A single disruption at a single location within a supply chain may effect many other locations of a the supply chain through the Ripple effect (Ivanov et al., 2014). In this article we focus on the role of inventory to mitigate the Ripple effect. Our main finding states that operational decisions (such as safety inventory level or service level) are highly interrelated with decisions that mitigate the Ripple effect (such as a high Resilience of the supply chain). Specifically, we find: 1) An increase in demand volatility may lead a firm to invest more in operational safety stock as well as in dedicated Risk Mitigation Inventory. 2) A high service level and high resilience of a supply chain can be conflicting objectives

Lucker, F.

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City Research Online

              City, University of London Institutional RepositoryCitation: Lucker, F. ORCID: 0000-0003-4930-9773 (2019). Using inventory to mitigate the Ripple effect. IFAC PAPERSONLINE, 52(13), pp. 1272-1276. doi: 10.1016/j.ifacol.2019.11.373 This is the accepted version of the paper. This version of the publication may differ from the final published version. Permanent repository link:  https://openaccess.city.ac.uk/id/eprint/23467/Link to published version: http://dx.doi.org/10.1016/j.ifacol.2019.11.373Copyright and reuse: City Research Online aims to make research outputs of City, University of London available to a wider audience. Copyright and Moral Rights remain with the author(s) and/or copyright holders. URLs from City Research Online may be freely distributed and linked to.City Research Online:            http://openaccess.city.ac.uk/            publications@city.ac.ukCity Research OnlineUsing inventory to mitigate the RippleeffectFlorian Lu¨cker ∗∗ City, University of London, 106 Bunhill Row, London EC1Y 8TZ,UK (e-mail: florian.lucker@city.ac.uk).Abstract: A single disruption at a single location within a supply chain may effect many otherlocations of a the supply chain through the Ripple effect (Ivanov et al., 2014). In this articlewe focus on the role of inventory to mitigate the Ripple effect. Our main finding states thatoperational decisions (such as safety inventory level or service level) are highly interrelatedwith decisions that mitigate the Ripple effect (such as a high Resilience of the supply chain).Specifically, we find: 1) An increase in demand volatility may lead a firm to invest more inoperational safety stock as well as in dedicated Risk Mitigation Inventory. 2) A high servicelevel and high resilience of a supply chain can be conflicting objectives. Copyright c© 2019 IFACKeywords: Supply Chain Management, Risk Management, Modeling1. INTRODUCTIONA single disruption at a single location within a supplychain may effect many other locations through the Rippleeffect (Ivanov et al., 2014). In this article we want todiscuss various levers that mitigate the Ripple effect.Broadly speaking there are two kind of operational riskmitigation levers available to a firm: (i) to hold additionalcapacity at a reliable source, so called reserve capacity(Lu¨cker et al., 2018b) and (ii) to hold inventory that canbe used to meet demand in the event of a disruption. Inthis article we want to further specify inventory as a riskmitigation lever. A broad stream of literature assumes thatfirms hold so-called Risk Mitigation Inventory (RMI). RMIis designed to be used to meet customer demand in theevent of a supply disruption (Lu¨cker et al., 2018a). As it isnot designed to be used to mitigate demand uncertainty,RMI is different from safety inventory and is often held inaddition to safety inventory. However, it is important torecognize that any available inventory will be used in theevent of a disruption to supply the disruption demand.In particular, safety inventory and cycle inventory (ifavailable) will be used as risk mitigation levers even thoughthey are not designed a priori to serve