<figure><img src="image_121.png" /></figure>

market, and on understanding the customers of
remanufactured products. Optimisation models are
developed to determine the pricing and profitability
of remanufactured products (e.g., Ovchinnikov,
2011; Abbey et al., 2015). Game-theoretic models
are developed to determine optimal market-seg-
ments (based on pricing) for new and remanufac-
tured products (e.g., Debo et al., 2005; Atasu et al.,
2008). The market for- and customers of- remanu-
factured products are mostly analysed using empir-
ical methods, e.g., using sales data from websites
selling used and remanufactured products, usually
accompanied by customer surveys (e.g., Guide & Li,
2010; Subramanian & Subramanyam, 2012).
Behavioural experiments are used to understand
consumer perceptions (e.g., quality, functionality),
their acceptance (and rejection), and willingness to
pay for remanufactured products (e.g., Abbey et al.,
2017, and references therein).

**3.26. Telecommunications<sup>73</sup>**

Operational Research plays a key role in the design
and management of telecommunication networks. A
large variety of applications of both exact methods
and heuristics can be found in the literature. We
focus here on the applications for wired networks.

**3.26.1. Topological network design**

The earliest works on telecommunication networks
focused on wired fixed line telephony. For the long-
term planning of these networks, clients' demands
are not known in advance, or with a lot of uncer-
tainty. This often gives rise to two-stage approaches
where only the fixed cost of opening links are con-
sidered first, and the decisions on routing and cap-
acity allocation taken in a second (later) stage. This
approach is relevant when the fixed costs are very
high compared to routing and capacity costs, and/or
when topological decisions do not affect capacity
decisions too much. For example, digging a trench
to install fiber optic cables is very costly, while
increasing capacity can be done by adding or
upgrading equipment into nodes, which is relatively
simple and cheap. The objective is to build a net-
work at minimum cost, considering only the fixed
cost associated with opening a link, ignoring cap-
acity and routing costs.

Two main issues appear in the planning process
of such networks: economy and survivability.
Economy refers to the construction cost, while sur-
vivability refers to the restoration of services in the
event of equipment failure. A network is called a
tree if it is connected (i.e., there exists a path
between all pairs of nodes), and removing any link
disconnects at least one pair of nodes. Trees satisfy

the primary goal of minimising the total cost while
connecting all nodes. The minimum cost spanning
tree problem therefore received a lot of attention,
see e.g., Magnanti and Wolsey (1995).

However, only one node or edge breakdown
causes a tree network to become disconnected and
therefore to fail in its main objective of enabling
communication between all pairs of nodes. This
means that some survivability constraints have to be
considered while building the network. Usually,
these constraints come in the form with *k*-connect-
ivity requirements, i.e., the ability to restore network
service in the event of a failure of at most *k* - 1
components of the network. In their earliest
work on the subject, Grötschel and Monma (1990)
introduced a general model for survivability
requirements, and studied the polytope associated
with an integer programming formulation of the
problem.

The minimum-cost two-connected spanning net-
work problem, that consists in finding a network
with minimal total cost for which two node-disjoint
paths are available between every pair of nodes, was
studied extensively, starting with the work of
Monma and Shallcross (1989). Such networks have
been found to provide a sufficient level of surviv-
ability in most cases, but it turns out that the opti-
mal solution of this problem is often very sparse. In
such a topology, primary routing paths and re-rout-
ing paths in case of failure might become very long,
introducing large delays in the network.

Two kinds of solutions have been proposed to
remedy this problem: The first one imposes a con-
straint on the length of the paths (in terms of num-
ber of links crossed), the so-called *hop-constrained*
models. The second approach consists of imposing
that each edge belongs to at least one cycle (or *ring*)
whose length is bounded by a given constant.

Hop-constraints were first considered by
Balakrishnan and Altinkemer (1992) in order to
generate alternative solutions for a network design
problem. Later on, Gouveia (1998) presented a lay-
ered network flow reformulation that has since been
used in many network design applications involving
hops-constraints.

The second approach to avoid long re-routing
paths in case of failure is based on the technology
of self-healing *rings*. These are cycles in the network
equipped in such a way that any link failure in the
ring is automatically detected and the traffic
rerouted by the alternative path in the cycle. Many
problems involve setting a bound on the length of
the ring including each edge. Network design prob-
lems with bounded rings were first studied in Fortz
et al. (2000).
<figure><img src="image_122.png" /></figure>

### 3.26.2. Location problems

Location problems play a central role in telecommunications network design. We focus here on problems arising in wired (optical) telecommunications networks. These problems are mostly concerned with decisions related to the placement of specific equipments into nodes of the network, and are closely related to hub location problems (Alumur & Kara, 2008).

The *Concentrator Location Problem* is probably the most basic application of equipment placement. The problem consists of determining the number and location of concentrators that are used to aggregate end-user demands before sending them on the backbone network. The allocation of end-users demands to the concentrators has also to be determined such that the capacities of the concentrators are not exceeded. This problem has received much attention in the literature, starting with the work of Pirkul (1987).

Another classical problem arises with the replacement of an old technology by a new one, e.g., when telecommunications companies replace outdated copper twisted cable connections by fiber optic connections. The *Connected Facility Location Problem* (ConFL) aims at optimising the building cost for networks involving the two technologies, which is modeled as *tree-star* networks: the core network, made of fiber optic connections, has a tree topology and interconnects multiplexers that switch traffic between fiber optic and copper connections. Each multiplexer is the centre of a star-network of copper connections to the customers. Early work on ConFL concentrated on approximation algorithms, such as the primal-dual procedures proposed by Swamy and Kumar (2004). The currently best-known constant approximation ratio is given by the 4-approximation algorithm of Eisenbrand et al. (2010). Heuristic approaches have been proposed by Ljubić (2007) and Bardossy and Raghavan (2010). Different Mixed Integer Programming models for ConFL were proposed by Gollowitzer and Ljubić (2011).

In addition to these long-term design problems, operational short-term decisions are related to the routing of demands in the network, with a focus on avoiding congestion. Most networks nowadays operate the *Internet Protocol*. The internet is a collection of inter-connected networks called autonomous systems, that operates under a hierarchy of layers. An *Autonomous System* (AS) is defined as a set of routers under a single technical administration, such as an internet service provider or a country. As of July 2022, over 100, 000 ASes were registered<sup>74</sup>, connecting over 5 billion internet users worldwide<sup>75</sup>.

### 3.26.3. Traffic engineering

Traffic engineering (TE) addresses the problem of efficiently allocating resources in the network so that user constraints are met. Several criteria can be used to measure the effectiveness of a routing configuration. The selection of the objective function may drastically change the quality of the resulting routing. This distinction has been illustrated in Pióro and Medhi (2004). Balon et al. (2006) discuss various TE objective functions and evaluate how well these objective functions meet TE requirements.

The internet routing protocols can be clustered into two main groups: *inter-domain* and *intra-domain*. While inter-domain are used to route traffic between ASes, *Interior Gateway Protocols* (IGPs) handle the routing within ASes. As inter-domain protocols are mostly governed by administrative and political considerations, there is not much room for Operational Research techniques to be applied for performing TE. On the other hand, the optimisation of IGP s have received a lot of attention. The most popular IGP s are based on shortest path routing: shortest paths are calculated using a *link metric system*, which corresponds to the set of link weights or link metrics that belong to the same AS. The network operator controls the routing of the traffic indirectly by setting the link metrics. This gives rise to very challenging optimisation that have mostly been tackled heuristically by many authors, starting with the seminal work of Fortz and Thorup (2000). Some exact models have also been proposed, e.g., by Pióro et al. (2000).

Recently, Filsfils et al. (2015) proposed Segment Routing (SR), a new routing protocol developed to address known limitations of traditional routing protocols in IP networks. SR offers the possibility to deviate from the shortest path by using detours in the form of nodes or links respectively called node segments and adjacency segments. Optimisation of SR is a very active field of research and has been already addressed in Bhatia et al. (2015); Hartert et al. (2015); Jadin et al. (2019).

### 3.26.4. Further readings

For surveys on survivable network design, we refer the reader to Christofides and Whitlock (1981); Kerivin and Mahjoub (2005); Fortz and Labbé (2006); Fortz (2021). Location problems in telecommunications are surveyed in Skorin-Kapov et al. (2006); Fortz (2015) and a unified view on location and network design problems was proposed by Contreras and Fernández (2012). For a detailed survey on the Concentrator Location Problem, see Chapter 2 in Yaman (2005). Traffic engineering with shortest paths routing protocols is covered in
<figure><img src="image_123.png" /></figure>

the surveys of Bley et al. (2009); Fortz (2011); Altın et al. (2013).

## 3.27. Timetabling<sup>76</sup>

Timetabling represents a particular subgroup of scheduling problems, namely the set of problems for which activities must be assigned to resources within a set of fixed timeslots. Nevertheless, the two disciplines, scheduling and timetabling, are tightly related and benefit from mutual advancements in both modelling and method development.

Practical timetabling problems appear in many sectors, for example, in education, healthcare, sports and public transportation. They have been drawing academic attention for a few decades, partly because they are easy to grasp but challenging to solve. The timetabling community gathered at its first international conference in Edinburgh in 1995, one year before the Association of European Operational Research Societies (EURO) established a EURO Working Group on the Practice and Theory of Automated Timetabling (EWG PATAT 1996). Ever since the third conference, which took place in 2000, the timetabling community has gathered every two years<sup>77</sup> to share ideas on both theoretical and practical aspects of timetabling.

This subsection provides a brief overview of timetabling history, while highlighting what makes timetabling problems computationally challenging, which initiatives have boosted timetabling research and how state-of-the-art knowledge, models and algorithms can be applied in practice. We restrict the discussion to timetabling problems involving human resources, such as students, teachers, healthcare workers and sports teams.

### 3.27.1. Problem definition

Let us consider a set of timeslots $T = \{1, ..., |T|\}$, a set of activities $A = \{1, ..., |A|\}$ and a set of resources $R = \{1, ..., |R|\}$. A timetabling problem then consists in assigning (all) the activities in $A$ to resources in $R$ and timeslots in $T$ in such a way that a set of constraints is met. Constraints may apply to resources, timeslots and activities. They usually restrict the number of assignments to certain resources within subsets of $T$.

