Evaluating Upstream Supply Chain Disruptions
with Partial Availability
by
Jennifer J. Yip
Bachelor of Applied Science (Honors) in Management Engineering
University of Waterloo (2013)
Submitted to the Engineering Systems Division
in partial fulfillment of the requirements for the degree of
Master of Science in Engineering Systems
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
© Massachusetts Institute of Technology 2015. All rights reserved.
Author……………………………………………………………………………….
Engineering Systems Division
May 18, 2015
Certified by…………………………………………………………………………..
Yossi Sheffi
Elisha Gray II Professor, Engineering Systems Division
Professor, Civil and Environmental Engineering Department
Director, MIT Center for Transportation and Logistics
Thesis Supervisor
Accepted by………………………………………………………………………….
Munther A. Dahleh
Professor of Electrical Engineering and Computer Science
Acting Director of MIT Engineering Systems Division
2
Evaluating Upstream Supply Chain Disruptions
with Partial Availability
by
Jennifer J. Yip
Submitted to the Engineering Systems Division
on May 18, 2015, in partial fulfillment of the
requirements for the degree of
Master of Science in Engineering Systems
Abstract
Globalization, outsourcing, and the emphasis on lean supply chains continue to shape the supply
chain industry. These trends have increased the prevalence and severity of disruptions to
upstream supply. Disruptions to upstream supply can delay and potentially halt the flow of
necessary materials and/or services to purchasing firms, often resulting in severe operational and
financial losses. This has created a growing need for effective risk assessment techniques to
evaluate the impact of disruptions and inform risk mitigation policies. As a result, many
methodologies have been developed to assess risk by estimating the likelihood and impact of
disruptions. Given the inherent difficulty in estimating the likelihood of disruptions, this thesis
focuses on assessing the risk of supply shortfall independent of the causes and likelihoods of
such disruptions. This thesis presents an optimization-based framework to assess the risk of both
complete and partial supply disruptions and comments on inventory and procurement mitigation
strategies. The framework is used to compare two allocation policies (fair allocation and
preferential product allocation) for the distribution of scarce inventory in times of disruption. The
framework is then applied to data from a food products manufacturer to determine the impacts of
a disruption in the supply of two components feeding dozens of products.
Thesis Advisor: Yossi Sheffi
Title: Elisha Gray II Professor of Engineering Systems and Civil and Environmental Engineering
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4
Acknowledgements
This thesis is a result of all the support and guidance I have received from the many people I
have met on my journey to and at MIT. I would like to thank the people who have made this
possible.
First and foremost, I must express my deepest gratitude to my advisor, Professor Yossi Sheffi.
Without him, none of this research would be possible and I am so grateful for the opportunity to
work on this research project. This thesis benefited from Professor Sheffi’s invaluable guidance
and wealth of experience. His efforts were instrumental in leading me in the direction of clarity,
improving my academic writing, and encouraging me to think as a researcher and an industry
practitioner. Professor Sheffi, your patience and insights has never failed to improve the quality
of this thesis, and your witty banter on the Boston Bruins has never failed to improve my mood.
I would also like to thank my colleagues and friends. A special thanks to Leisa Kirkaldy, without
your kindheartedness and advice I wouldn’t have made it this far; Hitheam Mohamed, you have
always been the first to encourage me to take risks and bet on myself; and, Nick Rypkema, you
have inspired me with your own research, listened to my endless discussions about this project,
and have provided daily support and understanding for which I am perpetually grateful.
Finally, I would like to thank my family. To my parents and my brother, thank you for your
unparalleled encouragement, strength, and love throughout the years. Thank you for all the
sacrifices you have made to make all that I have achieved possible. To my brother, thank you for
keeping me sane all of these years and encouraging me to keep a positive outlook. To my
parents, you have always been my greatest teachers. My success would not be possible without
all the lessons I have learned from you both. I would not be where I am today without you.
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Table of Contents
1
2
3
4
5
Introduction ........................................................................................................................... 13
1.1 Motivation ...................................................................................................................... 15
1.2 Objectives ....................................................................................................................... 15
Literature Review.................................................................................................................. 16
2.1 Risk Background ............................................................................................................ 16
2.2 Sources of Supply Risk .................................................................................................. 17
2.3 Methodologies for Evaluating Supply Risk ................................................................... 18
2.4 Supply Risk Mitigation Strategies ................................................................................. 22
2.4.1 Extra Inventory ....................................................................................................... 23
2.4.2 Strategic Sourcing ................................................................................................... 25
Risk Metrics .......................................................................................................................... 27
3.1 Supplier Time-to-Recovery ............................................................................................ 27
3.1.1 Factors Affecting Supplier Time-to-Recovery ....................................................... 27
3.2 Firm Time-to-Recovery.................................................................................................. 27
3.2.1 Factors Affecting Firm Time-to-Recovery ............................................................. 29
3.3 Customer Impact Time ................................................................................................... 29
3.3.1 CIT with Finished Product Inventory ..................................................................... 31
3.3.2 CIT with Finished Product and Component Inventory ........................................... 33
3.3.3 CIT with all Forms of Inventory ............................................................................. 35
3.4 Value-at-risk ................................................................................................................... 37
Preferential Allocation .......................................................................................................... 41
4.1 Types of Preferential Allocation .................................................................................... 41
4.1.1 Preferential Product Allocation............................................................................... 41
4.1.2 Preferential Customer Allocation ........................................................................... 41
4.1.3 Preferential Product-Customer Allocation .............................................................. 42
4.2 Inventory Allocation ...................................................................................................... 42
4.3 Metrics for Evaluating Risk under Preferential Allocation ............................................ 44
4.3.1
Modified Value-at-Risk .......................................................................................... 44
4.3.1.1
Linear Program ................................................................................................ 45
4.3.1.2
Greedy Algorithm ............................................................................................ 46
4.3.2 Mitigation Factor .................................................................................................... 47
4.3.3 Marginal Financial Return ...................................................................................... 48
4.4 Comparison of Risk under Fair and Preferential Allocation .......................................... 50
4.5 Risk Metric Relationships .............................................................................................. 53
4.5.1 MVAR Relationships ............................................................................................... 54
4.5.2 Mitigation Factor Relationships.............................................................................. 56
4.5.3 Marginal Financial Return Relationships ............................................................... 61
Case Study ............................................................................................................................ 64
5.1 Comp1 ............................................................................................................................ 64
5.1.1 Demand ................................................................................................................... 64
5.1.2 Component Inventory ............................................................................................. 65
5.1.3 Finished Product Days-of-Supply ........................................................................... 66
5.1.4 Usage Rate .............................................................................................................. 67
5.1.5 Financial Value ....................................................................................................... 68
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5.2 Comp2 ............................................................................................................................ 70
5.2.1 Demand ................................................................................................................... 70
5.2.2 Component Inventory ............................................................................................. 71
5.2.3 Finished Product Days-of-Supply ........................................................................... 72
5.2.4 Usage Rate .............................................................................................................. 73
5.2.5 Financial Value ....................................................................................................... 74
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Analysis & Results ................................................................................................................ 77
6.1 Analysis with No Inventory ........................................................................................... 77
6.1.1 Financial Loss ......................................................................................................... 78
6.1.2 Mitigation Factor .................................................................................................... 84
6.1.3 Marginal Financial Return ...................................................................................... 86
6.1.4 Risk Mitigation Recommendations......................................................................... 89
6.2 Analysis with Inventory ................................................................................................. 97
6.2.1 Financial Loss ......................................................................................................... 97
6.2.2 Mitigation Factor .................................................................................................. 100
6.2.3 Marginal Financial Return .................................................................................... 102
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Conclusions ......................................................................................................................... 105
7.1 Further Research .......................................................................................................... 106
Appendix A ................................................................................................................................. 108
CIT Formulation...................................................................................................................... 108
Critical Time ....................................................................................................................... 112
Accumulation Time ............................................................................................................ 115
VAR Formulation .................................................................................................................... 122
MVAR Formulation ................................................................................................................. 126
Marginal Financial Benefit Formulation ................................................................................ 131
Appendix B ................................................................................................................................. 132
MVAR Formulation for a Single-Component Disruption using Substitution ......................... 132
MVAR Formulation for a Multiple-Component Disruption .................................................... 136
References ................................................................................................................................... 138
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List of Figures
Figure 3-1: Case of Firm and Supplier Time-to-recovery Equivalence ....................................... 28
Figure 3-2: Case of Differing Firm and Supplier Time-to-recovery ............................................ 28
Figure 3-3: CIT with Finished Product Inventory only ................................................................ 33
Figure 3-4: CIT with Finished Product and Component Inventory .............................................. 35
Figure 3-5: CIT with all Forms of Inventory ................................................................................ 37
Figure 3-6: Unmet Demand after Fair Allocation of Partial Rate of Supply ................................ 40
Figure 4-1: Unmet Demand after Preferential Product Allocation of Partial Rate of Supply ...... 52
Figure 4-2: MVAR vs. Partial Rate of Supply ............................................................................... 55
Figure 4-3: Maximal Effect of Preferential Allocation................................................................. 56
Figure 4-4: Mitigation Factor vs. Partial Rate of Supply.............................................................. 60
Figure 4-5: Mitigation Factor vs. Partial Rate of Supply (at all partial rates of supply) .............. 61
Figure 4-6: Marginal Financial Return vs. Partial Rate of Supply ............................................... 62
Figure 4-7: Marginal Financial Benefit vs. Partial Rate of Supply (at all partial rates of supply) 63
Figure 5-1: Comp1 Monthly Demand........................................................................................... 65
Figure 5-2: Comp1 Monthly Component Inventory and Demand ................................................ 66
Figure 5-3: Comp1 Monthly Percentage of Finished Products by Day-of-Supply ...................... 67
Figure 5-4: Comp1 Percentage of Finished Products by Usage Rate ........................................... 68
Figure 5-5: Comp1 Percentage of Finished Products by Financial Value .................................... 69
Figure 5-6: Comp1 Percentage of Finished Products by Financial Value per Component .......... 70
Figure 5-7: Comp2 Monthly Demand........................................................................................... 71
Figure 5-8: Comp2 Monthly Component Inventory ..................................................................... 72
Figure 5-9: Comp2 Monthly Percentage of Finished Products by Day-of-Supply ...................... 73
Figure 5-10: Comp2 Percentage of Finished Products by Usage Rate ......................................... 74
Figure 5-11: Comp2 Percentage of Finished Products by Financial Value .................................. 75
Figure 5-12: Comp2 Percentage of Finished Products by Financial Value per Component ........ 76
Figure 6-1: Comp1 Financial Loss vs. Partial Rate of Supply ..................................................... 81
Figure 6-2: Comp2 Financial Loss vs. Partial Rate of Supply ..................................................... 82
Figure 6-3: Comp2 First Order Finite Difference of MVAR with respect to Partial Rate of Supply
....................................................................................................................................................... 83
Figure 6-4: Comp1 and Comp2 Mitigation Factor vs. Partial Rate of Supply ............................. 85
Figure 6-5: Comp1 Marginal Financial Return vs. Partial Rate of Supply .................................. 88
Figure 6-6: Comp2 Marginal Financial Return vs. Partial Rate of Supply .................................. 88
Figure 6-7: Comp1 Monthly Safety Stock to Mitigate 25% of Total Financial Loss ................... 91
Figure 6-8: Comp2 Monthly Safety Stock to Mitigate 25% of Total Financial Loss ................... 92
Figure 6-9: Comp1 and Comp2 Percentage of Total Financial Loss vs. Safety Stock ................. 93
Figure 6-10: Comp1 and Comp2 Safety Stock and Partial Rate of Supply to Mitigate 25% of
Total Financial Loss ...................................................................................................................... 97
Figure 6-11: Comp1 and Comp2 Financial Loss vs. Partial Rate of Supply [Analysis with
inventory] .................................................................................................................................... 100
Figure A-1: Allocation of Finished Product Inventory ............................................................... 111
Figure A-2: Critical Time ........................................................................................................... 115
Figure A-3: Accumulation Time ................................................................................................. 122
Figure A-4: Unmet Demand after Fair Allocation of Partial Rate of Supply ............................. 126
Figure A-5: Unmet Demand after Preferential Product Allocation of Partial Rate of Supply ... 129
9
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List of Tables
Table 3-1: Finished Product Characteristics (CIT Example) ........................................................ 32
Table 4-1: Inventory Allocation Schemes .................................................................................... 43
Table 4-2: Product Characteristics (Marginal Financial Return Example) .................................. 49
Table 4-3: Decreasing Marginal Financial Return under Preferential Product Allocation .......... 50
Table 4-4: Finished Product Characteristics (Preferential Allocation Example).......................... 51
Table 4-5: Product Characteristics (Mitigation Factor Example) ................................................. 58
Table 4-6: Product Set Mean, Variance, CVFC (Mitigation Factor Example) .............................. 58
Table 6-1: Comp1 Financial Loss (in 000s of dollars) ................................................................. 79
Table 6-2: Comp2 Financial Loss (in 000s of dollars) ................................................................. 80
Table 6-3: Comp1 Mitigation Factor ............................................................................................ 84
Table 6-4: Comp2 Mitigation Factor ............................................................................................ 85
Table 6-5: Comp1 Marginal Financial Return (in 000s of dollars) .............................................. 86
Table 6-6: Comp2 Marginal Financial Return (in 000s of dollars) .............................................. 87
Table 6-7: Comp1 Financial Loss (in 000s of dollars) [Analysis with inventory] ....................... 98
Table 6-8: Comp2 Financial Loss (in 000s of dollars) [Analysis with inventory] ....................... 98
Table 6-9: Comp1 Mitigation Factor [Analysis with inventory] ................................................ 100
Table 6-10: Comp2 Mitigation Factor [Analysis with inventory] .............................................. 101
Table 6-11: Comp1 Marginal Financial Return [Analysis with inventory] ................................ 102
Table 6-12: Comp2 Marginal Financial Return [Analysis with inventory] ................................ 103
Table A-1: Finished Product Characteristics (Single Component Disruption with Varied Finished
Product Inventory Example) ....................................................................................................... 110
Table A-2: Time Intervals Considered for Determining Critical Time ...................................... 114
Table A-3: Time Intervals Considered for Determining Accumulation Time ........................... 119
Table A-4: Finished Product Financial Value (Single Component Disruption with Varied
Finished Product Inventory Example) ........................................................................................ 124
Table A-5: Finished Product Financial Value per Component (Single Component Disruption
with Varied Finished Product Inventory Example) .................................................................... 128
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1 Introduction
Disruptions such as Hurricanes Isaac and Sandy in 2012, the Corn Belt drought of 2012, and the
Tōhoku earthquake and tsunami in 2011, have spurred a heightened interest and emphasis on
addressing supply chain risk. Supply chain disruptions, defined as unplanned events affecting the
flow of materials within the supply chain, are becoming increasingly prevalent (Svensson, 2002).
The increase in frequency and intensity of both human and man-made disasters has prompted
companies to address the risks associated with supply chain disruptions.
Several trends have increased the susceptibility of supply chains to disruptions. Increased
competition has forced supply chains to be leaner, holding low inventory levels and relying
heavily on just-in-time deliveries. Leaner supply chains and logistics operations, coupled with
outsourcing and globalization of supply chains, have not only increased the likelihood of supply
chain disruptions but also the severity of their consequences. Furthermore, the increased
complexity and length of supply chains cause supply chains to be slower to respond to changes,
resulting in greater vulnerability (Tang and Tomlin, 2008).
While disruptions can impact any part of the supply chain, it has become especially
important to apply risk evaluation and mitigation strategies to procurement functions and
suppliers. Suppliers impact all downstream operations. Any disruption which prevents firms
from obtaining certain materials, parts, and/or services may result in production delays, stockouts, and delayed product launches at downstream organizations. The prominence of off-shore
sourcing and the increased connectedness of supply networks make supply chains vulnerable to
the risks imposed by suppliers’ disruptions.
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The operational and financial losses from such disruptions can be substantial. Ericsson
reported a loss of $400 million in 2000 and subsequently exited from the mobile phone business
after its sub-supplier plant caught fire (Norrman and Jansson, 2004). In 2001, Ford was forced to
shut down five of its plants because they were unable to receive engine parts from Canada due to
border crossing restrictions after the 9/11 terrorist attacks (Martha, 2002). Disruptions caused
from natural disasters have had equally destructive effects. The 2007 earthquake in Tokyo
severely damaged major production facilities of Riken Corporation which led to ceased
production in twelve Toyota plants; three days of suspended production at Mitsubishi Motors
Corporation; and, a temporary shutdown of five of Suzuki Motor Company’s domestic plants
(Hayashi, Smith and Chozick, 2007). Disruptions caused by natural disasters alone, resulted in
losses of $380 billion dollars in 2011 (Munich Reinsurance America, Inc., 2012).
Although the immediate operational and financial losses caused by disruptions are
significant, non-financial consequences such as loss of reputation among customers and
suppliers, loss of market share, and loss of brand value have further increased the importance of
addressing supply chain risk. Hendriks and Signhal (2005) found that companies that suffered
from supply chain disruptions experienced 33-40% lower stock returns relative to their
respective industry benchmarks.
Current supply chain trends and the increase in the number and severity of disruptions
have led companies to focus on risk identification, evaluation, and mitigation strategies. This has
created a growing need for metrics that quantify supply chain risk, thereby allowing management
to prioritize and focus.
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1.1 Motivation
Traditional risk assessment methods typically consist of “expected value” analyses based on a
combination of the likelihood of a disruption and its impacts (Zsidsin, 2004). However, due to
the difficulty of estimating likelihoods, alternative approaches use a “worst-case” or “what if?”
scenario, focusing the analysis on the consequences of a supplier ceasing to operate and the
unavailability of a part (or material) for a period. Following examination of many actual
disruptions, the latter “what if?” approach is too conservative and may lead to over-investment in
resilience and inventories. In reality, it is very rare that a supplier is completely down (most have
multiple plants where they can increase production through overtime and delayed maintenance)
or that no alternate supplier is available. Usually, a fraction of the volume is available during the
“down time”. This gap motivates the need for metrics that quantify supply chain risk
independent of estimating disruption likelihood and also model realistic scenarios, where a
fraction of supply is available during a supplier’s downtown.
1.2 Objectives
The objective of this thesis is to provide a framework for evaluating risk imposed on a
manufacturer when one of its suppliers fails. The research aims to combine and extend current
risk metrics, such as time-to-recovery and value-at-risk, and develop new risk metrics that can
determine the risk associated with a supplier failure without quantifying the likelihood of such
events. The framework will be used to estimate the disruption-minimizing strategy following a
partial supply disruption. These metrics can be used to assess and prioritize a manufacturer’s
vulnerability to a supplier failure or disruption. The results can be used to facilitate supplier
selection and the allocation of resources for monitoring and auditing suppliers.
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2 Literature Review
This chapter establishes a basis for understanding supply risk and risk management. The chapter
provides an overview of the research conducted in this domain, focusing on literature addressing
supply chain risk evaluation methods and mitigation strategies.
2.1 Risk Background
The two common terms used in the discussion of risk management are uncertainty and risk.
There is a distinction between risk and uncertainty, although they are closely related. Knight
(1964) defines risk as being quantifiable whereas uncertainty is not. Risk is identified as
quantifiable since the probability distribution of outcomes is measurable although the outcomes
are unknown. On the other hand, uncertainty is characterized by unquantifiable probability
distributions and unknown outcomes.
Zsidisin (2004) extends the meaning of risk to define supply risk within the context of
supply chains. He defines supply risk as “the potential occurrence of an incident associated with
inbound supply from individual supplier failures or the supply market, in which its outcomes
result in the inability of the purchasing firm to meet customer demand…”. Another suggested
definition of supplier risk is “…an abnormal operational state that might occur in an upstream
supplier, spread downstream through the supply chain, and harm the purchasing firm” (Jung,
Lim and Oh, 2011). An abnormal operational state is considered, by the authors, as an event,
such as a bankruptcy or a strike, which affects the working process of the supply chain (Jung,
Lim and Oh, 2011). For the purpose of this research, supply disruption will be defined as the
financial loss that may be incurred from an inbound supply interruption causing an inability to
meet customer demand.
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Supply chain management, as defined by Norrman and Lindroth (2002), is the
collaboration of supply chain partners to handle risk and uncertainties impacting or caused by
logistics-related activities and resources. Tang (2006) suggests a slightly different meaning of
supply chain risk management, defining it as the collaboration and coordination of supply chain
partners to manage supply chain risks in order to ensure continuity and profitability of all
partners.
2.2 Sources of Supply Risk
Purchasing firms can be impacted by any of the disruptions that their upstream suppliers may
experience. While there are multiple and diverse sources of supplier risk, each of which can be
studied at length, the following literature review describes some of the more recent supplierrelated risks firms have encountered.
Finances have always been a prevalent source of risk to suppliers, in particular,
insolvency and bankruptcy. In 2008, Chrysler was forced to temporarily shut down four of its
plants due to the cash-flow and liquidity problems experienced by its supplier Plastech (Revilla
and Sáenz, 2013). In a similar situation, Land Rover had to make a multi-million pound
“goodwill” payment to its only supplier of chassis frames, UPF-Thompson, to prevent a ninemonth production halt of its products and a potential loss of 1,500 Land Rover jobs (Sheffi and
Rice, 2005).
Natural disasters, ranging from tsunamis, earthquakes, and hurricanes, to fires and floods,
have been the source of disruption in supply chains. Hurricane Katrina prevented inbound supply
of products to stores within the affected areas (Oke and Gopalakrishnan, 2009). In 1999, an
earthquake in Taiwan hit factories that were responsible for over half of global semiconductor
contract manufacturing (Sherin and Bartoletti, 1999). This led to a shortage of semiconductors,
17
which impacted revenue and earnings of major electronic product producers, such as HewlettPackard and Dell Inc.
Finally, the terrorist attacks on September 11, 2011 have caused security threats and
terrorism to be viewed as significant sources of supplier risk. Even when suppliers are not
directly affected, government actions, taken in response to terrorist threats, have greatly
impacted international suppliers and purchasing firms (Sheffi, 2002). During the attacks of 9/11,
US borders were sealed, preventing the flow of inbound parts. Following this attack, Ford and
Toyota halted production in U.S. manufacturing plants due to delivery delays of parts sourced
internationally (). However, security threats are not limited to terrorist attacks; they also include
vandalism, sabotage, theft, cyber-attacks, and attacks on infrastructure or information systems.
2.3 Methodologies for Evaluating Supply Risk
The following section provides a survey of both qualitative and quantitative supply chain risk
evaluation methodologies.
Ericsson developed a supplier risk assessment approach following the fire in a subcontractor plant in Albuquerque, New Mexico, which caused Ericsson to exit the handset
manufacturing business (Norrman and Jansson, 2004). Ericsson evaluates its sourced
components through their first, second, and third tier suppliers. Each component is evaluated in
terms of a business interruption value metric, similar to what is commonly known as value-atrisk. The business interruption value metric represents the financial consequences of a disruption,
calculated by multiplying gross margin and business recovery time. Business recovery time is the
estimated length of time Ericsson’s deliveries may be affected by a specific incident disrupting a
component’s supply. For Ericsson, the key factors affecting business recovery time are the
number of suppliers available in the marketplace and the current sourcing strategy of the firm
18
(whether the component is single-sourced or multi-sourced). Ericsson also qualitatively defines
the probability of each supplier facing disruptions. The probability and consequences (business
interruption value metric) are mapped into a standard likelihood/consequence matrix to help
prioritize risks.
Jung et al. (2011) developed a binary logit model to assess whether suppliers pose low or
medium levels of risk firm. The dependent variable was based on a Korean vehicle
manufacturer’s classification of its first-tier suppliers as posing low or medium levels of risk.
Data about the suppliers was collected from both the manufacturer and the suppliers and used to
determine independent variables for the model. The model’s independent variables were
operational and financial capability indicators, measured objectively, and variables
characterizing the market for sourced parts, measured subjectively. Each independent variable
category has a number of sub-characteristics, each with its own scale for measurement. The
independent variables act as proxies to estimate the likelihood of a supplier encountering a
disruption and the impact of a disruption. The authors suggest the logit model use five key
independent variables (switching cost, operating profit margin, asset turnover ratio, quality
capability, and technological capability). Switching cost, operating profit margin, and quality
capability variables are negatively correlated with supplier risk. Asset turnover ratio and
operation technology variables are positively correlated with supplier risk.
Another risk evaluation approach uses Bayesian networks to develop supplier risk
profiles. The methodology analyzes relationships between supplier attributes and the purchasing
firm to determine an expected value-at-risk, representing the supplier’s impact on the purchasing
firm’s revenues (Lockamy and McCormack, 2012). The expected value-at-risk is the product of
the probability of revenue impact and monthly revenue impact. The purchasing firm’s subjective
19
assessment of the likelihood of its suppliers encountering each of three types of risks (network,
operational, and external risks), are used as a basis for the probability of revenue impact. The
average value of the probabilities assigned to each of the three risk categories represents the
supplier’s overall likelihood of encountering a disruptive event. The monthly gross revenue of all
finished assemblies using parts sourced from the supplier represents the impact of the disruption
on the purchasing firm. The single value-at-risk metric can be used to assess and compare the
risk that each supplier poses to the purchasing firm. The framework can also be used to evaluate
how changes in the probabilities of risk categories can affect the supplier’s overall risk profile
and value-at-risk (Lockamy and McCormack, 2012).
Wu et al. (2006) developed an analytical hierarchy process (AHP)-based method to
evaluate supplier risk in single-tier supply chains. The model is based on a set of risk factors
derived from literature and interviews with managers from various industries, including
electronics and food service manufacturers. Each risk factor is classified within a hierarchy; the
first tier categorizes risks as internal or external and the second tier categorizes risks as
controllable, partially controllable, or uncontrollable. AHP is used to determine the relative
importance of the set of risk factors. Then, each risk factor is assigned a subjective probability
of impacting each supplier. A single risk index is computed using the weights and probabilities
of each risk factor for each supplier. Wu et al. (2006) define this metric as the integrated
uncertainty factor, representing the level-of-risk a supplier poses to the purchasing firm. The
integrated uncertainty factor ranges from 0 to 1; the greater the value, the greater the risk posed
by the supplier. The integrated uncertainty factor is a subjective measure of risk, based upon a
firm’s perception of the importance of a set of risk factors and their probability of occurrence.
20
Blackhurst et al. (2008) produced a similar multi-criteria risk assessment model to
evaluate components. The model was developed through analysis of suppliers to an automotive
manufacturer. Nine risk categories, each with individual risk subcategories, are defined. Each
risk category is classified as internal, where the supplier has control over the risk factor, or
external, where the supplier has limited or no control over the risk factor. Each risk category is
assigned a weighing based upon the probability of each category of disruption occurring or the
relative impact of the disruption of each category on supply. These weights are subjective, based
on what is important to the automotive manufacturer. Sourced components are then graded on a
100-point scale in each risk category; a higher score is indicative of poorer supplier performance.
Component risk scores are based upon the weights and grade in each risk category. The model
also computes supplier risk scores, based upon component scores, weighted by the percentage of
volume the supplier provides of the given component. This methodology does not account for
the criticality of the part but rather the volume of the part supplied by the supplier. Blackhurst et
al. (2008) suggest using their model to evaluate risk over time; this enables the firm to assess the
movement of part of supplier risk toward unacceptable levels, aiding in the prediction of
potential supply problems in the future.
The aforementioned inbound risk assessment models are “expected value” analyses,
involving techniques that enumerate risk sources; assign probabilities to risk factors; and,
evaluate the consequences of disruptions to the purchasing firm. The assignment of probabilities
to risk factors is largely subjective, while consequences are evaluated based on financial impact.
Most models compute a single risk score to characterize suppliers’ or components’ risk. The
underlying logic to compute these risk scores varies among these methodologies.
21
Another branch of supply risk assessment research focuses on evaluating the “worstcase” or “what if?” scenario, where supplier operations are completely ceased. Simchi-Levi et al.
(2014) use linear optimization to quantify the financial and operational impacts of an inoperative
supplier facility for a specified time period. The impacts of removing a supply node from the
purchasing firm’s network are calculated under the assumption that total loss (of revenues,
profits, or production) is minimized. The model accounts for inventory and production
dependencies. The approach assigns each node a risk exposure index, which is a normalization of
the impact of each node relative to the node with the greatest impact. The risk exposure index
provides a prioritization of the risk to the various supply nodes in a firm’s network.
Berger et al. (2004) use decision trees to assess single- and multi-sourcing strategies in
the presence of risk. The methodology combines risk assessment with the evaluation of risk
mitigation sourcing strategies. Decision trees are used to compare the expected cost of sourcing
strategies where either one or several suppliers are simultaneously out of commission. The
financial loss of disrupted suppliers and the operating costs of working with multiple suppliers
are integrated in the model.
Most inbound supply risk assessment models revolve around “expected value” analyses,
which have limited use given the difficulties in estimating disruption likelihood. A small number
of researchers have focused on quantifying supply risks under the “what if?” scenario – evaluate
the consequences in case a part or material cannot be supplied for a period.
2.4 Supply Risk Mitigation Strategies
Upon evaluating and assessing risks, various strategies can be implemented to mitigate these
risks. Some mitigation strategies specifically aimed at supply risk include holding additional
inventory, dual or multi-sourcing, increasing flexibility, and increasing capacity (Chopra and
22
Sodhi, 2004). This section focuses on literature regarding the mitigation of risk through holding
additional inventory and multiple sourcing. While the literature for each strategy will be
considered separately, it is important to note that use of a single strategy is unlikely to mitigate
all supply risk. Multiple mitigation approaches can be used in concert to provide a robust
mitigation strategy.
2.4.1 Extra Inventory
Holding inventory is an established strategy for mitigating many types of risk. The 2002 West
Coast port lockout provides an illustrative example of the significance of inventory during times
of disruption. This 10-day strike of unionized West Coast port workers was estimated to have
impacted the United States economy at a rate of $934 million per day (Wolk, 2002) in the first
week. The impact rose quickly thereafter. Many firms were unable to bring in parts to their
manufacturing plants. However, not everyone was affected in the same way. The New United
Motor Manufacturing Inc. was affected within less than a week because it had almost no
inventory. The company operated a just-in-time system that only allowed them to continue
operations for five to seven days (Isidore, 2002). At the other end of the spectrum was Playmates
Toys, who invested in inventory before the disruption and thus were largely unaffected by the
disruption (Tomlin, 2006).
Inventory, however, is expensive to hold so a firm should analyze how much inventory to
hold, when to hold additional inventory, what policies to use for managing additional inventory,
and where within the supply chain to hold the inventory. The following section summarizes
some of the relevant literature addressing these questions.
A number of factors play a role in determining the applicability of holding inventory as a
risk mitigation strategy. When disruptions are frequent but their length is short, holding
23
additional inventory may be better than dual sourcing and contingent rerouting tactics (Tomlin,
2006). In this case, inventory is not being held for long periods of time between disruptions and
since the disruptions are not large, the amount of extra inventory needed is limited. The function
of additional inventory to hedge against frequent but short disruptions is similar to that of safety
stock. Although not mentioned by the authors, if these frequent and short disruptions are of a
similar order of magnitude to demand fluctuations, it is possible to combine inventory held for
these disruptions with safety stock. Then, safety stock could be used to mitigate both demand
and supply fluctuations.
Inventory can be helpful when there are advanced warnings of disruptions. For example,
labor negotiations going badly may be indicative of imminent labor strikes. Weather alerts and
terrorism threats also offer advance warning of potential disruptions. In such cases, inventory
reserves can be built ahead of the disruption instead of holding additional inventory
continuously. A system that periodically reviews threat levels can help determine the need to
adjust inventory levels, prior to an anticipated disruption.
Another proposed strategy is to use a dual inventory system, which applies to large
anticipated threats (Sheffi, 2002). A dual inventory system involves holding strategic emergency
stock that is to be used only in times of extreme disruption. To ensure this stock is not used in
normal times, its use requires senior management or board approval.
In a multi-echelon supply chain, additional inventory is recommended to be held
downstream in a variety of sites for diversification (Snyder and Shen, 2006). This contrasts with
the strategy of holding additional inventory in an upstream, centralized location, which is
appropriate under demand uncertainty since it provides risk pooling. Schmitt et al. (2015)
maintain that decentralized inventory models are optimal when both demand uncertainty and
24
supply disruptions are prevalent in the system since risk-diversification benefits outweigh risk
pooling benefits.
An alternative classification, based on the value of products, can be used to determine
whether inventory should be held centrally or across multiple sites (Chopra and Sodhi, 2004). A
decentralized inventory model is suggested for lower value products with predictable demand
patterns. Higher value products that have less predictable demand patterns may require a more
centralized inventory model.
In industries characterized by high value products with uncertain demand and short
lifecycles, the cost of holding inventory is significant. Often, using inventory to mitigate risk is
more appropriate for products with low risks of obsolescence; these products have lower holding
costs, which increases the economic feasibility of holding inventory of these products (Chopra
and Sodhi, 2004). However, for higher value products or those with high rates of obsolescence,
using a redundant supplier is considered more appropriate (Chopra and Sodhi, 2004). The
following section examines how strategic sourcing decisions can mitigate supply risk.
2.4.2 Strategic Sourcing
Single sourcing is always more effective in terms of leveraging pricing and building
relationships with the supplier. However, single sourcing is risky even without a disruption. For
example, a single supplier may not have the capacity to meet surges in demand. While multisourcing can mitigate procurement risks, in some situations sole sourcing is the only option
because no alternative suppliers exist. A variety of conditions impact sourcing decisions that
determine which strategy is most appropriate to mitigate risk. These conditions include, but are
not limited to, current relationships with suppliers, the depth of supplier reliability information, a
purchasing firm’s perception of their suppliers’ vulnerability, and product characteristics.
25
The relationship between a purchasing firm and supplier can often govern which sourcing
strategy is best for the purchasing firm. A firm may choose to have a single supplier to have
greater access to innovation or greater influence upon the supplier. To make a single-sourcing
strategy effective, the relationship between the firm and supplier must be deep; significant
investment must be placed into developing a strong relationship with the supplier. In this
situation, it is important for the purchasing firm to keep a watchful eye on the supplier’s financial
indicators and its operations. A firm may also choose to source from multiple suppliers to foster
competition, decrease prices, or any other business reason. In that case, it may be too expensive
to maintain deep relationships with all suppliers, relying instead on the flexibility to switch
between suppliers during times of disruption.
While multi-sourcing provides flexibility to move between suppliers, it does not
guarantee full supply continuity. Sheffi (2005) provides three reasons that multiple suppliers
cannot offer continuous supply: regional disruptions that affect multiple suppliers, lack of
capacity by alternative suppliers, and the interconnection of commodity markets. Alternate
suppliers may not have the capacity or may be unwilling to ramp up to help a company that only
uses them in times of disruption. Moreover, disasters have the potential to impact multiple
suppliers. To make the multi-sourcing strategy more effective, the suppliers should be spread
geographically.
26
3 Risk Metrics
This chapter introduces established metrics that are typically used for risk evaluation, including
time-to-recovery, customer impact time, and value-at-risk. Factors affecting these metrics are
also discussed.
3.1 Supplier Time-to-Recovery
Time-to-recovery, TTR, is defined as the time it takes to fully recover after the occurrence of a
disruption. TTR can characterize the firm and/or the supplier. Supplier TTR is the duration of
time, starting at the time of the disruption, and ending at the point in time when the supplier can
once again provide the firm with the component at the same rate that the component was
supplied before the disruption.
3.1.1 Factors Affecting Supplier Time-to-Recovery
A supplier’s TTR is largely influenced by the characteristics of the disruption, specifically the
location and the type of disruption. Some disruptions, such as bankruptcy, do not affect an actual
location whereas natural disasters may affect a site or a small group of sites. The type of
disruption also determines a supplier’s TTR; different types of disruptions will affect the supplier
to different extents and may affect the entire business or just a portion of it.
3.2 Firm Time-to-Recovery
A firm’s TTR is the time it takes for the firm to fully recover from a disruption affecting its
supplier, not accounting for inventory available to the firm. This duration of time begins at the
time of the disruption to its supplier and ends when the firm can fulfill all its customers’
demands.
27
A firm’s TTR can differ from its supplier’s TTR. Before the supplier recovers, the firm
may be able to find and qualify an alternative supplier or component; it may also use dilution
solutions to reduce the amount of the disrupted component required in finished products; or, it
may implement other engineering solutions, such as qualifying other materials or creating new
processes, thereby eliminating the need for the component. In all these cases the firm’s TTR will
be less than the supplier’s TTR. When a firm’s supplier faces a disruption, the firm’s TTR is the
minimum amount of time among the aforementioned various activities and the time to wait for
its current supplier to recover. In the case that the supplier’s TTR is shorter than the time it takes
to find an alternate supplier or another engineering solution, the firm’s TTR is equivalent to the
supplier’s TTR (see Figure 3-1). In cases where the time to find an alternate supplier or another
engineering solution are shorter, the firm’s TTR is less than the supplier’s TTR (see Figure 3-2).
Figure 3-1: Case of Firm and Supplier Time-to-recovery Equivalence
Figure 3-2: Case of Differing Firm and Supplier Time-to-recovery
28
3.2.1 Factors Affecting Firm Time-to-Recovery
A firm’s TTR is mainly affected by its sourcing strategy; the speed at which the firm can conduct
contingency activities; the specific component whose supply has been disrupted; and, whether
the disruption is industry-wide. A firm that sources its disrupted component from a single
supplier will likely have a different TTR than a firm with multiple suppliers. In a multi-sourcing
situation, the firm has the potential to procure the component from another supplier who has not
been disrupted, as long as the other supplier has enough capacity.
The speed at which the firm can conduct contingency activities, such as qualifying an
alternate supplier or finding engineering solutions, can also affect a firm’s TTR. The speed at
which the firm can conduct these activities is a function of its pre-planning. For example, if a
firm has identified alternate suppliers and qualified them before a disruption, the firm must only
wait for the alternate supplier to ramp-up its production.
The specific component whose supply has been disrupted also affects the firm’s TTR. If
the component is unique, the firm may have no choice but to wait for its supplier to recover.
However, a common component may be easily procured by an alternate supplier, thereby
reducing the firm’s TTR.
An industry-wide disruption can significantly affect a firm’s TTR. In an industry-wide
disruption, competitors of the firm are also disrupted. In this situation, not only may the
component be in limited supply, but many firms may be competing for this limited resource.
This can affect how long it will take for a firm to obtain the required supply of components.
3.3 Customer Impact Time
Customer impact time, CIT, measured in the same units of time as TTR, represents the interval of
time during which a firm’s customers will be affected. The period begins when a firm can no
29
longer satisfy 100% of its expected demand and ends at the firm’s recovery time. In the model
described here and subsequent analysis CIT does not account for transit times, such as the
inbound transportation of components from the supplier to the firm or outbound transportation of
the finished goods to customers. This is a simplifying assumption that can be easily relaxed.
CIT is determined by accounting for inventory available to the firm. This includes
finished product inventory, component inventory at the firm, and components accumulated from
the on-going partial rate of supply while these inventories are expended. The calculation of CIT
depends on an assumption for allocating these inventories. Two potential assumptions are
described below.
1. Continued allocation of inventory among affected finished products according to demand
for as long as finished inventory and parts are available.
2. Continued allocation of inventory for the duration of the firm’s TTR.
Under the first assumption, original demand for all affected products is sustained until all
inventories are exhausted. This assumption allows the firm time to determine its allocation policy
during the CIT period, communicate its intended actions to customers, and provide its customers
some buffer time to make contingency plans and mitigate their own losses. Under the second
assumption, this inventory is added to incoming components from the on-going partial rate of
supply and spread throughout the firm’s TTR. For now, assume CIT is calculated based on the
first assumption.
CIT can be derived using the following variables:

