Let's solve each question step by step, beginning with **Question 1: Newsvendor Model**. --- ### **1. Newsvendor: Bread Dough for Dallas and Waco** #### **Given Data** - **Dallas:** - Mean demand (\(\mu_D\)) = 70 pizzas/day - Std dev (\(\sigma_D\)) = 15 - Selling price = $12 - Toppings cost = $4 - Dough cost = $2 - **Waco:** - Mean demand (\(\mu_W\)) = 40 pizzas/day - Std dev (\(\sigma_W\)) = 10 - Toppings cost = $4.50 - Dough cost = $2 #### **General Newsvendor Formula** Order quantity \(Q^*\) is found using the **critical ratio:** \[ Q^* = \mu + z^* \cdot \sigma \] where \(z^*\) is the z-value for the critical ratio. **Critical ratio (CR):** \[ CR = \frac{\text{Cost of Underage (Cu)}}{\text{Cost of Underage (Cu) + Cost of Overage (Co)}} \] **Cu (Underage):** Lost profit per unit short = Selling price - Topping cost - Dough cost **Co (Overage):** Cost per unsold unit = Dough cost (since toppings can be saved, only dough is wasted) #### **Step 1: Calculate Cu and Co** - **Dallas:** - Cu = $12 - $4 - $2 = **$6** - Co = $2 - **Waco:** - Cu = $12 - $4.50 - $2 = **$5.50** - Co = $2 #### **Step 2: Calculate Critical Ratio** - **Dallas:** \[ CR_D = \frac{6}{6+2} = \frac{6}{8} = 0.75 \] - **Waco:** \[ CR_W = \frac{5.5}{5.5+2} = \frac{5.5}{7.5} \approx 0.733 \] #### **Step 3: Find z-value for each CR** - For \(CR_D = 0.75\): \(z^* = 0.674\) (from z-tables) - For \(CR_W = 0.733\): \(z^* \approx 0.62\) (from z-tables) #### **Step 4: Calculate the order quantity** ##### **A. Dallas** \[ Q^*_D = \mu_D + z^* \cdot \sigma_D = 70 + 0.674 \times 15 = 70 + 10.11 = \boxed{80.1 \approx 80 \text{ units}} \] ##### **B. Waco** \[ Q^*_W = \mu_W + z^* \cdot \sigma_W = 40 + 0.62 \times 10 = 40 + 6.2 = \boxed{46.2 \approx 46 \text{ units}} \] --- ### **Final Answers** **1a. Dallas:** Jimmy should order **80 units** of bread dough per day. **1b. Waco:** Jimmy should order **46 units** of bread dough per day. --- Would you like step-by-step solutions for questions 2 and 3 as well?

Let's proceed with **Question 2: EOQ Model** and **Question 3: (Q,R) Policy**. --- ## **2. EOQ Model for Pencils** ### Given Data: - Annual demand, \(D = 3000\) cases - Cost per case, \(C = \$3\) - Holding cost per case per year, \(h = \$0.6\) - Ordering cost per order, \(S = \$100\) --- ### **a. Find the optimal order quantity (EOQ):** The EOQ formula: \[ EOQ = \sqrt{\frac{2DS}{h}} \] Plugging in the values: \[ EOQ = \sqrt{\frac{2 \times 3000 \times 100}{0.6}} = \sqrt{\frac{600,000}{0.6}} = \sqrt{1,000,000} = 1000 \] **Answer:** \(\boxed{EOQ = 1000 \text{ cases}}\) --- ### **b. How often to place an order?** Order frequency: \[ \text{Cycle Time} = \frac{EOQ}{D} \times \text{Number of periods per year} \] Since demand is annual, the number of orders per year: \[ N = \frac{D}{EOQ} = \frac{3000}{1000} = 3 \] Time between orders (in years): \[ T = \frac{1}{N} = \frac{1}{3} \text{ year} \approx 4 \text{ months} \] **Answer:** Jimmy should place **about 3 orders per year**, roughly every 4 months. --- ## **3. (Q,R) Inventory Policy for Gerald A. Butler & Sons** ### Given Data: - Weekly demand, \(\mu_{d} = 800\) - Weekly demand standard deviation, \(\sigma_{d} = 100\) - Lead time, \(L = 2\) weeks - Cost per order, \(S = \$200\) - Variable cost per unit, \(c = \$5\) - Carrying rate = 20% per year - Weeks per year = 50 --- ### **Step 1: Calculate annual demand** Total weekly demand: \[ D_{weekly} = 800 \text{ units} \] Annual demand: \[ D_{annual} = 800 \times 50 = 40,000 \text{ units} \] ### **Step 2: Demand during lead time** Demand during lead time: \[ \mu_{L} = \mu_{d} \times L = 800 \times 2 = 1600 \text{ units} \] Standard deviation during lead time: \[ \sigma_{L} = \sigma_{d} \times \sqrt{L} = 100 \times \sqrt{2} \approx 100 \times 1.414 = 141.4 \] ### **Step 3: Determine the desired service level** Probability of stockout \(\leq 5\%\) implies a service level of **95%**. From standard normal distribution: \[ z_{0.95} \approx 1.645 \] ### **Step 4: Calculate safety stock** Safety stock: \[ SS = z \times \sigma_{L} = 1.645 \times 141.4 \approx 232.8 \text{ units} \] ### **Step 5: Determine order quantity (Q)** Economic order quantity: \[ Q = \sqrt{\frac{2 D S}{h_c}} \] where - \(h_c\) = annual holding cost per unit = 20% of unit cost: \[ h_c = 0.20 \times 5 = \$1 \] Calculate Q: \[ Q = \sqrt{\frac{2 \times 40,000 \times 200}{1}} = \sqrt{16,000,000} = 4000 \] **Note:** Since demand is high, the EOQ is large (~4000 units). ### **Step 6: Determine reorder point (R)** Reorder point: \[ R = \text{Demand during lead time} + \text{safety stock} = 1600 + 233 \approx 1833 \text{ units} \] --- ## **Final Recommendations:** - **Order Quantity (Q):** **4000 units** each time. - **Reorder Point (R):** **1833 units**. - This policy ensures a 95% service level with a reorder point of approximately 1833 units, considering demand variability. --- **Summary:** | Question | Final Answer | |------------|--------------| | **2a. EOQ** | **1000 cases** | | **2b. Order frequency** | About **3 times per year** (every 4 months) | | **3. (Q,R) policy** | Order **4000 units** each time, reorder when inventory drops to **1833 units** | --- If you'd like further clarification or calculations, feel free to ask!

Certainly! Here are the final answers summarized: **Question 1: Newsvendor Model (Bread Dough)** - **Dallas:** Order approximately **80 units** of bread dough per day. - **Waco:** Order approximately **46 units** of bread dough per day. **Question 2: EOQ Model for Pencils** - **Optimal order size (EOQ):** **1000 cases** - **Order frequency:** About **3 times per year** (roughly every 4 months). **Question 3: (Q, R) Inventory Policy** - **Order quantity (Q):** **4000 units** - **Reorder point (R):** **1833 units** Let me know if you need anything else!