Sourcing Strategies to Manage Supply Disruptions

Abstract

Supplier diversification and backup sourcing offer alternatives to stockpiling inventory as a means of mitigating disruption risks. In this chapter, we introduce some simple models to analyze and explore the performance of these sourcing strategies. The effectiveness of diversification depends largely on the number of suppliers, the possibility of disruption correlation, and the available spare capacity at suppliers. Most of the benefits are achieved with a small number of suppliers. The effectiveness of backup sourcing depends largely on the cost and availability of the backup source, with availability being measured as response time and capacity provided. When choosing the appropriate strategy, managers need to account for all the significant factors that influence performance, including the disruption profile, inventory costs, the fixed and variable supplier costs, capacities, response times, and disruption correlation. Crafting the best strategy relies on sound judgment aligned with suitable analysis.

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3.1 Introduction

Supply chains depend on the successful flow of material in order to function and satisfy customer demand. When that flow is interrupted at a particular facility, alternate sources can be considered to keep material moving through the chain. We explore such alternatives in this chapter, focusing primarily on two sourcing strategies: diversification and emergency backup sourcing. Diversification means a firm uses multiple supply sources on an ongoing basis, which lowers the impact of any one source becoming unavailable. However, this requires ongoing investment in multiple supplier relationships and logistics. Emergency backup sourcing, on the other hand, is used only if a disruption occurs and thus may cost less on an ongoing basis; however, in the event of a disruption that emergency source may be more costly to use and have a slower response time than a routine supplier.

In the chapter we introduce some simple models to analyze these considerations. We focus on developing insights as to which strategies are appropriate in various settings. Our models reflect many important aspects of disruption mitigation but do not reflect all the possible complexities that might influence strategy implementation in a given firm’s setting. We would recommend a more tailored analysis for detailed implementation purposes.

While the term “supplier” typically refers to external sources, we use it throughout this chapter to mean either an external or an internal source of supply. Internal sources refer to supply chains with multiple levels within a single firm, where downstream locations depend on upstream sources for their material. Firms should evaluate sourcing strategies not only for external suppliers, but also for mitigating disruptions to their own upstream network locations.

As the focus of this chapter is on sourcing strategies to manage disruption risk, we ignore inventory in this chapter unless otherwise stated. We refer the reader to Chap. 5 for a full treatment of the inventory strategy. For the sake of completeness, we include a brief analysis of the inventory strategy in the appendix to this chapter.

The remainder of the chapter is organized as follows: in Sect. 3.2 we cover the acceptance strategy (no proactive mitigation) and introduce our modeling approach. In Sect. 3.3 we discuss diversification strategies, and in Sect. 3.4 we discuss emergency backup strategies. We compare these approaches along with acceptance and inventory in Sect. 3.5 We discuss other considerations for sourcing strategies in Sect. 3.6, and we summarize the chapter’s insights in Sect. 3.7. The appendix presents the mathematical development of the profit expressions shown in this chapter.

3.2 Acceptance Strategy

We start our analysis by studying the acceptance strategy in which the firm sources from a single supplier (prone to disruptions) and does not attempt to mitigate supply risk through inventory, supplier diversification, or backup supply. The acceptance strategy can stand as a benchmark against which other strategies can be measured. Also, it may be an appropriate strategy if disruption risk is low or if the mitigation costs are very high.

In what follows we introduce our modeling approach and develop an expression for the long-run average profit the firm obtains under the acceptance strategy. ¹ The other strategies will build upon the model introduced here. We assume that all unsatisfied demand is lost, but discuss backlogging of demand in Sect. 3.6. We adopt the following notation: each unit sold yields a profit margin of MARG, unsatisfied demand incurs a penalty cost of LOST per unit, and the supplier’s exogenous percentage uptime is denoted by UP, where $0 \leq {\it UP} \leq 1.$ For example, if a supplier has a probability of being available of 95%, then UP would be 0.95. We assume demand is constant and normalize the demand to be 1 in every period. ²

The acceptance strategy gives a per-period profit of MARG in a non-disrupted period, but in a disrupted period a penalty cost of LOST is incurred. Thus the acceptance strategy profit, denoted ${\it ACC}\_{\it PROF},$ is equal to ${\it MARG} \times {\it UP} - {\it LOST} (1-{\it UP}).$ Equivalently, this can be presented as:

$$ {\it ACC}\_{\it PROF} = {\it MARG} - ({\it MARG} + {\it LOST}) (1-{\it UP}) $$

(3.1)

If the supplier was perfectly reliable, i.e., 100% uptime, the profit would be MARG. Therefore the lost profit associated with the lack of reliability is $({\it MARG} + {\it LOST})(1-{\it UP}),$ and this lost profit increases (linearly) as the supplier becomes less reliable, i.e., as UP decreases.

3.3 Supply Diversification

Diversifying supply sources is a logical way to manage the risk of supply disruptions. Diversification means that multiple sources are used for the same product on a regular basis. Thus some material flow will still continue in the event of a disruption if at least one supplier is still operating. Diversification takes time and effort, though. Contracts and relationships with all external suppliers must be created and maintained, or investments must be made to create and maintain multiple internal sources for a product. Operational capability for receiving or picking up material from multiple sources must be put in place and maintained. There are clear savings available from making a firm’s supply base more lean (reducing the number of suppliers); thus the risk-mitigation benefits of supplier diversification need to be weighed against the cost of expanding the firm’s supply base.

Diversification is not a panacea for disruptions. Issues can arise if problems simultaneously disrupt multiple facilities or if spare capacity is not sufficient. Kellogg Co. recently experienced disruptions at two of its four plants that produce Eggo brand frozen waffles. The plant in Atlanta, Georgia, experienced a bacterial contamination (and unrelated flooding) that shut down the facility for most of September and October 2009 [6]. In the mean time, multiple production lines at the plant in Rossville, Tennessee, were closed for repairs. Because the other two plants did not have sufficient capacity to make up for the disrupted production, Kellogg forecasted that there would be a shortage of Eggo waffles on retail shelves until the middle of 2010 [11]. While the supplier diversification strategy helped Kellogg mitigate the disruptions it did not completely insulate the business from their effects: “The temporary supply disruption contributed to the 3% sales decline in the North American frozen and specialty channels unit last quarter, Kellogg chief executive officer David Mackay said on an October 29 earnings conference call” [3].

In what follows, we explore the impact of the number of suppliers, disruption correlation and spare capacity on the performance of the diversification strategy.

