a decomposition scheme for large-scale service network

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A decomposition scheme for large-scale Service Network Design

with asset management

Nicolas Teypaz, Susann Schrenk, Van-Dat Cung *

G-SCOP, Laboratoire des Sciences pour la Conception, lOptimisation et la Production, INPG-UJF-CNRS, 46, Avenue Flix Viallet, 38031 Grenoble, France

a r t i c l e i n f o

Article history:

Received 3 September 2008

Received in revised form 27 May 2009

Accepted 11 July 2009

Keywords:

Freight transportation

Service Network Design

Asset management

a b s t r a c t

In this paper, we address a large-scale freight transportation problem for maximizing the

profit of a carrier. We propose two solving algorithms using a decomposition of the prob-

lem into three main steps: construction of the network, filling vehicles with commodities

and construction of the vehicle plannings. The resolution of these steps involves heuristic

schemes, Mixed Integer Programming and Constraint Programming techniques. To evalu-

ate the model and the solution algorithms, we produce instances based on a study of real-

life data. The results show that the methods without transhipment provide solutions with a

good computation time/quality trade-off.

2009 Elsevier Ltd. All rights reserved.

1. Introduction

Freight transportation is one main activity at the core of the Supply-Chain of companies. According to the French trans-portation ministry and INSEE national institute of economics and statistics, this activity impacts strongly on the national

economy with more than 1 million workers, 120 billion euros of sales turnover. Crainic (2000) mentioned that the transport

represents a significant part of the national expenditures of any country. Todays increasing competition and rationalization

of resources put a high pressure on carriers to optimize their transportations. In this context, we are interested in a problem

raised from an industrial application. We deal with the determination of the transportation plan for regular operations. This

plan is constructed for a given planning horizon (often 1 day, 1 week or 1 month) from a set of selected transportation

services. It is replicated for each planning horizon. Those tactical planning problems are generally modeled as Service

Network Design Problems.

Crainic (2002) stated that Service Network Design Problems (SNDP) are essential in the construction of a transportation

network. Their formulations are associated with the long-term evolution of transportation infrastructures and services.

Service Network Design is often used to solve freight transportation problems with consolidation: by rail in Barnhart

et al. (2000), by air in Barnhart et al. (2002, 1996) and also by road. A recent survey is given by Wieberneit (2008).

The Service Network Designer is making decisions that affect both commodities and vehicles by answering suchquestions as:

What are the routes and schedules of the services?

Which vehicles should be used to operate all services?

How should the commodities be routed through this network of services?

Other more operational issues are often included in Service Network Design formulations, e.g. service frequencies, crews

assignments and customer service requirements.

1366-5545/$ - see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.tre.2009.07.003

* Corresponding author.

E-mail addresses: [email protected] (N. Teypaz), [email protected] (S. Schrenk), [email protected] (V.-D. Cung).

Transportation Research Part E 46 (2010) 156170

Contents lists available at ScienceDirect

Transportation Research Part E

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / t r e
http://dx.doi.org/10.1016/j.tre.2009.07.003mailto:[email protected]:[email protected]:[email protected]://www.sciencedirect.com/science/journal/13665545http://www.elsevier.com/locate/trehttp://www.elsevier.com/locate/trehttp://www.sciencedirect.com/science/journal/13665545mailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.tre.2009.07.003


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A Service Network Design solution is a transportation plan including both the schedules of vehicles and the flow of com-

modities in order to achieve a certain objective. In our case, the objective is to maximize the profit of the carrier. More pre-

cisely, our freight transportation problem is defined as follows. Given a physical network composed of terminals (harbors,

rail stations, airports or warehouses) and potential connections (sea routes, railways, airways, roads) and a planning horizon

divided into a set of periods, the carrier may satisfy globally or partially origindestination transportation demands (com-

modities). This characteristic is known as price collecting or team orienteering problems and was integrated by Feillet

et al. (2005) and Archetti et al. (2008) in different routing problems. These demands are periodic and repeated on every per-

iod (regular freight transport planning). To achieve this, the carrier has a set of vehicles of different types (their number is not

limited). He may organize transhipment operations to improve his profits by consolidation of commodity flows while

respecting the capacities of vehicles.

The transportation costs are twofold: a fixed cost to organize one transportation service (crew cost, handling cost and fuel

cost) and a variable cost depending on the tonnage of commodities on board. The variable cost is assumed linear with the

commodity flow. This assumption is classical in Service Network Design for freight transportation as in Barnhart and Schneur

(1996) and Crainic (2000). Every time the carrier takes charge of the transport of a freight unit, he has some revenue depend-

ing on the tonnage and the commodities. Thus, he has to make a trade-off between transportation costs and the revenues to

maximize his profit.

The freight transportation problem we deal with has two asset management constraints. The first concerns vehicle rota-

tions. To provide a regular freight transportation planning, exactly the same transportation service should be organized in

each time period. To guarantee this, vehicles should be spread out in an analogous way at the beginning and at the end

of a planning horizon. It is not necessarily the same vehicle which has to be in the same location, vehicles may rotate. Chou

et al. (2003) give an example of vehicle rotation for sea transportation, and Yan et al. (2005) for air media. This rarely occurs

for road transportation; the same vehicle can run several routes starting at the same terminal in one period. To ensure vehi-

cle rotation, it may be necessary to make provisions for repositioning some vehicles as in Crainic (2000). In literature, the

vehicle rotations are referred to as the design balance constraints. They ensure that there is an equal number of vehicles

entering and leaving each terminal in the network (Pedersen et al., 2007; Andersen et al., 2007a). Design balance constraints

have been modeled for an application of express freight service by air in Barnhart and Schneur (1996) and Barnhart et al.

