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The stochastic maintenance location routing allocation problem

for rolling stock

D.D. T¨onissen

School of Business and Economics, Vrije universiteit Amsterdam, Amsterdam, The Netherlands

d.d.tonissen@vu.nl

J.J. Arts

Luxembourg Centre for Logistics and Supply Chain Management, University of Luxembourg, Luxembourg, Luxembourg;

joachim.arts@uni.lu

June 8, 2020

Abstract

We study a problem where –given a railway network and diﬀerent ﬂeets– we have to locate

maintenance locations and allocate the ﬂeets to these locations. The allocation of ﬂeets to the

maintenance locations complicates the maintenance location routing problem. For each candidate

location diﬀerent facility sizes can be chosen and for each size there is an associated annual facility

cost that captures the economies of scale in facility size. Because of the strategic nature of facility

location, these facilities should be able to handle changes such as adjustments to the line plan and

the introduction of new rolling stock types. We capture these changes by discrete scenarios and we

formulate this two-stage stochastic problem as a mixed integer problem. Furthermore, we perform a

case study with the Netherlands Railways that provides novel managerial insights by showing that

the number of opened maintenance facilities highly depends on the allocation restrictions.

Keywords: facility location, maintenance routing, rolling stock, stochastic programming

1 Introduction

The maintenance of rolling stock is important to keep railway operations functioning. Without frequent

maintenance, many trains would break down leading to the cancellation of trains or even dangerous

situations. As a consequence, trains are maintained regularly and when such maintenance is required,

the train has to reach a suitable maintenance facility. The accessibility of such a maintenance facility

depends on the railway infrastructure and the line plan. A line plan consists of a set of train lines, where

each line is a path in the railway network that is operated with a certain frequency by one rolling stock

1

type. In this paper we study the problem of locating such maintenance facilities while also determining

their sizes and the allocation of the rolling stock types to the maintenance facilities.

Facility location decisions are long term obligations, while the line and ﬂeet plan are updated regularly

to meet changing passengers demand. As a consequence, any sensible facility location plan must work

well under a diverse range of line and ﬂeet plan scenarios. This includes changes in how lines run, up

and down-scaling of service frequencies on any given line, the rolling stock types assigned to the lines,

and the introduction of new rolling stock types.

To deal with these changes, we formulate the problem as a stochastic maintenance location routing

allocation problem for rolling stock (SMLRAP). In the SMLRAP, we seek the optimal locations and

sizes of maintenance facilities for rolling stock and the best allocation of the rolling stock types to the

maintenance facilities. The objective consists of minimizing the annual depreciation cost of the facilities

and the average annual transportation cost. The annual cost of a facility depends on its location and

size. The size of a facility must be chosen from a discrete set that model the economies in scales: a

facility which is twice as large costs less than twice as much. As a consequence, it is possible to open

a few large facilities to proﬁt from economies of scale or to open multiple smaller facilities to limit the

transportation cost.

The maintenance location routing problem for rolling stock was introduced by T¨onissen et al. (2019),

and extended by T¨onissen and Arts (2018). T¨onissen and Arts (2018) show that the best strategy is

to reduce transportation cost by locating many small facilities instead of opening a few large facilities

to proﬁt from economies of scale. However, their paper does not include allocation restrictions of the

rolling stock types to the maintenance facilities. These allocation restrictions are based on the fact that

each rolling stock type requires speciﬁc equipment and resources. Furthermore, a mechanic has to work

suﬃciently many hours on a speciﬁc rolling stock type to retain the qualiﬁcation for type maintenance.

Consequently, there is a restriction on the number of maintenance facilities that each rolling stock can

be maintained at. This paper shows that this restriction is very important by a case study with the

Netherlands Railways (NS). In addition, these allocation restrictions determine whether the best solution

has many small facilities or a few large facilities to beneﬁt from economies of scale.

The paper starts with a literature review. In Section 3, we formulate the SMLRAP as a mixed integer

problem. In Section 4, we explain how we generate our instances and give computational results. Finally,

we perform a case study for the NS and provide managerial insights in Section 5.

2 Literature review

For the traditional facility location literature, we refer to the reviews of Daskin (1995) and ReVelle

and Eiselt (2005). However, the traditional deterministic facility location literature studies a theoretical

problem that often cannot be used to solve real life problems. Most real life facility location problems

arise in the combination with other supply chain decisions and contain a great deal of uncertainty.

