Operational planning of a large-scale multi-modal transportation system

European Journal of Operational Research 156 (2004) 41–53

www.elsevier.com/locate/dsw

Operational planning of a large-scalemulti-modal transportation system q

Benjamin Jansen a,*, Pieter C.J. Swinkels a, Geert J.A. Teeuwen a,Babette van Antwerpen de Fluiter b, Hein A. Fleuren c

a Centre for Quantitative Methods CQM b.v., P.O. Box 414, 5600 AK Eindhoven, The Netherlandsb Altera Corp. 101 Innovation Drive, San Jose, CA 95134, USA

c Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands

Received 17 March 2001; accepted 20 February 2003

Abstract

This paper describes the operational planning system POP developed for Danzas Euronet, a merger of Deutsche

Post Transport and Danzas NTO. As of November 1997, the system has been used daily for the transportation

planning of on average 4000 container-orders a day on trains and trucks in Germany. An important feature is that the

future has to be taken into account: trucks have to return home within a couple of days, and empty containers have to

be available at the right time and the right place. These repositioning aspects are taken into account integrally with the

planning of the orders in order to get a cost-efficient solution. In addition, practical constraints play an important role,

and the system has to be flexible for new and modified constraints. The system has not only been used heavily for daily

planning, but also for many simulation studies, thereby supporting operations as well as commerce.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Transportation; Decision support systems; Large scale optimization; Network flows

1. Introduction

Deutsche Post World Net is a major player in

the transportation of parcels. Starting in the nine-teen nineties, Deutsche Post laid out a new network

in Germany consisting of about 35 freight centers

with very modern sorting facilities. Long-

haul transportation takes place overnight with

qThis paper was finalist for the EURO Excellence in Practice

Award 2001.* Corresponding author. Tel.: +31-40-2758702; fax: +31-40-

2758712.

E-mail address: [email protected] (B. Jansen).

0377-2217/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/j.ejor.2003.02.001

container transportation between the freight cen-

ters. In the morning and in the afternoon the par-

cels are being distributed to and collected from post

offices and clients. In the past five years, the Deut-sche Post network has been connected with the

networks of several other companies, partly sharing

the same sorting facilities, partly sharing the same

transportation capacity. The combined network

has been operated by Deutsche Post Transport. As

of summer 2000 a merger with Danzas NTO led to

the new company Danzas Euronet, now operating

the synergy of their respective networks.A planning system is indispensable to be able to

operate a network where transportation requests

ed.

mail to: [email protected]

42 B. Jansen et al. / European Journal of Operational Research 156 (2004) 41–53

vary and so a daily changing plan is required. In1997, Deutsche Post therefore decided to develop a

new operational planning system to support the

operations of the long haul. To give an impression

of size: the number of transport orders averages

over 4000 orders per day and is highly seasonal. It

has a dip during the summer of about 2000 con-

tainers a day and peaks the week before Easter and

Christmas with over 6000 per day. Transport isdone by over 1500 trucks as well as a number of

trains. The number of freight centers involved per

day is over a hundred, consisting of the basic

network of Deutsche Post combined with other

outlets like regional post offices.

The complexity of the planning system has been

growing steadily over the years. For instance,

increasing competition between the large postaland express companies in Europe gives rise to in-

creased pressure on service times and costs. As of

1998 Deutsche Post guarantees next day service for

parcels throughout Germany. An extra complica-

tion is that transport in Germany is typically

unbalanced. For instance, the number of transport

orders from west to east is much larger than from

east to west. Therefore, also the repositioning ofempty containers has to be taken into account. As

costs for repositioning cannot be directly charged

to a customer, it is evident that repositioning costs

should be avoided as much as possible.

The planning system has been developed by the

companies CQM and AMIS in close cooperation

with the client. CQM is specialized in the appli-

cation of quantitative models and methods incomplex planning environments. AMIS has

expertise in Oracle databases. The system is being

used daily since November 1997, hence it can be

regarded as proven technology. The system is also

used to support operations and commerce in var-

ious simulation studies. It gives a quick insight

in the cost and planning effects of network syner-

gies, customer offerings, and of changes in con-straints.

The savings obtained by using the system are

difficult to measure. We estimate the total cost

savings at least at 5%. This does not include the

costs of manpower saved by automating the

planning. Note that a 1% decrease in cost already

leads to a few million dollars cost savings per year.