3.3.1 Number of Suppliers

Suppose the firm uses multiple suppliers, with the number of suppliers denoted by ${\it NUM}\_{\it SUP}.$ Assume that all suppliers have the same percentage uptime UP. To start, assume that supplier disruptions are not correlated and that suppliers have infinite capacity (that is, any supplier can instantly satisfy 100% of the demand). Because any single supplier can satisfy all demand, the firm’s supply is only disrupted if all suppliers are simultaneously down, which happens with probability $(1-{\it UP})^{{\it NUM}\_{\it SUP}}\!.$ Therefore, the diversification strategy profit is given by ${\it DIV}\_{\it PROF}_{\it u}$ (where the u subscript denotes uncorrelated suppliers):

$$ {\it DIV}\_{\it PROF}_{\it u} = {\it MARG}-({\it MARG} + {\it LOST}) (1-{\it UP})^{{\it NUM}\_{\it SUP}} $$

(3.2)

We have purposely ignored any ongoing fixed costs associated with maintaining a supplier. A full accounting would subtract such costs from this expression. If ${\it NUM}\_{\it SUP}=1$ then, ${\it DIV}\_{\it PROF}_{\it u}={\it ACC}\_{\it PROF},$ i.e., Eq.  3.2 reduces to Eq.  3.1. Increasing the number of suppliers increases the profit, as we would expect. An important characteristic of the profit function is that it exhibits diminishing returns to the number of suppliers, that is, the incremental value of adding another supplier decreases as the number of suppliers increases.

We illustrate this using a particular numerical example, setting the profit margin, MARG, at 5, and the lost sales parameter, LOST, at 3. For different UP values, Fig. 3.1 shows the profit as ${\it NUM}\_{\it SUP}$ increases. (The profit is reported as a percentage of the profit if there are never any disruptions, i.e, MARG, which is 5 for this case). Observe the rapidly diminishing returns to adding suppliers. The benefits of diversification are largely achieved with a relatively small number of suppliers. Even at extremely low uptime percentages, e.g., ${\it UP}=0.6,$ any incremental risk-mitigation benefit of going beyond four suppliers will likely be outweighed by the additional fixed costs of maintaining more suppliers. For reasonable uptime percentages, firms may find that using two or three suppliers strikes an appropriate balance between risk mitigation and supply-base rationalization.

Fig. 3.1

Diminishing returns to adding suppliers

3.3.2 Correlated-Supplier Disruptions

The underlying cause of a supplier disruption may not be specific to the affected supplier. For example, a bankruptcy at one supplier may be due to a global recession, and so other suppliers may also have a heightened risk of bankruptcy. Or if suppliers are located in the same geographic region, then a natural disaster in that region may impact multiple suppliers. Indeed, as illustrated by the Kellogg example above, multiple suppliers might be simultaneously disrupted by different events. It is important to consider the impact of disruption correlation when evaluating the diversification strategy.

Let CORR denote the (pair-wise) correlation coefficient for supplier disruptions. Our model allows for no correlation or positive correlation, that is $0 \leq {\it CORR} \leq 1.\,{\it CORR}=0$ indicates independent disruptions and, at the other extreme, ${\it CORR}=1$ indicates perfectly positively correlated disruptions, i.e., if one supplier is down then all are down. The diversified profit (allowing for correlation, denoted with the subscript c) is:

$$ \begin{aligned} {\it DIV}\_{\it PROF}_{\it c}&={\it MARG}- ({\it MARG} + {\it LOST})\\ &\enskip\quad\times\left(1- \frac{{\it UP} \left(1 - \left[(1-{\it CORR}) (1-{\it UP})\right]^{{\it NUM}\_{\it SUP}}\right)} {1- (1-{\it CORR})(1-{\it UP})}\right) \end{aligned} $$

(3.3)

The development of this expression (and all other expressions) can be found in the appendix at the end of this chapter. We note that if ${\it CORR}=0,$ this expression collapses to our earlier expression Eq.  3.2 for ${\it DIV}\_{\it PROF}_{\it u}.$ The profit decreases as CORR increases, that is, diversification provides less value as disruption correlation increases. At ${\it CORR}=1,$ ${\it DIV}\_{\it PROF}_{\it c} = {\it MARG} - ({\it MARG} + {\it LOST}) (1-{\it UP})$ which is the acceptance profit given by Eq.  3.1, and so diversification provides no value if disruptions are perfectly positively correlated. We also note that ${\it DIV}\_{\it PROF}_{\it c}$ exhibits decreasing reductions to correlation, that is, the incremental profit reduction is lower at higher values of CORR; however, the profit decreases in an almost straight-line fashion unless the uptime percentage is low.

We illustrate these general findings in Fig. 3.2 which shows the profit (again as a percentage of the no-disruption profit) as a function of correlation for two different percentage uptime values: 0.85 and 0.95. As before we set ${\it MARG}=5$ and ${\it LOST}=3.\,$ Observe that as the correlation increases, the profit decreases and eventually equals the profit of a single-supplier strategy at ${\it CORR}=1.$ (When only one supplier is used, then obviously correlation has no affect on the profit). As before, we see that adding just a few suppliers to diversify sourcing can offer significant increases in profits, but the value of additional suppliers is lower at higher correlations.

Fig. 3.2

Profit decreases as disruption correlation increases

Disruption correlation is not readily measured and firms cannot design their supply chain to “dial in” a precise correlation number. However, firms can qualitatively assess the degree of correlation in their current or proposed supply base. Suppliers located in the same region may be prone to correlated natural-hazard or socio- political-related disruptions. Suppliers using the same raw material input pose the risk of simultaneous contamination-induced disruptions. As correlation reduces the value of diversification, firms can and should design their supply chains to reduce correlation when possible, e.g., by not concentrating suppliers in one region.

3.3.3 Limited Spare Capacity

To this point we have assumed that each supplier had sufficient capacity to produce the total demand if needed. While one would expect that suppliers can provide some additional capacity beyond their normal volume, it may not be reasonable to expect that they can produce the full demand amount. We now expand our model to consider capacity constraints at the suppliers. In doing so, we assume that order quantities are evenly split across the available suppliers and that all suppliers have the same spare capacity. We introduce the parameter ${\it SP}\_{\it CAP}$ to denote how much capacity any supplier can provide in excess of their normal order quantity; if 10 units are normally ordered from a supplier and its spare capacity is ${\it SP}\_{\it CAP} = 0.20,$ then the maximum volume that supplier can provide is 12 units. When suppliers can provide some spare capacity, but not infinite capacity, the mathematical formulation becomes complex. We include it in the appendix for reference, but discuss and demonstrate the impact of spare capacity here.