Constraints are usually divided into two categories: hard constraints, which must be strictly satisfied, and soft constraints, for which violations may be tolerated but should be avoided if possible. Weights may be set on the soft constraints, denoting their relative importance. A common timetabling objective is to minimise the weighted sum of soft constraint violations. This objective sometimes has to be combined with other timetabling objectives,

for example, to minimise the cost associated with the employed resources.

### 3.27.2. Educational timetabling

Educational timetabling problems can be split into three major groups: university examination timetabling, university course timetabling and high-school timetabling. In *examination timetabling*, the task is to assign examinations in $A$ to a limited number of timeslots in $T$ and rooms in $R$ such that no student has more than one exam at a time. Each student's exams should be spread out in time as much as possible. Additional constraints may include precedence constraints between exams, special room requirements, and limited room capacities. *Course timetabling* involves the assignment of course sections (lectures, tutorials, lab sessions, seminars) to specific days of the week and times of the day. Real-world problems may require sectioning, when students have to be split into separate subgroups for different sections. Typically, the objective is to minimise the number of students' conflicts. *High-school timetabling* assumes that students are split into classes and each class has to take a set of resources. Given a set of timeslots, each activity (involving both a student group and a teacher) must be assigned to a timeslot so that no teacher and no student group are participating in more than one activity at a time. Most practical problems have additional constraints; for example, teachers may have limited availability and some activities may require more than one timeslot. In general, educational timetabling problems are $NP$-hard (de Werra et al., 2002). Additionally, the constraints often pose a feasibility challenge.

The educational timetabling community made a considerable effort to create rich sets of benchmark instances to be used for comparing methods. The first set of examination timetabling instances was defined by Carter et al. (1996). Four competitions on educational timetabling, entitled ITC-2002 (McCollum, 2002), ITC-2007 (McCollum et al., 2007), ITC-2011 (Post et al., 2016) and ITC-2019 (Müller et al., 2018), further advanced the development of timetabling algorithms. Post et al. (2012) developed a general format and benchmark instances for high-school timetabling, which were extended later by Post et al. (2014). Ceschia et al. (2022) published a review of educational timetabling, presenting detailed characteristics of all benchmark instances and state-of-the-art results. OPTHUB<sup>78</sup> provides a common platform for storing problem instances and solutions to selected optimisation problems, including educational timetabling.
<figure><img src="image_124.png" /></figure>

### 3.27.3. Personnel timetabling

Personnel timetabling, also referred to as employee timetabling or rostering, concerns the construction of a timetable for personnel in *R* in such a way as to satisfy coverage constraints throughout a time horizon (Ernst et al., 2004). The timeslots in *T* often represent shifts, which correspond to tasks or duties in *A*. Some activities may require certain skills, and hence can only be conducted by a subset of *R*. Many work-rest-related objectives are formulated in terms of time-related constraints, restricting, for example, the number of hours worked, the number of weekends worked, the number of consecutive night shifts (Burke et al., 2004a). Additionally, personnel rostering problems typically consider personal preferences as regards working time or days off. Whereas the problem is generally considered NP-hard, Smet et al. (2016) showed that some personnel rostering problems are polynomially solvable, provided they do not contain a particular class of constraints. De Causmaecker and Vanden Berghe (2011) developed a categorisation of personnel rostering problems, based on the characterisation of resources, objectives and constraints. Kingston et al. (2018b) complemented this work by providing a unified notation for nurse rostering problems.

The Practice and Theory of Automated Timetabling (PATAT) community organised two International Nurse Rostering Competitions, entitled INRC I and INRC II. The problem definition of INRC I (Haspeslagh et al., 2014) was based on the instances collected by Burke and Curtois (2014). INRC II (Ceschia et al., 2019) incorporated real-world constraints concerning subsequent rostering horizons. The competition datasets have been collected and published⁷⁹.

Apart from the constraints and objective functions considered in the two INRCs, some sectors expect their personnel rosters to be cyclic (Musliu, 2006; Rocha et al., 2013). Recent trends also include objectives related to fairness (Gross et al., 2019) and well-being (Petrovic et al., 2020). Objective priorities set by the users may lead to unwanted solutions. To address this issue, Böðvarsdóttir et al. (2021) developed an approach to automatically set acceptable weights which avoid conflicting objectives from leading to poor solutions.

### 3.27.4. Sports timetabling

Sports timetabling problems often address tournament or competition scheduling. They require assigning sports activities in *A*, represented by pairs of teams in *R*, to timeslots in *T* in such a way that each team meets all the other teams. Constraints depend on the competition's rules, which may differ in different parts of the world (Ribeiro, 2012;

Durán, 2021). Specific sports timetabling constraints prescribe that teams must not meet the same opponent within consecutive timeslots, or that the number of consecutive home or away games is restricted. The travelling tournament problem (TTP), introduced by Easton et al. (2001), is an academic adaptation of the Major League Baseball competition in the United States. The objective of the TTP is to minimise the sum of travelling distances for each team. Travelling umpire scheduling (Trick et al., 2012) is subject to similar constraints, but it assumes that the tournament is fixed and that each game is assigned an umpire.

Rasmussen and Trick (2008) provided a review on round robin sports timetabling, where each team plays against each other team twice, once at home and once away. Drexl and Knust's (2007) review focused on graph-theoretical approaches to sports timetabling. Briskorn et al. (2010) investigated the complexity of several variants of the round-robin tournament problem, and similarly, de Oliveira et al. (2015) studied the complexity of travelling umpire scheduling problems. The characteristic sports timetabling constraints, which forbid the assignment of activities to subsets of *T*, can be challenging in terms of feasibility.

Trick (2001) and Toffolo et al. (2015) boosted sports timetabling research by publishing challenging benchmark instances and monitoring best known and/or optimal results. Van Bulck et al. (2021) organised the first international sports timetabling competition, for which the instances are available at the website of STT (2021).

### 3.27.5. Timetabling and related problems

Academic timetabling problems are often considered in isolation from other problems. However, many real-world situations face timetabling entangled with other optimisation problems. Solutions for one of them have an impact on the solution for the other problems. For example, the staffing problem is concerned with optimising a group of human resources and their characteristics such as skills and contracts in an organisation, across a relatively large time horizon. From a staffing perspective, the personnel structure should adequately cover the organisation's anticipated workload while respecting the available budget. On the other hand, from a rostering perspective, the personnel structure should enable computing good quality rosters across many subsequent rostering periods (Komarudin et al., 2020). Similarly, task scheduling usually assumes personnel rosters are fixed, but both problems can also be addressed in an integrated manner (Paul & Knust, 2015). The workforce routing and scheduling problem is related to vehicle routing. Apart from
<figure><img src="image_125.png" /></figure>

scheduling a fleet of vehicles to serve a set of cus-
tomers, timetabling issues, such as temporal con-
straints, contracts and skills are also imposed on the
problem (Castillo-Salazar et al., 2016). Some produc-
tion scheduling and inventory problems are subject
to additional timetabling restrictions which apply to
their employees (Sartori et al., 2021).

3.27.6. Where do we stand and what is the future

Academic timetabling has made good progress and
instances, models and algorithms have been shared
and published. For example, the heuristic search
strategies Step Counting Hill-climbing (Bykov &
Petrovic, 2016) and Late Acceptance Hill-climbing
(Burke & Bykov, 2008) were initially developed for
solving timetabling problems. Due to their simplicity
and effectiveness, they continue to be used in a
much wider application domain by many computa-
tional experts.

So long as some instances remain unsolved, or
solutions for instances have not been proven opti-
mal, algorithm development remains open for
improvement. Future challenges may also apply to
new combinatorial optimisation problems encom-
passing a timetabling component. They may not
necessarily map to any of the three timetabling cate-
gories detailed in this chapter. However, they may
gain importance due to either increased practical
need or academic initiatives, such as the publication
of benchmarks or the organisation of competitions.

Apart from these future computational chal-
lenges, timetabling research should also focus on
how to address *human resources'* considerations.
Besides the traditional work-rest constraints and
objectives, academia should also reconcile personnel
well-being with their perception of fair workload
within a team and with their level of autonomy in
determining their personal timetables. Research
should also focus on how to address the increasing
personnel resignation in human-centric working
environments such as education and healthcare.
Robust timetabling, for example, has a lot of poten-
tial and at the same time induces scientifically inter-
esting modelling questions.

**3.28. Transportation: Rail<sup>80</sup>**

The transportation of goods and passengers by rail
has played an important role in the evolution of
industrialised societies, contributing to their devel-
opment and prosperity. Rail freight transport still
holds critical importance in supporting the eco-
nomic growth of many countries around the world
due to its contribution to guaranteeing an efficient
flow of goods internally and across borders.
Furthermore, rail transportation is also essential for

the movement of people, being the preferred trans-
portation mode for commuters in many large urban
areas. This preponderant role also affects the
internal mobility of cities. First, a differentiation
must be made between freight and passenger trans-
port. Freight trains are longer and heavier than pas-
senger trains, and can often have multiple
propulsion units. Compared to that, passenger trains
are much lighter and have more horsepower per
tonne. There are also important planning and oper-
ational differences, whereas passengers decide freely
where they will travel, each load of freight must be
managed and routed from a specific origin to its
destination. These differences originate very differ-
ent problems in both areas. Even in passenger trans-
portation, different problems arise depending on the
type of service; long- and medium-distance, com-
puter rail, urban rapid transit, and scenic and sight-
seeing train transportation; see, for instance,
Caprara et al. (2007).

Despite all these differences, a set of common
hierarchical stages can be highlighted in the process
of planning and operating a rail transportation sys-
tem (Bussieck et al., 1997): network design and/or
line planning, timetabling, platforming, rolling stock
circulation, shunting, and crew planning.

At a strategic level, the problems are character-
ised by long planning horizons and typically involve
resource acquisition. This level includes network
design and line planning problems. The first refers
to the construction or modification of existing rail-
way infrastructure and mainly concerns urban rapid
transit systems. For a railway company or agency,
the line planning problem consists of defining a set
of lines and determining their frequencies, and it is
usually the first stage in planning medium and
long-distance passenger rail networks.

