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Rich arc routing problem in city logistics: Models and solution algorithms using
a fluid queue-based time-dependent travel time representation
Jiawei Lu
School of Sustainable Engineering and the Built Environment
Arizona State University, Tempe, AZ 85281, USA
Email: jiaweil9@asu.edu
Qinghui Nie
College of Architectural Science and Engineering
Yangzhou University, Yangzhou 225127, China
Email: nieqingh@yzu.edu.cn
Monirehalsadat Mahmoudi
School of Packaging, College of Agriculture and Natural Resources
Michigan State University, East Lansing, Michigan, 48824, USA
Email: mahmou18@msu.edu
Jishun Ou*
College of Architectural Science and Engineering
Yangzhou University, Yangzhou 225127, China
Email: jishun@yzu.edu.cn
Chongnan Li
School of Traffic and Transportation
Beijing Jiaotong University, Beijing 100044, China
Email: chongnanli1997@hotmail.com
Xuesong (Simon) Zhou*
School of Sustainable Engineering and the Built Environment
Arizona State University, Tempe, AZ 85281, USA
Email: xzhou74@asu.com
i
Abstract
City logistics, as an essential component of the city operation system, aims at managing the complex
flow of goods and services from providers to customers efficiently. Delays associated with peak-period
traffic congestion exists in both large and small metropolitan areas. As many of the service tasks in city
logistics are needed to be performed during peak hours, operators of urban management movement should
consider reducing the total trip time and delay when designing service plans. Equally important, the
congestion impact of service vehicles to other road users should also be considered. In this paper, we focus
on formulating and solving rich arc routing problems (RARPs) in city logistics with a congested urban
environment. We highlight the needs of embedding a structurally parsimonious time-dependent travel time
model in RARP for producing high-quality and practically useful solutions. A fluid queue model based
analytical approach is presented for link travel time calibration in the form of polynomial arrival rate
functions. Accordingly, system-wide (societal) impact of vehicles routing is analytically derived and
incorporated into the RARP models which enables traffic managers to systematically consider operation
costs and societal impacts when designing routing policies in real-life city logistics applications.
Additionally, we develop two new representation schemes for time-dependent travel time modeling in
RARPs, including a discretized time-expanded representation scheme and a nonlinear polynomial
representation scheme. Three modeling approaches for RARPs are proposed, with different perspectives of
capturing time-dependent travel time and formulating problem-specific constraints. With a real-life
sprinkler truck routing problem as the representative example of RARP, we develop two efficient exact
solution algorithms, including a Lagrangian relaxation-based method and a branch-and-price based method.
The latter one is embedded with an enhanced parallel branch-and-bound algorithm. Extensive numerical
experiments are conducted based on real-world networks and traffic flow data to demonstrate the
effectiveness of the proposed methods.
Keywords: City logistics, Rich arc routing problem (RARP), Time-dependent travel time, Societal impact,
Lagrangian relaxation, Branch and price solution method
1
1. Introduction
In a broader sense, city logistics refers to the management of the flow of goods and services from
providers to customers in urban areas. The urban management movement (UMM) problem (Cattaruzza et
al., 2017), as an example, aims to find an optimal set of routes for a fleet of vehicles to satisfy requests for
the development, public maintenance, and other functional needs in a city. An efficient city logistics system
helps to reduce operation cost, mitigate traffic congestion impact, protect the environment, respond to
climate change, connect underserved communities, and support economic vitality. Considering the traffic
congestion experienced in cities, this type of problems needs to be formulated as rich arc routing problems
(RARPs) under congested traffic conditions.
The word “rich” is used in RARP because besides normal constraints in the variants of the standard
ARP (e.g., capacity and time window constraint), some other problem-specific constraints are also
considered. For example, in the winter gritting problem, service requests on road links change with time
and weather conditions (Eglese and Li, 1992; Eglese, 1994; Li and Eglese, 1996; Tagmouti et al., 2007;
Tagmouti et al., 2010; Tagmouti et al., 2011), or in the snow plowing problem, truck routes are specified at
the lane level (Perrier et al., 2007a; Perrier et al., 2007b; Salazar-Aguilar et al., 2012; Dussault et al., 2013;
Dussault et al., 2014; Quirion-Blais et al., 2017; Castro Campos et al., 2020). Rich constraints considered
in RARPs have great practical significance and values while, at the same time, they bring additional
challenges, especially for large-scale instances. By fully recognizing rich features in transportation
networks, we aim to develop a modeling framework and solution approaches for RARPs that can
systematically examine traffic-oriented characteristics.
One of the most important “rich” features of transportation networks is time-varying traffic conditions.
Service vehicles may experience time-dependent travel times on roads when serving customers (Liu et al.,
2020; Yao et al., 2021). In early research on both vehicle routing problems (VRPs) and ARPs, travel times
on links are treated as constant or time-independent; however, in a congested urban environment, solutions
obtained with the constant travel time assumption may significantly underestimate the delay and could even
lead to infeasibility under tight schedules in real-life applications. Although many recent VRP studies
considered piecewise travel times to capture time-varying traffic conditions in a more realistic fashion, as
discussed by Vidal et al. (2021), additional efforts are still critically needed to offer more precise
approximations/representations of reality. On the other hand, in the ARP literature, time-dependent travel
times have been largely simplified or ignored (Gendreau et al., 2015). Vidal et al. (2021) first conducted
extensive studies on the ARP with time-dependent travel times at a network level and proposed methods
for quick travel and service time queries as well as quickest path queries, based on the travel speed function
definition given by Ichoua et al. (2003).
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Another important goal of RARPs on transportation networks is how to reduce the system-wide
(societal) congestion impact of service vehicle routings. RARP applications in city logistics are typically
fulfilled by large trucks, e.g., freight trucks and street sweeping trucks. A service truck with a much lower
driving speed could affect traffics on multiple lanes which leads to another important class of moving
bottleneck problems studied in the literature (see e.g., Li et al., 2020). Thus, RARP applications in city
logistics should minimize not only the total operating cost for meeting customer requests but also the
potential negative effects to the background transportation system.
1.1 Related studies
1.1.1 Time-dependent travel time modeling in vehicle routing problems and arc routing problems
The important study by Malandraki (1989) first formulated link travel time as a piecewise-constant
function of the departure time. Later on, this approach was adopted by Malandraki and Daskin (1992) and
Chen et al, (2006). Yet, this approach might not satisfy the First-In-First-Out (FIFO) property. The FIFO
property states that, among two identical vehicles that travel along the same path, the one that departs earlier
from the origin node could always arrive at the destination node earlier than the other vehicle. Two major
approaches are proposed to address the potential FIFO violation issue in the piecewise constant
representation. First, piecewise-linear functions are considered by Ahn and Shin (1991) and improved by
Fleischmann et al., (2004). In this approach, a linear transition function is built to connect travel times in
two adjacent time periods such that travel time changes slowly and smoothly. It can be proved that the FIFO
property holds if the slope of linear lines is less than 45. Another method proposed by Ichoua et al., (2003)
starts with time-dependent link travel speed in the form of a piecewise-constant function and
derives/computes piecewise-linear travel times satisfying the FIFO property. Table 1 summarizes the above
three major methods for time-dependent travel time modeling in VRP and ARP models. Xiao and Konak
(2016) offers a more detailed classification and illustration.
Figliozzi (2012) offered a comprehensive set of benchmarks for modeling time-dependent travel times
in VRP. The important modeling aspect of time discretization is further studied by Boland et al. (2017) for
a broader class of continuous-time shortest path and service network design problems. One can find the
related studies in Scherr et al. (2020), Belieres et al. (2021), He et al. (2021), Marshall et al. (2021), Vu et
al. (2022), Hewitt (2022), and Lagos et al. (2022). In particular, for the fundamentally important time-
dependent shortest path problem, the study by He et al. (2021) systematically considers degree 4 and degree
6 polynomial functions to approximate time-dependent travel time through the piecewise linear interpolants
sampled at integer points. The polynomial function can be calibrated for an extended time period (e.g., 24
hours of a day) using real world travel time data, while the related non-linear functional form could have
many local minima and maxima, which could greatly affect the computational efficiency of dynamic
3
discretization discovery algorithms (see Boland et al., 2017).
In fact, how to obtain reliable and accurate travel time estimates from field data has remained a
challenging problem for the VRP field, while many published papers still use randomly generated or
hypothetical travel time distributions for simplicity. By utilizing data from advanced traffic information
systems in Berlin, Fleischmann et al. (2004) proposed a smoothing method to ensure the resulting travel
time functions satisfy the FIFO property. By using traffic flow data collected from a Belgian highway, Jabali
et al. (2009) created a speed profile with five periods, including two periods for morning and afternoon
peak hours and three periods for the remaining non-peak hours, for each link within the research area.
Kritzinger et al. (2012) adopted 15-minute link travel time information from floating car data and developed
an extended version of Dijkstra’s algorithm to compute the distance matrices between points with various
departure times. In the paper by Gmira et al. (2021), the authors developed a discrete-event simulator that
generates travel speed updates for real-time vehicle routing applications. Different from the aforementioned
numerically driven approximation approaches, Woensel et al. (2007) first proposed an innovative queue-
theoretic modeling scheme for calibrating expected travel times which was adopted in the studies by
Woensel et al. (2008) and Lecluyse et al. (2009).
Table 1 Three major methods for representing time-dependent travel times in VRP/ARP.
Graphical illustration of methods
Description
Method type: piecewise-constant travel
time function
Modeling details: each link has a
constant travel time within each pre-
defined time period
FIFO property: not necessarily
satisfied
Method type: piecewise-linear travel
time function
Modeling details: each link has a
constant travel time within each pre-
defined time period; a linear transition
line is built between two adjacent periods
FIFO property: satisfied
Method type: piecewise-linear travel
time function
Modeling details: a piecewise-constant
travel speed function is constructed for
each link first; piecewise-linear travel
time function is then derived based on its
corresponding speed function
FIFO property: satisfied
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1.1.2 Rich arc routing problems
Considerable efforts have been devoted to the ARP and its variants. Interested readers are referred to
a number of survey papers (Wøhlk, 2008; Corberán and Prins, 2010; Corberán and Laporte, 2015; Mourão
and Pinto, 2017; Corberán et al., 2021). In this subsection, with the focus on mathematical modeling and
solution method development, we review the existing studies along two research lines, including real-life
applications of RARP on urban networks and ARP with time-dependent travel times.
(1) Three representative applications of RARP as the urban management movement problem
Sprinkler truck routing problem: Li et al. (2008) investigated the water truck routing problem in
open pit mines in which the travel time along a road for each truck is not fixed but relies on its leading truck
on the same road. The authors proposed minimum cost flow and set-partitioning based heuristics solution
algorithms. Huang and Lin (2014) formulated the street tree watering problem as the periodic arc routing
problem with refill points where the watering frequency of each street tree may not be fixed and needs to
be scheduled according to the period of watering activity. A graph transformation strategy was first adopted
to convert the original problem to a VRP with an ant colony heuristic algorithm. Riquelme-Rodríguez et al.
(2014) introduced periodic capacitated arc routing problem with inventory constraints that models the loss
of humidity on each road with an inventory consumption function. The quantity of water delivered was
considered to be fixed or variable. Two mathematical optimization models were proposed and solved by
the commercial solver CPLEX.
Street sweeping problem: Street sweeping requires a special vehicle equipped with a rotating brush
that can move along the roadside and sweep material into a container on the vehicle. Early studies (Bodin
and Kursh, 1978; Eglese and Murdock, 1991) modeled the street sweeping problem as the capacitated
Chinese postman problem and developed heuristics to find reasonable routes for the road-sweeping vehicles.
Blazquez et al. (2012) considered two extra “rich” constraints in sweeper route design: the sweepers must
visit each selected street exactly as many times as its number of street sides; certain types of turns are not
allowed to use. The original ARP was transformed into a VRP which was further solved by a nearest
neighbor heuristic algorithm.
Waste collection problem: Mourão and Almeida (2000) and Mourão and Amado (2005) examined
the waste collection problem on randomly generated networks and proposed lower bounds and a three-
phase heuristic that transforms one of the lower bound solutions to a feasible one. Maniezzo, 2004
considered additional bin compatibility, forbidden turns and one-way street constraints in the model, which
is solved by a local search-based heuristic. Ghiani et al. (2005) considered practical constraints in waste
collection, e.g., large vehicles are not allowed to use some narrow streets, or services at some sites have to
be scheduled at night to avoid traffic congestion. The problem was solved by a cluster-first route-second
based heuristic. Many other constraints include traffic regulations (Bautista et al., 2008), mobile depots
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(Del Pia and Filippi, 2006), intermediate facilities (Ghiani et al., 2001), and trip length restrictions (Ghiani
et al., 2010). Recently, Willemse and Joubert (2016a) integrated major problem features of early studies
and studied the mixed capacitated arc routing problem with time window and intermediate facilities
(MCARPTWIF). Four constructive heuristics were developed and comprehensively evaluated. Given
splitting procedures play a key role in giant tour-based heuristics and meta-heuristics, the authors further
proposed optimal and heuristic splitting procedures for the MCARPTWIF (Willemse and Joubert, 2016b).
In a follow-up study (Willemse and Joubert, 2019), three acceleration mechanisms for local search meta-
heuristics were developed to better cope with large-scale instances.
In general, there are three categories of approaches for solving RARPs, including constructive
heuristics, meta-heuristics, and exact approach (Corberán and Laporte, 2015). Constructive heuristics are
designed based on problem features and typically provide a single final solution, while meta-heuristics are
more general and can be applied to any problems. Both approaches cannot produce measurable quality of
solutions. Exact optimization approaches and related model reformulations have not been studied
extensively for RAPPs, especially with the consideration of time-varying traffic conditions.
