sharing costs, we vary (i) the number of nodes (|V|), (ii) the density (d), and (iii) the operating costs of the shipper (c0i;j ) and subcontractor (csi;j ). We use a parameter, referred to as Rc , to measure of the extent of overlap between the uniform supports used for generating the cost parameters c0i;j and csi;j . Three values of Rc are considered: Rc ¼ 1 (resp., Rc ¼ 0:5, Rc ¼ 0:25), corresponds to the case where c0i;j is generated from U[0.3,0.4] and csi;j is generated from U[0.3,0.4] (resp., U[0.25,0.35], U[0.225,0.325]). Intuitively, the parameter Rc measures the cost effectiveness of the subcontractor relative to the shipper. The generation of lane capacities and arc capacities on a lane and the value of lane-sharing cost is the same as in section 4.2.1. To represent practical problem instances, the shipping volume on each lane in the shipper’s network is generated from Γ[1,30]. The mileages of the lanes are generated from U[100,1000]. Table 2 indicates the values of the various parameters. For each combination of the parameters, we generate five instances of Problem PDL . Thus, there are a total of 3 9 3 9 3 9 5 = 135 instances in this test bed. 4.2.3. Test Bed 3: To Test the Impact of Merging Networks of Two Shippers on the DSN. As mentioned earlier, merging the networks of two shippers can possibly lead to a DSN that offers higher saving as compared with the sum of the savings from the two individual DSNs. To focus solely on the impact of merging and to avoid any undesired influence due to Table 2 Parameter Settings to Study the Impact of Density, Size, and Costs Parameter Number of nodes, |V| Network density, d Mileage of a lane, M(i,j) 0 Shipper’s operating cost, ci;j Extent of overlap, Rc

Values 100, 200, 300 0.2, 0.5, 0.8 U[100, 1000] U[0.3, 0.4] 1, 0.5. 0.25

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differences in the sizes and cost parameters of the two networks, we assume the same cost parameters for both the networks: c0i;j U½0:3; 0:4; csi;j U½0:3; 0:4; cdi;j ¼ 0:15csi;j . Demand is generated from Γ(2,30). All the other parameters are as described in section 4.2.1. The idea of merging the networks of two shippers is interesting only when there is a substantial overlap in the lanes of these networks. In the absence of such an overlap, the two networks are essentially disjoint (e.g., when the two shippers operate in different geographic areas) and the optimal DSN for the merged network is, for the most part, the union of the optimal DSNs for the two individual networks. Two key characteristics of the networks that are likely to dictate the impact of merging are the extent of overlap and their relative densities. Accordingly, we choose three levels for the number of overlapping lanes: 25%, 50%, and 75%, and six combinations of the densities of the two networks: (0.2,0.2) (0.2,0.5) (0.2,0.8) (0.5,0.5) (0.5,0.8) (0.8,0.8). To allow for different sizes of the two networks, we consider three values of the number of nodes, |V| = 100, 150, and 200. As before, we generate five instances for each parameter combination. Thus, this experiment considers 3 9 3 9 6 9 5 = 270 instances. 4.3. Results, Insights, and Intuition We first quantify the potential benefit from the optimal DSN on the real-world network and then justify the quality of heuristic FFI. The last two subsections summarize the useful insights gained from our computational experiments. 4.3.1. The Benefit of Subcontracting a DSN on the Real-World Network. A reasonable metric to indicate the benefit of extracting a DSN is the cost saving it offers the shipper, as a percentage of the total operating cost the shipper would have to incur (in the absence of the DSN) to satisfy all demand over the entire network. For the real-world network (described in section 4.1), we obtained the data on demands, mileages, and operating costs on the lanes from the shipper. The costs and the resource constraints of the

Table 3 Parameter Settings to Study the Impact of Subcontractor’s Cost and Lane Capacities on the DSN from the Real-World Network Parameter Subcontractor’s cost, Arc capacity, U a ði; j Þ

Values 0 0 U½ac i;j ; 1:1ci;j ,

a = 0.5,0.6,0.7 U l 0:5r; l þ br , b = 1,1.5,2

s ci;j

subcontractor were not available to us. Therefore, to understand the impact of the (i) subcontractor’s cost and (ii) lane capacities on the DSN, we vary the uniform supports used in the generation of above two parameters as shown in Table 3. The node capacities are set as discussed in section 4.2.1. We generated one instance for each parametric combination, for a total of 3 9 3 = 9 instances. When calculating the lane and node capacities, the shape parameter k of the Gamma distribution (section 3.3) was set to 1. For each of the nine instances, we identify a DSN using Heuristic FFI. Table 4 summarizes the impact of varying a and b on several useful metrics of the resulting DSN. We now provide the average statistics over all nine instances. For the clustered real-world network (see section 4.1), the average percentage cost saving (for the shipper) from the DSN after allowing for both deadheading and lane-sharing is about 10%. The maximum (resp., minimum) cost saving is 13.3% (resp., 6.3%). The average total volume (load miles) in the DSN is about 60% of the total volume in the shipper’s network. The average deadheading volume in the DSN is about 10%. The average deadheading cost, as a percentage of the total cost saving, is about 4%. The average number of lanes shared is 12. The average lane-sharing cost as a percentage of the total cost saving is about 4%. Since the network density is low (about 40%), the number of cycles is limited. Therefore, the DSN uses a significant amount of deadheading volume to extract useful cycles. For any fixed value of b, as the subcontractor’s cost effectiveness increases (i.e., as a decreases) the (i) percentage cost saving from subcontracting a DSN increases, (ii) percentage volume subcontracted

Table 4 Impact of Subcontractor’s Cost and Arc Capacities on the Extraction of a DSN

% Cost saving Subcontractor’s cost reduces↓ a = 0.7,b = 1 6.3% a = 0.6,b = 1 9.1% a = 0.5,b = 1 12.0% Arc capacity increases↓ a = 0.7,b = 1 6.3% a = 0.7,b = 1.5 6.8% a = 0.7,b = 2 7.0%

% Volume subcon. (load miles)

% Deadheading volume in the DSN

% Deadheading cost w.r.t. Cost Saving

% Lane-Sharing cost w.r.t. Cost Saving

54.4% 58.3% 61.6%

9.1% 10.2% 10.3%

4.0% 3.1% 2.4%

6.2% 5.2% 5.0%

54.4% 58.3% 60.4%

9.1% 8.6% 8.5%

4.0% 4.3% 4.6%

6.2% 4.0% 3.8%

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increases, and (iii) percentage deadheading volume decreases. To observe the impact of arc capacities on the percentage cost saving, for each value of b, we compute the percentage cost saving by averaging over the three instances obtained by varying the value of a. As b increases, the percentage cost saving also increases. Since tighter lane capacities result in lower volume in the DSN and possibly more lane-sharing, the cost saving suffers at lower values of b. As b increases, the ability to subcontract more volume on a given lane increases. Also, the solution has the ability to possibly eliminate lane-sharing (on the lanes where demand is higher than capacity). Hence, the solution uses lower lane-sharing, more volume, and more deadheading to achieve a higher cost saving. 4.3.2. Performance of the Heuristic FFI. Table 5 summarizes the performance of the heuristic for Problem PDL . CPLEX (version 11.1) could not solve any of the problem instances to optimality within an imposed CPU time limit of 3600 seconds. For each instance, we compare the heuristic solution with the best upper bound available from CPLEX within the time limit. We report the percentage gap, computed as the ratio of the difference between the upper bound and the heuristic solution to the heuristic solution. Each cell of Table 5 corresponds to a statistic (minimum, average, and maximum percentage gap) over nine instances (see section 4.2.1). Over all the instances, the average percentage gap is 1.88%. Recall that in Step 4, the heuristic solves a reduced MIP obtained after fixing the variables in Step 3. While the reduced MIP is typically solved to optimality within a couple of minutes, we impose a limit of 400 seconds to ensure that the time required by Step 4 never exceeds this bound. The total time required by the heuristic for an instance is never more than 400 seconds and, therefore, not reported. For 30% of the instances, the heuristic provided a better solution than the best solution available from CPLEX after 3600 seconds. For the remainder of this section, our focus will be on observing the trends—with respect to changes in the cost parameters and structural properties of the network—in the use of deadheading and lane-sharing in extracting an optimal DSN. Therefore, instead of

Table 5 The Percentage Gap between the Heuristic Solution and the Upper Bound Obtained from CPLEX (version 11.1) Nodes Density min gap avg gap max gap

100 0.4 0.3 0.7 2.3

200 0.6 0.5 1.8 2.8

0.4 0.3 1.4 5.6

300 0.6 0.2 0.7 2.3

0.4 3.8 4.0 4.3

0.6 2.5 2.6 2.8

reporting absolute values of the cost savings, we use appropriate normalizations. 4.3.3. Dense vs. Sparse Networks. Let the total cost saving offered by an optimal DSN (to PDL ) be SDL , and the corresponding deadheading and lanesharing costs be CD and CL , respectively. We examine the impact of network density, size (number of nodes), and cost effectiveness parameter Rc (see section 4.2.2) on the normalized ratios CD =SDL (marginal deadheading cost per unit cost saving) and CL =SDL (marginal lane-sharing cost per unit cost saving). Figure 7 illustrates the impact of the network density and the change in Rc on the marginal deadheading cost. Consider a fixed value of Rc . Then, (i) for a fixed value of network density, an increase in the size of the network implies an increase in the number of lanes and consequently, a reduction in the use of deadheading in the DSN. Thus, the ratio CD =SDL decreases with the number of nodes. (ii) for a fixed size, an increase in density implies an increase in the number of lanes and, again, a lesser need of deadheading. Consequently, the ratio CD =SDL also decreases as the density increases. For the instances in the test bed, the highest value of the ratio CD =SDL corresponds to the case when the number of nodes in the network is 100, the network density is 0.2 and the value of Rc is 1. For ease of comparison, we normalize this value to 1 and scale the others accordingly. Observe that at low densities, the ratio CD =SDL decreases as Rc decreases. A decrease in Rc implies an increase in the subcontractor’s cost effectiveness in operating the lanes. The number of lanes in the network is limited at a low density, resulting in fewer cycles. Therefore, deadheading becomes necessary for creating profitable cyclic routes for the optimal DSN. A decrease in Rc also implies a decrease in deadheading cost (since deadheading cost on a lane is a fraction of the subcontractor’s cost in operating that lane). Therefore, as Rc decreases, the use of deadheading increases. As this added deadheading facilitates the extraction of more cycles, the cost saving also increases. Since the density is low, this increase in deadheading results in the ability to combine more lanes efficiently to create profitable cycles. Thus, as Rc decreases, the cost saving increases at a higher rate than the increase in deadheading cost. This implies a decrease in the ratio CD =SDL as Rc decreases. At a high density, as network size increases, the number of lanes in the shipper’s network also increases significantly. There is a corresponding increase in the availability of cyclic routes. As Rc

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Figure 7 Behavior of Normalized Marginal Cost of Deadheading with Change of Network Density

|V|=200

|V| = 100

CD S DL

CD S DL

Network Density

Network Density

(a)

|V|=300

(b)

RC : Extent of overlap in the ranges of the costs

ci0, j and c is, j RC

CD S DL

RC RC

Network Density

(c) Notes: The ratio CD/SDL has been normalized with respect to its highest value corresponding to |V| = 100, d = 0.2, and Rc = 1.

decreases, deadheading becomes cheaper. Therefore the use of deadheading increases, resulting in an increase in total deadheading cost. Since the subcontractor can operate the lanes at a lower cost, the total cost saving also increases. Also, since the shipper’s network is dense, the option of additional deadheading does not result in a substantial increase in the total volume of the DSN. Thus, as Rc decreases, although the use of deadheading increases, the ability of the optimal DSN to increasingly utilize deadheading to obtain a significant improvement in cost saving progressively reduces. Consequently, when the density is high and the size of the network is large, the ratio CD =SDL increases as Rc decreases. This behavior can be clearly seen in Figure 7c. Figure 8 illustrates the impact of network density, number of nodes, and the subcontractor’s cost effectiveness parameter Rc on the ratio CL =SDL . For a fixed value of Rc , at a low density, deadheading can be effectively used to extract the cycles that are included in the DSN. Hence, as described in the explanation for Figure 7, the opportunity for utilizing deadheading to improve the total cost saving increases. Consequently, at a low density, the use of lane-sharing does not result in a significant improvement in the cost saving. At high densities, deadheading is limited in its ability to further

improve the cost saving. In contrast, lane-sharing becomes increasingly useful for achieving cost saving. Therefore, as density increases, the ratio CL =SDL first increases and then typically decreases. At low density values, as described above, deadheading continues to improve the cost saving from a DSN as Rc decreases. Correspondingly, the use of lane-sharing continues to decrease. This behavior is illustrated in Figure 8. At high densities, the ability of deadheading to improve the solution decreases. However, note that lane-sharing alone may not be sufficient to extract a significant number of cyclic routes. Hence, at high densities, lane-sharing can better impact cost saving if there is an opportunity to use cheap deadheading as well to complete the cycles. Therefore, as Rc decreases, a cheap deadheading cost increasingly supports lane-sharing. This complementary use of deadheading and lane-sharing is evident in Figure 8c. 4.3.4. Strategic Alliances. Our goal is to illustrate that merging two shipping networks may result in a better DSN. To this end, we examine the special case of fully merging the networks of two shippers, that is, the volume on each lane in the merged network is the sum of the volumes of the two shippers on that lane. We quantify the net percentage improvement in cost