First, we consider the case in which there is no spare capacity at the suppliers. The expected profit is ${\it DIV}\_{\it PROF}_{\it ns} = {\it MARG} - ({\it MARG} + {\it LOST}) (1-{\it UP}),$ with the subscript ns denoting no spare capacity. Perhaps surprisingly, this is the same as the acceptance strategy profit given in Eq.  3.1. In other words, diversification provides no value if there is no spare capacity. While diversification does provide protection if a supplier is down (because only a fraction of supply is lost), the likelihood of at least one supplier being down is higher when there is more than one supplier. These counteracting factors balance each other to eliminate any value. ³

Next we consider the case in which suppliers have some spare capacity. The profit increases in the spare capacity but it exhibits diminishing returns, that is, the incremental value of adding more spare capacity is lower at higher spare capacities. These general findings are illustrated for two different percentage uptime values (0.85 and 0.95) in Fig. 3.3. As before we set ${\it MARG}=5$ and ${\it LOST}=3.$ Observe that spare capacity has a large impact on profit and that there are diminishing returns to spare capacity.

Fig. 3.3

Diminishing return to increasing spare capacity

Recall that ${\it SP}\_{\it CAP}$ is the percentage a supplier can provide above and beyond its normal order quantity. For example, if demand is 100 units and ${\it SP}\_{\it CAP}$ is 50%, then with two suppliers each can provide at most 75 units (1.5 times their normal order quantity of 50), and with three suppliers each can provide at most 50 units (1.5 times 33.33). For a given ${\it SP}\_{\it CAP}$ value, the total capacity in the supply base is the same regardless of the number of suppliers, e.g., 150 in the example just given. Looking at the two-supplier case in Fig. 3.3, we see that the profit increases until the spare capacity reaches 100% (where either supplier can fully back each other up) and then remains level. When one supplier has enough spare capacity to meet all demand, then there is no additional value to adding more capacity. With three suppliers, the profit increases until each supplier has 50% spare capacity (at which point if one supplier goes down the other two can provide full demand coverage) and then increases at a lower rate until each supplier has 200% spare capacity (at which point each supplier can fully back up both other suppliers). The 4-supplier curve has a similar form, with distinct changes in slope when a capacity is reached where more suppliers may be disrupted without reducing material availability.

The amount of spare capacity in a firm’s supply base significantly influences the risk mitigation benefits of diversification. To ensure access to spare capacity, firms may need to invest in some “safety” capacity at internal suppliers and/or pay external suppliers to provide volume flexibility. Also, firms need to rapidly communicate the need for increased production in the event of a disruption elsewhere in their network. All this entails ongoing collaboration with suppliers.

3.4 Backup Supply

Rather than routinely source from multiple suppliers, a firm might instead single source under normal circumstances but rely on an emergency backup supplier in the event of a disruption to its primary supplier. If the emergency backup can respond rapidly when called upon, then an adequate flow of material can be maintained. Single sourcing eliminates the complexities and costs associated with routinely using multiple suppliers. However, emergency backup sourcing has its own set of complexities. Assuming that a backup source exists without validating availability and response time can leave a firm vulnerable to disruptions. Firms that rely on a spot market for emergency supply may discover that their competitors do also, causing issues with price and material availability in the event of an industry-wide disruption. Even similar plants within the firm’s own network may not be able to back each other up if they do not have available capacity, or if labeling, customs, or overlooked plant differences cause issues.

For example, we learned from a large consumer packaged goods firm that they had issues implementing their backup plan when customs went on strike in a South American country in which one of their plants was located. Managers had assumed that they would be able to source material from another plant in their network which made the same product and had some capacity flexibility. However, when trying to implement the plan, it was quickly realized that packaging and labeling was not identical for the two plants and therefore the emergency backup could not be put into effect. The firm lost over a million dollars while it waited for the strike to end so that the South American plant could receive raw materials again and try to catch up on backlogged demand. Effective backup sourcing requires proactive planning, and the firm is working to outline better plans to provide backups for certain critical facilities if they are disrupted. It put one such advance plan into action in 2009 when a tornado disrupted one of its distribution centers (DCs) and another DC was able to cover the disrupted DCs’ demand within a day or two.

In what follows, we explore the impact of the backup supplier cost, response time and capacity on the performance of the backup strategy.

3.4.1 Modeling Backup Supply

As sourcing from an emergency supplier typically costs more than a routine supplier we assume the unit profit margin when using the backup supplier, ${\it EM}\_{\it MARG},$ is less than unit profit margin, MARG, when using the regular supplier. The backup supplier will only be used if it makes economic sense, that is, using the backup supplier must be better than incurring a penalty for not filing demand, i.e., ${\it EM}\_{\it MARG} > -{\it LOST};$ otherwise the backup strategy is not viable. We assume that it takes time for the backup supplier to come on stream. This response time, ${\it EM}\_{\it RES},$ reflects the time taken to alert the emergency backup and to get material flowing from the backup. If a disruption occurs, the backup can only provide supply after ${\it EM}\_{\it RES}$ periods have passed. Also, we allow for the fact that the backup supplier might not have sufficient capacity to cover all demand; we denote the backup’s capacity limit as ${\it EM}\_{\it CAP},$ where $0 \leq {\it EM}\_{\it CAP} \leq 1$ is the percentage of demand that the backup supplier can fill. ⁴ Because we allow for a delayed response, the percentage uptime UP does not fully capture the disruption attributes relevant to the backup strategy (as it did for the acceptance and diversification strategies modeled earlier). In addition to the percentage uptime we also need to characterize the length of disruptions. For simplicity, we assume that (1) when the primary supplier is up there is a constant probability of a disruption occurring, and (2) when the primary supplier is down there is a constant probability of the disruption ending. That is, disruptions are geometrically distributed. We denote the average disruption length as ${\it DIS}\_{\it LEN}.$

The profit in a period when the regular supplier is up is MARG. If the regular supplier has been down ${\it EM}\_{\it RES}$ periods or less, then the profit in that period is $-{\it LOST}$ as the backup supply has yet to come on stream. If the regular supplier has been down for more than ${\it EM}\_{\it RES}$ periods, then the profit in that period is ${\it EM}\_{\it MARG} \times {\it EM}\_{\it CAP}-{\it LOST}\times (1-{\it EM}\_{\it CAP}),$ as the backup can only supply ${\it EM}\_{\it CAP}.$ This leads to the following profit for the emergency backup strategy, ${\it EM}\_{\it PROF}:$

$$ \begin{aligned} {\it EM}\_{\it PROF}&={\it MARG}-({\it MARG} + {\it LOST})(1-{\it UP})+ \\&\enskip\quad ({\it EM}\_{\it MARG}+{\it LOST})(1-{\it UP})({\it EM}\_{\it CAP}) \left(1-\frac{1} {{\it DIS}\_{\it LEN}}\right)^{{\it EM}\_{\it RES}} \end{aligned} $$

(3.4)

As one would expect, the profit decreases (in a straight line) as the emergency supplier becomes more expensive, i.e., as ${\it EM}\_{\it MARG}$ decreases. Interestingly, the profit increases as the average disruption length increases, i.e., as ${\it DIS}\_{\it LEN}$ increases. This is because for a given percentage uptime, the frequency of disruptions decreases as the average disruption length increases. Also, the fraction of the disruption time for which the backup is not producing (due to the delayed response) is shorter for longer disruptions, e.g., 50% for a disruption of length 6 and 25% for a disruption of length 12 if the response time is 3. Therefore, the backup strategy is more effective at mitigating rare-long disruptions than short-frequent ones, and this is why the profit increases as the average disruption length increases (for a given percentage uptime).

