Let's break down the problem step by step: ### **Given Data** - **Annual demand, \(D\)**: 20,000 units/year - **Order cost, \(S\)**: \$600 per order - **Holding cost rate, \(h\)**: 0.4 per year (i.e., 40% of unit cost per year) - **Price schedule**: - First 300 units in any order: \$10/unit - Units beyond 300: \$5/unit If you order \(Q\) units: - If \(Q \leq 300\): All units cost \$10/unit - If \(Q > 300\): First 300 units at \$10, remaining \(Q-300\) at \$5 --- ### **Step 1: Express Purchase Cost per Order** If you order \(Q\) units: - **If \(Q \leq 300\):** Total purchase cost = \(10Q\) - **If \(Q > 300\):** Total purchase cost = \(10 \times 300 + 5 \times (Q-300) = 3000 + 5(Q-300) = 5Q + 1500\) So, average unit price for \(Q > 300\): \[ \text{Average unit price} = \frac{5Q + 1500}{Q} \] --- ### **Step 2: Economic Order Quantity (EOQ) Approach** Let’s first ignore the price break and find EOQ as if the price per unit is constant. #### **Case 1: All units at \$10 (if order size ≤ 300 units)** - Holding cost per unit per year: \(hC = 0.4 \times 10 = \$4\) - EOQ formula: \[ Q^* = \sqrt{\frac{2DS}{hC}} = \sqrt{\frac{2 \times 20000 \times 600}{4}} = \sqrt{\frac{24,000,000}{4}} = \sqrt{6,000,000} \approx 2449.49 \] - But this is **much greater than 300**, so this case is not possible if we want to use EOQ. #### **Case 2: Price Break (\$5/unit for Q > 300)** - Holding cost per unit per year: \(hC = 0.4 \times 5 = \$2\) - EOQ formula: \[ Q^* = \sqrt{\frac{2DS}{hC}} = \sqrt{\frac{2 \times 20000 \times 600}{2}} = \sqrt{\frac{24,000,000}{2}} = \sqrt{12,000,000} \approx 3464.1 \] - This is **greater than 300**, so this order quantity is feasible for the discount. --- ### **Step 3: Total Cost Calculation** The **Total Cost per Year** is: \[ \text{Total Cost} = \text{Purchase Cost} + \text{Order Cost} + \text{Holding Cost} \] Where: - Purchase Cost = \(D \times \) (average price per unit) - Order Cost = \(\frac{D}{Q} \times S\) - Holding Cost = Average inventory \(\times\) Holding cost per unit = \(\frac{Q}{2} \times hC\) #### **Case 1: Order Q ≤ 300 (All at \$10/unit)** Try \(Q = 300\): - Purchase cost: \(20,000 \times 10 = \$200,000\) - Order cost: \(20,000/300 \times 600 = 66.67 \times 600 = \$40,000\) - Holding cost: \(300/2 \times 4 = 150 \times 4 = \$600\) - **Total cost:** \(200,000 + 40,000 + 600 = \$240,600\) #### **Case 2: Order Q = EOQ ≈ 3464 (Price break for units > 300)** - Average price per unit: \[ \text{Total purchase cost per cycle} = 300 \times 10 + (3464-300) \times 5 = 3000 + 3164 \times 5 = 3000 + 15,820 = \$18,820 \] \[ \text{Average price per unit} = 18,820/3464 \approx \$5.43 \] - Yearly purchase cost: \(20,000 \times 5.43 = \$108,600\) - Order cost: \(20,000/3464 \times 600 = 5.77 \times 600 = \$3,462\) - Holding cost: \(3464/2 \times 2 = 1,732 \times 2 = \$3,464\) - **Total cost:** \(108,600 + 3,462 + 3,464 = \$115,526\) #### **Case 3: Order Q just above 300 (say, 301 units)** - Total purchase cost per cycle: \(300 \times 10 + 1 \times 5 = 3,005\) - Average price per unit: \(3,005/301 \approx \$9.9867\) - Yearly purchase cost: \(20,000 \times 9.9867 = \$199,734\) - Order cost: \(20,000/301 \times 600 = 66.445 \times 600 = \$39,867\) - Holding cost: \(301/2 \times 2 = 150.5 \times 2 = \$301\) - **Total cost:** \(199,734 + 39,867 + 301 = \$239,902\) --- ### **Step 4: Conclusion** - Ordering at EOQ (3464 units) gives total annual cost of **\$115,526**. - Ordering at Q = 300 or Q = 301 costs over \$239,000. - **Optimal order quantity is 3,464 units, using the price break.** --- ## **Final Answers** ### **Optimal Order Quantity:** \[ \boxed{3,464 \text{ units (rounded to the nearest unit)}} \] ### **Corresponding Optimal Total Cost per Year:** \[ \boxed{\$115,526} \] --- ### **Summary Table** | Order Quantity | Unit Cost | Total Annual Cost | |:--------------:|:------------------:|:----------------:| | 300 | \$10 | \$240,600 | | 301 | \$9.99 | \$239,902 | | **3464** | **\$(5 + small avg)** | **\$115,526** | - **Order 3,464 units each time and pay \$5 for units above the first 300 each order.** - **Order cost, holding cost, and purchase cost sum to \$115,526 per year.** If you have follow-up questions or need a breakdown by cost component or cycle, let me know!