Reducing Truckload Transportation Costs ThroughCollaboration

Ozlem Ergun1, Gultekin Kuyzu, Martin Savelsbergh2

Industrial and Systems Engineering and The Logistics InstituteGeorgia Institute of Technology

Atlanta, GA 30332-0205 {oergun, gkuyzu, msavelsbergh@isye.gatech.edu}

Abstract

In the highly fragmented truckload transportation industry a substantial fractionof truck movements involves empty trucks, i.e., involves moves that reposition trucks.However, reducing the amount of truck repositioning is difficult because the need fora carrier to reposition its trucks depends on the interactions between the shippersthe carrier is serving. Through collaboration, shippers may be able to identify andsubmit sequences of continuous loaded movements to carriers, reducing the carriers’need for repositioning, and thus lowering the carriers’ costs. A portion of the carriers’cost savings may be returned to the shippers in the form of lower prices. We discussoptimization technology that can be used to assist in the identification of repeatable,dedicated truckload continuous move tours with little truck repositioning. Timingconsiderations are critical to practical viability and are a key focus of our efforts.We demonstrate the effectiveness of the algorithms developed on various randomlygenerated instances as well as on instances derived from data obtained from a strategicsourcing consortium for a $14 billion dollar sized US industry.

Key words: truckload transportation, collaboration, cycle cover, cycle generation.

1. Introduction

The growing interest in collaborative logistics is fuelled by an ever increasing pressure on

companies to operate more efficiently, the realization that suppliers, consumers, and even

competitors, can be potential collaborative logistics partners, and the connectivity provided

by the Internet.

In the trucking industry, shippers and carriers are continuously facing pressures to op-

erate more efficiently. Traditionally shippers and carriers have focused their attention on

controlling and reducing their own costs to increase profitability, i.e., improve those business

processes that the organization controls independently. More recently, shippers and carri-

ers have focused their attention on controlling and reducing system wide costs and sharing

1Ozlem Ergun was supported in part under NSF grants DMI-0238815 and DMI-0427446.2Martin Savelsbergh was supported in part under NSF grant DMI-0427446.

1

these cost savings to increase everyone’s profit. A system-wide focus, e.g., a collaborative

focus, opens up cost saving opportunities that are impossible to achieve with an internal

company focus. A good example is the empty repositioning of trucks. To execute shipments

from different shippers a carrier often has to reposition its trucks empty. Shippers have no

insight into how the interaction between their shipments affects a carrier’s need to reposi-

tion its trucks. However, shippers are implicitly charged for these repositioning costs. No

single participant in the logistics system controls empty repositioning costs, so only through

collaborative logistics initiatives can these costs be controlled and reduced.

Collaborative transportation networks, such as those managed by Nistevo (www.nistevo.com)

and Transplace (www.transplace.com), are examples of collaborative logistics initiatives fo-

cused on bringing together shippers and carriers to increase truck utilization and reduce

logistics costs. For example, analysts at Nistevo have been able to identify a repeatable

dedicated 2,500-mile continuous move tour for two of the members of the Nistevo network

visiting distribution centers, production facilities, and retail outlets [14]. The tour has re-

sulted in a 19% savings for both shippers (over the costs based on one-way rates) and

the carrier is experiencing higher margins through better truck utilization and lower driver

turnover through more regular driver schedules. Identifying tours minimizing empy truck

repositioning costs in a collaborative logistics network is no simple task. When the num-

ber of members of the network, and thus the number of truckload movements to consider,

grows, the number of potential tours to examine becomes prohibitively large. In that case,

optimization technology is needed to assist the analysts.

We discuss the development of optimization technology that can be used to assist in the

identification of repeatable, dedicated truckload continuous move tours. This technology is

of value for companies that regularly send truckload shipments, say several days of the week,

and are looking for collaborative partners in similar situations to cross-utilize a dedicated

fleet, or strengthen their negotiating position during transportation procurement. In situa-

tions where shippers regularly send truckload shipments, it is common to find contracts in

which carriers dedicate a portion of their fleet to the shipper, but then transfer responsibility

for all costs, including repositioning costs, to the shipper. Thus, it is in the interest of the

shipper to find partners to most effectively use the dedicated fleet.

Timing considerations are critical to the practical viability of continuous move tours

and are a key focus of our efforts. We have developed a highly effective and extremely

efficient heuristic that incorporates, among others, fast routines for checking time-feasibility

2

of a tour in the presence of dispatch time windows and for minimizing the duration of a

tour by appropriately selecting a starting location and departure time. We demonstrate the

effectiveness of the algorithms developed on various randomly generated instances and on

instances derived from data obtained from a strategic sourcing consortium for a $14 billion

dollar sized US industry.

The basis for the proposed optimization technology is a heuristic for the time-constrained

lane covering problem. The lane covering problem was introduced in Ergun et al. [5] and

represents the core optimization problem: covering a set of lanes, i.e., shipments, with a set

of cycles, i.e., continuous move tours, of minimum cost. The time-constrained lane covering

problem captures the additional complexities introduced when considering dispatch windows

on the lanes.

The benefits of continuous move tours depend on the pricing model employed by a carrier.

Different carriers have different schemes for incorporating repositioning costs in the price

charged for a one-way move between two locations. The simplest scheme applies a markup

factor to the cost of a one-way move, e.g., 20 percent, to cover anticipated repositioning

costs. More sophisticated schemes may take the destination location into account, because

the destination location is typically a good indicator of the expected repositioning costs.

Our proposed methodology is flexible and can accommodate a variety of carrier pricing

models. For convenience, we assume a simple pricing model for most of our presentation

and discussion. However, for the computational study with real-life data from a strategic

sourcing consortium, we have used a pricing model representative of those used in practice.

As mentioned above, we focus on optimization technology that can be used to assist in

the identification of repeatable, dedicated truckload continuous move tours. However, the

technology can be adapted straightforwardly to handle more dynamic situations, in which

continuous move paths (as opposed to tours) are constructed, extended, and modified based

on truckload shipments being revealed to a dispatcher over time.

Finally, we observe that even though the need for and the value of our optimization

technology is presented from a shippers’ perspective, it is in essence optimization technology

that solves a carrier’s problem, namely the problem of constructing a set of truck tours

serving a set of contracted lanes with as few empty repositioning moves as possible.

The remainder of the paper is organized as follows. In Section 2, we briefly review

related literature. In Section 3, the Lane Covering Problem and the Time-Constrained Lane

Covering Problem are introduced. In Section 4, we discuss various solution approaches. In

3

Section 5 we present the results of an extensive computational study. In Section 6 we discuss

the results of a real-world case study. Finally, in Section 7, we discuss possible extensions

and their associated challenges.

2. Related Literature

As minimizing truck repositioning is so important in truckload transportation, it is discussed

in other truckload transportation procurement studies as well. Much of the research reported

in the literature, though, has focused on truckload transportation procurement for a single

shipper. In that setting, the typical goal is to identify a set of carriers that can serve a ship-

per’s set of lanes at minimal cost. Moore et al. [15] develop a simulation and optimization

tool, which in real-time identify two lanes that can be served in sequence and a set of carriers

that can serve both of these lanes. The tool is used by a centralized transportation procure-

ment department at Reynolds Metal Company. Centralizing procurement and aggressive

searching for continuous move opportunities has resulted in $7 million of annual savings in

transportation costs. Caplice and Sheffi [4] describe a combinatorial auction run by a shipper

to determine the minimum cost allocation of its lanes to carriers. Here it is assumed that the

carriers will bid on bundles of lanes based on the operational synergies that such bundles will

create in their existing network. (Our optimization technology can easily be modified to as-

sist carriers in identifying such bundles of lanes and operational synergies.) Song and Regan

[18] also investigate combinatorial auctions as a means for a shipper to procure truckload

transportation services. They simulate such an auction to assess the benefits for both the

shipper and the carriers. Their study, however, uses a fairly simple model for representing

the way in which carriers identify tours minimizing the cost of covering lanes. Their model,

for example, ignores any temporal constraints, such as dispatch windows, associated with

lanes.

Another related stream of research focuses on truckload operations (as opposed to truck-

load transportation procurement). There, the focus is on how to best manage the fleet of

trucks of a single carrier in response to dynamic demand. A few representative examples

of that stream of research include Powell et al. [16], who describe the development of a

stochastic network model for assigning loads to drivers and trucks taking into consideration

a high level of demand uncertainty, Yang et al. [20] who study different on-line strategies for

assigning and reassigning trucks to transportation requests, as well as the value of advanced

4

information for such schemes, and Powell et al. [17], who discuss an online driver assignment

model and the issues and challenges arising when implementing such a model at a carrier.