(2) ARP with time-dependent travel times
Several VRP-related studies highlight the significant impact of time-dependent travel times on vehicle
routing and scheduling. To name a few, Malandraki and Daskin, 1992; Haghani and Jung, 2005; Chen et
al., 2006; Donati et al., 2008; Figliozzi, 2012; Dabia et al., 2013; Spliet et al., 2018; and Sun et al., 2018.
On the other hand, research on ARP with time-dependent travel times is still limited in the literature. Vidal
et al. (2021) recently offered an extensive study for the time-dependent capacitated arc routing problem
(TDCARP). Based on the piecewise-constant speed function proposed by Ichoua et al., (2003), the authors
derived a closed-form representation for link arrival time functions and developed a continuous
preprocessing approach for point-to-point quickest path query. A branch-cut-and-price exact algorithm and
a hybrid genetic search-based metaheuristic were proposed for solving the TDCARP.
1.2 Potential contributions and overview of the paper
In this paper, we focus on model reformulation and developing exact solution approaches for RARPs
in congested city transportation network. We have the following observations on the modeling challenges
in the related VRP and ARP literature. First, most studies treat travel speeds on roads as constant, while the
time-varying feature of transportation networks is largely simplified. A realistic, parsimonious and
mathematically rigorous model with a calibration workflow for time-dependent travel time is important for
VRP deployment and applications. Second, the system-wide impact of service vehicles in city logistics to
the entire transportation system (with other travelers and road users) has not been systematically studied.
6
Third, in most studies, a single model was developed for real-life RARP applications, which lacks a
comprehensive investigation and comparison on the effects of rich constraints. Finally, recognizing of the
high complexity of RAPRs, most studies developed heuristics for solving real-life problems, while exact
approaches are critically needed to offer theoretical benchmarks for quantifying the solution quality and the
degree of optimality. This paper aims to offer the following contributions to the growing list of modeling
and optimization literature for city logistics.
First, by fully recognizing finer resolution and inherent congestion in transportation networks, we
develop three different models for RARPs to handle rich constraints in real-life applications.
Second, a fluid queue model-based time-dependent travel time representation is introduced.
Accordingly, an analytical form of queue-based dynamic travel time and marginal impact is developed to
simultaneously evaluate vehicle operation costs and (societal) system impacts in RARP applications. This
leads to two time-dependent travel time representation schemes, which have not been systematically
examined in the VRP/ARP literature, including a discretized time-expanded representation scheme and a
continuous-time polynomial function representation scheme.
Third, with a real-life sprinkler truck routing problem (SRP) as the representative example of RARP,
we develop two exact solution algorithms, namely a Lagrangian relaxation based method and a branch-and-
price based method that are embedded with an enhanced parallel branch-and-bound algorithm.
The remainder of this paper is organized as follows. Section 2 presents the methodology for time-
dependent travel time modeling and congestion impact analysis on transportation networks. In Section 3,
the SRP, as an example of RARPs on transportation networks, is discussed in detail. In Section 4, three
mathematical models are proposed for the SRP from different network modeling perspectives, followed by
a detailed comparison. Two efficient solution methods, i.e., Lagrangian relaxation and branch-and-price
algorithm, are designed in Section 5 for solving the models constructed in Section 4. Comprehensive
computational experiments are conducted in Section 6 to evaluate the proposed models and solution
approaches.
2. Modeling time-dependent travel time and congestion impacts using fluid queue models
with polynomial arrival rates
To calculate the travel time of a specific link, one can divide link length by travel speed, which is in
turn derived from simulated/estimated flow/density based on empirically calibrated fundamental diagrams
(Greenshields et al., 1935). However, as pointed out by Woensel et al., (2007), in the context of vehicle
routing problems, instantaneous speed and density have to be embedded in another set of continuum flow
models (Kuhne and Michalopoulos, 1997) to characterize congestion evolution. As an alternative method,
7
queuing models hold the promise for modeling travel flows with the capability of offering analytical
evaluation and sensitivity analysis (Heidemann, 1996). However, the queueing-based approach is mainly
based on the stochastic queueing principle with under-saturated conditions such as M/M/1 or G/G/1
(Woensel et al., 2007). In this section, by adapting and extending the fluid queue model with quadratic
arrival rates proposed by Newell (2013) to represent time-dependent travel times at both link and path levels,
we introduce analytical forms that satisfy FIFO conditions and offer precise congestion impact measures
during a period of oversaturation.
2.1 Time-dependent travel time
2.1.1 Link travel time profile derived from fluid queueing model
Based on free-flow speed or cut-off speed, which is more precisely defined in congestion bottleneck
identification (Hale et al., 2016), traffic states on transportation networks can be classified into two distinct
classes: uncongested state and congested state. Under uncongested states or non-peak hours, vehicles travel
with their preferred speeds, and travel delays can be assumed to be zero without loss of generality. Under
congested states, vehicle travels are constrained by road capacity. Queues form when total inflow travel
demand exceeds road capacity. In this case, travel time of vehicles includes two parts: (1) free-flow travel
time, and (2) time spent in the queue (travel delay). Due to the dynamics of travel inflow demands, queue
length evolves for an extended congestion period, resulting in time-dependent travel time of vehicle trips.
In this study, we mainly focus on time-dependent vehicle travel time calculation during congested periods
and use constant or piecewise-linear travel times during uncongested periods.
Without loss of generality, each congested road link on transportation networks is modeled as a single
queuing system with a constant service rate that equals to the maximum link discharge rate. Fig. 1 shows a
graphical illustration of queue evolution for a road link in the classical work by Newell (2013).
8
t3
t0t2
t1
μ
λ(t)
Flow rate
t3
t0t2
t1
t3
t0t2
t1
Queue length
Cumulative number of
vehicles
Time
Time
Time
A(t)
D(t)
μ
t
Q(t)
( ) ( )w t Q t
=
Q(t)
P=t3-t0
(a)
(b)
(c)
Fig. 1. General graphical illustration of queue evolution for a road link, adapted from Newell (2013), with
a single maxima of queue length/waiting time reached at time during congestion duration .
In Fig. 1(a), the blue curve and red straight line represent the arrival rate and departure
(discharge) rate , respectively. , , and denote the time at which queue starts to form, queue starts
to dissipate, and queue disappears; is the time with the highest arrival rate. By using the second-order
Taylor approximation at time , arrival rate can be approximated by the following quadratic
function:
(1)
With the observation that , Eq. (1) can be simplified as
(2)
, where
( ) . Notice that passes two points and , arrival rate
can also be expressed by the following factored form:
(3)
With Eq. (3), time-dependent queue length can be derived as follows:
9
(4)
, where and denote the cumulative arrival and departure at time , respectively. By
introducing the queue clearance time (i.e., ), the following relationship between critical time
points can be derived from Eq. (4). The detailed derivation process can be found in Newell (2013) for the
quadratic arrival rates and cubic arrival rates by Cheng et al. (2022).
(5)
Then, Eq. (4) can also be written as:
(6)
The discharge rate , queue forming time , and queue clearance can be directly observed from field
data. Thus, time-dependent queue length in Eq. (6) only has one inflow demand curvature parameter
that needs to be calibrated from observed spatial queue length or link travel times. Interested readers can
refer to a recent paper by Cheng et al. (2022) for the detailed calibration process that connects the above
queuing model with the observations from a spatial queue representation. Finally, by integrating
, time-dependent delay can be expressed as:
(7)
In many city logistics applications, only a single data source of observed speed is available,
where observation time interval can be 5 minutes or 15 minutes. The following paragraphs describe four
steps for calibrating the key parameters of and in Eq. (7).
(1) Even without the flow count observations, we can still obtain an estimate of the ultimate road
capacity according to the facility types and speed limit. The cutoff speed can be determined
based on the well-established traffic fundamental diagram between flow, density and speed, such
10
as Greenshields model (Greenshields et al., 1935).
(2) One can determine the congestion duration , while and correspond to the timestamps at
which speed is dropping from or recovering to the cutoff speed.
(3) The average discharge rate can be estimated from the volume-speed curve and the average speed
during the congestion duration. A default value of for undersaturated links can be ultimate road
capacity .
(4) The space-mean speed can be converted to the virtual waiting time in Eq. (7), and one
can use nonlinear regression methods to calibrate parameter accordingly. Alternatively, as
, we can also quickly estimate based on the lowest speed and converted
highest waiting time.
It should be noted that, as derivations are based on the point queue model (Vickrey, 1963), in
Eq. (7) represents the delay of a link for vehicles arriving the downstream node (stop line) of the link at
time . With link free-flow travel time , time-dependent travel time of a link during congested
periods (denoted by ) can be calculated using Eq. (8) for vehicle leaving from the upstream node of
a link at time .
(8)
We next derive the range of feasibility for parameter in Eq. (8). For a queuing system on
transportation networks, the arrival and departure rates must be positive. As we have assumed departure
rate as a constant parameter, we also require that arrival rate during the analysis period be
nonnegative for any between and . The quadratic function has the lowest value at time
in the congestion period. That is, ensures , . Integrating Eq. (3) and Eq.
(5) yields Eq. (9):
(9)
, where represents the congestion duration.
Proposition 1. The time-dependent link travel time function in Eq. (8) satisfies the FIFO property
within the time period of interest .
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Proof. The FIFO property can be proved if holds for any
. Alternatively, it can be proved if
holds for any (Carey et al.,
2014). Based on the derivation of given in Eq. (8),
can be expressed as
(10)
Let , Eq. (10) can be rewritten as
(11)
It is easy to observe that
is a quadratic function of and reaches its minimum
at . That is,
(12)
By utilizing the range of feasibility for derived in Eq. (9), it is obvious that
, indicating
. Thus, function satisfying the FIFO property is proved.
2.1.2 Time-dependent path travel time
On the basis of the time-dependent link travel time function, travel time derivations and FIFO property
proof are further performed on a path level. Consider two arbitrary nodes and on a transportation
network, and there are paths from to . Take path as an example, the travel time of
path at time () can be calculated as
(13)
, where denotes the index of links in path ; is the total number of links in path ;
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represents the link travel time function of the th link in path ; is the arrival time at the upstream
node of the th link if departure from node at time .
,
when
.
Proposition 2. Path travel time function in Eq. (13) satisfies the FIFO property.
Proof. Consider two vehicles, and , departing along the same path at time and time
respectively, and . That is,
. As the travel time function of link 1 satisfies the FIFO property,
we have
. Note that all links along path satisfy the FIFO
property, we can obtain
, and apply the process recursively till
the last link , i.e.,
. Thanks to the FIFO
property on link , we can easily conclude that vehicle will arrive at node earlier than vehicle .
Thus, path travel time satisfying the FIFO property is proved.
It should be noted that the derived path travel time might still have multiple local minima and maxima,
but its structurally parsimonious form could balance the tradeoff between computational tractability and the
required level of details in representing real-world traffic congestion.
2.2 Analytical form of modeling system-wide congestion impacts to background traffic
As mentioned earlier, service vehicles used in RARP applications are typically slow-moving trucks,
thus may bring significant impacts to background traffics, especially during peak hours. In this subsection,
we further utilize calibrated queuing profile to analytically measure the system impact of service vehicles,
by following the approach proposed by Ghali and Smith (1995).
In Fig. 2, the blue and red solid lines denote the cumulative arrival and departure on a road link,
respectively. Let us assume that there is a service truck entering the link at time . Note that, similar to
Section 2.1, we still focus on the congested period, i.e., . The marginal delay arising from the
service truck is the blue dash area, which also equals to the grey area. The marginal delay includes two
parts and . is the delay experienced by the service truck, while denotes the
additional delay experienced on that link by every vehicle arriving between time and , due to the
arrival of the service truck at time , which is called system-wide (societal) congestion impact in this paper.
Therefore, with the derivation of in Section 2.1, can be calculated as follows:
(14)
13
With the consideration that service trucks typically move slower than passenger cars and use more
road resources, obtained in Eq. (14) further is multiplied by a passenger car equivalent (PCE), i.e.,
, to measure the system impact of a service truck entering a road link at time .
t3
t0
Cumulative number of
counts vehicles
Time
D(t)
t
A(t)
One service
vehicle
Link marginal delay equals this area,
which equals this grey area
w(t)SI(t)
Fig. 2. Illustration of road link marginal delay adapted from Ghali and Smith (1995).
With the link-level marginal cost formulation in Eq. (14), system-wide impact along a given path can
also be recursively derived similar to travel time derivations in Section 2.1.
3. Sprinkler truck routing: An arc routing application with rich constraints in a congested
traffic network
Vehicle travels as well as the action of wind could incur serious airborne particulate matters from roads
(Li et al., 2008), causing dust emissions that bring substantial negative effects to surrounding workers and
pedestrians. The dust in the air also reduces the visibility of roads and is therefore likely to cause traffic
accidents (Bhattachan et al., 2019). Street watering constitutes one of the most common and essential
services provided by municipal departments (Gambatese et al., 2001). In such service, sprinkler trucks, also
known as water carts or water trucks, are assembled into a fleet to spray water and wash the road surface
alongside streets in urban networks. As a crucial component in street watering operation systems, sprinkler
truck route design aims to determine a set of optimal routes for a fleet of sprinkler trucks such that total
cost is minimized while road cleaning tasks can be completed as required.
In this study, we use the SRP as an example to illustrate the modeling framework and solution
approaches for RARPs on urban networks. The SRP studied in this work is essentially a capacitated arc
routing problem with time window (CARPTW), with the consideration of following additional rich features:
14
(1) time-varying traffic conditions on urban transportation networks,
(2) turn delays at intersections,
(3) repeated cleaning services on certain links, and
(4) water refilling at water refilling stations.