(2002), by road in Smilowitz et al. (2003) and for an application of passengers transportation service by ferry in Lai and

Lo (2004), Yan and Chen (2002) and Wang and Lo (2008).

The second asset management constraint is that carriers want to use all their vehicles evenly. At the end of a given period

(month or year) no vehicle should be neither under- nor over-used. This constraint means that the length of any vehicle

route is between two bounds. In vehicle routing problems literature, this constraint is referred to as the time- or dis-

tance-constraint in Laporte (1992).

Recently, Andersen et al. (2007c) extended formulation of the Service Network Design to a time-space network. The pur-

pose of their paper is to compare solving models based on different Mixed Integer Programming formulations. The authors

also analyzed the effect of asset management issues on the optimal solutions. In general, solving the latter issues increases

the required computational effort. This study is the closest study to ours, but their test problems used are smaller than many

instances from real-world problems we deal with. In Andersen et al. (2007b), the greatest instance used to evaluate the per-

formance of different formulations contains five terminals, 200 demands with time windows and 25 time units per period.

Our smallest instance studied contains almost 40 terminals, 140 demands and 480 time units per period. We note that the

commodities is loaded during a period, it has to be delivered in the same period. In our case, we do not have additional time

windows. Thus, a commodity is only defined by its origin and destination terminals.

As a consequence, to tackle real-world problems, our main proposal is to decompose the problem into three steps as fol-

lows: construction of the network, choice of the transported commodities and construction of the vehicle plannings (routes

and scheduling). Although, even if the decomposition in three steps seems quite natural, we did never see it really applied. To

the best of our knowledge, most papers treat only one of the steps. Concerning applications, our approach can provide a glo-

bal transportation solution and can solve real-life size problems in freight transportation.

The main advantages of the decomposition method are as follows:

a step wise construction of the solution which permits to have rapidly an estimation of the potential profit;

a high flexibility to handle specific constraints, e.g. new time constraints can be easily integrated;

improvements of one of the steps can be added independently so as to boost the solving method;

in addition to this decomposition, we also studied how these steps can be performed successively. The order in which we

solve these three steps decides whether or not transhipment can be used;

in a few seconds, our algorithm provides feasible solution of high quality for large-scale instances.

Even if no new techniques have been proposed, our resolution scheme involves many different tools of Operations Re-

search to solve efficiently each step of the decomposition method: graph algorithms, Linear and Mixed Integer Programming,

column and row generation and Constraint Programming techniques. This variety of techniques and their combination re-

flects the creativity of the proposed solution method.

The paper is organized as follows. We present a model for this difficult problem in Section 2. Section 3 is dedicated to the

decomposition of the problem into three main steps which allow to determine the transportation services, choose which

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demands to satisfy and schedule the transportation services. To conclude, Sections 4 and 5 present computational results of

our method, its limits and some avenues for further study.

2. Model of the problem

This section presents a model for the freight transportation problem studied in this paper as a single objective linear

mixed integer programming model on a time-space network. This model is based on the formulation of the Service Network

Design with asset management Andersen et al. (2007c). We have completed this formulation by taking into account the par-ticularities of our problem:

the carrier can reject some demands completely or partially,

no vehicle must be neither under- nor over-used (time-constraints).

2.1. Notation

The periodic transportation demand k is as follows: for every planning horizon (week), transport the quantity wk unities

of commodities, from origin Ok to destination Dk. Each time the carrier satisfies one unit of demand k (one ton, one con-

tainer, one barrel, etc.), he earns quantity Pk of money. Let Kbe the set of all demands. Origin and destination are terminals

on which carrier has access. The distance between two terminals is di;j. We represent terminals V and their physical con-

nections E by a directed graph G V;

E.To transport the freight we have a set L of different vehicles which are classified by their types. We can assume that the

number of vehicles is large. We denote by Ltype the set of vehicles of types type, i.e. L UtypeLtype. We assume that the vehicles

are spread out as wished at the beginning of the planning horizon. Each vehicle l 2 L is able to transport a maximum of Cl

(capacity of vehicle l) units and needs time tli;j to go from origin i to destination j. Once arrived at j, the vehicle has to stay

there for at least s time units before continuing its trip. This is for the vehicle to be refueled and for cargo to be loaded and

unloaded. This break time may include a buffer to reduce lateness.

If a vehicle l carries w units (from several demands wk) directly from terminal i to terminal j, it costs the fixed cost Flfi;j

plus the variable cost Flv

w; i;j. The variable cost is assumed to be linear with the weight of carried freight: we denote Uli;j

the variable cost per unit of commodity transported between i and j: thus Flv

w; i;j Uli;j w. In fact, these costs are the

same for all vehicles of same type. Over a planning period, if a vehicle l is used, it has to fulfill a specific quota of hours defined

by the minimum Quotalmin and the maximum Quotalmax quota of hours. These quotas are the same for the vehicles of same type.

