Emergency Management Strategies for the Retail Industry by Kandace N. Ballard A dissertation submitted to the Graduate Faculty of Auburn University in partial ful llment of the requirements for the Degree of Doctor of Philosophy Auburn, Alabama December 12, 2011 Keywords: Inventory, minimax, minimax regret, disruption management, stochastic programming Copyright 2011 by Kandace N. Ballard Approved by: Emmett J. Lodree, Jr., Chair, Assistant Professor of Industrial and Systems Engineering Chan Park, Daniel F. and Josephine Breeden Professor of Industrial and Systems Engineering Ming Liao, Professor of Mathematics and Statistics Abstract Uncertain disruptions complicate inventory management for retailers because it is dif- cult to determine when and how to adjust order quantities to ensure high service levels before and after a possible disruption. In order to assist retailers with their decision-making, three di erent models have been developed to determine the importance and e ectiveness of considering proactive approaches to disruptions, speci cally, forecasted storms. The rst model addresses the time horizon when a storm is approaching and it is not known whether a demand surge for emergency items will occur. Minimax decision criterion highlighted the cir- cumstances that constitute increasing the order quantity or encouraging the retailer to adopt a proactive approach instead of the current strategy of \wait-and-see". Ordering strategies also a ect the response e orts as shown in the second model, which is comprised of two uncertain disruptions. Minimax and minimax regret decision criteria were used to evaluate the model and provided insight into the inventory management decisions during and after the storm. Minimax regret decision criterion supported holding inventory during the storm to ensure a higher service level after the storm while minimax decision criterion supported the opposite. However, both criteria advocated holding the same order quantity during the second disruption or demand surge after the storm. Lastly, a stochastic programming model was developed to determine pre-positioned quantities for a network of retailers in addition to post-storm shipments between retailers and the manufacturer to alleviate shortages. The results from the commercial software and solution methodology supported pre-positioning items at most retailers despite the manufacturer?s location. Overall, the research presented in this dissertation illustrates the importance and cost e ectiveness of incorporating inventory strategies into management decisions when a storm is on the horizon. ii Acknowledgments I would rst and foremost like to thank my Lord and Savior Jesus Christ because without Him, I would not have completed the process of obtaining a Ph.D. He re ned me and showed me that I can do anything with Him. Secondly, I want to thank my husband, Eric, for being so encouraging and supportive thoughout my studies. He continued to push me when I wanted to give up and listened to all of the issues I ran into even when it was not interesting. I am so grateful and thankful for him. I am also eternally grateful for my advisor, Dr. Emmett Lodree, for continuing to guide my dissertation from afar. He continued to encourage, support, and guide me even when it was uncertain where my dissertation was going. In addition, I would like to thank my committee, Dr. Chan Park and Dr. Ming Liao, for their support in the dissertation process. I would also like to thank my parents, Don and Diane, for always teaching me to strive to do my best in all things. I also appreciate my second set of parents, my in-laws, Randy and Sandy for their continued prayers, support, and understanding (especially during the holidays). Lastly, I hope that I can be an example to my daughter, Kya, to show her that she can do anything with God on her side, hard work, and perseverance. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Overview of Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Minimax EOQ Policies with Demand Surge, Lead-Time and Lost Sales 2 1.1.2 Minimax and Minimax Regret Inventory Control Policies Regarding Demand Disruptions and Damaged Inventory . . . . . . . . . . . . . 3 1.1.3 Pre-Positioning Hurricane Supplies in a Commercial Supply Chain . . 4 2 Minimax EOQ Policies with Demand Surge, Lead-Time and Lost Sales . . . . . 6 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Demand Disruptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Minimax Decision Criterion . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.2 General Cost Equation and Model Notation . . . . . . . . . . . . . . 12 2.3.3 Minimax Decision Criterion . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.4 Reactive Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.5 Proactive Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.1 Numerical Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 22 iv 2.4.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 Summary and Future Extensions . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Minimax and Minimax Regret Inventory Control Policies Regarding Demand Disruptions and Damaged Inventory . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.1 Demand Disruptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.2 Minimax Decision Criterion . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.3 Minimax Regret Decision Criterion . . . . . . . . . . . . . . . . . . . 34 3.3 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.1 General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.2 Model Assumptions and Notation . . . . . . . . . . . . . . . . . . . . 35 3.3.3 General Cost Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3.4 Inventory Policy: Hold Inventory During 1st Disruption . . . . . . . . 41 3.3.5 Inventory Policy: Do Not Hold Inventory During 1st Disruption . . . 46 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.1 First Experiment: =f0;1g . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.2 Second Experiment: 2[0;1] . . . . . . . . . . . . . . . . . . . . . . 49 3.4.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.5 Summary and Future Extensions . . . . . . . . . . . . . . . . . . . . . . . . 55 4 Pre-Positioning Hurricane Supplies in a Commercial Supply Chain . . . . . . . . 57 4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1.1 Disastrous Hurricanes and Emergency Management . . . . . . . . . . 57 4.1.2 Pre-Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 Problem Description: Commercial Pre-Positioning of Commodities . . . . . . 59 4.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 v 4.4.1 Stochastic Programming Model . . . . . . . . . . . . . . . . . . . . . 67 4.4.2 PDSA Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.5.1 Demand Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5.2 Case Study Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.5.3 PDSA Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.5.4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.6 Summary and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . 96 5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.1 Minimax EOQ Policies with Demand Surge, Lead-Time and Lost Sales . . . 99 5.2 Minimax and Minimax Regret Inventory Control Policies Regarding Demand Disruptions and Damaged Inventory . . . . . . . . . . . . . . . . . . . . . . 100 5.3 Pre-Positioning Hurricane Supplies in a Commercial Supply Chain . . . . . . 100 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 A Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 A.1 Derivation of Cost Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 A.2 Reactive Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 A.3 Proactive Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 B Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 vi List of Illustrations 2.3.1 Reactive Strategy with Disruption: Case 1 . . . . . . . . . . . . . . . . . . . 16 2.3.2 Reactive Strategy with Disruption: Case 3 . . . . . . . . . . . . . . . . . . . 17 2.3.3 Proactive Strategy with Disruption . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.4 Proactive Strategy with No Disruption: Case 1 . . . . . . . . . . . . . . . . . 20 2.3.5 Proactive Strategy with No Disruption: Case 3 . . . . . . . . . . . . . . . . . 21 2.4.1 The E ect of the Order Cost, A . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.2 Strategy Selection Considering No. of Orders with No Lost Sales . . . . . . . 27 2.4.3 The E ect of the Holding Cost, h . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.1 Time Horizon for Demand Disruptions . . . . . . . . . . . . . . . . . . . . . 31 3.3.1 Hold Inventory with Disruption: Case 1 . . . . . . . . . . . . . . . . . . . . . 41 3.3.2 Hold Inventory with Disruption: Case 2 . . . . . . . . . . . . . . . . . . . . . 42 3.3.3 Hold Inventory with Disruption: Case 3 . . . . . . . . . . . . . . . . . . . . . 43 3.3.4 Hold Inventory with No Disruption: Case 1 . . . . . . . . . . . . . . . . . . . 45 3.3.5 Do Not Hold Inventory with Disruption . . . . . . . . . . . . . . . . . . . . . 46 3.3.6 Do Not Hold Inventory with No Disruption . . . . . . . . . . . . . . . . . . . 47 4.5.1 Source: Rawls and Turnquist [53]. Network in southeast United States used for case study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5.2 The e ect of the unit production cost, c. . . . . . . . . . . . . . . . . . . . . 92 4.5.3 The e ect of the post-hurricane transportation cost, cw. . . . . . . . . . . . . 93 4.5.4 The e ect of the unit shortage cost, s. . . . . . . . . . . . . . . . . . . . . . 95 A.1.1Reactive Strategy with Disruption: Case 1 Appendix . . . . . . . . . . . . . 107 vii List of Tables 2.3.1 Chapter 2 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Chapter 2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.1 Values Tested for 28 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.2 Sample Results for 28 Experiment . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3.1 Chapter 3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.2 Chapter 3 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4.1 Values Tested for 29 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.2 Minimax Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4.3 Possible Minimum Cost Decisions for Each Outcome . . . . . . . . . . . . . 51 3.4.4 Regret Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.5 Possible Maximum Regret Outcomes for Each Decision . . . . . . . . . . . . 52 3.4.6 Minimax Regret Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.7 Sensitivity Analysis Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4.1 Summary of notations for stochastic programming model. . . . . . . . . . . . 69 4.4.2 Notations for PDSA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.4.3 Input data for numerical example. . . . . . . . . . . . . . . . . . . . . . . . . 76 4.4.4 Example demand scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4.5 Optimal pre-positioning strategy for numerical example. . . . . . . . . . . . 77 4.5.1 Cost data for case study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5.2 Demand information for each sample hurricane, adapted from [53]. . . . . . . 84 viii 4.5.3 PDSA performance relative to the optimal solution for each problem instance. 89 4.5.4 The e ect of the unit production cost, c. . . . . . . . . . . . . . . . . . . . . 91 4.5.5 The e ect of the post-hurricane transportation cost, cw. . . . . . . . . . . . . 92 4.5.6 The e ect of the unit shortage cost, s. . . . . . . . . . . . . . . . . . . . . . 94 ix Chapter 1 Introduction Recent natural disasters such as the terrorist attacks of 9/11, Hurricane Katrina, and the ooding along the Mississippi River during the spring of 2011, have revealed the shortcomings of the preparation and response procedures of public and private sectors. As a result, a multitude of academic disciplines have begun to explore ways to mitigate risks, prepare in advance for impending disasters, respond to supply disruptions caused by disasters, and rebuild socio-economic structures a ected by disasters. Private sector organizations? decisions regarding emergency management a ect resi- dents? abilities to prepare for a possible storm. Their decisions also a ect government and non-governmental organizations? execution of their policies to provide necessary supplies. Retailers face the decision of whether or not they should order additional supplies to prepare for a storm, and if the storm does not materialize, they will incur holding costs or be forced to sell goods at discounted prices. On the other hand, if they are not prepared, they will hinder the preparation and response processes and forfeit additional revenue. The uncer- tainty of events such as hurricanes make decisions regarding emergency management for the private sector challenging. On a personal note, my parents evacuated for Hurricane Ike in 2008, and it made me consider how to make the process of preparing for a storm seem less daunting. They, like many others, struggled to nd the necessary supplies and food to sustain them until the retailers reopened. Through my research, I am thankful that I can utilize my knowledge about inventory and operations research to solve meaningful problems that may alleviate the stress for coastal residents in the event that they have to evacuate or prepare to ride out a storm. 1 1.1 Overview of Proposal 1.1.1 Minimax EOQ Policies with Demand Surge, Lead-Time and Lost Sales Chapter 2 addresses an inventory control problem that seeks to determine an ordering strategy for an uncertain demand disruption considering emergency supplies. The model extends the model presented in Lodree [16] by including lead-time and lost sales. The demand disruption that is explored in this paper is a possible sudden surge in demand or an increase in the demand rate before a possible disaster. Retailers must decide if they will adopt a proactive approach meaning they will modify their batch size before they are certain the demand surge will occur, or a reactive approach which means that they will not modify their batch size until they are certain the demand surge will occur. If retailers choose a reactive ordering strategy, they may incur lost sales as a consequence of the lead-time. The typical approach for handling demand uncertainty is determining an optimal order quantity using long run approximations, which does not adequately address the problem because each storm has di erent characteristics that will a ect the strategy selection. In order to e ectively address the problem, minimax decision criterion is used to evaluate the ordering strategy decision. Many papers have addressed risk mitigation associated with supply or demand disruptions by determining optimal order quantities during disruptions caused by various events (e.g., [46], [23] and [65]) and predicting the supplier?s availability to determine optimal order quantities (e.g., [47], [47], and [55]). Within the demand disruption literature, Lodree and Taskin [35] and Taskin and Lodree [62] allowed lost sales for single- period problems when a demand surge occurs. Other papers in demand disruption literature address lead-time, but not lost sales because the disruption is de ned as a period of no demand (e.g., [51]). As previously discussed, this paper investigates ordering strategies for uncertain demand surges while considering lead-time and allowing lost sales. Minimax decision criterion has been utilized in the literature considering optimal ordering quantities (e.g., [49], [19], [28], and [18]), but minimax decision criterion has been applied to a disruption 2 problem in only one other paper ([16]) to the knowledge of the authors. We develop a framework to assist retailers with their decision-making before an uncertain demand surge using minimax decision criterion. This paper provides the following contributions: a136 Closed form solutions are determined for the number of orders and order quantities for items with non-zero lead-time during a demand disruption over a nite horizon. a136 An ordering strategy (proactive or reactive) is determined by applying minimax deci- sion criterion to inventory disruption problems. 1.1.2 Minimax and Minimax Regret Inventory Control Policies Regarding De- mand Disruptions and Damaged Inventory The model presented in Chapter 3 considers the time horizon that begins at the end of the demand disruption that is explored in Chapter 2. When the time horizon in Chapter 3 begins, a second and third demand disruption may occur and last for a known amount of time. The disruptions explored in Chapter 3 are a period of no demand where the retailer might be closed as a result of an uncertain event such as a hurricane and the demand surge that occurs after the e ects of the storm have subsided. The surge in Chapter 2 captures the preparation for the event, while Chapter 3 considers the time period during and after the storm. When retailers are faced with an uncertain event, they must decide if they will keep inventory on-hand to be prepared for a demand surge after the disruption is over. An additional source of uncertainty that is considered in this chapter is whether the inventory will be damaged during the disruption. Retailers may either hold inventory, which also includes the decision about the quantity to hold, or they may not hold inventory assuming that all of the inventory will be damaged. In addition to minimax decision criterion, minimax regret decision criterion will be utilized to determine the best strategy. Savage [57] rst explored the minimax regret decision rule to demonstrate its uses when making optimal decisions. Several authors have used 3 minimax regret decision criterion when considering the distribution free newsvendor problem (e.g., [43], [32], [64], and [69]). Perakis and Roels [49] found solutions for the optimal order quantity that minimize the maximum regret for not choosing optimally when the demand distribution has partial information. Jammernegg and Kischka [29] considered two other risk factors in their objective function to determine a robust solution for the order quantity. With regard to our model, the absolute regret will be minimized considering each inventory decision. This paper provides similar contributions to Chapter 2 and they are the following: a136 A model is developed for two possible disruptions that occur after the demand surge presented in Chapter 2. a136 A more general class of possible disruption cases with two sources of uncertainty are considered. a136 An ordering strategy is determined by applying minimax and minimax regret decision criteria to an inventory disruption problem. 1.1.3 Pre-Positioning Hurricane Supplies in a Commercial Supply Chain The fourth chapter considers the pre-positioning problem from a pro t-driven perspec- tive for a supply chain comprised of one manufacturer and a network of retailers. The manufacturer must make decisions regarding the location and quantity of emergency items that will be pre-positioned across a network of retailers when an observed storm might make landfall at one or more of the retailers. After the storm comes ashore, transshipments from other retailers and shipments directly from the manufacturer alleviate shortages at re- tailer locations a ected by the storm. The problem is formulated as a two-stage stochastic programming model, which is illustrated by a numerical example and case study scenarios derived by Rawls and Turnquist [53]. Commercial software and a solution approach that reduces the model to the well-known transportation problem are employed to solve the rst 4 and second stage decisions, which involve the pre-positioned quantities and post-storm ship- ments including transshipments and direct shipments from the manufacturer. As a result of several devastating hurricanes, earthquakes, and other disasters, pre-positioning has been researched extensively for governmental and non-governmental agencies. Location, inventory and distribution comprise the three aspects of pre-positioning mentioned in Richardson et al. [54], and only a few studies integrate all three into a two-stage stochastic programming model (e.g., [14], [38], [53], and [56]). The model presented in this chapter considers the inventory and distribution decisions similar to Barbaroso glu and Arda [6]. This paper provides the following contributions to the pre-positioning literature: a136 A two-stage stochastic programming model is developed considering the manufacturer?s perspective. a136 The model takes into account the additional production costs incurred during the rst and second stages, transshipments from other retailers in addition to direct shipments from the manufacturer. 5 Chapter 2 Minimax EOQ Policies with Demand Surge, Lead-Time and Lost Sales 2.1 Introduction Many organizations such as FEMA and the Red Cross have been instrumental in pro- viding disaster relief assistance to residents that have been a ected by ooding, hurricanes, and other catastrophic storms. These organizations set up shelters and pre-position supplies to respond to the a ected area as soon as possible after a storm has passed through. While these organizations have provided assistance to millions of disaster victims, the success of disaster relief e orts also depends heavily upon local retailers. Retailers are an important part of the humanitarian relief supply chain because they stockpile items that are consumed directly by victims. Consider that organizations such as FEMA recommend that coastal residents prepare a survival kit at the beginning of the hurricane season ([1]), which begins June 1st. However, most coastal residents do not gather these items until they are certain an observed storm will make landfall. In 2010, the National Hurricane Survival Initiative surveyed residents from the coast of Virginia to Texas and dis- covered that 47 percent of the polled residents did not have a hurricane survival kit, which is three day?s worth of water, food, and medicines for each family member [27]. Coastal resi- dents or consumers typically wait until the last minute to gather supplies, and unfortunately, many retailers adopt the same attitude regarding their preparation for a possible emergency supply demand surge. As a result, residents are more susceptible to the risks posed by se- vere storms or other hazards. Additionally, stockout of items penalize retailers in terms of lost sales costs and possibly long term e ects related to loss of customer goodwill. While preparation is the residents? responsibility, they can only prepare as well as the retailers that provide the supplies. 6 In order to improve preparation e orts for a possible spike in demand for emergency supplies that often accompanies disaster relief operations, this paper presents an approach that allows the retailer to compare reactive and proactive inventory strategies. We consider reactive and proactive inventory control policies for commodities such as food, water, and over the counter medicines and supplies such as batteries and ashlights. Since these types of items are typically characterized by a steady demand rate with minimal variation un- der normal conditions, we assume an underlying continuous review inventory control policy, particularly the economic order quantity. We do not speci cally address demand for gas- powered generators, which are slower moving items on a day-to-day basis. A newsvendor or periodic review policy would be more appropriate for this purpose. We also evaluate the e ectiveness of the continuous review inventory control policy under storm uncertainty according to the minimax decision criterion as opposed to the more mainstream expected value criterion. Disruption management strategies based on the expected value criterion are appropriate when planning for frequently occurring disruptions in which meaningful fore- casts could be generated based on su cient historical data. However, our focus in this paper is severe storms, namely hurricanes or tropical storms, as well as winter storms. For these and other extreme events, forecasts are less meaningful compared to more frequent disrup- tion events. Interestingly, distribution free approaches to proactive disruption management planning are a rare occurrence in the research literature. Existing proactive approaches to disruption management are almost exclusively based on expected value criterion, which is not appropriate for rare disruption events as described above. On the other hand, existing approaches to disruption management that do apply to rare events are almost exclusively re- active strategies. This paper investigates both reactive and proactive disruption management policies for rare events that do not depend on unreliable forecasts. We extend Lodree?s [16] work by incorporating lost sales and a non-zero lead-time into the model, and we seek to answer the following questions: 7 Question 2.1 How does the lead-time and lost sales a ect the strategy decision? Question 2.2 What circumstances dictate the strategy decision? Question 2.3 How does each parameter a ect the strategy decision? It should be noted that this paper does not intend to show how the retailers can always meet their customers? needs during a crisis, but we demonstrate that there are circumstances when it is bene cial for retailers to be prepared for a possible demand surge before a storm a ects their area. As a result, it might improve consumers? ability to obtain the supplies that they need to prepare for a storm. The paper is organized as follows: literature review, model formulation, results, and summary and future extensions. 