<figure><img src="image_83.png" /></figure>

Thompson (1994) as well as the “contractarian” ethical philosophy of Gauthier (1983) and has been recommended for wage negotiations and similar applications (Alexander, 1992).

Threshold criteria are of two types. One, based on an efficiency threshold, imposes a maximin objective until the efficiency cost becomes unacceptably great, at which point some stakeholders are switched to a utilitarian criterion. The other, based on an equity threshold, imposes a utilitarian criterion until inequity becomes unacceptably great, at which point a maximin criterion is introduced. Originally proposed for two stakeholders (Williams & Cookson, 2000), the threshold criteria have been extended to n persons, using an MILP formulation for the former (Hooker & Williams, 2012) and a linear programming model for the latter (Elçi et al., 2022). A parameter $\Delta$ regulates the equity/efficiency trade-off in both models, in that stakeholders with utility within $\Delta$ of the worst-off are given special priority. Thus, the parameter $\Delta$ may be interpretable in a practical situation in a way that $\alpha$ in the alpha fairness criterion is not. Both threshold criteria inherit a weakness of the maximin criterion, namely that they may be insensitive to the equity position of disadvantaged stakeholders other than the very worst-off. This has been addressed for the efficiency threshold by combining a utilitarian criterion with a leximax rather than a maximin criterion. McElfresh and Dickerson (2018) accomplish this by assuming there is a pre-existing priority ordering of stakeholders. Chen and Hooker (2022a) avoid this assumption by giving greater priority to stakeholders with utilities closer to the lowest, and by solving a sequence of MILP models to balance the leximax element with total utility.

Fairness modelling is a relatively recent research program in operational research that may forge new connections with other fields. Much as interactions between OR and economics, management, and engineering have been mutually beneficial on both a theoretical and practical level, collaboration with ethicists on the precise formulation of fairness concepts may bring similar benefits to both ethical philosophy and operational research.

### 3.9. Finance<sup>55</sup>

The use of mathematical models and numerical algorithms to solve an extensive range of problems in finance is widespread, by both researchers and practitioners. In this subsection, we offer an overview of some established models and discuss a selection of the corresponding OR approaches and techniques.

#### 3.9.1. Resource allocation models

As in any other industry, the optimal allocation of resources to activities is a central problem in finance. Prototype models include short-term cash flow management (a linear program), portfolio dedication and immunisation (linear programs), capital budgeting (knapsack problem), asset/liability management (stochastic program with recourse), and portfolio selection (quadratic program).

The portfolio selection model introduced in Markowitz (1952) and discussed in Markowitz and Todd (2000) is one of the best known optimisation models in finance. This mean-variance model consists of determining the composition of a portfolio of risky assets – a vector of weights – where the performance (to be maximised) is measured by the expected portfolio return, a linear function of the assets’ weights, while the risk (to be minimised) is measured by the variance of the portfolio return, a quadratic function of the weight vector. The resulting optimisation problem gives rise to a convex quadratic program. This model and its analytical properties led to a formalisation of diversification as a strategy to mitigate risk and to important developments in financial theory.

While the Markowitz model represents a considerable simplification of the portfolio management problem, mean-variance optimisation models are still very much applied in practice. Straightforward variations of the Markowitz model can account for various constraints on the asset weights (e.g., bounds, minimum participation, regulatory or operational restrictions, logical constraints, etc.), yielding mixed integer quadratic programs.

Mean-variance models rely on sets of parameters describing the expected returns and their correlation matrix in the universe of the set of considered assets. Various forecasting approaches (§2.10) have been proposed to obtain estimates of these parameters, often relying on some assumptions about the correlation structure. One important issue related to the use of mean-variance optimisation models is the sensitivity of their solutions to the estimated parameter values (Michaud, 1989), specifically when the feasible region is relatively unconstrained. Robust optimisation (see also §2.21) is increasingly used to limit the estimation risk of mean-variance portfolio solutions (Ismail & Pham, 2019; Yin et al., 2021; Blanchet et al., 2022).

Another related limitation of mean-variance models is the fact that they are static models, that is, expectations and correlations of asset returns are assumed to be known and constant over the planning horizon. In practice, estimations are updated periodically to reflect changes in data, and portfolios are rebalanced to the optimal composition
<figure><img src="image_84.png" /></figure>

corresponding to the new set of estimates. Small perturbations in the values of the input parameters may lead to significant changes in the composition of the portfolio from one period to the next (for instance, when groups of assets have similar characteristics). When the costs associated with changing the composition of the portfolio are significant, a static model may be far from optimal. The portfolio selection problem can be readily extended to a multi-period context, allowing to account for transaction costs and/or to use a dynamic model of the evolution of asset prices over time (Li & Ng, 2000). Dynamic models can also account for additional frictions, such as taxes on capital gains or losses (Dammon et al., 2001). The resulting dynamic portfolio selection problem may be a large-scale stochastic dynamic program (§2.9; §2.21). Moreover, risk measures based on portfolio variance are not additively separable, precluding the efficient use of dynamic programming. Steinbach (2001) proposes a solution approach based on scenario decomposition.

### 3.9.2. Risk management

While, in OR, the classical way to deal with decisions under risk is utility theory, finance models usually take a different approach by directly measuring and/or pricing risk. Various measures, such as variance, semi-variance, Value at Risk (VaR) or Conditional value at risk (CVaR) have been proposed to characterise risk⁵⁶. VaR is effectively concerned with computing quantiles of the predictive distribution (see also §2.10 and §3.19). In the following paragraphs, we present two contrasting families of approaches to financial risk management.

Diversification and hedging approaches are closely related to the resource allocation models presented above. They consist in setting up and managing portfolios of securities with desirable properties. Diversification is effective in reducing risk that is uncorrelated across securities, while hedging is used to reduce systematic risk, for instance by holding securities exposed to the same risk factors to eliminate uncertainty, or by buying insurance in the form of derivative contracts. In general, hedging positions must be continuously adjusted to account for the time evolution of risk factors and security prices. In addition, investment portfolios are often required to satisfy institutional or regulatory constraints. Risk mitigation portfolio planning problems give rise to dynamic stochastic mathematical programs. In recent years, CVaR has become prominent for measuring portfolio risk; CVaR is well-suited to measure down-side risk in skewed distribution and, as shown in Artzner et al. (1999), it has the desirable properties of a coherent risk measure.

Moreover, the use of CVaR in optimisation models gives rise to convex or linear programs, allowing to efficiently solve the large-scale problems encountered in practice (Rockafellar et al., 2000; Andersson et al., 2001; Rockafellar & Uryasev, 2002).

Risk pricing approaches rather seek to evaluate the consequence of unpredictable events and are notably used for the management of credit and counterparty risk, that is, the risk that the issuer of a security (for instance, a corporate bond) will not be able to meet its future obligations. A variety of models have been proposed to evaluate the VaR of debt instruments, mainly for the purpose of assessing regulatory requirements ensuring that financial institutions put aside sufficient capital to sustain eventual losses. Crouhy et al. (2000) presents a review of methodologies currently proposed by the industry to evaluate the probability and consequences of default events. Most approaches used in the industry to price credit and counterparty risk are based on probabilistic models or Monte-Carlo simulation (§2.19) and, as such, cannot account for strategic behaviour by the debtor or the lender (Breton & Marzouk, 2018).

### 3.9.3. Asset pricing

Most asset pricing models are founded on an absence of arbitrage assumption, which is usually motivated by the efficiency of markets. Under this assumption, the value of a financial asset is equal to the expected value of its future payoffs, under a suitable probability measure. One specific application is the valuation and replication of contingent claims, such as financial options. The contribution of OR to this area lies in the development and implementation of efficient numerical pricing methods for complex financial securities.

Starting from the binomial tree model of Cox et al. (1979), numerical methods for option pricing include Monte-Carlo (§2.19) and quasi-Monte-Carlo approaches (Acworth et al., 1998; L'Ecuyer, 2009); dynamic programming (§2.9) and approximate dynamic programming models accounting for optimal exercise strategies (Ben-Ameur et al., 2002; Longstaff & Schwartz, 2001); and robust control models (§2.21) accounting for transaction costs and model uncertainty (Davis et al., 1993; Bernhard, 2005; Bandi & Bertsimas, 2014). Numerical algorithms developed for option pricing have also been applied to the valuation of numerous instruments, including corporate bonds, credit derivatives, contracts, and, under the designation of real options, managerial flexibility (Trigeorgis, 1996; Schwartz & Trigeorgis, 2004; Dixit & Pindyck, 2009).

In the context of algorithmic trading, asset pricing algorithms have been revisited using artificial
<figure><img src="image_85.png" /></figure>

intelligence approaches, for instance by using
machine learning to identify factor models or
reinforcement learning to compute optimal exercise
strategies (Dixon et al., 2020; Gu et al., 2020), or by
augmenting the set of covariates with textual data
(Algaba et al., 2020).

**3.9.4. Strategic interactions**

Decisions made by investors, firms, financial institu-
tions, and regulators have a direct impact on asset
values, returns and risk. Players in the financial sec-
tor have competing interests and interact strategic-
ally over time, and these interactions are recognised
in many game-theoretic models of investment and
corporate finance (§2.11). Important issues include
market impact and market manipulation, option
games, strategic exercise of real options, agency con-
flicts, corporate investment, dividend and capital
structure policies, financial distress, and mergers
and acquisitions.

Optimal execution refers to the determination of a trading strategy minimising the expected cost of trading a given volume over a fixed period, accounting for the impact of the trades on the price of the security. This problem is addressed in Bertsimas and Lo (1998), using a stochastic dynamic program minimising the execution costs, and in Almgren and Chriss (2001), where a combination of volatility risk and transaction costs is minimised. Optimal execution and market impact are particularly significant issues in the context of algorithmic trading and have been addressed by the recently developed mean-field game theory, acknowledging the fact that price is impacted by the trades of many atomic players (Firoozi & Caines, 2017; Cardaliaguet & Lehalle, 2018; Huang et al., 2019).

*Option games* appear in asset pricing models when a security gives interacting optional rights to more than one holder, that is, when the exercise of an optional right by one holder modifies those of the others. Examples include callable, putable and convertible bonds, warrants, and, especially, instruments subject to credit or counterparty risk. In general, the pricing of such financial instruments corresponds to the solution of a non-zero-sum stochastic game where players use feedback strategies (Ben-Ameur et al., 2007).

*Financial distress* models are used to price cor-
porate debt, according to various assumptions about
strategic default, debt service and bankruptcy proce-
dures (Fan & Sundaresan, 2000; Broadie et al., 2007;
Annabi et al., 2012).

Finally, a large literature in corporate finance
uses game-theoretic models to deal with financial
decisions made by firms, such as the choice between
debt and equity when financing operations, the

amount of dividends paid out to shareholders,
and decisions about whether to invest in risky
projects.

**3.9.5. Further readings**

The recognition of finance as an thriving applica-
tion area for OR methods developed about thirty
years ago (see, for instance, Dahl et al., 1993a,
1993b, for an introduction to optimisation prob-
lems underlying risk management strategies and
instruments). A review of practical applications of
OR methods in finance appeared in Board et al.
(2003). For a comprehensive textbook covering
optimisation models in finance, we refer the
reader to Cornuejols and Tütüncü (2006). A uni-
fied framework for asset pricing can be found in
Cochrane (2009) and a review of applications of
dynamic games in finance in Breton (2018).
Recent discussions about the interface of opera-
tions, risk management and finance as a promis-
ing research area are presented in Wang et al.
(2021) and Babich et al. (2021).

**3.10. Government and public sector<sup>57</sup>**

This subsection will present some OR applications
within the UK's government operational research
service (GORS). GORS represents over 26 depart-
ments and agencies across Great Britain and
Northern Ireland with analysts working in multi-
disciplinary teams to find workable solutions to real
life problems. The outbreak of the Coronavirus pan-
demic in 2020 introduced a new global backdrop
and we were faced with the challenge of producing
appropriate analysis to answers questions during an
ever-changing landscape where time was of the
essence. This led to collaborations across a wide
range of departments across the nations.

A few examples of where this collaborative
approach was adopted successfully are highlighted
by the work carried out by the Department for
Transport (DfT) and the Office for National
Statistics (ONS). The ONS worked with other gov-
ernment departments such as Department of Health
and Social Care (DHSC) and schools across the UK
to monitor infection rates. They also applied their
expertise in artificial intelligence (AI) in the form of
semantic maps to gather insight into the pandemic.
Additionally, the DfT along with other government
departments used agent based modelling and dis-
crete event simulation to unpick the issues around
border disruptions and international travel.
<figure><img src="image_86.png" /></figure>

### 3.10.1. Coronavirus (COVID-19) infection survey and schools infection survey

The Office for National Statistics (ONS) played a vital role during the pandemic in monitoring infection rates. The Coronavirus (COVID-19) infection survey estimates how many people across England, Wales, Northern Ireland, and Scotland would have tested positive for a COVID-19 infection, regardless of whether they report experiencing symptoms. This study was a collaboration with academic partners and funded by Department of Health and Social Care. This major study involved asking people up and down the country to provide nose and throat swabs on a regular basis. These are analysed to see if they have contracted COVID-19. In addition, some adults are also asked to provide blood samples to determine what proportion of the population has antibodies to COVID-19. Further details of the methodology can be found in Office for National Statistics (2022a).

Estimates of the total national proportion of the population testing positive for COVID-19 are weighted to be representative of the population that live in private-residential households in terms of age (grouped), sex and region. The analysis for the infection study is complex, the model generates estimated daily rates of people testing positive for COVID-19 controlling for age, sex, and region. This technique is known as dynamic Bayesian multi-level regression post-stratification (MRP). Details about the methodology are also provided by Pouwels et al. (2021).

Estimates from the ONS survey are published weekly, a critical element was how best to communicate the uncertainty, for dissemination estimates were translated into, for example, 1 in 50 people, with appropriate visuals including the ONS insights tool (Office for National Statistics, 2022b). A complementary piece of work was monitoring transmission and antibody levels within schools, enabling the government to accurately assess the risk of different policy options.

The Schools Infection Survey (SIS) was a longitudinal study which collected data through polymerase chain reaction (PCR) tests, antibody tests and questionnaires. As well as monitoring transmission within a school environment, this data was used to assess the wider impacts of the pandemic and repeat lockdowns on our children and young people, including long covid, mental health and physical activity levels. Further detail can be seen in Office for National Statistics (2022c) and Hargreaves et al. (2022).

The Daily Contact Testing (DCT) trial was a blind medical trial which compared infection rates across two groups subject to different policies: the

control group where children were in “bubbles”, and after one child testing positive the entire bubble would be sent home from school, and the intervention group where after a child tested positive, close contacts would then test daily and were allowed to remain in school as long as their results were negative. The study was a success and led to a policy change that resulted in schools being kept open for longer. Further detail can be seen in Young et al. (2021).

### 3.10.2. Semantic maps and their use for understanding regional disparities

Semantic Maps are a type of knowledge graph, championed in the world of robotics and Artificial Intelligence as a way to provide infrastructure to exploit all kinds of potentially even crowdsourced data and information in such a way as to provide dynamic, online, interactive visualisations that support the controlled and secure use of live data. The can be geospatial in nature, but they can also reflect connections through semantic relationships. These maps populated with data would provide users with many different ways to consume the underlying data and help inspire citizens about the potential power of data to drive understanding and generate insights.

During the development of the Levelling Up White Paper the evidence base needed to be developed across government in order to define the key metrics and measures to focus policies in areas that would drive change. The white paper itself was delivered by the Levelling Up Taskforce in the Cabinet Office, however, ONS worked with the geospatial commission in convening a group of chief analysts from all departments on a regular basis. ONS and the Levelling Up Taskforce used the group to commission and collate existing evidence and then worked with officials in His Majesty's Treasury (HMT) to develop a systems thinking model from that evidence. This systems mapping was the basis of the theory of change that underpins the white paper. Subsequently, the metrics and missions were developed and refined with this group in recognition of the fact that there needs to be a focus across the system to reduce disparities that are often larger within areas such as local authorities or regions than they are between them.

ONS took this and developed a semantic map that identifies potential data sources for various aspects of the knowledge graph, using this to both prioritise filling evidence gaps where data and evidence do not currently exist and developing an integrated data asset for Levelling Up. This data is in the process of being acquired and engineered to be able to be easily linked through a set of linking
<figure><img src="image_87.png" /></figure>

‘spines’ referred to as the reference management database. This engineering and architecture is key for supporting the sharing of information in a way that ensures privacy. It is intended that in future this asset will be made available across government and in a secure integrated data service.

### 3.10.3. Agent based modelling

The COVID-19 pandemic presented challenges for the international travel community. Government officials in transport and health needed to model the preventative effect of the various policy options including testing and isolation on importation of infections from international travel.

The approach chosen in the Department for Transport to support this fast-moving policy area was agent based modelling which built upon the more scientific epidemiological modelling undertaken by colleagues in Health and academia. This allowed for the incorporation of the various differing parameters of international travellers including, where they were coming from and their risk of being infected and infectious, the uncertainty over incubation and infectious periods, and their likely behavioural response to various isolation and testing regulations.

Whilst not designed to be a scientific forecast, the modelling allowed the cross-government community to estimate the relative effectiveness of policy options. This work supported policy making during a highly uncertain and changing environment when Government had to balance risk with the wider impacts on the aviation sector and the second order impacts on the economy.

### 3.10.4. Discrete event simulation

As part of the EU Exit preparations, it was important for both the Department for Transport, the Home office and regional resilience teams to understand the impact of the expected border disruption at UK ports for roll-on roll-off freight traffic travelling to EU Member States. As the issue was around changes to the time taken and resource available to carry out additional border processes, the natural choice was discrete event simulation. Analysts in government developed a detailed model of the Short Strait crossings (Port of Dover and the Channel Tunnel to France accounting for 84% of accompanied heavy goods vehicles travelling to continental Europe in 2019; Department for Transport, 2022). Regional models were developed to cover other ports. These allowed government officials to understand the likely queues and flow of vehicles and to understand the impact of changes to the system, which was vital to supporting contingency planning.

### 3.10.5. Statistical analysis and forecasting

The COVID-19 lockdowns of 2020 accelerated the uptake of new and novel data sources for understanding mobility. Government analysts rapidly ingested new data sources such as that provided by Google mobility as open source data as well as procuring in additional anonymised and aggregated mobile network operator data. By analysing these new data sets alongside traditional demographic and geographic data sets it was possible to generate insights into the changes in mobility being seen across the country as a result of the various national and regional restrictions. Regression analysis was undertaken to produce a predictive model. It was then possible to forecast the impact of later changes to restrictions on population mobility.

### 3.10.6. Net zero – system thinking

In 2020 the UK Prime Minister's Council for Science and Technology advised on the following: 'a whole systems approach can provide the framework that government requires to lead change across public and private sectors and ... enables decision makers to understand the complex challenges posed by the net-zero target and devise solutions and innovations that are more likely to succeed' (Council for Science & Technology, 2020). The Prime Minister agreed.

As transport represents a huge portion of the challenge, ORs have run participatory systems mapping workshops in the Department for Transport with modal subject matter experts to identify the key causes and effects in the Transport Net Zero system. This aims to enable those working on Transport policy to explore the evidence, gain new insights and visibility of interdependencies within the system, and to understand the likely wider impact of their policy choices.

