affil0affil0affiliationtext: Delft University of Technology, Delft, The
Netherlands

# The Maintenance Location Choice Problem for Railway Rolling Stock

Jordi Zomer  Nikola Bešinović  Mathijs M. de Weerdt  Rob M.P. Goverde

(October 2020)

###### Abstract

Due to increasing railway use, the capacity at railway yards and maintenance
locations is becoming limiting to accommodate existing rolling stock. To
reduce capacity issues at maintenance locations during nighttime, railway
undertakings consider performing more daytime maintenance, but the choice at
which locations personnel needs to be stationed for daytime maintenance is not
straightforward. Among other things, it depends on the planned rolling stock
circulation and the maintenance activities that need to be performed. This
paper presents the Maintenance Location Choice Problem (MLCP) and provides a
Mixed Integer Linear Programming model for this problem. The model
demonstrates that for a representative rolling stock circulation from the
Dutch railways a substantial amount of maintenance activities can be performed
during daytime. Also, it is shown that the location choice delivered by the
model is robust under various time horizons and rolling stock circulations.
Moreover, the running time for optimizing the model is considered acceptable
for planning purposes.

##  1 Introduction

In many countries, the demand for rail transport increasing. For instance in
The Netherlands, the total number of passenger kilometers increased with over
30% in the last 20 years. To accommodate increasing travel demand, the
utilization of the available rolling stock units may increase and more rolling
stock units may be added to the network.

In order for rail transport to function properly, the rolling stock that
operates on the railway network needs to receive maintenance on a regular
basis. Rolling stock includes all units that move over railway tracks, such as
locomotives, passenger wagons and freight wagons. Rolling stock units are
fixed compositions that consist of various carriages. The aim of maintenance
is to ensure that the rolling stock remains available, safe and comfortable
for passengers [Dinmohammadi et al., 2016]. Maintenance activities can be
divided into two categories: regular maintenance, corresponding to maintenance
activities with higher frequencies (every 1 to 14 days) and shorter duration
(1-3 hours), and heavy maintenance, corresponding to maintenance types with
lower frequencies (every several months or less) and longer duration (up to
several days) [Andrés et al., 2015]. Since regular maintenance and heavy
maintenance have such a different nature, the way they are planned may also
differ: regular maintenance may be performed whenever a rolling stock unit has
a planned standstill; heavy maintenance, however, often requires a rolling
stock unit to be taken completely out of regular service. The current work
focuses on regular maintenance.

In general, the maximum interval between consecutive maintenance activities is
governed by strict rules that are imposed by railway authorities. Maintenance
activities are carried out at so-called maintenance locations, which are
railway yards with maintenance facilities, spread over the network. For a
maintenance location to be operational, it is necessary that personnel is
stationed at a maintenance location. The number of personnel stationed at each
location is the railway undertaking’s decision and determines, together with
the maintenance location design, the availability of a maintenance location.
An operator can choose to open a facility at certain moments of the day or
night by stationing personnel at this location. In particular, a distinction
can be made between daytime operations (which means that a location is opened
during the day) and nighttime operations (which means that a location is
opened during the night). This distinction is clearly visible in The
Netherlands where maintenance is usually carried out during nighttime because
the rolling stock is mostly in use during daytime.

Due to increasing use of the railway network, the capacity of maintenance
locations during nighttime is under pressure and reaching its limits. As a
result, the Dutch Railways (Nederlandse Spoorwegen, NS) are considering
performing more maintenance activities during daytime. This raises the
question at which existing maintenance locations teams need to be stationed to
perform daytime maintenance, referred to as the maintenance location choice.
These maintenance location choice decisions can be made several months before
operations, when the rolling stock circulation, which describes all planned
rolling stock movements for a given period, for the next period is usually
fixed. The challenge is to choose maintenance locations such that more
maintenance activities can be performed during daytime without modifying the
rolling stock circulation.

In planning, commonly three types can be distinguished being strategical
(long-term, typically multiple years before operations), tactical (medium-
term, from week to one year before operations) and operational (short-term,
within several days before operations). When a plan has to be adjusted during
operations due to certain disturbances or disruptions, then it is referred to
as (real-time) traffic management. In the problem under consideration, it is
assumed that the rolling stock circulation is already fixed, and hence it
concerns a tactical planning problem. Currently, the choice regarding which
locations to open for daytime maintenance is made at a moment in time when the
rolling stock circulation has already been determined, justifying the
assumption of a fixed rolling stock circulation.

This research introduces the Maintenance Location Choice Problem (MLCP), which
formulates an optimal maintenance location choice to facilitate timely regular
maintenance of all rolling stock, without changing their planned rolling stock
circulation. The aim of the model is to reduce the amount of work that needs
to be performed during nighttime. The model performance is demonstrated on
realistic instances provided by the Dutch railways (NS).

The main contributions of the current paper are the following:

  * •

It defines a new mathematical problem formulation for finding locations for
daytime maintenance to reduce nighttime activities without altering the
planned rolling stock circulation.

  * •

This addresses the location choice problem at a tactical level.

  * •

It investigates a new concept of daytime and nighttime maintenance for rolling
stock.

  * •

It demonstrates the performance of the proposed model on real-life experiments
on the Dutch railway network.

Section 2 examines the scientific literature. Section 3 introduces the problem
and Section 4 develops a model for solving it. The model’s performance is
investigated using various scenarios in Section 5. Section 6 gives the key
takeaways and identifies some interesting directions for future research.

##  2 Literature review

This section discusses papers on rolling stock maintenance location choice. It
covers both railway and aviation papers, given the systematic similarities of
the two domains related to vehicle (rolling stock and aircraft) maintenance.
The characteristics of papers are summarised and the differences are mapped
out. The existing gaps are defined in the final part of the section.

Tönissen et al. [2019] aimed at building new maintenance locations in the
railway network, which represents a decision problem at the strategic level.
Their focus was on determining optimal maintenance locations under
unknown/uncertain train lines and fleet size (and mix). They proposed a two-
stage stochastic mixed integer programming model, in which the first stage is
to open a facility, and in the second stage to minimize the routing cost for
the first-stage location decision for each line plan scenario.

Tönissen and Arts [2018] built on Tönissen et al. [2019] by including recovery
costs of maintenance location decisions, unplanned maintenance, multiple
facility sizes and economies of scale (providing that a location twice as big
is not twice as expensive). Since, as a result, the second-stage problem
becomes NP-hard, an algorithm was provided with the aim to avoid having to
solve the second stage for every scenario.

Canca and Barrena [2018] considered the simultaneous rolling stock allocation
to lines and choice for depot locations in a rail-rapid transit context. They
proposed a Mixed Integer Linear Programming (MILP) formulation which appears
hard to solve. Therefore they proposed a three-step heuristic approach
determining first the minimum number of vehicles needed for each line,
subsequently the routes of rolling stock on each line, and lastly the
circulation of rolling stock on lines over multiple days together with the
depot choice.

Zomer [2019] considered the railway network in The Netherlands and designed a
simulation model to estimate the expected effects of carrying out daytime
maintenance at given maintenance locations using historical data. Although the
use of historical rolling stock data has advantages, since it accurately
describes reality (as opposed to rolling stock planning data), it cannot be
used for situations in the future for which only planning data is available.
Also, although simulation offers the opportunity to investigate effects for
various scenarios of daytime maintenance locations, it cannot be used to
systematically optimize the maintenance location choice.