2.2 Literature Review Supply and demand disruptions caused by events such as strikes, natural disasters, unreliable suppliers, and machine breakdowns within the context of production, operations, and supply chain management have been extensively researched in the literature. As a result of disruptions, lost sales or backorders are common occurrences depending on the length of the disruption. Supply disruptions are characterized as an interruption in a supplier?s operations, which means that the supplier becomes unavailable or unable to ll orders for various reasons from strikes to machinery breakdowns. One of the earliest papers to address disruptions and lost sales computed the percentage of unmet demand in a production system as a result of a disruption [39]. Several papers have also concentrated on supply disruptions within a production system and determined periodic review (e.g., [42], [34]) and continuous review inventory policies ([41]). In relation to our model, Parlar and Berkin [46], Berk and Arreola-Risa [9], Snyder [60], and Heimann and Waage [23] developed economic order quantity inventory models considering supply disruptions. Recently, Qi et al. [50] found an optimal order quantity when the retailer and supplier both face disruptions where the 8 retailer?s disruption is a loss of inventory. The focus of this literature review will mainly be on inventory models that consider demand disruptions, lead-time, and lost sales as well as a brief review of minimax inventory models. 2.2.1 Demand Disruptions Demand disruptions have been addressed as periods of no demand ([65]), uctuating demand ([61], [11]), and a surge in demand ([16], [35], [55]). Brill and Chaouch [11] found results to support their claim that an adjustment should be made to an EOQ policy when a major disruption occurs, but if the disruption is small, no adjustment is necessary. Ross et al. [55] developed time-dependent policies for disruptions caused by a surge in demand. Within the disaster management literature, Lodree and Taskin [35] compared the newsvendor solution to the optimal inventory level they determined for a one-time decision for retailers that were considering stocking up before a storm. However, the model can be applied to any type of demand disruption that has a forecast associated with it. Using optimal control, Lodree and Taskin [35] determined when retailers should increase their order quantity of emergency supplies. In addition, Taskin and Lodree [62] determined how a retailer should adjust their inventory level in anticipation of a demand surge that will occur during hurricane season based on estimated demand that is a convolution of current demand and an estimate of demand for hurricane season. Lodree [16] presented a model to determine the retailers? ordering strategy for emergency supplies when they are faced with an uncertain demand surge caused by a forecasted storm. He used minimax decision criterion to determine the best ordering strategy, which is contrary to the approach in the demand and supply disruption literature. Typically, long run approximations are used to determine order quantities in order to alleviate the e ects of a demand or supply disruption (e.g., [46], [23]). Similar to the model presented in Lodree [16], we are considering infrequent, unpredictable storms, and it is not appropriate to use these types of approximations as previously mentioned. In addition, each storm is unique with di erent characteristics such as the severity of the 9 storm, how long people may potentially be con ned to their homes, etc. Therefore, the same approximation for an order quantity cannot be applied to each storm. We handle this issue by using Lodree?s [16] approach of employing minimax decision criterion to aid the retailer in determining whether they should consider adopting a reactive (maintain order quantity) or proactive (increase order quantity) strategy regarding inventory control policies before an uncertain demand surge. 2.2.2 Minimax Decision Criterion Lastly, we will brie y review a sample of minimax models in the inventory control lit- erature. Scarf [58] developed the rst distribution free model by determining the demand distribution with a known mean and variance that would maximize the minimum pro t. Several authors have extended Scarf?s research by increasing the number of observed periods ([2]), proving Scarf?s ordering rule is optimal for the newsboy problem ([19]), determining optimal (Q;r) policies ([28], [18]), and determining the ordering policy using linear program- ming ([20]). By applying minimax decision criterion, Yu [68] solved the EOQ problem with a random demand rate, order cost, and holding cost using an e cient linear time algorithm. He demonstrated the many advantages of the robustness approach compared to the stochas- tic approach. With respect to our model, the only other paper known to have used minimax decision criterion within the context of disruption management is Lodree [16]. The main contribution of this paper is to provide a general decision framework that retailers can utilize to determine when it is advantageous to increase their order quantity before an uncertain demand surge. We provide an ordering strategy supported by minimax decision criterion instead of a speci c inventory policy, which is contrary to other literature addressing demand disruptions and lost sales. If the retailer?s objective is to minimize the cost for the worst outcome, the results given by minimax decision criterion will be more useful than determining an inventory policy that minimizes average total cost. 10 2.3 Model Formulation 2.3.1 Model Assumptions The inventory model presented in this paper extends the model presented in Lodree [16], which is based on an underlying continuous review model, and more speci cally, the EOQ model. We assume that if the disruption does not occur, the retailer employs an EOQ ordering policy. The EOQ model is extended to consider how a change in the order quantity will improve the retailers? costs if a disruption occurs. If the disruption occurs, we determine the conditions that promote increasing the current order quantity (EOQ) to accommodate a surge in demand by using minimax decision criterion. In addition, lead-time is considered in the model, and we assume that retailer will choose a reorder point that will minimize lost sales. However, it should be noted that we are not focused on nding an optimal reorder point or quantity, but determining the strategy that minimizes the retailers? maximum cost. The retailer may decide to adopt a reactive, R, inventory control strategy meaning that the current order quantity will not be changed in response to learning about a potential demand disruption or surge. After the demand surge begins, the retailer will have the op- portunity to modify the order quantity, which will also alter the reorder point. On the other hand, she may choose to be proactive, P, where the order quantity is increased in anticipa- tion of a demand surge. If the disruption does not materialize, she will decrease the order quantity to the normal order quantity (EOQ) after it is certain the demand surge will not occur. It should be noted that the order quantity received before the demand disruption oc- curs is based on the strategy selected before the rst cycle begins. Please refer to Table 2.3.1 for the list of model assumptions in addition to the following EOQ model assumptions: con- stant lead-time, known demand for disruption and normal conditions, constant demand, and constant ordering cost. A few of the model assumptions generate further explanation, which we will address in this paragraph. We assume that the lead-time is short because the types of items that 11 Table 2.3.1: Chapter 2 Model Assumptions 1. Single, fast-moving, non-perishable item is considered. 2. Lead-time is short. 3. Backorders are not allowed. 4. Order quantity under normal conditions is EOQ. 5. Reliable and adaptable supplier 6. Demand surge occurs during rst inventory cycle. 7. Time horizon for disruption is known. 8. Time horizon for demand surge is long. 9. Retailers will not incur lost sales if they are proactive. 10. Order quantity (q) and number of orders (m) are non-integer valued. 11. Reordering decisions are based on strategy selection. 12. The one source of uncertainty is the occurrence of a disruption. are considered are fast-moving items such as bottled water. Backorders are not allowed because customers will not wait for the next order to arrive. In addition, we assume that the demand surge occurs during the rst inventory cycle meaning retailers make a decision beforehand to receive either a modi ed order quantity or the EOQ at the beginning (t = 0) of the cycle containing the possible demand surge. The time horizon considered is longer than a usual inventory cycle because retailers would not be concerned about a negligible disruption. An additional note about reordering is that the retailer may have to reorder before the disruption is expected to occur. Therefore, when a retailer selects the proactive strategy, she will reorder based on the assumption that the disruption will occur. On the other hand, if the retailer decides to maintain her order quantity, she will reorder based on the assumption that the disruption will not occur. This will be discussed further in the sections regarding the strategies. 2.3.2 General Cost Equation and Model Notation In order to determine the strategy supported by minimax decision criterion, cost equa- tions were developed for all of the possible cases considering the model assumptions. The cost 12 equations have the following notation (Please refer to Table 2.3.2 for the model notation): TCDS (~q) = A~m+hIDS +sYDS : (2.3.1) Table 2.3.2: Chapter 2 Notation Parameter Description D2f0;1g 0, no disruption occurs or 1, disruption occurs S2fR;Pg R, reactive strategy or P, proactive strategy ~q2fqi;qi+1;:::;qMg ordering policy qi2fqE;qR;qPg order quantity i qE economic order quantity qR disruption order quantity for R qP disruption order quantity for P mi sum of the number of orders for each qi A ordering cost h holding cost (per unit/per unit time) I average inventory level s lost sales cost (per unit) Y number of lost sales units T1 time when disruption begins T2 time when disruption ends normal demand D surge demand If the retailer chooses a reactive strategy and a demand surge does not occur, the retailer?s cost will be TC0R(~q); if a disruption does occur, the cost is TC1Rj(~q). If a reactive strategy is selected, there are four di erent cost equations for the outcome, D = 1, which are dependent upon the lead-time denoted by j2f1;2;3;4g. Similarly, the cost functions for the proactive strategy are TC1P(~q) and TC0Pk(~q) where k2f1;2;3;4g. 2.3.3 Minimax Decision Criterion We apply minimax decision criterion to the cost equations developed for each combina- tion of strategy and outcome to determine the strategy that minimizes the maximum cost. 13 The general minimax formulation is shown in Eq. (2.3.2). min S2fR;Pg max D2f0;1g min ~q2Q TCDS (~q); (2.3.2) where Q = MX i=1 qimi = DT and I(t) 08t2[0;T2] a136 Q = solution space for inventory policies a136 qi = order quantity a136 mi = number of orders of qi a136 M = number of times qi changes a136 DT = total demand ful lled during time horizon First, the inventory policies that minimize the cost for each of the cases are selected by nding ~q, which is based on the strategy. Then, the cost function for the outcome that maximizes the cost for each strategy is selected meaning that the choices are narrowed down to two, a maximum cost equation for the reactive strategy and a maximum cost equation for the proactive strategy. It should be noted that the cost equation used for the comparison between the reactive disruption case (j) and proactive no disruption case (k) is based on the parameters meaning two cost equations are compared for each strategy, which are no disruption and disruption. Lastly, the maximum cost for each strategy is compared, and the strategy with the smallest maximum cost is selected. 2.3.4 Reactive Strategy In this section, we develop cost equations for each possible case, give the optimal dis- ruption order quantity, and determine the maximum cost equation for the reactive strategy. Four demand disruption cases are presented and the costs are compared to the no disruption case, which is a nite horizon EOQ model. 14 Disruption Cases Based on our assumptions, the lead-time (L) and the time when the disruption begins (T1) create four cases when a demand surge occurs. In addition, it is assumed that the time horizon for the demand surge is long (i.e., T1 + L T2), so we only consider cases that require the retailer to place at least one order of the disruption order quantity meaning the lead-time is shorter than an inventory cycle regardless of whether the disruption occurs. Case 1: qE T1 D L qE T1 The inventory level and order quantities over the time horizon are shown in Figure 2.3.1 for Case 1. The inequality for the values of L gives the range of lead-time values that create the situation where the retailer normally reorders after the disruption has begun and the lead- time causes lost sales to occur. The lead-time is longer than the time between T1 and t1R (time when the order is depleted). In a typical EOQ model, the retailer would reorder when the inventory level is L, and in this case, the reorder point will occur after the disruption has begun indicating a short lead-time. We assume that the retailer places her order at T1 instead of waiting until the inventory level reaches L. If the retailer waits to place her next order when the inventory level reaches L, she will incur higher lost sales costs, which is why we assume the order is placed at T1. The retailer will increase her order quantity to qR at T1 in order to meet demand during the remainder of the disruption. When the rst order of qR arrives at T1 +L, it will be after the inventory has been depleted at t1R and the retailer will experience lost sales. The ordering policy consists of two components, which are the order quantity received at t = 0 (q1 = qE), and the order quantity received after the disruption has begun (q2 = qR). After the retailer receives the order of qR, she will not incur lost sales for the remainder of the disruption. We determined the number of orders, mi, by determining the total demand that will be met by the retailer (excluding lost sales) and dividing by the order quantity, qi. The values of m are as follows: m1 = T1+ D(t1R T1)q E and m2 = D(T2 T1 L)q R . After inserting 15 q E q R T 1 t 1R ? ? D ? D ? D ? D ? D T 2 Time Inventory Level T 1 +L t r } Lost Sales Or der of q R plac ed Figure 2.3.1: Reactive Strategy with Disruption: Case 1 the values for ~q and m into Eq. (2.3.2), we derived Eq. (2.3.3) in Appendix A.1. TC1R1(qE;qR) = A T 1 qE + D(t1R T1) qE + D(T2 T1 L) qR (2.3.3) +h q Et1R 2 + qR(T2 T1 L) 2 +s D(T1 +L t1R) Case 2: L qE T1 D In Case 2, the lead-time is such that the retailer does not experience lost sales, and her typical reorder point occurs after T1 meaning the lead-time is shorter than the lead-time in Case 1. Again, we assume that the retailer will place an order at T1 and receive it when the rst order has been depleted at t1R. This case is similar to Case 1 as shown in Figure 2.3.1 with the exception of no lost sales. Eq. (2.3.4) gives the cost equation for Case 2. TC1R2(qE;qR) = A T 1 qE + D(t1R T1) qE + D(T2 t1R) qR (2.3.4) +h q Et1R 2 + qR(T2 t1R) 2 Case 3: L qE D + qE T1 The third case represents when the lead-time causes the time to reorder to occur before T1. We assume that the retailer places another order of qE at her normal time to reorder 16 because she has selected a reactive strategy. She maintains her order quantity because she is ordering before it is known that the disruption will occur as shown in Figure 2.3.2. In addition, we assume that the retailer places an order of qR at T1, and she will receive it at T1 +L, which is after the second order has been depleted. This creates a second time period of lost sales. Once she receives her order of qR, she will not incur lost sales for the remainder of the disruption. The cost equation for Case 3 is shown in Eq. 2.3.5. q E q R T 1 t 1R ? ? D ? D ? D T 2 Time Inventory Level q E /? } Lost Sales T 1 +L } Lost Sales ? D t r 1st or der (q E ) is plac ed 1st or der (q E ) is r ec eiv ed 2nd or der (q R ) is plac ed 2nd or der (q R ) is r ec eiv ed Figure 2.3.2: Reactive Strategy with Disruption: Case 3 TC1R3(qE;qR) = A T 1 qE + D(t1R T1) qE + 1 + D(T2 T1 L) qR (2.3.5) +h q Et1R 2 + qE2 2 D + qR(T2 T1 L) 2 +s D T1 +L qE D t1R Case 4: qE T1 L qE D + qE T1 The fourth and nal case that we consider for the reactive strategy with a disruption is similar to the third case because the retailer places her second order of qE before T1. However, the lead-time is such that lost sales are not incurred after the second order is received. We assume that the retailer chooses to order qR so that she receives it when the second order is depleted at qE D + qE meaning the order will be placed after T1. Please refer to Eq. 2.3.6 for 17 the cost equation for Case 4. TC1R4(qE;qR) = A T1 qE + D(t1R T1) qE + 1 + D(T2 qE D qE ) qR ! (2.3.6) +h qEt1R 2 + qE2 2 D + qR(T2 qE D qE ) 2 ! +s D qE t1R We determine the optimal order quantity during the disruption by applying the rst order condition to each cost equation. The disruption order quantity is the same for each case, and it is similar to the EOQ (qE) with the disruption demand rate, D, replacing as shown in Eq. (2.3.7). qR = r 2A D h (2.3.7) No Disruption Case If the disruption does not occur, we have a nite horizon EOQ model. This means that the disruption does not occur at T1 and the demand rate is during the time horizon [0;T2]. Please refer to Eq. 2.3.8 for the cost equation. TC0R(qE) = A T 2 qE +h T 2qE 2 (2.3.8) For the following proposition, the proofs are provided in Appendix A.2. Proposition 1 When S = R, max D2f0;1g min ~q2Q TCDS (~q) = TC1Rj(~q)8j2f2;3;4g . Proposition 1 states that when the reactive strategy is selected, the maximum cost is incurred when a disruption occurs. There are a few conditions to this statement, though. For the disruption cases, j = f2;3;4g, it is always clear that the cost is greater than if there is no 18 disruption. If the disruption case j = 1 occurs, we are able to nd that the cost is greater than the case when there is no disruption if the lost sales cost satisfy the inequality, s hqR D . Overall, retailers can expect to incur the maximum cost if a disruption occurs, and a reactive strategy is selected. 2.3.5 Proactive Strategy Disruption Case When the retailer selects a proactive strategy and the demand surge occurs, there is only one case to consider. Regardless of whether the time to reorder is before or after T1, the retailer will order qP, which is the optimal order quantity during the disruption if a proactive strategy is selected. The retailer will base the ordering decision on the assumption that the disruption will occur. Please refer to Figure 2.3.3 for the representation of the disruption case for the proactive strategy. q E q P T 1 ? ? D ? D T 2 t 1P ? D ? D Time Inventory Level Figure 2.3.3: Proactive Strategy with Disruption Eq. (2.3.9) gives the cost equation for the proactive strategy with a demand surge. TC1P(qP) = A D T qP +h T 2qP 2 ; (2.3.9) where: DT = T1 + D(T2 T1) 19 By applying the rst order condition to Eq. (2.3.9), we are able to determine the optimal order quantity for the proactive strategy given in Eq. (2.3.10). qP = r2AD T hT2 (2.3.10) No Disruption Cases If no disruption occurs, four cases are associated with the proactive strategy. The four cases are dependent upon the lead-time, T1, and T2. Case 1: L qP T1 D and T2 2qP In this case, the lead-time is longer than the time between when the disruption begins and the rst order is depleted, which means that the retailer must reorder before being certain that the demand surge will occur. Because the retailer has selected a proactive strategy, she will reorder based on the assumption that the demand surge will occur as shown by the two orders of qP in Figure 2.3.4. In addition, the retailer may delay the order to be received at qP to decrease holding costs. We assume this change can be made at no penalty because the supplier is adaptable and reliable. However, the supplier is not be able to rush orders to prevent lost sales in previous cases. Also, the time horizon is such that the retailer will switch her order quantity after T1 to qE. The cost equation for Case 1 is given in Eq. (2.3.11). q E ? Time q P T 1 ? Inventory Level t r T 2 ? Figure 2.3.4: Proactive Strategy with No Disruption: Case 1 20 TC0P1(qP;qE) = A 2 + T2 2qPq E +h qP2 + qE T2 2qP 2 ! (2.3.11) Case 2: L qP T1 D and T2 2qP Case 2 is similar to Case 1 with the exception that the time horizon ends before the retailer receives an order of qE. Essentially, the retailer receives at most two orders of qP. The cost equation for Case 2 is given in Eq. (2.3.12). TC0P2(qP) = A T 2 qP +h q PT2 2 (2.3.12) Case 3: L qP T1 D and T2 qP In the third case, the lead-time causes the time to reorder to occur after T1. This leads the retailer to change her order quantity to qE, and she will reorder so that she will receive the order of qE at qP . Figure 2.3.5 illustrates Case 3, and the cost equation is given in Eq. (2.3.13). q E T 2 ? ? Time q P T 1 ? Inventory Level t r Figure 2.3.5: Proactive Strategy with No Disruption: Case 3 TC0P3(qP;qE) = A 1 + T2 qPq E +h qP2 2 + qE T2 qP 2 ! (2.3.13) 21 Case 4: L qP T1 D and T2 qP Case 4 is similar to Case 3 except there is only one order of qP because the disruption ends before the rst order is depleted. Eq. (2.3.12) can be used to determine the cost for Case 4. Proposition 2 When S = P, max D2f0;1g min ~q2Q TCDS (~q) = TC1P(~q) . Similar to the reactive strategy, we are able to nd that the maximum cost equation is always the disruption cost equation when it is compared to the no disruption cost equations. Please see Appendix A.3 for the proof of Proposition 2. 2.4 Results It is impossible to analytically show that one strategy is always preferred over the other based on minimax decision criterion, so we performed a numerical experiment and sensitivity analysis. Our analysis enabled us to better understand the conditions when it is advantageous to select either the reactive or proactive strategy when faced with a possible demand disruption. 