1. **State the problem:** We have three discount options for ordering skateboards with different quantity discounts and fixed order costs. We want to find the most cost-effective option plan, the adjusted order quantity for that plan, the annual holding cost, and the total annual inventory cost. 2. **Given data:** - Regular price $P = 40$ - Order cost $S = 95$ - Annual demand $D = 23750$ - Interest rate (holding cost rate) $i = 0.13$ - Discount schedules: - A: 1-999 units, 0% discount, price $P_A = 40$ - B: 1000-2299 units, 0.75% discount, price $P_B = 40 \times (1 - 0.0075) = 39.7$ - C: 2300+ units, 2.5% discount, price $P_C = 40 \times (1 - 0.025) = 39$ 3. **Formulas:** - Economic Order Quantity (EOQ): $$ EOQ = \sqrt{\frac{2DS}{H}} $$ where $H = i \times P$ is the holding cost per unit per year. - Total annual inventory cost: $$ TC = \text{Purchase cost} + \text{Ordering cost} + \text{Holding cost} $$ $$ TC = PD + \frac{DS}{Q} + \frac{HQ}{2} $$ 4. **Calculate EOQ and total cost for each option:** **Option A:** - $H_A = 0.13 \times 40 = 5.2$ - $EOQ_A = \sqrt{\frac{2 \times 23750 \times 95}{5.2}} = \sqrt{868269.23} \approx 931.8$ - Since EOQ_A is within 1-999, valid. - Total cost: - Purchase cost = $40 \times 23750 = 950000$ - Ordering cost = $\frac{23750 \times 95}{931.8} \approx 2420$ - Holding cost = $\frac{5.2 \times 931.8}{2} = 2420$ - $TC_A = 950000 + 2420 + 2420 = 954840$ **Option B:** - $H_B = 0.13 \times 39.7 = 5.161$ - $EOQ_B = \sqrt{\frac{2 \times 23750 \times 95}{5.161}} = \sqrt{873682.5} \approx 934.7$ - EOQ_B is less than 1000, so order quantity must be at least 1000 to get discount. - Adjust $Q_B = 1000$ - Total cost: - Purchase cost = $39.7 \times 23750 = 942875$ - Ordering cost = $\frac{23750 \times 95}{1000} = 2256.25$ - Holding cost = $\frac{5.161 \times 1000}{2} = 2580.5$ - $TC_B = 942875 + 2256.25 + 2580.5 = 947711$ **Option C:** - $H_C = 0.13 \times 39 = 5.07$ - $EOQ_C = \sqrt{\frac{2 \times 23750 \times 95}{5.07}} = \sqrt{889867.4} \approx 943.3$ - EOQ_C less than 2300, so order quantity must be at least 2300 to get discount. - Adjust $Q_C = 2300$ - Total cost: - Purchase cost = $39 \times 23750 = 923250$ - Ordering cost = $\frac{23750 \times 95}{2300} \approx 981.52$ - Holding cost = $\frac{5.07 \times 2300}{2} = 5830.5$ - $TC_C = 923250 + 981.52 + 5830.5 = 930062$ 5. **Rank options by total cost (lowest to highest):** - Option C: 930062 (most preferred) - Option B: 947711 - Option A: 954840 (least preferred) 6. **Adjusted order quantity for most preferred (Option C):** $$ Q_C = 2300 $$ 7. **Annual holding cost for Option C:** $$ H_C = \frac{5.07 \times 2300}{2} = 5831 $$ (rounded) 8. **Total annual inventory cost for Option C:** $$ TC_C = 930062 $$ (rounded) **Final answers:** - Preferred order: C, B, A - Adjusted order quantity (Option C): 2300 - Annual holding cost (Option C): 5831 - Total annual inventory cost (Option C): 930062