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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.
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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). 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,dD, 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 CNP.
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 lLddD, 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 SdDNd
L.
Arcs between the line nodes from Nd
PL dDare 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
IdD.
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 nNd
LdD. 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
PdD|Nd
L|+|NC|+ 2 and the number of arcs is equal to PdD(|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 dDthat 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 uR+. 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 nNCare denoted by the set Qn. A tuple iQn, 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 rR, 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} ∀nNC,iQnthat
is 1 when a facility of size iis opened at location nand 0 otherwise. Xcontains the binary decision
variables xnr ∈ {0,1} ∀nNC,rRthat 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
sR+
0is a restriction on the annual number of interchanges at end station sSdfor scenario dD.
The number of annual interchanges for scenario dis restricted by the parameter GdR+
0. The flow
through arc aassociated with the annual maintenance frequency from line lNd
L, in scenario dD,
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=SnNCδd
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) = SdDδd
out(n) for any nNM.
The cost of arc ais c(a), which is only defined for arcs in the set SdDAd
IAd
C. Finally, the weights
wddDdenote 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
nNC
X
iQn
cniyni +X
dD
wdX
aAd
IAd
C
c(a)z(a)
s.t. X
iQn
yni 1nNC,(1)
X
nNF
xnr Kr,rR, (2)
xnr X
iQn
yni nNC,rR, (3)
X
aδinPr(n)
z(a) + uX
aδinUr(n)
z(a)xnr|D|max
iQnqni nNC,rR, (4)
X
aδd
inP(n)
z(a) + uX
aδd
inU(n)
z(a)X
iQn
yniqni dD, nNC,(5)
7
X
aδin(n)
z(a) = X
aδout(n)
z(a)nNM\ {S,T },(6)
z(a) = md
ldD, lNd
l, a δd
in(nd
l)\AI,(7)
X
aAd
s
z(a)gd
sdD, sSd,(8)
X
aAd
I
z(a)GddD, (9)
xnr ∈ {0,1} ∀nNC,rR, (10)
yni ∈ {0,1} ∀nNC,iQn,(11)
z(a)0aAM.(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+Cx
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 )dD, 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
21
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
23
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.
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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 nNCfrom capacity ito capacity jas α(cnj cni), where α1.
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.
12
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
13
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
14
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
15
estimate but may be quite sizeable.)
Acknowledgment
We thank the NS for all the help they have given us while writing this paper.
References
E. ´
Alvarez-Miranda, E. Fern´andez, and I. Ljubi´c. The recoverable robust facility location problem.
Transportation Research Part B: Methodological, 79:93–120, 2015.
L. Anderegg, S. Eidenbenz, M. Gantenbein, C. Stamm, D. S. Taylor, B. Weber, and P. Widmayer. Train
routing algorithms: Concepts, design choices, and practical considerations. In Proceedings of the Fifth
Workshop on Algorithm Engineering and Experiments, volume 111, pages 106–118. SIAM, 2003.
J. R. Birge and F. V. Louveaux. Introduction to Stochastic Programming. Springer, 1997.
M. S. Daskin. Network and Discrete Location: Models, Algorithms, and Applications. Wiley, New York,
1995.
T. A. Feo and J. F. Bard. Flight scheduling and maintenance base planning. Management Science, 35
(12):1415–1432, 1989.
R. Gopalan. The aircraft maintenance base location problem. European Journal of Operational Research,
236(2):634–642, 2014.
R. Gopalan and K. T. Talluri. The aircraft maintenance routing problem. Operations Research, 46(2):
260–271, 1998.
K. Govindan, M. Fattahi, and E. Keyvanshokooh. Supply chain network design under uncertainty: A
comprehensive review and future research directions. European Journal of Operational Research, 263
(1):108–141, 2017.
S. Halfin and W. Whitt. Heavy-traffic limits for queues with many exponential servers. Operations
Research, 29(3):567–588, 1981.
N. Julka, T. Baines, B. Tjahjono, P. Lendermann, and V. Vitanov. A review of multi-factor capacity
expansion models for manufacturing plants: Searching for a holistic decision aid. International Journal
of Production Economics, 106(2):607–621, 2007.
¨
O. B. Kınay, F. Saldanha-da Gama, and B. Y. Kara. On multi-criteria chance-constrained capacitated
single-source discrete facility location problems. Omega, 2018.
16
Z. Liang, Y. Feng, X. Zhang, T. Wu, and W. A. Chaovalitwongse. Robust weekly aircraft maintenance
routing problem and the extension to the tail assignment problem. Transportation Research Part B:
Methodological, 78:238–259, 2015.
J.-R. Lin, L. K. Nozick, and M. A. Turnquist. Strategic design of distribution systems with economies
of scale in transportation. Annals of Operations Research, 144(1):161–180, 2006.
G. Mar´oti and L. Kroon. Maintenance routing for train units: The transition model. Transportation
Science, 39(4):518–525, 2005.
G. Mar´oti and L. Kroon. Maintenance routing for train units: The interchange model. Computers &
Operations Research, 34(4):1121–1140, 2007.
M. T. Melo, S. Nickel, and F. Saldanha-da-Gama. Dynamic multi-commodity capacitated facility loca-
tion: a mathematical modeling framework for strategic supply chain planning. Computers & Operations
Research, 33(1):181–208, 2006.
M. T. Melo, S. Nickel, and F. Saldanha-da-Gama. Facility location and supply chain management–a
review. European Journal of Operational Research, 196(2):401–412, 2009.
J. Penuel, J. C. Smith, and Y. Yuan. An integer decomposition algorithm for solving a two-stage facility
location problem with second-stage activation costs. Naval Research Logistics (NRL), 57(5):391–402,
2010.
Prorail and NS. Beter en Meer, 2014. URL https://www.rijksoverheid.nl/documenten/rapporten/
2014/03/28/bijlage-1e-operationele-uitwerking- prorail-en-ns-beter-en-meer.
C. S. ReVelle and H. A. Eiselt. Location analysis: A synthesis and survey. European Journal of Opera-
tional Research, 165(1):1–19, 2005.
H. E. Romeijn, T. C. Sharkey, Z.-J. M. Shen, and J. Zhang. Integrating facility location and production
planning decisions. Networks: An International Journal, 55(2):78–89, 2010.
A. Sarac, R. Batta, and C. M. Rump. A branch-and-price approach for operational aircraft maintenance
routing. European Journal of Operational Research, 175(3):1850–1869, 2006.
T. C. Sharkey, J. Geunes, E. H. Romeijn, and Z.-J. M. Shen. Exact algorithms for integrated facility
location and production planning problems. Naval Research Logistics (NRL), 58(5):419–436, 2011.
L. V. Snyder. Facility location under uncertainty: a review. IIE Transactions, 38(7):547–564, 2006.
D. D. T¨onissen and J. J. Arts. Economies of scale in recoverable robust maintenance location routing
for rolling stock. Transportation Research Part B: Methodological, 117:360–377, 2018.
D. D. T¨onissen, J. J. Arts, and Z.-J. Shen. Maintenance location routing for rolling stock under line and
fleet planning uncertainty. Transportation Science, 53(5):1252–1270, 2019.
17
T. Wu, H. Shen, and C. Zhu. A multi-period location model with transportation economies-of-scale and
perishable inventory. International Journal of Production Economics, 169:343–349, 2015.
S. Xie, X. Chen, Z. Wang, Y. Ouyang, K. Somani, and J. Huang. Integrated planning for multiple types
of locomotive work facilities under location, routing, and inventory considerations. Interfaces, 46(5):
391–408, 2016.
18
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