2.4.1 Numerical Experiment In order to determine the conditions associated with each strategy selection, a 28 ex- perimental design was created. Table 2.4.1 contains the set of values tested for each of the eight parameters. We ensured that the values tested did not violate any of the previously mentioned assumptions. For example, we checked that the lead-time was less than each type of inventory cycle (depending on the demand rate and order quantity) and ensured that T2 was greater than the time when the rst order is depleted (t1R and t1P), which depends on the strategy selected. In Table 2.4.2, we provide a sample of our results. The table contains 22 Table 2.4.1: Values Tested for 28 Experiment Parameter Low High D 50 100 T2 6 12 h 1 2 A 100 200 s 10 20 L 0.5 1 10 15 T1 2 2.5 information regarding the description of the value (low or high) for each parameter, the cost for each outcome associated with the strategy, and the best strategy for the given condi- tions. When we performed the numerical experiment with the values in Table 2.4.1, there Table 2.4.2: Sample Results for 28 Experiment D T2 h A L s T1 Decision L L L L L L L L REACTIVE L L L L L L L High PROACTIVE L L L L L L High L REACTIVE L L L L L L High High PROACTIVE L L L L L High L L REACTIVE L L L L L High L High PROACTIVE L L L L L High High L REACTIVE L L L L L High High High PROACTIVE were cases with lost sales and no lost sales. The reactive strategy was selected for about 8% of the tested combinations with lost sales. We found the reactive strategy combinations to have the following characteristics if lost sales were incurred: a136 High lost sales cost were allowed a136 Short lead-time a136 Small D and T1 a136 tr T1 After analyzing the characteristics of the reactive strategy combinations, we explored the explanations for each of the characteristics. Surprisingly, a high lost sales cost (s) did not automatically mean that a proactive strategy was more cost e ective. If the reactive strategy was selected when s was at its high value, the amount of lost sales was minimal. In addition, 23 the reactive strategy was predominantly chosen if the surge demand was at its smallest value, which resulted in less lost sales. As expected, if the lead-time was short, the time to reorder (tr) was after T1 supporting the reactive strategy. As a result, the retailer was able to change her order quantity to qR at T1 and ensure no additional lost sales after she received the next order. When no lost sales were incurred, the proactive strategy was selected almost 50% of the time. The ordering costs a ected the selection of the proactive strategy in these circum- stances. In our experiment, the reactive strategy always yielded higher ordering costs than the proactive strategy while the proactive strategy had higher holding costs. With the values tested, the reactive strategy was preferred when the di erence in the number of orders was small meaning there was a small di erence in the ordering costs. The ordering costs a ected the strategy selection more than the holding costs, which is to be expected because A h. In addition, when lost sales were not incurred, the time to reorder was after T1, which also means that the lead-time (L) was short. 2.4.2 Sensitivity Analysis In order to better understand the e ect of each parameter on the strategy selection, we performed a sensitivity analysis. The parameter in question was varied so that it did not violate the model assumptions while the other parameters were kept at their low values given in Table 2.4.1. We will rst review the results of the parameters that incurred lost sales for all values tested. Parameters that Incurred Lost Sales Only When T2 and s were tested, the reactive strategy was preferred for every tested case. Through the results, we concluded that the lost sales cost and the length of the disruption do not have a signi cant e ect on the strategy selection. 24 The time when the disruption begins (T1), the normal demand ( ), and the disruption demand ( D) yielded reactive strategies at the lowest values tested. Intuitively, one would expect the reactive strategy to be selected if the demand and disruption demand are both small because the retailer would experience a smaller number of lost sales. In addition, if T1 is small, this means that tR T1, so the retailer will be able to change her order quantity to qR and thus, only incur lost sales for one time period. Parameters that Incurred Lost Sales and No Lost Sales When the lead-time was tested at smaller values, there were no lost sales and mini- max decision criterion supported a reactive strategy. On the other hand, when the retailer incurred lost sales, the reactive strategy was selected if the lead-time was small meaning that the number of lost sales was small. Otherwise, as the lead-time increases, the proactive strategy is favored. The lost sales cases were separated from lost sales in Figure 2.4.1 in order to show the values of A where lost sales are present, which is when A is less than 100. In addition, there is a greater di erence between the total cost when lost sales are incurred, and we wanted to highlight this nding. According to our model, the retailer experiences lost sales when the order cost is small (or less than 100). As shown in Figure 2.4.1, the retailer should choose a proactive strategy until the order cost reaches 100. Then, the retailer should select a reactive strategy for a small range of values when lost sales are incurred. For these values of A, the number of lost sales is very small, which is why the retailer may select a reactive strategy. As the order cost increases, the retailer does not incur lost sales because the order quantities increase while the lead-time remains the same. In general, one would expect for the retailer to always choose a reactive strategy when they do not anticipate incurring lost sales. However, we found that as the order cost increases, the proactive strategy is more cost e ective. It is apparent that the di erence in the costs for each strategy is very small for most values of A as shown in Figure 2.4.1, so the retailer should select the strategy that will 25 300 400 500 600 700 800 900 1000 1100 50 100150200250300350400450 Total Cost Order Cost Reactive Lost SalesProactive Lost Sales Reactive No Lost SalesProactive No Lost Sales Figure 2.4.1: The E ect of the Order Cost, A best suit the circumstances. In all of the cases when the retailer does not incur lost sales, the reactive strategy always has a larger order cost than the proactive strategy. Figure 2.4.2 illustrates that as the di erence between the number of orders increases, it is better to select the proactive strategy when there are no lost sales. In contrast to the order cost, there were no lost sales incurred when the holding cost was small, but as the holding cost increased, lost sales were incurred. When there were no lost sales, the proactive strategy was preferred with the exception of a small range of h values that yielded a reactive strategy. When the value of h causes a switch from no lost sales to lost sales, there is a small range of h values that lead to a reactive strategy as shown in Figure 2.4.3. As h increases, the chosen strategy is proactive. The amount of lost sales increases because the order quantities decrease, thus, the proactive strategy is selected. The lost sales cost is more expensive than the holding cost, so the di erence in cost continues 26 0 0.5 1 1.5 2 2.5 3 0 50 100 150 200 250 300 350 400 450 No. o f Ord ers Order Cost Proactive Reactive R P Figure 2.4.2: Strategy Selection Considering No. of Orders with No Lost Sales to increase until tr T1, and then, it begins to decline as shown in Figure 2.4.3. When tr T1, the retailer will reorder based on her strategy and if she is reactive, she might incur two periods of lost sales. However, the values we tested did not lead to two time periods of lost sales. The decrease in the di erence in cost is attributed to the decrease in holding costs because with a reactive strategy, the retailer will hold two orders of qE, and then, a smaller number of qR orders than if she had only placed one order of qE. Plus, during the time period of lost sales, holding costs will not be incurred. 2.5 Summary and Future Extensions The numerical experiment and sensitivity analysis provided us with valuable conclusions about the conditions that warrant each strategy selection. The following is a summarized list of our ndings: 27 0 200 400 600 800 1000 1200 0 0.5 1 1.5 2 2.5 3 3.5 4 Total Cost Holding Cost Proactive No Lost Sales Reactive No Lost Sales Proactive Lost Sales Reactive Lost Sales Figure 2.4.3: The E ect of the Holding Cost, h 1. Lead-time has a greater e ect on strategy selection than lost sales cost. 2. It is possible to select the reactive strategy despite lost sales when lead-time is short. 3. It is possible to select the proactive strategy when there are no lost sales. Perhaps the most interesting nding is that the lead-time has a greater e ect than the lost sales cost on the strategy selection, which was proven in the sensitivity analysis. The values of T2 and s are arbitrary compared to the lead-time. As a practical application, the retailer can determine which strategy she should choose based on the lead-time. We also found that the proactive strategy is favored for most circumstances even when there are no lost sales. A retailer should select the proactive strategy if her holding or ordering costs are high regardless of whether lost sales occur. In summary, the retailer should adopt a proactive strategy unless the lead-time is short and the other parameters have small values. 28 In order to capture the essence of a disruption, the model could be extended in several ways such as including a variable lead-time. During the demand surge, some suppliers may experience many orders from other retailers, thus decreasing their ability to ful ll orders in a timely manner. In addition, we could explore optimizing the reorder point because we assume that it is optimal for the retailer to not incur lost sales if possible. Lastly, an ordering strategy based on a retailer?s target service level could be determined. Retailers play a key role in helping communities prepare for a forecasted storm, and their preparation is crucial. Through our research, we were able to show situations where it is bene cial for retailers to adopt a proactive strategy as well as a reactive strategy. In most cases, it is better for them to select a proactive strategy, which will in turn assist the community in their preparations because the retailers will be able to provide the necessary supplies. However, there are situations when it is best to select the reactive strategy, and if lost sales are incurred, it is a very small percentage, so most customers? needs will be met. All in all, our approach provides a practical tool for retailers to use when they are faced with an uncertain demand surge. 29 Chapter 3 Minimax and Minimax Regret Inventory Control Policies Regarding Demand Disruptions and Damaged Inventory 3.1 Introduction On September 13, 2008, residents in Galveston, Texas, assessed the damage that Hurri- cane Ike left in its wake. Many parts of Galveston were unrecognizable, and power outages were widespread on the island. The residents who stayed behind despite warnings to evac- uate were running low on basic commodities such as gas for generators and non-perishable food. Most retailers were unable to reopen quickly with the exception of one Kroger store that opened it doors 3.5 days later ([48]). The purpose of this paper is to assist retailers with inventory policy decisions during and after a forecasted storm. In this paper, we focus on two disruptions that occur consecutively: a period of no demand leading into a surge in demand for emergency items. Chapter 2 captures the disrup- tion before the storm and provides insight into the conditions that support increasing the order quantity before the disruption occurs using minimax decision criterion. In Chapter 3, we evaluate two uncertain disruptions that occur during and after the storm by applying minimax and minimax regret decision criteria to provide retailers with an inventory policy depending on the objectives they strive to meet. Figure 3.1.1 illustrates the time horizon considered for Chapters 2 and 3. Chapter 2 covers the time horizon between t = 0 and t = T2, while Chapter 3 covers the time horizon between t = T2 to t = T4. Please note that the demand rate for the disruption between T3 and T4 is greater than the demand rate for the disruption between T1 and T2. It is assumed that all customers will not stock up on emergency supplies before a possible storm, but if the storm occurs, all customers will be in need of emergency supplies. In this paper, we will refer to the disruption between T2 and 30 T3 as the rst disruption while the second disruption refers to the demand surge between T3 and T4. We assume that T2 = 0 because we are considering the last two disruptions separate from the disruption between T1 and T2. q E q D T 1 t 1 T 2 ? ? D ? D ? D Time Inventory Level q E /? T 3 Demand Disruption Before StormDemand Disruption During Storm ? S ? S T 4 Demand Disruption After Storm ? S q 2 (1-?)q 1 Figure 3.1.1: Time Horizon for Demand Disruptions In the supply disruption literature, it has been shown that retailers should take disrup- tions into account when determining inventory control policies (e.g., [46], [9], [50]). For the time horizon considered in this paper, there are key factors that must be examined to deter- mine inventory policies: the likelihood of the forecasted storm a ecting the area and causing the retailer to close, the likelihood of damaged inventory caused by a storm, the length of time the retailer will be closed, the lead-time for emergency items, and the length of time the demand surge will last after the storm. Our intention is to provide retailers with an inventory policy based on two di erent criteria: minimax and minimax regret. If the retailer decides to hold inventory during the rst disruption, the inventory policy will be determined based on the amount that the retailer assumes will be damaged. On the other hand, if the retailer anticipates that all of the inventory will be lost, she will not hold inventory during the rst disruption. We seek to answer the following research questions using minimax and minimax regret: Question 3.1 Under what conditions should retailers hold inventory if they expect a storm to a ect their area? 31 Question 3.2 If the retailer should hold inventory, how much inventory should be held considering a percentage may be damaged? Question 3.3 How does each parameter a ect the inventory policy? In the next section, we will review relevant disruption literature in addition to a brief summary of minimax and minimax regret literature. After the literature review, the general model is presented followed by the cost equations for each outcome associated with the inventory policies. Then, the results of the numerical experiment and sensitivity analysis give insight into the inventory policy supported by each decision criterion. In closing, a brief summary of the ndings and possible extensions are provided. 3.2 Literature Review Supply and demand disruptions caused by events such as strikes, natural disasters, unreliable suppliers, and machine breakdowns within the context of production, operations, and supply chain management have been extensively researched in the literature. As a result of disruptions, lost sales or backorders are common occurrences depending on the length of the disruption. Supply disruptions are characterized as an interruption in a supplier?s operations, which means that the supplier becomes unavailable or unable to ll orders for various reasons from strikes to machinery breakdowns. One of the earliest papers to address disruptions and lost sales computed the percentage of unmet demand in a production system as a result of a disruption ([39]). Several papers have also concentrated on supply disruptions within a production system and determined periodic review (e.g., [42], [34]) and continuous review inventory policies ([41]). In relation to our model, Parlar and Berkin [46], Berk and Arreola-Risa [9], Snyder [60], and Heimann and Waage [23] developed economic order quantity inventory models considering supply disruptions. Recently, Qi et al. [50] found an optimal order quantity when the retailer and supplier both face disruptions where the 32 retailer?s disruption is a loss of inventory. The focus of this literature review will mainly be on inventory models that consider demand disruptions, lead-time, and lost sales as well as a brief review of minimax and minimax regret inventory models. 3.2.1 Demand Disruptions Demand disruptions have been addressed as periods of no demand ([65]), uctuating demand ([61], [11]), and a surge in demand ([16], [35], [55]). Brill and Chaouch [11] found results to support their claim that an adjustment should be made to an EOQ policy when a major disruption occurs, but if the disruption is small, no adjustment is necessary. Ross et al. [55] developed time-dependent policies for disruptions caused by a surge in demand. Within the disaster management literature, Lodree and Taskin [35] compared the newsvendor solution to the optimal inventory level they determined for a one-time decision for retailers that were considering stocking up before a storm. However, the model can be applied to any type of demand disruption that has a forecast associated with it. Using optimal control, Lodree and Taskin [35] determined when retailers should increase their order quantity of emergency supplies. In addition, Taskin and Lodree [62] determined how a retailer should adjust their inventory level in anticipation of a demand surge that will occur during hurricane season based on estimated demand that is a convolution of current demand and an estimate of demand for hurricane season. Lodree [16] presented a model to determine the retailers? ordering strategy for emergency supplies when they are faced with an uncertain demand surge caused by a forecasted storm. He used minimax decision criterion to determine the best ordering strategy, which is contrary to the approach in the demand and supply disruption literature. Typically, long run approximations are used to determine order quantities in order to alleviate the e ects of a demand or supply disruption ([46], [23]). Similar to the model presented in Lodree [16], we are considering infrequent, unpredictable storms, and it is not appropriate to use these types of approximations. In addition, each storm is unique with di erent characteristics such as the severity of the storm, how long people may potentially 33 be con ned to their homes, etc. Therefore, the same approximation for an order quantity cannot be applied to each storm. We handle this issue by adopting the approach of employing minimax decision criterion in addition to minimax regret to aid the retailer in determining an inventory policy for two possible disruptions with two sources of uncertainty similar to the method presented in Lodree [16]. 3.2.2 Minimax Decision Criterion Scarf [58] developed the rst distribution free model by determining the demand distri- bution with a known mean and variance that would maximize the minimum pro t. Several authors have extended Scarf?s research by increasing the number of observed periods ([2]), proving Scarf?s ordering rule is optimal for the newsboy problem ([19]), determining optimal (Q;r) policies ([28], [18]), and determining the ordering policy using linear programming ([20]). By applying minimax decision criterion, Yu [68] solved the EOQ problem with a ran- dom demand rate, order cost, and holding cost using an e cient linear time algorithm. He demonstrated the many advantages of the robustness approach compared to the stochastic approach. With respect to our model, the only other paper known to have used minimax decision criterion within the context of disruption management is Lodree [16]. 3.2.3 Minimax Regret Decision Criterion In addition to using minimax decision criterion, we will also apply minimax regret decision criterion to the model. Minimax regret nds the minimum regret among the worst case scenarios for each decision. Savage [57] rst explored the minimax regret decision rule to demonstrate its uses when making optimal decisions. Several authors used minimax regret decision criterion when considering the distribution free newsvendor problem ([43], [32], [64], [69]). Perakis and Roels [49] found solutions for the optimal order quantity that minimize the maximum regret for not choosing optimally when the demand distribution has partial information. Jammernegg and Kischka [29] considered two other risk factors in their 34 objective function to determine a robust solution for the order quantity. With respect to our model, minimax regret decision criterion is applied to the objective function to determine the ordering strategy that minimizes the maximum regret. The contributions of this paper to the reviewed literature include: a general class of possible disruption cases with two sources of uncertainty are considered and an ordering strategy is determined by applying minimax and minimax regret decision criteria to an inventory disruption problem. In the next section, we present the inventory model that will be evaluated using the two decision criteria previously reviewed. 3.3 Model Formulation 3.3.1 General Model The model presented in this paper closely resembles the EOQ model in that many un- derlying assumptions for the EOQ model are included in the model presented in this section. If no disruption occurs, the retailer?s inventory policy is the EOQ. However, the model con- tains two sources of uncertainty that di erentiate it from the EOQ model, which are the occurrence of the two disruptions and the percentage of damaged inventory as a result of the rst disruption. In addition, the model includes lost sales if the retailer underestimates the percentage of damaged inventory or decides not to hold inventory during the rst disruption. 3.3.2 Model Assumptions and Notation In this section, the notation shown in Table 3.3.1 will be discussed in addition to the model assumptions. ~S, which is given in Eq. (3.3.1), represents the inventory policy selected by the retailer and contains two decisions: the decision to hold inventory, and the amount of inventory that will be held during the rst disruption with assumptions made about the amount of damaged inventory. S denotes the decision to hold (H) or not hold (NH) inventory, while represents the amount of inventory the retailer assumes will be damaged during the rst disruption. The list of possible decisions for ~S is given by . If < 1, 35 Table 3.3.1: Chapter 3 Notation Parameter Description ~S inventory policy ~D outcome ~q2fq1;q2g ordering policy q1 order quantity during rst disruption q2 order quantity during second disruption qE economic order quantity ~n2fn1;n2g number of orders of ~q A order cost (per order) h holding cost (per unit/per unit time) I average on-hand inventory z lost sales cost (per unit) X number of lost sales units y cost per unit of damaged inventory (purchase price) W number of units of damaged inventory percentage of damaged inventory percentage of inventory retailer assumes will be damaged T3 time when rst disruption ends T4 time when second disruption ends L lead-time normal demand rate S surge demand rate S = H, meaning the retailer assumes that all of the inventory will not be damaged, an order quantity will be determined to attempt to combat lost sales. The retailer will not hold (NH) inventory if = 1 because it means that the retailer assumes that all of the inventory will be damaged. ~S 2 fS; g (3.3.1) S 2 fH;NHg (3.3.2) 2 [0;1] (3.3.3) = f(H;0);(H; );:::;(NH;1)g (3.3.4) The notation for the rst disruption is represented by ~D. The vector, ~D, has two entities, D and (D). D represents the occurrence of the disruption and the outcome space is 36 D2fd0;d1g. If no disruption occurs, D = d0, and if a disruption occurs, D = d1. (D) denotes the percentage of inventory that is damaged as a result of the disruption. Eqs. (3.3.5)- (3.3.8) represent the outcome space for (D) and . (D) 2 f (d0); (d1)g (3.3.5) (d0) = 0 (3.3.6) (d1) = f0; ;:::;1g (3.3.7) = f(d0;0);(d1;0);(d1; );:::;(d1;1)g (3.3.8) Please also note that the ordering policy consists of two order quantities as shown in Table 3.3.1. We allow for di erent order quantities for each disruption if necessary. If a disruption occurs and < 1, we will explore two policies, q1 = q2 and q1 6= q2. If = 1, the retailer will not hold inventory, so q1 6= q2. As previously mentioned, if there is no disruption, we assume that the retailer will place an order for the economic order quantity, qE, to be received after it is certain that the rst disruption will not occur meaning q1 > 0 or q1 = 0 and q2 = qE. The number of orders of each quantity is given by ~n and is determined by the demand and order quantity; therefore, it is not a decision variable. The rst entity, n1, will either be 0 or 1 depending on the value for q1, and we allow n2 to be non-integer because it accounts for the number of orders of q2 between the time when q1 is depleted or the retailer receives q2 at L (depending on the inventory policy) and T4. The average on-hand inventory, I, is determined by the number of orders of each order quantity. The number of lost sales units given by X is determined by the length of time when the inventory level is zero and the demand rate is greater than zero. Lost sales will occur if q1 = 0 or < meaning the retailer underestimated . The number of damaged units, W, is determined by q1 and the percentage of units damaged, which is given by . The normal demand rate, , represents the demand rate when the disruptions do not occur while S represents the demand rate during the second disruption. 