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Mitigation Options for Supply Chain DisruptionsJesse Sowell

Abstract

Just-in-time manufacturing (JIT) introduces flexibility to more accurately meet dynamic demand requirements. The simple form of JIT exclusively maximizes cycle time and inventory levels, but is detrimental to resilience and recovery planning concerns that arise in the face of unexpected disruptions. Many of these problems are a direct consequence of having little to no warehousing capacity under the assumption that upstream materials will be consumed and shipped at an expected rate, typically the rate of production. An interesting problem in JIT supply chains is the dependence on second order effects. The availability of packaging and its affect on distribution capabilities is an instance of these second order effects. The scenario presented here is the case where the packaging supply network is disrupted, but the supply networks for the primary materials necessary for product manufacture are not affected. This finds the manufacturer with the capability to continue producing the product but unable to distribute it without proper packaging. Furthering the manufacturer’s conundrum is, stopping production could create a bullwhip effect that moves upstream. The manufacturer is now faced with problems at both ends: failure to meet downstream service requirements and the potential to create a bullwhip effect upstream, possibly affecting contractual agreements there as well.

As a first step to understanding this problem, possible real options analyses are applied. In particular, two possible real options are explored. The first is the application of functional symmetry, allowing for the redistribution of packaging manufacturing to other facilities in the event of a disruption. The second solution is the introduction of “on-demand” warehousing as a form of insurance that is intended to cost less than conventional warehousing but allows manufacturers to store a small amount of product while packaging manufacturing capacity is restored. These are evaluated using decision analysis and lattice analysis, followed by a discussion of the strengths and weaknesses of each of these approaches as applied to the unique nature of supply network disruptions.

Note on data: The problem described here is based on an ongoing research project. The problem is real, but the data used is fabricated because (1) the actual data is not available yet and (2) the fabricated data simplifies the problem.

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Table of Contents....................................................................................................................................................Introduction 3

..............................................................................................................Disruption Types and Uncertainties 5

.........................................................................................................................................Natural Disasters 5.......................................................................................................Natural Disaster as a Discrete Event 6

................................................................................................................................................Magnitude 6......................................................................................................................................Mitigation Costs 7

...............................................................................................Defects as a Continuously Recurring Events 7

..........................................................................................................................Two Stage Decision Analysis 9

..............................................................................................................................Defining System Designs 9

....................................................................................................................................Standard Operations 9

...................................................................................................................................................Disruptions 9

..................................................................................................................................Mitigation Strategies 10....................................................................................................Functional Symmetry (Fixed option) 10

...........................................................................................On-demand warehousing (Flexible option) 11

......................................................................................................................Results of Decision Analysis 12

............................................................................Lattice Valuation of Continuous Packaging Disruption 15

......................................................................................................................................................Discussion 19

..........................................................................................................Disruptions and Path Independence 19

.........................................................................................................Avoidance and Mitigation Strategies 20

........................................................................................................................The Role of Hybrid Models 20

................................................................................................................................Potential Future Work 20

...........................................................................................................................Appendix A: Decision Tree 22

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1 Introduction

“Just-in-Time” inventory management has emerged as a very efficient, lucrative, and competitive supply chain methodology. The essence of just-in-time (JIT) is to streamline the supply and manufacture process through strategies such as minimizing component stock and introducing the flexibility necessary to more accurately meet real demand. It is an excellent example of an analysis and implementation where flexibility options “in” the system serve to maximize efficiency. The problem that arises with the simple form of the JIT methodology is that it exclusively maximizes efficiency in terms of cycle time and inventory management. This is in contrast to conventional security and resilience strategies that prescribe managing backup stock for use in the event of a disruption. As such, JIT efficiency is detrimental to resilience and security concerns that arise in the face of unexpected disruptions.

The uncertainty lies in the possible disruption of the packaging supply network. This is an interesting problem because it illustrates how a second-order parameter (product packaging) can affect the distribution of the primary product. Even though the distribution channels for the product itself have not been disrupted, distribution cannot move forward without proper packaging. The aspect of this problem that makes it a real option is that the supply of the primary materials for making the product itself are operating as normal. Manufacture of the product itself can continue without interrupting these upstream supply flows or the manufacturing process itself. Further, if manufacturing is stopped due to inability to distribute, there is also the risk of causing a bullwhip effect upstream. In this scenario, packaging is necessary to distribution, but not to manufacture. Under JIT as applied to this product, packaging stock matches expected production and will be exhausted quickly in the event of a disruption to the packaging supply chain.

