00c_inventory_recap_2020_annotated.pdf

# Introduction to inventory and its purpose Inventory is a crucial component of supply chain management, representing the stock of goods that a company holds for various purposes, including meeting customer demand, production processes, and strategic positioning within the supply chain. It encompasses raw materials, work-in-progress, and finished goods that are held by a firm at a given time. The management of inventory is not inherently "bad"; rather, it involves strategic decisions to balance the costs associated with holding inventory against the costs of not having it when needed [3](#page=3) [4](#page=4). ### 1.1 The necessity of inventory Inventory is necessary in supply chains for several key reasons: * **Meeting customer demand:** Businesses need to have products available to fulfill orders promptly and avoid lost sales [3](#page=3). * **Production continuity:** Holding raw materials and components ensures that production processes can continue without interruption [3](#page=3). * **Economies of scale:** Producing or purchasing in larger batches can reduce per-unit costs, leading to the accumulation of inventory [7](#page=7). * **Strategic buffering:** Inventory acts as a buffer against uncertainties in supply and demand [3](#page=3). ### 1.2 Types of inventory Several distinct types of inventory exist, each serving a specific purpose and requiring different management strategies: #### 1.2.1 Seasonal inventory Seasonal inventory is held to meet predictable surges in demand that occur during specific times of the year. This type of inventory is often accumulated in anticipation of peak seasons, such as holidays or specific weather conditions [5](#page=5). > **Tip:** The decision to hold seasonal inventory involves a trade-off between the cost of carrying this extra stock and the cost of rapidly increasing capacity to meet peak demand [5](#page=5). #### 1.2.2 Safety inventory Safety inventory, also known as buffer stock, is maintained to protect against the inherent uncertainty in demand and lead times. Its primary purpose is to ensure a desired level of customer service by preventing stockouts when demand exceeds forecasts or when supply is delayed [6](#page=6). > **Example:** A retailer might hold extra units of a popular item to guard against unexpected spikes in customer purchases or a delay in replenishment from the supplier [6](#page=6). The management of safety stock involves a critical trade-off between the costs incurred from holding excess inventory (e.g., storage, obsolescence) and the costs associated with lost sales due to stockouts [6](#page=6). #### 1.2.3 Cycle inventory Cycle inventory arises when a company produces or orders goods in batches rather than producing or ordering them on a per-order basis. This is common in situations where there are fixed costs associated with setting up production runs or placing orders [7](#page=7). > **Key principle:** Cycle inventory allows businesses to exploit economies of scale by taking advantage of lower per-unit costs associated with larger order quantities or production runs [7](#page=7). The trade-off for cycle inventory is between the costs of holding this inventory and the fixed costs of ordering or setting up production [7](#page=7). --- # Little's Law and its application to inventory This topic introduces Little's Law and its fundamental application to understanding supply chain inventory dynamics. ### 2.1 Little's Law: The General Formula Little's Law is a general and fundamental principle in queuing theory that establishes a relationship between the average number of items in a system, the rate at which items enter the system, and the average time an item spends within the system. It is a powerful concept because it is independent of the arrival process and the system's internal workings, holding true for any stable system [8](#page=8). The general formula for Little's Law is: $$L = \lambda W$$ Where: * $L$ represents the long-term average number of items in the system [8](#page=8). * $\lambda$ (lambda) is the average arrival rate of items into the system [8](#page=8). * $W$ is the average time an item spends in the system [8](#page=8). > **Tip:** Remember that Little's Law applies to *average* values and requires a stable system, meaning the rate of items entering the system is less than the rate of items leaving it over the long term. ### 2.2 Little's Law Applied to Inventory In the context of supply chain management, Little's Law can be directly applied to understand inventory levels. Here, the "system" is the inventory within a specific part of the supply chain (e.g., a warehouse, in transit). The adapted formula for inventory is: $$I = D \times T$$ Where: * $I$ represents the average supply chain inventory [9](#page=9). * $D$ is the demand rate, which is analogous to the arrival rate ($\lambda$) in the general formula [9](#page=9). * $T$ is the average time the material spends in the system, analogous to the average time spent in the system ($W$) [9](#page=9). This formula highlights a direct