### 3.10.7. Conclusions

All these examples help to illustrate the breadth of analysis undertaken across central government during the global pandemic to tackle real life issues; and the ranges of techniques we have as OR analysts to find workable solutions in an ever changing world.

## 3.11. Healthcare⁵⁸

Why is the organisation and delivery of health and care services so difficult to manage, plan for and improve? Difficulties and delays in accessing care services, cancellations and increasing costs have a negative impact on all of us: patients, carers, and care professionals. Despite the attention and resources invested in addressing these problems, many
<figure><img src="image_88.png" /></figure>

health systems face increasing pressure to improve
the effectiveness and efficiency of their operations.
Part of the problem is the complexity inherent in
the organisation of care services and our limited
understanding of how changes will affect their deliv-
ery. Another problem is the intrinsic uncertainty
and variability in many aspects of care service deliv-
ery. Add the multifaceted dynamics arising in this
very complex socio-technical system involving pro-
fessionals, patients and existing and new technolo-
gies, against a background of increased demand and
budgetary constraints, and it is no surprise the effort
to improve healthcare has been termed ‘rocket sci-
ence’ (Berwick, 2005).

Operational research has a long established his-
tory in this area with the first application (schedul-
ing outpatient hospital appointments) reported in
the early 1950s (Bailey, 1952). Since then, there has
been a proliferation of OR applications reported in
the literature (Katsaliaki et al., 2010), and evidence
of use to support policy making and care delivery
(Royston, 2009). This is not a surprise given the
importance of healthcare in our lives and that many
of the problems faced by those managing and deliv-
ering care services are amenable to the methods and
ethos of OR (Utley et al., 2022). In the short review
that follows, which is by no means exhaustive and
draws primarily (but not exclusively) from the UK
academic community and the National Health
Service (NHS), I have attempted to give examples of
review and individual studies grouped in a few
broad areas of healthcare.

**3.11.1. Applications to hospital settings**

Hospital care has been the setting of a large number
of OR studies (Jun et al., 1999). Hospitals typically
survive reorganisations and funding cuts (unlike
management, policy and other statutory bodies),
and are large enough to be able to engage meaning-
fully in research projects (unlike, for example, many
primary care practices). Some (mainly teaching)
hospitals host large biomedical research centres,
with many of the professionals working in them
active researchers.

A specific area that has attracted the attention of
operational researchers is the Emergency
Department (ED). A recent review article identified
21 studies that used a computer simulation method
to capture patient progression through the ED of an
established UK NHS hospital, mainly focusing on
service redesign (Mohiuddin et al., 2017). Individual
studies have addressed the micro (single hospital)
level (Baboolal et al., 2012), as well as the meso-level
of emergency and on-demand healthcare within a
region (Brailsford et al., 2004). The study by Lane
et al. (2000) used System Dynamics to model the

interaction of demand patterns and resources
deployed in ED and other parts of the hospital to
examine the link between emergency and elective
operations in hospitals.

Another hospital area that has been the focal
point of OR is peri-operative care. Sobolev et al.
(2011) in their systematic review identified 34 stud-
ies modelling the flow of surgical patients. Various
forms of optimisation have also been applied to sur-
gical scheduling problems including operating room
(Fairley et al., 2019), staffing (Bandi & Gupta, 2020)
and nurse rostering (Xiang et al., 2015) among
others. Cardoen et al. (2010) identified almost 250
papers, with the rate of published studies accelerat-
ing at around the start of the new millennium (simi-
lar trends have been observed across many
disciplines). The review revealed that most of the
research was directed towards the planning and
scheduling of elective patients in highly stylised
scenarios – although many operational challenges
are triggered by factors such as the arrival of non-
elective (emergency) patients. More recently, the
problems tackled have become more realistic to
include considerations of downstream resource
availability such as critical care and general ward
beds (Fügener et al., 2014), and scheduling elective
operations in such a way that randomly arriving
emergency patients can be accommodated without
excessive delays (Jung et al., 2019).

An area OR has demonstrably made a beneficial
impact is the organisation of acute stroke services.
Several studies have attempted to address the rate
and speed with which patients with suspected acute
ischemic stroke go through the initial diagnostic
steps and receive treatment (Meretoja et al., 2014).
Monks et al. (2012) made a number of recommen-
dations for improving treatment rates in a rural hos-
pital. In a follow-up study that evaluated the results
of their recommendations, mean door-to-needle
times (a key performance metric with direct impact
on patient survival and recovery) fell from 100 min
to 55 min while thrombolysis rates increased to
14.5% (Monks et al., 2015). More recently, the focus
has shifted to supporting decision around the cen-
tralisation of regional acute stroke services (Wood &
Murch, 2020; Wood et al., 2022), as well as the sup-
porting the introduction of endovascular thrombec-
tomy, a new and very effective treatment for
ischemic stroke (Maas et al., 2022).

**3.11.2. Applications to non-acute hospital care
settings**

Much of healthcare is delivered outside of large hos-
pital facilities. Primary care, home and community
care, social care are significant components of the
healthcare ecosystem. Primary care, whether provided
<figure><img src="image_89.png" /></figure>

by physicians, nurse practitioners or pharmacists, is
typically concerned with providing a first contact and
principal point of continuing care and/or coordinat-
ing other specialist care. There are early examples of
theoretical work to assist primary care planning by
estimating the coverage achieved by staff and facili-
ties, using antenatal care as an example (Kemball-
Cook & Vaughan, 1983). More recently in the area
of maternity care provided in community as well as
hospital facilities, Erdoğan et al. (2019) developed
and empirically tested an open source facility location
solver to assist with a decision on the number and
location of regional maternity facilities.

Home-based care has attracted considerable attention from operational researchers (Grieco et al., 2021). This review identified studies proposing models and solution methods for operational decisions on staff rostering, the allocation of staff to patient visits, the scheduling of visits and the routing of staff. An example of impactful OR project is the Swedish study by Eveborn et al. (2009), where a set of algorithms and accompanying software tool were developed to provide solutions to staff-to-patient allocations, staff scheduling and staff routing problems. Having deployed the tool to more than 200 units/organisations, operational efficiency was increased by up to 15%, resulting in annual savings of 20-30 million euros. More recently, modelling work has supported the effort to address the timely discharging of hospital patients by using a combination of home-based and bedded ‘step-down’ community care (Harper et al., 2021).

Mental health, one of the leading causes of dis-
ease burden internationally, has also received the
attention of operational researchers (Long &
Meadows, 2018). Specific areas of application varied
from psychiatric ICUs (Moss et al., 2022) to system
design (Smits, 2010) and planning (Vasilakis et al.,
2013), medical decision making (Afzali et al., 2012),
and epidemiology (Ciampi et al., 2011).

3.11.3. Public health, health system preparedness and resilience, and pandemic response

Public health, the science and practice of helping people stay healthy and protecting them from threats to their health, is another area of OR applications. The review article by Fone et al. (2003), identified OR studies of infection and communicable disease, screening, and several epidemiological and health policy studies. Microsimulation, a type of simulation which models individual life trajectories through a number of healthy and disease states, has found wide applicability in the area of public health (Krijkamp et al., 2018), such as forecasting the long-term care needs of the older population in England (Kingston et al., 2018a).

Multicriteria decision analysis (MCDA) methods have also been used extensively to address questions of health policy or health technology assessment (Glaize et al., 2019).

An area that has seen increased attention over the last two decades is that of emergency preparedness and health system resilience (Tippong et al., 2022). Emergency preparedness studies include a study of red blood cell provision following mass casualty events (Glasgow et al., 2018). Examples of health system resilience studies include the paper by Crowe et al. (2014), which examined the feasibility of using modelling to assess the capacity of a care system to continue operating in the face of major disruption.

The COVID-19 pandemic not only gave rise to a large number of modelling studies, it also raised the profile of mathematical modelling with the general public and policy makers. Pagel and Yates (2022), in their excellent article on the role of modelling in the pandemic response, discussed the early lessons learnt from this experience including the poor understanding of policy makers and the public of key concepts such as exponential growth. They argue that infection disease modelling, which generated much of the evidence used to support decisions of pandemic response (Brooks-Pollock et al., 2021), is intrinsically difficult given the complex relationships between the model parameters, and the difficulties associated with quantifying these parameters.

The possible benefits of modelling in addressing the challenges presented by the pandemic were outlined by Currie et al. (2020). Indeed, several studies emerged early in the pandemic including, for example, an attempt to forecast the number of infected and recovered cases used univariate time series models (Petropoulos & Makridakis, 2020). Wood et al. (2020) published one of the first OR studies that examined the likely impact of increases in critical care capacity as a means to reduce the COVID-19 death toll. In a follow-up study, the sophistication of the model was increased to capture notions of triaging access of patients to critical care beds during periods of intense demand (Wood et al., 2021b). The operation of large vaccination centres was also the topic of several modelling studies, both theoretical (Franco et al., 2022) and empirical (Wood et al., 2021a; Valladares et al., 2022).

3.11.4. Concluding remarks

Despite the large body of literature, the role and
impact of OR on improving care systems is less
clear. Hospitals have "largely failed to use one of the
most potent methods currently available for
<figure><img src="image_90.png" /></figure>

improving the performance of complex organ-
isations" (Buhaug, 2002) and "staff may be largely
unaware of the potential applications and benefits of
OR" (Utley et al., 2022). A systematic review found
that only half of the included studies reported mod-
els that were constructed to address the needs of
policy-makers, and only a quarter reported some
involvement of stakeholders (Sobolev et al., 2011).
Recent positive developments include the introduc-
tion of guidelines to improve the reporting of OR
studies (e.g., Monks et al., 2019), studies that recog-
nise the importance of behavioural factors in
attempts to influence practice and decision making
with OR (Crowe & Utley, 2022) and attempts to
systematically generate evidence on the value and
impact of OR on patient and system outcomes
(Monks et al., 2015; Soorapanth et al., 2022). The
research agenda should continue to evolve with the
aim of addressing the challenges around engage-
ment, implementation and evidencing the impact of
OR applied to healthcare problems.

**3.12. Inventory<sup>59</sup>**

Inventories are the materials, parts, and finished
goods held by an organisation for future use or sale.
Not having enough inventory is costly. Shortages of
materials and parts cause interruptions in produc-
tion processes, delays in product delivery, and stock-
outs of finished goods. On the other hand, carrying
inventory is costly, too, involving the cost of capital
due to tied-up capital, storage cost, insurance, taxes,
and spoilage and obsolescence costs.

Inventory theory studies analytical models and
solution techniques to help organisations meet the
service requirement most cost-effectively or minim-
ise the total expected costs of ordering, inventory-
holding, and shortage. It does so by quantifying the
tradeoffs driven by economies of scale, lead time
(the time it takes to receive the ordered quantity
after placing an order), and supply and demand
uncertainties. It prescribes effective inventory-con-
trol policies that govern when to order an item
(called reorder point) and how much to order (called
order quantity).

Inventory models distinguish from each other
along several features: single or multiple planning
periods, discrete- or continuous-time inventory
monitoring, single- or multi-product, single- or
multi-stage (or location), demand nature (determin-
istic or stochastic, stationary or nonstationary, dis-
tribution known or unknown), product perishability,
lost sales or backlogging when shortages occur,
deterministic or stochastic lead time, supply system
(single- or dual-source, exogenous or endogenous, a
finite or infinite capacity), and cost structure (with

or without a fixed ordering cost, etc.). The following
research-based textbooks and handbooks offer more
detailed coverage and references: Arrow et al.
(1958), Axsäter (2006), de Kok and Graves (2003),
Graves et al. (1993b), Hadley and Within (1963),
Nahmias (2011), Porteus (2002), Silver et al. (1988),
Simchi-Levi et al. (2014), Snyder and Shen (2019),
Song (2023), and Zipkin (2000).

One class of models focuses on characterising the
optimal inventory-control policy under a given sup-
ply and demand environment and cost structure. A
common approach is formulating a multi-period
inventory decision problem as a dynamic program
and transforming the original formulation into a
simpler one through state reduction. Next, identify
the structural properties of the single-period cost
function to determine the optimal policy form for a
single-period problem. Then, show that these prop-
erties are preserved by the (Bellman) optimality
equation, so the policy form is optimal for each
period. The optimal policy parameters may not be
easy to compute; hence some works develop effi-
cient algorithms to calculate the optimal policy
parameters.

Another class of models focuses on developing
efficient performance evaluation tools for a given
type of inventory policy that is either commonly
used in practice or of simple structure and easy to
implement. This is particularly important for systems
where state reduction is not viable and the dimension
of the system state grows exponentially in the num-
ber of periods (the so-called curse of dimensionality),
so the optimal policy has no simple form. Typically,
this type of work analyses a continuous-review sys-
tem in which demand follows a stochastic process
and derives steady-state performance measures of any
given policy, such as average inventory, average back-
orders, and stockout rate, as well as the long-run
average cost. Then, optimisation tools can be devel-
oped to find the optimal policy parameters that min-
imise the long-run average cost.

The third class of models conducts asymptotic
analysis to establish asymptotic optimality of some
simple-structured policies for less tractable inventory
systems with unknown and complex optimal
policies.

The following are several classic models where
the optimal policies are shown to have simple
forms. Unless otherwise stated, the models assume a
single stage, a single source, and a single nonperish-
able product.

*The EOQ (Economic Order Quantity) Model* was first developed by Harris (1913) (see the reprint Harris, 1990) and popularised by Wilson (1934). It concerns the balancing of holding and ordering costs due to economies of scale in procurement or
<figure><img src="image_91.png" /></figure>

production. It is a continuous-review model over an
infinite planning horizon, assuming the annual
demand for the stocked item is a constant $\lambda$. There
is a fixed procurement cost $k$ independent of the
order size, accounting for administrative, material
handling, and transportation-related costs. The
annual per-unit inventory-holding cost is $h$. The
optimal order quantity (EOQ) that minimises the
annual order and holding costs equals $\sqrt{2k\lambda}/h$,
which is insensitive to small perturbations of the
model parameters. Variations of this model can
accommodate finite production rate, planned back-
logs, random yield, a quantity discount, and time-
varying demand (also known as the dynamic lot
sizing problem; see Wagner & Whitin, 1958; Silver
& Meal, 1973). It also forms the basis for the devel-
opment of efficient multi-item joint replenishment
policies and multiechelon coordinated replenish-
ment policies such as the power-of-two policies; see
Roundy (1985), Roundy (1986). For major develop-
ments and references, see Axsäter (2006), Muckstadt
and Roundy (1993), Silver et al. (1988), Simchi-Levi
et al. (2014), and Zipkin (2000).

The Newsvendor Model, which originated from Edgeworth (1888) in a banking application, was formalised by Arrow et al. (1951) in the general inventory context. It optimises the tradeoff between too much and too little inventory caused by demand uncertainty for a seasonal product. It is a single-period model with only one ordering opportunity before the selling season, assuming an estimated demand distribution. The fixed order cost is negligible. After the ordered quantity arrives, the selling season begins, and demand realises. At the end of the season, there will be either unsold units (overage) or unmet demand (underage). The unit overage cost ($o$) = purchasing cost less - salvage value, while the unit underage cost ($u$) is the lost profit. The optimal newsvendor order quantity equals to the fractile of the demand distribution at the critical ratio $u/(u+o)$. The model can be generalised in many ways, including random yield, different cost structures, pricing, and distribution-free bounds (Gallego & Moon, 1993; Petruzzi & Dada, 1999; Porteus, 1990; Qin et al., 2011) and multi-location with risk-pooling effect (Bimpikis & Markakis, 2016; Eppen, 1979).

*Dynamic Backlogging Models.* The most tractable and developed setting for multi-period models with stochastic demand and a constant lead time is full backlogging. When stockouts are rare, this model is a reasonable approximation for the lost-sales system. An important concept (due to state reduction) is *inventory position*, which is the sum of the on-hand inventory plus total pipeline inventory minus

backorders. This is the total system inventory avail-
able to satisfy future demand if we do not order
again.

Assume demand is independent over time. A *base-stock policy* is optimal if the order cost is linear (no fixed order cost). Each period has a target inventory position called the *base-stock level*. If the inventory position before ordering is below this level, order up to this level; otherwise, do not order. If the demand is stationary, the myopic base-stock level that minimises a single-period expected cost is optimal. The base-stock level has the same form as the newsvendor quantity, with the holding cost as the overage cost, the backorder cost as the underage cost, and the demand during a lead time replacing the single-period demand. For nonstationary demand, as long as the myopic base-stock levels are nondecreasing in time, the myopic base-stock level is still optimal. See Veinott Jr (1965) and Porteus (1990).

When the order cost is linear plus a fixed cost $k$, the optimal policy is an ($s, S$) *policy*. In each period, if the inventory position before ordering is below a threshold $s$, order up to $S$; otherwise, do not order. The key enabler of this result is that the single-period cost is $k$-convex, a property discovered by Scarf (1960a). When the demand is stationary, the policy is also stationary. In a continuous-review system with Poisson demand, the optimal policy is an (*r*, *q* ) *policy*: When the inventory position reaches *r*, order *q* units. It is equivalent to the (s, S) policy with *r* = *s* and *q* = *S* − *s*. A simple yet effective heuristic policy is to use the optimal base-stock level to approximate *r*, and use the EOQ formula to approximate *q*; see Zheng (1992) and Axsäter (1996).

These policy structures have been extended to
more complex models, such as Markov modulated
demand (Iglehart & Karlin, 1960; Song & Zipkin,
1993; Sethi & Cheng, 1997), exogenous and sequen-
tial stochastic lead times (Kaplan, 1970; Nahmias,
1979; Ehrhardt, 1984; Song, 1994; Song & Zipkin,
1996), capacity constraints (Federgruen & Zipkin,
1986a, 1986b), unknown demand distribution (Scarf,
1959, 1960b; Azoury, 1985), and a dual-source prob-
lem where the lead times of the two sources differ
by one period (Fukuda, 1964) or the lead times are
stochastic and endogenous (Song et al., 2017). See
Veinott Jr (1966), Perera and Sethi (2022b), Perera
and Sethi (2022a), Porteus (1990), and Zipkin
(2000) for more detail.

Multiechelon (or multi-stage) inventory systems
are common in supply chains where the stages are
interrelated, such as production facilities, ware-
houses, and retail locations. The literature focuses
<figure><img src="image_92.png" /></figure>

on understanding three basic system structures: series, assembly, and distribution systems.

In a series system with *N* stages and backlogging, random customer demand arises at stage 1, stage 1 orders from stage 2, and so on, and stage *N* orders from an outside supplier with ample supply. There is a constant transportation time between two consecutive stages. Define the echelon inventory of each stage to be the inventory at the stage plus all downstream inventories (including those in transit). Assuming no fixed order costs, Clark and Scarf (1960) establish that an echelon base-stock policy is optimal for all stages. That is, we can treat each echelon as a single location and order the echelon inventory position up to a target base-stock level. Axsäter and Rosling (1993) show that for any echelon base-stock policy, there is an equivalent local base-stock policy; therefore, the implementation of the optimal policy is simple. Federgruen and Zipkin (1984b) find that the optimal echelon base-stock policy for the infinite horizon problem can be efficiently obtained. Rosling (1989) proves that under certain mild conditions, an assembly system can be transformed into an equivalent series system, so the Clark-Scarf result applies. Chen and Zheng (1994) further streamline the proofs of these results. Shang and Song (2003) construct effective single-stage newsvendor solutions to approximate the optimal echelon base-stock levels. Chen and Song (2001) show that a state-dependent echelon base-stock policy is optimal for Markov-modulated demand. See Axsäter (1993), Axsäter (2003), Axsäter (2006), Angelus (2023), Federgruen (1993), Kapuściński and Parker (2023), and Shang et al. (2023) for more developments, including batch ordering, capacity limits, distribution systems, transshipment, and expediting.

