The freight routing problem of time-definite freight delivery common carriers

Transportation Research Part B 35 (2001) 525±547

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The freight routing problem of time-de®nite freight delivery common carriers Cheng-Chang Lin * Department of Transportation and Communication Management Science, National Cheng Kung University, 1 University Road, 701 Tainan, Taiwan, ROC Received 7 July 1999; received in revised form 20 November 1999; accepted 24 November 1999

Abstract The freight routing problem of time-de®nite common carriers is to minimize the sum of handling and transportation costs, while meeting service commitments and operational restrictions. There are two types of operational restrictions, capacity and directed in-tree rooted at each destination. Directed in-tree implicitly implies that there is a singular path for each origin±destination pair. The routing problem is an integrality constrained multi-commodity problem with side constraints. In this research, we study two approaches, the Lagrangian relaxation (LR) and implicit enumeration algorithm with e-optimality (IE-e). We use the third largest time-de®nite freight delivery common carrier in Taiwan for our numerical test. The result shows that the IE-e outperforms the LR, both quantitatively and qualitatively. In addition, two major shortcomings of the LR approach are shown: it may fail to ®nd any feasible solutions even though they exist, and it cannot determine whether the feasible set is empty or not. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Time-de®nite freight delivery industry; Hub-and-spoke network; Lagrangian relaxation; Implicit enumeration algorithm

1. Introduction Time-de®nite freight delivery common carriers are classi®ed under the less-than-truckload industry of transportation and warehousing sector. They provide time-guaranteed delivery of small shipments for shippers. They collect, consolidate and transport small shipments through their hub-and-spoke line-haul operations networks before delivery to consignees. Providing a variety of

*

Tel.: +886-6-275-7575; fax: +886-6-275-3882. E-mail address: [email protected] (C.-C. Lin).

0191-2615/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 1 - 2 6 1 5 ( 0 0 ) 0 0 0 0 8 - 4

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time-de®nite services with competitive prices, together with cost-e€ective operations are the keys to achieving maximum pro®t. To plan the most ecient operations involves the three hierarchical planning levels of strategic, tactical and operational. Long-term strategic planning sets up the delivery network. Medium-term tactical planning determines the size of various ¯eets. Lastly, operational planning develops the short-term operations. Hub-and-spoke is the most common delivery network structure for carriers, consisting of two types of facilities: centers and hubs. A two-hub and four-center network is shown in Fig. 1. Centers housed in hubs are in-hub centers, while stand-alone centers are remote centers. Each center has an assigned geographic service area, daily dispatching a ¯eet of small package cars to serve the customers in its service area. These centers deliver shipments to consignees during the day, and pick up new shipments from the shippers in the late afternoon before returning to the center. The former is considered delivery operations at the delivery center while the latter is considered pickup operations at the pickup center. All the centers perform both operations that interact with customers, and together they are part of overall local service operations. On the other hand, the line-haul operations move freight from pickup to delivery centers. A complete sequence is as follows. When a center completes its pickup operations, it runs a consolidation operation called a local (reload) sort. This operation unloads the freight from the package cars, sorts and loads it onto a ¯eet of line-haul tractors with various trailer sizes. Tractortrailers feed freight between facilities, thus, in practice are called feeders. However, hubs that house in-hub centers run a single twilight sort instead of one local sort for each in-hub center separately. The local or twilight sort normally starts at 5.30 pm and lasts 3 h. A ¯eet of feeders is then dispatched to various hubs for further consolidation. Only on rare occasions will centers make direct loads for other centers. Thus, the local sorts are where the volumes originate in the line-haul operations. They are the origins of all shipments. Before dawn, each center will receive its delivery volume for the day. The center runs another consolidation called preload sort, unloading freight from the feeders, then sorting and reloading

Fig. 1. Illustrative delivery operations network.

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the shipments onto its ¯eet of package cars. Of course, each hub runs a sunrise sort for its in-hub centers. The preload or sunrise sort typically starts at 4.30 am and lasts 3 h. Thus, the preload sorts are where the volumes are destined for in the line-haul operations. They are the destinations of all shipments. Thus, each piece of package has an associated origin local sort and destination preload sort. They together are called the origin±destination (OD) pair in the line-haul operations. Hubs, on the other hand, are intermediate transshipment facilities. Typically, there are several sorts that take place in a hub. Besides twilight and sunrise sorts, there are AM-day, day and night sorts depending on the hubÕs needs. Generally, these sorts start at 10 am, 1 pm and 11 pm, respectively, each lasting for 3 h. They are where the inbound freight, received from the local sort of centers and/or the hub sort of other hubs, is unloaded, sorted and reloaded onto a ¯eet of feeders. The ¯eet is then dispatched to the preload sort of centers for delivery, and/or the hub sort of other hubs for further consolidation. No shipments are left over in the building. However, the multiple sorts of a hub facility may share a common trailer for the same destination. The earlier processed volumes are loaded onto the trailer until the end of the last sort, when it is dispatched. All volumes besides the ones from the last sort are called holdover volumes. The multiple-sort operations increase the total handling volume by each hub building and substantially lower the unit overhead cost. This hub-and-spoke system greatly reduces the number of partial loads and thus, the total operating cost. Therefore, the most cost-e€ective line-haul operating plan is to determine how shipments should be consolidated through the hub facilities, and how many schedules of various types of feeders should be dispatched to transport freight between facilities so that the overall operating cost is minimized while meeting the following operational constraints: Service commitment. This is the time commitment to the customers, or the number of days after the day of pickup that the consignees will receive the shipment. Suppose the service commitment between remote centers B and C (in Fig. 1) is two days, the consignees served by center C will receive their shipments on Thursday if it is picked up on Tuesday by center B. Flow conservation. All the pickup volume must be shipped to the preload sort of delivery centers. All the delivery volume must depart from the local sort of pickup centers. All the volume handled at any hub sort is transported to the next facility without any left over. Capacity. The processing volume of any sort at either centers or hubs cannot exceed its design capacity. In addition, the total transport volume between any pair of sorts of two separate facilities cannot exceed the total available space of the trailers assigned to it. Directed in-tree rooted at each destination. Handlers at any sort of centers or hubs are trained to feed the same destination (preload of a center) volume, disregarding their origins, to an outbound area/door in the building, where a single outbound sort of that destination is assigned. It is then loaded onto feeders and fed to that sort. Therefore, at any sort, there is at most one outbound sort for each destination. The night sort of Hub 2 receives volume destined to center C from centers B and D (in Fig. 1). It may be fed either to the night sort of Hub 1 for further consolidation, or directly to the preload sort of center C for delivery, but not both. Thus, at most one of the two is the outbound sort at the night sort of Hub 2 for center C. This restriction also implicitly implies there is a unique path for each OD pair. Trailer balancing. The line-haul operations are repeated daily. Therefore, the number of inbound and outbound trailers of any type at any sort must be equal daily.