𝑡𝑑 – time of the disruption.

𝑡𝑟 – time of recovery.

𝐹. 𝑇𝑇𝑅 – firm time-to-recovery, equivalent to the difference between the time of recovery
and the time of the disruption (𝑡𝑟 − 𝑡𝑑 ).
30

𝐶. 𝐼𝑛𝑣 – component inventory-on-hand at the time of disruption (units: components).

𝑃. 𝐼𝑛𝑣𝑖 – inventory-on-hand of finished product 𝑖 at the time of disruption (units: finished
products).

𝐷𝑖 – average demand of finished product 𝑖 by all customers per day, from the date of the
disruption (𝑡𝑑 ) until the firm recovers (𝑡𝑟 ). Where 𝑖 is from 1 to n.

𝑢𝑖 – usage rate of components in finished products; the number of components required
to produce a single unit of finished product 𝑖 (units: components per single unit of
finished product).

𝑅 – the percentage of the normal rate of supply of the disrupted component available
during the firm’s TTR (units: percentage).
Calculations of CIT given varying availability of different types of inventories are detailed in
the following subsections. In all calculations, if total demand can be fulfilled throughout the
firm’s recovery period, the firm’s customers are unaffected and CIT = 0.
3.3.1 CIT with Finished Product Inventory
Given availability of only finished product inventory, CIT is defined in equation (3-1). When
only finished product inventory is available, CIT has will vary between products, depending on
the days of supply represented by this inventory. In equation (3-1), 𝑃. 𝐷𝑂𝑆𝑖 represents finished
product inventory days-of-supply, available to meet total demand.
𝐶𝐼𝑇𝑖 = max [(𝐹. 𝑇𝑇𝑅 − 𝑃.
𝐼𝑛𝑣𝑖
) , 0] = max[(𝐹. 𝑇𝑇𝑅 − 𝑃. 𝐷𝑂𝑆𝑖 ), 0]
𝐷𝑖
(3-1)
In the rest of this section, finished product days-of-supply will be assumed to be the same
for all affected products.
31
Example
The following example will be used throughout the chapter to illustrate the allocation of
inventory and how CIT changes with available inventory. Assume there are three products with
the following characteristics:
Total Daily Product Demand 𝑫𝒊
Usage Rate 𝒖𝒊
Finished Product Inventory on Hand
𝑷. 𝑰𝒏𝒗𝒊
Product 1
5,000
4
Product 2
2,000
8
Product 3
1,000
4
3,750
1,500
750
Table 3-1: Finished Product Characteristics (CIT Example)
The disruption is assumed to occur at time 0, and the firm recovers 6 days afterward.
𝑡𝑑 = 0, 𝑡𝑟 = 6
𝐹. 𝑇𝑇𝑅 = 𝑡𝑟 − 𝑡𝑑 = 6 − 0 = 6
Although the amount of finished product inventory differs between products, each
sustains average daily demand for three-quarters of a day. Thus, the CIT for each product is the
same, as depicted in Figure 3-3. The components usage rate is not relevant in the calculation of
CIT when only finished product inventory is considered.
𝐶𝐼𝑇1 = max (0, [𝐹. 𝑇𝑇𝑅 −
𝐶𝐼𝑇2 = max (0, [𝐹. 𝑇𝑇𝑅 −
𝐶𝐼𝑇3 = max (0, [𝐹. 𝑇𝑇𝑅 −
𝑃.𝐼𝑛𝑣1
𝐷1
𝑃.𝐼𝑛𝑣2
𝐷2
𝑃.𝐼𝑛𝑣3
𝐷3
3,750
]) = max (0, [6 − 5,000]) = max(0, [6 − 0.75]) = 5.25
1,500
]) = max (0, [6 − 2,000]) = max(0, [6 − 0.75]) = 5.25
750
]) = max (0, [6 − 1,000]) = max(0, [6 − 0.75]) = 5.25
32
Figure 3-3: CIT with Finished Product Inventory only
3.3.2 CIT with Finished Product and Component Inventory
Given the availability of both finished product inventory and component inventory, CIT is
defined in equation (3-2). Under the assumption that component inventory is fairly allocated
across all affected finished products and all affected products have the same days-of-supply,
component inventory days-of-supply is the same for all finished product.
𝐶. 𝐼𝑛𝑣
) , 0]
∑𝑛𝑖=1 𝐷𝑖 𝑢𝑖
= max[(𝐹. 𝑇𝑇𝑅 − 𝑃. 𝐷𝑂𝑆𝑖 − 𝐶. 𝐷𝑂𝑆𝑖 ), 0]
𝐶𝐼𝑇𝑖 = max [(𝐹. 𝑇𝑇𝑅 − 𝑃. 𝐷𝑂𝑆𝑖 −
(3-2)
Where:
 ∑𝑛𝑖=1 𝐷𝑖 𝑢𝑖 represents the total number of components required per day to fulfill total
demand of all affected products.

𝐶. 𝐷𝑂𝑆 represents the component inventory days-of-supply, the number of days that
component inventory can meet average demand of all products.
33
Example
Assume that at the time of disruption there are 30,000 components of inventory available
(𝐶. 𝐼𝑛𝑣 = 30,000). The average daily demand of components is (5,000 ∙ 4 + 2,000 ∙ 8 + 1,000 ∙
4 =) 40,000, so component inventory can meet average total demand for 0.75 days. This results
in finished product 1 being allocated 15,000 components, used to produce 3,750 finished
products. Finished product 2 is allocated 12,000 components, used to produce 1,500 finished
products. Finished product 3 is allocated 3,000 components, used to produce 750 finished
products. Figure 3-4 shows the CIT of each product given the allocation of component and
finished product inventory.
𝐶. 𝐷𝑂𝑆 =
𝐶. 𝐼𝑛𝑣
30,000
30,000
=
=
= 0.75
3
∑𝑖=1 𝐷𝑖 𝑢𝑖 5,000 ∙ 4 + 2,000 ∙ 8 + 1,000 ∙ 4 40,000
𝐶𝐼𝑇1 = max(0, [𝐹. 𝑇𝑇𝑅 − 𝑃. 𝐷𝑂𝑆1 − 𝐶. 𝐷𝑂𝑆]) = max(0, [6 − 0.75 − 0.75]) = 4.5
𝐶𝐼𝑇2 = max(0, [𝐹. 𝑇𝑇𝑅 − 𝑃. 𝐷𝑂𝑆2 − 𝐶. 𝐷𝑂𝑆]) = max(0, [6 − 0.75 − 0.75]) = 4.5
𝐶𝐼𝑇3 = max(0, [𝐹. 𝑇𝑇𝑅 − 𝑃. 𝐷𝑂𝑆3 − 𝐶. 𝐷𝑂𝑆]) = max(0, [6 − 0.75 − 0.75]) = 4.5
34
Figure 3-4: CIT with Finished Product and Component Inventory
3.3.3 CIT with all Forms of Inventory
During a disruption, it is common for some inbound supply to be available during a disruption.
This can be the result of the supplier having some ability to produce or a dual source having
capacity to meet a fraction of the firm’s demand. The available inbound supply, henceforth
referred to as the partial rate of supply (denoted by 𝑅), represents the fraction of the normal rate
of supply of the disrupted component that is available from the time of the disruption until the
purchasing firm recovers from the disruption. The available partial rate of supply is assumed to
be constant and available throughout the firm’s time-to-recovery. This, naturally, is a simplifying
assumption, as in most cases the partial rate of supply increases over time. The constant partial
rate of supply can be interpreted as the average availability over the disruption.
35
While finished product and component inventories are expended, components from the
partial rate of supply accumulate and extend the time over which total demand can be met. CIT,
given the availability of finished product inventory-on-hand, component inventory-on-hand, and
accumulated parts from the partial rate of supply, is defined in equation (3-3). Under the
assumption that the components accumulated from the partial rate of supply are allocated fairly
across all finished products, the CIT of each product will be extended by the same amount of
time. This equation is valid under the assumption that all finished products inventory is the same
in terms of days-of-supply, across all products.
𝐶𝐼𝑇 = max [(𝐹. 𝑇𝑇𝑅 −
𝑃. 𝐷𝑂𝑆 + 𝐶. 𝐷𝑂𝑆
) , 0]
1−𝑅
(3-3)
Where:
 𝑅 represents the partial rate of supply; the percentage of the normal rate of supply of the
disrupted component available during the firm’s TTR.