It should be noted that the rich constraints listed above can also be generalized to many other city
logistics applications. For example, for the emerging electric vehicle routing problem in green logistics, the
rich constraint (4) can be changed to charging vehicles at charging stations, without changing the essence
of the resulting problem.
For a given transportation network , where and denote the set of nodes and road
links respectively, the SRP studied in this paper is to find a set of routes for sprinkler trucks such that all
cleaning tasks can be fulfilled as required and the total cost is minimized. Each link is associated
with a cleaning task. That is, link must be cleaned times within its time window ,
where and denote the earliest and latest service starting time, respectively. To clean link
once, a sprinkler truck will consume unit of water. Some nodes belonging to also serve as water
refilling stations. For simplicity, in this paper, we consider the following three assumptions on water
consumption and refilling: (a) once a sprinkler truck starts to clean a road link, it must clean the whole link.
In other words, cleaning part of a link then going to refill water is not allowed; (b) a sprinkler truck is
always refilled to its maximum water tank capacity when visiting a water refilling station, and the time used
for refilling is fixed, e.g., 5 minutes, no matter how much water left before visiting a water refilling station;
(c) sprinkler trucks are full of water when departing from their origin depot. Due to the existence of
assumption (a), the water consumption of each link should not exceed the maximum water
capacity of sprinkler trucks. In the case that is larger than , link will be split into multiple
short links, with each of which meeting the requirements mentioned above.
The total cost to be optimized is calculated by , where and represent
total operating cost and total system-wide impact cost respectively; parameter is a user-defined weight
of to measure the importance of societal cost in routing solutions. The societal impact of using a
specific link has been derived in Eq. (14). The total operation cost consists of sprinkler truck
acquisition cost and total travel time cost. We assume all sprinkler trucks used are identical, and acquisition
cost will be applied for each used sprinkler truck. Besides, the number of available sprinkler trucks is
unlimited. The travel time cost of each sprinkler truck equals to the time difference between leaving depot
and returning back to the depot, which further consists of four parts: cleaning time, deadheading time,
waiting time, and water refilling time. Deadheading means a sprinkler truck traverses a road link without
15
cleaning the link. It may occur in two situations: (a) the link does not need to clean or has been cleaned; (b)
a sprinkler has used up water and is heading to a water refilling station. For each road link, its time-
dependent deadheading time and cleaning time are calculated as follows:
(15)
(16)
where is time-dependent link delay derived in Eq. (7); represent the length of the link, and
denote the maximum speed of sprinkler trucks in the deadheading mode and cleaning mode respectively.
Waiting time represents the time gap between the arrival time and service starting time of a sprinkler on
links (due to time window), rather than the waiting time at intersections caused by control delay. Note that
we do not allow sprinkler trucks to wait if they do not have to, e.g., keep staying at a link after cleaning
service is complete.
Fig. 3 shows an illustrative example with a simple network and a route of a sprinkler truck. The
network consists of 17 nodes (intersections) indexed from 0 to 16. The depot is located at node 0, and two
water refilling stations are located at node 11 and 12 respectively. The orange line denotes an example route,
which is composed of solid lines (truck in cleaning mode) and dash lines (truck in deadheading mode). In
Fig. 3, the sprinkler truck starts from depot node 0, cleans links (0,4), (4,1), (2,5), (5,8), refills water at node
11, cleans links (16,15), (15,14), (14,13), (13,7), (7,4), (4,0), and returns back to depot node 0.
Fig. 3. An illustrative network and the route of a sprinkler truck.
16
4. Mathematical formulations of the optimization problem
In this section, three models are developed for the proposed SRP from different perspectives.
Specifically, a discretized time-expanded network based arc routing model M1 in Section 4.1, an arc-based
node routing model M2 in Section 4.2, and a path-based node routing model M3 in Section 4.3. Section 4.4
offers a comprehensive comparison.
4.1 An arc routing model based on time-expanded network
In the first model M1, the SRP is formulated as a variant of CARPTW. To enable M1 to accommodate
the need of turn delay modeling, with the intersection expansion process similar to the approach discussed
by Kirby and Potts (1969), Ziliaskopoulos and Mahmassani (1996), and Pallottino and Scutella (1998), a
new network is first constructed from the network described in Section 3.
4.1.1 Time-expanded network representation
A standard VRP or ARP typically assumes, each customer must be served once and exactly once.
Benefitting from the assumption, vehicle states (arrival time, cumulative loads) can be associated with
customers, thus a concise physical network based model can be built (Cordeau, 2006). However, in the SRP
considered in this study, some road links may be required to be cleaned multiple times, implying a road link
may be serviced multiple times by sprinkler trucks, which makes vehicle states intractable if associating
them with physical networks. As a result, we adopt a time-expanded network based modeling approach. In
the literature, the time-expanded network based modeling approach has been successfully applied in solving
a wide range of transportation supply-side optimization problems, e.g., dynamic traffic assignment (Lu et
al., 2016), VRP (Yao et al., 2019), passenger flow state estimation (Shang et al., 2019), and train timetabling
problem (Zhang et al., 2019). By extending a physical network to a time-expanded network, nodes and road
links in the original physical network are extended to vertexes and arcs with an extra time dimension. With
the extended time dimension, the multiple-services requirements can be systematically modelled as well as
time-dependent travel times.
For a time-expanded network from , vertex is constructed from physical node
, where denotes time. Arc represents a space-time traveling activity from vertex
to vertex . As the time dimension is continuous, we evenly discretize the entire planning horizon
into short time intervals, e.g., 10 seconds, so that the number of vertexes and arcs is finite. As a result,
and both represent the indices of discretized time intervals. We use to denote the
set of indices of discretized time intervals, where is the index of the last time interval. Arcs in a time-
17
expanded network consist of two categories: traveling arc and waiting arc . A
traveling arc means a vehicle enters link at time and leaves at time , and
equals the travel time of the link at time . Note that link can either be a road link or a movement
link in network . A waiting arc corresponds to the waiting activity of a vehicle at node
for one time interval. Waiting arcs are used when a sprinkler truck arrives at a road link earlier than
the link’s earliest service starting time. Fig. 4 depicts a simple three-node network with its corresponding
time-expanded network, where the travel time of link and link are 1 and 2, respectively.
Fig. 4. A simple network and its corresponding time-expanded network.
With the basic concepts introduced above, we present the time-expanded network built for the SRP in
Fig. 5, with the following problem-specific highlights:
(1) Cleaning arc and deadheading arc: When a sprinkler moves on a road link, it has two possible
modes, i.e., cleaning mode and deadheading mode. In the cleaning mode, the sprinkler truck cleans the link
when traveling on it, but not in the deadheading mode. Note that sprinkler trucks typically have different
speeds in the cleaning mode and deadheading mode. Therefore, it is necessary to build cleaning arcs (Fig.
5(a)) and deadheading arcs (Fig. 5(b)) with different travel times for each physical link. For a cleaning arc
in Fig. 5(a), equals to cleaning time on link ; for a deadheading arc arc
in Fig. 5(b), equals to deadheading time on link . Another difference between cleaning arcs
and deadheading arcs is about water consumption. If we use to denote the water consumption of
arc , if the arc is a cleaning arc; otherwise, .
(2) Time-dependent travel time: For a deadheading arc , equals to in Eq.
(15); for a cleaning arc , equals to in Eq. (16). One can observe that each traveling
arc (including deadheading arc and cleaning arc) in Fig. 5 is associated with its own travel time, which can
be viewed as a fine discretized approximation of continuous functions in Eqs. (15) and (16).
(3) Service time window: In the SRP, each link is associated with a service time window, within which
cleaning services must be started. In other words, for a specific link, cleaning arcs outside its service time
18
window are not allowed to use. We can easily impose this constraint by deleting cleaning arcs outside
service time windows when building a time-expanded network. For example, in Fig. 5(a), the service time
window of link is , then the arcs with a red cross will be deleted.
(4) Waiting arc: According to the problem description in Section 3, waiting is only allowed when a
sprinkler truck arrives earlier than link service starting time. Therefore, compared with the illustrative
example in Fig. 4, tighter restrictions are considered in building waiting arcs for the SRP to avoid invalid
waiting. First, waiting arcs are only built for inbound nodes of road links that need cleaning services. Second,
waiting arcs with entering time later than the corresponding link’s service starting time will not be generated.
For road link with time window in Fig. 5, waiting arcs are only constructed
for .
(5) Water refilling arc: Water refilling arcs are built on each water refilling station ,
where denotes the time required to refill a sprinkler truck.
(6) Origin and destination: In the newly generated time-expanded network, vertex and
will serve as the origin and destination vertex of sprinkler trucks, where and denote the
origin node and destination node in network ; is the index of the last discretized time interval in the
whole planning horizon. Waiting arcs on destination are constructed to ensure that sprinkler trucks are
able to return back to and keep staying at the destination depot after finishing cleaning tasks.
(a) Physical network and its corresponding time-expanded network (with cleaning arcs)
(b) Physical network and its corresponding time-expanded network (with deadheading arcs)
Fig. 5. A simple network and its corresponding time-expanded networks in SRP.
19
To reduce the size of time-expanded networks and simplify the subsequent optimization model, we
can further remove vertexes and arcs outside the space-time prism between the origin vertex and
destination vertex . A space-time prism is an envelope that covers all possible paths between two space-
time vertexes. Interested readers are referred to Miller (2005) and Tong et al. (2015).
4.1.2 SRP formulation on time-expanded network (model M1-TEN)
With the preparation of intersection expansion and time-expanded network construction, we now
present the arc routing formulation on time-expanded network (model M1-TEN) for the proposed SRP.
Notations are summarized in Table 2.
Table 2 Notations used in model M1-TEN.
Symbols
Definition
Indices
Index of nodes in graph
Index of time intervals
Index of sprinkler trucks
Sets
Set of links in graph
Set of sprinkler trucks
Set of vertexes in graph
Set of arcs in graph
Cleaning arc set associated with link
Set of water refilling arcs in graph
Parameters
Origin (destination) vertex of sprinkler trucks in graph
Travel time cost of arc
System impact cost of arc
General cost of arc
Number of cleaning requests of link in graph
Water consumption of arc
Water tank capacity of sprinkler trucks
Water refilling time
Sprinkler truck acquisition cost
Variables
Binary variable.
if sprinkler uses arc ; otherwise,
Water level of sprinkler at vertex
Model M1-TEN:
Objective function
20
(17)
Subject to:
Flow balance constraint:
(18)
Cleaning request satisfaction constraint:
(19)
Sprinkler truck water level updating constraint:
(20)
Decision variables:
(21)
The objective function in Eq. (17) minimizes the total cost to complete cleaning tasks, which includes
three parts: travel time cost
, vehicle acquisition cost
, and system impact cost
. The travel time
cost of arc equals to for all arcs in set , except waiting arcs built at sprinkler
destination node . We let the cost of waiting arcs at node equal to 0 (i.e., if ).
Expression
denotes the number of sprinkler trucks used. By integrating three
types of coefficient of each
, Eq. (17) can be simplified as
(22)
, where represents the general cost of arc , consisting of travel time cost, vehicle
acquisition cost, and system impact cost. Constraint (18) guarantees that the incoming flow equals to
outgoing flow on vertexes. Note that we do not impose this constraint on the origin vertex and
21
destination vertex . Constraint (19) makes sure that links are cleaned as requested, while constraint (20)
is used to update the water level of sprinkler trucks. Finally, constraint (21) specifies decision variables
with their domains. Specifically,
are binary variables, and
are positive continuous variables
with an upper bound of , where represents the water tank capacity of sprinkler trucks.
4.2 Graph transformation and a slot-based time discretization node routing model (model M2-STD)
The adoption of intersection expansion and time-expanded networks can allow the consideration of
various rich constraints into the generated network, contributing to a concise form of model M1-TEN. Yet,
additional movement links created at intersections and high-dimension time-expanded networks
significantly increase the size of model M1-TEN, making it extremely challenging to solve on large-scale
instances. In this section, we further propose a node routing model by converting the original ARP to a node
routing problem (NRP). Solving an ARP by converting it to an NRP was first proposed by Pearn et al. (1987)
and was adopted and improved by Longo et al., (2006). The core idea behind the conversion process is
treating arcs to be served as activity nodes in the NRP and building virtual edges to connect each pair of
the resulting nodes based on the shortest paths in the original network.
In the proposed SRP, due to the consideration of additional rich features, there are three major
challenges during the conversion process. First, on a congested urban transportation network with time-
varying traffic conditions, the shortest path (in terms of travel time) between two nodes changes during the
day. Second, cost of a path consists of travel time cost and system impact cost, therefore a shortest path
with the least travel time between two activity nodes does not necessarily correspond to the best path with
the least total cost. Third, due to the existence of service time window, even the best path with the least total
cost does not guarantee an optimal solution. Consider two paths (e.g., path 1 and path 2) between a pair of
activity nodes, where path 1 has lower total cost than path 2. However, path 1 takes more travel time,
causing some customers that are served in the optimal solution cannot be visited. This may result in a
suboptimal routing solution. To address this issue, we keep paths between each pair of activity nodes in
the resulting NRP. The details of the transformation process are illustrated in Appendix A.
Due to the adoption of nonlinear functions for representing time-varying traffic conditions in the
converted graph , i.e., , , , and , one can expect that an optimization
model built on is highly nonconvex and extremely hard to solve (see Appendix B). Therefore, in model
M2-STD, nonlinear functions are approximated by piecewise-constant functions. That is, the entire
planning time horizon is split into multiple time slots with the same duration, e.g., 10 minutes, and
nonlinear functions are approximated by constants within each slot. Model M2-STD as well as additional
notations is presented as follows.
22
Table 3 Additional notations used in the node routing model.