Bussieck et al. (2004) considered the design of
line plans in public transport with the objective of
minimising the total cost. Goossens et al. (2006)
presented several models for solving line planning
problems in which lines can have different halting
patterns. Laporte et al. (2007) proposed a first rail-
way rapid transit network design model to maximise
the expected trip coverage. Gutiérrez-Jarpa et al.
(2013) presented a model to minimise travel cost
while maximising the captured demand. See also
Laporte and Pascoal (2015) for an extension where
the idea consists of first building a set of segments
within broad corridors connecting some vertex sets
to later assemble the segments into lines.

A different set of works pays attention to the for-
mulation of network design models from scratch.
Starting from an underlying network, these models
construct lines by joining edges, incorporating topo-
logical constraints to guarantee connectivity between
<figure><img src="image_126.png" /></figure>

consecutive edges of each line. This approach gives rise to complex models which are quite difficult to solve using exact procedures; see, for instance, the work by Szeto and Jiang (2014), or the recent works by Canca et al. (2017) and Canca et al. (2019) which concern the design of a railway rapid transit network.

For a comprehensive review of the different methodologies used in practice to solve this problem, the readers can consult the review of Guihaire and Hao (2008). More recent reviews of Schöbel (2012) and Ibarra-Rojas et al. (2015) present a systematic classification of problem variants, considered objectives and solving methodologies.

At the tactical level, the next stage in planning a railway system consists of several problems, starting with scheduling and timetabling, followed by rolling stock planning, crew rostering, and crew scheduling. The timetabling problem concerns the determination of the arrival and departure times of trains to stations. When overtaking and overlapping are allowed, the timetabling problem becomes a train scheduling problem. Timetables can be cyclic, regular, hybrid, and demand-driven. Concerning the design of cyclic timetables, Caprara et al. (2002) proposed a graph-theoretic formulation for the train timetabling problem using a directed multigraph in which nodes correspond to departures and arrivals at a certain station at a given time instant. Liebchen and Möhring (2002) used a Periodic Event Scheduling problem (PESP) with several add-ons concerning problem reduction and strengthening. Chierici et al. (2004) extended the classical timetabling model to take into account the reciprocal influence between the quality of a timetable and the transport demand captured by the railway with respect to alternative means of transport. Cacchiani et al. (2008b) proposed heuristic and exact algorithms for the (periodic and non-periodic) train timetabling problem on a corridor.

Regular timetables have been commonly used in the case of railway rapid transit systems, especially at relatively short time planning horizons where demand can be considered approximately constant. Canca et al. (2016) proposed a sequential optimisation approach to determine the best regular timetable for a railway rapid transit network where lines share tracks. Canca and Zarzo (2017) incorporated aspects of energy consumption in the design of a two-way rapid rail transit line. Later, Canca et al. (2018) extended the previous work to a full network, taking into account transfers between lines. Robenek et al. (2017) proposed a new type of timetable combining both the regularity of the cyclic timetables and the flexibility of the non-cyclic ones.

During recent years, starting from the works of Canca et al. (2014) and Niu and Zhou (2013) many

researchers have paid attention to the design of demand-driven timetables (see, for instance, Barrena et al., 2014a, 2014b). The design of a specific train timetable can be combined by using different acceleration strategies such as stop-skipping and short-turning. For example, given predetermined train skip-stop patterns, Niu et al. (2015) proposed a quadratic integer programming model with linear constraints to synchronise effective passenger loading and train arrival and departure times at stations. Zhou et al. (2022) proposed a mixed integer linear programming model to jointly optimise the train timetable and the rolling stock circulation plan, allowing rolling stock to change its composition through coupling/decoupling operations at the terminal stations of a metro line. Yuan et al. (2022) introduced a new integrated optimisation model for train timetabling that also considered rolling stock assignment and incorporated a short turn strategy on a bidirectional metro line.

Several authors have also proposed methods to increase the transport capacity of a given timetable (see Burdett & Kozan, 2009). Cacchiani et al. (2010) studied the problem of incorporating freight transport trains in railway networks, where both passenger and freight trains are running. To finish this description of the OR contributions for the train timetabling problem, a special mention of the work by Kroon et al. (2009) is convenient. In this research, the authors generated several real timetables using Operational Research techniques for the Dutch railway network.

Rolling stock management is probably the most complex stage in the classical sequential railway planning process and plays a key role in the efficient operation of railway networks. At a tactical level, the rolling stock circulation plan consists of a set of interrelated subproblems such as train composition decisions (coupling and decoupling operations involving locomotives and carriages), selection of rest locations, the design of vehicle circulations (specific paths that vehicles must follow to guarantee an efficient and safe operation), and the definition of maintenance policies (Caprara et al., 2007). In a general rolling stock circulation problem, every train circulation has a variable length (distance and number of days) and incorporates information about the allowed specific rolling stock types, composition, coupling/decoupling operations, maintenance and cleaning activities. (Maróti & Kroon, 2005, 2007). Other practical considerations such as rolling stock availability, depot capacity (Lai et al., 2015), coupling and decoupling activities (Fioole et al., 2006), turnaround times, maintenance (Maróti & Kroon, 2007), and track and platform capacities are simultaneously considered depending on the specific problem. Given the importance of this topic within the set of planning tasks, other contributions have
<figure><img src="image_127.png" /></figure>

been proposed for different problems concerning rolling stock management, as, for instance, determining a set of minimum cost equipment cycles such that the most convenient rolling material is assigned to each planned trip Cordeau et al. (2000) or obtaining the optimal circulation of rolling stock considering order in train compositions (Alfieri et al., 2006; Peeters & Kroon, 2008). Maintenance also plays an important role in several rolling stock management contributions, see, for instance, the works by Maróti and Kroon (2005); Giacco et al. (2014) and D’Ariano et al. (2019). Robustness is another topic of interest in the related literature. Interested readers can consult the works by Cacchiani et al. (2008a, 2012).

After rolling stock management, the crew scheduling process determines the set of duties that covers all programmed services (Caprara et al., 1998). Finally, the crew is assigned to serve the crew schedule and the corresponding train services (Huisman et al., 2005). The rostering process aims at determining an optimal sequencing of a given set of duties into rosters satisfying operational constraints deriving from union contract and company regulations (Caprara et al., 2003).

To finish this section, two important problems of rail freight transportation are briefly commented. The first concerns the strategic design of freight transport networks and the second concerns the tactical operation of marshalling yards. Concerning the design of service networks, Crainic et al. (1984) analysed the problems of routing freight traffic, scheduling train services, and allocating classification activities between yards on a rail network. Crainic et al. (1990) developed a model of rail freight transportation adapted for the strategic planning of freight traffic considering other transportation modes. Zhu et al. (2014) addressed the problem of scheduled service network design for freight rail transportation integrating service selection and scheduling, car classification and blocking, train makeup, and shipments routing based on a three-layer cyclic space-time network representation.

Shunting yards, also known as marshalling or classification yards, play a key role in rail freight transport networks, acting as hubs where inbound trains are first disassembled and the carriages are then to form new convoys, generating new trains which transport the load towards the correct destinations. This procedure allows carriages to be sent through the network according to their destinations without the need for many connections. Therefore, time savings in shunting operations (Jaehn et al., 2015) have a great impact on cost savings in the movement of freight through the rail network (Boysen et al., 2012). In passenger transportation, shunting operations focus on train units that are not necessary to operate a schedule and must be

parked at shunt yards. Since different types of trains use the rail infrastructure, the specific type of a unit restricts the set of shunt tracks where they can be parked. The aim of this problem is to assign train locations to the shunt tracks while minimising routing costs from platforms to the corresponding shunt tracks (Huisman et al., 2005; Kroon et al., 2008). For a more detailed description of the optimisation problems involved in shunting operations, we refer the reader to the works by Jaehn and Michaelis (2016) and Ruf and Cordeau (2021).

### 3.29. Transportation: Maritime⁸¹

Maritime transportation carries more than 80% of the world's trade and some 70% of the value of that trade (UNCTAD 2022). The spectrum of Operational Research (OR) applications in maritime transportation is broad. Following the classification of Christiansen et al. (2013), these problems can be broken down into three levels: *strategic, tactical and operational*. Some typical problems in each of these levels will be described in this section.

It is important to note that, in much of the OR maritime transportation literature, traditional economic criteria such as cost minimisation or profit maximisation are the norm, and environmental criteria (for instance emissions minimisation) are less frequent. However, with the quest to decarbonise shipping (IMO 2018), the body of knowledge that includes environmental criteria is growing very fast in recent years. Sometimes environmental criteria map directly into economic criteria: if for instance *fuel cost* is the criterion, and since it is directly proportional to ship *emissions*, if fuel cost is to be minimised as an objective, so will emissions, and the solution is *win-win*. However, for other objectives this direct relationship may cease to exist and one would need to look at environmental criteria in their own right.

In conceptual terms, if $x$ is a vector of the decision variables of the problem at hand, $f(x)$ is the fuel cost associated with $x$, $c(x)$ is the cost other than fuel and $m(x)$ are the associated maritime emissions (carbon, sulphur, or other), then a generic optimisation problem is the following:

$$ \text{Minimise} \quad \alpha(f(x) + c(x)) + \beta m(x) \\ \text{s.t.} \quad x \in X $$

where $\alpha$ and $\beta$ are user-defined weights (both $\ge 0$) representing the relative importance the decision maker assigns to cost versus emissions, and $X$ represents the feasible solution space, usually defined by a set of constraints.

