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 3 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. 5 6 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 7 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 6 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 7 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 8 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 10 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 11 12 1 Introduction Disruptions such as Hurricanes Isaac and Sandy in 2012, the Corn Belt drought of 2012, and the TΕhoku 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. 13 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. 14 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. 15 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. 16 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: ο· π‘π – time of the disruption. ο· π‘π – time of recovery. ο· πΉ. πππ – firm time-to-recovery, equivalent to the difference between the time of recovery and the time of the disruption (π‘π − π‘π ). 30 ο· πΆ. πΌππ£ – component inventory-on-hand at the time of disruption (units: components). ο· π. πΌππ£π – inventory-on-hand of finished product π at the time of disruption (units: finished products). ο· π·π – average demand of finished product π by all customers per day, from the date of the disruption (π‘π ) until the firm recovers (π‘π ). Where π is from 1 to n. ο· π’π – usage rate of components in finished products; the number of components required to produce a single unit of finished product π (units: components per single unit of finished product). ο· π – 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), π. π·πππ represents finished product inventory days-of-supply, available to meet total demand. πΆπΌππ = max [(πΉ. πππ − π. πΌππ£π ) , 0] = max[(πΉ. πππ − π. π·πππ ), 0] π·π (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 π«π Usage Rate ππ Finished Product Inventory on Hand π·. π°πππ 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. π‘π = 0, π‘π = 6 πΉ. πππ = π‘π − π‘π = 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. πΆπΌπ1 = max (0, [πΉ. πππ − πΆπΌπ2 = max (0, [πΉ. πππ − πΆπΌπ3 = max (0, [πΉ. πππ − π.πΌππ£1 π·1 π.πΌππ£2 π·2 π.πΌππ£3 π·3 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. πΆ. πΌππ£ ) , 0] ∑ππ=1 π·π π’π = max[(πΉ. πππ − π. π·πππ − πΆ. π·πππ ), 0] πΆπΌππ = max [(πΉ. πππ − π. π·πππ − (3-2) Where: ο· ∑ππ=1 π·π π’π represents the total number of components required per day to fulfill total demand of all affected products. ο· πΆ. π·ππ 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 (πΆ. πΌππ£ = 30,000). The average daily demand of components is (5,000 β 4 + 2,000 β 8 + 1,000 β 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. πΆ. π·ππ = πΆ. πΌππ£ 30,000 30,000 = = = 0.75 3 ∑π=1 π·π π’π 5,000 β 4 + 2,000 β 8 + 1,000 β 4 40,000 πΆπΌπ1 = max(0, [πΉ. πππ − π. π·ππ1 − πΆ. π·ππ]) = max(0, [6 − 0.75 − 0.75]) = 4.5 πΆπΌπ2 = max(0, [πΉ. πππ − π. π·ππ2 − πΆ. π·ππ]) = max(0, [6 − 0.75 − 0.75]) = 4.5 πΆπΌπ3 = max(0, [πΉ. πππ − π. π·ππ3 − πΆ. π·ππ]) = 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 π ), 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. πΆπΌπ = max [(πΉ. πππ − π. π·ππ + πΆ. π·ππ ) , 0] 1−π (3-3) Where: ο· π represents the partial rate of supply; the percentage of the normal rate of supply of the disrupted component available during the firm’s TTR. ο· π. π·ππ + πΆ. π·ππ represents the number of days that available finished product and component inventories can fulfill total demand. ο· π.π·ππ+πΆ.π·ππ represents the total number of days, from the time of the disruption, that total demand can be fulfilled with all forms of inventory. 1−π 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. πΆπΌπ = max (0, [πΉ. πππ − = 3.5 πΆ. π·ππ + π. π·ππ 1.5 ]) = max (0, [6 − ]) = max(0, [6 − 2.5]) 1−π 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, πΉπ , 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 π ππ΄π = (1 − π ) ∑(πΆπΌππ β π·π β πΉπ ) (3-4) π=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. π ππ΄π = (1 − π ) β πΆπΌπ β ∑ π·π πΉπ = (1 − 0.4) β (3.5) β (5,000 β 5 + 2,000 β 8 + 1,000 β 6) π=1 ππ΄π = 0.6 β 3.5 β 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, πΉπ , and the numbers of components used to make each product, π’π . 