To understand how different types of disruptions to the packaging supply network affect, two types of disruption event are modeled. The first type of disruption is a discrete, infrequent event, such as an earthquake. These are referred to as discrete disruption events. The second type of disruption event is the aggregate of many small disruptions that occur over a period. These are referred to as continuous disruption events.

This document proposes a simple analysis of these two categories of disruption events and possible options that could help preserve service levels. Two options are presented to mitigate the effects of a disruption event. Under the first, lost capacity will be redistributed to other facilities that (a) have the flexibility to perform the same task (functional symmetry) and (b) can be run overtime to make up for the capacity of the disrupted resource(s). Under the second, on-demand warehousing is introduced as a form of insurance that is intended to cost less than conventional warehousing but allows manufacturers to store some amount of product unpackaged product while packaging manufacturing capacity is restored. In contrast to the conventional strategy of

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maintaining warehousing and backup stock of packaging materials, these options attempt to introduce dynamic options that preserve service levels.

It is important to stress that these options are mitigation strategies, not avoidance strategies. Mitigation strategies assume that the disruption cannot be avoided. The objective of the strategy is to understand the nature of the disruption and, based on this information, minimize the costs necessary to recover from that disruption. Part of this process is the proactive investment in options that can be exercise when a disruption occurs and that ultimately reduce recovery costs. This model only focuses on reducing costs of a disruption. An interesting, but different problem, is reducing the duration of the disruption.

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2 Disruption Types and Uncertainties

A discussion of disruptions requires some disambiguation regarding the causes and effects of the disruption:

• Disruption event - the event that causes a disruption, such as an earthquake or packaging defects

• Disruption event magnitude - describes how severe a disruption event is• Disruption duration – amount of time packaging manufacturing capacity is not

available due to a disruption event

The assumption here is that duration is a direct function of magnitude1. Duration is used as a proxy for the magnitude of an event. Duration is thus the amount of time the packaging manufacturing capacity is unavailable. The packaging manufacturing capacity lost due to a disruption will be referred to as disrupted capacity. It is important to distinguish between the disruption event and the duration of the disruption. The disruption event and magnitude is the cause whose effects are disrupted capacity for some amount of time (the duration).

The two types of disruptions discussed here are discrete (for example, natural disasters like earthquakes) or continuous (for example, quality-based disruptions such as defects). The former, discrete disruptions, are discrete events—the event does not happen in each period and is essentially “instantaneous”. Discrete disruptions are modeled as path dependent. The latter, continuous disruptions, is an aggregate level of disruption (measured in duration) associated with each period. A continuous disruption is the result of a set of many small disruptions throughout the period, but whose aggregate level can be thought of as having a rate of occurrence and having a growth rate. As such, continuous disruptions can be modeled as path independent.

Costs in these scenarios fall into two categories: those intrinsic in the disruption event itself and those that arise in the subsequent recovery efforts. These uncertainties and costs will be referred to as disruption costs and uncertainties and mitigation costs and uncertainties, respectively.

2.1 Natural DisastersFor the project, the discrete disruption event is a natural disaster; the easiest example is an earthquake. Natural disasters follow the Power Law; distributions of accidents can be modeled using data from actual incidents and near misses. For this project, a simple distribution (see Figure 1) was generated to represent the categories and magnitudes of discrete disruption events.

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1 This is a gross oversimplification; in reality duration is a function of both magnitude and a number of exogenous factors that affect restoring lost capacity.

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2.1.1 MagnitudeNatural disasters follow a power law distribution. In the case of disruptions, this means that a small proportion of disruptions account for a large proportion of the magnitude effected and vice versa. This is generally likened to the 80-20 rule. If 80-20 is in fact the proportion2, 20% of events account for 80% percent of the effects (magnitude of disruption in this case). The other 20 percent of total magnitude is accounted for by 80 percent of the disruptions. This type of distribution is illustrated in Figure 1.

Figure 1 also illustrates the relationship between magnitude and frequency. High magnitude (measured in duration) events are category 1 and 2, quickly decaying to substantially smaller magnitude events in categories 3-6, events of in category 7-15, and those less than ~2 days in category 16-20. For this model, the four categories of event types will be high (event types 1-2), medium (event types 3-6), low (event types 7-15), and none (event types 16-20). Technically, in the none category does have a magnitude, but is set to zero to indicate that disruptions of less than two days are not significant

because there is sufficient capacity to store 2 days worth of product at the manufacturing facility. The probabilities of a discrete disruption will correspond to the simple proportion of number of event types to total events (20). Table 1 summarizes these. The following section describes the factors that contribute to the recovery efforts.

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2 It may be 70-30, or even 15-60. The ratio does not have to be 80-20 or even add up to 100, although it must be less than 100.