One can safely say and without loss of generality that if $d(x)$ is the amount of fuel consumed, $p$ is the fuel price, and $e$ is the emissions coefficient (kg of emissions per kg of fuel), then $f(x) = pd(x)$ and
$m(x) = ed(x)$. Therefore $f(x) = km(x)$ with $k = p/e$, as both $f(x)$ and $m(x)$ are proportional to the amount of fuel consumed $d(x)$. The cases that different fuels are used onboard the ship, for instance in the main engine vs the auxiliary engines, or if fuel is switched from high to low sulphur along the ship's trip, represent straightforward generalisations of the above formulation. Then the above problem can also be written as

$$ \begin{array}{ll} \text{Minimise} & \alpha c(x) + (\alpha k + \beta) m(x) \\ \text{s.t.} & x \in X \end{array} $$

The following special cases of the above problem are important:

1. The case $\alpha = 0, \beta > 0$, in which the problem is to minimise emissions.

2. The case $\alpha > 0, \beta = 0$, in which the problem is to minimise total cost.

3. The case $c(x) = 0$, in which fuel cost is the only component of the cost.

A solution $x^*$ is called *win-win* if both case (1) and case (2) have $x^*$ as an optimal solution. It is important to realise that such a solution may not necessarily exist.

It is also straightforward to see that in case (3), cost and emissions are minimised at the same time and we have a *win-win* solution. It is clear that $c(x) = 0$ is a *sufficient condition* for a *win-win* solution. But this is not a *necessary condition*, as it is conceivable to have the same solution being the optimal solution under two different objective functions. An interesting question is to what extent policy makers can introduce either (a) a Market Based Measure (MBM) such as a fuel tax and/or (b) a set of constraints, that would make win-win solutions possible.

Let us now examine some typical OR problems in the 3-level hierarchy.

**Strategic level problems** involve planning horizons of several years (from 1 to 25). Among them, *fleet size and mix* problems involve the basic questions, what is the best mix for a shipping company's fleet in the years ahead? How large should these ships be? How many should they be, and how fast they should go? See Alvarez et al. (2011), Zeng and Yang (2007) and Pantuso et al. (2014) for some work in this area.

**Network design** problems also belong to the strategic problem category and are special to liner shipping. They involve the design of a liner company's network, which comprises the ports it would serve, the routes it will use, which ports will be chosen as hub ports, how are the company's feeder networks configured, and whether the company will use the hub-and-spoke concept or direct calls. See Agarwal and Ergun (2008), Reinhardt and Pisinger (2012), and Brouer et al. (2014) for more on these problems.

**Tactical level problems** involve intermediate planning horizons, from a few days to a year. Among them, *ship routing and scheduling* is perhaps the most important problem class, mainly for tramp shipping, with works by Christiansen et al. (2013), Andersson et al. (2011), Fagerholt et al. (2010), and Lin and Liu (2011). Routing and scheduling of offshore supply vessels belongs also to this area (Halvorsen-Weare & Fagerholt, 2011; Norlund & Gribkovskaia, 2013). All of these problems call for the determination of the best set of ship routes under some predefined criteria.

**Fleet deployment** is also included in the class of tactical level problems, calling for the allocation of ships to routes (see Meng & Wang, 2011; Andersson et al., 2015; Lai et al., 2022, among others). *Speed optimisation* problems are also tactical level problems and have received increased attention in recent years, due to the pivotal role of ship speed with regard to both economic and environmental criteria. Due to the fact that fuel consumption is a nonlinear function of ship speed, these problems are typically nonlinear. Related formulations attempt to find best vessel speeds along the legs of the route, according to specific criteria (see Psaraftis & Kontovas, 2013; Fagerholt & Ronen, 2013; Magirou et al., 2015). These problems may also involve *flexible frequencies* (Giovannini & Psaraftis, 2019).

Speed and route decisions may also be combined (Psaraftis & Kontovas, 2014; Wen et al., 2017). One of the perhaps counter-intuitive results of these combined scenarios is that sailing the minimum distance route at minimum speed does not necessarily minimise fuel consumption and hence emissions. This may be so whenever the minimum distance route involves a heavier load profile for the ship. Depending on ship type, the difference in fuel consumption between a fully loaded and a ballast (empty) condition can be up to 40%. A result that is less surprising is that expensive cargoes sail faster and hence induce more emissions. This is to be expected if cargo in-transit inventory costs are taken into account.

**Modal split/discrete choice models** examine scenarios in which shippers may choose a transportation mode that is alternative to the maritime mode as a result of unfavourable time, cost, or other considerations. As a result, cargoes from the Far East to Europe may prefer the rail vs the maritime mode, or cargoes in European short sea trades may choose the road mode as opposed to shipping. Such modal shifts may increase the overall level of CO2 and may warrant mitigation measures by the shipping lines and the policy makers. Papers that look into this problem include Psaraftis and Kontovas (2010) and Zis and Psaraftis (2017, 2019). A multi-commodity network flow formulation in the context of China's Belt and Road initiative is given by Qi et al. (2022).
<figure><img src="image_128.png" /></figure>

*Operational level problems* concern problems with planning horizons from a few hours to a few days. Among them, a very important class of problems concerns *weather routing* scenarios. The important difference vis-à-vis the ship routing and scheduling problems described earlier is that weather routing problems are typically *path* problems defined as trying to optimise a ship's track from a specified origin to a specified destination, under a prescribed objective and under time varying and maybe also stochastic weather conditions. Decision variables include the selection of the ship's path and the speeds along the path, and typical objectives include minimum transit time and minimum fuel consumption. Several constraints such as time windows, or constraints to accommodate a feasible envelope on ship motions, vertical and transverse accelerations and ship loads such as shear forces, bending moments and torsional moments can be introduced. The influence of currents, tides, winds and waves, which may be varying in both time and space should be taken into account. See Perakis and Papadakis (1989), Lo and McCord (1998), and Zis et al. (2020) for some references on this topic.

*Disruption management* is also another important operational level problem class and typically refers to liner shipping. It entails actions that can help the shipping company manage its recovery from possible disruptions of its schedule. Such disruptions may be the result of bad weather, port strikes, equipment malfunction, or more recently, the COVID-19 pandemic that caused massive congestion in many ports worldwide or the Ever Given incident that disrupted traffic in the Suez Canal and the Far East to Europe route in 2021. See Qi (2015) and Asghari et al. (2023) for work in this area.

*Terminal management, berth allocation, and stowage planning* problems also belong to the class of operational level problems, as they deal with an important part of the overall maritime supply chain, that of the coordination between a ship and a port. See Moccia et al. (2006), Goodchild and Daganzo (2007), and Zhen (2015) for some related work.

To conclude, maritime transportation constitutes an important application area for OR, and the related problems are interesting and significant, both from a methodological perspective and from a business and policy perspective. This is so both for traditional economic performance criteria and for environmental criteria, the importance of the latter getting higher in recent years.

### 3.30. Transportation: Aviation<sup>82</sup>

According to the Air Transport Action Group, in 2019, the world's 1,478 airlines transported 4.5 billion passengers to 3,780 airports, generating 11.3

million direct jobs. Today's airlines are sophisticated businesses making aviation a worldwide economic engine. Yet, aviation is a competitive industry, vulnerable to exogenous shocks, e.g., oil prices, infectious diseases or terrorism. This leads to high costs, and low profit margins, even in the best of times. To tackle these challenges, the industry relies heavily on Operational Research (OR) for decision-making. Prominent OR application domains within aviation include revenue management, airline schedule planning, airline operations recovery, airport flight scheduling, and air traffic flow management. Additionally, some recent OR studies focus on modelling delay propagation through aviation networks

#### 3.30.1. Revenue management (RM)

RM is broadly defined as the strategies and tactics to increase revenues by optimally matching demand for products/services with the available capacity. Seat allocation and pricing are the two main decisions to control ticket sales of different fare-classes. Models using capacity allocation as the control variable are called quantity-based RM models. They allocate seats to fare-classes with exogenously determined prices. In contrast, price-based RMs uses pricing policies to maximise revenues. Early RM models focused on overbooking – the practice of selling more tickets than seats to hedge against cancellations or no-shows. Though various static and dynamic models have been presented since the pioneering work of Rothstein (1971), airlines mostly use simpler static policies in practice.

Static and dynamic models have been proposed for both single-leg and network-wide seat allocation. Static models optimise seat allocation at a certain time, typically the beginning of the booking period. Dynamic models monitor and adjust to the booking process over time. The earliest static leg-based approach (Littlewood, 1972) considered two fare-classes. Brumelle et al. (1990) relaxed the assumption of statistical independence between demands. For the multi-class problem, Belobaba (1987b) introduced the Expected Marginal Seat Revenue heuristic, a widely used approach in practice. Many studies (e.g., Brumelle & McGill, 1993) provided optimality conditions for static models, while others developed methods to compute optimal protection levels in the absence of demand information, using optimality conditions (Van Ryzin & McGill, 2000) or stochastic approximations (Kunnumkal & Topaloglu, 2009). Dynamic formulations allow time-based controls, but require restrictive demand assumption for tractability, limiting practical impact. Solving network models exactly is computationally hard. Accordingly, most studies on network models use approximations, based on deterministic linear
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programming (Talluri & Van Ryzin, 2004a), randomised linear programming (Talluri & Van Ryzin, 1999) or decomposition into single-resource problems, as well as solutions using simulation-based optimisation (Bertsimas & De Boer, 2005). Seat inventory control usually assumes capacity to be fixed, an assumption relaxed by Büsing et al. (2019) integrating capacity uncertainty in leg-based RM. Others integrated inventory control and pricing (You, 1999).

Simplest deterministic pricing models are price-sensitive versions of the well-known newsvendor problem (Gallego & Van Ryzin, 1994). This allows mathematical derivations of optimal prices. Several studies, such as, Feng and Gallego (1995), generalised this problem to include demand dynamics and/or multiple products. Stochastic dynamic programming is a natural way to tackle dynamic pricing. Dynamic models depict reality more accurately, but are harder to solve (Gallego & Van Ryzin, 1994). Interestingly, solutions to deterministic models are usually good approximations for their stochastic counterparts, and are often used in practice. Traditional RM assumed independent demand, ignoring product substitutability. With the seminal paper of Talluri and Van Ryzin (2004a), the RM field has shifted toward including customer choice behaviors within pricing and capacity decisions. §3.21 provides a detailed overview of RM concepts and trends beyond aviation.