37 Table 3.3.2: Chapter 3 Model Assumptions EOQ Assumptions 1. Lead-time is constant. 2. The retailer?s ordering policy does not include lost sales if q1 > 0. 3. Demand rate for surge is known. 4. All costs are constant. Additional Assumptions 5. Single, fast-moving, non-perishable item is considered. 6. Supplier is reliable and adaptable. 7. The retailer will not be charged a holding cost and damaged inventory cost. 8. No backorders will be allowed. 9. The second disruption only occurs if the rst one occurs. 10. If a disruption occurs, the retailer cannot place an order until t = T3. 11. The retailer will determine q2 after the rst disruption begins. 12. Order quantity under normal conditions is EOQ. 13. First and second disruption time horizon is known. 14. Time horizon for demand surge is long (T4 T3 +L). Table 3.3.2 provides a list of the assumptions for the model with the EOQ assump- tions listed rst followed by additional model assumptions. The EOQ model assumes all parameters are known and lost sales are not allowed. If the retailer selects based on the assumption that all of the inventory will not be lost, she determines q1 such that lost sales will not be incurred if = . This means that the retailer will order more than the an- ticipated demand to account for a percentage of the inventory being damaged during the storm. We also assume that customers will be stocking up on fast-moving items such as bottled water and canned goods to sustain them during the recovery period when drinking water may be unsafe or electricity lines may be down. Also, if any or all of the inventory is damaged, the retailer will not be charged a holding cost for the unusable items. Instead, she will be charged the purchase price, y, for the items that were damaged if q1 > 0. We will assume that the purchase price is greater than the holding cost. In addition, we do not allow backorders if lost sales are incurred because customers will need items immediately after the disaster is over, so if the items are unavailable, they will go to another retailer. Moreover, the second disruption, which is characterized as a demand surge, will only occur if the rst disruption occurs. It is also assumed that if the rst disruption occurs, the retailer will not 38 be able to place an order of q2 until T3 because the retailer may not know the extent of the damage or be able to place orders to the supplier until the disruption is over. Along the same lines, Assumption 11 assumes that the retailer will determine q2 after the rst disruption begins because the retailer will know how to adjust q2, so that lost sales are not incurred for the remainder of the time horizon. Lastly, the time horizon is assumed to be longer than the time it takes to receive the rst order of q2. Otherwise, the second disruption would be negligible. 3.3.3 General Cost Equation The general cost equation given in Eq. (3.3.9) is comprised of the ordering, holding, damaged inventory, and lost sales costs. TC~D~S (~q) = A~n+hI~D~S +yW~D~S +zX~D~S (3.3.9) where W = q1 q1 0 X = max[0; SL (1 )q1] q1 0 Before applying either of the decision criterion, we will nd the inventory policy that mini- mizes Eq. (3.3.9), and it is shown in Eq. (3.3.10). MinTC~D~S (~q) = min (~q)2 TC~D~S (~q) (3.3.10) where (~q) 2 ; and = f(~q) : n1q1 +n2q2 8 ; 2f ; Sg; 2fT4;T4 T3gg 39 represents the amount of time that the demand rate is greater than zero while represents the demand rate during . If the disruption occurs, = T4 T3 and = S, and if the disruption does not occur, = T4 and = . The sum of the amount ordered during the time horizon, n1q1 + n2q2, may be less than the total demand, which is given by . The solution space accounts for the decision to hold or not hold inventory during the rst disruption. If the retailer decides not to hold inventory, the sum of the amount ordered will be less than the total demand during the time horizon, and the retailer will experience lost sales. After the solution that minimizes the cost equation for each combination of outcome and inventory policy is determined, minimax and minimax regret decision criteria are applied. The general minimax formulation is presented rst and is represented by Eq. (3.3.11). First, the maximum cost equation is determined for each policy considering the outcomes in . Then, the minimum cost equation is found considering the maximum cost for each policy. min ~S2 max ~D2 MinTC~D~S (~q) (3.3.11) Next, the minimax regret formulation is presented. The rst step is to create a matrix with columns representing the possible outcomes and rows representing the possible decisions for q1 and q2 that minimizes the cost equation for each possible disruption outcome. If the retailer decides to hold inventory, the amount held will be a quantity associated with a disruption outcome meaning that q1 6= qE. The maximum regret formulation is given in Eq. (3.3.12). Let Min TC~D~S represent the minimum cost for each outcome, and please note that the maximum regret is chosen for each decision in ~S. Please also note that Eq. (3.3.12) nds the absolute value of the di erence because the largest regret is determined by the largest negative value. However, in order to dispel any confusion, we took the maximum of the absolute value of the di erence. MaxMinTC~D~S (~q) = max ~S jMin TC~D~S (~q) MinTC~D~S (~q)j (3.3.12) 40 Lastly, the minimum regret is selected among the maximum regrets for each policy in as shown in Eq. (3.3.13). min ~S2 MaxMinTC~D~S (~q) (3.3.13) 3.3.4 Inventory Policy: Hold Inventory During 1st Disruption The retailer bases her decision to hold inventory on the percentage of inventory that is expected to be damaged by the storm. If the retailer underestimates the percentage of damaged inventory, lost sales will be incurred. On the other hand, if the retailer has su cient on-hand inventory after the storm, she will meet customer demand and aid in the recovery e orts. Disruption Cases If the retailer decides to hold inventory, there are three cases to consider if the disruption occurs and two cases to consider if the disruption does not occur. The rst case describes the outcome where no lost sales are incurred, which occurs when (1 )q1 qL and qL = SL. The retailer will not incur lost sales if the on-hand inventory at T3 is at least qL to meet the demand until the rst order of q2 arrives. In this case, the retailer has an adequate amount of on-hand inventory to meet demand after the rst disruption ends and the second disruption immediately begins as shown in Figure 3.3.1. The cost equation is given in Eq. (3.3.14). (1-?)q 1 q 2 T 3 ? S ? S ? S T 4 Time Inventory Level T 3 +(1-?)q 1 /? S ? S Or der of q 2 is plac ed Figure 3.3.1: Hold Inventory with Disruption: Case 1 41 TC1(d1; )(H; )(q1;q2) = A 0 @1 + S T4 T3 (1 )q1 S q2 1 A+h(1 )q1T3 (3.3.14) +h 0 @(1 ) 2q 12 2 S + q2 T4 T3 (1 )q1 S 2 1 A+y q1 The second case addresses the outcome in which lost sales are incurred where (1 )q1 qL. The main di erence between Case 1 and Case 2 is the inclusion of lost sales, which is caused by the retailer ordering under the assumption that a smaller percentage of inventory will be damaged ( < ). Figure 3.3.2 illustrates the inventory levels for Case 2 when lost sales are incurred. The cost equation for Case 2 is given in Eq. (3.3.15). (1-?)q 1 q 2 T 3 ? S ? S ? S T 4 Time Inventory Level T 3 +L } Lost Sales ? S Figure 3.3.2: Hold Inventory with Disruption: Case 2 TC2(d1; )(H; )(q1;q2) = A 1 + S(T4 T3 L)q 2 +h((1 )q1T3) (3.3.15) +h (1 )2q 12 2 S + q2(T4 T3 L) 2 +y q1 +z L (1 )q1 S The third and nal case is shown in Figure 3.3.3. If the retailer anticipates that a large percentage of her inventory will be damaged, she might order a very large order quantity. However, the actual percentage of inventory lost may be much smaller than causing the retailer not to place another order before the second disruption is over as shown in Figure 42 3.3.3. The cost equation for Case 3 is given in Eq. (3.3.16). In this case, q2 is not a decision (1-?)q 1 q 2 T 3 T 4 Time Inventory Level ? S Figure 3.3.3: Hold Inventory with Disruption: Case 3 variable because the retailer will revert back to the EOQ inventory policy when the second disruption is over at T4. TC3(d1; )(H; )(q1) = A+h (1 )q1T3 + (1 )q1(T4 T3)2 (3.3.16) In order to utilize minimax and minimax regret decision criteria, the optimal inventory policy associated with each outcome for where = must rst be determined. As previously mentioned, it is assumed that the retailer either places an order for the quantity necessary to satisfy demand if a disruption occurs or does not place an order. Two di erent policies are considered for the special case fd1;0g: q1 = q2 and q1 6= q2. Eq. (3.3.17) gives the cost equation for the special case fd1;0g where q = q1 = q2. TC(d1;0)(H;0) (q;q) = A S(T4 T3) q +h T3q + q(T4 T3)2 (3.3.17) By applying the rst order condition, the closed form solution for q is derived and given in Eq. (3.3.18). q = s 2A S(T4 T3) h(T4 +T3) (3.3.18) 43 Even though we are able to nd a closed form solution similar to the EOQ, we are able to prove that the total cost can be improved if q1 6= q2. However, the optimal values for q1 and q2, considering the assumptions, must rst be found before it can be explained that the cost can be minimized by adopting an inventory policy where q1 6= q2. In order to determine the optimal values for q1 and q2, the optimal value for one variable is found and plugged into the cost equation before solving for the other variable. The optimal value for q2, which is similar to the result that was found in Chapter 2, is given in Eq. (3.3.19). We will refer to q2 as qS for the remainder of this section. q2 = r 2A S h (3.3.19) In order to solve for q1, we employed the use of commercial software since a closed form solution that guaranteed no lost sales could not be found. When the constraints for no lost sales and non-negativity were included, the same solution was consistently returned, and it is given in Eq. (3.3.20). Eq. (3.3.20) is the least amount that the retailer can hold to ensure no lost sales until the rst order can be received after the second disruption begins if = = 0. q1 = SL (3.3.20) Eq. (3.3.20) can be generalized to include other decisions for , not including = 1, which is shown in Eq. (3.3.21). In the results section, we will give insight into the conditions when Eq. (3.3.21) is the best ordering policy for q1. q1 = SL1 (3.3.21) If the cost equations when q = q1 = q2 and q1 6= q2 are compared, the value of q that does not allow lost sales has to be considered. Therefore, we employed commercial software to compare the answer for q to ensure that it did not allow lost sales if the amount of damaged 44 inventory is known. Again, we considered the case (d1;0) to solve for q and compared the answer to the total cost for q1 6= q2. We are able to prove that if q1 = qL (qL = SL) and q2 = qS, the cost is less than if q = q1 = q2 as long as q qL. This proof is shown in Appendix B. In addition, 64 combinations of eight parameters were tested to compare q1 = qL and q2 = qS to q = q1 = q2. The constraint q qL was used to ensure no lost sales, and the results showed that q = qL if q qL, and q = q if q qL. The proof in Appendix B shows that it is cheaper to hold two di erent quantities for q1 and q2 even when q was the optimal value for q. No Disruption Cases The retailer must consider the possibility that no disruption will occur if she decides to hold inventory. If the disruption does not materialize, the retailer will incur more expensive holding costs if q1 qE. There are two possible cases to consider if the retailer holds inventory during the rst disruption and the disruption does not occur. The rst case is illustrated in Figure 3.3.4, and represents the situation when the retailer places multiple orders before T4. The cost equation for Case 1 is given in Eq. (3.3.22) where q1 is based on q E T 4 ? ? Time q 1 T 3 ? Inventory Level q 1 /? Figure 3.3.4: Hold Inventory with No Disruption: Case 1 (H; ) and q2 = qE. TC4(d0;0)(H; )(q1;q2) = A 1 + T 4 q1 qE ! +h q12 2 + qE T4 q1 2 ! (3.3.22) 45 The fth and nal outcome associated with holding inventory during the rst disruption does not require any additional orders before T4 occurs. The gure is not provided because it is similar to Figure 3.3.4 with the exception that there is only one order, q1. The cost equation is given in Eq. (3.3.23). TC5(d0;0)(H; )(q1) = A+ hq1T42 (3.3.23) 3.3.5 Inventory Policy: Do Not Hold Inventory During 1st Disruption Based on forecasts, retailers may decide that it is better to not hold inventory during the rst disruption if they expect the storm to severely damage their property. However, forecasts are typically not accurate, so retailers will risk lost sales and dissatis ed customers if the disruption does not occur or if the disruption is not as extreme as predicted. Disruption Case If the disruption occurs, the retailer will experience lost sales until an order can be received at T3 +L as shown in Figure 3.3.5. As previously mentioned, it is assumed that the retailer is unable to place an order until the disruption is over at T3, which is why the rst order of q2 will not be received until T3 +L. The cost equation associated with Figure 3.3.5 is given (1-?)q 1 q 2 T 3 ? S ? S ? S T 4 Time Inventory Level T 3 +L Figure 3.3.5: Do Not Hold Inventory with Disruption in Eq. (3.3.24). Please note that if = 1, it would be better for the retailer to not hold inventory because the retailer would not incur the damaged inventory cost, y, or ordering 46 cost, A, for the damaged inventory if Eq. (3.3.24) is compared to Eq. (3.3.14). TC(d1; )(NH;1)(q1;q2) = A S(T4 T3 L) q2 +h q 2(T4 T3 L) 2 +z SL (3.3.24) Regarding the inventory policy, it is assumed that q1 = 0, and if the rst order condition is applied to Eq. (3.3.24), q2 = qS, which is the same result if the retailer decides to hold inventory during the rst disruption. Therefore, we can conclude that the retailer?s decision for q2 will be the same regardless of the decision for q1. No Disruption Case If no disruption occurs, the retailer will incur lost sales until she is able to receive an order at t = L. It is assumed that the retailer will place an order of qE at t = 0 after it is known that the disruption will not occur as shown in Figure 3.3.6. Depending on the holding costs resulting from q1 qE, it might be bene cial from a cost perspective to incur lost sales. q E T 4 ? Time L ? Inventory Level ? Figure 3.3.6: Do Not Hold Inventory with No Disruption The cost equation for the not holding inventory when no disruption occurs is given in Eq. (3.3.25). Please note that q1 = 0 and q2 = qE. TC(d0; )(NH;1)(q1;q2) = A (T 4 L) qE +h q E(T4 L) 2 +z L (3.3.25) 47 After identifying all of the possible outcomes considering the assumptions, we performed a numerical experiment to determine the conditions that support each inventory policy. The results of the numerical experiment and sensitivity analysis are provided in Section 3.4. 3.4 Results A 29 experimental design was created to determine the inventory policy supported by minimax and minimax regret decision criteria. Both decision criteria were applied because di erent retailers may approach uncertain disruptions with contrasting objectives, and we wanted to determine if the decision criteria would provide contradicting decisions. Table 3.4.1 gives the values tested in the numerical experiment. One constraint was considered when the values were selected, and it is based on the inequality, T3 +L T4. The constraint was necessary because retailers may consider the second disruption negligible if the rst shipment is received after the demand rate has returned to . It should be noted that the lead-time values are longer than the tested values in Chapter 2 because the supplier?s lead-time may increase after a disruption. Table 3.4.1: Values Tested for 29 Experiment Parameter Low High S 100 200 10 15 T3 2 4 T4 8 16 h 1 2 A 100 200 L 1 2 z 10 20 y 4 8 3.4.1 First Experiment: =f0;1g For the rst experiment, we assumed that only three outcomes might occur: no disrup- tion, disruption and all of the inventory was damaged, or disruption and none of the inventory was damaged. Minimax decision criterion supported the decision not to hold inventory if 48 a possible disruption may interfere with the retailer?s operations for all tested cases, and this result can be proven analytically. The maximum cost equation for the decision to hold inventory is always the cost equation associated with the outcome where all of the inventory is damaged, = 1. By referring back to Eqs. (3.3.14) and (3.3.24), it is obvious that the di erence between the two maximum cost equations is yqL, the cost of damaged inventory, and the order cost for q1, A. This leads us to the conclusion that minimax decision criterion supports the decision not to hold inventory for an uncertain disruption. When we performed the numerical experiment using the values from Table 3.4.1 for the special case = f0;1g, we found that q1 = qL when minimax regret decision criterion is applied. This is contrary to our results found using minimax decision criterion. Therefore, we can assume that there are conditions when it is favorable to hold inventory during the rst disruption if the retailer seeks to minimize the maximum regret regarding costs. 3.4.2 Second Experiment: 2[0;1] In order to understand the impact of on the retailers ordering decision, a numerical experiment including additional values of was performed. When ve di erent values of 2[0;1] were tested, minimax decision criterion advocated not holding inventory, which is the same result as the rst experiment. The total cost for each decision of q1 was compared considering all outcomes ( ), and the maximum cost for each decision occurred when = 1. After the maximum cost was found for each q1, the minimum cost considering all of the order quantities associated with the maximum cost was found to be q1 = 0. This result is intuitive because q1 = 0 minimizes the cost equation when = 1, which always gives the maximum cost for each decision. An example is given in Table 3.4.2 for ve values of and ve values of q1 based on Eq. (3.3.21). When we determined the order quantity, q1, associated with each value of , we assumed that the retailer ordered more than qL to accommodate the damaged inventory caused by the rst disruption meaning the retailer would have qL on-hand to satisfy demand after the disruption. It should be noted that when = 0:75, the 49 minimum cost occurred when q1 = 0 not q1 = qL1 0:75. We will explore the selection for q1 to minimize costs by considering a threshold value for later in this section. In Table 3.4.2, the bolded values represent the maximum cost for each decision for q1, and the minimum maximum cost occurred when q1 = 0 with a total cost of a361,707. Table 3.4.2: Minimax Costs q1 = 0 = 0:25 = 0:50 = 0:75 = 1 No Disruption qL a361,057 a361,335 a361,620 a361,910 a362,207 a36500 qL 1 0:25 a361,116 a361,190 a361,563 a361,946 a362,340 a36633q L 1 0:50 a361,266 a361,349 a361,457 a362,020 a362,607 a36900q L 1 0:75 a361,983 a361,974 a362,066 a362,257 a363,407 a361,700 0 a361,707 a361,707 a361,707 a361,707 a361,707 a36413 In order to gain more insight about the e ect of on the ordering decision, we compared two outcomes: one value of and no disruption. Again, we determined the possible solutions for q1 based on Eq. (3.3.21) for each value of < 1. The experiment demonstrated that there is a threshold value of when it is best not to hold inventory even if all of the inventory is not damaged. This means q1 = qL1 is not optimal for some problem instances even when = . We compared the total cost of not holding inventory and holding inventory such that qL would be available at T3 regardless of . The threshold value of was most sensitive to changes in the lost sales cost. As expected, if the lost sales cost was increased, the threshold value of increased. When the low values for each parameter were tested, the value of caused the decision to switch from q1 = qL1 to q1 = 0 when at least 61% of the inventory was lost. This explains the result we found for the minimum cost for = 0:75 in Table 3.4.2. If z = 40, or ten times the purchase price, y, the threshold value was 90%. We were able to nd a value of z that supports the threshold value as shown in Eq. 3.4.1. If z upholds the inequality in Eq. (3.4.1), it is best to hold q1 = qL1 when = . z hT3 + hL( 2)2 +y 1 1 1 (3.4.1) 50 Next, we applied minimax regret decision criterion considering the same ve values for . First, we found the lowest cost for each outcome (column), which is shown in Table 3.4.3, considering all of the combinations in the 29 experiment. Please note that q1 = 0 was a minimum cost alternative for every outcome. Also, the value of q1 that ensured no lost sales for each value of was also a minimum cost alternative for combinations in the experiment. When the lost sales cost was less than 2.5 times the purchase price, y, minimax decision criterion supported the decision to hold less inventory, thus resulting in lost sales in some situations when the holding cost and T3 were at their highest values. As expected, if the disruption damaged all of the inventory, the retailer should not hold inventory, which is the same result supported by minimax decision criterion. Table 3.4.3: Possible Minimum Cost Decisions for Each Outcome Outcome Possible Minimum Cost Decisions = 0 q1 = 0, q1 = qL = 0:25 q1 = 0, q1 = qL, q1 = qL1 0:25 = 0:50 q1 = 0, q1 = qL, q1 = qL1 0:50 = 0:75 q1 = 0, q1 = qL1 0:75 = 1 q1 = 0 No Disruption q1 = 0, q1 = qL In order to test the sensitivity of the decisions based on z, we performed the same experiment, but increased the values of z to be ve, ten and twenty times the purchase price, y, to see the e ect on the minimum cost. We found that the minimum cost occurred when q1 = qL1 for < 1 and q1 = 0 for = 1 if we assumed to be known. These results matched previous results when we found the value of q1 that minimized the cost equation. If no disruption occurred, the minimum cost was almost split evenly between q1 = 0 and q1 = qL. The di erence between the decision depended on the lead-time, lost sales cost, holding cost, and order cost. However, if the disruption does not occur, the retailer should not hold more than qL in order to minimize costs. We then compared the maximum regret for each row (decision) to determine the decision that supported the smallest maximum regret. In Table 3.4.4, the regret matrix is shown for 51 the costs given in Table 3.4.2, and the bolded values represent the maximum regret for each decision. The largest negative value in each row represents the largest regret for each decision. Table 3.4.4: Regret Matrix q1 = 0 = 0:25 = 0:50 = 0:75 = 1 No Disruption qL a360 -a36145 -a36163 -a36203 -a36500 -a3687 qL 1 0:25 -a3659 a360 -a36106 -a36239 -a36633 -a36220q L 1 0:50 -a36209 -a36159 a360 -a36313 -a36900 -a36487q L 1 0:75 -a36926 -a36784 -a36609 -a36550 -a361,700 -a361,287 0 -a36650 -a36517 -a36250 a360 a360 a360 Table 3.4.5 gives the possible outcomes that resulted in the maximum regret for each decision. As expected, the possible maximum regret outcomes were associated with the extreme scenarios (e.g., = 0) with the exclusion of q1 = qL. Table 3.4.5: Possible Maximum Regret Outcomes for Each Decision q1 Possible Maximum Regret Outcomes qL No Disruption, = 50%, = 1 qL 1 0:25 No Disruption, = 1q L 1 0:50 No Disruption, = 0, = 1q L 1 0:75 No Disruption, = 0, = 1 0 = 0 When we evaluated the eleven di erent possibilities for maximum regret, only six gave the minimum maximum regret for the 512 combinations tested. The decision, outcome, and percentage of occurrence is given in Table 3.4.6. It was best to hold qL for more than half of the combinations. On the other hand, 45% of the combinations tested supported the decision not to hold inventory. 