Solving the Integrated Shipment Routing Problem of a ...

Solving the Integrated Shipment Routing Problem of a Less-than-truckload Carrier

Abstract We study a less-than-truckload (LTL) transportation network and investigate the potential benefits of implementing an integrated solution approach for shipment routing throughout this network. In LTL transportation, shipments are first delivered to local hubs to be consolidated, then transferred to another hub, and finally sent to their destinations. This routing planning process involves three routing decisions: (i) from origins to hubs, (ii) between hubs, and (iii) from hubs to destinations. These decisions are usually handled in a sequential manner due to the complexity of these individual problems, which may result in suboptimal solutions. In this study, we develop an integrated framework for solving all of these decision problems simultaneously. We propose an optimization-driven approach that can efficiently handle large instances with thousands of loads and provides solutions with significant cost savings over the sequential approach.

Keywords: less-than-truckload shipping; vehicle routing problem; shipment routing; integrated routing decisions.

1.

Introduction

Truck transportation, compared to other modes of transportation, is quite fast, volume insensitive, and most importantly relatively cost-efficient. The last two features, volume insensitivity and low prices, are mainly due to different type of services offered by trucking industry customized to customers’ needs. One classification of these services is based on the volumes of the loads and the locations of the customers: Truckload (TL) transportation is the preferred method for customers with large-volume and/or short-distance shipments and less-than-truckload (LTL) transportation is for customers with small-volume and/or long-distance shipments. The key point in LTL transportation is the consolidation of the shipments. In general, LTL shipping handles shipments that weighs less than 10,000 pounds (lbs), hence most of these shipments occupy less than 10% of the truck capacity. In order to handle these shipments in a cost effective manner, LTL carriers need to bundle the freight from different consignors to increase truck utilization. This consolidation process takes place at terminals (hubs) of the LTL carrier. Hence, all the shipments are first collected from the shippers’ locations and then brought to the terminal. The collected shipments are sorted and combined into bundles and then sent to other terminals. Finally, each shipment is delivered to its destination from the final terminal location. In large LTL 1

networks, such as North American LTL networks, a shipment may pass through several terminals before reaching its destination. Also, there is a distinction between terminals with respect to their operations: (i) end-of-line terminals where the shipments from customers first arrive to or the shipments to customers last depart and (ii) breakbulk terminals where shipments are bundled/mixed by other shipments going through the same direction. In that aspect, end-of-line terminals serve as hubs for so called “local transportation” or “city operations” and breakbulk terminals serve as transhipment locations. The shipments are hauled via pickup/delivery trucks from customer locations to terminals or from terminals to customer locations and via large trucks or sometimes by another transportation mode such as rail between two terminals. The hubs in the network are serving both needs, shipment collection/distribution and shipment consolidation, at the same time. In LTL transportation, the total distance covered for handling the shipments and the truck utilizations are influenced by two major factors: the network structure of the LTL carrier (location of the terminals) and the shipment flow/routing decisions across this network. The former decision is a strategic decision for the LTL carrier, hence it is impossible to modify this decision to better serve a portfolio of shipments during an operational planning period. The latter decision, however, is a tactical/operational decision for the carriers, and can be customized for each portfolio of shipments. However, it requires solving several complex optimization problems with hundreds/thousands of shipments. Even though such a task is extremely challenging, solving these problems effectively may result in considerable savings for the carrier, which is the main motivation of our work. Shipment routing decision of an LTL carrier, also referred as load plan in the literature (Powell and Sheffi, 1983; Erera et al., 2013a,b), corresponds to how shipments are routed through the carrier’s network and involves two types of routing decisions: Routing between customer locations and hubs and routing between the hubs. The routing problem between customer locations and hubs is basically a vehicle routing problem (VRP) once the hubs that will be used to handle each particular shipment is determined. The routing problem between hubs is a combination of bin packing and network flow problems, which tries to identify the optimal bundling of the shipments and optimal flow of these bundles through the hubs. Both problems are NP-Hard problems and real-life instances are usually very large. Traditionally, the problem of finding the optimal flow of shipments through the hubs is solved using path-based approaches and column generation. Routing between customer locations and hubs is considered as a separate problem and assumed to be solved subsequent to LTL load plan design phase. Also, in large LTL networks, local transportation might seem relatively unimportant compared to the hauling of the shipments between terminals as the distances are considerably larger in the latter phase. Hence, most studies in the literature do not even mention local transportation phase. However, in local transportation, a large portion of the