Simple Natural Disaster Distribution

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Frequency ranking

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Figure 1 A simple, fabricated distribution of disruptions whose magnitude is measured in days

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2.1.2 Mitigation CostsThe components of the recovery effort are initial investments in mitigation alternatives (option premiums) and recovery operating costs. Different firms have different levels of costs but these two broad categories cover recovery efforts for the purpose of this analysis.

Investments in mitigation may incur costs (and uncertainties in costs) related to equipment update, upgrade or replacement costs to introduce functional symmetry or the optional capacity (warehousing) necessary to handle a predetermined class of disruptions. Investments may also include initial training and planning. The actual operating costs may have a great deal of uncertainty, depending on the type of disruption, disruption magnitude, disruption duration, equipment affected, availability of replacement equipment, availability of additional resources, and availability of labor. Many of these factors do not manifest in the simple models described in sections 3 or 4, but will be discussed when evaluating the models presented here in contrast to more sophisticated models such as simulation or hybrid models.

A number of exogenous factors (outside the enterprise) may also affect recovery operations. The market of materials in the markets of the supporting factories, the availability and cost of overtime labor, and the costs of potentially shifting materials routes from delivering to the disrupted plant to delivering to the supporting plants are all factors that will affect the cost and feasibility of mitigation alternatives.

2.2 Defects as a Continuously Recurring EventsDefects in packaging received from the packaging supplier reduce overall throughput (and subsequently affects the service level) because additional packaging may need be requested and some amount of product shipping will be delayed. In a JIT inventory management system, this creates problems because product cannot be stored for a long period of time.

A continuous disruption is a set of recurring events whose effects are aggregated over a period and has a net effect that may grow or shrink at some rate over the course of a number of periods. In this case, the individually recurring defects have the same net effect as 5 days of disruption. As such, there is a rate of disruption per period that results

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Category Event Types Magnitude (days) ProbabilityHigh 1-2 28.09 0.10

Medium 3-6 8.94 0.20

Low 7-15 3.04 0.45

None 16-20 0 0.25

Table 1 Categorization of discrete disruption events

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in ~6 days of disruption per period. Over the sequence of multiple periods, the number of defects per period may increase or decrease (growth or decay).

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3 Two Stage Decision Analysis

The following describes how a two-stage decision analysis can be applied to evaluate discrete disruptions.

3.1 Defining System DesignsThis decision analysis will describe two options for mitigating packaging supply chain disruption. This analysis also includes a third “do nothing” case that illustrates the cost of simply stopping production and foregoing sales for the duration of the disruption. The two mitigation plans are (1) an investment in functional symmetry and (2) on-demand warehousing. An important note regarding these plans is that they are literally mitigation plans, not avoidance plans. The natural disasters that cause these disruptions are considered unavoidable. These mitigation plans are intended to provide a set of second best solutions that will satisfy second-best service levels until the disrupted capacity is restored.

3.2 Standard OperationsThe standard operating procedure is to produce 5 million widgets a month. The total costs for a month is $2 M. The total sales yields $3 M per month. Thus, the total profit is $1 M a month.

For this model, a month is 30 days. One day’s worth of production costs 3 1/3% of the total costs or ~$67,000. Overtime costs are 1.5 that of normal operations, so the cost of a single day disruption is $100,000 and will be referred to as a nominal unit of disruption.

3.3 DisruptionsThe model for the total cost of disrupted capacity is based on the cost of the total units of disrupted capacity experienced for a given disruption. The base cost is the number of days of downtime and is equal to the cost of the corresponding number of units of disrupted capacity. It is assumed that additional supporting capacity is in overtime costs; the cost of supporting capacity is thus 1.5 times the cost of disrupted capacity, $100,000. The base cost of disrupted capacity is the sum of disrupted cost and supporting costs.

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Category Downtime Cost calculation Approximate Base cost ($)high 28.09 28.09(1.5)($67,000) 2,809,000.00

medium 8.94 8.94(1.5)($67,000) 894,000.00

low 3.04 3.04(1.5)($67,000) 304,000.00

none 0 0(1.5)($67,000) 0.00

Table 2 Costs of disruption categories

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As noted, disruptions are categorized by magnitude (measured in the resulting days of downtime) as none, low, medium, or high. Table 2 describes each level of disruption, the number of days of downtime corresponding to that level, the cost calculation, and the approximate base cost for each level of magnitude. When decomposing recovery costs, the costs in Table 3 emerge. Some, but not all, apply to each mitigation strategy.

Repair costs and replacement costs are the same for both strategies. This model does not include these in the cost calculations. Mitigation operation costs and transportation costs differ. These differences will be discussed in the context of each mitigation strategy.