In the area of aviation, Feo and Bard [1989] introduced the problem of
assigning aircraft to given flights and simultaneously optimizing the number
of maintenance facilities. They solved the problem as a minimum cost multi-
commodity flow problem. They come up with a two-phase heuristic approach to
solve it: firstly, many possible trip patterns for individual aircraft are
computed; secondly, the most possible patters are combined to solve the
complete problem.

Gopalan [2014] is closely related to the work of Feo and Bard [1989], and
assumed routes during the day are given for each aircraft, although these
routes are not yet assigned to specific aircraft. Each route needs to be
connected to a route on the next day in such a way that the routing passes
through a maintenance location with some given periodicity. The objective was
to minimize the number of maintenance locations (one of the differences from
Feo and Bard [1989], that considers cost minimization). They proposed four
heuristics to solve their problem.

The aforementioned research focused on maintenance location choice. Another
area of research in the field of rolling stock maintenance is maintenance
scheduling, which is relevant due to the fact that the optimal maintenance
location choice typically depends on the schedules for the rolling stock units
that need to be maintained. In the literature, maintenance scheduling is often
combined with the assignment of rolling stock units (or aircraft) to train
trips (or flight legs) such that maintenance constraints are satisfied. Some
examples in railways are Herr et al. [2017], Andrés et al. [2015], Wagenaar
and Kroon [2015], Van Hövell [2019] and some examples in aviation are Clarke
et al. [1997], Gopalan and Talluri [1998], Sarac et al. [2006], Gopalan
[2014]. Those problems may be solved using Mixed Integer Linear Programming
(MILP) models (e.g. Herr et al. [2017], Andrés et al. [2015], Wagenaar and
Kroon [2015]), sometimes extended with solution approaches such as column
generation (e.g. Clarke et al. [1997]). Other problems are solved using
heuristics (e.g. Gopalan and Talluri [1998]). The model developed in this
paper also contains important components of maintenance scheduling that
incorporate the timely performance of maintenance activities. In particular,
time slots are defined during which maintenance actions shall be performed,
i.e. maintenance opportunities (introduced in Section 3), while the exact
start time and end time of actions are not considered. The way in which the
current contribution includes these maintenance scheduling components differs
from what is typically found in the literature in at least two ways. Firstly,
the current contribution incorporates these maintenance scheduling components
in conjunction with location choice. Secondly, since the proposed problem is
solved on a tactical level, the maintenance scheduling component is addressed
assuming a fixed rolling stock circulation.

Table 1 summarises the discussed papers. It shows for each paper whether it is
written in the aviation (A) or in the railway (R) context, whether it
considers maintenance constraints, whether it considers locations choice,
whether it considers timing of maintenance for every individual MU, the
adopted solution approach and the planning stage to which the problem relates.

Some more explanation may be necessary on the column indicating maintenance
timing of individual rolling stock units. Even if a paper considers
maintenance (for example by requiring that each trip path shall not exceed
some specified length), it need not provide information on where and when each
maintenance activity shall be performed. When a paper indeed includes the
latter aspect, it is considered to address maintenance timing of individual
MUs.

|  A/R |  Maint. considered |  Location choice |  Indiv. maint. timing |  Solution approach |  Planning stage Tönissen et al. [2019] | R | x | x |  | Optimization | Strategic
---|---|---|---|---|---|---|---|---|---|---|---|---
Tönissen and Arts [2018] |  R |  x |  x |  |  Optimization |  Strategic
Canca and Barrena [2018] |  R |  |  x |  |  Heuristic |  Strategic
Feo and Bard [1989] |  A |  x |  x |  |  Heuristic |  Strategic
Gopalan [2014] |  A |  x |  x |  |  Heuristic |  Strategic
Van Hövell [2019] |  R |  x |  |  x |  Optimization |  Tactical
Zomer [2019] |  R |  x |  x |  |  Simulation |  Tactical
Current |  R |  x |  x |  x |  Optimization |  Tactical
Table 1: Overview of the aviation (A) and railway (R) maintenance literature
discussed in Section 2.

This literature review indicates that several aspects have not been addressed
in the currently existing literature. First, although variants of problems
relating to rolling stock maintenance location have been investigated before,
this problem has not been tackled at a tactical level, i.e. deciding on which
maintenance locations to open for given rolling stock circulation. Second, no
research is available that explicitly distinguishes between daytime
maintenance and nighttime maintenance. This distinction is relevant for the
situation in the Netherlands, where nighttime maintenance is standard and
recent developments have led companies to investigate daytime maintenance as
well. Third, although maintenance location choice for the Netherlands has been
addressed from a simulation perspective in Zomer [2019], no systematic method
has been developed yet to find a set of locations for daytime maintenance to
minimize the number of nighttime maintenance activities.

##  3 Problem description

The problem that is addressed in the current research is defined as the
Maintenance Location Choice Problem (MLCP). The input of the MLCP consists of
a rolling stock circulation, a set of potential maintenance locations, and a
set of maintenance activities that need to be scheduled. The circulation
describes the planned movements of the rolling stock. The maintenance
locations are locations where maintenance activities can be carried out. These
maintenance activities each have a given duration and constraints on the time
between consecutive ones. The goal of the MLCP is to find an optimal choice of
locations used for maintenance, from the set of potential maintenance
locations, such that all maintenance activities can be completed in time.
Given that rolling stock movements and out of service time in the network can
have a direct influence on where each rolling stock unit can be maintained, it
is important to consider the rolling stock circulation and some scheduling
aspects. The relevant terms and concepts for the MLCP are given in the
following.

##### Rolling stock circulation

A rolling stock circulation contains for each rolling stock unit a list of
trips that a rolling stock unit is scheduled to perform. These trips include
an origin and destination position (which are often train stations), and the
corresponding planned departure and arrival times. An example of a rolling
stock circulation is found in Figure 1. Squares represent arrivals or
departures of rolling stock. Inside the squares, the station abbreviation and
the departure or arrival time at this station is given. Solid lines represent
time intervals where a rolling stock unit is used for a train service. For
example, from 07.09 to 10.41, the depicted rolling stock unit is planned to be
used for a train service between Ekz (Enkhuizen) and Hrl (Heerlen). Dashed
lines represent time intervals where a rolling stock unit is not in service.
For example, between 10.41 and 16.19, the depicted rolling stock unit is not
in service and will be standing still at Hrl.

Figure 1: Example of a rolling stock circulation

##### Maintenance opportunities

It is assumed that maintenance can be carried out if and only if a rolling
stock unit stands still. These moments are referred to as maintenance
opportuntities (MOs). Table 2 indicates the MOs corresponding to the rolling
stock circulation from Figure 1. All MOs (even short ones) can potentially be
used for maintenance activities. When a maintenance activity is assigned to an
MO, this implies that the maintenance activity needs to be performed between
the start and end time of this MO. The exact scheduled time of the maintenance
activity is not determined in the scope of this research.

MO | day | location | start time |  end time 1 | 1 | Hrl | 10:41 | 16:19
---|---|---|---|---|---|---|---|---
2 | 1 | Ekz | 19:52 | 20:09
3 | 1 | Mt | 23:31 | 00:01
4 | 2 | Ehv | 01:06 |  05:34
Table 2: Example of the MOs corresponding to the rolling stock circulation
example from Figure 1.