3. Lane Covering Problems

The Lane Covering Problem (LCP) can be stated as follows. Given a set of lanes, find a

set of tours covering all lanes such that the total duration of the tours is minimized. More

formally, given a complete bi-directed Euclidean graph D = (V, A) with node set V , arc set

A, lane set L ⊆ A, and travel time ta for each a ∈ A, find a set of directed cycles covering

the lanes in L of minimum total duration. The LCP can be solved in polynomial time as

it can be formulated as a min-cost flow problem [5]. Unfortunately, as soon as additional

constraints are imposed on the cycles the associated lane covering problems become much

more difficult. For example, the duration constrained lane covering problem (DCLCP), in

which the duration of a cycle has to be less than or equal to a prespecified bound, is NP-hard

[5].

As mentioned in the introduction, we focus on developing technology for identifying

repeatable, dedicated truckload continuous move tours for companies that regularly send

truckload shipments and are looking for collaborative partners. Properly incorporating tim-

ing considerations in this technology is of critical importance to the practical viability. There

are two sources of temporal constraints: dispatch time windows on the loads to be trans-

ported and Department of Transportation Hours of Service regulations. Regularly scheduled

truckload movements are usually referred to as lanes and are specified by an origin, a des-

tination, and a dispatch window. The dispatch window indicates the time interval in which

the load to be moved should be dispatched, for example Mondays between 8am and 2pm.

A dispatch window incorporates information on the time at which the load to be moved is

ready as well as information on the time at which the load needs to depart to ensure it will

reach its destination on time. Department of Transportation Hours of Service regulations

limit driving and duty hours of truck drivers. Truck drivers may not drive more than 11

hours following 10 hours off-duty, may not drive beyond the 14th hour after coming on-duty,

and may not drive after 60/70 hours on-duty in 7/8 consecutive days. From an algorithmic

perspective dispatch windows pose more interesting challenges.

Consequently, we focus on the Time-Constrained Lane Covering Problem (TCLCP),

which we define as follows. For a given set of lanes, find a set of tours covering all lanes such

5

that the total duration of the tours is minimized and the dispatch windows are respected.

More formally, given a complete bi-directed Euclidean graph D = (V, A) with node set V ,

arc set A, and travel time ta for each a ∈ A, a period length T , and a lane set L, where

each lane l ∈ L is specified by an arc a ∈ A and a dispatch window [el, ll] (el, ll ≤ T ), find

a set of time-feasible directed cycles covering the lanes in L of minimum total duration. A

directed cycle C is time-feasible if we can identify a first arc in the cycle and we can assign

a departure time da to the origin of each arc a ∈ C in such a way that da + ta ≤ dsuc(a) for

all arcs in the cycle except for the last arc, where suc(a) denotes the immediate successor

of arc a in the cycle, and el ≤ da ≤ ll or el + T ≤ da ≤ ll + T if arc a is used to cover lane

l, and such that the duration of the cycle is less than or equal to T , i.e., the time elapsed

between the departure at the origin of the first arc and the return to the origin of the first

arc is less than or equal to T . The latter condition ensures that the cycle can be repeated

every period.

Observe that if a cycle is time-feasible, then it is time-feasible regardless of the arc chosen

as the first arc. To see this, suppose there exists an arc and a departure time at the origin of

that arc that results in a duration of the cycle of less than or equal to T . Since a single driver

can complete the entire cycle in a single period, that driver can serve the same cycle every

period. The driver visits each of the points (origins and destinations of the arcs) once in every

period. In other words, the driver returns to each of the points of the cycle within at most

time T from the departure from that point. But that means, that the cycle is time-feasible

regardless of the arc chosen as first arc. However, the duration of a time-feasible cycle does

depend on the arc chosen as the first arc. Therefore, the first arc of a time-feasible cycle

should be chosen such that the cycle has minimum duration. Since a cycle repeats every

period, this is equivalent to choosing the first arc of a time-feasible cycle in such a way that

the difference between the return to the origin of the first arc and the departure from the

origin of the first arc in the next period is the maximum possible.

We have been unable to find any literature specifically dealing with TCLCP. However,

there does exist a body of literature on related covering problems. The cycle covering problem

(CCP) looks for a least cost cover of a graph with simple cycles, each containing at least

three different edges. This constrained version of the Chinese Postman Problem (CPP) was

shown to be NP-hard on general graphs by Thomassen [19] and to be equivalent to the CPP

on planar graphs by Guan and Fleischner [8] and Kesel’man [12]. Itai et al. [10] provided

an upper bound for CCP on 2-connected unweighted graphs and gave a polynomial time

6

algorithm which finds such a cover. Improvements to this bound were proposed by Bermond

et al. [3], Alon and Tarsi [1], Fraisse [7], Jackson [11], and Fan [6], and a simple heuristic

was proposed and tested by Labbe et al. [13]. Most recently, Hochbaum and Olinick [9]

developed and tested heuristics for a constrained version of the CCP where no cycle in the

cover contains more than a prescribed number of edges.

4. Solution Approaches

The TCLCP has a natural set covering formulation. Let C represent the set of all time-

feasible cycles. We may assume that each cycle C ∈ C is such that C ∩L 6= ∅. Let cC denote

the cost of cycle C, let lC be 1 if lane l is on the cycle C and 0 otherwise, and let xC be a

0-1 variable indicating whether cycle C is selected or not. Then TCLCP can be formulated

as the following set covering problem

min∑

C∈CK

cCxC (1)

∑C∈CK

lCxC ≥ 1 ∀l ∈ L (2)

xC ∈ {0, 1} ∀C ∈ CK . (3)

Note that a lane may be covered by more than one cycle, but that in all but one cycle

the corresponding arc of the cycle represents asset repositioning (or deadheading). This

observation causes complications when the cost of moving loaded is different from moving

empty. In that case, the cost cC of a cycle is no longer uniquely determined as it depends

on the “role” of each arc in the cycle, i.e., whether it represents repositioning or not. By

explicitly defining, in advance, the role of an arc in a cycle, i.e., whether it covers a lane

or whether it represents repositioning, we can get around this issue at the expense of a

larger set of cycles. For each cycle, we can generate its “siblings” by replacing one or more

arcs covering lanes with arcs representing repositioning, ensuring that we never replace two

consecutive arcs. The process is illustrated in Figure 1. If the cycle in Figure 1(a) represents

the original cycle, then the cycles in Figure 1(b) represent its siblings (dashed lines represent

asset repositioning). The cost cC of each cycle can now be established upfront, because we

know precisely which arcs are used to cover lanes and which arcs are used for repositioning.

Doing so also facilitates the use of more complex carrier pricing models. (Note that this

results in a set partitioning formulation of the TCLCP.)

7

Figure 1: Cycle siblings.

We focus on developing an effective and efficient heuristic for TCLCP as instances en-

countered in practice are expected to be large. We have implemented a greedy heuristic

that generates a large number of time-feasible cycles (potentially all) and greedily selects a

subset of those cycles to cover the lanes based on some criterion measuring the desirability

or attractiveness of a cycle. After all lanes are covered we perform a local improvement step

to improve the solution.

A natural way to capture the desirability of a cycle is through the ratio of the sum of the

travel times of the lanes covered by the cycle and the duration of the cycle (the sum of travel

and waiting time). This cover ratio takes on values in (0,1] and a higher value indicates a

more desirable cycle.

The basic greedy heuristic iteratively selects a cycle with the highest cover ratio until all

lanes have been covered. This involves sorting the set of cycles, selecting a cycle, and then

deleting all cycles that cover one or more lanes of the selected cycle. These operations can

efficiently be implemented using heaps and reference lists.

4.1 Cycle Generation

We generate time-feasible cycles using a recursive procedure. For each lane ` ∈ L, we

construct all simple paths starting with ` and ending with a lane arc and consisting of lane

and repositioning arcs. We then connect the endpoints of the constructed path, if necessary,

with the appropriate repositioning arc to convert the path into a cycle. To control the

number of cycles generated, we employ two simple limits: (1) we restrict the travel time

of the repositioning arcs on the path to be smaller than a predefined value R (there is no

restriction on the travel time of the repositioning arc connecting the endpoints of the path),

and (2) we restrict the total number of lanes in a cycle. (More sophisticated schemes keeping

8

track of and considering the asset repositioning on the partial path are possible, but did not

appear to be necessary.) Each recursion starts with a base path, initially just the lane arc `.