𝑃. 𝐷𝑂𝑆 + 𝐶. 𝐷𝑂𝑆 represents the number of days that available finished product and
component inventories can fulfill total demand.

𝑃.𝐷𝑂𝑆+𝐶.𝐷𝑂𝑆
represents the total number of days, from the time of the disruption, that total
demand can be fulfilled with all forms of inventory.
1−𝑅
Example
Using the same example, now assume there is a partial incoming rate of supply of 40%. Each
day, 40% of the total required number of components per day is available for use in production.
Over the time that component and finished product inventory-on-hand are being expended, 1.5
days in total, 24,000 components accumulate. While these 24,000 components are being
expended, another 9,600 components have accumulated. The amount of components
accumulating continues to decrement until no more components accumulate from the partial rate
of supply. This accumulation enables another full day of demand to be met. Recall that initially
36
CIT was 4.5 days. After considering the accumulated components from the partial rate of supply,
CIT is 3.5 days.
𝐶𝐼𝑇 = max (0, [𝐹. 𝑇𝑇𝑅 −
= 3.5
𝐶. 𝐷𝑂𝑆 + 𝑃. 𝐷𝑂𝑆
1.5
]) = max (0, [6 −
]) = max(0, [6 − 2.5])
1−𝑅
1 − 0.4
Figure 3-5 shows the CIT of each product after allocation of three types of inventory:
component
inventory-on-hand;
finished
product
inventory-on-hand;
and,
components
accumulated from the partial rate of supply.
Figure 3-5: CIT with all Forms of Inventory
3.4 Value-at-risk
During the CIT period, financial losses accumulate as a result of unmet customer demand, where
unmet customer demand is assumed to be lost. During this time, the partial rate of supply is
37
allocated to fulfill demand to the extent possible and reduce financial loss. Value-at-risk, VAR,
represents the financial loss during the CIT period under a fair allocation policy. A fair allocation
policy allocates the partial rate of supply to affected products such that each product will receive
the same percentage of its demand during the shortage period.
VAR is based on the financial value, 𝐹𝑖 , of each affected product, representing the worth
of a single unit of the product to the firm. It can be based on revenue, profit, contribution margin,
or any other related metric. VAR is also based on the demand of each affected product.
Calculations of demand are assumed to be based on historical data. In actual disruption cases and
other short supply situations, many customers inflate their orders assuming that the firm will
allocate a set percentage under a fair allocation policy. The assumption is the firm is aware of
this inflation and any allocation will be based on the firm’s best estimate of actual customer
demand of affected products.
When there is no partial rate of supply, the value lost for each finished product is the
length of time where no customer demand is fulfilled, CIT, multiplied by the product’s demand
and its financial value. VAR is the summation of the values lost for all affected products. When a
partial rate of supply is available, it can be used during the CIT period to partially fulfill demand.
Once all available inventories have been expended (assuming that the firm continues to serve all
its customers’ demand as long as inventories are available), the firm can continue to make
products at a reduced rate, based on the partial rate of supply. The value lost is now dependent
upon the partially unmet demand for a product throughout the product’s CIT period. Given that
the partial rate of supply will be allocated fairly across all affected products, each affected
product will meet the same percentage of demand, which is equivalent to the percentage of
available components – the partial rate of supply. VAR is defined in equation (3-4).
38
𝑛
𝑉𝐴𝑅 = (1 − 𝑅) ∑(𝐶𝐼𝑇𝑖 ∙ 𝐷𝑖 ∙ 𝐹𝑖 )
(3-4)
𝑖=1
Example
Using the same example, now assume that the financial value of products 1, 2 and 3 are 5, 8 and
6 respectively. The VAR of this example is 98,700 financial units. This is based upon a partial
rate of supply of 40% and a CIT value of 3.5 days for all products.
𝑛
𝑉𝐴𝑅 = (1 − 𝑅) ∙ 𝐶𝐼𝑇 ∙ ∑ 𝐷𝑖 𝐹𝑖 = (1 − 0.4) ∙ (3.5) ∙ (5,000 ∙ 5 + 2,000 ∙ 8 + 1,000 ∙ 6)
𝑖=1
𝑉𝐴𝑅 = 0.6 ∙ 3.5 ∙ 47,000 = 98,700
The allocation of all types of inventory and the partial rate of supply is shown in Figure
3-6. Each affected product has been allocated 40% of the total number of components it requires
to meet customer demand. 60% of daily customer demand is unmet for each finished product.
39
Figure 3-6: Unmet Demand after Fair Allocation of Partial Rate of Supply
40
4 Preferential Allocation
In the previous section, risk metrics were established under the assumption of fair allocation,
where each product, affected by a component whose supply has been disrupted, receives
available components proportionate to the demand of the product. However, components can
also be allocated preferentially. Preferential allocation refers to allocating components to those
products or customers that have the highest financial contribution first. This chapter introduces
different preferential allocation policies, but focuses on one scheme (preferential product
allocation) and the methodologies to evaluate this policy.
4.1 Types of Preferential Allocation
Product, customer, and product-customer preferential allocation are different policies that can be
used to govern the distribution of inventory during a disruption.
4.1.1 Preferential Product Allocation
Under a preferential product allocation policy, components are allocated to those products with
the highest financial contribution per component first. Taking into account the number of
components used to make each product, the allocation should be based on the financial
contribution per component of the finished product.
The preferential product allocation policy is applicable when all customers are valued the
same. In this case, there is no benefit in allocating a given product to one customer over another.
4.1.2 Preferential Customer Allocation
A preferential customer allocation policy uses the value of customers to govern the allocation of
inventory during a disruption. Customers possessing the greatest value to the firm are allocated
41
components first. The value of the customer to the firm may be based on a number of factors
such as the depth of the relationship; potential for future business; customer vulnerability; etc.
The preferential customer allocation policy is applicable when all products are valued the
same. In this case, there is no benefit in allocating a given customer to one product over another.
4.1.3 Preferential Product-Customer Allocation
The preferential product-customer allocation policy is a hybrid of the preferential product
allocation and preferential customer allocation policies. Under a preferential product-customer
allocation policy, inventory is allocated to those product-customer pairs that have the greatest
value to the firm first. A product-customer pair represents a certain customer’s demand for a
specific finished product. The value of a product-customer can be the product of the financial
contribution per component of the product and the value of the customer.
The product-customer allocation policy is relevant when there is variation in the values of
products and customers. This indicates that providing a certain product to a certain customer can
be much more beneficial than producing the same product for a different customer or providing
the same customer a different product.
4.2 Inventory Allocation
Preferential allocation policies use the value of products, customers, and product-customer pairs
to specify which customers or products should be allocated available inventory and supplies first.
However, not all forms of inventory and supply must be allocated preferentially; some inventory
may still be allocated fairly, such that demand for all products or customers is sustained for the
same amount of time.
The three types of inventory considered for allocation are: component inventory-on-hand;
finished product inventory-on-hand; and, components accumulated from the incoming partial
42
rate of supply while other inventories are expended. The allocation of the partial rate of supply is
also considered.
Table 4-1 shows various combinations of allocating the three types of inventory and the
partial rate of supply fairly or preferentially. Inventory allocation scheme #1 specifies allocation
of all inventory and supply fairly, where demand of each customer or product is sustained for the
same amount of time. Inventory allocation scheme #16 lies at the other end of the spectrum,
where all types of inventory and the partial rate of supply are allocated preferentially. Firms may
opt for a hybrid inventory allocation scheme (#2 through #15), where some types of inventory
and the partial rate of supply are allocated fairly and other types allocated preferentially. These
hybrid schemes ensure that demand of every affected customer or product is fulfilled to some
degree.
Inventory Allocation Scheme #
Component
Inventory-onhand
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Fair
Fair
Fair
Fair
Preferential
Fair
Fair
Preferential
Fair
Preferential
Preferential
Fair
Preferential
Preferential
Preferential
Preferential
Type of Inventory
Finished
Accumulated
Product
Components
Inventory-onfrom Partial Rate
hand
of Supply
Fair
Fair
Fair
Fair
Fair
Preferential
Preferential
Fair
Fair
Fair
Fair
Preferential
Preferential
Fair
Fair
Fair
Preferential
Preferential
Fair
Preferential
Preferential
Fair
Preferential
Preferential
Fair
Preferential
Preferential
Fair
Preferential
Preferential
Preferential
Preferential
Table 4-1: Inventory Allocation Schemes
43
Partial Rate of
Supply
Fair
Preferential
Fair
Fair
Fair
Preferential
Preferential
Preferential
Fair
Fair
Fair
Preferential
Preferential
Preferential
Fair
Preferential
Inventory allocation scheme #2 will be the scheme studied henceforth. As mentioned
above, this allows the firm time to determine its allocation policy during the CIT period;
communicate its intended actions to customers; and, provide its customers some buffer time to
make contingency plans and mitigate their own losses. Only the partial rate of supply, used to
reduce the amount of unmet demand during the CIT period, will be allocated preferentially.
Since CIT is dependent only upon component inventory-on-hand, finished product
inventory-on-hand and accumulated components from the partial rate of supply, there is no
difference between the CIT used in the calculation of VAR (see Equation (3-4)), and the CIT
under a preferential allocation policy.
4.3 Metrics for Evaluating Risk under Preferential Allocation
Three metrics will be used to evaluate risk under preferential allocation. The first metric is
modified value-at-risk, MVAR, which measures financial loss as the result of unmet customer
demand under preferential allocation. The second metric is a mitigation factor, which measures
the relative benefit of allocating the partial rate of supply preferentially instead of fairly. The
third metric is marginal financial return, measuring the change in the firm’s financial results with
respect to a change in the partial rate of supply over the firm’s CIT period. These metrics and the
methodologies to determine are defined in this section. In addition, the example from section 3 is
used to evaluate the preferential product allocation policy and compare it with the results under
fair allocation.
4.3.1 Modified Value-at-Risk
Modified value-at-risk, MVAR, represents the financial loss during the CIT period, under a
preferential product allocation scheme. Under preferential product allocation, the partial rate of
44
supply is allocated to affected products so as to minimize the total financial impact to the firm.
This policy takes into account the financial value of each product, 𝐹𝑖 , and the numbers of
components used to make each product, 𝑢𝑖 . MVAR will always be less than VAR; only when all
affected products have the same financial contribution per component, or the partial rate of
supply is 0 or 100%, will the two values be the same.
MVAR is contingent upon the set of variables 𝑋𝑖 , representing the number of affected
products 𝑖, produced per day, given the allocation of components from the partial rate of supply
to each respective product.
MVAR, defined in equation (4-1), is the summation of the product of the amount of time
that total demand is unfulfilled, CIT; the average unmet demand during this period; and, the
financial value for each product. In the equation, 𝐷𝑖 − 𝑋𝑖 represents the unmet demand after the
allocation of the partial rate of supply.
𝑛
𝑀𝑉𝐴𝑅 = ∑(𝐶𝐼𝑇𝑖 ∙ (𝐷𝑖 − 𝑋𝑖 ) ∙ 𝐹𝑖 )
(4-1)
𝑖=1
There is no difference in the customer impact time, 𝐶𝐼𝑇𝑖 , or the financial value of
products, 𝐹𝑖 , in the calculations of VAR and MVAR. The only difference between the two metrics
is the allocation of the partial rate of supply during the CIT period.
4.3.1.1 Linear Program
A linear program, with the objective to minimize MVAR, can be used to determine the
preferential allocation of the partial rate of supply. For preferential product allocation, the
decision variable of the linear program {𝑋𝑖 }, representthe number of products produced per day
based upon the allocated partial rate of supply.
Under preferential product allocation, the linear program is as follows:
45
𝑛
min ∑(𝐶𝐼𝑇𝑖 ∙ (𝐷𝑖 − 𝑋𝑖 ) ∙ 𝐹𝑖 )
(4-2)
𝑖=1
Subject to:
𝐷𝑖 − 𝑋𝑖 ≥ 0,
(4-2.1)
∀𝑖
∑𝑛𝑖=1 𝑋𝑖 𝑢𝑖
𝑅− 𝑛
≥0
∑𝑖=1 𝐷𝑖 𝑢𝑖
𝑋𝑖 ≥ 0,
(4-2.2)
(4-2.3)
∀𝑖
Where:
 ∑𝑛𝑖=1 𝑋𝑖 𝑢𝑖 represents the total number of components per day allocated from the partial
rate of supply.


∑𝑛𝑖=1 𝐷𝑖 𝑢𝑖 represents the total number of components required per day to fulfill total
demand of all affected products.
∑𝑛
𝑖=1 𝑋𝑖 𝑢𝑖
∑𝑛
𝑖=1 𝐷𝑖 𝑢𝑖
represents the percentage of the total number of components per day required to
meet all demand that is satisfied with the allocated partial rate of supply
The first constraint ensures the supply allocated to fulfilling demand does not exceed
what is demanded. The second constraint ensures that the supply allocated does not exceed what
is available.
4.3.1.2 Greedy Algorithm
The linear program formulation falls into a class of linear programs that, as shown for example
by Hoffman (1985), could be solved using a simple greedy algorithm1. This is possible when the
costs of the decision variables follow a Monge sequence. A cost matrix, 𝐶 , of size 𝑚 × 𝑛,
follows a Monge sequence if it satisfies the condition that 𝐶𝑞𝑟 + 𝐶𝑠𝑡 ≤ 𝐶𝑞𝑡 + 𝐶𝑠𝑟 for all 2 × 2
subset matrices where 1 ≤ 𝑞 < 𝑠 ≤ 𝑚 and 1 ≤ 𝑟 < 𝑡 ≤ 𝑛. Given that the costs of the decision
1
Hoffman’s (1985) work specified that linear programs that could be solved optimally using a greedy algorithm had
maximizing objective functions. While the linear program formulation presented in the previous section minimizes
its objective function, it is equivalent to maximizing the financial returns during the CIT period.
46
variables in this case are the financial contribution per component2 and can be ordered in a 1 × 𝑛
matrix, the Monge sequence is satisfied when every 2 × 1 subset matrix meets the condition
𝐶1𝑟 ≤ 𝐶1𝑡 where 1 ≤ 𝑟 < 𝑡 ≤ 𝑛 . This condition is met when the financial contributions per
component are ranked in descending order as in expression (4-3).
𝐹(1) 𝐹(2) 𝐹(3)
𝐹(𝑛)
≥
≥
≥⋯≥
𝑢(1) 𝑢(2) 𝑢(3)
𝑢(𝑛)
(4-3)
The method operates in a single path through this hierarchy, allocating the supply to each
affected product until it the supply is exhausted. The amount of partial rate of supply allocated
to a given product cannot exceed the number of components required to meet total demand of the
product.
4.3.2 Mitigation Factor
The mitigation factor, MF, defined in equation ( 4-4 ), is a measure of the relative benefit of
allocating the partial rate of supply preferentially instead of fairly.
𝑀𝐹 = 1 −
𝑀𝑉𝐴𝑅
𝑉𝐴𝑅
( 4-4 )
The mitigation factor takes on values between 0 and 1, but cannot attain a value of 1 in
other words, 𝑀𝐹 ∈ [0,1). When the mitigation factor is 0, there is no benefit of allocating
inventory preferentially; the expected loss is the same under fair or preferential allocation
schemes. The mitigation factor value is highest when the coefficient of variation of the products’
financial contribution per component is highest, a situation occurring when there are some
products with very high financial contributions per component and others with very low financial
contributions per component.
2
In the objective function of the linear program, the costs that are incurred to the firm are represented as the
financial contributions per product. However, given that the constraint limits the number of components allocated to
affected products and that the number of components dictates how many of each of the affected products are made,
the costs of the linear program formulation are the financial contributions per component. The cost coefficients that
appear in the objective function are derived costs.
47
4.3.3 Marginal Financial Return
Marginal financial return represents the change in the firm’s financial results with respect to a
change in the partial rate of supply, over the firm’s CIT period. This can be considered the
derivative of financial return with respect to the partial rate of supply.
Equations (4-5) and (4-6) express financial returns under fair and preferential product
allocation policies, respectively.
𝑛
𝐹𝑅𝑉𝐴𝑅 = 𝑅 ∙ 𝐶𝐼𝑇𝑖 ∙ ∑ 𝐷𝑖 𝐹𝑖
𝑛
(4-5)
𝑖=1
𝐹𝑅𝑀𝑉𝐴𝑅 = ∑(𝐶𝐼𝑇𝑖 ∙ 𝑋𝑖 ∙ 𝐹𝑖 )
(4-6)
𝑖=1
Where:
 𝑅 is the partial rate of supply; the percentage of the normal rate of supply of the disrupted
component available during the firm’s TTR.

𝐶𝐼𝑇𝑖 is the customer impact time for finished product 𝑖.

𝐹𝑖 is the financial value per unit of finished product 𝑖.