Symbols
Definition
Indices
Index of nodes in graph
Index of edges between each pair of nodes in graph
Index of time slots
Sets
Set of nodes in graph
Set of service nodes in graph
Set of water refilling stations in graph
Set of edges in graph
Set of time slots
Parameters
Number of edges between each pair of nodes in graph
Service time of node in slot
Travel time of edge in slot
Congestion impact of serving node in slot
Congestion impact of using edge in slot
Water consumption of node
The earliest (latest) service starting time of node
Ending time of slot for node
Variables
Binary variable. if a sprinkler uses edge in slot ; otherwise,
Binary variable. if cleaning service starts on node in slot ; otherwise,
Water level of a sprinkler at node
Departure time of a sprinkler from node
Service starting time of node
Model M2-STD:
Objective function
(23)
Subject to:
Cleaning request satisfaction constraint:
(24)
Sprinkler truck spatial route constraint:
(25)
23
(26)
Sprinkler truck temporal route constraint:
(27)
(28)
(29)
(30)
(31)
(32)
Time window constraint:
(33)
Sprinkler truck water level updating constraint:
(34)
(35)
(36)
Decision variables:
(37)
The objective function in Eq. (23) aims to minimize the total cost, where represents the origin
node of sprinkler trucks in graph , and
denotes the copy of destination node for sprinkler
in graph . By creating a dummy copy
of the destination depot for each sprinkler truck , the
arrival time of sprinkler truck at the destination depot
is the time used by sprinkler truck . Note
that since service time at destination depots is zero, arrival times and departure times at destination depots
are the same, thus we can use departure time
in the objective function to avoid defining extra variables.
Constraint (24) ensures that each service node is serviced exactly once. Constraint (25)-(26) and (27)-(32)
formulate the spatial and temporal route constraints of sprinkler trucks, respectively. In constraint (29),
denotes the service time of node if service starts in time slot . For service nodes, equals to the
cleaning time of corresponding road links in slot . For water refilling stations and depot nodes,
24
equals to the water refilling time and zero, respectively. Constraint (33) makes sure that cleaning services
start within preset time windows. Constraints (34)-(36) are used to track the water level updating of
sprinkler trucks. Finally, constraint (37) specifies decision variables and their domains used in model M2-
STD.
In addition to the essential constraints (24)-(37), two sets of constraints are further constructed below
to tighten the linear relaxation of model M2-STD. Specifically, constraint (38) ensures that at least one
sprinkler is needed to complete cleaning tasks. Note that, as sprinkler trucks are allowed to refill water
during their trips, we cannot calculate the minimum number of sprinkler trucks using total water needed
divided by water tank capacity of sprinkler trucks. Although constraint (38) seems to be very loose, it does
serve as the lower bound of many instances in the numerical experiment section. Constraint (39) is used to
break the symmetry of model M2-STD.
(38)
(39)
4.3 Path-based set partition formulation with continuous-time representation (model M3-CTR)
In this section, we further propose a path-based model (set partition formulation) on the graph
constructed in Appendix A. Instead of time discretization, the original continuous polynomial form of
service time function and travel time function is adopted. As pointed by Boland et al. (2017), the granularity
of the discretization has an impact on both candidate solutions and the computational tractability. Let
denote a feasible sprinkler truck route, where represents the set of all feasible routes. A feasible route
should satisfy constraints (B3)-(B12) in Appendix B. The set partition formulation can be expressed as
follows:
Model M3-CTR:
Objective function
(40)
Subject to:
Cleaning request satisfaction constraint:
25
(41)
Decision variables:
(42)
The objective function (40) minimizes the cost of all selected routes, where is a binary variable
indicating whether route is chosen in a solution or not; is the cost of route , including operation
cost and system impact. Constraint (41) states that all service nodes are serviced once, where is a
mapping coefficient between service node and route (number of times that route passes activity
node ).
4.4 Model comparison in terms of time-varying travel time representation and model formulation
An illustrative example with two consecutive links is first used to show the difference in time-
dependent travel time modeling among the three models. Table 4 presents the sample values for the
parameters of polynomial travel time functions proposed in Section 2. The time unit in this example is
minute.
Table 4 Parameters of polynomial travel time functions.
Link
Link travel time function
1-2
3
16
15
1200
1.36
2-3
4
18
12
1000
1.54
Fig. 6 shows the travel time functions (blue lines) used in the three models, among which piecewise-
constant functions depicted in Fig. 6(b) correspond to the first method in Table 1, while functions in Fig.
6(a) and Fig. 6(c) are the two proposed approaches in this study. For model M1-TEN, the time dimension
is discretized into 0.2-minute time intervals. Therefore, both link travel times and node arrival (departure)
times are approximated with a 0.2-minute resolution. For model M2-STD, the entire time horizon is split
into four 5-minute slots, and travel times remain constant in each slot. For model M3-CTR, the original
polynomial form of travel time functions is used.
26
(a) Model M1-TEN (b) Model M2-STD (c) Model M3-CTR
Fig. 6. An illustrative example of time-dependent travel time modeling in three models, ranging from
high-fidelity discretization M1-TEN, semi-dynamic slot-based discretization M2-STD and continuous-
time representation M3-CTR.
In Table 5, a comprehensive summary of the three proposed models is further provided. For the
formulation with continuous-time representation M3-CTR, it is important to use a well-behaved travel time
function satisfying the FIFO property; otherwise, unnecessary and unrealistic waiting could be observed
from the final solutions. In Section 6, extensive numerical experiments will be performed to examine the
computational efficiency of the three models.
27
Table 5 Comparisons among the three proposed models.
Model M1-TEN
Model M2-STD
Model M3-CTR
Model type
Arc routing model (MILP)
Arc-based node routing model (MILP)
Path-based node routing model
(MILP)
Network type
Time-expanded networks from physical networks
Activity node-based networks from physical networks
Rich
constraint
handling
(1) TV
Embedded in arcs; approximated with a fine
resolution
Piecewise-constant approximation; extra slot
dimension
Directly using polynomial
travel time functions for an
entire congestion period
(2) TD
Intersection expansion
Embedded in edges between activity nodes
(3) RS
Allow multiple visits along extra time dimension
Dummy activity nodes
Dynamically considered when
constructing a path
(4) WR
Water refilling arcs
Dummy water refilling nodes
Main decision
variables
,
,
,
,
Side constraints
Flow balance constraint;
Cleaning request satisfaction constraint;
Sprinkler truck water level updating constraint
Sprinkler spatial route constraint;
Sprinkler temporal route constraint;
Cleaning request satisfaction constraint;
Time window constraint;
Sprinkler truck water level updating constraint
Cleaning request satisfaction
constraint
Features
Embed various rich constraints into time-
expanded networks, resulting in a concise model
formulation;
Fine time-dependent travel time modeling
Turn delays are considered in the activity node-based graph construction stage,
without increasing the complexity of subsequent optimization models
Less number of decision variables compared
with the other two models
Only one set of side constraint
Challenges
A large number of additional decision variables
are introduced due to intersection expansion and
the use of time-expanded network
Time-dependent travel times are approximated
with piecewise-constant functions and
modeled with an additional slot dimension
which increases the model size (complexity)
For large-scale instances,
feasible paths cannot be
explicitly numerated.
Notes: MILP – mixed integer linear programming, TV - time-varying traffic conditions on urban transportation networks, TD - turning delays at intersections, RS
- repeated cleaning services on certain links, WR - water refilling at water refilling stations.
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5. Exact solution methods
In this section, we develop two exact solution methods for efficiently solving the proposed models.
Specifically, a Lagrangian relaxation (LR) based method is proposed for solving model M1-TEN, while a
branch-and-price (BnP) based approach is proposed for solving model M3-CTR.
5.1 Lagrangian relaxation
The concise form of model M1-TEN provides the possibility of using decomposition techniques to
obtain good solutions quickly. Model M1-TEN consists of three sets of side constraints, among which only
cleaning request satisfaction constraint (19) couples different sprinkler trucks. If constraint (19) is relaxed,
model M1 can be decomposed into multiple easier sprinkler-specific sub-problems, which can be solved
independently. In this work, we propose a LR-based method for solving model M1-TEN.
Under the LR framework, coupling constraint (19) is relaxed, and the violation is penalized in
objective function (17) using Lagrangian multipliers . The relaxed problem is
Objective function
(43)
Subject to:
(18), (20), (21).
By reorganizing the coefficients of decision variable
, the objective function (43) can be further
simplified to Eq. (44). Given the value of , the relaxed problem needs to find the least cost shortest path
on the discretized time-expanded network with modified arc cost
for each sprinkler truck, which
can be exactly solved by the dynamic programming approach (Mahmoudi and Zhou, 2016; Yao et al., 2019).
(44)
We adopt the classical sub-gradient method to update the values of Lagrangian multipliers using
Eq. (45) and Eq. (46), where
and denote the value of and step length at iteration ,
respectively. It should be noted that, as constraint (19) is an equality constraint, Lagrangian multipliers can
be positive, negative, or 0. A positive Lagrangian multiplier means an additional penalty when serving the
29
corresponding arc, while a negative value means an additional bonus. Lagrangian iteration stops when
constraint (19) is satisfied on all links, i.e., no Lagrangian multipliers updating in Eq. (45).
(45)
(46)
5.2 Branch and price algorithm
For model M3-CTR, the number of feasible routes in set can be extremely large in large-scale
problems, making it nearly impossible to enumerate all routes and solve the resulting model using existing
solvers. In this section, we further propose a Branch and Price (BnP)-based exact solution approach to solve
model M3-CTR. The proposed BnP algorithm consists of two modules, i.e., column generation (CG) and
bound-and-bound (BnB). Model M3-CTR is denoted as the master problem (MP), then we need to use CG
module to solve the linear master problem (LMP), while the BnB module is needed to obtain integer
solutions based on the results from the CG module. The LMP is the same as MP except the domain of
decision variables. In LMP, we relax the integer requirement of decision variable , i.e., . Due
to the existence of constraint (41), we can further simplify the domain constraint to .
5.2.1 Column generation
The LMP is still hard to solve due to the large size of route set . Therefore, instead of directly solving
the LMP, we iteratively solve its corresponding restricted linear master problem (RLMP), whose route set
only contains part of feasible routes and is extended with new routes as needed. The RLMP is presented
as follows:
Objective function
(47)
Subject to:
Cleaning request satisfaction constraint:
(48)
Decision variables:
30
(49)
In this paper, the initial routes in set are generated by assigning one sprinkler for each service node.
Note that, when generating initial routes, if a node cannot be serviced due to the violation of time window
constraint or water consumption constraint, the original problem is then infeasible. The route set in
RLMP is extended by solving the so-called pricing problem, during which routes with negative reduced
cost are added into set .
(a) Formulation of the pricing problem
The pricing problem is formulated as follows:
Objective function
(50)
Subject to:
constraints (B3)-(B12) in Appendix B.
Objective function (50) minimizes the reduced cost of a feasible route, where is the reduced cost
associated with node . Essentially, the optimization problem presented above is an elementary shortest
path problem with resource constraint (ESPPRC).
(b) Dynamic programming algorithm for the pricing problem
Finding an elementary shortest path with resource constraint in a network is an NP-hard problem.
Therefore, finding the exact solution for the aforementioned pricing problem may be extremely time-
consuming in large-scale instances. In the literature, to reduce the computational complexity of the pricing
problem, following the pioneering work by Christofides et al. (1981), researchers started to develop
efficient algorithms for finding non-elementary shortest paths with resource constraint through relaxing the
elementary requirement, i.e., allowing visiting a node multiple times (Irnich and Villeneuve, 2006; Baldacci
et al., 2011; Martinelli et al., 2014). The use of non-elementary shortest paths will not affect finding the
optimal solution of the original problem but will weaken its linear relaxations. In this paper, we adopt the
dynamic programming ng-route algorithm proposed by Martinelli et al. (2014) with some modifications.
31
Specifically, in Martinelli et al. (2014), a vehicle only needs to collect goods during its trip, therefore the
load of a vehicle keeps increasing along with its trip. As a result, a two-dimension matrix with vehicle load
as one axis can be created, and the dynamic programming process can be performed in the order of vehicle
load increasing. However, in the SRP considered in this work, sprinkler trucks are allowed to refill water
during their trips, hence the water level changes on sprinkler trucks are not monotonous anymore. Another
challenge is that demands are assumed to be discrete in Martinelli et al. (2014), while it is continuous in the
proposed SRP, thus creating a matrix with sprinkler water level as an axis is impossible.
We first introduce the three key components of a dynamic programming algorithm, including state
(label) definition, extension rule, and dominance rule, followed by the modified ng-route algorithm
developed in this paper.
Label definition. Labels used in the pricing problem have the following form:
, where means current node label is on; , , denotes the
reduced cost, the actual service starting time, and the water level of the sprinkler at current node,
respectively; stands for the previous label of label ; represents the set of forbidden nodes that
are not allowed to extend from label , which will be introduced in the extension rule below.
Extension rule. The process of extending current label to node via edge is presented in
Algorithm 1. In Algorithm 1, line 2-4 check if node is reachable from current node ; line 7 checks if
the destination depot is reachable after visiting node . Line 12 creates a new label on node from
label via edge . Specifically, the ng-set of node is used when updating forbidden set . For
each node , a ng-set is predefined which includes a certain number of nearest nodes of (including
itself). A label on node only includes the intersection of visited nodes and into its forbidden set ,
resulting in the possibility of forming cycles. Generally, the larger the ng-sets are, the less likely to contain
cycles in routes, and also closer to the original ESPPRC (harder to solve). Water refilling stations will also
not be put into forbidden sets as they are always allowed to revisit in the SRP settings.
Algorithm 1 Label extension procedure.
