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126 NAW 5/9 nr. 2 June 2008 Flows in railway optimization Alexander Schrijver

Alexander SchrijverCWI and University of Amsterdam

Kruislaan 413

1098 SJ Amsterdam, Netherlands

lex@cwi.nl

Research

Flows in railway optimization

The Netherlands has an intricate network of railway tracks to connect its many closely situatedcities. Recently, the railroad timetable has been completely updated. Unlike the yearly manualadjustments of the past, this update was done using mathematical techniques. For the Dutchpublic, this was a rare opportunity to come into contact with mathematics. The researchthat resulted in the new timetable won a team of CWI consisting, besides Lex Schrijver, ofGábor Maróti and Adri Steenbeek the prestiguous Franz Edelman Award for Achievement inOperations Research and the Management Sciences.

In December 2006, Nederlandse Spoorwegen(NS, Dutch Railways) introduced the Spoor-boekje 2007 (Timetable 2007). It included anew structure of the train timetable and newplans for train circulation and crew schedul-ing. Several new connections were madeand train lengths were better matched to thenumber of passengers, while on the otherhand train times were scheduled less tight-ly, a number of transfers were cancelled andsome direct train connections were cut intoparts in order to reduce the propagation ofdelays.

Mathematicians were involved in design-ing the algorithms that created the new sched-ules. Until 2007, new timetables were madeby ‘manually’ adapting the timetable intro-duced in 1975. The increase in passengernumbers and the addition of extra infrastruc-ture (new lines and forks, four tracks, andfly-overs) was met mainly by shifting traintimes and inserting new trains between ex-isting trains. The need for a completely newtimetable arose since hardly any trains couldbe added to the existing schedule whereneeded. The schedulers at NS felt they need-ed algorithms to better exploit the capacity ofthe railway network, and that is why NS asked

mathematicians.While the Netherlands is not large, it is one

of the most densely populated countries ofthe world. The Dutch railway network is corre-spondingly rather dense, with many short tra-jectories, run by frequent trains, having sever-al transfer connections on the way. Moreover,space to extend infrastructure is limited in theNetherlands.

These characteristics mean that railwayoptimization in the Netherlands is faced withrather specific problems compared to mostother countries and no adequate off-the-shelfsoftware was therefore available. The algo-rithms for timetabling and train circulationwere made at the Center for Mathematics andComputer Science (CWI) in Amsterdam, forcrew scheduling at the University of Padova,and for routing trains through stations at theErasmus University in Rotterdam.

As might have been expected, the changesraised much public discussion. In an editorialcommentary, the Dutch nationwide newspa-per NRC Handelsblad characterized the newtimetable as “the only form of higher math-ematics that arouses furious emotions in thecountry”. In a reaction, the CEO of NS claimedthat the new timetable was the best possible

given the existing infrastructure.Journalists phoned to ask me if indeed

mathematically no better schedule was pos-sible. My answer amounted to the fact thatoptimality depends on the boundary condi-tions, that is, in this case, the input of thealgorithm: which trains and connections doyou want to have, how often, where do trainsstop, etc. This is not a question of math-ematics but a question of company policy,and not only passenger comfort plays a rolehere but also other objectives and constraintslike crew scheduling, profit, punctuality andinfrastructure. Given these constraints andthese choices made by the company, the al-gorithm searches for an optimum timetable.

Economically, the new timetable appearsto be successful. The yearly profit of NS is pro-jected to increase by 70 million euros due tothe new schedules. On trajectories with thelargest timetable improvements, the numberof passengers has increased by 10–15%. Theeffect of this on societal economics and wel-fare has been estimated tentatively at hun-dreds of millions of euros. Moreover, despitemore trains, the punctuality has increasedand the economical effect of this is of the or-der of tens of millions of euros.

As mentioned, Timetable 2007 means notonly the introduction of a new timetable butalso of new systems for rolling stock circu-lation, to increase seat availability for pas-sengers, for crew scheduling, to improvetrain personnel rosters, and for routing trainsthrough stations. At CWI we made algo-rithms for timetabling and for rolling stock

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circulation. Although the timetabling prob-lem (in particular finding the cyclic, hourlypattern) is also interesting from a networkpoint of view, in this article we just focus onrolling stock circulation, also because of itshistorical interest.