2

The combination of facility location with uncertainty is reviewed in the paper of Snyder (2006) and

the combination with supply chain decisions in Melo et al. (2009). Since those reviews many papers that

include uncertainty and/or supply chain decisions with facility location have been written e.g., Penuel

et al. (2010); ´

Alvarez-Miranda et al. (2015); Kınay et al. (2018). Furthermore, Govindan et al. (2017)

provide a review about the closely related problem of supply chain network design under uncertainty.

In this paper we look at facility location in combination with facility sizes that model economies

of scale, maintenance routing, and allocation restrictions. Many papers (e.g., Melo et al. (2006); Julka

et al. (2007); Xie et al. (2016)) consider facilities with diﬀerent facility sizes, but most of them focus on

capacity expansion models. Economies of scale in production (Romeijn et al., 2010; Sharkey et al., 2011)

or economies in scale for transportation (Lin et al., 2006; Wu et al., 2015) are studied in the literature.

However, economies of scale in facility size are only brieﬂy mentioned in Melo et al. (2006) and to our

best knowledge only studied in depth by T¨onissen and Arts (2018).

Maintenance routing for rolling stock is studied by Anderegg et al. (2003); Mar´oti and Kroon (2005,

2007) and for aviation by Gopalan and Talluri (1998); Sarac et al. (2006); Liang et al. (2015) and many

others. The combination of maintenance routing and facility location for aviation is studied by Feo and

Bard (1989) and Gopalan (2014), for locomotives by Xie et al. (2016), and for rolling stock by T¨onissen

et al. (2019) and T¨onissen and Arts (2018). The paper of T¨onissen et al. (2019), includes the maintenance

routing for rolling stock in an aggregate way into a facility location model. They model the maintenance

location routing problem as two-stage robust optimization and stochastic programming problems and

provide a Benders decomposition and a scenario addition algorithm to solve the models to optimality.

This problem was extended by T¨onissen and Arts (2018) to include unplanned maintenance, economies

of scale in facility size and recoveries of the facility location decisions. Our paper extends these papers

further by including allocation restrictions of the rolling stock types to the maintenance facilities. These

allocation restrictions are required to eﬃciently apply the model to practice and consequently the results

of this paper have important practical implications.

3 The maintenance location routing model

The two-stage stochastic maintenance location allocation routing problem consist of two stages. The ﬁrst

stage decisions are the locations of the facilities, their sizes and the allocation of rolling stock types to

facilities. The assignment of rolling stock to facilities is constrained because each rolling stock type can

be allocated to at most Kdiﬀerent maintenance facilities. These assignment constraints are imposed

because each facility needs speciﬁc resources to service a rolling stock type and, more importantly,

mechanics needs to retain their certiﬁcation to be allowed to work a speciﬁc rolling stock type. The

certiﬁcation can only be retained when mechanics work a suﬃciently large number of hours on a rolling

stock type.

The second-stage decisions consist of ﬁnding optimal routings of the train units to the maintenance

3

facilities. Planned maintenance typically occurs once every half year up to every month. The transport

from the train lines to the maintenance facility is done by interchanging the destinations of two train

units of the same rolling stock type that are at the same end station. The train units continue on each

other’s train line after such an interchange. Train units that require maintenance are interchanged until

they reach a train line connected to a maintenance facility. Whether such an interchange is possible

depends on the operational rolling stock schedule and the shunting infrastructure of the end stations. In

our two-stage stochastic model, these restrictions are modeled by putting a restriction on the number

of interchanges that can occur annually at any given station. A detailed description and operational

maintenance routing model for the NS that includes these restrictions can be found in Mar´oti and Kroon

(2005, 2007).

Deadheading, which is driving a train without passengers, is used for the remaining trip when a main-

tenance facility cannot be reached via these interchanges. Deadheading is expensive and the deadheading

cost consists of driving (train driver, fuel etc.) and disservice costs because the train is not available for

public transport. Unplanned maintenance occurs when a train unit fails in the ﬁeld. The failed train unit

has to deadhead to the maintenance facility to be repaired. The deadheading of unplanned maintenance

is even more expensive, because it cannot be planned in advance and because the train unit sometimes

has to be towed.

Like T¨onissen and Arts (2018), we formulate the SMLRAP described above as a ﬂow model. This

ﬂow model is based on a directed graph in which the lines and candidate facilities are represented by

nodes, and the interchanges and deadheading possibilities by arcs. In the next section, we explain how

such a graph can be built and how it can be extended to deal with multiple scenarios. In Section 3.2

this graph is used for our mixed integer formulation that provides us with the ﬁrst stage decisions and

the second stage decisions for each scenario.