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Figure 8 Behavior of the Normalized Marginal Cost of Lane-Sharing with Change of Network Density

|V|=100

|V|=200

CL S DL

CL S DL

Network Density

Network Density

(a)

|V|=300

(b)

RC : Extent of overlap in the ranges of the costs

ci0, j and

cis, j RC

CL S DL

RC RC Network Density (c) Notes: To enable a fair comparison with Figure 7, the ratio CL/SDL has been normalized with respect to the highest value of CD/SDL corresponding to |V| = 100, d = 0.2, and Rc = 1.

saving due to merging, that is, the percentage increase in cost saving offered by the DSN identified on the merged network as compared with the sum of the cost savings from the two individual DSNs. To obtain a baseline value for comparing the cost savings due to merging, we consider a sequential approach. This approach works as follows: We assume that the two shippers do not share any information. Therefore, both the shippers offer the subcontractor the DSNs for their individual networks in the following sequential manner. Assume that the subcontractor treats the shippers on a first-come firstserve basis. That is, the shipper who approaches the subcontractor first extracts a DSN based on the subcontractor’s available node/arc capacities. This DSN consumes some of the subcontractor’s resources. Therefore, the subcontractor updates its node/arc capacities and then the second shipper extracts a DSN based on these updated capacities. For the instances in our test bed, the highest net cost saving due to merging is obtained when the number of nodes in the two networks is 200, the overlap (i.e., the number of common lanes) is 75%, and the density of each network is 0.2. We normalize this value to 100% and scale the other values appropriately. When two networks are merged, the net percentage improvement in cost saving is not necessarily posi-

tive. In other words, merging does not always guarantee an optimal DSN with a better cost saving. The demands increase on the lanes in the overlapping portion of the merged network. These “fatter" lanes could result in better cycles and a corresponding decrease in deadheading/lane-sharing. On the other hand, the increased volumes could disturb the balance of flow (achieved in the two individual DSNs) and, in turn, necessitate more deadheading/lane-sharing. Depending on the amount of non-subcontracted volume on the overlapping lanes in the optimal DSNs for the individual networks and the availability of subcontractor’s resources, an optimal DSN in the merged network may utilize (1) more or less deadheading, (2) more or less lane-sharing, or (3) more or less subcontracted volume than that in the two individual networks. Thus, an optimal DSN in the merged network is influenced by (i) positive effects due to an increase in the subcontracted volume and a decrease in deadheading and lane-sharing and (ii) negative effects due to decrease in the subcontracted volume and an increase in deadheading and lane-sharing. We first study the impact of the extent of overlapping on net percentage improvement in cost saving due to merging (Figure 9). When the two networks have the same density, the positive effects surpass the negative effects for most of the instances, thereby

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Normalized Net Percentage N Im Improvement in Cost Saving

Normalized Net Percentage Improvement in Cost Saving

Figure 9 The Impact of Merging Two Shipping Networks on the DSN

Extent of Overlapping

Extent of Overlapping

(b)

Normalized zed Net Percentage Improvement ent in Cost Saving

(a)

Extent of Overlapping

(c) Notes: The figures plot the behavior of the net improvement in cost saving with the extent of overlapping: (a) d1 = d2 = 0.2, (b) d1 = d2 = 0.5, (c) d1 = d2 = 0.8, where d1 and d2 denote the densities of the two networks.

resulting in a positive net improvement in cost saving. The trade-off between the positive and negative effects not only depends on the common density but also on the size of the network. When the size is small to moderate, that is, number of nodes |V| = 100 and 150, (resp., large, i.e., |V| = 200), the impact of the negative (resp., positive) effects increase as the extent of overlapping increases, resulting in a consistent decrease (resp., increase) in the marginal increase of the net improvement in cost saving. Similarly, as the common network density increases, the number of overlapping lanes increases. Therefore, the subcontractor’s capacity restrictions are more likely to get violated at higher densities. Consequently, as the common density increases, at all overlapping levels and for all sizes, the net percentage improvement in cost saving due to merging decreases. Figure 10 illustrates the impact of varying densities on the net percentage improvement in cost saving due to merging. For illustration purposes, let us denote the two densities by d1 and d2 . For simplicity, we fix the overlapping level at 50%. When the density of the first network is high (d1 ¼ 0:5; 0:8), we note the following effects in the merged network as d2 increases: (i) the number of overlapping lanes increases, (ii) the ability to create more cyclic routes in the non-overlapping portion of the network increases,

and (iii) the capacity restrictions become tighter. As a result, (a) the total volume subcontracted increases, (b) deadheading increases to compensate for tighter lane capacities, and (c) lane-sharing increases due to tighter lane/node capacities. Therefore, as d2 increases, the total saving first increases due to the positive effect of increase of volume in the DSN. A further increase in d2 necessitates an increase in deadheading and lane-sharing due to capacity restrictions, resulting in a decrease in marginal increase of the net percentage improvement in cost saving. When the density of one network is low (d1 ¼ 0:2), the demand in that network is limited and merging does not significantly increase the volume of the DSN. Therefore, the negative effects of merging discussed above progressively surpass the positive effect due to increase in volume, resulting in a near-steady deterioration of the net percentage improvement in cost saving as d2 increases. Since the models in this study focus on the problem of identifying an optimal DSN, a natural question arises regarding the impact (if any) of the DSN on the remainder of the shipper’s network. We now discuss this issue and justify our simplification for practical purposes. Theoretically, this issue highlights a limitation of our work and suggests an important direction for future work.

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The Impact of Varying Densities on the Net Percentage Improvement in Cost Saving Offered by the DSN in the Merged Network 30%

|V| = 100

Normalized Net Percentage Improvement in Cost Saving

Normalized Net Percentage Improvement in Cost Saving

Figure 10

15%

0% 0.2

0.5 Density d2

0.8

Normalized Net Percentage Improvement in Cost Saving

(a)

|V| = 150

60%

40%

20%

0% 0.2

0.5 Density d2

0.8

(b)

|V| = 200

60%

d d11 = 0.2 d11 = 0.5 d

40%

d1 d 1 = 0.8

20% 0.2

0.5 Density d2

0.8

(c)

5. Impact of Network Synergies on Cost Savings: Discussion of a Limitation Recall that the shipper’s objective is to maximize the cost savings due to subcontracting a DSN. When calculating the overall savings, we focused on the possible savings/losses on the lanes that are included in the DSN (section 2). That is, we assumed that the operating costs on the remaining lanes (i.e., the lanes that are not in the DSN) are not affected due to the extraction of the DSN. If the extraction of the DSN influences the cost of the remainder of the network (hereafter referred to as the non-DSN network), then the global optimization problem should consider both criteria, that is, the cost savings from subcontracting the DSN and the cost of operating the remainder of the shipper’s network. Our work only considers an approximation of this global optimization problem. The main reasons that make this simplification reasonable in practice are as follows: 1. In practice, the non-DSN lanes that are affected due to the extraction of a DSN are those that are tightly connected with the lanes in the DSN. Here, by “tightly connected," we mean those lanes that have either the origin or the destination in common with the DSN. Our experience with real-world implementations

suggests that the volume on these lanes is small (typically less than 10%) compared with the total volume in the shipper’s original network. Furthermore, these tightly connected lanes in the non-DSN network are of two types: (i) lanes connecting metro regions and (ii) lanes that have either origin or destination in a remote, non-metro region. Typically, the total volume (in load-miles) on the former type of lanes is significantly higher than that on the latter type. This observation is important because the volume on the non-DSN lanes is handled by lane-bylane auctioning or by using one-way carriers. The cost-effectiveness of these approaches depends on several factors such as the ability to utilize the backhaul on a lane, the location of the origin and the destination, the extent of competition between one-way carriers, the efficiency of the auctioning mechanism, etc. Since a majority of the traffic connects metro regions, competition is healthy and there are plenty of opportunities for one-way carriers to utilize backhaul. Consequently, lane-by-lane auctioning and use of one-way carriers are cost-effective ways of handling the tightly connected lanes (of the non-DSN network) that connect metro regions. Since the volume on the lanes that include non-metro centers is small, the impact of any possible increase in cost on these lanes is limited.

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In summary, there are two reasons why the impact of extracting a DSN on the non-DSN network is small in practice: (i) the total volume on the lanes in the non-DSN network that are tightly connected to the DSN is small and (ii) there are cost-effective ways in handling these lanes subsequently. For the many DSNs that have been implemented over the past 2 years, the increase in the cost of the non-DSN network—with respect to the total operating cost of the shipper’s entire network before subcontracting the DSN—has been below 2%. Our simplification (i.e., ignoring the non-DSN lanes in the optimization) has allowed us to obtain solutions that closely approximate the optimal solution of the global optimization problem. 2. As established in Theorem 1 (section 3), since the problem of finding an optimal DSN is itself strongly NP-hard, the global optimization problem is an extremely challenging problem. The large sizes of real-world shipping networks further underlines the difficulty in obtaining a reasonable solution. As a combined consequence of the above two reasons, finding the optimal DSN (without considering the impact on the non-DSN network) is a good and practical heuristic for the global optimization problem, and can be viewed as an important milestone toward a comprehensive solution.

6. Future Research Directions While the idea of a DSN of a shipper’s transportation network has been mentioned in recent times in research articles and trade magazines, there have been no comprehensive models aimed at understanding the relevant trade-offs. In this first-of-its-kind study, we formulate four fundamental problems that, together, help in identifying a good DSN that could be subcontracted. Our models incorporate the capacity restrictions of the subcontractor and exploit two key concepts—deadheading and lane-sharing— to improve the shipping volume in the DSN. After formulating the problems, we first resolve their complexities and then derive a number of properties—based on the topological structure of the shipper’s network and the relationships between the parameters — that enable us to identify several efficiently solvable special cases. Then, we develop a heuristic that provides near-optimal solutions to practical problem instances within a reasonable time limit. Based on real-world data from a national shipper, we design a representative test bed to demonstrate (i) the substantial monetary impact of subcontracting a DSN and (ii) offer several useful

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guidelines based on the density and size of the shipper’s network and the subcontractor’s relative cost advantage over the shipper. We also investigate the suitability of a strategic alliance—combining the networks of two different shippers—to identify a better joint DSN. There are a number of directions in which future work can proceed. We offer the following suggestions: 1. Network synergies: As discussed in section 5, the simultaneous consideration of (a) the cost savings from subcontracting the DSN and (b) the cost of operating the remainder of the shipper’s network (i.e., the non-DSN network), leads to a more sophisticated model (which we referred to as the global optimization problem). The importance of this problem naturally depends on two factors: the extent to which the non-DSN lanes are affected by subcontracting the DSN and whether or not there are costeffective ways of handling the non-DSN lanes subsequently. While our interactions with practitioners indicate that the simplification we have assumed is reasonable in practice, the global optimization problem is both important and challenging. 2. Pricing issues: It would be interesting to address more sophisticated pricing of the subcontractor’s services. There are several characteristics of a DSN that could be incorporated in advanced pricing models: (i) the ratio of the volume of the DSN to the total shipping demand in the entire network, (ii) the relative proportion of deadheading in the DSN, and (iii) the structure and geographic locations of the nodes in the DSN. It is clear that these characteristics affect both the shipper and the subcontractor, albeit in different ways. As such, the problem of obtaining equilibrium prices should be both interesting and challenging. Furthermore, in anticipation of a mature market, competition can also be modeled where two or more subcontractors bid for a DSN in the shipper’s network. 3. Practical issues: The model proposed in this study can be viewed as the “base” model for identifying a DSN. There may be additional constraints and objectives that arise from practical considerations. For instance, the shipper may want to impose lower and/or upper bounds on the size of the DSN, the amount of deadheading, and the amount of lane-sharing. In turn, the subcontractor may also want to ensure that the DSN is neither too small (i.e., the shipping volume is substantial) nor too

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large (to avoid the lack of trucking and human resources for handling the shipping volume). As another example of a real-world consideration, the shipper may want to restrict the DSN to be in a specific geographic area. 4. Strategic alliances: In section 4.3.4, we showed that fully merging two shipping networks may result in a better DSN, that is, one with better total cost saving, compared with the sum of savings from the individual DSNs for the two networks. It is important to note, however, that fully merging two or more shipping networks might not be the best possible choice. For example, a cooperative solution, where the shippers partially combine the lanes (i.e., use only a part of their respective volumes on each lane), may provide a better overall solution. An arbitrary solution of this type may use excessive volume of one shipper’s network and little from the other shipper’s network. To avoid such undesirable solutions, the two shippers may agree to merge portions of their network, subject to a priori lower bounds on their individual volumes. In general, the issue of a strategic alliance between two or more shippers, to exploit benefits vis- a-vis the DSN, is an interesting and challenging optimization problem.