3.4 Mitigation StrategiesThere are two mitigation strategies, each with their own associated costs above and beyond the base cost of disruption. Functional symmetry is a fixed option whose associated costs are an upfront investment in common manufacturing equipment and processes and whose variable costs are largely overtime costs and the costs of redistributing disrupted capacity to supporting manufacturers. On-demand warehousing is a flexible option that allows for temporary storage of primary product until disrupted capacity can be brought back online.

3.4.1 Functional Symmetry (Fixed option)An investment in functional symmetry is a fixed investment that ensures packaging manufacturers implement the same processes with the same equipment. Although conceivable to introduce functional symmetry incrementally, it would require retooling packaging manufacturing facilities multiple times. The retooling process is itself a service disruption—in the context of a JIT manufacturer, even well planned disruptions can have adverse consequences. It is possible to stagger the introduction on a case by case basis, but that is beyond the scope of this project. Introducing functional symmetry

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Cost Description

mitigation operations Cost of running supporting factories overtime to meet mitigation service levels. Mitigation service level is the service level an organization determines is sufficient to preserve business continuity until disrupted capacity is fully restored.

repair costs Costs of repairing damaged equipment and infrastructure

replacement costs Costs of replacing irreparable equipment and infrastructure

transportation costs Transportation costs necessary to reroute raw materials for packaging from the original destination to supporting manufacturers.

Table 3 Components of recovery costs

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is simplified to a one-time capital cost. This initial capital cost includes equipment and training. For this exercise the initial outlay is $0.5 M.

Mitigation operations costs are the supporting costs (1.5 times disrupted capacity cost) plus the cost of running two different types of packaging from the same factory. This will be calculated similar to other costs, as a constant cost applied per day. For this exercise, it will be 0.1 disrupted capacity costs per day. These are essentially scaling factors used to map mitigation operations costs back to a common unit of cost.

In the case of functional symmetry, the transportation costs are the costs of rerouting raw packaging materials to supporting manufacturers and then transporting the finished packaging to the primary facility. This is relatively expensive compared to other, optimized logistics operations because the packaging manufacturers are typically located near the primary product manufacturing facility to minimize these costs. For this exercise, the cost of additional transportation will be 0.2 disrupted capacity costs per day.

3.4.2 On-demand warehousing (Flexible option)The flexible investment option is the purchase of on-demand warehousing, which is akin to insurance. On-demand warehousing is optional storage space provided by a third party. In this simple model a month-by-month contract is purchased for a given time to cover expected need. For example, one can purchase a contract that will guarantee a warehousing capacity sufficient to accommodate a given number of days of production. For example, 6 days of capacity will provide the space necessary to store 6 days worth of unpackaged product. Ideally, this contract would be flexible enough to be increased or decreased at the end of each period. For simplicity, the contract is modeled as having two levels, 6 days and 9 days. These levels are referred to as on-demand capacity. This is essentially an option that allows one to upgrade from the 6 to 9 day plan.

On-demand warehousing differs from functional symmetry in the transportation costs. It is assumed that the warehousing is nearby and transportation is minimal. In this scenario the product is simply stored. Production operations proceed until packaging manufacturing is restored or the on-demand capacity is met. Once back online, packaging manufacturing runs at 1.5 times capacity until the stored product is shipped. In the case of on-demand warehousing, there is a 25% loss in sales due to delays.

Each month, a premium of $75,000 covers warehousing capable of storing 6 days worth of production for 18 days (6 days of disruption plus 12 half days to catch up on lost time and distribute the stored inventory). If selected, the second month can be extended to cover 9 days of production for 27 days (9 days of disruption plus 18 half days to distribute) for a premium of $125,000.

The cost of a disruption under the on-demand warehousing plan is contingent on the duration of the disruption and level of on-demand capacity. If the disruption duration is

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less than the level of on-demand capacity, the mitigation cost is a nominal unit of disruption per day plus the additional costs outline above. If the disruption duration is greater than the level of on-demand capacity, the mitigation cost is based (1) on the cost nominal disruption cost of the total on-demand capacity (6 or 9 days) and (2) the amount of time disrupted over and above the on-demand capacity is a total loss (production stops). For instance, a high disruption may result in ~22 days worth of total capacity lost with the 6 day contract and ~19 days lost with the 9 day contract.

3.2 Results of Decision AnalysisThe decision tree is broken into three branches: functional symmetry (the fixed option), on-demand warehousing (the flexible option), and the do nothing case. The decision tree for this scenario is fairly large and is displayed in three figures in Appendix A.