3.1 Constructing the maintenance routing graph

Given is a physical rail network GP= (NP, EP), consisting of rails EP, all stations NP. Next we are

given a discrete set of scenarios D, in which each scenario deﬁnes a line plan. A line plan consists of a

set of lines Ld,∀d∈D, with for each line, two end stations, the type of rolling stock that operates the

line, and the unplanned and planned maintenance frequency of maintenance visits that originate from

this line. Furthermore, a line plan determines the unplanned and planned deadheading cost for each

line to each facility, and the set of possible interchanges with for each interchange a coordination cost.

Finally, we are given a set of candidate maintenance locations C⊆NP.

Figure 1 shows on the left-hand side an example of a physical rail network graph containing the end

stations and in the middle and on the right-hand side two line plans for two diﬀerent scenarios. There are

two train types in the example shown in Figure 1. The ﬁrst, denoted by a, is a regional train, stopping

at every station, while train type bis an intercity train that skips the small stations. An example of an

interchange in the right-hand side of Figure 1 is line (U, Z, a) to line (Z, V , a), while an interchange from

4

(U, Z, a) to line (Z, Y, b) is not possible because the rolling stock types do not match. Also note that

the number of end stations is diﬀerent between the line plans: station Zis an end station in the right

hand-side of Figure 1, while it is an in-between station of the line (U, X, b) in the line plan in the middle.

U V

W

XY

Z

U V

W

XY

a

a

a

b

b

a

b

b

a

U V

XY

Z

b

a

b

b

a

b

a

Figure 1: The physical rail network on the left and two possible line plans.

The maintenance routing graph is a directed ﬂow graph, GM= (NM, AM) that is constructed by the

following steps:

•For each line l∈Ld∀d∈D, two nodes are made one for the planned maintenance and one for the

unplanned maintenance. The set of planned maintenance nodes for scenario dis denoted by Nd

PL

and for the unplanned maintenance nodes we have Nd

UL. Furthermore, we deﬁne Nd

L=Nd

PL ∪Nd

UL.

•A source Sis made that is connected with a directed arc to each node in Sd∈DNd

L.

•Arcs between the line nodes from Nd

PL ∀d∈Dare created whenever an interchange between these

lines is possible (the lines have a common end station and there is positive interchange capacity at

that end station). The cost of these arcs are the interchange coordination costs. The set of these

interchange arcs is denoted by Ad

I∀d∈D.

•A node is made for every candidate facility and we denote the set of these nodes as NC. Further-

more, each node in NCis connected to the sink Twith an arc.

•An arc to each facility is created for each node n∈Nd

L∀d∈D. The cost of this arc is the

deadheading cost of the line to the facility. The cost of the arc can be 0 when the line is connected

to the facility and deadheading is not necessary to reach that facility. The set of all incoming

facility arcs for scenario dis denoted as Ad

C.

The number of nodes and arcs in the ﬂow graph GMis polynomial in the number of lines, end

stations and scenarios. This can be easily shown as the number of nodes in the ﬂow graph is equal to

Pd∈D|Nd

L|+|NC|+ 2 and the number of arcs is equal to Pd∈D(|Nd

L|+|Nd

L||NC|+|Ad

I|) + |NC|, where

|Ad

I|is bounded by |Nd

PL|(|Nd

PL|−1). Note that the ﬂow from diﬀerent scenarios can never mingle in the

graph GM, and as a result we do not have to distinguish ﬂow of the diﬀerence scenarios. Consequently,

there is a non-overlapping S-Tpath for each scenario d∈Dthat is equivalent to the total route of

interchanges and deadheading for an annual maintenance frequency from a train line to the maintenance

facility.

5

In Figure 2, we demonstrate how to create a maintenance routing ﬂow graph for a small example

with only one scenario. The line plan of that scenario is depicted on the left-hand side. In this example,

we again show the rolling stock type by using letters and we numbered the lines such that the associated

line node can easily be found in the maintenance routing ﬂow graph. We assume that an interchange is

possible between line 0 and 1, and NC={A,B,E}, where A and B are facilities located at end stations A

and B, while E is a candidate facility that is located elsewhere. The ﬁgure on the right-hand side depicts

the ﬂow network with unplanned (U) and planned (P) line nodes.

