00b_newsvendor_refresher_annotated.pdf

# Introduction to the newsvendor model The newsvendor model is a fundamental inventory management tool used to make optimal ordering decisions when demand is uncertain [2](#page=2). ## 1. Introduction to the newsvendor model The newsvendor model addresses the problem of deciding how much inventory to order for a single selling period when the demand is stochastic, meaning it is not known with certainty before the order must be placed. It helps in balancing the risks associated with having too much inventory versus too little [2](#page=2). ### 1.1 The core tradeoff The essence of the newsvendor model lies in managing the tradeoff between overage costs and underage costs [2](#page=2). * **Overage cost ($C_o$)**: This is the cost incurred for each unit of inventory that is ordered but not sold by the end of the period. This often includes the purchase cost of the item, plus any disposal or markdown costs, minus any salvage value [2](#page=2). * **Underage cost ($C_u$)**: This is the cost incurred for each unit of demand that cannot be met due to insufficient inventory. This typically represents lost profit or goodwill costs [2](#page=2). The objective of the newsvendor model is to determine the optimal order quantity ($Q$) that minimizes the expected total costs, which are comprised of the costs associated with overstocking and understocking [4](#page=4). The total cost can be expressed as: $$ \text{Total costs} = C_o \times E[\text{Overstock}] + C_u \times E[\text{Understock}] $$ where: * $C_o$ is the cost of overage per unit [4](#page=4). * $E[\text{Overstock}]$ is the expected number of units overstocked [4](#page=4). * $C_u$ is the cost of underage per unit [4](#page=4). * $E[\text{Understock}]$ is the expected number of units understocked [4](#page=4). ### 1.2 Key elements and considerations * **Single period**: The model is typically applied to situations where there is only one selling opportunity for the product [2](#page=2). * **Stochastic demand**: Demand is uncertain and can be represented by a probability distribution [2](#page=2). * **Decision timing**: The decision on the order quantity must be made before the actual demand is realized [2](#page=2). > **Tip:** Understanding the precise definition of overage and underage costs is crucial. For instance, if a vendor buys an item for 10 dollars and can sell it for 20 dollars (implying a profit of 10 dollars per unit), and the salvage value is 5 dollars, then the overage cost ($C_o$) would be the purchase cost plus any loss from not selling it, minus salvage value. If the purchase price is 10 dollars, and selling price is 20 dollars, and salvage value is 5 dollars: $C_o = \text{purchase price} - \text{salvage value} = 10 - 5 = 5$ dollars. The underage cost ($C_u$) is the profit lost from not meeting demand, which is the selling price minus the purchase price: $C_u = 20 - 10 = 10$ dollars. #### 1.2.1 Example scenario Consider a street vendor selling newspapers. The vendor buys newspapers for 2 dollars each and sells them for 4 dollars each. At the end of the day, unsold newspapers are worthless (salvage value is 0 dollars). If the vendor orders 100 newspapers, and the demand turns out to be 90 newspapers, there are 10 overstocked units. If the demand is 110 newspapers, there are 10 understocked units [3](#page=3). * Overage cost ($C_o$): The cost of ordering a unit that is not sold. In this case, it is the purchase cost minus the salvage value: $2 \text{ dollars} - 0 \text{ dollars} = 2 \text{ dollars}$ per unit [3](#page=3). * Underage cost ($C_u$): The cost of not having a unit to sell when there is demand. This is the profit lost per unit: $4 \text{ dollars} - 2 \text{ dollars} = 2 \text{ dollars}$ per unit [3](#page=3). The vendor's goal is to determine the optimal number of newspapers to order ($Q$) to minimize total costs, considering the uncertain demand. The expected total costs are calculated based on the costs of overage and underage, weighted by the probability of each occurring [3](#page=3) [4](#page=4). --- # Optimizing order quantity in the newsvendor model This section focuses on quantifying the trade-off between overstocking and understocking to determine the optimal order quantity in the newsvendor model. ### 2.1 The fundamental trade-off The core of optimizing order quantity in the