3.4.3 Sensitivity Analysis In the sensitivity analysis, we tested the same values for that were used in the numerical experiment for = [0;1]. All of the parameters were kept at the lowest value given in Table 3.4.1, and the parameter in question was increased or decreased. Table 3.4.7 gives the lowest 52 Table 3.4.6: Minimax Regret Outcomes q1 Outcome Percentage of Occurrence qL No Disruption 9% qL = 0:5 9% qL = 1 33% qL 1 0:25 No Disruption 0.2%q L 1 0:25 = 1 4% 0 = 0 45% and highest values tested for each parameter. We did not allow any of the parameters to equal zero because that would not be realistic, and if T3 = 0, there would be no disruption. For T3 and L, we did not test values that would cause T3 + L > T4 because the second disruption would be negligible. For h and y, we did not test values that were higher than z because the holding cost and purchase price would most likely not be greater than the lost sales cost. In addition, the values of z were all higher than y because the lost sales cost would probably be at least the purchase price. Table 3.4.7: Sensitivity Analysis Values Parameter Lowest Value Highest Value D 25 500 5 100 A 25 500 h 0.25 4 y 1 10 s 4 80 T3 1 6 T4 3 20 L 1 6 We discovered that z was the only cost that changed the decision given by minimax decision criterion if the lost sales cost was about 1.5 times y. The lost sales cost often encompasses the retail price of the item and loss of customer goodwill while y represents the purchase price, which in most cases, will be less than the retail price. Therefore, we can still conclude that minimax decision criterion will always support the inventory policy to not hold inventory during the rst disruption. 53 On the other hand, the sensitivity analysis for minimax regret decision criterion provided many insights into the e ect of each parameter. In the next four subsections, we separate the parameters according to how they a ect q1. q1 = qL)0 Four of the parameters, A, h, T3, and y, showed a change in the decision for q1 from qL to 0 as the parameters were increased. This result is expected because a low ordering cost, holding cost, and purchase price as well as a short rst disruption will constitute holding inventory when minimax regret decision criterion is applied. However, as the costs and duration of the rst disruption increase, it becomes more cost e ective not to hold inventory during the rst disruption, especially if the retailer expects a large percentage of the inventory to be damaged. q1 = 0)qL There were only two parameters, L and S, that caused a change in q1 from 0 to qL. As the lead-time and disruption demand rate were increased, the total cost increased as well. A short lead-time allows the retailer to order at T3 and incur minimal lost sales. In some cases, the total lost sales may be cheaper than the order and holding costs incurred for holding inventory during the rst disruption. If the disruption demand rate, S, is small, then the cost of lost sales will also be minimal similar to the result for the short lead-time. No Change Two parameters, and T4, supported holding qL during the rst disruption for all tested parameter values. Therefore, we can conclude that the normal demand and the length of second disruption do not have an a ect the decision for q1. This result is intuitive because neither parameter a ects lost sales or q1. 54 Multiple Changes in q1 The lost sales cost, z, altered the value of q1 four times for the tested values meaning all of the values of q1 beginning with q1 = 0 were utilized as z increased. The value for z was increased in increments of four with the rst value equaling y. If z 50 or more than 12 times the purchase price, the retailer will hold qL0:25 ( = 0:75), which is the largest amount of inventory we tested to be held by the retailer. This result indicates that the decision is very sensitive to changes in the lost sales cost as expected. Retailers must seriously consider how they will value lost sales because it will a ect their ordering decisions, particularly if they implement minimax regret decision criterion. 3.5 Summary and Future Extensions Minimax and minimax regret decision criteria provided di erent solutions in our numer- ical experiment, and the retailer must prioritize the objectives of the company when making decisions regarding inventory policies for possible disruptions caused by forecasted storms. The following is a summarized list of our ndings: 1. It is best to hold a di erent quantity for each disruption (q1 6= q2). 2. Minimax decision criterion always yielded q1 = 0 unless the lost sales cost was slightly higher than the holding cost. 3. Both decision criteria supported q2 = qS. 4. A threshold value of exists indicating that even if all of the inventory is not damaged, it is better to not hold inventory depending on the lost sales cost, z. 5. The minimax regret decision is very sensitive to changes in the lost sales cost, but it is also sensitive to changes in all other parameters except and T4. The results showed the importance of the retailer accurately estimating the lost sales cost because it will drastically change the decision for q1. However, it is di cult to assess the rami cations of not meeting the customers? needs. In addition, the forecast may play a 55 signi cant role in the retailers? inventory policy. If the storm is expected to be catastrophic, it is probably in the best interests of the retailer not to hold any inventory. On the other hand, most property structures for retailers such as grocers are able to withstand the e ects of storms. Based on the results, our recommendation is that the retailer carefully consider how lost sales are valued when selecting an inventory policy. Our results indicated that if the lost sales cost is estimated to be ve times or more than the purchase price, then the retailer should hold at least qL. Otherwise, the retailer should hold nothing. The model could be extended to include the disruption discussed in Chapter 2. It seems that if minimax decision criterion is used to solve the extended model, q1 would either be 0 or the amount of inventory at the retailer at T2. On the other hand, it might be worth exploring the problem using minimax regret decision criterion. In addition, a variable lead-time and exible reorder point could also be included to make the model more realistic. Customers are often in search of items such as bread, milk, and fruit after a storm, and the model could be extended to include perishable items. We did not consider perishable items because we assumed that the retailer would not stock up on such items because of the uncertainty surrounding the length of the disruption between T2 and T3. Also, if perishable items were included, the cost for the decision to hold inventory would only increase, resulting in the decision not to hold inventory. This is the same decision we found for minimax decision criterion considering non-perishable items. We also predict that the decision not to hold inventory would increase among the situations we tested considering minimax regret decision criterion because the total cost for holding inventory during the rst disruption would increase. In closing, our results indicated the importance of estimating cost parameters as well as selecting decision criterion to determine an inventory policy for disruptions. However, if retailers do not consider the impact of their inventory decisions before a disruption, they will not be able to capitalize on an opportunity to generate additional pro t as well as foster positive customer relationships during a time of crisis. 56 Chapter 4 Pre-Positioning Hurricane Supplies in a Commercial Supply Chain 4.1 Background 4.1.1 Disastrous Hurricanes and Emergency Management Hurricane Katrina was a daunting reminder that hurricanes represent one of Nature?s most destructive forces. This catastrophic storm, which left a trail of destruction along the United States Gulf Coast in August 2005, was responsible for more than 1,800 casualties, 250,000 displaced residents, and a36125 billion in expenditures making it the costliest disas- ter in United States history [21]. Although the signi cant environmental, socio-economic, and cultural repercussions of Hurricane Katrina are evident, disastrous hurricanes are cer- tainly not a new phenomenon. For example, the 18th century storm known as the \Great Hurricane" claimed 22,000 lives throughout the Caribbean and is the deadliest hurricane of recorded history [40]. The deadliest hurricane in United States history was responsible for an estimated 8,000 to 12,000 casualties in Galveston, Texas during the year 1900 [26]. More recently, hurricanes Ike, Gustav, and Dolly in 2008 and Dennis, Katrina, Rita, and Wilma in 2005 each accounted for over a361 billion in expenditures and numerous casualties [37]. The eld of Emergency Management outlines a framework for protecting civilization from the adverse e ects of natural and man-made disasters caused by hazards such as hur- ricanes, tornadoes, earthquakes, drought, and terrorist attacks. This framework entails the following functional areas: (i) mitigation, (ii) preparedness, (iii) response, and (iv) recovery. Mitigation refers to \a sustained action to reduce or eliminate risk to people and prop- erty from hazards and their e ects" [22]. Mitigation activities focus on long-term solutions such as construction design and structural control (e.g., strengthening levee systems), and 57 tend to be the most expensive emergency management options. In contrast to mitigation, preparedness targets short term activities directly related to response and recovery such as preparing large-scale evacuation plans or securing emergency supply items. The objective of the response function is to provide immediate short term relief following disaster by saving lives, protecting property, and meeting basic human needs. Activities include search and rescue, medical assistance, law and order, and immediate food and shelter. The nal stage of the disaster-relief lifecycle is the recovery e ort, which seeks to restore communities to pre- disaster conditions, and also to curtail future vulnerability. Examples of recovery activities include debris removal, rebuilding residential properties, and restoring businesses. 4.1.2 Pre-Positioning The focus of this paper will be strategic positioning of emergency supplies in anticipation of a threatening hurricane or other predictable hazard, which can be considered a cross- section of the preparedness and response activities of the Emergency Management framework described in Section 4.1.1. Pre-positioning is de ned as the \stockpiling of equipment and supplies at, or near the point of planned use" [15]. Although originally applied within military contexts (e.g., [8], [31]), public and private sector organizations have adopted pre- positioning strategies to prepare for civilian response and recovery operations. For example, the U.S. Federal Emergency Management Agency (FEMA) oversees a variety of permanent and temporary staging locations for pre-positioning commodities, equipment, and personnel. Permanent facilities include nine FEMA Logistics Centers, which are traditional distribution centers for commodities geographically dispersed throughout the Continental United States, as well as a number of commercial storage sites owned and operated by private rms [59]. On a smaller scale, locating ambulances and re stations can also be considered a form of public sector pre-positioning (e.g., [3], [52]). Private sector rms implemented a most noteworthy application of temporary staging and pre-positioning in response to Hurricane Katrina. In particular, The Home Depot and 58 Wal-Mart independently pre-positioned commodities at temporary staging locations near New Orleans prior to Katrina?s landfall in 2005 (e.g., [24]). As a result, emergency supplies, equipment, and personnel were readily available for initial response e orts in contrast to the government?s slow and uncoordinated response. The logistics response of Wal-Mart and The Home Depot to Hurricane Katrina accentuates the pivotal role of private sector rms in facilitating e ective disaster relief operations, which is a perspective that has been acknowledged in the humanitarian logistics research literature. 4.2 Problem Description: Commercial Pre-Positioning of Commodities In an e ort to encourage a critical mass of private sector rms to follow the lead of Wal-Mart and The Home Depot in responding to domestic disasters as described in Section 4.1, this study examines a humanitarian logistics problem from the perspective of a tradi- tional pro t-driven private sector rm. Speci cally, the problem presented in this paper is motivated by a real-world supply chain scenario consisting of one manufacturer, multiple retailers, and one or more observed storms that threaten the consumers served by these retailers. The manufacturer produces items such as bottled water, non-perishable foods, or portable electronic devices that are in high demand during the hurricane season, especially in the presence of an ominous hurricane or tropical storm. In particular, it is common for retailers to experience a spike in demand for these and other emergency supply items during the inventory cycles that precede an observed storm?s probable landfall. Such \pre-storm demand surge" is predominantly driven by consumers who are preparing for emergency evacuation, or by consumers planning to \ride out the storm" and risk the inconvenience of potentially pervasive power outages. A temporary spike in demand can also be observed subsequent the actual landfall of a major storm as a result of disaster relief and recovery activities. Based on conversations between one of the authors and supply chain managers from a manufacturing rm that encounters potential post-storm demand surge as described above, 59 orders for emergency supplies are typically initiated by retailers in response to a realized pre- or post-storm demand spike. In other words, the retailers served by this manufacturer exhibit a propensity for adopting reactive inventory policies that respond to an actual demand surge as opposed to proactive inventory policies that anticipate a likely demand surge. Such reactive approaches to stock control coupled with the delivery lead-times associated with order ful llment often lead to widespread stockout of emergency supplies during peak demand surge periods, which in turn exacerbates the vulnerability of the populace a ected by an approaching storm, inhibits post-storm response and recovery e orts, and proliferates lost sales and/or backorders for the above-mentioned supplier-retailer supply chain. In order to alleviate the negative e ects of supply shortages that often occur during pre- and post-storm demand surge periods, this paper will explore inventory pre-positioning strategies from the standpoint of the manufacturing rm described at the beginning of this section. The proposed pre-positioning strategy is characterized by a manufacturer that proactively pushes inventory upon geographically dispersed retailers in anticipation of pre- or post-storm demand surge, which is a strategy that contrasts the reactive wait-and-see approach indicative of current practice. After the storm subsides and actual demands become known, inventory is then transhipped among the retailers and manufacturer. The proposed pre-positioning strategy assumes that the manufacturer bears all the costs and risks of pre-positioning, similar to vendor-managed inventory (VMI) systems. These costs may be attributed to production at the manufacturing facility, material handling at the manufacturing facility, and transportation of products from the manufacturer to the retailers. On the other hand, risks are driven by the uncertainty of which retailers will experience a demand surge for emergency supplies as a result of a storm making landfall at their location. These risks include the expected costs due to inventory shortages or excess inventory at each retailer location, and the possibility of additional transportation costs for redistributing improperly pre-positioned supplies. If there is a shortage of inventory at a retailer location, the items are backordered, and the manufacturer will incur a shortage cost 60 representing loss of customer goodwill for not providing the item immediately after the e ects of the storm have subsided. Additionally, conversations with the representative manufacturer also revealed the risk of a prohibitively constrained post-storm logistics system characterized by a scarcity of available third-party logistics providers (3PLs). This environment inhibits post-storm redistribution e orts, or signi cantly increases the cost of doing so. Within the above-mentioned context, this investigation will be driven by the following research questions. Question 4.1 Given the possibility of a post-storm demand surge for an emergency supply item, how should the manufacturer optimally pre-position the commodity among a network of geographically dispersed retailers? In particular, what quantity should be stockpiled at each retailer prior to a potential post-storm demand surge? Question 4.2 How bene cial is pre-positioning relative to the existing wait-and-see approach currently used in practice? Note that Question 1 includes the possibility that the pre-position quantities for any one or more retailers could be zero, which suggests that pre-positioning will occur for a subset of the retailers. Also note that the manufacturer?s interests relative to these research questions will be represented solely in terms of expected cost, which is consistent with the commercial perspective of this study. The remainder of this manuscript is organized as follows. First, Section 4.3 surveys the relevant academic literature. Next, Section 4.4 introduces a stochastic programming based methodology. Speci cally, the manufacturer?s problem is represented as a two-stage stochas- tic programming model with recourse where the rst stage decision is the quantity of supplies to stockpile at each retailer situated in the projected path of an observed storm, and the recourse decision entails the redistribution of supplies throughout the supply chain network after all demand surge information (i.e., quantities and locations) is known with certainty. In addition to the stochastic programming methodology, an alternative solution methodology 61 is applied to the pre-positioning problem, which utilizes a heuristic to compute the pre- positioned quantities and reduces the model to the well-known transportation problem. The stochastic programming and alternative solution methodology are illustrated via numerical examples, including a case study based on real-world data in Section 4.5. The examples are solved using the scenario driven stochastic programming capability of the commercial opti- mization software, What?s Besta151. Then, the optimal solution found by What?s Besta151 and the solution approach are compared. A sensitivity analysis is also provided to illustrate the e ects of the cost parameters on the expected bene t of pre-positioning. The nal section of the paper, Section 4.6, summarizes key ideas and ndings, discusses limitations, and outlines an agenda for future research. 4.3 Literature Review As previously mentioned, the model presented in this paper resembles a VMI (vendor- managed inventory) system because the manufacturer assumes a majority of the costs and risk associated with the production and distribution of an item. VMI models also consider various costs associated with holding, transportation, shortage, and ordering costs incurred by the vendor that are similar to the costs considered in our model ([33], [67]). Moreover, many models within the VMI literature determine the optimal shipment quantity to the retailer(s) (e.g., [13], [17]), which is also determined in our model. However, the model presented in this paper di ers from a VMI system in several ways. Current VMI models assume a multi-period or in nite planning horizon ([13],[33]) in contrast to our model, which considers a short planning horizon with a one-time decision regarding inventory quantities at retailers. In addition, our model allows for transshipments between retailers if a shortage is experienced at a retailer, which is not characteristic of VMI models. Lastly, we consider an increased unit transportation cost after the extreme event (i.e., hurricane) has occurred. Our model is formulated as a two-stage stochastic program to determine the quantity that should be pre-positioned at each retailer in the network, which is similar to the models 62 presented in the pre-positioning literature. Therefore, the focus of the literature review will be pre-positioning models that are closely related to the model presented in this paper. The de nition of pre-positioning presented in [15] as the \stockpiling of equipment and supplies at, or near the point of planned use" suggests that pre-positioning activities entail both location and inventory related choices. In fact, the more recent de nition proposed in Richardson et al. [54] implies that pre-positioning encompasses location, inventory, and dis- tribution decisions, which spans the preparation and response dimensions of the emergency management framework. More speci cally, pre-positioning in preparation for potential dis- aster relief operations involves determining the number, location, and capacity of permanent or temporary warehouses used to coordinate distribution of emergency supplies to disaster areas following a catastrophic event. In addition, pre-positioning also concerns inventory decisions for these facilities as well as transportation policies (e.g., shipping and routing) for distributing commodities from storage facilities to a ected communities. From this perspec- tive, the academic literature related to inventory pre-positioning can be characterized as a cross-section among the facility location, inventory control, and distribution / routing liter- atures. Indeed, several studies address some of these areas individually within the context of disaster relief. For instance, Jia et al. [30] and Huang et al. [25] consider humanitarian relief chain design in terms of facility location with no explicit inventory or routing decisions, while inventory models related to disaster relief with no network design or distribution deci- sions include Beamon and Kotleba [7], Lodree and Taskin [36], and Taskin and Lodree [63]. Papers that focus exclusively on \last-mile distribution" with no location or inventory de- cisions include Ozdamar et al. [45] and Bal cik et al. [4]. A few studies also integrate the location, inventory, and distribution aspects of pre-positioning described in Richardson et al. [54], namely Chang et al. [14], Mete and Zabinsky [38], Rawls and Turnquist [53], and Salmer on and Apte [56]. Nevertheless, the focus of this review is a series of papers that consider at least two components of pre-positioning including location and inventory ([5], 63 [12]), inventory and distribution ([6]), and the above-mentioned papers [14], [38], [53], and [56]. One of the rst papers to investigate pre-positioning of commodities in preparation for potential disaster relief activities is Barbaroso glu and Arda [6], who address inventory and distribution decisions on an existing humanitarian logistics network. In particular, Bar- baroso glu and Arda [6] formulate a two-stage stochastic programming model where the rst stage represents a multi-commodity, multi-modal network ow problem in which commodi- ties are transported from a given set of supply nodes to a set of potential demand nodes with uncertain requirements. During the second stage, demand locations and requirements become known. Then, redistribution during the second stage based on known demand in- formation and the pre-positioning arrangement generated in the rst stage is optimized by solving another multi-commodity, multi-modal network ow problem. The model also ac- counts for vulnerability of supply and demand nodes as well as uncertain arc capacities, both of which become known in the second stage. The objective function of the stochastic pro- gramming model minimizes rst stage transportation cost and the expected transportation and inventory costs of the second stage. Commercial software was used to solve a numerical example based on data from an August 1999 earthquake that occurred in Turkey. Bal cik and Beamon [5] and Campbell and Jones [12] consider both humanitarian relief chain design and inventory decisions. Bal cik and Beamon [5] develop a variation of the max- imal covering problem in which the number of distribution centers (DCs) and their locations are chosen such that the total expected demand covered by the distribution centers across a set of probabilistic scenarios is maximized. Similar to Barbaroso glu and Arda [6], Bal cik and Beamon [5] illustrate their model by generating data sets that correspond to historical earthquake scenarios and use commercial software to conduct computational experiments. On the other hand, Campbell and Jones [12] introduce the rst scenario-free pre-positioning study. They develop an analytic framework with newsboy results and sensitivity analyses for a network with one supply node and one demand node, where the supply node faces 64 the possibility of being destroyed. Using results from their single supply / single demand node model, Campbell and Jones [12] also describes a procedure for optimally selecting a supply location from among a set of potential locations. In addition, a heuristic approach to locating multiple supply points for multiple demand locations is also described and explored in a computational experiment. Chang et al. [14], Rawls and Turnquist [53], Salmer on and Apte [56], and Mete and Zabinsky [38] incorporate all three aspects of pre-positioning into their models. Chang et al. [14] seems to be the rst study that location, inventory, and distribution decisions are con- sidered explicitly. In addition, the two-stage stochastic programming model is predicated on a preliminary stochastic model with chance constraints that partitions a multi-group, multi-echelon supply chain network into zones. The solution approach is a multi-step pro- cedure that entails sample average approximations of the stochastic problems. Rawls and Turnquist [53] also integrate network design, inventory, and distribution decisions for dis- aster relief purposes. In Rawls and Turnquist [53], a two-stage stochastic mixed integer programming model is presented in which DC locations, DC capacities, and inventory levels are determined during the rst stage based on probabilistic information concerning demand locations and requirements. The second stage decision concerns the distribution of com- modities from the DC locations established during the rst stage to disaster areas which are revealed at the beginning of the second stage. Rawls and Turnquist [53] also design a heuristic solution approach called the Lagrangian L-shaped method intended for large-scale problem instances, which is then illustrated in terms of a case study where disaster scenar- ios are generated based on historical hurricane data. Further details of the aforementioned case study are discussed in Section 4.5 of this paper. Moreover, the focus of Salmer on and Apte [56] is more on people as opposed to facilities and commodities, which is unlike other pre-positioning studies. The two-stage stochastic programming formulation is actually a hierarchical bi-objective model that minimizes casualties within the critical and stay-back populations, and minimizes unmet demand for the transfer population. Lastly, Mete and 65 Zabinsky [38] investigate location, inventory, and distribution decisions within the context of pre-positioning medical supplies. In the rst stage of their two-stage stochastic program- ming methodology, warehouse locations are determined. Then, after demands throughout a network of hospitals are revealed, medical supplies are transported from warehouses to hospitals. The second stage distribution problem is a mixed integer programming (MIP) model that represents a pseudo vehicle routing problem in which vehicles are assigned to predetermined routes. To illustrate the model, a case study based on earthquake scenarios in the Seattle Washington area is presented. For this case problem, the integrated stochastic programming / MIP models are solved using commercial software. It is evident from this review that the research literature has produced several frame- works that could be useful to government and relief organizations in a variety of real-world disaster relief environments. However, none of the existing studies speci cally address the manufacturer?s pre-positioning problem introduced in Section 4.2. For instance, since the manufacturer?s pre-positioning problem involves an existing network of retailers, there is no need to consider facility location decisions as in references [14], [38], [53], [56], [5], and [12]. From this perspective, our paper is similar to Barbaroso glu and Arda [6], who also address inventory and distribution decisions on an existing network. Similar to Barbaroso glu and Arda [6], the two-stage stochastic programming methodology described in Section 4.4 of this paper considers the transportation costs of pre-positioning commodities during the rst stage and the expected cost of transportation, inventory shortages, and excess inventory dur- ing the second stage. But in addition to the transportation and inventory costs considered in Barbaroso glu and Arda [6], the model described in Section 4.4 also takes into account the production costs incurred by the manufacturing facility during both the rst and sec- ond stages. Another di erence is that the underlying model in Barbaroso glu and Arda [6] is a multi-commodity, multi-modal network ow problem, whereas the underlying model presented in this paper is a transportation problem with possible transshipments among 66 retailers. Finally, the case study data applied to the model in this paper is derived from the hurricane scenarios generated by Rawls and Turnquist [53]. 4.4 Methodology 4.4.1 Stochastic Programming Model This section describes a two-stage stochastic programming model for the manufacturer?s inventory pre-positioning problem introduced in Section 4.2. The fundamental logic of the two-stage stochastic programming methodology entails a rst stage decision ~x2Rn in the presence of uncertain parameters ~D2Rn, followed by a second stage (or recourse) decision ~y2Rn after the realization ~ 2Rn of ~D has occurred (a complete list of notations is shown in Table 4.4.1). The general form of a two-stage stochastic program can be expressed as follows (e.g., [10]): Minimize: ~c T~x+ E h Q ~x; ~D i (4.4.1) Subject to: A~x =~b (4.4.2) ~x ~0; (4.4.3) where the recourse function is given by E h Q ~x; ~D i = min n ~q T~yjW~y = h T~x; ~y ~0 o : (4.4.4) The objective function (4.4.1) minimizes the cost of the rst stage decision (~c 2 Rn represents objective function coe cients or unit costs, and ~c T is the transpose of ~c ) plus the expected cost of the second stage decision, which is the recourse function. The recourse decision ~y is optimized based on both the rst-stage decision ~x and the observations ~ . W represents the xed recourse matrix, and T represents the technology matrix while h refers to the right-hand side in the second stage constraints. T and h become known when ~ is 67 realized. Decisions at each stage have their respective constraints, (4.4.2) and (4.4.3) for the rst stage and (4.4.4) for the second. Stage 1 Applying this framework to the manufacturer?s pre-positioning problem, the rst stage decision ~x = (x1;x2;:::;xn) represents the quantity of supplies that the manufacturer should pre-position across the network of retailers. That is, xi is the amount of inventory sent to retailer i prior to the landfall of an observed storm, where the demand at each retailer is a random variable Di 2 ~D. The demands, Di, are interpreted as additional demand at each retailer i that emerge because of the storm?s presence. It is assumed that ci = c for each ci2~c and i = 1;2;:::;n, where c is the cost attributed to producing and preparing one unit of inventory for shipment. The rst stage decision also incurs the transportation cost cvdixi for each retailer location i = 1;2;:::;n, where cv is the unit cost of transportation (a36 per mile) and di is the distance (in miles) between the manufacturer and retailer i. Therefore, the total rst stage cost is Production cost + pre-storm transportation cost = nX i=1 cxi + nX i=1 cvdixi: (4.4.5) Stage 2 The second stage cost is comprised of holding costs due to excess inventory at certain retailer locations, shortage costs due to inventory shortages (backorders) at other retailer locations, and the cost of redistributing inventories throughout the supply chain network. More precisely, the recourse function optimizes the recourse decision ~y in terms of the ex- pected costs due to excess inventory, shortages, and redistribution. In general, the stochastic programming methodology handles the uncertain parameters ~D = (D1;D2;:::;Dn) using scenarios as opposed to assuming a conventional probability distribution for each Di. In particular, suppose there are k scenarios such that ~D = ~ t under scenario t2f1;2;:::;kg, 68 Category Symbol Description Decision variables xi Quantity shipped to retailer i before hurricane ylmt Quantity transshipped from retailer l to retailer m in scenario t y mt Quantity transshipped from manufacturer to retailer m in scenario t Random variable Di Demand at retailer i Costs c Unit production cost cv Pre-hurricane transportation cost/mile cw Post-hurricane transportation cost/mile h Unit holding cost s Unit shortage cost bEPP(Bene t) Expected bene t of PDSA solution E PP(Bene t) Expected bene t of optimal pre-positioning strategy dECPP Expected cost of PDSA solution EC PP Expected cost of optimal pre-positioning strategy ECWS Expected cost of wait-and-see strategy Other notations i Retailer locations, i = 1;:::;n t Scenarios, t = 1;:::;k it Observed demand at retailer i if scenario t occurs I+ Set of retailers with excess inventory l Retailers with excess inventory, l2I+ I Set of retailers with an inventory shortage m Retailers with an inventory shortage, m2I Manufacturer location index Manufacturer location di Distance between manufacturer and retailer i dlm Distance between retailer l and retailer m d m Distance between manufacturer and shortage location Pt Probability of scenario t occurring Table 4.4.1: Summary of notations for stochastic programming model. 69 where scenario t occurs with probability Pt. That is, it represents the actual demand for retailer i if scenario t occurs. Then the scenario-based probability distribution for demand at retailer i is PrfDi = itg= Pt for each i = 1;2;:::;n and t = 1;2;:::;k. Now if xi it > 0 for a particular scenario t, then retailer i has excess inventory under scenario t. Similarly, xj jt < 0 suggests that a shortage has occurred at retailer j under scenario t. Clearly, two disjoint sets of retailers emerge for each scenario t for a given rst- stage pre-positioning arrangement: the set of retailers with excess inventory, denoted I+, and the set of retailers with inventory shortages, denoted I . Let h be the unit holding cost associated with excess inventory and s the unit cost of an inventory shortage, both of which are presumed to be homogeneous among the retailers. Since scenario t occurs with probability Pt, the expected cost due to inventory overages and shortages from the manufacturer?s perspective is Expected cost (holding + shortage) = kX t=1 Pt " X l2I+ h(xl lt) + X m2I s( mt xm) # : (4.4.6) Note that it is convenient to think of s as the unit cost associated with the loss of customer goodwill. In general, the components of shortage costs when backorders are con- sidered include the cost of processing one or more additional orders (possibly at an increased rate) and the loss of customer goodwill (e.g., [44]). In this paper, the redistribution costs described above can be interpreted as the former component of the conventional shortage cost, and s represents the latter. In addition to the expected holding and shortage costs shown in Eq. (4.4.6), second stage costs are also incurred for post-storm transportation and production activities. The transportation cost between a location l2I+ with excess inventory and a location m2I with inventory shortages is given by cwdlmylmt. Here, cw > cv is the unit cost of trans- portation during the post-storm redistribution stage, dlm is the distance between retailers l and m, and ylmt is the quantity of supplies transhipped from retailer l to retailer m if the 70 observed post-storm demand scenario is t. Note that the inequality cw > cv is assumed in order to re ect the scarcity of 3PLs available for post-storm redistribution relative to pre- storm pre-positioning as described in Section 4.2. Also note that ylmt for all l2I+, m2I , and t = 1;2;:::;k represent the recourse decision variables, which are optimized for each scenario t subject to the pre-positioned quantities xi and observed demands it. So far, we have assumed that all shortages (i.e., backorders) can be resolved through transshipments using excess inventory from the retailers l2I+. However, direct shipments from the manufacturer for the purpose of backorder replenishments will be necessary if the total network shortage associated with retailers m2I exceeds the total amount of excess inventory across the retailers l2I+. Furthermore, direct shipments from the manufacturer may sometimes be more cost e ective than transshipments from retailers. In order to account for direct replenishments from the manufacturer, let d m denote the distance between the manufacturer and shortage location m2I and y mt represent the quantity shipped from the manufacturer directly to retailer m2I under scenario t. Then the expected cost of transportation during the second stage, which includes both transshipments from retailers and direct shipments from the manufacturer, is given by Expected cost (transportation) = kX t=1 Pt " X l2I+[ X m2I cwdlmylmt # : (4.4.7) Finally, all products shipped directly from the manufacturer to shortage locations m2I during the transshipment stage incur production costs in addition to the transportation cost shown in Eq. (4.4.7). Assuming the unit cost of production, c, is preserved between the rst and second stages, the expected cost due to production during the second stage is given by Expected cost (production) = kX t=1 Pt " X m2I cy mt # : (4.4.8) 71 Model Formulation Equations (4.4.5), (4.4.6), (4.4.7), and (4.4.8) will be combined to formulate the objective function of the stochastic programming model. In addition, constraints related to overages and shortages will also be incorporated. The resulting stochastic programming model is as follows: Minimize nX i=1 cxi + nX i=1 cvdixi +E h Q ~x; ~D i (4.4.9) where, E h Q ~x; ~D i = miny lmt ( kX t=1 Pt " X l2I+ h(xl lt) (4.4.10) + X m2I s( mt xm) + X l2I+[ X m2I cwdlmylmt + X m2I cy mt #) Subject to: xi 0 8 i (4.4.11) xl lt X m2I ylmt; 8 l2I+ (4.4.12) mt xm = X l2I+[ ylmt; 8 m2I (4.4.13) xl lt; mt xm; ylmt 0 8 l2I+;m2I ; and t = 1;::;k:(4.4.14) Constraints (4.4.11) and (4.4.14) are non-negativity constraints. Speci cally, constraints (4.4.11) are non-negativity constraints for pre-positioned quantities during the rst stage, and constraints (4.4.14) ensure that overage, shortage, and transshipment quantities for the second stage are non-negative. Constraints (4.4.12) re ect the fact that it is impossible for any retailer with excess inventory to transship more than her surplus to other retailers during the post-storm redistribution stage. Lastly, constraints (4.4.13) ensure that all shortages in the network are replenished during the second stage either from retailers with excess inventory or directly from the manufacturing facility. 72 4.4.2 PDSA Solution Approach The computational e ort associated with solving the stochastic programming model de ned by equations (4.4.9) - (4.4.14) may become prohibitive for large scale problems, which turned out to be the case for the sensitivity analysis experiments presented in Section 4.5.4. In particular, computation times of over 30 hours were reported for problem instances solved used the commercial software What?s Besta151. In order to address this issue, we introduce an e cient solution approach that reduces the stochastic programming model to the well-known transportation problem (e.g., [66]), which can solve each scenario in seconds using Excel Solver. For the remainder of the paper, we refer to this solution approach as the Percentage of Demand Scenarios solution Approach or PDSA. The proposed approach entails two stages. In the rst stage, the pre-positioning quantities xi are determined. After demand realizations occur, the second stage is simply a deterministic transportation problem in which supply nodes are retailers with excess inventory as well as the manufacturer, and demand nodes are retailers with inventory shortages. The rationale behind the rst stage of PDSA is as follows. If the expected unit cost of excess inventory for no-demand scenarios is greater than the expected unit shortage cost for demand scenarios, then the pre-positioning strategy should be conservative. In this case, there is more risk associated with excess inventory compared to inventory shortages. Therefore, excess inventory should be avoided, which can be achieved by pre-positioning small quantities. Similarly, a more aggressive pre-positioning strategy would be appropriate if the expected unit cost of excess inventory for no-demand scenarios is less than the expected unit shortage cost of demand scenarios. In order to develop expressions for the expected unit cost of excess inventory for no- demand scenarios described above as well as the expected unit shortage cost for non-zero demand scenarios, it is necessary to partition the set of scenarios S =f1;2;::: kg into two mutually exclusive and collectively exhaustive sets for each retailer. Let Ni S denote the non-zero demand scenarios for retailer i, and Zi S the scenarios in which retailer i has 73 Symbol Description S =f1;:::; kg: Set of demand scenarios. Ni S: Non-zero demand scenarios for retailer i. Zi S: Scenarios with zero demand for retailer i. Di =f i1; i2;::: ikg: Demand values for retailer i for each scenario t2S. Di Di: Non-zero demand values for retailer i over all scenarios. mini = inf Di: Minimum value (in mum) of the set Di. tmin: Scenario that corresponds to mini . Nmini =ftming: The set that contains tmin. Table 4.4.2: Notations for PDSA. zero demand, where Ni[Zi = S (note that additional notations used for PDSA are shown in Table 4.4.2). Then E(Holding Cost j Zero Demand Scenarios) = X t2Zi hPt (4.4.15) E(Shortage Cost j Non-Zero Demand Scenarios) = X t2Ni sPt; (4.4.16) where Pt2ZiPt and Pt2NiPt are the percentage of zero-demand and non-zero demand sce- narios for retailer i, respectively. PDSA will also utilize the minimum non-zero demand value and its corresponding sce- nario to determine the actual pre-positioning quantities. Let Di = f i1; i2;::: ikg denote the demand values for retailer i under each scenario t2S, and Di Di denote the non-zero demand values for retailer i. Then the minimum non-zero demand value is inf Di, denoted mini . De ne tmin as the scenario that corresponds to mini and Nmini =ftming as the set that contains tmin. Then PDSA calculates the pre-positioning quantities xi as follows: Case 1: X t2Zi hPt > X t2Ni sPt. Case 1 corresponds to the case in which the expected unit cost of excess inventory for no demand scenarios is greater than the expected unit shortage cost of non-zero demand scenarios (see Equations 4.4.15 and 4.4.16). For this case, a conservative strategy is warranted and the PDSA pre-positioning quantities are given by 74 xi = 8 >>> >< >>> >: mini = inf Di; if X t2Zi Pt < X t2Ni Pt 0; if X t2Zi Pt X t2Ni Pt: (4.4.17) If the total probability associated with non-zero demand scenarios is greater than the total probability of zero-demand scenarios, then Equation (4.4.17) pre-positions the mini- mum non-zero demand for that retailer. Otherwise, the PDSA pre-positioning quantity is zero. Case 1 is further divided to consider the total probability of non-zero and no-demand scenarios, which is why the minimum amount demanded is the recommended amount when it accounts for a larger percentage of the demand scenarios. In Case 1, the expected unit holding cost is more expensive, therefore, the manufacturer would only want to pre-position the minimum amount of items if any at all. Case 2: X t2Zi hPt X t2Ni sPt. Case 2 corresponds to the case in which the expected unit cost of excess inventory for zero-demand scenarios is less than the expected unit shortage cost of non-zero demand scenarios, which warrants a more aggressive pre-positioning strategy relative to Case 1. For this case, the PDSA pre-positioning quantities are determined by xi = 8 >>> >>< >>> >>: X t2Ni=Nmini itPt; if X t2Nmini Pt < X t2Ni=Nmini Pt mini = inf Di; if X t2Nmini Pt X t2Ni=Nmini Pt: (4.4.18) The quantity Pt2Ni=Nmin i itPt in Equation (4.4.18) represents the expected demand for retailer i with respect to non-zero demand scenarios, but excluding the minimum non-zero demand scenario, tmin. This value is the PDSA pre-positioning quantity if total probability 75 associated with non-zero demand scenarios (not including tmin) is larger than the probability that scenario tmin occurs. Otherwise, the PDSA pre-positioning quantity is the minimum non-zero demand, mini . The non-zero demand scenarios are only considered for Case 2 because it is more expensive to incur shortages according to the calculated expected unit costs. As a result, the manufacturer should pre-position a non-zero quantity at the retailer location. Case 2 is further divided to consider the total probability of the smallest non-zero demand. For this reason, the recommended amount is the smallest non-zero demand when it accounts for a majority of the non-zero demand scenarios. 4.4.3 Numerical Example In this section, a numerical example is introduced to illustrate the stochastic program- ming formulation given by Eqs. (4.4.9) through (4.4.14) as well as the solution approach, PDSA, presented in Section 4.4.2. Hypothetical input data for a supply chain network con- sisting of n = 5 retailers and k = 3 possible demand scenarios is shown in tables 4.4.3 and 4.4.4. Observe that each scenario is assumed to be equally likely for this example. Also note that demands are interpreted as requests for additional emergency supplies caused by an observed storm?s presence. Therefore, a zero in Table 4.4.4 indicates that location i is not a ected by the storm under scenario t. Factor Notation Value Distance 1 2 3 4 5 Unit trans. cost (pre-storm) cv 2 0 8 9 5 7 11 Unit trans. cost (post-storm) cw 4 1 8 0 6 9 19 14 Unit holding cost h 4 2 9 6 0 6 12 15 Unit shortage cost s 5 3 5 9 6 0 6 5 Unit production/handling cost c 6 4 7 19 12 6 0 7 5 11 14 15 5 7 0 Table 4.4.3: Input data for numerical example. 