We now turn our attention to the role that response time and emergency capacity play in determining the effectiveness of the backup strategy.

3.4.2 Emergency Response Time

The backup profit decreases in the response time, and firms should make every effort to eliminate unnecessary delays in detecting the disruption and activating the emergency supply. Managers should ensure there is an agreed plan in place that sets out the actions and responsibilities required for rapid response. Emergency suppliers should be selected and validated in advance of any disruption. Effective planning can minimize unnecessary delays but it may take time to ramp up capacity at the backup and there may be physical constraints that preclude immediate production.

The backup strategy profit exhibits increasing returns to response time reductions; that is, the incremental benefit to reducing response times is higher the faster the response time. This is illustrated in Fig. 3.4 for two different disruption profiles: short-frequent and long-rare. The percentage uptime was set at ${\it UP}=0.95$ and, as before, we used ${\it MARG}=5$ and ${\it LOST}=3.$ The average disruption length was 4 periods for the short-frequent cast and was 20 periods for the long-rare case. Because the uptime was the same for both cases, disruptions are much more common in the short-frequent case - hence the name.

Fig. 3.4

Increasing returns to reducing response time

Looking at Fig. 3.4 we see that the nature of the disruption (short-frequent or long-rare) plays a crucial role. The profit falls off rapidly in the case of short-frequent disruptions; if the emergency supplier cannot respond very quickly then the backup strategy is not effective at mitigating short-frequent disruptions. The profit falls off much more slowly in the case of long-rare disruptions. In negotiating with a potential backup supplier, there may be a tradeoff between cost, response time, and capacity. In the case of long-rare disruptions it may make sense to sacrifice some response time to gain on the other dimensions. For short-frequent disruptions, the firm has to gain very significant concessions on cost and capacity to make up for response time degradation.

3.4.3 Emergency Capacity

The backup profit increases in the emergency capacity. Ensuring adequate surge capacity at an internal source (if that is the backup) may necessitate additional capital investment or a flexible workforce. Ensuring additional capacity at an external backup source might require the firm to pay an ongoing fee to reserve a desired level emergency capacity. If no one supplier can guarantee sufficient capacity, then the firm might contract with multiple suppliers to provide backup capacity.

We illustrate the impact of emergency capacity on the backup profit in Fig. 3.5, using the same parameters as used for Fig. 3.4. The profit increases in a straight line until the emergency capacity, ${\it EM}\_{\it CAP},$ equals 1, after which the profit would stay constant because unfilled demand is not backlogged (see earlier Footnote #4). Importantly, we again see that the nature of the disruption (short-frequent or long-rare) plays a significant role. For short-frequent disruptions, the profit is somewhat insensitive to capacity when the response time is long (e.g., 8, 9, or 10 in this figure) because additional capacity does not matter greatly if disruptions are almost over by the time the backup supplier comes on stream. In the case of rare-long disruptions, however, the profit increases significantly in capacity even at these longer response times.

Fig. 3.5

Profit increases (linearly) in emergency capacity (until EM_CAP  =  1)

Firms may face a tradeoff when selecting a backup supplier: one supplier might offer rapid response but only provide a limited capacity whereas another supplier might provide greater capacity but at the expense of a slower response time. When evaluating such tradeoffs, managers need to understand the type of disruption risk they face. Response time is a crucial concern for short-frequent disruptions whereas emergency capacity is important for long-rare disruptions. Moving beyond these generalities to explore a specific firm’s tradeoff curve between response time and capacity is possible but requires a more tailored analysis than provided here.

3.5 Choosing the Strategy

Having examined the acceptance, diversification and backup strategies in isolation, we now explore the firm’s strategy choice. That is, we determine which strategy best fits the firm’s situation. We now explicitly account for ongoing supplier maintenance costs. The firm incurs a per-period cost of FIXED for each supplier in the diversification strategy, where FIXED represents the operational, logistical and other volume-independent costs of maintaining a supplier. In addition to primary supplier maintenance cost of FIXED, we assume the firm incurs an ongoing cost of ${\it EM}\_{\it FIXED}$ to ensure access to the backup supplier in the backup strategy.

Because there are situations in which stockpiling inventory might be the most appropriate strategy, we allow this as a fourth option in this section. We use HOLD to denote the inventory holding cost per unit per period. We refer the reader to the appendix (heading Inventory model) for a brief analysis of the inventory strategy.

Let us examine how the nature of the disruption risk (percentage uptime and disruption length/fequency) influences the strategy choice. Holding the supplier percentage uptime constant for the moment, we first consider the impact of the average disruption length or, equivalently, disruption frequency on the performance of the four strategies. Now, based on our earlier analysis, the acceptance and diversification strategy profits do not depend on the average disruption length. That is, two disruption profiles with the same percentage uptime will result in the same profits even if the average disruption length differs. See the profit expressions Eq.  3.1 and Eq.  3.3. As discussed in Sect. 3.4.1, the backup strategy profit increases as the average disruption length decreases. In contrast, the inventory strategy profit decreases (for a given percentage uptime) as the average disruption length increases. The net effect is that the average disruption length has a profound impact on the preferred strategy, with inventory favored for short (more frequent) disruptions, backup favored for long (less frequent) disruptions, and diversification favored in between.

This is illustrated in Fig. 3.6, which presents the preferred strategy as a function of percentage uptime and average disruption length. ⁵ For short average disruption lengths, inventory (INV) is the preferred strategy unless the percentage uptime is very high. As the average disruption length increases (i.e., moving due north), inventory is initially displaced by diversification (DIV) [or acceptance (A)], but eventually the backup strategy (BACK) is preferred.