Many other features are much less tractable, such as lost-sales systems (Bijvank et al., 2023), censored demand data, perishable products (Li & Yu, 2023), general dual-sourcing systems (Xin & Van Mieghem, 2023), distribution systems, and assemble-to-order systems (Atan et al., 2017; Song & Zipkin, 2003; DeValve et al., 2023). Nonetheless, significant progress has been made in recent years on structural properties of the optimal policy, asymptotic optimal policies, and effective heuristics, thanks to more analytical tools such as discrete convexity, asymptotic analysis, and machine learning algorithms. See Chao et al. (2023), Cheung and Simchi-Levi (2023), Shi (2023), and other chapters in Song (2023).

### 3.13. Location⁶⁰

In the domain of operations research, location problems are concerned with determining the location of a facility or multiple facilities to optimise one or

more objective functions under constraints. Location problems seek answers to questions such as how many facilities should be located, where should each facility location be, how large should each facility be, and how should the demand for the facilities' services be allocated to these facilities (Daskin, 1995). An example of a facility to be located is a factory, distribution centre, warehouse, cross-dock, or hub, where demand can be for raw materials, components, products, passengers, data, etc.

Location decisions arise in a variety of public and private sector decision-making problems. Some examples from different sectors include locating landfills where demand is for disposal of household waste (Erkut & Neuman, 1989), ambulances where demand is for transporting emergency patients to hospitals (Brotcorne et al., 2003), warehouses where demand is for storing products arriving from factories (Aghezzaf, 2005), schools where demand is for students (Haase & Müller, 2013), regenerators in optical networks where demand is for data (Yıldız & Karaşan, 2017), shelter sites where demand is for refugees (Bayram & Yaman, 2018), and charging stations where demand is for electric vehicles that need to charge (Kınay et al., 2021). More applications of location problems from practice can be found in Eiselt and Marianov (2015).

Location decisions refer to the placement of a facility considering its interactions with demand points (e.g., customers, suppliers, retailers, households) and possibly with other facilities to be located. It includes selecting the location and determining how this location supports meeting a decision-maker or organisation's objective. It is important to note that facility location decisions are different from facility design decisions. Facility design decisions usually consist of facility layout and material handling systems design. The layout entails all equipment, machinery, and furnishing within the building, whereas material handling systems comprise the mechanism needed to satisfy the required facility interactions. Facilities planning and design are extensively discussed in Tompkins et al. (2010).

Several factors influence facility location decisions, the most prominent ones being transportation costs and the availability of the transportation infrastructure. Among other important factors are the availabilities and costs of land, market, labour, materials, equipment, energy, government incentives, and competitors as well as geographical factors and weather conditions. Distance is usually considered to be one of the most important criteria in facility location models. Several distance metrics can be used in location models such as Euclidean (straight-line), rectilinear (Manhattan), Cheybyshev (Tchebychev), and network distance. Network distance is the
<figure><img src="image_93.png" /></figure>

distance that is calculated on an existing transportation network, for example, through using Google or Bing maps.

An important criterion to be considered in location problems is how demands are to be satisfied by the facilities to be located. In some applications, the whole demand of each customer must be satisfied from a single facility ("non-divisible" demand) which is referred to as a *single source* or *single allocation*. Single allocation location problems are also referred to as *location-allocation problems* as each demand point is allocated to a single facility. On the other hand, in *multiple source/allocation* problems, the demand of a single customer can be served from several facilities.

Location decisions are usually classified according to their decision space. In *continuous* or *planar* location problems, the facilities can be located anywhere in the decision space. The search is for the optimal coordinates; i.e, latitude and longitude. In *discrete* location problems, a finite set of potential facility locations is provided, possibly determined through a pre-selection process. In *network* location problems, on the other hand, there is a given network and the facilities are to be located on this network. In network location problems, facilities can further be restricted to be placed only on the vertices or nodes of this network and not on the edges or arcs, referred to as *vertex-* or *node-restricted* location problems.

Continuous location problems focus on minimising some function related to the distance between the facilities to be located and the existing facilities or demand points, such as suppliers and customers, where minisum (minimising the total weighted distance) and minimax (minimising the maximum or worst weighted distance) are among the most commonly employed objectives. Special cases of continuous single-facility location problems with commonly used distance metrics (e.g., rectilinear and Euclidean) are well-studied and polynomial time solution algorithms exist (Francis et al., 2004). In the case of multi-facility continuous location problems, the facilities to be located can be homogeneous or non-homogeneous; in the latter, there are different types (e.g., a factory and a warehouse) or sizes of facilities to locate.

One of the most studied discrete location problems is the *p-median* problem. The goal is to pick a subset *p* of (homogeneous) facilities to open from among a given set of potential locations that minimise the total transportation cost of satisfying each demand point from the (nearest) facility it takes service from. There is a well-known node optimality theorem by Hakimi (1965) for the *p*-median problem on networks that proves that at least one

optimal solution to the *p*-median problem consists of locating the facilities only on the nodes of the network (even though a facility is allowed to be located anywhere on the network including any point on an edge between the nodes). Possible applications of the *p*-median problem are clustering, transit network timetabling and scheduling, placement of cache proxies in a computer network, diversity management, cell formation and much more (Marín & Pelegrín, 2019).

An important related problem is the *uncapacitated facility location problem* (UFLP) which is also referred to as the *simple plant location problem*. Unlike the *p*-median problem, in UFLP, the number of facilities to be located is no longer known and determined by optimising an objective function that considers the trade-off between the fixed costs of locating facilities and the transportation costs. Numerous extensions of UFLP with uncertainty, multiple commodities (e.g., products or services), multi-period planning horizon, multiple objectives, and network design decisions have been studied with applications in several domains such as supply chain and distribution systems design.

A nice structure of *p*-median and UFLP is that since the facilities to be located are assumed to have enough capacity (e.g., space or labour hour), all demands of each customer can be served from a single facility with minimum allocation costs. This is no longer the case for *capacitated* versions of the facility location problems, where single- and multiple-allocation versions are both extensively studied. For multiple allocation capacitated (fixed-charge) facility location problems, when the set of open facilities is given, the resulting subproblem of finding the best allocations is a transportation problem. In the single allocation case, on the other hand, when the set of open facilities is pre-determined the resulting allocation subproblem is a generalised assignment problem (Fernández & Landete, 2019).

When the worst-case is more important than the average, it might be better to consider the furthest or most disadvantaged demand point to ensure equity in servicing the demand. Accordingly, the *p*-centre problem aims to locate *p* facilities such that the maximum distance (or travel time/cost) from a demand point to its nearest facility is minimised (minmax). The *p*-centre problem can be used to locate public schools and various emergency service facilities such as police stations, hospitals, and fire stations. Different variations of this problem have been studied such as the capacitated, conditional, continuous, fault-tolerant, and probabilistic *p*-centre problems (Çalık et al., 2019).

In covering location problems, the aim is to locate facilities so as to cover demand. Typically, a
<figure><img src="image_94.png" /></figure>

demand point is considered to be covered if it is
within a certain distance or travel time of a facility.
Unlike in the previous models, the demand points
are not assigned to facilities in covering location
problems. The two most common covering location
problems are set covering and maximal covering
location problems. In the set covering location prob-
lem, the aim is to minimise the total cost of locating
facilities to cover all demand points, whereas, in the
maximal covering location problem, the aim is to
maximise the total demand covered subject to a
budget constraint or a constraint on the total num-
ber of facilities to locate. Continuous variants of
these covering location problems are also studied
(Plastria, 2002). Several different versions of cover-
ing location problems have been studied in the lit-
erature, including, but not limited to weighted,
redundant, hierarchical, backup covering problems
with applications in emergency services, crew sched-
uling, mail advertising, archaeology, metallurgy, and
nature reserve selections (García & Marín, 2019).

In general, facility location problems consider
and model only a single echelon; i.e., either the
flows of commodities (e.g., products, customers)
coming into or out from the facilities to be located
are negligible, for instance, when one of those trans-
portation costs is borne by another decision-maker
and somehow not related to the current decision-
making problem. An example would be a manufac-
turing company determining the location of its new
factory for delivering products to its customers with
minimum total cost, where the company is not dir-
ectly involved in the delivery of raw materials from
their suppliers to the factory. When the flow of
commodities coming into the facilities to be located
as well as the flow going out of those facilities are
simultaneously considered in the models, these loca-
tion problems are referred to as two-echelon location
problems. For example, while locating a distribution
centre, the transportation cost of products from the
factory to this distribution centre as well as the
transportation costs from the distribution centre to
the retailers may need to be considered in the
model. Sometimes there are facilities to be located
at several echelons where flows of commodities in
and out of all those facilities need to be considered.
These multi-echelon types of location problems are
encountered for several applications of supply chain
network design (Melo et al., 2009). Another related
category is when there is a hierarchical network
structure among the facilities to be located, referred
to as hierarchical facility location problems (Şahin &
Süral, 2007). An example of a hierarchical location
problem is designing a postal delivery network
where the locations of the sorting centres as well the

locations of the post offices that are to be allocated
to those sorting centres need to be determined.

There might also be interactions among the facili-
ties to be located. This is the case, for example, for
hub location problems where the demand is defined
between pairs of demand points (origin-destination
pairs) as opposed to having the demand of an indi-
vidual point. In that case, to satisfy the demand
from an origin to a destination point, flow can be
transported between the facilities to be located en
route to the destination, where those facilities can
act as switching, transshipment, sorting, connection,
consolidation, or break-bulk points. Hub location
models have several applications in passenger and
freight airlines, express shipment, postal delivery,
trucking, public transit, and telecommunication net-
work design (Alumur et al., 2021).

Location problems have been a testbed for many
algorithmic and methodological advances in opera-
tions research. Most discrete location problems
commonly belong to a class of NP-hard decision
problems (§2.5) and they can usually be formulated
with mixed-integer programming (MIP) models (see
§2.15). In addition to using commercial MIP solvers,
several exact and (meta)heuristic algorithms (§2.13)
have been developed and tested on benchmark
instances from the literature. Some of those bench-
mark instances can be obtained from Beasley
(1990), Posta et al. (2014), and Fischetti et al.
(2017b).

Location science is a very broad field of research
that encompasses geography, continuous and dis-
crete optimisation (§2.4), graph theory (§2.12),
logistics (§3.14), and supply chain management
(§3.24). This section only highlights the basic and
most well-known location models. For a more
detailed overview of the field of location science, we
refer the reader to several books written in this field,
such as Drezner and Hamacher (2004), Eiselt and
Marianov (2011), and Laporte et al. (2015).

**3.14. Logistics<sup>61</sup>**

Logistics refers to the organisation and implementa-
tion of the processes related to the procurement,
transport and maintenance of materials, personnel
and facilities. The application of operational
research to logistics dates back to 1930 (Schrijver,
2002), where Tolstoı (1930) solved to optimality the
problem of transporting salt, cement, and other
cargo on the railway network of the Soviet Union.
In general, the objective of logistics management
can be summed up as “getting the right thing/people
to the right place at the right time in the right
quantity at the right cost”. For materials, logistics
operations require the co-ordination of forecasting,
<figure><img src="image_95.png" /></figure>

purchasing, inventory control, warehousing, distribution, transportation, delivery and installation. Logistics management of personnel involves, in addition, skills matching, capabilities training, labour rules and worker preferences. At a strategic level, logistics involves the design of the transport network and facilities. In this subsection, we discuss several major domains of logistics applications, namely, military, inventory, time-sensitive, reverse and humanitarian logistics. We also mention some new technologies for emerging logistics applications.

### 3.14.1. Military logistics

Logistics play an important role in military operations. Indeed, the word “logistics” itself is derived from the position *Maréchal des logis* created in the French army in the 17th century, whose responsibilities of establishing camps and arranging transport/supplies were referred as “la logistique” (de Jomini, 1862). Many historians credited logistics as the success factor in wars from ancient to modern times. During World War II, the need for large-scale logistics planning accelerated the development of operational research. The ability to sustain the convoy of supply ships was a major factor in the Battle of the Atlantic (Kirby, 2003). The 1948-1949 Berlin Airlift, where over 2.3 million tons of goods were flown to besieged West Berlin, is well-known as the first use of logistics as a military and political strategy (Tine, 2005).

The North Atlantic Treaty Organisation defines logistics as the science of planning and carrying out the movement and maintenance of forces, and covers acquisition, transport, maintenance and evacuation of materiel, personnel and facilities, and provision of services and medical support. Operational research methodologies are extensively used (Scala and Howard II, 2020). Reliability and operability of the supply lines are a major concern in military logistics (McConnell et al., 2021), and simulation is much utilised. Cioppa et al. (2004) review agent-based simulation for military applications. Emerging technologies – such as additive manufacturing (den Boer et al., 2020) and unmanned transport (Jotrao & Batta, 2021) – have also sparked research in smart military logistics (Schütz & Stanley-Lockman, 2017). The reader is also referred to §3.16.

### 3.14.2. Inventory logistics

In modern logistics, most activities are related to products and goods, where their availability to customers or users is a key concern. Inventory, thus, plays an important role in this respect. A classic problem related to inventory logistics is the inventory-routing problem (IRP), introduced by Bell et al.

(1983) for the distribution of industrial gases. Since then, various IRP applications have been studied, including those related to automobile components (Blumenfeld et al., 1985), groceries (Gaur & Fisher, 2004), cement (Christiansen et al., 2011). Typically, IRP arises in vendor-managed inventory systems as the supplier monitors the inventory and makes replenishment decisions for its retailers (Archetti et al., 2007). Because inventory can be carried from one period to the next, IRP considers joint decisions of inventory and routing across multiple periods and aims to minimise the total transportation and inventory holding costs over the planning horizon, subject to all demands being satisfied. Speranza and Ukovich (1994) extended the IRP to settings with multiple products. When demands are uncertain, IRP becomes stochastic IRP (Federgruen & Zipkin, 1984a; Trudeau & Dror, 1992), where the objective function includes additional shortage cost. Coelho et al. (2014) investigated the stochastic dynamic IRP where decisions are made as customers’ demand become realised. Inventory logistics is even more timely today due to e-commerce (Archetti & Bertazzi, 2021). The main challenge of these inventory logistics problems is due to the computational complexity of solving multiple NP-hard problems simultaneously. The reader is also referred to §3.12.

### 3.14.3. Time-sensitive logistics

The quality and functionality of items, in storage or transit, deteriorate over time. For items such as fresh vegetables, the value degrades continuously. Other perishables, such as blood, have fixed lifetimes and cannot be used beyond expiry. Logistics management of time-sensitive goods must consider production, distribution and transport jointly. Federgruen et al. (1986) was one of the first papers to consider jointly inventory allocation and transportation for fixed-lifetime perishables with probabilistic demand. Since then, there has been much research exploring additional issues, such as freight consolidation (Hu et al., 2018), storage/transport capacities (Crama et al., 2022) and environmental concerns (Govindan et al., 2014). Shaabani (2022) gives a comprehensive literature review.

For continually decaying food items, delivery costs must be traded off with freshness-upon-arrival which may lead to lost sales or revenue (Mirzaei & Seifi, 2015). The overall network design – especially decisions on where along the supply chain processing occurs – is important, since deterioration rates differ for unprocessed vs. finished/packaged goods, and for items in transport vs. in storage (de Keizer et al., 2017).

An important category of perishable goods is blood. Integer-programming models were developed
<figure><img src="image_96.png" /></figure>

by Hemmelmayr et al. (2009) for collection and distribution of blood products to Austrian hospitals, and by Araújo et al. (2020) for blood delivery in south Portugal. Pirabán et al. (2019) survey research on blood supply chain management.

### 3.14.4. Reverse logistics

Due to increased sustainability awareness and legislation, reducing the environmental impact of production and distribution has become important. Twenty years ago, Beamon (1999) advocated that supply chains must be extended from one-way to a closed loop, where used products and materials are recovered for re-use, recycle or re-manufacture. Reverse logistics, thus, refer to the material flow from the point of consumption back upstream for regenerating value (Rogers & Tibben-Lembke, 2001). Compared to a forward supply chain, reverse logistics processes are more complicated. Firstly, the source, quality and quantity of recoverable used products/materials from end-users are highly unpredictable. There is an added decision-stage for inspection, evaluation and sorting of the collected materials, and streaming them into various processes (re-use, re-manufacture, disassembly, disposal, etc.). These re-purposing processes may be expensive, so trade-offs must be made between recovery cost and salvage value.

Re-manufacturing, where items are repaired to serviceable (like-new) condition, is an important aspect of reverse logistics. Simpson (1978) was the first to address a multi-period repairable inventory problem with random demand and returns supply; using dynamic programming, he found the optimal policy structure which specified the repair, purchase and scrap levels for each period. Later, the model was extended to consider side-sales (Calmon & Graves, 2017) and warranty demands (Lin et al., 2020a). Nowadays, the concept of reverse logistics is broadened holistically to closed-loop supply chains and the circular economy (Santibanez Gonzalez et al., 2019). See Van Engeland et al. (2020) for a recent review.

### 3.14.5. Humanitarian logistics

When disasters strike, speedy evacuation and prompt delivery of resources to affected areas are critical. From some sparse early studies (Sherali et al., 1991), humanitarian logistics research grew rapidly since 2000. The research stream yielded insights that have changed how humanitarian agencies plan and manage disaster relief. A key concept is inventory pre-positioning where depots are set up already stocked with supplies in anticipation of disaster occurrences, instead of scrambling for procurement in the aftermath. Duran et al. (2011)

developed a facility-location and supply pre-positioning plan for CARE. See also Rawls and Turnquist (2011). Many of the models used are large-scale mixed-integer-programming models.

Humanitarian logistics involve multiple objectives: costs, response urgency and fairness are all important. Huang et al. (2012) considered equity in last-mile distribution; Sheu (2014) incorporated perceptions of people awaiting rescue. Other researchers considered decision under uncertainty: Mete and Zabinsky (2010) developed a stochastic model for location and delivery of medical supplies. Yet other research took an interdiction approach and anticipated post-disaster deployment (O'Hanley & Church, 2011; Irohara et al., 2013). Recent technological advances have stimulated new research and practices. Maharjan et al. (2020) investigated pre-positioning of mobile logistics/ telecommunications hubs for Nepal. See Behl and Dutta (2019) for a survey. The reader is also referred to §3.4.