𝑋𝑖 is the number of products 𝑖 produced per day given the allocation of parts from the
available partial rate of supply.
Given these expressions, marginal financial return becomes the ratio between the
difference in financial gain at any two partial rates of supply (𝐹𝑅1 and 𝐹𝑅2 ) and the difference
between the two partial rates of supply (𝑅1 and 𝑅2 ), as shown in equation (4-7).
∆𝐹𝑅
𝐹𝑅2 − 𝐹𝑅2
=
∆𝑅
100 ∙ (𝑅2 − 𝑅1 )
(4-7)
Under preferential allocation, marginal financial return is always decreasing as the partial
rate of supply increases. It follows that increases in the partial rate of supply when it is low, pose
greater benefits than increasing the partial rate of supply when it is high. When the partial rate of
supply is low, only products with the greatest financial contribution are allocated components.
48
When the partial rate of supply is small, the value of having each component is very high. As the
partial rate of supply increases, more products are allocated components and the additional
products being allocated components do not have financial contributions that are as large,
thereby driving the overall value of each additional component down. The following example
illustrates this effect.
There are ten products, each demanded at the same rate of 10 products per day. Each
product requires one component, whose supply has been disrupted. The financial value of each
product differs. The characteristics of each product are provided in Table 4-2.
Product ID
1
2
3
4
5
6
7
8
9
10
Product Financial Value
𝑭𝒊
(value/product)
10
9
8
7
6
5
4
3
2
1
Product Demand
𝑫𝒊
(products/day)
10
10
10
10
10
10
10
10
10
10
Usage Rate
𝒖𝒊
(parts/product)
1
1
1
1
1
1
1
1
1
1
Table 4-2: Product Characteristics (Marginal Financial Return Example)
Given this information, on any day, 100 components would be required to satisfy demand
for all ten products. If there was a 10% partial rate of supply, then only ten components would be
available. Under preferential allocation, all ten of these components would be allocated to
product #1, which has the highest financial contribution per component. The total financial
return in this scenario is 100. The marginal financial return of having a partial rate of supply of
10%, as opposed to 0%, is 10. If the available partial rate of supply was 20%, then twenty parts
would be available for allocation. Under preferential allocation, all products would be allocated
49
to products #1 and #2 to satisfy their demand. The total financial return in this scenario is 190.
The marginal financial return of having a partial rate of supply of 20%, instead of 10%, is 9. The
marginal financial return for each 10% increase in the partial rate of supply from 0% to 60% is
shown in Table 4-3.
Partial rate
of Supply
𝑹
(parts/day)
Financial
Result
𝑭𝑹
(value/day)
0%
0
10%
Change in
Financial Result
∆𝑭𝑹
Change in Partial
Rate of Supply
∆𝑹
100
100 − 0 = 100
10% − 0% = 10%
20%
190
190 − 100 = 90
20% − 10% = 10%
30%
270
270 − 190 = 80
30% − 20% = 10%
40%
340
340 − 270 = 70
40% − 30% = 10%
50%
400
400 − 340 = 60
50% − 40% = 10%
60%
450
450 − 400 = 50
60% − 50% = 10%
Marginal Financial
Return
∆𝑭𝑹
∆𝑹
(value/partial rate)
100
= 10
10
90
=9
10
80
=8
10
70
=7
10
60
=6
10
50
=5
10
Table 4-3: Decreasing Marginal Financial Return under Preferential Product Allocation
4.4 Comparison of Risk under Fair and Preferential Allocation
The example in the previous chapter can be used to evaluate preferential allocation in terms of
the MVAR and mitigation factor metrics. The values of MVAR and VAR can also be compared to
determine the difference in the financial loss to the firm when fair allocation is used versus when
preferential allocation is used.
Recall that a disruption in the supply for a given component has affected three products.
At the time of disruption, component inventory-on-hand is 30,000 components. The partial rate
of supply is 40% and the firm’s TTR is 6 days. The characteristics of the affected products are
found in Table 4-4.
50
Total Product Demand 𝑫𝒊
Usage Rate 𝒖𝒊
Finished Product Inventory on Hand 𝑷. 𝑰𝒏𝒗𝒊
Financial Value 𝑭𝒊
𝑭
Financial Contribution per component 𝒖𝒊
𝒊
Product 1
5,000
4
3,750
5
Product 2
2,000
8
1,500
8
Product 3
1,000
4
750
6
1.25
1
1.5
Table 4-4: Finished Product Characteristics (Preferential Allocation Example)
Since
component
inventory-on-hand,
finished
product
inventory-on-hand,
and
components accumulated from the partial rate of supply will be allocated fairly, the CIT values
for each product in the VAR and MVAR calculations are identical. The CIT for all finished
products is 3.5 days.
When the preferential product allocation policy is applied, the partial rate of supply is
allocated first to the product with the highest financial contribution per component. In this case,
product 3 has the greatest financial contribution per component, followed by product 1, and then
product 2.
Using the greedy algorithm, the following logic is applied. The total number of
components required per day to completely fulfill demand is 40,000. Since the partial rate of
supply is 40%, there are 16,000 components available per day to be allocated to the finished
products. Product 3 has the greatest financial contribution per component and requires 4,000
components per day. After this allocation, 1,000 units of product 3 can be produced per day,
leaving no unmet demand during the CIT period. 12,000 components per day remain for
allocation to products 1 and 2. Product 1 has the next greatest financial contribution per
component and requires 20,000 components per day. Product 1 is allocated the remaining
available components, which enables 3,000 of product 1 to be produced per day. All of product
2’s demand remains unmet during the CIT period. Figure 4-1 depicts the allocation of the partial
51
rate of supply and the remaining unmet demand during the CIT period when preferential product
allocation is used in this example.
Figure 4-1: Unmet Demand after Preferential Product Allocation of Partial Rate of Supply
Using the following linear program, the same results can be attained.
𝑛
min 𝐶𝐼𝑇 ∙ ∑(𝐷𝑖 − 𝑋𝑖 ) ∙ 𝐹𝑖
𝑖=1
= min[3.5 ∙ ((5,000 − 𝑋1 ) ∙ 5 + (2,000 − 𝑋2 ) ∙ 8 + (1,000 − 𝑋3 ) ∙ 6)]
Subject to:
(1.1) 5,000 − 𝑋1 ≥ 0
(1.2) 2,000 − 𝑋2 ≥ 0
(1.3) 1,000 − 𝑋3 ≥ 0
(2) 40% −
4 ∙ 𝑋1 + 8 ∙ 𝑋2 + 4 ∙ 𝑋3
≥0
4 ∙ 5,000 + 8 ∙ 2,000 + 4 ∙ 1,000
52
(3.1) 𝑋1 ≥ 0
(3.2) 𝑋2 ≥ 0
(3.3) 𝑋3 ≥ 0
During the CIT period, the allocation of the partial rate of supply under preferential
product allocation enables the production of 3,000 units of product 1 (𝑋1 = 3,000), 0 units of
product 2 (𝑋2 = 0), and 1,000 units of product 3 (𝑋3 = 1,000). Thus, MVAR is 91,000 financial
units, as shown below.
𝑛
𝑀𝑉𝐴𝑅 = 𝐶𝐼𝑇 ∙ ∑(𝐷𝑖 − 𝑋𝑖 ) ∙ 𝐹𝑖
𝑖=1
= 3.5 ∙ ((5,000 − 𝑋1 ) ∙ 5 + (2,000 − 𝑋2 ) ∙ 8 + (1,000 − 𝑋3 ) ∙ 6)
= 3.5 ∙ ((5,000 − 3,000) ∙ 5 + (2,000 − 0) ∙ 8 + (1,000 − 1,000) ∙ 6)
= 3.5 ∙ (10,000 + 16,000) = 91,000
Recall that when fair allocation was used, VAR was 98,700, as shown below:
𝑛
𝑉𝐴𝑅 = (1 − 𝑅) ∙ 𝐶𝐼𝑇 ∙ ∑ 𝐷𝑖 𝐹𝑖 = (1 − 0.4) ∙ (3.5) ∙ (5,000 ∙ 5 + 2,000 ∙ 8 + 1,000 ∙ 6)
𝑖=1
= 0.6 ∙ 3.5 ∙ 47,000 = 98,700
Given the values of VAR and MVAR, the mitigation factor in this example is 7.8%
(calculations shown below). This indicates that value-at-risk is reduced by 7.8% when using
preferential product allocation instead of fair allocation.
𝑀𝐹 = 1 −
𝑀𝑉𝐴𝑅
91,000
= 1−
= 0.078
𝑉𝐴𝑅
98,700
4.5 Risk Metric Relationships
The risk metrics used to evaluate preferential allocation are affected by the partial rate of supply
and the coefficient of variation of the financial contribution per component for a set of products
53
(denoted by CVFC). The coefficient of variation is a measure of the dispersion of the financial
contribution per component.
To explore the sensitivity of the metrics outlined above and to develop some insight, a set
of data was simulated and the optimization and the metrics were applied to the simulated case.
The simulation involves a single component disruption affecting two hundred products. For
simplification, no finished product or component inventory-on-hand was assumed to be
available. Under these circumstances, CIT is equal to the firm’s TTR at all partial rates of supply.
The financial contribution per component of each of the affected products was assumed
to come from a truncated normal distribution, centered at 100. To evaluate the effects of different
coefficients of variation, five standard deviation values: 10, 20, 30, 40, and 50, representing
coefficients of variation between 0.1 and 0.5, were used. These values were somewhat
representative of an actual firm data set used later. For each distribution, with the same mean and
different standard deviation, five hundred samples were drawn; from each sample, two hundred
values were drawn, each representing the financial value per component of an affected product.
This number of repetitions was more than enough to get stable results. The average value of each
metric across the five hundred samples, at a specified partial rate of supply, were used to
determine the effects of the coefficient of variation and the partial rate of supply on the risk
metrics. Demand for the products and the usage rate of the component in each product were held
constant.
4.5.1 MVAR Relationships
Figure 4-2 shows the inverse relationship between financial loss and the partial rate of supply
under the preferential allocation policy. Note that when the CVFC = 0, MVAR and VAR are the
same at all partial rates of supply, meaning that for zero coefficient of variation, the results can
54
be interpreted as pertaining to fair allocation. Under both policies, financial loss decreases with
increasing partial rates of supply. Increases in the partial rate of supply result in more
components available for allocation, thereby fulfilling more demand and decreasing financial
loss.
Figure 4-2: MVAR vs. Partial Rate of Supply
Figure 4-2 also shows the inverse relationship between MVAR and the coefficient of
variation of financial contribution per component. When the partial rate of supply is held
constant, increases in CVFC result in a decrease in the value of MVAR. The preferential allocation
policy takes advantage of the spread and range of financial contributions per component under
different coefficients of variation. Distributions with a high CVFC have more products with
greater financial contributions per component compared to distributions with a low CVFC. When
few components are available, components are allocated to products with the greatest financial
55
contributions per component. This effect is more pronounced when the distribution has a high
coefficient of variation.
Figure 4-3 shows MVAR at all partial rates of supply for CVFC = 0 (where the results are
the same as with fair allocation) and for CVFC = 0.5. The strongest effect of preferential
allocation occurs when the difference between financial losses under preferential and fair
allocation is maximized. This difference is zero when the partial rate of supply is zero. The
difference then increases to a maximum value; beyond that point, the difference between
financial loss under fair and preferential allocation decreases until financial loss under both
policies is zero, when the partial rate of supply is 100%.
Figure 4-3: Maximal Effect of Preferential Allocation
4.5.2 Mitigation Factor Relationships
Recall the definition of the mitigation factor, which measures the relative benefit of allocating
the partial rate of supply preferentially instead of fairly. The mitigation factor increases with
56
increasing partial rates of supply and/or increasing coefficients of variation of the financial
contribution per component. The exception is when CVFC = 0, in which case the financial losses
are identical under both policies and the mitigation factor remains at 0 despite increases in the
partial rate of supply.
While both MVAR and VAR are inversely proportional to the partial rate of supply, as this
rate increases, MVAR decreases at a faster rate than VAR, thereby increasing the mitigation
factor as R grows. Under preferential product allocation, components are allocated to products
with the greatest financial contribution per component, which reduces financial loss by more
than allocating components fairly, thereby increasing mitigation factor values as the partial rate
of supply increases. Increases in CVFC also result in increases in the mitigation factor. Since
distributions with larger CVFC have more products with relatively higher financial contributions
per component, as compared with distributions with smaller coefficients of variations, mitigation
factor values are greater when CVFC is higher.
The effect of CVFC on the mitigation factor can be illustrated with the following example.
Assume that there are three sets of products, each set containing seven products. Assume that
one component is required to produce a single unit of each of the finished products. Under this
assumption, the financial contribution per component is equivalent to the financial contribution
per product. Also assume that demand for each product is ten units. The financial contribution
per component for each product in each set is shown in Table 4-5. The financial contribution per
component of each set of products is derived from a discrete uniform distribution.
57
Product #
1
2
3
4
5
6
7
Financial Contribution per Component
Set A
Set B
Set C
1
7
13
2
8
14
3
9
15
4
10
16
5
11
17
6
12
18
7
13
19
Table 4-5: Product Characteristics (Mitigation Factor Example)
The standard deviation of financial contribution per component is the same for each set of
products, but the mean varies as shown in Table 4-6.
Mean Financial Contribution per
Component
Standard Deviation of Financial
Contribution per Component
Coefficient of Variation of Financial
Contribution per Component
Set A
Set B
Set C
4
10
16
2
2
2
0.500
0.200
0.125
Table 4-6: Product Set Mean, Variance, CVFC (Mitigation Factor Example)
For each set of products, seventy products are required to completely fulfill demand,
assuming the CIT is one day. Assume that the partial rate of supply is 10% resulting in seven
available components. Under fair allocation, each product would receive one component and the
VAR of each set of products would be as follows:
7
𝑉𝐴𝑅 = (1 − 𝑅) ∙ 𝐶𝐼𝑇 ∙ ∑ 𝐷𝑖 𝐹𝑖
𝑖=1
𝑉𝐴𝑅𝑆𝑒𝑡𝐴 = (1 − 0.1) ∙ 1 ∙ (10 ∙ 1 + 10 ∙ 2 + 10 ∙ 3 + 10 ∙ 4 + 10 ∙ 5 + 10 ∙ 6 + 10 ∙ 7) = 252
𝑉𝐴𝑅𝑆𝑒𝑡𝐵 = (1 − 0.1) ∙ 1 ∙ (10 ∙ 7 + 10 ∙ 8 + 10 ∙ 9 + 10 ∙ 10 + 10 ∙ 11 + 10 ∙ 12 + 10 ∙ 13)
= 630
𝑉𝐴𝑅𝑆𝑒𝑡𝐶 = (1 − 0.1) ∙ 1 ∙ (10 ∙ 13 + 10 ∙ 14 + 10 ∙ 15 + 10 ∙ 16 + 10 ∙ 17 + 10 ∙ 18 + 10 ∙ 19)
= 1008
58
Under preferential allocation, the product with the highest financial contribution per
component will be allocated all seven available components. Seven units of finished product of
the product with the highest financial contribution are fulfilled, out of a demand of ten units, and
all other products have none of their demand fulfilled. MVAR for each set of products is shown
below:
7
𝑀𝑉𝐴𝑅 = 𝐶𝐼𝑇 ∙ ∑(𝐷𝑖 − 𝑋𝑖 ) ∙ 𝐹𝑖
𝑖=1
𝑀𝑉𝐴𝑅𝑆𝑒𝑡𝐴 = 1 ∙ [(10 − 0) ∙ (1 + 2 + 3 + 4 + 5 + 6) + (10 − 7) ∙ 7] = 231
𝑀𝑉𝐴𝑅𝑆𝑒𝑡𝐵 = 1 ∙ [(10 − 0) ∙ (7 + 8 + 9 + 10 + 11 + 12) + (10 − 7) ∙ 13] = 609
𝑀𝑉𝐴𝑅𝑆𝑒𝑡𝐶 = 1 ∙ [(10 − 0) ∙ (13 + 14 + 15 + 16 + 17 + 18) + (10 − 7) ∙ 19] = 987
Given the VAR and MVAR values, the mitigation factor for each set of products is as
follows:
𝑀𝐹𝑆𝑒𝑡𝐴 = 1 −
231
̅̅̅̅̅
= 0.08333
252
𝑀𝐹𝑆𝑒𝑡𝐵 = 1 −
609
̅̅̅̅̅̅̅
= 0.03333
630
𝑀𝐹𝑆𝑒𝑡𝐶 = 1 −
987
= 0.02083̅
1008
In this example, product set A had the greatest CVFC and the greatest mitigation factor.
Product set C had the lowest CVFC and the lowest mitigation factor.
The relationships between mitigation factor and the partial rate of supply and coefficient
of variation are seen in Figure 4-4. At any partial rate of supply, the greater the CVFC, the greater
the mitigation factor. Other than when CVFC = 0, an increase in the partial rate of supply results
in an increase in the mitigation factor for any given coefficient of variation. When CVFC = 0,
MVAR and VAR are equivalent, so the mitigation factor is zero at all partial rates of supply.
59
Figure 4-4: Mitigation Factor vs. Partial Rate of Supply
Figure 4-5 shows the mitigation factor at all partial rates of supply (Recall that the
mitigation factor is undefined when 𝑅=100%, because both VAR and MVAR are 0). As expected
the mitigation factor increases as the partial rate of supply increases at all partial rates of supply.
Across the entire range of partial rates of supply, the relationship between the mitigation factor
and CVFC still holds; the greater the CVFC, the greater the mitigation factor.
60
Figure 4-5: Mitigation Factor vs. Partial Rate of Supply (at all partial rates of supply)
4.5.3 Marginal Financial Return Relationships
As defined previously, marginal financial return represents the change in the firm’s financial
results with respect to a change in the partial rate of supply, over the firm’s CIT period.
Marginal financial return under fair allocation is unaffected by increases in the partial rate of
supply or the CVFC.
Under preferential allocation, marginal financial return decreases as the partial rate of
supply increases, as already discussed. In the case simulated here, the range is from 0% to 40%,
but will vary for different cases. The exact partial rate of supply, below which an increase in the
coefficient of variation of financial contribution per component yields an increase in marginal
financial return, is dependent on the distribution of financial contribution per component.
The relationship of marginal financial return with the partial rate of supply and CVFC is
shown in Figure 4-6. The relationship between the marginal financial return and partial rate of
61
supply is always decreasing when CVFC is greater than 0. Marginal financial return is greater
when CVFC is greater.
Figure 4-6: Marginal Financial Return vs. Partial Rate of Supply3
Figure 4-7 depicts the marginal financial return over the full range of partial rates of
supply. Under preferential allocation, the marginal financial return always decreases with
increasing partial rates of supply. Beyond a certain point, the marginal financial return decreases
as CVFC grows; at high partial rates of supply, components are used to satisfy demand of
products with relatively low financial contributions per components.
3
The marginal financial return versus partial rate of supply curve was developed based upon 1% increases in the
partial rate of supply
62
Figure 4-7: Marginal Financial Benefit vs. Partial Rate of Supply (at all partial rates of supply)4
The actual point of convergence of the marginal financial benefits in Figure 4-7,
occurring just before a partial rate of supply value of 50%, is due to the truncated normal
distribution from which the financial contribution per component values were sampled in this
analysis. While the distribution is not symmetric, the convergence point occurs close to a partial
rate of supply of 50% since the distributions used have relatively low CVFC. The highest CVFC is
0.5, and the probability of a negative financial contribution per component being sampled from
this distribution is less than 2.3%. Naturally, the convergence of marginal financial benefit
values across different coefficients of variation will depend on the underlying distribution.
4
The marginal financial return versus partial rate of supply curve was developed based upon 1% increases in the
partial rate of supply
63
5 Case Study
The proposed framework was used to analyze the impact of a supply disruption on a large food
products manufacturer. The firm has provided data on two components, henceforth referred to as
comp1 and comp2. Each component is sourced from a single supplier and is used in multiple
finished products. The firm has provided information regarding the financial value, demand, and
inventory-on-hand throughout the year for each product using these components. The company
also provided usage rates of each component in the finished products and component inventory
levels. None of the finished products use both components. Time-to-recovery data was not
available, owing to the analysis being conducted sans any disruption. Similarly partial rate of
supply data was not available for the same reason.
5.1 Comp1
This section details the demand and inventory of comp1 as well as the finished product
inventory, financial value, and usage rate of the products using comp1.
5.1.1 Demand
Total monthly demand for comp1 across the eight-five finished products using comp1 is shown
in Figure 5-1. Typically demand for components is between 150,000 to 450,000 units. The
average demand for comp1 across the year is 300,000 units. Fluctuations in demand patterns are
attributed to typical increases and decreases in demand for finished products.
64
Figure 5-1: Comp1 Monthly Demand
Of the eighty-five finished products, forty-two of them were demanded consistently in
every month of the year. On average, approximately 73% of products are demanded each month.
Most products have significant demand variations from month-to-month.
5.1.2 Component Inventory
The number of units of component inventory held at the beginning of each month is shown in
Figure 5-2. Total monthly demand, in components, is also shown in Figure 5-2 to compare
inventory and demand patterns. On average, the firm holds 235,000 components in inventory. In
the two months with the greatest demand, months 4 and 7, component inventory is far less than
what is demanded. Component inventory levels do not seem to be aligned with demand patterns.
65
Figure 5-2: Comp1 Monthly Component Inventory and Demand
5.1.3 Finished Product Days-of-Supply
The finished product days-of-supply varies widely between products and between months. In
month 1, the firm carries no finished product inventory. In the remaining eleven months, fortyfive products consistently have finished product inventory and nineteen never hold any finished
product inventory. The remaining twenty-one finished products fluctuate between having
inventory and not having any.
Figure 5-3 shows the percentage of finished products having days-of-supply between
specified ranges. Between month 2 and month 5, the breakdown of finished products falling into
each range of days-of-supply is roughly the same, with some changes in the number of finished
products with days-of-supply between 1,000 and 5,000. Between month 6 and month 9 there is a
decline in the percentage of products with no days-of-supply. The number of products that begin
to hold between 200 and 500 days-of-supply is greater in months 9, 10, and 11 compared to other
66
months. Between months 9 and 12 a few products have a stark increase in their days-of-supply,
where the days-of-supply is greater than 5,000.
Figure 5-3: Comp1 Monthly Percentage of Finished Products by Day-of-Supply
5.1.4 Usage Rate
The average number of units of comp1 used in affected finished products is 3.56 but varies from
0.3 to 12 units of comp1 per unit of finished product. Approximately 73% of products use four or
fewer units of comp1 in their finished products. 27% of products require between three and four
components per unit of finished product. There are seven products that require more than six
components per finished product. The percentage of finished products that fall into specified
ranges of usage rates is shown as a histogram in Figure 5-4. The actual usage rate of each
finished product is shown as a scatterplot, measured on the right hand side vertical axis.
67
Figure 5-4: Comp1 Percentage of Finished Products by Usage Rate
5.1.5 Financial Value
The financial value per unit of finished product varies between 0.1 and 8.25. The average
financial value is approximately 2.85. The majority of finished products have a financial value of
2.50. The percentage of finished products with the specified financial values per unit finished
product is shown in in Figure 5-5. The actual financial value per unit finished product for each
finished product is shown as a scatterplot, measured on the right hand side vertical axis.
68
Figure 5-5: Comp1 Percentage of Finished Products by Financial Value
The financial value per component ranges from 0.02 to 5.42. 67% of finished products
have financial values per component between 0.50 and 1.00, inclusively. Fewer than 10% of
finished products have financial values per component above 3.75. No products have a financial
value per component between 1.25 and 2.25. The standard deviation of average financial value
per component is 1.22 and the average is 1.26, yielding a coefficient of variation of 0.97. The
percentage of products that fall within specified ranges of financial value per component is
shown in Figure 5-6. The actual financial value per component for each finished product is
shown as a scatterplot, measured on the right hand side vertical axis.
69
Figure 5-6: Comp1 Percentage of Finished Products by Financial Value per Component
5.2 Comp2
This section details the demand and inventory of comp2 as well as the finished product
inventory, financial value, and usage rate of the products using comp2.
5.2.1 Demand
Total monthly demand for comp2 across the thirty-six finished products using comp2 is shown in
Figure 5-7. Average demand is 506,000 components, and varies between 356,000 and 660,000.
Demand between months is relatively stable, with some decreases in demand in months 2 and 6.
Aside from the increase in demand in month 10, between months 7 and 12, demand stays close
to the average demand.
70
Figure 5-7: Comp2 Monthly Demand
Sixteen finished products are demanded every month, but very few of them have stable
demand. Most products have large variations in demand between months. On average, 59% of
finished products are demanded each month, but this ranges from 45% to 70%.
5.2.2 Component Inventory
The number of units of component inventory held at the beginning of each month is significantly
less than total monthly demand for comp2. As shown in Figure 5-8, the magnitude of units of
component inventory is in the thousands. Demand, however, requires hundreds of thousands of
components. The number of units of comp2 available in inventory at the beginning of the month
varies widely from month-to-month. The average number of units held is 2,250 but can range
from 400 to 4,700. In months 1 and 3, the firm holds almost double the average inventory
amount. The lowest amount of inventory is held in month 6. Between months 9 and 12, there is a
gradual increase in the amount of inventory carried by the firm.
71
Figure 5-8: Comp2 Monthly Component Inventory
5.2.3 Finished Product Days-of-Supply
Finished product days-of-supply varies widely between products and between months. In month
1, the firm carries no finished product inventory. In the remaining eleven months, seventeen
products consistently have finished product inventory and nine products never have finished
product inventory.
Figure 5-9 shows the percentage of finished products having days-of-supply between
specified ranges. Between months 2 and 12, roughly the same the number of products that have
less than 100 but more than 0 days-of-supply is roughly the same. In months, 6, 7, and 8, fewer
products have no days-of-supply and a greater number of products hold more than 500 days-ofsupply compared to other months.
72
Figure 5-9: Comp2 Monthly Percentage of Finished Products by Day-of-Supply
5.2.4 Usage Rate
The number of components used per unit of finished product in affected products ranges from
0.38 to 0.63. The average usage rate is 0.47 components per unit of finished product. The
majority of finished products have a usage rate between 0.40 and 0.45. A large portion of
finished products, approximately 19%, also have usage rates between 0.55 and 0.60. The
percentage of finished products that fall into specified ranges of usage rates is shown as a
histogram in Figure 5-10. The actual usage rate of each finished product is shown as a
scatterplot, measured on the right hand side vertical axis.
73
Figure 5-10: Comp2 Percentage of Finished Products by Usage Rate
5.2.5 Financial Value
Finished products using comp2 have distinct financial values per unit of finished product of
either 0.75, 1.00, or 1.25. The average financial value per unit of finished is 1.05. 60% of
finished products have financial values of 1.00. The percentage of finished products having each
of the financial values per unit finished product is shown in Figure 5-11. The actual financial
value per unit finished product for each finished product is shown as a scatterplot, measured on
the right hand side vertical axis.
74
Figure 5-11: Comp2 Percentage of Finished Products by Financial Value
The financial value per component of affected finished products ranges from 1.35 to 2.85.
The majority of finished products have financial values per component between 2.25 and 2.50,
inclusively. The standard deviation of financial value per component is 0.41 and the average is
2.23, yielding a coefficient of variation of 0.19. The percentage of products that fall within
specified ranges of financial value per component is shown in Figure 5-12. The actual financial
value per component of each finished product is shown as a scatterplot, measured on the right
hand side vertical axis.
75
Figure 5-12: Comp2 Percentage of Finished Products by Financial Value per Component
76
6 Analysis & Results
The proposed framework was used to analyze the impact of a supply disruption to comp1 and
comp2. None of the finished products use both components so the analyses were conducted
independently. Since demand and inventory levels changed from one month to the next, it
provided an opportunity to test what will happen if disruptions were to take place at various time
through the year.
The first analysis conducted calculates the impact of a disruption for each component
assuming no component or finished product inventory, rather than using the actual inventories of
components and products. The firm’s TTR for both components is assumed to be 60 days. The
results were used to recommend the actual levels of component and finished product inventory
required to mitigate the firm’s financial risk. While the analyses of each component were
conducted independently, the results of each component were compared to validate the
sensitivity of metrics to the partial rate of supply and the CVFC determined in section 4.5.
A second analysis was conducted to evaluate the firm’s current risk to disruptions in
supply of comp2 and comp2 given the available inventory held by the firm. In this analysis, the
finished product inventory assumption is relaxed, to account for varying amounts of finished
product inventory in the data. The modified formulations can be found in Appendix A. For both
components, the firm’s TTR is assumed to 60 sixty days.
6.1 Analysis with No Inventory
Rather than using the actual inventories of components and products, the analysis described here
calculates the impact of a TTR=60 days’ disruption for each component assuming there is no
component or finished product inventory. The firm’s level of risk throughout the year was tested
by simulating a disruption occurring at the beginning of each month.
77
6.1.1 Financial Loss
Risk between months and between components differs significantly. The firm’s financial losses
under both fair and preferential allocation at varying partial rates of supply are shown in Table
6-1 and Table 6-2 for comp1 and comp2 respectively. The risk when the partial rate of supply is
zero is the worst case scenario for the firm; this is the financial loss incurred, assuming a TTR of
60 days, if the firm has no inventory and no partial rate of supply. The average financial loss,
across all months, in the worst case scenario is 541,000 financial units for a disruption in supply
of comp1 and 2,248,000 financial units for a disruption in supply of comp2. The month with the
lowest risk is month 5 for both components. The month with the highest risk is month 7 for
comp1 and month 9 for comp2. The month with the highest risk is considered the month with the
maximum potential for financial loss and the month with the lowest risk is considered the month
with the minimum potential for financial loss. Potential for financial loss is calculated as the
demand of affected products over the firm’s TTR multiplied by the financial value of the
products affected.
78
VAR
MVAR
VAR
2
MVAR
VAR
3
MVAR
VAR
4
MVAR
VAR
5
MVAR
VAR
6
MVAR
VAR
7
MVAR
VAR
8
MVAR
VAR
9
MVAR
VAR
10
MVAR
VAR
11
MVAR
VAR
12
MVAR
Fiscal Month
1
0%
472
472
417
417
564
564
576
576
412
412
575
575
632
632
489
489
582
582
617
617
580
580
572
572
10%
425
336
376
309
508
411
518
410
371
275
517
400
569
459
440
366
524
431
556
436
522
393
515
385
20%
378
287
334
265
452
351
461
350
330
234
460
340
505
392
391
313
466
368
494
372
464
333
457
327
30%
331
241
292
223
395
295
403
294
289
197
402
285
442
330
342
263
408
309
432
313
406
280
400
274
Partial Rate of Supply
40% 50% 60% 70%
283
236
189
142
200
163
126
90
250
209
167
125
186
151
117
83
339
282
226
169
245
199
153
110
346
288
230
173
244
198
153
109
247
206
165
124
163
132
103
74
345
287
230
172
236
189
144
103
379
316
253
190
273
218
166
119
293
245
196
147
218
176
135
96
349
291
233
175
257
207
159
114
370
309
247
185
259
210
161
115
348
290
232
174
232
187
144
103
343
286
229
172
228
184
141
101
Table 6-1: Comp1 Financial Loss (in 000s of dollars)
79
80%
94
57
83
52
113
70
115
70
82
47
115
66
126
76
98
62
116
73
123
74
116
67
114
65
90% 100%
47
0
27
0
42
0
26
0
56
0
34
0
58
0
34
0
41
0
22
0
57
0
30
0
63
0
35
0
49
0
30
0
58
0
35
0
62
0
36
0
58
0
33
0
57
0
32
0
Fiscal Month
VAR
1
MVAR
VAR
2
MVAR
VAR
3
MVAR
VAR
4
MVAR
VAR
5
MVAR
VAR
6
MVAR
VAR
7
MVAR
VAR
8
MVAR
VAR
9
MVAR
VAR
10
MVAR
VAR
11
MVAR
VAR
12
MVAR
0%
1,967
1,967
2,032
2,032
2,513
2,513
2,251
2,251
1,694
1,694
1,959
1,959
2,282
2,282
2,445
2,445
2,791
2,791
2,658
2,658
2,244
2,244
2,144
2,144
10%
1,770
1,729
1,829
1,783
2,262
2,204
2,026
1,968
1,524
1,477
1,763
1,711
2,054
1,991
2,200
2,139
2,512
2,442
2,392
2,328
2,019
1,969
1,930
1,881
20%
1,573
1,523
1,626
1,569
2,011
1,939
1,801
1,728
1,355
1,296
1,567
1,504
1,825
1,748
1,956
1,880
2,233
2,146
2,127
2,050
1,795
1,735
1,715
1,658
30%
1,377
1,321
1,423
1,359
1,759
1,681
1,576
1,496
1,186
1,120
1,372
1,303
1,597
1,514
1,711
1,630
1,954
1,860
1,861
1,778
1,571
1,509
1,501
1,439
Partial Rate of Supply
40% 50% 60%
1,180
983
787
1,122
925
732
1,219 1,016
813
1,152
949
749
1,508 1,257 ,1005
1,426 1,175
929
1,351 1,125
900
1,266 1,040
817
1,016
847
677
947
777
609
1,176
980
784
1,104
909
716
1,369 1,141
913
1,281 1,051
824
1,467 1,222
978
1,382 1,136
893
1,675 1,395 1,116
1,580 1,304 1,032
1,595 1,329 1063
1,513 1,250
990
1,346 1,122
897
1,285 1,062
841
1,286 1,072
858
1,225 1,012
802
70%
590
540
610
550
754
682
675
595
508
441
588
523
685
599
733
652
837
761
797
732
673
623
643
595
80%
393
349
406
351
503
437
450
374
339
274
392
332
456
375
489
412
558
492
532
476
449
405
429
388
90% 100%
197
0
158
0
203
0
163
0
251
0
201
0
225
0
166
0
169
0
120
0
196
0
153
0
228
0
173
0
244
0
186
0
279
0
222
0
266
0
219
0
224
0
188
0
214
0
181
0
Table 6-2: Comp2 Financial Loss (in 000s of dollars)
Financial losses under fair and preferential allocation policies for the month with the
highest risk and the month with the lowest risk are shown in Figure 6-1 and Figure 6-2 for
comp1 and comp2 respectively. The maximal effect of preferential allocation, when the
difference between MVAR and VAR is greatest, is marked with arrows on each figure.
From these figures, it is evident that a greater investment in mitigation strategies is
required to reduce the risk of comp2. The distance between the MVAR and VAR curves of comp1
is much greater than the distance between the MVAR and VAR curves of comp2. This suggests
that less partial rate of supply is required for comp1 to mitigate a greater amount of risk. The
results indicate that components with low CVFC pose the most risk. In contrast, components with
80
high potential for financial loss that also possess high CVFC pose lower risk because less
inventory and partial rates of supply would be required to mitigate avoid the same amount of
loss.
The results depicted in Figure 6-1 and Figure 6-2 are consistent with the relationships
between financial loss and the partial rate of supply and the CVFC seen using simulated data in
section 4.5.1. Financial loss under both allocation policies decreases as the partial rate of supply
increases. The effect of preferential allocation is more pronounced for comp1 than comp2 given
that the CVFC is greater for comp1 than for comp2; the distance between the fair and preferential
allocation curves is much greater for comp1 than comp2.
Figure 6-1: Comp1 Financial Loss vs. Partial Rate of Supply
81
Figure 6-2: Comp2 Financial Loss vs. Partial Rate of Supply
Points of pronounced change in the rate of impact of the partial rate of supply on
financial losses are visually seen as “kinks” in the MVAR curves. These points are the result of a
group of products having relatively high financial contribution per component and once that
demand is fulfilled using preferential allocation, other products have significantly lower
contribution per component. Visible kinks can be seen in Figure 6-1 at 5% partial rate of supply
in the highest risk month and 9% partial rate of supply in the lowest risk month.
Kinks in the MVAR curve of comp2 are not as obvious, but can be determined from
Figure 6-3, the plot of the first order finite difference of MVAR with respect to the partial rate of
supply. Each jump in the first order finite difference marks a point of pronounced change in the
rate of impact of the partial rate of supply on financial losses. In both the highest and lowest risk
months, a 10% partial rate of supply marks the first kink, representing the effect of fulfilling the
demand of products with the highest contributions.
82
Figure 6-3: Comp2 First Order Finite Difference of MVAR with respect to Partial Rate of Supply
Apart from visual inspection of the appropriate plot, the partial rate of supply required to
fulfill demand of the highest contributing products can be determined by taking the maximum of
the second order finite difference of MVAR with respect to the partial rate of supply, when the
partial rate of supply is less then what is required to achieve the maximal effect of preferential
allocation. (To achieve the maximal effect of preferential allocation, demand of all products with
the highest financial contributions per component must be satisfied. Thus, the initial pronounced
change in the reduction in loss must occur before the maximal effect of preferential allocation
occurs.) Bounding the range of partial rate of supply, over which the second order finite
difference of MVAR is taken, eliminates large variations in the financial contribution per
component for components with very low financial contribution.
The initial kink in the MVAR curve is significant in defining a mitigation strategy that
achieves the greatest effectiveness in mitigating financial loss.
83
6.1.2 Mitigation Factor
The mitigation factor shows again the benefits of using preferential allocation over fair
allocation. The mitigation factor at varying partial rates of supply is shown in Table 6-3 and
Table 6-4 for comp1 and comp2 respectively. A 90% partial rate of supply is required to achieve
the same benefit of preferential allocation for comp2 that is achieved with a 10% partial rate of
supply for comp1. This result is attributed to the CVFC of comp1 being much greater than that of
comp2. The slight variations in the mitigation factor values for the same component in each
month, at specified partial rates of supply, are attributed to variations in demand patterns of
Fiscal Month
affected products, which change from month to month.
1
2
3
4
5
6
7
8
9
10
11
12
0%
0
0
0
0
0
0
0
0
0
0
0
0
10%
0.21
0.18
0.19
0.21
0.26
0.23
0.19
0.17
0.18
0.21
0.25
0.25
20%
0.24
0.21
0.22
0.24
0.29
0.26
0.23
0.20
0.21
0.25
0.28
0.28
30%
0.27
0.24
0.25
0.27
0.32
0.29
0.25
0.23
0.24
0.28
0.31
0.31
Partial Rate of Supply
40%
50%
0.29
0.31
0.26
0.28
0.28
0.30
0.29
0.31
0.34
0.36
0.32
0.34
0.28
0.31
0.26
0.28
0.27
0.29
0.03
0.32
0.33
0.35
0.34
0.36
Table 6-3: Comp1 Mitigation Factor
84
60%
0.33
0.30
0.32
0.34
0.38
0.37
0.34
0.31
0.32
0.35
0.38
0.38
70%
0.36
0.33
0.35
0.37
0.40
0.40
0.37
0.34
0.35
0.38
0.41
0.41
80%
0.40
0.37
0.38
0.39
0.43
0.43
0.40
0.37
0.37
0.40
0.42
0.43
90%
0.43
0.39
0.39
0.41
0.46
0.48
0.45
0.39
0.39
0.42
0.44
0.45
Fiscal Month
1
2
3
4
5
6
7
8
9
10
11
12
0%
0
0
0
0
0
0
0
0
0
0
0
0
10%
0.02
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
0.03
20%
0.03
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.03
0.03
30%
0.04
0.04
0.04
0.05
0.05
0.05
0.05
0.05
0.05
0.04
0.04
0.04
Partial Rate of Supply
40%
50%
0.05
0.06
0.06
0.07
0.05
0.06
0.06
0.08
0.07
0.08
0.06
0.07
0.06
0.08
0.06
0.07
0.06
0.07
0.05
0.06
0.05
0.05
0.05
0.06
60%
0.07
0.08
0.08
0.09
0.10
0.09
0.10
0.09
0.08
0.07
0.06
0.06
70%
0.08
0.10
0.09
0.12
0.13
0.11
0.13
0.11
0.09
0.08
0.07
0.08
80%
0.11
0.14
0.13
0.17
0.19
0.15
0.18
0.16
0.12
0.11
0.10
0.10
90%
0.20
0.20
0.20
0.26
0.29
0.22
0.24
0.24
0.20
0.18
0.16
0.15
Table 6-4: Comp2 Mitigation Factor
Mitigation factor curves of comp1 and comp2 in the months with the highest and lowest
risk are shown in Figure 6-4. The “kink” in each mitigation factor curve is the same as that
identified in the corresponding financial loss graphic, but is more pronounced.
Figure 6-4: Comp1 and Comp2 Mitigation Factor vs. Partial Rate of Supply
85
Consistent with the results in section 4.5.2, the mitigation factor increases with increasing
partial rates of supply. The mitigation factor curves of comp1 lie completely above those of
comp2, owing to the effect of the CVFC. Differences in the mitigation factor curves for the same
component are caused by monthly variations in demand patterns of affected products. In spite of
this, mitigation factor values of comp1 in all months of the year are always greater than those of
comp2.
6.1.3 Marginal Financial Return
For both components, the marginal financial return metric depicts the extent to which
preferentially allocating the partial rate of supply yields greater financial results than fair
allocation. The marginal financial return, in thousands of dollars, under fair and preferential
allocation based on 10% increases in the partial rate of supply are shown in Table 6-5 and Table
6-6 for comp1 and comp2 respectively.
Fiscal Month
Fair
Allocation
1
2
3
4
5
6
7
8
9
10
11
12
4.7
4.2
5.6
5.8
4.1
5.7
6.3
4.9
5.8
6.2
5.8
5.7
0%
to
10%
13.7
10.8
15.4
16.6
13.7
17.4
17.3
12.4
15.1
18.1
18.7
18.6
10%
to
20%
4.9
4.4
6.0
6.0
4.1
6.0
6.8
5.3
6.3
6.4
5.9
5.8
20%
to
30%
4.5
4.2
5.6
5.6
3.7
5.5
6.2
5.0
5.9
5.9
5.4
5.3
Preferential Allocation
30%
40%
50%
60%
to
to
to
to
40%
50%
60%
70%
4.1
3.8
3.7
3.6
3.7
3.5
3.4
3.3
5.0
4.7
4.5
4.3
5.0
4.6
4.5
4.3
3.5
3.0
2.9
2.9
4.9
4.7
4.5
4.1
5.7
5.4
5.2
4.7
4.5
4.2
4.1
3.9
5.3
4.9
4.8
4.5
5.3
5.0
4.8
4.6
4.7
4.5
4.4
4.1
4.7
4.4
4.3
4.0
Table 6-5: Comp1 Marginal Financial Return (in 000s of dollars)
86
70%
to
80%
3.3
3.1
4.0
4.0
2.7
3.7
4.3
3.4
4.1
4.1
3.6
3.6
80%
to
90%
3.0
2.7
3.6
3.6
2.4
3.6
4.2
3.2
3.8
3.8
3.5
3.4
90%
to
100%
2.7
2.6
3.4
3.4
2.2
3.0
3.5
3.0
3.5
3.6
3.3
3.2
Fiscal Month
Fair
Allocation
1
2
3
4
5
6
7
8
9
10
11
12
19.7
20.3
25.1
22.5
16.9
19.6
22.8
24.4
27.9
26.6
22.4
21.4
0%
to
10%
23.7
24.9
30.9
28.3
21.6
24.8
29.0
30.6
34.9
33.0
27.5
26.3
10%
to
20%
20.6
21.4
26.5
24.0
18.1
20.8
24.3
25.9
29.5
27.8
23.4
22.3
20%
to
30%
20.2
21.0
25.8
23.2
17.6
20.1
23.4
25.0
28.6
27.2
22.6
21.8
Preferential Allocation
30%
40%
50%
60%
to
to
to
to
40%
50%
60%
70%
19.9
19.7
19.3
19.2
20.7
20.4
20.0
19.9
25.5
25.1
24.7
24.6
23.0
22.6
22.3
22.2
17.4
17.0
16.8
16.7
19.9
19.6
19.3
19.2
23.3
23.0
22.7
22.5
24.9
24.6
24.3
24.1
28.0
27.6
27.2
27.1
26.5
26.3
26.0
25.8
22.4
22.2
22.1
21.8
21.5
21.3
21.0
20.7
70%
to
80%
19.1
19.9
24.5
22.1
16.7
19.2
22.4
24.0
27.0
25.7
21.8
20.7
80%
to
90%
19.1
18.8
23.6
20.8
15.4
17.8
20.1
22.6
27.0
25.7
21.7
20.6
90%
to
100%
15.8
16.3
20.1
16.6
12.0
15.3
17.3
18.6
22.2
21.9
18.8
18.1
Table 6-6: Comp2 Marginal Financial Return (in 000s of dollars)
Marginal financial return curves under fair and preferential allocation in the months with
the highest and lowest risk are shown in Figure 6-5 and Figure 6-6 for comp1 and comp2
respectively. The intersection point of the marginal financial return curves under fair and
preferential allocation is where the difference between the two allocation mechanisms is
maximized (this is also when the difference between VAR and MVAR is maximized). For
comp1, this occurs when the partial rate of supply is 26%, in the highest risk month, and 17%, in
the lowest risk month. For comp2, this occurs when the partial rate of supply is 40% and 52%, in
the months with the highest and lowest risk respectively.
87
Figure 6-5: Comp1 Marginal Financial Return vs. Partial Rate of Supply
Figure 6-6: Comp2 Marginal Financial Return vs. Partial Rate of Supply
88
As in the simulation, marginal financial return under fair allocation is constant and
decreasing under preferential allocation. A relationship between marginal financial return and
CVFC cannot be inferred from the results of applying the framework to the firm’s data. CVFC
relies heavily on the usage rate (components per unit of finished product). The usage rate of each
affected products was held constant in the simulation when analyzing different distributions.
However, the usage rate and financial contribution per unit finished product differs between
comp1 and comp2, thus the relationship between marginal financial return and CVFC cannot be
determined from the results using the firm’s data.
6.1.4 Risk Mitigation Recommendations
The results can be used to determine desired safety stock levels (previously denoted as 𝐶. 𝐼𝑛𝑣).
To do so, the firm can define an acceptable level of financial loss in the event of a disruption
(denote this by ALOSS). The safety stock required to ensure that this financial loss does not
exceed any specified threshold depends on the firm’s allocation policy. Let TLOSS denote the
total financial loss to the firm – the loss from the disruption when there is no available inventory
𝐴𝐿𝑂𝑆𝑆
or partial rate of supply. (1 − 𝑇𝐿𝑂𝑆𝑆) represents the percent reduction in financial loss. Equation
(6-1) defines the safety stock, in components, required if the firm allocates its inventory fairly
across all products in order not to exceed the acceptable loss. Under fair allocation, any
percentage reduction in financial loss requires safety stock equivalent to the same percentage of
the total components demanded over the firm’s TTR. In comparison, equation (6-2) defines the
safety stock, in components, necessary to mitigate risk when inventory is allocated preferentially.
The parameter 𝑘 in equation (6-2) is determined by equation (6-3). The acceptable loss under
preferential allocation is the sum of the number of components included in the highest
89
contributing products over the firm’s TTR. To determine 𝑘 , recall the hierarchy defined in
expression (4-3), where the series 𝐹(1) , 𝐹(2) , 𝐹(3) is defined by the relationships
𝐹(3)
𝑢(3)
𝐹(1)
𝑢(1)
𝐹(2)
≥𝑢
(2)
≥
𝐹(𝑛)
≥ ⋯ ≥ 𝑢 . To find the value of 𝑘 that achieves the acceptable loss threshold, one can sum
(𝑛)
the demand, starting with the highest contribution, until the financial contribution sum equals the
acceptable loss. These equations suggest safety stock in components, but the firm can choose
whether to hold safety stock as components or finished products.
𝑛
𝐴𝐿𝑂𝑆𝑆
𝐶. 𝐼𝑛𝑣 under Fair Allocation = (1 −
) ∙ 𝑇𝑇𝑅 ∙ ∑ 𝐷𝑖 𝑢𝑖
𝑇𝐿𝑂𝑆𝑆
(6-1)
𝑖=1
𝑘
𝐶. 𝐼𝑛𝑣 under Preferential Allocation = 𝑇𝑇𝑅 ⋅ ∑ 𝐷𝑖 𝑢𝑖
(6-2)
𝑖=1
Where 𝑚 is the first (lowest) 𝑘 such that the following equation is satisfied:
𝑇𝑇𝑅 ∙ (𝐷(1) 𝐹(1) + 𝐷(2) 𝐹(2) + 𝐷(3) 𝐹(3) + ⋯ + 𝐷(𝑚) 𝐹(𝑚) ) ≥ 𝐴𝐿𝑂𝑆𝑆
(6-3)
Where:
 𝐴𝐿𝑂𝑆𝑆 represents the acceptable level of financial loss in the event of a disruption.