Transportation and circulation problemsbelong to the classical problems in operationsresearch and are motivated by application torailways. Flow techniques are basic to it andwe first describe two results that are of histor-ical interest. After that, we get back to railwaycirculation at NS.

Cargo transportation in the Soviet UnionThe earliest attributions to the mathematicalstudy of transportation problems are usual-ly to around 1940. The transportation prob-lem was formulated by Hitchcock [4] and acycle criterion for optimality was consideredaround the same time independently by Kan-torovich in the Soviet Union and Koopmans inthe USA. For political reasons, they publishedtheir results only later ([5,6]). When Kan-torovich found his method, publishing abouteconomics in the Soviet Union was very risky,as it was a politicized issue.

Koopmans, a Dutch mathematical econo-mist who fled to the USA at the beginning ofthe Second World War, found his methodswhen appointed at the Combined ShippingAdjustment Board, a British-American agencythat routed merchant ships during the SecondWorld War, as they had to sail in convoys un-der military protection. In 1975, Kantorovichand Koopmans received the Nobel Prize inEconomics for their work on transportation.

Less known is that earlier, in 1930, A.N.Tolstoı [7] found methods similar to those ofKantorovich and Koopmans. His article calledMethods of finding the minimal total kilo-metrage in cargo-transportation planning inspace was published in a book on transporta-tion planning issued by the National Commis-sariat of Transportation of the Soviet Union.

Tolstoı presented a number of approach-es for the transportation problem, includingthe now well-known idea that an optimum so-lution does not have any negative-cost cyclein its residual graph. This residual graph isobtained from the network by adding, to anyarc on which the flow is positive, an arc in thereverse direction, with cost equal to the nega-tive of the cost of the forward arc. Here ‘cost’can be true cost, length or anything similar.

Tolstoı seems to be the first to ob-serve that this cycle condition is neces-sary for optimality. Moreover, he assumed,but did not explicitly state or prove, the

fact that checking the cycle condition is alsosufficient for optimality.

Tolstoı illuminated his approach with ap-plications to the transportation of salt, ce-ment and other cargo between sources anddestinations along the railway network of theSoviet Union. In the paper, he explains thesolution of a concrete cargo transportationproblem along the Soviet railway network. Ithas 10 sources and 68 destinations, and 155links between sources and destinations, andis therefore quite large-scale for that time.

Tolstoı ‘verifies’ the solution by consider-ing a number of cycles in the network and heconcludes that his solution is optimum:

“Thus, by use of successive applications ofthe method of differences, followed by a veri-fication of the results by the circle dependen-cy, we managed to compose the transporta-tion plan which results in the minimum totalkilometrage.”

Checking Tolstoı’s problem with modernlinear programming tools shows that his so-lution is indeed optimum.

Max-flow min-cutThe Soviet rail system also aroused the in-terest of the Americans and again it inspiredfundamental research in optimization.

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128 NAW 5/9 nr. 2 June 2008 Flows in railway optimization Alexander Schrijver

Figure 1 Figure from Tolstoı [7] to illustrate a negative cycle

In their basic paper Maximal Flow through aNetwork (published in 1954), Ford and Fulker-son [1] mention that the maximum flow prob-lem was formulated to them by T.E. Harris asfollows:

“Consider a rail network connecting two citiesby way of a number of intermediate cities,where each link of the network has a numberassigned to it representing its capacity. As-suming a steady state condition, find a maxi-mal flow from one given city to the other.”

It inspired Ford and Fulkerson to their fa-mous Max-Flow Min-Cut Theorem: The maxi-mum amount of flow that can be sent alonga network from a set of sources to a set ofdestinations, subject to a given capacity up-per bound, is equal to the minimum capacityof the cuts of the network that separate allsources from all destinations.

In their 1962 book Flows in Networks, Fordand Fulkerson [2] give a more precise refer-ence to the origin of the problem:

“It was posed to the authors in the spring of

1955 by T. E. Harris, who, in conjunction withGeneral F. S. Ross (Ret.), had formulated asimplified model of railway traffic flow, andpinpointed this particular problem as the cen-tral one suggested by the model [11].”