Acknowledgments We thank the Departmental Editor, the Senior Editor, and two anonymous referees for several excellent suggestions that have collectively led to a significant improvement in the paper. We also thank Dr. Rajiv Namboothiri of the Inter-University Research Center on Enterprise Networks, Logistics and Transportation, Montreal, Canada, for many helpful discussions.

Note 1

We avoid revealing the name of the company to respect confidentiality.

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Supporting Information

Appendix: “Dedicated Transportation Design, Analysis, and Insights”

Subnetworks:

DOI 10.1111/poms.12029 © 2013 Production and Operations Management Society

Dedicated Transportation Subnetworks: Design, Analysis, and Insights Tharanga Rajapakshe Warrington College of Business Administration, University of Florida, Gainesville, Florida 32611,, USA, [email protected]

Milind Dawande Naveen Jindal School of Management, University of Texas at Dallas, Richardson, Texas 75080-3021,, USA, [email protected]

Srinagesh Gavirneni Johnson Graduate School of Management, Cornell University, Ithaca, New York 14853,, USA, [email protected]

Chelliah Sriskandarajah Mays Business School, Texas A&M University, College Station, Texas 77843,, USA, [email protected]

P. Rao Panchalavarapu Schneider Logistics Inc., 3101 South Packerland Drive, Green Bay, Wisconsin 54313-6187,, USA, [email protected]

dedicated subnetwork (DSN) refers to a subset of lanes, with associated loads, in a shipper’s transportation network, for which resources—trucks, drivers, and other equipment—are exclusively assigned to accomplish shipping requirements. The resources assigned to a DSN are not shared with the rest of the shipper’s network. Thus, a DSN is an autonomously operated subnetwork and, hence, can be subcontracted. We address a novel problem of extracting a DSN for outsourcing to one or more subcontractors, with the objective of maximizing the shipper’s savings. In their pure form, the defining conditions of a DSN are often too restrictive to enable the extraction of a sizable subnetwork. We consider two notions—deadheading and lane-sharing—that aid in improving the size of the DSN. We show that all the optimization problems involved are both strongly NP-hard and APX-hard, and demonstrate several polynomially solvable special cases arising from topological properties of the network and parametric relationships. Next, we develop a network-flow-based heuristic that provides near-optimal solutions to practical instances in reasonable time. Finally, using a test bed based on data obtained from a national 3PL company, we demonstrate the substantial monetary impact of subcontracting a DSN and offer useful managerial insights.

A

Key words: transportation network; dedicated subnetwork; deadheading; lane-sharing; heuristics History: Received: April 2012; Accepted: October 2012 by Michael Pinedo, after 1 revision.

freight transportation. Although railroads maintained their dominance until the early 1950s, improvements in productivity following the beginning of the “interstate" era resulted in a rapid growth in the trucking industry. Customers preferred trucking over railroads due to its flexibility and ability to customize services. Nevertheless, the trucking industry remained highly regulated until the early 1980s, with only a handful of national carriers. Thus, despite significant improvements in productivity, trucking companies priced their services considerably higher relative to railroads (Cambridge Systematics and Reebie Associates 2009). The deregulation of the trucking industry in the United States through the “Motor Carriers Act" of 1980 was a significant event that eliminated entry barriers and promoted the competitive pricing of

1. Introduction Dedicated networks operate semi-autonomously within the Schneider One-way truckload network. Using both third-party and internally developed optimization software, Engineering and Research sizes and designs these sub-networks to minimize operating costs, maximize synergy with the broader network, and reduce the impact of freight surges and flow imbalance. from The Schneider Enterprise, Ted Gifford, OR/MS Today, December 2007. Since the mid-1930s, the trucking industry in the United States has competed with the railroads for 138

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services. The existing carriers expanded their services, small trucking companies entered the growing market, and companies were allowed to publish their rates publicly. Consequently, competitive pressures forced companies to be efficient and productive to achieve profitability, resulting in comparatively low and flexible pricing along with customized service arrangements (Supply Chain Digest 2006). According to the American Transportation Research Institute (ATRI 2008), the trucking industry currently accounts for over 80% of revenue and 69% of tonnage of the freight-carrier market. A significant characteristic of the trucking industry is the high percentage of minor players (i.e., local 3PL providers) as compared with major operators (i.e., national 3PL providers). As reported by the ATRI, the number of interstate carriers grew from less than 20,000 in 1980 to more than 290,000 in 2006. In addition, there are about 504,000 private fleets and approximately 235,000 other interstate motor carriers registered with the US Department of Transportation. Although this is an overwhelmingly large number, about 97% of these operate with less than 20 trucks and about 89% with fewer than six trucks (ATRI 2008). It, therefore, comes as no surprise that the trucking industry is fiercely competitive and operates on thin profit margins. In the United States, more than 75% of the Fortune 500 companies use 3PLs to satisfy their logistics requirements (Armstrong Associates 2009a), while the global Fortune 500 3PL market has grown consistently over the recent past and was valued around US$199.7 billion in 2008 (Armstrong Associates, 2009b). In the United States, the trucking industry alone was responsible for about US$680 billion of the total estimated transportation cost of US $872 billion in FY 2008 (Burnson 2009). The large number of minor operators—and the resulting competitive pressures—leaves vulnerable those major operators with significantly owned assets (Baumann 2006). This is especially true during tough economic times (Field 2009). AAI predicts FY 2009 to be “the first recorded negative year in 3PL gross revenue growth" since 1996. According to Tom Sanderson, CEO of Transplace (a national 3PL provider), the 3PL business that is struggling somewhat is the owned fleets (Berman 2008). On the positive side, however, this structure of the industry presents an interesting business opportunity for the national trucking companies, namely, that of collaboration with the local players (Ozener 2008). In other words, national companies could subcontract a portion of their demand to local companies. Such a collaboration will not only benefit the local companies by reducing their market risk but also provide room for the national 3PL providers to shed their assets and concentrate more on investing resources in exploring new business opportunities.

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Industry experts believe that such a collaboration is indeed feasible: national 3PL providers cater to a large customer base and can, therefore, easily persuade the local trucking companies to work hand-in-hand to reap the benefits from the highly demanding and rapidly expanding market (Caplice and Sheffi 2003, Ergun et al. 2007). In light of these observations, the main goal of this paper is to study the problem of identifying a subnetwork in the network of a national 3PL provider (referred to henceforth as the shipper’s network) that is appropriate for such a collaboration. The identified subnetwork could then be operated independently— and, hence, referred to as a dedicated subnetwork—by one or more local trucking companies (Panchalavarapu 2010). We model the problem from the perspective of the national 3PL provider and measure the total cost saving that could be derived by subcontracting the identified subnetwork. Next, we develop the notion of a dedicated subnetwork more formally. 1.1. Dedicated Subnetworks A typical shipper’s network is defined by the set of origin–destination pairs, referred to as lanes, and the number of truck-loads (hereafter referred to as loads) that need to be shipped on each lane. The number of loads to be shipped in a lane over a certain time period (we discuss the planning horizon relevant to our problem in section 2) is typically referred to as the volume in that lane. We assume that the volume in a lane represents point-to-point traffic and ignore flow-routing decisions. These shipments can be made either by using the shipper’s own resources, that is, an owned fleet, or by using third-party resources, for example, small trucking companies who sell their services on a pre-determined set of lanes (Song and Regan 2003) or a common carrier such as UPS or JB Hunt (Caplice and Sheffi 2003; Sheffi 1990). Another novel alternative, motivated by the substantial presence of small 3PL companies, is to subcontract a part of the shipping network as a dedicated subnetwork (DSN). A DSN refers to a subset of lanes with associated loads in a shipper’s network for which a fleet of resources—trucks, drivers, and other equipment—is exclusively assigned to the subnetwork to carry out all its shipping requirements (Panchalavarapu 2010). The resources assigned to the DSN are not shared for carrying out shipping in the rest of the network. Hence, a DSN is an autonomously operated subnetwork, which could be subcontracted to a third party. We emphasize that the primary motivation behind identifying and subcontracting a DSN is to create an incentive for both the shipper and the subcontractor. On the one hand, subcontracting a DSN enables the shipper to shed its physical assets and expect dedicated service from the subcontractor. On the other

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hand, the requirement of flow balance in the DSN facilitates the efficient utilization of the subcontractor’s assets. Once the DSN is subcontracted, the remainder of the shipper’s network could also be either subcontracted or handled through lane-by-lane auctioning (Sheffi 2004; Song and Regan 2003). This approach has been proposed by practitioners: “[The dedicated subnetwork approach] identifies lanes that are suitable for dedicated contract carriage and those that are suitable for one-way moves. … After identifying the opportunities for dedicated moves, it is important to evaluate the implications for the remaining lanes, which will be served by other means.”(Panchalavarapu 2010). We assume that subcontracting a DSN does not impact the shipper’s operating cost of the remaining network (i.e., the lanes that are not included in the DSN). We discuss this assumption in detail in section 5 and provide a justification for our approach based on interactions with practitioners. The resources assigned to a DSN exclusively cater to the shipping requirements of that subnetwork. This exclusivity restriction necessitates the conservation of flow at each node in the DSN over the planning horizon: the total inflow of loads into a node equals the total outflow of loads from that node. This, in turn, implies that the flow requirements within a DSN can be decomposed into a collection of cyclic routes (see section 2). From a practical point of view, this is an important requirement. Since a subcontractor typically positions its fleet in diverse geographic locations, trucks are required to return to their respective bases after completing the assigned shipments (Chen and Vairaktarakis 2005, Golden et al. 2008, Li and Simchi-Levi 1990). In other words, trucks report to their corresponding home bases on the completion of the assigned deliveries. In an effort to maximize value-added activities, it is preferable to have a positive load on each lane of a vehicle route. Hence, it is natural to view each vehicle route in the subnetwork as a cycle. If the subnetwork does not satisfy flow conservation, then the subcontractor may have to incur additional costs due to a variety of consequences— empty travel of trucks, complex vehicle routing, compensation for the low quality of life of the truck drivers, higher driver turnover, etc.—to facilitate the routing of vehicles in the network. These non-valueadded activities, in turn, impact the pricing of the subcontractor’s services (Ozener and Ergun 2008). By identifying and subcontracting a sizable subnetwork that satisfies flow conservation, the shipper can (a) obtain a competitive price, (b) significantly reduce its owned fleet without diluting its customer base, and (c) focus attention on other non-asset-based businesses (e.g., logistics consulting) or on expanding its services to new markets (Baumann 2006).

Despite the above-mentioned benefits, several practical challenges can be expected in a real-world implementation of this idea.

•

•

While the requirement of conservation of flow at each node of a DSN is well-motivated, it may prevent the identification of a sizable subnetwork. For example, if the shipper’s network is sparse or the volume imbalance at the nodes (i.e., the difference between the total inflow and outflow of loads at the nodes) is high, then imposing flow conservation may result in no non-trivial DSN. In cases where non-trivial DSNs exist, such subnetworks may be too small (in terms of the total volume they address) to be of significance to the shipper and/or the subcontractor. To overcome this difficulty, in section 2, we use two notions— deadheading and lane-sharing—that aid in improving the size of the DSN. Deadheading has been used as an effective resource-repositioning tool in numerous studies; a discussion is provided later in section 2.1. Informally, deadheading allows for the introduction of dummy shipping volumes into the network (Dejax and Crainic 1987) while lane-sharing allows the shipping volume on a lane to be shared by both the shipper and the subcontractor (Caplice and Sheffi 2003). A DSN identified by the shipper may be infeasible for the subcontractor due to the latter’s inability to handle the required volume. Thus, it is natural for the problem of identifying a DSN to incorporate the capacity restrictions of the subcontractor. We consider two different capacities: 1.

2.