### 3.30.2. Airline schedule planning (ASP)

ASP is the process of designing airline schedules maximising profits subject to resource constraints. Taking demand, airport and aircraft characteristics, and maintenance and personal requirements as inputs, ASP outputs selected flight timetables, aircraft schedules and crew duty plans. Most ASP steps typically occur before RM actions and thus constrain the set of decisions available to RM systems. Key ASP steps include fleet planning, route planning, frequency planning, timetable design, fleet assignment, aircraft routing and crew scheduling. Fleet planning involves decisions regarding purchasing, selling, and leasing of aircraft fleet, while route planning selects airport pairs to operate nonstop flights. Early studies, e.g., Hane et al. (1995), matched a predetermined set of flights with aircraft types, developing a fleet assignment model (FAM). The FAM specifically focuses on fleet assignment, which is one particular step within the overall ASP process. The basic FAM, a mixed-integer linear program, minimised costs of operating aircraft and passengers unserved, given passenger demand for individual flight legs. This leg-based approach

ignores that passengers often fly on multiple flights in connecting itineraries.

Barnhart et al. (2002) overcame this limitation via an itinerary-based FAM to explicitly model network effects. Some studies developed tractable solution approaches. Barnhart et al. (2009) proposed a subnetwork-based decomposition for capturing FAM’s revenue implications, an approach recently extended by Yan et al. (2022a) to solve a FAM incorporating passenger choice. Others extended FAM by incorporating incremental timetable design decisions, e.g., changes to flight timings (Desaulniers et al., 1997) or selection of optional flights (Lohatepanont & Barnhart, 2004). Wei et al. (2020) developed a *clean slate* heuristic optimising entire timetables and fleet assignments under choice-based demand. Frequency planning, which optimises the number of flights operated during a day or part of a day, rather than deciding exact timetables, has also received attention, with an emphasis on capturing effects of competition from other airline and high-speed rail operators (e.g., Cadarso et al., 2017).

The last two steps in schedule planning are conceptually similar. Aircraft routing assigns individual aircraft to flights while ensuring that each aircraft undergoes periodic maintenance, and crew scheduling assigns crew to operate flights while satisfying a myriad of crew regulations. Early studies individually optimised aircraft routing (Gopalan & Talluri, 1998) or crew scheduling (Graves et al., 1993a). Lavoie et al. (1988) used column generation, an effective solution approach for both problems, to crew scheduling, while Cordeau et al. (2001) used Benders decomposition to jointly solve both problems.

Good schedules not only minimise planned costs, but are also robust to disruptions, to keep the actual costs low. Researchers in early 2000s optimised robustness proxies, e.g., station purity, short cycles, crew swapping opportunities, and crew schedule slack (Schaefer et al., 2005). Later studies directly minimised total planned and unplanned costs of aircraft routing (Lan et al., 2006) and crew scheduling (Yen & Birge, 2006) separately, and also jointly (Dunbar et al., 2012). Recent studies have used robust optimisation to solve the aircraft routing (Yan & Kung, 2018) and crew scheduling (Antunes et al., 2019) problems.

### 3.30.3. Airline operations recovery (AOR)

AOR encompasses the actions undertaken to repair schedules, when disruptive events such as inclement weather, equipment failures, etc., take place. Rosenberger et al. (2003) developed a model and a solution heuristic for repairing aircraft routing, whereas Lettovský; et al. (2000) tackled crew
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recovery. For the integrated recovery problem, Petersen et al. (2012) developed a decomposition strategy, while Maher (2016) used column-and-row-generation. Recent recovery studies incorporated other key elements, including flight planning (Marla et al., 2017) and passenger no-shows (Cadarso & Vaze, 2022).

### 3.30.4. Airport flight scheduling (AFS)

Beyond airline decision-making, OR is also used to improve decision-making of central authorities and air traffic managers. Research over the past decade demonstrated the potential for enhancing social welfare by constraining schedules at busy airports via slot-control mechanisms (Swaroop et al., 2012). Some studies balanced strategic cost of scheduling changes against tactical cost of delays, for a single airport (Jacquillat & Odoni, 2015) or multiple airports (Wang & Jacquillat, 2020). Zografos et al. (2012) used an integer program for allocating slots to airlines under administrative controls. Fairbrother et al. (2020) attempted to balance the often-conflicting goals of efficiency, equity and the incorporation of airline preferences in optimising slot-scheduling mechanism.

### 3.30.5. Air traffic flow management (ATFM)

The tactical side of airport and airspace capacity management has received considerable OR attention since the 1990s. ATFM is a broad term used to define key interventions, such as ground holding of airplanes, that ensure safe and efficient flight operations by restricting flow of aircraft into congested airspaces. Terrab and Odoni (1993) and Vranas et al. (1994) proposed the single-airport and multi-airport ground holding problems, respectively. The latter was extended to include enroute capacities by Bertsimas and Patterson (1998). Bertsimas et al. (2011b) additionally incorporated flight rerouting and solved larger-scale problems. Adoption of the collaborative decision-making (CDM) paradigm in practice ushered in a new era of research. Advocating increased agency to airlines, Vossen and Ball (2006) provided an integer program for slot trading mechanism design under CDM. Recent studies (e.g., Starita et al., 2020) are increasingly focused on explicit handling of uncertainty on both demand and capacity side within the ATFM optimisation problems.

### 3.30.6. Modelling delay propagation

Tightly coupled aviation networks make disruption management particularly challenging. Delays and disruptions in one part of the network propagate to other parts, through aircraft, crew and passenger connections. Recent studies quantified these

propagation effects. First, Pyrgiotis et al. (2013) proposed an analytical queuing and network decomposition model for aircraft-based delay propagation. Barnhart et al. (2014) presented discrete choice models for passenger itinerary estimation and a reaccommodation heuristic for passenger delay calculations. Wei and Vaze (2018) solved inverse optimisation for estimating crew itineraries and crew-based delay propagation. These studies attempted bridging the gap between sparse and aggregate public datasets, and the detailed and disaggregated data needs for aviation OR research.

### 3.30.7. Further reading

Readers interested in aviation OR are referred to the second edition of the book by Belobaba et al. (2015). In particular, Chapters 4 and 5 focus on pricing and RM, Chapters 8 and 10 on schedule optimisation, robustness and recovery, and Chapter 14 on air traffic management and control. Looking ahead, it is apparent that OR will keep finding natural applications within aviation, especially given the exciting disruptive innovations within urban air mobility. Rapidly growing fields of passenger air taxi operations and drone operations for parcel deliveries are giving rise to new variants of well-known OR problems, e.g., network design (Wang et al., 2022b), travelling salesperson (Roberti & Ruthmair, 2021), vehicle routing (Dayarian et al., 2020), and facility location (Chen et al., 2022).

## 3.31. Transportation: Network design⁸³

In a transportation context, the term *Network Design* (Magnanti & Wong, 1984) generally refers to planning the *supply* side of a transportation system so that it efficiently satisfies some estimate of *demand* within the quality standards of the customers using the system. The planning decisions typically prescribe the movements of vehicles, or convoys (e.g., a railroad train or tug and barges), between stations/terminals in the network to transport people or goods. Network design is typically undertaken for situations wherein what is transported, be it people or goods, is small relative to vehicle capacity. Thus, one primary measure of efficiency is vehicle utilisation, with high utilisation achieved through consolidation. Quality is typically measured based on on-time delivery.

Network design is relevant to passenger transportation systems such as urban public-transport (Mauttone et al., 2021) by bus (Ceder & Wilson, 1986) or light rail (Farahani et al., 2013), as well as systems providing interurban transport by train (Hooghiemstra et al., 1999) or airplane (Franke, 2017). It is also relevant to a wide range of goods
<figure><img src="image_131.png" /></figure>

transportation markets, such as parcel and small-package (Barnhart & Schneur, 1996) and less-than-truckload freight (Powell & Sheffi, 1989). A network design case study for a postal carrier can be found in Winkenbach et al. (2016). Transportation carriers serving these markets may rely on one or more modes, including motor carrier (Bakir et al., 2021), rail (Chouman & Crainic, 2021), ocean (Christiansen et al., 2020), and inland waterway (Konings, 2003). The planning of vehicle and goods movements by each mode and synchronisation of goods moving from one mode to the next (e.g., intermodal) can be assisted by network design (Arnold et al., 2004).

For different modes the scope of design decisions prescribed by network design models may be broadened in different ways. For example, modes such as rail and inland waterway involve multiple layers of consolidation. For rail (Zhu et al., 2014), goods are consolidated into rail cars, which are then consolidated into blocks that are transported by the same locomotive. For motor carriers, vehicles can not yet move without a driver, whose movements and schedules are restricted by governmental safety regulations and potentially labour management practices that dictate the driver return periodically to a specific physical location in the network (e.g., his/her domicile). Network design models for motor carriers may build schedules for drivers that observe safety regulations (Crainic et al., 2018) as well as determine how many drivers should be associated with each physical location (Hewitt et al., 2019).

The network design problem is typically modelled as a Mixed Integer Program (MIP) formulated on a directed graph (Crainic et al., 2021a). Nodes in such a graph model physical locations, potentially at different points in time. Directed edges between such nodes model transportation that begins in one physical location and ends at another. Edges may encode a scheduling dimension, such as when a vehicle departs from one location and arrives at another, that depends in part on the travel time required for the physical move (Erera et al., 2013). Associated with an edge is a function that maps the amount of vehicle capacity made available on that edge to cost. Typically, it is a step function with each step modeling an increase in capacity due to dispatching an extra vehicle. Commodities model people or goods that are to be transported; associated with each commodity is an origin node, a destination node, and a size.

The classical network design problem seeks to find a path for each commodity that begins at its origin node, ends at its destination node, and potentially visits one or more intermediate nodes. The problem evaluates these paths with respect to the

total cost of capacity made available to support them and seeks to minimise that total cost. Some network design models (Frangioni & Gendron, 2021) instead minimise costs that are a function of the amount of goods transported on an edge, as opposed to the capacity made available to transport them. Network design is an optimisation problem that has received significant attention both for its practical relevance and the computational challenges (Johnson et al., 1978) associated with solving it.