newsvendor model lies in quantifying the tradeoff between the costs associated with having too much inventory (overage) and too little inventory (underage). The goal is to find an order quantity that minimizes total expected costs [4](#page=4). The total expected costs ($TC$) can be expressed as the sum of the expected overage costs ($C_o$) and the expected underage costs ($C_u$) [4](#page=4). $$TC = C_o \cdot E[\text{Overage}] + C_u \cdot E[\text{Underage}]$$ Where: * $C_o$ is the cost of overage (e.g., cost of unsold inventory, salvage value loss) [4](#page=4). * $E[\text{Overage}]$ is the expected amount of inventory left unsold [4](#page=4). * $C_u$ is the cost of underage (e.g., lost profit due to stockout, lost goodwill) [4](#page=4). * $E[\text{Underage}]$ is the expected unmet demand [4](#page=4). ### 2.2 Deriving the optimal order quantity The optimal order quantity ($Q^*$) is achieved when the marginal benefit of stocking one more unit equals the marginal cost. Mathematically, this occurs when the probability of demand being less than or equal to the order quantity ($P(D \le Q^*)$) multiplied by the underage cost equals the probability of demand being greater than the order quantity ($P(D > Q^*)$) multiplied by the overage cost [5](#page=5). This can be represented by the following relationship: $P(D \le Q^*) \cdot C_u = P(D > Q^*) \cdot C_o$ [5](#page=5). Rearranging this equation, we arrive at the critical fractile (or critical ratio): $$\frac{C_u}{C_u + C_o} = P(D \le Q^*)$$ [5](#page=5). This critical fractile represents the probability that demand will be less than or equal to the optimal order quantity. ### 2.3 Determining $P(D \le Q^*)$ To find the optimal order quantity ($Q^*$), we need to determine the value of $Q^*$ such that the cumulative probability of demand up to $Q^*$ equals the critical fractile [6](#page=6). #### 2.3.1 Using the Normal Distribution If the demand ($D$) is assumed to follow a normal distribution with mean ($\mu$) and standard deviation ($\sigma$), i.e., $D \sim N(\mu, \sigma^2)$ we can use standard normal distribution tables or statistical software to find $Q^*$ [6](#page=6) [7](#page=7). The steps involved are: 1. **Normalize demand:** Convert the demand value to a standard normal variable ($Z$) using the formula $Z = \frac{Q^* - \mu}{\sigma}$ [6](#page=6). 2. **Find the critical Z-value:** Look up the critical fractile value in a standard normal distribution table (or use an inverse cumulative distribution function) to find the corresponding Z-score. This Z-score, let's call it $z^*$, satisfies $P(Z \le z^*) = \frac{C_u}{C_u + C_o}$ [6](#page=6). 3. **Calculate the optimal order quantity:** Once $z^*$ is found, rearrange the standardization formula to solve for $Q^*$: $$Q^* = \mu + z^* \sigma$$ [6](#page=6). > **Tip:** Always ensure your demand data is appropriately characterized to justify the use of the Normal distribution. If demand is highly skewed or has different distributional properties, other methods for finding $Q^*$ might be necessary. #### 2.3.2 Understanding the Cumulative Distribution Function (CDF) The probability $P(D \le Q^*)$ is the value of the cumulative distribution function (CDF) of the demand distribution evaluated at $Q^*$. For a continuous random variable $D$ with probability density function $f(x)$, the CDF is given by $F(Q^*) = P(D \le Q^*) = \int_{-\infty}^{Q^*} f(x) \, dx$ [7](#page=7). > **Example:** If demand is normally distributed $D \sim N(\mu, \sigma^2)$, then $P(D \le Q^*) = \Phi\left(\frac{Q^* - \mu}{\sigma}\right)$, where $\Phi(\cdot)$ is the standard normal CDF. By finding the $z^*$ such that $\Phi(z^*) = \frac{C_u}{C_u + C_o}$, we can then solve for $Q^*$ as $Q^* = \mu + z^*\sigma$ [7](#page=7). ### 2.4 The impact of costs The optimal order quantity is highly sensitive to the relative magnitudes of the overage and underage costs [5](#page=5). * **High underage cost ($C_u$) relative to overage cost ($C_o$):** This implies that the penalty for stocking out is severe. To avoid stockouts, the optimal order quantity will be higher, moving closer to the mean demand [5](#page=5). * **High overage cost ($C_o$) relative to underage cost ($C_u$):** This implies that the penalty for having excess inventory is severe. To avoid overstocking, the optimal order quantity will be lower, moving closer to a quantity that minimizes