2.2. Formulation as a mixed integer program

The formulation as a Mixed Integer Program (MIP) is adapted from Service Network Design problems. As stated in the

previous section, we need to index all variables with a time parameter. We denote by T the set of discretized time over

the planning horizon. For any terminal i; di denotes the set of predecessors of i and di the set of successors of i. The

variables we use are as follows:

At time tthe fraction xklijt of commodity k is charged in origin i for destination j by vehicle l. Variables xklijt are between 0 and

1 and represent flows of commodities.

The boolean variables ylijt indicate whether the vehicle l is going from origin i to destination j at time t. Variables ylijt choose

the service for the vehicle l (i.e. define the network).

The boolean variables dl indicate whether or not the vehicle l is used.

zk represents the fraction of demand k which is carried during one planning horizon, that is the fraction of demand k which

is satisfied.

The carrier wants to maximize his profit (which is the revenue minus the fixed and variable costs) by respecting all con-

straints as detailed below.

Maximize:

Profit = revenue fixed and variable costs:

Xk2K

Pk wk zk Xij2E

Xl2L

Xt2T

Flfi;j ylijt

Xk2K

Uli;j wk xklijt

!; 1

subject to:

Multicommodity flow constraints:

Xj2di

Xl2L

xklijt Xj2di

Xl2L

xkljittlj;is

0; 8i 2 V; i Ok; i Dk; 8k 2 K; t2T; 2

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Xt2T

Xj2di

Xl2L

xklijt Xt2T

Xj2di

Xl2L

xkljit zk if i Ok;

zk if i Dk;

(8i 2 V; k 2 K: 3

Vehicle route constraints:Xij2E

Xt0 2T:t06t


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The vehicles of the same type have to do cycles1 such that their transportation plannings can be repeated period after per-

iod without the need for an expensive repositioning of a vehicle by doing a trip with empty load. We will refer to these con-

straints (7) as the design-balance constraints. The Fig. 1 highlights the vehicle rotation permitted by these constraints. The

vehicle starting at terminal A in the first period will take the route starting at terminal D for the second period, and vice-versa.

2.3. An upper bound for our problem

As this formulation has too many variables and constraints, mainly due to time index, we cannot expect any response for

real transportation instances by solving this MIP directly. Nevertheless, the Service Network Design with asset balance

requirement formulation, without taking time into account, allows us to get an upper bound for global profit. The smallest

instance studied with this timeless model requires approximately 225,000 variables including 1600 integer variables and

approximately 7200 constraints.

Proposition 2.1. The Service Network Design with asset balance requirement formulation without taking time into account (that

is without time index for variables) is an upper bound for our problem.

Proof. The resolution of the time-free formulation provides only commodities to carry, vehicle types and related networks

and does not produce the transportation planning. There is no guarantee that a complete planning of vehicles, which fulfills

all these choices, exists. However, it does choose which commodities and transportation network would give the largest

profit. Thus the profit of any feasible solution cannot be greater than the profit of the found solution, even though it may

be infeasible. h

We calculated the upper bound by solving an aggregate form of Service Network Design formulation without time indexwith ILOG Cplex. The aggregated formulation reduces the smallest instance above to approximately 68,000 variables includ-

ing 1600 integer variables and approximately 3200 constraints.

The computed upper bound will be used in our computational study to measure the quality of our approach based on the

decomposition of the problem.

3. Solving algorithms based on a decomposition of the problem

A decomposition of the problem into three main steps is done to ease its resolution. We devise two solving algorithms

depending on the order of the steps according whether to allow transhipment. An overview of our approach follows which

will be further detailed in subsequent sections. This decomposition not only allows us to take into account all the constraints

of our problem but also to display solutions in progressive stages. Indeed, the solution contains two sets: the vehicle plan-

nings and the satisfied demand routes.

Step 1: Choose which transportation services will be open (i.e. construct the service network).

We proceed first by constructing the network of the most profitable vehicle type (the cheapest per transport unit per dis-

tance unit), then we iterate the network construction for the remaining vehicle types. The constructed network represents

the physical routes which will be operated. In some sense, this fixes variables yij which correspond to a service ylijt without

considering time dimension t and individual vehicle l. Thus, the fixed cost is completely determined after this step.

To ensure that design-balance constraints are respected to allow vehicle rotation from period to period, we care that the

number of in-coming services is equal to the number of out-going services for each terminal. Thus, the constructed network

is a Eulerian graph and it is well suited for the construction of Eulerian cycles in Step 3.

Two heuristics are proposed to solve this first step. One is based on the direct connection heuristic, the second is based on

a linear relaxation of a simplified MIP.

Step 2: Fill vehicles with commodities (i.e. choose which freight to carry).

From the transportation services network, we maximize the revenue, we determine the satisfied fraction of each demand

zk. Each transportation service a allows us to carry a volume xka for each commodity k. The service a is either the physical

route i;j operated by a vehicle type or the service i;j; t operated by the vehicle l if Step 3 fixing time tand vehicle l is per-

formed before Step 2. We have to take care that each commodity fraction is transported by only one vehicle and that capac-

ities of the vehicles are respected.

Profit is maximized in this step. The choice of which freight to carry is solved as a maximum flow problem on an appro-

priated network.