Most MIP formulations of the network design problem involve commodity flow variables that model the transportation of goods within the network and another set of edge-based variables that model the transportation of vehicles. Typically, commodity flow variables are continuous when a shipper's goods can be divided and routed on multiple paths or binary when they cannot. Commodity flow variables are typically edge-based, but some models involve paths from shipment origin to shipment destination. The use of a path formulation typically necessitates column generation (Hewitt et al., 2019). However, unlike the vehicle routing problem, extended, path-based formulations of the network design problem do not provide stronger linear relaxations than compact, arc-based formulations. Depending on the context and mode the vehicle edge variables may either be binary or integer. Linking constraints are included in the formulation to ensure sufficient vehicle capacity travels on an edge to carry the commodities making that transportation move. Typically, much larger cost coefficients are associated with vehicle edge variables than commodity flow variables.

The majority of literature on network design focuses on deterministic models wherein it is presumed all parameter values (costs, capacities, demands) are known with certainty. However, given that network design models are often solved as part of a tactical planning exercise, uncertainty has been studied (Hewitt et al., 2021). Much of that work focuses on uncertainty in commodity sizes and models such problems as two stage stochastic programs wherein vehicle movements are planned in the first stage and commodities are routed in the second stage given the vehicle movements prescribed in the first. There has been limited work on robust optimisation models (Koster & Schmidt, 2021) or those that view network design in a dynamic context (Al Hajj Hassan et al., 2022).

Both exact (Crainic & Gendron, 2021) and heuristic (Crainic & Gendreau, 2021) solution methods for deterministic network design models have been proposed. One challenge associated with solving MIP formulations of network design problems is that the linking constraints often lead to fractional
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vehicle edge variables. Thus, the linear programming relaxations of network design MIPs often yield weak bounds on the objective function value of an optimal solution to the MIP. As a result, much of the literature that focuses on speeding up the solution of MIP formulations of the network design problem focuses on strengthening formulations with valid inequalities (Nemhauser & Wolsey, 1988). Such inequalities are typically either based on classical ideas such as flow covers from integer programming (Gu et al., 1999) or leveraging the network structure of the problem (Raack et al., 2011). Another approach taken to solve network design problems is Benders decomposition (Benders, 2005; Costa, 2005), particularly when second stage variables are continuous and the optimisation problem resulting from fixing the network design is a linear program.

Another challenge associated with solving MIP formulations of the network design problem is due to the size of the network on which the MIP is formulated when that network encodes time. The classical approach to representing time in network design is to formulate a MIP on a network wherein multiple nodes represent the same physical location, albeit at different points in time (Crainic et al., 2016). Similarly, multiple edges represent the same physical transportation move, albeit at different departure and arrival times. Such networks are typically referred to as time-expanded networks and the overall solution procedure in contexts that require the modelling of time is to construct such a network, formulate a MIP on that network, and then solve that MIP. Boland et al. (2019) study the impact on solution quality of modelling time at different granularities and observe that the finer the representation the higher the quality of the resulting solution. However, such an approach can be computationally challenging when long planning horizons must be modelled or fine representations of time are used, as both cases lead to networks and resulting MIPs that are very large. An alternate approach, called *Dynamic Discretisation Discovery* (Boland et al., 2017; Hewitt, 2019) proposed to instead generate time-expanded networks in a dynamic and iterative manner.

Heuristic methods for deterministic network design models can be classified into one of two categories. The first category focuses on metaheuristics (Hussain et al., 2019) and neighbourhood structures. Early heuristics (Powell & Sheffi, 1983) proposed for network design models searched neighbouring solutions by reducing the capacity on one edge in the network and, if necessary, increasing the capacity on another. However, more recent and effective methods have proposed more complex neighbourhood structures such as cycles or paths (Ghamlouche

et al., 2003). The second category focuses on what is generally called matheuristics (Maniezzo et al., 2021). In these heuristics, a neighbourhood of a solution is searched by formulating and solving the MIP of the network design problem, albeit with the values of subsets of variables fixed to their values in the solution at hand (Hewitt et al., 2010). This is repeatedly done and with different mechanisms used for selecting subsets of variables to fix.

Similarly, both exact and heuristic solution methods have been proposed for stochastic network design models that take the form of scenario-based two stage stochastic programs. The vast majority of such stochastic programs studied to date involve continuous commodity flow variables in the second stage. As a result, the second stage subproblems are linear programs and the overall stochastic program is amenable to Benders decomposition (Birge & Louveaux, 2011). Thus, much of the methodological work on solving such stochastic programs has focused on techniques for speeding up or rendering more impactful different steps in the Benders scheme (Magnanti & Wong, 1981; Crainic et al., 2021c). While Progressive Hedging (Rockafellar & Wets, 1991) is an exact method for two stage stochastic programs with continuous variables in both stages, it has been used as the basis of heuristic methods for stochastic network design (Crainic et al., 2011, 2014).

Crainic et al. (2021b) contains deeper dives into the subjects touched on here as well as discussions of those not discussed.

### 3.32. Transportation: Vehicle routing⁸⁴

The Capacitated Vehicle Routing Problem (CVRP) was first proposed by Dantzig and Ramser (1959), and named the Truck Dispatching Problem. The goal was that of routing a fleet of identical gasoline delivery trucks from a central depot to service stations (often referred as 'customers'). Each truck had to return to the central depot, after visiting an ordered subset of the customers. All customers had to be visited once by a vehicle delivering all their gasoline requirements in the one delivery. The objective was the minimisation of the routing costs, as the sum of the travelling distances of every truck. The CVRP classical definition is the same as that proposed by Dantzig and Ramser (1959) more than 60 years ago. Introducing a capacitated fleet of vehicles makes the CVRP for a much harder generalisation of the Travelling Salesman Problem (Flood, 1956).

The CVRP definition has been enriched over the decades to take into account all the delivery requirements of the customers and of the transportation providers, as well as the characteristics of the available fleet of vehicles, and the increasing availability
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of technology (i.e., GIS and real time mapping, autonomous vehicles, shared mobility systems and so on). The research literature has flourished with new variants, as well as more sophisticated and flexible solution approaches. This chapter aims at providing pointers to key milestones achieved in the last 60 years of the CVRP literature, identifying the latest and most successful exact and metaheuristic algorithms, as well as referencing the most famous online challenges and standard techniques for benchmarking CVRP solution algorithms.

The CVRP 'classical' variants and solution approaches are well summarised in Toth and Vigo (2002). This book provides key references and definitions for critical application features, as for the CVRP with Time Windows, the CVRP with Backhauls and the CVRP with Pickup and Delivery, the CVRP with vehicle/site dependencies, the CVRP with inventory and the stochastic CVRP. Golden et al. (2008) extends the definition of the classical variants to routing problems with heterogenous fleets, periodic routing problems, split routing problems, dynamic and online routing problems. Toth and Vigo (2014) further widen the remit of application of routing algorithms to maritime applications, disaster relief distribution problems, and considers up-to-date objective functions different than minimising the distance travelled. More recently, fleets of electric vehicles (Pelletier et al., 2016), problems over time (Mor & Speranza, 2020), drones (Otto et al., 2018), cargo boats (Christiansen et al., 2013) and warehouse pickers (Schiffer et al., 2022) have been embedded in routing settings. The new dynamic environment inspired research on stochastic (Gendreau et al., 2016), dynamic (Soeffker et al., 2022) and time-dependent (Gendreau et al., 2015) routing problems.

An up-to-date survey on recent trends can be found in Vidal et al. (2020), in which the CVRP extensions due to richer objective functions, the integration with other optimisation problems, and application-oriented transportation requirements are surveyed. Partyka and Hall (2014) discuss routing algorithms from the practitioners' perspective, and surveys which are the requirements of a logistics company when they acquire a routing software.

Next the most successful CVRP solution algorithms are summarised, first discussing exact methods. Formulations with a polynomial number of variables and constraints were the first proposed mathematical models, as for the two-commodity formulation by Laporte (1992) and Baldacci et al. (2004). They have the advantage of being easy to use (as they just require encoding in the syntax of the solver). The disadvantage of them however is their poor performance due to high dimension of the formulations, and the weakness of the continuous

relaxation. Better results were obtained from formulations with an exponential number of constraints, such as those in which subtour elimination constraints are added dynamically to the formulations in a branch&cut fashion (Padberg & Rinaldi, 1991). The CVRPSEP library by Lysgaard et al. (2004) provides separation procedures for subtour elimination constraints, as well as other strengthening additional inequalities. The most successful exact solution framework is up-to-date the branch&cut&price (Desaulniers et al., 2006; Laporte, 2009). This method is based on the Dantzig-Wolfe decomposition (Desrosiers & Lübbecke, 2005). Binary variables model if a route is used or not in the solution, thus their corresponding set is exponential in size. As a consequence, a restricted set of variables is used to initiate the formulation and only profitable routes are iteratively generated solving a subproblem, called the pricing problem. The CVRP pricing problem is a shortest path with resource constrains, and it is typically solved through dynamic programming (Irnich & Desaulniers, 2005). Some of the most relevant milestones in developing branch&cut&price algorithms for the CVRP are combining branch&cut and column generation into the first branch&cut&price (Fukasawa et al., 2006), applying bi-directional search in the subproblem (Righini & Salani, 2008), introducing subset row cuts (Jepsen et al., 2008), using ng-routes to speed up the subproblem solution (Baldacci et al., 2011), using stabilisation techniques for dual values (Gschwind & Irnich, 2016; Pessoa et al., 2018), and proposing primal heuristics based on the restricted master problem (Sadykov et al., 2019). The reader might refer to Desaulniers et al. (2002) for the most widely used acceleration techniques for the solution of the pricing problem.

Lately, the work of Pessoa et al. (2020) provides an impressive open-source branch&cut&price algorithm, based on Pecin et al. (2017). This algorithm provides state-of-the-art exact solutions for the CVRP and, using a flexible solution representation, for most of the well known routing variants and other sequencing problems. The tool incorporates the algorithmic components previously mentioned, as well as other recent developments (see for example, Sadykov et al., 2021), and compares favourably to other branch&cut&price implementations. Some of the most powerful exact algorithms for the CVRP, available in different programming languages, are publicly available at Sadykov (2022).