Two time windows are considered: daytime and nighttime. An MO can be during
daytime or nighttime. This division is especially relevant for NS, since it is
currently considering a transition from performing maintenance only during
nighttime to performing maintenance during both nighttime and daytime [Zomer,
2019, Van Hövell, 2019]. The time window for daytime maintenance is set at
07.00-19.00, the time window for nighttime maintenance from 19.00-07.00. Some
MOs may be partly during daytime and partly during nighttime (this occurs, for
example, when an MO lasts from 18.00 to 20.00). For these MOs it is not
straightforward whether maintenance would be carried out during the day or
during the night (i.e. whether the MO should be considered to be during
daytime or during nighttime). For these reasons, the following simplification
is applied in the current problem: an MO is marked to be during daytime if and
only if both its start time and its end time are between 07.00 and 19.00 of
the same day; an MO is marked to be during nighttime in all other cases. Note
that, although this assumption is reasonable in most cases, there are some
occasions where it is not realistic: for example, an MO starting at 11.00
(during daytime) and ending at 19.01 (just after the start of the nighttime
time window) would be classified to be a nighttime MO whereas maintenance
scheduled in it can probably be performed during daytime.

##### Maintenance activities

It is assumed that a set of regular maintenance types of arbitrary size is
known, and that for each maintenance type a fixed duration and a fixed maximum
time interval between subsequent maintenance activities is given. For each
rolling stock unit, all maintenance types are applied within the respective
specified intervals. Specifically, maintenance activities are assigned to
maintenance opportunities, which is referred to as the maintenance scheduling.
It is assumed that maintenance activities are carried out in a subsequent
manner and cannot overlap. Furthermore, it is assumed that the number of hours
at the start of the planning horizon since the last maintenance activity for
each rolling stock unit is given.

##### Maintenance locations

Maintenance can be carried out at a potential maintenance location. The set of
potential maintenance locations is given. For each of these locations, it can
be decided whether a location should be opened or not. This decision is
referred to as the location choice. Each location can be opened during
nighttime or during daytime, that is, for each maintenance location, there are
four possible outcomes: the location is not used at all, the location is used
for daytime maintenance, the location is used for nighttime maintenance, or
the location is used for both. It is assumed that each maintenance type can be
applied at every location.

The reachability of maintenance locations is incorporated implicitly since the
current research takes the given rolling stock circulation as an input.
Maintenance activities can take place only at those locations where rolling
stock units are located according to the rolling stock circulation.

##### Planning horizon

The planning horizon in the current research is equal to the planning horizon
in the rolling stock circulation. In other words, a maintenance schedule is
determined for the entire time horizon of the rolling stock circulation, and
as a result, the optimal location choice is valid for the length of this time
horizon as well. In The Netherlands, rolling stock circulations are available
for periods of eight weeks. This implies that the planning horizon in the
current research is also fixed at eight weeks.

##### Objective

The objective of the MLCP is to assign maintenance activities to MOs and
determine for each location whether it is open during daytime and/or
nighttime, satisfying the intervals between maintenance activities and the
other constraints, in such a way that the number of nighttime maintenance
activities is minimized. This goal is relevant in practical contexts where the
capacity of maintenance locations at nighttime is under pressure. As a
technical aside, a small penalty applies for any maintenance activity, to
avoid the situation that more daytime maintenance activities are planned than
necessary.

##  4 Model development

The current section provides a Mixed Integer Linear Programming (MILP) model
to solve the MLCP. For models formulated as a MILP, sophisticated solvers
exist that deliver optimal solutions, and such models have been used for
similar problems in the past (e.g. Tönissen et al. [2019]). A downside of the
use of MILP in general is the fact that the computation time can be relatively
high. The aim therefore is to provide a MILP model that contains not too many
decision variables. Furthermore, given the fact that the MLCP is a problem on
the tactical planning level, moderate computation times can be accepted to
optimal solutions, as opposed to applications in for instance traffic
management (e.g. Wagenaar and Kroon [2015]), where short computational times
are the main goal.

###  4.1 Mathematical notation

Let $I$ be the set of rolling stock units considered in the current problem
and let $i\in I$ be the index used to indicate a specific rolling stock unit.
Let $T>0$ be the length of the planning horizon in hours. Let $L$ denote the
set of potential maintenance locations.

##### Maintenance opportunities

Let $J_{i}\equiv\\{1,...,|J_{i}|\\}$ denote the set of MOs for rolling stock
unit $i\in I$. The location of a rolling stock unit $i$ at MO $j\in J_{i}$ is
denoted by $l_{ij}\in L$. The start time of MO $j\in J_{i}$ is denoted by
$s_{ij}\in\mathbb{R}$ and the end time is denoted by $e_{ij}\in\mathbb{R}$,
where time is given as the hours that passed since midnight of the first day
in the planning horizon. For example, an MO $j$ of a rolling stock $i$ on the
fourth day starting at 9:30 and ending at 13:15 would be represented by
$s_{ij}=81.5$ and $e_{ij}=85.25$.

Each day is divided into two non-overlapping time windows: the daytime time
window and the nighttime time window. Each MO is classified as either a
daytime or a nighttime MO. Let $d_{ij}$ indicate whether an MO is classified
to be during daytime or during nighttime: let $d_{ij}=1$ if MO $j\in J_{i}$
for rolling stock unit $i$ is during daytime and let $d_{ij}=0$ if a MO $j\in
J_{i}$ for rolling stock unit $i\in I$ is nighttime. Let $\delta^{D}$ be the
hour of the day when the daytime maintenance window starts and let
$\delta^{N}$ be the hour of the day when the nighttime maintenance window
starts. Unless stated otherwise, $\delta^{D}=7.00$ and $\delta^{N}=19.00$. An
MO is classified to be during daytime if and only if both its start and end
time are during daytime of the same day, i.e.

$\displaystyle d_{ij}=\begin{cases}1&\text{ if }\delta^{D}\leq e_{ij}\text{
mod }24<\delta^{N}\\\ 0&\text{ else}\end{cases}.$

##### Maintenance types

Let $K$ be the set of maintenance types, $K\equiv\\{1,...,|{K}|\\}$. For each
maintenance type $k\in K$, let $v_{k}\in\mathbb{R^{+}}$ be its duration in
hours and let $o_{k}\in\mathbb{R^{+}}$ be the maximum interval between two
consecutive maintenance activities of maintenance type $k$ in hours.

##### Maintenance locations

A potential maintenance location can be opened during daytime (or not) and it
can be opened during nighttime (or not). Let $y_{l}^{D}\in\\{0,1\\}$ be a
binary variable equal to 1 if location $l\in L$ is available for daytime
maintenance and 0 otherwise. Let $y_{l}^{N}\in\\{0,1\\}$ be a binary variable
equal to 1 if location $l\in L$ is available for nighttime maintenance and 0
otherwise. The number of potential maintenance locations that can be opened
during daytime is restricted. Let $L_{max}^{D}$ denote the maximum number of
potential maintenance locations that can be opened during daytime.