Lane capacities: For efficient tactical (day-today) operations of the DSN, the subcontractor often needs to maintain several depots. The routing of trucks has to consider the resources at each depot while ensuring an appropriate repositioning of these resources. Excessive loads on lanes during a certain time window might result in infeasibility. Therefore, as a macro restriction, the subcontractor may impose upper bounds on the possible loads on each lane. Node (origin/destination) capacities: Once a truck arrives at a destination city, it typically requires maintenance before getting assigned to another trip. The maintenance capacity at a city imposes an upper bound on the total inflow to (or outflow from) a city.

Our aim is to analyze the problem of obtaining an optimal DSN when either deadheading or lane-sharing

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or both are allowed. We also examine the relative trade-offs between these two notions and provide transportation managers with guidelines—based on topological properties of the shipper’s network and the relative values of the parameters—to help identify an appropriate DSN.

2. Identifying Dedicated Subnetworks: Models and Assumptions We start by introducing the parameters of our models and justifying the primary assumptions. Then, we formally introduce the basic problem (Problem P) of identifying a DSN in a shipper’s network. Subsequently, in sections 2.1 and 2.2, we introduce the notions of deadheading and lane-sharing and the corresponding two Problems PD and PL , respectively, to enable the extraction of a better DSN. Our most general problem is Problem PDL , where both deadheading and lane-sharing are allowed. The shipper’s network is represented by a directed graph G(V,A), where the nodes in V correspond to origins and/or destinations of loads and A denotes the set of lanes. We assume that the volume from an origin to a destination is an integral number of truck loads. The volume and mileage on each lane (i,j) ∈ A are denoted, respectively, by non-negative integers D(i,j) and M(i,j). The volume D(i,j) is assumed to be the point-to-point traffic between nodes i ∈ V and j ∈ V, and flow routing is ignored. Let the operating cost per load-mile (currently incurred by the shipper) be c0i;j ; ði; jÞ 2 A and purchasing cost of dedicated service per load-mile be csi;j ; ði; jÞ 2 A. As discussed above, let U a ði; jÞ (resp., Uin ) be the upper bound on the subcontractor’s capacity of the arc (i,j) 2 A (resp., node i 2 V). We now justify our assumption of deterministic shipping volumes. We envision D(i,j) to represent the exogenouslyspecified total expected volume (in truck loads) on lane (i,j) over a time period of 2–3 weeks. In practice, it is reasonable to assume that the shipper will have fairly accurate estimates of these numbers. Recall that our models are developed from the perspective of a shipper (a national 3PL provider) that caters to a substantial clientele, usually under long-term contracts. Thus, the volume on each lane is typically generated by several customers. Although the shipping requirements of an individual customer on each lane may exhibit some variance, the variance of the aggregate load over a large number of customers over a reasonably long period of time (e.g., over 2–3 weeks) is negligible. Furthermore, the shipper not only has a good knowledge of the expected volume handled on behalf of each customer, but is also equipped with a substantial amount of historical data to enable the computation of the total volume on each lane over such a time

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period. Assuming a normal business environment, the aggregate demand on a lane over any window of 2–3 weeks is typically expected to remain stable. Therefore, once obtained, the structure of the optimal DSN can be expected to remain the same over a longer horizon (e.g., a year). There is another reason for preferring a window of 2–3 weeks. The purpose of our analysis is to identify a subnetwork which, as discussed earlier, decomposes into a set of cyclic routes. In practice, it is important to ensure that the resources assigned to a cyclic route return to their home base on the route within a reasonable time period (Powell et al. 2002). If aggregate demand over 2–3 weeks is used to identify the subnetwork, then it follows that (in steady state) the resources assigned to any cyclic route in the subnetwork will be typically away from their home base for up to 2–3 weeks, a duration which is common in practice (United States Department of Labor 2010). While the precise day-to-day routing of these resources is not a concern addressed in this study, we do want to ensure that the identified subnetwork lends itself to practically feasible cyclic routes (Bramel et al. 1993; Laporte et al. 2000; Lee and Qi 2009; Li et al. 1992). We are now ready to formally introduce the basic problem of identifying a DSN in the shipper’s network. Problem P: Extracting a DSN (without deadheading and lane-sharing) Instance: A directed network G(V,A), non-negative integers M(i,j), D(i,j), and U a ði; jÞ for each arc (i,j) 2 A, and non-negative integers Uin for each node i 2 V. The operating cost per load-mile is c0i;j ; ði; jÞ 2 A and purchasing cost of dedicated service per loadmile is csi;j ; ði; jÞ 2 A. Solution: A subnetwork SG ðN; EÞ of G(V,A) that satisfies (i) flow conservation at each node in SG ðN; EÞ, that is, for each node i ∈ N, we have P P j:ðj;iÞ2E Dðj; iÞ ¼ j:ði;jÞ2E Dði; jÞ, and (ii) arc and node capacities, that is, Dði; jÞ U a ði; jÞ; 8ði; jÞ 2 E and P n j:ðj;iÞ2E Dðj; iÞ Ui ; 8i 2 N. An example is illustrated in Figure 1. P 0 s Objective function: Maximize ði;jÞ2E ðci;j ci;j Þ Dði; jÞMði; jÞ, the total saving in operating cost due to subcontracting. As mentioned earlier in section 1.1, the primary requirement for a DSN is that there be conservation of flow at each node of the subnetwork. This immediately implies—via the flow decomposition theorem (see, e.g., Ahuja et al. 1993)—that the DSN can be decomposed into a union of cyclic routes. There are two main difficulties with the definition (of a DSN) above that might prevent us from extracting a DSN that offers a substantial cost saving. First, as defined, if an arc (i,j) 2 E, then the entire demand D(i,j) on this arc is required to be assigned to the DSN. In other

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Figure 1 Extracting a Dedicated Subnetwork (DSN) from the Shipper’s Network: (a) Shipper’s Network and (b) A DSN without Deadheading and Lane-Sharing, where Uin = 250; i = 1; 2; . . .; 9, and U a (i; j) = 250; ∀(i; j); i 6= j; i = 1; 2; . . .; 9; and j = 1,2,…,9

D(i , j)

(a)

words, it is not possible to assign a fraction of the demand D(i,j) to the DSN. Second, if the shipper’s network G(V,A) is a sparse network, then it may not have any cyclic routes. In this case, there is no nontrivial DSN. Furthermore, even if there are cyclic routes, there may not be enough of them to provide a DSN that carries a substantial amount of the shipper’s volume. Motivated by these shortcomings, we now discuss two notions—deadheading and lane-sharing— that allow us to guarantee feasibility and enable the identification of better DSNs. 2.1. Deadheading In order to achieve the conservation of flow at each node in a DSN, we have the option of introducing dummy shipping volumes into the network. These correspond to situations in which trucks go empty from one node to another, an operation commonly referred to as deadheading (Dejax and Crainic 1987). For example, backhauling, the empty return trip of a truck after fulfilling a delivery, could be viewed as a special case of deadheading. The concept of deadheading is more general in that empty trucks can be assigned to travel on any lane, irrespective of whether or not it currently has a positive volume. Clearly, deadheading is a nonvalue-added activity for the shipper, since no real loads get transported. However, it has been widely used as a repositioning tool (see, e.g., Chen et al. 2009, Ergun et al. 2007, Ozener 2008, Ozener and Ergun 2008). Apart from being significantly different from these studies in its context, our work also considers (wherever appropriate) the impact of

(b)

lane-sharing on the solution. For example, Chen et al. (2009) consider an auction mechanism where the shipper is interested in providing carriers with a set of lanes that minimizes the total cost associated with truck repositioning. In this study, while the winner determination problem incorporates several practically relevant constraints such as bounding the total volume assigned to each carrier, the number of carriers selected, and the maximum number of carriers assigned to each lane, the shipper incurs no cost for sharing the volume on a lane among several carriers. While the detailed routing of trucks within the DSN is out of the scope of our study, the work of Ergun et al. (2007), Ozener (2008), and Ozener and Ergun (2008) focuses on analyzing the operational/strategic issues involved in routing. Since the resources assigned to a DSN are to be used exclusively for its shipping requirements, the subcontractor should naturally be compensated for deadheading. Therefore, we assume that the subcontractor charges a per load mile price cdi;j for deadheading; clearly, cdi;j \ csi;j since cdi;j (resp., csi;j ) involves empty (loaded) travel for the subcontractor. Obviously, maintaining too high a deadheading traffic is an inefficient use of resources in the DSN and results in the shipper incurring a higher cost per load-mile for the real shipping volume transported. On the other hand, if we restrict deadheading to be too low, then we may not be able to identify a sufficiently large DSN (Figure 1). We refer to the problem of identifying a DSN when deadheading is allowed as Problem PD . To avoid repetition, we formally define this problem as a special case of Problem PDL below.

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2.2. Lane-Sharing Under lane-sharing, both the shipper and the subcontractor are allowed to share the shipping volume on a lane (Caplice and Sheffi, 2003). In other words, a portion, say X(i,j) < D(i,j), is handled by the subcontractor and the remainder (i.e., D(i,j) X(i,j) > 0) is handled by the shipper. Clearly, lane-sharing is a relaxation of the condition that the volume on a lane be handled entirely by the shipper or entirely by the subcontractor. Like deadheading, lane-sharing also has the ability of enabling flow conservation over a larger DSN (as compared with Problem P). To understand the negative consequences of lanesharing, it is instructive to examine the dispatching process on a lane. The management of loads on each lane—meeting customer requirements, checking resource availability, assigning a load to the corresponding transporter, routing of trucks—is typically handled by one or more dispatchers (Brown and Graves 1981, Spieck et al. 1995). Each dispatcher manages a pre-assigned set of lanes which allows her to gain expertise over these lanes. From the shipper’s viewpoint, subcontracting the entire volume on a lane eliminates the need for a dispatcher on that lane. However, consider a situation when a lane is shared, for example, 40% of the volume is handled by the shipper and the remaining 60% by the subcontractor. Here, it is important to note that the entire demand on the lane does not occur simultaneously, but is instead generated over a period of time. Therefore, when a customer demand arises on a shared lane, the shipper needs to evaluate a host of issues—availability of own shipper’s resources compared with those of the subcontractor, the amount of loads subcontracted so far on this lane, the pros and cons of subcontracting now versus later, etc.—to decide whether or not the load is to be assigned to the subcontractor. Note that these decisions have to be taken regardless of the total volume on the shared lane and regardless of the split in volume between the shipper and subcontractor. We, therefore, model the fixed cost (incurred by the shipper) corresponding to the additional resources expended in making these decisions to manage a shared lane as cl . In addition to the increase in the complexity of the dispatching process discussed above, having multiple carriers serve the same lane also necessitates other costs. In Caplice and Sheffi (2003), the authors identify three different components—the real-time tendering system handling cross-company moves, shippers coming to agreeable terms for payment and cost sharing, and the capability of a freight payment and audit system to handle cross-company moves—that result in additional costs required for the effective collaboration of multiple shippers on the same lane

in freight transportation. The fixed cost cl in our model is also an attempt to capture these costs. Under lane-sharing the problem of identifying a DSN is referred to as Problem PL . Below, we formally define this problem as a special case of Problem PDL . Deadheading and lane-sharing offer two fundamentally different approaches for improving the DSN obtained by solving Problem P. It may be beneficial to use both these approaches to further improve the DSN. Also, if a lane is not entirely subcontracted, deadheading and lane-sharing could substitute each other. In other words, (i) lane-sharing could be eliminated by introducing deadheading on the shared lane and (ii) deadheading on a lane (i,j), where D(i,j) > 0 and X(i,j) = 0, could be eliminated by introducing lane-sharing. We refer to the problem in which both deadheading and lane-sharing are allowed as Problem PDL . If both deadheading and lane-sharing are allowed, the problem of identifying a DSN (Problem PDL ) is defined as follows. Problem PDL : Design of a DSN with deadheading and lane-sharing Instance: A directed network G(V,A) and non-negative integers M(i,j), D(i,j), and U a ði; jÞ for each arc (i,j) 2 A, and non-negative integers Uin for each node i 2 V. The operating cost per load-mile is c0i;j ; ði; jÞ 2 A, purchasing cost of dedicated service per load-mile is csi;j ; ði; jÞ 2 A, deadheading cost per load-mile cdi;j ; ði; jÞ 2 A, and lane-sharing cost cl per lane-shared. Solution: A subnetwork SG ðN; EÞ of G(V,A) that satisfies (i) flow conservation at each node in SG ðN; EÞ, that is, for each node i 2 N, we have X X ðXðj; iÞ þ Yðj; iÞÞ ¼ ðXði; jÞ þ Yði; jÞÞ j:ðj;iÞ2E

j:ði;jÞ2E

and (ii) arc and node capacities, that is, Xði; jÞ P U a ði; jÞ; 8ði; jÞ 2 E and j:ðj;iÞ2E ðXðj; iÞ þ Yðj; iÞÞ Uin ; 8i 2 N, where X(i,j) is the amount of volume subcontracted on lane (i,j) and Y(i,j) is the amount of deadheading on lane (i,j). An example is illustrated in figure 2. Objective Function: Maximize X X ðc0i;j csi;j ÞXði; jÞMði; jÞ cdi;j Yði; jÞMði; jÞ ði;jÞ2E

cl

X

ði;jÞ2E

Zði; jÞ ;

ði;jÞ2E

where Z(i,j) = 1 if lane (i,j) is shared with the subcontractor and Z(i,j) = 0 otherwise. Next, we define two subcases of Problem PDL : Problems PD and PL . Problem PD corresponds to the problem with X(i,j) = D(i,j), ∀(i,j) 2 A. Problem PL corresponds to the problem with Y(i,j) = 0, ∀i,j 2 V.