Then, to enforce the limit on the travel time of the repositioning arc, we find the lane arcs

with an origin at travel time less than or equal to R from the destination of the last lane arc

on the base path. (These searches can be implemented very efficiently using k− d-trees [2].)

For all such lane arcs ˆ̀, we check if the cycle consisting of the base path, a repositioning arc

(if any) to the origin of lane arc ˆ̀, the lane arc ˆ̀, and the return repositioning arc (if any)

to the origin of the first lane arc of base path is time-feasible. If so, we add the cycle to the

list of generated time-feasible cycles and invoke a new recursion where the base path is the

newly generated cycle without the final repositioning arc, i.e., lane arc ˆ̀ is the last arc on

the extended base path.

For a cycle to be time-feasible, it has to respect the dispatch windows and its duration

has to be less than the period length. Because of the dispatch windows, the duration of a

cycle may include waiting time. The waiting time on a cycle depends on the lane arc chosen

to be the first arc of the cycle and the departure times at each of the lane arcs in the cycle.

Therefore, the first arc of the cycle and the departure times at each of the lane arcs in the

cycle should be chosen in such a way that the total waiting time along the cycle is minimum.

Note that the set of departure times resulting in the minimum total waiting along a cycle

is not unique. We set the departure time at the origin of the first arc to be the earliest

time which results in the minimum total waiting time along the cycle and the departure

times of the remaining lane arcs in the cycle to the earliest feasible departure time. This

set of departure times yields the earliest departure time at the last lane arc in the cycle

given that the waiting time along the cycle is minimum. Throughout the construction of

time-feasible cycles we monitor and update, if necessary, the departure times on the active

path to maintain this (invariant) property.

There may be flexibility with respect to the departure time at the origin of the first arc

in the path, in the sense that departing a little later may not affect the total waiting time

along the path. We define the difference between the latest and the earliest departure time

at the origin of the first arc in the path resulting in minimum total waiting time along the

path as the flexibility of the path, denoted φP . Figure 2 illustrates the flexibility of a path

with two lane arcs. (The arrows in the graph on the right represent the time required to

move between locations.)

Let σp denote the departure time at the first node of a path p and let τp denote the time

9

0 T

1

2

1 2

L2

L1

33

Figure 2: Flexibility of a path

required to traverse a path p. Let ˆ̀ be a candidate lane arc for expanding the base path P

and let δˆ̀ denote the width of the dispatch window of lane ˆ̀. Furthermore, let P̄ denote the

base path P extended with the repositioning arc required to reach the origin of lane arc ˆ̀,

i.e., P̄ = P ∪ (head(P ), tail(ˆ̀). The required time to traverse the cycle that forms when ˆ̀ is

added to the base path is computed as follows. First, we compute the beginning and end of

the dispatch window of ˆ̀ relative to σP , say t1 and t2 respectively. That is t2 = lˆ̀− σP if

σP + τP ≤ lˆ̀ or lˆ̀ + T − σP if σP + τP > lˆ̀ and t1 = t2 − δˆ̀. Then, the departure time of

ˆ̀ relative to σP , dˆ̀, is equal to max{τP̄ , t1}. If t1 is strictly greater than τP̄ , then there will

potentially be a waiting time at ˆ̀. Let wˆ̀ denote this potential waiting. However, some or

all of this waiting time may be eliminated by departing later at the origin of the first lane

arc of the base path. The departure time at the origin of the first lane arc of the base path

can be delayed by at most φP , the flexibility of the base path. So the minimum waiting time

at lane ˆ̀ is w̄ˆ̀ = max{wˆ̀− φP , 0}. Note that the waiting time along the base path does

not change and that the waiting time along the new path Pnew (Pnew = P̄ ∪ ˆ̀) is equal to

the waiting time along the base path plus w̄ˆ̀. We now have all the information we need to

compute the duration of new cycle, and, if necessary, the duration, departure, and flexibility

of the extended base path:

τPnew = τP̄ + w̄ˆ̀ + tˆ̀

τCnew = τPnew + t(head(Pnew),tail(Pnew))

σPnew = σP + (wˆ̀− w̄ˆ̀)

φPnew = min{φP − (wˆ̀− w̄ˆ̀), t2 − dˆ̀}.

The above arguments show that we can handle lane dispatch windows during cycle gen-

10

eration without increasing the time complexity, because testing feasibility and maintaining

path information takes constant time.

4.2 Cycle Duration Minimization

Recall that if a cycle is time-feasible, then it is time-feasible regardless of the lane arc chosen

as the first arc of the cycle. Therefore, the first arc of a time-feasible cycle should be

chosen such that the cycle has minimum duration. Since a cycle repeats every period, this is

equivalent to choosing the first arc of a time-feasible cycle in such a way that the difference

between the return to the origin of the first arc and the departure from the origin of the first

arc in the next period is the maximum possible.

The cycle generation procedure described above does not necessarily result in an ordering

of the arcs that results in the minimum cycle duration. To illustrate this, consider the

instance shown on the left in Figure 3 with four lanes L1 = (1, 2), L2 = (2, 3), L3 = (3, 4),

and L4 = (4, 1) having dispatch windows [e1, l1], [e2, l2], [e3, l3], and [e4, l4] (0 < e1 < l1 <

e2 < l2 < e3 < l3 < e4 < l4 < T ), respectively. The cycle generation procedure would

produce the time-feasible cycle (L1, L2, L3, L4) with a departure time at the origin of L1 at

l1 as shown in the graph on the right in Figure 3. Note that if L1 is chosen as the first arc

of the cycle, some waiting time is incurred at the origin of L3. However, if L3 is chosen as

the first arc of the cycle with a departure time at the origin of L3 at l3, no waiting time is

incurred along the cycle; see the graph on the right in Figure 4.

Figure 3: Constructed cycle with lane L1 as first lane arc

Consequently, for each generated time-feasible cycle, we have to determine the lane arc

that results in the minimum cycle duration when it is chosen as the first arc of the cycle. A

11

Figure 4: Constructed cycle with lane L3 as first lane arc

relatively simple O(n2) algorithm, where n is the number of lane arcs in the cycle, simply

traverses the cycle starting from each lane arc in the cycle, computes the duration, and in the

end selects the lane arc that resulted in the minimum duration. A more efficient linear time

algorithm exists, which is presented below. As the number of time-feasible cycles generated

may be large, an efficient algorithm to identify the lane arc that results in the minimum

cycle duration when chosen as the first arc is of utmost importance.

The linear-time algorithm first determines the earliest possible arrival time at a lane arc

and then determines the latest possible departure at that lane arc in the next period ensuring

minimum waiting time along the cycle. Consider time-feasible cycle C = (1, 2, ..., k, 1), i.e.,

cycle C consists of k lane arcs labelled, without loss of generality, 1, 2, ..., k. The cycle

generation procedure has already provided the duration with lane 1 as the first arc of the

cycle. Let Ci represent the same cycle, but with lane i chosen as the first arc, i.e., (i, i +

1, ..., k, 1, 2, ..., i − 1, i). The algorithm makes one forward and one backward pass through

the cycle. In the forward pass, we determine for each lane i ∈ {2, ..., k} the earliest arrival

time aii at the origin of i and the associated departure time di

1 at the origin of 1 that ensures

minimum waiting time along the path (1, ..., i− 1), see Algorithm 1.

Figure 5 illustrates the results of the forward pass of the algorithm on the example given

in Figure 3.

In the backward pass, we determine for each lane i ∈ {2, ..., k} the latest departure time

dii at the origin of i and the associated arrival time ai

1 at the origin of 1 ensuring minimum

waiting time along the path (i, ..., k, 1), see Algorithm 2.

Figure 6 illustrates how the backward pass of the algorithm proceeds on the example

12

Algorithm 1 Forward Pass

d ← e1

s̄ ← l1 − e1

s ← 0for i = 2, ..., k do

si ← 0ai

i ← d + ti−1,i

d ← aii

if d < ei thensi ← min(ei − d, s̄)s̄ ← min(li − ei, s̄− si)s ← s + si

d ← ei

end ifdi

1 = e1 + send for

Figure 5: Results of the forward pass of the cycle duration minimization algorithm.

given in Figure 3.

The difference ∆i between departure time and arrival (return) time at i can now be

computed as:

∆i = dii − ai

i − (ai1 − (di

1 + T ))+.