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Driver's ocial work rules. The government usually restricts the maximum number of working hours daily and weekly. Moreover, some labor unions may request speci®c rules, such as a break after a certain driving duration, and all drivers must return to their depots daily. For such a complicated line-haul operations problem, carriers generally adopt a sequential planning process, decomposing the planning process into three phases, freight routing, trailer assignment and balancing, and tractor±trailer scheduling problems. The freight routing problem determines routes for all shipments in a directed and capacitated network while meeting the service commitment, ¯ow conservation and directed in-tree constraints, thus minimizing the sum of handling and transportation cost. When the time commitment and directed in-tree constricts are ignored, the freight routing problem becomes a multi-commodity ¯ow problem (MCP), determining a minimum shipment for di€erent commodities in a directed and capacitated network. Each OD pair is treated as a commodity. The MCP has been extensively studied in the literature. Several approaches have been proposed, such as the Lagrangian Relaxation (LR) (Ahuja et al., 1993), Dantzig-Wolfe price-directive decomposition (Tomlin, 1966), and primal-dual (Papadimitriou and Steiglitz, 1982; Barnhart and She, 1993). On the other hand, when the time commitment and capacity constraints are excluded, the freight routing problem becomes a non-capacitated MCP with directed in-tree constraint problem. Powell and Koskosidis (1992) studied a similar problem, but with one additional constraint, a minimum trailer frequency between pairs of facilities. They developed a local improvement heuristic and several primal-dual algorithms, such as subgradient, multiplier adjustment and dual ascent. The result showed that the subgradient approach outperforms all other approaches both quantitatively and qualitatively. Upon the determination of the freight routes for all shipments, the trailer assignment and balancing determines a balanced trailer movement for each type of trailers to transport shipments between facilities in order to minimize the total transportation cost (Eckstein and She, 1987). However, the selection of trailer types and their routes may further a€ect the minimum freight routes. Thus, some scholars have suggested integrating the two problems into the load planning problem to simultaneously determine how small shipments should be consolidated through hub facilities in a directed network and how various types of trailers should be dispatched to minimize the overall handling and transportation cost. Powell and She (1989) studied the load planning for the time-insensitive less-than-truckload industry. On the other hand, Leung et al. (1990) and Lin (1998) studied the load planning problem for time-de®nite freight delivery industry. Lastly, the feeder scheduling problem is to construct the least number of schedules to transport those dispatched trailers while meeting the driversÕ work rules (Suter et al., 1996). The line-haul operations plan is normally revised bi-yearly or more frequently, as necessary. There have been few computational studies of the integrality constrained MCP which compare alternative algorithmic philosophies for real data sets. In this paper, we present two algorithms, the LR and an implicit enumeration with e-optimality (IE-e). Numerical experiments are presented to evaluate the e€ectiveness of their computational performance. The structure of this paper is as follows. In Section 2, we show that the carrierÕs hub-and-spoke line-haul operations network can be transformed as a directed and capacitated freight routing planning network. In Section 3, we present an integer program for the freight routing problem. Section 4 describes in detail the LR and IE-e algorithms for the problem. In Section 5, we identify some potential

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improvement strategies to speed up the computations. In Section 6, we use the third largest timede®nite carrier in Taiwan for our numerical test, and present our computational results. Our conclusions together with further research suggestions, are presented in Section 7. 2. Freight routing planning network A carrier's line-haul operations network can be represented by a directed and capacitated freight routing planning network. The freight routing network for the line-haul operations network in Fig. 1 is shown in Fig. 2. A node represents a sort of a facility. There are two nodes for a center, representing local and preload sorts. The local-sort node is also an origin node, where the volume originates. On the other hand, the preload-sort node is a destination node, where the volume is destined. Each OD demand is the volume picked up by its local sort and delivered by its preload sort. Each hub sort in a hub facility is represented by a hub-sort node. Each node in the network possesses three attributes, unit handling cost, duration and capacity. The unit handling cost is the sum of unloading, sorting, and reloading costs to handle a piece (a ton in Taiwan) in the facility. We assume that the unit handling cost is constant and independent of the handling volume. This is a relatively valid assumption, since the unit handling cost in the most productive handling range is relatively constant (Leung et al., 1990). The duration is the sort span of the node, which is the total available time to complete its consolidation operation. Lastly, the node capacity is the design capacity (pieces in US, tons in Taiwan) of the building. A link is a direct connection between two nodes with no intermediate nodes. Its head node is where the freight/feeder departs, and its tail node is where they arrive. There are links connecting all origin nodes to all hub-sort nodes, all hub-sort nodes to all destination nodes,

Fig. 2. Illustrative freight routing network.

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and all hub-sort nodes to all other hub-sort nodes. Each link has three attributes, unit transportation cost, time window and capacity. Since carriers have to run all the feeders, therefore, the unit transportation cost represents only the additional operating variable cost due to the weight increased by each additional unit of freight on the equipment. Fuel cost, tire wear, etc. are some of the cost elements. We assume that each additional unit increase will cause the same amount of cost. The time window is the elapsed time from the sort end time at the tail node to the sort start time at the head node, and should be sucient for the actual travel. Otherwise, 24 h will be incrementally added to its window, indicating that the load will arrive one day later. Lastly, the capacity represents the total available space on the trailers dispatched on the link. The freight routing planning network does not allow center-to-center direct loads, since these rarely happen. Alternatively, we may exclude center-to-center full loads in a preprocessing program and only deal with partial loads in the freight routing problem.

3. The freight routing problem and mathematical model Let (N, A) be a directed network. N is the set of nodes with generic elements i; j 2 N . Furthermore, let O  N be a set of origin nodes with generic elements m0 , while D  N is a set of destination nodes with generic element md . Every node i 2 N is associated with a unit handling cost Ci , a duration Ti (Tm0 for origin m0 ), and a capacity V^i (V^m0 for origin m0 ). Let A^  A be a set of non-holdover links. Every link ij 2 A, is associated with a unit transportation cost Cij , a time ^ there is no transportation window Tij , and a capacity V^ij . Note that for holdover links ij 2 A n A, cost Cij ˆ 0, and there is in®nite capacity (V^ij ˆ 1). The number of OD pairs is characterized by a vector M with a demand Qm (Qm0 md ) and a service commitment T^m for every OD pair m 2 M. There are at most jMj ˆ jN j  …jN j 1† OD pairs. 3.1. The freight routing model A link formulation for the freight routing problem is formally stated as follows: X X X X Cm0 Qm0 md ‡ …Cj ‡ Cij † Qm xmij …P† min K ˆ m0

md

ij2 A^

subject to: X Tm0 ‡ …Tij ‡ Tj †xmij 6 T^m

8m 2 M;

…1†

m

…2†

ij

XX Qm xmij 6 V^j

8j 2 N n O;

…3†

i:ij2 A^ m

XX Qm0 md xmm00 mj d 6 V^m0 j

md

8m0 2 O;

…4†

C.-.C. Lin / Transportation Research Part B 35 (2001) 525±547

X Qm Xijm 6 V^ij

^ 8ij 2 A;

531

…5†

m

X i2N fjg

2

1 if j 2 D xmji ˆ 4 1 if j 2 O fjg 0 if o=w

X

xmij

i2N

Xm zij d 6 1;

8j 2 N ; 8m 2 M;

8i 2 N n D; 8md 2 D;