in the event of adisruption.Pharmaceutical firms often produce large batch seizes ofthe active pharmaceutical ingredient (API) of a drug.Such batches may supply more than a year of demandfor the API. Clearly, in the event of a disruption, thepharmaceutical firm would use all available cycle inventoryto mitigate the disruption.With this explorative research we want to better under-stand the roles of safety and cycle inventory on the optimalrisk mitigation strategy.To illustrate our insights, this analysis focuses on a singleechelon where inventory is replenished according to thepopular (Q, R) policy. We conduct extensive numerical ex-periments to find that operational decisions (such as safetyinventory level or service level) are highly interrelated withdecisions that mitigate the Ripple effect (such as a highResilience of the supply chain). Specifically, we find:(1) An increase in demand volatility may lead the firmto invest more in operational safety stock as well asin RMI.(2) A high service level and high resilience of a supplychain can be conflicting objectives.Finding 1 reveals the counter-intuitive result that theoperational safety stock and RMI can sometimes be com-plementary as RMI and operational safety stock may bothbe increasing in demand volatility. Finding 2 underlinesthat a resilient supply chain may not necessarily operateat a high service level. This is interesting because such asupply chain may be capable to deal with severe supplydisruptions, but not necessarily with demand uncertainty.2. RELATED WORKThe research topic of supply chain risk management hasrecently been explored by Ivanov and Dolgui (2018) whosuggest to emphasize uncertainty in the supply chainrather than certain assumptions. In this line of research theauthors identify key characteristics of these low-certainty-need supply chains.The ripple effect was introduced by Ivanov et al. (2014).This effect suggests that the impact of a disruption propa-gates across an entire supply chain, affecting performancemetrics along various dimensions. In (Ivanov et al., 2019)the authors combine findings on the ripple effect withsupply chain digitization.The work of Scheibe and Blackhurst (2018) studies dis-ruption propagation in multi-tier supply chains. Based ongrounded theory, the authors identify key themes that arerelevant for decision-makers when being exposed to sup-ply disruption risk. Further related literature focuses onresilience in the procurement process. Based on developingresilience-based supplier selection criteria, Hosseini andBarker (2016) proposes a new Bayesian network paradigm.3. METHODSLet us briefly restate key elements of the (Q,R) policy:This continuous-review policy is based on ordering Q unitswhen on-hand inventory hits a reorder point R. Key as-sumptions include that demand is stochastic and station-ary and that the lead time T for placing an order is positiveand constant. Although exact solutions are known, oftenheuristics are applied. Hadley and Whitin (1963) find aniterative heuristic solution, where the inventory holdingcosts h are approximated and where they assume that thelead time demand does not exceed the order quantity Q.This heuristic is based on the expected average annual costgiven byG(Q,R) = h(Q2+R− λT )+K λQ+(h+pλQ)n(R), (1)where λ is the demand rate, K the setup cost for placingan order, and p the backlog cost per unit of unsatisfieddemand. The expected number of shortages that occur inone cycle isn(R) =∞∫R(x−R)fT (x)dx, (2)where fT (x) describes the probability density function ofthe lead time demand. The operational safety stock SS isSS = R− λT. (3)Let us extend this policy to include disruptions by combin-ing the total cost function for a disruption with the totalcost function for the (Q,R) policy:Gˆ(Q,R, a) =G(Q,R)(1− ωτ)+ωτQ+R∫R(p∞∫I+aτ(x− I − aτ)fτ (x)dx+hI∫0(I − x)fτ (x)dx+cAI+aτ∫I(x− I)fτ (x)dx+cAaτ(1− Fτ(I + aτ)))ξ(I)dI+cˆAa. (4)In above total cost function, the first