The last term ensures that we arrive at the origin of 1 at or before the time of the departure

in the next period. The duration of Ci is equal to T −∆i, therefore the lane that results in

the largest value of ∆i should be chosen as the first arc of the cycle.

Because the algorithm makes two passes through the cycle it has a linear time complexity.

13

Algorithm 2 Backward Pass

d ← l1 + Ts̄ ←∞s ← 0for i = k, ..., 2 do

si ← 0d ← d− ti−1,i

if d > li thensi ← (d− li)s ← s + min(si, s̄)d ← li

end ifs̄ ← min(d− ei, s̄)di

i = dai

1 = l1 + T − send for

Figure 6: Results of the backward pass of the cycle duration minimization algorithm.

4.3 Local Improvement

By adjusting the limit on the travel time of the repositioning arcs on the base path and the

limit on the number of lanes in a cycle, we have control over the number of cycles produced

during cycle generation. With a smaller limit on repositioning time and a smaller limit on

the number of lanes allowed in a cycle, fewer cycles are generated. However, with fewer

cycles to choose from, greedy selection may not produce high quality lane covers.

Therefore, we have developed a local improvement scheme that merges cycles from a

cycle cover by removing the longest repositioning arcs from each cycle and by optimally

reconnecting the two resulting directed paths to form another cycle. An example of a cycle

merge is shown in Figure 7. In this example, each of the two cycles consists of two lane

14

Figure 7: Example of merging two cycles

arcs plus a single repositioning arc. In the merged cycle, the two repositing arcs have been

deleted and replaced by two shorter repositing arcs.

Such a merge improves the current cycle cover if the sum of the durations of the initial

cycles C1 and C2 is more than the duration of the merged cycle C12. Given an initial cycle

cover C = {C1, C2, . . . , CN}, the optimal set of cycle merges can be obtained by finding

a minimum cost matching on the digraph DC = (VC, AC) where VC = {1, 2, ..., N}, AC =

{(i, j) : Cij is time-feasible and τCi+ τCi

> τCij} with arc weights wij = τCij

− (τCi+ τCi

)

for (i, j) ∈ AC. Unfortunately, finding an optimal matching on DC becomes computationally

expensive when DC gets large. Hence, we have also developed a greedy merge heuristic.

Our greedy merge heuristic first identifies all feasible and beneficial cycle merges and

stores them in a heap data structure with respect to the magnitude of their benefits. Then

the most improving cycle merge is implemented and the cycles that were involved are taken

out of consideration for future merges. (See Algorithm 3.)

Algorithm 3 GreedyMergeHeuristic(C)

for all cycles Ci in C doif Cij is time-feasible and τCij

< τCi+ τCj

thenInsert((i, j), τCij

− (τCi+ τCi

), ImprovementHeap)end if

end forwhile ImprovementHeap 6= ∅ do

(k, l) := Pop(ImprovementHeap)if cycles Ck and Cl are not marked as merged then

merge Ck with Cl and mark them as mergedend if

end while

Note that the local improvement algorithms are run iteratively. That is, after a set of cycle

15

merges has been identified and implemented, we repeat the local improvement procedure on

the new cycles by setting up and solving another matching problem or by re-applying the

GreedyMergeheuristic. Local improvement terminates when we can no longer identify a

beneficial cycle merge on the set of cycles created in a previous iteration.

5. Computational Experiments

We have conducted a set of computational experiments to assess the overall effectiveness of

the algorithms proposed and to analyze the impact of instance size (number of points and

number of lanes) and instance characteristics (presence of clusters, supply chain structure,

and dispatch time width) on the performance of the proposed algorithms in terms of quality

and efficiency.

5.1 Instances

We have evaluated the algorithms on randomly generated Euclidean instances. Random

instance generation is controlled by the following parameters: the number of points, the

fraction of points in clusters, the number of points per cluster, and the number of lanes.

Clusters are introduced to represent geographical concentrations of points, such as metropoli-

tan areas. For a set of instances, we also impose a supply chain structure by dividing the

set of points into three classes representing: suppliers, plants and distribution centers, and

customers. The fraction of points belonging to each class and the fraction of lanes between

points in any two classes are given. These instances will be referred to as supply chain

instances (SC-instances). Given the number of points, the fraction of points in clusters, and

the number points per cluster, the number of clusters and a cluster radius are determined.

As the number of points in an instance increases, we let the cluster radius decrease in order

to avoid overlapping clusters. Next, the centers of the clusters are determined uniformly

at random within a 1, 800 × 1, 800 miles square region. The points within each cluster are

generated by randomly determining their coordinates using a Normal distribution, N(0, 1),

and the cluster radius. Finally, the points that are not in clusters are generated by de-

termining their coordinates uniformly at random within the square region. If the instance

being generated is a SC-instance, then each point generated is assigned a type: it is either

a supplier, or a plant or distribution center, or a customers with probability 0.2, 0.1, and

0.7, respectively. Once all the points are generated the complete bi-directed Euclidean graph

16

defined by these points is formed. Next, the desired number of lanes are randomly selected

from among the arcs of the complete bi-directed graph ensuring that each point has at least

one lane incident to it and there are no lanes with an origin and a destination in the same

cluster. For the SC-instances, we only allow lanes between suppliers and plants and distri-

bution centers, among plants and distribution centers, and between plants and distribution

centers and customers. Furthermore, we ensure that each supplier has at least one outgoing

lane, each plant and distribution center has at least one incoming and one outgoing lane,

and each customer has at least one incoming lane. Finally, after all the lanes are generated

we determine the dispatch time window for each lane. The dispatch time window length is

set to one of 2, 4, 6, or 12 hours for all lanes in each experiment. Given a lane, a day of

the week is chosen randomly as the dispatch day. Once the dispatch day is known, then

the beginning of the dispatch window is picked randomly between 8am and (8pm - dispatch

time window length).

To properly analyze the performance of the algorithms, we have generated instances for

several different parameter settings. We have varied the number of points, the fraction of

points in clusters, the number of points per cluster, the number of lanes, and the width of the

dispatch windows. More specifically, we generated instances with 300, 400, and 500 points,

with a fraction of points in clusters of 0.5, 0.6, 0.7, and 0.8, and with an average number of

10 customers per cluster. The density of the instance is controlled by the lane-to-point ratio

(ratio of the number of lanes and the number of points) with values 2 and 5.

The resulting instances represent a wide variety of situations, with mean lane length and

standard deviation equal to 851.5 and 382.6 miles, respectively.

5.2 Results

With our first set of experiments, we aim to better understand the impact of the algorithm

parameters that affect the construction of an initial solution, i.e., the generation and selection

of time-feasible cycles. We use 48 randomly generated instances grouped into 6 sets based on

their size (i.e., number of points and number of lanes). In each set, there are instances with

and without a supply chain structure and with different cluster characteristics. Furthermore,

we assume that all lanes have a 12 hour dispatch window which starts at 8am in the morning.

We present the quality of a solution by means of the repositioning distance as a percentage

of the total distance, denoted by ρdist, and the non lane travel time as a percentage of the

total time (which also captures waiting time at pickup locations), denoted by ρtime.

17

Recall that to control the number of cycles generated (to reduce memory usage and

computing time), we restrict the travel time of all but one of the repositioning arcs on the

cycle and we restrict the total number of lanes in a cycle.

In the first experiment, we limit the number of lanes in a cycle to at most six and vary

the travel time limit on repositioning arcs, i.e., R is set to the driving times corresponding to

1, 10, 25, and 50 miles. The quality of the solutions produced, the number of cycles generated,

and the CPU time required for constructing the solutions are presented in Table 1, 2 and 3,

respectively.

Table 1: Quality of solutions produced by the greedy construction heuristic.

R=1 R=10 R=25 R=50#points #lanes ρtime ρdist ρtime ρdist ρtime ρdist ρtime ρdist

300 600 41.28 39.77 36.97 35.20 30.79 29.15 28.91 27.36300 1500 39.12 37.72 33.69 32.32 27.51 26.37 26.24 25.27400 800 41.45 40.02 35.69 34.08 29.66 28.09 27.95 26.38400 2000 39.05 37.66 32.35 31.00 26.62 25.48 25.10 24.03500 1000 41.48 39.98 35.16 33.39 29.45 27.74 27.94 26.35500 2500 39.15 37.75 31.56 30.17 25.80 24.66 24.68 23.61

Table 2: Number of cycles created by the greedy construction heuristic.