…6† …7†

j

Bzmij d

X m0

xmij 0 md P 0

xmij ; zmij d 2 f0; 1g

8ij 2 A; 8md 2 D;

…8†

8ij 2 A; 8m 2 M; 8md 2 D:

…9†

For the decision variables, let xmij …xmm00 mj d † ˆ 1 if link ij…m0 j† is on the path of OD pair m with origin m0 and destination md , otherwise xmij …xmm00 mj d † ˆ 0; also let zmij d ˆ 1, if j is the out bound node at node i for destination md , otherwise zmij d ˆ 0. The objective function represents the carrier's total operating cost. The ®rst term is the sum of handling costs of all origin nodes. Since holdover volumes are loaded onto trailers, thus, to avoid counting the handling cost of head nodes of holdover links, thus, in the second term we only sum the transportation and handling cost over all non-holdover links. Constraint (2) is the service commitment. It is the maximum elapsed time in hours from the sort start time at the local sort of pickup center to the sort end time at the preload sort of delivery center. Constraints (3) and (4) together state that the total processing volume cannot exceed the available capacity at any sort. Constraint (5) states that the ¯ow on any link cannot exceed its space available on the assigned trailers. Constraint (6) is the ¯ow conservation, indicating that: all the pickup volumes must depart from local sorts; all the preload sorts must receive their delivery volumes; and all the inbound volume to any hub nodes must be transported to other nodes. Constraint (7) is the restriction that there is at most one outbound node for each destination at any node. Constraint (8) states that only when there is an outbound node assigned to a destination at a node can paths be routes via that outbound node for that destination. Parameter B is a large number. Constraints (7) and (8) together restrict that there must be a directed in-tree rooted at any destination. Lastly, constraint (9) restricts two decision variables that the freight path and one outbound node per destination at any node are binary. 4. Two solving algorithms 4.1. The LR algorithm The LR algorithm relaxes the capacity constraint of nodes (3)±(4) and links (5) which are incorporated into the objective function with a set of associated Lagrangian multipliers, where ai is for node i while bij is for link ij. The relaxed problem becomes

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…LR-P†

X X X X min K…a; b† ˆ …Cm0 ‡ am0 † Qm0 md ‡ …Cj ‡ aj ‡ Cij ‡ bij † Qm xmij m0

m

d ij2 A^ 8 9 = X < X ˆ Qm …Cm0 ‡ am0 † ‡ …Cj ‡ aj ‡ Cij ‡ bij †xmij : ; m ^

!

m

…10†

ij2 A

subject to (2), (6)±(9). The capacity-relaxed freight routing problem is a much simpler problem. It is to ®nd the minimum shipment for all OD pairs, while meeting service commitments and a directed in-tree rooted at each destination. To solve it, we incrementally add constraints whenever they are violated. To begin with, we consider only constraint (6) for xmij 2 f0; 1g 8m 2 M; ij 2 A. This subproblem may be decomposed into each OD pair as a shortest path problem, to be solved by the Dequeue implementation (Ahuja et al., 1993). If necessary, we expand the constraint set by adding the service commitment constraint (2) to those OD pairs whose least cost paths are timeinfeasible. We adopt Handler and Zang's (1980) method to dualize constraint (2). We solve 8 X X < Qm …Cm0 ‡ am0 ‡ rm Tm0 † ‡ …rm Tj ‡ rm Tij †xmij min K…a; b; c† ˆ : m ij2An A^ 9 X

 = ‡ Cj ‡ aj ‡ rm Tj ‡ Cij ‡ bij ‡ rm Tij xmij …11† ; ^ ij2 A

subject to (6). The dual problem is to minimize the weighted sum of cost and time, where rm represents the relative importance of these two attributes. At every iteration t, we update the multiplier, t …rm † ˆ C mf C mc =T mc T mf , where …C mf ; T mf † is the least cost but time-feasible path (initially, the least time path is used), and …C mc ; T mc † is the least time but time infeasible path. Conceptually, we explore the frontier de®ned by cost as the X-axis, and time as the Y-axis. This continues until a new solution, either …C mf ; T mf † or …C mc ; T mc †, is obtained. Then, …C mf ; T mf † is the optimality. Now, we obtain the least cost among all the time-feasible paths for all OD pairs. We then verify whether or not this solution forms a directed in-tree rooted at any destination, that is, it meets constraints (7) and (8) for zmij d 2 f0; 1g; md 2 D; ij 2 A. Any violation means that there exist two paths originating at two origin nodes with the same destination intersecting at a common intermediate node, from which they take di€erent path segments to the same destination. Suppose there are two paths in the network shown in Fig. 2: local sort of center B (origin) ® night sort of Hub 2 (via) ® preload sort of center C (destination) and local sort of center D (origin) ® night sort of Hub 2 (via) ® night sort of Hub 1 (via) ® preload sort of center C (destination). These two paths intersect at a common intermediate node, which is the night sort of Hub 2. Here, they take di€erent path segments, directing to and via another hub sort before arriving at the destination, preload sort of center C. When this occurs, the shorter time taken by the two path segments from the intermediate node to the destination is chosen, e.g. night sort of Hub 2 ® preload sort of center C. The path using the other path segment is rerouted onto the shorter time path segment. Thus, the new path for local sort of center D (origin) and preload sort of center C (destination) pair is now

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via night sort of Hub 2 (via). In this way, we ensure that all the rerouted OD pairs still meet the service commitment constraint (2). When ®nished, we have completed one iteration. The objective value of LR-P provides a lower bound to the optimality. Of course, any solution that does not violate any node or link capacity constraints is a primal feasible solution to the original optimization problem (P). We update and maintain the solution that has the least total operating cost, which becomes the upper bound to the optimality. Unless the gap between the upper and lower bounds is no greater than a predetermined tolerant, the Lagrangian multipliers will be updated by the subgradient method (Held et al., 1974). Multipliers either increase for overcapacitated nodes or links, or decrease if not 0. Even though the LR is fairly easy to implement, it has several disadvantages. First, there is no guarantee of ®nding a feasible solution, even though it may exist. When the LR fails at termination, it is not known whether it is infeasible to the current capacitated network, or the algorithm is simply unable to identify a solution. Second, the algorithm works poorly for large-scale network problems in which there are many node and link capacity constraints. It is not easy for the LR to determine a ``right'' set of multipliers to penalize over-capacitated nodes or links in order to assign volumes on the ``right'' paths to obtain a feasible solution. Lastly, the algorithm cannot properly handle degeneracy. When some OD pairs having more than one identical cost path, the algorithm ¯ip-¯ops between two paths in two consecutive iterations. It will not split OD pairs proportionally in order to satisfy the capacity constraints. 4.2. An implicit enumeration with e-optimality algorithm The implicit enumeration algorithm with e-optimality (IE-e) is based on the path formulation of the freight routing problem. All the paths for OD pair m is characterized by a vector P m with a unit path cost Cpm , and a path time Tpm associated with every path p 2 P m . In addition, let 0 md † ˆ 1 if link ij…m0 j† is on the path p for OD pair m with origin m0 and destination md , dmp;ij …dmp;m 0j 0 md † ˆ 0. The equivalent path formulation of (P) is as follows: otherwise dmp;ij …dmp;m 0j XX …P2† min K ˆ Cpm Qm xmp …12† m

p

subject to: Tpm xmp 6 T^m

8p 2 P m ; m 2 M;