In this paper we describe the part of theplanning system that performs the automatic

planning of transport orders, called Planung

und Optimierung Programm (POP). It contains

many operations research aspects of interest for

practitioners as well as theoreticians. Some of

the key requirements put forth by the customer

were:

• It is an operational planning system, and has

to take all practically relevant constraints into

account.

• The planning system should be flexible with re-

spect to future developments at Deutsche Post

(new customers, new ways of planning, new

constraints, increasing size, etc.).

• The daily planning run should require a compu-tation time not more than 15 minutes on a �stan-dard� PC or workstation.

• Implementation should be platform indepen-

dent.

To conform to these requirements it is impor-

tant to stress the following contributions which

might be of interest to OR-practitioners.

• Much attention was paid to modeling this

complex problem. Due to the size of the

instances and the required runtime very many

modeling decisions had to be made. These

decisions all had to be based on careful data

analysis.

• Because of the previous point we have chosen ahybrid and iterative algorithmic approach that

enabled us to solve instances of this size with

all their requirements. The methods for some

of the subproblems may not be algorithmi-

cally innovative, but the total framework cer-

tainly is.

• Much attention was paid to data-structures. On

the one hand for runtime reasons, on theother hand for adaptability to other require-

ments.

It is our opinion that the results could not have

been obtained without combining excellence in

operations research and excellence in information

technology.

B. Jansen et al. / European Journal of Operational Research 156 (2004) 41–53 43

This paper is structured as follows. In Section 2we elaborate on the problem description. In Sec-

tion 3 we give an overview of literature on trans-

portation planning. In Section 4 we describe the

approach we have taken in decomposing and

solving the problem. Section 5 describes some as-

pects of the implementation, as well as experience

with the real-life use of the system.

2. Problem description

2.1. General information

The objective of the planning is to provide a

cost-efficient transportation plan for a given set of

orders, taking a large number of constraints intoaccount. An order typically consists of a container

of some type, and has a pickup and delivery

location, a pickup time window and a delivery

time window. Two modes of transportation can be

used: rail and road. For orders that are trans-

ported by rail also the road feeders from and to the

train station have to be planned.

Most orders are flexible in their routing andtiming, and we will concentrate on these orders in

the remainder of the paper.

Transportation is boarded out to contractors,

and the set of available trucks can vary on a day-

to-day basis. Most trucks can transport at most

two containers at the same time. Constraints in-

clude: transport capacity, freight center capacity,

timing, and container repositioning. The objectiveis to minimize cost.

The transportation plan consists of so-called

tours that are assigned to vehicles. Each tour is

divided up in so-called tour sections, such that

loading of containers only takes place at the

beginning of a tour section, and unloading only

takes place at the end of a tour section. For each

vehicle the sequence of tours assigned to it shouldbe feasible, which means that no geographic gaps

between consecutive tour sections may exist and

tour sections may not overlap in time, or have

a time gap that is too large.

The remaining sections give details about the

different types of constraints and the cost struc-

ture.

2.2. Timing

Order time windows. Each order has four time

windows: one in which the container should be

available at the pickup location, one in which the

loaded container should be picked up, one in

which the loaded container should be delivered,

and one in which the container should have beenunpacked at the delivery location. The time dif-

ference between earliest pickup time and latest

delivery time usually is either 6–12 hour (next day

service) or 30–36 hour (second day service). Note,

that this implies that orders of different days can

be combined to save costs. Preferred by the cus-

tomer is that a pickup happens as early as possible

in the pickup window.Opening hours. Freight centers have two types

of opening hours: opening hours for pickup and

delivery of containers, and opening hours for

processing, in which containers can be unpacked.

The latter opening times are called processing

intervals (cf. Section 2.3).

Travel times and distances. Locations are linked

via sections. Length and travel time of a sectiondepend on the truck type being used. Other timing

constraints include the regulations regarding

driving and breaks, maximal tour duration, max-

imal waiting times, availability of trucks in certain

time windows only, and the required return of

trucks to their home location within a certain

amount of time.

Planning horizon. The user can freely set theplanning horizon. All orders with a pickup time in

the planning horizon should be planned. More-

over, it is possible to extend the planning horizon

with a period for which the repositioning is taken

into account but for which the orders are not

planned (the repositioning horizon). The planning

horizon is typically one to three days; the reposi-

tioning horizon is typically a week.