##### Maintenance schedule

Maintenance activities are assigned to maintenance opportunities. Let
$x_{ijk}\in\\{0,1\\}$ be a binary variable equal to 1 if maintenance of type
$k$ is performed to rolling stock unit $i\in I$ at MO $j\in J_{i}$, and 0
otherwise. It is required that the total time available at MO $j$ is not
exceeded: $\sum_{k\in K}x_{ijk}v_{k}\leq e_{ij}-s_{ij}$.

Furthermore, an MO $j$ can only be used if the corresponding location is open
at the moment of the MO. Therefore, $d_{ij}=0$ and $y_{l_{ij}}^{N}=0$ implies
$x_{ijk}=0,\forall k\in K$. In the same way, $d_{ij}=1$ and $y_{l_{ij}}^{D}=0$
implies $x_{ijk}=0,\forall k\in K$.

Moreover, the intervals between successive maintenance activities of the same
type should satisfy the given criteria. The interval between two MOs
$j,j^{\prime}\in J_{i}$, $j\neq j^{\prime}$ is measured from the end time of
the first MO to the start time of another MO: $s_{ij^{\prime}}-e_{ij}$. If
activity $k\in K$ is scheduled for rolling stock unit $i\in I$ in MO $j$, then
the next maintenance activity should be scheduled such that the interval
constraints are satisfied. Let $V_{ijk}\subset J_{i}$ denote the set of
maintenance opportunities for rolling stock unit $i\in I$ that start after the
end of MO $j\in J_{i}$ but earlier than $o_{k}$ hours after the end of MO
$j\in J_{i}$. This set is $V_{ijk}=\\{p\in J_{i}:e_{ij}<s_{ip}\leq
e_{ij}+o_{k}\\}$ for $j\in J_{i}$. It is then required that for all $i\in
I,j\in J_{i}$, the following implication holds:

$\displaystyle x_{ijk}=1\implies\exists j^{\prime}\in
V_{ijk}:x_{ij^{\prime}k}=1.$

Observe that a next maintenance activity only needs to be scheduled if
maintenance needs to be carried out within the current planning horizon, that
is, if ${e_{ij}+o_{k}\leq T}$.

Let $b_{ik}$ be the number of hours since the last maintenance activity of
type $k$ for rolling stock unit $i$ at midnight of the first day. Then let
$V_{i0k}=\\{p\in J_{i}:s_{ip}\leq o_{k}+b_{ik}\\}$.

###  4.2 Model formulation

The MLCP model aims to find $x_{ijk}$ and $y_{l}$ satisfying the above
described constraints that minimize the number of maintenance activities
during nighttime. In this model, the decision variables are $x_{ijk}$ ($i\in
I,j\in J_{i},k\in K$) and $y_{l}^{D}$ ($l\in L$). The variables $y_{l}^{N}$
($l\in L$) are considered to be given in the input.

The model can then be formulated as follows:

$\displaystyle\min\sum_{i\in I}\sum_{j\in J_{i}}\sum_{k\in
K}x_{ijk}(1-d_{ij})+\varepsilon\sum_{i\in I}\sum_{j\in J_{i}}\sum_{k\in
K}x_{ijk}$ (1)

subject to

$\displaystyle 1$ $\displaystyle\leq\sum_{p\in V_{i0k}}x_{ipk}$
$\displaystyle\forall i\in I,k\in K$ (2) $\displaystyle x_{ijk}$
$\displaystyle\leq\sum_{p\in V_{ijk}}x_{ipk}$ $\displaystyle\forall i\in
I,j\in J_{i},k\in K:e_{ij}+o_{k}\leq T$ (3) $\displaystyle x_{ijk}$
$\displaystyle\leq y^{D}_{l_{ij}}\cdot d_{ij}+y^{N}_{l_{ij}}\cdot(1-d_{ij})$
$\displaystyle\hskip 14.22636pt\forall i\in I,j\in J_{i},k\in K$ (4)
$\displaystyle\sum_{k\in K}x_{ijk}v_{k}$ $\displaystyle\leq e_{ij}-s_{ij}$
$\displaystyle\forall i\in I,j\in J_{i}$ (5) $\displaystyle\sum_{l\in
L}y_{l}^{D}$ $\displaystyle\leq L_{max}^{D}$ (6) $\displaystyle
x_{ijk}\in\\{0,1\\},$ $\displaystyle\hskip 5.69046pty_{l}^{D}\in\\{0,1\\}.$
(7)

The objective function (1) minimizes the number of nighttime maintenance
activities. The second term penalizes every maintenance activity with an
arbitrarily small penalty cost $\varepsilon$ in order to avoid unnecessary
maintenance activities being performed. Constraints (2) and (3) enforce that
intervals between successive maintenance activities are satisfied. Constraints
(4) ensure that maintenance can only be executed at a location that is opened.
Constraints (5) take account of the requirement that the duration of
maintenance may not exceed the total time of an MO. The number of locations
for daytime maintenance is restricted by constraint (6). Constraints (7)
ensure that the integer decision variables are also binary.

##  5 Results

The main goal of the experiments performed in the current section is to
investigate the validity of the model by studying the outcomes under various
inputs and to make practical recommendations regarding the number of
maintenance locations to open and which maintenance locations to open. Also,
some numbers regarding computational efficiency are reported.

###  5.1 Experimental design

The proposed MLCP model is tested on real data from the Dutch railway network
[NS, 2018]. The data set contains planned rolling stock movements of trips
that are operated by NS. A rolling stock circulation covers an 8-week period,
which represents a basic planning period, and is issued several weeks before
the start of such a period. A data set for such a period is referred to as
Basic Day update (in Dutch BasisDag update, BDu). This data is an appropriate
choice for the current research since it offers a complete view of all rolling
stock movements as expected several weeks before operations. In the current
research, 10 different BDus have been used. Each of these BDus are individual
data sets on which the MLCP model can be run. The period covered by these 10
BDus starts on 10-12-2017 and ends on 1-9-2019. This data set includes all
rolling stock units operated by NS, meaning it is of a realistic size and
regards a large part of the Dutch rail transport network. The number of
rolling stock units in each BDu varies between 820 and 993. Due to the
acquisition of new rolling stock units, an increasing trend in the number of
rolling stock units operating on the network is visible, leading to the fact
that each BDu does not need to contain the same number of rolling stock units.

Two maintenance types are considered ($K=\\{1,2\\})$: maintenance type 1 has a
duration of 0.5 hours ($v_{1}=0.5$) and a maximum interval of 24 hours between
successive maintenance activities ($o_{1}=24$), and maintenance type 2 having
a duration of 1 hour ($v_{2}=1$) and a maximum interval of 48 hours between
successive maintenance activities ($o_{2}=48$). Rolling stock units are
assumed to be as-good-as-new at the start of the planning horizon ($b_{ik}=0$
for all $i\in I,k\in K$). The start of the daytime time window $\delta^{D}$ is
fixed at $\delta^{D}=07.00$ and the start of the nighttime maintenance window
$\delta^{N}$ is fixed at $\delta^{N}=19.00$. The technical parameter
$\varepsilon$ is fixed at $\varepsilon=0.001$.

The following parameters are varied to construct the experiments investigated
in Section 5.2.

  * •

We consider 10 BDus $\beta\in\\{1,...,10\\}$, each representing a period of
approximately 8 weeks. Unless stated otherwise, BDu set 10 is used, which is
valid in the period 9-6-2019 until 1-9-2019.