the chance of leftover stock [5](#page=5). The critical fractile $\frac{C_u}{C_u + C_o}$ directly reflects this balance. A higher critical fractile (driven by a higher $C_u$) leads to a higher $P(D \le Q^*)$, and thus a higher $Q^*$ [6](#page=6). --- # Newsvendor model application and example scenarios This section applies the newsvendor model to a practical example involving a sporting goods store deciding on the number of skis to order, exploring different cost scenarios for salvage and lost sales [10](#page=10) [8](#page=8). ### 3.1 Problem setup: The sportmart ski ordering decision The core of this topic is a detailed case study involving the sportmart sporting goods store, which needs to determine the optimal number of skis to order for the upcoming winter season [10](#page=10) [8](#page=8). #### 3.1.1 Demand forecasting Demand for skis is forecast to be normally distributed [10](#page=10) [8](#page=8). * Mean demand ($\mu$): 350 skis [10](#page=10) [8](#page=8). * Standard deviation of demand ($\sigma$): 100 skis [10](#page=10) [8](#page=8). #### 3.1.2 Cost structure The costs associated with ordering and selling skis are crucial for the newsvendor model [10](#page=10) [8](#page=8). * Cost per pair of skis to sportmart ($c$): 100 dollars [10](#page=10) [8](#page=8). * Retail price per pair of skis ($p$): 250 dollars [10](#page=10) [8](#page=8). #### 3.1.3 Salvage value The value of unsold skis at the end of the season [10](#page=10) [8](#page=8). * Salvage value per pair of skis ($s$): 80 dollars [10](#page=10) [8](#page=8). ### 3.2 Calculating critical newsvendor costs To apply the newsvendor model, we need to define the costs of overage and underage. #### 3.2.1 Cost of overage ($c_o$) This is the cost incurred for each unit stocked but not sold. It represents the loss from ordering one too many units [9](#page=9). * $c_o = c - s$ [9](#page=9). * In the sportmart example: $c_o = 100 \text{ dollars} - 80 \text{ dollars} = 20 \text{ dollars}$ [9](#page=9). #### 3.2.2 Cost of underage ($c_u$) This is the cost incurred for each unit of demand that cannot be met due to insufficient stock. It represents the profit lost from ordering one too few units [9](#page=9). * $c_u = p - c$ [9](#page=9). * In the sportmart example: $c_u = 250 \text{ dollars} - 100 \text{ dollars} = 150 \text{ dollars}$ [9](#page=9). ### 3.3 Applying the newsvendor model to different scenarios The newsvendor model helps determine the optimal order quantity by balancing the costs of overage and underage. The critical fractile (or service level) is used to find this quantity. #### 3.3.1 Scenario (a): Standard salvage scenario This is the base case where unsold items can be salvaged at a reduced price [10](#page=10) [8](#page=8). **Calculating the critical fractile:** The critical fractile represents the probability that demand will be less than or equal to the optimal order quantity. It is calculated as: $$ \text{Critical Fractile} = \frac{c_u}{c_u + c_o} $$ [9](#page=9). Using the values from the sportmart example: $$ \text{Critical Fractile} = \frac{150 \text{ dollars}}{150 \text{ dollars} + 20 \text{ dollars}} = \frac{150}{170} \approx 0.882 $$ [9](#page=9). **Determining the optimal order quantity ($Q^*$):** The optimal order quantity is found by finding the demand value that corresponds to the critical fractile in the cumulative distribution function (CDF) of the demand. For a normally distributed demand, this is: $$ Q^* = \mu + Z \cdot \sigma $$ where $Z$ is the Z-score corresponding to the critical fractile [9](#page=9). From a standard normal distribution table (or calculator), the Z-score for a cumulative probability of approximately 0.882 is about 1.18 [9](#page=9). Therefore, the optimal order quantity is: $$ Q^* = 350 + 1.18 \cdot 100 $$ [9](#page=9). $$ Q^* = 350 + 118 $$ [9](#page=9). $$ Q^* = 468 \text{ skis} $$ [9](#page=9). > **Tip:** The critical fractile is the target service level that balances the risk of stocking out against the cost of overstocking. #### 3.3.2 Scenario (b): No salvage value In this scenario, unsold skis are disposed of, meaning the salvage value is zero [10](#page=10) [8](#page=8). * New salvage value ($s$): 0 dollars * New cost of overage ($c_o$): $100 \text{ dollars} - 0 \text{ dollars} = 100 \text{ dollars}$ * Cost of underage ($c_u$): Remains 150 dollars. **Calculating the