𝑇𝐿𝑂𝑆𝑆 represents the total financial loss to the firm. This is the loss from the disruption
when there is no available inventory or partial rate of supply

∑𝑛𝑖=1 𝐷𝑖 𝑢𝑖 represents the total number of components required per day to fulfill total
demand of all affected products.

∑𝑘𝑖=1 𝐷𝑖 𝑢𝑖 represents the total number of components per day required to fulfill total
demand of the 𝑘 highest contributing products.
Any level of acceptable financial loss can be chosen; the value can be based on
mitigating risk up to the first “kink”, when the effectiveness of preferential allocation has the
largest change in magnitude, or achieving the highest effect of preferential allocation, when the
difference between VAR and MVAR is maximized. For the following example, the firm’s
acceptable loss is assumed to be 75% of its total loss; that is to assume that the firm wishes to
90
reduce its financial loss by 25%. To determine the safety stock required under preferential
allocation we also use the second inventory allocation assumption defined in section 3.3; that is,
we assume that the available safety stock is allocated throughout the firm’s TTR. (Recall that
under fair allocation both inventory allocation mechanisms will result in the same loss.)
Monthly total inventory levels required to mitigate total financial losses by 25% under
fair and preferential allocation are shown in Figure 6-7 and Figure 6-8 for comp1 and comp2
respectively. The figure demonstrates the impact of preferential allocation, which requires less
inventory to achieve the same mitigation of financial loss. The difference between inventory
levels under fair and preferential allocation is much greater for comp1 than for comp2. This is a
result of the CVFC of comp1 being much greater than that of comp2.
Figure 6-7: Comp1 Monthly Safety Stock to Mitigate 25% of Total Financial Loss
91
Figure 6-8: Comp2 Monthly Safety Stock to Mitigate 25% of Total Financial Loss
Recall that Figure 6-1and Figure 6-2 depicted financial loss vs. partial rate of supply. A
similar plot can be used to depict the percentage of total financial loss vs. safety stock in the
system for the case under consideration. Figure 6-9 depicts both the high and low risk months for
comp1 and comp2 under preferential allocation. The relationship between safety stock and the
percentage of total financial loss appears exponential for comp1 yet closer to linear for comp2.
The closer the relationship between safety stock and the percentage of total financial loss is to
linear, the greater the risk. Linearity suggests greater risk because more inventory is required to
reduce financial loss. When the relationship between inventory levels and the percentage
reduction in financial loss takes an exponential form, a significant amount of financial loss can
be reduced with a small amount of inventory. For comp1 in both months, 20% of financial loss
can be reduced by holding inventory equivalent to 5% of the total inventory required to eliminate
all financial losses. For comp2 in both months, 20% of financial loss is reduced by holding
92
inventory equivalent to almost 20% of the total inventory required to eliminate all financial
losses, negating the effect of preferential allocation. Naturally, this stems from the fact that
comp1 feeds products with a much higher coefficient of variation per component used.
Figure 6-9: Comp1 and Comp2 Percentage of Total Financial Loss vs. Safety Stock
The previous recommendations have focused on risk mitigation with safety stock.
However, at the time of the disruption, there may be some available partial rate of supply. In this
circumstance, the firm may choose to reduce the amount of safety stock it holds given its
expectation of some available partial rate of supply during a disruption.
Following the original assumptions of the framework, this dual mitigation strategy
(where the firm uses safety stock and also relies on some expected available partial rate of
supply) enables analysis of preferential and fair allocation schemes in concert. Recall that in the
original assumptions of the framework, safety stock is allocated fairly and the partial rate of
supply is allocated preferentially. Using a dual mitigation strategy, safety stock shortens the CIT
93
period and provides customers with some time to make contingency plans and the partial rate of
supply, allocated preferentially, increases the firm’s reduction in daily financial loss during the
CIT period.
Equation (6-4) defines the safety stock, in components, required to achieve an acceptable
level of financial loss (𝐴𝐿𝑂𝑆𝑆). In addition to defining an acceptable level of financial loss, the
firm must also specify the partial rate of supply it expects to be available. Given the specified
partial rate of supply (𝑅), parameter 𝑘 is determined by equation (6-5). The partial rate of supply
is the ratio between sum of the number of components included in the highest contributing
products and the number of components included in all affected products. To determine 𝑘, recall
the hierarchy defined in expression (4-3), where the series 𝑢(1) , 𝑢(2) , 𝑢(3) is defined by the
relationships
𝐹(1)
𝑢(1)
𝐹(2)
≥𝑢
(2)
𝐹(3)
≥𝑢
(3)
𝐹(𝑛)
≥ ⋯ ≥ 𝑢 . To find the value of 𝑘 that achieves the expected
(𝑛)
partial rate of supply, one can sum the number of components, starting with the highest
contribution, until the ratio of the number of components in the highest contributing products is
equal to the partial rate of supply. The parameter 𝑘 is also used to determined 𝑀𝑉𝐴𝑅𝑅 (see
equation (6-6)), the loss at the specified partial rate of supply, assuming there is only partial rate
of supply and no inventory. 𝑀𝑉𝐴𝑅𝑅 is the summation of the financial contributions of the
remaining 𝑛 − 𝑘 affected products over the firm’s TTR. The demand of these 𝑛 − 𝑘 will be
unfulfilled with the partial rate of supply
𝑛
𝐴𝐿𝑂𝑆𝑆
𝐶. 𝐼𝑛𝑣 = max [((1 −
) ∙ 𝑇𝑇𝑅 ∙ (∑ 𝐷𝑖 𝑢𝑖 ) ∙ (1 − 𝑅)) , 0]
𝑀𝑉𝐴𝑅𝑅
(6-4)
𝑖=1
Where 𝑚 is the highest value of 𝑘 such that the following equation is satisfied:
𝐷(1) 𝑢(1) + 𝐷(2) 𝑢(2) + 𝐷(3) 𝑢(3) + ⋯ + 𝐷(𝑚) 𝑢(𝑚)
≤𝑅
∑𝑛𝑖=1 𝐷𝑖 𝑢𝑖
94
(6-5)
𝑀𝑉𝐴𝑅𝑅 = 𝑇𝑇𝑅 ∙ (𝐷(𝑘+1) 𝐹(𝑘+1) + 𝐷(𝑘+2) 𝐹(𝑘+2) + ⋯ + 𝐷(𝑛) 𝐹(𝑛) )
(6-6)
Where:
 𝑀𝑉𝐴𝑅𝑅 represents MVAR at a specified partial rate of supply (when there is only a
partial rate of supply and no inventory).