  * •

The set of rolling stock units $\nu$ is varied. Smaller-sized sets are used to
avoid large computation times and thus to allow validating the model by
solving multiple instances; larger-sized sets are more realistic which are
used to demonstrate a practical performance of the model. In addition, the use
of sets of various sizes also enables to investigate the effect of the number
of rolling stock units on model outcomes. In most sets, rolling stock units of
type VIRM are used since these rolling stock units typically spread out over
the entire country visiting many different maintenance locations and therefore
tend to provide most interesting insights on the network level. Specifically,
the following sets of rolling stock units are considered:

    * –

Set 1, with the 10 first occurring rolling stock units of type VIRM4.

    * –

Set 2, with the 20 first occcurring rolling stock units of type VIRM4.

    * –

Set 3, with all rolling stock units of type VIRM4 (in BDu 10: 92)

    * –

Set 4, with all rolling stock units of type VIRM4 (in BDu 10: 92) and VIRM6
(in BDu 10: 68)

    * –

Set 5, with all rolling stock units used for intercity services (in BDu 10:
360)

  * •

The planning horizon $\tau\in\\{7,21,42\\}$. For meaningful interpretation,
$\tau$ should not exceed the length of the period for which the BDu considered
is valid (i.e. in general eight weeks).

  * •

The maximum number of daytime maintenance locations
$L^{D}_{max}\in\\{0,1,2,3,5,10,20\\}$.

Section 5.2 presents four experiments to investigate the performance of the
MLCP model provided in Section 4. Experiment 1 focuses on the maximum number
of locations for daytime maintenance: when more locations can be opened during
daytime, supposedly more activities can be performed during nighttime and
hence the goal of the MLCP model (minimizing the number of nighttime
maintenance activities) may be achieved better. Experiment 2 focuses on the
influence of the planning horizon. If a shorter planning horizon yields a
similar location choice, a shorter time horizon may suffice to determine good
maintenance locations, which in turn leads to a shorter running time.
Experiment 3 compares 10 different BDus, leading to 10 different scenarios
that apply for different time periods, allowing to investigate whether results
on the location choice are consistent over multiple time periods. Experiment 4
includes all rolling stock units that serve intercity (long-distance) lines,
while the results in the first three experiments considered only the rolling
stock types VIRM4 and VIRM6. This allows to investigate the performance and
results of the MLCP model in a scenario of realistic size. More experiments,
including experiments elaborating upon the sensitivity of various parameters
and the computation time of the MLCP model, can be found in Zomer [2020].

It must be noted that, from the ten available BDus, five resulted in
infeasible solutions when applied in combination with rolling stock sets 4 and
5. This becomes relevant in Experiment 3, where all 10 BDus are considered.
These five scenarios are infeasible due to the fact that the corresponding
data set contained some rolling stock units for which it is not possible to
create a feasible maintenance schedule, for example, no two suitable
successive MOs could be found within 24 hours to schedule maintenance
activities of type 1. Even if this is the case for only one rolling stock
unit, no feasible solution can be found. In reality, this problem could be
overcome by manually removing the rolling stock units that caused the
infeasible solution and run the model again. However, in the current research,
results are reported for the five BDus for which a feasible solution was found
only. This is expected to provide sufficient insights as to whether results of
one period carry over to multiple periods.

The MLCP is implemented implemented using Python and solved using Gurobi.
Zomer et al. [2020] provides the implementation code of the MLCP model used to
generate the results in this section. Due to confidentiality, the actual data
could not be provided, but a synthetic data set is made available instead.

###  5.2 Results

####  5.2.1 Experiment 1: influence of the maximum number of daytime
maintenance locations

For varying numbers of the maximum number of daytime maintenance locations,
this section presents two measures. Firstly, it presents the mean daytime
share, which is the percentage of hours of activity performed during daytime,
averaged per day, relative to the mean total hours of activity performed per
24 hours, averaged per day. A high mean daytime share indicates that
relatively more work is performed during daytime, and this is desired, since
it is an indication of a decrease in the pressure on maintenance location
capacity during nighttime. However, to check that not only the relative amount
of work but also the absolute amount of work during nighttime decreases, it is
necessary to verify that the total hours of activity remains more or less
constant. Therefore, secondly, this section reports the mean total hours of
activity per 24 hours, averaged per day, for multiple scenarios.

##### Mean daytime share

Scenarios have been run for all combinations of $L^{D}_{max}$, $\nu$ and
$\tau$, with $L^{D}_{max}\in\\{0,1,2,3,5,20\\}$, $\nu\in\\{1,2,3,4\\}$ and
$\tau\in\\{7,21,42\\}$ and mean daytime shares for each of these scenarios
have been computed. Since the mean daytime shares showed to be relatively
invariant for different values of $\nu$ and $\tau$, Table 3 shows average mean
daytime shares, averaged over all 12 combinations of $\nu$ and $\tau$, for
various values of $L^{D}_{max}$. It shows that the average mean daytime share
is increasing with $L^{D}_{max}$.

The fact that the mean daytime shares are increasing in $L^{D}_{max}$ is
expected. When more locations are opened for daytime maintenance, more
maintenance can be performed during daytime and hence the mean daytime share
increases. This is particularly interesting since the goal of NS is to move
hours of activity from nighttime to daytime. Apparently, when opening 20
locations, up to on average 42.0% of the work can be performed during daytime.

$L^{D}_{max}$ |  average mean daytime share 0 | 0.0%
---|---|---
1 | 4.2%
2 | 8.9%
3 | 13.3%
5 | 22.3%
20 |  42.0%
Table 3: The mean daytime share increases for increasing values of
$L^{D}_{max}$.

##### Mean total hours of activity

Scenarios have been run for all combinations of $L^{D}_{max}$, $\nu$ and
$\tau$, with $L^{D}_{max}\in\\{0,1,2,3,5,20\\}$, $\nu\in\\{2,3,4\\}$ and
$\tau\in\\{7,21,42\\}$. The set $\nu=1$ yields similar results but has been
excluded from the graph for visualization purposes. The mean total hours of
activity for each of these scenarios have been computed. Since the mean total
hours of activity showed to be relatively invariant over different values of
planning horizon $\tau$, Figure 2 shows mean total hours of activity per day,
averaged over all 3 choices of $\tau$, for various values of $L^{D}_{max}$ and
$\nu$.

Figure 2 shows the mean total hours of activity for various sets of rolling
stock units $\nu\in\\{2,3,4\\}$. Numbers are averaged over scenarios with
different values of planning horizon $\tau\in\\{7,21,42\\}$. It shows that
indeed the mean total hours of activity per day increases when the maximum
number of daytime maintenance locations increases: up to about 15% when 20
daytime maintenance locations are opened. This implies that, when more
possibilities for daytime maintenance arise, this leads to more maintenance in
total. From this figure it also becomes clear that this holds for various
numbers of rolling stock units, and (evidently) that the mean total hours of
activity also increases when more rolling stock units are added to the
analysis. The increase of the mean total hours of activity is however
convincingly outweighed by the increase of the mean daytime share (cf. Table
3), and hence the application of daytime maintenance decreases the pressure on
the capacity of maintenance locations during nighttime.

Figure 2: The mean total hours of activity increases when the number of
daytime maintenance locations $L^{D}_{max}$ increases.