A B

C

0a

2b

1a

S

2 P

1 P

0 P

0 U

1 U

2 U

A

E

T

B

Figure 2: Left a line planning possibility and right the resulting ﬂow graph (GM= (NM, AM)). The

arcs from and to the source and sink are dotted black, the interchange arcs (Ad

I) solid red and the arcs

to the facilities (Ad

C) are dashed blue.

Note that only interchanges followed by deadheading directly to the maintenance facility are allowed in

the graph GM. Deadheading followed by interchanges can easily be allowed in the graph by creating an arc

from every planned maintenance line node to every other planned maintenance line. That arc represents

the deadheading from one line to another, that can be followed by any combination of interchanges and

deadheading until the maintenance facility is reached. The reason that we exclude these kind of routes is

that they are very expensive because they cause imbalances in the number of train units per line, which

are quite diﬃcult ro resolve in practice.

3.2 The mixed integer programming formulation

The size of a facility is expressed by the number of maintenance visits that it can process annually. The

workload generated by a planned maintenance visit is thus set at 1 and that of an unplanned maintenance

visit as u∈R+. The total annual workload of the entire line plan for the current situation is denoted

6

by M. The sizes of a facility at location n∈NCare denoted by the set Qn. A tuple i∈Qn, consists

of a size qni that represents the annual workload that a facility can handle and the annual facility cost

cni for facility location n. Furthermore, each rolling stock type r∈R, where R is the set containing all

rolling stock types, can be maintained by at most Krdiﬀerent facilities. The ﬁrst-stage decisions are

represented by Yand X.Ycontains the binary decision variables yni ∈ {0,1} ∀n∈NC,i∈Qnthat

is 1 when a facility of size iis opened at location nand 0 otherwise. Xcontains the binary decision

variables xnr ∈ {0,1} ∀n∈NC,r∈Rthat is 1 when rolling stock type ris allocated to facility nand 0

otherwise.

The maintenance frequency for line land scenario dis deﬁned by the parameter md

l, and nd

lis the

node associated with line lfor scenario d. The set of end stations for scenario dis given by Sd, and

gd

s∈R+

0is a restriction on the annual number of interchanges at end station s∈Sdfor scenario d∈D.

The number of annual interchanges for scenario dis restricted by the parameter Gd∈R+

0. The ﬂow

through arc aassociated with the annual maintenance frequency from line l∈Nd

L, in scenario d∈D,

is represented by the second-stage decision variable z(a)∈R+

0. For example, z(1,7) represents the

frequency of interchanges from line 1 to line 7, while we also know to which scenario and rolling stock

type line 1 and 7 belongs.

We deﬁne δd

in(n) and δd

out(n) as the set of ingoing and outgoing arcs of node nfor scenario din graph

GM. In addition, we let the index P and U denote the planned and unplanned maintenance subset,

respectively, and when we use the index rwe only include the subset of arcs that belong to rolling

stock type r. As deﬁned in Section 3.1, Ad

Iis the set of interchange arcs and Ad

C=Sn∈NCδd

in(n), the

set of incoming candidate facilities arcs. Furthermore, we deﬁne Ad

sas the set of arcs representing the

interchanges at end station sfor scenario dand when we drop the index dfor a set, this is shorthand

notation for taking the union of the sets for all scenarios, e.g., δout(n) = Sd∈Dδd

out(n) for any n∈NM.

The cost of arc ais c(a), which is only deﬁned for arcs in the set Sd∈DAd

I∪Ad

C. Finally, the weights

wd∀d∈Ddenote the expected fraction of time that a scenario is used during the life time of the

facilities. We can now formulate the SMLRAP:

(SMLRAP) min X

n∈NC

X

i∈Qn

cniyni +X

d∈D

wdX

a∈Ad

I∪Ad

C

c(a)z(a)

s.t. X

i∈Qn

yni ≤1∀n∈NC,(1)

X

n∈NF

xnr ≤Kr,∀r∈R, (2)

xnr ≤X

i∈Qn

yni ∀n∈NC,∀r∈R, (3)

X

a∈δinPr(n)

z(a) + uX

a∈δinUr(n)

z(a)≤xnr|D|max

i∈Qnqni ∀n∈NC,∀r∈R, (4)

X

a∈δd

inP(n)

z(a) + uX

a∈δd

inU(n)

z(a)≤X

i∈Qn

yniqni ∀d∈D, ∀n∈NC,(5)

