myjournal manuscript No.
(will be inserted by the editor)
Workforce Scheduling and Routing Problems
Literature Survey and Computational Study
J. Arturo Castillo-Salazar
1
,
Dario Landa-Silva, Rong Qu
Received: date / Accepted: date
Abstract In the context of workforce scheduling, there are many scenarios in
which personnel must carry out tasks at different locations hence requiring some
form of transportation. Examples of these type of scenarios include nurses visit-
ing patients at home, technicians carrying out repairs at customers’ locations and
security guards performing rounds at different premises, etc. We refer to these
scenarios as Workforce Scheduling and Routing Problems (WSRP) as they usu-
ally involve the scheduling of personnel combined with some form of routing in
order to ensure that employees arrive on time at the locations where tasks need
to be performed. The first part of this paper presents a survey which attempts to
identify the common features of WSRP scenarios and the solution methods ap-
plied when tackling these problems. The second part of the paper presents a study
on the computational difficulty of solving these type of problems. For this, five
data sets are gathered from the literature and some adaptations are made in order
to incorporate the key features that our survey identifies as commonly arising in
WSRP scenarios. The computational study provides an insight into the structure
of the adapted test instances, an insight into the effect that problem features have
when solving the instances using mathematical programming, and some bench-
mark computation times using the Gurobi solver running on a standard personal
computer.
Keywords workforce scheduling, employee rostering, routing problems, mobile
workforce, mathematical programming, benchmark instances
1 Introduction
In recent times, employees often need to be more flexible regarding the type of
jobs they perform and similarly, employers need to make compromises in order to
1
The author acknowledges CONACYT for its financial support
J. Arturo Castillo-Salazar · Dario Landa-Silva · Rong Qu
E-mail: {psxjaca, dario.landasilva, rong.qu}@nottingham.ac.uk
ASAP Research Group, School of Computer Science, University of Nottingham, Jubilee Cam-
pus, Wollaton Road, Nottingham, United Kingdom, NG8 1BB
J. Arturo Castillo-Salazar et al
retain their best employees (Eaton, 2003; Mart´ınez-S´anchez et al, 2007). Moreover,
in some cases workforce should perform tasks at different locations, e.g. nurses
visiting patients at their home, and technicians carrying out repairs at different
companies, etc. Therefore, the scheduling of workforce with ‘flexible’ arrangements
and ‘mobility’ is of great importance in many scenarios. Many types of personnel
scheduling problems have been tackled in the literature (Baker, 1976; Miller, 1976;
Golembiewski and Proehl Jr, 1978; Cheang et al, 2003; Ernst et al, 2004; Alfares,
2004). We are interested in those workforce scheduling problems in which personnel
is considered flexible (in terms of tasks and working times) and mobile (travelling
is required in order to do the job). By mobility we refer specifically to those cases
in which moving from one location to another takes significant time and therefore
reducing the travel time could potentially increase productivity. To some extent,
this problem combines features from the general employee scheduling problem and
also from vehicle routing problems. The survey and computational study presented
here represent a step towards our longer term aim of formulating and tackling the
problem of scheduling flexible and mobile workforce. In the rest of this paper, we
refer to this as the workforce scheduling and routing problem (WSRP).
In Section 2 we describe the WSRP and identify some of the main character-
istics of this type of workforce scheduling problems. Section 3 is dedicated to the
vehicle routing problem with time windows (VRPTW) because it represents the
basic routing component of many of the problems discussed in this survey. Sec-
tion 4 outlines some workforce scheduling scenarios that have been investigated
in the literature and that in our view present a case of WSRP. Examples include
home care, scheduling of technicians, manpower allocation, etc. In each subsec-
tion of Section 4 we also review the different solution techniques (optimisation,
heuristics and hybrid approaches) that have been used to tackle these problems.
Section 5 contains the computational study. The experiments are performed using
the Gurobi solver, IP and MIP models. A set of problem instances are also pre-
sented in this section. Finally, Section 6 summarises our findings and outlines the
next steps in our research into workforce scheduling and routing.