2.3. Capacity

Transportation capacity. Transportation capac-

ity consists of capacity on rail and on road. The

rail system consists of a two-hub system, where

trains either drive to or from the hub via a number

of intermediate train stations, or drive from hub to


hub, or drive from train station to train station.Trains have a given, fixed schedule. The available

capacity per train is known in advance and typi-

cally in the order of 20–40 containers per train.

Per day over 1500 trucks are being used. There

are various types of trucks, which differ in the type

and amount of containers they can load, as well as

in their driving speed profiles. Typically, a truck

loads at most two containers. The trucks are hiredfrom contractors. Each contractor has a set of

home locations. Each truck initially departs from a

home location, and has to return to a home loca-

tion within a specified number of days (typically 2–

5). We note that in the planning no detailed crew

scheduling is involved. Instead it is assumed that a

driver change can happen in any of the home

locations of a contractor. Furthermore, regula-tions on driving and resting times are taken into

account during planning. Moreover, it is deter-

mined whether or not two drivers are required for

a tour.

For orders to be transported by rail also the

feeders have to be planned, which means that the

container has to be transported by road from

pickup location to train station (up feeder) andlater from train station to delivery location (down

feeder). For up feeders, orders that have different

delivery train stations should preferably not be

planned on the same truck, even if they have the

same pickup location. This has to do with the fact

that trains are composed of wagon groups and

that shunting is typically done with wagon groups.

At a train station the truck enters the line for awagon group, which means that transporting or-

ders from different wagon groups implies double

waiting time at the lines. The same holds for

planning of down feeders.

Processing capacity. At the freight center,

delivered containers have to be unpacked. It is

required that all containers that enter should be

unpacked before the end of the first possible pro-cessing interval. This has to do with the fact that

during the day the sorting facilities are used for

collection and distribution activities. The process-

ing capacity of sorting facilities is measured in

number of packages per minute, and each order

has a number of packages given. It is essential to

check feasibility of arrival times at freight centers

in view of the capacity of a sorting facility. Espe-cially in the top season, when freight centers are

heavily loaded.

2.4. Repositioning

Container repositioning. The order set is typi-

cally unbalanced; in Germany much more (loaded)

containers go from west to east than from east towest. This means that container repositioning is

needed. An exact balance of containers (in the

sense that the number of incoming containers

equals the number of outgoing containers) is not

required. Instead, container stock is allowed and

the number of containers at a location should be

between given bounds at any point in time. The

lower bound is also used to model unknown ordersthat will take place after the end of the reposi-

tioning horizon. The number of outgoing orders

for each location is estimated from historical data,

and added to the lower bound for that location.

2.5. Cost aspects

The objective is to minimize cost of transpor-tation, which is the sum of the cost of the tours.

The cost of a tour depends on the type of the tour,

the length of the tour and the contractor. The tour

type is related to the attractiveness of the tour for a

contractor. For instance a tour A–B–A is preferred

over a tour A–B, since it returns the truck to its

home location. Costs are measured via tariff ta-

bles. A tariff table divides the possible distances ofa tour into a sequence of consecutive intervals. For

each interval ½xi; yiÞ, it gives either a fixed cost Cfi ,

or a cost per kilometer Cki . Note that this cost

function may be discontinuous, not convex, and is

not necessarily increasing in the distance.

3. Literature

There is a lot of literature on traditional vehi-

cle routing problems. Starting in the early days

with simple single-depot problems, nowadays

complicated pickup and delivery problems are

being modeled and solved with various approaches

(see e.g. Savelsbergh and Sol [17] or Dumas et al.


[8]). We refer to Desrosiers et al. [6] for an over-view of the history of routing problems. In

the world of express and parcel transportation the

operational vehicle routing is just one of the

underlying problems. From Crainic and Laporte

[4] we cite the following design and planning

problems:

• service network design (selection of hubs,routes, service frequencies),

• traffic distribution (routing of requests),

• terminal policies (handling at terminals),

• empty balancing (repositioning of resources),

• crew and motive power scheduling (having re-

sources and crews at the right time at the right

place).

In our application we have to deal with the

aspects traffic distribution, empty balancing and

motive power scheduling. We believe that these

have to be taken into account integrally to obtain

good solutions.