####  5.2.2 Experiment 2: influence of the planning horizon

The goal of the second experiment is to investigate the sensitivity of the
solutions provided by the MLCP model to various choices of the planning
horizon $\tau$. A shorter planning horizon requires less computation time. If
a shorter planning horizon yields similar location choice, then the required
computation time to determine the optimal location choice could be
significantly reduced.

(a) $L^{D}_{max}=5$

(b) $L^{D}_{max}=20$

Figure 3: Location choice is consistent over various choices for the planning
horizon. Moreover, significant workloads are assigned to various locations.

Scenarios have been run for all combinations of $L^{D}_{max}$, $\nu$ and
$\tau$, with $L^{D}_{max}\in\\{5,20\\}$ and $\tau\in\\{7,21,42\\}$ with
$\nu=4$ (i.e. all VIRM4 and VIRM6), which means there are three scenarios for
each choice of $L^{D}_{max}$.

Figure 3 indicates for each location in how many of the investigated scenarios
the location was chosen. Moreover, for each location the mean hours of daytime
activity assigned to each location is indicated. provides insight in the
location consistency and the workload assigned to each location. Blue
represents the number of scenarios (out of 3) in which the respective location
was open during daytime. Orange represents the average daily number of hours
of daytime activity assigned to the respective location.

For $L^{D}_{max}=5$ (blue lines in Figure 3(a)), the locations Amr, Hdr, Hfdo
and Mt are chosen in all three scenarios. This justifies that these four
locations are good choices for daytime maintenance in the given subset, but
more importantly it shows that the location choice is robust under various
time horizons. For $L^{D}_{max}=20$ (blue lines in Figure 3(b)), the location
choice is for all three values of planning horizon $\tau$ exactly equal, i.e.
each time the same 20 locations are chosen.

Moreover, Figure 3 shows in orange color the mean hours of daytime activity,
averaged over all 3 combinations of $\tau$. When $L^{D}_{max}=5$ and all
rolling stock units are included, the hours of daytime activity is higher than
2 hours per day for these locations that are chosen in all 3 scenarios. In
other words, for the locations for which the choice is most consistent, also
the highest workload is found.

When comparing Figure 3(a) and Figure 3(b), the locations Amr, Hdr, Mt and
Hfdo appear to be four good candidate locations to be opened. They are chosen
in all three scenarios, and the workload assigned to these locations is
highest. Hence, also when more locations can be opened (20 in this case), the
locations found for a smaller maximum number of daytime maintenance locations
(5 in this case, see Figure 3) still remain good candidates. This is important
in practical situations where the number of locations is gradually expanded:
the locations found in a stage where only a few locations can be opened, still
remain good candidates even when more locations can be opened in a later
stage. This is not an obvious finding, for it may have been true that the
location choice would change drastically for an increasing value of
$L^{D}_{max}$.

####  5.2.3 Experiment 3: consistency over different BDus

Figure 4: Location choice is consistent over various BDus.

Scenarios have been run for $L^{D}_{max}=20$, $\nu=4$ (i.e. all VIRM4 and
VIRM6) and $\tau=42$. As indicated, out of the 10 availables BDus, only 5
resulted in a feasible solution and hence only these have been considered
here.

Figure 4, shows for each location that was opened in at least one of scenarios
(with a feasible solution), in how many of these five scenarios this location
is opened (blue color) and the mean hours of daytime activity at each opened
location, averaged over the scenarios in which the location was opened (orange
color).

It shows that most of the locations (14 in total) are chosen in all five
scenarios (blue lines). This implies that the optimal locations are consistent
across different BDus. From a managerial perspective, this is important, since
it means that once a location is chosen, it will most likely still be optimal
to choose this location in a next period, i.e. for a new rolling stock
circulation plan. This means that the applicability of the results, obtained
by applying the MLCP model, are in this case not only valid for one BDu, but
also carry over to multiple BDus and hence to multiple periods of time.

In addition, in some cases, a location was chosen in only a limited number of
scenarios: for example, Vs and Zl were chosen in only one out of five
scenarios. However, in general, these locations are not the locations to which
high workloads are assigned. The influence of choosing not to open such a
location may therefore be limited.

Looking at the mean hours of daytime activity of the opened locations (orange
lines), it can be observed that there are four locations with more than three
hours of activity and that these locations are consistently chosen over the
multiple scenarios. This is positive since it indicates that the choice on
whether a location is opened consistently or not correlates with the workload
assigned to that location. In other words, if a location is opened
consistently throughout multiple scenarios, this usually also means that a
relatively high workload is assigned to this location.

####  5.2.4 Experiment 4: Performance of scenario of realistic size

Experiments 1-3 have investigated the performance of the MLCP model under
various parameter settings. The largest data set considered so far, included
rolling stock units of one train type, VIRM. To assess the workings of the
MLCP model in practice, it is necessary to consider a larger and more
representative data set. To this end, this fourth experiment considers all
rolling stock units used for intercity services combining multiple train types
(rolling stock set $\nu=5$).

Table 4 gives the most important results for the scenarios with 10 and 20
locations available: it provides the mean hours of daytime activity, the mean
daytime share and the running time of the four different scenarios
investigated in the current experiment. It shows that, for $L^{D}_{max}=20$, a
mean daytime share of over 30% can be attained. The mean daytime share appears
to be higher for the scenarios with a maximum number of 20 locations opened
for daytime maintenance, which is consistent with the findings in Experiment 1
(in Section 5.2.1).

$\nu$ | $L^{D}_{max}$ | mean total hrs. activity | mean daytime share |  running time (s) 7 | 10 | 305.9 | 22.6% | 204
---|---|---|---|---|---|---|---|---
| 20 | 310.6 | 30.9% | 201 |
42 | 10 | 328.8 | 22.2% | 11,522 |
| 20 | 334.9 | 30.1% |  7,885  |
Table 4: In a realistic case, a substantial amount of work can be performed
during daytime. Figure 5: Good candidate locations for daytime maintenance.

Figure 5 provides the mean hours of daytime activity per location, for
planning horizon $\tau=42$, set of rolling stock units $\nu=5$, for
$L^{D}_{max}=10$ (blue) and $L^{D}_{max}=20$ (orange). It shows that, in the
situation with a maximum number of locations for daytime maintenance of 10, at
least 5 locations have a workload of more than 8 hours per day (Gvc, Bkd, Bkh,
Dv and Gn), which can be considered substantial since it is enough to provide
work to one maintenance team. Moreover, it shows that the addition of
maintenance locations does not seem to reduce the average workload on any of
the initial 10 locations. Hence, the initial 10 locations are still good
choices, even when daytime maintenance is possible at more locations. Instead,
it can be assumed that some additional maintenance work was allocated to these
newly opened locations for $L_{D}^{max}$. The added locations, however, are
assigned a much lower workload than the initial 10 locations. It may be
therefore questionable whether the addition of these locations is worthwhile.