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truck moves are relatively empty, which may cause inefficiencies. In that aspect, the transportation cost per lb of the shipment in local transportation might be comparable (or even larger, especially in case of small LTL networks) to that of interhub transportation given that pickup/delivery trucks’ capacities are lower than interhub trucks’ capacities and urban freight movements are much more expensive than freight movements on highways. Based on a report by Center for Transportation Analysis1 , Class 5 Trucks (trucks used for city deliveries) have a fuel consumption rate of 25.6 gallons per thousand ton-miles. On the other hand, Class 8b Trucks (trucks used for interhub transportation) have a fuel consumption rate of 6.5 gallons per thousand ton-miles. This supports our claim that the local transportation contributes significantly to the operating cost of the LTL carriers. Motivated by this, we analyze both routing problems of an LTL carrier and develop a comprehensive routing solution for the shipments. Even though simultaneously solving both of these problems might result in the highest benefit for the LTL carrier, due to their complexities, we first attempt to solve them sequentially. Based on this sequential model, we develop a simultaneous solution approach as well. In sequential solution approach, first the interhub transportation decision between the hubs is identified assuming that each shipment is brought to (or delivered from) terminals via direct shipment. Next, the routing decisions for pickups/deliveries are given. Note that the former decision precedes the latter as it is not clear that which shipment will be sent through which hub(s). Also, note that a particular shipment may not be sent through the nearest hub locations to the shipment’s origin and destination due to insufficient shipment volume along the same route. We illustrate this on a simple example. Consider an LTL carrier’s network (Figure 1) with three hubs and 6 customer locations. Suppose that the LTL carrier uses homogenous trucks for both pickup/delivery operations. Suppose that shippers located at A and C are sending half truckload of shipments to destination E and shippers located at B and D are sending half truckload of shipments to destination F . In this case, the optimal shipment routing is collecting the shipments from A to B together on a truck route and bringing to Hub 1, similarly collecting the shipments from C to D together and bringing to Hub 2, sending one full-truckload shipment both from Hub 1 and Hub 2 to Hub 3, and sending one full-truckload shipment from Hub 3 to both E and F . In this solution, even though the customer located at C is the closest to Hub 1, its shipment is hauled through Hub 2 due to truck utilization purposes. In fact, this is not only for truck utilization in interhub transportation, but also in pickup operations as well. Assigning C to Hub 1 along with A and B requires two trips in pickup operations, increasing the overall transportation costs. Hence, the solution to the first stage not only yields the consolidated flow of shipments across the 1

http://cta.ornl.gov/vtmarketreport/pdf/chapter3 heavy trucks.pdf

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LTL carrier’s network but also determines which hubs should be used by each particular shipment, which is the input for the second routing decision problem. In the second stage, routing decision from/to customer locations is determined by solving several VRPs and the final shipment routing decision is obtained. However, as the former decision heavily influences the latter decision, the sequential approach might yield solutions significantly worse than the optimal solution. Therefore, we attempt to identify the best overall shipment routing decision in an integrated manner. As both decisions affect each other in a cyclic manner, simultaneous approach has the potential to offer the maximum possible benefits for the LTL carrier. We propose an approach that first solves both routing problems iteratively and employs a final optimization step to obtain simultaneous solutions. This integrated framework can efficiently handle large instances with thousands of loads and provides solutions within acceptable computational time limits. We compare the integrated solutions with the solutions obtained by the sequential approach and show that our approach provides on average 8-10% cost reduction.

Figure 1: An LTL carrier’s network. The remainder of the paper is organized as follows. In Section 2, we discuss the related work in the literature. In Section 3, we provide a formal definition of the problem that needs to be solved to determine the optimal shipment routing across an LTL carrier’s network. In Section 4 and Section 5, we present the sequential and the integrated solution approaches, respectively. In Section 6, we computationally demonstrate how our proposed mechanisms perform under different

4

settings. Concluding remarks are provided in Section 7.

2.

Literature Review

In this section, we briefly review the related work in the literature. Our study is related to two main streams of research: (i) service network design and load plan design in LTL transportation and (ii) vehicle routing problem applications. There exists a large body of literature on service network design in freight transportation. We refer the reader to surveys by Crainic (2000) and Wieberneit (2008) for solution approaches proposed in the literature. The service network design problem has many applications in practice. For example, Wieberneit (2008) presents a generic service network design problem and reviews five tactical planning problems studied in the literature such as express shipment delivery problem, flight network for lettermail, and LTL operations in different regions/settings. The literature on LTL load plan design is more closely related to our work. Early studies include Powell and Sheffi (1983), Powell (1986), Powell and Sheffi (1989), and Powell and Koskosidis (1992). Powell and Sheffi (1983) propose a local improvement heuristic for the LTL network of a large national carrier. Powell (1986) formulates the problem as a fixed charge network design problem and solves using a local improvement heuristic. Powell and Sheffi (1989) formulate the problem as a mixed integer program and propose a decomposition strategy based on this formulation and real life considerations of the problem. Powell and Koskosidis (1992) solve LTL shipment routing problem with tree constraints, the paths from all origins into a destination should form a tree, using again local improvement heuristics. Jarrah et al. (2009) develop a column generation approach for the service network design problem in LTL freight operations and provide good solutions using slope scaling and load-planning tree generation within reasonable computational time. Barcos et al. (2010) propose a metaheuristic, ant colony optimization, for the routing design problem and test the performance of the heuristic on a real-life instance in Spain that has 49 terminals, with 6 acting as breakbulk terminals. Erera et al. (2013a) consider an LTL carrier setting with a pre-specified load plan, routing of the freight through the LTL’s network, and develop an approach for creating an operational schedule, a timedetailed plan for trucks and drivers. Developing such a schedule allows better estimation of the cost of the given load plan. Finally, Erera et al. (2013b) develop an improvement scheme based on a large neighborhood search using an integer program at each iteration. The local transportation planning, routing problem from customer locations to hubs, is a variant of the well-studied VRP. There is a vast literature on VRP and several solution techniques, both exact solution and heuristic based, have been proposed (some examples include Baldacci et al. 5

