Let's break down the question using the **Newsvendor model** and **inventory pooling** concepts: ### Given Data - **Number of stores:** 4 - **Daily demand per store:** Normally distributed - Mean (\(\mu\)) = 200 units - Standard deviation (\(\sigma\)) = 60 units - **Sales price:** $80 - **Wholesale cost:** $40 - **Salvage value:** $20 - Demand is **i.i.d.** across stores. - **Inventory pooling**: Central fulfillment can allocate stock dynamically across stores. ### What’s being asked? **Will pooling inventory increase or decrease the expected total sales, and by how much (closest answer)?** #### **Key Concept:** - **Expected Sales** = Expected Demand – Expected Lost Sales - **Pooling** can reduce lost sales due to demand variability being "smoothed" when aggregated. --- ## Step 1: Calculate expected demand For 4 stores: - **Total expected demand** = \(4 \times 200 = 800\) units --- ## Step 2: Calculate expected lost sales **without pooling** Each store faces demand \(N(200, 60^2)\). First, find the optimal order quantity per store. The critical ratio (\(CR\)) is: \[ CR = \frac{Selling\,price - Cost}{Selling\,price - Salvage} = \frac{80 - 40}{80 - 20} = \frac{40}{60} = 0.6667 \] For a normal distribution, the optimal order quantity is the quantile at \(CR\): \[ Q^* = \mu + z \cdot \sigma \] Where \(z\) is the standard normal value at 0.6667. Looking up \(z_{0.6667} \approx 0.43\). So, per store: \[ Q^* = 200 + 0.43 \cdot 60 \approx 200 + 25.8 = 225.8 \approx 226 \text{ units} \] Total across 4 stores: \(4 \times 226 = 904\) units --- ### **Expected lost sales per store** For a normal distribution, expected lost sales (per store): \[ ELS = \sigma L(z) \] Where \(L(z) = \phi(z) - z[1 - \Phi(z)]\), \(\phi(z)\) is standard normal pdf, \(\Phi(z)\) is cdf. - \(z = 0.43\) - \(\phi(0.43) = \frac{1}{\sqrt{2\pi}}e^{-0.43^2/2} \approx 0.364\) - \(1 - \Phi(0.43) \approx 1 - 0.6667 = 0.333\) \[ L(0.43) = 0.364 - 0.43 \times 0.333 \approx 0.364 - 0.143 = 0.221 \] So, per store: \[ ELS_{store} = 60 \cdot 0.221 \approx 13.3 \text{ units} \] Total lost sales (all stores): \(4 \cdot 13.3 = 53.2\) units --- ### **Expected total sales (without pooling)** \[ \text{Expected total sales} = \text{Expected demand} - \text{Expected lost sales} = 800 - 53.2 = 746.8 \text{ units} \] --- ## Step 3: Calculate expected lost sales **with pooling** **Pooled demand:** \(N(800, (4 \cdot 60^2)) = N(800, 240^2)\) - \(\mu_{pool} = 800\) - \(\sigma_{pool} = \sqrt{4} \cdot 60 = 120\) The **critical ratio** is the same, \(CR = 0.6667\), so \(z = 0.43\). Order quantity for the pooled system: \[ Q_{pool} = 800 + 0.43 \cdot 120 = 800 + 51.6 = 851.6 \approx 852 \text{ units} \] Expected lost sales: \[ ELS_{pool} = \sigma_{pool} \cdot L(z) = 120 \cdot 0.221 = 26.5 \text{ units} \] --- ### **Expected total sales (with pooling)** \[ \text{Expected total sales} = 800 - 26.5 = 773.5 \text{ units} \] --- ## Step 4: **Compare and answer** - Without pooling: **746.8 units** - With pooling: **773.5 units** - **Increase in expected sales:** \(773.5 - 746.8 = 26.7 \text{ units}\) --- ### **Conclusion** **Pooling the inventory will increase the expected total sales by about 27 units.** --- #### **Final Answer (Choose closest):** > **Pooling the inventory will increase expected total sales by about 27 units.**