In a series of papers Powell (e.g. [15]) investi-

gates the design and implementation of motor

carrier networks. Kim et al. [12] consider the ser-vice network design problem for UPS. Here the

question is which routes to use for which trans-

portation requests, i.e., in which hubs transship-

ment for a transportation request should take

place and which mode(s) of transportation should

be used. Note that this is a strategic problem, since

the resulting service network typically yields a

fixed (repeating) schedule between the hubs in thenetwork. The planning horizon considered is one

day. For postal and express companies the

underlying networks are huge, implying that more

traditional network design models and algorithms

(see e.g. Magnanti and Wong [14]) are not appli-

cable. Kim et al. [12] give computational results

for some realistic networks. Solving the design

problem took several hours for problems withabout 100 locations and 10 hubs. Kuby and Grant

[13] consider some variants of a similar problem

on a smaller section of the network of Federal

Express.

For more research on network design in freight

transportation we refer to Crainic [2] for a recent

paper, and to Roy and Crainic [16].

Gr€unert and Sebastian [11] consider the net-work design problem for the letter mail division of

Deutsche Post AG. As in [12] the emphasis is on

the strategic level to design a cost minimal air and

ground transportation schedule. Unfortunately,

the paper gives hardly any information about size

of the problems, algorithms and computational

results.

Savelsbergh and Sol [18] show the applicationof a branch-and-price algorithm for the replanning

activities at Van Gend and Loos, an express

company operating in the Benelux. Incorporation

of about 300 orders in a schedule with 100 trucks

is performed with computing time in the order of

5–10 minutes.

The problem of repositioning of transportation

capacity has been considered in many papers. Forinstance, the problem occurs in the design of

transportation schedules (e.g. Gr€unert and Sebas-tian [11], Kim et al. [12]), where the number of

trucks leaving should equal the number entering

each day. In the sea container business a similar

situation occurs as in ours: the number of con-

tainers is allowed to vary between certain bounds

and repositioning should use certain forecasts toprevent unnecessarily expensive empty container

transport at the last moment. We refer to Dejax

and Crainic [5] for an overview. In the land dis-

tribution we refer to Crainic et al. [3] who give

both dynamic and stochastic models for allocation

of empty containers, based on multi-commodity

network flow formulations. An approach using

inventory and stock-control policies is used by Duand Hall [7], albeit for networks with a very special

structure.

Related to the above mentioned literature we

believe that our application is rather unique. This

stems from the following facts:

• The problem we solve comes from opera-

tions and is not a strategic design problem.The transportation requests as well as the re-

quired vehicle capacity vary on a day to day

basis.

• The size of our problem is huge; given the

requirements on computation time we are not

aware of methods from the literature that can

be directly used.


• There are a number of difficult practical con-

straints. Moreover, the cost functions are not

convex, discontinuous, and not necessarily

increasing in the number of kilometers.

• The repositioning of empty containers is an

integral part of daily operations.

• The algorithm should be flexible with regard to

addition or modification of constraints and coststructures.

modality choice

repositioning

order combination

order planning

planningimprovement

tour test

Fig. 1. Structure of the algorithm.

4. Solution approach

4.1. General

As is clear from the problem description,the problem is too large to be solved in one

step. Therefore, we have decomposed the prob-

lem into a number of subproblems. This de-

composition is motivated by the following

requirements.

• Each of the subproblems should have an effi-

cient and effective algorithm, for the overallsolution to be cost-efficient.

• The decomposition allows for a flexible imple-

mentation in the sense that adding constraints

and changing cost structures can be done rela-

tively easily.

• Initial solutions should be easy to construct,

and when stopping the algorithm a partial solu-

tion should be available.

The subproblems we solve are the following:

• Multi-modality. Given a set of orders with pref-

erence for train transportation, assign each

order to a mode of transportation, and if this

mode is train, plan the train part of the trip

for this order, taking timing and capacity con-straints into account.

• Repositioning. Given a set of partially planned

orders and upper and lower bounds on the

number of containers at each location, deter-

mine a set of empty container orders that can

be transported efficiently, such that each loca-

tion has enough and not too many containers

for the next day(s).

• Combination of orders.Given a set of unplanned

orders, combine the orders in pairs such that

planning the pairs together on tours will give

a cost-efficient planning.

• Planning of (combinations of) orders. Given a

set of unplanned orders and combinations of

orders and a partial plan, plan the unplanned

(combinations of) orders in a cost-efficientway, taking timing and capacity constraints

into account.