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Branch Expected Value

Do Nothing $0.81 M

Functional Symmetry (fixed) $0.12 M

On-demand warehousing (flexible) - $1.07 M

Table 4 Expected Value of the do nothing option and mitigation options

Value at Risk and Gain for Discrete Disruptions

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Figure 2 Value at Risk and Gain for Discrete Disruptions, note the shape of the graph looks much like a power law distribution

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The do nothing costs less than the mitigation options. Of the two mitigation options, the optimal is functional symmetry, the fixed option. The expected values of each branch shown in Table 4. The value at risk and gain graph for this scenario is presented in Figure 2.

Not surprisingly, the shape of the S-curve in Figure 2 is similar to the shape of a power law curve. The largest losses appear in the left tail. This is correlated to the probability of high magnitude (duration) disruptions. As we move to the right, the more probable, lower duration disruptions show lower losses. As indicated above, the do nothing case is optimal, the whole curve being shifted to the right. Second is the fixed option (functional symmetry) and third is the flexible option (on-demand warehousing).

One reason the optimal strategy seems to be the do nothing case is because the mitigation strategies incur additional costs to reduce the impact of disruptions on service levels. This increases costs, but does preserve service levels. Comparing just the two mitigation plans, functional symmetry and on-demand warehousing, functional symmetry is superior. This is illustrated in both the VARG (Figure 2) and the expected values (Table 4).

Comparing the fixed and flexible options, the fixed option is the optimal solution (of the mitigation strategies) in this case. As a mitigation strategy, it saved approximately $1 M more than the on-demand warehousing. The intuitive reason for this is that it keeps production going. The cost for functional symmetry is the capital investment in common equipment and processes. The mitigation costs are transportation and overtime, but the functional symmetry option maintains the required service levels.

The flexible option (on-demand warehousing) is also fairly limited: the manufacturer can store 6 or 9 days of product but cannot ship it until the packaging manufacturer is back online. This delay is represented in a 25% loss of sales. Also, once the duration is greater than the on-demand capacity, production must stop until the packaging manufacturing comes back online. A more sophisticated model might increase the percentage of lost sales based on duration of disruption based on historical data on sales loss due to delays.

The positive effects of mitigation strategies can also be interpreted in terms of the power law distribution in Figure 1. Recall this power law distribution used for this project measures disruption magnitude in terms of duration. Thinking of the impact of the disruption only in terms of the base cost of disruption, valued in units (days) of disrupted capacity, the distribution in Figure 1 could also be interpreted as the base costs for a disruptions of a given magnitude. The objective of a mitigation strategy is to shift the entire curve down. Comparing this interpretation of Figure 1 with the shift in Figure 2, it is arguable that the effects of mitigation strategies to shift the curve to the right (in

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particular, comparing the fixed and the flexible options) also effectively shifts the curve in Figure 1 down, representing a decrease in the mitigation costs.

Another limitation of this model as a whole is that it only considers local operating costs and sales. In this case, local costs are only those experienced by the manufacturer, not by the enterprise as a whole, to which ongoing operations and service levels may be more valuable in the long run than the cost of either option or the immediate impact of lost sales. Supply chain operations are cross cutting—there are many more dependencies that exist outside the manufacturing plant and its local operation.

Following this a bit farther, a more complete model would incorporate a number of enterprise level variables that are technically external to the manufacturing plant. One example is the contractual obligations to fulfill service requirements. If a disruption causes a breach in this service level agreement, the losses are substantially more than simply lost profits for that period. Some contracts include fines for late delivery of products, others may simply cancel the contract altogether.

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4 Lattice Valuation of Continuous Packaging Disruption

The other type of disruption evaluated here is continuous disruptions. In the case of packaging disruptions, rather than a single discrete event that may or may not happen, there is a history of packaging disruptions that occurs each period at a recognizable rate. In this case, the rate of continuous disruptions is 5 days per period. The proposed solution to this problem is the purchase and maintenance of on-demand warehousing that can be increased from 6 days to 9 days depending on how much is deemed necessary based on the expected values for the next period.

A lattice is used to model this scenario. The value that evolves in the lattice is the number of days disrupted. The parameters that describe the continuous disruption (defects) modeled in this section are shown in Table 5.

The growth rate of -4% is intended to indicate the packaging manufacturer has identified the problem and is incrementally making progress to resolve the issue. The 15% standard deviation is intended reflect the difficulty in pinpointing the problem. Given these parameters, the following equations were used to define the lattice parameters:

u = e exp [ σ √(Δt) ]

d = e exp [ - σ √(Δt) ]

p = 0.5 + 0.5 ( v/σ )( √Δt )

where

u per period increment

d per period decrement

t period – one month

σ volatility (standard deviation)

p probability of increment

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Parameter Value

Disruption rate 5 days per month

Growth rate -4% per month

Volatility (standard deviation) 15%

Table 5 Lattice values for continuous disruptions

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v growth rate

Table 6 shows the resultant lattice parameters used to predicat total duration of continuous disruptions, starting at a rate of 5 days per month. The resulting lattice of days of packaging disruption is provided in Table 7.