X XX m dp;ij Qm xmp 6 V^j

i:ij2 A^ m

m

8m0 2 O;

…15†

^ 8ij 2 A;

…16†

p

X xmp ˆ 1 8m 2 M; p

…14†

p

XX m dp;ij Qm xmp 6 V^ij m

8j 2 N n O;

p

XXX m dp;m0 j Qm xmp 6 V^m0 j

…13†

…17†

534

X j

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zmij d 6 1

Bzmij d

8i 2 N n D; 8md 2 D;

XX m m dp;ij0 d xmp 0 md P 0 m0

8ij 2 A; 8md 2 D;

…18† …19†

p

xmp ; zmij d 2 f0; 1g

8p 2 P m ; m 2 M; ij 2 A; md 2 D

…20†

with decision variable, xmp ˆ 1, if path p is the delivery path for OD pair m, otherwise xmp ˆ 0. The objective is to minimize the total operating cost. The unit path cost of an OD pair is the sum of unit handling cost of nodes and unit transportation cost of links on the path. This path cost does not include the handling cost of the head node of holdover links if it is on the path. Constraint (13) is the service commitment. Constraints (14)±(16) are the node and link capacities. Constraint (17) is a single path for each OD pair. Constraint (18) limits to at most one outbound node per destination at any node. Constraint (19) is the bundle constraint, so that only when there is an outbound node for a destination may paths ¯ow via that outbound node for that destination. Constraints (18)±(19) together de®ne a directed in-tree for each destination. Constraint (20) is composed of binary constraints on the freight path and outbound node decision variables. To implement IE-e for the freight routing problem, we ®rst construct a two-dimensional search matrix with generic elements (m, p(m)) (see Fig. 4). Each row represents an OD pair, while each column represents a time-feasible path. All the paths in the search matrix meet the service commitment constraint (13). Furthermore, time-feasible paths of any OD pair are sequenced by their costs Cpm and arranged from left to right. Therefore, cell (m, p(m)) is the pth least cost timefeasible path of mth OD pair. Of course, the matrix may not necessarily be rectangular, since the number of time-feasible paths may vary by OD pair. To construct the search matrix, we need to identify all the time-feasible paths for all OD pairs. For a given set of hub nodes, we ®rst construct all the possible hub-to-hub path segments that originate, are destined to, and may ¯ow via none, one, some, or all of the other hub nodes. There are four hub-to-hub segments in the network of Fig. 2. They are: night sort of Hub 1, night sort of Hub 2, night sort of Hub 1 ® night sort of Hub 2, and night sort of Hub 2 ® night sort of Hub 1. Upon completion, we sequentially select an OD pair and determine all its time-feasible paths, and at the same time, calculate their unit path costs. For OD pair m, 0. We sequentially select possible hubinitially its number of time-feasible paths is set jP m j to-hub segments until they are exhausted. For each one, we connect the origin node to the ®rst hub node and also the last hub node to the destination. This completes a path for OD pair m. We calculate its path time and unit path cost. If the path time is greater than the service commitment T m , we discard it; otherwise, it is a time-feasible path. We increase the number of jP m j ‡ 1, denoting the path time and unit path cost as Tpm and time-feasible paths by jP m j m Cp , respectively. When the possible hub path segments are exhausted, we sort and order all the time-feasible paths according to their unit path costs, and place them on the mth row of the  search matrix. The ®rst cell of the mth row is Cpm ˆ minfCpm ; p 2 P m g. We repeat this process for all OD pairs.

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4.3. The search procedure We employ a depth-®rst search approach, initially starting at the top-left cell, which is the least ^ p^…m††, ^ we refer to cost time-feasible path of the ®rst OD pair. At any time, when we scan a cell …m; ^ is the current OD pair and p… ^ m† ^ is its current path. We refer to one it as the current cell, where m ^ 1, and one number greater number less than the current OD pair, as the previous OD pair, m ^ ˆ jP m j ‡ 1. To meet constraint (17), we can than the last cell of an OD pair, as the null cell, p^…m† only select exactly one cell of each row (OD pair). At the current cell, only when capacity constraints (14)±(16) and directed in-tree constraints (18)±(19) are met, can we calculate the lower bound for the current cell. In practice, planners may tolerate a small percentage higher than the optimum to gain time eciency. This concept is e-optimality, where e is the tolerance of planners. Of course, when optimality is desired planners may set e ˆ 0. Therefore, only when the lower bound in¯ated by e tolerance does not exceed the value of the incumbent solution can we accept it. In this case, we move the current OD pair ^ from the set of candidate OD pairs, denoted as S, to the set of scanned OD pairs, denoted as S. At the same time, we store the current path in the set of chosen paths, denoted as ^ P^ P^ [ f^ p…m†g. Fig. 3 is a ¯ow chart of the basic steps in the IE-e algorithm. The IE-e algorithm constantly performs loading and removal procedures at the current cell, as described below. ^ on Loading procedure. Update the network ¯ows by loading the volume of current OD pair m P ^ m^ ^ m ^ m ^ ^ Vmo Vmo Qm^ ; Vj Vj ‡ i dmp;ij Q ; 8j 2 N and V V ‡ d Q ; 8ij 2 A. the current path p; ij ij ^ ^ p;ij ^ to the set of Then, move the current OD pair from the set of candidate OD pairs, S S fmg, ^ scanned OD pairs S^ S^ [ fmg. Of course, also update the set of chosen paths by ^ P^ P^ [ f^ p…m†g. Removal procedure. This is the reverse of the loading procedure. the volume of current P Unload ^ m ^ m OD pair from its current path, Vmo Vmo Qm^ ; Vj Vj d Q ; 8j 2 N and Vij Vij ^ i p;ij ^ m ^ ^ Move the current OD pair from the set of scanned OD pairs set S^ ^ fmg, ^ Q ; 8ij 2 A. S to dmp;ij ^ ^ the set of candidate OD pairs, S S [ fmg. Also update the set of chosen paths, ^ P^ P^ f^ p…m†g. Step 0 (Initialization). Initialize the incumbent solution P  ˆ f/g, and its value K  ˆ 1. Set S ˆ M and S^ ˆ f/g. Set also the network ¯ows x ˆ f0g. Set the current node as the top-left cell ^ p… ^ m†† ^ of the search matrix, that is, the least unit path cost of the ®rst OD pair, …m; …1; 1†. ^ on p^…m†. ^ Step 1 (Perform loading procedure). Perform the loading to m ^ ) Vij > V^ij , conStep 2 (Verify capacity constraints). When 9j 2 N ; ) Vj > V^j or 9ij 2 A; ^ on p… ^ m†. ^ Make the next higher unit path straints (14)±(16) are violated. Perform the removal to m ^ to be the new current cell …m; ^ p^…m†† ^ ^ p^…m† ^ ‡ 1† (see Fig. cost of m …m; Go to step 7. P 4). md Step 3 (Ensure directed in-tree). When 9i 2 N nD and 9md 2 D; ) j zij > 1, then constraints ^ on p^…m†. ^ Make the next higher unit path cost of (18)±(19) are violated. Perform the removal to m ^ to be the new current cell …m; ^ p… ^ m†† ^ ^ p… ^ m† ^ ‡ 1† (see Fig. 4). Go to step 7. m …m; Step 4 (Calculate the lower bound). The lower bound with e ˆ 0 at any current cell is ^ p… ^ m†Š ^ ˆ l‰m;

jMj ^ m X X  Cpm^ Qm ‡ C m Qm : mˆ1

mˆm‡1 ^

…21†

536

C.-.C. Lin / Transportation Research Part B 35 (2001) 525±547

Fig. 3. Flowchart of the implicit enumeration algorithm.