𝐴𝐿𝑂𝑆𝑆
(1 − 𝑀𝑉𝐴𝑅 ) represents the proportion of the firm’s threshold of financial loss that
𝑅
requires mitigation with inventory.

𝑅 is the partial rate of supply; the percentage of the normal rate of supply of the disrupted
component available during the firm’s TTR.

∑𝑛𝑖=1 𝐷𝑖 𝑢𝑖 represents the total number of components required per day to fulfill total
demand of all affected products.

𝐷(1) 𝑢(1) + 𝐷(2) 𝑢(2) + 𝐷(3) 𝑢(3) + ⋯ + 𝐷(𝑘) 𝑢(𝑘) represents the total number of
components required per day to fulfill total demand of the 𝑘 highest contributing
products.

𝐷(𝑘+1) 𝐹(𝑘+1) + 𝐷(𝑘+2) 𝐹(𝑘+2) + ⋯ + 𝐷(𝑛) 𝐹(𝑛) represents the financial loss per incurred per
day by having a partial rate of supply that only fulfills demand of the 𝑘 highest
contributing products.
In equation (6-4), when the firm’s acceptable level of financial loss (𝐴𝐿𝑂𝑆𝑆) is equal to
the financial loss at a specified partial rate of supply (𝑀𝑉𝐴𝑅𝑅 ), the ratio between 𝐴𝐿𝑂𝑆𝑆
and 𝑀𝑉𝐴𝑅𝑅 is 1 and no safety stock is required to mitigate risk to an acceptable level since the
specified partial rate of supply, allocated preferentially, can mitigate all the acceptable risk. If the
ratio between the firm’s financial loss threshold and the financial loss at the specified partial rate
of supply is greater than 1, the selected partial rate of supply, allocated preferentially, mitigates
more risk than what is required to meet the firm’s acceptable financial loss. The required safety
stock to reduce financial loss to an acceptable level resulting from the equation (6-4) is zero.
The proportion of the firm’s acceptable amount of financial loss that requires mitigation
with safety stock is multiplied by TTR, to determine the number of days during the recovery
95
period that inventory must fulfill demand. This is then converted to components by multiplying
by the demand for components per day. Finally, the multiplication by (1 − 𝑅) takes into account
the components that accumulate from the partial rate of supply while inventory is expended,
which reduces the total number of components required in inventory.
For this example, assume again that the firm’s acceptable loss is 75% of its total loss; that
is, the firm wishes to reduce its financial loss by 25%. Figure 6-10 shows the combinations of
safety stock and the partial rate of supply required to reduce total financial loss by 25% in the
months with the highest and lowest risk for both components. The intersection of each of the
curves with the vertical axis indicates the amount of safety stock required to achieve acceptable
loss when there is no available partial rate of supply. This is the inventory requirement when the
firm chooses to mitigate its risk only through holding safety stock and allocates it fairly. The
intersection of each of the curves with the horizontal axis indicates the partial rate of supply
required to achieve the acceptable loss of the mitigation policy when there is no available
inventory.
96
Figure 6-10: Comp1 and Comp2 Safety Stock and Partial Rate of Supply to Mitigate 25% of Total Financial Loss
6.2 Analysis with Inventory
In this analysis, the firm’s current risk to disruptions in supply of comp1 and comp2 is evaluated.
The firm’s current component and finished product inventory levels are then included in the
analysis. To account for varying amounts of finished product inventory available in the data, the
finished product inventory assumption from the framework was relaxed. The modified
formulations for the framework can be found in Appendix A.
The firm’s level of risk throughout the year was evaluated by simulating a disruption
occurring at the beginning of each month. A TTR of 60 days was assumed for the analysis.
6.2.1 Financial Loss
Given the current levels of inventory the firm holds, the firm is susceptible to very little risk
when TTR = 60 days. The firm’s financial loss, in thousands of dollars, under fair and
97
preferential allocation at varying partial rates of supply are shown in Table 6-7 and Table 6-8 for
comp1 and comp2 respectively. The risk when the partial rate of supply is zero is the firm’s
expected financial loss during a disruption, assuming a TTR of 60 days, based on the firm’s
current inventory levels.
0%
1%
2%
3%
Partial Rate of Supply
4%
5%
6%
8%
9%
10%
200
195
191
186
181
177
172
167
162
200
187
175
165
155
147
139
134
129
Table 6-7: Comp1 Financial Loss (in 000s of dollars) [Analysis with inventory]
7%
158
125
152
121
Fiscal
VAR
1
Month
MVAR
0%
1
2
3
4
Fiscal Month
5
6
7
8
9
10
11
12
1%
2%
3%
Partial Rate of Supply
4%
5%
6%
7%
8%
1,957 1,938 1,918 1,898 1,879 1,859 1,839 1,820 1,800
1,957 1,932 1,908 1,883 1,858 1,833 1,809 1,784 1,762
209
189
169
149
128
108
88
68
48
209
183
158
132
107
88
71
55
39
244
213
181
150
118
87
55
24
0
244
212
180
149
117
85
54
22
0
74
58
41
25
7
0
0
0
0
74
52
34
19
5
0
0
0
0
117
104
90
77
64
50
37
24
10
117
97
79
64
50
39
28
18
7
216
197
178
159
140
121
102
83
63
216
193
171
149
129
109
89
70
51
141
122
102
83
64
44
23
2
0
141
115
92
69
49
32
16
1
0
71
43
14
0
0
0
0
0
0
71
42
13
0
0
0
0
0
0
153
121
88
54
20
0
0
0
0
153
121
87
53
19
0
0
0
0
158
129
99
67
36
4
0
0
0
158
127
96
65
34
4
0
0
0
208
186
163
140
117
94
71
47
23
208
182
157
131
106
82
61
40
19
114
91
68
45
21
0
0
0
0
114
89
65
42
19
0
0
0
0
Table 6-8: Comp2 Financial Loss (in 000s of dollars) [Analysis with inventory]
VAR
MVAR
VAR
MVAR
VAR
MVAR
VAR
MVAR
VAR
MVAR
VAR
MVAR
VAR
MVAR
VAR
MVAR
VAR
MVAR
VAR
MVAR
VAR
MVAR
VAR
MVAR
9%
10%
1,780
1,741
28
22
0
0
0
0
0
0
44
34
0
0
0
0
0
0
0
0
0
0
0
0
1,761
1,720
7
6
0
0
0
0
0
0
25
19
0
0
0
0
0
0
0
0
0
0
0
0
For comp1, the amount of component and finished product inventory held between
months 2 and 12 ensures that the firm will not have any financial loss for a disruption lasting 60
days. In month 1, the firm has some risk since it does not hold any finished product inventory.
98
In any month, a disruption to comp2, from which the firm recovers 60 days later, would
result in some financial loss. Similar to comp1, month 1 poses the greatest risk to the firm, since
the firm holds no finished product inventory in this month. Between months 2 and 12, a partial
rate of supply greater than 10% would ensure that all demand throughout the entire recovery
period is met.
Figure 6-11 shows the financial loss versus partial rate of supply under fair and
preferential allocation for both components. Financial loss for comp2 in month 2 is shown in the
figure to depict a case where there is both finished product and component inventory available.
Although there is no comparable case for comp1, where the firm has risk and holds both
component and finished product inventory, financial loss in month 1 for comp1 is included in the
figure. The results shown in the figure affirm the relationships of financial loss with the partial
rate of supply and with CVFC determined in section 4.5.1. The inclusion of inventory in this
analysis does not affect these relationships. Financial loss under both allocation policies
decreases with increases in the partial rate of supply. The effect of preferential allocation is more
pronounced for comp1 than comp2 given that CVFC is greater for comp1 than for comp2.
99
Figure 6-11: Comp1 and Comp2 Financial Loss vs. Partial Rate of Supply [Analysis with inventory]
6.2.2 Mitigation Factor
The mitigation factor at varying partial rates of supply for both components are shown in Table
6-9 and Table 6-10.
Fiscal
Month
1
0%
1%
2%
3%
0
0.043
0.082
0.114
Partial Rate of Supply
4%
5%
6%
0.143
0.167
0.190
7%
8%
9%
10%
0.199
0.203
0.207
0.211
Table 6-9: Comp1 Mitigation Factor [Analysis with inventory]
100
Fiscal Month
1
2
3
4
5
6
7
8
9
10
11
12
0%
0
0
0
0
0
0
0
0
0
0
0
0
1%
0.003
0.029
0.002
0.098
0.060
0.021
0.051
0.017
0.006
0.011
0.017
0.019
2%
0.005
0.064
0.004
0.175
0.121
0.041
0.102
0.045
0.010
0.02
0.038
0.046
3%
0.008
0.109
0.007
0.254
0.174
0.063
0.166
Partial Rate of Supply
4%
5%
6%
0.011 0.014 0.017
0.166 0.191 0.191
0.011 0.017 0.029
0.308
0.214 0.234 0.242
0.081 0.100 0.123
0.237 0.265 0.302
0.017
0.032
0.064
0.079
0.029
0.051
0.096
0.109
0.083
0.126
0.136
7%
0.020
0.191
0.056
8%
0.021
0.191
9%
0.022
0.191
10%
0.023
0.191
0.251
0.156
0.357
0.271
0.189
0.223
0.263
0.149
0.176
Table 6-10: Comp2 Mitigation Factor [Analysis with inventory]
The analysis with inventory results in greater variation in the mitigation factor values for
comp2 between months, at a specified partial rate of supply, than in the analysis without
inventory. In the analysis without inventory, the variations in mitigation factor values were
attributed to variations in demand patterns of affected products. While this is true for this
analysis, finished product inventory also affects the variation in mitigation factor. The amount of
finished product inventory for each affected product determines which products require
components from the partial rate of supply during the CIT period. Finished products with
inventory extending beyond the TTR will not need any components from the partial rate of
supply while other products will require components from the partial rate of supply. Depending
on the financial value per component of the products requiring components from the partial rate
of supply, the preferential allocation of the partial rate of supply may lead to greater reductions
in financial loss in some months compared to others.
The effect of finished product inventory on the mitigation factor also makes it difficult to
quantify the relationship between the mitigation factor and CVFC. When inventory is unavailable,
the greater CVFC values result in greater mitigation factor values. While comp1 has a greater
CVFC than comp2, the mitigation factor of comp2 can sometimes exceed that of comp1 at the
101
same partial rate of supply. For example, when the partial rate of supply is 5%, comp1 has a
mitigation factor of 0.17 in month 1. At the same partial rate of supply, in three out of eight
months, the mitigation factor for comp2 is greater than that of comp1. This is caused by varying
finished product inventories. The allocation of components from the partial rate of supply during
the CIT period is only to those products that no longer have inventory extending beyond the start
of the CIT period. This may only be a fraction of affected products, and the coefficient of
variation of only these products would be different than the whole set of products using the
component.
6.2.3 Marginal Financial Return
Marginal financial return, in thousands of dollars, under fair and preferential allocation, based on
1% increases in the partial rate of supply between 0% and 10% are shown in Table 6-11 and
Table 6-12 for comp1 and comp2 respectively.
Fiscal
Month
1
Fair
Preferential
0%
to
1%
1.97
10.32
1%
to
2%
1.92
9.39
2%
To
3%
1.86
7.31
Partial Rate of Supply
3%
4%
5%
6%
to
to
to
to
4%
5%
6%
7%
1.80 1.74
1.68 1.61
6.46 5.30
4.85 2.20
7%
to
8%
1.54
1.22
Table 6-11: Comp1 Marginal Financial Return [Analysis with inventory]
102
8%
to
9%
1.47
1.13
9%
to
10%
1.40
1.03
Fair
Preferential
Fair
2
Preferential
Fair
3
Preferential
Fair
4
Preferential
Fair
5
Preferential
Fair
6
Preferential
Fair
7
Preferential
Fair
8
Preferential
Fair
9
Preferential
Fair
10
Preferential
Fair
11
Preferential
Fair
12
Preferential
Fiscal Month
1
0%
to
1%
19.6
24.7
19.6
25.1
29.0
29.4
15.3
21.0
12.9
19.1
19.0
23.2
18.1
24.3
10.9
11.6
19.3
20.0
25.3
26.7
22.2
25.4
21.1
22.8
1%
to
2%
19.6
24.7
19.4
24.8
28.3
28.6
14.2
15.8
12.7
17.4
19.0
22.0
17.4
21.6
-1.4
-1.5
9.8
10.0
10.2
10.8
22.0
24.9
18.8
20.2
2%
To
3%
19.6
24.7
19.3
24.5
27.1
27.4
6.8
6.0
12.5
15.0
18.9
21.7
16.0
19.3
Partial Rate of Supply
3%
4%
5%
6%
to
to
to
to
4%
5%
6%
7%
19.6
19.6 19.6
19.6
24.7
24.7 24.6
24.6
19.0
18.6 17.9
16.7
24.2
17.9
14
12.8
25
21.1 12.2 -17.0
25.3
21.4 12.3 -17.3
-22.6
-26.7
12.1
11.3
9.4
3.2
12.3
9.4
6.6
0.2
18.9
18.8 18.6
18.3
20.1
19.5 19.1
18.6
11.4
-2.9 -16.9 -38.0
12.8
-6.3 -21.6 -44.4
2.2
2.3
4.9
5.0
21.4
24.2
3.3
3.7
-12.1
-12.5
-5.3
-5.6
20.2
22.5
-15.6
-16.8
-28.6
-30.1
16.7
17.3
-5.5
-7.7
-15.4
-18
7%
to
8%
19.6
21.8
14
10.2
8%
to
9%
19.6
20.9
6.6
2.7
9%
to
10%
19.6
20.6
-35.4
-39.3
-28.6
-31.7
17.3
16.3
11.9
9.9
-72.4
-75.7
-21.4
-24.4
Table 6-12: Comp2 Marginal Financial Return [Analysis with inventory]
There are two differences in the marginal financial returns resulting from an analysis
without inventory and an analysis with inventory. The first is negative marginal financial
benefits which occur under both allocation policies at certain partial rates of supply in the
analysis with inventory. The second difference between the analyses is the decreasing marginal
financial benefit under fair allocation when inventory is considered. (Recall that without
inventory, marginal financial benefit is constant under fair allocation as the partial rate of supply
increases.) These two differences are attributed to two factors. This is attributed to two factors.
The first is a reduction of CIT. As the partial rate of supply increases, there is a greater
accumulation of components while components and finished product inventory are expended.
103
This extends the amount of time that all demand can be fulfilled, thereby lessening the time over
which losses can result. This has the effect of not only reducing VAR and MVAR but also
reducing financial return, since the CIT period is shorter. The second factor causing negative
marginal financial benefits is the allocation of the partial rate of supply to products with lower
financial contributions per component. As the partial rate of supply increases, there are more
components to allocate during the CIT period, which enables more lower-valued products to be
allocated components. Since these products do not produce much value, there are only slight
changes in MVAR.
104
7 Conclusions
In this thesis, a framework for evaluating supplier risk, under fair and preferential allocation
schemes, was established. In the proposed framework, component and finished product inventory
are allocated fairly among all affected products while the available partial rate of supply is
allocated either fairly or preferentially. The framework uses four metrics to evaluate risk. Valueat-risk, VAR, provides a measure of the financial loss given the allocation of partial rate of
supply fairly across all products. Modified value-at-risk, MVAR, measures financial loss given
preferential allocation, where products yielding the greatest value per component are allocated
components first. The mitigation factor measures the relative benefit of allocating the partial rate
of supply preferentially instead of fairly. Marginal financial return represents the change in the
firm’s financial results with respect to a change in the partial rate of supply.
In an analysis of simulated data, the value of preferential allocation and its potential to
reduce risk were observed to be dependent upon the available partial rate of supply and the
coefficient of variation of the financial value per component (CVFC) of affected products. This
analysis was based on the assumption that no inventory is available. MVAR was found to
decrease as the CVFC and/or the partial rate of supply increased. The mitigation factor increased
as the CVFC and/or the partial rate of supply increased. Under preferential allocation, marginal
financial benefit decreases as the partial rate of supply increases. Over a certain range, increases
in the CVFC result in increases in the marginal financial benefit under preferential allocation. The
range over which this relationship holds is dependent on the distribution of the coefficient of
variation for financial value per component.
When the framework was applied to actual data, the results demonstrated the importance
of the coefficient of variation of the financial contribution per component in utilizing effective
105
safety stock policies for risk mitigation. In particular, it demonstrates that when a component
feeds a set of products where the CVFC is low, more inventory will be required to achieve a given
reduction in the financial impact of a disruption. This approach provides another metric for
assessing risky components and suppliers who provide them.
The framework can also be used to determine and evaluate risk mitigation strategies. By
defining an acceptable level of financial loss, a firm can determine the amount of safety stock it
requires under preferential or fair allocation. The firm can also determine necessary safety stock
levels to mitigate risk if a certain partial rate of supply is expected during an actual disruption.
7.1 Further Research
There are many potential extensions of this research. The proposed framework evaluates risk for
a single component disruption and depends on some key assumptions regarding the availability
of substitutes and finished product inventory. The risk evaluation framework assumed that there
were no substitutes that could be used in place of the affected products. However, firms can often
sell other products, whose component supply has been unaffected, and achieve some degree of
substitutability. Modeling substitution must account for the financial value of the substitute.
Should a product have a substitute whose financial value is less than the financial value of the
product, then the incremental gain of allocating components to the product must be traded-off
with the gain of using the substitute and allocating components to another product.
A disruption also has the potential to affect the supply of many components
simultaneously. Extending this research to the case where multiple components are disrupted
would be effective. Research can further be extended if substitution and varying finished product
inventory are accounted for in the multiple component disruption case.
106
Generalized formulations of MVAR for single component disruptions accounting for
substitutes and multiple component disruption assuming no substitutes and constant finished
product inventory are included in the appendix.
Further extensions of the framework can be made by accounting for different forms of
inventory; engineering dilution solutions; and varying partial rates of supply throughout the
firm’s recovery horizon. At the time of disruption, a firm may have some work-in-process
inventory that may or may not need additional components whose supply has been disrupted.
Further, the number of components required to produce a finished product may depend on how
far along the work-in-process inventory is. The usage rate of the component in the product may
change depending on the form of inventory.
A similar modification to the framework could be made if dilution solutions are found
during the firm’s time-to-recovery. If a firm is able to reduce the number of the disrupted
component required to produce finished products, then the usage rate changes at a discrete point
in time which would change the allocation of partial rate of supply.
Finally, within the framework, the partial rate of supply was assumed to be available
from the time of disruption and the rate was assumed constant throughout the firm’s time-torecovery. In most cases, the partial rate of supply increases over time and may not be offered at
the time of the disruption. To model a changing partial rate of supply, discrete points in time
when there are changes to the partial rate of supply must be identified and then the changes to the
allocation of the newly available partial rate of supply must be taken into account.
107
Appendix A
This appendix provides a modification of the framework established in sections 3 and 4. The
original framework assumed that finished product inventory of all affected products was the
same. The following formulations are for a single component disruption where affected finished
products have varying amounts of finished product inventory. Alternative formulations for CIT,
VAR, and MVAR under the relaxation of the finished product inventory assumption are provided.
An alternative calculation of the marginal financial benefit metric is also shown.
CIT Formulation
Recall the definition of CIT, which measures the interval of time during which a purchasing
firm’s customers will be affected. When finished product inventory varies between products, the
CIT period can be determined by finding two intervals of time. The first is the critical time
(𝑡𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 ) representing the interval of time beginning at the time of disruption and ending when
all component inventory has been expended. The second time interval is the accumulation time
(𝑡𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 ) representing the interval of time beginning when all component inventory has
been expended and ending when all accumulated components from the partial rate of supply
have been expended. It is still assumed that component inventory and components accumulated
from the partial rate of supply are allocated fairly. The CIT period is the difference between the
firm’s TTR and the summation of the critical time and accumulation time, as shown below.
𝐶𝐼𝑇 = 𝐹. 𝑇𝑇𝑅 − 𝑡𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 − 𝑡𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛
(A-1)
To find CIT, time must be segmented into 𝑛′ discrete points. At any point 𝑛′, the given
finished product(s) 𝑛′ has expended its inventory. This point in time is equivalent to finished
product days-of-supply, the number of days that the available finished product inventory can
108
meet average demand for the product. The initial point in time is at the time of the disruption,
denoted by 𝑡𝑑 . The final point in time is when the firm has recovered from the disruption,
represented as 𝑡(𝑛′+1) . These discrete points in time are ranked such that:
𝑡𝑑 ≤ 𝑡(1) ≤ 𝑡(2) ≤ 𝑡(3) … ≤ 𝑡(𝑛′) ≤ 𝑡(𝑛′+1)
(A-2)
While there are 𝑛 affected finished products that require the component whose supply has
been disrupted, there are only 𝑛′ points in time that need to be considered. 𝑛′ represents the set of
finished products whose days-of-supply is less than the firm’s time-to-recovery which is a subset
of all affected finished products 𝑛. 𝑛′ will be less than 𝑛 if some of the finished products’ daysof-supply are greater than the firm’s time-to-recovery. 𝑛′ can also be less than 𝑛 if multiple
finished products have the same days-of-supply. 𝑛′ will be equal to 𝑛 if all finished products
have fewer days-of-supply than the firm’s time-to-recovery and the days-of-supply are unique.
CIT can be derived using the following variables:

𝐶. 𝐼𝑛𝑣 is number of components available at the time of the disruption.

𝐶. 𝑅𝑒𝑚 is the number of components remaining that have not yet been expended.

𝑡𝑑 is the start of the disruption.

𝑡𝑠𝑡𝑎𝑟𝑡 is the start of the interval of time being considered.

𝑡𝑒𝑛𝑑 is the end of the interval of time being considered.

𝑡𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 is the interval of time being considered.

𝑡𝑖 is the point in time when finished product 𝑖 has expended its inventory.

𝑡(𝑛′+1) is the point in time when the firm recovers from the disruption, 𝑡𝑟 .

𝐷𝑖 is the average demand of finished product 𝑖 by all customers per day, from the date of
the disruption (𝑡𝑑 ) until the firm recovers (𝑡𝑟 ). Where 𝑖 is from 1 to n.

𝑢𝑖 is the usage rate of components in finished products; the number of components
required to produce a single unit of finished product 𝑖.