Fig. 3.6

Strategy choice depends on disruption profile

We now turn our attention to the influence of the percentage uptime. Different from the average disruption length, the percentage uptime has the same directional effect on all four strategies, with the profits all decreasing as the percentage uptime decreases. Looking at Fig. 3.6, we see that diversification is preferred over a large region but as the percentage uptime increases (i.e., moving due east), it is displaced by the backup strategy. At higher percentage uptimes the supply risk is by definition lower, and the supplier-related costs incurred by diversification to mitigate this lower risk become unattractive. Acceptance eventually becomes the preferred strategy because at very high uptimes the risk-cost tradeoff is such that it does not make economic sense to incur even the backup strategy’s supplier-related costs. Moving due west, i.e., as the percentage uptime decreases, diversification becomes less attractive because the likelihood of multiple suppliers being down increases. This could be alleviated with additional suppliers, but at the expense of higher supplier-related costs. If the average disruption length is short, then diversification is displaced by inventory (as short-frequent disruption risks do not require the firm to stockpile prohibitive amounts of inventory), but for higher average disruption lengths diversification is displaced by the backup strategy.

Summarizing Fig. 3.6, acceptance is preferred only if the percentage uptime is very high, and mitigating disruption risk through sourcing (diversification or backup) is preferable to mitigation through inventory unless disruptions are short/frequent and/or supplier-related fixed costs are prohibitive. (In certain capital intensive industries, e.g., pharmaceuticals, inventory might be used to protect against long/rare disruptions due to the high cost of building multiple plants.) In comparing the sourcing strategies, backup is preferred to diversification as disruptions become lengthier and more rare.

The disruption-risk profile is not the only determinant of the preferred strategy. For example, Fig. 3.7 illustrates the impact of inventory holding costs and supplier fixed costs. Figure 3.7a (top left quadrant) is identical to Fig. 3.6, but the other figures have a higher inventory holding cost (${\it HOLD}=0.2$ instead of ${\it HOLD}=0.05$) and/or higher supplier fixed costs (${\it FIXED}=0.08$ instead of ${\it FIXED}=0.04.$) Comparing Fig. 3.7a–c, we see that inventory is preferred over a smaller region as the holding cost increases. Comparing Fig. 3.7a–b, we see that diversification is less often preferred as the fixed costs of maintaining suppliers increases. Comparing Fig. 3.7a–d, we see that acceptance is preferred over a larger region as inventory holding and fixed supplier costs increase. When choosing the appropriate mitigation strategy, managers need to account for all the significant factors that influence performance, including the disruption profile, inventory costs, the fixed and variable supplier costs, capacities, response times, and disruption correlation.

Fig. 3.7

Influence of inventory and supplier fixed costs

For the sake of clarity, we have discussed the strategies as if they were mutually exclusive, that is, the firm can choose only of the four strategies. In fact, there may be situations in which it makes sense to deploy a combination of strategies. For example, a firm whose backup supplier has a longer-than-desired response time might hold some inventory to use during the early stages of a disruption. Or, a firm pursuing a diversification strategy might find that instead of maintaining three routine suppliers, it is better off maintaining two routine suppliers but having a backup in place.

3.6 Additional Considerations

The intent of this chapter was to explore the role of sourcing strategies in mitigating the risk of supply disruptions. To that end, we introduced some simple models that captured many salient features of sourcing strategies. However, by design, we did not address all possible complexities, as doing so might obscure some fundamental insights. We now briefly discuss some additional considerations that may be relevant when crafting a strategy to mitigate disruption risk. In doing so, we refer the interested reader to the relevant academic literature for a deeper treatment of these issues.

Backlogging of demand: we assumed that unfilled demand was lost rather than backlogged. While this might seem like a minor distinction it does have some important implications. For example, with lost sales, there was no benefit to the backup strategy having more capacity than demand. This is not the case with backlogged demand: the backup can reduce the backlog (that accumulated during the time it took to come onstream) only if its capacity exceeds normal demand. In general, capacity concerns are amplified in both the diversification and backup strategies if unfilled demand is backlogged rather than lost. The impact of a disruption is felt for longer in the case of backlogged demand as the firm may have to work through its backlog after the disruption ends. The lower the capacity, the longer it takes to eliminate the backlog. We refer the interested reader to [8] for a treatment of the inventory, diversification, and backup strategies when demand is backlogged.

Product life cycle: the models presented in this chapter implicitly assumed that the firm sells and replenishes the product on an ongoing basis. While no product lasts forever, an “infinite-horizon” model, as adopted in this chapter, is a reasonable approximation to reality if the product life cycle is significantly longer than procurement lead times or disruptions. However, a finite horizon model may be more appropriate if the product lifecycle is not significantly longer. If product life cycles are very short relative to procurement lead times, such that the firm only has one ordering opportunity, then inventory is not a viable strategy and single-period models are needed to analyze the sourcing strategies. We refer the interested reader to [4, 9] for this type of analysis.

Type of supply risk: we modeled supply risk by assuming suppliers where either fully operational, i.e., up, or temporarily completely unavailable, i.e., down. This is a reasonable model of disruptions but not the only one. Rather than complete failures, disruptions might entail the loss of a portion of the capacity at a facility. If the possible capacity loss is deterministic, then the models used here are easily modified. If the capacity loss is uncertain, then the analysis becomes more complicated. We refer the reader to [4, 12] for a treatment of diversification in a random capacity setting. In some industries, e.g., the semiconductor industry, significant yield loss is of more (or equal) concern to supply disruptions. We refer the reader to [1, 4, 5] for a treatment of diversification in a random yield setting. Chopra et al. [2] explores the backup strategy in which the primary supplier is subject to complete failures and yield variability, showing that it is important to correctly account for the underlying drivers of supply risk. Schmitt and Snyder [7] extend Chopra et al.’s model to consider multiple periods. Yano and Lee [13] provides an excellent review of the random-yield literature.

Nonidentical suppliers: for simplicity, we assumed that suppliers were identical in the diversification strategy. While this might be a reasonable approximation for some settings, oftentimes a firm’s supply base will contain suppliers that differ across costs, reliabilities, and capacities. The diversification question then moves beyond how many suppliers to use to include the questions of which suppliers to select and how much volume to allocate to suppliers. Tomlin [8] explores diversification with non-identical suppliers in an infinite-horizon setting with random disruptions. We refer the reader to [1, 4, 5, 12] for a treatment of non-identical suppliers in a single-period setting with random yield and/or random capacity.