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 ππ , representing the number of affected products π, 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, π·π − ππ represents the unmet demand after the allocation of the partial rate of supply. π πππ΄π = ∑(πΆπΌππ β (π·π − ππ ) β πΉπ ) (4-1) π=1 There is no difference in the customer impact time, πΆπΌππ , or the financial value of products, πΉπ , 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 {ππ }, 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 π min ∑(πΆπΌππ β (π·π − ππ ) β πΉπ ) (4-2) π=1 Subject to: π·π − ππ ≥ 0, (4-2.1) ∀π ∑ππ=1 ππ π’π π − π ≥0 ∑π=1 π·π π’π ππ ≥ 0, (4-2.2) (4-2.3) ∀π Where: ο· ∑ππ=1 ππ π’π represents the total number of components per day allocated from the partial rate of supply. ο· ο· ∑ππ=1 π·π π’π represents the total number of components required per day to fulfill total demand of all affected products. ∑π π=1 ππ π’π ∑π π=1 π·π π’π 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, πΆ , of size π × π, follows a Monge sequence if it satisfies the condition that πΆππ + πΆπ π‘ ≤ πΆππ‘ + πΆπ π for all 2 × 2 subset matrices where 1 ≤ π < π ≤ π and 1 ≤ π < π‘ ≤ π. 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 × π matrix, the Monge sequence is satisfied when every 2 × 1 subset matrix meets the condition πΆ1π ≤ πΆ1π‘ where 1 ≤ π < π‘ ≤ π . This condition is met when the financial contributions per component are ranked in descending order as in expression (4-3). πΉ(1) πΉ(2) πΉ(3) πΉ(π) ≥ ≥ ≥β―≥ π’(1) π’(2) π’(3) π’(π) (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. ππΉ = 1 − πππ΄π ππ΄π ( 4-4 ) The mitigation factor takes on values between 0 and 1, but cannot attain a value of 1 in other words, ππΉ ∈ [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. π πΉπ ππ΄π = π β πΆπΌππ β ∑ π·π πΉπ π (4-5) π=1 πΉπ πππ΄π = ∑(πΆπΌππ β ππ β πΉπ ) (4-6) π=1 Where: ο· π is the partial rate of supply; the percentage of the normal rate of supply of the disrupted component available during the firm’s TTR. ο· πΆπΌππ is the customer impact time for finished product π. ο· πΉπ is the financial value per unit of finished product π. ο· ππ is the number of products π 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 (πΉπ 1 and πΉπ 2 ) and the difference between the two partial rates of supply (π 1 and π 2 ), as shown in equation (4-7). βπΉπ πΉπ 2 − πΉπ 2 = βπ 100 β (π 2 − π 1 ) (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 ππ (value/product) 10 9 8 7 6 5 4 3 2 1 Product Demand π«π (products/day) 10 10 10 10 10 10 10 10 10 10 Usage Rate ππ (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 πΉ (parts/day) Financial Result ππΉ (value/day) 0% 0 10% Change in Financial Result βππΉ Change in Partial Rate of Supply βπΉ 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 βππΉ βπΉ (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 π«π Usage Rate ππ Finished Product Inventory on Hand π·. π°πππ Financial Value ππ π Financial Contribution per component ππ π 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. π min πΆπΌπ β ∑(π·π − ππ ) β πΉπ π=1 = min[3.5 β ((5,000 − π1 ) β 5 + (2,000 − π2 ) β 8 + (1,000 − π3 ) β 6)] Subject to: (1.1) 5,000 − π1 ≥ 0 (1.2) 2,000 − π2 ≥ 0 (1.3) 1,000 − π3 ≥ 0 (2) 40% − 4 β π1 + 8 β π2 + 4 β π3 ≥0 4 β 5,000 + 8 β 2,000 + 4 β 1,000 52 (3.1) π1 ≥ 0 (3.2) π2 ≥ 0 (3.3) π3 ≥ 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 (π1 = 3,000), 0 units of product 2 (π2 = 0), and 1,000 units of product 3 (π3 = 1,000). Thus, MVAR is 91,000 financial units, as shown below. π πππ΄π = πΆπΌπ β ∑(π·π − ππ ) β πΉπ π=1 = 3.5 β ((5,000 − π1 ) β 5 + (2,000 − π2 ) β 8 + (1,000 − π3 ) β 6) = 3.5 β ((5,000 − 3,000) β 5 + (2,000 − 0) β 8 + (1,000 − 1,000) β 6) = 3.5 β (10,000 + 16,000) = 91,000 Recall that when fair allocation was used, VAR was 98,700, as shown below: π ππ΄π = (1 − π ) β πΆπΌπ β ∑ π·π πΉπ = (1 − 0.4) β (3.5) β (5,000 β 5 + 2,000 β 8 + 1,000 β 6) π=1 = 0.6 β 3.5 β 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. ππΉ = 1 − πππ΄π 91,000 = 1− = 0.078 ππ΄π 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 ππ΄π = (1 − π ) β πΆπΌπ β ∑ π·π πΉπ π=1 ππ΄π πππ‘π΄ = (1 − 0.1) β 1 β (10 β 1 + 10 β 2 + 10 β 3 + 10 β 4 + 10 β 5 + 10 β 6 + 10 β 7) = 252 ππ΄π πππ‘π΅ = (1 − 0.1) β 1 β (10 β 7 + 10 β 8 + 10 β 9 + 10 β 10 + 10 β 11 + 10 β 12 + 10 β 13) = 630 ππ΄π πππ‘πΆ = (1 − 0.1) β 1 β (10 β 13 + 10 β 14 + 10 β 15 + 10 β 16 + 10 β 17 + 10 β 18 + 10 β 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 πππ΄π = πΆπΌπ β ∑(π·π − ππ ) β πΉπ π=1 πππ΄π πππ‘π΄ = 1 β [(10 − 0) β (1 + 2 + 3 + 4 + 5 + 6) + (10 − 7) β 7] = 231 πππ΄π πππ‘π΅ = 1 β [(10 − 0) β (7 + 8 + 9 + 10 + 11 + 12) + (10 − 7) β 13] = 609 πππ΄π πππ‘πΆ = 1 β [(10 − 0) β (13 + 14 + 15 + 16 + 17 + 18) + (10 − 7) β 19] = 987 Given the VAR and MVAR values, the mitigation factor for each set of products is as follows: ππΉπππ‘π΄ = 1 − 231 Μ Μ Μ Μ Μ = 0.08333 252 ππΉπππ‘π΅ = 1 − 609 Μ Μ Μ Μ Μ Μ Μ = 0.03333 630 ππΉπππ‘πΆ = 1 − 987 = 0.02083Μ 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 π =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 πΆ. πΌππ£). 