#points #lanes R = 1 R = 10 R = 25 R = 50300 600 1,163 2,411 14,086 33,987300 1500 5,501 29,754 733,299 2,946,147400 800 1,578 4,047 24,882 50,374400 2000 7,379 54,609 1,252,090 3,487,056500 1000 1,996 6,730 57,276 128,566500 2500 9,473 117,403 2,479,371 5,334,276

Table 3: CPU times in seconds for the greedy construction heuristic.

#points #lanes R = 1 R = 10 R = 25 R = 50300 600 0.01 0.03 0.23 0.68300 1500 0.05 0.44 16.86 79.70400 800 0.01 0.04 0.44 1.06400 2000 0.07 0.88 30.46 95.53500 1000 0.01 0.08 1.20 3.11500 2500 0.09 2.19 66.32 155.97

We observe that the solution quality increases steadily as the value of R increases, but

that, at the same time, the number of cycles created and the run times increase exponen-

tially. We also observe that ρdist and ρtime are 2-3 percentage points lower for instances

with the higher lane-to-point ratio 5. This is most likely the result of the fact the number

of opportunities for time-feasible continuous moves increases when the number of incoming

and outgoing arcs at a point increases.

18

Next, we limit the travel time of repositioning arcs to 25 and vary the maximum number

of lanes in a cycle, i.e., K = 3, 4, 5, and 6. The results can be found in Table 4.

Table 4: Quality of solution produced by the greedy construction heuristic for varying limitson the number of lanes in a cycle.

K=3 K=4 K=5 K=6#points #lanes ρtime ρdist ρtime ρdist ρtime ρdist ρtime ρdist

300 600 30.76 29.19 30.77 29.13 30.79 29.15 30.79 29.15300 1500 27.42 26.31 27.48 26.34 27.50 26.34 27.51 26.37400 800 29.75 28.16 29.68 28.10 29.67 28.09 29.66 28.09400 2000 26.51 25.41 26.60 25.45 26.63 25.50 26.62 25.48500 1000 29.39 27.77 29.42 27.69 29.44 27.72 29.45 27.74500 2500 25.88 24.81 25.98 24.87 26.01 24.90 25.80 24.66

We see that the quality of the solutions is almost identical for all values of K. This

suggests that high quality solutions typically involve only a small number of lanes. This may

be a consequence of the criterion for choosing cycles, i.e., the cover ratio. The cover ratio,

i.e., the ratio of the sum of the travel times of the lanes covered by a cycle and the duration

of that cycle, favors cycles with little empty repositioning and waiting time. It is more likely

to find cycles with a high cover ratio among cycles with only a few lanes. Therefore, the

results may be different when other criteria are used for cycle selection. On the other hand,

cycles with only a few lanes are preferred in practice as the chance for failed continuous

moves is smaller in cycles with only a few lanes.

Next, we investigate the effectiveness of local improvement and the impact of using

the greedy merge heuristic (GMH) as opposed to solving the matching problems optimally

(OM). Our computational experiments revealed that the initial solution characteristics did

not change the comparative performance of GMH and OM. Therefore, we compare the

performance of the two methods starting from an initial solution consisting of only trivial

2-cycles, i.e., cycles representing moving a load and returning empty. We use the same

instances as before. Note that local improvement is applied iteratively. After a set of cycle

merges has been identified and implemented, we continue the local improvement by setting

up and solving another matching problem with time-feasible and beneficial cycle merges.

The procedure terminates when no beneficial cycle merges can be identified on the set of

cycles from a previous iteration. The results can be found in Table 5, where we present the

value of the solution obtained by the greedy merge heuristic (GMH) and the value obtained

when the matching problem is solved to optimally (OM).

First, we observe that local improvement is very effective. In the starting solution, i.e.,

19

Table 5: Quality of solutions produced by the local improvement heuristics.

GMH OM cpu time (sec.)#points #lanes ρtime ρdist ρtime ρdist GMH OM

300 600 22.54 21.53 21.17 20.14 0.10 1.40300 1500 19.23 18.64 17.80 17.24 0.58 11.18400 800 21.50 20.59 20.12 19.23 0.17 2.60400 2000 18.39 17.85 16.99 16.46 1.03 22.82500 1000 20.86 19.94 19.53 18.65 0.26 4.57500 2500 17.50 16.96 16.11 15.58 1.65 36.53

the set of 2-cycles covering all lanes, ρdist and ρtime are both 50%. After local improvement,

they have dropped to around 20% and are even significantly better than the percentages

obtained with the greedy heuristic with R = 50. Second, we see that ρdist and ρtime obtained

by the optimal matching are about 2 percentage points less than those obtained by the greedy

merge heuristic. However, solving the matchings to optimality substantially increases the

running time, particularly for the larger instances. As in earlier experiments, we see that for

the dense instances, with lane-to-point ratio 5, the solutions obtained are about 3 percentage

points better, but that the running time is an order of magnitude larger. This is due to the

fact that the number of time-feasible and beneficial cycle merges is significantly larger. In

the rest of the paper, we use the greedy merge heuristic as our local improvement algorithm

due to its computational efficiency.

Table 6 shows ρdist and ρtime for the solutions produced by the greedy heuristic followed

by the greedy merge heuristic (for different values of R) and for the solutions produced by

the greedy merge heuristic when started from 2-cycle solutions.

Table 6: Quality of solutions for the construction and improvement heuristics combined.

R=1 R=10 R=25 R=50 2-cycle#points #lanes ρtime ρdist ρtime ρdist ρtime ρdist ρtime ρdist ρtime ρdist

300 600 26.41 23.46 26.22 23.33 24.69 22.43 24.12 22.07 22.54 21.53300 1500 22.36 20.10 21.90 19.90 20.83 19.30 20.73 19.43 19.23 18.64400 800 25.47 22.72 24.94 22.33 23.70 21.54 23.22 21.13 21.50 20.59400 2000 21.68 19.48 20.88 18.95 19.78 18.26 19.50 18.10 18.39 17.85500 1000 24.90 22.04 24.42 21.63 23.07 20.70 22.71 20.56 20.86 19.94500 2500 20.93 18.69 20.06 18.08 18.47 16.96 18.46 17.07 17.50 16.96

We can make the following observations regarding the results in Table 6. First and fore-

most that local improvement is very effective. The solutions reported for R = 1, 10, 25, and

50, are substantially better than those reported in Table 1. An additional 5-6 percentage

points is shaved off from even the best solutions obtained using the greedy construction

heuristic by itself. This indicates that to obtain high quality continuous move tours it is

20

necessary to include several relatively long asset reposition moves. This is a valuable obser-

vation, because planners and analysts without access to decision support technology may not

naturally consider such solutions. The presence of dispatch windows most likely increases

the need to include relatively long asset repositioning moves in high quality continuous move

tours as dispatch windows may invalidate connections that appear to be attractive geograph-

ically. The fact that we search for continuous moves tours (as opposed to continuous move

paths) probably also contributes to the need for relatively long empty repositioning moves.

Second, that the quality of the starting solution does not appear to have a major impact.

Even starting from a trivial 2-cycle solution results in high quality solution. In fact, the best

solutions are obtained when starting from a 2-cycle solution. Finally, we mention that the

increase in running times as a result of local improvement is negligible (at most 2 seconds).

To be able to truly assess the performance of our proposed solution approach, we have

tried to solve a number of small to medium size instances to optimality. To solve an instance

to optimality, we generated all time-feasible tours (i.e., no restriction on the travel time

of repositioning arcs and no restriction on the number of lanes in a cycle) and solved the

resulting set partitioning problem to optimality using XPRESS-Optimizer 2005B. For study-

ing the effects of working with a restricted set of cycles, we also solve the set partitioning

problem with only the cycles generated when R = 100. The results can be found in Table 7

and 8, respectively. The tables contain a few empty cells, as we were unable to solve some of

the larger instances to optimality due to either excessive memory requirements or excessive

computation time requirements.

Table 7: Comparison of the quality of solutions (ρdist).