7

X

a∈δin(n)

z(a) = X

a∈δout(n)

z(a)∀n∈NM\ {S,T },(6)

z(a) = md

l∀d∈D, ∀l∈Nd

l, a ∈δd

in(nd

l)\AI,(7)

X

a∈Ad

s

z(a)≤gd

s∀d∈D, ∀s∈Sd,(8)

X

a∈Ad

I

z(a)≤Gd∀d∈D, (9)

xnr ∈ {0,1} ∀n∈NC,∀r∈R, (10)

yni ∈ {0,1} ∀n∈NC,∀i∈Qn,(11)

z(a)≥0∀a∈AM.(12)

We minimize the cost of opening the facilities and the expected maintenance routing cost over all

scenarios. Constraints (1) guarantee that each facility can be opened with at most 1 size. Constraints

(2) ensure that rolling stock type rcan be maintained at most at Krfacilities. Constraints (3) guarantee

that we can only allocate rolling stock types to opened facilities and constraints (4) guarantee that rolling

stock type rcan only be maintained at a facility nwhen matching resources are installed. Constraints (5)

restrict the number of annual planned and unplanned maintenance visits that can be assigned to opened

facility (n, i) with size qni. Constraints (6) are the ﬂow conservation constraints, while Constraints (7)

guarantee that every maintenance visit is assigned to a facility. Constraints (8) and (9) are the end

station and budget interchange capacity constraints.

4 Computational results

4.1 Instance generation

We generate the instances based on data gained from the NS. We assume that the candidate locations

are always located at the end stations. We have 59 end stations which all can be used as candidate

locations. When we generate instances with a certain number of candidate facilities, these candidate

facilities are randomly chosen from these 59 end stations. The facility costs are an estimation of the

average annual cost of land, the necessary infrastructure and the maintenance facility itself including

all side buildings. Furthermore, we either decrease or increase the facility cost based on the average

land price of the province that a location is in. To create the cost for the diﬀerent sizes, we multiply

the cost estimation for each location with the factors depicted in Table 1. The factors for the sizes are

estimated with the square root safety staﬃng rule (Halﬁn and Whitt, 1981): When the size of a facility

is increased by a factor x, the needed safety size Cto deal with uncertainty is only increased by a factor

√x. Consequently, the cost increases by a factor x+C√x

1+C< x. The required safety stock for a standard

location of size 1/3 M is estimated to be approximately 0.21.

The network interchange budget Gdis U(0.25M, M )∀d∈D, the unplanned maintenance factor uis

set to 0.25, and Kis uniformly randomly generated between 1 and 4. All scenarios are based on four

8

Size 1/12 M 1/8 M 1/6 M 1/4 M 1/3 M 1/2 M 2/3 M M 4/3 M

Factor 0.29 0.42 0.54 0.77 1.00 1.45 1.90 2.78 3.65

Table 1: Cost increases for locations compared to a facility with size 1/3M.

basic line plans. These basic line plans are: the current situation (2015)1, an estimation of 2018, and

two possibilities for approximately 2025. The future line plans are based on the plan “Beter en Meer”

(Prorail and NS, 2014), a commercial plan made by the NS and Prorail. The purpose of the plan is

to cater to the growing numbers of passengers. These basic line plans contain all the lines (97, 97, 99,

and 100 lines), the rolling stock type serving the line, and an estimate of the number planned yearly

maintenance visits per line. Scenarios are made by picking such a basic line plan, and slightly altering

the planned maintenance frequency and rolling stock types. The altered planned maintenance frequency

for each line of the line plan is generated from a triangular distribution. The planned maintenance

frequency of the basic plan is the mode of this distribution. Furthermore, we assume that the number

of maintenance visits can decrease by 32.5% and increase by 75%, due to uncertainty in the number of

maintenance visits each train unit requires each year and the number of passengers using a certain line.

A maximum of 20% of the rolling stock types of the lines can be swapped with each other. Moreover,

the unplanned maintenance frequency for a line is the same as the planned maintenance frequency, as

they occur approximately equally often for the NS.

4.2 Benchmark experiments

Experiments show that the number of sizes of the facilities or the number of basic scenarios that are used

to create the scenarios have no signiﬁcant inﬂuence on the solution time of the instances. The solution

time is mainly impacted by the number of candidate facilities and to a lesser degree by the number of

scenarios.

