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Evaluating Supply Chain Resilience During Pandemic Using
Agent-based Simulation
Teddy Lazebnika,b,c
aDepartment of Mathematics, Ariel University, Ariel, Israel
bDepartment of Cancer Biology, Cancer Institute, University College London, London, UK
cCorresponding author: lazebnik.teddy@gmail.com
Abstract
Recent pandemics have highlighted vulnerabilities in our global economic systems, especially
supply chains. Possible future pandemic raise a dilemma for businesses owners between short-
term profitability and long-term supply chain resilience planning. In this study, we propose
a novel agent-based simulation model integrating extended Susceptible-Infected-Recovered
(SIR) epidemiological model and supply and demand economic model to evaluate supply
chain resilience strategies during pandemics. Using this model, we explore a range of supply
chain resilience strategies under pandemic scenarios using in silico experiments. We find that
a balanced approach to supply chain resilience performs better in both pandemic and non-
pandemic times compared to extreme strategies, highlighting the importance of preparedness
in the form of a better supply chain resilience. However, our analysis shows that the exact
supply chain resilience strategy is hard to obtain for each firm and is relatively sensitive to
the exact profile of the pandemic and economic state at the beginning of the pandemic. As
such, we used a machine learning model that uses the agent-based simulation to estimate a
near-optimal supply chain resilience strategy for a firm. The proposed model offers insights
for policymakers and businesses to enhance supply chain resilience in the face of future
pandemics, contributing to understanding the trade-offs between short-term gains and long-
term sustainability in supply chain management before and during pandemics.
Keywords: pandemic spread, supply chain, supply-and-demand model, machine learning
1. Introduction
Pandemics, such as the recent global COVID-19 crisis [1, 2] or more historical ones such
as the Spanish Influenza during World War I [3, 4], have starkly illuminated the vulnera-
bilities in our economic systems [5, 6]. In the short term, pandemics disrupt production,
significantly shift consumer demand, and strain healthcare resources, leading to immediate
economic downturns [7]. The long-term effects are equally concerning, with industries facing
restructuring [8, 9], labor market shifts [10, 11], and altered consumer behavior patterns [12],
all of which pose substantial challenges to economic recovery and growth.
arXiv:2405.08830v1 [cs.MA] 13 May 2024
In particular, supply chains, as the lifelines of global commerce, are especially sensitive
to the shocks induced by pandemics [13, 14]. The intricate web of interconnected suppliers,
manufacturers, distributors, and retailers can quickly unravel under the strain of widespread
disruptions, leading to shortages, price volatility, and logistical bottlenecks [15, 16]. The
fragility of these supply chains has been laid bare during recent crises, prompting a critical
reevaluation of resilience strategies [17].
While the inevitability of pandemics is widely acknowledged [18, 19], businesses face a
dilemma in balancing the need to prepare for future disruptions against the imperative to gen-
erate profits in the present [20]. This “greedy” mindset, driven by short-term financial goals,
often conflicts with the long-term resilience planning required to withstand future shocks [21].
This dilemma is similar to other dual-objective optimization tasks with conflicted agendas
such as food exploration by ants [22] to more complex ones such as investment portfolio op-
timization tasks [23]. Generally speaking, this tension between immediate profitability and
future preparedness underscores the complexity of decision-making in a volatile and uncertain
environment [24, 25].
Addressing this tension presents an intriguing computational challenge [26, 27, 28]. How
can businesses optimize their supply chain strategies to simultaneously maximize current
profits and enhance resilience against future pandemics? Previous studies tried to address
this question by analyzing strategies applied by businesses during previous pandemics and
analyzing which businesses better handled the crisis [29, 30]. These studies indeed provide a
fruitful ground for policy-makers and business owners to design their strategy but actually
susceptible to Lucas’s critique as well [31]. Namely, one would require a set of businesses to
try two or more strategies to see which one works best for multiple pandemic configurations
to be able to empirically establish a claim. Unfortunately, such an experiment is impossible
in practice.
To this end, mathematical models and computer simulation emerge as powerful tools to
overcome this challenge [32, 33, 34]. While limited in their expressiveness and accuracy in
predicting the real world, they commonly provide accurate enough predictions to establish a
reasonable replica (commonly referred to as “digital twin”) of the studied case [35]. Specifi-
cally, agent-based simulation (ABS) is a computational method to describe the dynamics that
occur due to the interaction of diverse agents [36]. In this case, the agents represent different
supply chain entities and the dynamic interactions between them [37, 38]. By simulating vari-
ous scenarios and resilience strategies, one can gain valuable insights into effective approaches
for navigating the delicate balance between short-term gains and long-term sustainability in
the face of pandemics [39, 40].
Indeed, previous studies investigate supply chain resilience during a crisis, in general, and
for cases of a pandemic, in particular [41, 42, 43]. For instance, [44] adopted the ABS method
to analyze the influence of shifts in supply and demand due to the COVID-19 pandemic on
the supply chain’s ability to deliver. The authors studied two main recovery strategies rel-
evant to building emergency supply and extra manufacturing capacity to mitigate supply
chain disruptions using their model. [45] suggested analyzing and designing supply chain
reliance from an immune system perspective as immune systems are safeguarding against
disruptions and facilitating recovery - properties also associated with a resilient supply chain.