76 Scenario Probability Retailer t Pt 1 2 3 4 5 1 1/3 15 150 200 0 0 2 1/3 0 150 200 50 0 3 1/3 0 0 200 50 90 Table 4.4.4: Example demand scenarios. Retailer 1 2 3 4 5 xi 0 150 200 50 0 Table 4.4.5: Optimal pre-positioning strategy for numerical example. Optimal Solution using Commercial Software The solution to the example problem that corresponds to the data shown in Table 4.4.3 and Table 4.4.4 was generated using the stochastic programming option of the com- mercial software program, What?s Besta151. According to Table 4.4.5, the optimal strategy pre-positions enough inventory to cover retailers 2, 3, and 4 for all three demand surge sce- narios. Consequently, the second-stage redistribution quantities among retailers 2, 3, and 4 are zero for all three scenarios. That is, the recourse decision variables are ylmt = 0 for l = ;2;3;4; m = 2;3;4; and t = 1;2;3. However, retailer 1 would experience an inventory shortage of 15 units if scenario t = 1 occurs (see tables 4.4.4 and 4.4.5) and would require a post-storm replenishment. Similarly, a shortage of 90 units would occur at retailer 5 if scenario t = 3 occurs, which would also warrant a post-storm replenishment. Thus the only nonzero recourse quantities in the optimal solution are y 11 = 15 and y 53 = 90. Note that the post-storm replenishments for retailers 1 and 5 under scenarios 1 and 3, respectively, would be direct shipments from the manufacturer. All other recourse variables ylmt for re- tailers l;m = 1;2;3;4;5, the manufacturer l = , and scenarios t = 1;2;3 are zero in the optimal solution. The optimal solution also leads to excess inventory under scenarios 1 and 3. Speci cally, retailer 4 will have 50 units of excess inventory if scenario 1 occurs and retailer 2 will have 150 units of excess inventory if scenario 3 occurs. Using our notational convention, we have 77 I+ =f4g and I =f1g if scenario t = 1 occurs, I+ = I =; if scenario t = 2 occurs, and I+ =f2g and I =f5g if scenario t = 3 occurs. The above information can be used to compute the expected cost of inventory overages and shortages. Recalling from Table 4.4.3 that the unit costs for overages and shortages are h = 4 and s = 5, respectively, and that each scenario is equally likely according to Table 4.4.4, the expected cost due to excess and shortage inventories is 1 3 [($5)(15) + ($4)(50)] + 1 3 0 + 1 3 [($4)(150) + ($5)(90)] = $441:67: (4.4.19) Using Eq. (4.4.7), the expected transportation cost associated with post-storm redistribution is Expected cost (transportation) = 13 cwd 1y 11 + 13 cwd 5y 53 = 13 ($4)(8)(15) + 13 ($4)(11)(90) = $1;480: (4.4.20) From Eq. (4.4.8), the expected production cost for direct shipments from the manufacturer during post-storm redistribution is Expected cost (production) = 13 cy 11 + 13 cy 53 = ($6)(15)3 + ($6)(90)3 = $210:(4.4.21) The sum of Eq. (4.4.19), Eq. (4.4.20), and Eq. (4.4.21) is the expected cost of the recourse function: E h Q ~x; ~D i = $441:76 + $1;480 + $210 = $2;131:67: (4.4.22) 78 From Eq. (4.4.5), Table 4.4.3 and Table 4.4.5, the actual cost of pre-storm pre-positioning can be computed as c(0 + 150 + 200 + 50 + 0) +cv 0 + (9)(150) + (5)(200) + (7)(50) + 0 = ($6)(400) + ($2)(2700) = $7;800: (4.4.23) The sum of Eq. (4.4.22) and Eq. (4.4.23) gives the expected cost of optimal pre-positioning: Expected cost (pre-positioning strategy) = $2;131:67 + $7;800 = $9;931:67: (4.4.24) The Expected Bene t of Pre-Positioning The solution to the two-stage stochastic programming model given by equations (4.4.9) through (4.4.14) in Section 4.4.1 and then illustrated via the numerical example presented in Section 4.4.3 answers the rst of two research questions introduced in Section 4.2. The focus of this section is the second of those two questions, which investigates the value of the proposed pre-positioning strategy relative to the reactive wait-and-see approach currently used in practice. First, an expression for the expected cost associated with the reactive wait- and-see approach is developed. Then the expected cost of the reactive strategy is calculated for the numerical example shown in Section 4.4.3. The expected bene t of pre-positioning is the di erence between these two expected costs, which will be positive if pre-positioning is bene cial and negative (or zero) otherwise. If no inventory has been pre-positioned prior to the observed storm, then all post-storm requests for emergency supplies will be direct shipments from the manufacturer to the af- fected retailers. Consequently, all shipments will incur both production and post-storm transportation costs. Also recall that the it?s are interpreted as additional demands neces- sarily induced by the storm, which suggests that it also represents the amount of inventory shortage at retailer i under scenario t whenever no inventory has been pre-positioned. The 79 expected cost of the wait-and-see strategy can then be expressed as Expected cost (prod, shortage, trans) = kX t=1 Pt " nX i=1 (c+s) it + nX i=1 cwdi it # : (4.4.25) From the example data in Table 4.4.3, we have c = 6, cw = 4, s = 5, and the distances di for i = 1;2;:::;5 are (8;9;5;7;11). The total demands P it for each scenario t = 1;2;3 can be obtained by summing across the rows of Table 4.4.4 to obtain (365;400;340). Incor- porating these values, the demands in Table 4.4.4, and Pt = 1=3 for each t into Eq. (4.4.25) leads to Expected cost (reactive) = $4;631:67 + $5;066:67 + $4;366:67 = $14;065:(4.4.26) The di erence between Eq. (4.4.26) and Eq. (4.4.24) yields the value of pre-positioning: Expected bene t of pre-positioning = $14;065 $9;931:67 = $4;133:33: (4.4.27) Since the quantity in Eq. (4.4.27) is greater than zero, it would be in the manufacturer?s best interest to implement the proposed pre-positioning strategy as opposed to the current reactive strategy. Speci cally, the manufacturer is $4;133:33 better o with pre-positioning relative to the wait-and-see approach. In general, the expected bene t of pre-positioning can be obtained by subtracting Eq. (4.4.9) from Eq. (4.4.25). PDSA Solution Based on the example data shown in tables 4.4.3 and 4.4.4, PDSA generated the op- timal solution shown in Table 4.4.5. Therefore, the expected cost and bene t will not be reproduced in this section. However, for illustration, PDSA calculations for determining the pre-positioning quantity for Retailer 2 will be summarized. First, the scenarios are parti- tioned as N2 = f1;2g and Z2 = f3g. Since each scenario occurs with probability 1/3 as 80 shown in Table 4.4.4, we have X t2Z2 hPt = 4 3X t=3 1 3 = 4 3 X t2N2 sPt = 5 2X t=1 Pt = 5 1 3 + 1 3 = 103 : The above calculations show that Case 2 of PDSA applies, which implies that the PDSA pre-positioning quantities are determined according to Equation (4.4.18). The minimum non-zero demand is min2 = 150 with tmin = 1 or 2. Without loss of generality, let tmin = 1 so that Nmin2 = f1g. Then Pt2Nmin 2 Pt = P1 = 1=3 and Pt2N2=Nmin 2 Pt = P2 = 1=3. Thus x2 = min2 = 150, which is consistent with the optimal solution. PDSA pre-positioning quantity calculations for the other retailers are similar. 4.5 Case Study This section examines the proposed pre-positioning strategy in a more realistic setting than the illustrative example presented in Section 4.4.3 by considering more representa- tive hurricane information, a larger number of scenarios, and a more extensive network of retailers. The scenario data is derived from the case study introduced in Rawls and Turn- quist [53], and 30 retailers are assumed to be situated according to the network depicted in Figure 4.5.1. Speci cally, a macro level view of the manufacturer?s pre-positioning problem is assumed where each city in the network represents a retailer location or collection of local retailers. In order to re ect the necessary input data for the stochastic programming model pre- sented in Section 4.4.1, the case study data in [53] is modi ed as follows. First, observe that the network shown in Figure 4.5.1 represents potential DC locations in [53]. However, this study interprets each node in the network as a retailer location, where exactly one of the nodes is assumed to be the location of the manufacturing facility. Next, recall that the 81 Figure 4.5.1: Source: Rawls and Turnquist [53]. Network in southeast United States used for case study. model in [53] accommodates multiple commodity types (i.e., water, food, and medical kits) whereas our model assumes a single commodity type. Therefore, this paper considers one of the product types (water) from [53] for the case study. Moreover, consider that the scenarios generated in [53] include demand locations and quantities as well as nodes and arcs that are destroyed by a hurricane or tropical storm. Since the pre-positioning model presented in this paper does not consider network reliability, all scenario information presented in [53] related to unusable nodes and arcs is ignored for the purposes of this study. Finally, the model in [53] does not include production costs or in ated post-storm transportation costs. Hence, this case study introduces hypothetical values for these costs to accompany the illustrative cost data described in [53]. A summary of the cost data used in this study is shown in Table 4.5.1. Note that one unit corresponds to 1,000 gallons of water, which explains the high cost of production and inventory shortages. Also, the post-storm transportation cost is assumed to be twice the pre-storm transportation cost. 82 Notation Description Value c Production cost a36323.85 / unit cv Pre-hurricane transportation cost a360.30 / mile cw Post-hurricane transportation cost a360.60 / mile h Holding cost for excess inventory a36161.93 / unit s Shortage cost a366,477 / unit Table 4.5.1: Cost data for case study. 4.5.1 Demand Scenarios The case study in [53] uses 15 historical hurricanes that have impacted the southeastern United States to create scenarios. For each of these 15 recorded hurricanes, aggregate de- mands for three commodities (water, food, and medical kits), a ected cities, and damaged roadways are documented in Table 3 of [53]. Since the stochastic programming model intro- duced in Section 4.4.1 is based on a single product type as mentioned previously, only the aggregate demands for water shown in Table 4.5.21 are considered for the purposes of this study. Rawls and Turnquist [53] generated 50 scenarios using the 15 hurricanes shown in Ta- ble 4.5.2 and combinations of these hurricanes. An additional scenario in which none of the nodes experience demand surge is also accounted for so that the total number of scenarios is actually 51. The probabilities associated with each scenario are \. . . based on approxi- mately matching aggregate historical characteristics of hurricanes in the region, but should be treated as simply illustrative values" [53]. For scenarios that represent a combination of two hurricanes, it is assumed that the hurricanes strike within a short time frame. Sce- narios based on one hurricane have a combined probability of 75% and the scenarios with a combination of two hurricanes account for the other 25%. Table 4 in [53] lists the ag- gregate demands for each commodity for each of the 51 scenarios along with the associated probabilities. 1A \-" in the landfall node column of Table 4.5.2 indicates no actual landfall for the observed hurricane. However, the referenced storm still managed to induce demand surge. 83 Hurricane Category Landfall Node Water Demand (1000 gallons) 1 3 5 350 2 5 14 560 3 2 22 861 4 2 22 9000 5 4 11,29 7500 6 3 15 1000 7 2 21 600 8 1 11 1500 9 5 13,29 1040 10 2 - 2250 11 3 21 5000 12 3 - 18000 13 3 - 2818 14 4 14,30 2239 15 4 22 4400 Table 4.5.2: Demand information for each sample hurricane, adapted from [53]. 4.5.2 Case Study Results The optimal pre-positioning strategy for the network shown in Figure 4.5.1, the cost data shown in Table 4.5.1, and the 51 demand scenarios derived in [53] was determined using the stochastic programming model presented in Section 4.4.1 and the stochastic program- ming solver of the commercial software package What?s Besta151. In addition, the PDSA was applied to the pre-positioning problem and compared to the optimal solution. For illustrative purposes, it was assumed that the location of the manufacturing facility was Birmingham Alabama. Manufacturer Location in Birmingham, Alabama Within the 51 scenarios, demand for water was reported at 26 of the 30 nodes shown in Figure 4.5.1. The optimal strategy pre-positioned inventories at all but one of these nodes (Brownsville, Texas). This is likely explained by the large distance between Birmingham and Brownsville relative to the other cities in the network. In addition, the expected demand for water in Brownsville across all scenarios is comparatively small. On the other hand, the 84 PDSA pre-positioned items at every location. The exorbitant shortage cost caused the PDSA to classify all of the locations as Case 2, meaning all of the locations will hold inventory. In some cases, pre-positioned quantities were su cient in terms of satisfying storm de- mands (more detailed results related to service level are discussed in Section 4.5.2) when the optimal solution was applied. However, most scenarios required post-storm transshipments. For example, the scenario that represents Hurricane Emily created demand for water in Wilmington NC that exceeded the pre-positioned quantity. Consequently, transshipments from Charlotte NC, Charleston SC, and Columbia SC were necessary to ful ll the unmet demand. As another example, consider Hurricane Fran which generated a demand surge for water in North Carolina (Charlotte and Wilmington), South Carolina (Charleston and Columbia), and Savannah GA. For this scenario, shortages materialized in Charleston and Savannah based on the optimal pre-positioning strategy. Transshipments from Wilmington NC and Orlando FL were used to replenish inventory in Charleston while the unful lled demand in Savannah was replenished with a transshipment from Orlando. Similar results were found when the PDSA was applied to the problem, but about 1,000 less items were transshipped considering all 51 scenarios. This can be explained by the larger quantity of to- tal pre-positioned items computed by the PDSA relative to the optimal solution. The PDSA focuses on the holding and shortage costs without regard to the cost of transportation, which most likely explains the di erence between the amount pre-positioned and transshipped. In Section 4.5.3, the PDSA solution is compared to the optimal solution based on expected costs. Expected Bene t of Pre-positioning The bene t of pre-positioning relative to the reactive wait-and-see approach currently used in practice was quite dramatic. The expected cost of the reactive policy with no pre- positioning was a3628.0 million and the expected cost of the pre-positioning optimal strategy was a3616.5 million while the expected cost of the PDSA solution was a3617.1 million. Thus the 85 corresponding expected bene t was approximately a3611.5 million and a3610.9 million, which was the average bene t of the optimal pre-positioning and PDSA strategies over all 51 scenarios. However, the wait-and-see approach was actually more cost e ective for about half of the scenarios considering the optimal solution. The scenarios that seemed conducive to the wait- and-see approach were characterized by a small number of retailers with nonzero demand. At the other extreme, the cost increase for not pre-positioning when a major storm impacts many large cities was astronomical, especially for the scenarios that represent two landfall hurricanes within a short period of time. In some cases, the bene t was more than a3650 million. When the PDSA approach is compared to the wait-and-see approach, it is more cost e ective for approximately 88% of the 51 scenarios and the associated probabilities. This can be attributed to lower shortage costs incurred by the PDSA approach. In comparison to the optimal solution, the maximum bene t was a3630 million less (a3620 million). The decrease in expected bene t is associated with the additional holding costs incurred by the PDSA. In addition, smaller quantities were pre-positioned at high impact retailers, in particular, Charleston and Columbia. Since the expected bene ts of pre-positioning for both strategies are positive for the case data, the pre-positioning strategy is recommended in lieu of the wait-and-see strategy if the manufacturer is located in Birmingham AL. Other Manufacturer Locations The expected bene t of pre-positioning also turned out to be positive if the manufacturer was situated in a city other than Birmingham. This claim is based on solving nine variations of the case problem in which the manufacturer location was a di erent city for each problem instance. The PDSA approach pre-positioned the same amount of items regardless of manufac- turer location because shortage and holding costs are only considered to determine the amounts, so it will not be discussed in this section. However, it should be mentioned that 86 shortages were met by overages at other retailers rst, and then, the manufacturer. This nding was consistent with all manufacturer locations. Pre-positioning and transshipment policies were quite di erent depending on the man- ufacturer?s location in the network for the optimal solution. For example, a manufacturer located in San Antonio TX, which is far removed from the majority of the other nodes in the network, would pre-position small quantities. Consequently, the majority of shortages replenished during the second stage would be direct shipments from the manufacturer as op- posed to transshipments from other retailers. At the other extreme, a manufacturer located in Tallahassee FL would pre-position larger quantities with few post-storm direct shipments. These somewhat counterintuitive ndings can be explained as follows. If the manufacturer is in close proximity to high impact cities, then the de nitive upfront cost of pre-positioning is low in comparison to the expected shortage and redistribution costs. On the other hand, a manufacturer far removed from high impact cities would be subjected to a prohibitive and certain pre-positioning expense compared to an excessive but uncertain shortage and direct shipment expense. In summary, the location of the manufacturer does not a ect the deci- sion of whether or not a pre-positioning strategy should be used, but it does a ect decisions related to pre-storm pre-positioning quantities and post-storm transshipment policies. Service Level In addition to expected costs and bene ts, service level is also an important performance metric for inventory systems, particularly within the context of disaster relief activities. In this paper, service level refers to the percentage of demand that is lled on time from pre- positioned quantities. Shortages replenished during post-storm transshipments from other retailers or direct shipments from the manufacturer are considered late, and consequently incur shortage penalty costs. This section reports the service level performance associated with our numerical experimentation. 87 The optimal pre-positioning strategies associated with 10 variations of the case problem, each of which corresponds to a distinct manufacturer location, induced service levels of at least 80%. Speci cally, for any given scenario, less than 20% of retailers required post storm replenishment even when other manufacturer locations were considered. This result can most likely be attributed to the exorbitant shortage costs shown in Table 4.5.1. The PDSA solution approach produced a slightly higher service level of 83% compared to the optimal solution. However, at most 35% of locations required post storm shipments for a given scenario, which can be explained by the smaller pre-positioned amounts at high demand retailer locations in Charleston and Columbia. The proposed pre-positioning strategy also represents an improvement in service level compared to the wait-and-see strategy. This observation is a direct consequence of interpret- ing retailer demands as the additional inventory requests due to the presence of a threatening storm (see the beginning of Section 4.4.3). From this perspective, the wait-and-see service level is 0%. Therefore, any pre-positioning e ort would be an improvement. 4.5.3 PDSA Results On average, PDSA (Section 4.4.2) produced results that were within 3.6% of the optimal solution as shown in Table 4.5.3. The maximum deviation was less than 5% (Charleston SC) and the minimum was just under 3% (Mobile AL). The performance of PDSA appears to be in uenced by the location of the manufacturer relative to high impact retailers. In particular, smaller deviations from optimality seemed to occur whenever the manufacturer was situated in close proximity to high impact retailers (e.g., manufacturer in Mobile AL). On the other hand, di erences between optimal and PDSA solutions were slightly more pronounced for larger distances between the manufacturer and high impact retailers (e.g., manufacturer in Charleston SC). Deviations between the PDSA and optimal solutions were most likely due to the fact that the rst stage of PDSA does not account for transportation costs. Thus, the performance of PDSA will deteriorate as the manufacturer?s distance from other retailers 88 increases. Computation times for PDSA were essentially negligible. The methodology for determining PDSA pre-positioning quantities outlined in Section 4.4.2 was instantaneous using a spreadsheet. Similarly, the resulting transportation problems associated with post- storm redistribution were solved instantaneously using Excel Solvera151. Computation times for obtaining optimal solutions are discussed in the next section. Manufacturer Location % Increase in Expected Cost Atlanta 3.71% Baton Rouge 3.28% Birmingham 3.31% Charleston 4.71% Jackson 3.20% Little Rock 3.41% Mobile 2.97% Nashville 3.62% San Antonio 4.39% Tallahassee 3.53% AVERAGE 3.61% Table 4.5.3: PDSA performance relative to the optimal solution for each problem instance. 