term G(Q,R)(1 −ωτ)represents the expected cost function if no disrup-tion occurs (costs for the undisrupted (Q,R) policy). Nodisruption occurs with probability (1 − ωτ ). The secondterm represents the the expected costs if a disruptionoccurs. The disruption probability is ωτ . We refer to theliterature (Lu¨cker et al., 2018b) for a detailed discussionof the expected costs in the event of a disruption. We notethat the first of these terms represents the penalty cost,the second the RMI holding costs, the third and fourththe reserve capacity production costs. The last term rep-resents the reserve capacity reservation cost and is alwaysincurred regardless of whether a disruption takes place.ξ(I) represents the probability distribution density for theinventory level I (e.g. on-hand inventory minus stockouts).Using the truncated normal distribution we can rewritethe terms as follows:Gˆ(Q,R, a) =G(Q,R)(1− ωτ)+ωτ1QQ+R∫R(p(µτ + στfτ(I+aτ−µτστ)1− Fτ(I+aτ−µτστ))+h(µτ + στfτ(−µτστ)− fτ( I−µτστ )Fτ(I−µτστ)− Fτ(−µτστ ))+cA(µτ + στfτ(I−µτστ)− fτ( I+aτ−µτστ )Fτ(I+aτ−µτστ)− Fτ( I−µτστ ))+cAaτ(1− Fτ(I + aτ)))dI + cˆAa. (5)To further evaluate this objective function, we performa numerical analysis of key parameters with MATLAB(fmincon optimizer).We measure supply chain resilience with the risk measureconditional value at risk CV aRα with threshold α. Weconsider two types of uncertainty in our supply chain:demand uncertainty and inventory uncertainty (only anuncertain amount of the total cycle inventory can beused in the event of a disruption). CV aR is then givenby the expected shortages beyond a threshold. A lowexpected shortfall (CV aR) corresponds to a high supplychain resilience, whereas a high expected shortfall (CV aR)corresponds to a low supply chain resilience.4. EXPERIMENTAL RESULTSIn the following we provide numerical results for the ex-tended (Qˆ, Rˆ) policy. We focus on realistic scenarios withparameters that were aligned with the pharmaceuticalcompany: K = 100, λ = 1, p = 40, h = 1, T = 1, cA =20, cˆA = 2, τ = 10, ωτ = 0.05, α = 0.95 and a normallydistributed demand with µ = 1 and σ = 0.3. Throughoutthe experiments we find consistent results along a widerange of parameter choices that support the observationsbelow. We analyze how the decision variables Qˆ, Rˆ and a∗as well as CV aRα depend on key model parameters.First, we describe the behavior of our decision variablesand CV aRα in various cost parameters:Proposition 1. Sensitivity in various cost parameters:• Qˆ increases in cA, cˆA and may decrease or increase inh, p.• Rˆ decreases in h and increases in cA, cˆA, p.• a∗ decreases in cA, cˆA and increases in h, p.• CV aRα increases in h, cA and cˆA, but may increaseor decrease in p.Coefficient of variation CV :CV = 0.1 CV = 0.15 CV = 0.2 CV = 0.25 CV = 0.3R 1.08 1.13 1.17 1.21 1.25Rˆ 1.19 1.28 1.37 1.45 1.54Q 13.57 13.52 13.47 13.43 13.40Qˆ 10.04 10.07 10.09 10.11 10.14Table 1. Sensitivity of R and Rˆ in the coeffi-cient of variation of demand CVThese observations are in line with the Hadley-Whitin(Q,R) policy.Next, we elaborate on the role of the coefficient of variationof demand CV :Proposition 2. Sensitivity in the coefficient of variation ofdemand CV :• Qˆ decreases in CV .• Rˆ and CV aRα increase in CV .• a∗ may decrease or increase in CV .Table 1 shows how Rˆ increases in CV . We also display theincrease in the reorder point R from the Hadley-Whitin(Q,R) policy. Interestingly, the reorder point increaseseven more strongly in the case of a disruption (Rˆ increasesby 29.8% over the range CV = 0.1 to CV = 0.3) thanwithout disruptions (R increases by 15.3% over the rangeCV = 0.1 to CV = 0.3). Since the lead time is constant,the increase in R entails an increase in the safety stock.At the same time, in the case of a disruption, Rˆ increasesmore strongly, indicating that safety stock and RMI canbe complements rather than substitutes.Table 1 also shows that Qˆ increases in CV , whereas Q isapproximately constant in CV .In contrast to previous analytical results (Lu¨cker et al.,2018b), the reserve capacity production rate a∗ may de-crease in the coefficient of variation of demand CV . Asdemand volatility increases, more safety stock is built up.Hence, a lower reserve capacity production rate is optimal.Next, we elaborate on the role of disruption probabilityωτ :Proposition 3. Sensitivity in the disruption probabilityωτ :• Qˆ, a∗ and CV aRα may increase or decrease in ωτ .