The author proposed a mathematical formalization for supply chain resilience based on the
biologically-inspired framework of immune systems. In a more general sense, [46] reviewed
supply chain resilience work with a focus on connecting the supply chain to other networks
such as command-and-control and transportation. The author identified three main model-
ing strategies - linear, branching, and graph-based and concluded that graph-based models
provide the most realistic and accurate modeling strategy of these.
In this study, we present a novel ABS-based model to explore how much a business should
focus its resources on supply chain resilience during a pandemic. The novelty of this work
lies in the integration of a well-established pandemic-spread model based on an extended SIR
(Susceptible-Infectious-Recovered) epidemiological model with theoretically and empirically
proven economic supply-chain model which emerges from supply-and-demand dynamics and
their shifts due to the pandemic. We first show how pandemics influence firms for different
supply chain resilience strategies. Then, we explore the sensitivity of these strategies to
economic and epidemiological changes, and finally, we use machine learning to estimate the
supply chain resilience strategy a firm should adopt.
The rest of this paper is organized as follows. Section 2 provides a quick overview of
how supply chains are designed and utilized. In addition, an overview of the ABS method
is presented alongside its usage for both epidemiological and supply chain models. Section 3
presents the formalization of the proposed ABS simulation to explore supply chain resilience
and experimental setup. Section 4 outlines the obtained results from the experiments. Fi-
nally, Section 5 discusses the results in an economic context with its possible application and
suggests possible future work.
2. Related Work
In this section, we provide an overview of how supply chains are designed, established,
and managed. These properties are later taken into consideration in the modeling phase.
In addition, an overview of the ABS method is presented in general and in the context of
both epidemiological and supply chain models. These models are later used as the modeling
fundamentals for our model.
2.1. Supply chains
Supply chains play a pivotal role in modern economies, encompassing the design, estab-
lishment, and management of interconnected networks that facilitate the flow of goods and
services [47, 48]. Indeed, in the modern economy, individual companies no longer compete
solely based on their unique brand identities. Instead, they operate as interconnected parts
of supply chains. Success now hinges on a firm’s ability to manage and coordinate the com-
plex network of relationships within these supply chains [49]. A supply chain functions as an
integrated system, coordinating processes from sourcing raw materials to delivering finished
products, adding value, promoting and distributing them, and facilitating information ex-
change among various entities like suppliers, manufacturers, distributors, logistics providers,
and retailers. The primary goal is to boost operational efficiency, profitability, and competi-
tive advantage for both the firm and its supply chain partners [49]. Supply chain management
is succinctly defined as the integration of crucial business processes, spanning from end-users
to original suppliers, which adds value for customers and stakeholders [49].
Multiple approaches have been suggested for modeling supply chains. According to [50],
these can be categorized into four groups: deterministic models with known parameters,
stochastic models with at least one unknown parameter following a probabilistic distribution,
economic game-theoretic models, and simulation-based models assessing supply chain strat-
egy performance. Most of these models are steady-state, focusing on average performance or
steady conditions [51]. However, static models fall short in capturing the dynamic nature of
supply chain systems affected by demand variations, lead-time delays, sales forecasting, to
name a few [51]. Multiple works show that a combination of supply and demand model with
network analysis is appropriate to capture the complexity of supply chain management [52].
This mathematical framework is often solved using ABS [53].
2.2. Agent based simulation
Agent based simulation (ABS) is a computational method of capturing (spatio-)temporal
dynamics occurring for multiple agents [54, 55]. An ABS more often than not contains two
main components - an environment and a population of agents which can be homogeneous
or heterogeneous [56, 57]. ABS is based on three types of interactions between the agents
and their environment - spontaneous, agent-agent, and agent-environment. Spontaneous
interactions are interactions between an agent and itself that only depend on the agent’s
current state and time. Agent-agent interactions are interactions between two or more agents
that alter the state of at least one of the agents taking part in the interaction. Agent-
environment interactions are interactions between agents and their environment that change
either or both the agent’s state or the environment’s state. Interestingly. ABS can be
computationally reduced to the population protocol model [58] and therefore it is Turing-
complete [59, 60] meaning the ABS can express any dynamics solvable by a computer.
A growing body of work utilizes the ABS computational method for a wide range of tasks
[61]. In particular, we below examine the usage of ABS in epidemiological and supply chain
models.
2.2.1. Epidemiological models
ABS is commonly used in the context of epidemiological models to model and solve het-
erogeneous population dynamics as other methods such as differential equations or functional
models are limited in their ability to efficiently describe such dynamics [62, 63]. For exam-
ple, [64] proposed an ABS of pedestrian dynamics to evaluate the behavior of pedestrians
in public places in the context of contact transmission of airborne infectious diseases. The
authors used a continuous space with a random direct walk of agents and infectious dynamics
as a function of distance between the agents. [65] described a ABS based model of pandemic
spread in facilities which is based on the popular SIR epidemiological model [66] that as-
sumes three epidemiological states - susceptible, infected, and recovered. In their model,
agents had heterogeneous movement dynamics which were accomplished by three rules that
take into consideration the agent’s epidemiological state, as well as its local environment and
the agent’s in this environment. [67] proposed an ABS based model with an SEI (susceptible-
exposed-infected) epidemiological model operating in a single room with three-dimensional
geometry. The authors combined airflow dynamics using the Computational Fluid Dynamics
(CFD) model with the epidemiological dynamics for airborne pathogens where agents are
spatially static but have heterogeneous breathing patterns.
2.2.2. Supply chain models
Similar to the ABS usage in epidemiological models, ABS is used for supply chain models
to capture “economic players” which different objectives, capabilities, or roles in an econ-
omy, in general, and in a supply chain, in particular [68, 69]. For instance, [70] proposed
an ABS based model of four three-level supply chains that apply different types of com-
bined contracts by taking into account the effects of vertical and horizontal competition
between supply chains. The authors show that the simulated results agree with previous
socio-economic knowledge from the literature. [71] introduced an integrated framework for
ABS inventory–production–transportation modeling and distributed simulation of supply
chains. The authors show that their framework produces predictions that agree with previ-
ous known dynamics such as utilization of machines in the manufacturer and quantity change
of products. [72] investigate retail stockouts using an ABS based model. The authors con-
sider the change in market share as a measure of resilience for both the manufacturer and
the retailer to examine the impact of the stockout and using the model, explore the effect of
different scenarios on these metrics.
3. Methods and materials
In this section, we initially introduce the proposed epidemiological-economic model for
supply chain resilience management based on the ABS method. Afterward, we describe a
supply chain resilience formalization with strategies firms can adopt. Next, we outline a
machine learning strategy to obtain an approximation to the right balance of supply chain
resilience and profit using a machine learning algorithm. Finally, an experimental setup for
the model is outlined.
3.1. Model definition
The proposed epidemiological-economic model for supply chain resilience management is
based on the ABS approach and constructed from three interconnected sub-models: epidemi-
ological, economic, and supply chain (spatial). These sub-models are structured on top of
three types of agents - consumers (which also function as workers), firms, and products.
We define the model as a tuple M:= (C, F, P, G), where Cis a set of consumers, Fis a
set of firms, Pis a set of products, Gis a graph of locations that the consumers, firms, and
products are physically located in and interacting with each other and between themselves.
The components of the tuple are described below in detail. Fig. 1 provides a schematic view
of the model’s components and the interactions between them.
Spatial sub-model
location (v)
Supply
chain
link
Epidemiological sub-model
susceptible exposed
infected recovered
dead
Economic sub-model
Dynamics
Consumer Firm Product
Agnets
Avalible money (m)
Salary (s)
Product demand (η)
pandemic-related demand (ν)
epi-state (e)
epi-state clock (et)
Avalible products (χ)
Avalible money (φ)
Operational cost (δ)
Workers (w)
Price (ι)
Preparation time (π)
Ingredients (υ)
product selling
supply chain
product preparation
Figure 1: A schematic view of the proposed epidemiological-economic model with supply chains.
Following the ABS method, we first formally define the three types of agents. In our
model, all agents are represented by a timed finite state machine [73]. The consumer agent,
c∈Cis defined by the tuple c:= (m, s, η, ν, e, et) where m∈[0,∞) is the currently available
money of the consumer, s∈[0,∞) is the amount of salary the consumer gets in each step in
time, η∈N|P|is the demand for products, ν∈R|P|is the change for products’ demand due
to the pandemic state, e∈ {S, E, I, R, D}is the consumer’s epidemiological state, and et∈N
is the number of steps in time passed since the last time the consumer’s epidemiological state
changed. A firm f∈Fis defined by the tuple f:= (χ, φ, δ, w) where χ∈N|P|is the number
of currently available products to supply, φ∈[0,∞) is the currently available money of
the firm, δ∈R+is the operational cost of the firm, w⊂Cis the set of consumers that
are also workers of the firm. A product p∈Pis defined by the tuple p:= (ι, π, υ) where
ι∈[0,∞) is the price of the product, π∈[0,∞) is the preparation time of the product from
its ingredients, and υ∈N|P|is the vector of ingredients required to prepare a product.
The interaction between these agents are the core of the model and is captured by the
three sub-models. Since there are many moving parts in the model, let us consider a simple
example to capture the underlying behavior of the model. Let us consider two locations such
that the first one has two firms and no consumer population while the other has also two
firms and some consumer population. In this scenario, three out of the four firms will be
factories as they produce products that do not have any ingredients and sell them to the
fourth firm which operates as a store. As each firm (i.e. factory) has different processes, the
price of its product is different. For this example, let us assume each of the firms operating
as factories are able to supply all the demand the firm operating as a store has in times of
no pandemic. Here, we focus on the firm which operates as a store. If this firm aims to make
as much profit as possible and ignores the supply chain realisance, it should buy the product
from the firm that offers it for the cheapest price. On the other extreme case where this firm
is only worried by the pandemic, establishing all possible supply chains will ensure the ability
to satisfy demand. A balanced objective would cause a more diverse supply chain strategy
while also considering profits. Fig. 2 presents a schematic representation of this example.
3.1.1. Epidemiological model
For each location, the epidemiological sub-model is an extension of the SIR model and
considers a constant consumer population with a fixed number of consumers (Ci) of size Ni
for the ith location. In the context of studying the immediate economic influence of the
pandemic on the supply chain, the time horizon of interest is relatively short, ranging for up
to several months, and as such, population growth can be neglected. Each consumer in the
population belongs to one of the five epidemiological groups: susceptible (Si), exposed (Ei),
infected (Ii), recovered (Ri), and dead (Di), such that Ni=Si+Ei+Ii+Ri+Di.
Consumers in the susceptible group have no immunity and are susceptible to infection.
When an individual in the susceptible group (Si) is exposed to the pathogen through an
interaction with an infected consumer, the susceptible consumer is transferred to the ex-
posed epidemiological group (Ei) at a rate corresponding to the average interaction between
infected consumers and susceptible consumers, denoted by β. Each consumer stays in the
exposed group on average θdays, after which the consumer is transferred to the infected
epidemiological group (Ii). Infected consumers stay in this group on average γdays, after
which they are transferred to the recovered epidemiological group (Ri) or the dead (Di) epi-
demiological group with probability ρand 1 −ρ, respectively. The recovered consumers are
again healthy, no longer contagious, and immune from future infection. The epidemiological
dynamics are formally described using a system of ordinary differential equations, as follows:
Location 1 Location 2
No preparation for pandemic (ω1=0.0)
Location 1 Location 2
Preparation for pandemic equals profit (ω1=0.5)
Location 1 Location 2
Only preparation for pandemic (ω1=1.0)
Figure 2: A simple example of three supply chain strategies obtained for different objectives of a firm from
the perspective of a store (firm) for a case of only two locations and four firms where the consumer population
is present only in the same location as the store. The solid-green arrows indicate the store firm established
these supply chains while dashed-red arrows indicate the store firm do not established these supply chains.
dSi(t)
dt =−βSi(t)Ii(t),
dEi(t)
dt =βSi(t)iI(t)−θEi(t),
dIi(t)
dt =θEi(t)−γIi(t),
dRi(t)
dt =ργIi(t),
dDi(t)
dt = (1 −ρ)γIi(t).
(1)
S
susceptible
E
exposed
I
infected
D
dead
R
recovered
β θ
γρ
γ(1-ρ)
Figure 3: A schematic view of the epidemiological model which is divided into five states - susceptible,
exposed, infected, recovered, and dead.
Importantly, θ, γ, and ρare biological-clinical properties and therefore are constant between
locations while βis highly affected by the population density, culture, and other properties
making it unique for each location [74]. Fig. 3 presents a schematic view of the epidemiolog-
ical model.
3.1.2. Local supply and demand model
For each location, the local supply and demand sub-model is based on the classical supply
and demand model. Nonetheless, for simplicity, we assume the products’ prices do not alter
due to change over time. At each step in time, each consumer has a demand for products (η)
which it can buy from firms in her location. We assume the consumer is well-informed and
prefers to buy each product from the firm that provides it at the lowest price. In addition, η
changes due to the pandemic state as a function of its sensitivity to each product as indicated
by a vector ν. Formally, a consumer’s overall demand at some point in time, tis η+νIi
Ni−Di.
Hence, at each step in time, a consumer first obtains salary, s, which adds to its current
available money (s) and uses it to buy products according to its current demand η+νIi
Ni−Di.
If some product is not available, the consumer just do not buy it. Moreover, if the consumer’s
available money is not enough to buy all the products according to its own demand, a random
subset of products such that the consumer has enough available money to buy, is chosen.
In addition, firms buy and sell products to other firms and sell products to consumers.
Each firm is operating using consumers from the location that operates as workers. Therefore,
its operational capacity is a function w
w−wi−wdwhere wiand wdare the subset of workers that
are infected or dead, respectively. The operational capacity is multiplied by the duration it
takes the firm to prepare a product from its ingredients. It takes the firm πsteps in time
to make produce from its ingredients (υ) are available to it. At each step in time, a firm
performs five actions. First, the firm pays its operational cost, δ. Second, the firm buys
products from other firms, if any. Third, the firm established new supply chains, if any.
Fourth, the firm prepares products from the products it acquired, if any. Finally, the firm
sells its ready products to consumers.
3.1.3. Supply chain and spatial model
Let us assume each location represents a single community and local economy and that
products are moved between locations via supply chains, while consumers are spatially static
in their communities over time. Formally, let G:= (V, L) be a directional, multi-edge, and
non-empty graph where Vis the set of locations and L⊂V×Vis the set of supply chains.
Each supply chain, l∈L, is defined by two firms (f1, f2) located in the same or different
locations and is defined by a tuple l:= (p, d, a) where p∈Pis the product firm f1sell to
firm f2,d∈R+is the cost to initially establish the supply chain, and a∈Nis the number
of steps in time that takes since firm f2buys the products and f1provides it once these are
ready for delivery.
3.1.4. Integrating into a single framework
In order to numerically solve the proposed model, we implement it using the ABS approach
[75, 76]. The simulation is developed using the Python [77] programming language (version
3.9.2). The simulation occurred in rounds marked by t∈[0, T ] such that T < ∞. In the
first round (t= 0), the environment in the form of the Gsub-model is set where locations
with their initial consumer population and set of firms are generated such that the firms do
not have supply chain links between them (|L|= 0). The set of products is obtained from
a pre-defined distribution. In order to allow a reasonable simulation, we assume that before
the pandemic, products were produced and delivered to at least satisfy the demand for each
location. As such, a random subset of the firms is assumed to generate products without
any ingredients (i.e., υ= [0,...,0]). In addition, supply chains are established at random
between firms until the overall demand of the consumers is met. Finally, prices are allocated
to the products without any ingredients and the selling prices of each firm are established
to be the cost of all the products to generate its own with some random profit margin that
allows it to end up each step in time with a profit of x∈[1%,50%] from its volume.
Afterward, for each round (t > 0), the following processes occur and are performed by the
agents. For the epidemiological dynamics, for each location (v∈V) the susceptible consumers
become infected due to interaction with infectious consumers. Exposed consumers become
infectious once et=θ. Infectious consumers transform to either the recovered or dead
epidemiological state once et=γ. Consumers that are in the dead epidemiological state
(i.e., e=D) are removed from the population. In addition, for the economic dynamics,
the consumers buy the products they want after obtaining a salary. Moreover, firms pay
their operational cost, buy products from other firms, establish new supply chains, prepare
products from the products they acquire, and sell products to consumers.
3.2. Supply chain resilience
Each firm aims to make as much profit as it can (Op). In our case, it means, each firm aims
to fulfill the demand in all the locations it is located at over time. However, it also wishes
not to bankrupt during a pandemic as long as possible (Ob). Intuitively, by establishing
cost-optimal supply chains, a firm optimizes for the first objective while establishing supply
chains with all relevant firms in the economy such that their cost is smaller than the selling
price of the finished product optimizes for the latter objective. One can notice a trade-off
between the two objectives. As such, a supply chain reliance strategy aims to solve the
following optimization problem:
max
SC ω1Op+ω2Ob(2)
where SC is the set of supply chains established by the firm to acquire products and ω1, ω2∈
[0,1] are the weights of the two objectives, respectively. Formally, Op=1
TPT
t=0 mand
mintI(t, m < 0) where I(x, y) is a predict function returning xwhen condition yis satisfied.
In order to solve the optimization task, we adopted the Monte Carlo approach [78].
Namely, we sample at a random manner the combination of possible connections for each
given firm. This process occurs for ζ >> 1 times, and the configuration that established the
highest value for Eq. (2) is chosen.
We focus on four supply chain strategies where no preparation for a pandemic occurs
(ω2= 0), where firms do not wish to make profit (ω1= 0), both objectives are identically
important (ω1=ω2= 0.5), and random case where both making profit and preparation are
important to each firm in different manner (ω1>0, ω2>0).
3.3. Approximating supply chain strategy using machine learning
Since the right values of ω1and ω2are strongly dependent on the economic status of
the firm, its dependency on other firms in the economy, the consumers’ behavior, and the
magnitude of the pandemic, it is only reasonable that different firms will adopt different
ω1, ω2values as part of their supply chain reliance strategy. Nonetheless, solving such a
question analytically is unrealistic as one would be required to solve for the entire economy
at once and would be highly sensitive to any change.
Thus, in order to find the (near) optimal ω1, ω2values for each firm, we adopted a data-
driven approach. Namely, using experimental data, one can use a machine learning (ML)
based model to fairly approximate the ω1, ω2values of a firm without actually solving for
the exact scenario due to the generalization capabilities of ML models [79, 80, 81]. Formally,
this is a one-dimentional regression problem since ω1+ω2= 1 which infers that by finding
ω1, one can directly compute ω2. In order to use a ML model, one is first required to
collect representative data. To this end, we run the simulation multiple times with different
parameter values. For each such run, we use the MC approach for the values of ω1, ω2for
each firm. As a target variable for the ML to predict, we use Op+Ob, ignoring the values
of ω1, ω2in the objective to obtain a consistent evaluation of the firm’s performance over
different runs.
Using the obtained dataset, we use the Tree-based Pipeline Optimization Tool (TPOT)
[82], an automated machine learning tool that uses genetic algorithms [83] to optimize ML
models. TPOT tries multiple ML models on the dataset to find the one that performs
best. In order to make sure the results are robust, we adopted the k-fold cross-validation
method with k= 5 [84]. Since the obtained model is black-box [85], we used two methods
to explore how the model allocates ω1values to firms. First, we use the information gain
feature importance method [86] to learn how much each feature of the economy influences
the model’s prediction. Second, SHapley Additive exPlanations (SHAP) analysis was used
to gain insight into the influence of various features [87]. The SHAP values can be used to
explain the output of a ML model by attributing the contribution of each individual feature
to a particular prediction [88].
3.4. Experimental setup
Due to the difficulty of obtaining realistic data on firms’ supply chain and financial deci-
sions, one can explore a large number of synthetic economies to obtain statistically represent-
ing dynamics. Thus, we explore the influence of a pandemic in different levels of magnitude
on an arbitrary economy. For simplicity, for each sample configuration of an economy, we
run the ABS simulation for n= 100 time to obtain a statistically represented sample. For
the epidemiological-related parameter values, we used values associated with the COVID-19
pandemic [89]. Table 1 shows the parameter values used in the experiments. Importantly,
we assume all days are working days (i.e., not considering weekends and holidays).
4. Results
In this section, we present the results of the experiments. First, we show the pandemic
effect on firms over time for four supply chain resilience strategies. Second, we show a
sensitivity to important pandemic-related parameters. Finally, we show the obtained ML
model to estimate ω1for firms.
4.1. Pandemic effect of supply chain over time
We start by investigating the four configurations of supply chain resilience over time
during a pandemic. Fig. 4.prsents this analysis where the x-axis is the time in days that
passed since the beginning of the pandemic and the y-axis is the normalized firm performance
as outlined in Eq. (2) where ω1and ω2are agnostic to make allow the comparison between
the four different strategies. One can notice that firms that only consider the preparation for
the pandemic (ω1= 0) are very inefficient overall as these start around 0.3 while less affected
by the pandemic as after a year the average performance is around 0.2. Unalike, when firms
do not prepare their supply chains for a pandemic at all (ω1= 1), the firm’s performance is
near optimal but after only two months of the pandemic the average performance drops to
around 0.04 which is almost full economic collapse. The strategy that all firms balance the
two strategies results in an average performance between the two previous cases where the
initial performance is around 0.55 and after 40 days drops to around 0.2 performance followed
by a further slower decline towards 0.1 after around 200 days. For the heterogeneous case
where each firm aims to find its balance of ω1and ω2, the initial performance is the second
Parameter Symbol Value range
simulation duration T365
Duration of a step in time ∆t1 day
Monte Carlo repetition count ζ1000
Number of consumers in a location |Ni|500-5000
Number of firms in a location |Fi|5-50
Number of locations |V|1-20
Initial amount of money m(t= 0) 1 ·102−5·106$
Salary s5.5·101−5.5·103$
Number of unique products in the economy |P|10-250
Firms initial available money φ; (t= 0) 1 ·104−5·107$
Firms operational cost δ0.0025φ−0.025φ$
Workers in a firm w1−1000
Price of a product ι1·100−1·104$
Location’s average infection rate β5·10−5−1·10−2$
During from exposed to infectious θ5−9 days $
During from infection to recovered/dead γ10 −18 days $
Recovery rate ρ0.975 −0.995 $
Ingredients per product 1-20
Number of products a firm sell to consumers 1-10
TPOT population size 50
TPOT number of generations 20
Number of simulations for the machine learning model 500
Number of ω1, ω2configurations for each machine learning sample 20
Table 1: The model’s parameter value ranges used in the experiment.
highest with a score around 0.7. During the time of the pandemic, the performance decreases
relatively slowly and balanced after around 240 days around 0.35 - the highest performance
of the four strategies.
4.2. Pandemic-related sensitivity analysis for supply chain resilience
Let us focus on the most realistic case of the four strategies out of the four - where each
farm has its own values for ω1and ω2. Since finding the optimal value for ω1is extremely
computationally challenging, compute a sensitivity analysis such that each case shows n=
100 unique runs, taking the average best ω1value. Fig. 5 shows the results of the sensitivity
analysis such that the x-axis indicates the parameter under investigation and the y-axis
indicates the optimal average ω1value. The results are shown as the mean ±standard
deviation of the n= 100 runs for each parameter value. Specifically, Fig. 5a shows the
change of the optimal average ω1as a function of the average infection rate (β). One can
notice a sigmoid function with the standard deviation growing alongside the value of ω1. Fig.
0 30 60 90 120 150 180 210 240 270 300 330 360
Time since the beginning of the pandemic in days [t]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Normalized firm performance (
Op
+
Ob
)
1= 0.0, 2= 1.0
1= 1.0, 2= 0.0
1= 0.5, 2= 0.5
1[0, 1], 2= 1 1
Figure 4: Analysis of four supply chain resilience strategies as the course of a one-year-long pandemic. The
results are shown as the mean ±standard deviation of n= 1000 simulations.
5b focus on the change due to the average recovery rate (γ) where a higher γvalue results
in higher ω1values in a somewhat linear fashion. Similarly, Fig. 5c shows that ω1is linearly
increasing with the increase in the average population size but this outcome is increasingly
less accurate as the population grows which is indicated by the increasing in the error bars’
size. Fig. 5d reveals a sharp decrease in the value of ω1between one and two locations while
afterward the value of ω1decreases logarithmically with respect to the value of the average
number of locations.
0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5
Average infection rate in year ( )
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Optimal average SC preparation ( 2)
(a) Average infection rate in year (β).
4 5 6 7 8 9 10 11 12 13 14
Average recovery rate in days ( )
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Optimal average SC preparation ( 2)
(b) Average recovery rate in days γ.
100 200 300 400 500 600 700 800 900 1000 1100
Average population size (|
Ni
|)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Optimal average SC preparation ( 2)
(c) Average population size in a location (|Ni|).
1234567891011
Average number of locations (|
V
|)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Optimal average SC preparation ( 2)
(d) Number of locations (|V|).
Figure 5: A sensitivity analysis of the model’s main epidemiologically-related parameters. The weight of
profit rather than supply chain resilience parameter (ω1) is obtained as an average ±standard deviation of
n= 100 simulations with the rest of the parameters are sampled according to Table 1.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Near-optimal value for 1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Model's prediction for 1
Sampled data
Optimal prediction
Figure 6: The machine learning model’s performance for 1000 randomly picked predictions for near-optimal
value of ω1as obtained by the Monte Carlo method with the agent-based simulation compared to the obtained
machine learning model’s prediction. The dashed line indicates the locations of perfect predictions.
4.3. Supply chain resilience machine learning model
Overall, we trained the ML model on 2.2 million samples of companies ω1value, Op+Ob,
and the economy’s initial state before the beginning of a pandemic. An additional 0.55 million
samples are used to evaluate the obtained model’s performance. The error analysis of the
model shows that the obtained model has a mean absolute error of 0.047 and a coefficient of
determination of R2= 0.816. Fig. 6 presents the ML model’s prediction for 1000 randomly
picked samples. The x-axis indicates the near-optimal value for ω1as obtained using the
MC process using the ABS simulation while the y-axis is the ML model’s prediction for ω1
for the same firm given the same initial conditions used by the ABS simulation. One can
notice that for ω1<0.2 and ω1>0.7 the ML model makes very accurate predictions while
for 0.2< ω1<0.7 the model has a larger error, on average. In addition, some firms have a
relatively large error but these are a relatively small portion of the entire dataset.
|
Ni
| |
Fi
|
m
(
t
= 0)
s
(
t
= 0) |
w
| |
SC
|/|
P
|
i
0
2
4
6
8
10
12
14
16
18
20
Relative feature importance percent
Figure 7: Feature importance analysis of the obtained machine learning model. The results show the average
value of a 5-fold cross-validation analysis. These features are responsible for 87.13% of the model’s decision
making process.
Fig. 7 shows the feature importance distributions of the top 10 most important features
which are responsible for 87.13% of the properties the ML uses to make a prediction. The
most important feature with 19.23% is the number of consumers in a location the firm
operates in, followed by the number of firms in the location (15.44%). Afterward, the initial
available money of the consumers and the initial available money of the company are the third
and sixth most important features with 12.32% and 6.08%, respectively. The operational cost
and consumer’s average salary are the forth and fifth most important features with 8.31%
and 6.34%, respectively. The pandemic-related parameters, the average infection rate (βi),
and recovery rate (γ) are the ninth and tenth most important features with 4.65% and 3.44%,
respectively. The operational features - the number of workers and the number of supply
chains divided by the number of products in the economy are the seventh and eights most
important features with 5.87% and 5.45%, respectively.
Fig. 8 shows the Shap analysis of the obtained ML model. On the x-axis, dots on the
right side indicate a contribution towards the model predictions of ω1= 0 while left side dots
indicate a contribution towards the model predictions of ω1= 0. One can notice a mostly
consistent behavior for |Ni|and |Fi|while the other feature demonstrate a more chaotic
pattern.
5. Discussion and Conclusion
In this study, we proposed a novel agent-based simulation (ABS) based model for the
influence of pandemic spread on supply chain resilience strategies. The spatio-temporal model
combined an extended SIR epidemiological model (SEIRD) integrated with a supply-demand
economic model for multiple physical locations. This model is used to evaluate the efficiency
of different supply chain resilience on a wide range of scenarios using in silico experiments,
focusing on balancing between profit maximization and preparation for a pandemic in terms
of supply chain resilience strategy.