For 5, 10, 20, 40, and 59 candidate facilities the average solution time over 10 instances is shown

in Figure 3. Note that throughout this paper we use Lg to represent the binary logarithm (log2). The

number of scenarios is increased by a factor of 2, each time that 8 or more out of 10 instances could be

solved within an hour. Otherwise, the experiments were stopped. With those conditions we can solve

up to 128 (27) scenarios when we have 5 candidate facilities, but only instances with 1 scenario when we

1The data was gathered in 2015.

9

include all 59 candidate facilities.

2021222324252627

2−1

21

23

25

27

29

211

Lg scenarios

Lg time (sec.)

5 Candidate facilities

10 Candidate facilities

20 Candidate facilities

40 Candidate facilities

59 Candidate facilities

Figure 3: Computational time for the MIP for diﬀerent number of scenarios and candidate facilities.

When the additional assignment constraint of a maximum of K facilities is removed (and consequently

also the associated binary decision variables) the instances are more easier to solve (more than one order

of magnitude decrease in solution time). These results are shown in Figure 4 and in this case we can

solve up to 512 (29) scenarios when there are 5 candidate locations and 8 (23) scenarios when there are

59 candidate locations. Consequently, these allocation decisions signiﬁcantly increase the solution time

of the instances.

20212223242526272829

2−3

20

23

26

29

212

Lg scenarios

Lg time (sec.)

5 Candidate facilities

10 Candidate facilities

20 Candidate facilities

40 Candidate facilities

59 Candidate facilities

Figure 4: Behavior of the computational time for the MIP without the K constraint for diﬀerent number

of scenarios at diﬀerent scales.

10

5 Case study: NS

We see from the benchmark instances that instances with many candidate facilities are hard to solve.

However, many of the 59 end stations are not serious candidates for a maintenance facility. When we

limit the list of candidate facilities to the most likely candidates, a list of 12 end stations remains. This

“shortlist” contains end stations that are on strategic locations throughout the country. These strategic

locations are around an urban agglomeration in the Netherlands where most train traﬃc is concentrated

and at the extremities of the rail network. These 12 end stations are used as candidate locations for the

remainder of the case study.

5.1 Scenarios

We start with 10 instances with one scenario, and double the number of scenarios in each next set.

In Table 2 we report the minimum and maximum number of opened facilities, followed by the average

number of opened facilities between parentheses. Furthermore, we denote the average total cost in

millions per year, the average solution time, and the number of instances that could not be solved within

180 minutes. When an instance is not solved within the 180 minutes time limit, we report the best found

integer solution, and report 180 minutes as solution time.

Scenarios opened cost (M/year) time (min.) fails

1 2-5 (3.8) 22.2 0.06 0

2 2-5 (3.4) 21.9 0.23 0

4 2-4 (3.0) 22.6 1.3 0

8 2-4 (2.9) 23.2 5.0 0

16 2-4 (2.7) 23.3 27.9 0

32 2-4 (2.5) 23.4 73.0 2

64 2-4 (2.6) 23.9 152.2 5

128 2-4 (2.9) 24.2 180.0 10

Table 2: Scenario results for the NS instances.

The average number of facilities seems to slightly decrease with the number of scenarios. We expect

that the reason for this is that a solution with only a few large facilities has a more stable performance

for the diﬀerent scenarios than a solution with many small facilities. The trade-oﬀ between solution

stability and solution time seems to be best at 16 scenarios. Consequently, we use 16 scenarios for our

remaining experiments.

11

5.2 The value of the stochastic solution

The value of the stochastic solution (VSS) is a common measures within the stochastic programming

literature (Birge and Louveaux, 1997). The VSS is deﬁned as the diﬀerence between the expectation

of the expected value solution and the optimal objective value of the two-stage stochastic programming

problem. In our problem, where each scenario contains a graph, the best equivalent to the expected

value solution is the ﬁrst-stage solution of solving the SMLRAP with one single scenario, which is the

best estimate of the current situation. We then use this ﬁrst-stage solution as input for the SLMRAP

with multiple scenarios, i.e., we ﬁx the allocation and facility decisions and based on these decisions

an optimal maintenance routing is found for each scenario. We deﬁne the optimal objective value of

this program as the expected value of the current situation (EVCS). Consequently, we now have: VSS

= EVCS - SMLRAP. Because, the VSS depends on the scale of the objective value, we increase the

interpretability by deﬁning the percentage value of the stochastic solution (PVSS): VSS

EVCS ·100%. The

PVSS can be seen as the expected percentage of cost savings of solving the SMLRAP with a suﬃcient

number of scenarios instead of a deterministic model based only on the current situation.