##  6 Conclusion

This paper introduces the Maintenance Location Choice Problem (MLCP) and
models it as a mixed integer program, taking as its main input a rolling stock
circulation and provides for this rolling stock circulation an optimal
maintenance location choice that minimizes the total number of maintenance
activities during nighttime, thereby reducing the capacity pressure during the
night. For the NS case with all rolling stock units of type VIRM4 and VIRM6,
up to 22.3% of the work can be performed during daytime if five locations are
opened, to even 42.0% if 20 locations are opened during daytime. The location
choice is consistent for different lengths of planning horizons and for
different input data sets. The four locations with the highest assigned
workload (in hours of daytime activity) are Amr, Hdr, Mt and Hfdo. For the
largest scenario (including all rolling stock units used for intercity
services), a mean daytime share of at least 30.1% of the activities can be
achieved if 20 maintenance locations are opened. In this case, Gvc, Bkd, Bkh
en Dv show to be the locations with the highest workloads. Moreover, the model
is efficient for planning purposes: for the largest scenario run (with all
rolling stock units used for intercity services) for a planning horizon of 42
days, the computation time was 3.2 hours. For operational purposes, however,
this running time is not sufficiently efficient.

Some limitations to the current approach can be identified. First, the current
research uses planned rolling stock data. Although this is common practice in
the railway industry and therefore a logical choice, it does not take into
account the potential occurrence of disruptions. Due to disruptions, the
rolling stock circulation is often altered during operations and as a result,
the provided optimal maintenance schedule and optimal maintenance location
choice may not be the best anymore at the moment of execution. The current
model can provide a basis for more robust decisions on daytime maintenance
locations and/or the online adjustment of a maintenance schedule. Second, the
current research uses a simplified definition of maintenance types. Their
durations and intervals may not exactly correspond to practical situations.
Moreover, in the current research these are assumed to be equal for each
rolling stock unit, whereas in reality these values often differ per rolling
stock type. We expect that enriching the formulation with these types would
not be very complex and lead to similar results. Third, the capacity of
maintenance locations is not restricted. According to expert judgment at NS,
this is acceptable: currently, the primary goal of NS is to reduce the
pressure on maintenance locations during nighttime and hence it is not
relevant to restrict the capacity of maintenance locations during nighttime.
Also, the work assigned to daytime maintenance locations seems to be not
excessive and hence it is deemed unnecessary to restrict the capacity of
daytime maintenance locations. However, there may be applications in which it
is desirable to restrict the capacity of maintenance locations as well, which
is an interesting direction for further work.

The MLCP model is intended for use on a tactical level (i.e. months before
operations), since it assumes that a rolling stock circulation is already
given. Hence, it answers the question at which already existing locations
personnel needs to be stationed during daytime and/or night time. However,
when applied on multiple rolling stock circulations, the MLCP model can also
serve more strategic goals (i.e. years before operations), regarding for
example the question at what positions new locations could best be built.

Several directions for future research can be recommended. First, an
interesting next research topic is how to improve the computational
performance of the MLCP. Using the structure of the problem, the problem may
potentially be decoupled into multiple smaller sub-problems that are much
easier to solve, e.g. solving rolling stock types independently, or
considering a rolling horizon framework, thereby first optimizing a few days
in ahead and iteratively adding more days. Second, as indicated, some of the
input data sets appeared to result in infeasible solutions. Since the model is
not capable of allowing (minor) violations of constraints, for some input sets
no solution can be found at all. However, in some contexts it may be better to
obtain a slightly violated solution than obtaining no solution at all. This
could for instance be achieved by accepting and penalising constraint
violations such as reducing the time for some maintenance activities by a
small amount. Third, the MLCP model can be applied to other domains as well.
Examples of such extensions include the applicability of the research to serve
other railway undertakings with potentially different objectives and policies,
the investigation of other cost structures and the inclusion of exchange
opportunities of rolling stock units so that more rolling stock units can be
maintained during daytime.

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11institutetext: Department of Physics and Astronomy, Wayne State University,
Detroit, Michigan 48201, USA 22institutetext: Department of General Education,
Faculty of Engineering, Osaka Institute of Technology, Osaka 535-8585, Japan
33institutetext: Physics Department, Brookhaven National Laboratory, Upton, NY
11973, USA 44institutetext: RIKEN BNL Research Center, Brookhaven National
Laboratory, Upton, NY 11973, USA

# Tracing baryon and electric charge transport
in isobar collisions

Gregoire Pihan  (Speaker)11<EMAIL_ADDRESS>Akihiko Monnai 22 Björn
Schenke 33 Chun Shen 1144

###### Abstract

It is of fundamental interest to understand the carrier of conserved quantum
charges within protons and nuclei at high energy. Preliminary data from isobar
collisions at RHIC reveal a scaled net-baryon to net-electric charge ratio
($B/\Delta Q\times\Delta Z/A$) at midrapidity between 1.2 and 2, consistent
with string junction model predictions. Here, we compute the initial stage
scaled net-baryon to net-electric charge ratio for isobar collisions. Our
model incorporates a realization of the string junction model and models the
nuclear structure. Our predictions identify the baseline expectations for such
measurement and quantify the impact of the nuclear structure.

##  1 Introduction

A gauge-invariant approach to constructing a proton’s baryon charge is to
connect its three valence quarks through a three-gluon vertex known as the
string junction Montanet:1980te . It was later proposed that the effective
carrier of baryon number within a baryon could be the string junction itself
Kharzeev:1996sq . In that case, it is anticipated that in relativistic nuclear
collisions the net baryon number ($B$) will experience more stopping at
initial impacts than the net electric charge number ($Q$), assigned to the
valence quarks. The precise measurement of the baryon-to-electric-charge
stopping ratio in isobar collisions (Ru+Ru: $A,Z=96,44$, Zr+Zr: $A,Z=96,40$)
at RHIC STAR:2021mii  offers an unprecedented opportunity to test this
assumption experimentally. Any deviation from global charge conservation
($Q\sim Z/A\times B$) at mid-rapidity may signify distinct longitudinal
dynamics for $B$ and $Q$ during the collision or the possibility that $B$ is
not carried by the valence quarks, supporting the string junction model
Lewis:2022arg . In this study, we compute the initial conditions for
$\sqrt{s_{\text{NN}}}=200$ GeV isobar collisions, utilizing a 3D-Glauber model
Shen:2017bsr ; Shen:2022oyg . These initial conditions can then be used in the
iEBE-MUSIC framework to produce final-state particle observables to be
compared to the STAR measurements. Understanding the initial conditions for
isobar collisions is an essential step in studying the baryon-to-electric
charge-stopping ratio. Indeed, all key features observed at the initial stage
are expected to be seen at the final stage if the longitudinal dynamics are
similar for different flavors of conserved charges.

##  2 The Theoretical Framework

In the 3D-Glauber model, nucleon-nucleon collisions decelerate incoming
partons and produce strings as energy-momentum source currents for the
hydrodynamic medium Shen:2017ruz ; Shen:2017fnn ; Shen:2017bsr . The baryon
numbers from the incoming nucleons can either remain in the wounded beam
remnants or fluctuate inside the string pulled by the colliding parton pairs
Shen:2022oyg . If the baryon number is assigned to the string, its rapidity
$y^{B}_{P/T}$ follows a probability distribution given by

$P(y^{X}_{P/T})=(1-\lambda_{X})y_{P/T}+\lambda_{X}\frac{e^{(y^{X}_{P/T}-(y_{P}+y_{T})/2)/2}}{4\sinh{((y_{P}-y_{T})/4)}},$
(1)

where $P$ and $T$ represent the projectile and target, respectively and $X=B$
Kharzeev:1996sq ; Shen:2022oyg . Following this probability, the baryon number
may be placed at the string end with a probability of ${(1-\lambda_{B})}$,
consistent with the assumption of baryon number carried by valence quarks, or
toward mid-rapidity based on predictions from the string junction, with a
probability $\lambda_{B}$. The STAR measurements on midrapidity net-proton
yields in Au+Au collisions in the Beam Energy Scan program favor
$\lambda_{B}=0.2$ STAR:2017sal ; Shen:2022oyg .

