Complexity

Volume 2018 (2018), Article ID 1724125, 13 pages

https://doi.org/10.1155/2018/1724125

## A Network-Based Impact Measure for Propagated Losses in a Supply Chain Network Consisting of Resilient Components

^{1}Computing Science Department, Institute of High Performance Computing, Agency for Science, Technology and Research, 1 Fusionopolis Way, 16-16 Connexis North, Singapore 138632^{2}School of Electrical and Electronic Engineering, College of Engineering, Nanyang Technological University, Block S2.1, 50 Nanyang Avenue, Singapore 639798

Correspondence should be addressed to Jesus Felix Bayta Valenzuela

Received 3 May 2017; Accepted 1 January 2018; Published 19 February 2018

Academic Editor: Pietro De Lellis

Copyright © 2018 Jesus Felix Bayta Valenzuela et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The topology of a supply chain network affects the impacts of disruptions in it. We formulate a network-based measure of the impact of a disruption loss in a supply chain propagating downstream from an originating node. The measure takes into account the loss profile of the originating node, the structure of the supply network, and the resilience of the network components. We obtain an analytical expression for the impact measure under a beta-distributed initial loss (generalizable to any continuous distribution supported on the interval ), under a breakthrough scenario (in which a fraction of the initial production loss reaches a focal company downstream as opposed to containment upstream or at the originating point). Furthermore, we obtain a closed-form solution for a supply chain network with a -ary tree topology; a numerical study is performed for a scale-free network and a random network. Our proposed approach enables the evaluation of potential losses for a focal company considering its supply chain network structure, which may help the company to plan or redesign a robust and resilient network in response to different types of disruptions.

#### 1. Introduction

In supply chain networks, the mitigation of production losses arising from upstream disruptions is an important aspect of their management. Another important aspect is identifying which components of the network are bound to play a key role in either spreading or mitigating such an impact, based on their positions in the network as well as their inherent risk profiles; a component’s critical position in the supply chain network may amplify the effects arising from its disruption.

The impacts of disruptions on supply chain networks can be seen in the aftermath of the floods which affected Thailand in 2011 and the earthquake and tsunami which affected the Tohoku region of Japan in the same year. Haraguchi and Lall [1] made an assessment of the impacts of the floods on both industries and the local economy, as well as overseas companies in both the automobile and the electronics sectors. Saito et al. [2] examined the impacts of the Tohoku earthquake and tsunami on the sales growth and transaction relationships of firms outside the affected areas (but have suppliers and customers within the affected areas) and found significant negative effects propagated downstream to firms as far as five degrees of separation away.

In an analysis of a survey conducted among various organizations and firms in 2017, the Business Continuity Institute reported that 65% of the respondents experienced at least one disruption within the past 12 months [3], a decrease from 70% in the previous year’s survey [4]. The same series of surveys reveals patterns in the origins of disruptions in supply networks. Disruptive events arose predominantly in “Tier 1,” or direct, suppliers (44% in 2017; 41% in 2016), and “Tier 2,” those suppliers’ own suppliers (24% in 2017; 17% in 2016). Cumulative losses of at least € 1,000,000 became less frequent (22% in 2017; 34% in 2016). However, the frequency of events costing the same amount increased over the same period (23% in 2017; 9% in 2016).

These results illustrate the need for awareness by companies of upstream conditions and effective anticipation and compensation for disruptions arising there. This need is more and more keenly felt: respondents without origin-identification mechanisms for disruptive events drastically decreased from 40% in 2016 to 22% in 2017. As the intercomponent relationships of a supply chain can be mapped onto a network, network theory offers a natural way to help fulfill such a need. By examining how an initial loss of an entity propagates downstream in the network, the impacts of the entity’s loss on the other entities in the chain can be measured.

Aspects of network theory have been applied in supply chain management, specifically in the context of analyzing the relationships between the components of the chain [5]. Network theory also provides a natural context in which risks associated with a network’s components or the whole network itself can be determined: a loss sustained by a component due to some adverse event may propagate downstream in a cascade of losses, unless mechanisms to mitigate losses had been put in place. Such a process is not restricted to supply chain networks alone but can be found elsewhere, such as in banking and financial networks [6, 7], organizational networks [8], production and input-output networks [9], infrastructure networks [10], and indeed any network consisting of interconnected, vulnerable entities [11–13].