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Period 0 Period 1 Period 2 Period 3 Period 4 Period 5 Period 6

5.00 5.81 6.75 7.84 9.11 10.59 12.30

4.30 5.00 5.81 6.75 7.84 9.11

3.70 4.30 5.00 5.81 6.75

3.19 3.70 4.30 5.00

2.74 3.19 3.70

2.36 2.74

2.03

Table 7 Lattice evolution of duration of continuous disruption, measured in total days of disrupted capacity

Period 0 Period 1 Period 2 Period 3 Period 4 Period 5

NO / 6 YES / 9 YES / 9 YES / 9 YES / 9 YES / 9

NO / 6 NO / 6 YES / 9 YES / 9 YES / 9

NO / 6 NO / 6 NO / 6 YES / 9

NO / 6 NO / 6 NO / 6

NO / 6 NO / 6

NO / 6

Table 8 On-demand warehousing upgrade decision lattice

probability up (p) 0.37

probability down (1-p) 0.63

upside increment (u) 1.16

downside decrement (d) 0.86

Table 6 Lattice parameters for continuous disruptions

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The objective of the on-demand warehousing (option) is to implement a put option. The demand of warehousing necessary for the next period is based on the projected days of disruption (expected value). This is used determine whether 6 days of on-demand warehousing is sufficient for the expected disruption or whether the organization should upgrade to the 9 day package. Table 8 contains the resulting decision lattice. The format is the decision to increase and the number of days of on-demand warehousing for that period.

As described when the on-demand warehousing option was introduced, the costs of a disruption is affected in part by having the appropriate level of warehousing available to store product rather than having to stop production and take a more substantial loss. In this case, the option increases the amount of coverage (indicated by “Yes / 9” in Table 8). This has the effect of either eliminating or decreasing the days for which there is no warehousing space. The difference in days of disruption with and without flexibility can be seen in the Tables 9 and 10. Table 9 shows the number of days over the 6 day on-demand capacity the projected disruptions will run. Table 10 shows the number of days

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Period 0 Period 1 Period 2 Period 3 Period 4 Period 5

0.00 0.00 0.75 1.84 3.11 4.59

0.00 0.00 0.00 0.75 1.84

0.00 0.00 0.00 0.00

0.00 0.00 0.00

0.00 0.00

0.00

Table 9 Difference between projected disruption duration and days covered by initial (fixed) on-demand warehousing contract (6 days)

Period 0 Period 1 Period 2 Period 3 Period 4 Period 5

0.00 0.00 0.00 0.00 0.11 1.59

0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00

0.00 0.00 0.00

0.00 0.00

0.00

Table 10 Difference between projected disruption magnitude (in days) and days covered by flexible on-demand warehousing contract (6 or 9 days, see corresponding cell in Table 8)

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over the on-demand capacity assuming the option to buy more warehousing. Note the flexible option reduces the scenario where there are days over the on-demand capacity to only the two worst case scenarios in periods 4 and 5.

Finally, pushing these disruption magnitudes through the cost function for these manufacturing options results in the lattices in Tables 11 and 12. Table 11 is the cash flows without the option to expand (fixed), Table 12 is the cash flows with the option to expand (flexible).

As illustrated by the expected net present values (ENPV) in the grey cells of Tables 11 and 12, the flexibility does in fact decrease losses. The value of the option is ~$0.6 M. These lattices also illustrate the general idea behind a mitigation option: losses are not eliminated, but they are reduced. For disruptions of substantial magnitude (the upper right portions of the above two lattices), there will be inevitable losses. Making investments in flexible options that allow the system to “absorb the shock” better can reduce, but not necessarily eliminate the losses.

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418,511 152,308 239,296 759,587 1,376,844 2,093,995 1,276,987

626,137 418,511 152,308 239,296 759,587 753,570

804,842 626,137 418,511 152,308 143,568

958,655 804,842 626,137 208,333

1,091,044 958,655 413,519

1,204,991 565,524

678,132

Table 11 Expected net present value of on-demand warehousing option with fixed contract (6 days)

1,001,457 247,094 503,229 1,118,218 1,551,495 1,759,904 1,401,987

1,364,739 711,449 109,680 407,241 670,926 578,570

1,535,339 940,597 417,211 17,436 193,638

1,534,132 969,019 482,955 133,333

1,373,693 815,474 338,519

1,061,809 490,524

603,132

Table 12 Expected net present value of on-demand warehousing option with flexible contract (6 or 9 days)

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5 Discussion

The following discussion describes the value gained from modeling packaging supply network disruptions using real options methods. From the outset, this project was about both applying real options and gaining a better understanding of the two types of disruption, discrete and continuous. As a method for exploring the problem by structuring options and identifying solutions, the real options approach proved valuable and has given the author new insights into the immediate problem, related issues, and the overall approach.