After local improvementR=100 all cycles R=100 all cycles 2-cycle

#points #lanes IP heur IP heur heur heur heur100 200 30.25 32.97 26.86 30.47 31.01 30.47 29.52100 200 29.95 32.20 27.66 30.02 30.51 30.02 29.83100 200 27.54 30.67 25.76 28.52 29.15 28.52 27.89100 200 25.50 28.65 24.56 27.22 28.07 27.20 27.92100 500 22.53 26.63 20.50 22.78 24.61 22.71 23.74100 500 22.51 24.46 19.57 22.45 23.13 22.44 22.60100 500 21.99 23.45 19.34 21.74 22.58 21.64 21.92100 500 18.37 21.70 16.26 20.57 21.41 20.56 20.77200 400 29.78 32.36 26.37 29.10 30.05 29.08 28.81200 400 27.97 31.20 24.50 28.10 29.41 28.10 27.73200 400 26.81 30.82 23.31 27.25 29.27 27.21 27.13200 400 26.15 29.84 24.72 27.13 28.18 27.10 27.58200 1000 21.40 26.21 18.60 23.09 24.48 23.05 22.55200 1000 20.78 24.70 21.70 23.07 21.68 21.82200 1000 23.48 21.86 21.49200 1000 21.21 19.91 19.31

21

Table 8: Comparison of the number of cycles generated and IP cpu times in seconds.

R=100 All Cycles#points #lanes #cycles IP time #cycles IP time

100 200 3,660 0.17 49,324 0.34100 200 9,837 0.22 267,763 0.75100 200 9,386 0.24 466,221 0.48100 200 5,251 0.22 56,330 0.30100 500 62,384 1.90 1,991,406 249.79100 500 467,039 185.14 27,228,768 741.73100 500 619,215 264.46 80,193,577 2,281.11100 500 203,569 96.08 2,566,438 287.77200 400 11,342 0.69 645,845 17.15200 400 28,638 0.53 2,449,117 132.05200 400 86,204 7.27 9,032,150 617.09200 400 48,272 1.55 5,293,459 46.16200 1000 287,041 2,475.46 27,346,535 232,565.55200 1000 1,622,206 6,988.97 392,827,574200 1000 4,712,970200 1000 2,310,338 339,233,302

A number of observations can be made regarding the results. First, and most importantly,

our approach produces near optimal solutions. Starting from the trivial 2-cycle solution and

repeatedly performing the greedy merge heuristic results in solutions that are never more

than 4 percentage points higher than the optimal solution and in many cases the difference

is only 2-3 percentage points. This is due, most likely, to the fact that the number of

lanes covered by the cycles in an optimal solution tends to be very small. Of the cycles in

optimal solutions approximately 57% cover two lanes, approximately 27% cover three lanes,

while only less than 5% cover four lanes or more. Second, as expected, that the number

of time-feasible cycles grows rapidly with the instance size and that the set partitioning

problems become very difficult when the number of cycles is large. On the other hand, for

all the instances used in this experiment, performing the local improvement starting from

the trivial 2-cycle solution never takes more than one second.

Next, we analyze the effect of the width of the dispatch windows on the number of time-

feasible cycles and the solution quality. For these experiments, we start the greedy merge

heuristic from an initial solution generated by the greedy heuristic with R = 25 and from

the trivial 2-cycle covers. We use the same randomly generated instances as before except

for the dispatch window widths. We use dispatch window widths of 2, 4, 6, and 12 hours

and choose the start of the dispatch windows randomly for instances with dispatch window

widths less than 12 hours. Once chosen, the start of the dispatch windows for each lane is

kept constant for all instances. The results are presented in Tables 9 and 10.

As expected, we observe that as the dispatch window widths get narrower the quality

22

Table 9: Quality of solutions for the construction and improvement heuristics combined forvarying dispatch window widths.

width = 2hrs width = 4 hrsGMH 2-cycle GMH 2-cycle

# points # lanes ρtime ρdist ρtime ρdist ρtime ρdist ρtime ρdist

300 600 47.49 35.31 41.56 34.92 42.93 33.99 37.22 32.99300 1500 37.67 29.16 32.96 28.56 33.64 27.76 29.40 27.00400 800 45.07 33.46 39.38 33.05 40.20 31.63 35.11 31.12400 2000 35.60 27.53 31.36 27.32 31.68 26.29 27.95 25.60500 1000 43.67 32.28 37.88 32.04 39.25 30.88 33.57 30.12500 2500 34.18 26.03 29.85 25.95 30.19 24.71 26.40 24.20

width = 6 hrs width =12 hrsGMH 2-cycle GMH 2-cycle

# points # lanes ρtime ρdist ρtime ρdist ρtime ρdist ρtime ρdist

300 600 39.28 32.47 34.09 31.19 32.90 29.06 29.22 27.57300 1500 30.90 26.66 27.14 25.58 26.55 24.24 24.05 23.18400 800 37.13 30.47 32.13 29.41 31.13 27.56 27.46 26.01400 2000 29.07 25.02 25.76 24.27 24.87 22.59 22.73 21.94500 1000 35.94 29.52 30.74 28.36 30.04 26.17 26.40 24.97500 2500 27.37 23.40 24.23 22.82 22.76 20.56 21.34 20.56

Table 10: Number of cycles generated by the construction heuristic with varying time windowwidths.

#points #lanes width = 2hrs width = 4hrs width = 6hrs width = 12hrs300 600 3,917 5,256 6,780 14,086300 1500 84,493 138,524 218,569 733,299400 800 6,513 8,788 11,641 24,882400 2000 145,007 240,355 376,604 1,252,090500 1000 10,863 15,733 22,254 57,276500 2500 289,811 513,722 857,076 2,479,371

of the solutions decreases substantially. In fact, ρdist for instances with 2 hour dispatch

windows is about 5 percentage points higher for low density instances and about 4 percent-

age points higher for high density instances than ρdist for instances with 12 hour dispatch

windows. Similarly, ρtime for instances with 2 hour dispatch windows is about 7 percentage

points higher for low density instances and about 5 percentage points higher for high den-

sity instances than ρtime for instances with 12 hour dispatch windows. We see again that

the impact of changes, in this case dispatch window width, is less severe for high density

instances where the number of opportunities for beneficial continuous moves is larger. The

results show, as expected, that there is both an increase in empty mileage and in waiting

time. We also observe that as the dispatch window width increases the number of cycles

generated by the greedy heuristic increases and so does the number of cycles that can be

merged (not presented in the tables). Finally, we note that for all dispatch window widths

starting the local improvement from the 2-cycle solution, as opposed to starting from the

solution generated by the greedy algorithm, performs best.

23

Next, we analyze the effects of geographical instance characteristics. Instances are char-

acterized by the number of points, the fraction of points in clusters, the number of points

in a cluster, the number of lanes, and whether or not there is a supply chain structure. In

Tables 11 and 12 we present results for instances grouped by whether or not a supply chain

structure is present (Yes/No), their density (i.e., the ratio of number of lanes to number of

points (2 or 5)), and the fraction of points in a cluster (0.5, 0.6, 0.7, or 0.8). Each of these

16 groups has three members with 300, 400, and 500 points. The dispatch window widths

are set to 12 hours starting at 8am.

Table 11: Effect of instance characteristics on the solution quality.

R=1 R=10 R=25 R=50 2-cycleGroup SC Density Frac in Clst ρtime ρdist ρtime ρdist ρtime ρdist ρtime ρdist ρtime ρdist

1 Yes 2 0.5 24.98 23.43 25.38 23.56 25.00 23.34 24.74 23.15 23.25 22.542 Yes 2 0.6 25.02 23.30 25.33 23.48 24.99 23.18 24.75 23.01 23.16 22.333 Yes 2 0.7 24.61 23.11 25.08 23.20 24.69 22.99 24.26 22.70 22.52 21.694 Yes 2 0.8 25.20 23.65 25.61 23.56 25.03 23.40 24.67 23.27 23.26 22.395 Yes 5 0.5 22.08 21.52 22.78 21.91 22.73 21.87 22.76 21.89 21.63 21.226 Yes 5 0.6 21.48 20.87 22.17 21.20 22.20 21.29 22.19 21.34 20.89 20.477 Yes 5 0.7 21.27 20.62 21.97 20.98 21.86 21.00 21.79 20.98 20.73 20.328 Yes 5 0.8 21.48 20.89 22.52 21.43 21.88 21.02 21.78 21.02 20.96 20.569 No 2 0.5 27.20 23.06 26.26 22.47 23.86 20.96 23.23 20.46 20.95 19.9710 No 2 0.6 26.19 22.19 25.19 21.69 23.43 20.43 22.75 20.08 20.41 19.2711 No 2 0.7 25.90 22.05 24.53 21.03 22.20 19.67 21.80 19.33 20.13 19.0112 No 2 0.8 25.61 21.13 24.17 20.45 21.35 18.46 20.60 18.01 19.36 18.2813 No 5 0.5 22.53 18.71 20.91 17.89 19.05 16.83 18.65 16.63 16.76 16.1214 No 5 0.6 22.04 18.25 19.88 16.91 17.97 15.73 17.37 15.37 16.10 15.4115 No 5 0.7 21.64 17.80 19.17 16.35 16.80 14.88 16.31 14.61 15.43 14.7616 No 5 0.8 20.74 16.74 18.20 15.14 15.45 13.21 15.00 13.05 14.48 13.67

Table 12: Effect of instance characteristics on number of cycles generated during the con-struction phase.