In this work, we examine a different aspect of the propagation of the impact of a disruption event in a supply network. Specifically, we look at the production* losses* sustained by a member of the network due to a disruption occurring elsewhere. In addition, unlike previous studies, we consider the network as composed of resilient members. In other words, the elements of the network have the capability to partially or fully cushion losses which they otherwise would have sustained. The propagation of losses is thus potentially attenuated due to the resilience of individual components. Under such a scenario we seek the conditions where an initial loss sustained by a network component succeeds in propagating, in the face of attenuation at intervening components. We situate our work in this paper in the context of social network theory-based perspectives in supply chain risk management [14], as well as under supply chain risk assessment (specifically, within generic risk assessment), under the framework proposed by Ho et al. in 2015 [15].

The rest of this work is divided as follows. We review some of the related literatures in the second section. In the third section, we consider the downstream propagation of an initial disruption (measured as a fraction of the production lost) in a directed, acyclic network from an initial node, where each node has a certain degree of resilience to the disruption. We make a simplifying approximation to the equations we obtain and show that the simplification does not incur loss of accuracy. We then combine this propagated loss with a communicability-based measure first introduced by Estrada and Hatano [16] to yield an analytic expression for the measure of impact a production loss in a node has over its downstream nodes. Afterwards, we obtain an expression for the distribution and the moment of this impact measure, assuming that the production loss fraction follows a beta distribution.

The fourth section contains calculations using the impact measure for three different network structures. We first obtain exact results for a perfect -ary tree, with the assumption that the nodes in the tree are identically resilient to production loss. Afterwards, we obtained computational results for two acyclic networks: one is a scale-free network, and the other is an Erdős-Rényi network. The work is then summarized in the fifth and last section.

#### 2. Related Literature

##### 2.1. Supply Chain Risk Management Definitions and Frameworks

Consensus for the definition of “supply chain risk management (SCRM),” as well as frameworks for it, had been slow to form. This is in part due to the relative nascence of the field, the diversity of definitions among researchers and practitioners, and perspective gaps between the latter two groups. In surveys of both research teams and company executives, Sodhi et al. [17] uncovered research gaps in SCRM, ranging from a lack of clear consensus in its definition (a* definition gap*) to a lack of studies in response to supply chain risk incidents (a* process gap*) and to a shortage of empirical research (a* methodology gap*). Within the same work, in a review of research articles covering SCRM up to that point (with the exception of most works which used mathematical modeling as their main methodology [18]), they also found an abundance of literatures on risk identification, but a surprisingly low number of works specifically focusing on risk assessment. Other works dealt with risk assessment and mitigation on a conceptual basis, or in the context of the wider SCRM framework. Furthermore, most of the surveyed literatures on risk response only covered high-frequency, low-impact events* (operational risks)*.

Tang’s 2006 review [18], focusing on quantitative models for managing supply chain risks, sketched out a framework for classifying the SCRM articles as dealing with supply management (including supply network design, with a focus on mixed-integer models), demand management (dealing with strategies to shift demand across time, markets, or products), product management, and information management, respectively. He then provided a discussion of robust strategies for mitigating operational and disruption risks for each of the four management subdivisions. He outlined two properties of robust strategies:* efficiency*, enabling a company to manage operational risks in spite of occurrence of major disruptions, and* resiliency*, enabling sustained operations during and quick recovery after major disruptions.