### 3.14.6. Emerging technologies

As technologies advance, the role of logistics has become more important in the Industry 4.0 era (Tang & Veelenturf, 2019). Tracking and locating technologies (RFID, GPS, IoT, etc.) enable organisations and companies to acquire information in real time. Powerful computing facilities can perform analytics of massive volumes of historical data to support near real-time solutions for large-scale problems – essential for city logistics involving thousands of orders to fulfil within a day, or even an hour. Exciting emerging applications include TSP/VRP for routing of drones (Masmoudi et al., 2022) and/or autonomous vehicles (Reed et al., 2022), risk analysis powered by block-chain technology (Choi et al., 2019), flow-based optimisation for crowd-sourcing logistics (Sampaio et al., 2020) and cargo hitching (Fatnassi et al., 2015), demand-driven optimisation for car/bike sharing systems (Wang et al., 2022e), and queuing and simulation for robotic warehouses (Fragapane et al., 2021).

These more complex and larger-scale problems with tighter response times require new solution methodologies. Most of these new approaches are combinations of operational research and data science techniques, for example, robust optimisation (Zhang et al., 2021), reinforcement learning (Yan et al., 2022c) and other machine learning-based optimisation approaches (Bengio et al., 2021).

### 3.15. Manufacturing⁶²

Manufacturing is the production process from materials to goods. Such goods can be finished goods sold to end consumers or components sold to other
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manufacturers for the production of other more complex products. Manufacturing has gone through several different phases (Industry 2.0 to 4.0) in the twentieth and twenty-first centuries. Here we offer an overview of important manufacturing topics in different time periods.

In Industry 2.0 (from the end of nineteenth century to the 1980s), demand was relatively stable. Important manufacturing systems include the Toyota production system (TPS) and cellular manufacturing. The aim of these systems is to increase productivity with lower production cost, which fits the needs of a stable market during this time period.

Taichi Ohno published Toyota Seisan Hoshiki describing the TPS in 1978. TPS is an integrated production system that can supply products to meet both requirements of product volumes and product varieties. Research and practical papers, reports, and books were published in various media to describe TPS. The underlying management principles and theoretical mechanisms of TPS are well-known. A TPS is an integrated production system that generates products to satisfy requirements of volumes and varieties simultaneously with minimum resource waste. A large number of TPS enablers have been reported and include just-in-time material system (JIT-MS), seven wastes, heijunka, multi-skilled workers, quick set-up and changeover, and keiretsu. Excellent analysis and review papers on the TPS are de Treville and Antonakis (2006), Hines et al. (2004), and Narasimhan et al. (2006).

Cellular manufacturing (CM) uses group technology to efficiently produce a high variety of parts. Cells are converted from job shops with functional layouts to improve efficiency (Yin & Yasuda, 2006). A cell consists of a machine group and a part family. The first step in CM system design is cell formation. Part families and machine groups are identified to form manufacturing cells such that the intercell movements of parts are minimised. Parts in the same family have similar machining sequences. Machines in a cell are arranged to follow this sequence. In this way, parts flow from machine to machine in their processing sequence, resulting in an efficient machining flow that is similar to an assembly line. For each part family, the volume of any particular part type may not be high enough to utilise a dedicated cell. The total volume of all part types in a part family should be high enough to utilise a machine cell well. CM attempts to flexibly accommodate high variety and simultaneously efficiently take advantage of flow lines (Celikbilek & Süer, 2015).

In Industry 3.0 (from the 1980s to today), demand is relatively volatile because of technological

innovations, higher product variety, and shorter product life cycles. The important manufacturing topics include flexible manufacturing systems (FMSs) and seru production system. The theme of these topics is to meet the increased demand for high variety and short delivery time. Product life cycles decreased during this time period, which drives manufacturers to focus on responsiveness and delivery time. Short changeover time between different product types is useful.

An FMS is an integrated, computer-controlled manufacturing system of automated material handling and computer numerically controlled machine tools that can simultaneously process medium-sized volumes of a variety of part types. A fully automated FMS can attain the efficiency of well-balanced, machine-paced transfer lines, while utilising the flexibility that job shops have to simultaneously machine multiple part types (Stecke & Solberg, 1981; Stecke, 1983; Browne et al., 1984).

A seru production system is an assembly system that has been adopted by many Japanese electronics companies. Yin et al. (2008) is the first English language paper on seru production. They describe and analyse the success of seru production systems in Canon and other Japanese companies. It is more flexible than TPS, which cannot achieve the required responsiveness in this innovative time period. A seru production system consists of one or more serus. Serus within a seru system are quickly reconfigurable, i.e., they can be constructed, modified, dismantled, and reconstructed frequently in a short time. There are three types of serus, called divisional serus, rotating serus, and yatais. They represent the evolutionary development of serus. A divisional seru is a short, often U-shaped, assembly line staffed with several partially cross-trained workers. Tasks within a divisional seru are divided into different sections. One worker is in charge of each section. A rotating seru is often arranged in a U-shaped short line with several workers. Each worker performs all required tasks from start to finish without interruption. Tasks are performed on fixed stations, so workers walk from station to station. A worker follows the worker ahead of her or him, and is also followed by the worker behind her or him. A seru with only one worker is called a yatai. An important performance of the seru production system is that it can quickly respond to product varieties with fluctuated volumes. By applying seru, delivery time is reduced. Variety and volume are easily handled.

The TPS-based assembly line became inefficient because of an inability to change very frequently to produce small-volume demands. The typical seru creation process in Sony and Canon can be summarised as follows (Yin et al., 2017). Assembly lines were
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dismantled and replaced with divisional seru systems through resource co-location and removal/replacement, cross training, and autonomy. The technique of karakuri (involves procedures to discover and appropriate the useful functions of expensive equipment into inexpensive self-made equipment) is applied to replace expensive dedicated equipment by inexpensive self-made and/or general-purpose equipment that can be duplicated and redeployed as needed by serus. As cross-training progresses, divisional serus evolve into rotating serus and yatais.

More details about the underlying management and control principles of seru can be found in Stecke et al. (2012), Yin et al. (2008), Yin et al. (2017), and Liu et al. (2014). Roth et al. (2016) reviews the last 25 years of OM research and provides eight promising research directions, one of which is seru production systems.

Today, manufacturing has entered a new age (Industry 4.0) because of the emergence of disruptive technological innovations. Examples of important manufacturing topics include smart manufacturing, mass-customisation, sustainable manufacturing, and additive layer manufacturing. Strozzi et al. (2017) examines the evolution, trends, and emerging topics of a smart factory and provides topics for future research. Hughes et al. (2022) provides a review for manufacturing in the Industry 4.0 era.

Smart manufacturing refers to flexible and adaptable manufacturing processes through integrated systems and using advanced technologies such as sensors, IoT, cloud computing, big data, artificial intelligence, automation, robots, cyber-physical systems, and additive layer manufacturing. Some detailed discussions can be found in Ivanov et al. (2016), Kersten et al. (2017), Liao et al. (2017), Theorin et al. (2017), Thoben et al. (2017), and Hughes et al. (2022).

One important benefit of smart manufacturing is that it aids the capability of mass customisation and short lead time to quickly meet changing demands. Zawadzki and Żywicki (2016) suggested smart product design and production control for efficient operations in a smart factory to enable mass customisation. Brettel et al. (2014) show that self-improving smart manufacturing systems can utilise data and quickly react (e.g., reconfigure) to personalised customer orders, which helps realise mass customisation. Some efficient mathematical models that use big data to manage and control manufacturing processes can be applied in smart factories (Ivanov et al., 2016, 2017).

Sustainable manufacturing aims to minimise negative environmental impacts while conserving energy and natural resources. Sustainable

manufacturing also enhances employee, community, and product safety. The emergence of blockchain technology and its potential disruption within the manufacturing and supply chain industries present opportunities for greater levels of sustainability in Industry 4.0. The immutability and smart contract capability of blockchain technology allow the provenance and integrity of products to be monitored more effectively. These factors contribute to reducing verification costs and the provision of real-time status information on the quality of materials throughout the supply chain (Ko et al., 2018). The disintermediation attributes of blockchain can directly contribute to manufacturing sustainability by effectively reducing complexity, and improving efficiency with less waste via the streamlining of the supply chain (Hughes et al., 2019).

Additive layer manufacturing may generate a disruptive and revolutionary impact on manufacturing (Garrett, 2014). It enables a manufacturer to further increase responsiveness by reducing lead time and increasing customisation levels. Long et al. (2017) provide a definition, characteristics, and mainstream technologies of 3D printing. Dong et al. (2016) compared the optimal assortment strategies under traditional flexible technology and 3D printing to find that 3D printing may allow a larger set of product assortment. Song and Zhang (2020) and Ivan and Yin (2017) examined the use of 3D printing on a logistics system for spare parts inventory design. 3D printing tends to be slower than other manufacturing methods, which currently limits its use in practice.

For a detailed encyclopedic overview of the manufacturing field, both in terms of theory and practice, see Yin et al. (2017). They discuss and compare production systems from Industry 2.0 to Industry 4.0. The demand dimensions of each industry era are analysed and provided as the driving force for the changes in the production systems over time.

### 3.16. Military and homeland security⁶³

The birth of OR is related to the use of optimisation modelling for military operations and resource planning during the Second World War. The early linear programming (§2.14) problems ranged from the efficient use of weapon systems to logistics and strategy planning. Today, the arena of defence has expanded extensively with new areas including information and cyber warfare. The need to counter terrorism has created the field of homeland security. OR has a role in all these emerging topics. One can say that all OR methods are applied in military and homeland security problems.
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Optimisation methods are used in a wide range of defence and security applications. For instance, assigning weapons to targets (Kline et al., 2019) using integer programming (§2.15; §2.4) has been addressed with a variety of optimisation algorithms. Other integer programming studies include, for example scheduling of training for military personnel (Fauske & Hoff, 2016) as well as military workforce and capital planning (Brown et al., 2004). Mixed integer linear programming is utilised in diverse applications such as path planning of unmanned ground and areal vehicles including drones, mission planning, acquisition decisions of military systems as well as load planning in transportation. Optimisation of vehicles' routes is also carried out by solving network optimisation problems (§2.12) with shortest path algorithms (Royset et al., 2009). Network optimisation is also used, e.g., in developing military countermeasures. Examples of bilevel and robust optimisation (§2.21) formulations cover positioning of defensive missile interceptors (Brown et al., 2005) and design of a supply chain for medical countermeasures against bioattacks (Simchi-Levi et al., 2019). Multiobjective optimisation has been applied, for example in optimising boat resources of coast guard (Wagner & Radovilsky, 2012) and planning of airstrikes against terrorist organisations (Dillenburger et al., 2019). Inherent structures of specific military optimisation problems have motivated the development of new solution techniques (Boginski et al., 2015) including, for example, metaheuristics (§2.13). Such techniques are used, e.g., for solving nonlinear military optimisation tasks (§2.16) such as design of projectiles.

Game theoretic modelling (§2.11) is used in many defence studies. Information related topics include misinformation in warfare (Chang et al., 2022) and public warnings against terrorist attacks (Bakshi & Pinker, 2018). Examples of game theoretic strategy design problems cover the optimal use of missiles and the validation of combat simulations (Poropudas & Virtanen, 2010). Designing security and counter strategies against enemies, terrorists and adversarial countries naturally lead to the use of game models. Interdiction network game models arise in security applications (Holzmann & Smith, 2021), and they are used, e.g., in route planning through a minefield.

Military simulation models (§2.19) are classified into constructive, virtual and live simulations (Tolk, 2012). Constructive simulations do not involve real-time human participation. They are based on well-known modelling methodologies such as Monte Carlo, discrete event and agent-based simulations. Applications of constructive models cover, e.g., the development and use of weapons, sensor and

communications systems, planning of operations and campaigns, improving maintenance processes of military systems, and evaluating effects of fire. In addition, cyber-defence analyses have been conducted (Damodaran & Wagner, 2020). Constructive simulations have also been used in simulation-optimisation studies such as scheduling maintenance activities of aircraft, military workforce planning, and aircraft fleet management (Mattila & Virtanen, 2014; Jnitova et al., 2017).

The complexity of modelling human behaviour generates a major challenge for constructive simulation. This issue is avoided in virtual simulations, i.e., simulators in which real people operate simulated systems and in live simulations where real people operate real systems with simulated weapon effects. These practices are typically used, e.g., in military exercises and training of personnel. An emerging trend is to combine live, virtual and constructive simulations into a single simulation activity (Mansikka et al., 2021b). Applications of this simulation type vary from training to testing large-scale systems and mission rehearsals (Hodson & Hill, 2014). In a combined simulation, new ways to measure performance are introduced (Mansikka et al., 2021a) by complementing traditional measures such as loss exchange or kill ratio by human measures such as participants' situation awareness, mental workload and normative performance (Mansikka et al., 2019).

Features of virtual simulation can be recognised in wargaming (Turnitsa et al., 2021) that has been used for military training and educating since the early 19th century. Other wargaming areas are, for example, examination of warfighting tactics as well as evaluation of military operations and scenarios. Nowadays, wargames are also applied in studies of international relations and security as well as in analyses of government policy, international trade, and supply-chain mechanics (Reddie et al., 2018). The implementations of wargames range from manual tabletop map exercises to computer-supported setups in which different OR and artificial intelligence techniques are utilised (Davis & Bracken, 2022).

Dynamic phenomena regarding military and defence are often represented with differential or difference equations. Examples of these models are Lanchester attrition equations that describe the evolution of strengths of opposing forces in gunfire combat (e.g., Jaiswal, 2012). There are also several modifications of these equations aiming to model, e.g., asymmetrical combat, tactical restrictions and even morale issues. Another example of simple combat models is the salvo model that represents naval combat of warships involving missiles
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(Hughes, 1995). Optimal control (see also §2.6) has been utilised, for example in planning optimal paths of military vehicles as well as in guidance systems of unmanned aerial vehicles, drones and missiles (Karelahti et al., 2007). For a recent overview, see for example Israr et al. (2022). Another type of optimal control application is the assignment of resources to counter-terror policies and measures (Seidl et al., 2016). Markov decision processes and approximate dynamic programming (§2.9) have recently emerged as important techniques for analyzing dynamic military decision-making problems related to, e.g., missile defence interceptors and military medical evacuation (Jenkins et al., 2021).

The need for multicriteria evaluation is common in military decision-making. Example applications of multi-criteria decision analysis (MCDA; see also §2.8) are acquisition of military systems and equipment procurement, military unit realignments and base closures, locating military bases, and assessment of future military concepts and technologies (Ewing et al., 2006; Geis et al., 2011; Harju et al., 2019). Public procurement even for the military is regulated in many countries, and directives require it to consider multiple criteria (Lehtonen & Virtanen, 2022). It is interesting to notice that the recent acquisition decision of a 5th generation multi-role fighter aircraft in Finland was, indeed, supported by MCDA (Keränen, 2018). MCDA weighting methods have also been used to create measures of mental workload in military tasks (Virtanen et al., 2022). In portfolio decision analysis problems, the goal is to find the best set of components, e.g., in weapons systems or in force mix for reconnaissance, with respect to multiple criteria (Burk & Parnell, 2011). The evaluation of the effectiveness of military systems calls also for the use of cost-benefit analysis (§2.18; Melese et al., 2015). Data envelopment analysis (§2.7) is a multicriteria approach helping to seek efficiency also in military problems such as personnel planning.

MCDA studies in homeland security is a broad area ranging from the design of countermeasure portfolios to threat analysis and cyber-security (Wright et al., 2006). The questions of interest include, e.g., identification of terrorists' goals and preferences, estimation of attacks' consequences, and comparison of countermeasure actions (Abbas et al., 2017). Cost-benefit models are also relevant in terrorism research (Hausken, 2018).

Today, we are witnessing the vast growth of the use of machine learning and artificial intelligence (§2.1) in military and security problems (Dasgupta et al., 2022; Galán et al., 2022). Such problem areas are, e.g., wargaming and simulation, command and control of autonomous unmanned vehicles including

drones, air surveillance, and cyber-security only to mention a few. Data analytics (see also §2.3) is naturally also used in military OR (Hill, 2020), e.g., for supporting logistics planning. Considering uncertainty is essential, e.g., in intelligence analysis and risk analysis related to terrorism (see also §2.18). Adversarial risk analysis (Rios Insua et al., 2021) uses Bayesian approaches (see also §2.18) for taking into account information, beliefs and goals of adversaries. A similar approach is also applied in the modelling of pilot decision-making in air combat with influence diagrams (Virtanen et al., 2004). Markov models and Bayesian networks are used to evaluate risks and conduct time dependent probabilistic reasoning related to military missions (Poropudas & Virtanen, 2011). Kaplan (2010) studies the infiltration and interdiction of terror plots using queueing theory (§2.17).

In the future, combat models need to include socio-cultural and behavioural factors (Numrich & Picucci, 2012). We are also likely to see an increase in the modelling of individual and group behaviour as well as the consideration of behavioural issues in military and homeland security contexts. Behavioural game theory can give insights into military strategy and conflict situations. Behavioural OR (§2.2), which studies the impacts of the human modeller and model users including cognitive biases in decision support, is likely to receive increasing attention in military applications as well.

For further readings, we refer to the military OR textbooks by Fox and Burks (2019) and Jaiswal (2012). The recently edited volume by Scala and Howard II (2020) describes various OR methods and how to apply them in military problems. Abbas et al. (2017) and Herrmann (2012) focus on homeland security modelling.

### 3.17. Natural resources⁶⁴

Climate change and natural resource management require different quantitative and qualitative models that support public policy (Ackermann & Howick, 2022). One of the early papers on the use of modelling for natural resource utilisation describes a resource analysis simulation procedure to assess the environmental impact of human activities (Bryant, 1978). The procedure comprises a structural model to express the complex network of interacting human activity systems and a parametric model to determine the scale of the activity being modelled.

An integrated decision support system for water distribution and management was built to generate alternative water allocation and agricultural production scenarios for a semi-arid region (Datta, 1995). The model considers ground and surface water
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sources as the supply. The water demand is a combination of the need for drinking, irrigation, household and public utility, natural vegetation, industrial use, and ecological balance. The decision support tool visualises water allocation to competing crops under a range of simulation scenarios, providing a wider set of options to the department taking decisions how water is distributed.

A web-based decision support system developed for the US Fish and Wildlife Service and the US Geological Survey initiative facilitates cross-organisational data sharing and performs analyses to improve conservation delivery (Hunt et al., 2016). Situation-specific management actions such as controlled burn or prescribed graze required by this decision support tool improves ecological outcomes of other conservation efforts. Buffelgrass is an invasive species that causes significant damage to the native desert ecosystem. A multi-period multi-objective integer programming model was proposed to find optimal treatment strategies to control the buffelgrass population in the Arizona desert (Büyükaftakin et al., 2014). The multiple objectives minimise damage to threatened resources: a native cactus species (saguaros), buildings, and vegetation subject to budget and labour constraints. The results show the necessity of cooperation between different interest groups to establish reasonable treatment strategies and the need for a policy change because current resources cannot stop an ecological disaster in the future.

A mixed-integer programming model is developed to evaluate fishery management policies over an infinite horizon by incorporating steady state levels of variables into a multi-period analysis framework (Glen, 1997). This model is intended to be used annually with updated stock estimates to set a dynamic total allowable catch per year depending on the stock estimates over several years. Statistical and fuzzy multiple criteria analysis establishes which materials contribute the most to the presence and the abundance of species in artificial reef structures (Shipley et al., 2018). Managers of fisheries can use this model to screen different species without loss of rigour and validity of results. Multiple-criteria analysis is used in conjunction with integer programming to assist complex management plans in ecology and natural resources (Álvarez-Miranda et al., 2020). A case study on the Mitchell River catchment (Australia) shows the trade-offs between ecological, spatial, and cost criteria, enabling decision-makers to explore and analyse a broad range of conservation plans. The use of catastrophe theory in management of natural resource systems are described with cases on forestry and fishery management (Wright, 1983). Catastrophe theory applies

the mathematical theory of structural stability to practical systems. It allows modelling of ecosystems with low and high levels of predator and prey populations. It helps model a catastrophic jump from one level to the other, supporting decision making for management of natural resources.

As a natural resource, wind provides clean, renewable, and sustainable energy. A multi-objective model minimises cost and idle time under reliability thresholds, maintenance priority, and opportunism (Ma et al., 2022). Reliability thresholds trigger maintenance activity. Maintenance priority indicates which maintenance tasks need to be performed under limited maintenance resources. Opportunistic approach indicates whether additional maintenance should be performed when a maintenance team is already out to service several turbines. The proposed multi-objective optimisation model is tested using a stochastic simulation model of a wind farm and confirmed to keep the wind system at a higher performance level with lower cost and higher availability.

Natural resource exploration is frequently subject to real options analysis (Nishihara, 2012; Martzoukos, 2009). A stochastic mixed integer nonlinear programme is proposed to incorporate geological and market uncertainty into mineral value chain optimisation (Zhang & Dimitrakopoulos, 2018). Simulation of mine deposits and commodity market informs the profitability of strategic and tactical plans. A range of real options applied to natural resource management include investments in infrastructure, use of land, and management of natural resources (Trigeorgis & Tsekrekos, 2018). Firms require high output price levels to invest in environmental technologies, because they would not want to commit to an investment that could turn out to be unprofitable in the event of a price fall (Cortazar et al., 1998).

Several papers are published on the use of operational research for natural resource management. Typical operational research problems and actors in agricultural supply chains informs strategic investment and operations management under increased pressure on natural resources (Plà et al., 2014). The contribution of operational research applications to agricultural value chain sustainability and resilience call for applications of complex systems methods such as agent-based modelling, systems modelling, and network analysis (Higgins et al., 2010). A review of environmental management and sustainability papers in major management science/operational research and systems journals revealed dominance of hard optimisation methods (Paucar-Caceres & Espinosa, 2011).

The environment-development problem concerns reconciling industrial development and environmental protection. A methodological framework is proposed to model the environmental impact of
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development under uncertainty arising from the degree of unpredictability arising from decision makers and environmental processes (Dzidonu & Foster, 1993). Natural resource development contracts depend on the bargaining power of transnational corporations and host country governments (Anandalingam, 1987). Contracts that stipulate sharing of the net income from resource development between the developing corporation and the government show that government would receiver higher income if many corporations are involved and if the government agrees to contribute to production costs. A review of Operational Research in mine planning reports optimisation and simulation applied to surface and underground mine planning problems, including mine design, long- and short-term production scheduling, and equipment selection (Newman et al., 2010). The operational research on mining is evolving to solve larger and more detailed and realistic models.

A series of cases studies from Asia, Africa, and Latin America presents principles and applications of an integrated approach to natural resources management, including the complexity of systems and redirecting research towards participatory approaches, multi-scale analysis, and tools for systems analysis, information management, and impact assessment (Campbell & Sayer, 2003). A specialised book on the Baltic region presents scientific research on activities depleting natural resources, emissions from energy use, pollution, and strategies for environmental management (Fenger et al., 1991). Stochastic Models and Option Values: Applications to Resources, Environment and Investment Problems presents a collection of research papers on the use of control theoretic methods to address problems that arise in natural resource development (Lund & Oksendal, 1991). Strategic Planning in Energy and Natural Resources contain innovative and methodologically rigorous operational research applications (Lev et al., 1987). The Handbook of Operations Research in Natural Resources collate research papers that address optimal allocation of scarce resources in agriculture, fisheries, forestry, mining, and water resources (Weintraub et al., 2007). Operations Research and Environmental Management organises its content by regional and global policies (Carraro & Haurie, 1996). Models help local and regional authorities optimise their energy distribution and minimise natural resource waste.

**3.18. Open-source software for OR⁶⁵**

Commercial solvers for solving Operational Research (OR) problems have been used for several decades and have provided both practitioners and

academics with access to the state-of-the-art OR techniques. Mathematical programming solvers IBM ILOG CPLEX (IBM 2022), Gurobi (Gurobi, 2022), BARON (Sahinidis, 1996), and discrete event simulation software Arena (Rockwell Automation, 2022) and Simul8 (Simul8 Corporation, 2022) are among the best-known examples. There also exists ad-hoc software for particular problems raising in manufacturing (e.g., AIMMS; AIMMS, 2022), healthcare (e.g., SimCAD Pro Health; CreateASoft Inc., 2022), and logistics (e.g., AnyLogistix The AnyLogic Company, 2022). The strength of commercial software is primarily based on the fact that they provide users with a simple interface to declare a problem, utilise state-of-the-art solution algorithms, and visualise the result with minimal effort.

These software do not only solve problems but also provide modelling, debugging, and scenario analysis to improve the solution process (Dagkakis & Heavey, 2016). However, the lack of access to the source code and knowledge of how these tools work internally inhibit users from customisation. It is difficult to contribute to the development of commercial software as it is a black-box to the end-user. The high licence costs of those software has been one of the most prominent factors blocking many companies, especially small and medium enterprises, from integrating them into their tactical and operational planning (Linderoth & Ralphs, 2005). Dagkakis and Heavey (2016) argued that the lack of reusability and modularity have been the additional factors impeding the use of commercial OR software.

Open-source software, on the other hand, enable users to solve OR problems without a significant initial investment. Although using open-source software does not require licensing fees, the effort to deploy it may require a significant amount of effort and time. Nevertheless, the opportunity to access the core components of a solver (or simulator) and ease of development has driven the OR community to shift from a strict, slow-pace black-box software development to modular, flexible, and quick open-source software development.

In this section, we discuss open-source OR software, categorising them into two main groups: (i) open-source solvers and (ii) open-source simulators. The former category covers the software focusing on solving mathematical programming problems. The latter includes software for simulating a real-world environment and helping decision makers to understand and analyse the system without consuming physical resources. Note that we neither provide the specific features of such software nor the characteristics in terms of programming languages, etc. Interested readers are referred to the comprehensive reviews in Linderoth and Ralphs (2005) and
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Dagkakis and Heavey (2016) for some of the software we mention below.

### 3.18.1. Open-source solvers

A solver can be defined as a set of computationally efficient analytical tools that can find optimal (or near-optimal) solutions to a mathematical programming model. In 2000, a public initiative was built by the IBM Research Division (Pulleyblank et al., 2000) to promote and support community-driven development of open-source solvers that utilise the state-of-the-art research in OR. Subsequently, a public project called COIN-OR (COIN-OR Foundation, Inc. 2022) has been initiated to host a range of open-source solvers with an open-source interface that enables contributors, users, and developers to implement their own algorithms. The repository has been expanded to provide different open-source solvers for different programming problems such as CLP (Forest et al., 2022) and HiGHS (Huangfu & Hall, 2018) for linear programming (LP); ABACUS (Jünger & Thienel, 2000), BCP (Lad'anyi, 2004), CBC (Forrest et al., 2022), Pyomo (Bynum et al., 2021), and SYMPHONY (Ralphs & Guzelsoy, 2005) for mixed integer linear programming (MILP); Bonmin (Bonami et al., 2005), Couenne (Belotti et al., 2009), DisCO (Bulut et al., 2019), Ipopt (Wächter & Biegler, 2006), and SHOT (Lundell et al., 2022) for linear and mixed integer nonlinear programming (MINLP); SMI (King, 2022) and Pyomo (Bynum et al., 2021) for Stochastic Programming (SP). COIN-OR also includes several other projects that would help users to improve their experience with modelling like PuLP (Mitchell et al., 2022) and visualising such as GiMPy (Ralphs et al., 2022).

SCIP Optimization Suite (Bestuzheva et al., 2021) can be used as a framework for mixed-integer linear or nonlinear programming as well as a standalone solver for such problems. A recent initiative commenced by the introduction of Julia language (Bezanson et al., 2017), which is a high-level, high-performance dynamic language for technical computing, is JuMP (Dunning et al., 2017) which helps users to solve a variety of problem classes including linear, mixed-integer linear, and nonlinear programming. It allows developers to use its framework and introduce new open-source solvers for particular problem classes.

We would like to also mention GLPK (Makhorin, 2020), which is the default linear programming solver behind some of the aforementioned mixed integer linear programming solvers. GLPK can also be used as a standalone linear programming solver. Finally, a suite of open-source solvers has been developed by Google (Google, 2022) to tackle integer programming and constraint programming problems. The OR-Tools provide a modelling

interface and allow users to select different commercial or open-source solvers to generate solutions.

### 3.18.2. Open-source simulation software

Simulation software can be categorised into three based on the methods that they use to define the system and its resources. We should note that we cover the software that has been applied particularly in OR domains. We opt to omit open-source software that focus on specific domains, e.g., OMNet++ (Andras, 2010) for communication networks, for the sake of brevity. We refer interested readers to the works of (Dagkakis & Heavey, 2016) and (Lang et al., 2021b) and references therein for a broader review. An experimental comparison of some of the software presented here can be found in Kristiansen et al. (2022).

The first method, Discrete Event Simulation (DES), is based on the processes of a system. In DES, the processes are defined as hosts of resources that run different operations on entities. For instance, one can define a part to be manufactured as an entity and create a manufacturing process with three machines to shape the part. DES software can be used to model and visualise complex queuing systems in order to help decision makers better understand the interactions between entities and processes.

JaamSim (JaamSim Development Team, 2016) is one of the most popular open-source DES with its user-friendly interface, easy-to-use drag and drop facilities, and continuous maintenance support. JaamSim provides a standalone executable which allows users to start using the software without technical knowledge on installations. Another DES framework is SimPy (Scherfke, 2021) which is based on standard Python functions. Its simple structure enables users to quickly obtain results for their simulation problems. SimPy has also initiated two other DESs, SimSharp (Beham, 2020) and SimJulia (Lauwens, 2021), which are the implementations of SimPy on C# and Julia languages, respectively. The last DES we would like to mention is Facsimile (Facsimile Simulation Library, 2021) which uses Scala as its basis scripting language. The main purpose of Facsimile, is to provide a high-quality discrete-event simulation library that can be used for industrial projects.

The second method, known as System Dynamics (SD), is based on representing a system as a *causal loop diagram* to define interactions between different components of a system. Some of the open-source SD software are PySD (Martin-Martinez et al., 2022), InsightMaker (InsightMaker, 2016), SysDyn (Simantics System Dynamics, 2017), and OpenModelica (Fritzson et al., 2020). PySD can convert the well-known commercial SD software
<figure><img src="image_104.png" /></figure>

Vensim (Ventana Systems Inc. 2022) input and allow user to configure. SysDyn uses the OpenModelica environment for simulation but provides an alternative built-in environment to speed up the simulation process. All these software have their own visualisation and reporting tools.

The third method is called Agent Based Simulation (ABS) and focuses on the autonomous individuals, i.e., agents, in a system. Each agent in ABS has its own characteristics and its way to interact with the other agents and the surrounding environment can differ. One of the open-source ABS software is Gama (Taillandier et al., 2019), which provides users a modelling language, a cross-platform to reproduce simulations, as well as a visualisation tool. InsightMaker (InsightMaker, 2016) is another open-source ABS software that allows users to create their own model on a web-based interface. Lastly, NetLogo (Wilensky, 1999) provides a modeling environment together with different applications to interact with other scripting languages.

We would like to complete this section with a brief summary of the application areas of both solvers and simulation software. Table 4 provides examples of areas on which the OR software can be used.

### 3.18.3. Discussion

For the sake of completeness, we should also mention that there are also several ad-hoc software that address specific OR problems. For instance, OptaPlanner (Red Hat, 2022) can solve staff rostering, scheduling, timetabling, and Vehicle Routing Problems (VRPs). Another example is VRP Spreadsheet Solver (Erdoğan, 2017), which is an Excel-based solver. Although these software provide easy and fast access to solutions, the lack of generalisation to more complex OR problems and limited development opportunities can be seen as barriers to widespread impact.

As a final discussion point, we would like to list some of the essential features of an open-source OR software. First and foremost is the performance of the software. A user would expect a comparable level of performance from an open-source software with respect to commercial software. Secondly, the scalability of a solver, i.e., its performance when the problem size increases, is one of the factors desired by practitioners. Finding an optimal solution to a

VRP instance with 20 customers does not guarantee that a VRP solver will achieve the same performance when the number of customers increases to 2000. Thirdly, technical support for a software has a crucial role in attracting users. Continuous development, documentation, and clear descriptions to change requests are some of the aspects that an open-source software should address to improve its maintainability. Finally, integration with existing libraries would help an open-source software widen its community and attract more developers to contribute.

### 3.19. Power markets and systems<sup>66</sup>

The energy industry relies on forecasts (§2.10) and decision support tools (§2.8) for operations and planning. While long-term demand forecasts – with lead times measured in months, quarters or years – have been used for planning purposes for over a century, contemporary energy forecasting literature focuses more on the short- (minutes, hours) and mid-term (days, weeks) horizons (Hong et al., 2020). Since the late 1990s, the workhorse of power trading and a typically used reference point for long-term contracts is the day-ahead market (Mayer & Trück, 2018), where prices for all load periods of the next day are determined at the same time during a uniform-price auction (Weron, 2014, see also §3.1). No wonder, the majority of studies focus on predicting intermittent generation from *renewable energy sources* (RES), electric load (or demand) and prices for the 24 hours of the next day (Maciejowska et al., 2021). Two classes of approaches dominate: regression-based models and *artificial neural networks* (ANN; Lago et al., 2021, see also §2.1).

Regression and ANN models of the 1990s and 2000s were built on expert knowledge, often independently for each hour of the day. Their inputs included past values of (depending on the context) RES generation, loads or prices from the last few days, day-ahead forecasts of exogenous variables (e.g., temperatures for load, load for prices) and a seasonal component captured by sinusoidal functions or weekday dummies (Hong, 2014; Weron, 2014). Their sub-optimal performance could be readily improved by combining forecasts across different models (Bordignon et al., 2013), calibration

<table><caption>**Table 4.** Application areas of solvers and simulation software.</caption><thead><tr><th>Subject</th><th>Methodology</th><th>Application areas</th></tr></thead><tbody><tr><td rowspan="5">Solver</td><td>LP</td><td>Transportation, agriculture, manufacturing</td></tr><tr><td>MILP</td><td>Logistics, healthcare, network design, pooling, disaster response</td></tr><tr><td>MINLP</td><td>Scheduling, telecommunication, energy systems, layout design, network design, portfolio optimization, water systems</td></tr><tr><td>SP</td><td>Supply chain planning, production planning, process control and optimisation</td></tr><tr><td>DES</td><td>Manufacturing, network design, healthcare operations, financial systems, inventory management</td></tr><tr><td rowspan="3">Simulation</td><td>SD</td><td>Telecommunication, macro- and micro-economics, social systems, ecological systems</td></tr><tr><td>ABS</td><td>Stock markets, robotics, epidemiology, game theory, evacuation planning</td></tr></tbody></table>
<figure><img src="image_105.png" /></figure>

sets (Nowotarski et al., 2016) or calibration windows (Hubicka et al., 2019). Interestingly, combining is not only a remedy for time-varying point forecasting performance. Together with quantile regression it provides a simple, yet powerful tool for probabilistic predictions – *Quantile Regression Averaging* (QRA; Nowotarski & Weron, 2018). During the Global Energy Forecasting Competition 2014, teams using variants of QRA were ranked 1st and 2nd in the price track (Gaillard et al., 2016; Maciejowska & Nowotarski, 2016). QRA can be also used to construct dynamic strategies aiming at finding the optimal trade-off between risk and return when trading the intraday and day-ahead markets (Janczura & Wójcik, 2022).

With the advent of easily accessible computational power, the models became more complex and expert knowledge was no longer enough to handle them. A major breakthrough came with the introduction of regularisation methods to energy forecasting in the 2010s. Although regularisation is a much older concept, its use in load (Chae et al., 2016; Ziel & Liu, 2016), price (Uniejewski et al., 2016; Ziel, 2016; Ziel & Weron, 2018), wind (Messner & Pinson, 2019) and solar forecasting (Yang, 2018) began only recently. Ridge regression has not seen many applications in energy forecasting, however, the *least absolute shrinkage and selection operator* (LASSO) and elastic nets (Hastie et al., 2015) have been shown to yield extremely competitive predictive models. *LASSO-estimated autoregressive* (LEAR) models often have hundreds of inputs, e.g., spanning all hours of the past week, but LASSO can shrink redundant coefficients to zero and, thus, perform variable selection. Despite their ability to handle only linear relationships between variables, LEAR models tend to be only slightly inferior to the much more complex and much harder to estimate deep ANNs (Lago et al., 2021).

The availability of high-performance GPUs and advances in optimisation algorithms made it possible to efficiently train ANNs with hundreds of inputs and outputs, multiple hidden layers and recurrent connections (§2.1). This led to a wave of deep learning and hybrid energy forecasting models in the late 2010s (Gao et al., 2019; Wang et al., 2017). A prominent, yet relatively parsimonious example is the deep neural network (DNN) model proposed for price forecasting (Lago et al., 2018). It uses a feedforward architecture with two hidden layers, 24 outputs (one for each hour of the next day) and ca. 250 inputs: past prices from the previous week, day-ahead forecasts of fundamental variables (demand, RES generation) and weekday dummies. To decrease the computational burden, its hyperparameters (number of neurons per layer,

activation functions, optimisation algorithm, etc.) and inputs (treated as binary hyperparameters – either selected or discarded) are jointly optimised once every few weeks, while the weights are recalibrated every day to account for the most recent market data. Despite this simplification, daily recalibration of the DNN model is two orders of magnitude slower than of the LEAR model with the same inputs (minutes vs. seconds on a quadcore i7 CPU; see Lago et al., 2021).

The increased complexity of deep ANNs is a major obstacle in understanding the underlying processes. Partial remedy provide recently proposed architectures like the neural basis expansion analysis for interpretable time series forecasting (NBEATS; Oreshkin et al., 2021; Olivares et al., 2023), which project the time series onto basis functions in the fundamental blocks of the network structure. The final forecasts can be decomposed into interpretable components returned by groups of blocks (called stacks). Separate stacks can account for the trend, seasonality and exogenous variables. Another recent innovation in energy forecasting is a distributional ANN (Mashlakov et al., 2021). It only requires a small change in the architecture – instead of the 24 hourly forecasts, the network can return the parameter sets of 24 probability distributions (e.g., the mean and standard deviation for the Gaussian). The benefits are clear. The downside, however, is that the distribution itself has to be estimated (it is a hyperparameter). Somewhat surprisingly, distributional ANNs not only can yield more accurate probabilistic predictions, but also better point forecasts (Marcjasz et al., 2022).

For horizons beyond the next 48 hours other approaches have been proposed (Weron, 2014), not necessarily forecasting per se. Structural models define the functional relationships between physical (weather, generation, consumption, etc.) and economic (bidding, trading) variables that set the price, then utilise – typically – parsimonious statistical or machine learning techniques to provide the stochastic inputs. Due to the nature of fundamental data, often of weekly or monthly granularity, such models are more suitable for medium-term risk management, portfolio optimisation and derivatives pricing (Kiesel & Kusterman, 2016), than for short-term forecasting (Mahler et al., 2022). In the class of multi-agent approaches, Ventosa et al. (2005) identify three trends: equilibrium, simulation and optimisation models. The former (Nash-Cournot framework, supply function equilibrium, strategic production-cost models) have seen limited application in oligopoly markets (Ruibal & Mazumdar, 2008). Agent-based simulations are used when the
<figure><img src="image_106.png" /></figure>

problem is too complex to be addressed within an
equilibrium framework (Fraunholz et al., 2021).

Optimisation models address profit maximisation
from the point of view of a firm competing in the
market. One of the simplest settings is that of the
price clearing process being exogenous to electricity
generation optimisation – as the price is fixed, the
market revenue is a linear function of the production
and linear programming (§2.14) or mixed integer lin-
ear programming (MILP) can be employed (Ventosa
et al., 2005). On the other hand, Virtual Power Plant
(VPP) operations constitute a more complex problem
of decision-making under uncertainty. A VPP is a
cluster of dispersed generating units (e.g., intermit-
tent rooftop solar panels on residential houses), flex-
ible loads and battery storage that operates as a
single entity. Robust optimisation and stochastic pro-
gramming can be used to derive the optimal VPP
trading strategy (Morales et al., 2014).

To support broader regulatory decisions at the
firm or country level, frontier analysis methods are
employed. Such methods aim to estimate the efficient
frontier of the evaluated production units, to measure
their relative efficiency (against the frontier) and to
provide targets that can support policymakers. The
benchmarking nature of these methods has estab-
lished them as a flexible and multifaceted decision-
making tool. In particular, Data Envelopment
Analysis (DEA, §2.7) has been employed in a wide
spectrum of energy applications. Early DEA studies
relied only on a few factors (labour, fuel, capital, elec-
tricity production) to assess the technical efficiency of
electric utilities (Färe et al., 1983). Later studies took
into account sustainable practices by including envir-
onmental variables. Such factors are commonly
treated as undesirable outputs that arise as by-prod-
ucts of the production process (Färe et al., 1996) or
as non-controllable variables, which reflect external
factors that the unit under evaluation cannot control
(Hattori et al., 2005). When price information is
available, DEA allocation models can be used to
evaluate revenue, cost and profit efficiency. Ederer
(2015) argued that sophisticated cost efficiency assess-
ment methods should be employed to evaluate RES,
and relied on DEA models to assess the capital and
the operating cost efficiency of offshore wind farms.
Notably, DEA is often combined with multi-criteria
decision-making techniques to incorporate decision
maker's preferences into the assessment (Lee et al.,
2011; Wang et al., 2022a) and econometric techni-
ques to study causal effects (Shah et al., 2022).

For a review and outlook into the future of energy
 load, price, wind, solar) forecasting see Hong et al.
 (2020). Hong and Fan (2016) offer a tutorial review on
 probabilistic load forecasting. The standard reference
 for electricity price forecasting is Weron (2014). Lago

et al. (2021) offer a more recent viewpoint focusing on
deep learning and hybrid models. They also provide a
set of guidelines/best practices and make freely available
the epftoolbox with Python codes for two highly com-
petitive benchmark models (LEAR, DNN). Two thor-
ough treatments of probabilistic price forecasting are
Nowotarski and Weron (2018) and Ziel and Steinert
(2018). Sweeney et al. (2020) present a brief overview
of the state-of-the-art in RES forecasting, whereas Yang
et al. (2022) jointly discuss atmospheric science and
power system engineering in the context of solar fore-
casting. Finally, for detailed literature reviews on energy
related applications of DEA see Mardani et al. (2017),
Sueyoshi et al. (2017) and Yu and He (2020).

3.20. Project management⁶⁷

Operational Research methods play a fundamental
role in managing a portfolio of projects, in project
selection and in the management of each individual
project. *Project portfolio management* is concerned
with the optimal mix and prioritisation of proposed
projects in order to maximise the organisation's
overall goals (Levine, 2005). At the strategic level,
*project selection* deals with the selection of and
resource allocation among a group of projects
(Kavadias & Loch, 2004). Static models rely on
mathematical programming, scoring and sorting,
financial modelling, graphical and charting techni-
ques. Dynamic models for selecting projects from a
stream of arrivals may rely on queueing theory
($2.17$), simulation ($2.19$), decision theory ($2.8$)
and stochastic dynamic programming ($2.9$; $2.21$).

At the tactical and operational levels, *project management* (Meredith & Mantel, 2003) basically involves the planning, scheduling and control of project activities to achieve performance, cost and time objectives for a given scope of work, while using resources efficiently and effectively. The scope of a project is the magnitude of the work that must be performed to make sure that the product or items to be provided (the project result or the project deliverables) meet the requirements or acceptance criteria agreed upon at the onset of a project. Once the project is properly defined in terms of its scope and objectives, the planning phase may start through the identification of the project activities, the estimation of time and resources, the identification of relationships and dependencies between the activities and the identification of the schedule constraints. The activities can be graphically portrayed in the form of a project network showing the necessary interdependencies of the activities. Based on the type and quantities of resources required, cost estimates can be made. *Project scheduling* (Demeulemeester & Herroelen, 2002) then involves
<figure><img src="image_107.png" /></figure>

the construction of a project base plan which specifies for each activity the precedence and resource feasible start and completion dates, the amounts of the various resource types that will be needed during each time period, and as a result the budget. Once a baseline schedule has been established, it must be implemented. This involves performing the work according to the plan and controlling the work by monitoring the progress and taking necessary corrective action when the project is on its way to run behind schedule, to overrun the budget, or to violate the original technical specifications.

### 3.20.1. Construction of the project network

A project network is a graph consisting of a set of nodes and a set of arcs. In the activity-on-arc representation (AoA), the nodes represent the events and the arcs represent the activities. AoA networks form the basis of the Project Evaluation and Review Technique (PERT; Malcolm et al., 1959) and the Critical Path Method (CPM; Kelley, 1961). The precedence relationship used is the finish-start relationship with a zero time lag: an activity can start as soon as all its predecessor activities have finished. In the mostly used activity-on-node representation (AoN) the nodes represent the activities and the arcs denote the precedence relations. The AoN representation allows for the specification of generalised precedence relations of four types: start-start, start-finish, finish-start and finish-finish with minimal and/or maximal time lags. A minimal time lag specifies that an activity can only start (finish) when its predecessor activity has already started (finished) for a certain time period, whereas a maximal time lag specifies that an activity should be started (finished) at the latest a number of time periods beyond the start (finish) of another activity.

### 3.20.2. Temporal analysis for deterministic unconstrained project scheduling

In this case, a single deterministic duration estimate is used for the activities. Basically, the temporal analysis then involves the computation of the activity start times under the objective of minimising the project duration. In the presence of strict finish-start precedence relations, this can be achieved by simple forward and backward pass calculations. Generalised precedence relations with maximal time lags call for the use of graph algorithms for computing the longest path (critical path) in networks.

The temporal analysis may also be performed under the objective of maximising the net present value of the project. The deterministic max-npv problem can be formulated as a nonlinear problem. An efficient recursive solution procedure has been developed for AoN networks and has been extended

to deal with the case of time-dependent cash flows (Vanhoucke et al., 2001b).

Another non-regular performance measure is the minimisation of the weighted earliness-tardiness penalty costs of the project activities, where activities have an individual due date with associated unit earliness and unit tardiness penalty costs. The problem can be solved by an exact recursive search procedure (Vanhoucke et al., 2001c).

### 3.20.3. The deterministic resource-constrained project scheduling problem

Project activities require resources for their execution. Renewable resources (e.g., manpower, machines) are available on a per-period basis. Their introduction into the analysis complicates matters considerably. Computing a precedence- and resource-feasible deterministic schedule that minimises the project duration, the resource-constrained project scheduling problem (RCPSP) is NP-hard in the strong sense (§2.5). Both exact and suboptimal procedures have been presented in the literature.

Many mathematical programming formulations (§2.15), either binary or mixed integer linear programs, have been developed (Demassey, 2008). The RCPSP may also be solved through constraint-based scheduling (Laborie & Nuijten, 2008). Also a number of branch-and-bound algorithms have been presented for optimally solving the RCPSP (Brucker et al., 1998; Demeulemeester & Herroelen, 1992).

Heuristic procedures broadly fall into two categories: constructive heuristics and improvement heuristics. Constructive heuristics start from an empty schedule and add activities one by one until a feasible schedule is obtained. Activities are ranked by priority rules which determine the order in which the activities are added to the schedule. Improvement heuristics start from a feasible schedule that was obtained using a constructive heuristic. Operations are then performed on a schedule which transforms a solution into an improved one. These operations are repeated until a locally optimal solution is obtained.

Project scheduling metaheuristics come in a wide variety and broadly include tabu search (Baar et al., 1999), simulated annealing (Bouleimen & Lecocq, 2003), genetic algorithms (Hartmann, 2002), ant colony optimisation (Merkle et al., 2002), scatter search and electromagnetic approaches (Debels et al., 2006).

### 3.20.4. Resource problem variants and generalisations

Branch-and-bound may be used for solving the RCPSP with generalised precedence relations (Demeulemeester & Herroelen, 1997; De Reyck & Herroelen, 1998), when activities may be preempted (Demeulemeester &
<figure><img src="image_108.png" /></figure>

Herroelen, 1996), when the problem has to be solved under the objective of maximising the net present value (Vanhoucke et al., 2001b) or with the earliness-tardiness objective (Vanhoucke et al., 2001a).

The resource levelling problem aims at completing the project within its deadline with a resource usage that is leveled as much as possible over the entire project duration. Exact solution procedures based on integer or dynamic programming and branch-and-bound as well as heuristic procedures have been developed (Neumann & Zimmermann, 2000). The resource availability cost problem, that consists of scheduling the project activities such that the total cost of acquiring the necessary resources is minimised, assuming that a resource is assigned to the project for the total project duration, can be solved optimally (Demeulemeester, 1995). The resource renting problem (Nübel, 2001) which assumes that resources can be added or removed from the resource pool over the project life, can be solved optimally using branch-and-bound or heuristically using genetic algorithms and scatter search (Ballestín, 2007a; Kerkhove et al., 2017).

The multi-mode RCPSP assumes limited availability of renewable and nonrenewable (e.g., money) resource types and assumes that a project activity may be executed in multiple modes, where an activity mode corresponds to the assignment of a mode-specific number of units of a (non)renewable resource type to the activity with correspondingly resulting activity duration. The project decisions then involve the decisions to start and perform the activities in a specific mode in order to minimise the project duration. Branch-and-bound (Hartmann & Drexel, 1998), branch-and-cut, local search (De Reyck & Herroelen, 1999) and genetic algorithms (Hartmann, 2001) are available.

For projects with a flexible project structure, where activities to be performed are not known in advance, decisions for the implementation of optional activities can be made using genetic algorithms and tabu search (Kellenbrink & Helber, 2015; Servranckx & Vanhoucke, 2019a, 2019b).

### 3.20.5. Dealing with uncertainty

Risk analysis involves the identification of the qualitative and quantitative assessment of the risk factors for the project through the estimation of the probability of the risk factors (activity duration, cost and resource requirement increases, start time delays) as well as their potential impact. The impact of each risk is best assessed individually and mapped to the duration of a project activity (Creemers et al., 2014). Risk responses may then involve risk avoidance by performing an alternative approach without the risk, taking actions to reduce the risk, and risk impact

reduction by switching to a different execution mode, adding additional resources, etc.

Stochastic scheduling does not generate a baseline schedule before the start of the project, but deals with time uncertainty by viewing the scheduling problem as a multi-stage decision process where scheduling policies are used to decide at each of the stages which occur serially through time at random decision points, which activities selected from the set of precedence and resource feasible activities have to be started under the objective of minimising the expected project duration (Demeulemeester & Herroelen, 2011).

Proactive project scheduling generates a robust baseline schedule through solving the RCPSP and subsequently tries to protect it as well as possible against time and resource disruptions that may occur during project execution. This protection can be achieved by deciding on a clever way to transfer the renewable resources between the activities (Leus & Herroelen, 2004). Both branch-and-bound and heuristics are available for the minimisation of the weighted sum of the difference between the planned and the realised activity start times (Van de Vonder et al., 2008; Lambrechts et al., 2008). Another way involves the insertion of time buffers that should prevent the propagation of distortions throughout the schedule. The critical chain methodology (Goldratt, 1997; Herroelen & Leus, 2001; Newbold, 1998) defines the critical chain as that set of tasks which determines the overall project duration. Protection is then realised through the insertion of feeding buffers and resource buffers in combination with a project buffer at the end of the critical chain.

When during the actual execution of the project disruptions occur that cause deviations from the protected baseline schedule or even render this schedule infeasible, reactive scheduling procedures may be deployed.

For reviews and comprehensive textbooks on project management and scheduling we refer the reader to Demeulemeester and Herroelen (2002), Demeulemeester and Herroelen (2011), Hartmann and Briskorn (2010), Herroelen (2007), Herroelen and Leus (2005), Meredith and Mantel (2003), Neumann et al. (2003), Shtub et al. (2004), and Vanhoucke (2018).

### 3.21. Revenue management⁶⁸

The discipline of revenue management (RM) deals, in the widest sense, with demand management decisions to improve overall revenue or profit. Demand management decisions aim at influencing demand, such as pricing and availability control. Occasionally, demand management decisions can also take
<figure><img src="image_109.png" /></figure>

different forms like ranking lists (e.g., when showing customers of a meal delivery platform a rank order of restaurants that offer home delivery) or green van icons to denote which time slots for grocery home delivery are more environment-friendly (because there is a delivery planned to take place already anyway). RM is about IT-supported decision-making, mostly on the operational level, in contrast to strategic pricing theory encountered in the marketing domain.

Such decision support systems, referred to as RM systems, have been first developed in the airline industry in the 1970s when deregulation introduced competition in the US airline market. They were so successful that the practice of RM soon spread to other industry domains, particularly to those that sell services or perishable goods (perishability creates pressure to sell within a limited selling horizon). Examples include restaurant tables, hotel rooms, rental cars, or airline seats, among many other applications. In these industries, the supply is usually fairly inflexible, fixed costs are high and variable costs are relatively low (which also makes revenue maximisation mostly equivalent to profit maximisation, hence the name *revenue* management).

In this section, we first outline recent research trends on demand models (and their estimation) that are required to provide an input to the RM optimisation system. Then, we present recent research on efficient optimisation of demand management decisions. Finally, we outline further reading suggestions including some current popular application areas. We mostly use the passenger airline industry throughout this subsection to illustrate developments in the field of RM.

### 3.21.1. Modelling demand

In order to make good demand management decisions, we first need to have a model of demand to describe the response to specific RM actions (such as pricing or changing the availability of products or services). The first demand models used in RM assumed that demand for a given product is independent of what else is offered. These so-called 'independent demand models' are relatively easy to estimate. However, this independence assumption usually only holds in applications where rate fences (such as advance purchase requirements) strongly limit customers' abilities to substitute a product with one another.

One way to relax the – often unrealistic – independent demand assumption is by considering that the customer looks at all alternatives available and chooses one. The requires modelling of customers' choice behaviour; the seminal paper by Talluri and Van Ryzin (2004a) introduced discrete choice

modelling to the domain of revenue management. In choice-based RM, demand for a product is assumed to depend on the available purchase alternatives and their attributes. These models tend to be more accurate in predicting demand if the independent demand assumption is not met, at the cost of being more difficult to estimate and implement (Klein et al., 2020). Much research has been carried out on choice-based RM since 2005; for a recent review, see Strauss et al. (2018).

Among the most recent trends – building on the aforementioned choice-based RM literature with fixed choice model parameters – is a stream of work on personalisation and choice model parameter learning. For example, Cheung and Simchi-Levi (2017) solve an online personalised assortment optimisation problem formulated as a multi-armed bandit problem. Demand learning models balance the trade-off between gathering new samples (and thereby learning more about the true customer behaviour) and applying the RM decision that, based on the current belief of customer behaviour, looks to be the best. In the short term, this means that we occasionally make decisions that seem not very promising, yet that will gain us insights into customer behaviour (for instance, by offering price points that were never offered before). For our airline example, a potential application is the ongoing learning of model parameters governing the choice of ancillary products (like seat upgrades, extra luggage, etc.). Models like the one by Agrawal et al. (2019) can be used for this purpose.

Demand models in RM may be biased if they are estimated on constrained data, meaning that the sales data does not necessarily reflect the actual demand. For example, if a flight is fully booked, we observe no further sales transactions for that flight. Yet demand may well exceed flight capacity and, as such, should be estimated somehow. Methods to statistically unconstrained demand data are reviewed in Guo et al. (2012).

Another dimension of demand modelling is represented by strategic versus myopic customer behaviour. One of the earliest papers on this topic is the work by Su (2007). He considers customers who may delay their purchase when they expect lower-priced offers in the future. With RM mostly focusing on myopic customers (meaning customers who do not anticipate future developments in their purchase decision), the behaviour of strategic customers leads to inefficiency. Su (2007) proposes an intertemporal pricing component to adjust the offering based on the market composition between these customer types, and more work has built on this since.
<figure><img src="image_110.png" /></figure>

### 3.21.2. Optimisation advances

A central element of an RM system is the decision of what to offer whenever a customer arrives. Decision policies (essentially a mapping from the state space of available information to the action space) are used to determine which products are made available (i.e., availability control), or at which price (i.e., dynamic pricing) – and sometimes, a combination of both. Using dynamic prices to manage demand can be very similar to availability control: when there are products defined with identical features but different price tags such that there is a discrete set of prices to choose from for a product, it can be considered a special instance of the aforementioned availability control (Strauss et al., 2018). This can be observed in airline's implementations of differently priced booking classes for the same seat such that a customer can only purchase that seat for the fare of the booking class made available to them. The groundbreaking papers by Gallego and van Ryzin (1994, 1997) also featured a dynamic pricing concept for a single-commodity and a network-level problem, respectively. Both papers also considered the effect of significant cancellations and no-shows (meaning bookings that are not actually being used in the end). In that context, it can be valuable to accept more reservations than physical capacity would allow. This practice is called overbooking and is widespread in many industries where the risk of having to reject a customer with a valid reservation is not overly costly (with examples such as Simon, 1968, showing it being applied already before RM but now usually integrated into systems to manage demand for available capacity). For an overview of recent contributions on this matter, see Klein et al. (2020).

Decision policies in RM are trading off the immediate reward of having a customer buy a product versus the so-called opportunity cost associated with this purchase, stemming from having to commit some resources to a given sale. For example, a resource might be a flight with a specific seat capacity. Selling a ticket for a seat on this flight requires us to commit a seat to this customer, which otherwise might have still gotten sold in the future at a potentially larger fare. Therefore, by having a customer buy the product, we incur the cost of losing the opportunity to sell the associated resource units in the future. This value (or at least an approximation thereof) is sometimes used as a revenue threshold defining which products shall be shown as available; such special versions of availability control are known as bid price policies, with the bid price being this threshold, and only products with revenues that exceed the bid price being shown.

There are two major challenges in deriving optimal decision policies: first, we need to somehow obtain the opportunity costs involved with having a customer book a particular product at a given point in time; second, we need to solve the resulting optimisation problem to give us the actual decision to be implemented.

The latter decision problem, given opportunity cost, may be as simple as a comparison of two numbers (traditionally used in independent-demand settings), but can be non-trivial in the presence of sophisticated models of customer behaviour (dependent demand settings). Much research over the past few years has been devoted to studying properties of choice models that can be exploited to efficiently solve – or at least closely approximate – the online RM decision problem. This work is further motivated by the need to solve these RM decision problems quickly to ensure an acceptable user experience. Within availability control, assortment optimisation under various choice models has received particularly much attention because this problem becomes NP-hard for many customer choice models unless a certain structure can be exploited; for a review, see Strauss et al. (2018).

The other challenge in deriving optimal decision policies is the computation of opportunity costs. This is usually the more difficult task for real applications because the opportunity costs depend on time, the current state (of inventories), and future demand and actions. Dynamic programming (DP; §2.9) is usually being applied to solve or at least characterise the optimal decision policy over a given booking horizon.

However, obtaining opportunity costs using DP is often only possible when dealing with problems that have a single resource (like optimising for a single flight only, for example in Wollmer, 1992). When there are products that use more than one resource (like a itinerary of multiple flights connecting in a hub), then we speak of network RM problems. These require much more effort to solve (so as to get opportunity cost estimates for our decision policy) due to the fact that decisions on one product may affect many others that are using the same resource. Therefore, to reach at least an approximate solution for a network RM problem, one usually needs to resort to deterministic linear programming (Liu & van Ryzin, 2008, §2.14) or approximate dynamic programming (Gallego & Topaloğlu, 2019, describe how approximate dynamic programming can be used in RM). In practical applications, the network-level optimisation problem is often decomposed into a collection of single-resource problems such as in Kemmer et al. (2012) who were motivated by methods deployed by Lufthansa Systems in
<figure><img src="image_111.png" /></figure>

their RM optimisation module. In older RM sys-
tems, booking control was typically implemented
using versions of the so-called Expected Marginal
Seat Revenue (EMSR) heuristics (Belobaba, 1987a),
which are in turn rooted in the work of Littlewood
(2005), originally written in 1972.

Once a decision policy has been obtained (by first
obtaining opportunity costs and then solving the corre-
sponding decision problem), we then need to evaluate
the decision policy using simulation or even in real-
world trials. Bertsimas and De Boer (2005) give an
overview of different decision rules for airline RM that
are evaluated with a simulation study. Further details
on simulation techniques can be found in §2.19. An
example of testing a dynamic pricing policy in live tri-
als is the work by Fisher et al. (2018).

**3.21.3. Further reading**

RM is also applied in retailing, both for e-commerce
and offline shopping. Agatz et al. (2013) provide a
practical overview of ways online retailers imple-
ment RM in their business. But there are also retail
RM applications outside of online shopping. For
example, Caro and Gallien (2012) describe how
brick-and-mortar fashion stores optimise their price
 markdowns during season clearing sales.

In particular, linking RM to general transportation problems has received significant attention over recent years. An overview of advances in that field is given by Fleckenstein et al. (2022). Applications thereof can be seen here especially in business models with delivery constraints, such as same-day deliveries (Ulmer, 2020) or for attended home delivery (AHD) which is common for online grocery shopping. Customers' desire for short and guaranteed time windows in AHD leads to less than optimal routings. Yang et al. (2016) show how RM can be used to steer demand for delivery time slots towards a routing solution closer to the optimum, thereby increasing overall profit for businesses shipping goods that require AHD. Another example of applying RM ideas to new transportation problems is the work by Künnen and Strauss (2022). They analyse how an air traffic network manager could reduce overall delays for all airspace users (i.e., airlines) by offering dynamically priced flight trajectories.

For more detailed readings about the develop-
ment of the RM domain, the techniques being used,
and more applications, we refer the reader to the
books by Talluri and Van Ryzin (2004b) and
Gallego and Topaloğlu (2019).

**3.22. Service industries⁶⁹**

*Service industry from the perspective of operations:* Service industry is a concept from economics originally defined by what it is not. It is not a

manufacturing industry that produces tangible
goods (cars, clothes, equipment), but industry that
provides intangible outputs, such as hospitality,
healthcare, and education. In service research, serv-
ices are also defined by additional characteristics. In
addition to intangibility, the so-called IHIP charac-
teristics, recognises heterogeneity, inseparability, and
perishability as the defining characteristics of service
industries.

In Operations and Operational Research, service industry is approached not from its characteristics but operationally to support actionable insights (Burger et al., 2019). For Operational Research applications, service industry can be approached through the FTU-framework, defining services as a particular type of transformation (Figure 2). Service industries are distinguished from goods industries through the direct provision and integrative decision making. In services the decision making of customers and providing companies is intertwined, while in goods industries customer and providers make autonomous decisions. In service industries the value is directly provided to the customer, while in goods industries indirectly through the product.

However, operational reality is not this clear-cut.
In manufacturing industries, through servitisation
(Kohtamäki et al., 2018), some companies seek to
make their products more like services to differenti-
ate their offering, and directly create value to their
customers. In service industries, managers seek to
make services more like goods, to be able to run
service facilities more like factories and improve
productivity (Levitt, 1972; Schmenner, 2004). OR
methods that were initially developed in manufac-
turing industries (e.g., forecasting, queuing, schedul-
ing, simulation), are increasingly applied in service
industries to make service facilities operate more
like factories (cf. Eveborn et al., 2009). For servitisa-
tion, OR presents a more limited range of methods.
Methods supporting the servitisation of products are
for example, value constellation modelling
(Holmström et al., 2010; Brax & Visintin, 2017),
and ecosystem modelling to support innovation and
new business model design in an open environment
(Talmar et al., 2020).

The challenge in service industries is that service
systems tend to be open, problems wicked, and opti-
mising solutions difficult to develop and apply.
Value provision often requires interaction with cus-
tomers (customisation) limiting the situations where
facilities can be organised for flow and efficiency as
service factories. Also, servitisation of products
occurs in an open systems environment, requiring
responsiveness to influences and disturbances from
the outside, as will be seen for our application
<figure><img src="image_112.png" /></figure>

<figure><img src="image_113.png" /><figcaption>Figure 2. An actionable framework for service industries: facilities-transformation-usage (adapted from Moeller, 2010)</figcaption></figure>

examples from industrial services and home healthcare.

*Service industry applications:* In the following we will present two examples on the use of OR methods for creative insight and novel solutions in service provision. The examples are homecare of elderly patients (Groop et al., 2017), and line maintenance of commercial aircraft (Öhman et al., 2021). In the first example systems thinking, in the form of soft OM methods from Theory of Constraints (Davies et al., 2005), is used in combination with design science research (implementation and evaluation). In the second example design and simulation are used in combination, uncovering an unexpected new way of simultaneously improving resilience and reducing costs in a commercial airline.

*Homecare of elderly patients* (Groop et al., 2017): Nurses, team leaders, and healthcare management had distinct and diverging views on what is the problem with the homecare operations. Strongly held and divergent views are an indication of a possibly wicked problem (Sydelko et al., 2021). The divergent views in the case were uncovered through engagement (following actors in their work) and interviewing, with the purpose of articulating what different stakeholders identified as undesirable effects (UDE) of the current solution. UDE is thinking tools terminology from Theory of Constraints (Dettmer, 1997). These UDEs of the current operation were pruned for overlaps and narrowed down to a list of seven (including the seeming contradiction between low utilisation of full-time employed nurses, stressed-out nurses, and chronic under-staffing requiring frequent use of temporary staff). Using effect-cause logic, the interconnections between effects, and mechanisms behind the effects were specified and then evaluated by all stakeholder groups in joint workshops. In this case, the first effect-cause analysis pointed towards a core problem, a contradiction, which when addressed, would

improve efficiency. The needed change was in the way the nurse visits are scheduled to improve effectiveness. Instead of scheduling nearby patients after each other to save travelling time, the home care organisation should focus on only scheduling nurses for time-critical visits (visits that must be performed at a specific time) at the peak-demand in the morning. This way the time of full-time nurses will be more effectively used.

However, when implemented, the scheduling change had next to no effect. With the initial solution a failure, the evaluation of the implemented change pointed to issues with the problem framing. Going back, considering the stated problems (UDEs) and initial solution, the field researchers found that they had missed an important undesirable effect originating from the way the organisation operated. In the mapping nobody had raised as a problem that full-time nurses, when not busy, do not help-out in other teams. When nurses stay within their teams, work is evenly divided between everybody in the team, which is not a problem for nurses, nor for team leaders. Instead, when there is need for more nurses, outside temporary nurses are called in, and they move between teams if needed, but not the full-time employed nurses.

Management, for whom the low utilisation of full-time employed nurses is a cost issue (with payment of salaries both for idling full-time nurses and busy temporary nurses), were not aware of how nurses staying with their team was a mechanism behind the low utilisation. Nor had the researchers working in the field realised this before failing with the first solution design. Re-framing the problem once more, another solution changing the scheduling for employed nurses was proposed. Instead of dividing work equally between all nurses in their teams, the team leaders should seek to schedule work so that one, or even two nurses in their teams
<figure><img src="image_114.png" /></figure>

have no work, and can be made available to help-
out in other teams.

*Line-maintenance of commercial aircraft* (Öhman et al., 2021): The second example illustrates the use of simulation in problem reframing and finding a new type of solution. The service operations are line-maintenance of aircraft in an airline. Initially the problem was framed by management as improving departure reliability without increasing the number of maintenance technicians. The intended solution was introducing lean in the turn-around of aircraft.

However, in line maintenance there are no mater-
ial and time buffers for which lean approaches have
been so impactful in manufacturing. The minimum
frequency and content of maintenance tasks are regu-
lated. Departures are delayed by technical problems
that add unplanned tasks, which need to be carried
out. Here, lean principles can increase productivity
but not reduce the unplanned tasks. To reduce the
delays caused by unplanned tasks a resource buffer of
maintenance technicians appears necessary.

In this example, the same method of engagement
was applied as in the homecare example. Observing
and interviewing different actors involved, field
researchers sought to articulate a set of undesirable
effects of the current way of operating. However, no
agreement on a core problem to address could here
be reached. Instead, problem framing ended with a
question and a puzzling response that pointed in a
new direction. Line-maintenance scheduling
assumed that maximising the interval for planned
tasks is optimal, also when there are unplanned
tasks and constrained resources. Engaging and inter-
viewing maintenance planners for both long-haul
and short-haul fleets and operations, production,
and resource planning, field researchers began to
gain an in-depth understanding of the airline main-
tenance planning function. Heuristics and principles
related to dealing with over-maintenance not visible
in the operational documentation were encountered.

To explore the implications, the researchers first modelled the relationship between over-maintenance and planned workload variance in a deterministic setting, focusing solely on scheduled maintenance. The model indicated a promising relationship: an increase of one percent in the total planned workload (over-maintenance) could result in up to a six percent reduction in workload variance. Next, simulation of the airline operation and maintenance included the unplanned events according to their historical distribution. The simulation surprisingly indicated that increasing over-maintenance could reduce over-all costs and improve departure reliability, if combined with a re-scheduling solution for maintenance task. Re-scheduling introduces a new type of time buffer, a frontlog of planned maintenance tasks that can be

postponed to allow technicians to address unplanned
tasks without disruptions to departure schedule.

*Summary and conclusion:* In service industry applications problem framing methods are particularly important. The openness of service operations and wicked problems often require the Operational Researcher in service industry applications to go outside their comfort zone regarding methods (Mingers, 2011b, 2015) to search for actionable insight (Burger et al., 2019). In the examples provided, a combination of approaches, tools, and methods were continuously employed in the search for a good problem framing as the basis for an effective solution design. For the application of OR methods in service industries, the homecare example illustrates the use of a soft OR method in framing the problem (from the practice of Theory of Constraints, Davies et al., 2005), the use of scheduling from hard OR as a solution component, and implementation as a method of design science for evaluation and redesign (Holmström et al., 2009; Sein et al., 2011). The second example illustrates the use of simulation as a method of explorative design. In the empirical grounding of the simulation model we encountered the good problem, which is the key to success in simulation projects (Law, 2003). Through simulation we developed and explored the effect of the dynamic re-scheduling and buffer management approach, with surprising outcomes. Before the simulation study nobody knew about the opportunity to both improve departure reliability and reduce costs.

The example multi-method approach combined
soft OR, simulation, and systems thinking for fram-
ing the problem. As in cross-agency problem solving
in government and public administration, service
industry problem solving benefits from mapping dif-
ferent actor perspectives, as the purposes, perspectives
and values of the service supply chain actors can eas-
ily be in conflict (Sydelko et al., 2021). However, in
addition to methods for actionable insight, methods
for turning insights into solution proposals are also
needed. For proposing and developing a solution
design, the two examples relied on explorative design
science (Holmström et al., 2009), relying on OR
methods in evaluation when implementation is pos-
sible, and simulation for substituting implementation.
In the search of effective solution designs, OR meth-
ods such as scheduling, and forecasting were applied
as potential solution components in both examples.

3.23. *Sports*<sup>70</sup>

Moneyball (Lewis, 2003) told the story of how the
Oakland Athletics Major League Baseball team was
able to leverage an inefficiency in the labour market
for baseball players, and perform above expectations
<figure><img src="image_115.png" /></figure>

(given the team's salary spend). Its impact on how quantitative analysis is viewed within sport and wider society is unprecedented. We have moved from an age when society tended to undervalue quantitative skills to a post-Moneyball era where analytics is generally accepted as being “cool”. Told in both a best-selling book (Lewis, 2003), and a Hollywood movie of the same name, Moneyball has driven a rapid expansion of interest in the field of sports analytics. For an analysis of Moneyball, see, for example, Hakes and Sauer (2006).

The history of quantitative analysis in sports dates back to centuries before the Moneyball story, and to the conception of probability itself. The concepts of chance are as old as the first dice games, but they did not evolve into the mathematical principles of probability until the 17th century when Pascal and Fermat exchanged ideas in a series of letters during 1654. The letters were written in response to the following problem: *two players, A and B, each stake 32 pistoles on a first-to-three-point game. When A has 2 points and B has 1 point, the game is interrupted and cannot continue. How should the stakes of 64 pistoles be fairly distributed?* Fast forward three centuries and the similarities of this problem with the problem encountered in limited-overs cricket, when a match is cut short because of rain, are uncanny. Indeed, the solution offered by Duckworth and Lewis (1998) is one of the great success stories of OR in sport, or arguably OR in any field. That sports fans routinely use the names of a statistician, and an operational researcher should be the source of great pride to the OR community.

The field of sports analytics now boasts specialist journals, regular special issues in top-rated mainstream journals, large departments in sports teams, and many stories of success and over-achievement in professional sport.

### 3.23.1. What is 'sports analytics'?

Analytics is largely an umbrella term for data science, statistics, operational research, and nowadays, machine learning. A simple definition of sports analytics is *the use of analytics to gain a competitive edge in sport*. A wider definition would be *the use of analytics to improve decision making in sport*.

Research has been published on the use of analytics in almost all popular sports including: football, tennis, cricket, golf, American football, baseball, motor sport, martial arts, and many more. Rather than review the field by sport, it is more logical to consider the field by task. The following is not a comprehensive list of such tasks, but provides an overview of the more common objectives of sports analytics.

### 3.23.2. Ranking and rating

Ranking of competitors is, to a large extent, the entire purpose of organised sport, and rating is a popular area of research. There are several families of models used to rate competitors. Paired comparisons models are used when two competitors play in each contest. For example, Elo ratings were first developed for use in chess, but have since been used by, for example, Hvattum and Arntzen (2010) for forecasting football results. Glickman (2001) presented a more general Elo model based on a Bayesian updating system and applied it to the problem of dynamic ratings of chess players. Another paired comparisons model is the Bradley-Terry model and this was used in McHale and Morton (2011) to forecast the results of tennis matches. Multiple comparisons models are used when several competitors play in each contest and Baker and McHale (2015) use a time-varying multiple comparisons model to rate golfers from different eras. Langville and Meyer (2013) provide an excellent overview of rankings models.

Rating individuals in team sports is a somewhat more complex task than the examples given above, especially when the individuals have different objectives, as is the case in football for example, where some players are mainly responsible for defending, whilst others are mainly responsible for attacking. Basketball, ice-hockey and football all fall into this category. In such circumstances plus-minus ratings are useful. At its most basic level, a player's plus-minus rating is a comparison of a team's performances with and without the player. Rosenbaum (2004) presented a method for calculating plus-minus player ratings in basketball, before extensions were added by Macdonald (2012) and Kharrat et al. (2020) to account for the intricacies of ice-hockey and football, respectively.

The availability of more granular data, such as event data (detailing each and every event in a game, e.g., the timing, coordinates and players involved in a pass) and player tracking data (the coordinates of all players on the field of play recorded at several times per second), has enabled more advanced measures of player performance to be calculated. One such measure is that of *expected value of possession (EVP)* for valuing individual actions in team sports. The concept of EVP was first presented in Cervone et al. (2016) and asks the question "what is the probability of the objective happening before an action, compared to the probability after an action?". The objective may be to score a goal in football. If an action is a good one – the probability of a goal should increase, whilst if it is a bad one, the probability of a goal will likely decrease. The change in the probability of the
<figure><img src="image_116.png" /></figure>

objective occurring is then the value of the action. Recent applications of deep reinforcement learning have seen EVP calculated for football (e.g., Liu et al., 2020; Fernández et al., 2021). Indeed, it is likely that the EVP concept will be used in many sports in the future.

Akhtar et al. (2015) uses the change in probability of a team winning a Test match to rate cricketers. The idea is similar to the EVP idea proposed by Cervone et al. (2016), but uses multinomial regression to calculate probabilities of a team winning/drawing/losing the match, before and after each ball of the match.

The idea of monitoring the change in an expected value is also used in golf's 'strokes gained' metric (Broadie, 2012). Strokes gained measures how good an individual shot is, and by aggregating over many shots, one can identify how good a player is overall, or how good certain areas (e.g., putting and driving) of a player's game are.

### 3.23.3. Decision making

A core tenet of sports analytics is that it should drive improvement, indeed improving decision making is central to the OR paradigm. There are many papers looking to use analytics to improve decision making in a sports context.

Perhaps the costliest consequences of decision making in sport concern the recruitment of athletes. Indeed, the Moneyball premise is built on the idea of avoiding overpaying for talent.

Football clubs exchange huge sums of money to acquire the services of players. These transfer fees were studied in Coates and Parshakov (2022) who consider the issue of the wisdom of the crowd in estimating the fees. McHale and Holmes (2022) use machine learning techniques to model transfer fees as a function of performance metrics and contract status, amongst other things.

In lucrative team sports such as American football, football and basketball, recruitment of young talent with high potential is of potentially great value, but it appears a relatively little researched area. In one of only a handful of papers on this issue, Craig and Winchester (2021) present a model to predict the potential of college quarterbacks to one day play in the NFL.