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Figure 2 Extracting a Dedicated Subnetwork from the Shipper’s Network in Figure 1: (a) With Deadheading, (b) with Lane-Sharing, and (c) with Both Deadheading and Lane-Sharing

D(i , j)

(a)

(b)

(c)

Notes: Deadheading is indicated by broken lines, M(i, j ) = 10, ∀(i,j ), i 6¼ j, i = 1,2,…,9, and j = 1,2,…,9, Uin = 250, i = 1,2,…,9, and Ua(i,j ) = 250, ∀(i, j ), i 6¼ j, i = 1,2,…,9, and j = 1,2,…,9.

To illustrate the advantage of solving Problem PDL , consider the shipper’s network in Figure 1a under the following cost parameters: c0i;j ¼ 1; csi;j ¼ 0:8; cdi;j ¼ 0:2; U a ði; jÞ ¼ 250; 8ði; jÞ 2 A, Uin ¼ 250; 8i 2 V, and cl ¼ 50. Figure 2 illustrates the optimal solutions for Problems PD , PL , and PDL . Without the DSN, the total operating cost over the entire network of Figure 1a for the shipper is 10,250. The savings in cost corresponding to Problems PD , PL , and PDL are, respectively, 900, 720, and 1020. Thus, Problem PDL has the potential to substantially improve savings. Our next goal is to theoretically analyze the four problems: P, PD , PL , and PDL . In particular, we resolve their computational complexities and explore several interesting special cases arising from simplifications in the parametric relationships and the topological properties of the shipper’s network.

3. Problems P, PD , PL , and PDL: Theoretical Results and Heuristics To precisely define the problems discussed in the previous section, we formulate them as mixed-integer programs (MIPs). The formulations are provided in section A.1 of the Online Appendix. In section 3.1, we show that all these problems are strongly NP-hard as well as APX-hard. Furthermore, these problems remain hard for practically common topological structures such as Hub-Spoke and Trees. In section 3.2, we discuss some interesting polynomially solvable special cases that are derived from two sources: (i) simplifications resulting from the relationships between the parameters of the

models (section 3.2.1) and (ii) simplifications in the structural properties of the shipper’s network (section 3.2.2). Using the insights obtained from the theoretical analysis, we develop a heuristic that can provide efficient solutions to practical problem instances within a reasonable time limit. The heuristic is described in section 3.3 The heuristic is later evaluated on a comprehensive test bed and the results are discussed in section 4.3.2. The proofs of all technical results are placed in section A.2 of the Online Appendix. 3.1. Hardness Results for Problems P, PD , PL , and PDL In section 3.1.1 we first show that Problem P is strongly NP-hard on hub-spoke networks. The strong NP-hardness of the four problems on general networks then follows from this result. Furthermore, in Theorem 3, we establish that these four problems are APX-hard as well. 3.1.1. Hardness Results for Hub-Spoke Networks and Trees. Hub-Spoke and Trees are two commonly observed network structures in transportation (Hu 2010, Kim 1998, Kunnumkal and Topaloglu 2010, Pirkul and Schilling 1998). In a hub-spoke network (also referred to as a pool-point network—see Figure 3a), a set of (well-separated) hubs act as depots, collecting and redistributing loads to nearby destinations; the link between a hub and such a nearby destination is referred to as a spoke. Typically, a hub is linked by spokes to several destinations; these destinations can then only be served by the assigned hub. Hence, it is clear that the flow of loads between a

Rajapakshe, Dawande, Gavirneni, Sriskandarajah, and Panchalavarapu: Dedicated Transportation Subnetworks Production and Operations Management 23(1), pp. 138–159, © 2013 Production and Operations Management Society

Figure 3

Special Network Topologies: (a) Hub-Spoke and (b) Tree

Hub

(a)

hub and a corresponding destination point is bidirectional. In practical hub-spoke networks, the volume of transportation between the hubs is typically much more than that between a hub and a spoke (Elhedhli and Hu 2005, Geismar et al. 2011, Pirkul and Schilling 1998). Some other practically common transportation structures such as a star network (introduced later in section 3.2.2) are special cases of a general hub-spoke structure. A hub-spoke topology is capable of offering fast delivery and a high level of customer service (Elhedhli and Hu 2005, Pirkul and Schilling 1998). A tree (or a hierarchical network; see Figure 3b) is a more general acyclic network. For example, a hierarchical network of plants, warehouses, and retailers in which unidirectional transportation occurs from plants to warehouses and from warehouses to retailers, is a tree. Such a structure typically occurs in import distribution channels (Kim 1998). Theorems 1, 2, and Corollary 1 comment on the hardness of our problems for these two special network structures. THEOREM 1. Problem P is strongly NP-hard for hubspoke networks. COROLLARY 1. NP-hard.

Problems PD , PL , and PDL are strongly

THEOREM 2. For tree networks, Problems P and PL are infeasible while Problems PD and PDL are strongly NPhard. Theorem 3 below shows that all four problems are APX-hard as well. Hence, unless P = NP, there is no polynomial time approximation scheme for any of the four problems. THEOREM 3. hard.

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Problems P, PD , PL , and PDL , are APX-

Supply/ Demand Location

(b)

3.2. Polynomially Solvable Special Cases In this section, we identify several polynomially solvable special cases of the four problems: P, PD , PL , and PDL . These special cases are derived from two sources: (i) relationships between the parameters of the models (section 3.2.1) and (ii) characteristics of the topological properties of the shipper’s network (section 3.2.2). In our discussion, we will often use the well-known minimum cost flow problem on a directed network G(V,A). The formulation of the minimum cost flow problem is provided in section A.1 of the Online Appendix. 3.2.1. Simplifications Due to Relationships Between the Parameters of the Model. When a shipper caters to a wide variety of customers from diverse geographic markets, it is possible that the shipper’s network does not exhibit any specific topological structure (Kleywegt et al. 2006). In such a general network, one can exploit the relationships between the various parameters of the model to derive efficient algorithms for the four problems. We start by discussing a basic property (Lemma 1), which is used in our subsequent discussion in this section. LEMMA 1. In any optimal DSN to Problem PDL , on a given lane (i,j), either Y(i,j) = 0 or Xði; jÞ ¼ minfDði; jÞ; U a ði; jÞg. As a consequence of the above result, we conclude that if an optimal solution to PDL has deadheading on lane (i,j), then the total volume that can be subcontracted (without violating the subcontractor’s resource requirement) on this lane is already in the DSN (i.e., Xði; jÞ ¼ minfDði; jÞ; U a ði; jÞg). For Problems P and PL , the subcontracted volumes (i.e., the values of X(i,j)) satisfy flow conservation at each node, while for Problems PD and PDL , the sum of the

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subcontracted and deadheading volumes (i.e., X(i,j) + Y(i,j)) satisfy flow conservation at each node. It is, therefore, clear that any feasible solution to these problems can be decomposed into a union of distinct maximal cyclic flows in the corresponding variables (i.e., X(i,j) for P and PL ; X(i,j) + Y(i,j) for PD and PDL ). Recall from section 2.2 that the lane-sharing cost cl is the fixed cost corresponding to the additional resources expended in managing dispatching decisions on a shared lane. If additional resources are already available or if such resources are cheap to acquire, then the lane-sharing cost will typically be negligible (and can, therefore, be ignored) or be insignificant as compared with the operating and deadheading costs. In such cases, Lemma 2 shows that an optimal or near optimal solution to Problems PL and PDL can be obtained in polynomial time.

In this case, we can obtain a polynomial-time solution to Problems PD , PL , and PDL with an additive guarantee that is a linear function of r.

3.2.2. Simplifications Due to Structural Properties of the Transportation Network. We focus our attention on some widely observed network structures in practice: a star network together with its variants and a hub-spoke network. The special characteristics of these networks allow for several polynomially solvable special cases. In several practical settings, shippers are involved in transporting goods originating from a single point. For instance, consider a shipper responsible only for transporting imported goods from a harbor to several main warehouse locations serving a wide customer base (Horizon Logistics 2009, 2010). In this setting, loads originate from the harbor and the destination points (warehouses) can be considered as pure demand points (Horizon Logistics 2010). Such a network is typically referred to as a star network (Figure 4a). In such a network, since a node is either a pure supply or a pure demand point, deadheading is necessary to ensure the conservation of flow required by a DSN. Furthermore, since each destination point is exclusively connected to the origin, it is reasonable to assume that the distance between any two destination points is significantly larger than the distance between any origin-destination pair (Chen 2010). We, therefore, assume this to be the case. This assumption implies that, in an optimal solution, deadheading can occur only on the origin-destination lanes of the network. Accordingly, we assume that there is no transportation between two destination points of a star network. Two other more-sophisticated variants of the star topology are also observed in practice: (i) a connected-star network, where two or more stars are connected through one or more common destination points (Figure 4b) and (ii) a hybrid-star network, where the destination points of a star act as origin points to other star networks (Figure 4c). In turn, a hybrid-star is a special case of the more-general

LEMMA 2. When the lane-sharing cost cl ¼ 0, Problems PL and PDL can both be solved in polynomial time. More generally, if 0 \ cl e, then a near-optimal solution, with an additive guarantee that is a linear function of e, can be obtained in polynomial time. The following two results address the situation when (i) the demand on each lane of the shipper’s network is (approximately) the same and (ii) the subcontractor has the capacity to handle any amount of volume. Lemma 3 is used for proving the more general statement of Lemma 4. LEMMA 3. When U a ði; jÞ ¼ 1; 8ði; jÞ 2 A; i 6¼ j and Uin ¼ 1; 8i 2 V, and the demand on each lane in the shipper’s network is a constant, that is, D(i,j) = D, ∀(i,j) 2 A, Problems P, PD , PL , and PDL can be solved to optimality in polynomial time. Furthermore, there exists optimal solutions to PL and PDL that do not have lanesharing. LEMMA 4. Let U a ði; jÞ ¼ 1; 8ði; jÞ 2 A; i 6¼ j, Uin ¼ 1; 8i 2 V, and the demands in the shipper’s network D(i,j) 2 [(1 r)D,(1 + r)D], ∀(i,j) 2 A, where 0 < r 1. Figure 4

Special Network Topologies: (a) Star, (b) Connected-Star, and (c) Hybrid-Star

Origin

(a)

Destination

(b)

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hub-spoke topology (section 3.1.1). Again, we assume that, for these topologies, there is no transportation between two points that are destination points of the origin node of some star network. For these special structures, since deadheading is necessary for a feasible DSN, Problems P and PL are infeasible. We, therefore, comment only on Problems PD and PDL . Theorem 4 comments on the computational complexities of the star, connected-star, and hybrid-star network topologies. THEOREM 4. Problems PD and PDL are weakly NP-hard for star, connected-star, and hybrid-star network topologies. When the node capacity at the origin is unlimited, both the problems can be solved to optimality in polynomial time. Next, consider a hub-spoke network (section 3.1.1) in which there is no hub-to-hub transportation. In this case, the shipper’s network decomposes into multiple independent hub-spoke networks, each with a single hub. Note that these single-hub networks are different from star networks since the transportation between a hub and its destination points is bidirectional (unlike the unidirectional flow in a star network). Recall that each spoke location exclusively communicates with the corresponding hub and there is no transportation between spoke locations. Therefore, it is sufficient to examine the volumes between the hub and a spoke location to decide on the subcontracting of the corresponding lane. As before, we assume that there is no deadheading between spoke locations. The following lemma addresses the complexity of our four problems on such a network. LEMMA 5. Problems P, PD , PL , and PDL are weakly NP-hard on a hub-spoke network, even if there is no hubto-hub transportation. When there is no hub-to-hub transportation and no capacity restriction at the hubs, these problems are all polynomially solvable.