Two more recent reviews attempted to cover subsequent works and to develop frameworks for SCRM. In a review focusing on SCRM enablers, Kilubi and Haasis [14] performed a review of 80 articles from 2000 to 2015, covering definitions, research methodologies, and linkages between SCRM and performance. They found disparities in definitions of SCRM and proposed a definition of it as “the identification, assessment, monitoring and evaluation of risks and potential threats within and outside supply chain networks with all members and entities involved.” They also identified 12 main enablers in the literature:* visibility, flexibility, relationships, redundancy, coordination, postponement, multiple sourcing, collaboration, risk awareness, agility, avoidance, contingency planning, risk monitoring*, and the* transferring and sharing of risks*. Of these, they label six (visibility, relationships, collaboration, coordination, multiple sourcing, and postponement and redundancy) as* preventive* enablers, aiming at reducing the probability of occurrence of risk events. Five SCRM enablers (visibility, flexibility, multiple sourcing, redundancy, and coordination) were labeled as* responsive* enablers, focusing on mitigating adverse effects of risk events. Visibility, multiple sourcing, and redundancy are labeled both preventive and responsive and thus are of central importance among enablers. Lastly, they point to incorporating insights from social network theory as an avenue for future research. Some work on this front had been done by Hearnshaw and Wilson [19], who proposed that a supply chain network is efficient if it has a scale-free structure (a short characteristic path length, a high clustering coefficient, and a power-law degree distribution). Here, “efficiency” is from a focal company’s point of view: its supply chain is efficient if it can rapidly fulfill a customer’s product order even in the absence of stockpiles.

Ho et al. [15] sought to bring together insights from a large corpus of recent articles (224 articles between 2003 and 2013), for the purposes of classification, identification of recent developments, and exploration of potential research gaps. In doing so, they proposed new definitions for supply chain risk (“the likelihood and impact of unexpected macro and/or micro level events or conditions that adversely influence any part of a supply chain leading to operational, tactical, or strategic level failures or irregularities”) and SCRM (“an interorganisational collaborative endeavour utilizing quantitative and qualitative risk management methodologies to identify, evaluate, mitigate, and monitor unexpected macro and micro level events or conditions, which might adversely impact any part of a supply chain”). Reviewed articles were classified into the SCRM processes they cover:* risk identification*,* assessment*,* mitigation*, and* monitoring*. Identification-focused papers focused on developing methods for identifying potential risks, however without strong efforts on evaluating the impacts of such risks. Assessment-oriented papers focused on quantifying the impacts of various risk types (e.g., macrorisk, demand risk, manufacturing risk, supply risk, transportation risk, financial risk, and information risk) as well as generic risk assessment. They also include risk modeling, assessments of the relationships between supply chain risk and strategies, and evaluations of supply chain resilience. Their classification of risk mitigation articles follows the same schema as for risk assessment articles, using the same risk subtypes. The articles reviewed employed various modeling techniques, from linear programming models to simulations. In contrast, however, risk monitoring seems to have received less attention, with fewer articles classified as pertaining to it.

Somewhat different from the previous articles, a review by Olson and Swenseth [20] examined supply chain risk from a systems-theoretic perspective, involving tradeoffs in balancing costs, risk, and, increasingly, environmental considerations, for green supply chain management, risk, and efficiency. The authors discussed the usefulness of applying systems thinking to the decision-making process in supply chain management, to more effectively handle these tradeoffs.

##### 2.2. Systemic Vulnerabilities, Disruptions, and Propagation in Networks

Network approaches are a natural fit to the examination of multiple interconnected and interacting agents, both as individuals and as a whole. Thus, they provide a natural framework for considering systemic risks and propagation of shock events.

Acemoglu et al. [11] proposed a framework for the study of propagating microeconomic shocks via network interactions and how these shocks translate to macroeconomic ones. Their framework also enables the characterization of an economy’s macrostate performance stemming from its characteristic network interactions, as well as identification of the key network actors using centrality considerations.

Another treatment of propagating, or cascading, shocks, or failures in interdependent networks was made by Buldyrev et al. [12] in 2010. The authors developed a framework for evaluating the robustness of interacting networks in the face of cascading failures. With this, they demonstrated the presence of a critical fraction of network nodes (agents), upon whose failures will cause network fragmentation into disconnected components. Contrary to the established results for single networks (robust to failures of random nodes but susceptible to targeted disruptions), they show that the opposite holds true for interdependent networks (more vulnerable to random failures than systematic disruptions).