4.5.4 Sensitivity Analysis This section further investigates Question 2 (see Section 4.2) by examining the e ects that various cost parameters have on the expected bene t of pre-positioning determined by the optimal solution and the PDSA. The solution approaches were also compared in terms of the expected bene t with respect to the wait-and-see strategy. Regarding the optimal solution, the commercial software What?s Besta151 generated glob- ally optimal solutions within 2 to 15 minutes for a majority of the computational experi- ments. However, for some problem instances in which shortage cost was varied, the solver was manually interrupted before nding an optimal solution after several hours of compu- tation, sometimes as many as 30 hours. In these cases, What?s Besta151 managed to nd a feasible solution, which we used as a proxy for the optimal solution in our sensitivity analysis experiments. On the other hand, the PDSA was able to provide a solution for the instances 89 that resulted in a prolonged computation time. All results presented in this section assume that the manufacturer location is Birmingham, Alabama. The cost parameters that were re ected in the sensitivity analysis include (i) the unit cost of production, (ii) the post-storm unit cost of transportation, and (iii) the unit cost for inventory shortages (the notations for these parameters are c;cw, and s, respectively). Two of the cost parameters actually had no e ect on ECWS. In particular, ECWS was not in uenced by changes in the pre-storm transportation cost cv since there is no pre-storm transportation process associated with the wait-and-see approach. Additionally, the wait- and-see strategy does not produce any excess pre-positioned inventory after landfall, which suggests that h also does not a ect ECWS. Therefore, cv and h will not be included in the sensitivity analysis experiments. The forthcoming analyses will examine the following measures with respect to changes in c, cw, and s: (i) the expected cost of the optimal pre- positioning solution, ECPP; (ii) the expected cost of the wait-and-see strategy, ECWS; (iii) the expected cost of the PDSA solution, ECPD; (iv) the expected bene t of pre-positioning, E(Bene t), for both solution approaches; and (v) the percentage increase in expected cost associated with the wait-and-see solution relative to the optimal pre-positioning and PDSA solutions. Observation 1 E(Bene t) decreases as c increases. Table 4.5.4 and Figure 4.5.2 illustrate the e ects of increasing production costs. In Figure 4.5.2, ECWS and dECPP increase linearly while EC PP appears to be a concave function. However, as c increases, the slopes of EC PP and dECPP are greater than the slope of ECWS, which results in a decrease in E(Bene t) for both strategies. Table 4.5.4 also shows that the expected bene t of pre-positioning for the optimal solution (E PP(Bene t)) is extraordinarily signi cant for smaller production costs: over a3613 million if the nominal production cost is cut in half and over a3615 million if production is free, which represent 94% and 148% increases, respectively. The PDSA produces an expected bene t of a3612.5 million if the production cost is cut in half and a3614 million if the production is free, representing an increase of 85% and 90 116%, respectively. It should also be noted that as c increases, the performance of the PDSA improves until c is equal to the hypothetical production value in the Case Study. The ndings associated with Observation 1 are a result of the fact that the manufac- turer?s total production quantity associated with pre-positioning is usually larger than the production quantity associated with wait-and-see. In particular, the production quantity for pre-positioning is determined based on probabilistic demand information, which often leads to the production of additional inventory for the purpose of avoiding expensive short- age costs. On the other hand, the wait-and-see production quantity corresponds to exact demand requirements and zero safety stock. Thus, as c increases, EC PP and dECPP increase faster than ECWS because of the corresponding production quantities. c E PP(Bene t) bEPP(Bene t) Di in E(Bene t) % Cost Inc. Opt. % Cost Inc. PDSA a360.00 a3615,962,733 a3614,345,336 a361,617,397 148.31% 115.87% a36161.93 a3613,254,631 a3612,595,650 a36658,981 93.92% 85.27% a36323.85 a3611,438,729 a3610,890,449 a36548,280 69.04% 63.62% a36485.78 a369,941,756 a369,190,852 a36750,904 53.15% 47.24% a36647.70 a368,713,214 a367,491,403 a361,221,812 42.35% 34.37% Table 4.5.4: The e ect of the unit production cost, c. Observation 2 E(Bene t) increases as cw increases. The bene t of pre-positioning increases as post-storm transportation costs increase. In this case, it becomes more cost e ective to avoid post-storm transportation as cw increases, which can be accomplished by more aggressive pre-positioning. Table 4.5.5 and Figure 4.5.3 also show that the e ect of cw is not as dramatic as the e ect of c. In particular, E(Bene t) and the corresponding percentage cost increases do not uctuate as much for changing values of cw as they do for changing values of c. When the optimal solution and PDSA solution are compared, it is evident that the PDSA performance remains consistent in relation to the optimal solution. The PDSA is based solely on the expected unit shortage and expected unit holding costs, so its performance should not be greatly a ected by changes in cw. 91 $0 $5 $10 $15 $20 $25 $30 $35 $0 $200 $400 $600 $800 Total Expe cted Cost (milli ons) Production Cost (c) Pre-PositionNo Pre-Position PDSA Figure 4.5.2: The e ect of the unit production cost, c. cw E PP(Bene t) bEPP(Bene t) Di in E(Bene t) % Cost Inc. Opt. % Cost Inc. PDSA a360.30 a3611,031,237 a3610,398,386 a36632,850 67.15% 60.95% a360.45 a3611,234,743 a3610,604,436 a36630,306 68.10% 61.91% a360.60 a3611,438,729 a3610,810,487 a36628,242 69.04% 62.86% a360.75 a3611,643,005 a3611,016,539 a36626,466 69.97% 63.81% a360.90 a3611,848,971 a3611,222,592 a36626,379 70.92% 64.74% a361.05 a3612,059,257 a3611,428,636 a36630,621 71.90% 65.67% Table 4.5.5: The e ect of the post-hurricane transportation cost, cw. Observation 3 E PP(Bene t) increases as s increases, and is most sensitive to s. Pre-positioning reduces the risk of shortages relative to wait-and-see. In fact, all demands incur shortage penalty costs based on the wait-and-see policy. In Figure 4.5.4, it can be observed that ECWS increases linearly as s increases and that EC PP increases as s increases. However, the rates of change in the increase of EC PP are decreasing (i.e., EC PP appears concave in s) so that the distance between ECWS and the optimal pre-positioning strategy increases in s. 92 $0 $5 $10 $15 $20 $25 $30 $35 $0.00$0.20$0.40$0.60$0.80$1.00$1.20 Total Expe cted Cost (mill ions) Transportation Cost (cw) Pre-PositionNo Pre-Position PDSA Figure 4.5.3: The e ect of the post-hurricane transportation cost, cw. Table 4.5.6 and Figure 4.5.4 also show that the bene t of pre-positioning is insigni cant for small values of s relative to the other values of s and the other cost parameters. In particular, the smallest value of s in Table 4.5.6 represents the minimum expected bene t for the optimal strategy (just over a361.2 million) and corresponding cost increase (14.39%) for all cost parameters and problem instances considered in the case study. On the other hand, E(Bene t) and the percentage cost increase rapidly as s increases, which is likely a result of extremely large shortage cost values relative to the other cost parameters. Observation 4 The performance of PDSA deteriorates as s decreases. When the PDSA is compared to the wait-and-see approach, the bEPP(Benefit) is negative for small values of the shortage cost as shown in Table 4.5.6. The large amount of pre- positioned items cause expensive holding costs resulting in dECPP >ECWS for these values. However, when the shortage cost is increased, the PDSA provides a solution relatively close 93 to the optimal as shown in Table 4.5.6. It should be noted that the optimal solution should be employed when the value of s is similar to the purchase price. The PDSA provides a negative expected bene t for these smaller values of s leading the decision maker to choose the wait-and-see approach, which is contradictory to the optimal solution. As mentioned at the beginning of this section, What?s Besta151 was unable to obtain glob- ally optimal solutions for sensitivity analysis experiments that involved varying the shortage cost, s. These problem instances resulted in prohibitive computation times of up to 30 hours for reasons that are not apparent. For these problem instances, the solver was manually interrupted and a feasible solution was used for the purposes of this analysis. Speci cally, a feasible solution was used as a proxy for the optimal solution in Table 4.5.6 and Figure 4.5.4 whenever s was 5 times c or less. If the actual optimal values for EC PP were available, then a more dramatic increase in E PP(Bene t) and departure of the curves shown in Figure 4.5.4 would be observed. On the other hand, the PDSA was able to provide solutions for these problem instances, and it should be noted that in practice, this is the advantage of using PDSA instead of commercial software. s E PP(Bene t) bEPP(Bene t) Di in E(Bene t) % Cost Inc. Opt. % Cost Inc. PDSA a36323.85 a3673,069 -a365,854,090 a365,927,159 2.04% -61.52% a36647.70 a36208,743 -a364,977,007 a365,185,750 4.41% -50.17% a361,295.40 a36656,918 -a363,222,841 a363,879,759 9.59% -30.04% a361,943.10 a361,266,876 -a361,468,674 a362,735,551 14.39% -12.73% a362,590.80 a362,052,275 a36285,492 a361,766,784 19.40% 2.31% a363,238.50 a363,309,799 a362,039,658 a361,270,141 27.85% 15.51% a363,886.20 a364,723,495 a363,793,824 a36929,671 36.24% 27.17% a364,533.90 a366,325,856 a365,547,990 a36777,866 45.21% 37.56% a365,181.60 a367,947,087 a367,302,156 a36644,931 53.21% 46.87% a365,829.30 a369,669,318 a369,056,322 a36612,996 61.29% 55.26% a366,477.00 a3611,438,731 a3610,810,488 a36628,242 69.04% 62.86% Table 4.5.6: The e ect of the unit shortage cost, s. 94 $0 $5 $10 $15 $20 $25 $30 $0.00 $2.00 $4.00 $6.00 $8.00 Total Expe cted Cost (milli ons) Shortage Cost (thousands) Pre-PositionNo Pre-Position PDSA Figure 4.5.4: The e ect of the unit shortage cost, s. Observation 5 For all of the problem instances solved optimally, the expected bene t of pre-positioning relative to the wait-and-see approach was quite signi cant. For the optimal solution, the smallest expected bene t and corresponding percentage cost increase were a3673,069 and 2.04%, respectively, which corresponds to the problem in- stance where the shortage cost and production cost were equal. On the other hand, the largest bene t and percentage increase (excluding the case when production was free) were a3613.25 million and 93.92%, respectively. The largest bene t occurred for the problem in- stance with the smallest production cost, c. The PDSA produced a negative bene t when the shortage cost was small, but it performed well for all other problem instances with the largest bene t, a3612.5 million, occurring when the production cost was half of the tested pro- duction cost. Overall, this study shows that the optimal pre-positioning always outperforms the wait-and-see approach currently used in practice whenever probabilistic information re- garding landfall of an observed hurricane is available. 95 4.6 Summary and Future Directions This paper is motivated by the problem that emerges when a commercial manufacturer responds to orders for emergency supply items placed by retailers situated in the path of an observed hurricane or tropical storm. In practice, retailers often postpone the decision to initiate emergency orders from manufacturers until the demand requirements associated with pre-storm demand surge or post-storm disaster relief activities are known with cer- tainty. A typical consequence of this reactive \wait-and-see" policy is widespread stockout of emergency supplies and equipment, which compromises the e ectiveness of disaster relief operations and proliferates lost sales for the commercial supply chain. In an e ort to en- courage higher service levels from the private sector in support of disaster relief e orts, this paper proposes a proactive inventory pre-positioning strategy from the perspective of the manufacturer. The proposed pre-positioning strategy is characterized by a single manufac- turer that pushes inventory across a network of retailers prior to the landfall of an observed storm, and assumes responsibility for the resulting costs and risks similar to vendor man- aged inventory systems. The manufacturer?s pre-positioning problem is represented as a two-stage stochastic programming model. During the rst stage, supplies are distributed from the manufacturer to selected retailer locations within the supply chain network. The corresponding second stage is characterized by transshipment of inventories among retailers, including possible direct shipments from the manufacturer, after demand information be- comes known with certainty. To illustrate the model, the case study introduced in reference [53] which is derived from historical hurricane data, was solved. Solutions were generated using the stochastic programming capability of the commercial software program, What?s Besta151. In addition, a heuristic solution methodology (PDSA) was introduced for the pur- pose of reducing the computational e ort associated with solving large-scale problems. The proposed approach produced solutions within 3% - 5% of optimality and was executed with negligible computation time in a spreadsheet. Computational results and sensitivity analysis based on the case study suggest that the optimal pre-positioning strategy is always more 96 cost e ective than the wait-and-see approach often used in practice whenever the landfall locations and demand requirements associated with an observed storm are uncertain. The stochastic programming model presented in this study can be extended in several ways to re ect additional issues that may arise when commercial manufacturers attempt pre-positioning strategies in practice. In fact, some of these issues have been addressed in the pre-positioning literature based on the perspectives of government and non-government organizations and humanitarian relief supply chains. For instance, network reliability in which pre-positioned inventories and roadways are partially or completely destroyed could be considered as in [53], [12], and [6]. Also, it would be useful from a practical standpoint to expand the model to include multiple product types as in [14], [38], [53], [5], and [6]. There are also extensions which have not been considered in the literature that would be particularly relevant to the commercial pre-positioning problem introduced in this study. Before describing these extensions, rst recall that one of the unique features of the pre- positioning model developed in Section 4.4.1 is that it represents the manufacturer?s per- spective and cost structure. However, a limitation of the model from a production standpoint is that in nite production capacity is assumed and that only the variable cost of production is considered. It would therefore be of interest to expand the model to include more details regarding the manufacturer?s production process and cost measures. For example, a nite production rate could be incorporated into the model as opposed to assuming that produc- tion is instantaneous. Similarly, the model could also be extended such that production rate is a decision variable and the corresponding production cost is a function of the selected rate. Another useful extension would be a multiple product model as previously mentioned. This would introduce opportunities for scheduling production to reduce the costs associated with changeovers. Di erentiating pre- and post-storm production costs might also be relevant to the proposed pre-positioning strategy in practice. In a post-storm environment, constrained resources may lead to overtime labor and other in ated costs pertaining to production. In 97 terms of distribution, only the variable cost of transportation is considered. Therefore, it would be interesting to examine the e ect of xed transportation costs in a future e ort. The model introduced in this paper characterizes the role of the manufacturer in a short-term pre-positioning environment. This basic model, along with the above-mentioned extensions related to the manufacturer?s problem within the context of pre-positioning, rep- resents a promising new direction for disaster relief research. 98 Chapter 5 Summary In closing, the proposed research exhibits the potential to provide meaningful strategies that can be applied to the private sector with an emphasis on the retail industry when faced with an uncertain disruption such as a hurricane. This dissertation stresses the importance of considering the e ects of disasters when retailers make decisions regarding how they will approach inventory management for emergency supplies. The private sector can bene t from employing the strategies discussed in this dissertation by providing better customer service and minimizing costs. Additionally, the strategies presented can assist residents by ensuring that they are able to obtain the necessary supplies before and after the storm. 5.1 Minimax EOQ Policies with Demand Surge, Lead-Time and Lost Sales The model presented in Chapter 2 provides a solution for how retailers can determine an ordering strategy when faced with an uncertain demand surge or demand disruption. Retailers may select a reactive strategy meaning they will maintain their order quantity until it is certain that the surge will occur, or they may select a proactive strategy, which means they will modify the order quantity before it is certain that the surge will occur. A closed form solution was found for the optimal order quantity during the disruption. Minimax decision criterion was used to determine the best ordering strategy considering a 28 experimental design. In the numerical experiment, most of the tested cases constituted selecting the proactive strategy, which is contrary to the approach employed by most retailers when faced with a forecasted storm. On the other hand, minimax decision criterion supported the reactive strategy despite lost sales when lead-time was short. The sensitivity analysis revealed that the lead-time has a greater e ect on the strategy selection than the lost sales 99 cost. All of these ndings illustrate that the retailer must strongly consider the lead-time and the objectives (i.e., minimizing maximum costs) when selecting an inventory ordering strategy for handling uncertain disruptions. 5.2 Minimax and Minimax Regret Inventory Control Policies Regarding De- mand Disruptions and Damaged Inventory The model presented in Chapter 3 is an extension of the model presented in Chapter 2 by considering a di erent time horizon with two disruptions and two sources of uncertainty. The rst disruption represents a possible period of no demand caused by a storm while the second disruption symbolizes the surge in demand for emergency supplies after the storm. The rst source of uncertainty is whether the disruption, a period of no demand, will occur while the second source is whether the inventory will be damaged if the rst disruption occurs. Minimax and minimax regret decision criteria were applied to the model to determine the quantity that the retailer should hold during the rst and second disruptions. Minimax decision criterion did not support holding inventory during the rst disruption while minimax regret advocated holding inventory during 55% of the tested cases. However, the order quantity for the second disruption was the same for both decision criteria. All in all, the retailer should consider the decision criterion that best suits the cost objectives when selecting a strategy for the time horizon during and after the storm. 5.3 Pre-Positioning Hurricane Supplies in a Commercial Supply Chain Pre-positioning has not been considered in the literature from a pro t-driven perspec- tive, and Chapter 4 provides a model that minimizes the costs of the manufacturer when a hurricane has been observed. A stochastic programming model was developed to determine the quantity of emergency supplies that should be pre-positioned at each retailer as well as the necessary transshipments and direct shipments from the manufacturer after the storm to satisfy shortages. In addition, a solution methodology termed PDSA reduces the stochastic 100 programming model to the transportation problem by employing a heuristic to solve the rst stage decisions (i.e., pre-positioned quantities). The commercial software package What?s Besta151 was utilized to solve the stochastic programming model, and the optimal solution revealed that it was best to pre-position supplies at most retailers regardless of the man- ufacturer?s location. The manufacturer?s location did a ect the amount pre-positioned at each retailer meaning the retailers located closest to the manufacturer generally held larger amounts of pre-positioned quantities. In most cases, at least 80% of the retailers? demand for emergency items was met by the pre-positioned amounts illustrating that the retailers did not rely solely on post-storm shipments and the pre-positioning strategy is more cost e ective considering the numerical example and case study data from ([53]). The sensitivity analysis indicated that the lost sales cost had the greatest e ect on the decision to pre-position. 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D T 2 Time Inventory Level T 1 +L t r } Lost Sales Or der of q R plac ed A 1 A 3 A 2 A 3 A 3 A 3 The number of orders given in Chapter 2 are m1 orders of qE and m2 orders of qR, which can be found by dividing the total demand during the time period by the order quantity. The m1 orders of qE are found using A1 and A2 while the m2 orders encompass the m2 orders of qR. The total number of orders is given in Eq. A.1.1. m1 +m2 = T1 + D(t1R T1)q E + D(T2 T1 L)q R (A.1.1) The holding cost is determined by computing the average inventory level for A1, A2 and A3. The average inventory level for A1 and A2 is found by Eq. (A.1.2) where the number of orders of qE is multiplied by the area of A1 and A2. I1 = T 1 qE q E2 + D(t1R T1) qE q E2 D = qEt1R2 (A.1.2) 107 The average inventory level is computed in Eq. (A.1.3), which entails multiplying m2 by the area for A3. I2 = m2 (Area of A3) = D(T2 T1 L)q R q R2 D = qR(T2 T1 L)2 (A.1.3) The total number of lost sales is found by multiplying the demand rate by the time horizon when the inventory level is zero. In the case considered here, Eq. (A.1.4) gives the total number of lost sales (L). L = D(T1 +L t1R) (A.1.4) The cost equation presented in Eq. (2.3.3) is the sum of Eqs. (A.1.1)-(A.1.4) with each part multiplied by its respective cost. A.2 Reactive Strategy Case 1 Consider the following assumptions: T2 T1 +L, T1 qE L, L qE T1 D , and s qR D . TC1R1(qE;qR) TC0R(qE) 0 A T 1 qE + D(t1R T1) qE + D(T2 T1 L) qR +h q Et1R 2 + qR(T2 T1 L) 2 +s D (T1 +L t1R) A T 2 qE h T 2qE 2 0 Substitute t1R = qE T1 D +T1, qE = q 2A h , qR = q 2A D h and simplify. A 1 + D + p 2hA D(T2 T1 L) T2 p 2hA +T1 r hA 2 1 D +s D 0 @L q 2A h T1 D 1 A 0 108 Substitute T2 = T1 +L and L = p2A h T1 D . A 1 + D p 2hA 0 @T1 + q 2A h T1 D 1 A+T1 r hA 2 1 D 0 A 1 + D T1 p 2hA 1 D 2A D +T1 r hA 2 1 D 0 A 1 D T1 r hA 2 1 D 0 A T1 r hA 2 r 2A h T1 qE T1 We can say that TC1R1(qE;qR) TC0R(qE) 0 because T1 qE . Case 2 Consider the following assumptions: T2 t1R, T1 qE , and t1R = T1 + qE T1 D TC1R2(qE;qR) TC0R(qE) 0 A T 1 qE + D(t1R T1) qE + D(T2 t1R) qR +h q Et1R 2 + qR(T2 t1R) 2 A T 2 qE h T 2qE 2 0 109 Substitute t1R = qE T1 D +T1, qE = q 2A h , qR = q 2A D h and simplify. A 1 + D 2 r D ! +T2 p 2hA D +T1 r hA h 1 D T1 p 2hA D 1 D 0 Substitute T2 = T1 + qE T1 D and qE = q 2A h . A 1 + D 2 r D ! +T1 p 2hA D 1 D T1 p 2hA 1 D +2A r D ! 2A D T1 p 2hA D 1 D +T1 p 2hA 1 D 0 A 1 D T1 p 2hA r hA 2 ! 1 D 0 Substitute T1 = qE with qE = q 2A h . A 1 D r 2A h p 2hA r hA 2 ! 1 D 0 A 1 D A 1 D = 0 Since we substituted the smallest value for T2 and largest value for T1, the inequality should be 0 thus proving that TC1R2(qE;qR) TC0R(qE) 0. Case 3 Consider the following assumptions: T1 +L qE D + qE , T2 T1 +L. TC1R3(qE;qR) TC0R(qE) 0 110 A T 1 qE + D(t1R T1) qE + 1 + D(T2 T1 L) qR +h q Et1R 2 + qE2 2 D + qR(T2 T1 L) 2 +s D T1 +L qE D t1R A T 2 qE h T 2qE 2 0 Substitute t1R = qE T1 D +T1, qE = q 2A h , qR = q 2A D h and simplify. 2A 1 + D + p 2hA D(T2 T1 L) T2 p 2hA +T1 r hA 2 1 D +s D L 2qE T1 D 0 Substitute T2 = T1 +L, T1 = qE + qE D L, and M = L 2qE T1 D and simplify. 2A 1 + D q E + qE D p 2hA + q E + qE D L rhA 2 1 D +s DM 0 2A 1 + D 2A 1 + D + A D +A L r hA 2 ! 1 D +s DM 0 A D +A L r hA 2 ! 1 D +s DM 0: This inequality holds because A D +A L q hA 2 0 and M 0. Case 4 Consider the following assumptions: T2 qE D + qE and qE T1 L qE D + qE T1. TC1R4(qE;qR) TC0R(qE) 0 111 A T1 + D(t1R T1) qE + 1 + D(T2 qE D + qE ) qR ! +h qEt1R 2 + qE2 2 D + qR(qE D + qE ) 2 ! +s D qE t1R A T 2 qE h T 2qE 2 0 Substitute t1R = qE T1 D +T1, qE = q 2A h , qR = q 2A D h and simplify. 2A 1 + D T2 p 2hA + p 2hA D T2 s 2A h D2 r 2A h ! +T1 r hA 2 1 D +s D r 2A h T1 1 D s 2A h D2 ! 0 Substitute T2 = qE D + qE . 2A 1 + D p 2hA s 2A h D2 + r 2A h ! + p 2hA D s 2A h D2 + r 2A h s 2A h D2 r 2A h ! +T1 r hA 2 1 D +s D r 2A h T1 1 D s 2A h D2 ! 0 2A 1 + D p 2hA s 2A h D2 + r 2A h ! +T1 r hA 2 1 D +s D r 2A h T1 1 D s 2A h D2 ! 0 2A 1 + D 2A 1 + D +T1 r hA 2 1 D +s D r 2A h T1 1 D s 2A h D2 ! 0 T1 r hA 2 1 D +s D r 2A h T1 1 D s 2A h D2 ! 0 112 We know that the lost sales are positive, so q 2A h T1 1 D q 2A h D2 0. Also, T1 q hA 2 1 D 0. A.3 Proactive Strategy Case 1 TC1P(qP) TC0P1(qP;qE) 0 A D T qP +h T 2qP 2 A 2 + T2 2qPq E h qP2 + qE T2 2qP 2 ! 0 Substitute qE = q 2A h and qP = q 2ADT hT2 and simplify. 2A 2 rD T T2 1 DT T2 ! + p 2hA p DTT2 T2 p 0 p 2hA p DTT2 T2 p 2A r DT T2 1 !2 0 p 2hA p DTT2 T2 p 2A r DT T2 1 !2 T2 p 2hA r DT T2 1 ! 2A r DT T2 1 !2 T2 r h 2A r DT T2 1 ! To simplify the inequality, change the left side of the inequality to q DT T2 because it is greater than q DT T2 1 . T2 r hT 2 2ADT 1 T2 1 qP 1 T2 qP 113 We know that T2 qP because the assumption for this case is that T2 2qP . Case 2 and Case 4 TC1P(qP) TC0P2(qP;qE) 0 A D T qP +h q PT2 2 A T 2 qP h q PT2 2 0 A D qP (T2 T1) A qP (T2 T1) 0 D 0 Because we know that D , TC1P(qP) TC0P2(qP;qE). The same is true for Case 4. Case 3 TC1P(qP) TC0P3(qP;qE) 0 The proof for Case 3 is very similar to the proof for Case 1 with the exception that only one order of qP is placed. The inequality reduces to the following: p 2hA p DTT2 T2 p A r DT T2 1 !2 0 T2 p 2hA r DT T2 1 ! A r DT T2 1 !2 T2 r 2h A r DT T2 1 ! 114 To simplify the inequality, change the left side of the inequality to q DT T2 because it is greater than q DT T2 1 . T2 r2hT 2 ADT 1 T2 2 qP 1 T2 qP2 We know that T2 qP2 because the assumption for this case is that T2 qP . 115 Appendix B Chapter 3 Consider the cost equation where q1 = qL, q2 = qS, ~D = (d1;0), and ~S = (H;0). TC(d1;0)(H;0) (qL;qS) = A 0 @1 + S T4 T3 qL S qS 1 A+hqLT3 + hqL22 S + hqS T4 T3 qL S 2 Consider the cost equation where q1 = q2 = q , ~D = (d1;0), ~S = (H;0), and q SL. TC(d1;0)(H;0) (q ;q ) = A 0 @1 + S T4 T3 q S q 1 A+hq T3 + hq 22 S + hq T4 T3 q S 2 Now, we will show the following: TC(d1;0)(H;0) (q ;qS) TC(d1;0)(H;0) (q ;q ) For the proof, we will substitute q = qL, which is the smallest value of q . If we simplify the inequality using the cost equations, we have: A S qS + hqS 2 A S qL + hqL 2 h 2 (qS qL) A S 1 qL 1 qS h 2 (qS qL) A S q S qL qLqS h 2 A S qLqS qLqS 2A Sh qLqS qS2 116 We know this to be true because qS qL, and if this is true for the smallest value, then it will be true for all values of q because q qS and q qL. 117