The following discussion first contrasts decision analysis and lattice based analysis, clarifying which is most appropriate for realistic approaches to these problems. In particular, the lattice analysis, while unrealistic for discrete disruptions, does provide an excellent illustration of the difference a mitigation strategy can make. This type of result, especially the demonstration that the costs of a disruption, while unavoidable, can be mitigated with appropriate options investments, is a very useful tool for risk analysis and making the argument for investing in processes with no conventional return on investment structure. Following this discussion, an outline of an ideal method will be discussed, along with the hybrid nature of the method necessary to implement it. Finally, the discussion concludes with future work.

5.1 Disruptions and Path IndependenceFor exploratory purposes, this project modeled disruptions as both discrete, instantaneous events that may or may not happen during a given period and continuous events that did happened every period and were governed by a rate of occurrence. In the case of natural disasters, the former, discrete events, is the reality. A common misconception of the power law is that it tells you something about when “the next big one” is going to occur. In reality, small events do indicate a greater likelihood of a larger event, but that event could occur any time. As such, the sequence of these disruptions and their effects on cash flows is path dependent.

From the outset, this project assumed natural disasters as the disruption. While these are one category of disruption, appealing to the broader sense of the term, other, more systematic events can also cause disruptions. In this project, rather than assuming the continuous disruption event is some form of natural disaster, the continuous event is the aggregate number of defective packages received from a supplier where losses are in terms of operations costs necessary to acquire replacements. Defects in products are commonly tracked by quality management engineers and have associated rates of occurrence, variance, and growth and decay rates suitable as parameters to lattice based analyses. As such, systematic disruptions are prime candidates for lattice analysis and loss mitigation strategies.

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5.2 Avoidance and Mitigation Strategies The introduction indicates that the options presented here are specifically mitigation strategies, not avoidance strategies. This is a key distinction, predicated on accepting that certain types of disruptions, natural disasters in particular, cannot be avoided. Thus, the next best option is to recognize the effects of these disruptions and identify options in the system that can reduce both the impact of the disruption and minimize the recovery costs and time necessary to return to normal operations. Functional symmetry is an example of an option in the system that modifies the existing manufacturing infrastructure to provide redundancy via flexibility rather than replication. On-demand warehousing is an option on the system, using an external service to help mitigate losses.

One of the major hurdles faced by risk managers is demonstrating the value of a risk mitigation investment. Tables 11 and 12 are great illustrations of the value of proactive risk mitigation. As noted in the lattice analysis discussion, these not only illustrate the inevitable loss, but that it is possible to make option investments that can reduce the costs of disruptions. While this type of lattice valuation is not realistically applicable to natural disasters, it would be an effective tool for demonstrating the value of options that hedge systematic disruptions such as defects.

5.3 Applicability to Natural DisastersAs this project evolved, both decision analysis and lattice valuation proved to be insufficient for modeling options that would mitigate natural disaster based disruptions. In the case of path dependent decision analysis models, the model would quickly become intractable due to the state-explosion problem. As noted above, lattice valuation is also insufficient because disruptions of this type may or may not occur and the notion of predicting these disruptions based on a rate is not applicable.

5.4 Potential Future WorkPerhaps a more appropriate alternative is to use simulations to model discrete disruptions. One approach might be to design a simulation that modeled functional symmetry that would model disruptions of varying magnitudes at random periods. This would create a model that could effectively exercise the functional symmetry option over a reasonable number of scenarios in linear time rather than exploring all possible permutations using decision tree analysis.

One idea, which certainly needs revision and exploration and will be part of the author’s IAP activities, is to organize the possible combinations of disruptions into groups of disruptions categorized as classes. For instance, one class would represent all the permutations of a large disruption and two small disruptions over 12 periods and another class may represent all the permutations of three large disruptions and no small disruptions over 12 periods. To group these together and assume all the instances of a class had the same characteristics would require substantial understanding of the nature

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and effects of the type of disruption. The classes would not be based on unique combinations, but rather on the characteristics of the disruption and investments necessary to mitigate the impact. Assuming these classes can be identified, the next step is to develop a simulation that will exercise the effectiveness of an option when exposed to one or more of these classes of disruption.