Step 3: Construct the vehicles plannings (i.e. schedule the different routes).

1 For the sake of simplicity, all cycles considered in this paper are directed.



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We assign each operated transportation service (yij

fixed in Step 1) to one vehicle l and one departure time t. This means

that the set of consecutive transport services ylijt is decided.

Fig. 2. Algorithm without transhipment.

Fig. 3. Algorithm with transhipment.



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In this step we consider the vehicles individually. We construct the vehicle routes and the corresponding schedules,

which we call the vehicle planning. In order to facilitate planning, we start by construction of the Eulerian cycle of each fleet

network (and alternatively the Eulerian cycles of the connected components). On a given cycle, time quotas are easily taken

into account. We use Constraint Programming to solve this step.

The two proposed algorithms differ by the orders of resolution steps as described in Fig. 2 (steps in order 1-3-2) and (steps

in order 1-2-3) (see Fig. 3).

In any case, the last two steps (2 and 3) are not independent. For each step of both algorithms, we detail the points which

must not be overlooked. The first algorithm performs the steps in order 1-3-2: after designing the network it gives indepen-

dent transportation plannings respecting time quotas and finally it fills the plannings with freight. There is no interaction

between the different vehicle types, no transhipment is considered.

In the second algorithm, we swap the order of last two steps. Thus the second decision is the filling of the network with

freight. This has to be done carefully and ensures that the chosen commodities can be carried and delivered in the same plan-

ning horizon. Interactions between vehicles exist: some commodities may be transferred from one vehicle to another imply-

ing that transportation plannings must allow transfer operations. Thus, transhipment complicates construction of vehicle

plannings, and creates some precedences between transportation services. Besides, time quotas still have to be respected.

3.1. Construction of the service network

For the first step of our algorithms, we propose two different methods to design the service network. The first one is based

on an heuristic which searches for profitable cycles with commodities carried directly from their origin to their destination.

In the second method we solve a linear relaxation of a simplified formulation.

3.1.1. Direct trip policy heuristic

We construct the service networks successively for the different vehicle types. Let type be the first vehicle type for which

we construct the service network. In practice we start by constructing the service network of the most profitable type (that is

the vehicle, which, once filled, is the least expensive per transport unit per distance unit). Then, once the service network is

constructed for the vehicles of type type, we remove the commodities it deals with and apply the same procedure to con-

struct the networks for the other vehicle types.

We suppose that any commodity is transported directly from its origin to its destination by a vehicle of type type. We

denote by Ctype the capacity of the vehicles of Ltype. As there is no stop-over for any commodity, we need Wk

Ctype

l mvehicles of

type type to transport the total demand Wk of product k between its origin Ok an its destination Dk. Each of these virtual

transport operations of a fraction of demand k induces a transportation cost (fixed + variable) and brings in the revenue of

the satisfied fraction of the demand k. Thus, each transport operation corresponds to a profit (which is positive or negative).

We consider a graph in which the vertices are terminals, and the description above explains how we construct weightedarcs. We complete this graph with arcs corresponding to empty transports (their cost is only the fixed transportation cost)

such that there are exactly the same number maxk Wk

Ctype

l m of arcs between each pairs of vertices. These extra arcs will allow

repositioning of vehicles with empty load.

In this graph, some arcs are weighted by a positive value (which corresponds to profit) and others by a negative value

(loss). As transportation plannings have to be copied from one planning horizon to the other and due to vehicle repositioning

constraints, the chosen arcs (which will compose the network) have to form one or more cycles. Moreover, to maximize prof-

it we choose cycles with global positive weight. Now, to solve this graph problem optimally, we modeled it as a maximum

cost flow problem and solved it using the minimum mean cycle cancellingalgorithm defined by Ahuja et al. (1989). This algo-

rithm is polynomial On2m3 logn in size of the constructed graph (n vertices and m arcs), but the number of arcs of this

graph (depending on maxkWk

Ctype

l m) is pseudo-polynomial in size of the freight transportation problem.

Proposition 3.1. In the simplified problem without time quotas (constraints (5)), the profit of the solution of the direct trip policy

provides a lower bound for the solutions of both of our algorithms based on the direct trip policy heuristic for Step 1 (construction of

the network).

As time quotas are not too restrictive in our problem, this bound is a practical lower bound for our problem including them.

Proof. We first show that in the simplified problem without time quotas, the solution of the direct trip policy can be eas-

ily transformed in a solution of our problem: The solution of the direct trip policy is a set of profitable cycles for vehicles

and direct trips for chosen commodities. We can assign vehicles to any cycle with the following algorithm: take a cycle,

cut it in enough pieces of length less than the length of the period and assign one vehicle to each of these pieces. The

obtained vehicle routes respect design-balance constraints because they are built on cycles or pieces of cycles. Departure

and arrival times of vehicles at terminals can be easily included as no synchronization between vehicles is necessary. Thus

with the direct trips for chosen commodities, the obtained vehicle routes provide a solution of our problem without time

quotas.