Systemic risks in banking networks were examined by Haldane and May [6] in a 2011 article. Drawing inspiration from ecological and epidemic networks, they identify possible mechanisms by which the initial shock caused by a single bank’s collapse can propagate across the network. Firstly, a bank failure potentially leads to a collapse in turn of its creditors, causing a cascade of interbank loan-driven failures. This, however, tends to be attenuated due to losses being subdivided among a failing bank’s creditors. Secondly, market liquidity shocks can generate losses in the value of a bank’s external assets and potentially propagate the shock. In contrast with loan-driven failures, liquidity shocks tend to amplify with more banks failing, causing even small liquidity shocks to contribute strongly to systemic risk. A third mechanism,* liquidity hoarding*, can arise from hoarding of liquidity in interbank funding markets, causing a decrease in the availability of interbank loans. Liquidity hoarding can cascade through a banking network, resulting in a shock not subject to attenuation.

Laszka et al. [7] proposed a framework for estimating the systematic risk in networks. In their model, the risk of a node being compromised (or failing) may come from outside the network system (“direct compromise”) or propagate to it from a neighboring node (“indirect compromise”), with direct compromise being the only propagatable risk. The authors then calculated the network’s loss distribution (the probability that a given fraction of the nodes in the network becomes compromised) and the loss distribution of a subset of nodes (similar to the former distribution but restricted to a given subset of nodes in the network) in accordance with the model rules. They tested the model on two large real-life networks (a network of common-policy groups of IP addresses called* autonomous systems* participating in the Internet’s routing system, and the network of Facebook users), as well as scale-free models of the two networks. As their research was made with an eye on quantifying the insurability of the components of a network, they also computed the* safety loading*, a measure of the expensiveness in insuring a subset of the network nodes. They showed that while predicting the risk to a network using data from a subset is very challenging due to underestimation, it is nevertheless still possible.

Shabnam et al. [8] proposed a methodology for the measurement of risk in an organizational network utilizing (an agent-oriented conceptual modeling framework) and BPMN (business process modeling notation) frameworks, combined with a simple propagation of a node’s vulnerability measures to its dependencies. Blöchl et al. [9] examined* input-output networks* (networks of flows of goods and services between economic sectors) and defined node centrality measures (based on random walks on the network) suitable in identifying the key sectors in an economy. Using these metrics, they find commonalities in the network structures of economic networks which share geographical proximity and similar developmental status. Chopra and Khanna [10] also used the input-output network framework to examine the resilience of the United States’ economy and found that it is vulnerable due to greater interdependencies of its critical infrastructure sectors (CIS).

Nagurney and Qiang [21] reviewed developments in analytical tools for the assessment of network vulnerability and robustness. In the review they showed how appropriate network measures can capture, besides network topology, underlying behavior, network flows, and induced costs. Furthermore, they proposed ways to measure the synergy associated with network integration, focusing on topological changes in supply chain networks (such as those induced by corporate mergers and acquisitions, as well as teams and partnerships for disaster relief).

More recently, Garvey et al*.* [22] proposed a model for the propagation of risk in a supply chain network. They made use of the structure of the supply chain and the nodes’ individual risk profiles to construct a Bayesian Network (BN) of the causal risk relationships in the supply chain. They then developed risk measures based on the model and performed simulations to verify these measures. Käki et al. [23] also utilized the BN framework in modeling the propagation of risks across a supply chain network but combined it with probabilistic risk assessment (PRA) to evaluate supply network risks. A different approach was proposed by Yildiz et al. [24], where the reliability of a supply chain network (using a metric that takes into account both a network component’s intrinsic reliability and the reliability of its upstream) is used as a target for optimization, along with calculated cost, in a multiobjective nonlinear programming model. This model is then solved using a novel fusion of a genetic algorithm for the network design and linear programming for the optimization of the network flow. A third approach is examined by Xu et al. [25], who modeled the propagation of losses in a three-tier supply chain where information sharing and multiple sourcing for a focal company are present and the resulting value-at-risk (VaR).