Metaheuristics are capable of solving very large CVRP instances in limited computing time, however there is no proof of optimality for the solutions found. They are typically initialised with solutions generated by constructive heuristics (the Clarke and Wright is a famous example, Clarke & Wright, 1964).
<figure><img src="image_134.png" /></figure>

Metaheuristics rely heavily on local search procedures to improve the solution quality and intensify the search, and on a metaheuristic framework to obtain a good balance of diversification and intensification (Gendreau & Potvin, 2010). In chronological order, popular CVRP frameworks have been the Tabu Search (Cordeau & Laporte, 2005), the Adaptive Large Neighbourhood Search (Pisinger & Ropke, 2007), the Iterated Local Search (Subramanian et al., 2013), and the Hybrid Genetic algorithm (Vidal, 2022b). The latter two examples of metaheuristic frameworks are particularly relevant to the CVRP literature due to their high performance, their flexibility in solving effectively many VRP variants, and because their code had been made publicly available to the research community (the code presented in Vidal, 2022b is, for example, available at Vidal, 2022a). Vidal et al. (2013) provide a very good summary of the features that make a CVRP metaheuristic successful.

More recently, examples of algorithms producing very high quality solutions for the CVRP have been:

* Arnold and Sörensen (2019): data mining is used to identify solution features, and these features are used to effectively guide the search algorithms;

* Christiaens and Vanden Berghe (2020): SISRs is a ruin and recreate algorithm based on an innovative string removal operator;

* Queiroga et al. (2021): POPMUSIC is a metaheuristic that iteratively solves smaller subproblems by means of the branch&cut&price by Pessoa et al. (2020);

* Accorsi and Vigo (2021): FILO is an Iterated Local Search with acceleration techniques and annealing-based neighbour acceptance criteria;

* Máximo and Nascimento (2021): AILS-PR is an Iterated Local Search metaheuristic hybridised with Path Relinking; and,

* Cavaliere et al. (2022): a refinement heuristic using a penalty-based extension of the Lin and Kerninghan heuristic is combined with a restricted column generation to iteratively select meaningful routes.

Clear standards have been set by the CVRP community around which benchmark instances should be used for testing the performance of an algorithm, and which are ways of testing a computer code for a fair comparison with other previously proposed algorithms. Uchoa et al. (2017) discuss the most widely used instances and provides a link to the repository, in which the input data, as well as the best known solutions, are provided and kept up-to-date by the authors. A more recent set of instances and best known solutions is available in Queiroga et al. (2022), where the authors provide data enabling the use of machine

learning approaches to solve the CVRP. Accorsi et al. (2022) present the standard practices to test CVRP algorithms: how to determine computing time (typically on a single thread), common ways of tuning parameters, and providing best and average solutions on a specified number of executions, among others.

Finally, another popular and flourishing avenue for boosting research on the development of effective solutions approaches for the CVRP and variants is represented by competitions. Some of the most famous CVRP and routing challenges are:

* the DIMACS challenge (DIMACS 2021), where the goal was to promote research on challenging routing problem variants;

* the Amazon Last Mile Routing Research Challenge (Amazon last mile routing, 2021), where a specific problem was tackled, namely, the challenge of embedding driver knowledge into route optimisation;

* the recently launched EURO Meets NeurIPS 2022 Vehicle Routing Competition (EURO Meets NeurIPS 2022 2022), with the goal of developing and comparing machine learning techniques for the CVRP.

The Vehicle Routing problem has inspired an incredible amount of research. This is due to the challenges it poses when it comes to solving it, to the many variants related to it and to the relevant practical applications. Despite the decades of research efforts and achievements, interest continues to grow mainly thanks to the emerging topics raised by the ever changing application environment. This chapter provides a brief, but hopefully sufficiently comprehensive overview of the techniques, problem variants and emerging trends which will inspire further research.

## 4. Conclusions<sup>85</sup>

This encyclopedic article, dedicated to the 75th anniversary of the *Journal of the Operational Research Society*, is made up of an *Introduction* and two distinct though related sections: *Methods* and *Applications*. The introduction section gives an interesting overview of OR with an emphasis on its origin in the UK and highlights the methods and applications that are covered in this paper. A brief summary of the two sections is given below.

In the first main section (§2), 24 OR-based methods are presented by experts in their respective areas. These methods, which are given in alphabetical order, are concisely described, each starting with the basics, then moving to advanced and contemporary aspects. The authors also pinpoint
<figure><img src="image_135.png" /></figure>

challenging limitations while highlighting promising research directions.

As OR is rooted in the need to solve decision problems either through optimisation, statistics, visualisation and information technology tools, or through soft system methodologies, we aim to retain this historical flavor in summarising these methods by adopting a simple three-group categorisation.

The first category covers optimisation-related topics and includes 10 out of the 24 subsections. It ranges from the original optimisation model of linear programming (LP; §2.14) in the late 1940s to its various extensions. One is obtained by restricting the decision variables to discrete elements including binary ones (§2.15), allowing uncertainty in the input (§2.21), or relaxing the objective function or the constraints not to be necessarily linear (§2.16). An interesting area that had been dormant for more than 30 years was revived in the late 1970s and early 1980s by studying a special case of fractional LP which defines relative efficiency and is known as data envelopment analysis (§2.7). Combinatorial optimisation (§2.4), a topic that has fascinated and intrigued many mathematicians of the 18th century, seeks an optimal subset or values from a large finite set of elements. These problems can be defined and solved through graphs and networks (§2.12), some of which are relatively more difficult than others. To measure the performance of algorithms in terms of time and space complexity, computational complexity (§2.5) emerged as a solid foundation for distinguishing between classes defined as *P* and *NP* and studying α-approximation algorithms. One methodology can be traced back to the Ancient Greek times, and is based on the 'find and discover principle', now known as 'heuristic search' (§2.13), which has experienced a phenomenal growth in the late 1980s and early 1990s. This is a major development since these methodologies provide the best way to reduce not only the risk of getting stuck at poor local optima, but also have the power to yield practical solutions for complex discrete and global optimisation problems that could not have been solved otherwise. A methodology that is free from restrictions of linearity and convexity is the study of multi-stage process, as given in §2.9.

The next category includes statistics and decision-based tools and also covers 10 of the 24 subsections. For example, business analytics (§2.3), decision analysis (§2.8) and visualisation (§2.24), though they previously existed under different names, have grown significantly while retaining their simplicity. Machine learning, including artificial intelligence (§2.1), which borrowed its principles from heuristic search and statistics, has taken off very rapidly in teaching, research and applications.

This is mainly due to computer power, sophisticated algorithms, freely available computer languages such as R and Python, and their ability to handle massive amount of data that are now easily available to the public. Other older topics, though still relevant and widely applicable, have also seen a surge in new developments. These include queueing (§2.17), forecasting (§2.10), control theory (§2.6), and game theory (§2.11). Given the uncertainty and risk involved in many decisions, risk analysis (§2.18) is evolving fast so as to handle such environments alongside computer simulation (§2.19), especially discrete event simulation. The latter, which has a wide spectrum of applications in both the private and public sectors, has recently been enriched by incorporating multi-objective optimisation within its evaluation component.

The last category covers the remaining four subsections. Although some of these research areas existed in other fields such as system engineering in the 1950s, they have become contemporary OR topics especially in the UK in the late 1970s. Soft OR and problem structuring methods (§2.20) question the problem definition and aim to involve stakeholders for a better understanding, with system thinking (§2.23) analysing the interactions between people, machines and systems while also questioning the system boundaries. A related area is system dynamics (§2.22) where the dynamism is incorporated throughout and found to suit better applications with limited but plausible scaling. An interesting, though relatively recent OR area, but with a long history rooted in social psychology, is behavioural OR (§2.2), where people's behaviour and culture are incorporated into the decision making process. Although the methodologies included in this category usually do not directly aim to solve problems, they can be complementary to the harder OR techniques.

The second section covers applications that have been, since the very beginning, strongly interconnected with the development of OR methodologies. This section is very rich in examples coming from many fields. For the sake of brevity, we will not refer to each subsection individually but mention just a few. By reading the section it is evident that, on the one hand OR provides appropriate modelling and solution tools to practical problems that arise in the real world and are nowadays crucial in the design and management of most systems, from healthcare and other public services, to transportation and manufacturing. On the other hand, the complexity and size of practical problems has always stimulated the progress of OR towards more efficient and flexible techniques which are capable to cope with the challenges posed by the applications.
<figure><img src="image_136.png" /></figure>

This mutual and virtuous connection is well reflected by the richness of the Applications section of this work. It highlights not only the traditional areas which saw tremendous research efforts and successful implementations, as the traditional fields of transportation, manufacturing, cutting and packing, and inventory management, but also relatively new and interesting sectors such as sports and education.

It is worth noting that in the *Applications* section (§3), several dimensions of OR impact in the real world clearly emerge. The first one is the broad range of fields to which OR techniques have already successfully been applied and offer an even larger potential yet to be exploited. These range from *vertical* sectors, such as supply chain management, disaster relief and recovery, or military applications, where a wide array of problems are defined and solved through appropriate and varied methodologies, to more *horizontal* domains which may impact several vertical sectors, like vehicle routing or facility location, for which highly specialised methods have been developed. The second dimension is related to the great variety of methodologies applied to the different contexts. These span the whole tool set of OR, including exact and heuristic methods developed to solve specific optimisation problems, to techniques created to handle uncertainty and multi-criteria and, more recently, integrating artificial intelligence methods. Indeed, the great improvements achieved in the last decades in integer and nonlinear programming now allow to effectively model and solve many problems arising at the operational and tactical levels, where data are more available and reliable. The uncertainty in the data and the modelling typical of strategic decisions are successfully handled by a variety of methodologies that have proven to be effective in the solution of real applications which are well reviewed in this work. A third very interesting dimension is represented by the development of new broad research perspectives which may have a strong impact in all fields of OR and are deeply motivated by applications. An excellent example is the inclusion of fairness and ethics in optimisation which, on the one hand allow for considering important issues favouring the acceptability and usability of the results, and on the other hand pose new methodological challenges.