Risk attitudes: we assumed that management’s goal in crafting the disruption strategy was to maximize the long-run average profit. In other words, we assumed managers were risk neutral. If managers are risk averse, then disruption mitigation is more easily justified. More than that, risk attitudes will influence the preferred strategy as different strategies result in different profit/loss distributions. If disruptions are short and frequent, then a mean-variance framework might be reasonable for evaluating strategies. However, such a framework might not be desirable if the firm is facing low-probability, high-impact events such as rare-but-long disruptions. In this case, maximizing the average profit subject to some constraint on downside risk might be preferred. A Variance-at-Risk (Var) or Conditional Variance-at-Risk (CVaR) approach could be used. In addition to a risk-neutral approach, Tomlin [8] considers both a mean-variance and a CVaR approach in selecting a disruption strategy. In single-periods setting, Tomlin [9] explores a risk neutral objective and a loss-averse objective in which managers value profits less than they fear losses. These papers show that the preferred strategy can be heavily influenced by risk attitudes.

Multiple products: the unit of analysis in this chapter was a single product rather than a portfolio of products. With regard to the strategies presented here, this is appropriate if the products do not share any supply chain resources (e.g., inventories, facilities, or suppliers) and also provides a reasonable “first-cut” analysis if they do share resources. However, a portfolio perspective is recommended if any resources are shared. Disruption to a shared resource will impact multiple products and this interaction needs to be evaluated as it has direct implications for the spare or backup capacity available to any individual product. The issue of multiple products introduces a tension between demand and supply risks. Component commonality and flexible facilities are well-established approaches to managing demand risk by taking advantage of demand pooling. However, these approaches can also concentrate supply risk such that multiple products are now at risk if a particular component supplier or facility fails. Tomlin and Wang [10] examines this tension by exploring dual sourcing and flexibility. Firms selling multiple products may have the option of influencing customer demand in the direction of a particular product. This demand management (or shaping) can be used during a disruption to direct customers from constrained-supply products to less constrained ones. We refer the interested reader to [9] which explores demand management and its interaction with the diversification and backup strategies.

Supplier leadtimes: we were silent on procurement (production and/or transportation) lead times in this chapter. If all suppliers have the same lead times, then the analyses presented here are perfectly adequate. However, if suppliers differ in their lead times, then a more sophisticated analysis would be required to fully capture the lead time differentials.

Other situation-specific considerations might come to light when evaluating a particular firm’s supply chain. While the models introduced in this chapter provide a good starting point for evaluating different mitigation strategies, we highly recommend that managers carefully determine the relevant considerations for their supply chain and conduct a rigorous analysis that adequately captures those considerations.

3.7 Conclusions

Supplier diversification and backup sourcing offer alternatives to stockpiling inventory as a means of mitigating disruption risks. The effectiveness of diversification depends largely on the number of suppliers, the possibility of disruption correlation, and the available spare capacity at suppliers. Most of the benefits are achieved with a small number of suppliers; oftentimes firms may find that using two or three suppliers strikes an appropriate balance between risk mitigation and supply chain rationalization. However, disruption correlation and spare capacity considerations can limit the mitigation benefit of diversification. If these concerns cannot be alleviated, firms may need to increase the number of suppliers to adequately mitigate disruption risk.

Backup supply may be an attractive alternative to supplier diversification as a means of mitigating disruption risk. The effectiveness of backup sourcing depends largely on the cost and availability of the backup source, with availability being measured as response time and capacity provided. Effective backup supply requires the firm to have agreed emergency plans and protocols in place before a disruption occurs. In addition to cost, potential backup suppliers should be evaluated along the dimensions of response time and capacity. While a firm will of course prefer faster response and higher capacity, it may be forced to make a tradeoff between these two dimensions and/or pay higher price to improve one or both. The nature of the disruption risk (short-frequent versus long-rare) influences the value of response time and magnitude, and so managers need to account for this when developing their backup plan.

The disruption profile (uptime and frequency/severity) plays a major role in determining the most appropriate mitigation strategy, with, for example, backup sourcing being appropriate for long/rare disruptions but inventory being appropriate for short/frequent ones. As there is no one-size-fits-all solution to mitigating disruption risk, firms should choose the strategy that best aligns with their internal and external operating environment, recognizing that this may mean different strategies for different parts of the business. Choosing and crafting the best strategy relies on sound judgment aligned with suitable analysis; guesswork is neither required nor recommended. Regardless of the chosen strategy, successful implementation depends on proactive planning. Selecting and validating additional or alternative suppliers can be time consuming and cannot be left until a disruption occurs. Likewise, disruption detection and notification protocols should be designed, agreed upon, and documented in advance.

We offer the following thoughts in closing. One, be prepared. As the adage goes, failing to plan is planning to fail. Senior executives should ensure that there is a systematic approach to identifying, evaluating, and managing supply risk throughout their organization. Two, be vigilent. Delays in detecting and responding to a disruption can dramatically amplify its impact, especially if competitors preempt any backup supply options. Ongoing supplier communication and threat monitoring can aid in rapid detection. Three, be flexible. Risk identification is an inexact science; a disruption may occur at unanticipated location due to some unexpected cause. Supply chains that can rapidly detect anomalies and that are flexible enough to divert flows to other parts of the network are best able to react to unforeseen events.

Appendix

In this appendix we present some of the underlying analysis for the profit expressions used in this chapter.

3.1.1 Acceptance

• Profit in a period when supplier is up: ${\it MARG}$

• Profit in a period when supplier are down: ${\it -LOST}$

• Steady-state probability supplier is up: ${\it UP}$

• Steady-state probability supplier is down: $1-{\it UP}$

• Average profit: ${\it ACC}\_{\it PROF}={\it MARG} \times {\it UP} - {\it LOST} (1-{\it UP})$

• Rearranges to: ${\it ACC}\_{\it PROF} = {\it MARG} - ({\it MARG} + {\it LOST}) (1-{\it UP}),$ i.e., Eq. 3.1.

3.1.2 Supply Diversification

3.1.2.1 Basic Model

• There are ${\it NUM}\_{\it SUP}$ identical suppliers.

• Each supplier has enough capacity to meet all demand.

• Profit in a period when at least one supplier is up: ${\it MARG}$

• Profit in a period when all suppliers is down: $-{\it LOST}$

• Steady-state probability at least one supplier is up: $1-(1-{\it UP})^{{\it NUM}\_{\it SUP}}$

• Steady-state probability all suppliers are down: $(1-{\it UP})^{{\it NUM}\_{\it SUP}}$

Average profit: ${\it DIV}\_{\it PROF}_{\it u} = {\it MARG}(1-(1-UP)^{{\it NUM}\_{\it SUP}}) - {\it LOST} (1-{\it UP})^{{\it NUM}\_{\it SUP}}$

Rearranges to: ${\it DIV}\_{\it PROF}_{\it u} = {\it MARG} - ({\it MARG} + {\it LOST}) (1-{\it UP})^{{\it NUM}\_{\it SUP}},$ i.e., Eq.  3.2.