Group SC Density Frac in Clst R=1 R=10 R=25 R=501 Yes 2 0.5 1,596 3,602 12,974 23,8112 Yes 2 0.6 1,551 4,097 28,656 57,1943 Yes 2 0.7 1,610 4,901 34,226 69,5794 Yes 2 0.8 1,602 5,932 77,522 193,4335 Yes 5 0.5 4,901 29,301 368,195 1,041,8176 Yes 5 0.6 4,729 37,888 1,100,413 2,742,0387 Yes 5 0.7 4,937 57,204 1,418,979 4,404,7268 Yes 5 0.8 4,953 92,221 1,974,056 4,926,8319 No 2 0.5 1,540 3,250 13,819 25,87310 No 2 0.6 1,568 3,915 21,916 50,60211 No 2 0.7 1,573 4,367 27,645 60,10312 No 2 0.8 1,592 5,103 39,895 87,21113 No 5 0.5 9,591 47,051 563,984 1,533,12814 No 5 0.6 10,118 70,509 1,273,310 4,046,83315 No 5 0.7 10,060 90,686 1,983,492 5,783,19416 No 5 0.8 10,314 113,183 3,167,453 7,046,333

Table 11 shows that when a supply chain structure is present in instances, the solution

24

quality is worse. We believe this suggests that when a supply chain structure is present an

increase in reposition miles is inevitable. The reason for the increase in reposition miles is the

fact that a large fraction of the points has no incoming arcs (points representing suppliers)

and an even larger fraction of the points has no outgoing arcs (points representing customers).

Therefore, fewer natural connections are present. This view is supported to some extend by

the results in Table 12, which show that the number of cycles generated in the construction

phase is smaller for instances with a supply chain structure. The number of cycles generated

in the construction phase also suggests an explanation for the higher quality solutions we

have observed for high density instances. The presence of dispatch windows invalidates

many continuous moves that look good from a geographical perspective, i.e., involve only a

short repositioning move. Relatively speaking, this happens less frequently in high density

instances because the in- and out-degree of the points are higher. This is reflected in the

number of time-feasible cycles generated. When the density is high, the number of cycles

generated is much higher. Combined these observations clarify the differences in gap sizes

between low and high density instances.

6. A Real-World Case Study

Group purchasing organizations have become common place in the modern business world.

A group purchasing organization seeks to achieve cost reductions by combining purchasing

power to negotiate discounts. Their success in the area of product procurement, e.g., raw

materials, has lead to companies to explore whether a group purchasing organization may

also be effective when it comes to procuring services, e.g., truckload transportation.

To assess the potential value of collaborative purchasing of truckload transportation

services and to prepare itself for negotiations with truckload carriers a group purchasing

organization has to solve a time-constrained lane covering problem.

We have conducted a study for one such organization in which we assessed the potential

value of collaborative transportation procurement for individual member companies, by coor-

dinating purchasing of inbound and outbound truckload transportation services, and across

member companies, by collaboratively purchasing all truckload transportation services.

To estimate the savings associated with offering collaborative continuous moves to a

carrier, a carrier pricing model had to be developed. The carrier pricing model we developed

takes as input a continuous move path (as opposed to a tour) with an origin, a destination,

25

the distance traveled between the origin and the destination, and the time elapsed between

departing from the origin and arriving at the destination, and computes the associated

charge. The charge is computed as the sum of a fixed cost component, based on the time

elapsed between origin and destination, and a variable cost component, based on the distance

traveled between origin and destination. The fixed cost per week, set at $1,600, covers tractor

and trailer depreciation and interest, driver compensation and fringes, insurance, tags, taxes,

and licenses. The fixed cost of a path is computed by multiplying $1,600 by the fraction of

the week used by the path. The variable operating cost per mile, set at $0.45, covers fuel

and oil, tractor and trailer maintenance, and tractor and trailer tires.

In addition to the incurred direct costs, it is assumed that the carrier accounts for poten-

tial asset reposition to the next pickup location and for potential delay at the next pickup

location. Asset repositioning and delay depend on many factors, such as the location of the

destination, the time of arrival at this location, and the carriers demand patterns. For the

study, asset reposition of a 100 miles and a delay of 8 hours are assumed, i.e., 100 miles

are added to the distance traveled between origin and destination and 10 hours are added

to the time elapsed between departing from the origin and arriving at the destination (8

hours plus two hours for traveling the additional 100 miles). Finally, the carrier is assumed

to charge overhead and profit as a percentage of revenue, set at 15% and 10% respectively.

Consequently, the formula for computing the charge for a path P , cP , to a shipper is

cp =4

3

(1600

(duration(P ) + 10

168

)+ 0.45 (mileage(P ) + 100)

). (4)

Given the carrier pricing model, the savings associated with a collaborative continuous

move path are estimated as the difference between the sum of the charges for the lanes on

the path and the charge for the path. The savings represent (are the result of) elimination

of some of the carrier’s charges in anticipation of asset repositioning and delay.

In addition to using the carrier pricing model developed in the above section, to provide

realistic estimates of time elapsed between departure at the origin and arrival at the desti-

nation, we properly account for the Hours of Service regulations (those in effect in 2004),

i.e., a mandatory rest of 8 hours after driving for 10 hours or being on duty for 15 hours. As

a result, the travel time between two locations is no longer a constant because it depends on

the history of the driver. We also include a one hour rest after every four hours of driving.

In addition, each location has a time window specifying a period of the day during which a

pickup or delivery could take place. Furthermore, some locations require life loading whereas

26

other locations used spot trailers (when spot trailers are used loading or unloading is fast as

it only involves a change of trailers; when life loading is required no change of trailers takes

place and a driver has to wait until loading or unloading is completed).

A typical all-member instance for a single week involves about 750 locations and 5500

lanes. The potential savings due to continuous moves were estimated to be in the order of

9-10%. (Due to confidentiality agreements, we are unable to provide more details.)

Since we are unable to present detailed results for the instances of the real-world case

study, we have conducted a set of computational experiments on slightly simplified instances,

in which we do not account for loading and unloading times, and where we use a version of

the algorithm that ignores Hours of Service regulations.

In the first experiment, we investigate the potential savings and whether the obtained

savings are sensitive to the algorithm settings for the construction phase (local improvement

is applied in all variants tested). The results can be found in Table 13, where the savings

are computed asone-way charges - continous move charges

one-way charges× 100 percent.

Table 13: Savings from continuous moves.

Instance #locations #lanes R = 1 R = 10 R = 25 R = 50 2-cyclei1 103 1078 5.39% 5.36% 5.44% 5.34% 4.24%i2 65 140 6.97% 7.28% 6.94% 6.99% 6.81%i3 215 2346 9.66% 10.02% 10.04% 10.10% 8.81%i4 404 2074 12.42% 12.20% 12.45% 12.71% 11.58%i5 737 5445 11.87% 12.04% 12.02% - 11.33%i6 244 2889 10.47% 10.51% 10.65% 10.66% 9.58%i7 427 2524 13.04% 12.92% 13.24% 13.58% 12.17%i8 240 2828 10.23% 10.52% 10.73% 10.68% 9.24%i9 215 2346 9.66% 10.02% 10.04% 10.10% 8.81%

i10 244 2889 10.47% 10.51% 10.65% 10.66% 9.58%i11 240 2828 10.23% 10.52% 10.73% 10.68% 9.24%i12 236 2352 8.69% 8.83% 8.90% 8.92% 7.93%i13 130 1503 5.03% 5.31% 5.24% 5.28% 4.49%

We see that the savings range from about 5.5 percent to a little over 13 percent, where

the savings tend to be larger when the size of the instance is larger. It also appears that for

these instances the quality of the initial continuous move path has an impact on the quality

of the final continuous move path. The quality of the continuous move path obtained after

local improvement when starting from the individual lanes is universally worse than the

quality of the continuous move path obtained after local improvement when starting from

a continuous move path constructed with our greedy heuristic. One instance could not be

solved for R=50 because the number of tours generated was too large to fit into memory.