Input: label , node service time function , edge travel time function ,
system impact functions and
Output: new label on node
1:
label
2:
if and then
3:
arrival time at node :
4:
if then
5:
the earliest service starting time at node :
6:
the earliest time to return back to the depot after visiting node :
32
7:
if then
8:
9:
10:
11:
12:
label
13:
return label
Dominance rule. For two labels and on the same node, label is dominated by label if the
following conditions are satisfied and at least one of them is not equal:
(i) ,
(ii) ,
(iii) ,
(iv) .
It should be noted that condition (i) is much tighter than the condition that is commonly used in VRPs with
the objective of minimizing total travel distance, i.e., . A stricter dominance rule means less
labels can be dominated, thus there are more labels to be processed (the pricing problem is harder to solve).
If label is dominated by other labels, label can then be safely discarded without affecting the final
optimal results.
The dynamic programming ng-route algorithm is presented in Algorithm 2. Algorithm 2 finds the best
route with given graph and ng-sets , which may be very time-consuming in large instances. When
used under CG, to save computing time, Algorithm 2 can be terminated as soon as a certain number of paths
with negative reduced cost have been found.
Algorithm 2 Dynamic programming ng-route algorithm.
Input: node-based graph , ng-set for each node
Output: the best ng-route
1:
create a root label
2:
3:
while do
4:
choose a label with the minimum service starting time from , and delete from
5:
for do
6:
for do
33
7:
if label can be extended to node via edge (feasibility checking in Algorithm 1) then
8:
create a new label on node from label
9:
10:
for label on node do
11:
if label dominates then
12:
13:
break
14:
else if label dominates then
15:
delete
16:
if then
17:
18:
best label the label with the least reduced cost on the destination depot node
19:
retrieving the best route from label
20:
return the best route
(c) Acceleration techniques
Although finding non-elementary shortest paths using Algorithm 2 is faster than finding the exact
solution for elementary shortest path problem, the pricing problem is still hard to solve, especially in large-
scale instances. In this work, we further utilize the following three techniques to speed up Algorithm 2.
Decremental state-space relaxation. It is observed that the larger ng-set of nodes are, the harder it is to solve
the pricing problem. Therefore, instead of directly using the complete ng-set , Algorithm 2 can start with
a subset of ng-set , i.e., , for each node and iteratively add nodes from to until a valid ng-
route is found. A node to be added to should satisfy the following requirement: the absence of that node
in leads to a cycle in the final route that prevents the route from being an ng-route.
Heuristic domination. The first condition in the dominance rule presented above is by far stricter than the
condition used in VRPs with the objective of minimizing total travel distance, which is . Using
the condition is of course helpful to dominate more labels, then accelerates the pricing
process. However, some valid labels will also be discarded, which may affect the final solution. In our
experiments, we found that only a small portion of valid labels are discarded due to the use of the loose
domination condition. In other words, a relatively good solution can still be obtained with condition
in a much shorter time. Therefore, before using the exact dominance condition, i.e.,
, the loose condition will be used until no valid route can be found.
Truncated labels. As in Yao et al. (2021), a truncated version of dynamic programming is used before calling
the exact dynamic programming presented in Algorithm 2. In the truncated version, each node only
maintains a limited number of promising labels, e.g., 100 labels with the least reduced cost. Note that, as
the number of labels in the truncated version is limited, running one iteration of the truncated version is
very fast, therefore we do not allow node revisiting and do not use heuristic domination such that routes
34
with higher quality can be obtained.
5.2.2 Branch and bound
An optimal solution obtained from the CG module may contain fractional path usages, making it
infeasible for the MP (model M3-CTR). Therefore, BnB module is further utilized to produce an integer
optimal solution based on results from the CG module.
(a) Branching rule
If an optimal solution obtained from the CG module is fractional, the BnB module will perform
branching on an edge with a fractional usage. The branching process is presented in Algorithm 3.
Algorithm 3 Edge-based branching procedure.
Input: path usages from a CG solution
Output: two child nodes of the current node
1:
Calculate edge usages based on path usages from a CG solution
2:
Select an edge with its usage closest to 0.5
3:
(Child node one) remove edge from the graph used in the pricing problem
4:
(Child node two) remove edge set , edge set
, and edge set from the graph used in the pricing
problem, but keep edge .
5:
return child node one, child node two
Child node one forces sprinkler trucks not to use edge , while child node two forces sprinkler trucks
to use edge if service node is in their routes.
(b) Searching strategy
A searching strategy determines the sequence of solving active BnB nodes, and different strategies
may affect the performance of the BnB module for a specific problem. Breadth-first search (BFS) and
depth-first search (DFS) are the two widely used searching strategies. Generally, BFS is helpful to improve
the global lower bound (GLB) of a problem, while DFS is useful to quickly find a feasible solution thus
contributes to improving the global upper bound (GUB) of the problem. Benefitting from mature parallel
computing technologies, both BFS and DFS are implemented and run in parallel in this work. The overall
framework of the BnB module designed in this paper is presented in Fig. 7.
From Fig. 7, in our implementation, in addition to performing BnB-DFS loop and BnB-BFS in parallel,
as nodes in the same layer in BnB-BFS are independent from each other, solving nodes in Step 2b.1 is also
conducted in a parallel manner.
35
start Step 1 Initialization
1. Initialize the global lower bound (GLB) and upper bound (GUB)
2. Create a root node
Step 2a BnB-DFS loop Step 2b BnB-BFS loop
Step 2a.2 Get a node from ANL and solve
1. If time limit is reached or BnB-BFS loop
has finished, jump to step 3
2. get the last node from ANL
3. solve the node using CG module
Step 2a.3 Update BnB tree
if obj < GUB
if ns is integer
GUB = obj
else
create two child nodes and put them to
the end of ANL
endif
endif
Step 2a.1 Create an active node list ANL,
and put root node into ANL Step 2b.1 Create a layer node list LNL,
and put root node into LNL
Is ANL empty
Step 2b.2 Solve nodes in LNL in parallel
1. If time limit is reached or BnB-DFS loop
has finished, jump to step 3
2. solve each node in LNL using CG module
Is LNL empty
Step 2b.3 Update BnB tree
1. let LB = GUB
2. create a new layer node list NLNL
3. for each solved node in LNL
if obj < GUB
if ns is integer
GUB = obj
else
if obj < LB: LB = obj
create two child nodes and put
them into NLNL
endif
endif
3. Update GLB with LB, LNL=NLNL
Step 2 BnB loop (do step 2a and 2b in parallel)
Step 3 Results
Output the final global lower bound GLB and global upper bound GUB
Output the final best feasible solution
No
Step 3 Step 3
No
Yes
Yes
Note: obj means the objective value of a BnB node; ns represents the solution state of a BnB node, it can be integer or fractional
Fig. 7. The overall framework of the BnB module.
6. Computational experiments
In this section, extensive numerical experiments are conducted to evaluate the methods developed in
this paper. Specifically, 12 corridors in the Washington DC metropolitan area are first selected to
demonstrate the suitability of the proposed time-dependent travel time modeling method. Based on
calibrated travel time functions, three sets of SRP instances with different sizes are designed to examine the
performance of the three models, followed by an analysis on the system impact of vehicle routings. Finally,
sensitivity analysis of solution methods is performed.
36
6.1 Modeling of time-dependent travel time
Fig. 8 presents the 12 corridors selected for evaluating the proposed time-dependent travel time
modeling method, including 1 expressway corridor, 3 freeway corridors, and 8 arterial corridors that
experience different levels of traffic congestion. Table 6 summarizes the characteristics of the selected
corridors. The time horizon of interest is set as 6:00 – 20:00. Note that transportation networks typically
experience two or three congestion periods across a day (morning peak hours, afternoon peak hours, and
possibly midday hours), while the method proposed in Section 2 is suitable for a single congestion period
analysis. Hence, we split the analysis time horizon into three periods, including am (6:00-10:00), md (10:00-
14:00), and pm (14:00-20:00), then sequentially apply the method for each period. The dataset used for
method evaluation was collected from the Regional Integrated Transportation Information System (RITIS).
The raw dataset contains 5-minute space-mean traffic speed data from loop detectors and probe vehicles
for a majority of links along each corridor. We first calibrate congestion duration , average discharge rate
, and inflow demand curvature parameter using the four-step method proposed in Section 2.1.1. Then
smoothed time-dependent travel times are estimated based on Eq. (8) for each link along corridors.
US15 South_S
VA267 West_W
RT7 North_W
I66 outside Beltway_W
VA659_S
VA28 North_N
US29 West_W
US29 Middle_W
VA234_N
I395_E
US1 South_N
VA286 South_N
US1 Middle_N
US50 West_W
Fig. 8 Corridors used for time-dependent travel time modeling evaluation.
Table 6 Characteristics of the 12 corridors.
37
Corridor name
Type
# of
links
Corridor length
(mile)
LLS
(mph)
HLS
(mph)
ALS
(mph)
ALCD
(hour)
US15 South_S
Arterial
39
28.6
8.5
60.8
39.41
2.73
VA267 West_W
Freeway
20
13.0
16.6
74.7
63.80
0.49
VA659_S
Arterial
17
10.3
8.9
45.1
31.18
2.39
RT7 North_W
Arterial
36
19.7
14.0
62.1
44.76
3.68
VA28 North_N
Expressway
28
13.8
19.2
68.1
60.59
1.10
I66 outside Beltway_W
Freeway
50
27.9
15.0
74.7
58.47
2.93
US29 West_W
Arterial
27
13.1
14.0
55.9
42.19
2.82
US29 Middle_W
Arterial
17
11.8
11.5
51.2
26.45
6.76
VA234_N
Arterial
25
21.8
20.1
57.2
43.40
2.60
VA286 South_N
Arterial
27
19.1
14.4
58.1
47.15
1.40
I395_E
Freeway
28
9.7
6.1
65.8
43.26
5.63
US1 Middle_N
Arterial
14
13.7
8.8
52.7
36.88
4.74
Notes: HLS – highest link speed, LLS – lowest link speed, ALS – average link speed, ALCD – average link congestion
duration for an entire day
Table 7 compares the observed speed and modelled speed calibrated from the fluid-queue model with
polynomial arrival rates. The measure in terms of travel speed is selected as it is more intuitive than link
travel time depending on a specific link length. First, reasonable modeling accuracy was achieved on all
corridors with varied degrees of congestion. The heavily congested corridor I395_E has the highest mean
link MAE (7.55 mph) and mean link MAPE (27.27%), as it has a relatively long congestion duration (5.63
hours for all three congestion periods during the entire day in Table 6). It should be remarked that, under a
low average traffic speed associated with a heavy congestion, a typical deviation in absolute speed values
could still result in a relatively large percentage difference, due to the smaller value in the denominator. In
this regard, the mean link-based MAPE of corridor I395_E is still explainable. Second, for each corridor,
max link MAE are not significantly worse than corresponding mean link MAE, indicating that the
approximation error is reasonably bounded along each corridor. In short, better calibration results are
observed on the corridors with a shorter congestion duration, whereas the corridors with severer congestion
have larger modeling errors (e.g., corridor US29 Middle_W and I395_E).
Table 7 Comparison between observed and modeled speed across all links and 15-min interval resolution
on 12 selected corridors.
Corridor name
Mean link MAE (mph)
Mean link MAPE (%)
Max link MAE (mph)
US15 South_S
2.77
8.83
4.77
VA267 West_W
2.86
4.52
3.22
VA659_S
2.38
8.48
3.62
RT7 North_W
4.09
11.04
5.91
VA28 North_N
2.47
4.55
5.03
I66 outside Beltway_W
4.35
9.10
7.52
US29 West_W
2.91
7.75
6.34
US29 Middle_W
4.02
16.92
7.64
VA234_N
3.20
8.14
7.53
VA286 South_N
3.05
7.47
5.94
I395_E
7.55
27.27
12.50
38
US1 Middle_N
3.41
11.08
6.99
Notes: MAE – mean absolute error, MAPE – mean absolute percentage error
To further examine the travel time modeling accuracy, Fig. 9 and Fig. 10 present the heatmap
comparison between the observed and modeled travel speed on two corridors with a relatively poor
modeling accuracy in Table 7, i.e., corridor US29 Middle_W and I395_E. On both corridors, the overall
modeled speed pattern is closed to the corresponding observed one. Specifically, areas affected by
congestion propagation (grey dash rectangles) have been realistically captured.
6:00 8:00 10:00 12:00 14:00 16:00 18:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00
Time Time
Traffic flow direction
40
30
20
10
0
50
(mph)
(a) Observed speed (b) Modeled speed
Fig. 9. Heatmap comparison between observed and modeled speed on corridor US29 Middle_W.
0
10
20
30
40
50
60
6:00 8:00 10:00 12:00 14:00 16:00 18:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00
Time Time
Traffic flow direction
(mph)
(a) Observed speed (b) Modeled speed
Fig. 10. Heatmap comparison between observed and modeled speed on corridor I395_E.
Focusing on two links with different traffic congestion patterns from corridor I395_E and corridor
US1 Middle_N, Fig. 11 provides the time-dependent observed and modeled speed across the analysis time
horizon. The congestion occurs in the periods of am and md on link 32609, whereas only am experiences
significant delay on link 31948. One can observe that, for the link with a single congestion period (link
31948), the modeled speed closely matches the observed speed. However, for the link with multiple
congestion periods (link 32609), there are still obvious deviations between two lines, which highlights the
challenges in modeling complex and heavy traffic congestion.
39
(a) Link 32609 on corridor I395_E (b) Link 31948 on corridor US1 Middle_N
Fig. 11. Modeled speed and observed speed comparison on two links.