^ while the second term is the The ®rst term of Eq. (21) is the operating cost of OD pairs in S, least operating cost of OD pairs in S. To allow a tolerance e, the lower bound becomes ^ p^…m†Š ^ ^ p^…m†Š ^  …1 ‡ e†. l‰m; l‰m; Step 5 (Check whether the lower bound exceeds the upper bound). When the lower bound ^ p^…m†Š ^ > K  , this implies that although the exceeds the value of the current incumbent solution l‰m; freight of all unassigned OD pairs in S are routed on their least unit path cost paths, the total operating cost is still greater than the value of the current incumbent solution. Therefore, ^ on p^…m†. ^ Since time-feasible paths of all OD pairs are in an ascending we perform removal to m ^ Make the order by cost, it is meaningless to search any cell higher than the current cell of m. ^ p^…m†† ^ next higher unit path cost of the previous OD pair to be the new current cell …m; ^ 1; p^…m ^ 1† ‡ 1† (see Fig. 4). Go to step 7. …m

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Fig. 4. The strategic selection of new current cell.

^ ˆ jMj, Step 6 (Update a new incumbent solution when feasible). When S is an empty set, i.e. m we have found a feasible solution better than the incumbent. Update the incumbent solution by PjMj m m P^, and its value by K  C Q , and continue our search for optimality. Perform the P mˆ1 p^ ^ on p… ^ m†, ^ and make the next higher unit path cost of the previous OD pair to be the removal to m ^ p^…m†† ^ ^ 1; p… ^m ^ 1† ‡ 1†, which is the same case as when the lower is new current cell, …m; …m ^ < jMj, greater than the upper bound (see Fig. 4). Go to step 7. However, when S is not empty, m only a partially feasible solution is determined. Make the ®rst cell of the next OD pair to be the ^ p… ^ m†† ^ ^ ‡ 1; 1† (see Fig. 4), and also proceed to step 7. new current cell …m; …m Step 7 (Perform the null cell procedure). If the current cell is not a null cell ^ p^…m†† ^ 6ˆ …m; ^ jP m^ j ‡ 1†, go to step 1. Otherwise, make the next higher unit path cost of the …m; ^ p^…m†† ^ ^ 1; p^…m ^ 1† ‡ 1†. Of course, the previous OD pair to be the new current cell …m; …m new current cell could also be a null cell. Therefore, we continue step 7 until a new current cell is not a null cell (see Fig. 5). Then return to step 1. Of course, when none of the non-null cells can become the current cell, we have completed our search procedure. If the upper bound is ®nite K  < 1, then P  is the e optimal solution, and its value is K  . Otherwise, there are no feasible solutions for the freight routing problem in the current capacitated network, and we terminate the program.

538

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Fig. 5. The null cell procedure.

5. Improvement strategies The IE-e algorithm increases network ¯ows by sequentially loading OD volumes, and the sequence of OD volumes may in¯uence the algorithmic eciency. We test the IE-e algorithm with several experiment strategies as follows. Building ID number (BN). The do-nothing strategy is to sequence OD demands in an ascending order by their ID numbers. It is the most commonly-used strategy by carriers. Number of paths ®rst, cost second (PFCS). This strategy sequences OD pairs by the number of time-feasible paths in an ascending order ®rst. When there is a tie, they are sequenced in ascending order according to their least unit path costs. This is to determine an incumbent solution with the lowest possible operating cost in as few iterations as possible. This may make it more likely for the lower bound to exceed the value of the incumbent solution and fathom more unassigned OD pairs than otherwise. Number of paths ®rst, volume second (PFVS). This strategy sequences the number of timefeasible paths in ascending order as the ®rst step, but arranges OD volumes in descending order when there is a tie. Loading OD pairs with higher volumes may enable us to detect capacity violation in a relatively few OD pairs. On the other hand, if we can accommodate higher volumes, it is potentially easier to accommodate smaller ones in the leftover capacities. Center volume (CV). The last strategy uses the center instead of individual OD pairs as the sequencing unit. It sequences the center load factor in descending order. For each center, its pickup OD pairs appear prior to the delivery. The pickup load factor is the pickup volume divided by the total outbound trailer capacity of the center local node. Likewise, the delivery load factor is

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the total delivery volume divided by the total inbound trailer capacity of the center preload node. The center load factor is the smaller of the two. This concept is based on the fact that the capacity violation is generally caused by an insuciency of either the center inbound or outbound capacity. Again, on earlier realization of the insucient capacity may enable more unassigned OD pairs to be fathomed than otherwise. 6. Numerical examples with computational results 6.1. The network generation We use the third largest freight common carrier in Taiwan for our numerical testing. Taiwan is a small island, and all commercial activities and the population are heavily concentrated linearly along the west coast. Thus, center-to-center direct feeds are more economical than indirect feeds that require at least one additional hub consolidation and a longer travel distance. However, a pair of centers normally do not have sucient volume to ®ll up a whole trailer. Thus, carriers perform stopover with center directs operations to increase the trailer utilization. A few pairs of centers are assigned to a feeder run while meeting work rules. The trailer space is divided into several staging areas, one for each center. At each center, the arrival freight is unloaded and departure freight of centers down the route is loaded. However, expecting a continuous growth in volume, together with a lack of space for expansion in most of the centers, thus new hubs may locate in suburbs where less expensive land is available. This makes the hub-and-spoke network for line-haul operations to be more cost e€ective. This carrier plans to build three hubs near the three largest centers, located in three major metropolitan areas, Taipei, Taichung and Tainan, which are in the north, center, and south of Taiwan, respectively. The precise locations are not known as yet, so we use the location of existing centers as the hub locations to measure their distances to other centers. The carrier uses three types of trailer combinations, 20, 35 and 42 tons, each with carrying capacities of 10, 14, and 25 tons, respectively. The largest ¯eet has a weight of 20 tons with a capacity of 10 tons; we use this for our numerical tests. The unit handling cost is NT$ 600 per ton on average. The starting time is 5 pm for local sort and 4 am for preload, both of which last for 2 h. On the other hand, the hub night sort is assumed to start at 11 pm and last for 3 h. The time window of a link is the elapsed time between two sorts. The demand of each OD pair is the carrierÕs current average daily pickup volume. Lastly, the service time windows are the carrierÕs current commitment to its customers. Normally, the service commitment is one day, except for a few remote areas in the far north and south ends which will require 2 days for deliver. We test algorithmic performance on three network sizes: 5 centers, 10 centers, and all 68 centers. Each of these consists of 3 hubs. However, the freight routing problem requires a predetermined balanced trailer network that is not currently available. Therefore, for each network size, we randomly generate three sets of balanced freight routing networks with variations in node or link capacities. Thus, there are nine freight routing networks, each denoted as a base network. The procedure to randomly generate a base network is as follows. First, we randomly generate a number of trailers on each link. Second, we balance the trailer network by adding empty trailers. By assuming that the carrier uses only one trailer type for line-haul operation, the trailer balance