• Plan improvement. Given a set of mostly

planned orders, try to find a better planning

by moving orders and tours around.

• Tour test. Given a sequence of tours on a truck

and a set of constraints to be tested, calculate

departure and arrival times of each tour section,evaluate the specified set of constraints, and

compute the costs of the tour sequence.

The flow of the complete algorithm is depicted

in Fig. 1. The modality choice is a preprocessing

step and is performed only once. After that, the

planning of orders on the road and the generation


and planning of empty container orders is done ina sequence of iterations. This sequence starts with

repositioning the empty containers. After this,

orders are combined in the order combination

step. Next, the combinations of orders and

uncombined orders are planned with a construc-

tive algorithm. Finally, the planning is improved

with a local search algorithm. In the next iteration,

all the empty container orders that were generatedin the previous repositioning step are removed,

and a new set of empty container orders is com-

puted, now using information from existing tours

in the plan that are not filled to full capacity. Note

that this iterative approach allows the user to stop

the algorithm after each iteration having a partial

feasible plan.

The tour test is a basis for each of the steps inthe algorithm. It is used to test feasibility and

quality of the planning and of potential tours. As

the tour test is invoked a large number of times,

one test should be (and can be) done very fast.

We now give a more detailed description of the

various subproblems.

4.2. Modality choice

Each order has a preferred modality, which can

be either rail or road. Orders with preference for

road have to be planned on trucks, whereas orders

with preference rail can be planned on rail, but on

road as well if there is not enough capacity on

trains. The modality choice step is concerned with

assigning rail orders to specific trains, with theobjective of using the available trains in the most

efficient way, and the constraints of train capacity,

availability of trucks for transport to and from

train stations, and timing constraints. This is done

via a straightforward assignment technique, as the

prerequisites set by the client do not give more

optimization freedom. Of all the orders with rail

preference, those that would need to drive thelongest distance by vehicle have priority when

assigning them to trains. This is because trains have

a low cost per distance unit, and thus are more

attractive for long-distance orders than for short-

distance orders. Hence, the algorithm just proceeds

by sorting the orders to decreasing distance (from

pickup to delivery location) and assigning them

one by one to the first possible train; a path-searchalgorithm is used. For each location it is prescribed

which train station should be used. A priority is

given to direct trains, then to a train that passes one

hub, then to a train that passes two hubs. When-

ever no train with free capacity is available, the

order is transported by road.

From the assignment of orders to trains, a set

of up- and down feeder �orders� is derived. Theseorders are planned simultaneously with other road

orders.

4.3. Repositioning

The goal of repositioning is to generate a set of

orders for transporting empty containers in such

a way that

• at each moment in time, every location has a

stock of containers between given bounds, and

• the transportation of these orders can be done

cost-efficiently.

We model the problem as a min cost flow

problem (see e.g. Ahuja et al. [1]). A similar strat-egy is followed in Crainic et al. [3]. We build a time

space network. Here a node in the network means

a location (freight center or train station) at a

certain point in time in a certain status. Existence

of a link means movement of empty containers is

possible from one location to another location

(possibly the same) at certain times. A unit of flow

over such a link corresponds to the movement ofone empty container.

The entire network models both road and rail

transportation, and the subnetworks for these two

modalities are linked by the feeder links. There are

three types of nodes in the network:

• Train station nodes. Nodes that represent a train

station at a given point in time.• Freight center nodes. Nodes that represent

freight centers at a given point in time with a

given status. The statuses are the following (in

sequence): INCOMING, STOCKING, OUT-

GOING. The first and last are used to be able

to bound the number of incoming and outgoing

empty containers. The stocking node is used to


bound the amount of stock as well as to prevent

infeasibilities.

• A source and a sink that are used to model flow

conservation and infeasibilities.

Fig. 2 shows the model structure at two freight

center locations (upper and lower) at two consec-

utive time points (left and right).For freight center nodes, we restrict the number

of points in time per day to two. This choice im-

plies that all arrivals and departures of containers

are aggregated to these time points. We have

chosen for the fixed two time points because the

time windows of most orders tend to be close to

these two time points, and hence our approach will

be accurate and efficient. Alternatively, we couldhave made a network in which each node has the

time of an arrival or a departure of a specific

container. This would make the model more

accurate, but the size of the network would ex-

plode. For train station nodes, there is no need to

aggregate over time, because the number of trains

is limited. Hence for each train station, there is a

node for each departure time of a train and eacharrival time of a train in the repositioning horizon.

