Abhijit Sau

Kalaivanan Nagarajan

Bianca Patrahau

Lucas Lethuillier-Karl

Robrecht M A Vergauwe

Anoop Thomas

Joseph Moran

Cyriaque Genet

Thomas W Ebbesen

Modifying Woodward-Hoffmann Stereoselectivity under Vibrational Strong Coupling

Keywords: pericyclic reaction, Woodward-Hoffmann, strong coupling, orbital symmetry, electrocyclic ring opening 10

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Introduction:

The Woodward-Hoffmann (WH) rules elaborated in the 1960's epitomizes the importance of symmetry in chemical reactivity. 1-6 In essence, the conservation of orbital symmetry between reactants and products affect the reaction landscape and thereby the products that are favoured. In ideal cases, such as for cis-3,4-disubstitued cyclobutene, the products that are forbidden in the ground states are allowed in the excited state and vice-versa: cis-3,4-disubstitued cyclobutene undergoes conrotatory electrocyclic ring opening under thermal conditions to form cis-trans-1,3-butadiene, while under photochemical condition, trans, trans-1,3-butadiene is obtained through the disrotatory pathway, as illustrated in Figure 1a. However, in most cases, a mixture of products is obtained due to effects that mitigate the symmetry-based outcome, such as secondary effects of substituents, even if overall the WH rules are followed. 2-6 Chemistry under strong coupling has received much attention since the first demonstration of a modified chemical landscape. 7-34 Recently, we reported that vibrational symmetry plays a key role in modifying the charge-transfer complexation equilibrium between I2 and mesitylene under vibrational strong coupling (VSC). 12 To further explore the role of symmetry under VSC, we have chosen the well-studied thermal electrocyclic ring-opening of cyclobutene into butadiene, where the reaction path can be predicted by the principle of orbital symmetry conservation according to the WH rules. Among the various cyclobutene derivatives reported for this kind of reaction, we have synthesised cyclobutene-cis-3,4dimethylcarboxylate (CB), which is suitable for studies under VSC. As expected from the WH rules, [1] the molecule under study undergoes thermal ring-opening at 90 ºC to yield mostly the corresponding symmetry-allowed cis-trans product, but also some symmetry-forbidden transtrans 1,3-butadiene product. The formation of symmetry-forbidden trans-trans butadiene can be attributed to secondary orbital interaction from the carbonyl group, which reduces the activation energy of the disrotatory transition state. 3-6 In addition, it is a liquid at room temperature, the reaction is monomolecular and gives no products other than to those expected from the WH rules, all features that simplify the study.

Vibrational strong coupling (Figure 1b) is achieved by placing the molecular solution inside a Fabry-Perot optical cavity, which is tuned to have a mode in resonance with a selected vibrational mode of the molecule (Figure 1c). When the conditions for strong coupling are satisfied, the molecular transition splits into two new hybrid states known as P+ and P-, The Fabry-Perot cavity was fabricated by separating two parallel Au (10 nm) mirrorcoated ZnSe substrates with a 6 m Mylar spacer, and having a hollow channel designed to inject the liquid sample. Such cavities will generate many resonances separated the free spectral range or FSR and one of them needs to be tuned to a vibration (Figure S1, Supporting Information). To prevent the reaction mixture from coming in contact with the metal surface, 200 nm of SiOx was sputtered on top of the Au. For the so-called non-cavity condition, the same SiOx coated windows were used but without the Au coating. Electrocyclic ring-opening

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of CB was performed by heating the sample at 90 ºC for two hours (Figure 2a). To begin, we followed the reaction under non-cavity conditions (Figure S2, Supporting Information). Neat CB was injected into the preheated chamber at 90 ºC and the reaction was monitored in the FT-IR (Figure 2b). The first-order nature of the reaction was confirmed by the exponential growth of the peak at 1602 cm -1 , which corresponds to the C=C stretching mode of the butadiene derivative (Figure S3a, Supporting Information). After two hours, the reaction mixture was allowed to cool to room temperature (RT) and dissolved in CDCl3 to perform the 1 H-NMR measurement. The 1 H-NMR spectrum shows the formation of the symmetry-allowed cis-trans butadiene and the symmetry-forbidden trans-trans derivative, along with the unreacted starting material CB (Figure 2e). The trans-trans butadiene is characterised by the signals a and b, whereas the cis-trans derivative is identified by the signals c, d, e and f, as shown in Figure 2.

As predicted by the WH rules, the symmetry-allowed cis-trans butadiene is obtained as the major product (92%) while the symmetry-forbidden trans-trans derivative is present only with a 8% yield. This leads to a ratio between symmetry-allowed conrotatory (cis-trans) vs symmetry-forbidden disrotatory (trans-trans) product ratio of 12.7 ±0.5. No other products or intermediates were observed in the NMR spectra. The overall reaction rate (kobs) at 90 ºC under non-cavity conditions was 0.0120 min -1 (Figure 2h). The formation of symmetry-forbidden trans-trans butadiene product is attributed to the secondary orbital interactions arising from the carbonyl groups at C3 and C4 of cyclobutene derivative CB. 3-6 To perform the reaction under VSC, the FSR of Fabry-Perot cavities were carefully tuned at 90 °C to have a mode to couple either carbonyl stretching band at 1740 cm -1 or the inplane C-H bending at 1436 cm -1 . Cavity tuning is easily performed by adjusting the screws that hold the assembly together, as detailed in the Supporting Information (Figure S1, Supporting Information). As with the non-cavity experiment, the cavity assembly was equilibrated at 90 ºC for two hours before injecting the sample.

In a first set of experiments, the 5 th mode of the Fabry-Perot cavity was tuned to the C=O stretching mode of CB (Figure 2c) at 1740 cm -1 at 90 ºC. Neat CB was injected and the reaction progress was monitored by FT-IR. The appearance of equally spaced P+ and P-with a Rabi splitting of 160 cm -1 , corresponding to the molecular transition at 1740 cm -1 , confirms the strong coupling. The other peaks in Fig. 2c are the 4 th and 6 th cavity modes and are not interacting with any vibrations. The kinetics of the reaction was followed by monitoring the shift in the higher order cavity modes, which reflects the difference in refractive index of the reactant and products (Figure S3, Supporting Information). It should be noted that both the

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reactant and the products have the same C=O group and therefore as the reaction proceeds the system remains coupled and the Rabi splitting hardly changes. Under VSC of C=O stretching, the overall reaction rate (kobs) was 0.0080 min -1 . After two hours, the reaction mixture was allowed to cool down to RT and the products were quantified by 1 H-NMR (Figure 2f). Surprisingly, we observed significant reduction in the symmetry-forbidden trans-trans butadiene derivative (1.5%) when compared to the non-cavity condition (8%, Figure 2i). This leads to a ratio between the symmetry-allowed and the symmetry-forbidden products of 28 ±2 under VSC of the C=O stretching mode. As detailed in the Supporting Information, only ~ 60% area of the Fabry-Perot cavity is homogeneous and tuned under VSC. By incorporating the area correction, the ratio of cis-trans to trans-trans butadiene is actually 38 ±2 (Table 1). In other words, coupling the C=O stretching mode increases the selectivity for the symmetry-allowed product.