6.2 Performance evaluation of the solution methods
As summarized in Table 8, three sets of instances with different sizes are designed to evaluate the three
optimization models developed in this paper. Instance networks consist of expressway, freeway and arterial
link. Travel time functions calibrated on the 12 representative corridors are applied to links in benchmark
instances according to their road types.
The first instance set includes 15 small-size instances. The average number of road links to be cleaned
is 16. The length of links in the first set ranges from 0.07 to 0.74 mile. The second set contains 15 medium-
size instances, where the number of road links ranges from 28 to 38. The last set includes 13 large-size
instances, with 62 links to be cleaned on average for each instance network. For all of the instances, the
planning horizon is 3 hours. The vehicle capacity is 1200 gallons of water, and it is assumed that cleaning
1 mile of road consumes 400 gallons of water. The speed limit of sprinkler trucks in the cleaning mode and
deadheading mode is set as 5 mph and 14 mph, respectively. We assume that there is no limit on sprinkler
fleet size, but each sprinkler has an acquisition cost of equivalent 10 units of travel time. Note that, in this
subsection, as we mainly focus on efficiency evaluation on the three proposed models, the weight of system
impact cost is set as 0. The tradeoff between societal impact and private operating cost will be performed
in Section 6.3.
Table 8 Characteristics of three benchmark instance sets.
Small-size benchmark instance set
Medium-size benchmark instance set
Large-size benchmark instance set
Instance
Links
MinLL
MaxLL
Instance
Links
MinLL
MaxLL
Instance
Links
MinLL
MaxLL
S1
16
0.19
0.53
M1
28
0.14
0.68
L1
72
0.04
0.68
S2
14
0.22
0.28
M2
36
0.11
0.65
L2
66
0.11
0.65
S3
16
0.09
0.29
M3
28
0.09
0.31
L3
60
0.09
0.54
S4
16
0.07
0.24
M4
30
0.14
0.42
L4
54
0.08
0.42
S5
16
0.13
0.50
M5
30
0.15
0.5
L5
60
0.13
0.5
40
S6
14
0.10
0.46
M6
38
0.13
0.44
L6
74
0.13
0.54
S7
18
0.18
0.50
M7
32
0.18
0.5
L7
62
0.18
0.63
S8
14
0.07
0.18
M8
34
0.07
0.49
L8
62
0.07
0.49
S9
16
0.13
0.37
M9
30
0.13
0.56
L9
50
0.13
0.56
S10
16
0.08
0.31
M10
28
0.15
0.33
L10
62
0.09
0.62
S11
15
0.09
0.68
M11
32
0.07
0.23
L11
60
0.07
0.7
S12
14
0.10
0.44
M12
30
0.09
0.41
L12
60
0.09
0.38
S13
18
0.09
0.23
M13
28
0.07
0.7
L13
58
0.07
1.43
S14
18
0.09
0.27
M14
32
0.09
0.37
S15
16
0.15
0.74
M15
28
0.1
0.74
Average
16
0.12
0.40
Average
31
0.11
0.49
Average
62
0.10
0.63
Notes: MinLL and MaxLL represent minimum link length and maximum link length respectively, with the unit of
mile
In the results presented below, model M1-TEN is solved by the LR method presented in Section 5.1,
M2-STD is directly solved by a commercial MILP solver (CPLEX), M3-CTR is solved by the BnP
algorithm presented in Section 5.2. Both LR method and BnP algorithm proposed in this paper are coded
in C++. The RMLP in BnP is solved with CPLEX by calling its built-in C++ API. We use CPLEX solver
version 12.10 throughout the experiments. All numerical experiments conducted in this study are evaluated
on a 64-bit Linux server with Intel Xeon Gold 6230R processor @ 2.10 GHz and 180 GB RAM.
The results of the proposed models on the three sets of real-life instances (small-size, medium-size,
large-size) are shown in Table 9, Table 10, and Table 11, respectively. For each instance, Table 9 - Table 11
provide lower bound (LB), upper bound (UB), Gap, solution time (ST), and memory usage (MU) of results
obtained from the proposed models on the three sets of instances, respectively. The Gaps in tables are
relative gaps, which are calculated by
. The solution time limit is set as 15
minutes for small-size instances, 30 minutes for medium-size instances, 60 minutes for large-size instances.
The time discretization resolution of model M1-TEN is set as 0.2 min, while a detailed comparison between
different resolutions is provided in Section 6.4 and Appendix C. For model M3-CTR, the size of ng-set is
set as 2, and the impact of different ng-set size is presented in Appendix D.
Table 9 Performance comparisons of three proposed models on small-size instances.
Instance
M1-TEN (LR)
M2-STD (MILP Solver)
M3-CTR (BnP)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
S1
90.1
131.4
31.43
52
0.1
23.34
112.38
79.23
900
22.33
112.38
112.38
0.0
1
0.0
S2
60.8
78.4
22.45
8
0.0
15.15
78.07
80.59
900
21.74
75.65
75.65
0.0
1
0.0
S3
49.0
76.2
35.7
39
0.26
10.0
73.03
86.31
900
22.46
73.03
73.03
0.0
13
0.0
S4
28.0
53.6
47.76
56
0.33
11.28
51.87
78.25
900
26.97
51.85
51.85
0.0
125
1.08
S5
61.67
84.0
26.59
52
0.19
12.59
83.24
84.87
900
22.68
82.36
82.36
0.0
3
0.0
S6
56.07
81.6
31.29
50
0.17
21.09
77.05
72.63
900
20.71
77.05
77.05
0.0
2
0.0
41
S7
96.11
132.0
27.19
71
0.15
15.32
131.87
88.38
900
21.72
131.87
131.87
0.0
2
0.0
S8
26.8
48.8
45.08
38
0.26
11.76
45.84
74.35
900
17.79
45.84
45.84
0.0
165
2.27
S9
68.8
82.8
16.91
42
0.16
13.24
80.28
83.5
900
13.67
80.02
80.02
0.0
1
0.0
S10
53.6
71.4
24.3
57
0.23
12.44
71.12
82.5
900
20.15
71.12
71.12
0.0
12
0.0
S11
81.05
130.8
38.04
53
0.08
62.99
130.69
51.8
900
19.88
130.69
130.69
0.0
1
0.0
S12
48.0
72.4
33.7
51
0.2
15.4
72.27
78.69
900
19.02
59.48
59.48
0.0
1
0.0
S13
46.8
80.4
41.79
138
0.31
11.32
78.9
85.65
900
27.19
78.53
78.53
0.0
29
0.0
S14
34.0
57.2
40.56
159
0.54
10.0
56.35
82.25
900
4.1
54.49
56.29
3.2
900
0.6
S15
89.84
116.0
22.55
174
0.21
26.66
113.35
76.48
900
26.75
111.4
111.4
0.0
1
0.0
Average
32.4
69
0.21
79.03
900
20.48
0.21
84
3.95
Table 10 Performance comparisons of three proposed models on medium-size instances.
Instance
M1-TEN (LR)
M2-STD (MILP Solver)
M3-CTR (BnP)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
M1
200.07
391.6
48.91
1800
0.61
151.12
212.76
28.97
1800
77.36
206.64
211.4
2.25
1800
89.2
M2
184.14
383.4
51.97
1800
1.21
38.14
-
-
1800
17.23
221.28
221.28
0.0
25
0.0
M3
26.0
147.4
82.36
1800
1.17
10.0
146.64
93.18
1800
62.52
130.94
146.64
10.7
1800
65.06
M4
92.9
212.6
56.3
439
0.79
53.32
193.07
72.39
1800
24.98
190.07
190.07
0.0
8
0.0
M5
111.24
259.6
57.15
1800
1.29
232.48
232.48
0.0
1186
9.19
232.48
232.48
0.0
1
0.0
M6
54.8
227.4
75.9
1800
1.49
10.0
253.17
96.05
1800
11.4
202.23
215.27
6.06
1800
56.06
M7
201.07
476.4
57.79
1800
0.72
15.2
-
-
1800
14.71
225.52
225.52
0.0
14
0.0
M8
4.94
160.4
96.92
1800
1.28
10.03
153.4
93.46
1800
28.81
135.54
150.14
9.72
1800
2.42
M9
70.42
201.2
65.0
1800
0.76
22.0
194.04
88.66
1800
35.02
192.76
192.76
0.0
322
11.7
M10
146.84
358.6
59.05
1800
0.7
25.78
155.64
83.44
1800
24.82
153.03
153.03
0.0
20
0.0
M11
38.4
116.4
67.01
238
1.2
10.0
112.2
91.09
1800
93.41
103.99
112.32
7.42
1800
3.75
M12
91.93
242.0
62.01
636
0.63
13.29
222.67
94.03
1800
23.09
205.63
215.16
4.43
1800
116.2
M13
46.31
189.0
75.5
633
0.71
10.02
186.37
94.62
1800
118.91
172.61
183.1
5.73
1800
77.92
M14
222.14
495.2
55.14
1800
0.9
10.0
197.49
94.94
1800
74.56
197.49
197.49
0.0
1336
27.99
M15
83.27
230.2
63.83
580
0.74
31.31
222.55
85.93
1800
30.59
204.3
222.55
8.2
1800
159.71
Average
64.99
1368
0.95
81.12
1759
43.11
3.63
1075
40.67
Table 11 Performance comparisons of three proposed models on large-size instances.
Instance
M1-TEN (LR)
M2-STD (MILP Solver)
M3-CTR (BnP)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
L1
10.0
632.0
98.42
3600
9.55
10.0
-
-
3600
25.26
530.61
546.25
2.86
3600
22.16
L2
10.0
540.4
98.15
3600
8.91
10.0
-
-
3600
21.36
450.28
465.22
3.21
3600
17.15
L3
10.0
629.6
98.41
3600
8.79
10.0
-
-
3600
13.5
550.76
558.19
1.33
3600
9.05
L4
10.0
346.8
97.12
3600
7.97
10.0
431.64
97.68
3600
19.71
326.57
359.26
9.1
3600
4.61
L5
10.0
789.8
98.73
3600
10.97
10.0
-
-
3600
15.8
371.99
404.0
7.92
3600
4.79
L6
10.0
647.0
98.45
3600
11.91
10.0
-
-
3600
17.99
465.19
500.53
7.06
3600
15.02
L7
10.0
1111.2
99.10
3600
8.22
186.99
-
-
3600
14.08
503.43
518.02
2.82
3600
14.12
L8
10.0
625.8
98.40
3600
8.75
10.0
-
-
3600
22.12
304.09
317.13
4.11
3600
17.3
L9
10.0
772.4
98.71
3600
4.77
105.18
-
-
3600
13.59
602.83
602.83
0.0
14
0.0
42
L10
10.0
637.4
98.43
3600
8.1
10.0
-
-
3600
16.8
382.14
390.75
2.2
3600
5.36
L11
10.0
694.0
98.56
3600
8.9
10.0
-
-
3600
17.55
320.05
337.58
5.19
3600
3.7
L12
10.0
1022.2
99.02
3600
8.06
10.0
-
-
3600
15.8
514.32
545.89
5.78
3600
18.59
L13
10.0
935.8
98.93
3600
7.29
10.0
-
-
3600
15.32
637.2
678.12
6.03
3600
18.64
Average
98.49
3600
8.63
99.82
3600
17.61
4.43
3324
11.58
From Table 9 - Table 11, model M3-CTR can obtain optimal solutions on 14 small-size instances
within 15 min. For the medium size instances with a 30-min limit, 7 of 15 instances can be solved to
optimality, while the relatively small gap is less than 10% on the remaining 8 instances. Furthermore, it was
also able to obtain the solutions that are very closed to optimal solutions within 1 hour of CPU time for the
large-size instances, with an average relative gap of 4.43%.
In contrast to model M3-CTR, models M1-TEN and M2-STD show poorer performance on all three
instance sets. Specifically, the average gap of M1-TEN and M2-STD are 32.4% and 79.03% on small-size
instances, and 64.99% and 81.12% on medium-size instances. Model M3-CTR significantly outperforms
model M1-TEN and M2-STD in both small-size and medium-size instances. Comparing models M1-TEN
and M2-STD, in terms of average gap, model M1-TEN achieves better solutions in both instance sets.
Model M1-TEN typically provides tighter lower bounds and weaker upper bounds, while model M2-STD
provides weaker lower bounds and tighter upper bounds. For model M1-TEN, its weak upper bounds can
be explained by the symmetry issue, as discussed by Niu et al., (2018) and Yao et al., (2019). As all sprinkler
trucks are assumed to be identical, the upper bounds are generated by sequentially finding shortest paths
for sprinkler trucks, which may result in large gaps with optimal solutions. For model M2-STD, which is
directly solved by CPLEX with built-in linear relaxation and branch-and-bound capability. We find that
linearly relaxing a path-based model (M3-CTR) provides tighter lower bounds than simply relaxing an arc-
based model (M2-STD).
On large-size instances, model M2-STD is significantly outperformed by the other two models (the
upper bound estimate is only obtained on instance L4). This is due to the extremely high model complexity
from the large size of target networks and the addition of time slot dimension for time-dependent traffic
condition modeling. In Table 12, both Gap and BestGap are further provided to comprehensively measure
the quality of solutions obtained from model M1-TEN and M3-CTR. BestGap represents the relative
difference between upper bound and the best lower bound of the two models. The Gaps and BestGaps of
model M3-CTR are exactly the same on all instances, while, for model M1-TEN, BestGaps are significantly
smaller than the corresponding Gaps. The average BestGap of model M1 is 33.51%. This means that
although the solution gaps of model M1-TEN are relatively large, the upper bounds can actually serve as
good feasible solutions in practice.
43
Table 12 Optimality gaps of models M1-TEN and M3-CTR on large-size instances.