To end the proof, we now state that if the service network is built with direct trip policy heuristic then both of our

algorithms improve profit of our problem (if time quotas are ignored): the constructed network permits for both algorithms

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to carry directly from origin to destination all commodities chosen with direct trip policy. However, if more profitable

commodities can be taken instead, our algorithms which allow stopovers or transhipments in Step 2 will favour this

commodities. Thus, profit will increase. h

3.1.2. Linear relaxation heuristic

In order to measure the quality of our heuristic, we compare our result with the network constructed by a linear relax-

ation of the aggregated Service Network Design formulation with the design-balance constraints without time index, as for

the upper bound (Proposition 2.1).To obtain the integer solution, after each resolution step of the relaxation of the MIP we round out variables with a frac-

tional part less than 0.25 or greater than 0.75. If no variable can be rounded, we round the variable which is nearest to an

integer.

3.2. Choice of the freight to carry

Once the service network is constructed by the direct trip policy or linear relaxation, we have to choose which commod-

ities will be carried. We have two possibilities: either we allow transhipment or not. We look at both to measure how tran-

shipment increases profit.

3.2.1. Without transhipment

Without transhipment (steps in order 1-3-2) the choice of freight is done at the last step and corresponds to optimally

loading the planned vehicles. At the first step, a profitable network is constructed considering only direct trips. However,more profits could be achieved by considering stopovers and possibilities of loading and unloading at each terminal. This

is done by solving a maximum cost flow problem. From the planning of one vehicle l, we build one particular instance of

the maximum cost flow problem (see Fig. 5) in order to fill this vehicle with demands.

The maximum cost flow will be computed in a graph constructed as follows. The vertices of the graph are terminals (A, B,

etc.) visited by the vehicle, and each arc links the origin to the destination of one demand. For demand k, the capacity of the

arc corresponds to the demand size, wk; and the cost of the arc is the unit profit of the demand (Pk UlOkDk for a direct trip).

We also consider empty loads represented by the dashed arcs 1; 0. We add two vertices: one source s and one hole t. In

order to respect capacity of vehicle l, the arcs linking the source and the hole to the other vertices, have a capacity equal to

the capacity Cl of vehicle l and a zero cost. Fig. 4 gives an example with four demands, the maximum flow problem for one

vehicle l is represented in Fig. 5.

As, by construction, no interaction exists between the different plannings, we can fill them at the same time by solving a

maximum flow problem: we append the different flow problems and introduce a super-source and a super-hole. This max-

imum cost flow problem contains some specific arc capacity constraints in order not to exceed certain demands. In theexample, these specific constraints must be set for the demand from A to B: The flows over the two arcs AB must not globally

exceed the demand size w1.

Thus this step is done in polynomial time in size of the Service Network built in Step 1.

Fig. 4. Demands.

Fig. 5. Maximum flow problem for one vehicle l.



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3.2.2. With transhipment: time incoherences may occur

To make transhipment possible, we swap the order of the last two steps (steps in order 1-2-3). Once the service network

is designed we determine which commodities to transport on each arc. This is done by solving a multicommodity flow prob-lem in a multidigraph, where the objective is to maximize profit and each arc of the graph corresponds to one transportation

service. This problem is polynomial in size of inputs, it can be solved either by using Linear Programming (LP), or by using

approximation algorithm as by Garg and Konemann (1998) and Fleischer (2000).

However, when choosing which freight to carry, no time constraints are taken into account, and some time incoherences

may occur which would interfere with the construction of plannings in the Step 3. Time incoherences have to be identified

and, as a result, some commodity choices may have to be revised. This is managed with constraint and variable generations

into the multicommodity flow formulation. The so modified problem is then solved again to revise the first choices we made

when filling the network with commodities. All these are detailed and illustrated on an example in the following sections.

3.2.2.1. Description of time incoherences. Time is not considered neither when we construct the service networks nor when

we fill service networks with commodities. Thus, let A, B, C and D denote four terminals, the situation represented in Fig. 6

may occur:

The same vehicle has to transport the different represented commodities from their origin to their destination. Obviously,

we cannot transport the commodities 1, 2 and 3 from their origin to their destination by visiting all terminals only once, even

if we use different vehicles.

We can highlight this impossibility by using a precedence graph constructed as follows. Each transportation service can

be seen as a task. If the same commodity is carried on successive services, the first transportation service has to be done first

and so on for the next transportation services. Let us construct the precedence graph of these different commodity tasks, we

denote by TAB the task transportation service from A to B00.

The first commodity implies the following precedences: TDA !1

TAB !1

TBC The second gives precedence: TAB !

2TBC !

2TCD

For the third we have: TCD !3

TDA The other commodities do not imply any precedence, because they are transported directly.

Thus we need to schedule the task of precedence graph G represented in Fig. 7.

This graph has a cycle: it is impossible to schedule these tasks by respecting all time precedences.

3.2.2.2. How to find and eliminate time incoherences. To know which commodities are responsible for the appearance of a cy-

cle and on which transportation arcs, we construct a directed graph G0 as follows. The vertices of this graph are the trans-

portation demands k, the arc k; k0 means that demand k is in conflict with demand k

0. In the precedence graph G, we

consider that demand k is in conflict with demand k0

if and only if k0

is loaded after k and k is unloaded before k0

at the

end of an operation belonging to G.

Proposition 3.2. The graph G0 has a cycle if and only if the precedence graph G has a cycle.