(2004), Zeng et al. (2005), Fukasawa et al. (2006), Isleyen and Baykoc (2009), and Hollis and Green (2012)). We refer the reader to the surveys by Toth and Vigo (2002a), Cordeau et al. (2002), Golden et al. (2008), Laporte (2009), and Toth and Vigo (2002b). Contrary to the studies reviewed above, we study a shipment routing problem on a modest-sized LTL network as we focus on developing an integrated solution for both local transportation and interhub transportation phases of the problem. Except Barcos et al. (2010), other studies focus on North American LTL networks with hundreds of hubs and ten thousands of loads. However, these studies ignore the local transportation phase completely and assume the shipments are already available at the end-of-line terminals. Hence, if we were to assume a one-to-one correspondence between the end-of-line terminals in those studies and the customer locations in our study as both represent the origin/destination locations of the shipments, and similarly a one-to-one correspondence between the breakbulk terminals in those studies and the terminals (hubs) in our study, then both problems are comparable in size and characteristics with two exceptions. First, they assume directed moves from the end-of-line terminals to the breakbulk terminals whereas we assume VRP tours from the customer locations to the terminals. Second, in some of those studies, they develop a shipment plan for thousands of loads whereas we consider at most one thousand loads. In order to implement our proposed approach on a large-sized LTL network, we may use certain decomposition approaches. Even though these approaches may result in a certain level of sub-optimality, the computationally proven benefits of our approach would be more than enough to compensate such losses due to sub-optimality. However, such decomposition methods are out of scope of this work and may be considered as a future research direction.

3.

Problem Definition

In this section, we provide a formal definition of the problem and present the mathematical model that determines the best routing of shipments from their origins to their destinations through the LTL carrier’s network. Even though solving this combined routing problem would yield the minimum cost of hauling all the shipment requests, the size of the problem is quite large to be solved efficiently. We also list our assumptions in this section. In our setting, there are several shipment requests (loads) to be handled by an LTL carrier. Each load should be transported from its origin to its destination through the network of the carrier. We assume that there is only one type of hub where the loads are first brought to from their origins, bundled together with other loads based on their destinations and volumes, and transported to other hubs and finally sent to their destinations. The routing operations can be categorized as: (i) local transportation representing the transportation of the loads from origins to 6

hubs and from hubs to destinations, and (ii) interhub transportation representing the consolidated transportation of the loads between hubs. The transportation costs both for local transportation and interhub transportation depend on the distance traveled, however, the cost rate per mileage for local transportation is assumed to be higher than that of interhub transportation to represent the fuel efficiency difference between urban and highway freight movement. Moreover, we assume that pickup/delivery trucks have lower volume compared to the trucks hauling loads between hubs. We assume that each load visits at least one and at most two hubs before arriving to its destination. Finally, we assume that pickups and deliveries are separate operations, hence trucks do not perform different type of operations along a single route. We define the problem on a complete graph G = (V, E), where V is the set of customer and hub locations and E is the set of edges connecting the locations. Let L be the set of loads to be handled by the LTL carrier. We use Ol and Dl to denote the origin and the destination of load l, respectively. Let al be the volume of load l. Let H be the set of hubs, each located at one of the nodes on the graph. We assume that the carrier operates an infinite number of vehicles in local transportation, each with a capacity of A, where al ≤ A for all l ∈ L and the load splitting is not allowed. We also assume that the carrier operates only Q vehicles (as this number affects the ˜ complexity of the bin packing problem) between any two hub locations, each with a capacity of A, where al ≤ A˜ for all l ∈ L. The transportation cost incurred by a vehicle from hub h to hub f is denoted by bhf , which we assume to be independent of the loaded volume on the vehicle. We assume that all feasible VRP tours for pickups and deliveries starting from a single hub and visiting several customer locations are generated in a preprocessing step. Let Plh be the set of feasible VRP tours originating at hub h and handling load l. Let P is the set of all feasible VRP tours and let cp be the cost of tour p. Note that the number of such VRP tours may be exponential based on the load volume and truck capacity values. We use the following decision variables in our formulation:

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  1, if tour p is executed for picking up loads vp =  0, otherwise

∀p ∈ P

  1, if tour p is executed for delivering loads wp =  0, otherwise

∀p ∈ P

xlh

  1, if load l is first transported to hub h =  0, otherwise

∀l ∈ L, ∀h ∈ H

yhl

  1, if load l is transported from hub h to its destination =  0, otherwise

∀l ∈ L, ∀h ∈ H

ulhf

  1, if load l is transported using two different hubs h and f =  0, otherwise

slhf q =

    1, if load l is transported between hubs h and f on vehicle q

  

mhf q

∀l ∈ L, ∀h, f ∈ H, h 6= f

∀l ∈ L, ∀h, f ∈ H, h 6= f, ∀q ∈ {1...Q}

0, otherwise

  1, if vehicle q is used between hubs h and f =  0, otherwise

∀h, f ∈ H, h 6= f, ∀q ∈ {1...Q}

thf = Number of vehicles that have to be used by the carrier ∀h, f ∈ H, h 6= f

between hubs h and f

Using these variables, we formulate our problem (called LT LSP , which stands for LTL simultaneous planning problem) as an integer linear program as follows:

LT LSP : z = min

X

cp (vp + wp ) +

XX

bhf thf ,

(1)

xlh = 1, ∀l ∈ L,

(2)

h∈H f ∈H

p∈P

s.t.

