Resilience of Supply Chain Networks to Labor Disruptions

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QUESTIONS
The COVID-19 pandemic has demonstrated the importance of labor to supply chain network economic activities from production and transportation to storage and the ultimate distribution of products to the demand markets.With workers getting ill from the coronavirus and many sadly perishing from the disease, the negative impacts of higher product prices and unfulfilled demand became all too common in economic sectors such as the food sector, the household product sector, lumber supplies, as well as high tech and healthcare (Helper and Soltas 2021;Nagurney 2022).The recognition that labor is a critical resource in supply chains and that disruptions to labor can have unforeseen global consequences has led to the development of both optimization and game theory supply chain network models with the inclusion of labor needed for supply chain activities and with the productivity of labor incorporated, along with relevant constraints as to the availability of labor (Nagurney 2021a;Nagurney 2021b;Nagurney 2022).
This study builds on the earlier contributions of Nagurney and Qiang (2009) in quantifying the efficiency of critical infrastructure networks as well as those of Qiang, Nagurney, and Dong (2009) and Li and Nagurney (2017) in assessing the performance of supply chain networks specifically.The framework constructed here, however, allows one to quantify the resilience of a supply chain network subject to the reduction of labor availability (capacities) or under a reduction in labor productivity.The former situation can arise, for example, as a consequence of illness, death, being unable or unwilling to work, labor strikes, or being called to war or other types of service.The latter situation can arise because of the need for social distancing, a decrease in productivity due to long COVID or other illness or stressors, or fatigue.Our framework, in particular, helps answer the following questions: • Question 1: What is the impact on efficiency and on resilience of allowing workers to perform different tasks in a supply chain network, with the constraint represented by a single bound on labor, as opposed to bounds on labor on each supply chain network link?
• Question 2: Does resilience with respect to labor availability yield similar results to resilience with respect to labor productivity?
• Question 3: What can be the effect of a modification in the supply chain network topology, for example, as in the case of the introduction of electronic commerce, on network efficiency and resilience?

METHODS
We consider a supply chain network with the general topology depicted in Figure 1, with the topology being adapted for the specific supply chain under consideration.The topology is represented by the graph G = [N, L], where N is the set of nodes and L is the set of links.
The efficiency of a supply chain network, E, is formally defined by Equation 1.Here, the demands, d * , and the incurred demand market prices, are evaluated at the optimal value of the firm profit, subject to supply chain network flow constraints in existing markets, and with the labor constraints of interest formulated in Supplemental Information.A supply chain is evaluated as performing better if, on the average, it can handle higher demands at lower prices.
Table 1: Nomenclature a: A typical link p: A typical path w: A typical demand market n M : Possible production locations n D : Distribution centers for storage J: Number of demand markets W : Set of demand markets ρ w : Price at a demand market w ρ: A vector of prices ĉa : Total cost on link a ĉ: A vector of total costs α a : A positive productivity factor of link a reflecting how much an hour of labor will yield in terms of product flow on that link α: A vector of link productivity factors l: Bound on labor, which will be a vector if the constraints are on individual links (Nagurney, 2021a;Nagurney, 2021b) π a : Wage on link a π: A vector of wages that are paid for an hour of labor on a link Using ideas in Nagurney and Qiang (2009) and in Nagurney and Li (2016) for supply chains, the importance of a component g (e.g., node, link, a combination of nodes and links), I(g), is defined by Equation 2. I(g) represents the efficiency drop when g is removed from the network.
Following the definition of supply chain network efficiency, we propose two resilience measures with respect to labor availability disruptions (R lγ ) and labor productivity disruptions (R αγ ) formulated by Equation 3 and Equation 4, respectively.Here, lγ denotes the reduction of labor availability with γ ∈ (0, 1].For example, if γ = 0.9 this means that the labor availability associated with the labor constraints is 90% of the original labor availability as in E. The closer the value of our resilience measures to 100%, the greater the resilience. m The Firm We calculate E, E lγ , E αγ and the resilience measures R lγ and R αγ when γ = 0.9, 0.7, 0.5, 0.3, 0.1 for five examples with supply chain network topology depicted in Figure 2 and Figure 3. Table 2 represents the characteristics of each example.It has the identical data to that in Example 1 -Baseline in Nagurney (2021a) with the exception that the link labor bounds la = 10, 000 for all links a ∈ L.

SCN 2
It has the same data as SCN 1, except that now there is a single bound on labor l = 70, 000.In other words, labor is free to work on any link, provided that the sum of the labor hours does not exceed 70, 000.Note that 70, 000 is the sum of the labor bounds on all the links in SCN 1.