We can identify the demands which involve the cycles in G0 and, resulting from our proposal, these demands are also

responsible for the cycle in the precedence graph G. We detect the cycles in G and G0 with Floyds algorithm. These demands

are called the incriminated demands. We add new constraints to the multicommodity flow problem so as to discard at least

one precedence between transportation tasks, and to eliminate at least one cycle in the precedence graph. These constraints

impact the transportation arcs on which at least two incriminated demands are transported.

Fig. 6. Time incoherences.

Fig. 7. Precedence graph on transportation tasks.



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Let fk1; k2; . . . ; km1 k1g be a cycle in G0. The last transportation arcs in the graph G shared by the demands ki and ki1

(involved in this cycle) are denoted as aki . As we need to discard the cycle we have to reroute at least one commodity; in

other words, we need to choose which commodities will keep the same route and which will not. Thus we introduce a binary

variable ukij which determines whether or not commodity k will be carried on transportation arc i;j. We need these variables

only for the demands involved in the cycle. The added constraints are as follows:

xki

aki6 wki u

ki

aki; 8i 2 1 . . . m; 13

xki1

aki6 wki1 u

ki1

aki; 8i 2 1 . . . m; 14

Xi21...m

ukiaki u

ki

1

aki

6 2m 1; 15

ukiaki

2 f0;1g; 8i 2 1 . . . m; 16

uki1

aki2 f0;1g; 8i 2 1 . . . m: 17

Constraints (13) and (14) guarantee that commodity ki is carried on transportation arc aki only ifthis arc is kept for it(that is

ifukiaki

1). The constraint (15) enforces that at least one conflict between 2 demands belongingto cycle in G0 will be cancelled.

Thus after constraint generation the incriminated demands will not all follow the same paths as before, and at least the cycle

fk1; k2; . . . ; km1g in G0 will be eliminated. So the incriminated demands will no longer produce the same precedences in G.

3.2.2.3. Example. The three commodities 13 are responsible for the cycle in the precedence graph of Fig. 8. These commod-

ities follow, respectively, the paths: DAABBC, ABBCCD and CDDA. In order to break the cycle in the precedence graph, at

least one of the these commodities has to follow another path or even not be carried. The graph of incriminated demands G0

is represented in Fig. 8, and the critical transportation arcs are specified over the arcs of G0. The constraints we added are

detailed in Fig. 9.

3.2.2.4. How to more efficiently eliminate time incoherences. The number of introduced variables is limited by the number of

commodities jKj and the size of the network, but these are binary variables. Furthermore, we have no guarantee that the

number of constraints (15) we introduce is polynomial: in some iterations no new variables may be introduced but a new

constraint (15) may be added. Thus, the elimination of time incoherences may take a long time.

So as to speed up this step, we try to discard more than one cycle before refilling arcs with commodities. We used an

aggregate formulation which allows the reduction of the number of introduced binary variables and, consequently, the num-

ber of constraints (15). Thus, discarding a cycle in this aggregate version allows us to eliminate the development of a lot of

similar cycles at the same time. Moreover, so as to eliminate more time incoherences at each step, we try to find several

cycles in the graph G0 and in the precedence graph at each step. Even with these improvements, we have no time guarantee.

We will see in the empirical results (Section 4) how often elimination of time incoherences prevents us from getting a solu-

tion in a reasonable time.

Fig. 8. Graph of incriminated demands.

Fig. 9. Constraints to avoid cycles on incriminated demands.

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3.3. Construction of the vehicle plannings

In this step we care about time dimension and time constraints in the designed network and we assign vehicles to the

services. The plannings can be seen as a scheduling problem in which each transportation service is a task whose duration

is its transportation time. Vehicles are resources which execute these tasks with respect to following constraints.

Eulerian cycle: Starting with the service network which is Eulerian by construction, an Eulerian cycle is con-

structed by Eulers algorithm. Vehicles are assigned to this cycle or to part of it. This is to

respect the design-balance constraints including time dimension.

Geographical constraints: Two consecutive tasks on a resource must verify that the arrival terminal of the first task is thesame as the departure terminal of the second task.

Time quotas: The transportation tasks have to be assigned evenly between the different vehicles over one

period. Any used vehicle l should be used at least Quotalmin but less than Quotalmax time units.

This problem can be seen as a graph decomposition problem. This problem is detailed in Tey-

paz and Rapine (2008). Similar problems have been proved NP-Complete by Dor and Tarsi

(1997).

Transhipment operations: If transhipment is considered, including time becomes increasingly difficult because of prece-

dences between transportation arcs. When the same commodity is transported on successive

arcs, these arcs have to be scheduled in the same order.

To construct the vehicle plannings, we choose Constraint Programming which is a classical and efficient search technique

to solve difficult scheduling problems. In particular, we are interested in the intelligent backtrack ability among others (con-

straints propagation in a node, instantiation order of variables, etc.) offered by Constraint Programming Kumar, 1992.Considering an Eulerian cycle, a simple algorithm can solve the problem if the chosen cycle allows it, i.e. when the cycle

can be decomposed into paths with lengths between Quotalmin and Quotalmax. Otherwise, if the chosen cycle does not bring a

solution, we may branch to another alternative (backtracking) for the construction of this cycle. Moreover, Constraint Pro-

gramming makes it easier to integrate some secondary constraints for our problem such as: transit times, regular departure

times and time windows. We use GNU-Prolog compiler which is a constraint solver on finite domains described in Diaz

(2007).