3.3. The Flow-Fix-Improve Heuristic In this section, we provide a detailed description of a heuristic, referred to as Flow-Fix-Improve (FFI), for Problem PDL . The efficiency and the robustness of the heuristic is demonstrated later in section 4.3. The corresponding heuristics for Problems PD and PL can be obtained in a similar manner. We first describe the basic intuition behind FFI and then provide a formal procedure. Let I denote an instance of PDL . As shown in Lemma 3, when the shipper’s network has the same demand on each lane, then I can be solved optimally by transforming it to a minimum cost flow problem. Furthermore, this solution does not have any shared

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lanes. Motivated by this result, our first attempt is to extract a subproblem, say I1 , of I, which can be used to identify a set of lanes on which lane-sharing is absent. This task, however, poses a challenge. When the demands on the lanes in the shipper’s network vary significantly, the solution obtained using the transformation in Lemma 3 may not be a good approximation to PDL since it ignores the impact of lane-sharing cost. By ignoring lane-sharing cost, the minimum cost flow solution is expected to have more unsaturated lanes as compared with the optimal DSN. To overcome this drawback, the first step of the heuristic identifies a subset of lanes with small variation in demand. Note that our ability to extract cycles with no lane-sharing also improves with the density of the subnetwork corresponding to I1 . Thus, there is a need to balance the following two properties: the lanes under I1 should have limited variation in demand and the density of the subnetwork should be sufficiently high. Assuming that the demands in I follow a Gamma distribution, we select (for inclusion in I1 ) the demands that fall within the range [min{0, (k1)har},(k1)h + ar], where a 0 is a constant that decides the range. This assumption is later justified using real-world data in section 4.1. Note that a Gamma distribution is characterized by two parameters: scale h > 0 and shape k > 0; the mean of the distribution is kh and the mode is (k 1)h for k 1. A tentative DSN is obtained by solving a minimum cost flow problem on I1 . Clearly, this solution is also feasible for the original instance I. Due to the manner in which the instance I1 is extracted, it is reasonable to expect that there are only a few lanes shared between the shipper and subcontractor in the minimum cost flow solution corresponding to I1 . The second step of the heuristic creates a new instance, say I2 , using the original instance I and the solution obtained for I1 . To get I2 , we first fix the flow on the lanes with no lane-sharing in the solution to I1 . Then, we include all the other lanes from I with the corresponding node and lane capacities. The generated instance I2 is solved by CPLEX using the MIP formulation provided in the Appendix. Two points concerning the second step deserve mentioning. Since we fix the flow on a set of lanes (identified in the first step), the number of remaining lanes, on which flow is to be decided, in I2 is typically small. Therefore, the MIP can be solved effectively. Also, the inclusion of the lanes on which demands vary significantly and the consideration of the lane-sharing cost (i) improves the size of the DSN and the cost saving and (ii) possibly eliminates unnecessary lane-sharing through efficient utilization of the deadheading. Thus, the second step can be considered as one that improves the solution of the first step. We now formally describe the heuristic.

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The Flow-Fix-Improve Heuristic (FFI) Input: The network G(V,A), along with its parameters cl ; Dði; jÞ; Mði; jÞ; c0i;j ; csi;j ; cdi;j ; U a ði; jÞ; Uin ; the parameters of the Gamma distribution k,h,r, and the range constant a. Step 1: Define G0 ðV 0 ; A0 Þ such that V0 ¼ V; A0 ¼ ði; jÞ : minf0; ðk 1Þh arg Dði; jÞ ðk 1Þh þ ar; ði; jÞ 2 A : Step 2: (flow solution). Solve for the minimum cost flow solution on G0 ðV 0 ; A0 Þ. Let f be the flow vector obtained after solving for the minimum cost flow solution. Step 3: (fixing step). Fix the flows on the arcs in A00 A0 A, where A00 ¼ ði; jÞ : fði; jÞ ¼ Dði; jÞ, Dði; jÞ U a ði; jÞ; ði; jÞ 2 A0 . Step 4: (improvement step). Solve the Problem PDL on G(V,A) with the constraint Xði; jÞ ¼ Dði; jÞ; ði; jÞ 2 A00 , where X(i,j) is the volume subcontracted on lane (i,j). Output: The subcontracted and deadheading volumes X(i,j) and Y(i,j), respectively.

4. Understanding the Trade-off between Deadheading and Lane-Sharing Our aim in this section is to (i) test the performance of the heuristic, (ii) examine the monetary impact of subcontracting a DSN, and (iii) offer managers insights into the influence of density and size of the shipper’s network on the relative extents of deadheading and lane-sharing in an optimal DSN. To focus our investigation on realistic instances, we utilize real-world data obtained from a national 3PL provider. Before proceeding further, we briefly summarize the main issues addressed in our computational study: 1. Performance of the heuristic: In section 4.3.2, we evaluate the efficiency as well as the effectiveness of the FFI heuristic on a comprehensive test bed. 2. Monetary impact: In section 4.3, we demonstrate the attractiveness of the DSN concept by computing the cost saving offered by an optimal DSN as a percentage of the shipper’s total operating cost. 3. Dense vs. sparse networks: The density of the shipper’s network, that is, the fraction of the number of lanes in the network to the total possible number of lanes, is expected to be a fundamental factor in the extent to which deadheading and lane-sharing are utilized. In

section 4.3.3, we analyze DSNs on networks with varying densities to understand the relative behavior of deadheading and lane-sharing. We also study the impact of the subcontractor’s cost effectiveness (relative to the shipper) on the amount of deadheading and lane-sharing in the DSN. 4. Strategic alliances: Fierce competition has forced logistics providers to investigate novel approaches to lower costs. One emerging thought is that two or more providers could engage in a strategic alliance to benefit from the resulting increase in shipping volumes in the combined network (Ergun et al. 2007). In section 4.3.4, we examine the impact of merging the networks of two shippers on the DSN.

4.1. A Real-World Instance Data obtained from a national 3PL provider1 helped us in designing a test bed that is based in reality. We first briefly summarize the details of the real-world logistics network of the provider and then compute the cost saving (section 4.3.1) from subcontracting a DSN for this network. The provider serves a diverse set of customers (companies) spread across the continental United States. The total number of distinct customers is about 600, with each customer generating demand on several lanes spread over multiple states. We provide aggregate statistics over the 6-month period from March to August 2007. The actual demand is generated incrementally over this time period. The network has about 12,000 distinct lanes with positive demand, with about 5000 distinct origin/destination points. Figure 5 shows a snapshot of the origin/destination points. The mileages of the lanes vary from 10 miles to about 3000 miles; over all lanes, the average mileage is 668.50 miles. For a broader view, it is convenient to cluster all origin/destination points from the same state to a single node. Thus, this clustered, state-based network has 50 nodes and a density of about 35%. The volumes on the lanes in the clustered network vary from 1 to 3199 truck loads. Figure 6 provides a better understanding of the distribution of demand in the shipper’s network. As illustrated in the figure, the frequencies of the low-demand numbers are significantly higher than the high-demand numbers. We observe the same behavior under other clustering methods based on cities and zip codes. Moreover, this pattern is consistent even when we consider the demands over shorter time windows. We now provide an explanation for the demand behavior in Figure 6. The average traffic on a lane in the clustered network is around 15 truck loads. A careful analysis

Rajapakshe, Dawande, Gavirneni, Sriskandarajah, and Panchalavarapu: Dedicated Transportation Subnetworks Production and Operations Management 23(1), pp. 138–159, © 2013 Production and Operations Management Society

Figure 5

Figure 6

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A Snapshot of the Origin/Destination Points for the Real-World Example

Demand Distribution in the Real-World Network of a National Shipping Company

200

Frequency

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100

50

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 More

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Demand (in Truck Loads)

reveals that the states with high to-and-from traffic (e.g., exceeding 200 truck loads) represent some of the main population centers, industrial hubs, and states in which the company has a substantial presence. There is significant inter-state traffic between Indiana, Texas, Illinois, Virginia, Georgia, Missouri, California, Colorado, Pennsylvania, and Ohio. These represent major demographic/industrial hubs, and we observe high demands between these states from transportation of goods between key geographic sites (of the customers), for example, between ports and

warehouse locations, between distribution centers. The shipper has a service hub in Texas, with around 45% of the total traffic in the network either entering or leaving this state. The shipper’s reputation and efficiency in serving such geographical locations (i.e., those where it has significant presence) attracts considerable business. Therefore, we observe high to-and-from traffic associated with such locations. The demand originating from the states mentioned above to those with moderate populations and industrial activities are in the medium range (around

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75–200 truck loads per lane). Some examples include Georgia–Arkansas, Texas–Florida, and Georgia–Florida. A majority of the small customers are geographically dispersed and generate small demand volumes. For example, Massachusetts, Idaho, Maine, Montana, North Dakota, etc. have significantly lower to-andfrom traffic. The above observations indicate that the demand distribution of Figure 6 —where frequencies of lower demands are significantly larger than higher demands—can be expected for a typical national shipping network. This justifies our earlier assumption in section 3.3 that the demand on a lane follows a Gamma distribution. 4.2. Test Beds for Computational Experimentation To investigate the four issues mentioned at the beginning of this section, we now describe four separate test beds. 4.2.1. Test Bed 1: To Test the Performance of Heuristic FFI. The efficiency and the effectiveness of the heuristic is tested by varying (i) the number of nodes, (ii) the density of the shipper’s network, and (iii) the shape parameter k of the Gamma distribution. For the real-world data, about 50% of the demands are less than 60 truckloads. Hence, to anchor our test bed on practice, we set the scale parameter h = 60/k. We assume that fuel cost can vary in the range of USD 3–4 per gallon and a truck can travel about 10 miles per gallon. Therefore, as a base case, the cost parameters c0i;j and csi;j are both generated from U[0.3,0.4]; the ranges for these two costs are varied later for our other experiments. The deadheading cost shared by the shipper for a lane is set at 30% of the operating cost of the subcontractor for the same lane. Based on the structure of the clustered network described in section 4.1, the mileages of the lanes are generated from U[100,1000]. Recall that the lane-sharing cost, cl , represents the fixed cost incurred for the additional resources required to manage dispatching decisions on a shared lane. Based on the information received from the 3PL provider, we assume that one extra dispatcher, together with the required infrastructure, would be enough to manage this additional workload on a shared lane. We assume that the annual salary of a dispatcher is US$60,000 (http://swz.salary.com) and scale it appropriately (to be consistent with the time period of 2–3 weeks for the shipping volumes on the lanes, network density, and the typical workload assigned to a dispatcher per day). The lane capacities are generated from U l þ 2r; l 0:5r and the node capacities are generated pﬃﬃﬃﬃﬃ from U jVjdl þ r; jVjdl r where l ¼ kh; r ¼ kh, and d is the network density. Table 1 summarizes the parametric setting used in this experiment. For each combination of the

Table 1. Parameter Settings to Evaluate the Performance of the Heuristic Parameter

Values

Number of nodes, |V| Network density, d Demand on a lane, D(i,j) Mileage of a lane, M(i,j) 0 Shipper’s operating cost, ci;j Subcontractor’s operating cost, ci;js Lane-sharing cost, cl

100, 200, 300 0.4, 0.6 Γ[k,h],k = 1,2,3 U[100, 1000] U[0.3, 0.4] U[0.3, 0.4] 3600

parameters we generate three instances of PDL , for a total of 3 9 2 9 3 9 3 = 54 instances. 4.2.2. Test Bed 2: To Test the Impact of Density, Size, and Costs on the DSN. To understand the trade-off between deadheading and lane-sharing costs, we vary (i) the number of nodes (|V|), (ii) the density (d), and (iii) the operating costs of the shipper (c0i;j ) and subcontractor (csi;j ). We use a parameter, referred to as Rc , to measure of the extent of overlap between the uniform supports used for generating the cost parameters c0i;j and csi;j . Three values of Rc are considered: Rc ¼ 1 (resp., Rc ¼ 0:5, Rc ¼ 0:25), corresponds to the case where c0i;j is generated from U[0.3,0.4] and csi;j is generated from U[0.3,0.4] (resp., U[0.25,0.35], U[0.225,0.325]). Intuitively, the parameter Rc measures the cost effectiveness of the subcontractor relative to the shipper. The generation of lane capacities and arc capacities on a lane and the value of lane-sharing cost is the same as in section 4.2.1. To represent practical problem instances, the shipping volume on each lane in the shipper’s network is generated from Γ[1,30]. The mileages of the lanes are generated from U[100,1000]. Table 2 indicates the values of the various parameters. For each combination of the parameters, we generate five instances of Problem PDL . Thus, there are a total of 3 9 3 9 3 9 5 = 135 instances in this test bed. 4.2.3. Test Bed 3: To Test the Impact of Merging Networks of Two Shippers on the DSN. As mentioned earlier, merging the networks of two shippers can possibly lead to a DSN that offers higher saving as compared with the sum of the savings from the two individual DSNs. To focus solely on the impact of merging and to avoid any undesired influence due to Table 2 Parameter Settings to Study the Impact of Density, Size, and Costs Parameter Number of nodes, |V| Network density, d Mileage of a lane, M(i,j) 0 Shipper’s operating cost, ci;j Extent of overlap, Rc

