Let's solve each part step by step: ### **Given Data** - **Annual demand (D)** = 8,320 units - **Weekly demand** = 160 units - **Std dev of weekly demand (σ_w)** = 60 units - **Lead time (L)** = 4 weeks --- ### **a. To provide a 90% service probability, what must the reorder point be?** #### 1. **Find the mean demand during lead time (μ_L):** \[ \mu_L = \text{Weekly demand} \times \text{Lead time} = 160 \times 4 = 640 \text{ units} \] #### 2. **Calculate the standard deviation during lead time (σ_L):** \[ \sigma_L = \sqrt{L} \times \sigma_w = \sqrt{4} \times 60 = 2 \times 60 = 120 \text{ units} \] #### 3. **Find the z-value for 90% service probability:** - For a 90% service probability, the alpha level is 0.90. - Using Excel's **NORM.S.INV(0.90)** or a z-table: \[ z = NORM.S.INV(0.90) \approx 1.28 \] *(Rounded to 2 decimal places as requested)* #### 4. **Calculate the reorder point:** \[ \text{Reorder point} = \mu_L + z \cdot \sigma_L = 640 + 1.28 \times 120 = 640 + 153.6 = 793.6 \] - **Round to the nearest whole number:** \[ \text{Reorder point} = \boxed{794} \text{ units} \] --- ### **b. If safety stock is reduced by 90 units, what is the new service probability?** #### 1. **Find the new reorder point:** - Original reorder point = 794 units - New reorder point = 794 - 90 = 704 units #### 2. **Find the new z-value:** \[ z = \frac{\text{New reorder point} - \mu_L}{\sigma_L} = \frac{704 - 640}{120} = \frac{64}{120} \approx 0.53 \] #### 3. **Find the new service probability:** - Use Excel's **NORM.S.DIST(0.53, TRUE)** or a z-table: \[ \text{Service probability} = NORM.S.DIST(0.53, TRUE) \approx 0.7019 \] - Convert to percentage and round to the nearest whole number: \[ \text{Service probability} = 70\% \] --- ## **Final Answers** **a. Reorder point:** \[ \boxed{794 \text{ units}} \] **b. Service probability:** \[ \boxed{70\%} \] Let me know if you'd like to see the calculations in Excel as well!

Certainly! Let's go through each part step-by-step with detailed calculations, explanations, and final answers. --- ## **Given Data Recap:** | Parameter | Value | |-------------|---------| | Annual demand \(D\) | 8,320 units | | Weekly demand | 160 units | | Standard deviation of weekly demand \(\sigma_w\) | 60 units | | Lead time \(L\) | 4 weeks | | Inventory carrying cost per unit | $0.85 (not needed for this part) | | Ordering cost | $145 (not needed here) | --- ## **Part a: Determine Reorder Point for 90% Service Level** ### **Step 1: Find the mean demand during the lead time \(\mu_L\)** - The mean demand during lead time = weekly demand \(\times\) lead time \[ \boxed{ \mu_L = 160 \times 4 = 640 \text{ units} } \] **Explanation:** This is the expected number of units consumed during the 4-week lead time. --- ### **Step 2: Find the standard deviation of demand during lead time \(\sigma_L\)** - Since demand variances for independent periods add, the standard deviation during lead time is: \[ \sigma_L = \sqrt{L} \times \sigma_w \] - Calculate: \[ \sigma_L = \sqrt{4} \times 60 = 2 \times 60 = 120 \text{ units} \] **Explanation:** The variability during the lead time accounts for weekly demand fluctuations over 4 weeks. --- ### **Step 3: Find the z-value for 90% service probability** - For a service probability of 90%, the critical z-value from the standard normal distribution: \[ z = \text{NORM.S.INV}(0.90) \] - Using Excel: ```excel =NORMSINV(0.90) ``` - Approximate value: \[ z \approx 1.28 \] *(Rounded to 2 decimal places as instructed)* --- ### **Step 4: Calculate the Reorder Point (ROP)** \[ \text{ROP} = \mu_L + z \times \sigma_L \] Substitute the values: \[ \text{ROP} = 640 + 1.28 \times 120 = 640 + 153.6 = 793.6 \] - Round to the nearest whole number: \[ \boxed{ \textbf{Reorder Point} = \boxed{794 \text{ units}} } \] --- ## **Part b: Effect of reducing safety stock by 90 units on service probability** ### **Step 1: Find the new reorder point** - The safety stock is essentially the safety component: \[ \text{Original safety stock} = \text{ROP} - \mu_L = 794 - 640 = 154 \text{ units} \] - Reduced safety stock: \[ \text{New safety stock} = 154 - 90 = 64 \text{ units} \] - New reorder point: \[ \text{New ROP} = \mu_L + \text{new safety stock} = 640 + 64 = 704 \text{ units} \] Alternatively, directly subtract 90 units from the old ROP: \[ 794 - 90 = 704 \text{ units} \] --- ### **Step 2: Compute the new z-value** \[ z = \frac{\text{New ROP} - \mu_L}{\sigma_L} = \frac{704 - 640}{120} = \frac{64}{120} \approx 0.5333 \] ### **Step 3: Find the new service probability** - Using Excel: ```excel =NORM.S.DIST(0.5333, TRUE) ``` - Approximate value: \[ \text{Service probability} \approx 0.702 \] - Convert to percentage: \[ 70.2\% \] - Rounded to the nearest whole number: \[ \boxed{70\%} \] --- ## **Final Summary:** | Part | Answer | Explanation | |---------|--------|-------------| | **a** | **794 units** | Reorder point to maintain a 90% service level considering demand variability during lead time. | | **b** | **70%** | Service probability after safety stock reduction by 90 units. | --- Feel free to ask for further clarifications!