Instance
M1-TEN (LR)
M3-CTR (BnP)
LB
UB
Gap (%)
BestGap (%)
LB
UB
Gap (%)
BestGap (%)
L1
10.0
632
98.42
16.04
530.61
546.25
2.86
2.86
L2
10.0
540.4
98.15
16.68
450.28
465.22
3.21
3.21
L3
10.0
629.6
98.41
12.52
550.76
558.19
1.33
1.33
L4
10.0
346.8
97.12
5.83
326.57
359.26
9.1
9.1
L5
10.0
789.8
98.73
52.90
371.99
404
7.92
7.92
L6
10.0
647
98.45
28.10
465.19
500.53
7.06
7.06
L7
10.0
1111.2
99.1
54.69
503.43
518.02
2.82
2.82
L8
10.0
625.8
98.4
51.41
304.09
317.13
4.11
4.11
L9
10.0
772.4
98.71
21.95
602.83
602.83
0.0
0.0
L10
10.0
637.4
98.43
40.05
382.14
390.75
2.2
2.2
L11
10.0
694
98.56
53.88
320.05
337.58
5.19
5.19
L12
10.0
1022.2
99.02
49.68
514.32
545.89
5.78
5.78
L13
10.0
935.8
98.93
31.91
637.2
678.12
6.03
6.03
Average
98.49
33.51
4.43
4.43
In terms of solution time, we only compare model performances on small-size and medium-size
instances, as the preset time limit is reached on almost all large-size instances (except M3-CTR on L9). For
small-size instances, the average solution times of models M1-TEN and M3-CTR are very close (69s and
84s) and are significantly shorter than the time usage of model M2-STD. Similar findings can be observed
on medium-size instances. Another important finding from this comparison is the memory usage of each
model, especially for real-time applications. Overall, model M1-TEN consumes much less memory than
models M2-STD and M3-CTR. Even for large-size instances, the average and maximum memory usage of
model M1-TEN are 8.63 GB and 11.91 GB (instance L6), which highlight the applicability of model M1-
TEN in on-line computing. Model M3-CTR consumes more memory than model M1-TEN. This is due to
the use of parallel computing in the branch and bound module, which involves a large number of branch
and bound nodes being processed simultaneously. For model M2-STD, the average memory usage on
medium-size instances is 43.11 GB, which is close to model M3-CTR. Note that the memory statistics for
model M2-STD was not reported for large-size instances, as it is still in the pre-solving stage when reaching
the time limit.
Overall, models M1-TEN and M3-CTR with customized solution algorithms outperform model M2-
STD that relies on an off-the-shelf solver. Model M3-CTR has a balanced performance in terms of solution
quality and solution time, while model M1-TEN is able to produce high-quality feasible solutions with less
memory consumption.
44
6.3 System impact and private cost of vehicle routing
Table 13 presents the optimal solution costs of small-size instances for and . Note that,
to exactly measure the changes of optimal operation cost and system impact under different weights of
system impact, we only select instances that can be solved to optimality, thus instance S14 is not included
in Table 13. Overall, after putting more priorities on the marginal cost of vehicle routing, the system impact
obviously decreases while the private operation cost increases. For each instance, OCPI (SIPD) denotes the
percentage increase (decrease) of the instance’s optimal operation cost (system impact) when varying
from 0.0 to 1.0. Due to the diversity of selected instances, OCPI and SIPD may vary from one instance to
another. Specifically, for instance S1, its operation cost increases by 4.23% when changing from
0.0 to 1.0, while it increases by 69.35% in instance S4.
Table 13 Optimal routing costs of small instances under different weights of system impact .
Instance
Weight of system impact
Weight of system impact
OCPI
(%)
SIPD
(%)
Total cost
Operation cost
System impact
Total cost
Operation cost
System impact
S1
112.38
112.38
297.55
391.51
117.13
274.38
4.23
7.79
S2
75.65
75.65
320.44
335.83
88.63
247.20
17.16
22.85
S3
73.03
73.03
350.95
348.17
88.26
259.91
20.85
25.94
S4
51.85
51.85
328.06
328.10
87.81
240.29
69.35
26.76
S5
82.36
82.36
347.14
350.75
118.10
232.65
43.43
32.98
S6
77.05
77.05
289.92
318.01
89.25
228.76
15.83
21.09
S7
131.87
131.87
396.73
449.78
148.25
301.53
12.42
24.00
S8
45.84
45.84
318.53
292.49
58.60
233.89
27.84
26.57
S9
80.02
80.02
339.51
362.46
89.43
273.03
11.76
19.58
S10
71.12
71.12
343.32
339.80
88.73
251.07
24.76
26.87
S11
130.69
130.69
259.40
360.36
146.78
213.58
12.31
17.66
S12
59.48
59.48
278.47
297.54
89.42
208.12
50.34
25.26
S13
78.53
78.53
385.28
389.30
89.67
299.63
14.19
22.23
S14
-
-
-
-
-
-
-
-
S15
111.40
111.40
343.32
396.13
147.65
248.48
32.54
27.63
Notes: OCPI – operation cost percentage increase, SIPD – system impact percentage decrease
As an example, Fig. 12 presents the Pareto frontier of operation cost and system impact on instance
S5 with ranging from 0.0 to 1.0. Thus, a pareto-optimal solution should be systematically selected by
decision makers and planners to balance these two important criteria.
45
=1.0
=0.0
Fig. 12. Pareto curve of operation cost and system impact on instance S5 with ranging from 0.0 to 1.0.
6.4 Impact of time resolution in time-expanded network based model M1-TEN
For model M1-TEN, a suitable time discretization resolution (TR) is needed when constructing time-
expanded networks from physical networks. Generally, a higher resolution means lower approximation
errors and higher model complexity, and vice versa. We choose three different TR settings, i.e., TR=0.1
min, TR=0.2 min, and TR=0.5 min for solving model M1. Fig. 13 provides the solution quality comparisons
of M1-TEN on small and medium instances. The results on large instances are not presented since the model
M1-TEN cannot be solved within the time limit in most cases. To properly compare the solution quality,
we also show solutions obtained from model M3-CTR (labeled as ‘BnP’ in Fig. 13). A triangle means the
corresponding instance is solved to optimality, while two endpoints of a line represent the LB and UB of
the corresponding instance. Besides, the average solution time and memory usage on three instance sets are
also reported, as seen in Fig. 14. The detailed experimental results are listed in Appendix C.
It can be observed that the solutions of M1-TEN with TR=0.1 min are similar to that of M1-TEN with
TR=0.2 min on most of the small and medium instances, which are closer to the ones founded by the BnP
approach. In contrast, M1-TEN with TR=0.5 min is always far away from the better solutions obtained by
BnP. This may imply that, with a lower time resolution, the solution quality of M1-TEN is affected. It is
worth noting that the gaps of M1-TEN with TR=0.1 on medium-sized instance M3 and M5 are still quite
large, which are even significantly larger than that of M1-TEN with TR=0.5 min. In addition, there is no
feasible solution that can be obtained for M1-TEN with TR=0.5. The above observations suggest that M1-
TEN with TR=0.5 min could be problematic when handling practical SRPs.
Based on the above analysis, we can see that it is important to examine the trade-off between solution
quality and solution efficiency (i.e. solution time and memory usage) when selecting the time resolution for
M1-TEN. As a result, we set TR=0.2 min in the comparison experiments of Section 6.2.
46
(a) Solution quality on small-size instances
(b) Solution quality on medium-size instances
Fig. 13. Impact of time resolution on solving model M1-TEN (solution quality).
(a) Solution time on different size instance sets
(b) Memory usage on different size instance sets
Fig. 14. Impact of time resolution on solving model M1-TEN (solution time and memory usage).
7. Conclusions
In this study, focusing on formulating and solving RARPs in city logistics with a congested urban
environment, we propose a comprehensive modeling framework and exact solution algorithms. Specifically,
based on the fluid queuing model with a polynomial functional assumption for arrival flow rates, time-
dependent link travel time as well as system-wide (societal) congestion impact is analytically derived. Two
new time-dependent travel time representation schemes are investigated. With the SRP as an example, three
optimization models, including a time-expanded network based arc routing formulation (model M1-TEN),
an arc-based node routing formulation (model M2-STD), and a path-based node routing formulation (model
47
M3-CTR), are constructed from different perspectives of capturing time-dependent travel time and
formulating problem-specific constraints. In addition, two exact solution methods, i.e., Lagrangian
relaxation and branch-and-price algorithm, are developed for efficiently solving model M1-TEN and model
M3-CTR.
With real-world traffic flow data collected from the Washington DC metropolitan area, 12
representative corridors are selected to demonstrate the effectiveness of the proposed time-dependent link
travel time modeling method. Results show that the proposed method can reasonably capture travel time
evolution profiles under different levels of traffic congestion. Based on calibrated travel time functions,
three sets of SRP instances with different sizes are designed to examine the performance of the three
optimization models and their corresponding solution algorithms. It is found that, overall, model M3-CTR
produces high-quality solutions in a shorter time, while model M1-TEN is able to provide good feasible
solutions with less memory consumption. Furthermore, routing solutions with different priorities on
system-wide (societal) congestion impact are systematically analyzed.
This study can be extended along the following directions in the future. First, this paper considers
fixed departure time of service vehicles at the origin depot. Future studies could relax this restriction to
allow the selection of proper departure times/schedules so as to minimize the overall cost. One interesting
yet challenging topic along this line is how to produce optimal continuous-time path solutions without
explicitly performing time discretization. This theoretically important and practically useful question was
discussed in a recent paper by Boland et al., (2017) for the service network design problem. Second, with
simplified fluid queue models, the system-wide (societal) congestion impact of routing solutions is
considered in this paper. The tradeoff among additional costs such as energy use, emissions reduction, and
location congestion reduction (Lam and Hentenryck, 2016) could be further systematically investigated
under more realistic traffic flow modeling frameworks to accommodate different practical needs in real-life
applications.
Appendix A. Graph transformation
The key idea of converting the original ARP to an NRP is treating each link to be cleaned as a customer
(activity node) and connecting different customers with virtual edges. In Fig. A1, we use an illustrative
network with 4 intersections and 8 directed road links as an example to introduce the process of converting
the original ARP to an NRP. Among the four intersections in Fig. A1(a), intersection 1 and 3 also serve as
the depot of sprinkler trucks and water refilling station, respectively. The workflow of transforming Fig.
A1(a) to Fig. A1(b) is presented in Algorithm A1.
48
(a) Physical transportation network (b) Activity node based service network
Fig. A1. A simple transportation network and its activity node based service network.
Algorithm A1 Graph transformation process.
Input: physical transportation network
Output: activity node based network , node service time function , edge travel time
function
1:
(Prepare Underlying Network)
2:
Step 1: Build an intersection-expanded underlying network from the physical network
3:
Build an intersection-expanded underlying network (the network in grey in Fig. B1(b)) from Fig.
B1(a).
4:
(Construct Activity Node Set )
5:
Step 2: Construct origin depot and destination depot
6:
Create a virtual origin depot and a virtual destination depot from intersection 1.
7:
Step 3: Construct service nodes
8:
Create service nodes for each road link with service requests in . The number of service nodes
created for a specific road link should be equal to its number of service requests. We build two service
nodes (node 1 and node 5) for road link (4,1) with two cleaning requests, and one service node for each
other road link.
9:
Step 4: Construct water refilling nodes
10:
Create dummy water refilling nodes for each intersection that serves as water refilling station, e.g.,
node and for intersection 3 in our example. Dummy water refilling nodes are created to avoid
multiple visits at one water refilling station. The number of dummy water refilling nodes needed for
each physical water refilling station equals ceiling(total water needed to complete all cleaning tasks /
water tank capacity of sprinkler trucks).
11:
(Construct Edge Set )
12:
Step 5: Construct virtual edges between activity nodes
13:
. For each service node, build virtual links with origin
depot, destination depot, other service nodes, and water refilling stations. As an example, we show the
virtual edges created for service node 1 in Fig. B1(b).
14:
return
49
In Step 3, the service time of service node () equals to the cleaning time of its corresponding
road link. That is, , where is the link cleaning time in Eq. (16).
In Step 5, up to edges will be constructed between each pair of activity nodes, with each edge
corresponding to a path in the physical network. Paths can be obtained from candidate path sets in real life
or searched by k-shortest path algorithms (e.g., Scano et al., 2015; Lu et al., 2018). Time-dependent path
travel time and system impact can be calculated using the method proposed in Section 2.
To reduce the size of the resulting graph as much as possible, we only build edges that is possibly used
in an optimal solution. For example, as we require that sprinkler trucks are full of water when departing
from the origin depot, an optimal solution will not contain an edge from the origin depot to a water refilling
station. Therefore, edges from the origin node to water refilling stations will not be built.
Appendix B. Model M2 with nonlinear functions
Table B1 Notation list.
Symbols
Definition
Indices
Index of nodes in graph
Index of edges between each pair of nodes in graph
Sets
Set of nodes in graph
Set of service nodes in graph
Set of water refilling stations in graph
Set of edges in graph
Functions
Time-dependent service time function of node
Time-dependent travel time function of edge
Time-dependent system impact function of serving node at time
Time-dependent system impact function of using edge at time
Parameters
Number of edges between each pair of nodes in graph
Water consumption of node
The earliest (latest) service starting time of node
Variables
Binary variable. if a sprinkler uses edge ; otherwise,
Water level of a sprinkler at node
Departure time of a sprinkler at node
Service starting time of node
Model M2 with nonlinear functions:
Objective function
50
(B1)
Subject to:
Cleaning request satisfaction constraint:
(B2)
Sprinkler spatial route constraint:
(B3)
(B4)
Sprinkler temporal route constraint:
(B5)
(B6)
(B7)
Time window constraint:
(B8)
Sprinkler truck water level updating constraint:
(B9)
(B10)
(B11)
Decision variables:
(B12)
Appendix C. Performance comparisons of model M1-TEN with different time resolutions
Table C1 Performance comparisons of the arc routing model (M1-TEN) with different time resolutions on
small size instances.