Values 100, 200, 300 0.2, 0.5, 0.8 U[100, 1000] U[0.3, 0.4] 1, 0.5. 0.25

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differences in the sizes and cost parameters of the two networks, we assume the same cost parameters for both the networks: c0i;j U½0:3; 0:4; csi;j U½0:3; 0:4; cdi;j ¼ 0:15csi;j . Demand is generated from Γ(2,30). All the other parameters are as described in section 4.2.1. The idea of merging the networks of two shippers is interesting only when there is a substantial overlap in the lanes of these networks. In the absence of such an overlap, the two networks are essentially disjoint (e.g., when the two shippers operate in different geographic areas) and the optimal DSN for the merged network is, for the most part, the union of the optimal DSNs for the two individual networks. Two key characteristics of the networks that are likely to dictate the impact of merging are the extent of overlap and their relative densities. Accordingly, we choose three levels for the number of overlapping lanes: 25%, 50%, and 75%, and six combinations of the densities of the two networks: (0.2,0.2) (0.2,0.5) (0.2,0.8) (0.5,0.5) (0.5,0.8) (0.8,0.8). To allow for different sizes of the two networks, we consider three values of the number of nodes, |V| = 100, 150, and 200. As before, we generate five instances for each parameter combination. Thus, this experiment considers 3 9 3 9 6 9 5 = 270 instances. 4.3. Results, Insights, and Intuition We first quantify the potential benefit from the optimal DSN on the real-world network and then justify the quality of heuristic FFI. The last two subsections summarize the useful insights gained from our computational experiments. 4.3.1. The Benefit of Subcontracting a DSN on the Real-World Network. A reasonable metric to indicate the benefit of extracting a DSN is the cost saving it offers the shipper, as a percentage of the total operating cost the shipper would have to incur (in the absence of the DSN) to satisfy all demand over the entire network. For the real-world network (described in section 4.1), we obtained the data on demands, mileages, and operating costs on the lanes from the shipper. The costs and the resource constraints of the

Table 3 Parameter Settings to Study the Impact of Subcontractor’s Cost and Lane Capacities on the DSN from the Real-World Network Parameter Subcontractor’s cost, Arc capacity, U a ði; j Þ

Values 0 0 U½ac i;j ; 1:1ci;j ,

a = 0.5,0.6,0.7 U l 0:5r; l þ br , b = 1,1.5,2

s ci;j

subcontractor were not available to us. Therefore, to understand the impact of the (i) subcontractor’s cost and (ii) lane capacities on the DSN, we vary the uniform supports used in the generation of above two parameters as shown in Table 3. The node capacities are set as discussed in section 4.2.1. We generated one instance for each parametric combination, for a total of 3 9 3 = 9 instances. When calculating the lane and node capacities, the shape parameter k of the Gamma distribution (section 3.3) was set to 1. For each of the nine instances, we identify a DSN using Heuristic FFI. Table 4 summarizes the impact of varying a and b on several useful metrics of the resulting DSN. We now provide the average statistics over all nine instances. For the clustered real-world network (see section 4.1), the average percentage cost saving (for the shipper) from the DSN after allowing for both deadheading and lane-sharing is about 10%. The maximum (resp., minimum) cost saving is 13.3% (resp., 6.3%). The average total volume (load miles) in the DSN is about 60% of the total volume in the shipper’s network. The average deadheading volume in the DSN is about 10%. The average deadheading cost, as a percentage of the total cost saving, is about 4%. The average number of lanes shared is 12. The average lane-sharing cost as a percentage of the total cost saving is about 4%. Since the network density is low (about 40%), the number of cycles is limited. Therefore, the DSN uses a significant amount of deadheading volume to extract useful cycles. For any fixed value of b, as the subcontractor’s cost effectiveness increases (i.e., as a decreases) the (i) percentage cost saving from subcontracting a DSN increases, (ii) percentage volume subcontracted

Table 4 Impact of Subcontractor’s Cost and Arc Capacities on the Extraction of a DSN

% Cost saving Subcontractor’s cost reduces↓ a = 0.7,b = 1 6.3% a = 0.6,b = 1 9.1% a = 0.5,b = 1 12.0% Arc capacity increases↓ a = 0.7,b = 1 6.3% a = 0.7,b = 1.5 6.8% a = 0.7,b = 2 7.0%

% Volume subcon. (load miles)

% Deadheading volume in the DSN

% Deadheading cost w.r.t. Cost Saving

% Lane-Sharing cost w.r.t. Cost Saving

54.4% 58.3% 61.6%

9.1% 10.2% 10.3%

4.0% 3.1% 2.4%

6.2% 5.2% 5.0%

54.4% 58.3% 60.4%

9.1% 8.6% 8.5%

4.0% 4.3% 4.6%

6.2% 4.0% 3.8%

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increases, and (iii) percentage deadheading volume decreases. To observe the impact of arc capacities on the percentage cost saving, for each value of b, we compute the percentage cost saving by averaging over the three instances obtained by varying the value of a. As b increases, the percentage cost saving also increases. Since tighter lane capacities result in lower volume in the DSN and possibly more lane-sharing, the cost saving suffers at lower values of b. As b increases, the ability to subcontract more volume on a given lane increases. Also, the solution has the ability to possibly eliminate lane-sharing (on the lanes where demand is higher than capacity). Hence, the solution uses lower lane-sharing, more volume, and more deadheading to achieve a higher cost saving. 4.3.2. Performance of the Heuristic FFI. Table 5 summarizes the performance of the heuristic for Problem PDL . CPLEX (version 11.1) could not solve any of the problem instances to optimality within an imposed CPU time limit of 3600 seconds. For each instance, we compare the heuristic solution with the best upper bound available from CPLEX within the time limit. We report the percentage gap, computed as the ratio of the difference between the upper bound and the heuristic solution to the heuristic solution. Each cell of Table 5 corresponds to a statistic (minimum, average, and maximum percentage gap) over nine instances (see section 4.2.1). Over all the instances, the average percentage gap is 1.88%. Recall that in Step 4, the heuristic solves a reduced MIP obtained after fixing the variables in Step 3. While the reduced MIP is typically solved to optimality within a couple of minutes, we impose a limit of 400 seconds to ensure that the time required by Step 4 never exceeds this bound. The total time required by the heuristic for an instance is never more than 400 seconds and, therefore, not reported. For 30% of the instances, the heuristic provided a better solution than the best solution available from CPLEX after 3600 seconds. For the remainder of this section, our focus will be on observing the trends—with respect to changes in the cost parameters and structural properties of the network—in the use of deadheading and lane-sharing in extracting an optimal DSN. Therefore, instead of

Table 5 The Percentage Gap between the Heuristic Solution and the Upper Bound Obtained from CPLEX (version 11.1) Nodes Density min gap avg gap max gap

100 0.4 0.3 0.7 2.3

200 0.6 0.5 1.8 2.8

0.4 0.3 1.4 5.6

300 0.6 0.2 0.7 2.3

0.4 3.8 4.0 4.3

0.6 2.5 2.6 2.8

reporting absolute values of the cost savings, we use appropriate normalizations. 4.3.3. Dense vs. Sparse Networks. Let the total cost saving offered by an optimal DSN (to PDL ) be SDL , and the corresponding deadheading and lanesharing costs be CD and CL , respectively. We examine the impact of network density, size (number of nodes), and cost effectiveness parameter Rc (see section 4.2.2) on the normalized ratios CD =SDL (marginal deadheading cost per unit cost saving) and CL =SDL (marginal lane-sharing cost per unit cost saving). Figure 7 illustrates the impact of the network density and the change in Rc on the marginal deadheading cost. Consider a fixed value of Rc . Then, (i) for a fixed value of network density, an increase in the size of the network implies an increase in the number of lanes and consequently, a reduction in the use of deadheading in the DSN. Thus, the ratio CD =SDL decreases with the number of nodes. (ii) for a fixed size, an increase in density implies an increase in the number of lanes and, again, a lesser need of deadheading. Consequently, the ratio CD =SDL also decreases as the density increases. For the instances in the test bed, the highest value of the ratio CD =SDL corresponds to the case when the number of nodes in the network is 100, the network density is 0.2 and the value of Rc is 1. For ease of comparison, we normalize this value to 1 and scale the others accordingly. Observe that at low densities, the ratio CD =SDL decreases as Rc decreases. A decrease in Rc implies an increase in the subcontractor’s cost effectiveness in operating the lanes. The number of lanes in the network is limited at a low density, resulting in fewer cycles. Therefore, deadheading becomes necessary for creating profitable cyclic routes for the optimal DSN. A decrease in Rc also implies a decrease in deadheading cost (since deadheading cost on a lane is a fraction of the subcontractor’s cost in operating that lane). Therefore, as Rc decreases, the use of deadheading increases. As this added deadheading facilitates the extraction of more cycles, the cost saving also increases. Since the density is low, this increase in deadheading results in the ability to combine more lanes efficiently to create profitable cycles. Thus, as Rc decreases, the cost saving increases at a higher rate than the increase in deadheading cost. This implies a decrease in the ratio CD =SDL as Rc decreases. At a high density, as network size increases, the number of lanes in the shipper’s network also increases significantly. There is a corresponding increase in the availability of cyclic routes. As Rc

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Figure 7 Behavior of Normalized Marginal Cost of Deadheading with Change of Network Density

|V|=200

|V| = 100

CD S DL

CD S DL

Network Density

Network Density

(a)

|V|=300

(b)

RC : Extent of overlap in the ranges of the costs

ci0, j and c is, j RC

CD S DL

RC RC

Network Density

(c) Notes: The ratio CD/SDL has been normalized with respect to its highest value corresponding to |V| = 100, d = 0.2, and Rc = 1.

decreases, deadheading becomes cheaper. Therefore the use of deadheading increases, resulting in an increase in total deadheading cost. Since the subcontractor can operate the lanes at a lower cost, the total cost saving also increases. Also, since the shipper’s network is dense, the option of additional deadheading does not result in a substantial increase in the total volume of the DSN. Thus, as Rc decreases, although the use of deadheading increases, the ability of the optimal DSN to increasingly utilize deadheading to obtain a significant improvement in cost saving progressively reduces. Consequently, when the density is high and the size of the network is large, the ratio CD =SDL increases as Rc decreases. This behavior can be clearly seen in Figure 7c. Figure 8 illustrates the impact of network density, number of nodes, and the subcontractor’s cost effectiveness parameter Rc on the ratio CL =SDL . For a fixed value of Rc , at a low density, deadheading can be effectively used to extract the cycles that are included in the DSN. Hence, as described in the explanation for Figure 7, the opportunity for utilizing deadheading to improve the total cost saving increases. Consequently, at a low density, the use of lane-sharing does not result in a significant improvement in the cost saving. At high densities, deadheading is limited in its ability to further

improve the cost saving. In contrast, lane-sharing becomes increasingly useful for achieving cost saving. Therefore, as density increases, the ratio CL =SDL first increases and then typically decreases. At low density values, as described above, deadheading continues to improve the cost saving from a DSN as Rc decreases. Correspondingly, the use of lane-sharing continues to decrease. This behavior is illustrated in Figure 8. At high densities, the ability of deadheading to improve the solution decreases. However, note that lane-sharing alone may not be sufficient to extract a significant number of cyclic routes. Hence, at high densities, lane-sharing can better impact cost saving if there is an opportunity to use cheap deadheading as well to complete the cycles. Therefore, as Rc decreases, a cheap deadheading cost increasingly supports lane-sharing. This complementary use of deadheading and lane-sharing is evident in Figure 8c. 4.3.4. Strategic Alliances. Our goal is to illustrate that merging two shipping networks may result in a better DSN. To this end, we examine the special case of fully merging the networks of two shippers, that is, the volume on each lane in the merged network is the sum of the volumes of the two shippers on that lane. We quantify the net percentage improvement in cost

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Figure 8 Behavior of the Normalized Marginal Cost of Lane-Sharing with Change of Network Density

|V|=100

|V|=200

CL S DL

CL S DL

Network Density

Network Density

(a)

|V|=300

(b)

RC : Extent of overlap in the ranges of the costs

ci0, j and

cis, j RC

CL S DL

RC RC Network Density (c) Notes: To enable a fair comparison with Figure 7, the ratio CL/SDL has been normalized with respect to the highest value of CD/SDL corresponding to |V| = 100, d = 0.2, and Rc = 1.

