Let's break down the problem and solve for the **optimal order quantity** and **total annual cost**. ### **Known Data** - **Annual demand (\(D\))** = 20,000 air filters/year - **Fixed order cost (\(S\))** = \$600/order - **Inventory holding cost rate (\(h\))** = 0.4 per year (this is likely 40% of unit price per year) - **Price schedule:** - **First 300 units**: \$10/unit - **Additional units**: \$5/unit For an order of \( Q \) units: - If \( Q \leq 300 \): all units at \$10/unit. - If \( Q > 300 \): first 300 at \$10 each, rest at \$5 each. --- ## **Step 1: EOQ for Each Price Bracket** ### **Case 1: All units at \$10 (if order ≤ 300)** #### **EOQ Formula:** \[ EOQ = \sqrt{\frac{2DS}{hC}} \] Here, \( C \) is the unit cost. \[ EOQ_{10} = \sqrt{ \frac{2 \times 20000 \times 600}{0.4 \times 10} } = \sqrt{ \frac{24,000,000}{4} } = \sqrt{6,000,000} \approx 2449.49 \] But this number is **greater than 300**, so **Case 1 is not feasible** (for all units at \$10, Q must be ≤ 300). --- ### **Case 2: First 300 at \$10, rest at \$5 (if order > 300)** Average unit cost for \( Q > 300 \): \[ \text{Total purchase cost} = 300 \times 10 + (Q-300) \times 5 = 3000 + 5(Q-300) = 5Q + 1500 \] \[ \text{Average unit cost} = \frac{5Q + 1500}{Q} \] However, **holding cost per unit per year** is 40% of the unit price paid for each unit. For calculation, we use the marginal cost (\$5 per unit) for units beyond 300. But for EOQ calculation, we use the lower price bracket since the EOQ is above 300. So, for \( Q > 300 \), **unit cost for holding cost** is \$5. \[ EOQ_{5} = \sqrt{ \frac{2DS}{hC} } = \sqrt{ \frac{2 \times 20000 \times 600}{0.4 \times 5} } = \sqrt{ \frac{24,000,000}{2} } = \sqrt{12,000,000} \approx 3464.1 \] This is > 300, so in this bracket, EOQ is feasible. --- ## **Step 2: Calculate Total Annual Cost at EOQ** Total annual cost = Purchasing + Ordering + Holding Where: - **Purchasing cost/year** = \( D \times \) **Average Unit Cost** - **Ordering cost/year** = \( \frac{D}{Q} \times S \) - **Holding cost/year** = Average inventory \(\times\) holding cost per unit per year ### **Case A: Order Q = 300 (all at \$10/unit)** \[ \text{Purchasing cost} = 20,000 \times 10 = \$200,000 \] \[ \text{Ordering cost} = \frac{20,000}{300} \times 600 \approx 66.67 \times 600 = \$40,000 \] \[ \text{Average inventory} = 300 / 2 = 150 \] \[ \text{Holding cost per unit} = 0.4 \times 10 = \$4 \] \[ \text{Holding cost} = 150 \times 4 = \$600 \] \[ \text{Total annual cost} = 200,000 + 40,000 + 600 = \$240,600 \] --- ### **Case B: Order Q = EOQ = 3464 (first 300 at \$10, rest at \$5)** - **Purchase cost per order**: \( 300 \times 10 + (Q-300) \times 5 \) - **Average unit cost:** \( \frac{3000 + (Q-300)\times5}{Q} \) \[ \text{Average unit cost} = \frac{3000 + (3464-300)\times5}{3464} = \frac{3000 + 3164 \times 5}{3464} = \frac{3000 + 15820}{3464} = \frac{18820}{3464} \approx 5.43 \] \[ \text{Purchasing cost per year} = 20,000 \times 5.43 = \$108,600 \] \[ \text{Ordering cost} = \frac{20,000}{3464} \times 600 \approx 5.77 \times 600 = \$3,462 \] \[ \text{Average inventory} = 3464 / 2 = 1732 \] \[ \text{Holding cost per unit} = 0.4 \times 5 = \$2 \] \[ \text{Holding cost} = 1732 \times 2 = \$3,464 \] \[ \text{Total annual cost} = 108,600 + 3,462 + 3,464 = \$115,526 \] --- ### **Step 3: Check at Order Quantity Just Above 300 (Q = 301)** \[ \text{Purchase cost per order} = 300 \times 10 + 1 \times 5 = 3000 + 5 = 3005 \] \[ \text{Average unit cost} = \frac{3005}{301} \approx 9.99 \] \[ \text{Purchasing cost per year} = 20,000 \times 9.99 = \$199,800 \] \[ \text{Ordering cost} = \frac{20,000}{301} \times 600 \approx 66.45 \times 600 = \$39,870 \] \[ \text{Average inventory} = 301/2 = 150.5 \] \[ \text{Holding cost per unit} = 0.4 \times 5 = \$2 \] \[ \text{Holding cost} = 150.5 \times 2 = \$301 \] \[ \text{Total cost} = 199,800 + 39,870 + 301 = \$240,000 \] --- ## **Conclusion** - **Order Q = 300 (all at \$10):** \$240,600/year - **Order Q = 301 (first 300 at \$10, 1 at \$5):** \$240,000/year - **Order Q = 3464 (EOQ, mostly at \$5):** \$115,526/year ### **Optimal Order Quantity and Cost** - **Optimal Order Quantity \(Q^*\) = 3,464** - **Optimal Total Cost per Year** = **\$115,526** --- ## **Final Answers** ### **Optimal order quantity:** \[ \boxed{3,464} \] ### **Corresponding optimal total cost per year:** \[ \boxed{\$115,526} \] --- **If you need the answer expressed as cost per unit time (per year), it is \$115,526.**