In practice, at each step of the search, we schedule one task either on the current resource or on a new one. The domain of

time variables of each task (indicating the beginning of the tasks) are updated according to the graph of task precedence. At

each node of the search tree, a task is scheduled. We introduce different scheduling rules. They are applied following a given

order which favours to find quickly a feasible solution. In case of failure, we backtrack by changing the applied rule or by

using a new Eulerian cycle.

Table 1 gives some mathematical details of the rules for the simple case without transhipment. We denote by T1 the last

transportation task assigned to a vehicle, and by T2 the next transportation task to assign. The clause EulerT1;

T2 states thattask T2 is after task T1 in the current Eulerian cycle. AT1 DT2 means that the arrival of transportation task T1 is equal to

the departure of transportation task T2. Clause Quotalmin guarantees that the current vehicle has reached the minimum uti-

lization time.

The order in which we apply the different rules is essential to find a solution. Nevertheless, it may happen that no feasible

solution respecting time quotas and delivering all commodities in the same period exists or can be computed in reasonable

time. In this case, no solution will be provided. To take transhipment into account, we introduce new scheduling rules which

consider precedences between transportation tasks. This makes this scheduling problem even harder to solve as it will be

presented in computational results.

4. Tests and computational results

We present results obtained using an homogeneous fleet, i.e. a fleet of identical vehicles. Some remarks on the heteroge-

neous fleet will also be given.

Table 1

Constraint Programming scheduling rules without transhipment.

Rules Clauses Decision

EulerT1 ; T2 AT1 DT2 Quotalmin

Rule no 1 True T2 after T1 on same vehicle

Rule no 2 False True T2 after T1 on same vehicle

Rule no 3 True True T2 on a new vehicle

Rule no 4 False True True T2 on a new vehicleRule no 5 False False True T2 on a new vehicle



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To compare the profit values of the algorithms, we compute the relative gain (RG) defined as the difference between the

profit value of a solution and the lower bound, divided by the difference between the upper and the lower bound. We ex-

pressed this relative gain as a percentage. A negative relative gain means that the profit is worse than the lower bound, a

relative gain of 100% means that the profit is equal to the upper bound. The last column gives the relative gain of the best

solution of the aggregated formulation computed by Cplex in 20 min. We also specify the percentage of satisfied demands in

the columns labelled Sat. To summarize results from the 12 data profiles, average and standard deviation r of the relative

gains are given at the end of the table.

We observe that most of the algorithms improve profits. Only the method without transhipment and based on a network

construction using linear relaxation is often worse than the lower bound. Indeed, linear relaxation constructs networks

privileging transhipment; this prevents construction of independent plannings and less profitable commodities are carried.

Table 4

Relative gain and satisfied demands (all in percentage).

Instances Direct trip policy Linear relaxation Cplex

NoT T NoT T RG

RG Sat RG Sat RG Sat RG Sat

Small demands ( vehicle size

1 hub 12.63 68.19 31.96 69.11 2.41 68.70 31.46 71.92 81.31

8.75% 5.83 64.74 25.56 64.53 11.73 64.40 19.23 65.76 81.67

3 hubs 18.59 72.86 49.01 76.06 6.59 73.75 57.66 79.32 90.21

16% 10.36 69.95 40.95

a

72.34 7.78 73.16 50. 81

b

83.03 90.36Demands % vehicle size

1 hub 16.68 86.99 42.33 88.57 7.49 84.94 43.25 87.70 79.93

8.75% 7.29 80.74 37.92 81.68 17.91 81.44 33.86c 82. 66 79.48

3 hubs 16.81 86.41 51.15 89.27 3.32 84.72 68.26 88.29 89.64

16% 10.46 83.06 47.82 84.63 12.87 82.43 63.36d 86. 60 89. 84

Big demands ) vehicle size

1 hub 16.38 87.26 41.63 88.23 4.26 86.68 45.87 87.55 83.10

8.75% 8.55 86.87 36.84 86.95 14.55 87.02 39.38 87.75 82.92

3 hubs 21.88 87.78 53.15 88.74 7.83 87.51 70.21 88.74 90.30

16% 13.59 88.21 48.91 86.95 7.82 88.46 61.98 89.66 90.65

Average 13.25 42.31 5.76 48.61 85.78

r 5.78 9.41 11.95 17.26 5.00

a Average of six results.b Only one result.c Average of seven results.d Average of five results.

Table 3

Computation time (in seconds) and resolution failures.

Instances Direct trip policy Linear relaxation

NoT T NoT T

Time Timea Fail2 Fail3 Time Timea Fail2 Fail3

Small demands ( vehicle size

1 hub 0.32 1.25 0 2 61.83 62.67 0 2

8.75% 0.33 22.33 0 1 58.34 127.36 1 2

3 hubs 0.44 40.26 0 3 71.35 151.20 0 3

16% 0.43 65.18 4 1 87.71 130.25b 9 0

Demands % vehicle size

1 hub 0.46 0.88 0 1 64.10 62.73 0 3

8.75% 0.44 10.94 0 0 62.79 132.59 3 0

3 hubs 0.60 29.44 0 1 73.59 83.97 0 1

16% 0.62 209.30 0 1 78.58 218.83 5 1

Big demands ) vehicle size

1 hub 0.78 1.95 0 2 62.09 74.33 0 0

8.75% 0.73 2.15 0 2 66.20 66.79 0 1

3 hubs 1.47 2.83 0 2 84.27 177.80 0 0

16% 1.48 7.95 0 1 81.06 116.12 0 0

Total 4 17 18 13

a If resolution fails, its computation time is not used to compute the average.b Only one result.