saving due to merging, that is, the percentage increase in cost saving offered by the DSN identified on the merged network as compared with the sum of the cost savings from the two individual DSNs. To obtain a baseline value for comparing the cost savings due to merging, we consider a sequential approach. This approach works as follows: We assume that the two shippers do not share any information. Therefore, both the shippers offer the subcontractor the DSNs for their individual networks in the following sequential manner. Assume that the subcontractor treats the shippers on a first-come firstserve basis. That is, the shipper who approaches the subcontractor first extracts a DSN based on the subcontractor’s available node/arc capacities. This DSN consumes some of the subcontractor’s resources. Therefore, the subcontractor updates its node/arc capacities and then the second shipper extracts a DSN based on these updated capacities. For the instances in our test bed, the highest net cost saving due to merging is obtained when the number of nodes in the two networks is 200, the overlap (i.e., the number of common lanes) is 75%, and the density of each network is 0.2. We normalize this value to 100% and scale the other values appropriately. When two networks are merged, the net percentage improvement in cost saving is not necessarily posi-

tive. In other words, merging does not always guarantee an optimal DSN with a better cost saving. The demands increase on the lanes in the overlapping portion of the merged network. These “fatter" lanes could result in better cycles and a corresponding decrease in deadheading/lane-sharing. On the other hand, the increased volumes could disturb the balance of flow (achieved in the two individual DSNs) and, in turn, necessitate more deadheading/lane-sharing. Depending on the amount of non-subcontracted volume on the overlapping lanes in the optimal DSNs for the individual networks and the availability of subcontractor’s resources, an optimal DSN in the merged network may utilize (1) more or less deadheading, (2) more or less lane-sharing, or (3) more or less subcontracted volume than that in the two individual networks. Thus, an optimal DSN in the merged network is influenced by (i) positive effects due to an increase in the subcontracted volume and a decrease in deadheading and lane-sharing and (ii) negative effects due to decrease in the subcontracted volume and an increase in deadheading and lane-sharing. We first study the impact of the extent of overlapping on net percentage improvement in cost saving due to merging (Figure 9). When the two networks have the same density, the positive effects surpass the negative effects for most of the instances, thereby

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Normalized Net Percentage N Im Improvement in Cost Saving

Normalized Net Percentage Improvement in Cost Saving

Figure 9 The Impact of Merging Two Shipping Networks on the DSN

Extent of Overlapping

Extent of Overlapping

(b)

Normalized zed Net Percentage Improvement ent in Cost Saving

(a)

Extent of Overlapping

(c) Notes: The figures plot the behavior of the net improvement in cost saving with the extent of overlapping: (a) d1 = d2 = 0.2, (b) d1 = d2 = 0.5, (c) d1 = d2 = 0.8, where d1 and d2 denote the densities of the two networks.

resulting in a positive net improvement in cost saving. The trade-off between the positive and negative effects not only depends on the common density but also on the size of the network. When the size is small to moderate, that is, number of nodes |V| = 100 and 150, (resp., large, i.e., |V| = 200), the impact of the negative (resp., positive) effects increase as the extent of overlapping increases, resulting in a consistent decrease (resp., increase) in the marginal increase of the net improvement in cost saving. Similarly, as the common network density increases, the number of overlapping lanes increases. Therefore, the subcontractor’s capacity restrictions are more likely to get violated at higher densities. Consequently, as the common density increases, at all overlapping levels and for all sizes, the net percentage improvement in cost saving due to merging decreases. Figure 10 illustrates the impact of varying densities on the net percentage improvement in cost saving due to merging. For illustration purposes, let us denote the two densities by d1 and d2 . For simplicity, we fix the overlapping level at 50%. When the density of the first network is high (d1 ¼ 0:5; 0:8), we note the following effects in the merged network as d2 increases: (i) the number of overlapping lanes increases, (ii) the ability to create more cyclic routes in the non-overlapping portion of the network increases,

and (iii) the capacity restrictions become tighter. As a result, (a) the total volume subcontracted increases, (b) deadheading increases to compensate for tighter lane capacities, and (c) lane-sharing increases due to tighter lane/node capacities. Therefore, as d2 increases, the total saving first increases due to the positive effect of increase of volume in the DSN. A further increase in d2 necessitates an increase in deadheading and lane-sharing due to capacity restrictions, resulting in a decrease in marginal increase of the net percentage improvement in cost saving. When the density of one network is low (d1 ¼ 0:2), the demand in that network is limited and merging does not significantly increase the volume of the DSN. Therefore, the negative effects of merging discussed above progressively surpass the positive effect due to increase in volume, resulting in a near-steady deterioration of the net percentage improvement in cost saving as d2 increases. Since the models in this study focus on the problem of identifying an optimal DSN, a natural question arises regarding the impact (if any) of the DSN on the remainder of the shipper’s network. We now discuss this issue and justify our simplification for practical purposes. Theoretically, this issue highlights a limitation of our work and suggests an important direction for future work.

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The Impact of Varying Densities on the Net Percentage Improvement in Cost Saving Offered by the DSN in the Merged Network 30%

|V| = 100

Normalized Net Percentage Improvement in Cost Saving

Normalized Net Percentage Improvement in Cost Saving

Figure 10

15%

0% 0.2

0.5 Density d2

0.8

Normalized Net Percentage Improvement in Cost Saving

(a)

|V| = 150

60%

40%

20%

0% 0.2

0.5 Density d2

0.8

(b)

|V| = 200

60%

d d11 = 0.2 d11 = 0.5 d

40%

d1 d 1 = 0.8

20% 0.2

0.5 Density d2

0.8

(c)

5. Impact of Network Synergies on Cost Savings: Discussion of a Limitation Recall that the shipper’s objective is to maximize the cost savings due to subcontracting a DSN. When calculating the overall savings, we focused on the possible savings/losses on the lanes that are included in the DSN (section 2). That is, we assumed that the operating costs on the remaining lanes (i.e., the lanes that are not in the DSN) are not affected due to the extraction of the DSN. If the extraction of the DSN influences the cost of the remainder of the network (hereafter referred to as the non-DSN network), then the global optimization problem should consider both criteria, that is, the cost savings from subcontracting the DSN and the cost of operating the remainder of the shipper’s network. Our work only considers an approximation of this global optimization problem. The main reasons that make this simplification reasonable in practice are as follows: 1. In practice, the non-DSN lanes that are affected due to the extraction of a DSN are those that are tightly connected with the lanes in the DSN. Here, by “tightly connected," we mean those lanes that have either the origin or the destination in common with the DSN. Our experience with real-world implementations

suggests that the volume on these lanes is small (typically less than 10%) compared with the total volume in the shipper’s original network. Furthermore, these tightly connected lanes in the non-DSN network are of two types: (i) lanes connecting metro regions and (ii) lanes that have either origin or destination in a remote, non-metro region. Typically, the total volume (in load-miles) on the former type of lanes is significantly higher than that on the latter type. This observation is important because the volume on the non-DSN lanes is handled by lane-bylane auctioning or by using one-way carriers. The cost-effectiveness of these approaches depends on several factors such as the ability to utilize the backhaul on a lane, the location of the origin and the destination, the extent of competition between one-way carriers, the efficiency of the auctioning mechanism, etc. Since a majority of the traffic connects metro regions, competition is healthy and there are plenty of opportunities for one-way carriers to utilize backhaul. Consequently, lane-by-lane auctioning and use of one-way carriers are cost-effective ways of handling the tightly connected lanes (of the non-DSN network) that connect metro regions. Since the volume on the lanes that include non-metro centers is small, the impact of any possible increase in cost on these lanes is limited.

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In summary, there are two reasons why the impact of extracting a DSN on the non-DSN network is small in practice: (i) the total volume on the lanes in the non-DSN network that are tightly connected to the DSN is small and (ii) there are cost-effective ways in handling these lanes subsequently. For the many DSNs that have been implemented over the past 2 years, the increase in the cost of the non-DSN network—with respect to the total operating cost of the shipper’s entire network before subcontracting the DSN—has been below 2%. Our simplification (i.e., ignoring the non-DSN lanes in the optimization) has allowed us to obtain solutions that closely approximate the optimal solution of the global optimization problem. 2. As established in Theorem 1 (section 3), since the problem of finding an optimal DSN is itself strongly NP-hard, the global optimization problem is an extremely challenging problem. The large sizes of real-world shipping networks further underlines the difficulty in obtaining a reasonable solution. As a combined consequence of the above two reasons, finding the optimal DSN (without considering the impact on the non-DSN network) is a good and practical heuristic for the global optimization problem, and can be viewed as an important milestone toward a comprehensive solution.

6. Future Research Directions While the idea of a DSN of a shipper’s transportation network has been mentioned in recent times in research articles and trade magazines, there have been no comprehensive models aimed at understanding the relevant trade-offs. In this first-of-its-kind study, we formulate four fundamental problems that, together, help in identifying a good DSN that could be subcontracted. Our models incorporate the capacity restrictions of the subcontractor and exploit two key concepts—deadheading and lane-sharing— to improve the shipping volume in the DSN. After formulating the problems, we first resolve their complexities and then derive a number of properties—based on the topological structure of the shipper’s network and the relationships between the parameters — that enable us to identify several efficiently solvable special cases. Then, we develop a heuristic that provides near-optimal solutions to practical problem instances within a reasonable time limit. Based on real-world data from a national shipper, we design a representative test bed to demonstrate (i) the substantial monetary impact of subcontracting a DSN and (ii) offer several useful

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guidelines based on the density and size of the shipper’s network and the subcontractor’s relative cost advantage over the shipper. We also investigate the suitability of a strategic alliance—combining the networks of two different shippers—to identify a better joint DSN. There are a number of directions in which future work can proceed. We offer the following suggestions: 1. Network synergies: As discussed in section 5, the simultaneous consideration of (a) the cost savings from subcontracting the DSN and (b) the cost of operating the remainder of the shipper’s network (i.e., the non-DSN network), leads to a more sophisticated model (which we referred to as the global optimization problem). The importance of this problem naturally depends on two factors: the extent to which the non-DSN lanes are affected by subcontracting the DSN and whether or not there are costeffective ways of handling the non-DSN lanes subsequently. While our interactions with practitioners indicate that the simplification we have assumed is reasonable in practice, the global optimization problem is both important and challenging. 2. Pricing issues: It would be interesting to address more sophisticated pricing of the subcontractor’s services. There are several characteristics of a DSN that could be incorporated in advanced pricing models: (i) the ratio of the volume of the DSN to the total shipping demand in the entire network, (ii) the relative proportion of deadheading in the DSN, and (iii) the structure and geographic locations of the nodes in the DSN. It is clear that these characteristics affect both the shipper and the subcontractor, albeit in different ways. As such, the problem of obtaining equilibrium prices should be both interesting and challenging. Furthermore, in anticipation of a mature market, competition can also be modeled where two or more subcontractors bid for a DSN in the shipper’s network. 3. Practical issues: The model proposed in this study can be viewed as the “base” model for identifying a DSN. There may be additional constraints and objectives that arise from practical considerations. For instance, the shipper may want to impose lower and/or upper bounds on the size of the DSN, the amount of deadheading, and the amount of lane-sharing. In turn, the subcontractor may also want to ensure that the DSN is neither too small (i.e., the shipping volume is substantial) nor too

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large (to avoid the lack of trucking and human resources for handling the shipping volume). As another example of a real-world consideration, the shipper may want to restrict the DSN to be in a specific geographic area. 4. Strategic alliances: In section 4.3.4, we showed that fully merging two shipping networks may result in a better DSN, that is, one with better total cost saving, compared with the sum of savings from the individual DSNs for the two networks. It is important to note, however, that fully merging two or more shipping networks might not be the best possible choice. For example, a cooperative solution, where the shippers partially combine the lanes (i.e., use only a part of their respective volumes on each lane), may provide a better overall solution. An arbitrary solution of this type may use excessive volume of one shipper’s network and little from the other shipper’s network. To avoid such undesirable solutions, the two shippers may agree to merge portions of their network, subject to a priori lower bounds on their individual volumes. In general, the issue of a strategic alliance between two or more shippers, to exploit benefits vis- a-vis the DSN, is an interesting and challenging optimization problem.

Acknowledgments We thank the Departmental Editor, the Senior Editor, and two anonymous referees for several excellent suggestions that have collectively led to a significant improvement in the paper. We also thank Dr. Rajiv Namboothiri of the Inter-University Research Center on Enterprise Networks, Logistics and Transportation, Montreal, Canada, for many helpful discussions.

Note 1

We avoid revealing the name of the company to respect confidentiality.

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Supporting Information

Appendix: “Dedicated Transportation Design, Analysis, and Insights”

Subnetworks: