<figure><img src="image_46.png" /></figure>

nonlinear functions. Mathematically, a nonlinear programming problem is represented as

$$ (P) \quad \min\{f(x) : x \in S\}. $$

Here, $S = \{x \in \mathbb{R}^n : g_i(x) \le 0, \ i = 1, \dots, m\}$ denotes the feasible region, where, $g_i : \mathbb{R}^n \to \mathbb{R}$, $i = 1, \dots, m$, and $f : \mathbb{R}^n \to \mathbb{R}$ denotes the objective function.

In comparison with linear programming, nonlinear programming problems have much more expres- sive power. As such, nonlinear programming problems naturally arise in almost every setting, ranging from investment planning to machine learn- ing; from engineering to medicine; and from energy to sustainability (§3.14; §3.5; §3.19; §3.9; §3.11; §3.13).

In this subsection, we will give a brief overview of theory and algorithms. While we will not cover the modelling aspect, we will mention some classes of optimisation problems with desirable properties, which should imply that using an optimisation model from such classes would significantly increase the likelihood of solving it.

The difficulty of the generic optimization prob- lem (P) is largely determined by the properties of the objective function $f : \mathbb{R}^n \to \mathbb{R}$ and of the func- tions $g_i$, $i = 1, \dots, m$ that define the feasible region $S \subseteq \mathbb{R}^n$. Generally speaking, increasingly more restrictive assumptions on $f$ and on $g_i$, $i = 1, \dots, m$ give rise to increasingly more structured optimisa- tion problems with stronger and more desirable properties. For instance, the special case in which each of $f$ and $g_i$, $i = 1, \dots, m$ is a linear function, referred to as linear programming (§2.14), is argu- ably the most structured class of optimization prob- lems with very appealing theoretical properties, which lay the groundwork for several effective solu- tion methods such as the simplex method (see, e.g., Dantzig, 1990) and interior-point methods (Karmarkar, 1984; Wright, 1997; Ye, 1997). In con- trast, general nonlinear programming problems usu- ally enjoy fewer desirable properties.

The class of convex optimisation problems is com- prised of optimisation problems in which each of $f : \mathbb{R}^n \to \mathbb{R}$ and $g_i$, $i = 1, \dots, m$ is a convex function, which implies that $S \subseteq \mathbb{R}^n$ is a convex set, and includes linear programming as a special case. Any optimisation problem that does not belong to this class is a nonconvex optimisation problem. On the other hand, (P) is called an unconstrained optimisa- tion problem if $S = \mathbb{R}^n$, and a constrained optimisa- tion problem otherwise.

A useful notion in nonlinear programming is that of local optimality. A point $\hat{x} \in \mathbb{R}^n$ is said to be a local minimiser of (P) if there exists an open ball $B \subset \mathbb{R}^n$ of positive radius centred at $\hat{x}$ such that

is a minimiser of $f$ over the potentially smaller feasible region $B \cap S$. In contrast, $\hat{x}$ is a global minimiser of (P) if $\hat{x}$ is a minimiser of $f$ over the entire feasible region $S$. Note that a global minimiser is also a local minimiser.

We next briefly give an overview of optimality conditions for each aforementioned class of optimisation problems. We start with unconstrained optimisation problems in the one-dimensional setting (i.e., $n=1$). If $\hat{x} \in \mathbb{R}$ is a local minimiser of (P), then $f$ should be neither decreasing nor increasing at $\hat{x}$. Assuming that $f$ is a continuously differentiable function, we therefore obtain $f'(\hat{x}) = 0$. This geometric interpretation carries over to the higher-dimensional setting (i.e., $n \ge 2$) by simply viewing a mutivariate function as a collection of one-dimensional functions along feasible directions at each $\hat{x} \in \mathbb{R}^n$, i.e., directions along which one can move starting from $\hat{x} \in \mathbb{R}^n$ and still remain in the feasible region. In the unconstrained case, every direction $d \in \mathbb{R}^n$ is a feasible direction at every $x \in \mathbb{R}^n$. Using the result from the one-dimensional case, if $\hat{x} \in \mathbb{R}^n$ is a local minimiser of (P), then the partial derivatives of $f$ with respect to each variable should be zero, or equivalently, that $\nabla f(\hat{x}) = 0 \in \mathbb{R}^n$, where $\nabla f : \mathbb{R}^n \to \mathbb{R}^n$ is the gradient of $f$. Such a point is called a stationary point.

For the special case of convex unconstrained optimisation problems, the convexity of the objective function $f : \mathbb{R}^n \to \mathbb{R}$ implies that the aforementioned necessary conditions are also sufficient, i.e., a point is a local minimiser if and only if it is a stationary point. Furthermore, for convex functions, every local minimiser is, in fact, a global minimiser. Therefore, we obtain the equivalence between global minimisers and stationary points. On the other hand, for a nonconvex optimisation problem, there may be stationary points that may not correspond to a local minimiser of (P) (e.g., if $f(x) = x^3$, then $\hat{x} = 0$ is a stationary point but not a local minimiser). As illustrated by this example, the complete characterisation of global optimality does not carry over from convex optimisation to nonconvex optimisation, even in the unconstrained setting.

For constrained optimisation problems, we first consider the convex optimisation case. By the convexity of the feasible region $S \subseteq \mathbb{R}^n$, for any $\hat{x} \in S$, the set of all feasible directions is given by $\tilde{x} - \hat{x} \in \mathbb{R}^n$, where $\tilde{x} \in S$. Arguing similarly to the unconstrained case and using the convexity of $f$, a point $\hat{x} \in S$ is a global minimiser of (P) if and only if $f$ does not decrease along any feasible direction, i.e., if and only if $\nabla f(\hat{x})^T(\tilde{x} - \hat{x}) \ge 0$ for all $\tilde{x} \in S$. Therefore, as in the unconstrained case, we once again have the equivalence between local and global minimisers.
<figure><img src="image_47.png" /></figure>

Next, consider a nonconvex constrained optimisation problem. If the feasible region $S \subseteq \mathbb{R}^n$ is a convex set but $f$ is a nonconvex function, a similar argument as in the convex case gives rise to the following necessary condition: If $\hat{x} \in S$ is a local minimiser of (P), then $\nabla f(\hat{x})^T(\tilde{x} - \hat{x}) \ge 0$ for all $\tilde{x} \in S$. As in the unconstrained case, simple examples show that this condition is no longer sufficient for local optimality. If, on the other hand, $S \subseteq \mathbb{R}^n$ is a non-convex set, then we instead rely on a more general notion of tangent directions to the feasible region $S$ at $\hat{x}$. Therefore, if $\hat{x} \in S$ is a local minimiser, then $\nabla f(\hat{x})^T d \ge 0$ for every tangent direction $d \in \mathbb{R}^n$ to the feasible region $S$ at $\hat{x}$. In general, the set of such tangent directions may not be easy to characterise. Under certain additional assumptions about the geometry of the feasible region $S \subseteq \mathbb{R}^n$, referred to as constraint qualifications (Bazaraa et al., 2005, Chapter 5), explicit necessary optimality conditions can be derived.

Having reviewed optimality conditions, we finally give a brief overview of methods for solving optimisation problems. Nonlinear optimisation algorithms are generally iterative in nature, i.e., they generate a sequence of points $x_k \in \mathbb{R}^n$, $k = 1, 2, ...$ that satisfies certain properties. For instance, the sequence may either converge to a local or global minimiser of an optimisation problem, or may simply have a limit point that satisfies the necessary conditions for local optimality. As illustrated by the discussion on optimality conditions, one can establish considerably weaker properties for nonconvex optimisation problems in comparison with convex optimisation problems. In fact, most classes of non-convex optimisation problems are provably difficult in a formal complexity sense, even when restricted to minimising a quadratic function over a polyhedron (Murty & Kabadi, 1987; Pardalos & Vavasis, 1991). As such, it would not be reasonable to expect an algorithm to solve every optimisation problem to global optimality in a reasonable amount of time.

Therefore, different performance metrics are employed for assessing algorithms for different classes of optimisation problems. While, for convex optimisation problems, one usually expects a “good” algorithm to compute a global optimal solution, an algorithm for a nonconvex optimisation problem could be deemed “effective” if it always converges to a local (rather than a global) optimal solution.

In the unconstrained case, given an iterate $x_k \in \mathbb{R}^n$, the main idea is to identify a feasible direction $d \in \mathbb{R}^n$ along which the objective function will decrease. Such a direction $d \in \mathbb{R}^n$, called a descent direction, would necessarily satisfy $\nabla f(x_k)^T d < 0$. Then, a step size in this direction is determined according to certain criteria that would guarantee a

decrease in the objective function. Therefore, this family if algorithms is referred to as *gradient descent methods* and includes *steepest descent* as a special case (i.e., the case where $d = -\nabla f(x_k)$). Under mild assumptions, this class of algorithms converges to a stationary point of $f$. Recall that such a point is a global minimiser if $f$ is a convex function. Other methods in this class are Newton methods, conjugate gradient methods, and quasi-Newton methods, each of which generates iterates that converge to a stationary point under appropriate assumptions.

Considering the constrained case, while general convex optimisation problems do not retain all desirable properties of the simpler class of linear programming problems, they still have a sufficiently rich structure that pave the way for provably efficient solution algorithms. In fact, every convex optimisation problem, in theory, can be solved to global optimality by the ellipsoid method (Yudin & Nemirovskii, 1976; Shor, 1977) or by interior-point methods in polynomial time (Nesterov & Nemirovskii, 1994). Furthermore, a variety of highly effective commercial and non-commercial solvers are available for solving several classes of convex optimisation problems such as linear programming, second-order cone programming, and semidefinite programming that frequently arises in applications (see, e.g., https://neos-server.org/neos/solvers/index.html).

For the nonconvex constrained case, one approach is based on approximating a constrained optimisation problem by a sequence of unconstrained optimisation problems by either using a penalty function, based on penalising violation of constraints (*penalty methods*), or using a barrier function, based on preventing the violation of constraints by keeping the iterates strictly in the relative interior of the feasible region $S \subseteq \mathbb{R}^n$ (*barrier methods*). Other methods include Augmented Lagrangian methods, based on combining Lagrangian relaxation with penalty methods, and Sequential Quadratic Programming methods, based on approximating the optimisation problem by a quadratic programming problem.

Finally, various real-life applications in machine learning and data science give rise to very large-scale problems that are beyond the capability of current solvers and computing platforms. For such problems, there exist a variety of heuristic optimisation methods that can be employed to find a good solution in a reasonable amount of time (§2.13). However, in contrast with exact methods, such methods usually do not provide any guarantees on the quality of the solution.

Nonlinear optimisation is a very active area of research. The reader is referred to excellent
<figure><img src="image_48.png" /></figure>

textbooks for further information (e.g., Fiacco &
McCormick, 1968; Mangasarian, 1994; Bazaraa
et al., 2005; Nocedal & Wright, 2006; Bertsekas,
2016; Luenberger & Ye, 2016).

## 2.17. Queueing²⁷

Queueing systems arise in many real life applica-
tions including production, service systems, finance,
logistics and transportation. As mentioned in
Stidham (2002), many queueing models have been
studied even before the introduction of Operational
Research in the 1950s. We provide a brief overview
of methodologies used in queueing systems analysis.
We start with exact methods and then continue
with approximations and asymptotic analysis.

The classical analysis of queueing systems
involved modelling the single stage Markovian
queues as birth-death processes and computing their
steady state performances using Markov chain the-
ory. These earlier theoretical contributions were ini-
tially summarised in Feller's two volume books
Feller (1957a, 1957b), then in classical textbooks
such as Cooper (1972), Gross and Harris (1974) and
Kleinrock's two volumes Kleinrock (1975a, 1975b),
and more recently in Gautam (2012), Harchol-Balter
(2013), and many other books. Takács (1962)
focused on using transforms and generating func-
tions for steady state and transient behaviour of
queueing systems. In the early days, transforms and
generating functions were considered to be exact
expressions but one had to invert these generating
functions in order to obtain the actual performance
measures which is in general difficult. Marcel Neuts
was the first one who approached this inversion
problem algorithmically. In his 1981 book, Neuts
(1981) focused on queues that generalise the G/M/1
structure, whereas in his second book, Neuts (1989)
generalised the structure of the M/G/1 queue. The
main idea in these books is to approximate the non-
exponential distributions with a phase type distribu-
tion (convolution and mixture of exponentials)
which yields a continuous time Markov chain model
for the original system that could be analysed, at
least numerically. This line of research resulted in
many contributions on the so-called *matrix-geomet-
ric* methods (see also Latouche & Ramaswami,
1999). Arguably the most well known result in
queueing theory is Little's law ($L = \lambda W$ or its gener-
alisation $H = \lambda G$) which provides a relationship
between the mean steady state number of customers
and the mean sojourn time in a system. For a thor-
ough survey of the Little's result and its extensions,
the reader is referred to Whitt (1991). There are
numerous proofs of Little's law but El-Taha and
Stidham (1999) provide an elegant sample path

proof. On the other hand, Bertsimas and Nakazato
(1995) relate the steady-state distribution of the
number in the system (or in the queue) to the
steady state distribution of the time spent in the sys-
tem (or in the queue) in a queueing system under
FIFO (First In First Out).

While there has been a lot of interest in station-
ary queues, Massey's, 1981 dissertation Massey
(1981) drew attention to the analysis of non-station-
ary queues (i.e., queues with time dependent arrival
and service processes). Massey's dissertation started
with the analysis of M(t)/M(t)/1 queue and then
extended to other non-stationary Markovian sys-
tems. Many subsequent papers, such as Massey and
Whitt (1998), focused on queueing models with
time-dependent arrival rates, especially infinite-ser-
ver “offered-load” models which describe the load
that would be on the system if there were no limit
to the available resources. The main idea of these
papers is to provide algorithms (approximations) to
solve the Poisson equation. On the other hand,
Bertsimas and Mourtzinou (1997) derived a set of
transient distributional laws that relate the number
of customers in the system (queue) at time t to the
system (waiting) time of a customer that arrived to
the system (queue) at time t.

Networks of queues have been of interest to
researchers since 1950s. Jackson (1957) was the first
one to observe that joint steady state distribution of
the number of customers at the nodes of a network
of Markovian queues with single server (at each
node) is the product of individual distribution of
M/M/1 queues. Jackson (1963) generalised this
result to networks of queues with multiple servers at
the nodes. Gordon and Newell (1967) discovered
that the stationary distribution again has a product
form in closed Markovian networks but in this case
a normalisation constant is required. Baskett et al.
(1975) proved that the product form is insensitive
to the service time distribution if the service discip-
line satisfies certain assumptions. This and other
insensitivity results in networks were also consid-
ered by Kelly (1979) and Serfozo (1999) which also
has results on other networks such as those with
blocking and rerouting. Daduna (2001) focused on
obtaining explicit expressions for the steady behav-
iour of discrete time queueing networks and gave a
moderately positive answer to the question of
whether there can be a product form calculus in dis-
crete time. In recent years, a number of models
involving different compatibilities between jobs and
servers in queueing systems, or between agents and
resources in matching systems, have been studied,
and, under Markovian assumptions and appropriate
stability conditions, the stationary distributions were
again shown to have product forms (see Gardner &
<figure><img src="image_49.png" /></figure>

Righter, 2020, and the references therein). Baccelli et al. (1992) modelled a class of networks using the so-called (max,+} linear systems. In their pioneering work, using (max,+} algebra techniques, Baccelli and Schmidt (1996) derived Taylor series expansions for the mean waiting times in Poisson driven queueing networks that belong to the class of (max,+} linear systems. Even though these expansions are sometimes referred to as light traffic approximations, in some cases all coefficients of the series expansion can be computed yielding an exact expression. These results were generalised to transient performance measures by Baccelli et al. (1997) and joint characteristics by Ayhan and Baccelli (2001).

Exact analysis of general queueing systems is often challenging, making the characterisation of performance measures difficult. Thus, asymptotic analyses are commonly carried out via various approximation methods. We next provide an overview of such methods.

Many of the earlier works on the asymptotic ana-
lysis of queueing systems focused on heavy traffic
and many server approximations for single stage
queues. In his pioneering work, Kingman (1961,
1962, 1965) have asymptotically characterised the
waiting time distribution for the single server queue
with general interarrival and service time distribu-
tions under heavy traffic conditions (i.e., when the
traffic intensity ρ → 1). Several others have devel-
oped heavy traffic approximations for the G/G/s
queue, where a sequence of systems with fixed num-
ber of servers and traffic intensities {ρ_n} approach-
ing one are considered (see, for example,
Köllerström, 1974). In these approximations, the
sequence of normalised (i.e., scaled) queue length
processes converge to a reflected Brownian motion
with negative drift (see Whitt, 2002), and the associ-
ated sequence of scaled stationary queue-length dis-
tributions (i.e., the stationary distribution of the
limiting diffusion process) converges to an exponen-
tial distribution. We refer the reader to Harrison
(1985) for a detailed technical treatment of heavy
traffic limits and diffusion approximations.
Asymptotic analysis was also considered for multi-
class and multi-stage queueing networks. Defining
the stability region of these networks using fluid
limit analysis was considered in Chen (1995); Dai
(1995); Dai and Meyn (1995). Many of the works
that considered heavy traffic analysis of multi-class
queueing networks focus on achieving the so-called
state space collapse. Bramson (1998) demonstrated
the state space collapse for first-in first-out queueing
networks of Kelly type and head-of-the-line propor-
tional processor sharing queueing networks. His
framework has been used to prove state space

collapse results in several other works including
Stolyar (2004) and Mandelbaum and Stolyar (2004).
For a more comprehensive review of heavy traffic
analysis of multi-class queueing networks, we refer
the reader to Chen and Yao (2001).

Many-server approximations were also considered for asymptotic analyses of queueing systems. In these approximations, the traffic intensity can be kept constant while letting the arrival rate and the number of servers go to infinity. Iglehart (1965) showed that the resulting sequence of normalised queue length processes converges to an Ornstein-Uhlenbeck process in the many server setting when the service time distributions are exponential. Later on, Whitt (1982) generalised this result for systems with non-exponential service times. For a more comprehensive overview of results in this area, see Whitt (2002). In their seminal work, Halfin and Whitt (1981) defined the so called Halfin-Whitt regime for the GI/M/s queue where the traffic intensities converge to one from below, the number of servers and arrival rates tend to infinity, but steady-state probability that all servers are busy remains fixed. They showed that under the appropriate scaling, the queue length processes converge to a diffusion process. In the past decades, many other asymptotic results have been obtained for many server queues in the Halfin-Whitt regime. Reed (2009) studied the G/GI/s queue and obtained fluid and diffusion limit results for the queue length process. We refer the reader to van Leeuwaarden et al. (2019) for a further review of the various asymptotic results obtained in the Halfin-Whitt regime.

Although heavy traffic approximations for queues
have been popular in recent decades, light traffic (as
the traffic intensity ρ → 0) and interpolation approx-
imations have also been developed. Bloomfield and
Cox (1972) developed light traffic approximations for
a single server queue. Burman and Smith (1983)
developed approximations for the expected delay in
M/G/s queue both for heavy and light traffic, and
showed that as traffic intensity goes to zero, probabil-
ity of delay depends only on mean service time dis-
tributions. Daley and Rolski (1992) used light traffic
approximations to study the limiting properties of
the waiting time in many-server queues. Light traffic
approximations have also been used to study the lim-
iting processes in queueing networks (see, for
example, Simon, 1992).

As mentioned earlier, approximation methods were
commonly used in the asymptotic analysis of time-
varying (i.e., non-stationary) queues. Mandelbaum
et al. (1999) developed a fluid approximation for the
queue length process in time-varying multiserver queue
with abandonments and retrials. Pang and Whitt
<figure><img src="image_50.png" /></figure>

(2010) have developed heavy traffic approximations for
infinite server queues with time-varying arrivals. The
reader is referred to Whitt (2018) for a recent review
of the literature on non-stationary queues.

Due to the interest in communication/telecommu-
nication systems, in late 1990s and early 2000s, there
was a lot of research on queues with heavy tailed
interarrival and/or service times. Intuitively, heavy
tailed distributions decay slower than an exponential
distribution (see Resnick, 2007, for a thorough discus-
sion). Boxma and Cohen (2000) provided an overview
of results for single service queues with heavy tailed
interarrival and/or service time distributions. Baccelli
et al. (1999) and Ayhan et al. (2004) showed that the
asymptotics of response time was dominated by the
station with the heavy tailed service time in a class of
open and closed networks, respectively. Foss and
Korshunov (2012) developed upper and lower bounds
on the tail distribution of the stationary waiting time
in the GI/GI/s queue with heavy tailed service times.

**2.18. Risk analysis<sup>28</sup>**

Risk analysis is a discipline that seeks to inform
people about what might happen and how to reduce
the probability and severity of undesired outcomes.
It draws on decision analysis, game theory, and
other areas of Operational Research but is distin-
guished from them by the questions it asks, the
frameworks it provides for answering them, and the
uses to which its answers are put (Aven, 2020;
Greenberg et al., 2020). Where decision analysis
focuses on principles for identifying logically coher-
ent choices that make preferred outcomes more
likely based on a decision maker's beliefs and value
trade-offs, risk analysis seeks to inform *analytic-
deliberative decision-making* by multiple stakehold-
ers—possibly with conflicting worldviews, values,
and beliefs—for managing critically important mat-
ters ranging from the safe operation of nuclear
power plants to priority-setting for public and occu-
pational health and safety measures.

Risk analysis is often subdivided into risk perception, risk assessment, risk communication, risk management, and risk governance and policy-making (Greenberg & Cox, 2021). The following sections describe these components.

**2.18.1. Risk perception**

Public concerns and political appetite to address
them are shaped by *perceived* risks, whether or not
they are accurate. Several frameworks have been
developed to help understand the technical, psycho-
logical, and social drivers of risk perceptions
(Siegrist & Árvai, 2020). The *psychometric paradigm*
(Slovic, 2000) explains many aspects of risk

perceptions in terms of a few underlying factors
such as as *dread risk* (associated with a lack of con-
trol, dreaded consequences, catastrophic potential,
inequity in the distribution of risks, risks increasing
over time, and fatal consequences) and *unknown
risk* (associated with unobservability, novelty,
unknown exposure, being unknown to science, and
delayed consequences). The cognitive *heuristics and
biases* literature positions risk perceptions within a
“dual process” framework in which rapid emotional
evaluations (“System 1”) can be modified by slower,
more effortful cognition (“System 2”) (Kahneman,
2011; Skagerlund et al., 2020). The *cultural theory of
risk* (Douglas & Wildavsky, 1983; McEvoy et al.,
2017; Bi et al., 2021) posits that individual percep-
tions of risk are shaped by social and ideological
processes that emphasise or suppress perceptions of
risks depending on the respondent’s values and pre-
ferred form of social order. The *social amplification
of risk framework* (SARF) (Kasperson et al., 2022)
describes the social amplification or attenuation of
perceived risks as risk information is communicated
among people with different worldviews.

Major lessons from the study of risk perception are
that experts and members of the public often view
risks quite differently; that experts often focus on the
probability and consequence severity dimensions of
risk while members of the public consider many other
aspects; that most people tend to overestimate the fre-
quencies of rare but vivid events (e.g., terrorist attacks,
murders) and underestimate the frequencies of com-
mon but familiar ones (e.g., car accidents, heart attack
fatalities); and that risk perceptions of both experts
and lay people are predictably shaped and distorted by
cognitive heuristics and biases and are amplified or
attenuated by media reports and other communica-
tions in ways that reflect the recipients’ worldviews.
System 1 tends to be innumerate, responding emo-
tionally to possibilities and categories of harm while
underweighting or ignoring relevant frequencies and
magnitudes. System 2 often fails to sufficiently adjust
or correct the promptings of System 1 leading to deci-
sions with predictable regrets. These findings help to
explain why expert and actuarial assessments of risk
often differ from lay perceptions of risk. In a demo-
cratic society, perceptions affect decisions. A major
challenge for risk analysis is to assess and communi-
cate risks to help inform and improve collective deci-
sions in ways that understand and respect the realities
of risk perception.

**2.18.2. Risk assessment**

Risk assessment addresses how large and uncertain
risks are. It begins with qualitative questions about
what might go wrong and proceeds to quantitative
assessments of how likely adverse events are to
<figure><img src="image_51.png" /></figure>

occur and what their possible consequences and their probabilities would be (Kaplan & Garrick, 1981). Probabilistic risk assessment (PRA) and quantitative risk assessment (QRA) methods apply probability models and statistical methods to data and modelling assumptions to quantify or bound the predicted frequencies and severities of losses and to estimate how their joint probability distribution would be changed by different risk management policies or interventions. Quantitative measures of risk can be derived from the full probability distribution or stochastic process descriptions of uncertain outcomes (Smidts, 1997), including dynamic coherent risk measures used in financial risk analysis (Bielecki et al., 2017). Stochastic process models of the occurrence frequencies of adverse events (such as accidents at power plants or tornadoes in cities) and the probability distribution of losses for each event can also be used to estimate entire cumulative probability distributions for losses over a stated time interval for different scenarios or sets of assumptions (Kaplan & Garrick, 1981, see also §2.10 and §3.19).

In the past decade, PRA techniques such as causal Bayesian networks (BNs), dynamic Bayesian networks (DBNs), and related probabilistic graphical models have increasingly been used to predict the probabilistic effects caused by interventions in engineering systems (Ruiz-Tagle et al., 2022) and public health applications (Butcher et al., 2021). They have largely supplanted older and less general PRA techniques such as fault tree analysis and Markov decision processes (Hanea et al., 2022; Cox et al., 2018). Together with discrete-event stochastic simulation models and continuous (systems dynamics) simulation, they provide constructive methods to predict how risk management interventions would change the probabilities of outcomes over time. This information can be used for simulation-optimisation of risk management decisions (Better and Glover, 2011).

PRA techniques have been extended to address adversarial risks in which intelligent adaptive adversaries rather than chance events threaten the safety and values that a risk manager seeks to protect (Banks et al., 2022); and *unknown risks* (or risks under radical uncertainty, sometimes called *Knightian uncertainty* by economists), in which relevant probabilities are unknown, e.g., by using uncertainty sets that replace precise probability distributions by (usually convex) sets of possible probability distributions or by scenarios of possibilities that are not necessarily exhaustive (Gilboa et al., 2017). Recent artificial intelligence and machine learning (AI/ML) methods are now being applied to natural hazards and disasters (Guikema,

2020), cybersecurity (Nifakos et al., 2021), power markets (Marcjasz et al., 2022), and financial portfolio risk management problems where new, changing, and unknown conditions make it necessary to learn effective risk prediction and management decision rules from data and experience without the guidance of well-validated PRA models (Cox, 2020).

### 2.18.3. Risk management, governance, communication, and risk-cost-benefit analysis

Given public perceptions and technical estimates of risks, what should be done about them? Who should decide, and how? Managing risks to human health, safety, or the environment often involves “wicked” decision problems and “deep” uncertainties, meaning that there are no clear, widely agreed-to definitions of the decision problem and solutions to it (Lempert & Turner, 2021). Although multiobjective and risk-sensitive or risk-constrained optimisation problems can be formulated for some risk management problems, such as routing hazardous cargo, for many wicked risk management problems, relevant decision variables, constraints, possible outcomes, and objective functions may be unknown or not widely agreed to. Risk management in such challenging cases usually involves issues of causation (what can be done and how much difference in outcome probabilities would different feasible choices make?), *collective choice* (how should the disparate perceptions and preferences of individuals be resolved or aggregated for purposes of collective decision-making?) and *risk governance* (who should be responsible for making, implementing, obeying, enforcing, and revising risk management decisions; how should stakeholders participate in risk management decisions; what institutions and processes should guide, restrain, and integrate collective risk management; and how should conflicts be resolved and collective decisions be made at individual, organisational, community, local, national, and international levels?) (Klinke & Renn, 2021).

The most immediate decisions for risk managers responding to potential or actual crises are often about *risk communication*. For example, if a pandemic or natural disaster such as a tsunami, volcanic eruption or hurricane, seems possible but not necessarily imminent or certain, then what should scientists and government officials tell policy-makers and the public about the uncertain risks? Who should say what to whom, and how soon? Different risk communication goals such as informing and empowering individual and community decisions, persuading individuals to change their behaviours, instructing citizens what to do, and informing or shaping policy deliberations and decisions, require different communication approaches. Risk
<figure><img src="image_52.png" /></figure>

communication frameworks for addressing these challenges overlap with risk perception frameworks but also emphasise the roles of trust in information sources and of outrage in mobilising public engagement and changing behaviours (Malecki et al., 2020).

Risk-cost-benefit analysis provides a simple-sound-
ing approach to collective risk management: take risk
management actions to maximise expected social util-
ity or net societal benefits (expressed as expected net
present value, possibly with a risk-adjusted discount
rate; Eliashberg & Winkler, 1981; Hammond, 1992).
However, mathematical impossibility theorems have
shown that when different people have sufficiently
different beliefs and preferences, there may be no
coherent way to aggregate them to make collective
decisions that respect normative principles such as
Pareto efficiency (Nehring, 2007). Trade-offs between
measures of accuracy and fairness have recently been
identified for machine-learning algorithms used in
risk assessment in areas such as mortgage lending
and criminal justice (Corbett-Davies et al., 2017).
Societal risk management is now often viewed less as
a top-down or centralised decision and control pro-
cess in which experts provide estimated probabilities
and social utilities or net benefits to use in risk-cost-
benefit or social utility maximisation calculations
than as a participatory democratic deliberative pro-
cess (Rad & Roy, 2021). Experts in risk analysis can
provide useful technical information in this process
but should not dominate it (Pellizzoni & Ungaro,
2000; Greenberg & Cox, 2021; Klinke & Renn, 2021).

Risk analysis poses intellectual, technical, and
practical implementation challenges that are likely
to engage and challenge Operational Research and
risk analysis professionals for the foreseeable future.
A more detailed review of the accomplishments,
current state, and remaining challenges for much of
the field of risk analysis can be found in the 40th
Anniversary Special Issue of Risk Analysis
(Greenberg et al., 2020), in specialised books such as
Aven (2015), and in online resources²⁹.

**2.19. Simulation³⁰**

A simulation aims to reproduce the important
behaviour of a real system. Our focus here is on the
use of computer simulation models within oper-
ational research (OR), whilst acknowledging that the
field is much wider and ranges from computer
models of sub-atomic particles to simulations
involving real human actors, particularly prevalent
in medicine and health sciences. Three main fla-
vours of simulation are used in this context: discrete
event simulation, agent-based modelling and system
dynamics. After discussing the uses of simulation,

we continue this subsection by introducing these
three main flavours before going on to discuss four
important areas in simulation research: conceptual
modelling, input modelling and parameterisation,
simulation optimisation, and finally the newer area
of data-driven simulation, linking to Industry 4.0
and digital twins. A selection of open source tools
for simulation are given in §3.18.

A simulation model, built on a computer, has a
number of potential functions. Principally it is used
for experimentation because testing out new settings
or ways of working on a simulation results in fewer
negative implications than experimenting with the
real system. This can allow simulation to be used
for optimisation of complex stochastic systems and
there has been considerable research in this area in
recent years, as we discuss below. Simulation can
also be used for predicting future behaviour, and
the COVID-19 pandemic showcased the predictive
power of simulation modelling in a very high-profile
situation (e.g., the agent-based model used to advise
the UK government and described in Ferguson
et al., 2020). The process of building a simulation
model results in a better understanding of the real
system because of the need to identify and model
the important relationships between different enti-
ties. Running the model can also help with estimat-
ing the sensitivity of model outputs to system
parameters. Another use of simulation models is for
training. Within the OR context, this most often
takes the form of strategic game-playing (e.g., the
beer game developed by MIT and described in
Sterman, 1989) to practice decision-making under
different scenarios in a safe environment.

*Discrete event simulation (DES)* is typically used to model systems in which entities move through a set of activities. Where these activities require resources, entities will queue until the resource becomes available. Such simulations are described as discrete event because the system state only varies at discrete time points, known as events. For example, a DES model might be used to describe a manufacturing line and in this case the events could include an item starting or finishing processing by a machine on the line. Usually in DES, the simulation clock will jump from one event to the next rather than moving in equal time steps.

*System dynamics (SD; §2.22)* was first developed in the 1950s by Jay Forrester to help with the understanding of industrial problems. SD models deal with stocks and flows, where the dynamics are dictated by a set of connected differential equations. Describing a system using an SD model can help with detecting feedback loops and delay effects and SD modelling is useful for strategic decision-making.
Agent based modelling (ABM) describes the behaviour of individual entities or agents within a population. As Macal (2016) states, one of the key differences between ABM and both DES and SD is that it takes an agent perspective of a system. Each agent in the simulation will follow its own set of rules dictating its behaviour and how it interacts with the environment and other agents. Agent behaviour is typically stochastic, allowing natural variability to be included in the model. An ABM can be used as a bottom up approach to determine emergent behaviour where individual actions lead to a system level response.

Regardless of the simulation modelling technique, the first part of any simulation project is to gain an understanding of the system being modelled, the objectives of the work, and the key components that should be included, referred to as conceptual modelling. There is some discussion of the exact definition of conceptual modelling in Robinson (2008) but some key points are made to support the process. First, the conceptual model can be thought of as separate from the final computer model that is built and serves as an abstraction of the real system that describes what is going to be modelled and gives an indication of how that might be done. Second, the development of the conceptual model requires input from the modeller and the system owners. Third, a conceptual model does not remain constant through a simulation project but is revisited and adapted as the project continues. Recent research in conceptual modelling is reviewed by Robinson (2020) and has focused on designing modelling frameworks. The work described tends to be related to DES models but the core principles can also apply to the building of both ABS and SD models.

Like any other model, the utility of a simulation is very much dependent on its inputs: the garbage-in-garbage-out principle holds true here. Setting up the probability distributions used for inputs of a stochastic simulation model or parameterisation of a deterministic simulation model is referred to as input modelling and is typically achieved through fitting statistical models to available data and eliciting expert opinion. When estimating the inputs for a simulation model from data there is some uncertainty in their true values. With a different set of data, the estimates of the inputs would likely be different. Any uncertainties setting the model inputs will propagate through to the model outputs, resulting in input uncertainty. This is influenced both by the accuracy of the estimates of the inputs and the sensitivity of the model output to that particular distribution or parameter. Corlu et al. (2020) provide a review of the current state of the art in input uncertainty research for simulation, while Song et al.

(2014) provide practical suggestions on how to estimate the impact of uninput uncertainty on the output results.

Often simulation models are used to experiment with different system set-ups. Simulation optimisation, sometimes referred to as optimisation via simulation describes the use of a simulation model to find the optimal value for one or more decision variables. Typically it is used in the design of stochastic systems that are too complex to be effectively described by an analytical model. Practical examples of problems that can be solved using simulation optimisation include finding the optimal number and configuration of beds in a hospital ward; determining the appropriate number of repair staff on a production line; choosing between a selection of different configurations for a system.

The problem can be represented mathematically as

$$ming(\mathbf{x}), \mathbf{x} \in \mathcal{O},$$

where the function we are optimising $g(\mathbf{x})$ is generally the expected value of the output of a stochastic simulation model, $g(\mathbf{x}) = E[Y(\mathbf{x}, \xi)]$; $\mathbf{x}$ is a vector of decision variables; $\mathcal{O}$ denotes the feasible region for $\mathbf{x}$; and $\xi$ indicates the randomness inherent in the model. The majority of research in simulation optimisation aims to improve the efficiency of the optimisation algorithms by reducing the number of simulation replications needed to estimate the optimal values of $\mathbf{x}$. Where a complex and slow-running simulation model is used to generate the $Y(\mathbf{x}, \xi)$ this efficiency is particularly important. Hong and Nelson (2009) classify simulation optimisation problems into three categories:

1. **Small number of solutions: $\mathcal{O}$** contains only a small number of solutions and the decision variable $\mathbf{x}$ might define a particular system configuration. In this case the problem is one of ranking and selection.

2. **Decision variables are continuous: $\mathcal{O}$** is a convex subset of the set of $d$-dimensional real numbers and the problem is continuous optimisation via simulation.

3. **Decision variables $\mathbf{x}$ are discrete and ordered: $\mathcal{O}$** is a subset of the set of $d$-dimensional integers and the problem is discrete optimisation via simulation.

A set of algorithms exists for solving each category of problem. There has also been significant work on multi-objective optimisation via simulation; for example, see Hunter et al. (2019) for a detailed description of the problem and different solution approaches.
<figure><img src="image_53.png" /></figure>

In recent years, sensing has become more wide-
spread and the transfer of data from physical sys-
tems to control systems is now happening in close
to real time. This has allowed modellers to design
simulation models that are automatically fed data
from the real system allowing them to either predict
the future (e.g., prediction of emergency depart-
ments crowding Hoot et al., 2008) or to use the
simulation models to optimise system parameters as
part of a dynamic control process. Such models are
sometimes referred to as a digital twin or symbiotic
simulation (see Onggo, 2019, for a definition). Xu
et al. (2016) describe how simulation can be incor-
porated into the Industry 4.0 framework using the
example of a semiconductor fab operation. The use
of simulation in dynamic control is still in its
infancy and requires fast and reliable simulation
optimisation algorithms as well as mechanisms for
enabling the simulation model to evolve based on
new input data. Industry 5.0 is intended to comple-
ment Industry 4.0 by putting societal goals at the
heart of industrial decision-making³¹. This is a
potential growth area for simulation optimisation,
particularly multi-objective simulation optimisation
(e.g., see Hunter et al., 2019) which enables solu-
tions to be found that support several competing
objectives.

Being a huge topic with many different facets,
there is no single article that provides an overview
of simulation but there are several excellent text-
books covering simulation techniques including Law
(2015) and Banks et al. (2004). The archive of the
Winter Simulation Conference³² is also an extensive
resource in understanding the state-of-the-art in the
field and its tutorial papers provide more basic
tuition in the effective implementation of simula-
tion. Recently, the history track at the conference
has also provided an overview of the evolution of
the simulation field.

## 2.20. Soft OR³³ and problem structuring methods³⁴

Problem Structuring Methods (PSMs) are concerned
with addressing problem formulation in OR.
Following definitions of Mingers and Rosenhead
(2004) and Rosenhead (1996) they consist of a set of
rigorous but not mathematical methods based on
qualitative, diagrammatic modelling. They allow for
a range of distinctive stakeholder views of a problem
to be expressed, explored and accommodated. They
encourage active participation of stakeholders in the
modelling process, through facilitated workshops
and the cognitive accessibility of the modelling
approach. PSMs afford negotiating a joint agenda
and ownership of actions. The aim is for

exploration, learning and commitment from stake-
holders, rather than optimisation or prediction.
PSMs thus are vital and constitute a significant
developmental direction for OR. See Smith and
Shaw (2019) and Franco and Rouwette (2022) for
recent reviews.

Understanding the contribution of PSMs to OR
requires some knowledge of their evolution. We
characterise the development of the field into three
phases: (i) origins, (ii) growth (only noted here
through the increased publication rate of PSM
related articles), and (iii) maturity, covering the dif-
fusion of PSMs to fields outside of OR, and re-inte-
gration of problem structuring into mainstream OR.
Looking at the last first, we see PSMs at an impor-
tant turning point, as recent work by Dyson et al.
(2021) specifically identify the centrality of problem
structuring in the origins of OR and lead us towards
the important question of why PSMs are not seen as
an essential element of every OR engagement.

The origins of PSMs as a set of formal methods
in OR arose as a consequence of the broad critique
of the process of OR in the 1970-80s; the label itself
was pioneered by Woolley and Pidd (1981). Ackoff
became a trenchant critic of the sole pursuit of
objectivity and optimisation in OR describing it as
an “opt-out” (Ackoff, 1977) and set out an agenda
for reconceptualising OR practice (Ackoff, 1979a).
Dando and Bennett (1981) described the situation
as a “Kuhnian crisis in management science”. In
Rosenhead’s “Rational Analysis for a Problematic
World: Problem Structuring Methods for Complexity,
Uncertainty and Conflict” their prescription in OR
engagements was associated with dealing with prob-
lem contexts identified variously as wicked, messy,
or swampy (Rosenhead, 1989, pp. 3-11). These can
be summarised as problem situations that are not
well-defined, involving many interested parties with
different perspectives (worldviews), where there is
difficulty agreeing objectives and the meaning of
success, and that require creating agreement
amongst the parties involved for actions to be taken.
The implication of the dichotomous framing of
problem contexts – i.e., wicked/tame, swamp/high
ground, hard/soft, tactical/strategic – was to set out
a clear critique for the whole field of OR and to
suggest that to retain its relevance in dealing with
the messiness of real-world problems PSMs were
required to bring some rigour – and indeed a
reminder of the importance – to the process of
problem formulation. Importantly, the pioneers of
PSMs were concerned that traditional (i.e., ‘Hard
OR’) processes for problem formulation were practi-
tioner-free (Checkland, 1983; Rosenhead, 1986).

The main PSMs set out by Rosenhead (1989)
were Strategic Options Development and Analysis
<figure><img src="image_54.png" /></figure>

(SODA; Eden, 1989), arising from cognitive mapping; Soft Systems Methodology (SSM; Checkland, 1989), emerging from the failure of Hard Systems Thinking approaches (e.g., Systems Engineering, RAND-style Systems Analysis) when applied to messy problems; and the Strategic Choice Approach (SCA; Friend, 1989), arising from planning. In addition, Robustness Analysis, Metagame Analysis, and Hypergame Analysis were also included. However, setting the boundary of PSMs has always been an open question (Mingers, 2011b). In the main, the core methods (SODA, SSM, SCA) are seen as exemplary and provide sufficient coherence for OR scholars and practitioners to be provided with a clear view of a common theme.

The methodology of PSMs has long been associated with contextual matters through Systems Thinking (§2.23 Checkland, 1983; Mingers & White, 2010), Community OR (Johnson et al., 2018; Jones & Eden, 1981; Parry & Mingers, 1991), and large group processes (Shaw et al., 2004; White, 2002). Methodological individualism has been addressed through Behavioural OR (§2.2; Franco & Hämäläinen, 2016; White, 2016). There has also been long-standing relevance to Multi-Criteria Decision Analysis (MCDA; Marttunen et al., 2017), value-focused thinking (Keeney, 1996b), policy analysis (Eden & Ackermann, 2004), and strategy making (Ackermann & Eden, 2011; Dyson, 2000). Bridging between PSMs and other techniques in OR has been developed as multimethodology (Mingers & Brocklesby, 1997); for example, integration with Simulation (§2.19; Kotiadis & Mingers, 2006; Tako & Kotiadis, 2015). Some approaches to using the Viable Systems Model (VSM), System Dynamics (§2.22), and Decision Analysis (§2.8) would also be considered as PSMs too (Rosenhead & Mingers, 2001, pp. 267-288), e.g., VSM (Lowe et al., 2020) and System Dynamics (Lane & Oliva, 1998). We also see developments in Group Model Building (GMB) from the System Dynamics community making a significant contribution to PSMs (Andersen et al., 2007). In their growth and mature phase, applications of SSM, SCA, and SODA have extended the reach of PSMs into, e.g., project management (Franco et al., 2004) and environment, sustainability, and energy policy, e.g., SCA (Fregonara et al., 2013), SODA (Hjortsø 2004), SSM (Pahl-Wostl, 2007), and the Drivers, Pressures, State, Impact and Response framework (DPSIR Bell, 2012).

The state of the art and research agenda for PSMs has been the subject of periodic reflection e.g., reviews by Rosenhead (1996) and Mingers and Rosenhead (2004). A Special Issue of JORS in 2006 questioned where PSMs were heading (Rosenhead, 2006) – variously argued as a “grassroots revolution”

(Westcombe et al., 2006), an appeal to common principles (Eden & Ackermann, 2006), and observations that “form and content have evolved through interaction between the ideas and their practical use” (Checkland, 2006). A more recent viewpoint debate “whither PSMs” again questioned their direction of travel Harwood (2019); Lowe and Yearworth (2019).

The qualitative nature of PSM methods raises questions about evaluating both effectiveness and value. Midgley et al. (2013), White (2006), and Franco and Rouwette (2022) have addressed the question of effectiveness, and whilst White goes some way towards defining the value of PSMs it is important to note the reservations expressed by Checkland and Scholes (1990, p. 299) – that measuring value in a unique problem context, the ‘messy’ realm of PSMs, is unlikely to be meaningful. Tully et al. (2019) examine this conundrum in depth from the perspective of a consulting business and make some practical suggestions for its resolution.

Theory provides an important basis for PSM development. The range of PSM practice reported has been explained by the constitutive rules that underpin specific methods. First articulated by Checkland (1981, pp. 252-254), constitutive rules are generative of method rather than prescriptive and account for the range of practices that emerge, even when adopting a specific methodology such as SSM i.e., adaptation is always necessary to address the specifics of the application context. The idea was developed further by Jackson (2003, pp. 307-311) and then by Yearworth and White (2014) into a generic constitutive definition for PSMs. Another significant development has been a focus on PSMs as practice and drawing on practice theories. These theories provide a means of understanding OR practices by “zooming-in” to the detailed, fine-grained, scale and by “zooming-out”, looking at how specific practices affect the broader context (Ormerod et al., 2023). Together these theoretical strands provide sufficient basis on the one hand, to liberate PSMs from the pigeon-hole of the dichotomous framing of their origins, and on the other, to address OR practice as a whole and to see problem structuring as a normal, indeed necessary, part of every OR intervention. For instance, Actor Network Theory (ANT) provides a lens to look at the processes of problematisation (i.e., problem formulation) in OR practice (White, 2009). Callon (1981) draws specific attention to the “abundance of problematisations” facing expert practitioners – that there is no single specific way of problematising. Strands of ANT focus on the performative idiom; (Ormerod, 2014a) draws attention to the “mangle of practice” and the need for more informative case studies of OR
<figure><img src="image_55.png" /></figure>

practice. Other theoretical underpinnings are rele-
vant to PSM developments e.g., PSMs as technology
(Keys, 1998), Critical Realism (Mingers, 2000),
Activity Theory (White et al., 2016), and the specific
role of models as boundary objects (Franco, 2013)
in facilitated workshops (Franco & Montibeller,
2010).

From a practitioner point of view, the recent
report “Reinvigorating Soft OR for Practitioners” by
Ranyard et al. (2021) to the Heads of OR and
Analytics Forum (HORAF) and the inclusion of the
knowledge requirement “How to select and apply, a
range of problem structuring methods to understand
complex problems” in the Operational Research
Specialist Degree Apprenticeship specification by the
Institute for Apprenticeships and Technical
Education (2021) are a welcome development.

In conclusion, for PSMs we see a return to the
roots of OR as a discipline – encompassing both
practice and academic scholarship – through the
centrality of problem formulation to the process of
OR (Churchman et al., 1957, p. 13) and a reminder
that the seeds of problem structuring can be seen in
the work of the ‘founders’ of OR as uncovered by
Dyson et al. (2021). We have identified a number of
research gaps that indicate future research directions
for the development of PSMs. In the area of the
impact of new digital technologies, Yearworth and
White (2019) propose greater use of online “same
time/different places” problem structuring work-
shops in order to meet the requirements for fast
meeting setup times, reducing carbon emissions,
enabling the scale-up to large group participation,
and supporting new post-pandemic working pat-
terns. The need to address complex policy issues in
the context of wicked problems is highlighted by
Howick et al. (2017) and Ferretti et al. (2019), who
argue for a re-invigorated engagement for PSM
practice in policy analysis. Finally, Ormerod
(2014b), Ranyard et al. (2015) and Ormerod et al.
(2023) remind us that we need to see a renewed
practitioner-led orientation for OR scholarship that
grounds future development in solid empirical
work.

**2.21. Stochastic models**³⁵

Many decision problems involve uncertainty, e.g.,
network design with disruption risk, portfolio selec-
tion with uncertain return, resource planning with
unknown resource availability, crop planting with
uncertain yield, inventory control with varying
demand, and project scheduling with random task
duration, etc. While the effect of uncertain parame-
ters on the optimal solution and objective value can
be studied through the well-known sensitivity

analysis, or what-if analysis, in a deterministic opti-
misation approach, such post-optimality analysis
does not prescribe solutions under uncertainty a pri-
ori. This subsection provides an overview of a suite
of optimisation models and methods that seek to
obtain optimal or near-optimal solutions for the
class of decision problems where some parameters
are stochastic with known probability distribution³⁶.

Originated in the seminal work of Dantzig (1955), stochastic programming is one of the earliest and most prominent approaches to deal with optimisation problems with stochastic parameters. The basic stochastic programming model has a two-stage framework, called two-stage stochastic programming with recourse (2S-SPR; Birge & Louveaux, 2011). In the first stage, the *here-and-now* decision is made. Then in the second stage, the *recourse* decision is prescribed for each scenario of stochastic parameters after their realisation. The objective function minimises the total cost as the summation of the first-stage cost and the expected second-stage cost given the probability distribution of the stochastic parameters. It is often insightful to compute the value of stochastic solution (VSS; Birge, 1982) as the difference between the optimal objective function value of the deterministic counterpart (by substituting the stochastic parameters with their point estimates) and that of the stochastic programming model. We refer to Birge and Louveaux (2011) and Shapiro et al. (2021) for a systematic and updated treatment on the modelling and theory of stochastic programming, and to Wallace and Ziemba (2005) for a collection of applications of stochastic programming. Recent applications include disaster relief management (Grass & Fischer, 2016), transit network design (Zhao et al., 2017), portfolio selection (Masmoudi & Abdelaziz, 2018), treatment plant placement in drinking water systems (Schwetschenau et al., 2019), process systems (Li & Grossmann, 2021), multi-product aggregate planning (Gómez-Rocha & Hernández-Gress, 2022), and resource allocation for infrastructure planning (Zhang & Alipour, 2022), among others.

A stochastic programming model may also
include a constraint that is satisfied with a probabil-
ity. This model is is known as the chance con-
strained programming model introduced by
Charnes et al. (1958). The probabilistic constraint
can often be transformed into a deterministic con-
straint given the known probability distribution of a
stochastic parameter. Detailed coverage on the
chance constrained programming models and meth-
ods is available in Prékopa (2013). Notable applica-
tions include farm management (Moghaddam &
DePuy, 2011), broadband wireless network design
(Claßen et al., 2014), supply chain network design
<figure><img src="image_56.png" /></figure>

(Shaw et al., 2016), equity trading server allocation (Sun & Hassanlou, 2019), and power system planning (Geng & Xie, 2019), among others.

A stochastic programming model can be formulated as a deterministic mathematical programming model by associating its decision variables with the scenarios of stochastic parameters, an approach often referred to as deterministic equivalent formulation (DEF). Solving a stochastic programming model via its DEF can be computationally challenging as the size of DEF grows rapidly with the number of scenarios of stochastic parameters. Thus custom designed algorithms are often needed to obtain quality solutions for medium- and large-size stochastic programming models. Assuming the set of random parameters has finite support, the DEF of a 2S-SPR has a block structure with L-shape, which motivates the well-known L-shape method (Van Slyke & Wets, 1969) based on Benders decomposition (Benders, 1962). For problems with a large number of random scenarios, it can be computationally intractable for the exact decomposition method to obtain an optimal solution. One may resort to various sampling-based methods to obtain approximate solutions. The stochastic decomposition method proposed by Dantzig and Infanger (1991) and Higle and Sen (1991) employs Monte Carlo simulation and importance sampling to compute sampling cuts instead of generating the exact cuts in the L-shape method. The other successful approach is sample average approximation (SAA; Kleywegt et al., 2002; Shapiro, 2003), which approximates the second-stage objective function by an expected value function corresponding to a set of scenarios of the random parameters. Numerical experiments and results of the SAA method on various benchmark instances are available in Linderoth et al. (2006).

Another well-known stochastic modelling and solution approach is the integrated simulation-optimisation (Fu et al., 2005), especially used for solving problems involving discrete decision variables, widely encountered in applications in management science, operations and supply chain management. A typical integrated simulation-optimisation framework consists of two inter-related components: *search* and *sampling*. The search component deals with the solution space, often combinatorial in nature with large size, for which various metaheuristics (Glover & Kochenberger, 2003) can be applied. These include local search based metaheuristics such as simulated annealing (Kirkpatrick et al., 1983), tabu search (Glover & Laguna, 1997) and scatter search (Glover, 1998), as well as population-based metaheuristics, e.g., genetic algorithm (Holland, 1975). The sampling component evaluates a candidate solution via simulation, e.g., Monte Carlo or

discrete event simulation. Thus an integrated simulation-optimisation approach can be viewed as an augmented deterministic metaheuristic that employs simulation to evaluate/estimate solutions in an uncertain environment. Recent applications include maritime logistics (Zhou et al., 2021), pooled ride-hailing operators (Bischoff et al., 2018), and staffing for service operations (Solomon et al., 2022).

Many real-world applications need decisions to be made sequentially under uncertainty, e.g., production planning, inventory control, resource allocation, and project scheduling, etc. One approach to this type of applications is the multi-stage stochastic programming (Birge & Louveaux, 2011), which is a generalisation of the 2S-SPR. In a typical multi-stage stochastic programming framework, a decision is made in a stage, based on the observed realisation of random parameters and the decisions made in the previous stage, to minimise the total expected future cost. The random parameters are assumed to evolve according to some known stochastic process. We refer to Zhang (2023) for an updated and comprehensive treatment on various stochastic processes. A nested decomposition as a generalisation of the L-shape method can be applied to obtain exact solutions to a multi-stage stochastic programming model (Birge, 1985). Conceptually, it applies Benders decomposition or the L-shape method recursively to a series of nested two-stage subproblems. Although theoretically sound, it can be computationally challenging to handle reasonably large instances as the number of scenarios grows exponentially with the number of stages and random parameters. Thus one often resorts to various approximation algorithms for obtaining quality solutions efficiently, including value function approximation, constraint relaxation, scenario reduction, and Monte Carlo methods, among others (Birge & Louveaux, 2011).

An alternative approach to sequential decision making under uncertainty is stochastic dynamic programming (Ross, 1983) or as a Markov decision processes (MDP; Puterman, 2014). See §2.9 for more details.

There are two general approaches for solving an MDP model: open-loop and closed-loop (Bertsekas, 2012a). An open-loop approach obtains a solution to all the decision variables upfront, which is *static* in nature without updating during execution of the sequential decision-making process. The integrated simulation-optimisation approach introduced above is a successful way to obtain an open-loop policy, e.g., using genetic algorithms (Ballestín, 2007b), tabu search (Tsai & D. Gemmill, 1998), or the greedy randomised adaptive search procedure (GRASP; Ballestín & Leus, 2009) with simulation for the
<figure><img src="image_57.png" /></figure>

Stochastic Resource-Constrained Project Scheduling Problem (SRCPSP).

Instead of optimising the entire problem upfront, a closed-loop approach seeks to obtain an optimal decision rule (*policy*) to map the state at a stage to a decision, given the information available to the decision-maker at the current stage. A closed-loop policy is *dynamic* and *adaptive* in nature, thus is more flexible than an open-loop policy (Dreyfus & Law, 1977; Bertsekas, 2012a). Although theoretically attractive, to obtain an optimal closed-loop policy through the well-known Bellman equation in recursive way (Bellman, 1957) is computationally intractable due to the curse-of-dimensionalities of MDP in state space, solution space of decision variables, and sample space of random parameters.

Recent advances advocate the design and implementation of approximate dynamic programming (ADP) for solving large-scale MDPs. ADP has its roots in neural dynamic programming (NDP; Bertsekas & Tsitsiklis, 1996) and reinforcement learning (RL; Sutton & Barto, 2018). Its key idea is to replace the exact cost-to-go function with some sort of approximation. We refer to Si et al. (2004) and Powell (2011) for comprehensive coverage on ADP and its applications. There are two approximation paradigms for the design of an ADP algorithm. The value function approximation approach works directly on the cost-to-go function to replace it with an alternative functional form that is computationally tractable. Using sample path simulation, a forward iteration procedure can be implemented to solve a deterministic sub-problem with the approximated objective function subject to the set of constraints corresponding to the state of the current stage (Powell, 2011). This approach has been successfully applied to the multicommodity network flow problem (Topaloglu & Powell, 2006), dynamic fleet management (Simão et al., 2009), and dynamic resource planning (Solomon et al., 2019).

While the value function approximation approach works well for problems with structures amenable to efficient mathematical programming methods such as linear programming or network optimisation, many combinatorial optimisation problems are *NP*-hard themselves, and can be computationally demanding for mathematical programming to handle. We refer to §2.5 for a review on the topic of computational complexity and *NP*-hardness. This calls for an alternative approximation paradigm known as the rollout policy (Bertsekas et al., 1997). A rollout policy estimates the cost-to-go function using some heuristic via simulation, which can be either an efficient problem-specific heuristic or a custom-designed metaheuristic for the problem at hand. It can be viewed as a look-ahead policy that

estimates the cost of a decision-state pair under uncertainty about the future, which can be in contrast to the lookup table approach in RL (Sutton & Barto, 2018) where the cost of a decision-state pair is learned through simulation in a look-back fashion. A hybrid look-ahead and look-back ADP algorithm has been developed by Li and Womer (2015) to take advantage of the complementary strengths of the pure rollout approach and the lookup table approach alone. Successful applications of rollout policy have been reported for stochastic vehicle routing (Secomandi, 2001; Goodson et al., 2013), SRCPSP with stochastic activity durations (Li & Womer, 2015), RCPSP with multiple-overlapping modes (Chu et al., 2019), ride-hailing system planning (Al-Kanj et al., 2020), and attended home delivery (Koch & Klein, 2020).

All the aforementioned models and methods assume that the probability distribution of random parameters is known or can be properly estimated. This assumption may not hold in some situations where there is lack of knowledge about the uncertain parameters, or error in measurement or implementation. Optimisation with uncertain parameters without probability distribution calls for the robust optimisation (RO) approach. Although the origin of RO can be dated back to the 1970s (Soyster, 1973), RO has been growing as an active research field since the last two decades. In an RO model, one assumes that uncertain parameters are within a user-specified uncertainty set. A robust feasible solution satisfies the set of uncertain constraints for all realisations of the uncertain parameters in the uncertainty set. One main technique to solve an RO model is the robust reformulation approach to obtain a computationally tractable robust counterpart (RC) with a finite number of deterministic constraints (Bertsimas et al., 2011a). When choosing the type of uncertainty set for the model, one often needs to trade-off between robustness against realisations of the uncertain parameters and computational tractability, i.e., size of the uncertainty set (Gorissen et al., 2015). We refer to Ben-Tal and Nemirovski (2002) and Ben-Tal et al. (2009) for systematic treatment on robust optimisation. RO has been applied in various fields including finance (Georgantas et al., 2021), energy and utility (Sun & Conejo, 2021), supply chain (Ben-Tal et al., 2005; Pishvaee et al., 2011), healthcare (Meng et al., 2015), and marketing (Wang & Curry, 2012).

## 2.22. System dynamics³⁷

System Dynamics (SD), founded by Forrester (1961), is a “rigorous method for qualitative description, exploration and analysis of complex systems in terms
<figure><img src="image_58.png" /></figure>

of their processes, information, organisational boundaries and strategies; which facilitates quantitative simulation modelling and analysis for the design of system structure and control" (Wolstenholme, 1990). SD modelling involves (as extracted from the System Dynamics Society website - www.systemdynamics.org):

* "Defining problems dynamically, in terms of graphs over time.

* Striving for an endogenous, behavioural view of the significant dynamics of a system, a focus inward on the characteristics of a system that themselves generate or exacerbate the perceived problem.

* Thinking of all concepts in the real system as continuous quantities interconnected in loops of information feedback and circular causality.

* Identifying independent stocks or accumulations (levels) in the system and their inflows and outflows (rates).

* Formulating a behavioural model capable of reproducing, by itself, the dynamic problem of concern. The model is usually a computer simulation model expressed in nonlinear equations or can be left without quantities as a diagram capturing the stock-and-flow/causal feedback structure of the system.

* Deriving understandings and applicable policy insights from the resulting model.

* Implementing changes resulting from model-based understandings and insights."

SD can be employed for both qualitative and quantitative modelling. On the one hand, tools and methods employed for qualitative SD modelling are also considered Soft Operational Research or Problem Structuring methods. On the other hand, quantitative SD modelling shares many aspects of traditional simulation methods or Hard Operational Research. Using SD quantitatively implies the development of a 5-steps process (Sterman, 2000) that

starts with a dynamic hypothesis about a structure responsible for the performance over time observed in the system followed by the model formulation, testing and experimentation. The next section discusses both approaches in detail.

One interesting characteristic of SD models is the spectrum of model fidelity they cover (Morecroft, 2012). Figure 1 illustrates a spectrum of model fidelity and realism. Models range in size from large and detailed to small and metaphorical. On the left-hand side are analogue, high-fidelity models epitomised by aircraft flight simulators used to train pilots and to rehearse crisis scenarios. They are constructed with realistic detail and accurate scaling to provide a vivid and lifelike experience of flying the aircraft they represent. People typically expect business and social models to be similarly realistic; the more realistic the better. Realistic high-fidelity models are discussed later in this subsection. But very often smaller models are extremely useful, particularly when their purpose is to aid communication and to build shared understanding of contentious problem situations in business and society. As Figure 1 suggests, the spectrum of useful models can include illustrative models (of limited detail yet plausible scaling) or even tiny metaphorical models (of minimal detail yet transferable insight).

At the other end of the spectrum, on the far right, is a low fidelity Romeo and Juliet simulator (Morecroft, 2010). This particular simulation model contains just four main concepts: Romeo's love for Juliet, Juliet's love for Romeo and the corresponding rates of change of their love. It is used as a metaphorical model or transitional object to help undergraduates and high school students to better understand something complex and abstract – differential equations or even Shakespeare's play. Clearly, a simulator cannot possibly replicate Shakespeare's play, but it can encourage students to study the play more closely than they otherwise would. It is this metaphorical property of small models – to attract

<figure><img src="image_59.png" /><figcaption>**Figure 1.** Modelling and realism: a spectrum of model fidelity. Adapted from Morecroft (2015), Chapter 10.</figcaption></figure>
<figure><img src="image_60.png" /></figure>

people's attention, to encourage them to reflect and debate – that often underpins their value to model users. Sometimes metaphorical models enable client engagement. One could say that 'small is beautiful' in the world of policy and strategy modelling. Over the years SD studies have included models and simulators that cover the entire range. See Kunc (2017b) for a sample of papers published in the Journal of the Operational Research Society and Kunc et al. (2018) for a study on published SD models.

## 2.22.1. Qualitative system dynamics

The main objective of qualitative SD involves discovering the structure, in terms of feedback loops, driving the dynamic behaviour of key variables, usually with clients through facilitated workshops. The main tool employed in qualitative SD modelling is causal loop diagram (CLD). The steps for developing a CLD are (based on Kunc, 2017a):

1. Understanding the direction of causality between two variables. Interestingly, it is a source of important discussion among participants in facilitated qualitative SD modelling.

2. Defining polarities involves identifying the relationships between two variables as either positive (same sense of direction of change) or negative (opposite sense of direction of change).

3. Identifying feedback processes responsible for the dynamic behaviour of variables. They originate from connecting variables in a circular chain of cause-and-effect. There are two types of feedback process: reinforcing and balancing.

Finally, Table 1 shows description of the modelling process

<table><caption>Table 1. Qualitative SD modelling process based on Kunc (2017a).</caption><thead><tr><td>Modelling process</td><td>Qualitative SD</td></tr></thead><tbody><tr><td>Objective</td><td>Understand the feedback structure of the system.</td></tr><tr><td>Inputs</td><td>Text data obtained through facilitated face-to-face meetings, interview or the interpretation of causal mechanism in reports and from theories.</td></tr><tr><td>Process</td><td>The modelling process implies the construction of CLD to represent individual and/or group-level interpretations of causal links. Facilitation processes are critical to uncover the causal links.</td></tr><tr><td>Outputs</td><td>There are three main outputs: learning about the structure of the system, changes in participants perspectives, and agreement on future policies.</td></tr></tbody></table>

<table><caption>Table 2. Quantitative SD modelling process based on Kunc (2017a).</caption><thead><tr><td>Modelling process</td><td>Quantitative SD</td></tr></thead><tbody><tr><td>Objective</td><td>Test a hypothesis about the structure driving the reference mode of a variable.</td></tr><tr><td>Inputs</td><td>Text data obtained through facilitated face-to-face meetings, interview or the interpretation of causal mechanism in reports and from theories to determine the structure. Numerical data for the model can come from three sources: judgement from experts or managers, numerical data sets and facilitation processes for nonlinear functions.</td></tr><tr><td>Process</td><td>After defining the boundary of the model, a stock and flow diagram is developed to represent the structure of the system. Equations are formulated for each variable and parameters entered. Testing of the structure and outputs are performed to confirm the model structure replicates the behaviour observed in the key variables.</td></tr><tr><td>Outputs</td><td>There are three outputs: time series showing performance over time; performance over time of policies or interventions in the system; and learning about the dynamic behaviour of the system.</td></tr></tbody></table>

## 2.22.2. Quantitative system dynamics

Quantitative System Dynamics characterises the system behaviour using a set of accumulation processes linked through feedback processes. The structure of the model is represented through stocks and flows diagrams. The numerical results, which are deterministic and continuous, aim to replicate past system behaviour through calibration and testing processes before the model is used to test interventions in the system. Table 2 presents a summary of the modelling process.

## 2.22.3. Application areas

a. **Behavioural modelling:** There are three main areas of application. Firstly, research in decision making under dynamic complexity focused on identifying and documenting systematic misperceptions of feedback in decision making processes across multiple industries and environmental conditions using SD models (Gary et al., 2008; Atkinson & Gary, 2016). Secondly, experimental studies explore decision making and performance using management flight simulators or microworlds based on SD models (Gary et al., 2008; Sterman, 2014). Thirdly, individual experimental work using SD models examines how differences in mental model accuracy and decision rules lead to differences in the performance (Torres et al., 2017). Recently, scholars have advocated for a practice of behavioural system dynamics (Lane & Rouwette, 2023).

b. **Group model building:** There is a wide body of research on model conceptualisation in groups; see Rouwette et al. (2010), where the outcome is either qualitative or quantitative SD models. Researchers have assessed the effects on communication, learning, consensus and commitment
<figure><img src="image_61.png" /></figure>

in the behaviour of groups, as well as measuring
the changes in mental models and understanding
the impact of group model building in terms of
persuasion and attitudes (Rouwette, 2016).

c. *Multi-scale high-fidelity systems modelling*: SD modellers are embracing new approaches to improve the scale and fidelity of their models to move from aggregate conceptual models into realistic detailed models supported with specific data. There are multiple considerations to develop high-fidelity models (Sterman, 2018). Firstly, these models represent heterogeneous actors in the system, which involves disaggregating single stocks into multiple stocks reflecting their differences across dimensions (e.g., age). While this solution increases granularity in the model, it also implies increasing computational burden and long simulation times, which can limit the ability to perform sensitivity analysis, change structure and test interventions by stakeholders. Secondly, high-fidelity models reflect business and social processes in detail fitting the data. Therefore, models may move from the traditional representation of time, continuous, to discrete; from state continuous variables or discrete variables; and include uncertainty using stochastic variables. Consequently, SD models can employ ordinary differential equations, stochastic differential equations, discrete event simulations, agent-based models and dynamic network models. Thirdly, multiscale modelling involves integrating models working at different temporal scales (e.g., fast and slow dynamics). Fourthly, since SD models tend to employ qualitative data (e.g., decision making rules), modellers should identify and mitigate biases in sample selection and data elicitation to collect robust qualitative data. Fifthly, quantitative data should have a clear purpose in terms of the model, so specific data related to the problem the model is solving has to be collected rather than accepting only available numerical data. Sixthly, high-fidelity models should consider parameter estimation and model analysis extremely necessary to replicate historical data.

d. *SD and Artificial Intelligence (AI):* Since abundant information is available in different forms (images, text, and numbers), there is a need for technologies that not only predict data but also learn from the environment such as AI (Baryannis et al., 2019). AI can be used for cognitive thinking, learning from behaviour, recalling, and drawing inferences (Min, 2010). SD models use inferences of the casual structure in system to predict future trends or test interventions (e.g., new policies). SD can combine with

AI to generate AI-driven simulations based on machine-learned and mathematical rules to make more accurate models (Li et al., 2022b). Another use is the employment of AI methods to interpret the results of simulations, especially feedback loop dominance in complex SD models.

**2.22.4. Future of system dynamics research**

The future of SD may be driven by developments in
several on different areas. Firstly, SD can be used as
a problem structuring or systems thinking method
(in terms of qualitative SD) so improvements in
terms of facilitation will be critical. Secondly, when
SD is an aggregated simple model that helps model-
ler and client to learn about dynamic complexity,
improvements in terms of impact on behaviour
from using the model (Kunc et al., 2020) will be
expected. Thirdly, SD can be high fidelity systems
models using all the toolkit available in terms of
simulation methods and AI. The next section on
systems thinking (§2.23) looks at other systems
methodologies for different purposes.

**2.23. Systems thinking<sup>38</sup>**

'Systems thinking' involves us viewing complex problem situations and possible human responses to them using systems theories, methodologies, methods and concepts. We will start this section by presenting a contemporary understanding of what a 'system' is. An explanation of how systems thinkers use this understanding to support action to address or prevent complex problems will then follow. Subsequently, we will review 70+ years of systems thinking to show how we got to this contemporary understanding via three 'waves' of methodological development.

**2.23.1. What is a system?**

A system is made up of a set of *interrelated parts*, with *emergent properties* (Emmeche et al., 1997). An emergent property is a feature that cannot be traced back to any single part of the system, so can only be understood as arising from the whole (all the parts and interrelations together). Systems have *boundaries*: we can say what is inside and outside the system (Ulrich, 1994), although some interactions may cross these boundaries (von Bertalanffy, 1968). However, systems are always seen from the *perspective* of an observer/participant (Churchman, 1979; Cabrera et al., 2015). Indeed, there can be multiple perspectives on the boundaries of the system, what interrelationships (within the system and with its environment) need to be considered, what emergent properties matter, and what other perspectives should be heard.
<figure><img src="image_62.png" /></figure>

### 2.23.2. What is systems thinking?

Based on the above understanding of systems, we can now explain *systems thinking*. It is taking a systems approach to *rethinking the taken-for-granted assumptions* of decision makers, OR practitioners and stakeholders on what perspectives, boundaries, interrelationships and/or emergent properties matter in a given situation, and what the implications are for action. Many systems thinkers use the adjective 'critical-systemic': thinking is *systemic* because of the use of the above systems concepts, but it is also critical because it involves rethinking options for understanding and action in relation to the deployment of these concepts (Ulrich, 1994; Gregory et al., 2020).

### 2.23.3. Three waves of systems methodology

Since the 1950s, there have been three 'waves' (or successive paradigms) of systems methodology, although the second and third waves didn't fully replace their predecessors: some groups of practitioners stuck with earlier ideas. The first wave was typified by early work in systems engineering (e.g., Hall, 1962; Jenkins, 1969), systems analysis (e.g., Miser & Quade, 1985, 1988), system dynamics (e.g., Forrester, 1961, see also §2.22), and organisational cybernetics (e.g., Beer, 1966, 1981). The first wave emphasised quantitative computer modelling by experts serving clients. These experts explained emergent properties of systems by understanding interrelatedness, and then deployed these explanations to make recommendations to clients on the possible consequences of strategic and tactical decisions.

In terms of the definition of a system presented earlier, systems were seen as real-world entities; the emphases were on interrelationships and emergent properties; boundaries were relevant because modelling had to account for all the parts and interrelationships in a system that are needed to understand given emergent properties; but multiple perspectives were often bypassed, rather than listened to, in the interests of objectivity or impartiality.

Almost all the first-wave methodologies regarded models as representations of reality, with people often being viewed as deterministic parts of systems being modelled rather than self-conscious actors who can change their purposes (Ackoff, 1979a). Indeed, stakeholder purposes can differ significantly from those of the systems modeller and his/her client, and ignoring this can create conflict that undermines an OR project (Checkland, 1985). Some critics (e.g., Hoos, 1972; Lee, 1973) argued that massive investments in large-scale modelling were wasted because systems practitioners tried to be comprehensive (e.g., modelling all interacting

problems at the city scale), yet they didn't sufficiently account for the actual questions that decision makers wanted to address—more modest modelling for specific purposes would have been better. Worse, the typical response to project failures was to say that the models were *not comprehensive enough*, so the ideal of comprehensiveness remained unquestioned (Lilienfeld, 1978).

These criticisms led to a second wave of systems methodologies focused on stakeholder participation, qualitative modelling and dialogue for collaborative learning. The idea of producing expert recommendations was replaced by a facilitation role for the practitioner, so multiple stakeholders could develop and integrate their ideas into proposals for change. Modelling shifted from a focus on real-world systems to understanding stakeholder perspectives, which could help people develop better mutual understanding and agree broadly-acceptable ways forward. Second-wave methodologies included soft systems methodology (Checkland, 1981), strategic assumption surfacing and testing (Mason & Mitroff, 1981), interactive planning (Ackoff, 1981) and interactive management (Warfield, 1994). Several earlier, first-wave methodologies were transformed in the second wave to become more participative, most notably system dynamics (e.g., Vennix & Vennix, 1996) and organisational cybernetics (e.g., Espejo & Harnden, 1989).

It was during the second wave that the definition of a system was expanded to recognise that all systems are understood *from a perspective*. Boundaries were no longer considered the real-world edges of systems, but instead marked what people include in or exclude from their deliberations (Churchman, 1970). There was a shift away from seeing systems as real-world entities to viewing them as useful *ways of thinking* to structure interpretations, either of the world or of prospective actions to change that world (Checkland, 1981).

However, this second wave came to be critiqued by a third wave of systems thinkers. Two issues came to the fore. First, a bitterly-entrenched paradigm war between first- and second-wave systems thinkers was sparked by the emergence of the second wave (Jackson & Keys, 1984). In response, there were many third-wave proposals for *methodological pluralism*: drawing creatively from both first- and second-wave methodologies, and reinterpreting methods through new frameworks or guidelines for choice. The idea was to refuse the forced choice between first- and second-wave thinking, and embrace the best of both. This gave us a more flexible and responsive practice than either of the previous two waves could deliver (e.g., Jackson, 1991; Mingers & Gill, 1997). Much of the work on methodological
<figure><img src="image_63.png" /></figure>

pluralism was developed under the banner of ‘critical
systems thinking’ (Flood & Jackson, 1991b; Flood &
Romm, 1996; Jackson, 2019).

The second issue identified in the third wave was
that earlier approaches were relatively naïve with
respect to power relations. The first-wave assumption
that the practitioner and/or client knows best could
result in the coercive imposition of ‘solutions’ and/or
a lack of stakeholder buy-in, which would frustrate
implementation of recommendations for change (e.g.,
Jackson, 1991; Rosenhead & Mingers, 2001). Also,
there was a second-wave, practice-limiting belief that
stakeholder participation in dialogue, in and of itself,
allows the better argument to prevail. This overly
minimises problems of bias, coercion, groupthink,
deceit, ideological framing and disempowerment
(Mingers, 1980, 1984; Jackson, 1982).

A seminal, third-wave response to the power
issue was Ulrich's (1987; 1994) critical systems heu-
ristics. Ulrich's central idea is being critical of the
boundary judgements made by decision makers,
including the OR practitioner him/herself. Nobody
can have a comprehensive viewpoint, so boundaries
are inevitably set with reference to the purposes and
values of decision makers. However, too often,
boundary judgements are taken for granted, so deci-
sion makers (often unknowingly) foist their norma-
tive assumptions on those affected by their
decisions, and the latter's voices are not heard.
Ulrich encourages those involved in and affected by
an OR project to reach agreement in dialogue on
the key assumptions upon which that project should
be based. However, when dialogue is avoided by
decision makers, those affected by their ideas have
the right to make a 'polemical' case to embarrass
the decision makers into accepting discussion. The
key principle is preventing powerful stakeholders
(decision makers and 'experts', including the OR
practitioner) from simply taking their boundaries
and values for granted and imposing them on
others.

Following Ulrich (1994), Midgley et al. (1998)
then reviewed all the second- and third-wave work
on boundaries, and proposed a broader theory and
practice of *boundary critique*. This encourages the
practitioner to explore different possible boundaries,
purposes and values in an OR project, and also to
uncover conflicts (Midgley & Pinzón, 2011) and
processes of marginalisation (Midgley, 1992).
Midgley et al. (1998) argue that boundary critique is
necessary in all projects dealing with complex issues,
as there are likely to be initially-hidden elements of
the situation that need to be accounted for. Indeed,
even deciding whether a problem situation should
be viewed as complex or not requires some up-front
boundary critique.

In terms of the definition of a system given ear-
lier and its implications for systems thinking, this
work deepened our understanding of boundaries:
taken-for-granted boundaries can reflect the struc-
tural entrenchment of power relations in our organ-
isations, institutions and wider society (Jackson,
1985), which can cause major socio-political and
environmental issues (Midgley, 1994). Therefore,
third-wave systems thinkers started talking about
evolving stakeholder perspectives *and* structural
relationships: doing either without the other can
result in systemic resistance to change (Gregory,
2000). However, the starting point for intervention
(following an initial boundary critique) is usually
stakeholder perspectives because it is the stakehold-
ers themselves who can then turn their attention to
structural reform (Boyd et al., 2007). Here we see the
co-existence of both the first-wave understanding of
real-world systems and the second-wave emphasis on
stakeholder perspectives. Methodological pluralism
makes perfect sense in this context, as some
approaches are particularly useful for evolving stake-
holder perspectives (e.g., Checkland, 1981), and
others support intervention in organisational and
institutional structures (e.g., Beer, 1981). Both can be
integrated into an OR project design (e.g., Sydelko
et al., 2021, 2023).

Eventually, research on methodological pluralism
and boundary critique was integrated into a new
'systemic intervention' approach by Midgley (2000).
He recognised that boundary critique could support
deep diagnoses of problem contexts, and these diag-
noses could then inform the design of OR projects,
drawing creatively upon methods from both previ-
ous waves of systems thinking *and* from other tradi-
tions. This work unified the different strands of
third-wave methodology.

Recently, however, there have been discussions
about whether a fourth wave is forming. Current
research foci include whether a universal theory of
systems thinking is possible and necessary (Cabrera
et al., 2023); how to construct a simple narrative of
systems thinking to effectively communicate our
work (Midgley & Rajagopalan, 2021); how arts-
based methods can enhance practice (Rajagopalan,
2020); and what we can learn from neuroscience to
inform methodological development (Lilley et al.,
2022). It remains to be seen whether addressing
these issues will extend the third wave or launch a
fourth wave of systems thinking.

## 2.24. Visualisation³⁹

Visualisation, the graphic (and often interactive) dis-
play of quantitative or qualitative information, has
established itself not only as a powerful working
<figure><img src="image_64.png" /></figure>

modality for many management and engineering contexts (Basole et al., 2022; Lindner et al., 2022), but also as a research field (and research method) in its own right (Eppler & Burkhard, 2008). In this subsection we briefly review the visualisation field, its relevance for Operational Research (including its benefits and caveats), its various types and application contexts, its theoretical perspectives and approaches, as well as its likely future evolution.

Why care about the graphic representation of information, especially for Operational Research? The short answer that research has provided over the last decades to this question is that it provides numerous cognitive, emotional, and social benefits and thus improves our individual and collective ability to make use of information. These benefits include a quicker comprehension of information (Kress & Van Leeuwen, 2006), the detection of important patterns (Bendoly & Clark, 2016), the ability to better discuss information (Bendoly & Clark, 2016; Meyer et al., 2018), or the greater recall of information (Paivio, 1990; Childers & Houston, 1984).

The visual representation of information is not, however, without risks or potential disadvantages (see Bresciani & Eppler, 2015; Basole et al., 2022). Visualisations can be misleading, manipulating, oversimplified, biased, or simply confusing or overwhelming. If, for example, the y-axis of a line chart has been cropped, a small improvement may be mistakenly interpreted as a substantial one. The correct interpretation of information may also require what is often referred to as visual literacy (including data and information literacy, see Locoro et al., 2021) on behalf of the viewers. A diagram is sometimes worth ten thousand words (Larkin & Simon, 1987), but at times it requires that many words of explanation to properly understand it.

To avoid such risks, professionals need to choose the right visualisation format for the task at hand and use it diligently and in line with existing guidelines (such as those made popular by Tufte, 2001) and our perceptual preferences (Ware, 2020). There is research on both of these questions, i.e., on the available types of visualisation (Shneiderman, 1996; Chi, 2000) and on their proper use (see for example Ware, 2020).

In terms of segmenting the different kinds of graphic representations for operations management contexts, one can, at the highest level, differentiate between quantitative and qualitative information visualisation. This distinction is based on the type of information that is represented: in the case of numbers or data this is referred to as quantitative visualisation. Typical examples of this genre of visualisation are business intelligence dashboards or

simple overhead slides with bar and pie charts. Pie charts, however, are perceptually problematic, as we cannot visually distinguish pie section sizes accurately, let alone compare them in different pie charts. In the case of concepts, arguments, ideas, or issues this is often labelled as qualitative visualisation. Argument mapping (Bresciani & Eppler, 2018) is one approach within this group that is already used in different management contexts. Whereas quantitative visualisation is mostly software-based, qualitative visualisation can be done on paper, walls, flipcharts, and other physical media.

There are, of course, also instances of mixed visualisations that combine quantitative information and qualitative insights in a single image (see Eppler & Kernbach, 2016, for such combined representations). An example of such a hybrid visualisation would be a business intelligence dashboard (consisting of charts) that reveals conceptual diagrams through mouse-over comments (or vice versa).

The aforementioned distinctions are part of one tradition of visualisation research, namely the classification or taxonomic approach (see, for example, Shneiderman, 1996). This research stream or visualisation perspective aims at providing a systematic and comprehensive overview on all forms of information visualisation that are useful for the engineering or management sector.

Another theoretical framing of the visualisation field comes from the literature on graphic representations as boundary objects that span professional frontiers and connect expertise across disciplines – through the help of joint visual displays (Black & Andersen, 2012). This stream of literature emphasises the dual nature of visualisations to be simultaneously fixed and fluid, clear and open to multiple interpretations or functions (for example the blueprint chart of a building or the Gantt chart for a project).

A third influential approach to make sense of the use and impact of visualisation in workplace settings is the cognitive or collaborative dimensions approach (Green et al., 2006; Bresciani & Eppler, 2018). This theoretical lens sheds light on the different qualities of graphic representation that make them more or less suited to be collaboration catalysts—for example based on their (procedural or representational) clarity, unevenness or facilitated insight.

A similar theoretical perspective is the affordance approach (Meyer et al., 2018), that highlights the different cognitive ‘invitations’ or incentives that visualisations can provide, such as their attention grabbing effect, their interpretive flexibility or their story telling potential.
<figure><img src="image_65.png" /></figure>

Another influential research stream focuses not on how images are best designed for their application contexts, but on how they are appropriated and interpreted. Many researchers with this research stream employ a semiotic approach to the study of visualisation based on the seminal work by Kress and Van Leeuwen (2006). This approach is also informed by (research-based) insights into our perception of visual information, but additionally enriched by the conventions and (cultural) traditions that govern our interpretation processes of graphic symbols.

There are of course many other research streams discussing the design and use of visualisation in management or engineering contexts. While some of them focus on particular visualisation formats, such as diagrams, maps, 3D models, or sketches, others focus on certain application contexts, such as (big data) analytics, creativity and innovation, production simulation, or planning. This brings us to the actual application contexts of visualisation.

When are the visualisation formats and perspectives described above mostly used? Typical application contexts for information visualisation are strategising (Eppler & Platts, 2009) and planning sessions of managers and experts (for example with the help of Gantt charts or technology roadmaps (Blackwell et al., 2008), risk analysis (Eppler & Aeschimann, 2009), ideation and problem solving workshops (using mind mapping, argument mapping, or simple sketching) in product development and business model innovation contexts (using canvases and other visual frameworks), training and development (including knowledge transfer and retention), as well as for performance management, simulations and forecasting or scenario workshops. Last but not least, visual methods are also used as research tools in their own right (Comi et al., 2014) to enable better access to practitioners' expectations, experiences, or priorities (Bell & Davison, 2013).

Many new application contexts are currently emerging within the realm of Operational Research and management, including new forms of visualisation. These novel forms include trainings and simulations in three dimensional immersive settings such as the Metaverse or augmented reality visualisations for simulations or assisted on-site decision making or operations. Another fascinating recent phenomenon consists of images created by artificial intelligence based on user instructions (such as DALL-E or similar systems). Such artificially created, at times photo-realistic images, can help (for example) in the ideation, service innovation, or product development context. The rise of artificial intelligence also impacts the interpretation of information visualisation: A case in point are data visualisation packages (such as Tableau or

PowerBI) that (through AI) already assist the user in the exploration and interpretation of the provided data charts and suggest areas for deeper analysis. The visualisation field is thus a highly dynamic area with great promise, both in terms of its methodological repertoire as well as its application scope.

# 3. Applications

## 3.1. *Auctions and bidding*<sup>40</sup>

The 2020 Nobel Prize in Economics was awarded to Paul Milgrom and Robert Wilson for their improvements to auction theory and inventions of new auction formats. Their theoretical discoveries have improved auctions in practice and benefited sellers, buyers and taxpayers around the world (RSAS 2020).

An *auction* is usually a process of selling and/or buying goods or services that are up for bids. A *bid* is a competitive offer of a price and/or quantity tag for a good or service. Auction is a particular way to determine prices and allocation of goods or services.

Auctions have been used since antiquity for the sale of a variety of objects. Today, both the range and the value of objects sold by auction have grown to staggering proportions (Krishna, 2010). The contexts within which auctions are applied include art objects, antiques, rare collectibles, expensive wines, numerous kinds of commodities, livestock, radio spectrum, used cars, real estate, online advertising, vacation packages, wholesale electricity and emission trading, and many more.

In the basic economic model, the price of a good or service is obtained when the supply and demand meet and it is normally an equilibrium value after adjustments over time. However, in some situations such adjustments cannot be made to reach an equilibrium. As Haeringer (2018) points out, auctions are commonly used when (a) sellers and/or buyers have little knowledge of what would be the “right” price (e.g., a tract of land with an unknown amount of oil underground); (b) the supply is scarce (e.g., an art painting); (c) the quantity or quality of the good changes very frequently (e.g., electricity or fish); and (d) transaction frequency is low (e.g., radio spectrum).

Bidders behave strategically. Based on the available information, what they know themselves and what they believe other bidders to know, it is difficult to analyse the outcomes of different bidding rules. This is where auction theory comes in, which is closely linked to many other domains of operational research, such as game theory (§2.11), behavioural OR (§2.2), combinatorial optimisation (§2.4), computational complexity (§2.5), linear programming (§2.14) and integer programming (§2.15).
<figure><img src="image_66.png" /></figure>

### 3.1.1. Key concepts and results

As detailed by Haeringer (2018), an auction consists of the following component rules: (a) bidding format (e.g., a price, a price and a quantity, a quantity only, or a list of items if more than one item are for sale); (b) bidding process (e.g., auction stopping criteria and information for bidders); and (c) price and allocation (i.e., auction winner(s) and the final price(s)).

In studying auctions, it is important to bear in mind the underlying model of valuation, the values attached to the objects by individual buyers and/or sellers. If the value, though unknown at the time of bidding, of an object is the same for all bidders, then the evaluation is of a common value. More generally, in situations of private values, the value of an object varies from one bidder to another. These values can be independent or interdependent.

If there is only one item to be sold, we have the most basic auction. Some common forms of such simple auctions are well known. In an open-outcry auction, an auctioneer takes bids from the participants and at some point of time a winner is declared, who will then pay for the item at some price related to the bids. If all bids follow the dynamics of ascending prices and the winner is the highest bidder, who pays his bidding price, then we have an English auction. In the case of private values, the English auction is strategically equivalent to the second-price sealed-bid auction (Krishna, 2010), in which bidders submit written bids without knowledge of other bids. The highest bidder wins but pays the price that is the second highest in the auction. On the other hand, if the auctioneer in an open-outcry auction begins with a high asking price (in the case of selling) and lowers it until some participant accepts the price (or until it reaches a predetermined reserve price), then we have a Dutch auction. This type of open-outcry descending-price auction is most commonly used for goods that are required to be sold quickly such as flowers, or fresh produce (Mishra & Parkes, 2009), as it has the advantage of speed since a sale never requires more than one bid. The Dutch auction is strategically equivalent to the first-price sealed-bid auction (Krishna, 2010), which is the same as the second-price sealed-bid auction except that the winner pays his bidding price.

If there are multiple homogeneous (resp., heterogeneous) items to be sold, we have a multiunit (resp. combinatorial) auction.

One of the most important results in auction theory is the revenue equivalence theorem (Heydenreich et al., 2009; Nisan, 2007), which in its simple form states that when bidders' valuations are private and uniformly distributed, the expected revenue of the

seller is the same in the English (or second-price) and Dutch (or first-price) auctions.

In a forward auction, a number of buyers compete to obtain goods or services from one seller (e.g., spectrum auction). In contrast, in a reverse auction, a number of sellers compete to obtain business from one buyer (e.g., electricity capacity market). In a double auction, there are multiple sellers and multiple buyers (e.g., wholesale electricity market). Potential buyers submit their bid prices and potential sellers submit their ask prices to the market institution, which then chooses the price that clears the market. At this price *p*, all the sellers who asked no more than *p* sell and all buyers who bid at least *p* buy.

The main issues that guide auction theory involve a comparison of the performance of different auction formats. Naturally revenue is by far the most common yardstick from the seller's perspective. However, if the auction concerns the sale of a publicly held asset to the private sector, such as the case of spectrum auction, efficiency may be more important – the object ends up in the hands of whoever values it most ex post, or in the more general case of multiple items, the sum of realised values for all participants is maximised. Besides, simplicity and susceptibility to collusion among bidders are among other criteria for the choice of an auction format (Krishna, 2010).

### 3.1.2. Some best practices

One of the most important applications of auction theory is the implementation of spectrum auctions to allocate licenses to mobile phone carriers, who act as buyers in the forward auction. One of the auction formats introduced by Milgrom (1987) and Wilson (1998) was first used in 1994 by the US authorities to sell radio frequencies. This practice has since spread globally and led to great benefit to society.

There can be many ways for allocating licences in general. In addition to an auction, it can proceed either with a lottery in which any interested party would just have to sign up possibly with an entry fee, or with a beauty contest in which all those wishing to obtain a licence are required to present a case and the final winners would be selected by a committee. Haeringer (2018) provides a detailed argument why a lottery or a beauty contest is inappropriate in the case of radio frequencies and why an auction offers a more attractive solution. There are a number of issues in selling licences of spectrum, such as those concerning collusion, demand reduction and lack of entry. Of particular relevance for common-value auctions is the so-called winner's curse – the winner pays too much
<figure><img src="image_67.png" /></figure>

and loses out. Haeringer (2018) discusses how a suitable format of an auction can be used to address these issues.

A wholesale electricity market exists when competing generators offer their electricity output to retailers. Double auctions are normally used for such a market (Mayer & Trück, 2018). By its nature electricity is difficult to store and has to be available on demand. Consequently, unlike other products, it is not possible, under normal operating conditions, to keep it in stock, ration it or have customers queue for it, so the supply should match the demand very closely at any time despite the continuous variations of both (Weron, 2014, §3.19). The supply uncertainty becomes particularly relevant with an increased use of green energy (such as solar, tidal, wind energy).

The electricity capacity market becomes necessary to build and maintain electricity capacity that may be called upon in time of need to maintain the grid balance. In the UK's system for purchasing Short Term Operating Reserve (STOR) for electricity supply (National Grid ESO 2022), the National Grid maintains a reserve generation ability in case of sudden demand or supply variations. Part of the operating reserve is made up by contracts through auctions. In this market, the bids come as electricity capacity, so the National Grid determines the right amount of capacity to reserve from a competitive tender. Tenders are assessed on the basis of availability prices and utilisation prices together with a consideration of response times and geographical locations. In this reverse auction, the National Grid acts as the buyer, while individual electricity operators act as sellers. Extensive studies on such auctions can be found in Chao and Wilson (2002) and Schummer and Vohra (2003). More recently, Anderson et al. (2017, 2022) investigate the problem under more general settings. They show that a natural equilibrium is not only efficient but also optimal for individual bidders.

The Internet is a new exciting venue for auctions and eBay is certainly the most well-known auction place on the Internet. Auctions on eBay face new challenges due to the nature of the Internet, where an auction can take days or even weeks and potential buyers can bid whenever they want. In response, eBay uses proxy bidding wherein a computer programme is used to bid on behalf of the bidder, who enters an auction effectively with a maximum bid. The computer programme raises rival bids by the minimum increment set beforehand as long as it is below the maximum bid. It is easy to see that such an auction is effectively a second-price auction in which the amount entered by the bidder serves as the bidding amount. Ariely and Simonson (2003)

propose an analytical framework for studying bidding behaviour in online auctions. Chothani et al. (2015) provide an overview of online auctions. Hickman (2010) analyses significant differences between electronic auctions and non-electronic auctions.

### 3.1.3. Closing remarks

There are many excellent surveys of auction theory and applications. Milgrom (1985) and McAfee and McMillan (1987) provide a cogent account of the theory of single-object auctions and explain many extensions and applications of the theory. Milgrom (2004) provides a comprehensive introduction to modern auction theory and its important new applications. Samuelson (2014) examines the use of auctions, paying equal attention to theory and practice. Haeringer (2018) and Kagel (2020) give respectively an overview of empirical and experimental studies on auctions. Cassady (1967) provides a colourful and insightful overview of real-world auction institutions.

## 3.2. Community operational research⁴¹

Community Operational Research (COR) reflects the aspirations of OR's early theorists and practitioners of “science helping society” (Cook, 1973/1984, p.36). There is a long tradition of COR practice that includes Ackoff's, 1970 engagement with members of the Mantua ghetto in Philadelphia, Cook's projects with inner-city community organisations (Cook, 1973; Luck, 1984), Beer's work with the Allende Government in Chile (Beer, 1981), and numerous projects undertaken from the University of Bath (Jones & Eden, 1981; Sims & Smithin, 1982). See Jackson (2004) and Rosenhead (1993) for a discussion of such work. Although these early examples of COR are significant, they were far from the norm as a focus on “science helping the establishment” predominated (Cook, 1973, 1984, p.36). In recognition of this, Rosenhead (1986) posed the question of “who O.R. worked for (“custom”)” (p.335) in his inaugural address as President of the UK’s Operational Research Society. Rosenhead answered his own question in stating that the customers were, in the main, “big business, public utilities, the military and central government departments, with a thin scatter of local governments and health and other public authorities” (p.336) to the neglect of other groups “located outside the power structure” (p.337). Rosenhead (1986) not only discussed the custom of OR, but also tackled the related issue of practice in asserting that “The evolved forms of tools reflect the circumstances of their use” (p.338). Hence, mainstream OR's
<figure><img src="image_68.png" /></figure>

focus on quantification and modelling reflected its
customers' privileging of technical matters over dia-
logue between stakeholders and issues of emancipa-
tion (Rosenhead, 1993, drawing on Habermas,
1972), and involved the use of OR methods "beyond
the comprehension of most people" (Rosenhead,
1986, p.339), effectively masking the social and
value-laden nature of much decision making. In
contrast, concerns for better mutual understanding
in society and freedom from oppressive power rela-
tionships inspired the call for a more transparent
OR to support "a more lively, complex and elabor-
ate social process of decision-making" (Rosenhead,
1986, p.339).

Such was the impact of Rosenhead's inaugural
speech and his efforts within the OR Society that
engagement with non-traditional clients quickly
became legitimised and formalised through the
founding of a research centre, the Community
Operational Research Unit, located at Northern
College, which later moved to its present location at
the University of Lincoln, UK. In the 1980s, the OR
Society also provided support for the Centre for
Community OR at the University of Hull (later to
be merged into the Centre for Systems Studies), and
the Community OR Network of around 300 OR
practitioners. The universities of Lincoln and Hull
continue to actively practice and promote COR.
More recently, in 2011, the OR Society created a
Pro Bono OR scheme that connects volunteer ana-
lysts with good causes⁴²,⁴³.

Given the multi-faceted and often complex nature
of COR projects, there appears to be no one par-
ticular OR approach that has emerged as dominant.
There are, though, three streams of complementary,
sometimes overlapping, approaches that have proven
useful in multiple reported cases of COR:

1. Problem Structuring Methods (PSMs) are a collection of approaches that offer decision support by “way of representing the situation (that is, a model or models) that will enable participants to clarify their predicament, converge on a potentially actionable mutual problem or issue within it, and agree commitments that will at least partially resolve it” (Mingers & Rosenhead, 2004, p.531). The modelling effort may involve clarification of normative agendas through dialogue, as PSMs are largely founded on interpretivist or social constructivist epistemologies (Jackson, 2006). For more on PSMs see §2.20.

2. Critical Systems Thinking (CST) and Critical Systems Practice (CSP) focus on the distinction of a broad range of problem contexts and the development of systems-based methods appropriate to those contexts (Flood & Jackson,

1991a, 1991b; Mingers & Brocklesby, 1997). Having a broad range of systems methodologies to draw on is necessary but not sufficient for good practice. Consequently, Jackson (2000) encapsulated the notion of good practice in his statement of three commitments of CSP: critical awareness, relating to critique of the different systems methodologies, and social awareness of the societal and organisational context; improvement, referring to the achievement of something beneficial, reflecting a cautious approach to the aspiration of universal liberation; and pluralism, the need to work with multiple paradigms without recourse to some unifying metatheory. For more on systems thinking, see §2.23.

3. Systemic Intervention (SI) developed out of CST and took as its two primary concerns critical reflection on boundaries of inclusion and exclusion (Churchman, 1970; Ulrich, 1983, 1987; Midgley, 2000) and methodological pluralism. Midgley (2000) defines SI thus: “If intervention is purposeful action by an agent to create change, then systemic intervention is purposeful action by an agent to create change in relation to reflection on boundaries” (p.129). He shows how exploring boundaries informs the methodological design of a project, with the meaningful engagement of communities built in. For more on SI see section §2.23.

These three streams of approaches have much in
common with action research (AR; Levin, 1994;
Midgley, 2000; Mingers & Rosenhead, 2004) and,
perhaps not surprisingly, AR has been a focus of a
lot of COR work. Indeed, the Community
Operational Research Unit explicitly articulated a
working philosophy of AR following the traditions
established in Latin America and Scandinavia
(Thunhurst, 1992). Over the years, a considerable
and diverse body of COR work has amassed, with
some contemporary and notable examples including
conference papers (e.g., Wong & Hiew, 2020), case-
based research papers (e.g., Rosenhead & White,
1996; Deutsch et al., 2022; Paucar-Caceres et al.,
2022; Pinzon-Salcedo & Torres-Cuello, 2022;
Chowdhury et al., 2023), journal special issues (e.g.,
Johnson et al., 2018), project reports (e.g., Stephens
et al., 2018) and edited books (e.g., Bowen, 1995;
Ritchie et al., 1994; Midgley & Ochoa-Arias, 2004;
Johnson, 2012a).

What counts as COR is not a simple matter
though, and there have been several papers over the
years that have critically discussed this (see for
example the different understandings reflected in
Midgley et al., 2018, and White, 2018). Importantly,
<figure><img src="image_69.png" /></figure>

Johnson and Smilowitz (2012) suggest that some examples of COR might be more appropriately classed as capacity-building instead of “applications based on analytic models intended to provide specific policy and operational guidance to decision-makers in a way that extends existing theory and methods” (p.39). While some COR might indeed be classed as capacity building (for example, Boyd et al., 2007, are explicit that capacity building was part of their project), it is important not to confuse such interventions with those that are based on the use of models of a qualitative rather than quantitative nature. Indeed, the commitment to knowledge being embedded within the client organisation (Klein et al., 2007), handing over tools and techniques (Gregory & Jackson, 1992a, 1992b; Boyd et al., 2007; Gregory & Ronan, 2015) and self-organised learning (Herron & Mendiwelso-Bendek, 2018) serve to bring about capability-building alongside model building and the use of analytical approaches at the local level, which does not rule out modelling and data analysis. The needs and skills of citizens and associated groups have moved on since the 1980s, such that the tools of OR (data and models) are not so incomprehensible as they might previously have been regarded, and are familiar to most if not all citizens (Caulkins et al., 2008). Indeed, Hindle and Vidgen's (2018) work with the Trussell Trust on mapping food bank data demonstrates that charities can make good use of big data and data visualisation.

Although there are examples of COR projects being undertaken world-wide, sustained organised support has been most evident in the UK and US. Johnson (2012a, 2012b) brought renewed interest to the field in the US with his promotion of a stream of activity that goes by the title of Community-Based Operations Research (CBOR). Johnson and Smilowitz (2012) define CBOR as “a subfield of public-sector OR… that emphasizes most strongly the needs and concerns of disadvantaged human stakeholders in well-defined neighborhoods. Within these neighborhoods, localized characteristics vary over space and exert a strong influence over relevant analytic models and policy or operational prescriptions” (p.38). Complementary to the remit of CBOR is the Institute for Operations Research and the Management Sciences (INFORMS) Pro Bono Analytics® initiative⁴⁴.

Whilst COR, CBOR and pro-bono OR may be said to have a related remit, it is worth mentioning a key difference, “COR takes as its remit to work with (i.e., to take as its clients) disadvantaged community groups themselves” (Rosenhead, 2013, p.610), whereas CBOR and pro-bono OR are more focused on making OR and analytics available to third sector and public organisations. This

distinction is not undisputed (Midgley et al., 2018), but the important thing is that such efforts, geared to meaningful community engagement, have not only enabled community access to OR, but have also provided a strong impetus for its theoretical and methodological development in a way that honours the legacy of OR's early founders.

As we have an increasing number of ways to connect with others and form communities, it would be easy to assume that, going forwards, COR merely needs to develop new forms of practice to support communities in these different realms. But, in a VUCA (volatile, uncertain, complex and ambiguous) world (Bennis & Nanus, 1985), we must be alert to the need to challenge simple assumptions. Rather, there is a good argument for a critical turn in COR involving the explicit examination of underpinning values and ethics (Córdoba & Midgley, 2006; Jackson, 2006). Midgley and Ochoa-Arias (2004) have already claimed that “if practitioners do not reflect on the different visions that it is possible to promote, then there is a danger that they will default to the understanding of community that is implicit in the liberal/capitalist tradition” (p.259). This brings with it missed opportunities to pursue more challenging and empowering practices that enable political activism and give some of the most marginalised people in our society a meaningful voice in OR projects. Much of COR has arguably been quite tame, doing good in local communities without challenging the political status quo (Wong & Mingers, 1994), but in an era of climate change, biodiversity loss, rising nationalism, insecure employment, mass migration, and increasing wealth inequality, a new, more critical agenda for COR is urgently needed.

### 3.3. Cutting and packing⁴⁵

Cutting and packing (C&P) problems are geometric assignment problems, in which small items are assigned to large objects such that a given objective function is optimised and two basic geometric feasibility conditions hold, specifically containment and non-overlap. They appear in a wide range of settings, but are most commonly investigated for applications in manufacturing and transportation. For example, cutting pattern pieces from material or packing boxes into containers. These are combinatorial optimisation problems and NP-hard. Depending on the size or geometry of the problem, there exists strong formulations that can be solved using exact methods. However, there remains many open problems that have instances that cannot be solved to optimality, or computational times are impractical for applications in practice. Moreover,
<figure><img src="image_70.png" /></figure>

there are problems where bounds are weak and only toy instances can be solved to optimality. As a result, heuristics remain an important tool in C&P.

Given the wide variety of C&P problems, Dyckhoff (1990) and later Wäscher et al. (2007) defined a typology of problems using the following dimensions:

### 3.3.1. Objective function

* Output maximisation: packing the greatest value of items in a given fixed dimension finite number of large object(s).

* Input minimisation: pack all items using the minimal number of fixed dimension objects or the minimum size large object with at least one unconstrained dimension.

### 3.3.2. Assortment of small items (items to be packed)

* All items are identical.

* Weakly heterogeneous: few item types given the total number of items.

* Strongly heterogeneous: many item types that are unique or have few copies.

### 3.3.3. Assortment of large items

* Single large object: fixed dimension for output maximisation, open dimension(s) for input minimisation.

* Multiple large objects: fixed dimension, either identical or heterogeneous.

These distinctions lead to named problem types, e.g., bin packing problems (BPP) are input minimisation problems, with strongly heterogeneous small items and multiple large objects, while a knapsack problem (KP) shares the same characteristics but is an output maximisation problem. Note that problem names and their definitions are not universally accepted or consistently used, so researchers should check the articulated problem definition in the paper when selecting literature.

The following focuses on two-dimensional (2D) and three-dimensional packing (3D) as these include the unique challenges of the geometric constraints associated with C&P problems. One-dimensional (1D) problems remain interesting and challenging (see Martinovic et al., 2018; Munien & Ezugwu, 2021). For an introduction to C&P, see Scheithauer (2017).

### 3.3.4. Geometry

Handling the geometric characteristic of C&P problems adds significantly to the computational burden and the number of variables needed in a model. These increase with the spatial dimensions and with the irregularity of the shape of the small items. For

1D problems, the geometric constraints of overlap and containment are trivial. Regular shapes (rectangles, boxes, circles, spheres) add complexity through additional item location variables x, y (and z) coordinates, and dimensions: length, width (and depth). However, the common characteristics of the shape mean these are straightforward to model. Pairwise constraints between items and between each item and the boundary of the large object ensure feasibility.

In the case of irregular shaped items, accurate non-overlap constraints cannot be reduced to comparing a set of common dimensions. While the item location is still determined by a defined origin, the arbitrary nature of the shape significantly increase the complexity of assessing geometric feasibility. At a basic level, it requires testing for edge intersections between items and containment of one item inside another. Methods to reduce the complexity include the nofit polygon, raster method and phi-functions in 2D and voxels and phi-functions in 3D. Bennell and Oliveira (2008) provide a tutorial in geometric methods for 2D nesting problems. Lamas-Fernandez et al. (2022) describe approaches for modelling 3D geometry. Developing solution methods for irregular packing problems requires a comprehensive, fast and robust geometry library.

### 3.3.5. Constraints

There exists a wide range of practical constraints arising from the applications. These may relate to the material being cut having defects or quality variability, the cutting tool requiring space between items or constraints on the types of cut. There may be sequencing constraints or assignment constraints that include precedence or prevent/require the packing of items together. In 2D rectangle C&P, a common requirement is guillotine cuts where all cuts must be orthogonal and span the entire width or breadth of the rectangular material sheet. Furthermore, the number of alternate cuts (e.g., a switch from vertical to horizontal cuts) may be restricted. Applications in 3D container loading provide a challenging set of constraints on the arrangement of boxes to ensure weight distribution, horizontal and vertical stability of load and consider the weight baring strength of the stacked boxes. Bortfeldt and Wäscher (2013) describe the different types of constraints and their adoption by researchers.

### 3.3.6. Two-dimensional problems

In two dimensions, research has focused on rectangle packing problems, and irregular shape packing problems, often called nesting problems. There is
<figure><img src="image_71.png" /></figure>

also a smaller body of literature on circle packing (Hifi & M'Hallah, 2009).

Exact solution approaches to the 2D rectangle packing problem are reviewed by Iori et al. (2021) and cover the main problem types. The paper evidences the recent advances in exact methods for these problems while identifying a number of open problems that remain very challenging. Specifically, they identify the open-dimension problem, some specific instances of BPP, and problems with multiple heterogeneous large objects. Moreover, nesting problems remain a rich area of research for developing high fidelity and scalable exact methods.

Heuristics and metaheuristics are a natural choice, particularly for problems with a large number of small items. Early research focused on placement heuristics such as First Fit and Next Fit for BPP (Coffman Jr. et al., 1980), Bottom Left and Bottom Left Fill for open dimension problems (Albano & Sapuppo, 1980; Chazelle, 1983; Burke et al., 2004b). These place items into the large object in a given sequence according to the placement rule and observing any additional placement constraints. A natural evolution of this approach is to apply a metaheuristic to re-sequence the packing order to obtain better solutions, of which there are many examples.

The 2D rectangle identical item packing problem is known as the manufacturer's pallet loading problem. Research is mature with exact methods and heuristics that perform well across benchmark instances. Silva et al. (2016) comprehensively review these problems. Fewer papers have looked at the case where the small item is irregular, for example cutting metal blanks (e.g., Costa et al., 2009).

Output maximisation problems focus on the 2D rectangle knapsack problem, with few articles considering problems with weakly heterogeneous data. Problems cover guillotine and non-guillotine cutting and item values may be equivalent to area or have an assigned value. Furthermore, the constrained variant places upper and lower bounds on the number of each item type placed. The guillotine variant where area and value align has a fast exact method (Oliveira & Ferreira, 1990). Non-aligned item values and non-guillotine cutting is still challenging. Cacchiani et al. (2022b) includes a summary of 2DKP. Packing a single large object is often a component of a larger practical problem, where the decision problem of whether a set of rectangles will fit into a fixed dimension rectangle, referred to as a 2D orthogonal packing problem, is of interest (Clautiaux et al., 2007).

Cutting stock problems have been studied for over 60 years with the seminal paper by Gilmore and Gomory (1965) that described the column generation approach for 2D rectangle cutting stock

problem with guillotine cuts. Most papers focus on ILP/MILP approaches. A notable feature of these problems and how they differ from the BPP, is the way solutions are constructed arising from the data instances. Since there are few item types, but many items of each type, the solutions are composed of pattern types that are repeated across multiple stock sheets leaving a residual problem of unmet demand.

Bin packing problems have been extensively studied and include the guillotine and non-guillotine variant. Lodi et al. (2014) provide a review of BPPs. Early heuristic approaches (Berkey & Wang, 1987) include two-phase algorithms that pack multiple bin width strips and then solve a 1D BPP where the item size is the height of the strip, while single-phase algorithms pack directly into the bins either using a level packing approach or a placement heuristic such as bottom-left. Increasingly, researchers are focusing on exact methods; see for example Pisinger and Sigurd (2007) who use branch and price for variable size and fixed size bins. There are very few examples of bin packing with irregular shapes, where one example is glass cutting (Bennell et al., 2018).

The open dimension problem variant is often called the strip packing problem. This can be formulated as a linear mixed integer programme, although practical size problems are still very challenging. Martello et al. (2003) develop bounds by relaxing the problem so it can be solved as a one dimensional BPP. Placement heuristics (bottom left and bottom left fill) based on sequencing of items within a (meta)heuristic framework are widely used. Hopper and Turton (2001) undertook an extensive analysis of this type of approach.

Nesting problems, where the small items are irregular, are commonly formulated as open dimension problems. Solution approaches are dominated by the use of heuristics and metaheuristics. Bennell and Oliveira (2009) provide a review of these methodologies including using exact models to improve local optima. This approach is also used by Stoyan et al. (2016) who use phi-functions, which allow orientation as a decision variable. In the last decade, researchers have developed formulations that can be solved to a global optimum for small problems. Toledo et al. (2013) approximates the items and the packing area to a discrete set of points allowing the problem to be solved as a MIP model. Alvarez-Valdes et al. (2013) used the nofit polygon to define a finite set of convex spaces and used binary variables to activate constraints.

### 3.3.7. Three dimensional problems

These problems are solved across the range of problem types, largely considering single container
<figure><img src="image_72.png" /></figure>

output maximisation, multi-container input mini-
misation and the open dimension problem. For
packing boxes, the mix of constraints addressed
across the literature is inconsistent and frequently
not congruent with industry standards. Solutions to
the problem focus on building walls, layers or blocks
of identical boxes. See Zhao et al. (2016) for a com-
parative review of algorithms including exact meth-
ods. Recent papers are now looking at the
additional constraints arising from a vehicle, such as
axle load and stability under breaking and acceler-
ation (see Ali et al., 2022). 3D packing of irregular
shapes is an open problem that had increasing rele-
vance in areas such as additive manufacturing.
Efficient handling of the geometry is a significant
factor in the solution approach along with the level
of fidelity required for the application.

3.3.8. Data

Across all problem types there are standard data
sets and data generators that provide a useful means
to test the effectiveness of solution approaches.
Many of these are listed on the EURO Special
Interest Group on Cutting and Packing (ESICUP)
website⁴⁶.

**3.4. Disaster relief and humanitarian logistics<sup>47</sup>**

Humanitarian logistics (HL) is one of the key appli-
cation areas that Operational Research (OR) has
been offering solutions to improve the welfare of
the society under difficult circumstances.
Humanitarian logistics problems are highly relevant
in today's world due to various challenges including
but not limited to, climate change and its conse-
quences (increases in extreme weather events such
as heatwaves, floods), natural disasters (e.g., earth-
quakes, tsunamis), man-made conflicts (e.g., Syria
and Ukraine crises) and health-related catastrophic
events (e.g., pandemics). Humanitarian logistics
operations involve complex systems with multiple
stakeholders such as victims, planners, public/private
service providers, volunteers, general public and
media, each with their own preferences and prior-
ities; and the inherently challenging decisions of
scarce resource allocation have to be made over a
long time span, under high uncertainty. We use
humanitarian logistics as an umbrella term, which
covers relief logistics, disaster logistics, and develop-
ment logistics. The humanitarian literature uses all
these terms interchangeably. Actually, disaster logis-
tics and relief logistics should refer to the cases
where a disaster is/was/is expected to be in action
whereas development logistics refers to cases which
aim to improve daily life.

In relief logistics, the operations require advanced
planning, hence the authorities are constantly facing
challenges in the four main stages of: mitigation,
preparedness, response, and recovery (Altay and
Green III 2006; Çelik et al., 2012; Kara & Savaşer,
2017). In Table 3, we list some of the most fre-
quently considered problems, categorised based on
the phase that they arise. As seen in the table, miti-
gation and preparedness phases mostly consist of
activities related to planning, which involves net-
work design, location, allocation and routing opera-
tions as well as provisioning processes that include
inventory and other supply chain-related decisions
(see, e.g., Balcik & Beamon, 2008; Rawls & Turnquist,
2010, for applications in location and prepositioning,
respectively).

Response activities occur after the crisis or the
disaster hits. In this phase, the aim is providing a
rapid response, prioritising the survival needs. This,
however, does not preclude considering efficiency
in these operations as the system requires scarce
resource allocation such as personnel, equipment
and supplies across demand points, invoking a need
for good decision support mechanisms. In line with
this need, a large body of work is devoted to the
application problems arising in this phase. Finally,
recovery phase focuses mainly on debris manage-
ment and infrastructure repair and restoration. Most
of the mentioned operations involve additional deci-
sions regarding workforce planning and scheduling
and require structured methods for data manage-
ment, information sharing and coordination, which
are key for effective response (Altay & Labonte,
2014). Some of these models require quantification
of human suffering due to lack of services or goods
and the deprivation cost can be used for this pur-
pose, as discussed in detail in Holguín-Veras et al.
(2013).

Not all humanitarian operations are triggered by
challenges stemming from a single well-defined event.
There are crises that can not be attributed to a single
cause e.g., famine in under-developed countries.
There are well-established efforts in the development
logistics literature to alleviate the effects of such cri-
ses. Some examples are global health projects for
increasing access to health coverage and fighting dis-
eases that occur in low and middle-income countries
on a wide-scale, such as malaria and AIDS, through
distribution of effective tools and/or medication.
Vaccine development and distribution to poorer
regions as well as distribution of other basic needs to
the deprived populations: food aid distribution
(Rancourt et al., 2015; Mahmoudi et al., 2022), clean
water network design and distribution (Laporte et al.,
2022), energy, education and hygiene provision are
also widely considered. A recent trend is utilising
<figure><img src="image_73.png" /><figcaption>**Table 3.** Problems in relief logistics.</figcaption></figure>

<table><thead><tr><th>Phase</th><th>Main problems considered</th><th>Main decisions involved</th><th>Main concerns</th></tr></thead><tbody><tr><td rowspan="10">Preparedness and mitigation</td><td>Infrastructure and network structuring</td><td>Network design<br>(mainly for strengthening purposes)</td><td>Connectivity</td></tr><tr><td>Risk assessment</td><td>Prioritization<br>(of roads, buildings, arcs)</td><td>Data preparation<br>for further analysis</td></tr><tr><td>Evacuation planning</td><td>Location of gathering points<br>Allocation of evacuees<br>Routing</td><td>Traffic<br>Accessibility<br>Evacuation Time<br>Behavioral factors</td></tr><tr><td>Shelter location (and allocation)</td><td>Selecting among potential locations</td><td>Ensuring accessibility<br>Shelter utilization<br>Infrastructure (Risk)<br>Behavioral factors</td></tr><tr><td>Prepositioning</td><td>Locating point of distributions<br>Provisioning (of supplies)<br>Related supply chain decisions<br>(supplier selection, allocation and routing)</td><td>Budget<br>Accessibility<br>Fairness<br>Speed<br>Efficiency</td></tr><tr><td>Supply chain (procurement) planning</td><td>Contract design</td><td>Quality<br>Efficiency</td></tr><tr><td>Damage assessment</td><td>Demand assessment</td><td>Speed</td></tr><tr><td>Search and rescue operations</td><td>Infrastructure assessment<br>Team formation and allocation</td><td>Accuracy<br>Speed<br>Fairness<br>Capacity</td></tr><tr><td>Evacuation</td><td>Location<br>Allocation<br>Routing</td><td>Speed<br>Risk</td></tr><tr><td rowspan="4">Response</td><td>Shelter management</td><td>Location<br>Allocation<br>Related supply chain decisions<br>(distribution, routing)</td><td>Physical environment<br>Risk<br>Accessibility<br>Fairness<br>Utilization</td></tr><tr><td>Donation management/<br>Resource allocation</td><td>Allocation<br>Routing<br>Inventory Management</td><td>Fairness<br>Efficiency</td></tr><tr><td>Debris management</td><td>Network design<br>Prioritization and scheduling<br>(deciding which nodes/arcs to clean first)</td><td></td></tr><tr><td>Recovery</td><td></td><td></td></tr></tbody></table>

cash and voucher distribution whenever possible, since it is a method that respects human dignity, avoids complications of relief item logistics and supports the local market (Karsu et al., 2019).

The recent COVID-19 pandemic has also motivated a wide range of HL applications such as personnel protective equipment allocation and distribution, frontline workforce planning and system design for testing, tracing and vaccination (Farahani et al., 2023).

The HL literature also integrates newer technologies to the delivery systems: There has been recent attempts to use drones in the last mile distribution as they constitute a convenient tool to reach remote areas in short time (examples include delivery of blood samples, vaccines and food aid; see also Gentili et al., 2022; Ghelichi et al., 2021; Alfandari et al., 2022).

OR offers decision support for humanitarian settings based on a wide range of quantitative and qualitative tools. Mathematical modelling and optimisation is used in almost all problems arising in HL to make the related location, allocation, routing and network design decisions. The models are shaped by the priorities in the phase and constraints imposed by the physical infrastructure, resource

availability as well as the social, economic and cultural environment. As the underlying technical problems are difficult to solve, various mathheuristic and metaheuristic approaches are employed. There is also an increasing trend in using system dynamics (Besiou & Van Wassenhove, 2021) and empirical analysis (Pedraza-Martinez & Van Wassenhove, 2016).

There is no one-size-fits-all methodology but some key properties require specific methods to be used. In most relief logistics problems the environment is highly stochastic, calling for applications of forecasting and stochastic programming. The uncertain factors include but are not limited to the number of affected individuals, the extent of the effect, types of needs, and usability status of the infrastructure and other resources. Multiple stakeholders and conflicting criteria are involved, requiring multicriteria decision making (Ferrer et al., 2018) approaches. Unlike commercial logistics systems, fairness is a key concern in humanitarian settings. Fairness or equity, however, is hard-to-quantify and is context dependent: a rule that is considered fair under some circumstances may not be deemed so in others. The policy makers may want to prioritise beneficiaries based on attributes such as socio-economic status
<figure><img src="image_74.png" /></figure>

and hence ensure vertical equity or may consider them indistinguishable and seek horizontal equity (Karsu & Morton, 2015).

HL has been receiving attention with an increas-
ing rate, which led to many review studies that the
interested reader can refer to: see, e.g., Luis et al.
(2012); Çelik (2016); Aringhieri et al. (2017); Besiou
et al. (2018); Baxter et al. (2020); Dönmez et al.
(2021). See also Kunz et al. (2017) for a discussion
on how to make humanitarian research more
impactful for humanitarian organisations and
beneficiaries.

Next we categorise the future of the humanitarian
applications to motivate and direct new researchers:

* OR is responsive to the difficulties the world is facing and humanitarian challenges are no exception to this. The recent COVID-19 pandemic has shown that relief logistics can be applied in health-related crisis management to provide quick, effective, efficient and fair responses to health care problems. WHO “urges countries to build a fairer, healthier world post-COVID-19” and this is doable with good humanitarian practices relying on OR. The recent efforts in designing fair and efficient systems using OR would contribute in addressing inequities in health and welfare, which have been exacerbated by the pandemics. We believe that there is still room for improvement in adopting a holistic approach and conducting multidisciplinary work when designing such systems. One example is the vaccine implementation and roll-out problem: conceptualising this problem as a sole logistics problem may not be the best practice as the success of any design highly depends on human behaviour. People have different views, risk attitudes and preferences over available options, which affects how any proposed policy will perform. Incorporating such behavioural factors is an important yet scarcely studied issue.

* The underlying technical problems in the HL domain are hard to solve due to uncertainty in various parts of the system, lack of (reliable) data and multiple criteria that are involved. Moreover, a significant portion of these problems are combinatorial optimisation problems, i.e., they require choosing from a prohibitively large set of solutions that are implicitly defined by constraints of the system. Therefore advances in OR methodology to obtain better, quicker solutions to optimisation problems, and in data analytics on handling big data such as the one obtained through geographic information systems (GIS), would pave the way for quicker and better response. Effective data analysis would especially

help when learning from past practice. Indeed,
lessons learned from humanitarian supply chain
practice can also be used in managing supply
chain disruptions in other sectors, as discussed
in (Kovács & Falagara Sigala, 2021).

* UN's Sustainable Development Goals emphasise the global challenges faced including poverty, inequality, climate change, environmental degradation, peace and justice (SDG 2022). As stated in (Street et al., 2016), "Increases in extreme weather events and climate change can compound risks of international food shocks, water insecurity, conflict and other humanitarian emergencies and crises. Difficulty of access to critical resources such water and food may trigger migrations or exacerbate conflict risks." All these areas are, by definition, related to humanitarian operations, hence humanitarian logistics has a lot to offer in these domains.

* The Turkey/Syria earthquakes in February 2023 have clearly demonstrated the importance of effective coordination and strategic planning. Thus, we would like to emphasise the need for collaborative research that brings field expertise (of e.g., municipalities, NGOs and volunteers) and academic know-how together.

3.5. E-commerce<sup>48</sup>

3.5.1. What is E-commerce about?

E-commerce deals with the transactions of goods and services through online communications (computers, tablets, smartphones, etc.). Both business-to-business (B2B) and business-to-consumer (B2C) realisations are observed in practice. In B2B, companies operate their supply chains through online networks. In B2C, products and services are sold directly to consumers. E-commerce sales steadily increased for years and amounted to $5,211 billion worldwide in 2021, with the pandemic being a major contributor.⁴⁹,⁵⁰

E-fulfilment describes all fulfilment activities for e-commerce. All necessary steps for a customer to receive an order after placing are thus referred to as the e-fulfilment process. Due to the nature of the e-commerce domain, these e-fulfilment activities often occur in a city context (Savelsbergh & Van Woensel, 2016). E-fulfilment processes are planning intensive, and creating a profitable business in this environment is challenging. Customer service expectations are high, however, and the customer is more and more in the lead on how and where their orders need to be delivered (the “logsumer” takes an active role in time, price, quality, and sustainability decisions of logistic services DHL, 2013).
<figure><img src="image_75.png" /></figure>

The e-fulfilment process can be divided into three steps, namely (i) order acceptance, (ii) order assembly, and (iii) order delivery (Campbell & Savelsbergh, 2005). For most online companies, these steps take place separately, one after the other. However, new on-demand companies have considerably shortened lead times and perform these steps simultaneously (Waßmuth et al., 2022).

During *order acceptance*, customer requests arrive on a retailer's website and ask for service. As fulfilment capacities are limited (for example, delivery capacities), the retailer wants to accept the most profitable subset of all customer requests. However, customer requests arrive one at a time. Thus, the retailer does not know the total delivery costs until all customers are accepted and the final delivery route is planned. In addition, when a request is accepted, the retailer does not know whether requests with higher revenues will arrive afterward for which capacity should have been reserved. To estimate costs, vehicle routing methods are adapted for usage as customer acceptance mechanisms (e.g., Ehmke & Campbell, 2014; Köhler & Haferkamp, 2019). Revenue management methods are used to allocate capacities to high revenue requests (e.g., Cleophas & Ehmke, 2014; Klein et al., 2019). However, since decisions in the online environment must be made instantly, the use of complex and, thus, computationally intensive solution methods is limited.

The warehouse picking and consolidating ordered goods are summarised under *order assembly*. Before this, the retailer must decide on the location and design of the warehouses. Choosing the location is closely linked to the fulfilment capacity of the retailer and must be well-planned. The design of the warehouse determines the efficiency of picking the ordered items. Finding efficient picking strategies to reduce retailer costs is studied in, for example, Schiffer et al. (2022). Lastly, the retailer must determine the optimal stock level of items. Given the short lead times in e-commerce, this task must be completed before customer requests arrive. The task is closely linked to the research field of inventory management, where techniques such as forecasting (Ulrich et al., 2021) or artificial intelligence (Albayrak Ünal et al., 2023) are commonly used to address this challenge effectively.

For *order delivery*, routes are planned for all accepted orders. For e-commerce, the last-mile delivery is usually towards the customer's location, i.e., the consumer's home or company site (Agatz et al., 2008), leading to a magnitude of fragmented delivery locations with small drop sizes. Significant challenges arise from how these last-mile deliveries (routes) are designed. Delivery route planning is

closely related to the established field of vehicle routing, and approaches are being adapted for use in e-fulfilment (e.g., Emeç et al., 2016). Two-echelon routing systems are often considered to maintain economies of scale and satisfy the emission zone requirements in the cities (Sluijk et al., 2022a, 2022b)). In most cases, delivery is made by conventional delivery vehicles. However, individual retailers are also starting to bring orders to customers in the city centre using bikes. We also see drones (e.g., Ulmer & Thomas, 2018; Dayarian & Savelsbergh, 2020)) and robots (e.g., Simoni et al., 2020).

### 3.5.2. E-fulfilment challenges

E-fulfilment processes present several challenges. For *unattended deliveries*, delivery is possible without the customer being present. Pick-up point delivery enhances the efficiency of the delivery operations via consolidation opportunities. Consumers can also find it a more convenient delivery option than waiting for the delivery at home. There is a need for incentive mechanisms to increase the attractiveness of pick-up points (e.g., reduced delivery price). Galiullina et al. (2022) study this problem as a trade-off between routing cost savings gained from steering the customer demand and the investments required to influence customer behaviour. Another challenge is to find the optimal locations for pick-up points, such that delivery costs are minimised and customers still have convenient access, which is, for example, considered in Lin et al. (2020b) and Wang et al. (2020). The customer must accept the delivery herself for *attended deliveries*, e.g., to prevent grocery spoiling. To avoid delivery failures, the customer and the retailer usually agree on a delivery time window.

Customers expect short time windows, which increase the retailer's delivery costs (Köhler et al., 2020). As the time windows assignment to orders is crucial for the retailer's profitability, several approaches consider balancing demand along the offered time windows. One possibility is to withhold specific time windows from customers and only offer a subset of beneficial time windows. Campbell and Savelsbergh (2005) and Cleophas and Ehmke (2014) consider routing costs and customer value and only offer time windows to customers that are expected to maximise the profit. Another possibility is to assign prices to time windows to nudge customers to specific time window options (Campbell & Savelsbergh, 2006; Yang et al., 2016; Klein et al., 2019). Some approaches consider adapting the time window design to increase routing flexibility. Köhler et al. (2020) only offer short time windows to customers when it does not impact the routing costs too much, and Strauss et al. (2021) hand out time window
<figure><img src="image_76.png" /></figure>

bundles to customers that are only narrowed down to one option once more customers requests are known.

Recently, many online retailers began offering on-demand deliveries, so customers can receive their orders the same day (some grocery stores promise delivery times within a few minutes⁵¹). Shortening lead times poses another challenge as there is almost no time for planning or consolidation of orders available. The approach presented by Klapp et al. (2020) hence supports retailers in deciding which customers can be promised an immediate delivery and which can only be served from the next delivery day. Ulmer and Thomas (2018) investigate how the number of same-day deliveries can be increased if delivery is not only done by vehicles but additionally by drones. In Banerjee et al. (2022), the authors examine how retailers must allocate their delivery capacity to cover same-day delivery needs per service area.

For delivery in an urban context, high demand in densely populated areas often goes hand in hand with high traffic and unreliable travel times, and vice versa. Ehmke and Campbell (2014) therefore create acceptance mechanisms that present the customer with a time window offer that is as reliable as possible so that the customer does not notice an unforeseen change in travel times. Köhler and Haferkamp (2019) test the suitability of customer acceptance mechanisms for more and less densely populated areas to derive how well different routing mechanisms approximate delivery times.

Another ongoing challenge is the increasing prevalence of customers being granted the option to return ordered items free of charge by many companies. As a result, the e-fulfillment process expands beyond the three steps outlined earlier to include the management of returns. Despite the typically high return rates that result in substantial additional costs for retailers, offering a return option is still profitable due to the subsequent improvement in customer satisfaction and retention (Rintamäki et al., 2021). The management of returns can be perceived as reverse order delivery, leading to routing challenges related to those presented earlier. To mitigate costs, several studies, such as Mahar and Wright (2017) and Yan et al. (2022b), explore the implementation of in-store returns.

### 3.5.3. Operational research challenges: Time, timing, and data

The *time* dimension involves all dimensions to how key elements are (conceptually) modelled with regards to the time (e.g., travel times or handling times). Identifying the time features in modelling and solution methodologies are essential qualifiers for realistic model representations.

The *timing* dimension involves all actions at a particular point or in a period when something happens (e.g., a new order arrives). Timing considers synchronisation issues where, for example, vehicles need to meet at a certain point in time and geographical location. Drexl (2012) presents a survey of vehicle routing problems with multiple synchronisation constraints. Synchronisation requirements between the vehicles relate to spatial, temporal, and load aspects. Synchronisation is a challenge, for example, in heterogeneous fleets (Ulmer & Thomas, 2018) or, in the case of battery-powered vehicles that must be charged in time.

* *Offline* means that we do the planning and scheduling before the execution, often assigned to tactical planning. Data is estimated (forecast) based on past observations, and the operations are planned based on that. For example, Agatz et al. (2011) use expected demand to decide which time windows should be offered within different parts of the delivery area. Lang et al. (2021a) propose a preparation offline phase that serves as input to speed up decisions during later online customer acceptance. The data considered could be either time-independent (i.e., independent of time) or time-dependent (i.e., the data has a time-stamp). For example, travel times can be modelled time-independent (i.e., constant speed) or time-dependent (e.g., Spliet et al., 2018).

* *Online* refers to the optimisation in real-time, where revealing new data and planning and scheduling operations happen simultaneously. The terms “dynamic” or “operational planning” is also often used. As time is critical in online planning, methods are always limited by their solution time. Instead of finding a routing solution, delivery costs are approximated (e.g., Yang & Strauss, 2017; Lebedev et al., 2021) or a simple routing heuristic is applied (e.g., Mackert, 2019; Klein et al., 2018)). Alternatively, customer choice is estimated simply (e.g., Campbell & Savelsbergh, 2006) instead of complex and time-consuming customer choice modelling. van der Hagen et al. (2022) uses a machine learning approach to fasten up feasibility checks of time windows offered during order acceptance.

The *data dimension* refers to how the data and observations are modelled. The data can be handled deterministic or stochastic, or we observe the realised data. Most models assume deterministic data and build their solution approach around this notion. More and more researchers, however, recognise the challenge of adequately representing reality in their models. Yang et al. (2016) use booking data of an online grocer to
<figure><img src="image_77.png" /></figure>

estimate realistic customer behaviour. Köhler et al. (2022) investigate how to accept high revenue requests by applying a sampling procedure with booking data from an e-grocer in Germany.

### 3.5.4. Relevant literature

Agatz et al. (2013) provide the first overview of how retailers can manage e-fulfilment processes. A recent review on e-fulfilment for attended home deliveries can be found in Waßmuth et al. (2022). We refer the reader to Fleckenstein et al. (2022) and Snoeck et al. (2020) for a focus on routing and revenue management methods in e-fulfilment, respectively.

### 3.6. Education⁵²

Education spans activity from kindergarten, through primary and secondary schooling, to higher education. The earlier years of education are often compulsory reflecting the premise that an educated workforce is crucial to economic performance. The extent to which education is publicly funded varies from one level of education to another, as well as from one country to another depending on the local view concerning the social return on investment. Public funding for education alongside the role education and training play in the performance of an economy therefore make education a prime context for application of operational research (OR) tools. This section provides a brief overview of some of the main areas.

Many OR methods can be useful to the policy maker for macro-planning and financial allocation purposes. Forecasting student numbers can be done using Markov chain models (Nicholls, 2009; Brezavšček et al., 2017) or machine learning (ML) and artificial intelligence (AI) (Yan & Wang, 2021) – the importance of AI applications to education will be further expanded later. Allocation of finances is typically supported by multi-objective decision analysis (Cobacho et al., 2010).

One important aspect of resource allocation relates to the efficient use of resources. Availability of published education data in many countries provides an opportunity to examine the “black box” of education production. Consequently, there is a long-standing literature surrounding efficiency in education, typically a not-for-profit context where conventional measures of performance are inappropriate.

Early studies of efficiency in higher education applied deterministic ordinary least squares methods to university-level data to examine efficiency in the production of specific outputs (Jauch & Glueck, 1975; Johnes & Taylor, 1990) while schools adopted multilevel modelling methods to derive performance

insights from pupil-level as opposed to school-level data (Woodhouse & Goldstein, 1988). But the multi-product nature of production in education establishments means that looking at inputs separately provides only a partial picture. The tools of multiple-criteria decision analysis such as principle components, the analytic hierarchy process and co-plot have therefore been adopted to examine and visualise the many dimensions more easily (Johnes, 1996; Paucar-Caceres & Thorpe, 2005; Mar-Molinero & Mingers, 2007).

Two frontier estimation approaches to analysing efficiency, both of which derive from Farrell (1957), have evolved to address various shortcomings of early approaches. The non-parametric data envelopment analysis (DEA) easily handles the multi-input multi-output nature of production observed in education and provides easily-interpreted measures of efficiency (Charnes et al., 1978). DEA shows each observation in its best possible light (in efficiency terms) by computing a distinct set of input and output weights. This permits the derivation of benchmark observations for each inefficient institution, i.e., the establishment(s) the observation should be looking to emulate to become more efficient. Non-parametric frontier estimation techniques have been applied in the context of education at all levels, providing management information at the institution level, and policy insights at the macro-level (Thanassoulis et al., 2011; Portela et al., 2012; Burney et al., 2013).

Network DEA provides a more forensic examination of the “black box” (Färe & Grosskopf, 2000) by breaking down the production process into its component parts, and overall efficiency can be decomposed into efficiency in each of the stages (Wang et al., 2019b; Lee & Johnes, 2022).

When longitudinal data are available, DEA can be used to analyse changes in efficiency using the Malmquist (1953) productivity index which decomposes productivity change into efficiency and technological change components Wolszczak-Derlacz (2018). The method can be used to make comparisons between groups rather than (or as well as) between time periods (Aparicio et al., 2017).

The deterministic non-parametric nature of DEA has been addressed in numerous extensions including by introducing bootstrapping and significance tests (Johnes, 2006; Essid et al., 2010; Papadimitriou & Johnes, 2019). Second stage analyses which examine the determinants of efficiency also abound (Haug & Blackburn, 2017). This approach is only valid if the hypothesis of separability holds i.e., the variables used in the second stage should only influence the efficiency scores and not the determination of the efficiency frontier (Simar & Wilson, 2011).
<figure><img src="image_78.png" /></figure>

The development of separability tests (Daraio et al., 2018) and the robust conditional estimation approach address these issues (Daraio & Simar, 2007); their application in education provide more robust and insightful results (López-Torres et al., 2021).

Stochastic frontier analysis (SFA) provides both parameter estimates (with significance tests) and efficiency estimates which allow for stochastic errors (Aigner et al., 1977; Meeusen & van den Broeck, 1977; Jondrow et al., 1982). Compared to DEA it is more difficult to model multi-input, multi-output production; most SFA applications in education therefore focus on cost efficiency (Agasisti, 2016), or a single output model (Kirjavainen, 2012), although there are some exceptions (Abbott & Doucouliagos, 2009; Johnes, 2014). The parameter estimates have made SFA popular in the cost function context as scope and scale economies can be estimated and these have useful policy implications (Johnes et al., 2005; Johnes & Johnes, 2013).

In its basic form, SFA parameter estimates apply to every observation in the dataset. Extensions of the technique include latent class SFA and random parameters SFA which allow the parameters to vary by specific groups (latent class) or by each observation (random parameter). These approaches benefit from the advantages of DEA and SFA although are computationally demanding but have been applied in education to interesting effect (Johnes & Schwarzenberger, 2011; Johnes & Johnes, 2016).

The interested reader is referred to comprehensive reviews of the relevant literature (Kao, 2014; Thanassoulis et al., 2016; De Witte & López-Torres, 2017; Johnes, 2022).

All these OR methods can be applied in the contexts of macro- and micro-level planning and budget allocation. One area at the micro-level for which OR techniques are useful is timetabling. Timetabling of examinations and/or teaching is most complex at secondary and tertiary levels and can be viewed as a scheduling problem whereby resources, limited in supply, are allocated to a constrained number of times and locations, with the allocation satisfying stated objectives. Timetabling differs from scheduling in that the resources (staff members) are typically specified in advance rather than being a part of the allocation problem; and while scheduling aims to minimise costs, the objective of timetabling is to realise desirable objectives (e.g., no clashes) as closely as possible (Petrovic & Burke, 2004). Timetablers face both hard and soft constraints in constructing the timetable (Asmuni et al., 2009) and this is therefore a problem which lends itself to solution by various possible OR

techniques in the field of combinatorial optimisation. The main approaches are briefly summarised below.

Mathematical programming (particularly integer linear programming) is commonly used in timetabling (Cataldo et al., 2017) but often leads to computationally demanding problems. Heuristics (see below) are introduced for increased efficiency (Dimopoulou & Miliotis, 2001). Case-based reasoning approaches use a past solution (stored in the case base) as the starting point for a new timetable and use similarity measures to identify the optimal solutions (Burke et al., 2006). These approaches are often problem-specific making them non-transferable. Their computational demands can be addressed by using heuristics (Petrovic et al., 2007). The multi-criteria approach assumes that there are solutions to the timetabling problem satisfying the hard constraints and then the quality of these solutions is assessed on the basis of how well each one satisfies the soft constraints (Burke & Petrovic, 2002). As with other methods it is often combined with heuristics.

Heuristics are an increasingly common method for application to timetabling either on their own or in combination with other methods. Low level construction heuristics include largest degree, largest weighted degree, largest colour degree, largest enrolment, saturation degree and random. Extensions include meta-heuristics which work in the search space guiding neighbourhood moves to a solution (Qu et al., 2015); fuzzy heuristics which can find a best approach in the initial timetable construction phase (Asmuni et al., 2009); and hyper-heuristics which find or generate appropriate heuristics to solve complex search problems as encountered in timetabling (Qu et al., 2015). Given their focus, hyper-heuristics have the potential to provide more generalised solutions to timetabling problems than other approaches (see Pillay, 2016, for a review).

The interested reader is referred to reviews of educational timetabling approaches (Oude Vrielink et al., 2019; Tan et al., 2021).

Finally, an emerging area of interest is the application of AI and ML to education. AI and ML are, as already highlighted, useful for forecasting as they can analyse rich data on, for example, student numbers, retention, achievement, teaching and quality to derive better predictions and/or understanding of the challenges (Alyahyan & Düstegör, 2020; Bates et al., 2020; Teng et al., 2022). They can also be used in the teaching and learning process itself by personalising each student's experience for example through use of chatbots, by creating exercises for students which address their weaknesses, and by reviewing assessments highlighting strengths and
<figure><img src="image_79.png" /></figure>

weaknesses (Teng et al., 2022). In the growing distance learning education arena where it is more difficult to manage participants who have more freedom to learn when they want and may encounter more distractions, AI can be used to support teachers in gauging student engagement. Thus AI algorithms can be used to develop an online education classroom management system (Wang, 2021). AI and ML have much to offer in education but their potential across all disciplines has yet to be properly explored (Bates et al., 2020). See Zawacki-Richter et al. (2019) for further literature.

### 3.7. Environment⁵³

Environmental problems are at the centre of societal concerns, and of many research activities, also in Operational Research (OR). It is impossible to comprehensively present this literature. Instead, we first introduce characteristics of environmental problems, then present some insights from specific OR fields, mainly citing review articles. Thereafter, we discuss *Decision Analysis* (§2.8) methods applied to environmental problems.

Environmental problems are usually multi-faceted and complex (French & Geldermann, 2005; Gregory et al., 2012; Reichert et al., 2015). Since 50 years, such public policy issues are known as “wicked problems” (Rittel & Webber, 1973). In many environmental cases, uncertainties are high. It may be difficult to establish scientific knowledge and adequately model environmental systems. They usually span all sustainability dimensions, which requires making trade-offs between achieving environmental, economic, and societal objectives. Various decision-makers and stakeholders with different world-views are affected, sparking conflicts of interest. Any action may have irreversible or far-reaching consequences over long time horizons. Additionally to the temporal dimension, spatial considerations over varying geographic regions may be important. As wicked problems are typically unique, we might need to find new solutions in each case. OR methods can be highly suitable to disentangle and structure complex environmental problems, and can certainly contribute to problem solving. Below, we present some viewpoints.

Soft OR methodologies, and *problem structuring methods* (PSMs; see also §2.20) have been developed to tackle complex real-world problems in interaction with stakeholders (Rosenhead & Mingers, 2001; Smith & Shaw, 2019). However, most (review) articles are not specific to environmental problems. Using an applied example, White and Lee (2009) explored the potential of soft OR for a city development case. Marttunen et al. (2017) reviewed the

combination of PSMs with Multi-Criteria Decision Analysis (MCDA) methods. More complex PSMs seem to be under-utilised, suggesting that their benefits cannot sufficiently inform real-world issues, including environmental decision-making. Similarly, French (2022) argued that literature of quantitative and qualitative OR approaches has developed in silos, and that an intertwined, cyclic understanding of soft and hard OR methods is needed to address complex problems. This author was also concerned that behavioural issues are less well understood in qualitative compared to quantitative model building. Related to problem structuring, *stakeholder analysis and participation* is central to environmental problems. Such research is recently gaining increased interest by OR (de Gooyert et al., 2017; Gregory et al., 2020; Hermans & Thissen, 2009). *Behavioural OR* (BOR; §2.2) is also gaining momentum (Franco et al., 2021). BOR strongly focuses on interventions, and could increase the understanding of societal and psychological issues in environmental problems. However, to date an environmental perspective is rarely taken. One exception is a conceptual paper about behavioural issues in environmental modelling (Hämäläinen, 2015). A meta-analysis of 61 environmental and energy cases analysed patterns and biases that may occur in the problem structuring phase of decision-making (Marttunen et al., 2018).

*Sustainable supply chains* (see also §3.24) have been recently reviewed by Barbosa-Póvoa et al. (2018). These authors took a multi-stakeholder perspective along the supply chain to achieve sustainability goals. They found a predominance of *optimisation* methods applied to strategic decision levels. Most of the 220 reviewed articles focused on economic and environmental aspects, leaving behind the social aspects. Similarly, another review focused on *combinatorial optimisation* (§2.4), integrating *reverse logistics* (see also §3.14) and waste management (Van Engeland et al., 2020). Among other aspects, the authors emphasised the importance of environmental, social and performance indicators, and stakeholder integration, when dealing with flows of waste products. Taking a life-cycle perspective, usually addressed with *life cycle sustainability assessment* (LCSA), Thies et al. (2019) reviewed advanced OR methods for *sustainability assessment of products*. While most articles used ecological indicators, the integration of economic and social indicators is emerging. They concluded that improved systematic procedures for uncertainty treatment are needed, and better integration of qualitative social indicators as well as spatially explicit data.

Other authors reviewed specific OR methods. For instance, Zhou et al. (2018) reviewed *Data Envelopment Analysis* (DEA; §2.7) for sustainability
<figure><img src="image_80.png" /></figure>

assessments. Again, economic and environmental measures were well included, but the literature lacked social measures such as customer satisfaction. New DEA methods should be developed that include *social network relationships*. *Mathematical programming* and *optimisation* methods to support biodiversity protection were reviewed by Billionnet (2013). Some of these difficult combinatorial optimisation problems were well solved, but further research is needed to satisfactorily address real-world biodiversity issues. For conservation management, spatial aspects are central, for example creating biological corridors in the landscape to increase biodiversity. Future research should include the temporal dimension and needs of practitioners. *Robust optimisation* (§2.21) could be a research avenue to handle uncertainty. A review of invasive species also took a mathematical perspective (Büyükaftakın & Haight, 2018). Among other conclusions, research should develop more realistic models to capture spatial and temporal dynamics of invasive species, improve uncertainty treatment and coordination among stakeholders, and include holistic approaches for addressing trade-offs between conservation management and costs of such programs.

*Multi-criteria decision analysis* (MCDA; §2.8) provides a rich literature addressing environmental decision problems. French and Geldermann (2005) discussed properties of wicked environmental problems from a conceptual point of view (introduced above), and implications for decision support. Cinelli et al. (2014) analysed MCDA methods for sustainability assessments. They voiced some concern that choosing the MCDA methods is rather based on preferences, not analytic considerations. Indeed, text-mining of 3,000 articles provided little evidence that particular environmental application fields used certain methods more frequently, possibly because researchers are unaware of specific method merits (Cegan et al., 2017). To overcome this, Cinelli et al. (2014) classified five MCDA methods using ten criteria important for sustainability assessments, e.g., uncertainty management and testing robustness of results, software, and user-friendliness. We know of two general articles for systematically choosing a suitable MCDA method (Cinelli et al., 2020; Roy & Slowiński, 2013). Several articles reviewed *decision support systems* (DSS) to identify features and best practices for supporting environmental problems (Mustajoki & Marttunen, 2017; Walling & Vaneeckhaute, 2020). Moreover, there are many reviews of MCDA applied to a specific environmental field, but only few were published in OR journals (e.g., Colapinto et al., 2020; Kandakoglu et al., 2019). There is a pronounced

increase of articles applying MCDA in all environmental areas (e.g., water, air, energy, natural resources, and waste management; Cegan et al., 2017; Huang et al., 2011). Below, we introduce some important findings from *decision analysis*.

Some authors defined frameworks for environmental assessments taking a method perspective. Gregory et al. (2012) proposed *structured decision making* (SDM) to tackle real-world environmental decision problems. Based on *multi-attribute value theory* (MAVT), SDM can be applied without much (mathematical) formalisation. This textbook discusses many practical environmental issues, highlighting solutions from international decision cases. Reichert et al. (2015) proposed a framework for environmental decisions that emphasises uncertainty of scientific knowledge and societal preferences. They argued that theoretical requirements are best met by combining *multi-attribute utility theory* (MAUT) with *scenario planning* and *probability theory*, illustrated with a river management case. Scenario planning has been advocated by various authors for tackling wicked problems (Wright et al., 2019). The combination of scenario planning with MCDA has been reviewed by Stewart et al. (2013), and applied to e.g., nuclear remediation management (Geldermann et al., 2009), coastal engineering under climate change (Karvetski et al., 2011), or water infrastructure planning (Scholten et al., 2015). Scenario analysis has also been combined with probabilistic statements and *mathematical optimisation* for *risk assessment* (see also §2.18) of nuclear waste repositories (Salo et al., 2022). A climate policy review illustrates the importance of integrating various OR methods to effectively support decision-making (Doukas & Nikas, 2020). The currently predominant evaluation of policy strategies with *climate-economy* or *integrated assessment models* (IAMs) fails to incorporate all relevant uncertainties and stakeholders, and sufficiently address system complexity. These authors proposed integrated approaches, including participatory stakeholder processes with *fuzzy cognitive maps*, combined with MCDA and *portfolio analysis* (PA). PA is especially useful as meta-analysis, and has been reviewed by Liesiö et al. (2021). A PA-framework for environmental decision-making has been proposed by Lahtinen et al. (2017).

To address *spatial aspects* of environmental problems, *geographic information systems* (GIS) are often combined with MCDA, sometimes also developing DSSs (Keenan & Jankowski, 2019). *Risk analysis* (§2.18) and OR research increasingly focuses on spatial planning (Ferretti & Montibeller, 2019; Malczewski & Jankowski, 2020). One example is the
<figure><img src="image_81.png" /></figure>

axiomatic foundation of spatial multi-attribute value
functions (Harju et al., 2019; Keller & Simon, 2019).

Many reviews found that *stakeholder integration* throughout the decision-making process was insuffi- ciently considered, e.g., in flood risk management (de Brito & Evers, 2016) or nature conservation (Esmail & Geneletti, 2018). This reflects generally found deficits in *problem structuring* ($2.20), for instance insufficient consideration of social objec- tives (Kandakoglu et al., 2019), or systematic under- estimation of the importance of economic objectives (Marttunen et al., 2018; Walling & Vaneeckhaute, 2020). Moreover, there is a tendency to choose too many objectives in environmental cases (Diaz- Balteiro et al., 2017), potentially inducing *biases* in later stages of MCDA (Marttunen et al., 2019).

Many reviews emphasised the importance of *uncertainty analyses* in environmental decisions, but this is strongly ignored in practice. One review found that only 19% of 271 articles included uncertainty analysis, 17% using *fuzzy techniques* to capture imprecise numbers (Diaz-Balteiro et al., 2017). In another review, 34% of 343 articles dealt with the imprecision of predictions, 70% using fuzzy sets, and 20% *stochastic modelling* (Kandakoglu et al., 2019). In both reviews, only 20%–30% of the articles per- formed *sensitivity analysis*. Additionally, only 5% of 343 reviewed papers included *temporal aspects* of the environmental decision (Kandakoglu et al., 2019).

As conclusion, OR researchers are widely engaging
in environmental problems. Environmental problems
are intriguingly complex, thus offering opportunities
for inspiring research. Although our evaluation is
neither comprehensive nor systematic, some general
research needs appear across all OR fields. Many
articles emphasised the importance of better integrat-
ing practitioners and stakeholders in environmental
problems, and of better considering societal objec-
tives. Various fields require improved methods to
address the complexities of environmental problems,
including appropriately dealing with many types of
uncertainties, time, and space. Combining soft with
hard OR, improving problem structuring, and inte-
grating questions from behavioural OR will increase
the chances of finding sustainable solutions for our
worlds' environmental problems. This can also spark
cross-disciplinary research over different fields of OR.

**3.8. Ethics and fairness⁵⁴**

There is substantial literature on the ethical practice
of operational research, surveyed in Brans and Gallo
(2007), Ormerod and Ulrich (2013), Tsoukias
(2021), and Bellenguez et al. (2023). While this is a
vitally important discussion, it is useful to consider
how the science of operational research can

contribute to ethics, as well as how ethics can con-
tribute to the practice of operational research. It has
accomplished this primarily through the develop-
ment of modelling techniques and algorithms that
embody ethical concepts, notably distributive justice.

An operational research model that aims simply
to minimise total cost or maximise total benefit may
unfairly distribute costs or benefits across stakehold-
ers. This concern arises in a number of application
areas, including healthcare ($3.11), disaster relief
($3.4), facility location ($3.13), task assignment, tele-
communications ($3.26), and machine learning
($2.1). It poses the problem of finding a suitable
formulation of equity or fairness that can be incor-
porated into a mathematical model.

For example, if donated organs are allocated in
the most economically efficient fashion, patients
with certain medical conditions may wait far longer
for a transplant than other patients (McElfresh &
Dickerson, 2018). If earthquake shelters are located
so as to minimise average distance from residents,
persons living in less densely populated areas may
have much further to travel (Mostajabdaveh et al.,
2019). If a machine learning algorithm awards mort-
gage loans so as to maximise expected earnings,
members of a minority group may find themselves
unable to obtain loans even when they are
financially responsible (Saxena et al., 2020). If traffic
signals at intersections are timed to maximise traffic
throughput, motorists on side streets may have to
wait forever for a green light (Chen et al., 2013).

We provide here a brief overview of mathemat-
ical formulations of fairness that have been pro-
posed for OR and AI models. Comprehensive
treatments can be found in Karsu and Morton
(2015) and Chen and Hooker (2022b). In addition,
Ogryczak et al. (2014) review formulations devel-
oped for telecommunications and facility location,
two major users of fairness models. Recent years
have seen an enormous surge of interest in fairness
criteria for machine learning, many of which are
surveyed in Mehrabi et al. (2022).

We suppose that the model into which one
wishes to incorporate fairness allocates *utilities* to a
collection of *stakeholders*, and we are concerned
about the fairness of this allocation. Utility could
take the form of wealth, resources, negative cost,
health outcomes, or some other type of benefit.
Stakeholders can be individuals, organisations,
demographic groups, geographic regions, or
other entities for which distributive justice is a
concern.

Fairness models can be divided into three broad
categories. *Inequality measures* are normally used to
constrain the degree of inequality in solutions
obtained by maximising total benefit or minimising
<figure><img src="image_82.png" /></figure>

total cost. Some of these focus on inequalities across individuals, and others on inequalities across groups. Various statistics for measuring the former are discussed in Cowell (2000) and Jenkins and Van Kerm (2011). Perhaps the best known is the *Gini coefficient*, widely used to measure income or wealth inequality (Gini, 1912; Yitzhaki & Schechtman, 2013). The *Hoover index* (Hoover, 1936) is proportional to the relative mean deviation of utilities and represents the fraction of total utility that must be redistributed to achieve perfect equality. Both the Gini coefficient and the Hoover index can be given linear formulations (§2.14) in an optimisation model by means of linear-fractional programming (Charnes & Cooper, 1962). *Jain's index* (Jain et al., 1984), well known in telecommunications, is a strictly monotone function of the coefficient of variation.

Inequality between groups, generally referred to as *group disparity*, is by far the most discussed type of inequality metric in the machine learning field (§2.1; Verma & Rubin, 2018; Mehrabi et al., 2022). It assesses whether AI-based decisions (e.g., mortgage loan awards, job interviews, parole, college admission) are biased against a designated group, perhaps defined by race, ethnic background, or gender. Fairness implementations in machine learning typically strive to minimise loss (due to defaults on loans, etc.) while placing a bound on some measure of resulting group disparities. The best known measures are *demographic parity* (Dwork et al., 2012), *equalised odds* (Hardt et al., 2016), and *predictive rate parity* (Dieterich et al., 2016; Chouldechova, 2017), and *counterfactural fairness* (Kusner et al., 2017; Russell et al., 2017). The first two have mixed integer/linear programming (MILP) formulations (§2.15), and the third a mixed integer/nonlinear formulation. Weaknesses of group parity measures include a lack of consensus on which one is suitable for a given application (Castelnovo et al., 2022), as well as on which groups should be monitored for bias.

A second category of models is concerned with *fairness for the disadvantaged*. They strive for equality, but with greater emphasis on the lower end of the distribution. The *maximin* criterion, based on the famous difference principle of John Rawls, maximises the welfare of the worst-off individual or social class (Rawls, 1971). It is defended with a social contract argument that has been intensely discussed in the philosophical literature (as surveyed in Freeman, 2003; Richardson & Weithman, 1999). A more sophisticated form of the principle is lexicographic maximisation (*leximax*), which maximises the worst-off, then the second worst-off, and so forth. The *McLoone index* compares the total utility

of stakeholders at or below the median utility to the utility they would enjoy of all were brought up to the median. It is based on a concern that no one be disadvantaged but tolerates inequality in the top half of the distribution. It has been used to assess the allocation of public services, particularly education (Verstegen, 1996) and can be given an MILP formulation (Chen & Hooker, 2022b).

Criteria that *balance efficiency and fairness* can be placed in three categories: convex combinations of efficiency and fairness, criteria from classical social choice theory, and threshold criteria. Convex combinations provide the simplest approach, as for example a combination of total utility and a fairness measure (e.g., Mostajabdaveh et al., 2019). Other formulations are given by Yager (1997), Ogryczak and Śliwiński (2003), and Rea et al. (2021). Convex combinations and other weighted averages pose the general problem of justifying a choice of weights, particularly when utility and equity are measured in different units, although Argyris et al. (2022) propose a means of avoiding this issue.

The task of balancing fairness and efficiency gave rise to one of the oldest research streams in social choice theory, beginning with the *Nash bargaining solution*, also known as *proportional fairness* (Nash, 1950a). Proportional fairness has seen application in such engineering contexts as telecommunication and traffic signal timing (Mazumdar et al., 1991; Kelly et al., 1998) and elsewhere. Nash gave an axiomatic argument for the criterion, while Harsanyi (1977), Rubinstein (1982), and Binmore et al. (1986) have shown that it is the outcome of certain bargaining procedures. *Alpha fairness* generalises proportional fairness by introducing a parameter $\alpha$ that governs the importance of fairness, where $\alpha = 0$ corresponds to a purely utilitarian criterion, $\alpha = 1$ to proportional fairness, and $\alpha = \infty$ to the maximin criterion (Mo & Walrand, 2000; Verloop et al., 2010). Alpha fairness has been derived from a set of axioms (Lan et al., 2010; Lan & Chiang, 2011), including an “axiom of partition” that is largely responsible for the result. It provides an objective function to be maximised that is nonlinear but concave (§2.16). Another criterion, *Kalai-Smorodinsky bargaining*, likewise has an axiomatic defence (Kalai & Smorodinsky, 1975) and addresses what one might see as a weakness in Nash bargaining, namely that it can result in reduced utility for some stakeholders when the feasible set is enlarged. The Kalai-Smorodinsky criterion can be viewed as a kind of normalised maximin, as it calls for allocating to each stakeholder the largest possible fraction of his or her potential utility (ignoring other stakeholders) on the condition that this fraction be the same for everyone. This criterion has received support from