For our experiments we use the instances with 16 scenarios from Section 5.1. However, for these

instances the EVCS cannot be calculated, as it is infeasible for each of the instances. This is not a

surprise as most of our scenarios are growth scenarios and many of the scenarios introduce new rolling

stock types. Consequently, infeasibility is caused by capacity problems and by the new rolling stock

types that are not assigned to any facility. To deal with this, we allow limited recovery to the ﬁrst-stage

solution that is used as input for calculating the EVCS. We allow recovery by upgrading opened facilities

to a higher capacity and we allow the allocation of the new rolling stock types to these facilities. We

deﬁne the cost of upgrading a facility n∈NCfrom capacity ito capacity jas α(cnj −cni), where α≥1.

The αrepresents the potential additional cost of upgrading an existing facilities compared to building a

facility of that size immediately. This value largely dependents on the time that upgrading the facilities

and installing the resources for the new rolling stock types takes. Consequently, when recovery is required

and it is executed poorly, e.g., because it is not incorporated into the planning at all, the potential cost

and damage (e.g., deﬀered maintenance and loss of public image) could be enormous.

The current situations ﬁrst-stage solution opens two facilities with a capacity of 1/8Mand M.

For the 10 instances with the 16 scenarios, these capacities are upgraded to at most 1/3Mand 4/3M,

respectively. We report the average transportation cost (Tcost), average facility cost (Fcost), the average

total cost in millions per year, and the average PVSS, for the SMLRAP solution followed by the EVCS

results for α∈ {1.0,1.5,2.0,3.0}in Table 3

From Table 3 it can be seen that even when we can recover without additional cost (which is very

unlikely) we can already save 9.2%. When we increase αto 3.0 the cost savings increase to 19.8%. Finally,

it can be seen that the transportation cost for these instances are independent of α. Consequently, the

cost savings are 9.2% plus the additional cost of recovery.

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Name Tcost (M/year) Fcost (M/year) cost (M/year) PVSS

SMLRAP 17.8 5.8 23.3 0

EVCS (α= 1.0) 19.3 6.4 25.7 9.2

EVCS (α= 1.5) 19.3 7.3 26.6 12.1

EVCS (α= 2.0) 19.3 8.1 27.4 14.8

EVCS (α= 3.0) 19.3 9.8 29.1 19.8

Table 3: PVSS for the NS instances.

5.3 Varying the number of allocated facilities

We compare the current situation (1 maintenance facilities for every rolling stock type except the VIRM

which can be allocated to two), with a situation where the rolling stock types with the largest number

of train units (VIRM, ICM, SLT, and SNG) can also be allocated to two maintenance facilities (G2).

Furthermore, we compare it to the situations where all rolling stock types can be maintained at respec-

tively 2, 3, and an inﬁnite number of maintenance facilities. Note that in our analysis, we only increase

Krin constraints (2) and that consequently any additional cost for increasing the capabilities of the

maintenance facilities is outside scope and not taken into consideration.

We generate for each case 10 instances and report the minimum and maximum number of opened fa-

cilities, followed by the average number of opened facilities between parentheses in Table 4. Furthermore,

we denote the average transportation cost (Tcost), the average facility cost (Fcost), and the average total

cost in millions per year.

Name opened Tcost (M/year) Fcost (M/year) cost (M/year)

K= 1 2-3 (2.4) 18.8 6.1 25.0

Current 2-4 (2.7) 17.5 5.8 23.3

G2 3-6 (4.6) 16.0 5.3 21.1

K= 2 5-6 (5.1) 14.5 5.4 19.7

K= 3 5-8 (6.6) 12.5 5.1 17.5

∞8-12 (10.2) 10.5 5.2 15.6

Table 4: Sensitivity analysis K for the NS instances.

In Table 4 we see that the number of opened facilities is highly dependent on K. When Kis large,

many small facilities are opened to limit the transportation cost. However, when Kis small, we are

forced to open only a few large facilities with economies of scale. It is interesting to note that even

though the larger facilities have economies of scale, the total facility cost is cheaper when we build many

small facilities. The reason for this is that the few facilities with economies of scales are built in the

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busy, central and expensive areas of the Netherlands, while in the case of many small facilities some of

them are built in the less expensive areas of the Netherlands.