Next, we investigate the impact of the size of the dispatch windows at the locations on

27

the quality of the tour. The results when starting from a continuous move path obtained by

the greedy construction heuristic with R = 25 and when starting from the individual lanes

can be found in Table 14.

Table 14: Savings from continuous moves for different dispatch window widths.

width = 2hrs width = 4hrs width = 6hrs width = 12hrsInstance R = 25 2-cycle R = 25 2-cycle R = 25 2-cycle R = 25 2-cycle

i1 4.15% 3.02% 4.57% 3.33% 4.89% 3.56% 5.44% 4.24%i2 5.12% 4.74% 6.85% 6.15% 6.79% 6.69% 6.94% 6.81%i3 7.61% 6.85% 8.50% 7.63% 8.96% 8.18% 10.04% 8.81%i4 10.23% 9.77% 11.48% 10.77% 11.86% 11.21% 12.45% 11.58%i5 9.42% 9.28% 10.51% 10.25% 11.26% 10.78% 12.02% 11.33%i6 7.84% 7.44% 9.01% 8.27% 9.64% 8.93% 10.65% 9.58%i7 10.77% 10.26% 11.86% 11.35% 12.63% 11.72% 13.24% 12.17%i8 7.79% 7.09% 8.81% 7.99% 9.34% 8.61% 10.73% 9.24%i9 7.61% 6.85% 8.50% 7.63% 8.96% 8.18% 10.04% 8.81%

i10 7.84% 7.44% 9.01% 8.27% 9.64% 8.93% 10.65% 9.58%i11 7.79% 7.09% 8.81% 7.99% 9.34% 8.61% 10.73% 9.24%i12 6.78% 6.05% 7.66% 6.89% 8.27% 7.36% 8.90% 7.93%i13 4.27% 3.51% 4.75% 3.94% 5.08% 4.23% 5.24% 4.49%

As expected, we see that the width of the dispatch windows has an impact on the savings

that can result from the use of continuous moves. The smaller the dispatch window width

the smaller the savings. This is due to the increased chance of waiting time between two

consecutive moves. A large waiting time between two consecutive moves renders a continuous

move path unprofitable.

Finally, we investigate the impact of a limit on the number of lanes in a continuous

move path. In practice, the chance for breaking a continuous move, i.e., being unable to

execute the complete sequence of moves as planned, increases when the number of lanes in

a continuous move path is large. Therefore, in practice it is preferable to achieve the bulk

of the savings from continuous move paths with a relatively small number of lanes. We

limited the number of lanes allowed in a continuous move path to 3, 4, and 5 respectively.

The results when starting from a continuous move path obtained by the greedy construction

heuristic with R = 25 and when starting from the individual lanes can be found in Table 15.

We see that limiting the number of lanes in a continuous move path has little impact

on the savings. Allowing at most three lanes in a continuous move path results in savings

that are about 0.5 percent less than when at most four lanes are allowed. The decrease in

savings is even smaller when we compare allowing at most four lanes to at most five lanes

in a continuous move path, less than 0.1 percent (for all instances that could be solved).

28

Table 15: Savings from continuous moves for different limits on the number of lanes in thepath.

K=3 K=4 K=5Instance R = 25 2-cycle R = 25 2-cycle R = 25 2-cycle

i1 5.38% 4.17% 5.44% 4.24% 5.45% 4.25%i2 6.94% 6.74% 6.94% 6.81% 6.94% 6.81%i3 9.74% 8.45% 10.04% 8.81% 10.13% 8.87%i4 11.89% 10.52% 12.45% 11.58% - 11.72%i5 11.52% 10.34% 12.02% 11.33% - 11.46%i6 10.30% 9.21% 10.65% 9.58% 10.66% 9.65%i7 12.66% 11.16% 13.24% 12.17% - 12.27%i8 10.33% 8.93% 10.73% 9.24% 10.62% 9.33%i9 9.74% 8.45% 10.04% 8.81% 10.13% 8.87%

i10 10.30% 9.21% 10.65% 9.58% 10.66% 9.65%i11 10.33% 8.93% 10.73% 9.24% 10.62% 9.33%i12 8.69% 7.71% 8.90% 7.93% 8.96% 7.95%i13 5.24% 4.41% 5.24% 4.49% 5.29% 4.49%

7. Conclusions and Future Research

In this paper, we discussed the development of optimization technology that can be used

to assist in the identification of repeatable, dedicated truckload continuous move tours.

Although we presented the technology primarily from a shippers perspective, it can also be

used by carriers.

Timing considerations were a key focus of our efforts. We showed that a highly effective

and extremely efficient heuristic can be designed and implemented. We demonstrated the

effectiveness of the algorithms developed on various randomly generated instances simulating

real-life supply chain structures, and on instances derived from data obtained from a strategic

sourcing consortium for a $14 billion dollar sized US industry.

In many auction models used in truckload transportation procurement, it is assumed

that carriers will price bundles of lanes. The optimization technology we have developed can

be used by the carriers to create and price these bundles for such procurement auctions.

References

[1] Alon, N. , M. Tarsi. 1985. Covering multigraphs by simple circuits. SIAM Journal on

Algorithmic and Discrete Methods 6 345-350.

[2] Bentley, J. L. . 1990. K-d trees for semidynamic point sets. Proc. 6th Ann. ACM Sympos.

Comput. Geom. 187-197.

29

[3] Bermond, J.C., B. Jackson, F. Jaeger. 1983. Shortest covering of graphs with cycles.

Journal of Combinatorial Theory, Series B 35 297-308.

[4] Caplice, C., Y. Sheffi. 2003. Optimization-based procurement for transportation ser-

vices. Journal of Business Logistics 24 109-128.

[5] Ergun, O., G. Kuyzu, M. Savelsbergh. 2003. The shipper collaboration problem. To

appear in Computers and Operations Research, Odysseus 2003 Special Issue.

[6] Fan, G. 1992. Covering graphs by cycles. SIAM Journal on Discrete Mathematics 5

491-496.

[7] Fraisse, P. 1985. Cycle covering in bridgeless graphs. Journal of Combinatorial Theory,

Series B 39 146-152.

[8] Guan, M., H. Fleischner. 1985. On the minimum weighted cycle covering problem for

planar graphs. Ars Combinatoria 20 61-68.

[9] Hochbaum, D.S., E.V. Olinick. 2001. The bounded cycle cover problem. INFORMS

Journal on Computing 13 104-119.

[10] Itai, A., R.J. Lipton, C.H. Papadimitriou, M. Rodeh. 1981. Covering graphs by simple

circuits. SIAM Journal on Computing 10 746-750.

[11] Jackson, B. 1990. Shortest circuit covers and postman tours in graphs with a nowhere

zero 4-flows. SIAM Journal on Computing 19 659-665.

[12] Kesel’man, D.Y. 1987. Covering the edges of a graph by circuits. Kibernetica 3 16-22.

[13] Labbe, M., G. Laporte, P. Soriano. 1998. Covering a graph with cycles. Computers and

Operations Research 25 499-504.

[14] Lynch, K. Collaborative Logistics Networks – Breaking Traditional Performance Barri-

ers for Shippers and Carriers. (http://www.nistevo.com/v1/downloads/index.html)

[15] Moore, E.W., J.M. Warmke, L.R. Gorban. 1991. The Indispensable Role of Mangement

Science in Centralizing Freight Operations at Reynolds Metal Company. Interfaces 21

107-129.

30

[16] Powell, W.B., Y. Sheffi, K. Nickerson, K. Butterbaugh, S. Atherton. 1988. Maximizing

profits for North American Van Lines’ truckload division: a new framework for pricing

and operations. Interfaces 18 21-41.

[17] Powell, W.B., A. Marar, J. Gelfand, S. Bowers. 2002. Implementing Real-Time Opti-

mization Models: A Case Application From The Motor Carrier Industry. Operations

Research 50 571-581.

[18] Song, J., A. Regan. 2002. Combinatorial auctions for transportation service procure-

ment: the carrier perspective. Transportation Research Record. Journal of the Trans-

portation Research Board 1833, 40-46.

[19] Thomassen, C. 1997. On the complexity of finding a minimum cycle cover of a graph.

SIAM Journal on Computing 26 675-677.

[20] Yang, J., Jaillet, P., and Mahmassani, H.S. 1999. On-Line Algorithms for Truck Fleet

Assignment and Scheduling under Real-time Information. Transportation Research

Record. Journal of the Transportation Research Board 1667, 107-113.

31