2 Workforce Scheduling and Routing Problems
2.1 Description of the Problem
Workforce Scheduling and Routing Problem (WSRP) refers to those scenarios in-
volving the mobilisation of personnel in order to perform work related activities at
different locations. In such scenarios, employees use diverse means of transporta-
tion, e.g. walking, car, public transport, bicycle, etc. Also, in these scenarios there
are more than one activity to be performed in a day, e.g. nurses visiting patients
at their homes to administer medication or provide treatment (Cheng and Rich,
1998), care workers aiding members of the community to perform difficult tasks
(Eveborn et al, 2006), technicians carrying out repairs and installations (Cordeau
et al, 2010), and security guards performing night rounds on several premises
(Misir et al, 2011). The number of activities across the different locations is usu-
ally larger than the number of employees available, hence employees should travel
between locations to perform the work. This results into combination of employee
scheduling and vehicle routing problems. The number of activities varies depend-
Workforce Scheduling and Routing (WSRP)
ing on the duration of the working shift, but assuming that each activity needs to
be performed at a different location, a routing problem also arises. A route is a
sequence of locations that need to be visited (Raff, 1983) but we exclude problems
in which workers need to move across work stations within the same factory for
example. Work activities which need to be performed within a given time require
scheduling in addition to routing. Tackling WSRP scenarios could potentially in-
volve many objectives like: reducing employees travel time, guaranteeing tasks to
be performed by qualified people only, reducing the cost of hiring casual staff,
ensuring contracted employees are used efficiently, etc.
We assume employees should rather spend more time doing work than travel-
ling, particularly in settings in which travelling time is counted as paid working
time, hence reducing travel time is valuable (Fosgerau and Engelson, 2011; Jara-
D´ıaz, 2000). Like in many workforce scheduling problems, the set of skills that
an employee has for performing a task is of great importance in WSRP scenarios
(Cordeau et al, 2010) so that employees perform activities at customer premises
more efficiently. Many papers in the literature assume that the workforce is ho-
mogeneous regarding skills but in many scenarios, a diverse set of skills is the
predominant environment. We should note that scenarios like the pick-up and de-
livery of goods (parcels) is not considered here as a WSRP because no significant
‘work’ (in terms of time) is carried out at customers’ premises. Although, one could
argue that the action of delivering a parcel is a task, it does not take a significant
amount of time once the worker gets to the destination. This type of pick-up and
delivery problems are usually defined as routing problems and are not covered in
our study of workforce scheduling and routing problems.
2.2 Main Characteristics
We outline here the main characteristics of WSRP. Some of these characteristics
are ‘obvious’ since they are in the nature of the problem while others were identified
during our survey. We include the characteristics that appear the most in the
literature.
Time Windows for performing a task (duty, job) at a customer premises. It is as-
sumed that employees can start the work as soon as they arrive at the location.
Time windows can be flexible or tight and in accordance to contractual arrange-
ments. In some cases, no time window is defined as employees work based on
annualised hours. Also, in some cases employees can benefit from over-time
payment, making compliance with the time window a soft constraint.
Transportation Modality refers to employees using different means like car, bicycle,
walking or public transport. We assume that time and cost of transportation
is not the same for each employee.
Start and End Locations can be from one location, where all employees start at the
main office (Eveborn et al, 2006), up to many locations (perhaps as many as
the number of employees) assuming each employee may start from their home.
In some cases company’s policy enforces employees starting their working time
at the main office but returning home directly after the last job performed.
Skills and Qualifications act as filters on who can perform a task and there are two
main cases. 1) All employees have the same skills and qualifications so anyone
J. Arturo Castillo-Salazar et al
can perform every task. This tends to be expensive for the organisation. 2)
Employees with diverse levels of abilities. This is common in industries such
as consulting and healthcare. Matching employees’ skills to the tasks assigned
has been tackled for complex organisations (Cordeau et al, 2010).
Service Time corresponds to the duration of the task and it varies depending on
the employee who performs it and the type of task. Most models in the liter-
ature assume a fixed duration. If service times are long enough so that they
restrict each worker to perform only one job, then the problem reduces to task
allocation since every route would consider only one job per employee.
Connected Activities refer to dependencies among two or more activities. Sequen-
tial dependency occurs when one activity must be performed before or after
another. Temporal dependencies are as defined by Rasmussen et al (2012).
Synchronisation is when two or more activities need to start at the same time.
Overlap occurs when at any point in time, activities happen simultaneously.
In minimum difference dependency, the second activity starts after some given
time has passed since the end of the first activity. A maximum difference de-
pendency establishes a limit for the start on the second activity from the end
of the first activity. A min+max difference dependency is a combination of the
previous two by creating an additional time window for the start of the second
activity based on the end of the first activity.