Ford-Fulkerson’s reference 11 is a secret re-port by Harris and Ross [3] entitled Funda-mentals of a Method for Evaluating Rail NetCapacities, dated 24 October 1955 and writ-ten for the US Air Force. At our request, thePentagon downgraded it to ‘unclassified’ on21 May 1999.

In fact, the Harris-Ross report solves arelatively large-scale maximum flow problemcoming from the railway network in the West-ern Soviet Union and Eastern Europe (‘satel-lite countries’). And the interest of Harris andRoss was not to find a maximum flow butrather a minimum cut (‘interdiction’) of theSoviet railway system. (Recall that the reportwas written for the Air Force.) We quote:

“Air power is an effective means of interdict-ing an enemy’s rail system, and such usage isa logical and important mission for this Arm.

As in many military operations, however, thesuccess of interdiction depends largely onhow complete, accurate, and timely is thecommander’s information, particularly con-cerning the effect of his interdiction-programefforts on the enemy’s capability to move menand supplies. This information should beavailable at the time the results are beingachieved.

The present paper describes the funda-mentals of a method intended to help thespecialist who is engaged in estimating rail-way capabilities, so that he might more read-ily accomplish this purpose and thus assistthe commander and his staff with greater ef-ficiency than is possible at present.”

The Harris-Ross report stresses that spe-cialists remain needed to make up the mod-el (which is always a good tactic to get newmethods accepted):

“The ability to estimate with relative accuracythe capacity of single railway lines is large-ly an art. Specialists in this field have noauthoritative text (insofar as the authors areinformed) to guide their efforts, and very fewindividuals have either the experience or tal-ent for this type of work. The authors assumethat this job will continue to be done by thespecialist.”‘

Whereas experts are needed to set up themodel, to solve it is routine (when having the‘work sheets’, which were added to the re-port).

The Harris-Ross report describes an appli-cation to the Soviet and East European rail-ways. For the data it refers to several se-cret reports of the Central Intelligence Agency(CIA) on sections of the Soviet and East Euro-pean railway networks. After the aggregationof railway divisions to vertices, the networkhas 44 vertices and 105 (undirected) edges.

The report applies flow techniques to ob-tain a maximum flow from sources in the So-viet Union to destinations in East European‘satellite’ countries (Poland, Czechoslovakia,Austria and East Germany) but the main objec-tive was to find the corresponding minimumcut separating the sources from the destina-tions. In the report, the minimum cut is indi-cated as ‘The bottleneck’ (see Figure 4).

While Tolstoı and Harris-Ross had thesame railway network as object, their objec-tives were dual.

Figure 2 A ‘koploper’

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Alexander Schrijver Flows in railway optimization NAW 5/9 nr. 2 June 2008 129

Figure 3 Timetable Amsterdam-Vlissingen and Vlissingen-Amsterdam

Rolling stock circulation in the NetherlandsWe will finally describe a more recent (andmore peaceful) application of flow methodsto railways, as used by Nederlandse Spoor-wegen for Timetable 2007.

NS runs an hourly train service on its routeAmsterdam-Rotterdam-Roosendaal-Vlissingenand vice versa, with the timetable shownabove.

The trains have more stops but for our pur-poses only those given in the table are of inter-est since at the stations given train sectionscan be coupled or separated. For each of thestages of any scheduled train, NS has esti-mated the number of passengers, as given inthe table on the next page (all data concernsweekdays and 2nd class seats).

The problem to be solved is:

What is the minimum amount of train stocknecessary to perform this train service in sucha way that at each stage there are enoughseats?

In order to answer this question, oneshould know a number of further character-istics and constraints. In a first version of theproblem considered, the train stock consist-ed of one type of two-way train units (‘koplop-ers’), each consisting of three carriages. Eachunit has 163 seats.

Each unit has at both ends an engineer’scabin and units can be coupled together up toa certain maximum length (often 15 carriages,meaning in this case 5 train units).