new critical fractile:** $$ \text{Critical Fractile} = \frac{150 \text{ dollars}}{150 \text{ dollars} + 100 \text{ dollars}} = \frac{150}{250} = 0.6 $$ [9](#page=9). **Determining the new optimal order quantity:** The Z-score for a cumulative probability of 0.6 is approximately 0.25 [9](#page=9). $$ Q^* = 350 + 0.25 \cdot 100 $$ [9](#page=9). $$ Q^* = 350 + 25 $$ [9](#page=9). $$ Q^* = 375 \text{ skis} $$ [9](#page=9). > **Observation:** When the salvage value decreases, the cost of overage increases, leading to a lower optimal order quantity to reduce the risk of holding excess inventory. #### 3.3.3 Scenario (c): Lost sales penalty cost This scenario introduces a penalty for unmet demand due to loss of customer goodwill [10](#page=10) [8](#page=8). * Cost of overage ($c_o$): Remains 20 dollars (salvage value is still 80 dollars). * New cost of underage ($c_u$): This now includes the lost profit plus the penalty. * Lost profit: 150 dollars * Penalty cost for lost goodwill: 50 dollars * Total $c_u = 150 \text{ dollars} + 50 \text{ dollars} = 200 \text{ dollars}$ [9](#page=9). **Calculating the new critical fractile:** $$ \text{Critical Fractile} = \frac{200 \text{ dollars}}{200 \text{ dollars} + 20 \text{ dollars}} = \frac{200}{220} \approx 0.909 $$ [9](#page=9). **Determining the new optimal order quantity:** The Z-score for a cumulative probability of approximately 0.909 is about 1.34 [9](#page=9). $$ Q^* = 350 + 1.34 \cdot 100 $$ [9](#page=9). $$ Q^* = 350 + 134 $$ [9](#page=9). $$ Q^* = 484 \text{ skis} $$ [9](#page=9). > **Observation:** Introducing a penalty for lost sales significantly increases the cost of underage, thus leading to a higher optimal order quantity to meet more of the potential demand and avoid goodwill penalties. --- ## 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 | |------|------------| | Newsvendor model | A single-period inventory model used to determine the optimal quantity of a product to stock when demand is uncertain (stochastic). It balances the risks and costs associated with having too much inventory (overage) versus too little (underage). | | Stochastic demand | Demand for a product that is uncertain and varies randomly over time. In the context of the newsvendor model, this refers to the unpredictable number of units customers will want to purchase in a single selling period. | | Overage cost ($C_o$) | The cost incurred for each unit of inventory that remains unsold at the end of the selling period. This typically includes the difference between the purchase cost and the salvage value, or the full purchase cost if the item is worthless. | | Underage cost ($C_u$) | The cost incurred for each unit of demand that cannot be met because of insufficient inventory. This represents the lost profit from a missed sale and can also include costs related to customer goodwill. | | Salvage value | The residual value of an unsold item at the end of the selling period. This value is typically lower than the original purchase or retail price and helps offset some of the overage cost. | | Expected profit | The anticipated profit from an inventory decision, calculated as the sum of potential profits weighted by their probabilities. The newsvendor model aims to maximize this expected profit. | | Normal distribution | A continuous probability distribution that is symmetric around its mean, indicating that data near the mean are more frequent than data far from the mean. It is often used to model demand in inventory management. | | Mean ($\mu$) | The average value of a set of numbers, representing the central tendency of a probability distribution. In the newsvendor model, it is the expected demand. | | Standard deviation ($\sigma$) | A measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. | | Critical ratio (CR) | A key metric in the newsvendor model that represents the optimal service level. It is calculated as the ratio of underage cost to the sum of underage and overage costs ($CR = C_u / (C_u + C_o)$). | | Lost sales | Occurs when a customer wants to purchase a product, but it is out of stock. This leads to a lost sale, representing a direct loss of potential revenue and profit. | | Goodwill cost | A penalty cost associated with unmet demand due to insufficient inventory, representing the potential long-term negative impact on customer loyalty and future sales. |