Teaming is necessary sometimes due to the nature of the work to be carried out (Li
et al, 2005). If team members remain unchanged then the team can be treated
as a single entity and we can assume that all start and end the joint activity
at the same time. Nevertheless, if teams change according to the task, then
synchronising the arrival of employees to the location of the activity is required.
Within teaming, synchronisation refers to employees and not to activities like
Connected Activities. Also, when teams change frequently then skill matching
must be ensured every time teams change to perform a job requiring multiple
skills not present in a single employee (Cordeau et al, 2010).
Clusterisation may be necessary for several reasons. One is that employees may
prefer not to travel more than a number of miles. Another reason is when
companies assign employees to perform work only in certain geographical areas.
Clusters may also be created just to reduce the size of the problem by solving
a number of clustered sub-problems.
3 Vehicle Routing Problem with Time Windows
The routing part in many of the problems considered here as examples of WSRP
is based on the vehicle routing problem with time windows (VRPTW). In this
problem the main objective is to minimise the total distance travelled by a set of
vehicles serving customers spread across different locations. Every customer must
be visited once by one vehicle. Each customer specifies a time window when the
visit should take place. The delivery vehicle must arrive at the location within
that specified time window. If the vehicle arrives before the time window, it must
wait until the time window opens to perform the delivery (Desrochers et al, 1992;
Kallehauge et al, 2005). Extensions of the VRPTW include other features such as
multiple trips, multiple depots and synchronisation of vehicles.
Workforce Scheduling and Routing (WSRP)
In the VRPTW extension that covers multiple depots, the fleet of vehicles is
distributed across multiple depots allowing vehicles to return to the closest depot
once all the deliveries by that vehicle have been completed. This VRPTW variant
(Desaulniers et al, 1998) is relevant to our study because its formulation is appli-
cable to workforce scheduling and routing. Many papers in the literature dealing
with WSRP scenarios use this VRPTW variant and associate every employee’s
starting and ending point to a depot. It is also possible for every employee to start
at the same location (depot) but to end their working day at a different location
(home).
Another extension of the VRPTW allows multiple trips (Brand˜ao and Mercer,
1998). This applies to the case when employees could perform more than one trip
on a day to visit the same location. A trip in this context involves a series of tasks
before going back to the depot. In WSRP scenarios, an employee is assumed to
have a means of transportation. Sometimes the employee might need to go back
to the main office (depot) to replenish resources. The type of vehicles that can be
used to access a particular customer’s location might also be restricted as pointed
out by Brand˜ao and Mercer (1997). Vehicles have different capacities which can
be associated to model an heterogeneous workforce. Vehicles can also be hired for
some time which is associated to hiring casual staff.
Finally, another extention of VRPTW which is important to WSRP is the
synchronisation of vehicles. Two or more workers executing a task can be modelled
in the same way as when two or more vehicles need to arrive simultaneously at the
same customer location (Bredstr¨om and R¨onnqvist, 2007). Precedence constraints
in WSRP are also related to synchronisation of vehicles (Bredstr¨om and R¨onnqvist,
2008). Assuming a client should be visited more than once per day, the order of
visits might matter. For example, before technicians install a satellite TV, the
antenna might need calibration and then a demodulator is set. These activities
could be performed by different people at different times but the order matters
and must be respected.
There are many solution methods proposed to tackle the VRPTW. When us-
ing exact approaches, researchers tend to model the problem as multi-commodity
network flow problems (Desaulniers et al, 1998; Salani and Vaca, 2011) or fol-
lowing a set partitioning/covering formulation (Bredstr¨om and R¨onnqvist, 2007).
Such models have been tackled using constraint programming, branch and bound,
and branch and price (column generation) (Barnhart et al, 1998; Desrosiers and
L¨ubbecke, 2005). Other researchers use hybrid methods that employ heuristics
for the generation of columns within a column generation setting (Bredstr¨om and
R¨onnqvist, 2008) or use heuristics to improve an initial solution found with math-
ematical programming (Fischetti et al, 2004). Alternative approaches include di-
viding the problem (clustering) into smaller sub-problems and then obtaining a
global solution. This approach does not guarantee finding the overall global opti-
mal solution but it is sufficient if the objective is to quickly find feasible solutions
(Landa-Silva et al, 2011).
There are a few benchmark data sets for the VRPTW, here we refer to the
ones by Solomon (1987) and Castro-Gutierrez et al (2011) which we adapt for our
computational experiments.