The train length can be changed, by cou-pling or decoupling units, at the terminal sta-tions of the line, that is at Amsterdam andVlissingen and en route at the intermediatestations Rotterdam and Roosendaal. Anytrain unit decoupled from a train arriving atplace p at time t can be linked up to anyother train departing from p at any time laterthan t (the Amsterdam-Vlissingen schedule is

such that in practice this gives enough timeto make the necessary switchings).

A last condition is that for each placep ∈ {Amsterdam, Rotterdam, Roosendaal,Vlissingen}, the number of train units stay-ing overnight at p should be constant duringthe week (but may vary for different places).This requirement is made to facilitate survey-ing the stock and to equalize at any placethe load of overnight cleaning and mainte-nance throughout the week. It is not re-quired that the same train unit, after a nightin Roosendaal, for example, should return toRoosendaal at the end of the day. Only thenumber of units is of importance.

Given these problem data and characteris-

Figure 4 From Harris and Ross [3]: Schematic diagram of the railway network of the Western Soviet Union and East Euro-pean countries, with a maximum flow of value 163,000 tons from Russia to Eastern Europe and a cut of capacity 163,000 tonsindicated as ‘The bottleneck’

tics, one may ask for the minimum number oftrain units that should be available to performthe daily cycle of train rides required.

It is assumed that if there is sufficient stockfor Monday till Friday then this should alsobe enough for the weekend services since atthe weekend a few early trains are cancelledand on the remaining trains there is a smallerexpected number of passengers. Moreover, itis not taken into consideration that stock canbe exchanged during the day with other linesof the network. In practice this will happenbut initially this possibility is ignored.

A network modelIf only one type of railway stock is used, clas-

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Figure 5 Number of required seats

sical minimum-cost flow techniques (root-ed in the work of Tolstoı) can be applied.To this end, a directed graph G = (V,A)

is constructed as follows. For each placep ∈ {Amsterdam, Rotterdam, Roosendaal,Vlissingen} and for each time t at which anytrain leaves or arrives at p, we make a vertex(p, t). So the vertices of G correspond to all198 time entries in the timetable (Figure 3).

Figure 6 The graph G. All arcs are oriented clockwise.

For any stage of any train ride, leaving placep at time t and arriving at place q at timet′, we make a directed arc from (p, t) to(q, t′). For instance, there is an arc from(Roosendaal,7.43) to (Vlissingen,8.38).

Moreover, for any placep and any two suc-cessive times t, t′ at which any train leaves orarrives at p, we make an arc from (p, t) to(p, t′). Thus in our example there will be arcsfor example from (Rotterdam, 8.01) to (Rot-terdam, 8.32) and from (Rotterdam, 8.32) to(Rotterdam, 8.35).

Finally, for each place p there will be anarc from (p, t) to (p, t′), where t is the lasttime of the day at which any train leaves orarrives at p and where t′ is the first time ofthe day at which any train leaves or arrives atp. So there is an arc from (Roosendaal,23.54)

to (Roosendaal,5.29).We can now describe any possible routing

of train stock as a functionf : A −→ Z+, wherefor any arc a, f (a) denotes the following. If

a corresponds to a ride stage, then f (a) isthe number of units deployed for that stage.So if a is the arc from (Roosendaal, 7.43)to (Vlissingen, 8.38), then f (a) = 4 meansthat 4 coupled train units will run this trainstage. If a corresponds to an arc from (p, t)to (p, t′), then f (a) is equal to the numberof units present at place p in the time periodt–t′ (possibly overnight).

First of all, this function is a circulation.That is, at any vertex v ofG one should have:

∑a∈δ+(v)

f (a) =∑

a∈δ−(v)

f (a), (1)

the flow conservation law. Here δ+(v) de-notes the set of arcs of G that are enteringvertex v and δ−(v) denotes the set of arcs ofG that are leaving v.

Moreover, in order to satisfy the demandand capacity constraints, f should satisfy thefollowing condition for each arca correspond-ing to a stage:f (a) ≤ 5 and 163f (a) ≥ d(a).

Here, d(a) is the ‘demand’ for that stage,that is, the lower bound on the number ofseats given in Figure 5 (and 163 is the numberof seats of a Type 3 ‘koploper’).

To find the total number of train units usedby a given circulation f , one may add up theflow values on the four ‘overnight arcs’. So ifwe wish to minimize the total number of unitsdeployed, we minimize

∑a∈A◦ f (a).

Here A◦ denotes the set of overnight arcs.So |A◦| = 4 in the Amsterdam-Vlissingen ex-ample.

It is easy to see that this fully models theproblem. Hence determining the minimumnumber of train units amounts to solving aminimum-cost circulation problem, where thecost function is quite trivial: we have cost(a) =

1 if a is an overnight arc and cost(a) = 0 forall other arcs.

Having this model, standard minimum-cost flow algorithms give an optimum rollingstock circulation in a fraction of a second. Itturns out that for the circulation describedabove, 22 train units are needed.

It is quite direct to modify and extend themodel so as to contain several other prob-lems. Instead of minimizing the number oftrain units one can minimize the amount ofcarriage-kilometres that should be made ev-ery day, or any linear combination of bothquantities.

In addition, one can put an upper boundon the number of units that can be stored atany of the stations.

Instead of considering one line only, onecan more generally consider networks of linesthat share the same stock of railway materi-al, including trains that are scheduled to besplit or combined. Nederlandse Spoorwegenhas hourly trains from Amsterdam, The Hagueand Rotterdam to Enschede, Leeuwarden andGroningen that are combined into one train ona common trajectory. This network is called‘The North-East’.

If only one type of unit is employed for thatnetwork, each unit having the same capacity,the problem can be solved quickly even forlarge networks.

Mixing several types of train unitsThe problem becomes harder if there are sev-eral types of train units of different capaci-ty that can be deployed for the train serviceand for that NS asked for the help of math-ematicians. Clearly, if for each scheduledtrain we would prescribe the type of unit thatshould be deployed, the problem could bedecomposed into separate problems of thetype above. But if we do not make such aprescription and if different types can be cou-pled together to form a train of mixed com-position, we should extend the model to a‘multi-commodity circulation’ model.

Let us restrict ourselves to the caseAmsterdam-Vlissingen again, where now wecan deploy two types of two-way train unitsthat can be coupled together. The two typesare type 3, each unit of which consists of 3 car-riages, and type 4, each unit of which consistsof 4 carriages. Indeed, NS has ‘koplopers’ ofboth lengths. Types 3 and 4 have 163 and 218seats, respectively.

Again, the demands of the train stages are

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given in Figure 5. The maximum number ofcarriages that can be in any train is again 15.This means that if a train consists ofx units oftype 3 andy units of type 4 then 3x+4y ≤ 15

should hold.It is quite direct to extend the model above

to the present case. Again we consider thedirected graph G = (V,A) as above. At eacharca, let f (a) be the number of units of type 3on the stage corresponding to a and let g(a)

similarly represent type 4. So bothf : A −→ Z+ and g : A −→ Z+ are circulations,that is, each satisfies the flow circulation law(1). For each stage a, the capacity constraintis now 3f (a) + 4g(a) ≤ 15 and the demandconstraint is 163f (a)+218g(a) ≥ d(a), whereagain d(a) denotes the number of requiredseats on stage a (Figure 5).

Let cost3 and cost4 represent the cost ofpurchasing one unit of type 3 and one unitof type 4, respectively. Although train unitsof type 4 are more expensive than those oftype 3, they are cheaper per carriage; that is,cost3 < cost4 < 4

3 cost3. This is due to thefact that engineer’s cabins are relatively ex-pensive.

Then we must minimize∑a ∈A◦

(cost3f (a) + cost4g(a)

).

However, there is an important further con-straint that makes the problem much harder:it is required that at any of the four stationsgiven (Amsterdam, Rotterdam, Roosendaaland Vlissingen) one may either couple unitsto or decouple units from a train but not bothsimultaneously. Moreover, one may couplefresh units only to the front of the train anddecouple laid off units only from the rear. Soif one must decouple a unit of type 3 from thetrain, it should be at the rear. Similar rulesapply at the terminal stations.