3.1.2.2 Correlated Disruptions

We model correlated disruptions as follows. Let there be two types of disruptions: systemic disruptions and supplier-specific disruptions. A systemic disruption causes all suppliers to be down. For example, a natural diaster might shut down all suppliers if they are located in the same region. A supplier-specific disruption only causes that specific supplier to be down. For example, an internal strike might disrupt one supplier but not disrupt other suppliers. Denote the steady-state probability that the overall system is up as ${\it SYS}\_{\it UP}.$ Denote the steady-state probability that an individual supplier is down due to a supplier-specific disruption as $1-{\it SUP}\_{\it UP}.$ Therefore, all suppliers are down if either the system is down or the system is up but all suppliers are down. The probability of this event is $1-{\it SYS}\_{\it UP}+{\it SYS}\_{\it UP}(1-{\it SUP}\_{\it UP})^{{\it NUM}\_{\it SUP}}.$ Following a similar logic to that presented for the basic model above, we have ${\it DIV}\_{\it PROF}_{\it c} = {\it MARG} - ({\it MARG} + {\it LOST}) \left(1-{\it SYS}\_{\it UP}+{\it SYS}\_{\it UP}(1-{\it SUP}\_{\it UP})^{{\it NUM}\_{\it SUP}}\right).$ Now, any particular supplier is up only if the system and the supplier is up. Therefore, the steady-state probability of a supplier being up (i.e., the percentage uptime) is ${\it UP} = {\it SYS}\_{\it UP} \times {\it SUP}\_{\it UP}.$ The correlation coefficient, denoted as ${\it CORR},$ for any two suppliers being up can be shown to be ${\it CORR}=\frac{{\it SUP}\_{\it UP} (1-{\it SYS}\_{\it UP})} {1-{\it SYS}\_{\it UP} \times {\it SUP}\_{\it UP}}.$ Using these expressions to write the profit expression in terms of UP and CORR instead of ${\it SYS}\_{\it UP}$ and ${\it SUP}\_{\it UP}$ gives the ${\it DIV}\_{\it PROF}_{\it c}$ expression Eq.  3.3.

3.1.2.3 Limited Spare Capacity

We assume that order quantities are evenly split across the available suppliers. Therefore the normal volume (i.e., when all suppliers are up) sourced from a supplier is $1 / {\it NUM}\_{\it SUP}.$ (Recall that we scale demand to equal 1.) Each supplier has spare capacity to produce more than its normal volume. Spare capacity, denoted by ${\it SP}\_{\it CAP} \geq 0,$ is measured as the percentage a supplier can provide above and beyond its normal order quantity. Therefore, the maximum quantity that a supplier can produce is given by $(1 / {\it NUM}\_{\it SUP})(1+{\it SP}\_{\it CAP}).$ Let n denote the number of suppliers that are up. If $n \geq \frac{{\it NUM}\_{\it SUP}} {1+{\it SP}\_{\it CAP}},$ then these n suppliers together can meet all demand, otherwise only $\frac{n(1+{\it SP}\_{\it CAP})} {{\it NUM}\_{\it SUP}}$ of demand can be filled, and the remainder is lost. Define $\hat{n} = \left\lceil\frac{{\it NUM}\_{\it SUP}} {1+{\it SP}\_{\it CAP}}\right\rceil$ where $\lceil x \rceil$ is the smallest integer greater than or equal to x (i.e., round up x to the nearest integer). Therefore, the profit in a period for which $0 \leq n < \hat{n}$ suppliers are up is ${\it MARG} \times \frac{n(1+{\it SP}\_{\it CAP})} {{\it NUM}\_{\it SUP}}-{\it LOST} \times \left(1-\frac{n(1+{\it SP}\_{\it CAP})} {{\it NUM}\_{\it SUP}}\right),$ and the profit in a period for which $n \geq \hat{n}$ suppliers are up is MARG. As before (see correlated disruptions above), the steady state probability that $n=0$ suppliers are up is $1-{\it SYS}\_{\it UP}+{\it SYS}\_{\it UP}(1-{\it SUP}\_{\it UP})^{{\it NUM}\_{\it SUP}}.$ The probability that exactly $0 < n \leq {\it NUM}\_{\it SUP}$ of the ${\it NUM}\_{\it SUP}$ suppliers are up is ${\it SYS}\_{\it UP} {\it NUM}\_{\it SUP}{n} ({\it SUP}\_{\it UP})^{n} (1-{\it SUP}\_{\it UP})^{{\it NUM}\_{\it SUP}-n}.$ (These probabilities can be expressed in terms of UP and CORR instead of ${\it SYS}\_{\it UP}$ and ${\it SUP}\_{\it UP}$ by using the transformations shown above.) Knowing the probability of there being exactly n suppliers up and knowing the associated profit, the average profit can then be evaluated. A closed-form expression for the profit will not typically exist, however.

Consider the special case where ${\it SP}\_{\it CAP} = 0.$ Then, $\hat{n}={\it NUM}\_{\it SUP},$ and when $0 \leq n \leq {\it NUM}\_{\it SUP}$ suppliers are up, the profit is ${\it MARG} \times \frac{n} {{\it NUM}\_{\it SUP}}-{\it LOST} \times \left(1-\frac{n} {{\it NUM}\_{\it SUP}}\right),$ or equivalently, $\frac{n} {{\it NUM}\_{\it SUP}} ({\it MARG}+{\it LOST}) - {\it LOST}.$ The long-run average profit is then

$$ {\it DIV}\_{\it PROF}_{\it ns} = \sum_{n=0}^{{\it NUM}\_{\it SUP}}\left(\frac{n} {{\it NUM}\_{\it SUP}} ({\it MARG}+{\it LOST}) - {\it LOST}\right) P[n], $$

where P[n] is the probability of n suppliers being up. Rearranging terms gives ${\it DIV}\_{\it PROF}_{\it ns} = \left(\frac{{\it MARG}+{\it LOST}} {{\it NUM}\_{\it SUP}}\right) \times \sum_{n=0}^{{\it NUM}\_{\it SUP}}\left(nP[n]\right) - {\it LOST}.$Now, $\sum_{n=0}^{{\it NUM}\_{\it SUP}}{\it nP}[{\it n}]$ is the average number of suppliers up, and this equals ${\it NUM}\_{\it SUP} \times {\it UP}.$ Substituting this into the profit expression, we then have ${\it DIV}\_{\it PROF}_{\it ns} = ({\it MARG}+{\it LOST}){\it UP} - {\it LOST}= {\it MARG}-({\it MARG}+{\it LOST})(1-{\it UP}).$