X h∈H

8

X

yhl = 1, ∀l ∈ L,

(3)

vp = xlh , ∀l ∈ L, ∀h ∈ H,

(4)

wp = yhl , ∀l ∈ L, ∀h ∈ H,

(5)

xlh + yf l − 1 ≤ ulhf , ∀l ∈ L, ∀h, f ∈ H, h 6= f,

(6)

h∈H

X p∈Plh

X p∈Plh

Q X

slhf q = ulhf , ∀h, f ∈ H, h 6= f, ∀l ∈ L,

(7)

˜ ∀h, f ∈ H, h 6= f, ∀q ∈ {1...Q}, slhf q al ≤ mhf q A,

(8)

q=1

X l∈L

Q X

mhf q = thf , ∀h, f ∈ H, h 6= f,

(9)

xlh ∈ {0, 1} ∀l ∈ L, ∀h ∈ H,

(10)

yhl ∈ {0, 1} ∀l ∈ L, ∀h ∈ H,

(11)

ulhf ∈ {0, 1} ∀l ∈ L, ∀h, f ∈ H, h 6= f,

(12)

slhf q ∈ {0, 1} ∀l ∈ L, ∀h, f ∈ H, h 6= f, ∀q ∈ {1...Q},

(13)

q=1

mhf q ∈ {0, 1}

∀h, f ∈ H, h 6= f, ∀q ∈ {1...Q},

(14)

thf ∈ Z+ , ∀h, f ∈ H, h 6= f.

(15)

The objective is to minimize the sum of the transportation costs including the transportation cost to, from and between hubs. Constraints (2-3) ensure that each load should be transported from its origin to one of the hubs and from one of the hubs to its destination. Note that the loads are not always routed through the hubs that are closest to their origins or destinations due to geographic and consolidation related efficiency considerations. Constraints (4-5) ensure that the loads are picked up/delivered along a VRP tour initiating at the corresponding hub location. Constraints (6) make sure that if a load is brought to one hub from its origin and delivered to its destination from another hub, then the load should have been transferred from the former hub to the latter hub. Constraints (7) make sure that each load is handled by exactly one vehicle and constraints (8) guarantee that the capacity restrictions for the vehicles are not violated. Finally, Constraints (9) ensure that a sufficient number of vehicles is assigned to each hub-to-hub link in the network to handle the required volume of the loads. The rest of the constraints are binary or integrality restrictions of the variables. Solving this problem would determine the optimal flow of the loads through the LTL carrier’s network. The next step is to evaluate the complexity of this problem, which is NP-Hard. Consider

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a special case of our problem where we have only one hub. Suppose that the destinations of all the loads are exactly at the same location with that particular hub. Hence, every load has to be collected from its origin and be brought back to that hub. In this case, VRP polynomially reduces to our problem. As VRP is known to be NP-Hard, this establishes that our problem is also NP-Hard.

4.

Sequential Approach

In this section, we describe our sequential approach for solving the routing problem of the LTL carrier. This sequential approach first employs a procedure that solves the load flow problem assuming only direct moves across LTL network, hence milk run type consolidation is not considered in local transportation. By solving this restricted problem, we determine not only the flow of the loads among the hubs, but also the assignment of each load to the hubs. In the second phase, we solve a separate VRP problem for each hub and for each distinct operation type (pickups and deliveries). At the end of this phase, we obtain an overall solution for the routing problem of the LTL carrier in a sequential manner. This solution will serve as a benchmark for evaluating our integrated solutions both with respect to solution quality and computational time.

4.1

Solving the load flow problem based only on the direct moves

In the first phase of our sequential solution, we solve a slightly modified version of the integer program described above. In this modified version (called LT LSP M , which stands for modified LTL simultaneous planning problem), we only consider direct moves between the customer locations and the hubs. Hence, we remove the variables representing the VRP tours (vp and wp ) and the corresponding constraints (Constraints (4-5)). Instead, we introduce a new parameter representing the cost of direct moves in local transportation. To this end, let dih denote the cost of transporting a load of any volume from location i to hub h (similarly let dhj denote the transportation cost from hub h to location j). Finally, we modify the objective function as follows:

z = min

XX h∈H l∈L

dOl ,h xlh +

XX h∈H l∈L

dh,Dl yhl +

XX

bhf thf .

h∈H f ∈H

This modified problem is solvable in acceptable computational time using commercial solvers and doing so would yield the optimal flow of the loads among the hubs. Also, the cost of interhub transportation in this solution would be equal to:

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zIH =

XX

bhf thf .

h∈H f ∈H

4.2

Solving the local transportation problem

After the first phase, the loads are assigned to hubs for both pickup and delivery operations. The remaining routing problem is decomposable by both the type of operation and the hubs. Hence, we need to solve several VRPs, to be exact twice the number of hubs. Even though this reduces the problem size and the complexity considerably, the remaining problems are still challenging considering the number of loads to be handled by the carrier. In order to solve each of these VRPs, we first employ a clustering procedure, a modified K-means clustering approach, and then solve individual TSPs based on these clusters. As the number of customers along a route is limited to the truck capacity restrictions, we are able to solve these TSPs using exact methods and commercial solvers. Let zLC be the cost of local transportation. The details of the algorithm are as follows: Procedure: Solving local routing problem For h = 1...|H| For e = 1, 2 (1: Pickups, 2: Deliveries) Let Leh be the set of loads assigned to hub h for operation e Initiate a set of tours R = {} Apply modified K-means clustering approach for loads in set Leh and obtain R For r = 1...|R| Solve a TSP to determine the order of the visits and calculate the cost of the tour, αr End For Calculate the local transportation cost for the subproblem, βeh =