Instance
M1 with time resolution 0.1 min
M1 with time resolution 0.2 min
M1 with time resolution 0.5 min
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
51
S1
112.5
112.5
0.0
605
10.67
112.6
112.6
0
62
4.13
138.46
141
1.8
900
7.53
S2
70.0
75.7
7.53
900
14.05
71.23
75.8
6.03
900
7.35
83
83
0
16.46
0.00
S3
65.58
73.2
10.41
900
13.23
66.28
73.2
9.45
900
7.96
78.5
78.5
0
41
3.14
S4
48.33
51.7
6.51
900
13.45
48.32
51.6
6.35
900
7.41
58
58
0
2
0.00
S5
78.17
82.4
5.14
900
13.65
80.34
82.4
2.5
900
6.88
103
103
0
748
4.61
S6
71.76
77.2
7.04
900
11.26
73.11
77
5.05
900
6.9
84
84
0
13
0.00
S7
121.15
131.9
8.15
900
12.36
131.8
131.8
0
805
8.86
148
148
0
4
0.00
S8
38.96
45.7
14.75
900
11.83
40.68
45.6
10.79
900
7.77
49
49
0
57
2.69
S9
75.74
80
5.32
900
10.93
79.8
79.8
0
797
6.9
89.5
89.5
0
1
0.00
S10
62.81
71.5
12.15
900
12.01
62.99
71.4
11.77
900
8.34
79.5
79.5
0
24
0.00
S11
125.74
131.2
4.16
900
12.85
126.05
131
3.78
900
7.71
162
162
0
10
0.00
S12
59.6
59.6
0.0
15
0.00
59.4
59.4
0
6
0.00
75.13
78
3.68
900
5.29
S13
69.86
78.9
11.45
900
13.05
69.7
78.6
11.32
900
7.39
88.5
88.5
0
7
0.00
S14
54.37
56.1
3.08
900
16.92
54.47
56
2.74
900
9.54
63.28
72.5
12.72
900
6.29
S15
109.93
111.4
1.32
900
13.02
111.4
111.4
0
71
5.51
136
136
0
163
3.61
Average
6.47
821
11.95
4.65
716
6.84
1.21
252
2.21
Notes: LB – Lower bound, UB – Upper bound, ST – Solution time, MU – Memory usage
Table C2 Performance comparisons of the arc routing model (M1-TEN) with different time resolutions on
medium size instances.
Instance
M1 with time resolution 0.1 min
M1 with time resolution 0.2 min
M1 with time resolution 0.5 min
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
M1
200.58
213.5
6.05
1800
27.97
200.83
212.4
5.45
1800
17.41
*
*
*
*
*
M2
217.88
237.7
8.34
1800
35.13
217.02
230.8
5.97
1800
24.55
246
254.5
3.34
1800
14.96
M3
127.57
840
84.81
1800
6.93
126.71
150.4
15.75
1800
3.09
141.1
152
7.17
1800
9.14
M4
185.48
192.6
3.7
1800
28.37
186.29
191.4
2.67
1800
18.45
227
227
0
71
4.94
M5
232.4
851.6
72.71
1800
7.2
228.63
232.4
1.62
1800
18.04
267.5
267.5
0
103
1.48
M6
199.37
-
-
1800
22.09
200.29
-
-
1800
21.07
240.86
257.5
6.46
1800
10.17
M7
216.68
230.3
5.92
1800
32.5
216.88
225
3.61
1800
18.87
274.5
274.5
0
124
7.55
M8
135.04
-
-
1800
7.51
134.81
159
15.21
1800
21.4
176.86
189.5
6.67
1800
9.56
M9
185.15
-
-
1800
28.91
185.16
193.8
4.46
1800
17.97
218.35
227
3.81
1800
9.7
M10
149.93
153.1
2.07
1800
29.53
150.58
153.4
1.84
1800
18.9
185.97
193
3.64
1800
9.78
M11
109.7
-
-
1800
19.27
102.29
118.8
13.9
1800
18.37
130
142.5
8.77
1800
9.26
M12
200.84
-
-
1800
30.11
200.65
218.2
8.04
1800
17.7
254.06
267
4.85
1800
9.67
M13
168.26
-
-
1800
28.11
168.96
187.8
10.03
1800
17.2
218.58
226
3.29
1800
9.26
M14
196.34
-
-
1800
7.32
195.23
236.4
17.42
1800
10.84
233.83
235.5
0.71
1800
8.53
M15
195.66
-
-
1800
28.24
197.14
224.8
12.31
1800
17.18
268.5
268.5
0
1507
8.03
Average
65.57
1800
22.61
14.55
1800
17.4
3.48
1415
8.72
Notes: LB – Lower bound, UB – Upper bound, ST – Solution time, MU – Memory Usage, * – Infeasible
Table C3 Performance comparisons of the arc routing model (M1-TEN) with different time resolutions on
large size instances.
52
Instance
M1 with time resolution 0.1 min
M1 with time resolution 0.2 min
M1 with time resolution 0.5 min
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
LB
UB
Gap
(%)
ST
(s)
MU
(GB)
L1
0
-
-
3600
45.38
176.47
-
-
3600
31.57
594.93
-
-
3600
7.34
L2
0
-
-
3600
48.09
398.46
-
-
3600
33.79
451.34
-
-
3600
22.52
L3
0
-
-
3600
46.12
545.72
-
-
3600
18.83
625.34
1753
64.33
3600
7.98
L4
0
-
-
3600
38.3
326.4
-
-
3600
15.31
355.58
1254
71.64
3600
5.42
L5
0
-
-
3600
46.34
372.19
-
-
3600
19.87
427.08
1753.5
75.64
3600
8.25
L6
0
-
-
3600
60.33
427.87
-
-
3600
23.71
487
-
-
3600
6.26
L7
0
-
-
3600
51.1
439.56
-
-
3600
20.8
*
*
*
*
*
L8
272.13
-
-
3600
42.69
275.43
-
-
3600
17.65
320.39
-
-
3600
17.85
L9
602.8
-
-
3600
30.18
602.2
-
-
3600
14.12
679.5
-
-
3600
6.41
L10
0
-
-
3600
45.41
325.81
-
-
3600
19.91
405.5
-
-
3600
7.92
L11
0
-
-
3600
41.04
303.84
-
-
3600
16.38
418.67
-
-
3600
7
L12
509
-
-
3600
35.87
512.13
-
-
3600
15.03
690.5
-
-
3600
6.67
L13
622.25
-
-
3600
36.26
619.23
-
-
3600
14.23
*
*
*
*
*
Average
100
3600
43.62
100
3600
20.09
91.96
3600
9.42
Notes: LB – Lower bound, UB – Upper bound, ST – Solution time, MU – Memory usage, * – Infeasible
Appendix D. Impact of the ng-set size on the solutions from CG module
In general, the larger ng-set is, the tighter lower bounds can be obtained, at the same time, also harder
to solve. In Table D1, with different sizes of ng-set, CG results for the LMPs of all instances are presented.
ngx means the size of ng-set is x. In our implementations, we choose x nearest nodes (Euclidean distance)
as the ng-set for each node. OBJ and ST represent the objective value of LMPs and solution time,
respectively. “-” means no solution is obtained within the time limit, which is 7200 seconds. An obvious
finding from Table D1 is that, for each instance, OBJs with a larger ng-set are higher than or equal to those
with a smaller ng-set. The reason is that larger ng-sets prevent more cycles; thus, tighter lower bounds can
be obtained. In terms of solution time, a trend that solution time increases with the increase of ng-set size
can be observed. This trend is more obvious in large-scale instances. Specifically, for instances L5 and L11,
they are no longer solvable within the preset time limit when the size of ng-set is larger than 4 and 8,
respectively. In this paper, considering the tradeoff between computation efficiency and lower bound quality
of the pricing problem, we set the size of ng-sets as 2 for all instances.
Table D1 Results of CG with different size of ng-sets.
Instance
ng1
ng2
ng4
ng8
ng16
ng24
OBJ
ST (s)
OBJ
ST (s)
OBJ
ST (s)
OBJ
ST (s)
OBJ
ST (s)
OBJ
ST (s)
S1
108.029
0.06
108.029
0.06
108.029
0.08
108.029
0.08
108.029
0.09
108.029
0.06
S2
69.888
0.09
70.0238
0.09
70.9043
0.1
70.9043
0.1
70.9043
0.1
70.9043
0.07
S3
65.288
0.37
65.31
0.77
65.328
0.42
65.328
0.42
65.6978
0.72
66.0244
1.33
53
S4
48.3182
6.26
48.3409
15.94
48.5236
5.21
48.6318
8.39
48.8493
40.4
48.9767
87.92
S5
78.44
0.18
78.44
0.18
78.44
0.28
78.44
0.25
78.9467
0.45
78.9467
0.49
S6
71.296
0.19
71.296
0.2
71.296
0.15
71.6471
0.25
72.04
0.32
72.04
0.41
S7
121.195
0.07
121.239
0.09
121.239
0.08
121.239
0.08
121.239
0.07
121.239
0.1
S8
39.0128
113.87
39.0796
28.4
39.0831
32
39.2578
24.71
39.972
2059.19
39.972
1899.03
S9
74.8906
0.19
75.156
0.24
75.173
0.18
75.1946
0.21
75.29
0.27
75.29
0.23
S10
62.1868
0.34
62.8888
0.39
62.8888
0.3
63.2029
0.67
63.5043
0.86
63.5043
0.84
S11
123.032
0.05
123.032
0.05
123.032
0.03
123.032
0.03
123.032
0.05
123.032
0.03
S12
59.48
0.34
59.48
0.29
59.48
0.37
59.48
0.59
59.48
0.72
59.48
0.72
S13
69.5325
0.5
69.5589
0.68
70.0774
0.56
70.0774
0.57
70.0774
0.59
70.0774
0.55
S14
54.4759
1646.83
54.4881
523.45
54.5064
646.67
54.5242
1390.39
-
-
54.7112
7130.58
S15
108.9
0.09
108.9
0.1
110.216
0.14
110.216
0.11
110.216
0.11
110.216
0.11
M1
200.91
0.43
201.281
0.74
201.406
0.83
201.406
0.88
201.449
1.19
201.491
0.76
M2
217.711
1.08
217.738
1.13
217.775
1.62
217.775
1.88
217.775
1.71
217.775
1.9
M3
127.228
31.31
128.768
25.76
129.268
18.16
129.712
41.08
129.719
49.23
129.719
56.68
M4
184.893
0.62
184.968
0.62
184.979
0.62
184.979
0.58
184.979
0.59
184.979
0.57
M5
232.48
0.34
232.48
0.34
232.48
0.49
232.48
0.49
232.48
0.51
232.48
0.48
M6
199.284
2.77
199.558
6.16
200.201
7.56
200.21
5.16
200.346
11.78
200.449
12.52
M7
219.806
0.66
219.838
0.65
220.111
0.92
220.111
0.91
220.111
0.98
220.111
0.95
M8
135.22
618.23
135.464
892.36
135.684
629.39
135.739
1030.65
136.315
3605.41
136.315
3884.25
M9
185.176
0.95
185.525
0.94
185.972
1.67
186.36
1.45
186.535
2.03
186.535
1.99
M10
149.708
2.42
149.708
3.43
149.715
2.54
149.803
1.99
149.836
1.93
149.855
1.83
M11
103.546
94.54
103.878
305.52
104.189
210.63
104.189
281.04
104.189
306.43
104.205
232.02
M12
199.124
0.5
199.128
0.62
199.128
0.66
199.297
0.81
199.733
0.99
199.733
0.98
M13
168.924
2.18
169.093
6.02
169.105
6.27
169.111
7.05
169.111
7.42
169.137
7.1
M14
195.959
3.13
195.959
2.4
196.473
2.77
196.473
2.98
196.473
3.3
196.473
3.31
M15
198.034
1.01
198.163
1.16
199.476
1.08
199.48
1.06
199.523
1.63
200.763
1.58
L1
530.124
607.56
530.151
795.78
530.269
809.19
530.269
586.33
530.269
578.46
530.429
654.29
L2
448.089
117.5
448.668
170.79
448.926
373.43
448.932
487.83
448.932
501.66
448.932
446.07
L3
549.765
1147.19
550.748
1345.2
550.748
1438.24
551.303
1144.81
551.321
2162.1
551.321
1245.25
L4
326.572
701.45
326.572
620.78
326.572
562.34
326.572
637.25
326.572
938.29
326.572
534.64
L5
371.434
180.62
371.993
272.2
-
-
-
-
-
-
-
-
L6
464.255
199.29
464.79
494.17
465.996
620.48
466.142
671.9
466.22
521.86
466.22
634.64
L7
499.571
47.83
501.104
119.58
501.523
137.9
501.525
123.07
501.579
146.96
501.579
148.16
L8
302.927
173.68
303.094
535.89
303.245
654.31
303.256
1116.18
303.509
660.22
303.587
852.03
L9
602.83
15.14
602.83
15.18
602.83
14.81
602.83
14.87
602.83
16.07
602.83
15.25
L10
382.122
1943.08
382.141
3373.92
382.256
3312.17
382.258
2922.25
-
-
382.311
6728.26
L11
319.703
1480.31
320.049
1993.82
320.288
4319.85
-
-
-
-
-
-
L12
514.324
204.71
514.324
181.39
514.324
177.38
514.324
180.85
514.324
175.39
514.324
180.69
L13
633.217
156.12
633.317
172.75
637.141
938.58
637.396
1201.52
637.473
1752.91
638.252
1389.4
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