540

Case Strategy Problem

First feasible solution

Capacity Iteration multiplier (%) 1

BN PFCS PFVS CV LR

2

BN PFCS

210 220 230 210 220 230 210 220 230 210 220 230 210 220 230

N/F 20 20 N/F 20 20 N/F 20 20 N/F 20 20 U/D 1 1

210 220 230 210 220 230

N/F 20 20 N/F 20 20

Total cost

Optimal solution (NT) Optimal gap (%)

179,486 179,486

0 0

179,486 179,486

0 0

179,486 179,486

0 0

179,486 179,486

0 0

179,486 179,486

0 0

179,486 179,486

0 0

179,486 179,486

0 0

Iteration

N/F 20 20 N/F 20 20 N/F 20 20 N/F 20 20 U/D 1 1 N/F 20 20 N/F 20 20

Handling cost (NT)

Transport cost (NT)

Total cost (NT)

172,620 172,620

6866 6866

179,486 179,486

172,620 172,620

6866 6866

179,486 179,486

172,620 172,620

6866 6866

179,486 179,486

172,620 172,620

6866 6866

179,486 179,486

172,620 172,620

6866 6866

179,486 179,486

172,620 172,620

6866 6866

179,486 179,486

172,620 172,620

6866 6866

179,486 179,486

Program termination

CPU Time (s)

Iteration

Ratio

By Average problem by strategy

1606 33 33 3 33 33 3 33 33 3 33 33 200 1 1

9.80e)2 2.00e)3 2.00e)2 1.80e)4 2.00e)3 2.00e)3 1.80e)4 2.00e)3 2.00e)3 1.80e)4 2.00e)3 2.00e)3 N/A N/A N/A

0.17 0.04 0.03 0.03 0.04 0.03 0.03 0.04 0.04 0.04 0.04 0.03 0.71 0.03 0.03

0.08

1606 33 33 3 33 33

9.80e)2 2.00e)3 2.00e)3 1.80e)4 2.00e)3 2.00e)3

0.17 0.03 0.04 0.02 0.03 0.04

0.08

0.03 0.04 0.04 0.26

0.03

C.-.C. Lin / Transportation Research Part B 35 (2001) 525±547

Table 1 Computation results for three 3-hub 5-center base networksa

PFVS CV LR

BN PFCS PFVS CV LR

a

N/F 20 20 N/F 20 20 U/D 1 1

70 80 90 70 80 90 70 80 90 70 80 90 70 80 90

N/F 24 23 N/F 23 23 N/F 23 23 N/F 25 23 U/D 2 2

179,486 179,486

0 0

179,486 179,486

0 0

179,486 179,486

0 0

180,141 179,705

0 0

180,468 180,468

0.18 0.42

180,468 180,468

0.18 0.42

180,467 179,705

0.18 0

180,583 180,583

0.24 0.49

N/F 20 20 N/F 0.20 0.20 U/D 1 1 N/F 24 23 N/F 90 104 N/F 74 72 N/F 93 23 U/D 2 2

172,620 172,620

6866 6866

179,486 179,486

172,620 172,620

6866 6866

179,486 179,486

172,620 172,620

6866 6866

179,486 179,486

172,620 172,620

7521 7085

180,141 179,705

172,620 172,620

7521 7085

180,141 179,705

172,620 172,620

7521 7085

180,141 179,705

172,620 172,620

7521 7085

180,141 179,705

172,620 172,620

7963 7963

180,583b 180,583c

Note: N/F: no feasible solutions; U/D: undetermined; N/A: not applicable. A local optimal solution with the optimality gap of 0.24%. c A local optimal solution with the optimality gap of 0.49%. b

3 33 33 3 33 33 200 1 1

1.80e)4 2.00e)3 2.00e)3 1.80e)4 2.00e)3 2.00e)3 N/A N/A N/A

0.04 0.04 0.03 0.03 0.03 0.04 0.72 0.03 0.02

323 156 82 3 137 128 3 117 96 3 128 115 200 200 200

1.97e)2 9.50e)3 5.00e)4 1.80e)4 8.40e)3 7.80e)3 1.80e)4 7.10e)3 5.90e)3 1.80e)4 7.80e)3 7.00e)3 N/A N/A N/A

0.06 0.04 0.04 0.03 0.04 0.05 0.03 0.04 0.04 0.03 0.04 0.04 0.74 0.79 0.75

0.04 0.03 0.26

0.47 0.04 0.04 0.04 0.76

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3

210 220 230 210 220 230 210 220 230

541

542

Case

Strategy

Problem

First feasible solution

Capacity Iteration multiplier (%) 1

BN PFCS PFVS CV LR

2

BN PFCS

160 170 180 160 170 180 160 170 180 160 170 180 160 170 180

N/F N/F 90 N/F N/F 90 N/F N/F 90 N/F N/F 90 U/D U/D 1

100 110 120 100 110 120

N/F 97 92 N/F 94 92

Optimal solution

Total cost Optimal (NT) gap (%)

662,263

0

662,263

0

662,263

0

662,263

0

662,263

0

662,639 662,264

0.018 0

663,146 662,489

0.095 0.034

Iteration HC (NT)

N/F N/F 90 N/F N/F 90 N/F N/F 90 N/F N/F 90 U/D U/D 1 N/F 3295 92 N/F 494,047 17,446

Program termination TC (NT)

Total cost Iteration (NT)

638,100

24,163

662,263

638,100

24,163

662,263

638,100

24,163

662,263

638,100

24,163

662,263

638,100

24,163

662,263

638,100 638,100

24,419 24,164

662,519 662,264

638,100 638,100

24,419 24,164

662,519 662,264

833,573,538 1,409,599,414 394 5 14 272 5 13 274 95 298 379 200 200 1 162,913 3693 286 6,379,232 494,342 17,596

Ratio

CPU Time (s) By problem

Average by strategy

3.22e)10 158,953.79 142,221.16 5.44e)10 267,709.56 1.52e)16 0.12 1.93e)18 0.08 0.09 5.41e)18 0.06 1.05e)16 0.12 1.93e)18 0.06 0.08 5.02e)18 0.06 1.06e)16 0.12 3.67e)17 0.07 0.11 1.15e)16 0.12 1.46e)16 0.13 N/A 2.77 1.88 N/A 2.81 N/A 0.05 6.29e)14 1.43e)15 1.10e)16 2.46e)12 1.91e)13 6.79e)15