#### 3. Materials and Methods

##### 3.1. Network Loss Propagation

Let us consider , a directed, acyclic network (a directed, acyclic graph, DAG) of size . This is a representation of, for example, the flow of materials in a supply chain network from the raw materials, through intermediate processed materials, and down to the finished product. We assume that there is a single type of flow through the network; a treatment of multiple flow types is beyond the scope of this work.

Furthermore, implicit in the acyclic formulation of a supply chain network is the assumption that the processing of a material at a given stage would not require material from* downstream* inputs, that is, from further processed stages of the material produced in the same supply network. This is a reasonable assumption for a wide variety of materials and products. The acyclic formulation also simplifies some of the mathematical steps in the formulation of the impact measure, as discussed below.

Given two nodes, and , we call a* supplier* of and a* customer* of , if a path from to exists. Node is a* direct* supplier of if the path is of (unweighted) length 1 (i.e., if a directed edge from to exists) and an* indirect* supplier otherwise. We use analogous definitions for direct and indirect customers.

The weight of the edge between any two adjacent nodes and is normalized such that the sum of all weights incident on equals 1. Equivalently, if is the (weighted) adjacency matrix of , then we require that . We can think of as the fraction of ’s input provided by a direct supplier . Furthermore, as is* acyclic*, loops are excluded and downstream inputs are nonexistent. These include self-loops, edges which lead from a node to itself (). This implies that nodes do not produce materials which are their own inputs but strictly receive them from upstream suppliers.

, the weighted adjacency matrix of , has an important property which we will use in subsequent sections. In , a path of length exists between two different nodes and if and only if , the th element of the power of , is nonzero [26, pp. 136-137]. To see this, let us explicitly write out the sums for and , as well as the general case:

We see that for a path of length 2 to exist between and (i.e., two directed edges exist such that one goes from to an intermediate node and the other from the intermediate node to ), by (1), there must be at least one node for which both and are nonzero, as otherwise zero indicates the absence of a corresponding edge. Similarly an path exists between and if, by (2), at least one pair of nodes and , different from , , and each other, can be found such that the three factors on the right hand side are all nonzero. For the general case of a path with length between and , (3) implies that other distinct intermediate nodes must be present. The cases where any node index equals another correspond to self-loops, which are assumed to be absent and thus do not contribute to the sum. We further observe that the right hand side of (3) requires exactly nonzero factors, each corresponding to a directed edge along the path from to .

Let us consider a scenario where node (the* originating node*) incurs a production loss. Depending on the properties of and intervening nodes, this loss might propagate through the network and reach node downstream, resulting in a potential loss at the latter node. In this context, we take as the* production loss fraction* sustained by due to the disruption, as the* potential* loss fraction of downstream from which can be sustained from a loss of , and as the* actual* loss fraction at due to .

Each node (including and ) also has a resilience threshold, , which represents the extent to which a node can compensate for loss in its total inputs. A node with can compensate for up to a loss in its input without its total output changing. A resilience of 1 indicates that the node can completely absorb any potential loss propagated from , while a resilience of 0 means that any loss cannot be absorbed, and hence the actual loss equals the potential loss. This can represent the aggregation of the node firm’s mechanisms in place in-house to compensate for input loss (such as safety stocks), which will allow it to partially or completely compensate for production losses upstream.

The loss may be propagated through or attenuated by intervening nodes, depending on their individual resilience thresholds. Let the index denote the immediate suppliers of node (’s upstream neighbors) and be ’s resilience, as previously defined. Then is the sum of the* actual* loss fractions of ’s immediate suppliers, , multiplied by their respective contributions to ’s input, . Consequently, is either zero (if ) or difference otherwise. Similarly, the potential and actual losses of ’s immediate suppliers , , and are defined identically for each of their own immediate suppliers, and so on. Thus, we can recursively define and by the following:where the summation in (4) is taken over ’s immediate suppliers and is the ramp function which stands for taking the enclosed argument to be zero if it is negative. and can be interpreted as the input and output losses, respectively, of node due to . Taken together, (4) and (5) recursively define the loss sustained by a node by those of its suppliers and ultimately by the loss at .

Consequently, the terminating condition (at the starting node ) is given byFigure 1 shows a schematic of a generic network and the downstream propagation of loss from the originating node.