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 π΄πΏπππ or partial rate of supply. (1 − ππΏπππ) 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 π 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 π , recall the hierarchy defined in expression (4-3), where the series πΉ(1) , πΉ(2) , πΉ(3) is defined by the relationships πΉ(3) π’(3) πΉ(1) π’(1) πΉ(2) ≥π’ (2) ≥ πΉ(π) ≥ β― ≥ π’ . To find the value of π that achieves the acceptable loss threshold, one can sum (π) 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. π π΄πΏπππ πΆ. πΌππ£ under Fair Allocation = (1 − ) β πππ β ∑ π·π π’π ππΏπππ (6-1) π=1 π πΆ. πΌππ£ under Preferential Allocation = πππ ⋅ ∑ π·π π’π (6-2) π=1 Where π is the first (lowest) π such that the following equation is satisfied: πππ β (π·(1) πΉ(1) + π·(2) πΉ(2) + π·(3) πΉ(3) + β― + π·(π) πΉ(π) ) ≥ π΄πΏπππ (6-3) Where: ο· π΄πΏπππ represents the acceptable level of financial loss in the event of a disruption. ο· ππΏπππ 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 ο· ∑ππ=1 π·π π’π represents the total number of components required per day to fulfill total demand of all affected products. ο· ∑ππ=1 π·π π’π represents the total number of components per day required to fulfill total demand of the π 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 (π΄πΏπππ). 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 (π ), parameter π 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 π, recall the hierarchy defined in expression (4-3), where the series π’(1) , π’(2) , π’(3) is defined by the relationships πΉ(1) π’(1) πΉ(2) ≥π’ (2) πΉ(3) ≥π’ (3) πΉ(π) ≥ β― ≥ π’ . To find the value of π that achieves the expected (π) 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 π is also used to determined πππ΄π π (see equation (6-6)), the loss at the specified partial rate of supply, assuming there is only partial rate of supply and no inventory. πππ΄π π is the summation of the financial contributions of the remaining π − π affected products over the firm’s TTR. The demand of these π − π will be unfulfilled with the partial rate of supply π π΄πΏπππ πΆ. πΌππ£ = max [((1 − ) β πππ β (∑ π·π π’π ) β (1 − π )) , 0] πππ΄π π (6-4) π=1 Where π is the highest value of π such that the following equation is satisfied: π·(1) π’(1) + π·(2) π’(2) + π·(3) π’(3) + β― + π·(π) π’(π) ≤π ∑ππ=1 π·π π’π 94 (6-5) πππ΄π π = πππ β (π·(π+1) πΉ(π+1) + π·(π+2) πΉ(π+2) + β― + π·(π) πΉ(π) ) (6-6) Where: ο· πππ΄π π represents MVAR at a specified partial rate of supply (when there is only a partial rate of supply and no inventory). ο· π΄πΏπππ (1 − πππ΄π ) represents the proportion of the firm’s threshold of financial loss that π requires mitigation with inventory. ο· π is the partial rate of supply; the percentage of the normal rate of supply of the disrupted component available during the firm’s TTR. ο· ∑ππ=1 π·π π’π represents the total number of components required per day to fulfill total demand of all affected products. ο· π·(1) π’(1) + π·(2) π’(2) + π·(3) π’(3) + β― + π·(π) π’(π) represents the total number of components required per day to fulfill total demand of the π highest contributing products. ο· π·(π+1) πΉ(π+1) + π·(π+2) πΉ(π+2) + β― + π·(π) πΉ(π) represents the financial loss per incurred per day by having a partial rate of supply that only fulfills demand of the π highest contributing products. In equation (6-4), when the firm’s acceptable level of financial loss (π΄πΏπππ) is equal to the financial loss at a specified partial rate of supply (πππ΄π π ), the ratio between π΄πΏπππ and πππ΄π π 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 − π ) 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 (π‘πΆπππ‘ππππ ) 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 (π‘π΄πππ’ππ’πππ‘πππ ) 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. πΆπΌπ = πΉ. πππ − π‘πΆπππ‘ππππ − π‘π΄πππ’ππ’πππ‘πππ (A-1) To find CIT, time must be segmented into π′ discrete points. At any point π′, the given finished product(s) π′ 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 π‘π . The final point in time is when the firm has recovered from the disruption, represented as π‘(π′+1) . These discrete points in time are ranked such that: π‘π ≤ π‘(1) ≤ π‘(2) ≤ π‘(3) … ≤ π‘(π′) ≤ π‘(π′+1) (A-2) While there are π affected finished products that require the component whose supply has been disrupted, there are only π′ points in time that need to be considered. π′ 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 π. π′ will be less than π if some of the finished products’ daysof-supply are greater than the firm’s time-to-recovery. π′ can also be less than π if multiple finished products have the same days-of-supply. π′ will be equal to π 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: ο· πΆ. πΌππ£ is number of components available at the time of the disruption. ο· πΆ. π ππ is the number of components remaining that have not yet been expended. ο· π‘π is the start of the disruption. ο· π‘π π‘πππ‘ is the start of the interval of time being considered. ο· π‘πππ is the end of the interval of time being considered. ο· π‘πππ‘πππ£ππ is the interval of time being considered. ο· π‘π is the point in time when finished product π has expended its inventory. ο· π‘(π′+1) is the point in time when the firm recovers from the disruption, π‘π . ο· π·π is the average demand of finished product π by all customers per day, from the date of the disruption (π‘π ) until the firm recovers (π‘π ). Where π is from 1 to n. ο· π’π is the usage rate of components in finished products; the number of components required to produce a single unit of finished product π. ο· π′ represents the number of unique days-of-supply of finished products whose days-ofsupply is less than the firm’s time-to-recovery. 109 ο· π§ and π 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. ο· π‘πΆπππ‘ππππ 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. ο· πΉππππ₯π‘ 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. ο· π is the partial rate of supply of the disrupted component available during the firm’s time-to-recovery. ο· π΄. πΆπππ‘ππππ is number of components that have accumulated from the time of disruption to the critical time. ο· π΄. π ππ is the remaining number of accumulated components that have not yet been expended. ο· π‘π π’π π‘πππ is the number of days that the available accumulated parts can sustain the demand at the interval being considered. ο· π‘ππππ’ππ’πππ‘πππ 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 π«π Usage Rate ππ Finished Product Inventory on Hand π·. π°πππ 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. π‘π = 0, π‘π = 7 πΉ. πππ = π‘π − π‘π = 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. π. πΌππ£π΄ 20 = =1 π ∑π=1 π·π΄,π 20 π. πΌππ£π΅ 20 π. π·πππ΅ = π = =2 ∑π=1 π·π΅,π 10 π. πΌππ£πΆ 80 π. π·πππΆ = π = =4 ∑π=1 π·πΆ,π 20 π. πΌππ£π· 120 π. π·πππ· = π = =4 ∑π=1 π·π·,π 30 π. πΌππ£πΈ 120 π. π·πππΈ = π = =6 ∑π=1 π·πΈ,π 20 π. π·πππ΄ = Figure A-1: Allocation of Finished Product Inventory 111 In this example, there are six time points to consider, ranked as follows. π‘π ≤ π‘(1) ≤ π‘(2) ≤ π‘(3) ≤ π‘(4) ≤ π‘(π′+1) The first point in time is at the time of disruption, π‘π = 0. The second point in time is when finished product A expends its finished product inventory, π‘(1) = π. π·πππ΄ = 1. The third point in time is when finished product B expends its finished product inventory, π‘(2) = π. π·πππ΅ = 2. The third point in time is when both finished products C and D expend their finished product inventory, π‘(3) = π. π·πππΆ = π. π·πππ· = 4. The fifth is when finished product E expends its finished product inventory, π‘(4) = π. π·πππΈ = 6. The final point in time is when the firm recovers from the disruption, π‘(π′ +1) = π‘π = 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 πΆ. π ππ = πΆ. πΌππ£ π‘π π‘πππ‘ = π‘π For π§ = 1 to (π′ + 1) π‘πππ = π‘π§ π‘πππ‘πππ£ππ = π‘πππ − π‘π π‘πππ‘ If (∑π§π=1 π·π π’π ) β π‘πππ‘πππ£ππ ≤ πΆ. π ππ Then πΆ. π ππ = πΆ. π ππ − (∑π§π=1 π·π π’π ) β π‘πππ‘πππ£ππ π‘π π‘πππ‘ = π‘πππ Else πΆ.