P

r∈R αr

End For End For Calculate the total local transportation cost, zLC =

P

h∈H

P2

e=1 βeh

We propose a variant of the K-means clustering algorithm, which is an efficient method in clustering customers in routing problems (Ganesh and Narendran, 2007; Kim et al., 2006). This iterative algorithm is quite easy to implement and it is also computationally very efficient. As the location assignment subroutine of this clustering algorithm, we implement a variant of the set partition problem (VSPP) as a subroutine to determine partitioning of the loads’ origins/destinations into the clusters. VSPP partitions the loads into the clusters while ensuring that the total volume

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of loads assigned to a cluster does not exceed the truck capacity. The inclusion of VSPP increases the complexity of the clustering algorithm, however, also increases the quality of the solution. Note that we need a pre-specified number of cluster centers, K, in this clustering approach. If K is not sufficiently large, then the VRP clusters become infeasible as there will not be enough vehicle tours (hence capacity) to haul the all the loads. On the other hand if it is too large, the efficiency of P the VRP solution decreases. To this end, we specify K = d

l∈Leh

A

al

e + Constant and choose the

constant to be equal to 0 at first, and increase this constant if we do not get a feasible solution and reiterate the procedure. The details of the modified K-means clustering approach is as follows:

Procedure: Modified K-means clustering P approach Calculate the number of cluster centers: K = d

l∈Leh

A

al

e + Constant.

For k = 1...K Initiate cluster centers by randomly choosing K locations out of all possible customer locations End For For i = 1...I Assign customers to cluster centers by solving VSPP If i < I For k = 1...K Calculate the centroid for all locations assigned to this cluster center Relocate the cluster center to the location closest to center of gravity of the cluster End For End If End For

The overall solution to the combined routing problem of the LTL carrier is determined by both stages of this sequential approach and the total cost of this solution is equal to z = zIH + zLC . Note that the cost to the second phase, zLC , might be significantly larger than it should be, as while solving the interhub transportation problem in the first phase, the consequences for the local transportation phase are completely ignored. This is the drawback of the sequential approach. Therefore, we propose the integrated solution approach to remedy this situation.

5.

Integrated Approach

In this section, we describe our integrated approach for solving the combined routing problem of the LTL carrier. Our approach solves both interhub trasportation and local transportation problems 12

iteratively and attempts to improve total transportation costs of the LTL carrier. Our motivation in using such an iterative procedure is to align LTL interhub solutions with local transportation solutions. Note that a very good solution for the first phase may produce undesirable results for the second phase. Hence in this iterative process, we need to transfer some information (certain modifications, restrictions etc.) from the local transportation phase back to the interhub transportation phase. By doing so, we may solve the interhub transportation problem in a way that benefits the local transportation problem as well. After a certain number of iterations is carried out, we apply a simultaneous solution approach, which yields the ultimate solution of our integrated mechanism.

5.1

Iteratively solving the combined routing problem

In order to iteratively solve both routing problems and improve the solution in each iteration, we require a signalling mechanism from the local transportation phase to interhub transportation phase, indicating that some of the load flow decisions between hubs actually lead to costly VRP routes. In that case, we would reassign the corresponding load(s) to other hubs, thereby reducing the total pickup/delivery costs provided that these new assignments would result in better VRP solutions. Unfortunately, this is not a straight-forward task. It is trivial to identify the costly VRP tours immediately, yet it is quite difficult to determine which loads are responsible for these costly routes. Even if we could, we still cannot tell whether it would be any better if we change the load plan decision in the interhub transportation stage. Having this in mind, we first calculate the cost contribution of each load to the VRP tour it is included in, in other words, the actual cost allocation/responsibility of each load in the VRP tour. One may think that calculating the marginal cost of adding a load to the VRP tour will be enough for this task. Unfortunately, this is not the case as the marginal cost calculation will not yield the actual cost contribution. We illustrate this on a simple example. Suppose we have two loads originating at the same customer location. In that case, for a pickup/delivery, the marginal cost of adding both of these loads to the VRP tour would be zero. However, this does not mean that their cost contributions are zero especially when the origin of these two loads is isolated from the rest of the customer locations. Hence, we need a much better cost allocation/determination method. To this end, we apply a cost allocation mechanism designed for the traveling salesperson problem (TSP) proposed by Faigle et al. (1998). In this mechanism, we assume that there exist several moats of constant widths surrounding nodes or group of nodes. Hence, each moat separates a group of nodes from the remaining ones within a TSP tour. An illustrative example is provided in Figure 2 (Figure is acquired from http://www.math.uwaterloo.ca/tsp/problem/index.html). As can be seen in Figure 2, “red” moats have a circular shape whereas the “blue” moats have a belt-like shape. At

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a given node, in order to reach the nodes outside, we have to cross all the moats surrounding this particular node. As in an TSP setting, we have to return to each node to complete the TSP tour, we have to cross the same moats at least one more time. In each TSP tour, there exists a nested moat packing, a non-intersecting moats surrounding locations. A maximum moat packing problem provides a lower bound on the total tour length of the optimal TSP tour. As mentioned before, in this setting, a TSP tour has to cross these moats twice and hence, twice the sum of widths of the moats will provide this lower bound. In order to visit the nodes outside a particular moat, a vehicle has to cross the moat twice. Therefore, a good cost allocation method is to allocate the cost of crossing the moat to the nodes outside the moat.

Figure 2: A nested moat packing. Let N ∪ {0} be the set of nodes of the TSP, where “0” denotes the depot. Let S be a subset of ¯ We assume, the nodes, S¯ be the complement of S, and let M be the set of all partitions {S, S}. ¯ Let w ¯ be the width of the moat surrounding the nodes set without loss of generality, that 0 ∈ S. S,S

S. The moat widths need to be non-negative and for any two nodes, the sum of moats separating these two nodes cannot exceed the distance between these two nodes. Therefore, the moat packing (MP) with maximum total width can be obtained by solving a linear program given below (see Faigle et al. (1998) for details).