In a second series of experiments, we coupled the cyclobutene ring C-H in-plane bending mode (Figure 2d), which overlaps with the strong CH3 deformation modes at 1436 cm - 1 . After injecting the neat sample into the equilibrated cavity at 90 ºC, FT-IR spectra show P+ and P-with a Rabi splitting of 62 cm -1 with the 4 th mode of the cavity. It should be noted that an apparent splitting appears between the 5 th mode and the C=O stretching mode but closer analysis shows that it is detuned by 60 cm -1 . It is now well-known that at such detuning, there is no strong coupling effect on the chemistry 12,15,16,22,23 . Again, after two hours, the reaction mixture was analysed by 1 H-NMR. Similar to VSC of C=O stretching mode, the overall reaction rate (kobs) was found to be 0.0083 min -1 . Interestingly, we observed a significant enhancement in the formation of symmetry-forbidden trans-trans butadiene (15%) when compared to the non-cavity condition (Figure 2g and2i). This results in a symmetry-allowed vs symmetry forbidden product ratio of 5 ±1, after correcting the values for the tuned area (Table 1).

As a final reference, similar experiments were performed at off-resonance condition (Figure S4, Supporting Information) where none of the cavity modes are strongly coupled with any of the molecular vibrations. The mere presence of cavity modes in the off resonance experiments did not show any modification in the product ratio (Table 1), which demonstrates the importance of strong coupling between the cavity mode and the molecular transition to observe the changes in the reaction.

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Individual rate constants for the symmetry-allowed conrotatory (kcon) and symmetryforbidden disrotatory (kdis) ring-openings were estimated from the observed rate constant (kobs) of the reaction (Figure 2h) and the product ratio, by combing equations 1 and 2:

𝑘 𝑐𝑜𝑛 + 𝑘 𝑑𝑖𝑠 = 𝑘 𝑜𝑏𝑠 ( 1
)
𝑘 𝑐𝑜𝑛 𝑘 𝑑𝑖𝑠 = [𝑐𝑖𝑠-𝑡𝑟𝑎𝑛𝑠] [𝑡𝑟𝑎𝑛𝑠-𝑡𝑟𝑎𝑛𝑠] (2)
The rate of formation of cis-trans and trans-trans butadiene under non-cavity conditions at 90 ºC was 0.011 min -1 and 0.001 min -1 respectively. Under VSC, kcon and kdis varies depending on the coupled vibration and the values are given in the Supporting Information (Table S1).

The difference between the free energies of activation of the symmetry-allowed conrotation and symmetry-forbidden disrotation could be calculated from the product ratio by using the following equation, and the results are shown in the Table 1:

∆∆𝐺 ‡ = 𝑅𝑇𝑙𝑛 [𝑐𝑖𝑠-𝑡𝑟𝑎𝑛𝑠] [𝑡𝑟𝑎𝑛𝑠-𝑡𝑟𝑎𝑛𝑠] (3)
Table 1. Cis-trans to trans-trans product ratio (conrotation/disrotation) obtained from 1 H-NMR measurements of electrocyclic ring opening of CB at 90 ºC in non-cavity and cavity tuned to different vibrational modes. Relative Gibbs free energy change (G ‡ ) between the con and disrotatory transition states are obtained from the product ratio using equation 3.

Experiment cis-trans/trans-trans G ‡ (kJ mol -1 )

Non-cavity 12.7 +0.5 7.7 ±0.2

Cavity off resonance 12.4 +0.5 7.6 ±0.2

Cavity on resonance

C=O stretching mode (1740 cm -1 ) 38.0 +2 11.0 ±0.5

C-H bending mode, in-plane (1436 cm -1 ) 5.0 +1 4.9 ±0.5

In non-cavity conditions, the average free energy change between con-and disrotatory transition states of CB at 90 ºC is 7.6 ±0.2 kJ mol -1 , which is much lower when compared to cis-3,4-dimethylcyclobutene (35  40 kJ mol -1 ). 2 This significant reduction in the G ‡ can be attributed to the stabilisation of the disrotatory transition state through secondary orbital interactions with the carbonyl group. 3-6 VSC of the C=O stretching mode exhibits increased G ‡ (11 ±0.5 kJ mol -1 ), resulting in reduced trans-trans product formation, which presumably occurs due to the perturbation of the secondary orbital interaction of the C=O with the 10.1002/anie.202013465

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cyclobutene ring. On the other hand, VSC on C-H bending shows decreased G ‡ (4.9 ±0.5 kJ mol -1 ), which enhances the symmetry-forbidden trans-trans product.

Such modifications can be interpreted by looking at the additional symmetry constraints that VSC puts on the orbital correlation diagram. Let us start with the case of VSC involving the in-plane C-H bending mode. The cavity mode strongly coupled to the molecules puts the orbitals in-phase with respect to the intracavity field, and thus forces the phases of the newly formed p orbitals to remain symmetric with respect to the mirror plane of the cis-cyclobutene moiety (Figure 1a). Such symmetries are compatible with the electrocyclic ring-opening through a disrotation, explaining why VSC on C-H bending favours disrotatory processes and increases kdis in the reaction path.

To understand the effects of VSC of the C=O stretching mode, it should be recalled the peculiar role played by the carbonyl groups in the ring-opening process. 3-6 It mediates secondary, but efficient, orbital interactions with the cyclobutene ring and thereby affecting the WH rules.

where kB, h and R are the Boltzmann, Plank's and gas constants respectively.H ‡ and S ‡ are extracted from the slope and intercept from the plot ln(k/T) vs 1/T (Figure S5, Supporting Information). The results are summarized in Table 2. When compared to non-cavity conditions, VSC on both C=O and C-H vibrations shows reduced H ‡ and S ‡ for cis-trans and trans-trans products. However, the relative Gibbs free energy change between the con-and disrotatory transition states is increased upon VSC on C=O (10.8 ±0.5 kJ mol -1 ), while it is reduced under VSC on C-H (4.9 ±0.5 kJ mol -1 ), when compared to non-cavity condition (7.6 ±0.5 kJ mol -1 ). These thermodynamic changes are represented in Figure 4 and are entirely consistent with the cavity effects discussed above, in that VSC leads to stabilizing either symmetry-allowed conrotatory or symmetry-forbidden disrotatory products depending on which C=O or C-H vibrational band is coupled. In conclusion, the modification of the stereoselectivity of a pericyclic reaction as studied here confirms the key role of symmetry in chemistry under VSC. The cavity vacuum field imposes symmetry constraints that facilitate either the con-or disrotation pathways in such cyclic ring-opening reaction and thereby the selectivity of the Woodward-Hoffmann rules.

Further such studies should enrich our understanding of the details of chemistry under VSC.

Figure 1 .

 1 Figure 1. a) Electrocyclic ring-opening reaction of cis-3,4-disubstitued cyclobutene which undergoes conrotation under thermal condition to form cis-trans butadiene and disrotation in the presence of light to give trans-trans butadiene ; b) Schematic representation of vibrational strong coupling between the molecular vibrational transition and Fabry-Perot cavity mode; c) FT-IR spectra of cyclobutene-cis-3,4-dimethylcarboxylate, with inset showing the molecular structure. vibration and square root of the concentration of the coupled molecules inside the cavity. VSC occurs in the dark due to the coupling of the zero-point energy of the cavity mode with that of the vibrational transition. 9

  10.1002/anie.202013465

  10.1002/anie.202013465 Accepted Manuscript

Figure 2 .