π ππ π‘πΆπππ‘ππππ = π‘π π‘πππ‘ + ∑π§ π· π’ − π‘π 11 12 13 14 πΉππππ₯π‘ = π§ Exit For End If Next π§ π=1 π π 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 πΉππππ₯π‘ , to be used for determining the accumulation time. 113 Example Assume that at the time of disruption there are 450 components of inventory available (πΆ. πΌππ£ = 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 π§ π‘π π‘πππ‘ π‘πππ π‘πππ‘πππ£ππ (∑ π·π π’π ) β π‘πππ‘πππ£ππ Components Remaining (at start of time interval) Components Remaining (at end of time interval) πΆ. π ππ πΆ. π ππ 450 450 450 350 350 - π=1 0 1 1 2 1 1 2 4 2 0 (π·π΄ π’π΄ ) β 1 = 20 β 5 β 1 = 100 (π·π΄ π’π΄ + π·π΅ π’π΅ ) β 2 = (20 β 5 + 10 β 10) β 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: π‘ππππ‘ππππ = π‘π π‘πππ‘ + πΆ. π ππ 350 350 − π‘π = 2 + −0=2+ = 2.875 π§ ∑π=1 π·π π’π 20 β 5 + 10 β 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 π΄. πΆπππ‘ππππ = (π‘ππππ‘ππππ − π‘π ) β π β (∑ππ=1 π·π π’π ) π΄. π ππ = π΄. πΆπππ‘ππππ π‘π π‘πππ‘ = π‘ππππ‘ππππ For π = πΉππππ₯π‘ to (π′ + 1) π‘πππ = π‘π π‘πππ‘πππ£ππ = π‘πππ − π‘π π‘πππ‘ π΄.π ππ π‘π π’π π‘πππ = ∑π π 8 9 10 11 12 13 14 15 16 If π‘π π’π π‘πππ > π‘πππ‘πππ£ππ or π‘π π’π π‘πππ < 0 π‘ππππ’ππ’πππ‘πππ = π‘ππππ’ππ’πππ‘πππ + π‘πππ‘πππ£ππ π΄. π ππ = π΄. π ππ − (∑ππ=1 π·π π’π − π β ∑ππ=1 π·π π’π ) β π‘πππ‘πππ£ππ π‘π π‘πππ‘ = π‘πππ Else π‘ππππ‘ππππ = π‘ππππ’ππ’πππ‘πππ + π‘π π’π π‘πππ Exit For End If Next π π=1 π·π π’π −π β∑π=1 π·π π’π 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, π‘π π’π π‘πππ , 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 π‘π π’π π‘πππ exceeds the interval being considered, then accumulation time will extend beyond the interval time. If π‘π π’π π‘πππ 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 π‘π π’π π‘πππ is less than the interval being considered, accumulation time is only extended by π‘π π’π π‘πππ . 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 π§ π‘π π‘πππ‘ π‘πππ π‘πππ‘πππ£ππ 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 (∑ π·π π’π ) β π‘πππ‘πππ£ππ π΄. π ππ π΄. π ππ π‘π π’π π‘πππ π‘ππππ’ππ’πππ‘πππ 287.5 175.0 2.3000 1.1250 175.0 0 0.2975 1.2596 π=1 2.875 4 1.125 (π·π΄ π’π΄ + π·π΅ π’π΅ ) β 1.125 = (20 β 5 + 10 β 10) β 1.125 = 225 (π·π΄ π’π΄ + π·π΅ π’π΅ + π·πΆ π’πΆ + π·π· π’π· ) β 2 4 6 2 = (20 β 5 + 10 β 10 + 20 β 10 + 30 β 10) β 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. π‘π π’π π‘πππ = π΄. π ππ ∑2π=1 π·π π’π − π β (∑5π=1 π·π π’π ) 287.50 (20 β 5 + 10 β 10) − 0.1 β (20 β 5 + 10 β 10 + 20 β 10 + 30 β 10 + 20 β 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 π΄. π ππ = π΄. π ππ − (∑ π·π π’π − π β (∑ π·π π’π )) β π‘πππ‘πππ£ππ π=1 π=1 = 287.50 − ((20 β 5 + 10 β 10) − 0.1 β (20 β 5 + 10 β 10 + 20 β 10 + 30 β 10 + 20 β 15)) β 1.125 = 287.50 − (200 − 0.1 β 1000) β 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. π‘π π’π π‘πππ = π΄. π ππ ∑2π=1 π·π π’π − π β (∑5π=1 π·π π’π ) 175 (20 β 5 + 10 β 10 + 20 β 10 + 30 β 10) − 0.1 β (20 β 5 + 10 β 10 + 20 β 10 + 30 β 10 + 20 β 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. π‘ππππ’ππ’πππ‘πππ = π‘ππππ’ππ’πππ‘πππ + π‘π π’π π‘πππ = 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. πΆπΌπ = πΉ. πππ − π‘πΆπππ‘ππππ − π‘π΄πππ’ππ’πππ‘πππ = 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: π′ +1 ππ΄π = ∑ [(1 − π β (∑ππ=1 π·π π’π ) π =ππΆπΌπ ∑π π=1 π·π π’π π ) β (π‘π − π‘π −1 ) β ∑ π·π πΉπ ] (A-3) π=1 Where: ο· ππΆπΌπ represents the first finished product to have its inventory expended during the CIT period. When π = ππΆπΌπ , π‘π −1 = πΆπΌπ; this is the initial time interval from the start of the CIT period until a finished product has its inventory expended. ο· π′ + 1 is defined such that π‘π′ +1 is the point in time when the firm recovers. ο· π β (∑ππ=1 π·π π’π ) ο· ∑π π=1 π·π π’π 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 π‘π for the time interval beginning at π‘π −1 and ending at π‘π . 123 ο· π β(∑ππ=1 π·π π’π ) ∑π π=1 π·π π’π represents the percentage of demand, required by those finished products whose finished product days-of-supply is less than π‘π 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. ο· π‘π − π‘π −1 represents the time interval. ο· ∑π π=1 π·π πΉπ represents the total financial value of all products whose finished product daysof-supply is less than π‘π . 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 ππ 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. π′ +1 ππ΄π = ∑ [(1 − π =3 ππ΄π = (1 − ππ΄π = (1 − π β (∑ππ=1 π·π π’π ) ∑π π=1 π·π π’π π ) β (π‘π − π‘π −1 ) β ∑ π·π πΉπ ] π=1 0.1 β (20 β 5 + 10 β 10 + 20 β 10 + 30 β 10 + 20 β 15) ) β (6 − 4.2917) (20 β 5 + 10 β 10 + 20 β 10 + 30 β 10) β (20 β 140 + 10 β 200 + 20 β 220 + 30 β 120) 0.1 β (20 β 5 + 10 β 10 + 20 β 10 + 30 β 10 + 20 β 15) + (1 − ) β (7 − 6) (20 β 5 + 10 β 10 + 20 β 10 + 30 β 10 + 20 β 15) β (20 β 140 + 10 β 200 + 20 β 220 + 30 β 120 + 20 β 450) 100 100 ) β (1.7083) β (12,800) + (1 − ) β (1) β (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 ππ,π representing the number of products of