There are different types of links in the network,

as described below. For each link, we specify the

lower and upper bound on the amount of flow

over the link, and the cost per unit of flow over the

link (see Table 1).

Since network models notoriously are degener-

ate, a solution returned by a network solver istypically �unbalanced�, or �extreme�. However, inpractice one likes solutions with certain regularity.

For instance, when 10 empty containers have to be

shipped from A to B before tomorrow night, one

Fig. 2. Network structure at some of the freight center nodes.

could have 10 today, or 10 tomorrow, or 5 todayand 5 tomorrow. The latter is typically preferred,

but will not be returned by the network solver. By

using multiple arcs with different costs between the

same nodes this is accounted for.

Several nodes in the network have supplies or

demands. The STOCKING nodes have a demand,

which equals the difference between the number of

outgoing (customer) containers between the cur-rent and the next time point, and the number of

incoming (customer) containers between the pre-

vious and the current time point. The first

INCOMING node in time of each freight center

has as supply the number of containers in stock at

that location in the beginning of the repositioning

horizon. The demand on the sink equals the total

supply that enters the network in the first timeperiod. The network flow model always has a

feasible solution.

For network flow problems standard algo-

rithms are available. In POP the excellent imple-

mentation �cs2� of Goldberg [10] is used.

4.4. Combination of orders

The way in which orders are combined on

trucks is very important for the quality of the

planning. Therefore, we create an optimal set of

combinations of orders before starting the actual

planning of the orders. These combinations serve

as a hint to the planning algorithm: the algorithm

tries to plan combinations of orders on the same

truck, but it can untangle a combination if thisdoes not provide a good solution. As most trucks

have a capacity of two containers, we solve the

problem of combining orders in pairs, such that

the overall costs of the pairs are low. When costs

are equal, the following preferences apply:

• Combine orders on up (down) feeders that

share the same delivery (pickup) location and/or delivery (pickup) train station.

• Combine orders that have pickup time window

close to each other.

Both stem from practical considerations. The

following mathematical model is used. For each

pair of orders i and j, we have a variable xij for

Table 1

Nodes and arcs in network flow model

Purpose From node To node Lower bound Upper bound Cost per unit

Model the containers

staying in stock

during a time period

STOCKING STOCKING; at same

location and next time

point

Minimum re-

quired stock

Maximum allowed

stock

Cost of having one

container in stock

Model the containers

over maximum

allowed stock

STOCKING STOCKING 0 Inf Cost of one con-

tainer over capacity

Model containers under

minimum stock

STOCKING STOCKING; at same

location and previous

time point

0 Minimum required

stock

Cost of being under

stock

Road transport OUTGOING INCOMING; nodes at

different freight centers

and different time

points, provided that

transport between these

locations can be done

0 Inf From tariff table

Bound the number of

empty containers that

arrives at a freight

center at a given

point in time

INCOMING STOCKING; at the

same location-time

0 Upper bound on

the total amount of

in-coming orders

with that delivery

time

0

Bound the number of

empty containers that

departs from a freight

center at a given

point in time

STOCKING OUTGOING; at the

same location-time

0 Transport capacity

of the location at

that time point

0

Transport from freight

center to train station

OUTGOING Train station, provided

transport physically

possible


Transport from train

station to freight

center

Train station INCOMING, provided

transport physically

possible


Transport by train Train station Train station if there is

a tour section on a train

corresponding to the

two stations

0 Capacity of train Cost of one con-

tainer on the train

Flow conservation OUTGOING Sink Minimum re-

quired stock at

the end of the

horizon

Inf 0

Flow conservation Sink Source 0 Inf 0

Feasibility Source STOCKING of every

location-time point

0 Inf High cost to obtain

a solution that is �asfeasible as possible�.Cost decreasing in

time

Use of planned road

transport

OUTGOING INCOMING if a tour is

already planned to drive

this stretch

0 Available capacity

on the truck

0

Use of existing empty

container orders

OUTGOING INCOMING if empty

container order exists

and must be preserved

0 Maximum amount

of orders to keep

on this stretch

0



i < j, with value 1 when orders i and j are com-bined, zero otherwise. Define parameter Fij to bethe gain obtained by combining orders i and j inone tour as opposed to transporting them in sep-

arate tours. For every two orders i and j that canbe combined, let

Fij ¼ Di þ Dj � Dij þ BRij þ BT

ij; ð1Þ

where for each order i, j:

Di minimal distance covered to pickup anddeliver order i

Dij minimal distance covered to pickup and

deliver orders i and j in one tourBRij bonus for feeder priority of orders i and j

BTij bonus for similarity of pickup time of

orders i and j.