As a general conclusion, thanks to the advances in computer technology, the availability of massive amount of live data, and novel developments, in both optimisation and statistics, effective optimisation software, powerful machine learning techniques and visualisation tools now exist to solve problems that were considered practically unsolvable just a

decade ago. Applications have always been a main driver for OR development, and the successes achieved increase the appetite for further improvements.

In the more classical area of exact and heuristic techniques, there is clearly a need to improve the capability of handling efficiently large and very large-scale instances to cope with more complex and demanding scenarios. This increase in scale is not only generated by the need to solve larger problems, but also to incorporate various steps of the planning processes into integrated and more comprehensive methods. A field that still deserves further research efforts is the consideration of uncertainty in OR methods. Important methodological obstacles have yet to be surmounted and there is clearly a need for the development of simple and pragmatic methods, possibly resulting from the integration of artificial intelligence techniques, which can be applied to the solution of large-scale problems arising in several important application domains. However, it is also worth stressing that these advances, though they are welcome, may suffer from shortcomings, such as the local optimality trap, biased data, and impractical assumptions. These hidden aspects could yield poor outcomes on which academics and practitioners ought to keep an open eye.

## Notes

1. This section was written by Gilbert Laporte.

2. This subsection was written by Laurent Charlin and Andrea Lodi.

3. This subsection was written by L. Alberto Franco and Raimo P. Hämäläinen.

4. We use the term ‘intervention’ to describe a structured process comprised of designed OR-related activities such as, for example, modelling, model use, data collection, interviews, meetings, workshops, and presentations.

5. To date, the practice of studying OR-supported intervention as a system of interest has been somewhat overlooked by behavioural modellers, with the notable exception of behavioural forecasting researchers; see §2.10, and also Arvan et al. (2019) and Petropoulos et al. (2016).

6. It should be noted that a variance approach could also be implemented through field research designs where pre and post intervention measures of key variables are used to assess changes in behaviour or surrogates of behaviour. Studies adopting this approach are common in the System Dynamics field (see, for example, Scott et al., 2013).

7. https://www.euro-online.org/websites/bor/

8. https://www.theorsociety.com/get-involved/society-groups/special-interest-groups-and-networks/behavioural-or/

9. This subsection was written by John E. Boylan.

10. This subsection was written by Silvano Martello and Paolo Toth.
<figure><img src="image_137.png" /></figure>

11. This subsection was written by Ulrich Pferschy and Clemens Thielen.

12. This subsection was written by Xun Wang.

13. This subsection was written by Sebastián Lozano.

14. This subsection was written by Matthias Ehrgott and Salvatore Greco.

15. This subsection was written by Dong Li and Li Ding.

16. This subsection was written by Fotios Petropoulos.

17. This subsection was written by Georges Zaccour.

18. This subsection was written by Ivana Ljubić.

19. www.math.uwaterloo.ca/tsp/history/milestone.html

20. akira.ruc.dk/~keld/research/LKH/

21. www.math.uwaterloo.ca/tsp/concorde

22. www.math.uwaterloo.ca/tsp/star

23. This subsection was written by Ceyda Oğuz and İstenç Tarhan.

24. This subsection was written by Jean-Marie Bourjolly.

25. This subsection was written by Adam N. Letchford and Andrea Lodi.

26. This subsection was written by E. Alper Yıldırım.

27. This subsection was written by Hayriye Ayhan and Tuğçe Işık.

28. This subsection was written by Louis Anthony Cox, Jr.

29. For example, https://www.sra.org/risk-analysis-introduction/

30. This subsection was written by Christine S.M. Currie.

31. https://msu.euramet.org/current_calls/documents/EC_Industry5.0.pdf

32. www.wintersim.org

33. Henceforth in this subsection we just refer to PSMs to avoid the obvious dual of ‘Hard OR’ and therefore to manufacture, or at least to continue to propagate, an unhelpful distinction. We retain its use here for signposting.

34. This subsection was written by Mike Yearworth and Leroy White.

35. This subsection was written by Haitao Li.

36. Decision problems with *incomplete information* about random parameter's probability distribution go beyond the scope of this review.

37. This subsection was written by Martin Kunc and John D.W. Morecroft.

38. This subsection was written by Gerald Midgley.

39. This subsection was written by Martin J. Eppler.

40. This subsection was written by Bo Chen.

41. This subsection was written by Amanda J. Gregory.

42. https://www.theorsociety.com/get-involved/probono-or/, accessed 2023-01-19

43. https://www.theorsociety.com/get-involved/society-groups/special-interest-groups-and-networks/or-in-the-third-sector/, accessed 2023-01-19

44. https://connect.informs.org/probonoanalytics/home, accessed 2023-02-08

45. This subsection was written by Julia A. Bennell.

46. https://www.euro-online.org/websites/esicup/data-sets/

47. This subsection was written by Bahar Y. Kara and Özlem Karsu.

48. This subsection was written by Charlotte Köhler and Tom Van Woensel.

49. https://bit.ly/3dfiwDW, 2022-08-08

50. https://bit.ly/3SzAWzf, 2022-08-08

51. https://bit.ly/3B1H1ww, 2022-09-09

52. This subsection was written by Jill Johnes.

53. This subsection fn-type="endnote" tion was written by Judit Lienert.

54. This subsection was written by John N. Hooker.

55. This subsection was written by Michèle Breton.

56. VaR is a quantile of the distribution of investment losses over a specified period, while CVaR is the amount of the expected losses, conditional to being above the VaR threshold

57. This subsection was written by Katherine Kent and Sam Rose.

58. This subsection was written by Christos Vasilakis.

59. This subsection was written by Jing-Sheng Song.

60. This subsection was written by Sibel A. Alumur.

61. This subsection was written by Janny Leung and Yong-Hong Kuo.

62. This subsection was written by Kathryn E. Stecke and Xuying Zhao.

63. This subsection was written by Kai Virtanen and Raimo P. Hämäläinen.

64. This subsection was written by Emel Aktas.

65. This subsection was written by Cihan Tugrul Cicek and Güneş Erdoğan.

66. This subsection was written by Dimitrios Sotiros and Rafał Weron.

67. This subsection was written by Willy Herroelen and Erik Demeulemeester.

68. This subsection was written by Arne K. Strauss and Jens Frische.

69. This subsection was written by Jan Holmström and Lauri Saarinen.

70. This subsection was written by Ian G. McHale.

71. This subsection was written by Stephen M. Disney.

72. This subsection was written by Akshay Mutha.

73. This subsection was written by Bernard Fortz.

74. https://www-public.imtbs-tsp.eu/~maigron/RIR_Stats/RIR_Delegations/World/ASN-ByNb.html

75. https://www.statista.com/statistics/617136/digital-population-worldwide/

76. This subsection was written by Greet Vanden Berghe and Sanja Petrovic.

77. The 25th anniversary of PATAT conferences had to be postponed till 2022, due to the COVID-19 pandemic.

78. https://OPTHUB.uniud.it

79. https://patatconference.org/

80. This subsection was written by David Canca.

81. This subsection was written by Harilaos N. Psaraftis.

82. This subsection was written by Virginie Lurkin and Vikrant Vaze.

83. This subsection was written by Mike Hewitt.

84. This subsection was written by Claudia Archetti and Maria Battarra.

85. This section was written by Daniele Vigo and Said Salhi.

Disclaimer

The views expressed in this paper are those of the authors and do not necessarily reflect the views of their affiliated institutions and organisations.

Acknowledgements

Fotios Petropoulos would like to thank all the co-authors of this article for their very enthusiastic response and participation in this initiative. He is also indebted to his lead advisor for this project, Gilbert Laporte, as well as
<figure><img src="image_138.png" /></figure>

Christos Vasilakis, Güneş Erdoğan, Stephen Disney and Maria Battarra for their help and suggestions. Finally, Fotios is grateful to John Boylan and the other Editors-in-Chief of the *Journal of the Operational Research Society* for inviting this paper to be part of the 75th issue of the journal. Fotios dedicates this article to Professor John Boylan: John, your kindness will always be remembered.

Maria Battarra's work reported in this paper was undertaken as part of the Made Smarter Innovation: Centre for People-Led Digitalisation, at the University of Bath, University of Nottingham, and Loughborough University.

David Canca's work was supported by the University of Sevilla, the Regional Government of Andalucia (Spain) and the European Regional Development Fund (ERDF) under grant US-1381656.

Laurent Charlin and Andrea Lodi would like to thank Didier Chételat and Mizu Nishikawa-Toomey for reading and commenting on drafts of their subsection (§2.1) and the CIFAR AI Chair and the CERC programs for funding.

Salvatore Greco wishes to acknowledge the support of the Ministero dell'Istruzione, dell'Università e della Ricerca (MIUR) - PRIN 2017, project "Multiple Criteria Decision Analysis and Multiple Criteria Decision Theory", grant 2017CY2NCA.

Katherine Kent and Sam Rose thank Mithu Norris for the help and coordination with $\$3.10$ but also Emma Hickman and Ffion Lelii for their contribution.

Silvano Martello, Paolo Toth and Daniele Vigo were supported by Air Force Office of Scientific Research under Grants no. FA8655-20-1-7012, FA8655-20-1-7019, FA9550-17-1-0234 and FA8655-21-1-7046.

Dimitrios Sotiros's work was partially supported by the National Science Center (NCN, Poland) grant no. 2020/37/B/HS4/03125.

Greet Vanden Berghe and Sanja Petrovic acknowledge the advice provided by Andrea Schaerf (University of Udine).

Rafał Weron's work was partially supported by the National Science Center (NCN, Poland) grant no. 2018/30/A/HS4/00444.

## Disclosure statement

No potential conflict of interest was reported by the authors.

## ORCID

Emel Aktas http://orcid.org/0000-0003-3509-6703

John E. Boylan http://orcid.org/0000-0001-5036-5137

David Canca http://orcid.org/0000-0002-8769-234X

Bo Chen http://orcid.org/0000-0001-7605-9453

Özlem Karsu http://orcid.org/0000-0002-9926-2021

Adam N. Letchford http://orcid.org/0000-0002-3191-5006

Haitao Li http://orcid.org/0000-0001-7609-2819

Sebastián Lozano http://orcid.org/0000-0002-0401-3584

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