finished product π produced per day during time interval that begins at π‘π −1 and ends at π‘π . The products are produced based upon the allocation of the partial rate of supply to the products. π′ +1 π min πππ΄π = min ∑ [(π‘π − π‘π −1 ) β ∑ ((π·π − ππ,π ) β πΉπ )] π =ππΆπΌπ (A-4) π=1 subject to: π ∑(π·π,π ) − ππ,π ≥ 0, ∀π, π (A-4.1) π=1 ∑π π=1 ππ,π π’π π − π ≥ 0, ∀π ∑π=1 π·π π’π (A-4.2) ππ,π ≥ 0, ∀π, π (A-4.3) ο· ππΆπΌπ represents the first finished product to have its inventory expended during the CIT period. When π = ππΆπΌπ , π‘π −1 = πΆπΌπ; this is the initial time interval from the start of the CIT period until a finished product has its inventory expended. ο· π′ + 1 is defined such that π‘π′ +1 is the point in time when the firm recovers. ο· ∑π π=1(ππ,π β π’π ) represents the total number of components allocated from the partial rate of supply for the time interval beginning at π‘π and ending at π‘π −1 . ο· ∑ππ=1(∑π π=1(π·π,π ) β π’π ) represents the total number of components required to meet demand for all affected products per day. ο· ∑π π=1(ππ,π βπ’π ) π (∑π π=1(∑π=1(π·π,π )βπ’π )) represents the percentage of total demand for components per day satisfied with the allocated partial rate of supply. ο· π‘π − π‘π −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 ππ Usage Rate ππ Financial Contribution per component Product A 140 5 Product B 200 10 28 20 ππ Product Product Product C D E 220 120 450 10 10 15 22 12 30 ππ 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. π′ +1 π πππ΄π = ∑ [(π‘π − π‘π −1 ) β ∑ ((π·π − ππ,π ) β πΉπ )] π =3 π=1 πππ΄π = (6 − 4.2917) β ((20 − 20) β 140 + (10 − 0) β 200 + (20 − 0) β 220 + (30 − 0) β 120) + (7 − 6) β ((20 − 0) β 140 + (10 − 0) β 200 + (20 − 0) β 220 + (30 − 0) β 120 + (20 − 6.67) β 450) πππ΄π = (1.7083) β (10,000) + (1) β (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 ππ΄,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 ππΈ,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) β ((20 − ππ΄,4 ) β 140 + (10 − ππ΅,4 ) β 200 + (20 − ππΆ,4 ) β 220 + (30 − ππ·,4 ) β 120) + (7 − 6) β ((20 − ππ΄,5 ) β 140 + (10 − ππ΅,5 ) β 200 + (20 − ππΆ,5 ) β 220 + (30 − ππ·,5 ) β 120 + (20 − ππΈ,5 ) β 450)] subject to: (1.1) 20 − ππ΄,4 ≥ 0 (1.2) 10 − ππ΅,4 ≥ 0 (1.3) 20 − ππΆ,4 ≥ 0 (1.4) 30 − ππ·,4 ≥ 0 (1.5) 20 − ππ΄,5 ≥ 0 (1.6) 10 − ππ΅,5 ≥ 0 (1.7) 20 − ππΆ,5 ≥ 0 (1.8) 30 − ππ·,5 ≥ 0 (1.9) 20 − ππΈ,5 ≥ 0 ππ΄,4 β 5 + ππ΅,4 β 10 + ππΆ,4 β 10 + ππ·,4 β 10 (2.1) 0.1 − ≥0 20 β 5 + 10 β 10 + 20 β 10 + 30 β 10 + 20 β 15 ππ΄,4 β 5 + ππ΅,4 β 10 + ππΆ,4 β 10 + ππ·,4 β 10 + ππΈ,4 β 15 (2.2) 0.1 − ≥0 20 β 5 + 10 β 10 + 20 β 10 + 30 β 10 + 20 β 15 (3.1) ππ΄,4 ≥ 0 (3.2) ππ΅,4 ≥ 0 (3.3) ππΆ,4 ≥ 0 (3.4) ππ·,4 ≥ 0 (3.5) ππ΄,5 ≥ 0 (3.6) ππ΅,5 ≥ 0 130 (3.7) ππΆ,5 ≥ 0 (3.8) ππ·,5 ≥ 0 (3.9) ππΈ,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: π′ +1 π ∑ [(π‘π − π‘π −1 ) β ∑(ππ,π β πΉπ )] π =ππΆπΌπ (A-5) π=1 ο· ππΆπΌπ represents the first finished product to have its inventory expended during the CIT period. When π = ππΆπΌπ , π‘π −1 = πΆπΌπ; this is the initial time interval from the start of the CIT period until a finished product has its inventory expended. ο· π′ + 1 is defined such that π‘π′ +1 is the point in time when the firm recovers. ο· ∑π π=1(ππ,π β πΉπ ) represents the total financial gain of all products whose finished product days-of-supply is less than π‘π , based upon the allocation of the partial rate of supply to those products under preferential product allocation. ο· π‘π − π‘π −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, ππ , representing the expected percentage of demand of product π that can be substituted by a non-disrupted product, such that 0 ≤ ππ ≤ 1 given there is enough capacity to produce the substitute. In certain situations, no substitute will be available and ππ = 0. In other cases, a substitute will be available (0 < ππ < 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: πΉ(1) πΉ(2) πΉ(3) πΉ(π) ≥ ≥ ≥β―≥ π’(1) π’(2) π’(3) π’(π) 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: π πππ΄π = min (πΆπΌπ β ∑ πΉπ β (π·π β (1 − ππ ) − ππ )) (B-1) π=1 subject to: π·π (1 − ππ ) − ππ ≥ 0, ∀π (B-1.1) ∑ππ=1 ππ π’π π − π ≥0 ∑π=1 π·π π’π (B-1.2) ππ ≥ 0, ∀π (B-1.3) Where: ο· πΆπΌπ is the customer impact time given available component and finished product inventory and the accumulated partial rate of supply. ο· π·π – average demand of finished product π by all customers per day, from the date of the disruption (π‘π ) until the firm recovers (π‘π ). Where π