Using this model, we started by exploring the “edge” cases of supply chain resilience as
well as the average and heterogeneous cases to evaluate the possible range of strategies, as
shown in Fig. 4. Unsparingly, for the case where firms adopted a supply chain resilience
strategy of doing nothing, a pandemic results in near to the collapse of the entire economy
while the other end of the rope results in the least effect by the pandemic but provides a
strongly unappealing and inefficient economic environment. A more balanced case performs
better for times without a pandemic but is still almost half as good as the first strategy while
in the end decreases to only slightly than twice as good performance from the first case just
after half a year. A heterogeneous case provides both a more realistic and better-performing
strategy for firms in an economy. This outcome, while unsurprising, empirically supports
the further analysis of this supply chain resilience strategy and aligns to previous works that
highlighted the benefits of heterogeneous decision-making of agents to solve complex tasks
[90, 91].
When considering the most realistic case of these where each firm has a unique strategy
of supply chain resilience for pandemics, the model predicts somewhat expected results, as
revealed by Fig. 5. Initially, as the infection rate growth, firms are required to be more
prepared to a pandemic to overcome it. This result is intuitive and very well established
[92, 93]. In a similar manner, increased in the recovery rate requires more preparation as the
overall number of infected individuals is higher at the same time which further contributes to
the pandemic spread [94]. Population size increases cause more drastic shifts in an economy
for the same pandemic, on average, which requires more supply chain resilience and also
generates more uncertainty as more infection paths can occur [95]. Interestingly, one can
observe a sharp decrease in the supply chain resilience between one and two locations, as
shown in Fig. 5d. This drop in the value of ω2can be associated with the fact that not
the entire pandemic occurs at once and influences the entire economy and therefore, a less
dramatic supply chain resilience strategy can be adopted by firms to overcome the same
-0.4 -0.2 0.2 0.40.0
SHAP value (impact on model output)
Low
High
Feature value
|Ni|
|Fi|
m (t=0)
δ
s
φ (t=0)
|w|
|SC|/|P|
βi
γ
Figure 8: A Shap analysis of the ML model. Dots on the right side indicate a contribution towards the model
predictions of ω1= 0 while left side dots indicate a contribution towards the model predictions of ω1= 0.
pandemic, on average. This effect is greatly reduced between two and more locations as
visible in the same plot.
We aimed to provide an approximation tool for firms to manage their supply chain re-
silience strategy with respect to their state and the economy’s state using an ML model. As
illustrated in Fig. 6, the obtained ML model is extremely accurate for firms that are not sig-
nificantly affected by a pandemic (of a small-medium size), and its performance is reduced as
the firm is more sensitive to the pandemic. That said, for very sensitive firms, the ML model
is again performing relatively well. This outcome can be explained by the fact that for the
more extreme cases, it is clearer to ignore pandemic risks or be extremely prepared for them
while the intermediate cases are more chaotic in nature, resulting in a wide range of possible
outcomes [96]. Interestingly, the ML model predicts, as shown in Fig. 7, that the economy
size, as reflected by the number of consumers and firms, is the most important parameter
to the individual firm strategy - agreeing with previous studies about supply chain resilience
[97, 98]. The firm’s size and economic state, as indicated by the number of workers and the
operational cost, are very important but less than the overall economy’s dynamics as also
found by [99]. However, this average importance value hides a more chaotic picture which is
revealed by Fig. 8. While the economic size has mostly a positive effect on the number of
how much a firm should prepare for a pandemic, the rest of the features show inconsistent
behavior when considered individually. This outcome can be expected due to the complex
connections between the features in the dynamics that dictate a firm’s future and therefore
optimal strategy before a pandemic starts. Nonetheless, it highlights the importance of other
studies to explore multiple features for supply chain resilience at once rather than one at a
time.
The proposed model is not without limitations. The proposed model assumes a pandemic
originated with a single pathogen that does not mutate over time. While this assumption is
commonly used [100, 101, 102, 103] it is known to be false for even medium-size pandemics,
and future works should use multi-strain with re-infection mechanism epidemiological sub-
model to obtain more realistic epidemiological dynamics. Second, we assume that the prices
of products from the firms are static over time to focus on the pandemic spread influence
on the supply chain rather than other factors. Nonetheless, natural price fluctuations as
well as high-order pandemic-related price fluctuations may play a central role in supply
chain management and should be included in future work to make it more realistic. Third,
population size and the number of firms as well as their spatial distribution are assumed to
be static over time. Adding consumer migration dynamics as well as the introduction of firm
establishment and closer are also promising venues for future work. Fourth, the pandemic-
related demand assumes all consumers are aware of the pandemic state in their community
at any given point in time. A relaxation of this assumption by adding delay and only an
estimation for the pandemic spread would make the model more realistic and reflect the
actual available information consumers have during a pandemic.
Taken jointly, the proposed model and its agent-based simulator provide policymakers
and business owners with a computational tool to evaluate their supply chain preparedness
for the event of a large-scale pandemic, such as COVID-19. Our results show that for even
relatively large pandemics, well-prepared businesses are theoretically able to overcome the
challenge and thrive during and after the pandemic ends.
Declarations
Funding
This research did not receive any specific grant from funding agencies in the public,
commercial, or not-for-profit sectors.
Competing interests
The authors have no relevant financial or non-financial interests to disclose.
Acknowledgment
The author wishes to thank Labib Shami for motivating this research and providing
valuable economic advice.
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