30.27 0.69 0.11 992.32 78.41 2.85

10.36 357.86

C.-.C. Lin / Transportation Research Part B 35 (2001) 525±547

Table 2 Computational results for three 3-hub and 10-center base networksa

PFVS CV LR

BN PFCS PFVS CV LR

a

N/F 94 92 N/F 316 92 U/D U/D 37

110 120 130 110 120 130 110 120 130 110 120 130 110 120 130

N/F 103 94 N/F 92 91 N/F 92 91 N/F 92 91 U/D 2 2

663,146 662,489

0.095 0.034

663,472 662,489

0.14 0.034

662,264

0

662,422 662,377

0 0.01

662,422 662,307

0 0

662,422 662,307

0 0

662,422 662,307

0 0

662,422 662,307

0 0

N/F 281,007 16,498 N/F 2197 493 U/D U/D 37 N/F 103 242 N/F 92 91 N/F 92 91 N/F 92 91 U/D 2 2

638,100 638,100

24,419 24,164

662,519 662,264

638,100 638,100

24,419 24,164

662,519 662,264

638,100

24,164

662,264

638,100 638,100

24,322 24,207

662,422 662,307

638,100 638,100

24,322 24,207

662,422 662,307

638,100 638,100

24,322 24,207

662,422 662,307

638,100 638,100

24,322 24,207

662,422 662,307

638,100 638,100

24,322 24,207

662,422 662,307

Note: N/F: no feasible solutions; U/D: undetermined; N/A: not applicable.

8,572,808 281,258 16,650 46 2545 627 200 200 37

3.31e)12 1.09e)13 6.43e)15 1.78e)17 9.83e)16 2.42e)16 N/A N/A N/A

1,329.05 44.46 2.7 0.06 0.53 0.18 3.44 2.79 0.55

458.74

109,976,702 2006 1076 13 1018 710 13 1098 766 9 268 271 200 2 2

4.25e)11 7.75e)16 4.15e)16 5.02e)18 3.93e)15 2.74e)15 5.02e)18 4.24e)16 2.96e)16 3.47e)18 1.03e)16 1.05e)16 N/A N/A N/A

20,588.25 0.40 0.24 0.06 0.22 0.18 0.07 0.23 0.21 0.07 0.11 0.11 2.89 0.07 0.06

6,862.93

0.26 2.26

0.15 0.27 0.10 1.01

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3

100 110 120 100 110 120 100 110 120

543

544

Table 3 Computational results for three 3-hub, 65-center, and 1961-OD-pair base networksa Case

PFCS PFVS CV LR

2

PFCS PFVS CV LR

3

PFCS PFVS CV LR

a

Problem

First feasible solution

e ˆ 1% optimal solution

Program termination

CPU time (s)

Capacity multiplier (%)

Iteration

Total cost (NT)

e gap (%)

Iteration

HC (NT)

TC (NT)

Total cost (NT)

Iteration

By problem Average by strategy

80 90 100 80 90 100 80 90 100 80 90 100

2064 2045 2029 2064 2045 2029 25,896 13,673 12,000 U/D U/D U/D

3,173,822 3,173,822 3,173,822 3,173,822 3,173,822 3,173,822 3,173,822 3,173,822 3,173,822

0 0 0 0 0 0 0 0 0

2064 2045 2029 2064 2045 2029 25,896 13,673 12,000 U/D U/D U/D

3,063,240 3,063,240 3,063,240 3,063,240 3,063,240 3,063,240 3,063,240 3,063,240 3,063,240

110,582 110,582 110,582 110,582 110,582 110,582 110,582 110,582 110,582

3,173,822 3,173,822 3,173,822 3,173,822 3,173,822 3,173,822 3,173,822 3,173,822 3,173,822

3413 3402 3398 3412 3401 3397 27,219 15,017 13,361 200 200 200

27.24 26.95 26.93 26.84 26.99 26.92 217.58 124.01 108.31 131.90 127.32 126.21

100 110 100 110 100 110 100 110

2041 2033 2041 2033 5535 6812 U/D U/D

3,170,478 3,170,317 3,170,487 3,170,317 3,170,505 3,170,317

0 0 0 0 0 0

2041 2033 2041 2033 5535 6812 U/D U/D

3,062,160 3,062,160 3,062,160 3,062,160 3,062,160 3,062,160

108,318 108,157 108,327 108,157 108,345 108,157

3,170,478 3,170,317 3,170,487 3,170,317 3,170,505 3,170,317

3409 3402 3410 3403 6887 10,894 200 200

26.91 26.86 26.77 26.82 54.09 65.22 125.28 122.15

90 100 90 100 90 100 90 100

2084 2043 2084 2043 12,351 9128 U/D U/D

3,172,749 3,171,610 3,172,749 3,171,610 3,172,749 3,171,610

0 0 0 0 0 0

2084 2043 2084 2043 12,351 9128 U/D U/D

3,063,360 3,063,360 3,063,360 3,063,360 3,063,360 3,063,360

109,389 108,250 109,389 108,250 109,389 108,250

3,172,749 3,171,610 3,172,749 3,171,610 3,172,749 3,171,610

3415 3400 3414 3399 13,653 10,000 200 200

26.90 26.85 27.06 26.82 110.10 85.20 124.34 122.10

Note: N/F: no feasible solutions; U/D: undetermined; N/A: not applicable.

27.04 26.92 149.97 128.48

26.89 26.80 59.66 123.72 26.88 26.94 97.65 123.22

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Strategy

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545

problem becomes a classic Transshipment Problem, which can be converted into a minimum-cost ¯ow problem. We implement the successive shortest path algorithm (Ahuja et al., 1993) to balance the trailer network. Finally, the node capacity is calculated. The capacity of local sort and preload sort of a center and the capacity of a hub sort is equal to the total capacity of their outbound trailers (which is the same as the inbound trailers). 6.2. Computational results The program is coded in C language. Since all the base networks are generated randomly, therefore, some of the base networks may or may not have more than sucient capacity to accommodate the daily pickup volume. Thus, we increase (decrease) the capacity of any base networks by an increment of 10% until the feasible set is not (is) empty. This strategy allows us to evaluate the algorithmic performance at three di€erent scenarios: an empty feasible set, with capacity is slightly or greatly larger than the daily demand. We conducted the computational test on a Pentium Pro PC with a CPU speed of 200 MHz under the Linux operating environment, because the PC is the common platform for freight carriers in Taiwan. The initial upper bound for the LR approach is the smaller of 100 times of the initial dual value or the largest integer value of the Ôlong integerÕ in C compiler. The initial step size is 2. Whenever the dual value fails to improve by more than 5% in four consecutive iterations, the step size reduces by half. The program terminates either when the duality gap is no greater than 5% or when the maximum of 200 iterations has completed. We evaluate and compare the two algorithms, LR and IE-e both quantitatively and qualitatively, using two quantitative criteria. The ®rst criterion is the computational time, and the other is the ratio of the number of cells scanned prior to the program termination, compared to the number of routing combinations. Whenever a new cell is scanned, we have completed one iteration. Note that in the LR, solving the dual problem is one iteration. In addition, we use two qualitative criteria. The ®rst is the gap between the solution and the optimality. The other is whether or not the algorithm is capable of detecting whether the feasible set is empty or not. We use identical quantitative and qualitative criteria to evaluate di€erent volume sequencing strategies for the IE-e. The testing results are organized in three tables, one for each network size. The ®rst freight routing network consists of 3 hubs, 5 centers and 20 OD pairs. There are 34 cells in the search matrix, therefore, 1.64e4 routing combinations whether they are feasible or not. In terms of computational time, the IE with e ˆ 0 outperforms the LR. On an average, the IE with e ˆ 0 takes no more than one-®fth of the computational time compared with the LR to determine the optimality. In the third base network, the LR runs for totally 200 iterations even though it determines the optimum solution in the second iteration. This is because the duality gap cannot be sequentially narrowed to less than the tolerance of 5%. Moreover, when the IE with e ˆ 0 determines no feasible solutions, the LR fails to do so for any of the networks. When we compare the impact of di€erent OD volume ordering strategies on IE with e ˆ 0, none of them obviously outperforms the others. The last observation is that the objective value of the ®rst feasible solution is no greater than 1% of the optimality for all IE with e ˆ 0 tests. This implies that if we allow a small margin o€ the optimum, which is a positive e, the algorithm may terminate in a fairly short computational time.