is from 1 to n. ο· π’π – usage rate of components in finished products; the number of components required to produce a single unit of finished product π. ο· πΉπ is the financial value per unit of finished product π. 133 ο· π is the partial rate of supply of the disrupted component available during the firm’s time-to-recovery. ο· ππ is the number of product π produced per day, given the allocation of parts from partial rate of supply during the CIT period. ο· π π is the maximum fraction of product π that can be expected to be substituted by nondisrupted products. ο· π·π (1 − ππ ) is the remaining unmet demand per day of product π across all customers π after the use of the substitute. ο· π·π (1 − ππ ) − ππ ≥ is the remaining unmet demand per day of product π across all customers π after use of the substitute and the allocation of the partial rate of supply. ο· ∑ππ=1 ππ π’π is the total number of components per day allocated from the partial rate of supply. ο· ∑ππ=1 π·π π’π 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. ο· ∑π π=1 ππ π’π ∑π π=1 π·π π’π 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, ππ , of product π 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 ππ representing the number of finished products per day produced given the allocation of partial rate of supply during the CIT period. The second is ππ , representing the portion of the total fraction of product π that can be expected to be substituted by an unaffected product that is actually used to reduce an affected products demand. π πππ΄π = min (πΆπΌπ β ∑(πΉπ (π·π − ππ ) − πΉππ π·π ππ )) (B-2) π=1 subject to: π·π (1 − ππ ) − ππ ≥ 0, ∀π π − (B-2.1) ∑ππ=1 ππ π’π ≥0 ∑ππ=1 π·π π’π (B-2.2) ππ ≥ ππ , ∀π (B-2.3) ππ ≥ 0, ∀π (B-2.4) ππ ≥ 0, ∀π (B-2.5) Where: ο· πΉπ is the financial value per unit of finished product π. ο· πΉππ is the financial value per unit of substitute used for finished product π. ο· ππ is the number of product π produced per day, given the allocation of parts from partial rate of supply during the CIT period. ο· ππ is the maximum fraction of product π that can be expected to be substituted by nondisrupted products. ο· ππ is the fraction of product π that is substituted by non-disrupted products. ο· πΉπ (π·π − ππ ) is the value-at-risk per day for product π after the allocation of partial rate of supply during the CIT period. ο· πΉππ π·π ππ is the value of using the substitute to meet some demand of affected product π. 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 π 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 π components have been disrupted, the parts are numbered π = 1,2,3 … πΎ. Each component has a partial rate of supply π π . These components go into π products numbered i = 1,2,3,… n. The financial contribution of each product is πΉπ and the demand for each product across all customers π·π . Each product requires a certain number of any of the disrupted components to produce a single unit of the affected product. Let π’π,π represent the number of components π required to produce a single unit of affected product π. Let ππ represent the number of products that are produced per day given the allocation of disrupted components. π πππ΄π = min (πΆπΌπ β ∑ πΉπ β (π·π − ππ )) ( B-3 ) π=1 subject to: π·π − ππ ≥ 0, ∀π ( B-3.1) ∑ππ=1 ππ π’π,π π π − π ≥ 0, ∀π ∑π=1 π·π π’π,π ( B-3.2) ππ ≥ 0, ∀π (B-3.3) 136 Where: ο· π’π,π is the number of disrupted parts π required to produce a single unit of finished product π. ο· π π is the partial rate of supply of disrupted component π available during the firm’s timeto-recovery. ο· ππ is the number of products produced per day given the allocation of components from the partial rate of supplies available. ο· π·π − ππ is the remaining unmet demand per day of product π across all customers π after the allocation of the partial rate of supply. ο· ∑ππ=1 ππ π’π,π is the total number of disrupted component π allocated from the partial rate of supply per day. ο· ∑ππ=1 π·π π’π,π is the total number of components π 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 References Berger, P.D., Gerstenfeld, A. and Zeng, A.Z. 2004. “How many suppliers are best? A decisionanalysis approach.” Omega (Oxford) 32 (1): 9-15. Blackhurst, J.V., Scheibe, K.P., and Johnson, D.J. (2008). Supplier Risk Assessment and Monitoring for the Automotive Industry. International Journal of Physical Distribution & Logistics Management. 38(2), 143-165. Chopra, S. and Sodhi, M.S. (2004). Managing Risk to Avoid Supply-Chain Breakdown. MIT Sloan Management Review. 