 2 Figure 2. a) Electrocyclic ring opening reaction of cyclobutene-cis-3,4-dimethylcarboxylate (CB) under thermal conditions. FT-IR spectra in transmission mode of b) non-cavity conditions, c) under VSC of the C=O stretching mode (1740 cm -1 ) and d) under VSC of the C-H bending mode (1436 cm -1 ). e), f) and g) 1 H-NMR spectra showing alkenyl proton signals of cis-trans and trans-trans butadiene products along with the unreacted cyclobutene starting material obtained respectively from non-cavity conditions, VSC at 1740 cm -1 and VSC at 1436 cm -1 . Note that the 1 H signal labelled * is from CDCl3 and CB is from unreacted starting material. h) Observed kinetics (kobs) of the reaction under non-cavity (black circle) and under VSC conditions (C=O, blue triangles and C-H, red square. i) Superimposed 1 H-NMR signals normalised to the cis-trans signal e showing the change in the yield of the disrotation product trans-trans (signal a).

  10.1002/anie.202013465

Figure 3 .

 3 Figure 3. Relative orientation of carbonyl groups in the transition state expected in a) symmetry allowed conrotation and b) symmetry forbidden disrotation.

  This article is protected by copyright. All rights reserved.The thermodynamics of the electrocyclic ring-opening reaction under VSC of both the C=O and the C-H vibrations were determined by measuring the reaction rate as a function of temperature (FigureS5, Supporting Information). The thermodynamic parameters H ‡ , S ‡ and G ‡ were then determined from the Eyring plot, ln(k/T) vs 1/T:

Figure 4 .

 4 Figure 4. Energy level diagram for the con-and disrotatory electrocyclic ring-opening of CB under non-cavity and VSC conditions. The relative Gibbs free energy change (G ‡ ) between symmetry-allowed conrotatory and symmetry-forbidden disrotatory transition states is altered by coupling to C=O (blue line) and C-H (red line), when compared to the non-cavity condition.

Table 2 .

 2 Thermodynamic parameters in non-cavity and cavity conditions tuned to different vibrational modes.

	Experiment	H ‡ (kJ mol -1 ) cis-trans trans-trans cis-trans trans-trans cis-trans S ‡ (J mol -1 ) G ‡ (kJ mol -1 ) trans-trans
	Non-cavity	93	92	5	-18	91	99
	C=O coupling	79	82	-24	-46	88	99
	C-H coupling	85	84	-21	-37	93	98

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Acknowledgements

We acknowledge support of the International Center for Frontier Research in Chemistry (icFRC, Strasbourg), the ANR "ERA-NET QuantERA" -Projet "RouTe" (ANR-18-QUAN-0005-01), the ANR Equipex Union (ANR-10-EQPX-52-01), CSC (ANR-10-LABX-0026 CSC) within the Investissement d'Avenir program ANR-10-IDEX-0002-02. TWE and JM both acknowledge the support of the ERC (project no 788482 MOLUSC and no 639170 CarbonFix, respectively). A.S. thanks the EU for a H2020 Marie Skłodowska Curie Fellowship (792101).

Conflict of Interest

The authors declare no conflict of interest.


Dmitry Ivanov
email: divanov@hwr-berlin.de

Alexandre Dolgui
email: alexandre.dolgui@imt-atlantique.fr

OR Methodologies and Managerial Insights for Coping with Disruption Propagations in Supply Chains Amid the COVID-19 and Withstanding Future Pandemics

Keywords: supply chain, COVID-19, pandemic, disruption propagation, ripple effect, structural dynamics, recovery, managerial insights

The COVID-19 pandemic unveils unforeseen and unprecedented fragilities of the supply chains (SC). One strong stressor to SCs and their numerous collapses comes from disruption propagations through the networks. The existing knowledge in modelling disruption propagations in SCs is multi-faceted and deserves to be analysed for the unique set of factors shaping SC adaptations during and after a global pandemic. We scrutinize OR studies published in international journals and dealing with disruption propagation in SCs. This streamlining of the literature allowed revealing several new research tensions and novel categorizations/classifications. Our study pursues two major contributions subject to two research questions. First, we collate the state-of-the-art in research on disruption propagations in SCs and identify a methodical taxonomy and theories displaying the value and applications of OR methods to coping with pandemic impacts on the SCs. Second, we reveal and systemize managerial insights from the theory that can be used for operating (adapting) amid a pandemic and at the times of recovery along with withstanding future pandemics. In particular, we find that outcomes of the OR contributions and the resulting managerial insights can be categorized into three levels, i.e., network, process, and control. Our outcomes in both managerial and theoretical domains are structured into five stages (i.e., Anticipation; Early detection; Containment; Control and Mitigation; and Elimination) following the WHO classification. Our study can be used by industry and researchers alike in order to progress the decision-support systems guiding the SCs amid the COVID-19 pandemic and recovering them thereafter. Suggestions for future research directions are offered and discussed.

Introduction

COVID-19 was first reported in late 2019 in Wuhan, China. By May 4, 2020, over 2.4 million people were infected and approximately 240,000 people died. The COVID-19 pandemic has created a lot of uncertainty in all areas of life and in supply chains (SC) in particular. SCs experience unprecedented vulnerabilities in lead-times and order quantities, disruptions in networks structures, and severe demand fluctuations. Adversely, many of these vulnerabilities are encountered simultaneously. 94% of the Fortune 1000 companies have been reported seeing coronavirus-driven SC disruptions (Fortune 2020). A recent survey by ISM of about 600 US companies revealed that in the mid of April 2020, average lead times were at least twice as long as compared to "normal" operations, for Asian (222% for China, 217% for Korea, and 209% for Japan), European (201%) and domestically sourced inputs (200%) (ISM 2020). The same report says that Chinese and European manufacturing is at about one-half normal capacity, 53% and 50% respectively.

The outbreak of COVID-19 virus and an associated global pandemic have shed light on the scope and scale of the ripple effect seen in disruption propagations in the global SCs (Ivanov et al. 2014a;[START_REF] Dolgui | Ripple Effect in the Supply Chain: An Analysis and Recent Literature[END_REF]. [START_REF] Haren | How Coronavirus Could Impact the Global Supply Chain by Mid-March[END_REF] observed two examples of the COVID-19-triggered ripple effect immediately after the epidemic outbreak. Fiat Chrysler Automobiles NV was halting production at a car factory in Serbia, because it can't get parts from China. Hyundai "decided to suspend its production lines from operating at its plants in Korea … due to disruptions in the supply of parts resulting from the coronavirus outbreak in China." While these observations were done in the second half of February 2020, the scaling of the ripple effects in March-May 2020 have been exponential, driven by the closures of manufacturing facilities, stores and logistics activities and adversely affecting almost all industries and services worldwide [START_REF] Choi | Coronavirus is a Wake-Up Call for Supply Chain Management[END_REF][START_REF] Choi | Innovative "Bring-Service-Near-Your-Home" Operations under Corona-Virus (COVID-19/SARS-CoV-2) Outbreak: Can Logistics Become the Messiah? Transportation Research Part E: Logistics and Transportation[END_REF][START_REF] Ivanov | Predicting the impact of epidemic outbreaks on the global supply chains: A simulation-based analysis on the example of coronavirus (COVID-19 / SARS-CoV-2) case[END_REF][START_REF] Ni | How China can rebuild global supply chain resilience after COVID-19[END_REF]. The World Economic Forum -WEF (2020) emphasized the need for firms and organizations to adapt their SCs amid the COVID19 pandemic and in light of future trade challenges. In its totality, the COVID-19 pandemic seriously wreaks havoc on SCs and poses a series of novel decision-making settings for SC professionals and novel questions for researchers which are relevant amid the pandemic and in the course of future recoveries.