3.1.3 Backup Supply

The backup supplier is not available until ${\it EM}\_{\it RES}$ periods of a disruption have passed, after which the backup can provide a capacity of $0 \leq {\it EM}\_{\it CAP} \leq 1.$ Units sourced from the backup supplier give a profit of ${\it EM}\_{\it MARG}<{\it MARG},$ where MARG is the profit obtained using the regular supplier. Therefore, the profit in a period when the regular supplier is up is MARG. If the regular supplier has been down ${\it EM}\_{\it RES}$ periods or less, then the profit in that period is $-{\it LOST}$ as the backup supply has yet to come on stream. If the regular supplier has been down for more than ${\it EM}\_{\it RES}$ periods, then the profit in that period is ${\it EM}\_{\it MARG} \times {\it EM}\_{\it CAP}-{\it LOST}\times (1-{\it EM}\_{\it CAP}),$ as the backup can only supply ${\it EM}\_{\it CAP}.$ Let $P[t]$ represent the steady-state probability that the regular supplier has been down for $t=1, 2, 3, \ldots$ periods. Then, the average profit is

$$ \begin{aligned} {\it EM}\_{\it PROF}&= {\it MARG}\times {\it UP} - \sum\limits_{t=1}^{{\it EM}\_{\it RES}}{\it LOST} \times P[t] \\ &\enskip\quad+\sum\limits_{t={\it EM}\_{\it RES}+1}^{\infty}\left({\it EM}\_{\it MARG} \times {\it EM}\_{\it CAP}-{\it LOST}\times (1-{\it EM}\_{\it CAP})\right) P[t]. \end{aligned} $$

To evaluate this profit expression, we need to know $P[t],$ the steady-state probabilities of being up and of being down for $t=1, 2, 3, \ldots$ periods. This requires us to make an assumption about the disruptions. For simplicity, we assume that (1) when the primary supplier is up there is a constant probability, denoted by FAIL, of a disruption occurring, and (2) when the primary supplier is down there is a constant probability, denoted by RECOVER, of the disruption ending. With these assumptions, it can be shown that ${\it UP}=\frac{{\it RECOVER}} {{\it RECOVER}+{\it FAIL}}$ and $P[t]=(1-{\it UP})({\it RECOVER})(1-{\it RECOVER})^{t-1}.$ Substituting these into the profit expression above, followed by some algebra, gives

$$ \begin{aligned} {\it EM}\_{\it PROF}&={\it MARG} - ({\it MARG} + {\it LOST}) (1-{\it UP}) +\\ &\enskip\quad({\it EM}\_{\it MARG} + {\it LOST}) (1-{\it UP}) ({\it EM}\_{\it CAP}) \left(1-{\it RECOVER}\right)^{{\it EM}\_{\it RES}}, \end{aligned} $$

which is the same as Eq.  3.4 because the average disruption length is given by ${\it DIS}\_{\it LEN}= 1 / {\it RECOVER}.$

3.1.4 Inventory Model

In the inventory strategy, the firm single sources but stockpiles a quantity of inventory, denoted by INV, to use during a disruption. There is a per-unit cost of HOLD to hold inventory for a period and the inventory is accounted for at the end of a period. For simplicity we assume the supplier has infinite capacity and, therefore, the inventory returns to INV as soon as a disruption ends. Therefore, the firm has INV units of inventory at the start of any disruption. The firm can fill demand from production if the supplier is up or from inventory if the supplier has been down for less than or equal to INV consecutive periods. (Recall that we scale demand equal to 1.) Demand cannot be filled if the supplier has been down for more than INV consecutive periods. Therefore, the profit in a period in which the supplier is up is ${\it MARG}-{\it HOLD} \times {\it INV}.$ The profit in a period in which the supplier has been down for $t \leq {\it INV}$ periods is ${\it MARG}-{\it HOLD} \times ({\it INV}-t)$ as t units of inventory will have been used to fill demand. The profit in a period in which the supplier has been down for $t > {\it INV}$ periods is $-{\it LOST}$ as the inventory is gone and demand cannot be filled. Then, the average profit is

$$ \begin{aligned} {\it INV}\_{\it PROF} &= ({\it MARG}-{\it HOLD} \times {\it INV}) {\it UP} \\ & \enskip\quad+ \sum_{t=1}^{{\it INV}}\left({\it MARG}-{\it HOLD} \times ({\it INV}-t)\right) P[t] - \sum_{t={\it INV}+1}^{\infty}{\it LOST} \times P[t]. \end{aligned} $$

To evaluate this profit expression, we need to know $P[t],$ the steady-state probabilities of being up and of being down for $t=1, 2, 3, \ldots$ periods. For simplicity, we will assume geometric disruptions as we did for the backup strategy. Therefore, ${\it UP}=\frac{{\it RECOVER}}{{\it RECOVER}+{\it FAIL}}$ and $P[t]=(1-{\it UP})({\it RECOVER})(1-{\it RECOVER})^{t-1}.$ Substituting these into the profit expression above, followed by some algebra, gives

$$\begin{aligned}[b] {\it INV}\_{\it PROF} &={\it MARG} - {\it HOLD} \times {\it INV} + (1-{\it UP}) \Bigg( {\it HOLD} \times {\it DIS}\_{\it LEN} -\\ & \quad \left({\it MARG} + {\it LOST} + {\it HOLD} \times {\it DIS}\_{\it LEN} \right) \left(1-\frac{1} {{\it DIS}\_{\it LEN} }\right)^{\it INV} \Bigg), \end{aligned} $$

(3.5)

where we have also used the fact that ${\it DIS}\_{\it LEN}= 1 / {\it RECOVER}.$ In deciding how much inventory to stockpile, the firm faces a tradeoff between increasing sales during a disruption by carrying more inventory and incurring the additional cost for holding more inventory. Using the ${\it INV}\_{\it PROF}$ expression above, the value of ${\it INV}$ that maximize the firm’s profit is given by ${\it OPT}\_{\it INV} = \max \left\{ 0, \lceil \frac{{\it ln A}} {ln B} \rceil \right\}$ where $A=\frac{{\it HOLD}} {(1-{\it UP}) \left({\it HOLD} + \frac{{\it MARG} + {\it LOST}} {{\it DIS}\_{\it LEN}}\right)},$ $B=1-\frac{1} {{\it DIS}\_{\it LEN}},$ and $\lceil x \rceil$ is the smallest integer greater than or equal to x (i.e., round up x to the nearest integer).