46(1), 53-61. Hekerman, D., Geiger, D., and Chickering, D.M. (1995). Learning Bayesian Networks: The Combination of Knowledge and Statistical Data. Machine Learning 20, 197-243. Hendricks, K.B. and Singhal, V.R. (2005). An Empirical Analysis of the Effect of Supply Chain Disruptions on Long-Run Stock Price Performance and Equity Risk of the Firm. Production and Operations Management. 14(1), 35-52. Hoffman, A.J. (1985). On Greedy Algorithms that Succeed. In I. Anderson (Ed.), Surveys in Combinatorics 1985 (Cambridge University Press, Cambridge, UK), pp. 97-112. Isidore, C. (2002). Bite of Port Lockout to be Felt Soon. CNN/Money. Retrieved from http://money.cnn.com/2002/09/30/news/economy/port_impact/ Jung, K., Lim, Y., and Oh, J. (2011). A Model for Measuring Supplier Risk: Do Operational Capability Indicators Enhance the Prediction Accuracy of Supplier Risk? British Journal of Management. 22, 609-627. Knight, F.H. (1964). Risk, Uncertainty, and Profit. Sentry Press, New York, NY. (Original work published 1921). Lockamy, A., III and McCormack, K. (2012). Modeling Supplier Risks using Bayesian Networks. Industrial Management & Data Systems. 112(2), 313-333. Martha, J. (2002). Just-In-Case Operations. Warehousing Forum. 17(2). Munich Reinsurance America. (2012). 2011 Natural Catastrophe Year in Review [PDF Document]. Retrieved from http://www.ctnow.com/media/acrobat/2012-01/67158951.pdf. Norrman, A. and Jansson, U. (2004). Ericsson’s Proactive Supply Chain Risk Management Approach after a Serious Sub-Supplier Accident. International Journal of Physical Distribution & Logistics Management. 34(5), 434-456. 138 Norrman, A. and Lindroth, R. (2002). Supply Chain Risk Management: Purchasers’ vs Planners’ Views on Sharing Capacity Investment Risks in the Telecom Industry. Proceedings of the 11th International IPSERA Conference, Holland, pp.577-595. Oke, A. and Gopalakrishnan, M. (2009). Managing Disruptions in Supply Chains: A Case Sutyd of a Retail Supply Chain. International Journal of Production Economics. 118, 168-174. Revilla, E. and Sáenz, M.J. (2013). A Taxonomy of Supply Chain Risk Management Strategies: Antecedents and Performance. International Decision Science Institute Conference 2013. Schmitt, A.J., Sun, S.A., Snyder, L.V, and Shen, Z.J.M. (2015). Centralization versus Decentralization: Risk Pooling, Risk Diversification, and Supply Chain Disruptions. Omega 52, 201-212. Simchi-Levi, D., Schmidt, W., and Wei, Y. 2014. “From Superstorms to Factory Fires: Managing Unpredictable Supply-Chain Disruptions.” Harvard Business Review 92 (1/2): 96–101. Sheffi, Y. (2002). Supply Chain Management under the Threat of International Terrorism. The International Journal of Logistics Management. 12(1), 1-11. Sheffi, Y. (2005). The Resilient Enterprise. MIT Press, Cambridge, MA. Sheffi, Y. and Rice, J.B., Jr. (2005). A Supply Chain View of the Resilient Enterprise. MIT Sloan Management Review. 47(1), 41-48. Sherin, B. and Bartoletti, S. (1999). Taiwan’s 921 Quake: Effect on the Semicnoductor Industry and Recommendations for Preparing Future Earthquakes. [PDF Document]. Retrieved from http://www.semiconductorsafety.org/proceedings00/sherinbrian.pdf Snyder, L.V., and Shen, Z-J.M. (2006). Supply and Demand Uncertainty in Multi-Echelon Supply Chains. Working Paper, Lehigh University and University of California, Berkeley. Svensson, G. (2002). A Conceptual Framework of Vulnerability in Firms’ Inbound and Outbound Logistics Flows. International Journal of Physical Distribution & Logistics Management. 32(2), 110-134. Tang, C.S. (2006). Perspectives in Supply Chain Risk Management. International Journal of Production Economics. 103, 451-488. Tang, C. and Tomlin, B. (2008). The Power of Flexibility for Mitigating Supply Chain Risks. International Journal of Production Economics. 116, 12-27. Tomlin, B. (2006). On the Value of Mitigation and Contingency Strategies for Managing Supply Chain Disruption Risks. Management Science. 52(5), 639-657. 139 Wolk, M. (2002). Behind the West Coast Port Lockout. NBC News. Retrieved from http://www.nbcnews.com/id/3073398/ns/business-us_business/t/behind-west-coast-portlockout/ Wu, T., Blackhurst, J., and Chidambaram, V. (2006). A Model for Inbound Supply Risk Analysis. Computers in Industry. 57, 350-365 Zsidisin, G.A. (2004). An Analysis of Supply Risk Assessment Techniques. International Journal of Physical Distribution & Logistics Management. 34(5), 397-413. 140