relationship between inventory levels and the time material spends in the supply chain. A system can achieve the same demand fulfillment with either high inventory and low flow time, or low inventory and high flow time, or a combination of both [9](#page=9). > **Tip:** This equation is crucial for inventory management. It shows that reducing the time inventory spends in the system (e.g., through faster transportation or quicker processing) can lead to lower average inventory levels while still meeting demand. Conversely, increasing flow time will naturally lead to higher inventory levels for the same demand rate. ### 2.3 Examples of Little's Law in Inventory Management #### 2.3.1 Example (i): Beer in Transit **Scenario:** Beer is transported from Leuven to New York City. The average transportation time is 2 weeks, and the demand for this beer in NYC is approximately 100 barrels per day. **Question:** What is the average amount of beer in transit to NYC at any given time? **Application of Little's Law:** We use the inventory version of Little's Law: $I = D \times T$. * Demand rate ($D$): 100 barrels/day. * Time in transit ($T$): 2 weeks. We need to convert this to days for consistency with the demand rate. 2 weeks = 14 days. **Calculation:** $$I = 100 \text{ barrels/day} \times 14 \text{ days}$$ $$I = 1400 \text{ barrels}$$ **Conclusion:** At any given time, there are, on average, 1400 barrels of beer in transit to NYC [10](#page=10). > **Example:** This calculation demonstrates how to quantify the inventory tied up in the transportation process. Understanding this allows for better cash flow management and risk assessment related to goods in transit. #### 2.3.2 Example (ii): Beer in a Warehouse **Scenario:** The Stella brewery in Leuven has, on average, 5000 barrels of beer in its warehouse on any given day. On average, 1000 barrels of beer are shipped to bars and distributors every day. **Question:** What is the average time a barrel of beer spends in the warehouse? **Application of Little's Law:** We use the inventory version of Little's Law: $I = D \times T$. This time, we need to solve for $T$. * Average inventory ($I$): 5000 barrels. * Demand rate ($D$): 1000 barrels/day. **Rearranging the formula to solve for T:** $$T = \frac{I}{D}$$ **Calculation:** $$T = \frac{5000 \text{ barrels}}{1000 \text{ barrels/day}}$$ $$T = 5 \text{ days}$$ **Conclusion:** On average, a barrel of beer spends 5 days in the warehouse before being shipped out [11](#page=11). > **Example:** This example shows how Little's Law can be used to calculate the velocity of inventory. A shorter time in the warehouse (low $T$) indicates faster inventory turnover, which is generally desirable as it reduces holding costs and the risk of obsolescence. --- # Cycle inventory analysis and optimization Cycle inventory analysis focuses on understanding and managing the inventory held due to ordering or producing in batches, with the goal of reducing overall inventory levels and associated costs [12](#page=12). ### 3.1 Understanding cycle inventory Cycle inventory is the average amount of inventory held to satisfy demand between replenishments. It arises from ordering or producing in discrete lot sizes rather than continuously [12](#page=12). #### 3.1.1 Generation of cycle inventory When a batch of size $Q$ is ordered or produced, the inventory level gradually decreases to zero as demand is met. The average cycle inventory is then half of the lot size, represented by $\frac{Q}{2}$. This is because the inventory level varies from $Q$ to 0, with the average being the midpoint [12](#page=12) [13](#page=13). The cycle time, which is the time it takes to satisfy demand with one lot size, can be calculated by dividing the lot size $Q$ by the demand rate $D$. This can be expressed as: $$ \text{Cycle Time} = \frac{Q}{D} $$ The average inventory is then the demand rate multiplied by half of the cycle time, or $\frac{D}{2} \times \frac{Q}{D} = \frac{Q}{2}$ [15](#page=15). #### 3.1.2 Impact of lead time If there is a positive lead time (LT) between placing an order and receiving it, the inventory management strategy might involve an $(s, Q)$ model. In this model, an order of size $Q$ is placed when the inventory position (on-hand inventory plus on-order inventory) drops to a reorder point $s$. This means that while the average inventory can still be calculated based on the lot size $Q$, the inventory never actually reaches zero due to the lead time [14](#page=14). #### 3.1.3 Example calculation Consider a warehouse with a demand rate $D$ of 1000 barrels per day and an average inventory of 5000 barrels. To find the lot size $Q$, we use the formula: $$ \text{Average Inventory} = \frac{Q}{2} $$ $$ 5000 = \frac{Q}{2} $$ $$ Q = 10000 \text{ barrels} $$ The cycle time is then calculated as: $$ \text{Cycle Time} = \frac{Q}{D} = \frac{10000 \text{ barrels}}{1000 \text{ barrels/day}} = 10 \text{ days} $$ Therefore, the cycle inventory is 5000 barrels, and the cycle time is 10 days [15](#page=15). ### 3.2 Strategies to reduce cycle inventory Cycle inventory can be reduced by