This makes the problem more combinato-rial, as the order of the different units in atrain does matter. This does not fit directlyin the circulation model described above and

requires an extension.To this end, one describes a train composi-

tion on a stagea by a vector z(a) in RCa . HereCa is the set of compositions that are allowedfor stage a. This takes the number of seatsrequired for stage a into account and also themaximum train length. So for a certain stagea, one might have

Ca = {3333,33333,3334,3343,3433,

4333,3344,3434,4334,3443,4343,

4433,444,4443,4434,4344,3444}

Here 3433 (for instance) means that the trainconsists of four units, of types 3, 4, 3 and3, respectively, seen from the front of thetrain. Then z(a)c is equal to 1 for precise-ly one c ∈ Ca (namely, the composition crunning stage a) and z(a)c = 0 for all otherc ∈ Ca. This can be described by integer lin-ear inequalities, and the values of f (a) andg(a) are linear functions of z(a). So we havean integer linear programming model.

A first attempt to include the coupling con-ditions is to add linear constraints onz(a) andz(a′), where a and a′ are consecutive stagesof a train ride (like Amsterdam-Rotterdam andRotterdam-Roosendaal of train 2127) so as toexclude transitions that are forbidden by therule that units should be added only at thefront of a train or removed only at the rear ofthe train and not both at the same stop.

Figure 7 The North-East

This however turned out to be computational-ly infeasible. For those who understand inte-ger linear programming, the linear relaxationof this problem is too loose to obtain goodbounds in a branch-and-bound approach.

It took us a long time to realize that byadding even more variables, the problem canbe solved relatively fast. For every allowedtransition, say composition c ∈ Ca on stageais allowed to be followed by composition c′ ∈Ca′ in the subsequent stage a′, we introducea variablewa,c,a′,c′ ≥ 0 and require for everypair of two consecutive stages a and a′:

z(a)c =∑

c′∈Ca′wa,c,a′,c′ for all c ∈ Ca and

z(a′)c′ =∑c∈Ca

wa,c,a′,c′ for all c′ ∈ Ca′ .

This describes all constraints and, important-ly, one does not need to require the new vari-ables wa,c,a′,c′ to be integers. Therefore, theextra variables do not add to the complex-ity (the branching tree) but rather serve asoil to make the integer programming machin-ery work. With this, the problem turns outto be solvable with standard integer linearprogramming software in a few minutes. Thenumber of extra variables is huge but this justhelps you to find a solution within a few min-utes.

It turns out that for the Amsterdam-Vlissingen problem, one needs 7 units of type3 and 12 units of type 4. Comparing this so-lution with the solution for one type only, thepossibility of having two types gives both adecrease in the total number of train unitsand in the total number of carriages needed.The method can be extended to include com-binations and splits of trains at intermediatestations and thus also applies to the largernetwork ‘The North-East’, where similar sav-ings have been obtained and more passen-gers can find a seat. k

References1 L. R. Ford, D. R. Fulkerson, Maximal Flow through

a Network, Research Memorandum RM-1400,The RAND Corporation, Santa Monica, Califor-nia, 1954.

2 L. R. Ford, Jr., D. R. Fulkerson, Flows in Networks,Princeton University Press, Princeton, New Jer-sey, 1962.

3 T. E. Harris, F. S. Ross, Fundamentals of aMethod for Evaluating Rail Net Capacities, Re-search Memorandum RM-1573, The RAND Cor-poration, Santa Monica, California, 1955.

4 F. L. Hitchcock, ‘The distribution of a productfrom several sources to numerous localities’,Journal of Mathematics and Physics 20 (1941)224–230.

5 L. V. Kantorovich, M. K. Gavurin, ‘The applica-tion of mathematical methods to freight flowanalysis’ (in Russian), in: Collection of Prob-lems of Raising the Efficiency of TransportPerformance, Akademiia Nauk SSSR, Moscow-Leningrad, 1949, pp. 110–138.

6 Tj. C. Koopmans, ‘Optimum utilization of thetransportation system’, Econometrica 17 (Sup-plement) (1949) 136–146.

7 A. N. Tolstoı, ‘Methods of finding the minimaltotal kilometrage in cargo-transportation plan-ning in space’ (in Russian), in: TransportationPlanning, Volume I, TransPress of the NationalCommissariat of Transportation, Moscow, 1930,pp. 23–55.