Going from the current situation to a situation where all larger rolling stock types are maintained by

two facilities, the number of facilities would increase almost twofold and the cost reduction is 9.4%. A

comparison with the cases where K= 3 and K=∞, gives cost savings of respectively 24.9% and 33.0%.

5.4 Varying unplanned maintenance visits

In this section we increase the number of unplanned maintenance visits with the following factors: 0,

0.25, 0.5, 0.75, 1, 1.33, 2, and 4. The number of planned maintenance visits remains unchanged. Again

we generate 10 instances for each situation and we report our results in Table 5.

unplanned factor opened cost (M/year)

0 2-3 (2.2) 7.8

0.25 2-3 (2.5) 11.8

0.5 2-4 (2.7) 15.8

0.75 2-4 (2.8) 19.5

1 2-4 (2.7) 23.3

1.33 2-4 (2.8) 28.4

2 2-4 (3.0) 38.5

4 2-5 (3.6) 68.6

Table 5: Sensitivity analysis unplanned maintenance visits for the NS instances.

We see that large cost savings (excluding the cost of the actual maintenance) can be made by decreas-

ing the number of unplanned maintenance visits. Furthermore, the number of unplanned maintenance

visits aﬀects the number of facilities. When there is no unplanned maintenance, the number of mainte-

nance facilities is approximately two, while it is almost four when the annual maintenance frequency is

multiplied by 4.

5.5 Rolling stock types

The more rolling stock types there are, the less likely it becomes that a train unit can reach a mainte-

nance facility by interchanges. Consequently, by decreasing the number of rolling stock types, the total

deadheading cost can be decreased and fewer facilities may be required. Currently, the NS has 5 intercity

types and 6 regional rolling stock types that needs to be maintained. We look at the eﬀect of decreasing

this number to 3 intercity types and 3 regional types, and 1 intercity type and 1 regional type. We

compare these results with the current Kand with the case that every rolling stock type is allowed to

be maintained by two facilities. Once more, we generate 10 instances for each case and we report our

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results in Table 6.

Cost factor opened cost (M/year)

5 - 6 (current) 2-4 (2.7) 23.3

5 - 6 (K=2) 5-6 (5.1) 19.7

3 - 3 (current) 2-3 (2.2) 23.4

3 - 3 (K=2) 3-5 (4.3) 19.9

1 - 1 (current) 2-3 (2.1) 21.15

1 - 1 (K=2) 2-4 (2.9) 19.0

Table 6: Sensitivity analysis number of rolling stock types for the NS instances.

When there are more rolling stock types, the diﬀerence between the number of facilities is larger

between the diﬀerent K’s. The diﬀerences in cost are respectively 15.4%, 15.0%, and 10.2%. The total

diﬀerence from the current situation to a situation with only 1 rolling stock type for both regional and

intercity transport is 9.2% and 18.5% for the current situation and K= 2, respectively.

6 Conclusion

We added the allocation of rolling stock types and allocation restrictions to the two-stage stochastic

maintenance location routing problem. This is an important extension because in practice there are

restrictions to which rolling stock types can be maintained by which maintenance facilities. These

restrictions are caused by the fact that each rolling stock type requires special equipment and matching

resources. Sensitivity analysis shows that these allocation restrictions are indeed important as they

highly inﬂuence the solution. The number of facilities decreases from an average of 10.2 facilities to only

2.4 facilities. Increasing the number of facilities where rolling stock can be maintained yields cost savings

up to 33%. Consequently, a trade-oﬀ should be made between the cost savings caused by relaxing the

allocation restrictions and the required cost to increase the capabilities of the facilities.

The number of rolling stock types only has a small inﬂuence on the number of facilities that are

opened and it only decreases the cost by approximately 9.2%. Furthermore, decreasing the unplanned

maintenance frequency by a factor 2 does not inﬂuence the number of opened facilities. However, it does

decrease the costs by 32.3%, without taking the additional cost savings in the maintenance cost into

consideration. Consequently, decreasing the unplanned maintenance frequency should be a priority.

Finally, our research shows that it is important to include multiple scenarios when locating main-

tenance facilities. The optimal allocation and facility decisions for the current situation at the NS are

infeasible for our 10 test instances. Consequently, a feasible solution can only be achieved by adding

(expensive) recovery actions. In our experiments we estimate that the cost savings of including these

scenarios is 9.2% plus the additional cost needed for the recovery actions. (These costs are diﬃcult to

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estimate but may be quite sizeable.)

Acknowledgment

We thank the NS for all the help they have given us while writing this paper.

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