decreasing the lot size $Q$ [16](#page=16). * **Reducing lot size:** A smaller lot size directly leads to a lower average cycle inventory ($\frac{Q}{2}$). For instance, if the lot size is halved, the average cycle inventory is also halved. This also results in a shorter cycle time [16](#page=16). > **Tip:** While reducing lot size effectively lowers cycle inventory, it can increase the frequency of orders or production runs, potentially leading to higher fixed ordering or setup costs. The optimal strategy involves balancing these trade-offs. ### 3.3 Trade-offs in lot sizing The decision on the optimal lot size $Q$ involves balancing various costs. * **Large lot sizes (large $Q$):** * **Benefits:** Reduce fixed costs associated with ordering or setup per unit. May allow for quantity discounts from suppliers [17](#page=17). * **Drawbacks:** Lead to higher average cycle inventory and associated holding costs (financial, warehousing, perishability, obsolescence, etc.) [17](#page=17). * **Small lot sizes (small $Q$):** * **Benefits:** Significantly lower average cycle inventory and thus lower holding costs [17](#page=17). * **Drawbacks:** Increase the frequency of orders or setups, leading to higher total fixed costs [17](#page=17). The best strategy is to find a lot size that minimizes the total inventory costs, which include holding costs and fixed ordering/setup costs [17](#page=17). --- # The Economic Order Quantity (EOQ) model The Economic Order Quantity (EOQ) model is a foundational inventory management tool used to determine the optimal quantity of goods to order at a time to minimize total inventory costs [18](#page=18). ### 4.1 Core concepts and trade-offs The EOQ model is a simplification of reality, designed to offer a deterministic and constant demand scenario. It seeks to balance two primary opposing costs [19](#page=19): * **Ordering Costs (or Setup Costs):** These are fixed costs incurred each time an order is placed or a production batch is set up. Examples include administrative costs for processing an order, shipping fees, or costs associated with setting up manufacturing equipment. The EOQ model assumes this cost is a fixed amount per order, denoted by $S$ [19](#page=19) [20](#page=20). * **Holding Costs (or Carrying Costs):** These are costs associated with storing inventory over a period of time. They include expenses like warehousing, insurance, obsolescence, spoilage, and the opportunity cost of capital tied up in inventory. The model defines a holding cost per unit per year, denoted by $H$. This is often expressed as a fraction ($h$) of the product cost ($C$), so $H = hC$ [19](#page=19) [20](#page=20). The model assumes that the product cost ($C$) per unit is independent of the order quantity and that there is zero replenishment lead time [19](#page=19). ### 4.2 The EOQ formula and its components The total cost ($TC(Q)$) for a given order quantity ($Q$) is the sum of the annual material costs, annual ordering costs, and annual holding costs [20](#page=20). The components of the total cost function are: * $D$: Annual demand rate [20](#page=20). * $S$: Fixed setup cost or ordering cost per order [20](#page=20). * $C$: Product cost per unit [20](#page=20). * $h$: Holding cost per year as a fraction of product cost [20](#page=20). * $H$: Holding cost per unit per year, where $H = hC$ [20](#page=20). * $Q$: Lot Size or order quantity [20](#page=20). The total cost function is expressed as: $$TC(Q) = CD + \frac{D}{Q}S + \frac{Q}{2}H$$ Where: * $CD$ represents the annual material cost. * $\frac{D}{Q}S$ represents the annual ordering cost. * $\frac{Q}{2}H$ represents the annual holding cost. ### 4.3 Calculating the optimal order quantity ($Q^*$) The EOQ model aims to find the quantity $Q$ that minimizes the total cost $TC(Q)$. This minimum occurs where the ordering costs and holding costs are equal. By taking the derivative of the total cost function with respect to $Q$ and setting it to zero, we can derive the optimal order quantity, $Q^*$ [22](#page=22) [23](#page=23). The formula for the optimal order quantity is: $$Q^* = \sqrt{\frac{2DS}{H}}$$ This formula indicates that the optimal order quantity increases with higher demand ($D$) and higher setup costs ($S$), and decreases with higher holding costs ($H$) [24](#page=24). The optimal frequency ($n^*$) of ordering (number of orders per year) is calculated as: $$n^* = \frac{D}{Q^*} = \sqrt{\frac{DH}{2S}}$$ The optimal frequency increases with higher demand and higher holding costs, and decreases with higher setup costs [24](#page=24). > **Tip:** The EOQ model demonstrates a non-linear relationship between the variables and the optimal quantity. Small changes in $S$ or $H$ can have a significant impact on $Q^*$ [24](#page=24). ### 4.4 Key insights from the EOQ model * The optimal lot size ($Q^*$) is determined by the trade-off between holding costs and setup costs; product costs ($C$) do not directly influence $Q^*$ [25](#page=25). * Total inventory costs are relatively stable around the optimal order quantity ($Q^*$). This means that ordering quantities