• Rˆ increases in ωτ .In order to provide intuition behind this Proposition, weplot how the batch size (Figure 1), the reorder point (Fig-ure 2), the reserve capacity production rate (Figure 3), andCV aRα (Figure 4) depend on the disruption probabilityωτ . In Figure 1 we observe that Qˆ varies strongly withthe disruption probability ωτ . Clearly, as the disruptionprobability increases, it is helpful to hold more inventoryin the supply chain. As long as the disruption probabilityremains low (below 6%) it is cheaper to substantiallyincrease the batch size rather than substantially increasingsafety inventory. However, once we pass a threshold (6% inthe numerical example), it becomes cost-efficient to sub-stantially increase the reorder point (see Figure 2) becauseRMI and safety inventory provide the best buffer againstthe very likely disruptions (rather than the highly variable0 0.05 0.1 0.15 0.2 0.25 0.399.51010.51111.51212.51313.514Disruption probability ωτBatchsizeQˆFig. 1. Batch size Qˆ0 0.05 0.1 0.15 0.2 0.25 0.3123456789Disruption probability ωτReorderpointRˆFig. 2. Reorder point Rˆcycle inventory). However, since it is optimal to hold largeamount of RMI and safety inventory, it is optimal to reducethe batch size for high disruption probabilities. This dis-cussion shows that supply chain resilience can effectivelybe improved by not only holding RMI and safety inventory,but also by setting optimal batch sizes. These resultscontrast with the practice of some companies, which focusexclusively on stocking RMI without taking the batch sizeinto account.Regarding the reserve capacity, we observe that the reservecapacity is switched on only for disruption probabilities ofωτ ≥ 0.2. Then a∗ increases in ωτ until it reaches its max-imum at ωτ ≈ 0.06 because low probability disruptionsare better mitigated with reserve capacity than inventory.Once we pass the threshold of ωτ ≈ 0.06, it is optimalto reduce the reserve capacity because it is cheaper tomitigate disruptions with inventory rather than reservecapacity.The dependency of CV aRα on the disruption probabilityωτ can be explained as follows. CV aRα increases inthe disruption probability ωτ until ωτ ≈ 0.02 as the0 0.05 0.1 0.15 0.2 0.25 0.300.10.20.30.40.50.60.7Disruption probability ωτReservecapacityproductionratea∗Fig. 3. Reserve capacity production rate a∗0 0.05 0.1 0.15 0.2 0.25 0.300.0050.010.0150.020.0250.030.0350.040.045Disruption probability ωτCVaRαFig. 4. CV aRα for α = 0.95likelihood of not being able to supply in the case of adisruption increases. We have the maximum CV aRα ≈0.042. Afterwards CV aRα decreases in ωτ until ωτ = 0.1as more RMI and reserve capacity are built up. For ωτ >0.1 CV aRα increases slightly in ωτ .Next, we elaborate on the role of the disruption time τ :Proposition 4. Sensitivity in the disruption time τ :• Qˆ and Rˆ increase in τ .• a∗ and CV aRα may increase or decrease in τ .In order to provide intuition behind this Proposition, weplot how the batch size (Figure 5), the reorder point(Figure 6), the reserve capacity production rate (Figure7), and CV aRα (Figure 8) depend on the disruption timeτ . We find that the batch size Qˆ and reorder point Rˆincrease in τ throughout the parameter range. It is veryintuitive that more inventory of any kind is needed whendisruptions tend to last longer. Note that when disruptionstend to be short (τ < 20), it is optimal to increase thereserve capacity substantially, whereas the reorder pointincreases only slightly because it is more cost-efficient0 10 20 30 40 5051015202530Disruption time τBatchsizeQˆFig. 