Let's tackle each part step by step with explanations and calculations. --- ### (1) **Economic Order Quantity (EOQ)** **Formula:** \[ EOQ = \sqrt{\frac{2DS}{H}} \] #### **Given:** - Annual demand, \(D = 12,000\) units - Ordering cost per order, \(S = \$100\) - Unit purchase cost = \$10 - Annual holding cost rate = 20% of unit cost \(\rightarrow\) Annual holding cost per unit, \(H = 0.2 \times \$10 = \$2\) #### **Calculation:** \[ EOQ = \sqrt{\frac{2 \times 12,000 \times 100}{2}} \] \[ EOQ = \sqrt{\frac{2,400,000}{2}} \] \[ EOQ = \sqrt{1,200,000} \] \[ EOQ \approx 1095.45 \] **Rounded to nearest whole unit:** \[ \boxed{1095\ \text{units}} \] --- ### (2) **Reorder Point (ROP) Without Safety Stock** **Formula:** \[ ROP = \left(\frac{\text{Annual demand}}{\text{Working days per year}}\right) \times \text{Lead time} \] #### **Given:** - Annual demand = 12,000 units - Working days per year = 300 - Lead time = 5 days #### **Calculation:** \[ \text{Daily demand} = \frac{12,000}{300} = 40\ \text{units/day} \] \[ ROP = 40 \times 5 = 200\ \text{units} \] \[ \boxed{200\ \text{units}} \] --- ### (3) **Safety Stock and ROP with Safety Stock** #### **Formulas:** - **Safety Stock:** \[ \text{Safety Stock} = Z \times \sigma \times \sqrt{L} \] - **ROP with Safety Stock:** \[ ROP = (\text{Average demand during lead time}) + (\text{Safety stock}) \] #### **Given:** - \(Z = 1.645\) (for 95% service level) - \(\sigma = 5\) units (std. dev. of daily demand) - \(L = 5\) days (lead time) #### **Calculation:** \[ \text{Safety Stock} = 1.645 \times 5 \times \sqrt{5} \] \[ \sqrt{5} \approx 2.236 \] \[ \text{Safety Stock} = 1.645 \times 5 \times 2.236 \approx 1.645 \times 11.18 \approx 18.38 \] **Rounded to nearest whole unit:** \[ \text{Safety Stock} = \boxed{18\ \text{units}} \] **Average demand during lead time:** \[ 40 \text{ units/day} \times 5 \text{ days} = 200 \text{ units} \] **ROP with Safety Stock:** \[ ROP = 200 + 18 = \boxed{218\ \text{units}} \] --- ### (4) **Total Annual Inventory Cost** #### **Components:** - **Annual ordering cost:** \((D/EOQ) \times S\) - **Annual holding cost:** \((\text{Average inventory}) \times H\) - Average inventory = \(EOQ/2 + \text{Safety stock}\) #### **Given:** - D = 12,000 - EOQ = 1095 - S = $100 - H = $2 - Safety stock = 18 #### **Ordering Cost:** \[ \text{Annual ordering cost} = \frac{12,000}{1095} \times 100 \approx 10.9589 \times 100 = \$1,095.89 \] #### **Average Inventory:** \[ \text{Average inventory} = \frac{1095}{2} + 18 = 547.5 + 18 = 565.5 \] #### **Holding Cost:** \[ \text{Annual holding cost} = 565.5 \times 2 = \$1,131.00 \] #### **Total Cost:** \[ \text{Total annual inventory cost} = 1,095.89 + 1,131.00 = \boxed{\$2,226.89} \] --- ### (5) **Inventory Replenishment Flowchart** Here’s a simple conceptual flowchart in text form (using standard shapes): ``` [Start] | [Monitor Inventory Level] <--- Rectangular (Process) | [Is Inventory <= ROP?] <--- Diamond (Decision) | \ Yes No | | [Place Order] [Continue Monitoring] | [Receive & Inspect Shipment] | [Update Inventory Records] | [Return to: Monitor Inventory Level] ``` **Steps:** 1. **Monitor Inventory Level** (rectangle/process) 2. **Is Inventory ≤ ROP?** (diamond/decision) - If **No**: continue monitoring - If **Yes**: place order 3. **Place Order** (rectangle/process) 4. **Receive and Inspect Shipment** (rectangle/process) 5. **Update Inventory Records** (rectangle/process) 6. **Loop back to monitoring** --- ## **Summary Table** | Step | Value | |-----------------------------|--------------| | EOQ | 1095 units | | ROP (no safety stock) | 200 units | | Safety Stock | 18 units | | ROP (with safety stock) | 218 units | | Total Inventory Cost | $2,226.89 | --- **Let me know if you need a graphic version of the flowchart!**