Note that to compute the F -value of a combi-nation we have to do (part of) the tour test. Then

the model to be solved is

maxP

ij Fij � xijs:t:

Pj xij 6 1 8i;

xij ¼ 0 8i; j that cannot be combined;xij 2 f0; 1g:

Note that this is a weighted matching problem. Allcombinations of couples of orders are first tested

to determine the F -values. Then the algorithm of

Gabow [9] is applied to find the optimum solution.

In the case of very many orders (our largest test set

contained 20,000 orders), the Oðn2Þ time that isneeded to find the F values is too slow, and we useonly combinations of orders having pickup and/or

delivery in each others neighborhood. Simulationsshowed that the quality of the planning is only

slightly influenced by this restriction.

4.5. Order planning

In this subproblem the planning of the road or-

ders is done. The planning of road orders proceeds

with a sliding window technique. The width of thesliding window depends on the width of the time

windows of the orders. The orders having earliest

pickup time within the window are being collected,

combined, and partly planned. Then the reposi-

tioning model is again solved to generate a new set

of repositioning orders (note, that the links includedin the model with their bounds and costs depend on

the current planning). Then again, the orders hav-

ing earliest pickup time in the current sliding win-

dow are being planned. A local search algorithm

improves this part of the planning. Then the plan-

ning in the first part of the sliding window is being

fixed, and the window is moved to the next position.

The order planning consists of two phases, thateach have a different approach:

• Phase 1: a transporter (tour) oriented planning

approach.

• Phase 2: an order oriented approach integrated

with a tour oriented approach.

Transporter oriented planning. In the trans-porter oriented approach the idea is that a trans-

porter should be efficiently and effectively used.

For instance, trucks that are far away from their

home locations should be used to transport orders

to a location that is closer to their home, since it is

expensive to let them drive home without a pay

load. This is especially true for trucks with two

drivers. The cost functions do not enforce this tohappen in the order-oriented approach.

The transporters that should be used are sorted

by attractiveness via number of drivers, tour type

and tour length. Running down the list of (com-

binations of) orders a planning on the list of tours

is done. Given an order only tours that satisfy

certain criteria (depending on the tour type and on

the order) are taken into account. A first fitstrategy is used for most tour types, as our

experiments showed that a best fit strategy did not

automatically lead to a better overall solution.

Order and tour oriented planning. The order

oriented planning proceeds in three outer loops. In

the first one, only combinations of orders can be

planned, in the second and third outer loops also

single orders can be planned. Moreover, in the firstouter loop no long empty mileage tours can be

made. Between the second and the third outer loop

transporters that cannot be sent home in time with

existing orders are sent home empty. Each outer

loop consists of a series of inner loops. In an inner

loop (combinations of) orders are being planned,

starting at the top of the list and running down-


wards. When an order has been planned, thealgorithm tries to extend the tour in which the

order has been planned, by searching for (combi-

nations of) orders that can improve upon the plan.

For instance, by searching for a similar order or by

finding an order for the way back.

Given an unplanned order the search for a tour

able to transport the order proceeds as follows.

First existing tours are tested, where tours havingempty places are preferred. In case there is no tour

that can transport the order without changing its

route, a best-fit strategy can be used to find a

suitable tour for the order. In case a new tour is

needed for the transporter, again a best-fit ap-

proach can be used to select an attractive truck.

Implementation details. The number of orders

and tours is very large, and it would be too slow tosimply try to place each order on each tour.

Therefore, we use balanced binary search trees to

keep track of unplanned orders at each location.

The orders are sorted by pickup time. Also, we use

balanced binary search trees to keep track of tour

sections with available space for each location.

These are sorted by start time. Both trees are

maintained dynamically. To further speed up thesearch for interesting orders and tours some

neighborhood structures are used. The neighbor-

hoods can be computed beforehand and do not

change during the planning.