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MP :

X

max

wS,S¯

¯ S,S∈M

s.t.

X

wS,S¯ ≤ dij ∀i, j ∈ N ∪ {0}

i∈S,j∈S¯

¯ ∈ M. wS,S¯ ≥ 0 ∀{S, S} Note that the moat packing with maximum total width provides the best lower bound that uses moat packing approach to the corresponding TSP. Therefore, the resulting allocation provides the most accurate cost contribution of each node in the given tour. However, the solutions to the linear program above with non-nested structures will not correspond to a moat packing as in a nested structure where the moats do not intersect. That is if wS1 ,S¯1 , wS2 ,S¯2 > 0 and S1 6⊆ S2 , S2 6⊆ S1 , then S1 ∩ S2 should be the empty set. Fortunately, (Faigle et al., 1998) proved that there always exists a maximum MP with a nested structure. Given a nested MP with maximum total width, the cost allocation mechanism distributes twice the width of any moat among the nodes on the outside of the moat, and hence, γl = 2

X ¯ {S,S}∈M,l∈S,0∈ S¯

∗ wS, S¯

|S|

.

Note that as the number of nodes along a TSP tour increases, the moat allocation process becomes more complicated. Fortunately due to truck capacity restrictions, this never becomes an issue in our computations. After determining the cost allocations using the moat allocation method, we still need to find a way to estimate the opportunity cost of not assigning a particular load to a particular hub. Without knowing the respective clustering and the routing solution after the reassignment of all the loads, which is yet to be decided at this particular stage, it is not possible to come up with a good estimate for this opportunity cost. Therefore, we use a simple rule of thumb and calculate the cost of assigning the load any hub other than the current one as a constant coefficient times the minimum distance of the load’s origin/destination to any hub other than the current hub. In our calculations, we assume this coefficient to be equal to 1.5 as it needs to be less than 2 (corresponding to a single load pickup/delivery) and a smaller value would lead to too many reassignments. Note that this a very rough estimation as at this stage we do not even know whether the load will be assigned to this particular hub in the second iteration of the interhub flow phase. Even if we did, we still do not know anything about the other reassignments, and hence the final VRP tours. However, even this rough estimation works satisfactorily. After calculating the estimated cost differences, we sort all loads for both pickup and delivery operations 15

in the descending order of their cost difference and we force the assignment of the top δ loads for each operation type to the hubs other than their current hub. We continue with the load plan design stage with the new hub restrictions, and repeat this iterative process for several times. Procedure: Determining the costly load-hub assignments For e = 1, 2 (1: Pickups, 2: Deliveries) For h = 1...|H| Let Reh be the set of VRP tours originated from hub h for operation e For r = 1...|Reh | Apply the moat allocation method and calculate the cost allocation for each load l, γl End For End For For l = 1...|L| Initiate θl = ∞ For h = 1...|H| Calculate the estimated cost of assigning load l to hub h, if pickup θ˜l = 1.5dO ,h , if delivery θ˜l = 1.5dh,D l

l

If θl > θ˜l then θl = θ˜l End For Calculate the potential cost savings estimation for load l, σl = γl − θl End For Sort the loads in order of descending σl Restrict the hub assignment of first δ loads to the other hubs End For

The hub restriction mentioned above is carried out by including a new constraint set to the load plan design integer programming formulation forcing the corresponding xlh or yhl to be equal to zero. The steps of the overall iterative process is as follows (where J represents the iteration limit): Procedure: Iterative solution approach For j = 1...J Solve LT LSP M Procedure: Solving local routing problem Procedure: Determining the costly load-hub assignments End For

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5.2

Simultaneously solving the combined routing problem

Our motivation for analyzing the integrated routing problem is to obtain best overall solution to both interhub and local transportation problem. Even though the iterative solution method might improve the solution obtained by the sequential approach, the decisions for the individual routing problems are still given separately. Therefore, we propose a solution approach that gives simultaneous routing decisions for both problems. In this approach, we first accumulate “good” VRP tours during the iterative solution process. It is clear that some of these VRP tours will be identical for consecutive iterations, and hence, we remove duplicate VRP tours to reduce the problem size as much as we can. Next, we solve the integer program presented in Section 3, LT LSP , with only the selected VRP tours. As the number of tours is limited, the integer program is now solvable within acceptable computational time. The solution obtained by solving the integer program will be the integrated routing solution for the LTL carrier. Finally, the details of the procedure are as follows: Procedure: Simultaneous solution approach Initiate a set of tours R = {} For j = 1...J Solve LT LSP M ˜ Procedure: Solving local routing problem and collect the set of selected tours, R S˜ Update the set of tours, R = R R Procedure: Determining the costly load-hub assignments End For Remove duplicates in the tour set, R Solve LT LSP to obtain the overall routing solution

The schematic illustration of the overall process is presented in Figure 3.

6.

Computational Study

We carried out a computational study to demonstrate the performance of our proposed solution approaches. We performed our experiments on randomly generated instances with varying characteristics. These instances are all generated on a region of 10,000 × 10,000 unit square. We generate a certain portion of the customer locations in clustered regions, representing metropolitan areas.