In addition to making good decisions around player recruitment, sports teams must make good decisions about their coaches. Peeters et al. (2020) consider the impact of coaches on the performance of Major League Baseball teams, whilst Muehlheusser et al. (2018) rate coaches in German football. Identifying good coaches is just one dimension of decision-making surrounding running a sports team, and it is often the case that team

owners are faced with the decision of whether or not to fire a coach. The impact of managerial dismissals has been the focus of attention in the economics literature. In football, Tena and Forrest (2007) measure the consequences of mid-season managerial dismissals on a team's performance and find that there is a short-term improvement in results, but only in home matches.

The final area of decision making we note is that of team selection. In cricket the ordering of the batting line-up was considered in Perera et al. (2016), whilst Watson et al. (2021) use machine learning to optimise team selection in rugby union. Cao et al. (2022) look at optimising team selection in soccer.

### 3.23.4. Other areas of sports analytics

Sport has attracted the attention of quantitative analysis in numerous other areas, though some do not have the objective of improving performance and/or decision making. For example, OR has been used to inform scheduling of tournaments (see also §3.27).

The popularity of sports betting means forecasting results has received a great deal of attention in the literature. As the sport with the largest global betting market, football has attracted the most attention in the forecasting literature. A notable contribution was that of Dixon and Coles (1997), whose Poisson model has been used as the basis of subsequent work for over two decades. More recently, machine learning techniques have begun to outperform Poisson-type models. See Dubitzky et al. (2019) for details of the results of the 'Soccer Prediction Challenge'.

Tournament design has been the subject of research in, for example, Scarf et al. (2009). The idea is that tournaments should maintain excitement. On a similar theme, Friesl et al. (2017) and Scarf et al. (2019) looked at the rules of ice-hockey and rugby and considered how they might be adjusted to increase excitement. By lowering scoring rates, the outcome of a game is more uncertain, and according to the uncertainty of outcome hypothesis this is what drives interest. However, there is conflicting evidence on the uncertainty of outcome hypothesis (see, for example, Forrest & Simmons, 2002). Understanding what drives the interest of fans was the subject of Buraimo et al. (2020) who looked at how suspense, surprise and shock during a match drives in-match television viewing figures.

To find more articles on sports analytics, the interested reader has several options including specialist journals (the Journal of Quantitative Analysis in Sports, the Journal of Sports Economics, and the Journal of Sports Analytics), and discipline journals such as the European Journal of Operational Research, the Journals of the Royal Statistical Society,
<figure><img src="image_117.png" /></figure>

the *Journal of the Operational Research Society*, and
the *International Journal of Forecasting*, together
with a plethora of blogs and websites all focused on
sports analytics.

### 3.24. Supply chain management<sup>71</sup>

The field of supply chain management (SCM) is concerned with the information, material, and cash flows within and between supply chain members. Materials generally flow down a supply chain (like water in a river); information and money flow up the supply chain. The way we design, source, produce, move, store, schedule, communicate, collaborate, and compete are important factors in SCM.

#### 3.24.1. Lean production

SCM is built on the foundations of good industrial engineering. The pioneering industrial engineers Frank and Lilian Gilbreth provided us with time and motion studies (Gilbreth, 1911), human factors, and scientific management. During the 1920s scientific management techniques were imported into Japan's Imperial Navy's shipyards and factories to improve efficiency and quality. Initially, some table top management games learnt from the Gilbreths in the United States were taken to the Kure Navel Arsenal (a navel shipyard) in 1923 (Robinson & Robinson, 1994). The table top management games were used to demonstrate the efficient flow, and organisation, of work. Robinson and Robinson (1994) claims these table top games facilitated Japan in general, and Toyota in particular, to become highly efficient at producing high quality, low cost, reliable products. The Toyota Production System (TPS) became the world standard in the highly efficient lean production technique (Ohno, 1988). Western companies soon sought to emulate the success of TPS (Womack et al., 1990), hunting high and low for the seven lean wastes (Hines & Rich, 1997). Holweg (2007) provides an excellent summary of the genealogy of lean production.

#### 3.24.2. Value stream mapping

One of the best ways to document and understand a
supply chain is to draw a *value stream map* (VSM;
Rother & Shook, 1999). VSMs detail how the mater-
ial flow is controlled by the information flow and
decision-making activities. Key is to determine the
point in the material flow where the customer order
directly regulates the production cadence. This point
is known as the *pacemaker process* or the *customer
order decoupling point* (CODP; Olhager, 2010).
The pacemaker is often the process that separates
the work that is pulled through the system by a

Kanban system, and the work that flows out to the
customer in a *first-in-first-out* (FIFO) queue.

#### 3.24.3. Agile and leagile supply chains

Lean supply chains are characterised by *just-in-time* inventories and *high capacity utilisation*. But not all supply chains should be lean. Some supply chains need to be responsive, with extra inventory and spare capacity held in reserve so the system can quickly respond to unexpected demand (Fisher, 1997). This has become known as *agile production*. The lean and agile paradigms can be integrated in together in a concept known as *leagility* (Naylor et al., 1999). In leagile supply chains, the material flow is set up to follow lean principles upstream from the CODP; downstream from the CODP, agile principles are followed.

### 3.24.4. Bullwhip and supply chain dynamics

The *bullwhip effect* is one of the biggest areas of SCM research. The moniker, coined by Lee et al. (1997), refers to the tendency of the slowly changing consumer demand (the bullwhip handle) to create wildly fluctuating fast moving demand at the raw material processors (the bullwhip popper). This *variance amplification effect* is caused by the decision-making activities (Forrester, 1958). The seminal paper by Lee et al. (1997) highlights four causes of the bullwhip effect: *demand signal processing, order batching, shortage gaming, and price fluctuations*.

Demand signal processing has been the most studied cause of the bullwhip effect. Demand signal processing refers to the activity of *forecasting* the demand over the lead time (and review period), so that one may determine production and/or replenishment order quantities to maintain *finished goods inventory* and *raw material* levels close to a target. Setting target inventory levels is a problem similar to the *newsvendor problem* (Churchman et al., 1957). As orders eventually turn into the inventory here is a *feedback loop* in the decision; there is also a *work-in-progress* feedback loop in the system (Sterman, 2000). Both these feedback loops contain delays. This creates a complex system whose dynamics are in part driven by the external demand, but are mostly an internally generated effect caused by the fundamental structure of the supply chain (Sterman, 2000).

Control engineers have developed a large toolkit to understand and manipulate the dynamics of feedback systems. Towill (1982) and John et al. (1994) studied the dynamics of continuous time replenishment rules with the *Laplace transform*. Dejonckheere et al. (2003) studied discrete time replenishment rules via the *z-transform* and the *Fourier transform*. They showed the *order-up-to* replenishment policy with *moving average* and
<figure><img src="image_118.png" /></figure>

exponential smoothing forecasts, for all lead-times and all possible demand patterns, always created bullwhip.

Michna et al. (2020) studied stochastic lead times, revealing the forecasting of lead times is an important cause of the bullwhip effect. Gaalman et al. (2022) explores the interaction between the lead time and bullwhip under general order auto-regressive moving average demand. They reveal the interaction between demand, lead times, and bullwhip is complex and subtle; bullwhip does not always increase in the lead time. Wang and Disney (2016) provides a recent review of the bullwhip effect, its causes, solution approaches, and thoughts on future research directions.

### 3.24.5. Location and localisation

The number, and location, of distribution centres (DC) is an important problem in distribution network design. Too few DCs result in longer travel distances (and times) to customers; too many result in high amounts of distribution inventory. The square root law for inventory (Maister, 1976) shows the amount of inventory in a distribution network falls by $1/\sqrt{n}$ when n DC's are consolidated into a single DC. The transportation costs involved in delivering customer demand from n DCs can be accurately modelled using transportation planning software (Hammant et al., 1999). This software typically includes: road maps, speed limits, tolls, congestion, as well as various methods for modelling transport costs.

Postponement can also reduce inventory in supply chains. Postponement involves delaying final assembly until demand reveals itself; products are then quickly customised to meet the consumer's desires. For example, HP build generic printers in Mexico to ship to Europe. Upon arrival, they are assigned to a country and the correct power pack is “assembled” into the product (Feitzinger & Lee, 1997). With postponement HP holds less generic inventory to buffer the shipping lead time compared to the amount of country specific inventory it would need if the power packs were assembled in Mexico.

Another important SCM decision is where to produce? Should you produce locally where perhaps labour cost is high, or should you outsource, or offshore, to a low labour cost country? Sometimes, offshore production is supplemented with a local factory or a near-shored supplier in a dual sourcing arrangement (Allon & Van Mieghem, 2010). A tailored base surge policy sends constant orders to the offshore supplier with the long lead time, while the near-shore supplier flexes production quantities with a short lead time. A small local SpeedFactory may be able to correct for the forecast errors and gain

enough of an inventory advantage to offset the increased local labour costs (Boute et al., 2022).

### 3.24.6. Information flows in supply chains

Changing the information used in replenishment decisions can improve the dynamics of supply chains. The sharing of demand information with upstream suppliers is often referred to as the information sharing (Lee et al., 2000), or information enrichment strategy (Dejonckheere et al., 2004). Knowing the end consumer demand allows upstream members to base their demand forecasts on the real demand information, removing one of the potential causes of the bullwhip effect. Indeed, information sharing allows for a linear, rather than a geometrically, increasing bullwhip effect as orders go echelon-to-echelon up the supply chain (Chen et al., 2000). Kaipia et al. (2017) considers the practicalities of implementing the information sharing strategy.

Sharing both demand and inventory information with your supplier can enable the vendor managed inventory (VMI) strategy (Dong & Xu, 2002). In the VMI strategy, the consumer demand and downstream inventory information is used by the supplier (the vendor) to make replenishment decisions on behalf of his customer. This allows two supply chain echelons to behave dynamically as one echelon, removing a bullwhip generating decision from the supply chain (Holweg et al., 2005).

### 3.24.7. Coordinating supply chain contracts

Supply chains often consist of many different organisations, each operating to maximise their own profit. Due to the double marginalisation problem, if each player acts solely in their own interests, the supply chain will not be able to reach the first best solution; money will be left on the table. Sometimes, the first best solution can be reached by a centralised decision-maker coordinating the supply chain; at other times the altruistic behaviour of one supply chain member, in return for a transfer payment, can coordinate.

There are many different types of contracts (Cachon & Lariviere, 2005): revenue sharing, buy-back, price-discount, quantity-flexibility, sales-rebate, franchise, and quantity discount contracts to name just a few. All have their strengths and weakness and are applicable in different settings. Many contracts are based on newsvendor principles (Lariviere, 2016). Another important concept in contract design is the idea of Pareto improving contracts, where no player is worse off than the (locally optimised) base case, but at least one other player is better off. Other contracts allow for the arbitrary allocation of profits between players, and for the
<figure><img src="image_119.png" /></figure>

delegation of decision-making activities to others (Chintapalli et al., 2017).

### 3.24.8. Emerging topics in the field of supply chain management

Emerging topics in the SCM field include:

* The distributed ledger technology behind *block chains* (Babich & Hilary, 2020) and *cryptocurrencies* (Choi, 2020) can be used to create a permanent record of provenance and ownership. Ensuring your cotton has not been produced by slaves, your diamonds are not conflict, and children did not mine your Lithium is vital now as UK Directors can face prison time under the Proceeds of Crime Act for crimes committed in their supply chains.

* *Opaque pricing* is a technique used to sell last minute travel industry inventory (e.g., hotel rooms) at discounted prices. The traveller books a room without knowing the exact hotel brand. Cost sensitive travellers are happy because they get a bargain. The hotel is happy because they get extra income without damaging their brand. Opaque pricing can be used for products as well; for example, a red pen sells for $10, and a blue pen sells for $10, but if you don't care which colour you have, a red-or-blue pen is offered at the opaque price of $8. The vendor is able to use the customer's lack of preference to reduce inventory requirements (Ren & Huang, 2022).

* *Quantum computing* allows one to solve *NP*-hard problems (such as the *travelling salesman problem*) to optimality instantaneously, rather than waiting for months with regular computers (Srinivasan et al., 2018). This technology has the potential to make supply chains more efficient.

### 3.25. Sustainability⁷²

In this subsection, we focus on the area of sustainable operations from the perspective of *closed-loop supply chains* (CLSC). We consider literature that focuses on product-, module/part-, and material-level recovery and reuse activities. These activities provide economic and environmental benefits. CLSC entail transportation and acquisition of used products; sorting, grading and disposition for different recovery methods; disassembly and reassembly (i.e., remanufacturing operations); and marketing of remanufactured products. Guide and Van Wassenhove (2003) and Ferguson and Souza (2010) provide comprehensive overviews of the strategic, tactical, and operational aspects of CLSC.

The supply side in CLSC differs from traditional supply chains in the following ways. The quantity of used products being returned is uncertain; the timing of when they are returned is uncertain; and the condition (quality) in which they are returned is also uncertain. These differences lead to uncertain recovery rates and processing times, uncertain cost of recovery, and imperfect matching between supply of used products and demand for remanufactured products, and hence the subsequent demand for new parts needed to make the remanufactured (finished) product. Below, we provide a brief overview of some of the methods used to optimise the different activities in CLSC, while managing these uncertainties.

The *reverse logistics (RL) network* (see also §3.14) handles the collection of used products from end-users, and their transportation between collection points, consolidation centres, testing, sorting, and grading facilities, and recovery (e.g., remanufacturing, reuse, recycling) facilities and landfill locations. Stylised and game-theoretic models are developed to determine the optimal collection strategy for producers (if they choose to, or are required to collect used products). The collection strategy includes decisions on whether producers should collect directly from end-users, or use the retail network as collection points, or use third-party collectors (e.g., Savaskan & Van Wassenhove, 2006). In further analysis of the collection strategy, the continuous approximation method is used to determine whether the producer (or business) should offer to pick-up, or have end-users drop-off the used product (e.g., Fleischmann, 2003).

Several quantitative models are developed to determine the optimal RL network design. An extensive discussion of these models and solution approaches can be found in Akçalı et al. (2009). Linear programming, mixed-integer linear programming (MILP), and stochastic programming are widely used to determine optimal network structures. Fleischmann et al. (2004) provide and excellent overview of MILP and stochastic programming models for facility location and network design for dedicated reverse, and integrated (forward and reverse) logistics networks. Mixed-integer nonlinear programming models are also sometimes used to determine the optimal RL network structure (e.g., de Figueiredo & Mayerle, 2008). In addition to optimal network design, vehicle routing models (§3.32) are used to determine optimal collection and pick-up routes. These vehicle routing problems are often *NP*-hard, and are based on location of demand: node, arc, and general. The models are extended to include vehicle routing with backhaul, routing with simultaneous delivery and pick-up, and routing with
<figure><img src="image_120.png" /></figure>

partially mixed deliveries and pick-ups (see Beullens et al., 2004, and references therein).

One way of managing the supply uncertainty in CLSC is to forecast the return of used products. Different methods are used to compute the product return probability. These include modelling returns as a function of past sales (via a known delay distribution), regression models (Samorani et al., 2019), simulation models, and queueing models (e.g., Toktay et al., 2000).

Buyers of used products (i.e., producers or their contract-remanufacturers, and third-party remanufacturers) actively manage the supply uncertainty (timing, quantity, and quality) by using incentive mechanisms such as quality-based pricing, trade-ins, and buybacks. Buyers acquire used products either in sorted (i.e., known quality-levels) or unsorted (i.e., unknown quality-levels) form. Lot-sizing models are developed to determine the optimal acquisition quantity when the used products are available in unsorted form (e.g., Galbreth & Blackburn, 2006); and when they are available in sorted (continuous or discrete quality-levels) form (e.g., Mutha et al., 2016). The acquisition process has also been analysed in the context of buyer-supplier contracts. The objective of these analyses is to determine the optimal contract structure with known or unknown quality levels, e.g., quality-dependant acquisition costs and quantities (Mutha et al., 2019), and coordination mechanisms (Debo et al., 2004; Vedantam & Iyer, 2021; Li et al., 2023). Several models are also developed to determine the optimal acquisition cost for used products and selling price for remanufactured products for an exogenous set of discrete quality-levels of used products (e.g., Guide et al., 2003).

Testing, sorting, and grading of the acquired used products are important activities in product recovery operations. Hahler and Fleischmann (2017) provide a detailed description of these operations for used consumer electronics. Sorting defective, and economically and technically infeasible-to-remanufacture units from the acquired quantity streamlines the subsequent operations (e.g., transportation, disassembly, and reassembly). Knowing the quality of the incoming units before scheduling recovery operations significantly improves the performance of the system. The benefit of yield information (i.e., information on the quality distribution of the incoming units) has been analysed using lot-sizing models and simulation models (e.g., Ketzenberg et al., 2003). Several models are developed to optimise the different decisions in the grading process, e.g., the optimal number of grades (e.g., Ferguson et al., 2009), the resulting optimal grade-wise remanufacturing cost and selling price (e.g., Mutha & Bansal, 2023),

and the optimal location and timing of the sorting and grading process, for example at the point of collection/return or at the disassembly stage (e.g., Guide et al., 2006; Zikopoulos & Tagaras, 2008).

The disposition of sorted and graded used products typically involve a problem of optimal assignment of the economically and technically recoverable units to different recovery options, e.g., product-level recovery (i.e., remanufacturing); module/part-level recovery (i.e., reuse for making remanufactured and new products, or for spares); and the non-recoverable products for material-level recovery (i.e., recycling). The assignment decisions are usually based on considerations of supply (yield information, processing times, and costs) and demand (revenue, opportunity cost, and inventory cost). Optimal control models (e.g., Inderfurth et al., 2001) and revenue management-based models (e.g., Pinçe et al., 2016; Calmon & Graves, 2017; Calmon et al., 2021) are widely used to determine optimal disposition decisions. Depending on the type of the product, single-period models (for products with short lifecycles, e.g., cellphones) and multiperiod models (for products with long lifecycles, e.g., engines) are used in the disposition analyses. For example, Özdemir-Akyıldırım et al. (2014) formulate the optimisation problem as a multiperiod Markov decision process (MDP) and provide a linear-programming model for solving the deterministic approximation of the MDP model.

Within the production planning and control literature in CLSC, a relatively small part has focused on disassembly planning and sequencing, and material requirement planning (MRP). Inderfurth et al. (2004) provide an extensive overview of the various optimisation models developed to optimise these elements, including shop floor control rules, in remanufacturing-only and hybrid (joint manufacturing and remanufacturing) systems. Disassembly sequencing is mainly analysed using direct graphs (see Lambert, 2003, and §2.12), and MRP decisions are analysed from an inventory control perspective (e.g., Inderfurth & Jensen, 1999; Ferrer & Whybark, 2001). A significant part of the literature on CLSC has focused on inventory management. Optimal inventory control policies are derived using periodic-review models (e.g., Teunter et al., 2004; Zhou et al., 2011) and continuous-review models (e.g., van der Laan et al., 1999; Toktay et al., 2000; Jia et al., 2016). The single-period newsvendor-like models are largely analysed as acquisition lot-sizing models (discussed in the preceding paragraphs).

The research on market (selling)-related aspects of CLSC is focused around understanding the profit and pricing implications due to the co-existence of new and remanufactured products in the (same)