Operational Research (OR) community possessуы an enormous scope of methods and tools that can be used for decision-making support under uncertainty. In particular, advanced modelling approaches have been established for decision-making environments with uncertain supply quantities and lead-times (Prak et al. 2017), supplier and facility disruptions [START_REF] Snyder | Reliability models for facility location: The expected failure cost case[END_REF][START_REF] Klibi | Scenario-based supply chain network risk modeling[END_REF][START_REF] Qi | A continuous-review inventory model with random disruptions at the primary supplier[END_REF][START_REF] Paul | Real time disruption management for a two-stage batch production-inventory system with reliability considerations[END_REF][START_REF] Silbermayr | Dual sourcing under disruption risk and cost improvement through learning[END_REF], transportation uncertainties [START_REF] Amiri-Aref | Equitable location of facilities in a region with probabilistic barriers to travel[END_REF], and demand uncertainties [START_REF] Rezapour | Resilient supply chain network design under competition: A case study[END_REF][START_REF] Amiri-Aref | The multi-sourcing location inventory problem with stochastic demand[END_REF][START_REF] Mohamed | Designing a two-echelon distribution network under demand uncertainty[END_REF]. We refer to comprehensive surveys on these operational and disruption risks [START_REF] Klibi | The design of robust value-creating supply chain networks: A critical review[END_REF][START_REF] Snyder | OR/MS Models for Supply Chain Disruptions: A Review[END_REF][START_REF] Govindan | Supply chain network design under uncertainty: A comprehensive review and future research directions[END_REF][START_REF] Shen | Market disruptions in supply chains: a review of operational models[END_REF], Hosseini et al. 2019). Alike literature on epidemics and humanitarian disasters with SC and logistics context delivers a rich body of inspiring methods and outcomes [START_REF] Altay | OR/MS research in disaster operations management[END_REF][START_REF] Van Wassenhove | Humanitarian aid logistics: Supply chain management in high gear[END_REF], Dasaklis et al. 2012[START_REF] Gupta | Disaster management from a POM perspective: mapping a new domain[END_REF], Dubey et al. 2019[START_REF] Besiou | Humanitarian operations: A world of opportunity for relevant and impactful research[END_REF].

Our study is devoted to one specific and strong stressor to SCs during a pandemicthe disruption propagations through the networks (i.e., the ripple effect) and the subsequent changes in the SC structures (i.e., structural dynamics). Adversely, SC disruptions are stimulated by simultaneous disruptions and uncertainties in demand and supply. The existing knowledge in modelling the SC ripple effect and structural dynamics is multi-faceted and deserves to be analysed for the unique set of factors shaping SC adaptations when facing a global pandemic.

We scrutinize 40 quantitative studies published in 15 international journals (cf. Appendix 1) and dealing with disruption propagation and structural dynamics in SCs. This streamlining of the literature allowed revealing several new research tensions and novel categorizations/classifications. To this end, our study pursues to address two central research questions (RQ): RQ1: How did the OR literature address issues related to the ripple effects and structural dynamics in SCs in terms of methodologies, problem settings, outcomes and managerial insights? RQ2: How the existing OR knowledge can be used to support the SC managers in adapting the supply networks amid the COVID-19 pandemic and what are future research opportunities?

In particular, we find that outcomes of the OR contributions and the resulting managerial insights can be categorized into three levels, i.e., network, process, and control. Our outcomes in both managerial and theoretical domains are structured into five stages (i.e., Anticipation, Early detection, Containment, Control and Mitigation, and Elimination) following the WHO classification. Our study can be used by industry and researchers alike in order to progress the decisionsupport systems guiding the SCs amid the COVID-19 pandemic and recovering them thereafter. Suggestions for future research directions are offered and discussed.

The rest of this study is organized as follows. In section, we presented methodic of our study. Section 3 is devoted to analysis of OR theories, applications and managerial insights. In Section 4, we organize the discussion around an extrapolation of the existing knowledge on pandemic settings. We discuss both managerial implications and future research angles. We conclude the paper in Section 5 by summarizing this study.

Methodic of our study

Purpose of our study

SC disruptions have become a visible research avenue over the last two decades. In several settings, disruptions can be localized without cascading through a network. Though, in other situations upstream disruptions are propagating downstream the SCs adversely impacting performance of individual firms and overall networks. According to [START_REF] Dolgui | Does the ripple effect influence the bullwhip effect? An integrated analysis of structural and operational dynamics in the supply chain[END_REF], the ripple effect "refers to structural dynamics and describes a downstream propagation of the downscaling in demand fulfilment in the supply chain (SC) as a result of a severe disruption." [START_REF] Ivanov | The ripple effect in supply chains: Trade-off 'efficiency-flexibility-resilience' in disruption management[END_REF] state that "Ripple effect describes the impact of a disruption on SC performance and disruption-based scope of changes in the SC structures and parameters". These definitions imply that the ripple effect is concerned with multi-stage networks and triggering failures in the network elements as a domino effect.

In 2010-2014, first studies appeared in the area of the ripple effect and an increased interest in disruption propagation and correlated disruptions has been observed [START_REF] Liberatore | Hedging against disruptions with ripple effects in location analysis[END_REF], Mizgier et al. 2012, Ghandge et al. 2013, Ivanov et al. 2014a). The first explicit definition of the ripple effect has been undertaken by [START_REF] Ivanov | The ripple effect in supply chains: Trade-off 'efficiency-flexibility-resilience' in disruption management[END_REF] as indicated above. Thus far, much progress has been done in this area deploying different methodologies and obtaining relevant managerial outcomes and recommendations [START_REF] Scheibe | Supply chain disruption propagation: a systemic risk and normal accident theory perspective[END_REF]. [START_REF] Dolgui | Ripple Effect in the Supply Chain: An Analysis and Recent Literature[END_REF] and [START_REF] Mishra | Evolution of supply chain ripple effect: a bibliometric and meta-analytic view of the constructs[END_REF] reviewed progress in the ripple effect research for the last years mostly focusing on classifications of the resilience aspects and risk type categorization. However, there is no published survey comprehensively encompassing the disruption propagations in SCs and the resulting structural dynamics from the OR methodology point of view.

Our study pursues two contributions. First, we collate the state-of-the-art in the research on SC ripple effect and structural dynamics and identify a methodical taxonomy and theories displaying the value and applications of OR methods to coping with pandemic impacts on the SCs. Second, we reveal and systemize managerial insights from the theory that can be used for recovery from the COVID-19 and withstanding future pandemics.

Literature selection process

The literature for analysis of recent methodical contributions to managing the ripple effect has been selected based on a thorough authors' work in the ripple effect area for the last decade and observing the relevant publications along with editing several related special issues in prestigious international journals. This selection was supported by a supplementing search in most common databases (e.g., Scopus Database, ScienceDirect (Elsevier), Emeraldinsight (Emerald), Wiley Online Library (Wiley), Taylor & Francis Online (Taylor & Francis), Springer Link (Springer), and Informs PubsOnline according to the following protocol (on the example of SCOPUS):

( supply AND chain AND disruption ) AND ( ripple ) OR ( cascade ) OR ( cascading ) OR ( propagation ) OR ( correlated AND disruption ) OR ( structural AND dynamics ) OR ( transmission )

In total, we obtained a list of 142 journal papers in the areas of operational and supply chain research that have been manually processed and narrowed to meet the scope and scale of our analysis. We do not claim that the literature analysed in this paper is a full collection of all most influential contributions; though we believe to be very close to this setting. We underline that we do not follow classical structured literature review scheme but rather analyse most representative studies in terms of theoretical tensions and managerial applications. The details of our literature review protocol are given in Fig. 1.

[Insert Fig. 1 here]

We followed five major inclusion criteria. The search has been performed on May 2, 2020; the papers published by this date in international peer-reviewed journals have been included. We considered only papers with SC topics related to keywords "ripple effect", "disruption propagation", "cascading", "structural dynamics". Moreover, we restricted ourselves to the papers utilizing OR methods (e.g., empirical studies have not been analysed). Further, we considered only papers that clearly display the mechanisms of disruption propagations. In particular, we excluded papers on two-echelon problems since a meaningful disruption propagation can be treated in the setting with three echelons as a minimal complexity level. Finally, the papers without or with insufficient information about managerial insights have not been included.

OR theories, major outcomes and managerial insights

Methodical perspectives

Our analysis revealed numerous OR theories that have been successfully applied to the SC problems with disruption propagations (Table 1) [Insert Table 1 here] We found applications of the following OR theories and methods (in alphabetical order): Agent-Based Simulation; Bayesian Networks; Complexity theory; Discrete-Event Simulation; Entropy Analysis; Graph theory; Linear / Mixed-Integer Programming; Markov Chains; Monte-Carlo Simulation; Optimal Control; Petri Nets; Reliability Theory; Robust Optimization; Statistical Analysis; Stochastic Optimization; and Systems Dynamics. The highest number of publications can be seen in mixed-integer and linear programming and Bayesian networks (6 papers respectively), optimal control, complexity and graph theories (5 papers respectively), reliability theory and discrete-event simulation (4 papers respectively).

When aggregating different methods at a larger scale, the largest number of studies can be observed in the area of network and complexity theories (24 papers); with regards to mathematical optimization 11 papers in total can be observed; simulation studies count to 8 papers, and 5 papers are related to control theory.

Analysis of such aggregated categories lead us to a proposition to classify the existing studies into three levels, i.e., network level, process level, and control level. Such a classification appeared to be logical and convenient to develop further categorizations of main outcomes, managerial insights and future research directions. We considered the following criteria to articulate the differences between the network, process, and control levels. The network level is characterized by a macro-view on SC structures and disruption propagations focusing on the structural properties and relations. This level does operate in terms of networks and graphs from a much generalized perspective of structures and does not consider operational parameters. These parameters are in the scope of the process level which organizes the debate around the parametrized structures to balance demands, processing capacities, and supply. Typical problems at the process level are related to production-distribution planning in terms of flow optimization. Though, these models do not go into details of inventory control and production-ordering policies which are accommodated at the control level.

Major outcomes and managerial insights

Now we are drawing the reader's attention towards the analysis of major outcomes and managerial insights. A detailed paper-by-paper analysis is offered in Table 2.

[Insert Table 2 here] Table 2 summarizes the titles, authors, journals, central research questions, methods and outcomes & managerial insights of each paper analysed. We focus now on major outcomes and managerial insights and generalize the insights from individual paper analyses at an aggregated scale according to the previously introduced classification into the network (N), process (P), and control (C) levels. The major outcomes and managerial insights that can be deduced from the existing OR studies are categorized and presented in Table 3.

[Insert Table 3 here] The detailed analysis follows.

Network level

The studies at the network level are mostly looking at unlocking associations between network structures and risk propagations [START_REF] Li | Network Characteristics and Supply Chain Resilience under Conditions of Risk Propagation[END_REF]. For example, [START_REF] Basole | Supply Network Structure, Visibility, and Risk Diffusion: A Computational Approach[END_REF] show that small-world supply network topologies (i.e., networks where each node is connected to several of its neighbors and a few distant nodes) consistently outperform supply networks with scale-free characteristics (networks where nodes are connected to a few other nodes, while a small number of nodes are connected to many other nodes). The network-and graph-theoretic studies allow to understand potential weaknesses in the SC designs taking into account the structure, connectivity, and dependence within the SC [START_REF] Blackhurst | Supply chain vulnerability assessment: A network based visualization and clustering analysis approach[END_REF]). An important contribution can be seen in detecting disruption scenarios and identifying critical nodes (or combinations of nodes) the failure of which would lead to SC discontinuities and operations collapses [START_REF] Zeng | Modelling of cluster supply network with cascading failure spread and its vulnerability analysis[END_REF][START_REF] Tang | Complex interdependent supply chain networks: Cascading failure and robustness[END_REF][START_REF] Deng | Risk propagation mechanisms and risk management strategies for a sustainable perishable products supply chain[END_REF][START_REF] Pavlov | Integrated detection of disruption scenarios, the ripple effect dispersal and recovery paths in supply chains[END_REF]. Another important application area is measuring of the SC robustness and resilience under disruption propagations and structural dynamics [START_REF] Han | Evaluation mechanism for structural robustness of supply chain considering disruption propagation[END_REF], Sokolov et al. 2016, Hosseini and Ivanov 2019[START_REF] Li | Network Characteristics and Supply Chain Resilience under Conditions of Risk Propagation[END_REF]. Along with the stress-testing of the existing SC designs, the network level analyses suggest directions to enhance the resilience, e.g., through supplier diversifications [START_REF] Lei | Modelling of risk transmission and control strategy in the transnational supply chain[END_REF]. Episodically, the issues beyond the mere economic performance such as sustainability have been examined [START_REF] Levner | Entropy-based model for the ripple effect: managing environmental risks in supply chains[END_REF]. Moreover, the macroproblems of SC economies such as bankruptcies of suppliers (Mizgier et al. 2012) and retail dynamics [START_REF] Osadchiy | Systematic risk in supply chain networks[END_REF]) have been studied. The studies with the use of Bayesian networks allow to model dependencies and inter-dependencies in supply networks; moreover, the robustness and resilience analyses with consideration of both vulnerabilities and recovery become possible [START_REF] Garvey | An analytical framework for supply network risk propagation: A Bayesian network approach[END_REF][START_REF] Ojha | Bayesian network modelling for supply chain risk propagation[END_REF][START_REF] Cao | An Ontology-based Bayesian network modelling for supply chain risk propagation[END_REF]). An integration of Markov chains into the Bayesian networks enables an additional and valuable contribution, i.e., to model the node's behaviors along with the overall network dynamics (Hosseini et al. 2019).

Process level

The studies at the process level are positioned at a less generalized perspective as compared to the network level. These studies build upon parametrized structures to balance demands, processing capacities, and supply. Production-distribution planning in terms of flow optimization under disruption propagation and structural dynamics is the focus of the process level analysis. The analysis at the process level is mostly grounded in mathematical optimization and system dynamics simulation.

The process level studies help to analyse measures for disruption propagation mitigation. The mathematical optimization studies are usually organized around an SC design which may vary structurally and parametrically over time and optimize flow reconfigurations under disruption propagation. For example, Bueno-Solano and Cedillo-Campos (2014) show that protective measures against the ripple effect can drastically increase the inventory levels in the SC. [START_REF] Garvey | The Rippled Newsvendor: A New Inventory Framework for Modelling Supply Chain Risk Severity In The Presence of Risk Propagation[END_REF] argue that "managers should focus more of their attention on control or mitigation of exogenous events […], and spend less of an effort and resources on mitigating the propagation of exogenous risk..". [START_REF] Ghadge | A systems thinking approach for modelling supply chain risk propagation[END_REF] show how systems dynamics simulation can help in prediction of potential failure points in the SC along with the overall impact of the ripple effect on the performance. Details differ across the studies but most of them share a common set of outcomes and managerial insights such as joint optimisation of SC capacities and recovery capabilities for new and existing SCs; trade-offs between investments in increased recovery capability and redundant capacity provision; decision-making support on safety stock management, reconfiguration of production and inventory plans after disruptions, and recovery scheduling [START_REF] Ivanov | Integration of aggregate distribution and dynamic transportation planning in a supply chain with capacity disruptions and ripple effect considerations[END_REF][START_REF] Sinha | Measuring and Mitigating the Effects of Cost Disturbance Propagation in Multi-Echelon Apparel Supply Chains[END_REF], Goldbeck et al. 2020). As the most desired outcome, process level analysis seeks to identify and test resilient SC designs to sustain disruptions which range from optimistic and pessimistic scenarios [START_REF] Ivanov | Optimal distribution (re)planning in a centralized multi-stage network under conditions of the ripple effect and structure dynamics[END_REF], probabilitybased disruptions [START_REF] Pariazar | Supply chain design considering correlated failures and inspection in pharmaceutical and food supply chains[END_REF]) to worst-cases in robust optimization [START_REF] Zhao | Robust Sourcing from Suppliers under Ambiguously Correlated Major Disruption Risks[END_REF]Freeman 2019, Özçelik et al. 2020). In some settings, the authors solve inverse problems and search for elements in the SC structures that should be strengthened to withstand the disruption propagations [START_REF] Liberatore | Hedging against disruptions with ripple effects in location analysis[END_REF][START_REF] Ivanov | Dual problem formulation and its application to optimal redesign of an integrated production-distribution network with structure dynamics and ripple effect considerations[END_REF]. Some extensions can be seen in including the recovery costs [START_REF] Ivanov | Disruption-driven supply chain (re)-planning and performance impact assessment with consideration of pro-active and recovery policies[END_REF]) and sustainability issues [START_REF] Pavlov | Optimization of network redundancy and contingency planning in sustainable and resilient supply chain resource management under conditions of structural dynamics[END_REF].

Control level

The studies at the control level are distinctively characterized by inclusion of details of inventory control and production-ordering policies into the analysis. At this level, simulation methods dominate. They allow analyzing dynamic SC behaviors and time-dependencies in disruption propagations and responses. One interesting observation from these studies is an insight about "disruption tails". Several works [START_REF] Ivanov | Disruption tails and revival policies: A simulation analysis of supply chain design and production-ordering systems in the recovery and post-disruption periods[END_REF], Dolgui et al. 2020) observed that noncoordinated ordering and production policies during a disruption period may result in backlog and delayed orders, the accumulation of which causes post-disruption SC instability, resulting in further delivery delays and non-recovery of SC performance. These residues have been named "disruption tails". The extant literature suggests that specific "revival" policies must be developed for the transition from the recovery to the disruption-free operations mode to avoid the "disruption tails". Interestingly, the first research work on the impacts of COVID-19 pandemic on the SC has utilized the simulation methodology revealing several unique features which make the pandemic a specific and very severe risk type for SCs (Ivanov 2020a).

Directions for managerial applications and future research in the pandemic settings

In this section, we focus on the articulation of the state-of-the-art knowledge in OR about the disruption propagations and structural dynamics for a pandemic setting. In particular, we extrapolate the outcomes and managerial insights revealed in Section 3 on the COVID-19 pandemic and elaborate on future research directions (Table 4).

[Insert Table 4 here]

Our analyses of outcomes in both managerial and theoretical domains in Table 4 are structured into five stages (i.e., Anticipation; Early detection; Containment; Control and Mitigation; and Elimination) following the WHO classification (WHO 2018).

Anticipation and early detection

The pandemic cycle usually begins with the stage of anticipation and early detection. At this stage, SCs should keep an eye out and enable preparedness measures. The OR methods can help in a number of areas such how to identify critical scenarios of disruption propagations according to an epidemic outbreak dynamics and forecast the impact of possible propagating disruptions on SC performance. Moreover, OR theories can be efficiently used to predict the time periods during which the SCs can sustain the disruption propagations and survive despite the discontinuities, identify critical suppliers and facilities for maintaining SC operations, and select and fortify SC designs to sustain epidemic outbreaks. Overall, the decision on the anticipation and early detection stage target at implementing of "Design-for-Resilience" network structures.

Nonetheless, further research directions arise for OR community in light of pandemic settings.

There are crucial opportunities to develop theories and models for disruption propagation analysis in supply networks with specific consideration of pandemics, to visualize the ripple effect and structural dynamics, and to extend towards a multi-categorical analysis spanning dimensions of resilience and sustainability. Moreover, researchers can examine new analysis categories such as network viability (Ivanov andDolgui 2020, Ivanov 2020). Another important direction is to investigate the data analytics and digital technology capabilities to early detect the disruption propagations following epidemic outbreaks.

Containment

At the containment stage, the environment becomes increasingly vulnerable followed by quarantine times, interruptions of logistics due to variations in containment timing and scaling in different geographical areas, and some lock-downs. At this stage, SCs are experiencing first misbalances of supply and demand due to longer-lead times, demand drops and supply unavailability due to facility closures. OR methods can support SC managers at this stage by stresstesting of the existing and alternative SC configurations and production-distribution plans for some scenarios of structural dynamics in anticipation of or as a reaction to facility/market closure due to quarantines and lock-downs. OR methods can also help optimise contingency / preparedness plans for their efficient and timely deployment under different scenarios of epidemic propagation.

The new research opportunities for OR community are very rich for the containment pandemic stage. For example, there is an urgent need to examine new understandings, theories, and novel approaches concerning SC preparedness and disruption mitigation during the beginning epidemic outbreaks. This can help articulate antecedents, drivers, and economic and social performance implications of simultaneous disruption and epidemic propagations. One specific and underexplored are is re-designing of the SCs for production switches to unusual products (e.g., mask production at car manufacturers).

Control and mitigation

Amid the control and mitigation stage, SCs need to adapt to a "new normal" and start preparing for a recovery. For example, OR models can help to identify the balanced levels of capacity utilization and production rates at different firms in the SC to achieve maximum possible performance. It is now very relevant to COVID-19 pandemic since the SCs are misbalanced and it is difficult to decide at which level of capacity the firms should start with and scale during a recovery thereafter. The OR models can help identify the optimal material flows in a multiperiod mode when the SC structures change through these periods. This is very relevant to modeling the SC flows amid a pandemic and during a recovery. Another relevant issue is consideration of backlog accumulations over the disruption time which can become a major driver of disruption propagations during the production and logistics ramp-up activities. At the control and mitigation stage, the role of digital twins is increasing since SC visibility and information coordination are the key capabilities to cope with the ripple effect. OR methods can help analyse the impacts of disruption propagation in dynamics with adaptations of ordering, production and inventory control policies and to simulate operations policies amid a pandemic. Moreover, OR theories can be used to explore re-allocations of supply and demand during a pandemic given simultaneous disruptions in upstream and markets. In addition, OR methods can be applied to propose recovery plans along with analysis of timing and scaling of facility/market closures and openings in different geographical regions.

As for future research, we point to the opportunities for substantial contributions to develop and examine digital SC twins to map the network elements and adapt the SCs according to disruption propagations and structural dynamics. Another promising research direction is to explore the role of timing and scaling of the production and logistics ramp-ups after the quarantines and lock-down eliminations.

Elimination or eradication

Exiting a pandemic can be even more challenging as stepping inside. During the elimination stage, the SCs need to be recovered and adapted to new, post-pandemic realities. OR methods can help incorporate after-pandemic environments in the re-designing of the SCs. OR can also be of value to examine the existing and potential SC configurations under post-pandemic conditions in the supply base and the markets. Furthermore, OR techniques can be used to analyse the "disruption tails" and long-term stabilization of production-inventory systems.

The elimination stage contains a variety of novel research problem settings. For example, there is a research gap in how to find optimal scaling and timing of production and logistics rampups during the "exit" after the lock-downs. It is also a timely and crucial issue to examine SC re-design methods for sever structural changes in supply and demand after a pandemic (e.g., supplier bankruptcies and demand drops/shifts). Finally, we point to the needs to explore the concept of SC viability as a long-term maintaining of survivability under different and ever changing environmental conditions.

4.5. Note on a danger of jeopardizing SC resilience models for the pandemic settings Undoubtedly, OR knowledge in SC disruption risks and resilience will be the leading perspective guiding researchers and industry leaders alike through the pandemic and a recovery thereafter. With that being said, there is a danger of jeopardizing SC resilience models for the pandemic settings. OR community has developed a mature body of literature on coping with different types of disruption risks. A pandemic is one specific type of disruption risks and it has quite unique implications for SCs which are not encountered in other types of SC disruptions.

In contrast to geographically centered natural/industrial disasters with singular occurrence, a pandemic is not limited to a particular region or confined to a particular time period [START_REF] Ivanov | Coronavirus (COVID-19 / SARS-CoV-2) and supply chain resilience: A research note[END_REF]. Different SC components are affected sequentially or concurrentlymanufacturing, DCs, logistics, and markets can become paralyzed within subsequent or overlapping time frames. Pandemic causes long-term disruption existence and its unpredictable scaling. Other specific issues are simultaneous disruption propagation (i.e., the ripple effect) and epidemic outbreak propagation, and simultaneous severe disruptions in supply, demand, and logistics infrastructure (Ivanov 2020a). Under pandemic conditions, it might be difficult to directly apply the most know SC resilience mechanisms such as risk mitigation inventories, subcontracting capacities, or backup supply and transportation infrastructures. As such, the studies on SC resilience should explicitly present pandemic-specific settings to be classified as a contribution to adapt through the pandemic and recover. Otherwise, literally each study on supply disruptions can be adapted to the pandemic background which is fundamentally wrong.

Conclusions

The COVID-19 pandemic unveils fragility of the SCs at an unforeseen scale. One specific and strong stressor to SCs amid a pandemic and during the post-pandemic recoveries comes from the disruption propagations through the networks (i.e., the ripple effect) and the subsequent changes in the SC structures (i.e., structural dynamics).

This paper deduced managerial implications from the existing OR literature on disruption propagations in SC and revealed future research directions. We collated for the first time the existing OR knowledge in modelling the SC ripple effect and structural dynamics. We believe that such an overview would be useful for industry leaders and researchers alike shaping SC adaptations during and after a global pandemic. On one hand, we collated the state-of-the-art in the research on SC disruption propagation and identified a methodical taxonomy. On the other hand, we revealed and systemized managerial insights from the theory that can be used for recovery from the COVID-19 and withstanding future pandemics. These results can be used by industry and researchers alike in order to progress the decision-support systems guiding the SCs amid the COVID-19 pandemic and recovering them thereafter.

In particular, we found that outcomes of the OR contributions and the resulting managerial insights can be categorized into three levels, i.e., network, process, and control. Our outcomes in both managerial and theoretical domains are structured into five stages (i.e., Anticipation; Early detection; Containment; Control and Mitigation; and Elimination) following the WHO classification.

Limitations exist, as with any study. We have narrowed our analysis on the disruption propagation literature related to commercial SCs. Obviously, a rich variety of knowledge in the area of humanitarian logistics and SCs can enrich the findings of our study. We also do not pretend to be encyclopedic; we assume that some relevant studies might have not been uncovered and remained outside of our review.

As for future research, we pointed numerous opportunities for substantial contributions to develop and test new theories, models, and resilience mechanisms for control and mitigation of disruption propagation in SCs with specific consideration of pandemic features, such as:

-long-term disruption existence and its unpredictable scaling; -simultaneous disruption propagation and epidemic outbreak propagation; -simultaneous severe disruptions in supply, demand, and logistics infrastructure

We hope that the novel systematizations/categorizations proposed in this study will be of value for researchers and practitioners alike in guiding the SCs through the pandemic times and preparing them for recovery thereafter. Along with the constructed generalized perspectives, our study can be of value for researchers and industry professional alike to cope with the existing COVID-19 pandemic, recover thereafter, and most importantly to create a valuable reference source for future pandemics or pandemic-like disruptions. -long-term disruption existence and its unpredictable scaling; -simultaneous disruption propagation and epidemic outbreak propagation; -simultaneous severe disruptions in supply, demand, and logistics infrastructure  Develop and examine digital SC twins to map the network elements and adapt the SC according to disruption propagations and structural dynamics  Explore the role of timing and scaling the production and logistics ramp-ups after the quarantines and lock-down eliminations Elimination  Incorporate after-pandemic environments in the re-designing of the SCs  Examine the existing and potential SC configurations under post-pandemic conditions in the supply base and the markets  Analyse the "disruption tails" and long-term stabilization of productioninventory systems  Find optimal scaling and timing of production and logistics ramp-ups during the "exit" after the lock-downs  Examine SC re-design methods for sever structural changes in supply and demand after a pandemic (e.g., supplier bankruptcies and demand drops/shifts)  Explore the concept of SC viability as a long-term maintaining of survivability under different and ever changing environmental conditions

Fig. 1 .