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Solve LTLSPM to determine interhub transportation solution

Apply Moat Allocation to determine costly load-hub assignments

Apply Modified K-Means for clustering

Solve TSPs to determine the optimal routes

not reached

iteration limit

reached

Solve LTLSP to determine the final solution

Figure 3: The schematic illustration of the integrated solution approach. There are several characteristics that define a particular instance, such as, the number of loads to be handled by the LTL carrier, the number of customer locations, the number of hubs, and the number of clusters. Even though all of these characteristics are important, the problem complexity is more affected by the number of loads compared to the other factors. We generate a total of 20 instances using different parameter configurations. The first five of the generated instances have 250 loads, 3 clusters, 100 locations, and 5 hubs, the second five instances have 500 loads, 4 clusters, 133 locations, and 6 hubs, the third five instances have 750 loads, 4 clusters, 166 locations, and 6 hubs, and finally the fourth five instances have 1000 loads, 5 clusters, 200 locations, and 7 hubs. Moreover, the number of trucks operated by the LTL carrier between any two hub locations is five. Loads are created by randomly picking an origin and a destination and the volume of a load is uniformly distributed between 5% and 25% of the pickup/delivery truck capacity, which is assumed to be 1000. The capacity of the interhub trucks is assumed to be twice the capacity of these smaller trucks, and hence is equal to 2000. The cost of traveling between the customer locations and the hub locations is assumed to be equal to the Euclidean distance between the locations. The transportation cost between the hub locations is assumed to be 50% of the Euclidean distance between the locations representing the efficiency in the highway freight movement. Other than the different parameter combinations selected to generate the instances, we analyze

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the performance of our algorithms under three types of instances with distinctive characteristics to test the effect of each characteristic. The first type of instances are our base instances, which correspond to the instances described above. In high volume instances, we increase the volume of the loads so that it is uniformly distributed between 10% and 30% of the pickup/delivery truck capacity. Finally, in less dense instances, we reduce the percentage of the number of locations that fall within a cluster to 50%, whereas the same percentage is 70% in the base instances. A major decision in our computational analysis is the number of iterations in our iterative solution procedure. Each additional iteration might improve the solution quality, however, it also increases the computational time of the algorithm as well as the complexity of the final integer program to be solved as the number of VRP tours is increased. In our computations, we set the iteration limit to 10 iterations. We also perform another set of runs where we set the iteration limit to 5, obtain an integrated solution and reiterate the whole simultaneous solution procedure for 5 more iterations starting from the first integrated solution. Even though the overall computational time might be slightly higher, the efficiency in solving the final integer program is significantly increased. Moreover, in second set of runs, the sixth iteration probably starts from a better initial solution as this solution is obtained by solving LT LSP after 5 iterations as opposed to being a solution of an iteration. For all the subroutines that solve an integer program, we set the time limit to 1800 seconds (0.5 hours) and the optimality gap to 1% for the termination of the run. In the final LT LSP , we allow an additional 600 seconds for polishing the solution. Finally, all the computational experiments are carried out on a 64-bit Windows Server with two 2.4 Ghz Intel Xeon CPU’s and 24 GB RAM. The algorithms are implemented using C++ and CPLEX Concert Technology. As mentioned before, our benchmark for evaluating the performance of the integrated solution approach will be the solution obtained by the sequential approach with respect to both the solution quality and the computational time. It is clear that the computational effort required to obtain the integrated solutions would be higher than that of the sequential solutions, however, the cost reductions provided by the integrated solutions justify this extra effort. Note that as the integrated solution, we do not present the intermediate solutions, and hence only the final simultaneous solution is compared with the benchmark solution. Table 1 provides the results for the base instances. The first column indicates the instance number, thus each of the first twenty row corresponds to a specific instance. The second column presents the number of loads to be handled by the LTL carrier. The third and fourth columns present the total transportation cost obtained by the sequential approach and the CPU time (in seconds) of the sequential approach. Similarly, the fifth and sixth column presents the percentage

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improvement in total transportation cost and the CPU time of the integrated approach when the number of iterations is set to 5 + 5. The last two columns present the same values for the integrated approach when the number of iterations is set to 10. Finally, the last three rows present the average, the minimum and the maximum values of the respective columns over all the instances. Table 1: Results for the base instances. Ins.

# of Loads

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Ave Min Max

250 250 250 250 250 500 500 500 500 500 750 750 750 750 750 1000 1000 1000 1000 1000

Sequential Cost($) CPU(secs) 339757.5 3 333354.5 3 316557.6 2 315886.4 3 378897.4 2 584144.3 20 602366.6 32 514211.7 36 610399.9 13 593764.0 38 808353.3 25 813103.3 52 993743.5 28 792481.3 44 869242.5 8 845616.1 64 1032814.5 60 914945.5 510 1103675.6 135 1006749.8 81 57.95 2 510

Integrated 5 + 5 Iterations Imp.(%) CPU(secs) 10.18 168 7.71 206 9.46 189 14.07 171 13.38 407 10.76 2993 10.55 3104 11.32 2053 15.51 1183 9.09 5814 8.61 2362 6.12 4476 6.40 2316 5.47 3086 7.99 1584 11.10 7063 10.35 7435 8.84 9291 7.17 8027 4.95 7381 9.45 3465.45 4.95 168 15.51 9291

Integrated 10 Iterations Imp.(%) CPU(secs) 11.88 176 8.96 2591 13.17 189 14.76 231 14.32 177 10.03 3024 11.25 3184 11.64 3022 15.96 2258 9.65 3773 6.93 3487 5.84 4039 6.78 3303 5.43 3693 7.95 3283 10.41 4721 8.96 4861 8.60 6319 0.00 4716 3.86 4894 9.32 3097.05 0.00 176 15.96 6319