slightly different from $Q^*$ may not lead to significant cost increases [25](#page=25). * To reduce the lot size ($Q$) without increasing costs, it is necessary to reduce the fixed ordering or setup costs ($S$) [25](#page=25). * When demand increases, it is optimal to increase both the optimal order quantity ($Q^*$) and the optimal order frequency ($n^*$) [25](#page=25). * A higher demand generally leads to a larger inventory that moves faster, which aligns with Little's Law ($I = DL$) where $I$ is inventory, $D$ is demand, and $L$ is lead time [25](#page=25). #### 4.4.1 Example Consider an example where: * Annual demand ($D$) = 1000 barrels [21](#page=21). * Holding cost per unit per year ($H$) = 10 euros per month [21](#page=21). * Setup cost per order ($S$) = 10,000 euros per batch [21](#page=21). * Product cost ($C$) = 10 euros per barrel [21](#page=21). First, we need to convert the holding cost to an annual figure. Assuming there are 22 working days in a month: $H_{annual} = 10 \text{ euros/month} \times 12 \text{ months/year} = 120 \text{ euros/barrel/year}$ [21](#page=21). Now, we can calculate the optimal order quantity ($Q^*$) using the EOQ formula: $$Q^* = \sqrt{\frac{2DS}{H}} = \sqrt{\frac{2 \times 1000 \times 10000}{120}}$$ $$Q^* = \sqrt{\frac{20000000}{120}} \approx \sqrt{166666.67} \approx 408.25 \text{ barrels}$$ [21](#page=21). The total cost per year for this order quantity would be: $$TC(Q^*) = CD + \frac{D}{Q^*}S + \frac{Q^*}{2}H$$ $$TC(Q^*) = (10 \times 1000) + \frac{1000}{408.25} \times 10000 + \frac{408.25}{2} \times 120$$ $$TC(Q^*) = 10000 + 2449.55 + 24495 \approx 36945 \text{ euros/year}$$ [21](#page=21). The document also provides a calculation for total cost per month for a different scenario, yielding 24,200 euros/month. This suggests the example in the document might be demonstrating a specific monthly cost calculation rather than the annual EOQ cost derived from the given parameters. However, the principle of summing ordering and holding costs is illustrated [21](#page=21). --- ## Common mistakes to avoid - Review all topics thoroughly before exams - Pay attention to formulas and key definitions - Practice with examples provided in each section - Don't memorize without understanding the underlying concepts

| Term | Definition | |------|------------| | Inventory | Goods or raw materials kept in stock by a business or organization, typically for use in production or for sale to customers. It represents a value that a company holds. | | Seasonal Inventory | Stock held to meet anticipated peak demand during specific periods, such as holidays or periods of high consumer interest. This type of inventory helps businesses manage fluctuations in demand throughout the year. | | Safety Inventory | Stock held to buffer against unexpected fluctuations in demand or delays in supply. Its primary purpose is to ensure a desired service level and prevent stockouts, thereby mitigating the risk of lost sales due to unpredictability. | | Cycle Inventory | Inventory that arises when a company produces goods in batches rather than on demand. This type of inventory is maintained to achieve economies of scale in production by taking advantage of fixed ordering costs. | | Aggregate Planning | A medium-term strategic planning process that aims to balance supply and demand for resources like labor and inventory. It is used to determine production levels and workforce sizes to meet anticipated demand, often involving trade-offs between inventory and capacity costs. | | Economies of Scale | The cost advantages that arise with increased output of a product. When production volume increases, the average cost per unit decreases, making larger batch production more cost-efficient. | | Production to Order | A manufacturing strategy where production is initiated only after a customer places an order. This approach minimizes finished goods inventory but can lead to longer lead times for customers. | | Batch Production | A manufacturing method where a group of products are produced together in a set quantity. This inherently creates inventory as a quantity of goods is completed before being moved to the next stage or sold. | | Holding Cost | The expense associated with storing unsold inventory. This includes costs such as warehousing, insurance, obsolescence, and potential spoilage. | | Fixed Ordering Cost | The cost incurred each time an order is placed or a production batch is set up. These costs are independent of the quantity ordered or produced. | | Service Level | A metric that quantifies the probability of not stocking out. It indicates the degree to which customer demand can be met from existing stock, often used as a target for safety inventory management. | | Lost Sales | The potential revenue that is forgone when a customer’s demand cannot be met due to insufficient inventory. This is a key consequence that safety inventory aims to prevent. | | Demand Uncertainty | The unpredictability or variability in the quantity of a product that customers will wish to purchase within a given period. This variability