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Each of the three 3-hub and 10-center base networks has 152 cells on the search matrix and therefore 2.59e18 possible routing combinations, whether they are feasible or not. The performance of IE with e ˆ 0 is in most cases satisfactory. This testing again demonstrates that the LR is not capable of determining whether or not the feasible set is empty. Not only does it fail to draw any conclusions when the feasible set is empty, but it also fails to identify feasible solutions when they exist. Especially, when the capacity is barely enough over (10%) the demand is in the second base network. Among various OD volume ordering strategies, the BN performs most poorly. It takes more than 39.5 and 1.9 h to determine no feasible solutions for the ®rst and the third base networks, respectively. On the other hand, the CV strategy performs the best for the cases when the feasible sets are empty. But when there are feasible solutions the PFCS and PFVS outperform the CV strategy. This may indicate that the infeasibility in most of the cases is caused by insucient capacities of either center inbound or outbound trailers. Lastly, each of the three 68-center and 3-hub base networks has 3,853 cells on the search matrix. The number of routing combinations, whether feasible or not, is larger than the maximum value of a ÔdoubleÕ (1.79E + 308) in C compiler. Therefore, in Tables 1±3 no ratio of number of cells scanned prior to the program termination to the number of routing combinations is provided. Moreover, since the BN strategy does not perform well on small networks, it is excluded in this test. We set e ˆ 1% which means that the objective value of the solution is within 1% of the optimum. The LR is not able to determine any feasible solutions on all tests of the three base networks. This further demonstrates that when there are many capacity constraints, the approach is not capable of determining a set of proper multipliers to derive a feasible solution. In addition, it fails to produce any solutions even though they exist. This inability to ®nd feasible solutions prolongs the computations. Thus, quantitatively, the LR takes longer to run than IE with e ˆ 1%. The performance of IE with e ˆ 1% is satisfactory overall. In most of the cases, it takes less than a minute on the Pentium Pro-200 to run. This result is highly acceptable to the planners. A similar conclusion to the previous one is that, when the feasible set is not empty, the PFCS and PFVS strategies outperform the CV strategy. In this testing, we do not show the results for empty feasible sets. This is because, when we decrease the base capacity by 10%, some of the centers do not have sucient capacities to accommodate their pickup or delivery volumes. This causes the IE-e to terminate instantly. 7. Conclusions and future research Time-de®nite common carriers need to plan their operations more eciently. The freight routing problem for the carriers is to determine a freight routing plan to minimize its overall operating cost, while meeting the operational restrictions and service commitments. It is an integrality constrained multi-commodity problem with additional operational side constraints, and the LR approach may be employed for it. However, it fails to identify whether or not the feasible set is empty. It also performs poorly for large networks, since it is not easy for this approach to determine a set of proper multipliers to derive feasible solutions if they exist. The situation is worsened when the demand is fairly close to the carrying capacity.

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Thus, we propose an implicit enumeration with e-optimality algorithm for this problem. We test the algorithm on the third largest freight common carrier in Taiwan with three network sizes of various randomly generated carrying capacities. Quantitatively and qualitatively, the IE-e outperforms the LR for almost all the cases, especially for large networks. Not only is it capable of determining whether or not the feasible set is empty, but it also obtains an e-optimal solution in less computational time than the LR algorithm. This result is very encouraging. We also evaluate di€erent strategies of ordering OD volumes. When the feasible set is empty, the volume ordering based on the centerÕs total handling volume (CV) is superior to other strategies. However, when feasible solutions exist, ordering strategies based on the number of available time-feasible paths and also OD least unit path cost (PFCS) or volume (PFVS) are better. The volume ordering based on the building ID number (BN) is overwhelmingly outperformed by other strategies. One of several possible extensions to this research is to design a strategy to construct a smaller set, but preserving all the meaningful routing combinations on the search matrix. We may exclude those paths for some OD pairs that, though time feasible, are not operationally intelligent to run. Alternatively, we may only consider the most inexpensive time-feasible paths for all OD pairs in the beginning of the search. Only when no feasible solution is determined or when the objective value of the incumbent solution is not within the e-optimality can we gradually add additional but more expensive time-feasible paths to the search matrix. References Ahuja, R.K., Magnanti, T.L., Orlin, J.B., 1993. Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cli€s, NJ. Barnhart, C., She, Y., 1993. A network-based primal-dual heuristic for the solution of multicommodity network ¯ow problems. Transportation Science 27 (2), 102±117. Eckstein, J., She, Y., 1987. Optimization of group line-haul operations for motor carriers using twin trailers. Transportation Research Record 1120, 12±23. Handler, G., Zang, I., 1980. A dual algorithm for the constrained shortest path problem. Networks 10, 293±309. Held, M., Wolfe, P., Crowder, H.P., 1974. Validation of subgradient optimization. Mathematical Programming 6, 62±88. Leung, J.M., Magnanti, T.L., Singhal, V., 1990. Routing in point-to-point delivery systems: formulations and solution heuristics. Transportation Science 24 (4), 245±260. Lin, C.-C., 1998. The load planning of time-de®nite freight delivery common carriers. Transportation Planning Journal 27 (3), 371±406. Papadimitriou, C.H., Steiglitz, K., 1982. Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cli€s, NJ. Powell, W.B., Koskosidis, I.A., 1992. Shipment routing algorithms with tree constraints. Transportation Science 26 (3), 230±245. Powell, W.B., She, Y., 1989. Design and implementation of an interactive optimization system for network design in the motor carrier industry. Operations Research 37 (1), 12±29. Suter, M.J., Nuggehalli, R.S., Zaret, D.R., 1996. Feeder schedules optimization system. Presented at the INFORMS Atlanta Fall Conference, Atlanta, USA. Tomlin, J.A., 1966. Minimum-cost multicommodity network ¯ows. Operations Research 14, 45±51.