5. Batch size Qˆ0 10 20 30 40 500510152025Disruption time τReorderpointRˆFig. 6. Reorder point Rˆmitigating short disruptions using reserve capacity ratherthan inventory. In contrast, when disruptions tend to belonger (τ > 20), the reorder point increases whereasreserve capacity decreases. Note that these results are inline with the analytical results of Lu¨cker et al. (2018b).CV aRα increases in τ until τ = 5 as the potential lossincreases‘. Afterwards CV aRα decreases in τ because thepotential total penalty cost increases, and it therefore paysoff for the firm to invest in RMI and reserve capacity.Finally, we elaborate on the role of the fixed costs K:Proposition 5. Sensitivity in the fixed costs K:• Qˆ increases in K.• Rˆ and a∗ decrease in K.• CV aRα may increase or decrease in K.As with the Hadley-Whitin (Q,R) policy, the batch sizeincreases in K and the reorder point decreases in K. Thedecrease in Rˆ also entails a decrease in the service level asthe lead time is kept constant. Given that CV aRα mayincrease or decrease, keeping a high service level and highsupply chain resilience as measured by CV aRα can be0 10 20 30 40 5000.10.20.30.40.50.60.70.8Disruption time τReservecapacityproductionratea∗Fig. 7. Reserve capacity production rate a∗0 10 20 30 40 5000.010.020.030.040.050.060.070.08Disruption time τCVaRαFig. 8. CV aRα for α = 0.95conflicting objectives. In particular, keeping a high servicelevel does not necessarily entail high supply chain resilienceas measured by CV aRα.5. CONCLUSIONSThis research suggests that operational decisions such asbatch size or safety inventory levels are highly interrelatedwith decisions that deal with disruption risk (RMI andreserve capacity).As avenues for future research we suggest to developfurther insights on how these dependencies affect eachother in more complex settings (multi-echelon, multi-product supply chains) and to provide a more rigorousmodeling foundation.REFERENCESHadley, G. and Whitin, T.M. (1963). Analysis of inventorysystems. Prentice-Hall, Englewood Cliffs, NJ.Hosseini, S. and Barker, K. (2016). A bayesian networkmodel for resilience-based supplier selection. Interna-tional Journal of Production Economics, 180, 68 – 87.Ivanov, D., Sokolov, B., and Dolgui, A. (2014). The rippleeffect in supply chains: trade-off ‘efficiency-flexibility-resilience’ in disruption management. InternationalJournal of Production Research, 52, 2154–2172.Ivanov, D. and Dolgui, A. (2018). Low-certainty-need (lcn) supply chains: a new perspective inmanaging disruption risks and resilience. In-ternational Journal of Production Research, 0(0),1–18. doi:10.1080/00207543.2018.1521025. URLhttps://doi.org/10.1080/00207543.2018.1521025.Ivanov, D., Dolgui, A., and Sokolov, B. (2019). Theimpact of digital technology and industry 4.0 on theripple effect and supply chain risk analytics. In-ternational Journal of Production Research, 57(3),829–846. doi:10.1080/00207543.2018.1488086. URLhttps://doi.org/10.1080/00207543.2018.1488086.Lu¨cker, F., Chopra, S., and Seifert, R. (2018a). Mitigatingproduct shortages due to disruptions in multi-stagesupply chains. Working paper, Swiss Federal Instituteof Technology EPFL, Lausanne.Lu¨cker, F., Seifert, R., and Bic¸er, I. (2018b). Roles ofinventory and reserve capacity in mitigating supplydisruption risk. International Journal of ProductionResearch.Scheibe, K.P. and Blackhurst, J. (2018). Sup-ply chain disruption propagation: a systemic riskand normal accident theory perspective. Inter-national Journal of Production Research, 56(1-2),43–59. doi:10.1080/00207543.2017.1355123. URLhttps://doi.org/10.1080/00207543.2017.1355123.

Using inventory to mitigate the Ripple effect

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Using inventory to mitigate the Ripple effect

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