 1 Fig. 1. Literature selection criteria

Table 1 . Theories used in the studies on the ripple effect and structural dynamics

 1

	Theory	Number of	Study
		studies
	Agent-Based Simulation	1	Mizgier et al. (2012)
	Bayesian Networks	6	Cao et al. (2019), Garvey et al. (2015), Garvey and Carnovale
			(2020), Hosseini and Ivanov (2019), Hosseini et al. (2019), Ojha
			et al. (2018)
	Complexity theory	5	Basole and Bellamy (2014), Deng et al. (2019), Lei et al. (2020),
			Levner and Ptuskin (2018), Zeng and Xiao (2014)
	Discrete-Event Simula-	4	Dolgui et al. (2020), Ivanov (2017, 2019, 2020)
	tion
	Entropy	2	Levner and Ptuskin (2018), Zeng and Xiao (2014)
	Graph theory	5	Basole and Bellamy (2014), Li et al. (2019), Li and Zobel
			(2020), Sinha et al. (2019), Sokolov et al. (2016)
	Linear / Mixed-Integer	6	Liberatore et al. (2012), Ivanov et al. (2013, 2014, 2015, 2016),
	Programming		Pavlov et al. (2019)
	Markov Chains	1	Hosseini et al. (2019)
	Monte-Carlo Simulation	1	Pariazar et al. (2017)
	Optimal Control	5	Ivanov et al. (2010, 2013, 2014, 2015, 2016)
	Petri Nets	1	Blackhurst et al. (2018)
	Reliability Theory / Sta-	4	Hand and Shin (2016), Osadchiy et al. (2016), Pavlov et al.
	tistical Analysis		(2020), Tang et al. (2016)
	Robust Optimization	3	Lu et al. (2015), Özçelik et al. (2020), Zhao and Freeman (2019)
	Stochastic Optimization	2	Goldbeck et al. (2020), Pariazar et al. (2017)
	Systems Dynamics	2	Bueno-Solano and.Cedillo-Campos (2014), Ghadge et al. (2013),

Table 2 . Operational Research studies on the disruption propagations in the SCs Table 3. Outcomes and managerial insights from OR contributions to the ripple effect and struc- tural dynamics

 2

	Level of Anal-	OR methods	Outcomes	Managerial insights
	ysis
	Network Level Graph theory	 Associations between network	 Identification of disruption
		Complexity theory	structures and risk propagation;	propagation scenarios of dif-
		Entropy	 Analysis of critical network ele-	ferent severity
		Petri Nets	ments leading to supply chain	 Stress-testing of SC designs
		Bayesian Networks	discontinuities and collapses	 Proneness of specific SC de-
		Markov Chains	through the cascading failure ef-	signs to disruption risk prop-
		Reliability Theory /	fects;	agation
		Statistical Analysis	 Modelling of interdependencies	 Identification of critical sup-
			in SCs;	pliers and facilities for main-
			 State dynamics within SC nodes;	taining SC operations
			 Assessment of SC robustness	 Selection and proactive en-
			and resilience to disruptions with	hancements of SC designs to
			the ripple effect considerations	sustain certain levels of dis-
				ruption propagations and
				structural dynamics
				 Adaptation of SC designs
				according to the environ-
				mental changes
	Process Level	Stochastic Optimi-	 Optimal reconfigurations of ma-	 Stress-testing of SC produc-
		zation	terial flows according to disrup-	tion-distribution plans within
		Robust Optimiza-	tion propagation scenarios	differently disrupted net-
		tion	 Impacts of the ripple effect and	work designs
		Linear / Mixed-In-	structural dynamics on service	 Analysis of contingency /
		teger Programming	level and costs	preparedness plans
			 Optimal re-allocation of supply	 Recovery plan selection
			and demand under conditions of
			disruption propagation and struc-
			tural dynamics
	Control Level	Optimal Control	 Impacts of the disruption propa-	 Building resilient SCs for
		Systems Dynamics	gations on service level, inven-	new, after-pandemic busi-
		Agent-Based Simu-	tory levels, and costs	ness models
		lation	 Time-dependent effect of disrup-	 Analysis of the disruption
		Discrete-Event	tion propagation on SC behav-	propagation in dynamics
		Simulation	iors and performance in dynam-	with considerations of pro-
			ics	duction and inventory con-
			 Individual behaviors of firms in	trol policies
			SCs	 Simulation of operations
				policies during the disrup-
				tion, in transition to recov-
				ery, and in the post-recovery
				periods

Table 4 . Suggestions for future research and applications in the pandemic settings Pandemic stages Suggestions for future research and applications of OR methods in the pan- demic settings

 4

		Managerial applications	Future research directions
	Anticipation and	 Identify critical scenarios of disruption	 Develop theories and models for dis-
	early detection	propagation according to an epidemic	ruption propagation analysis in supply
		outbreak dynamics	networks with specific consideration of
		 Forecast the impact of possible propa-	pandemics
		gating disruptions on SC performance	 Visualize the ripple effect and struc-
		 Predict the time periods during which	tural dynamics
		the SCs can sustain the disruption	 Multi-categorical analysis spanning di-
		propagations and survive despite the	mensions of resilience and sustainabil-
		discontinuities	ity
		 Identify critical suppliers and facilities	 Examine new analysis categories such
		for maintaining SC operations	as network viability
		 Implement "Design-for-Resilience"	 Investigate the data analytics and digi-
		network structures	tal technologies to early detect the dis-
		 Select and fortify SC designs to sus-	ruption propagations following epi-
		tain epidemic outbreaks	demic outbreaks
	Containment	 Stress-testing of SC configurations	 Examine new understanding, theories,
		and production-distribution plans for	and novel approaches concerning SC
		some scenarios of structural dynamics	preparedness and disruption mitigation
		in anticipation of facility/market clo-	during the beginning epidemic out-
		sure due to quarantines	breaks
		 Optimise contingency / preparedness	 Articulate antecedents, drivers, and
		plans for deployment under different	economic and social performance im-
		scenarios of epidemic propagation	plications of simultaneous disruption
			and epidemic propagations
	Control and mitiga-	 Analyse the impacts of disruption	 Develop and test new theories, models,
	tion	propagation in dynamics with adapta-	and resilience mechanisms for control
		tions of ordering, production and in-	and mitigation of disruption propaga-
		ventory control policies	tion in SCs with specific consideration
		 Simulate and articulate operations pol-	of pandemic features, such as:
		icies amid the pandemic
		 Explore re-allocations of supply and
		demand during the pandemic given
		simultaneous disruptions in upstream
		and markets
		 Re-design the SCs for production
		switches to unusual products (e.g.,
		mask production at car manufacturers)
		 Propose recovery plan selection with
		analysis of timing and scaling of facil-
		ity/market closures and openings in
		different geographical regions

Managers should focus more of their attention on control or mitigation of exogenous events that directly impact their own firm, and spend less of an effort and resources on mitigating the propagation of exogenous risk from a supplier to an exogenous risk of the firm itself.