The computational experiments reveal that the integrated solution approach provides significantly better results compared to the sequential approach. The average cost reduction over all instances is equal to 9.45% when the number of iterations is set to 5 + 5 and 9.32% when the number of iterations is set to 10. The maximum improvement values are 15.51% and 15.96%, respectively. The minimum improvement values are 4.95% and 0%. Thus we observe that for one of the instances, the integrated solution method failed to identify a better solution than the sequential method when the iteration limit is 10. This means that the final IP could not be solved in the given time limit. Note that the intermediate solutions obtained in the iterative approach might still have better total transportation cost values, however, we choose to ignore these results so as to be consistent in comparisons. The performance of the integrated method slightly diminishes as the number of loads handled by LTL carrier increases. However in 1000-load instances, we still obtain an average percentage improvement equal to 8.48%, which is significant especially considering the magnitude of the total transportation cost of the LTL carrier. As expected, the computational

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times of the integrated approach are much higher than that of the sequential approach, yet still within the acceptable time limits. In the “5+5 iterations” setting, the percentage of the total time spent in the simultaneous solution phase on average is 27% (ranging from 2% to 51%). In the “10 iterations” setting, the percentage of the total time spent in the simultaneous solution phase on average is 56% (ranging from 10% to 93%). Table 2: Results for the high volume instances. Ins.

# of Loads

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Ave Min Max

250 250 250 250 250 500 500 500 500 500 750 750 750 750 750 1000 1000 1000 1000 1000




Tables 2 provides the results and the summary of these results for the high volume instances. As expected, the total transportation costs in these instances are higher than the normal instances. The performance of integrated algorithm is expected to diminish slightly as the opportunities for better VRP tours are fewer with the increased volume of the loads. Based on the results, we observe that this is in fact the case, and the average percentage improvement values are decreased to 8.10% and 8.60%. Nevertheless, the improvements are still significant. Also, the results of the integrated approach with 10 iterations is slightly better than 5 + 5 iterations as it is more likely to identify better opportunities from a more comprehensive perspective. Table 3 provides the results and the summary of these results for the less dense instances. In these instances, the number of customer locations that fall within a cluster is lower compared to the base instances. Consequently, the customer locations are far apart from each other, which may increase the overall transportation costs. Hence, one may think that the performance of the

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Table 3: Results for the less dense instances. Ins.

# of Loads

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Ave Min Max

250 250 250 250 250 500 500 500 500 500 750 750 750 750 750 1000 1000 1000 1000 1000




integrated solution would improve due to higher potential for cost savings. Unfortunately, this is not the case. The reason is that as we decrease the number of clustered locations, the opportunities for efficient VRP tours also decrease since the customer locations are far away not only from the hubs but also from each other. Therefore, the performance of integrated algorithm diminishes slightly and the average percentage improvement values become to 7.92% and 8.10%. In order to provide a more detailed analysis of the computational times, we present the breakdown of the computational times of the base instances in Table 4. Based on our detailed computational study, we conclude that the integrated solution approach provides significantly better solution to the combined routing problem of the LTL carrier. The computational effort to obtain the integrated solutions are much higher than the sequential approach, yet still reasonable. Finally, the performance of the integrated solution approach is consistently good under difference instance characteristics.

7.

Concluding Remarks

In this paper, we investigate the potential of integrated solution approaches in LTL transportation and propose an integrated mechanism to solve local and interhub transportation problem of an LTL carrier simultaneously to reduce total transportation costs. The sequential solution approach first 22

Table 4: Breakdown of the Computational Times (CPU Secs.). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

5+5 Iterations 158 170 179 163 398 1880 1531 1105 898 3384 1356 3734 1437 2228 1156 4605 4998 6871 5597 4960

Simultaneous Sol. 10 36 10 8 9 1113 1573 948 285 2430 1006 742 879 858 428 2458 2437 2420 2430 2421

Total 168 206 189 171 407 2993 3104 2053 1183 5814 2362 4476 2316 3086 1584 7063 7435 9291 8027 7381

10 Iterations 155 159 165 163 159 618 778 607 465 1361 1062 1622 888 1280 871 2291 2418 3625 2296 2468

Simultaneous Sol. 21 2432 24 68 18 2406 2406 2415 1793 2412 2425 2417 2415 2413 2412 2430 2443 2694 2420 2426

Total 176 2591 189 231 177 3024 3184 3022 2258 3773 3487 4039 3303 3693 3283 4721 4861 6319 4716 4894

considers the interhub transportation problem and assumes that local transportation consists of direct moves between customer locations and hub locations. After the flow of the loads through the LTL carrier’s network is identified local transportation decisions are given based on the solution of the individual VRPs for each hub location. Due to the fact that first phase of this approach ignores the local transportation decisions in the second phase, this approach yields sub-optimal solutions. We propose an optimization-driven approach and solve the overall shipment routing problem first iteratively then simultaneously to improve the solution obtained by the sequential approach. Based on a detailed computational study, we show that our proposed method is able to identify significantly better solutions and reduces the cost of the sequential solutions by an average of 8-10%. Moreover, we show that the performance of our algorithm is robust under different instance characteristics including instances with varying number of loads, instances with higher volume of loads, and instances with geographically more disperse locations. We also show that the computation times of the proposed approach are within an acceptable limit.

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