𝑛′ represents the number of unique days-of-supply of finished products whose days-ofsupply is less than the firm’s time-to-recovery.
109

𝑧 and 𝑘 represent counters that iterates through the finished products who days-of-supply
is less than the firm’s time-to-recovery, based upon ascending finished product days-ofsupply.

𝑡𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 is the time at which all component inventory has been expended, given that the
components have been allocated fairly to any finished products that require components.

𝐹𝑃𝑛𝑒𝑥𝑡 represents the subsequent finished product whose finished product days-of-supply
will be expended after the end point of the critical time interval. This is a variable that
will be used to determine the accumulation time.

𝑅 is the partial rate of supply of the disrupted component available during the firm’s
time-to-recovery.

𝐴. 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 is number of components that have accumulated from the time of disruption
to the critical time.

𝐴. 𝑅𝑒𝑚 is the remaining number of accumulated components that have not yet been
expended.

𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛 is the number of days that the available accumulated parts can sustain the
demand at the interval being considered.

𝑡𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 is the time at which all accumulated components from the partial rate of
supply have been expended, given that they have been allocated fairly to any finished
products that require components.
Example
The following is an example that will be used to illustrate how CIT is calculated when finished
product inventory differs among finished products. Assume there are five products with the
following characteristics:
Total Daily Product Demand 𝑫𝒊
Usage Rate 𝒖𝒊
Finished Product Inventory on
Hand 𝑷. 𝑰𝒏𝒗𝒊
Product
A
20
5
Product
B
10
10
Product
C
20
10
Product
D
30
10
Product
E
20
15
20
20
80
120
120
Table A-1: Finished Product Characteristics (Single Component Disruption with Varied Finished Product Inventory
Example)
110
The disruption is assumed to occur at time 0, and the firm recovers 7 days afterward.
𝑡𝑑 = 0, 𝑡𝑟 = 7
𝐹. 𝑇𝑇𝑅 = 𝑡𝑟 − 𝑡𝑑 = 7 − 0 = 7
Finished product inventory differs between products, and each sustains average daily
demand for a different number of days. The finished product days-of-supply for each affected
product is shown below and in Figure A-1.
𝑃. 𝐼𝑛𝑣𝐴
20
=
=1
𝑚
∑𝑗=1 𝐷𝐴,𝑗 20
𝑃. 𝐼𝑛𝑣𝐵
20
𝑃. 𝐷𝑂𝑆𝐵 = 𝑚
=
=2
∑𝑗=1 𝐷𝐵,𝑗 10
𝑃. 𝐼𝑛𝑣𝐶
80
𝑃. 𝐷𝑂𝑆𝐶 = 𝑚
=
=4
∑𝑗=1 𝐷𝐶,𝑗 20
𝑃. 𝐼𝑛𝑣𝐷
120
𝑃. 𝐷𝑂𝑆𝐷 = 𝑚
=
=4
∑𝑗=1 𝐷𝐷,𝑗
30
𝑃. 𝐼𝑛𝑣𝐸
120
𝑃. 𝐷𝑂𝑆𝐸 = 𝑚
=
=6
∑𝑗=1 𝐷𝐸,𝑗
20
𝑃. 𝐷𝑂𝑆𝐴 =
Figure A-1: Allocation of Finished Product Inventory
111
In this example, there are six time points to consider, ranked as follows.
𝑡𝑑 ≤ 𝑡(1) ≤ 𝑡(2) ≤ 𝑡(3) ≤ 𝑡(4) ≤ 𝑡(𝑛′+1)
The first point in time is at the time of disruption, 𝑡𝑑 = 0. The second point in time is when
finished product A expends its finished product inventory, 𝑡(1) = 𝑃. 𝐷𝑂𝑆𝐴 = 1. The third point
in time is when finished product B expends its finished product inventory, 𝑡(2) = 𝑃. 𝐷𝑂𝑆𝐵 = 2.
The third point in time is when both finished products C and D expend their finished product
inventory, 𝑡(3) = 𝑃. 𝐷𝑂𝑆𝐶 = 𝑃. 𝐷𝑂𝑆𝐷 = 4. The fifth is when finished product E expends its
finished product inventory, 𝑡(4) = 𝑃. 𝐷𝑂𝑆𝐸 = 6. The final point in time is when the firm recovers
from the disruption, 𝑡(𝑛′ +1) = 𝑡𝑟 = 7.
Critical Time
The critical time represents the interval of time beginning at the time of disruption and ending
when all component inventory has been expended. It can be determined by iterating through each
interval in time that begins when a finished product expends its inventory and ends when the
subsequent finished product expends its inventory. The algorithm to determine critical time can
be programmatically represented as follows.
1
2
3
4
5
6
7
8
9
10
𝐶. 𝑅𝑒𝑚 = 𝐶. 𝐼𝑛𝑣
𝑡𝑠𝑡𝑎𝑟𝑡 = 𝑡𝑑
For 𝑧 = 1 to (𝑛′ + 1)
𝑡𝑒𝑛𝑑 = 𝑡𝑧
𝑡𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 = 𝑡𝑒𝑛𝑑 − 𝑡𝑠𝑡𝑎𝑟𝑡
If (∑𝑧𝑖=1 𝐷𝑖 𝑢𝑖 ) ∙ 𝑡𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 ≤ 𝐶. 𝑅𝑒𝑚 Then
𝐶. 𝑅𝑒𝑚 = 𝐶. 𝑅𝑒𝑚 − (∑𝑧𝑖=1 𝐷𝑖 𝑢𝑖 ) ∙ 𝑡𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
𝑡𝑠𝑡𝑎𝑟𝑡 = 𝑡𝑒𝑛𝑑
Else
𝐶.𝑅𝑒𝑚
𝑡𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = 𝑡𝑠𝑡𝑎𝑟𝑡 + ∑𝑧 𝐷 𝑢 − 𝑡𝑑
11
12
13
14
𝐹𝑃𝑛𝑒𝑥𝑡 = 𝑧
Exit For
End If
Next 𝑧
𝑖=1
𝑖 𝑖
112
Lines 1 and 2 are initializing statements. The first line initializes the remaining number of
components to the number of components available at the start of the disruption. The second line
initializes the start of the first interval being considered as the time of the disruption.
The counter z iterates through each of the finished products in order of ascending finished
product days-of-supply. The end of the time interval is set to the time at which the given finished
product has expended its finished product inventory.
In line 6, there is a check to see if the total number of components required to satisfy
demand over the time interval being considered is less than or equal to the remaining
components in inventory. If this is true, then total number of components required to satisfy
demand over the time interval being considered is deducted from the remaining components to
determine the new number of components remaining. The start of the next interval is then set to
the end of the current interval being considered.
If the total number of components required to satisfy demand over the time interval being
considered is greater than the remaining components in inventory, the critical time can be
computed as the summation of the start of the interval and the duration of time that the remaining
components can satisfy the average daily demand, less the time of the disruption. The duration of
time that the remaining components can satisfy average daily demand is the ratio of remaining
components to the average daily demand during the time interval being considered. The finished
product whose days-of-supply will be expended after the end of the critical time interval is stored
in the variable 𝐹𝑃𝑛𝑒𝑥𝑡 , to be used for determining the accumulation time.
113
Example
Assume that at the time of disruption there are 450 components of inventory available (𝐶. 𝐼𝑛𝑣 =
450). The critical time interval can be determined by iterating through time intervals. Details of
the demand and remaining component inventory at each time interval are shown in Table A-2.
Interval Interval
Time
Start
End
Interval
Time
Time
Demand for Components
Over Time Interval
𝑧
𝑡𝑠𝑡𝑎𝑟𝑡
𝑡𝑒𝑛𝑑
𝑡𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
(∑ 𝐷𝑖 𝑢𝑖 ) ∙ 𝑡𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
Components
Remaining
(at start of
time interval)
Components
Remaining
(at end of
time interval)
𝐶. 𝑅𝑒𝑚
𝐶. 𝑅𝑒𝑚
450
450
450
350
350
-
𝑖=1
0
1
1
2
1
1
2
4
2
0
(𝐷𝐴 𝑢𝐴 ) ∙ 1 = 20 ∙ 5 ∙ 1 = 100
(𝐷𝐴 𝑢𝐴 + 𝐷𝐵 𝑢𝐵 ) ∙ 2
= (20 ∙ 5 + 10 ∙ 10) ∙ 2 = 400
Table A-2: Time Intervals Considered for Determining Critical Time
The first time interval begins at the start of the disruption and ends when the finished
product A expends its finished product inventory. Since all finished products have at least 1 dayof-supply, none of the initial component inventory is used. At the next time interval, starting
when finished product A expends its finished product inventory and ending when finished
product B expends its finished product interval, only finished product A requires components.
Finished product A requires 100 components per day and the time interval is 1 day. By the end
of the time interval, 350 components from the original component inventory available remain.
The subsequent time interval starts when finished product B expends its finished product
inventory and ends when finished products C and D expend their finished product inventory.
Over this period, both finished product A and B require components. Finished products A and B
each required 100 components per day and the time interval is 2 days. Over the time interval of
2 days, a total of 400 components are required to fulfill 100% of demand until the end of the time
interval. This is not possible as only 350 components are available at the start of the interval.
114
The critical time can now be determined as follows:
𝑡𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = 𝑡𝑠𝑡𝑎𝑟𝑡 +
𝐶. 𝑅𝑒𝑚
350
350
− 𝑡𝑑 = 2 +
−0=2+
= 2.875
𝑧
∑𝑖=1 𝐷𝑖 𝑢𝑖
20 ∙ 5 + 10 ∙ 10
400
The critical time is the start of the interval, the second day, and the additional amount of
time that the remaining components can fulfill 100% of demand. In this case, 0.875 additional
days of demand can be fulfilled. Component inventory is fairly allocated between finished
products A and B, such that each can fulfill 100% of demand for an additional 0.875 days. Thus
the critical time interval is 2.875 days. The critical time is depicted in Figure A-2.
Figure A-2: Critical Time
Accumulation Time
Accumulation time represents the interval of time beginning when all component inventory has
been expended until the time where all components that have accumulated from the partial rate
of supply have been used. Accumulation time can be determined by iterating through intervals of
115
time. The time horizon considered will be from the critical time to the point in time when the
firm has recovered. The algorithm to determine accumulation time can be programmatically
represented as follows.
1
2
3
4
5
6
7
𝐴. 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = (𝑡𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 − 𝑡𝑑 ) ∙ 𝑅 ∙ (∑𝑛𝑖=1 𝐷𝑖 𝑢𝑖 )
𝐴. 𝑅𝑒𝑚 = 𝐴. 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙
𝑡𝑠𝑡𝑎𝑟𝑡 = 𝑡𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙
For 𝑘 = 𝐹𝑃𝑛𝑒𝑥𝑡 to (𝑛′ + 1)
𝑡𝑒𝑛𝑑 = 𝑡𝑘
𝑡𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 = 𝑡𝑒𝑛𝑑 − 𝑡𝑠𝑡𝑎𝑟𝑡
𝐴.𝑅𝑒𝑚
𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛 = ∑𝑘
𝑛
8
9
10
11
12
13
14
15
16
If 𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛 > 𝑡𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 or 𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛 < 0
𝑡𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 = 𝑡𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 + 𝑡𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
𝐴. 𝑅𝑒𝑚 = 𝐴. 𝑅𝑒𝑚 − (∑𝑘𝑖=1 𝐷𝑖 𝑢𝑖 − 𝑅 ∙ ∑𝑛𝑖=1 𝐷𝑖 𝑢𝑖 ) ∙ 𝑡𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
𝑡𝑠𝑡𝑎𝑟𝑡 = 𝑡𝑒𝑛𝑑
Else
𝑡𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = 𝑡𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 + 𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛
Exit For
End If
Next 𝑘
𝑖=1 𝐷𝑖 𝑢𝑖 −𝑅∙∑𝑖=1 𝐷𝑖 𝑢𝑖
The first line of the program establishes the total number of parts that have accumulated
during the critical time interval, from the time of disruption until component inventory has been
expended. The number of components that have accumulated is the product of the partial rate of
supply, total average daily demand, and the difference between the critical time and the time of
the disruption.
Lines 2 and 3 of the program are initialization statements. The second line initializes the
remaining number of accumulated components to the number of components that have
accumulated during the critical time interval. The third line initializes the start of the first interval
being considered as the end of the critical time interval.
116
The counter k iterates through each of the finished products having days-of-supply
greater than the critical time in ascending order days-of-supply. The end of the time interval is
set to the time at which the given finished product has expended its finished product inventory.
In line 7, the number of days that the accumulated components can sustain the demand at
the current interval, 𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛 , is determined. This number of days is the ratio between the number
of components that have accumulated and the difference between the components demanded per
day and the number of incoming parts per day the firm receives. The denominator inherently
accounts for additional accumulation that may occur while the components that accumulated
between the time of disruption and the critical time are being used.
If 𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛 exceeds the interval being considered, then accumulation time will extend
beyond the interval time. If 𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛 is negative, the daily demand at the current interval is less
than the number of components that the firm receives each day from the partial rate of supply.
Thus, more components accumulate than can be expended. In both cases, the accumulation time
is extended by the time interval being considered. The number of components that have
accumulated is updated based upon the difference between the total number of components
demanded over the time interval and the total number of components that have accumulated over
the time interval. When daily demand is less than the number of components that the firm
receives each day, there will be an increase in the number of components that have accumulated.
The start of the next interval is then set to the end of the current interval being considered.
If 𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛 is less than the interval being considered, accumulation time is only extended
by 𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛 . Only a fraction of total demand across the time interval being considered is fulfilled
with the remaining accumulated components.
117
Example
The total required number of components per day is 1,000. Assume that there is a partial
incoming rate of supply of 10%; each day, from the time of the disruption until the firm recovers
from the disruption, the firm receives 100 components. Over the time that component and
finished product inventory-on-hand are being expended, 2.875 days in total, 287.5 components
have accumulated.
The accumulation time interval can be determined by iterating through time intervals.
Details of the demand and remaining accumulated components at each time interval are shown in
Table A-3.
118
Interval Interval
Time
Start
End
Interval
Time
Time
Demand for Components
Over Time Interval
𝑧
𝑡𝑠𝑡𝑎𝑟𝑡
𝑡𝑒𝑛𝑑
𝑡𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
Time
Accumulation Accumulation Accumulated
Current
Remaining
Remaining
Components
Accumulation
(at start of
(at end of time Can Sustain
Time
time interval)
interval)
Demand
over Interval
(∑ 𝐷𝑖 𝑢𝑖 ) ∙ 𝑡𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
𝐴. 𝑅𝑒𝑚
𝐴. 𝑅𝑒𝑚
𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛
𝑡𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛
287.5
175.0
2.3000
1.1250
175.0
0
0.2975
1.2596
𝑖=1
2.875
4
1.125
(𝐷𝐴 𝑢𝐴 + 𝐷𝐵 𝑢𝐵 ) ∙ 1.125
= (20 ∙ 5 + 10 ∙ 10) ∙ 1.125 = 225
(𝐷𝐴 𝑢𝐴 + 𝐷𝐵 𝑢𝐵 + 𝐷𝐶 𝑢𝐶 + 𝐷𝐷 𝑢𝐷 ) ∙ 2
4
6
2
= (20 ∙ 5 + 10 ∙ 10 + 20 ∙ 10 + 30 ∙ 10) ∙ 2
= 1,400
Table A-3: Time Intervals Considered for Determining Accumulation Time
119
The first time interval begins at the end of the critical time interval and ends when
finished products C and D expend their finished product inventory. During this interval, lasting
1.125 days, only finished products A and B require components from the accumulated number of
components. Finished products A and B each require 100 components per day. Over the time
interval of 1.125 days, a total of 225 components are required to fulfill 100% of demand until the
end of the time interval. Given that there are also 100 components per day still received by the
firm from the partial rate of supply, the total number of days of that this amount of demand can
be sustained, given the number of components that have accumulated at the start of the interval,
is 2.3 days. The calculation is shown below.
𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛 =
𝐴. 𝑅𝑒𝑚
∑2𝑖=1 𝐷𝑖 𝑢𝑖
− 𝑅 ∙ (∑5𝑖=1 𝐷𝑖 𝑢𝑖 )
287.50
(20 ∙ 5 + 10 ∙ 10) − 0.1 ∙ (20 ∙ 5 + 10 ∙ 10 + 20 ∙ 10 + 30 ∙ 10 + 20 ∙ 15)
287.50
=
= 2.3
200 − 100
=
Since the amount of time that the demand at this interval can be sustained, 2.3 days, is
greater than the length of the time interval, 1.125 days, the accumulation time is equal to the
interval time. At the end of the interval, the number of accumulated components is 175, which
accounts for what has been used during the interval to fulfill demand and what has accumulated
from the partial rate of supply while demand was fulfilled (calculation shown below).
2
5
𝐴. 𝑅𝑒𝑚 = 𝐴. 𝑅𝑒𝑚 − (∑ 𝐷𝑖 𝑢𝑖 − 𝑅 ∙ (∑ 𝐷𝑖 𝑢𝑖 )) ∙ 𝑡𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
𝑖=1
𝑖=1
= 287.50 − ((20 ∙ 5 + 10 ∙ 10) − 0.1 ∙ (20 ∙ 5 + 10 ∙ 10 + 20 ∙ 10 + 30 ∙ 10 + 20 ∙ 15)) ∙ 1.125
= 287.50 − (200 − 0.1 ∙ 1000) ∙ 1.125 = 287.50 − 112.50 = 175
The next time interval begins when finished product C and D have expended their
finished product inventory and ends when finished product E expends its finished product
120
interval. During this time interval, products A through D require accumulated components to
fulfill their demand. Each day, these four products require 700 components. 1,400 components
are required to fulfill demand completely over the duration of the time interval, lasting 2 days.
Given that there are also 100 components per day still received by the firm from the
partial rate of supply, the total number of days of that this amount of demand can be sustained is
0.2917 days. The calculation is shown below.
𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛 =
𝐴. 𝑅𝑒𝑚
∑2𝑖=1 𝐷𝑖 𝑢𝑖 − 𝑅 ∙ (∑5𝑖=1 𝐷𝑖 𝑢𝑖 )
175
(20 ∙ 5 + 10 ∙ 10 + 20 ∙ 10 + 30 ∙ 10) − 0.1 ∙ (20 ∙ 5 + 10 ∙ 10 + 20 ∙ 10 + 30 ∙ 10 + 20 ∙ 15)
175
=
= 0.2917
700 − 100
=
Since the amount of time that the demand at this interval can be sustained, 0.2917 days, is
less than the length of the time interval, 2 days, the accumulation time is extended by the number
of days that of the demand over the current interval that can be sustained. The accumulation time
is now 1.2596 days as calculated below and shown in Figure A-3.
𝑡𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 = 𝑡𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 + 𝑡𝑠𝑢𝑠𝑡𝑎𝑖𝑛 = 1.125 + 0.2917 = 1.4167
121
Figure A-3: Accumulation Time
The total amount of time that the partial rate of supply can fulfill 100% of demand is
1.4167 days from the end of the critical time interval. Component inventory and components that
have accumulated from the partial rate of supply are able to be used to fulfill 100% of demand
for 4.2917 days from the time of disruption. The CIT period, where 100% of demand of affected
products cannot be fulfilled is 2.7083 days.
𝐶𝐼𝑇 = 𝐹. 𝑇𝑇𝑅 − 𝑡𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 − 𝑡𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 = 7 − 2.875 − 1.4167 = 2.7083
VAR Formulation
The formulation for VAR varies slightly from the formulation for VAR given finished product
inventory of all affected products is the same (see equation (3-4)). The factors of the formulation
that change are the percentage of demand fulfilled and the interval of time where a certain loss is
incurred.
When finished product inventory of all affected products was assumed the same, demand
throughout the CIT period was the same; all affected products could not fulfill any demand
122
without the partial rate of supply. Moreover, the partial rate of supply reflected the percentage of
demand that was capable of being fulfilled. However, when finished product inventory varies,
demand throughout the CIT period can also vary. Some finished products may expend their daysof-supply during the CIT period and thereby may not initially require some of the partial rate of
supply at the start of the CIT period but then require the partial rate of supply toward the end of
the CIT period. Thus, as more finished products expend their days-of-supply during the CIT
period, the percentage of demand that can be fulfilled changes.
As the percentage of demand that can be fulfilled changes, the allocation of the partial
rate of supply also changes, thereby changing the value lost to the firm. The duration that a given
value is lost to the firm depends on the interval of time when the allocation scheme is valid. An
allocation scheme is only valid for the interval of time when one finished product expends its
days-of-supply until the subsequent finished product expends its days-of-supply.
VAR can be expressed as follows:
𝑛′ +1
𝑉𝐴𝑅 = ∑ [(1 −
𝑅 ∙ (∑𝑛𝑞=1 𝐷𝑞 𝑢𝑞 )
𝑠=𝑛𝐶𝐼𝑇
∑𝑠𝑖=1 𝐷𝑖 𝑢𝑖
𝑠
) ∙ (𝑡𝑠 − 𝑡𝑠−1 ) ∙ ∑ 𝐷𝑖 𝐹𝑖 ]
(A-3)
𝑖=1
Where:

𝑛𝐶𝐼𝑇 represents the first finished product to have its inventory expended during the CIT
period. When 𝑠 = 𝑛𝐶𝐼𝑇 , 𝑡𝑠−1 = 𝐶𝐼𝑇; this is the initial time interval from the start of the
CIT period until a finished product has its inventory expended.

𝑛′ + 1 is defined such that 𝑡𝑛′ +1 is the point in time when the firm recovers.

𝑅 ∙ (∑𝑛𝑞=1 𝐷𝑞 𝑢𝑞 )

∑𝑠𝑖=1 𝐷𝑖 𝑢𝑖
represents the number of components received by the firm each day.
represents the number of components demanded from finished products whose
finished product days-of-supply is less than 𝑡𝑠 for the time interval beginning at 𝑡𝑠−1 and
ending at 𝑡𝑠 .
123

𝑅∙(∑𝑛𝑞=1 𝐷𝑞 𝑢𝑞 )
∑𝑠𝑖=1 𝐷𝑖 𝑢𝑖
represents the percentage of demand, required by those finished products
whose finished product days-of-supply is less than 𝑡𝑖 during the time interval being
considered, that can be fulfilled using the partial rate of supply. The partial rate of supply
represents the percentage of components of the total average daily demand across the
firm’s recovery period. However, at any time interval, the number of components
demanded is not necessarily equivalent to total average daily demand across the firm’s
recovery period; in certain cases, some finished products may still have some finished
product inventory. Thus the percentage of unfulfilled demand must be calculated based
upon what is demanded within the time interval.

𝑡𝑠 − 𝑡𝑠−1 represents the time interval.