This work considers the electric charge number to fluctuate toward mid-
rapidity following the same probability as in Eq. (1) with $X=Q$ and an
independent model parameter $\lambda_{Q}$. While this is not motivated by a
specific physical picture for electric charge stopping, it enables control
over the baryon-to-electric-charge stopping ratio at the initial stage. If
$\lambda_{Q}=0$, the electric charge is associated with valence quarks.
Suppose $\lambda_{Q}=\lambda_{B}$, both the baryon and the electric charge
numbers stop in the same way. Throughout the remainder of this work, we will
consider the baseline calculations for the $\lambda_{Q}=\lambda_{B}$ case.
This procedure yields the initial rapidity distribution of baryon and electric
charge numbers for isobar collisions.

The 3D-Glauber model incorporates modeling nuclear structure through deformed
Woods-Saxon distributions for protons and neutrons, including possible finite
neutron skins for the colliding nuclei. The Woods-Saxon densities in a nucleus
are expressed in spherical coordinates $(r,\theta,\phi)$ as

$\displaystyle\rho_{i}(r,\theta,\phi)=\frac{\rho_{0,i}}{1+e^{[r-R_{i}(\theta,\phi)]/a_{i}}},$
(2)

where $i=p,n$ denotes protons and neutrons, $\rho_{0,i}$ is the central
density and $a_{i}$ is the diffusivity parameter. Special deformation can be
introduced via the radius parameter $R_{i}$ with spherical harmonic functions
$Y_{lm}(\theta,\phi)$ as

$\displaystyle
R_{i}(\theta,\phi)=R_{0,i}\Big{[}1+\beta^{WS}_{2}[\cos(\gamma)Y_{20}(\theta,\phi)+\sin(\gamma)Y_{22}(\theta,\phi)]+\beta^{WS}_{3}Y_{30}(\theta,\phi)+\beta^{WS}_{4}Y_{40}(\theta,\phi)\Big{]},$
(3)

where $\beta^{WS}_{2,3,4}$ and $\gamma$ are the deformation parameters that
define the nucleus shape. The Ruthenium (Ru) and Zirconium (Zr) nuclei exhibit
different shapes, and the parameters used in this study are summarized in
Table 1 based on a recent Density Functional Theory calculation Xu:2021vpn .
The neutron skin is modeled with $R_{0,n}=R_{0,p}+dR$ and $a_{n}=a_{p}+da$.

Table 1: Woods-Saxon density parameters for the Ruthenium and Zirconium nuclei Xu:2021vpn . |  $R_{0,p}$ (fm) |  $a_{p}$ (fm) | $\gamma$ | $\beta^{WS}_{2}$ | $\beta^{WS}_{3}$ | $\beta^{WS}_{4}$ |  $da$ (fm) |  $dR$ (fm)
---|---|---|---|---|---|---|---|---
Ruthenium | 5.09 | 0.46 | 0 | 0.16 | 0 | 0 | 0.01 | 0.015
Zirconium | 5.02 | 0.52 | 0 | 0.06 | 0.2 | 0 | 0.05 | 0.1

##  3 Results

Figure 1: The baryon to electric charge stopping ratio from the 3D-Glauber
model with $\lambda_{B}=\lambda_{Q}=0.2$ for different nuclear structure
configurations.

The 3D-Glauber model provides the initial-state baryon and electric charge
rapidity distributions. We then derive the stopping ratio observable as
described in Ref. Lewis:2022arg . The isobar stopping ratio, denoted as
$B/\Delta Q\times\Delta Z/A$, involves the net baryon number $B$ measured at
mid-rapidity ($-0.5<y<0.5$) in Ru+Ru collisions, and $\Delta
Q=Q_{\text{Ru}}-Q_{\text{Zr}}$, representing the net electric charge
difference between Ru+Ru and Zr+Zr collisions at mid-rapidity. According to
global charge conservation, if the baryon and the electric charge number stop
equivalently, this ratio is expected to be close to unity.

We consider the following three scenarios to quantify the effects of the
nuclear structure. First, the nuclear structure and neutron skins of Ru and Zr
nuclei are defined by the parameters in Table 1. Second, the nuclear
deformation is considered for both nuclei, but neutron skins are removed
(${\rm d}a={\rm d}R=0$ fm). Third, the nuclear structure is set to be
identical for both nuclei (the Ru parameters in Table 1) and no neutron skin.
Figure 1 presents the predicted baryon-to-electric-charge ratio from the
3D-Glauber model for the equal stopping configuration
($\lambda_{B}=\lambda_{Q}=0.2$) for the three nuclear structure configurations
described above.

In the first scenario (depicted by the blue dashed-dotted line), the ratio
consistently remains below unity, and it increases with $N_{\text{part}}$,
demonstrating that even in an equal stopping situation, the baseline for the
ratio is not expected to be equal to one and flat in centrality. The reasons
for the observed behavior become clearer when examining the other two cases.
When the Ru and Zr nuclei are set to have identical shapes (illustrated by the
green solid line), the ratio exhibits a flatter trend and is close to unity.
This observation underscores the nuclear structure’s significant impact on the
ratio’s centrality dependence in the full calculation. When we remove the
neutron skin while keeping the different nuclear deformation in Ru and Zr
nuclei, the result is close to the full calculation in central collisions. The
different sizes of neutron skins in Ru and Zr nuclei give a 20% smaller ratio
in the peripheral collisions, at which collisions are more dominant by the
proton-to-neutron ratio at the edge of the colliding nuclei.

In an equal stopping configuration, where both baryon and electric charge have
an equal probability of stopping at the initial stage, the isobar-stopping
ratio significantly deviates from unity because of the difference in nuclear
structure between the Ru and Zr nuclei. Suppose an experimental study of the
isobar-stopping ratio reveals an increasing centrality dependence. A
comparison with our predictions, considering the full nuclear shape, is
essential to disentangle the effects from flavor-dependent stopping powers at
the initial stage.

##  4 Conclusions

In this study, we used the 3D-Glauber initial-state model to simulate isobar
collisions at $\sqrt{s_{\text{NN}}}=200$ GeV at RHIC, with a specific emphasis
on tracing the flavor-dependence dynamics for conserved charges. Our findings
reveal that, even with an equal probability of stopping for baryon and
electric charge, the expected ratio deviates from unity during the initial
stage, primarily attributed to nuclear structure effects.

## Acknowledgments

This work is supported by the U.S. Department of Energy (DOE), Office of
Science, Office of Nuclear Physics, under DOE Contract No. DE-SC0012704
(B.P.S.) and Award No. DE-SC0021969 (G.P. and C.S.). A.M. was partially
supported by JSPS KAKENHI Grant Number JP19K14722. C.S. acknowledges a DOE
Office of Science Early Career Award. This research was done using
computational resources provided by the Open Science Grid (OSG) Pordes:2007zzb
; Sfiligoi:2009cct , which is supported by the NSF award #2030508 and
#1836650.

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