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6 Appendix A: Decision Tree

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Figure A.1 - Functional Symmetry (fixed) branch of discrete disruption decision tree

Dm1 C

P(high)=0.1

C

Oll = -5.24 M

Olm = -2.94 M

Ols = -2.17 M

P(high)=0.1

P(medium)=0.2

P(low)=0.45

EV = 0.12 M

Ols = -1.87 M

P(none)=0.25

P(medium)=0.2

C

Oll = -2.94 M

Olm = -0.65 M

Ols = 0.12 M

P(high)=0.1

P(medium)=0.2

P(low)=0.45

Ols = 0.43 M

P(none)=0.25

P(low)=0.45

C

Oll = -2.18 M

Olm = 0.12 M

Ols = 0.89 M

P(high)=0.1

P(medium)=0.2

P(low)=0.45

Ols = 1.19 M

P(none)=0.25

P(low)=0.45

C

Oll = -1.87 M

Olm = 0.43 M

Ols = 1.19 M

P(high)=0.1

P(medium)=0.2

P(low)=0.45

Ols = 1.5 M

P(none)=0.25

Value = -2.56

Value = -0.26

Value = 0.50

Value = 0.81

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Figure A.2 - On-demand warehousing (flexible) branch of discrete disruption decision tree

Dm1 C

P(high)=0.1

P(medium)=0.2

P(low)=0.45

C

C

C

Oll = -12.26 M

Olm = -5.96 M

Ols = -5.13 M

Osl = -6.52 M

Osm = -0.21 M

Oss = 0.62 M

Oml = -7.70 M

Omm = -1.39 M

Oms = -0.56 M

P(high)=0.1

P(medium)=0.2

P(low)=0.45

P(high)=0.1

P(medium)=0.2

P(low)=0.45

P(high)=0.1

P(medium)=0.2

P(low)=0.45

Dm2

C

Oll = -12.81 M

Olm = -6.49 M

Ols = -5.08 M

P(high)=0.1

P(medium)=0.2

P(low)=0.45

C

Oml = -8.25 M

Omm = -1.93 M

Oms = -0.51 M

P(high)=0.1

P(medium)=0.2

P(low)=0.45

Dm2

Dm2

C

Osl = -7.07 M

Osm = -0.75M

Oss = 0.67 M

P(high)=0.1

P(medium)=0.2

P(low)=0.45

increasecoverage

maintaincoverage

increasecoverage

maintaincoverage

increasecoverage

maintaincoverage

Value = -6.05

Value = -5.92

EV = -5.92

Value = -1.36

Value = -1.48

EV = -1.36

Value = -0.26

Value = -0.39

EV = -0.26

EV = -1.07

Ols = -4.79 M

P(none)=0.25

Ols = -4.74 M

P(none)=0.7

Oms = -0.23 M

P(none)=0.25

Oms = -0.18 M

P(none)=0.25

Oss = 0.95 M

P(low)=0.25

Oss = 1.00 M

P(low)=0.25

P(none)=0.25

C

Osl = -6.17 M

Osm = 0.14 M

Oss = 0.97 M

P(high)=0.1

P(medium)=0.2

P(low)=0.45

Dm2

C

Osl = -7.07 M

Osm = -0.75M

Oss = 0.67 M

P(high)=0.1

P(medium)=0.2

P(low)=0.45

increasecoverage

maintaincoverage

Value = 0.17

Value = 0.04

EV = 0.17

Oss = 1.30 M

P(low)=0.25

Oss = 1.00 M

P(low)=0.25

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Dm1 C

P(high)=0.1

P(medium)=0.2

P(low)=0.45

C

C

C

Oll = --3.62 M

Olm = -1.70 M

Ols = -1.06 M

Osl = -1.06 M

Osm = 0.85 M

Oss =1.49 M

Oml = -1.70 M

Omm = 0.21 M

Oms = 0.85 M

P(high)=0.1

P(medium)=0.2

P(low)=0.45

P(high)=0.1

P(medium)=0.2

P(low)=0.45

P(high)=0.1

P(medium)=0.2

P(low)=0.45

Ols = -0.81 M

P(none)=0.25

Oms = 1.11 M

P(none)=0.25

Oss =1.75 M

P(none)=0.25

P(low)=0.25

C

Osl = -0.81 M

Osm = 1.11 M

Oss =1.75 M

P(high)=0.1

P(medium)=0.2

P(low)=0.45

Oss =2.00 M

P(low)=0.25

Value = 0.81

Value = -1.81

Value = 0.25

Value = 1.43

Value = 1.17

Figure A.3 - Do Nothing branch of discrete disruption decision tree