SCN 3
It has the identical data to that of SCN 1 except that now electronic commerce links h and i are added.The additional data for SCN 3 associated with the electronic commerce links are .00, π i = 10.00,α h = 1.00, α i = 1.00, lh = 10, 000.00, li = 10, 000.00.

SCN 4
It has the same topology and data as SCN 3 except that the labor availability constraint is for the entire supply chain with l = 90, 000.The value of 90, 000 is chosen since there are 9 links in SCN 3, with each link having a bound of 10, 000 and, hence, there would be a total labor availability of 90, 000 under the assumption that laborers would be free and interested in doing whichever tasks that are needed in the supply chain network with the productivity factors being as in SCN 3.

SCN 5
It has the same data as SCN 4, but now the labor hours available are no longer 90,000, rather there are only 70,000 hours available.Hence, the results for SCN 5 allow us to make a comparison with SCN 2 in terms of the impact of adding electronic commerce and having the same total amount of labor in the supply chain network available as before but having additional supply chain activities of electronic commerce.• First, comparing the results of SCN 1 with SCN 2 and SCN 3 with SCN 4, it is found that having labor be free to move across the supply chain network results in (1) a higher efficiency of the supply chain with the same total number of labor hours available and (2) a higher resilience with respect to both resilience measures and at different values of γ.
• Second, the value of R lγ is very similar to the corresponding R αγ for the same value of γ for each supply chain network example.They are, indeed, identical in many cases, until the value of γ becomes 0.5 or lower.In this condition, the supply chain network resilience with respect to labor availability exceeds the resilience with respect to labor productivity as shown in SCN 1, SCN 2, SCN 4, and SCN 5.This suggests that the firms should take care of their workers since a reduction in labor productivity can impact the supply chain network efficiency and the corresponding resilience.
• Third, comparing the results of SCN 5 with SCN 2, it is clear that the efficiency of the supply chain network with electronic commerce options (SCN 5) is consistently higher than that for the supply chain network without electronic commerce (SCN 2) at the same value of labor availability and disruption and at the same level of disruption to labor productivity on the links with the exception of the respective values of E α0.3 (and those respective values are equivalent to two decimal points).In addition, both these supply chain networks, under the specific data, retain their efficiency under even restrictive disruptions to labor availability.However, that is not the case when there are disruptions to labor productivity.Interestingly, as can be seen from the values of R αγ , SCN 2 that without electronic commerce is more resilient than SCN 5 for γ = 0.5, 0.3, 0.1.This can be explained by noting that the available labor hour amount is divided among fewer supply chain network economic activities in the case of SCN 2. Again, we see, from the investigation of results for SCN 5 versus those for SCN 2, that labor productivity on the links, when disrupted, can have an even bigger impact on resilience than a disruption to labor availability.

Acknowledgments
This paper is dedicated to essential workers, who, in the pandemic, helped humanity, and to freedom-loving Ukrainians now fighting for peace in their sovereign nation against the Russian aggressors.

SUPPLEMENTAL INFORMATION
The firm seeks to maximize its profits given the following objective function, where d is the vector of product demands and f is the vector of product link flows, subject to the The demand at a demand market is equal to the sum of the product flows of the firm to the demand market: where x p is the path flow on path p, d w is the demand at w, and P w is the set of paths from node 1 in Figure 1 to w.
The product flow on a link, f a , is equal to the sum of flows on paths that contain that link: where δ ap = 1, if link a is contained in path p, and is 0, otherwise.
The path flows must all be nonnegative: where P is the set of all paths from node 1 in Figure 1 to the demand markets.
The following equation relating labor hours on a link a, l a , with the product volume on a link must hold for each link: f a = α a l a , ∀a ∈ L. (5)

Figure 1 :
Figure 1: The Supply Chain Network Topology

FirmFigure 3 :
Figure 2: Supply Chain Network Topology for SCN 1 and SCN 2 conservation of flow equations relating the path flows to the demands; the link flows to the path flows, and the non-negativity constraints on the path flows, along with the link labor and link flow expressions and the appropriate constraints under consideration on labor.The conservation of flow equations and the equation relating labor hours to product flow on a link are as follows.Maximize w∈W ρ w (d)d w − a∈L ĉa (f ) − a∈L π a l a ,

Table 2 :
Description of Supply Chain Network (SCN) examples used for the analysis Example Description SCN 1