4.6. Planning improvement

In this step three types of local search proce-dures have been implemented, being order based,

transporter based, and tour based. In these itera-

tive improvement steps further cost savings in the

plan are attained. Note, that a change to the plan is

only accepted when a cost saving is being achieved;

techniques from simulated annealing or tabu

search have not been implemented because of

running time issues.In the order based local search algorithm a

search for empty space or time in tours is done that

can be filled up with orders from other tours. After

some steps, hopefully, tours or detours can be re-

moved hence saving money. In the transporter

based approach tours are moved from one trans-

porter to the other. Possible cost effects are:

• a cheaper contractor is being used;

• the moved tour can be combined with a tour at

the new transporter, thereby making two shorter

ones into one long one;

• the tour type of the moved tour can be more

attractive.

In the tour based approach tours are swappedbetween transporters. A parameter states whether

all tours are taken into account or only the last

tours per transporter. A quick check determines

whether a swap is likely to be attractive (for in-

stance the contractors of the tours could be dif-

ferent).

Once again, to speed up the local search we

heavily use neighborhood structures, and severalimplementation ideas.

4.7. Tour test

As mentioned previously, the tour test is used to

evaluate a given set of constraints and to compute

arrival and departure times and costs. It can be run

on a single tour, a sequence of tours, or the wholeplan. A binary pattern basically consisting of

flags is used to determine which constraints should

be checked. In this way the addition of a new

type of constraint implies that an extra check

should be implemented in the tour test, and

maybe the default binary pattern should be chan-

ged. No changes in the rest of the system are

necessary.As the tour test is used very often, it should run

fast. The most time-consuming part of the tour

test, is the determination of times (departure,

driving, arrival, etc.) and breaks. Note, that to

determine the position of a break the times are

required. However, to compute the times we al-

ready need to know where to position breaks. As

there is a flexibility in the positioning of thebreaks, tackling this �cycle� is not easy. We do it byfirst computing bounds on times as well as �ten-tative� times in a series of loops through the toursunder consideration. In the final loop, the times

and breaks are really set. The overall run time of

one tour test is linear in the number of tour sec-

tions plus the number of containers that is trans-

ported.


5. Implementation and real-life experience

Although the system was primarily developed

for the long-haul of Deutsche Post AG, it was

designed in such a way that new customers, plan-

ning types, constraints, etcetera, could be incor-

porated relatively easily. Throughout the years,

this design has proven its usefulness: a lot ofsmaller and larger changes have been made,

without the need for a complete redesign of the

system. This is both due to the flexibility in the

algorithm and the flexibility offered by the imple-

mentation. The planning algorithm has been

implemented in ANSI C. The program is platform

independent; at the development site typically

Windows is used, in the daily operations the pro-gram runs under Unix. On our standard PCs

computation times range from a few minutes to 15

minutes during peak days.

We paid much attention to the data structures.

The implementation heavily uses balanced binary

search trees, which are very efficient for quickly

checking and updating the plan.

Since November 1997, about 850 daily planningruns have been done with POP. Moreover, it has

often been used to support simulation studies for

answering questions like:

• What is the influence of smaller or larger pro-

cessing capacities at the freight centers?

• What synergy effects can be expected from mix-

ing the networks of customers X and Y ?• What price offering can be made to potential

customer Z?

From a theoretical point of view various

interesting questions remain. For instance, is it

possible to solve the planning and repositioning

problems simultaneously? Can such a model be

tackled with techniques as branch-and-cut,branch-and-price or set partitioning in reasonable

runtimes? A related question is whether it is

possible to compute a lower bound on the cost

value of a planning. Intriguing challenges for us

practitioners are to use detailed digital maps of

Germany as more and more diverse customers

are added to the network, to investigate and

implement the use of transshipment of road-

transported orders, or to make the techniques inPOP available for an on-line planning support

system.

Acknowledgements

We are grateful to Dr. Dieter P€utz of DanzasEuronet for giving us the opportunity to write thispaper. We are greatly indebted to Frank Cruse,

Gerd Erb and their colleagues for their support in

the development of the planning system. Thanks

are also due to our business partner AMIS. Last

but not least, we thank our colleagues Karin Lim,

Jacques Verriet, Lonneke Driessen and Judith

Lamers for their help in specific parts of the

development of POP.

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Operational planning of a large-scale multi-modal transportation system