∑𝑠𝑖=1 𝐷𝑖 𝐹𝑖 represents the total financial value of all products whose finished product daysof-supply is less than 𝑡𝑖 .
VAR is the summation of the value lost over all the time intervals between the start of the
CIT period until the firm’s recovery time. During each time interval, the value lost is the product
of the time interval, the daily financial value of the products without inventory during the
interval and the percentage of total demand during the time interval that is unmet. The
percentage of total demand during the time interval that is unmet is one less the ratio of the
incoming parts per day to the parts required per day by those products without inventory.
Example
Assume that products A through E have financial values that are shown in Table A-4.
Product Financial Value 𝑭𝒊
Product
A
140
Product
B
200
Product
C
220
Product
D
120
Product
E
450
Table A-4: Finished Product Financial Value (Single Component Disruption with Varied Finished Product
Inventory Example)
Based upon the financial values of the affected products and the partial rate of supply of
10%, VAR is 38,363 units of financial value. In the calculation below, there are two intervals
where potential loss can occur. The first interval begins at the start of the CIT period and ends
when finished product E expends its finished product inventory. Over this interval, 700
124
components are required to fulfill demand for products A through D. Since the firm only receives
100 components per day from the partial rate of supply, approximately 14% of each product’s
demand can be fulfilled during this interval when fairly allocated. 86% of demand is unmet.
During this interval, 12,800 financial units could be lost if there was no partial rate of supply.
With the partial rate of supply, 86% of this value is lost to the firm.
The second interval is from the point in time when finished product E expends its
finished product inventory until the firm recovers from the disruption. During this time, all
products need the components available from the partial rate of supply to fulfill their demand.
During this interval, 1,000 components are required to fulfill demand. Given that only 100
components per day are received by the firm from the partial rate of supply, only 10% of each
product’s demand can be fulfilled during this interval when fairly allocated. Of the potential
21,800 financial units that could be lost over the interval, 90% of this is actually lost to the firm.
𝑛′ +1
𝑉𝐴𝑅 = ∑ [(1 −
𝑠=3
𝑉𝐴𝑅 = (1 −
𝑉𝐴𝑅 = (1 −
𝑅 ∙ (∑𝑛𝑞=1 𝐷𝑞 𝑢𝑞 )
∑𝑠𝑖=1 𝐷𝑖 𝑢𝑖
𝑠
) ∙ (𝑡𝑠 − 𝑡𝑠−1 ) ∙ ∑ 𝐷𝑖 𝐹𝑖 ]
𝑖=1
0.1 ∙ (20 ∙ 5 + 10 ∙ 10 + 20 ∙ 10 + 30 ∙ 10 + 20 ∙ 15)
) ∙ (6 − 4.2917)
(20 ∙ 5 + 10 ∙ 10 + 20 ∙ 10 + 30 ∙ 10)
∙ (20 ∙ 140 + 10 ∙ 200 + 20 ∙ 220 + 30 ∙ 120)
0.1 ∙ (20 ∙ 5 + 10 ∙ 10 + 20 ∙ 10 + 30 ∙ 10 + 20 ∙ 15)
+ (1 −
) ∙ (7 − 6)
(20 ∙ 5 + 10 ∙ 10 + 20 ∙ 10 + 30 ∙ 10 + 20 ∙ 15)
∙ (20 ∙ 140 + 10 ∙ 200 + 20 ∙ 220 + 30 ∙ 120 + 20 ∙ 450)
100
100
) ∙ (1.7083) ∙ (12,800) + (1 −
) ∙ (1) ∙ (21,800) ≈ 38,363
700
1000
The resulting unmet demand during the CIT period under fair allocation is shown in
Figure A-4. Finished products A through D have proportionally less unmet demand during the
first time interval than during the second. This is because when product E finishes expending its
finished product inventory, its demand must also be accounted for when allocating the partial
125
rate of supply. This results in products A through D being allocated fewer components during the
second time interval.
Figure A-4: Unmet Demand after Fair Allocation of Partial Rate of Supply
MVAR Formulation
The formulation for MVAR varies slightly from the formulation for MVAR given finished product
inventory of all affected products is the same (see equation (4-1)). The factors of the formulation
that change are the percentage of demand fulfilled and the interval of time where a certain loss is
incurred. As previously stated, as finished products expend their days-of-supply during the CIT
period, demand for the partial rate of supply changes. This in turn will alter the allocation of the
partial rate of supply under preferential product allocation and change the value lost over
different time intervals.
MVAR can be determined with a linear program that can be reduced to a greedy
algorithm that is solved for each time interval. The decision variable of the linear program is the
126
set of variables 𝑋𝑖,𝑠 representing the number of products of finished product 𝑖 produced per day
during time interval that begins at 𝑡𝑠−1 and ends at 𝑡𝑠 . The products are produced based upon the
allocation of the partial rate of supply to the products.
𝑛′ +1
𝑠
min 𝑀𝑉𝐴𝑅 = min ∑ [(𝑡𝑠 − 𝑡𝑠−1 ) ∙ ∑ ((𝐷𝑖 − 𝑋𝑖,𝑠 ) ∙ 𝐹𝑖 )]
𝑠=𝑛𝐶𝐼𝑇
(A-4)
𝑖=1
subject to:
𝑚
∑(𝐷𝑖,𝑗 ) − 𝑋𝑖,𝑠 ≥ 0, ∀𝑖, 𝑠
(A-4.1)
𝑗=1
∑𝑠𝑖=1 𝑋𝑖,𝑠 𝑢𝑖
𝑅− 𝑛
≥ 0, ∀𝑠
∑𝑞=1 𝐷𝑞 𝑢𝑞
(A-4.2)
𝑋𝑖,𝑠 ≥ 0, ∀𝑖, 𝑠
(A-4.3)

𝑛𝐶𝐼𝑇 represents the first finished product to have its inventory expended during the CIT
period. When 𝑠 = 𝑛𝐶𝐼𝑇 , 𝑡𝑠−1 = 𝐶𝐼𝑇; this is the initial time interval from the start of the
CIT period until a finished product has its inventory expended.

𝑛′ + 1 is defined such that 𝑡𝑛′ +1 is the point in time when the firm recovers.

∑𝑠𝑖=1(𝑋𝑖,𝑠 ∙ 𝑢𝑖 ) represents the total number of components allocated from the partial rate
of supply for the time interval beginning at 𝑡𝑠 and ending at 𝑡𝑠−1 .

∑𝑛𝑞=1(∑𝑚
𝑗=1(𝐷𝑞,𝑗 ) ∙ 𝑢𝑞 ) represents the total number of components required to meet
demand for all affected products per day.

∑𝑠𝑖=1(𝑋𝑖,𝑠 ∙𝑢𝑖 )
𝑚
(∑𝑛
𝑞=1(∑𝑗=1(𝐷𝑞,𝑗 )∙𝑢𝑞 ))
represents the percentage of total demand for components per day
satisfied with the allocated partial rate of supply.

𝑡𝑠 − 𝑡𝑠−1 represents the time interval.
The first constraint ensures the supply allocated to fulfilling demand does not exceed
what is demanded. The second constraint ensures that the supply allocated does not exceed what
is available. The constraints are identical to those of the linear program formulated under the
127
assumption that all finished product inventory is the same, see equation (4-2). The difference is
the decision variable must be solved each time a finished product expends its inventory during
the CIT period.
Example
Over each time interval, a subset of all affected products will require components from the
partial rate of supply. Under a preferential product allocation policy, the partial rate of supply is
allocated to the products with the highest financial contribution per component that do not have
any remaining finished product inventory. The financial contribution per component of each
affect product is shown in Table A-5.
Product Financial Value 𝑭𝒊
Usage Rate 𝒖𝒊
Financial Contribution per component
Product
A
140
5
Product
B
200
10
28
20
𝑭𝒊
Product Product Product
C
D
E
220
120
450
10
10
15
22
12
30
𝒖𝒊
Table A-5: Finished Product Financial Value per Component (Single Component Disruption with Varied Finished
Product Inventory Example)
The allocation of the partial rate of supply under preferential product allocation must be
determined for two intervals. The first interval begins at the start of the CIT period and ends
when finished product E expends its finished product inventory. Only products A through D
require components during this interval. Product A has the highest contribution per component
and thus is allocated the entire partial rate of supply until the end of the interval. This allocation
results in all 20 demanded units to be fulfilled.
The second interval begins when finished product E expends its finished product
inventory and ends when the firm recovers from the disruption. All products require component
during this interval to fulfill demand. Product E has the highest contribution per component
128
during this interval and thus is allocated the entire partial rate of supply during this time, which
fulfills one-third of its demand, enabling approximately 6.67 products per day to be produced.
MVAR based on these allocations is 35,883, as shown in the calculation below. The
unmet demand over the CIT period is depicted in Figure A-5. Recall that under fair allocation
VAR is 38,363. Thus using preferential product allocation reduces the firm’s financial loss by
approximately 2,480 financial units.
𝑛′ +1
𝑠
𝑀𝑉𝐴𝑅 = ∑ [(𝑡𝑠 − 𝑡𝑠−1 ) ∙ ∑ ((𝐷𝑖 − 𝑋𝑖,𝑠 ) ∙ 𝐹𝑖 )]
𝑠=3
𝑖=1
𝑀𝑉𝐴𝑅 = (6 − 4.2917)
∙ ((20 − 20) ∙ 140 + (10 − 0) ∙ 200 + (20 − 0) ∙ 220 + (30 − 0) ∙ 120)
+ (7 − 6)
∙ ((20 − 0) ∙ 140 + (10 − 0) ∙ 200 + (20 − 0) ∙ 220 + (30 − 0) ∙ 120
+ (20 − 6.67) ∙ 450)
𝑀𝑉𝐴𝑅 = (1.7083) ∙ (10,000) + (1) ∙ (18,800) ≈ 35,883
Figure A-5: Unmet Demand after Preferential Product Allocation of Partial Rate of Supply
129
The following linear program can also be used to solve for MVAR. The same results are
attained when using the linear program. When the linear program is solved, variable 𝑋𝐴,4 takes
on a value of 20, indicating that 20 units of finished product A are produced per day during the
interval that begins when the critical time period begins and ends at time unit 4. Variable 𝑋𝐸,5
takes on a value of 6.67, indicating that 6.67 units of finished product E are produced per day
during the interval that begins when finished product E expends its inventory and ends at time
unit 5, which is when the firm recovers from the disruption. All other variables take on a value of
0, indicating that they have not been allocated any of the partial rate of supply.
min [(6 − 4.2917)
∙ ((20 − 𝑋𝐴,4 ) ∙ 140 + (10 − 𝑋𝐵,4 ) ∙ 200 + (20 − 𝑋𝐶,4 ) ∙ 220 + (30 − 𝑋𝐷,4 )
∙ 120) + (7 − 6)
∙ ((20 − 𝑋𝐴,5 ) ∙ 140 + (10 − 𝑋𝐵,5 ) ∙ 200 + (20 − 𝑋𝐶,5 ) ∙ 220 + (30 − 𝑋𝐷,5 )
∙ 120 + (20 − 𝑋𝐸,5 ) ∙ 450)]
subject to:
(1.1) 20 − 𝑋𝐴,4 ≥ 0
(1.2) 10 − 𝑋𝐵,4 ≥ 0
(1.3) 20 − 𝑋𝐶,4 ≥ 0
(1.4) 30 − 𝑋𝐷,4 ≥ 0
(1.5) 20 − 𝑋𝐴,5 ≥ 0
(1.6) 10 − 𝑋𝐵,5 ≥ 0
(1.7) 20 − 𝑋𝐶,5 ≥ 0
(1.8) 30 − 𝑋𝐷,5 ≥ 0
(1.9) 20 − 𝑋𝐸,5 ≥ 0
𝑋𝐴,4 ∙ 5 + 𝑋𝐵,4 ∙ 10 + 𝑋𝐶,4 ∙ 10 + 𝑋𝐷,4 ∙ 10
(2.1) 0.1 −
≥0
20 ∙ 5 + 10 ∙ 10 + 20 ∙ 10 + 30 ∙ 10 + 20 ∙ 15
𝑋𝐴,4 ∙ 5 + 𝑋𝐵,4 ∙ 10 + 𝑋𝐶,4 ∙ 10 + 𝑋𝐷,4 ∙ 10 + 𝑋𝐸,4 ∙ 15
(2.2) 0.1 −
≥0
20 ∙ 5 + 10 ∙ 10 + 20 ∙ 10 + 30 ∙ 10 + 20 ∙ 15
(3.1) 𝑋𝐴,4 ≥ 0
(3.2) 𝑋𝐵,4 ≥ 0
(3.3) 𝑋𝐶,4 ≥ 0
(3.4) 𝑋𝐷,4 ≥ 0
(3.5) 𝑋𝐴,5 ≥ 0
(3.6) 𝑋𝐵,5 ≥ 0
130
(3.7) 𝑋𝐶,5 ≥ 0
(3.8) 𝑋𝐷,5 ≥ 0
(3.9) 𝑋𝐸,5 ≥ 0
Marginal Financial Benefit Formulation
Variations in the inventory of finished products can result in different allocations of the partial
rate of supply during distinct time intervals over the CIT period. While the marginal financial
benefit metric still represents the change in the firm’s financial gain with respect to a change in
the partial rate of supply over the firm’s CIT period, the financial gain over each time interval
changes. The calculation must account for the allocation at each time interval.
The modified marginal financial benefit calculation sums the financial gain over each of
the time intervals during the CIT period. During each time interval, the financial gain is based
upon the allocation of the partial rate of supply to the finished products that do not have any
more finished product inventory. The financial gain over the entire CIT period given varying
finished product inventory is as follows:
𝑛′ +1
𝑠
∑ [(𝑡𝑠 − 𝑡𝑠−1 ) ∙ ∑(𝑋𝑖,𝑠 ∙ 𝐹𝑠 )]
𝑠=𝑛𝐶𝐼𝑇
(A-5)
𝑖=1

𝑛𝐶𝐼𝑇 represents the first finished product to have its inventory expended during the CIT
period. When 𝑠 = 𝑛𝐶𝐼𝑇 , 𝑡𝑠−1 = 𝐶𝐼𝑇; this is the initial time interval from the start of the
CIT period until a finished product has its inventory expended.

𝑛′ + 1 is defined such that 𝑡𝑛′ +1 is the point in time when the firm recovers.

∑𝑠𝑖=1(𝑋𝑖,𝑠 ∙ 𝐹𝑠 ) represents the total financial gain of all products whose finished product
days-of-supply is less than 𝑡𝑖 , based upon the allocation of the partial rate of supply to
those products under preferential product allocation.

𝑡𝑠 − 𝑡𝑠−1 represents the time interval.
131
Appendix B
This appendix provides formulations that extend upon the framework established in the thesis.
Formulations for MVAR under a single-component disruption with substitution and a multicomponent disruption are provided. Both of these formulations still hold the original assumption
that finished product days-of-supply of all affected products are the same. These formulations are
also under the assumption that finished product inventory-on-hand, component inventory-onhand, and components accumulated from the partial rate of supply as inventory-on-hand is used
are fairly allocated. The partial rate of supply is allocated preferentially during the CIT period.
MVAR Formulation for a Single-Component Disruption using
Substitution
In some cases, a firm may be able to use substitution, where one of the firm’s non-disrupted
products can be used in place of a disrupted product. In this case, assume that there is a given
fraction, 𝑆𝑖 , representing the expected percentage of demand of product 𝑖 that can be substituted
by a non-disrupted product, such that 0 ≤ 𝑆𝑖 ≤ 1 given there is enough capacity to produce the
substitute. In certain situations, no substitute will be available and 𝑆𝑖 = 0. In other cases, a
substitute will be available (0 < 𝑆𝑖 < 1) and preferential allocation must be modified. There are
two different cases that can occur with the use of substitutes. The first is when the financial
values of all substitutes that can be used for affected products are equivalent to the financial
values of the affected products. The second case is when the financial value of the substitute
differs from that of the affected product.
In the first case, demand of the affected product is modified by the rate of substitution.
That is, the demand for the affected product is reduced by the expected percentage of demand
that can be substituted by a non-disrupted product. This does not change the financial
132
contribution of the product but only the amount of the disrupted component that must be
allocated to the product given that its demand has been reduced from use of the substitute. When
using the greedy algorithm to solve for this, products are still ranked in descending order of their
financial contribution per product as follows:
𝐹(1) 𝐹(2) 𝐹(3)
𝐹(𝑛)
≥
≥
≥⋯≥
𝑢(1) 𝑢(2) 𝑢(3)
𝑢(𝑛)
The greedy algorithm will still go in a single path through the hierarchy and allocate
components to the products with the greatest financial contribution per component first. The only
difference is that fewer components need to be allocated to a product with an available substitute.
The linear program used to solve this case is as follows:
𝑛
𝑀𝑉𝐴𝑅 = min (𝐶𝐼𝑇 ∙ ∑ 𝐹𝑖 ∙ (𝐷𝑖 ∙ (1 − 𝑆𝑖 ) − 𝑋𝑖 ))
(B-1)
𝑖=1
subject to:
𝐷𝑖 (1 − 𝑆𝑖 ) − 𝑋𝑖 ≥ 0, ∀𝑖
(B-1.1)
∑𝑛𝑖=1 𝑋𝑖 𝑢𝑖
𝑅− 𝑛
≥0
∑𝑖=1 𝐷𝑖 𝑢𝑖
(B-1.2)
𝑋𝑖 ≥ 0, ∀𝑖
(B-1.3)
Where:
 𝐶𝐼𝑇 is the customer impact time given available component and finished product
inventory and the accumulated partial rate of supply.

𝐷𝑖 – average demand of finished product 𝑖 by all customers per day, from the date of the
disruption (𝑡𝑑 ) until the firm recovers (𝑡𝑟 ). Where 𝑖 is from 1 to n.

𝑢𝑖 – usage rate of components in finished products; the number of components required
to produce a single unit of finished product 𝑖.

𝐹𝑖 is the financial value per unit of finished product 𝑖.
133

𝑅 is the partial rate of supply of the disrupted component available during the firm’s
time-to-recovery.

𝑋𝑖 is the number of product 𝑖 produced per day, given the allocation of parts from partial
rate of supply during the CIT period.

𝑅𝑖 is the maximum fraction of product 𝑖 that can be expected to be substituted by nondisrupted products.

𝐷𝑖 (1 − 𝑆𝑖 ) is the remaining unmet demand per day of product 𝑖 across all customers 𝑗
after the use of the substitute.

𝐷𝑖 (1 − 𝑆𝑖 ) − 𝑋𝑖 ≥ is the remaining unmet demand per day of product 𝑖 across all
customers 𝑗 after use of the substitute and the allocation of the partial rate of supply.

∑𝑛𝑖=1 𝑋𝑖 𝑢𝑖 is the total number of components per day allocated from the partial rate of
supply.

∑𝑛𝑖=1 𝐷𝑖 𝑢𝑖 is the total number of components per day required to meet all demand of
affected products across all customers, without considering the availability of substitutes.

∑𝑛
𝑖=1 𝑋𝑖 𝑢𝑖
∑𝑛
𝑖=1 𝐷𝑖 𝑢𝑖
is the percentage of the total number of components per day required to meet all
demand that are satisfied with the allocated partial rate of supply
The first constraint ensures the supply allocated to fulfilling demand does not exceed
what is demanded. The second constraint ensures that the supply allocated does not exceed what
is available.
In the second case, the financial value of an affected product may be different than its
substitute. In some cases, it is more optimal to use the total fraction, 𝑆𝑖 , of product 𝑖 that can be
expected to be substituted by unaffected products. In other cases, it may be more profitable to
allocate the partial rate of supply before use of the substitute. This is beneficial when an affected
product has a substitute with a lesser value than the subsequent product with the next greatest
partial rate of supply and the affected product yields a greater financial value per component then
the subsequent affected product. A linear program can be used to determine the incremental
benefits of using substitutes versus allocating components to an affected product and thus not
134
having those components to allocate to another affected product. Two sets of decision variables
are used in this linear program. The first is 𝑋𝑖 representing the number of finished products per
day produced given the allocation of partial rate of supply during the CIT period. The second is
𝑍𝑖 , representing the portion of the total fraction of product 𝑖 that can be expected to be
substituted by an unaffected product that is actually used to reduce an affected products demand.
𝑛
𝑀𝑉𝐴𝑅 = min (𝐶𝐼𝑇 ∙ ∑(𝐹𝑖 (𝐷𝑖 − 𝑋𝑖 ) − 𝐹𝑖𝑆 𝐷𝑖 𝑍𝑖 ))
(B-2)
𝑖=1
subject to:
𝐷𝑖 (1 − 𝑍𝑖 ) − 𝑋𝑖 ≥ 0, ∀𝑖
𝑅−
(B-2.1)
∑𝑛𝑖=1 𝑋𝑖 𝑢𝑖
≥0
∑𝑛𝑖=1 𝐷𝑖 𝑢𝑖
(B-2.2)
𝑆𝑖 ≥ 𝑍𝑖 , ∀𝑖
(B-2.3)
𝑋𝑖 ≥ 0, ∀𝑖
(B-2.4)
𝑍𝑖 ≥ 0, ∀𝑖
(B-2.5)
Where:
 𝐹𝑖 is the financial value per unit of finished product 𝑖.

𝐹𝑆𝑖 is the financial value per unit of substitute used for finished product 𝑖.

𝑋𝑖 is the number of product 𝑖 produced per day, given the allocation of parts from partial
rate of supply during the CIT period.

𝑆𝑖 is the maximum fraction of product 𝑖 that can be expected to be substituted by nondisrupted products.

𝑍𝑖 is the fraction of product 𝑖 that is substituted by non-disrupted products.

𝐹𝑖 (𝐷𝑖 − 𝑋𝑖 ) is the value-at-risk per day for product 𝑖 after the allocation of partial rate of
supply during the CIT period.

𝐹𝑖𝑆 𝐷𝑖 𝑍𝑖 is the value of using the substitute to meet some demand of affected product 𝑖.
135
The first constraint ensures the supply allocated to fulfilling demand does not exceed
what is demanded. The second constraint ensures that the supply allocated does not exceed what
is available. The third constraint ensures that the fraction of substitution used does not exceed the
maximum fraction of product 𝑖 that can be expected to be substituted by non-disrupted products.
The fourth and fifth constraints are non-negativity constraints.
MVAR Formulation for a Multiple-Component Disruption
The following formulation for a multiple-component disruption assumes no availability of
substitutes and all finished products have the same amount of inventory. Assume that 𝑘
components have been disrupted, the parts are numbered 𝑘 = 1,2,3 … 𝐾. Each component has a
partial rate of supply 𝑅𝑘 . These components go into 𝑛 products numbered i = 1,2,3,… n. The
financial contribution of each product is 𝐹𝑖 and the demand for each product across all customers
𝐷𝑖 . Each product requires a certain number of any of the disrupted components to produce a
single unit of the affected product. Let 𝑢𝑖,𝑘 represent the number of components 𝑘 required to
produce a single unit of affected product 𝑖. Let 𝑋𝑖 represent the number of products that are
produced per day given the allocation of disrupted components.
𝑛
𝑀𝑉𝐴𝑅 = min (𝐶𝐼𝑇 ∙ ∑ 𝐹𝑖 ∙ (𝐷𝑖 − 𝑋𝑖 ))
( B-3 )
𝑖=1
subject to:
𝐷𝑖 − 𝑋𝑖 ≥ 0, ∀𝑖
( B-3.1)
∑𝑛𝑖=1 𝑋𝑖 𝑢𝑖,𝑘
𝑅𝑘 − 𝑛
≥ 0, ∀𝑘
∑𝑖=1 𝐷𝑖 𝑢𝑖,𝑘
( B-3.2)
𝑋𝑖 ≥ 0, ∀𝑖
(B-3.3)
136
Where:
 𝑢𝑖,𝑘 is the number of disrupted parts 𝑘 required to produce a single unit of finished
product 𝑖.

𝑅𝑘 is the partial rate of supply of disrupted component 𝑘 available during the firm’s timeto-recovery.

𝑋𝑖 is the number of products produced per day given the allocation of components from
the partial rate of supplies available.

𝐷𝑖 − 𝑋𝑖 is the remaining unmet demand per day of product 𝑖 across all customers 𝑗 after
the allocation of the partial rate of supply.

∑𝑛𝑖=1 𝑋𝑖 𝑢𝑖,𝑘 is the total number of disrupted component 𝑘 allocated from the partial rate of
supply per day.

∑𝑛𝑖=1 𝐷𝑖 𝑢𝑖,𝑘 is the total number of components 𝑘 required per day to meet all demand of
affected products across all customers.
The first constraint ensures the supply allocated to fulfilling demand does not exceed
what is demanded. The second constraint ensures that the supply allocated does not exceed what
is available.
137
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