necessitates strategies like safety inventory to ensure availability. | | Little's Law | A general formula that states the long-term average number of items in a stationary system (L) is equal to the average arrival rate of items to the system ($\lambda$) multiplied by the average time an item spends in the system (W). | | System Inventory | The total quantity of goods or materials present within a defined system at any given time. | | Arrival Rate | The average rate at which new items enter a system, often measured in units per time period. | | Time in System | The average duration an item spends within a system from its arrival to its departure. | | Demand Rate | The average rate at which customers or downstream processes require products or services from the system. | | Flow Time | The total time a unit of product spends within a supply chain, from the point of origin to the point of final consumption or use. | | Transit Inventory | The quantity of goods or materials that are in motion between different locations within a supply chain. | | Warehouse Inventory | The quantity of goods or materials stored in a warehouse facility, awaiting shipment or distribution. | | Lot Size | The quantity of an item ordered or produced in a single batch. Adjusting the lot size is a primary strategy for managing and reducing cycle inventory levels. | | Order Quantity | The number of units of a product ordered at one time. In the context of cycle inventory, this directly influences the size of the batches and the subsequent inventory levels. | | Average Inventory | The typical amount of inventory held over a given period. For cycle inventory, it is often calculated as half of the order quantity. | | Cycle Time | The time it takes to use up one batch of inventory. It is directly related to the demand rate and the order quantity, typically calculated as `$Q / D$`, where `$Q$` is the order quantity and `$D$` is the demand rate. | | Lead Time | The duration between the placement of an order and its receipt. In cycle inventory analysis, lead time can affect when replenishment orders are triggered and how inventory levels fluctuate. | | Replenishment | The process of ordering or producing more inventory to replace what has been used or sold. This action directly impacts the cycle inventory levels by creating new batches. | | Holding Costs | The expenses associated with storing inventory, including financial costs, warehousing, insurance, and potential spoilage or obsolescence. Reducing cycle inventory helps lower these costs. | | Fixed Costs | Costs that do not change with the level of production or sales, such as setup costs for machinery. Purchasing or producing in larger lot sizes can reduce the impact of fixed costs per unit. | | Economic Order Quantity (EOQ) Model | A model used in inventory management to determine the optimal order quantity that minimizes the total costs associated with ordering and holding inventory. It seeks a balance between the costs incurred from placing frequent small orders and the costs of holding large quantities of stock. | | Annual Demand Rate (D) | The total number of units of a product expected to be sold or consumed over a one-year period. This is a key input for the EOQ model, assuming it is deterministic and constant. | | Fixed Setup Cost or Ordering Cost (S) | The cost incurred each time an order is placed or a production run is set up. This cost is considered fixed regardless of the quantity ordered and contributes to the ordering cost component in the EOQ calculation. | | Product Cost Per Unit (C) | The direct cost of purchasing or producing a single unit of the item. In the standard EOQ model, this cost is assumed to be independent of the order quantity. | | Holding Cost Per Year as a Fraction of Product Cost (h) | The cost of holding one unit of inventory for one year, expressed as a proportion of the product's unit cost. This is used to calculate the absolute holding cost per unit per year. | | Holding Cost Per Unit Per Year (H) | The total monetary cost of holding one unit of inventory for one year. It is calculated by multiplying the product cost per unit (C) by the holding cost fraction (h), so $H = hC$. | | Lot Size (Q) | The quantity of an item ordered or produced in a single batch or order. The EOQ model aims to find the optimal lot size, denoted as $Q^{\ast}$. | | Total Cost Function (TC(Q)) | An equation that represents the sum of all relevant inventory costs as a function of the order quantity (Q). For the EOQ model, it includes the total annual product cost, total annual ordering cost, and total annual holding cost. The formula is $TC(Q) = CD + \frac{D}{Q}S + \frac{Q}{2}H$. | | Optimal Order Quantity ($Q^{\ast}$) | The specific order quantity that results in the minimum total inventory cost. It is calculated using the EOQ formula: $Q^{\ast} = \sqrt{\frac{2DS}{H}}$. | | Optimal Order Frequency ($n^{\ast}$) | The optimal number of orders to be placed per year to minimize total inventory costs. It is calculated as $n^{\ast} = \frac{D}{Q^{\ast}}$ or $n^{\ast} = \frac{H}{2S}D$. |