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Transhipment allows us to increase profits significantly. With direct trip policy, the average of the relative gain increases byaround 30%. With linear relaxation, the increase of the average is even improved.

We note that when the data instances contain hubs, the relative gain are higher from 5% to 10%. Indeed, efficient networks

are easier to construct and higher rates of satisfied demands are achieved.

When demands are randomly generated, of small or medium size, the use of direct trip policy is more attractive: The rel-

ative gains are good, and furthermore, this method provides a solution for nearly all instances. Also, this method is less influ-

enced by the profile of the instance (the standard deviation r is lower). For big size demands, the method with transhipment

using linear relaxation is the best according to the relative gains. This is explained by the fact that linear relaxation con-

structs networks privileging transhipments and that big demands favour transhipments of full loads, thus dependencies be-

tween vehicles are limited.

4.3.2. Vehicle utilization

For different methods, we compute the average utilization time of the vehicles (Table 5). These times are similar for thedifferent profiles of instances.

Without transhipment, vehicle utilization times are close to Quotamax 325, thus vehicles are almost optimally used. The

methods with transhipment are less efficient on vehicle utilization times which are close to Quotamin 130. In spite of re-

spect of time quotas and a better profit, methods with transhipment are unsatisfactory: they use 1.5 times more vehicles

than if no transhipment is used.

4.4. Heterogeneous fleet

We tested the same instances with an heterogeneous fleet containing the same type of vehicles as homogeneous fleet and

two other types with smaller capacities (100 and 60). These vehicles are competitive compared to the bigger vehicles espe-

cially for small demands. The network constructed by using linear relaxation is nearly the same as for the homogeneous fleet,

but as the linear model is bigger, it takes a longer time to obtain the network.

The lower bounds and the results for small size demands are improved but the benefits are quite small. Indeed, for med-ium and big size demands, using an heterogeneous fleet gives worse results!

The networks constructed for small vehicles in Step 1 are often not profitable after Step 2. Because, the commodities are

taken by larger vehicles in Step 2 when filling decision is made. Thus, the profitable cycles from Step 1 for the small vehicles

becomes unprofitable with the empty trips. We did not further investigate heterogeneous fleet with our decomposition

methods.

5. Conclusion and perspectives

In this paper, we formulate a large-scale freight transportation problem involving both the fleet assignment and Service

Network Design. The objective is to maximize the profit of a carrier. We developed a 3-step decomposition heuristic to solve

this problem. We used graph algorithms (flow problems), Linear and Mixed Integer Programming and Constraint Program-

ming techniques. The main contributions of the paper to the literature are to address real-life size instances, and to build step

by step a comprehensible solution whose representation is complex.

Major findings of the numerical tests for the two solving methods with or without transhipment are as follows. The one

without transhipment using direct trip policy for network construction provides the best trade-off between solution quality

and computational time: Computation is very fast (a few seconds), vehicles are well used and the profits are good. Methods

with transhipment get better profits, but vehicles are under-used. Furthermore, 20 min of resolution time is not always en-

ough to find a solution.

For operators, the contributions of our decomposition algorithms are: a fast estimation of the potential profit (after the

first step), a high flexibility w.r.t. new time constraints (to include in the constraint programming), and very fast computa-

tion of high quality solutions. Actually, our algorithm provides a decision support which helps our industrial partner to eval-

uate an ideal fleet for maximizing its profit. Furthermore, it is enough flexible concerning the transportation costs and

transhipment. For scholars, the specificity of our method is a step wise construction of a solution w.r.t. its complex structure.

Each step involves specific Operations Research techniques to provide efficiently high quality solutions in a low computation

time even for large-scale instances. These size instances had never been tackled including neither asset management issues,

nor price collecting aspects before.

Table 5

Vehicle utilization.

Instances Direct trip policy Linear relaxation

NoT T NoT T

Average utilization time 291.3 188 292.7 187.5



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The perspectives are twofold. First, our decomposition scheme may be enhanced by improving each of its step. Second, we

did not consider the fleet size and heterogeneous fleet which are natural in an industrial context. These constraints should

already be considered within the network construction and cannot be easily integrated in our resolution scheme. Further-

more, there is no backtrack on the choices we made in this constructive method. A formulation of our problem based on

routes (in this case, a route is a complete planning of a vehicle) combined with Service Network Design formulation may

be more suitable for this problem. These aspects are still under investigation.

Acknowledgments

This research has been supported by the following grants and contracts: BQR INPG Optimisation du transport de fret par

lutilisation de plateformes logistiques; Cluster de Recherche GOSPI, Gestion et Organisation des Systmes de Production et

de lInnovation, Rgion Rhnes-Alpes; ANR-05-CIGC-006-04 CHOC, Challenge en Optimisation Combinatoire.

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