Let’s solve the EOQ with **quantity discounts** and a **known defective rate**. --- ### **Step 1: Gather and Clarify Data** - **Annual demand (D)** = 10,000 units (usable units) - **Ordering cost (S)** = $300 per order - **Annual holding cost per usable unit (h)** = $0.40 - **Purchase price per unit (P):** - 0–3,000 units: $4.00 - 3,001–6,000 units: $3.90 - >6,000 units: $3.80 - **Defective rate (d)** = 2% - **Disposal cost per defective unit (cₚ)** = $0.20 - **All defective units are disposed of, not returned** - **Assume all calculations are based on usable units** --- ### **Step 2: Adjust for Defective Rate** For every order, only 98% of items received are usable. Let: - **Q** = Order quantity (total received per order) - **Usable units per order** = 0.98Q But **annual demand D is for usable units**, so: \[ \text{Number of orders per year} = \frac{D}{0.98Q} \] --- ### **Step 3: Calculate EOQ at Each Price Level** #### **A. $4.00 per unit (Q ≤ 3,000)** \[ \text{EOQ formula:}\quad Q^* = \sqrt{\frac{2DS}{h \cdot 0.98}} \] - Because holding cost is per usable unit, and there are only 98% usable units per order. Plug in values: \[ Q^* = \sqrt{\frac{2 \times 10,000 \times 300}{0.4 \times 0.98}} \] \[ = \sqrt{\frac{6,000,000}{0.392}} \] \[ = \sqrt{15,306,122.45} \] \[ \approx 3,913 \text{ units} \] But for this price bracket, **max Q = 3,000 units**. So, use **Q = 3,000**. --- #### **B. $3.90 per unit (3,001 ≤ Q ≤ 6,000)** Using EOQ formula: \[ Q^* = \sqrt{\frac{2 \times 10,000 \times 300}{0.4 \times 0.98}} \] As above, **Q\* ≈ 3,913 units**, which is within this range. So, **Q = 3,913**. --- #### **C. $3.80 per unit (Q > 6,000)** Use lower limit, **Q = 6,001**. --- ### **Step 4: Compute Total Annual Cost at Each Price Break** **Total Cost = Purchase Cost + Ordering Cost + Holding Cost + Defective Disposal Cost** Where: - **Purchase Cost:** \( P \times \text{total units ordered annually} \) - **Ordering Cost:** \( S \times \text{number of orders per year} \) - **Holding Cost:** \( h \times \text{average usable inventory} \) - **Disposal Cost:** \( c_p \times \text{number of defectives per year} \) #### **A. Q = 3,000, P = $4.00** - Orders/year: \( \frac{10,000}{0.98 \times 3,000} = 3.401 \) - Units ordered/year: \( 3,000 \times 3.401 = 10,204 \) - Purchase cost: \( 10,204 \times 4 = \$40,816 \) - Ordering cost: \( 3.401 \times 300 = \$1,020 \) - Avg. usable inventory: \( \frac{0.98 \times 3,000}{2} = 1,470 \) - Holding cost: \( 1,470 \times 0.4 = \$588 \) - Defective units/year: \( 10,204 \times 0.02 = 204 \) - Disposal cost: \( 204 \times 0.2 = \$40.8 \) **Total = \$40,816 + \$1,020 + \$588 + \$40.8 = \$42,464.8** --- #### **B. Q = 3,913, P = $3.90** - Orders/year: \( \frac{10,000}{0.98 \times 3,913} = 2.597 \) - Units ordered/year: \( 3,913 \times 2.597 = 10,164 \) - Purchase cost: \( 10,164 \times 3.90 = \$39,639.6 \) - Ordering cost: \( 2.597 \times 300 = \$779 \) - Avg. usable inventory: \( \frac{0.98 \times 3,913}{2} = 1,917 \) - Holding cost: \( 1,917 \times 0.4 = \$766.8 \) - Defective units/year: \( 10,164 \times 0.02 = 203 \) - Disposal cost: \( 203 \times 0.2 = \$40.6 \) **Total = \$39,639.6 + \$779 + \$766.8 + \$40.6 = \$41,225.98** --- #### **C. Q = 6,001, P = $3.80** - Orders/year: \( \frac{10,000}{0.98 \times 6,001} = 1.701 \) - Units ordered/year: \( 6,001 \times 1.701 = 10,206 \) - Purchase cost: \( 10,206 \times 3.80 = \$38,782.8 \) - Ordering cost: \( 1.701 \times 300 = \$510.3 \) - Avg. usable inventory: \( \frac{0.98 \times 6,001}{2} = 2,940.5 \) - Holding cost: \( 2,940.5 \times 0.4 = \$1,176.2 \) - Defective units/year: \( 10,206 \times 0.02 = 204 \) - Disposal cost: \( 204 \times 0.2 = \$40.8 \) **Total = \$38,782.8 + \$510.3 + \$1,176.2 + \$40.8 = \$40,510.1** --- ### **Step 5: Compare and Make Decision** | Order Qty | Price | Total Cost | |-----------|-------|------------| | 3,000 | $4.00 | $42,464.8 | | 3,913 | $3.90 | $41,225.98 | | 6,001 | $3.80 | $40,510.1 | **Lowest annual total cost:** \[ \boxed{Q^* = 6,001 \text{ units},\quad \text{Total Cost} = \$40,510.10} \] --- ## **Summary Table** | Order Quantity | Price per unit | Total Cost | |----------------|---------------|-------------| | 3,000 | $4.00 | $42,464.80 | | 3,913 | $3.90 | $41,225.98 | | 6,001 | $3.80 | $40,510.10 | ### **Optimal Policy:** Order **6,001 units** each time, pay **$3.80 per unit**, and incur a total annual cost of **$40,510.10**. --- ## **References and Further Reading** - Nahmias, S. (2013). *Production and Operations Analysis* (7th Ed.), McGraw-Hill. (See EOQ with defective items and quantity discounts) - Silver, E.A., Pyke, D.F., & Thomas, D.J. (2017). *Inventory and Production Management in Supply Chains* (4th Ed.), CRC Press. (Section: EOQ with non-perfect quality items) - Chopra, S., & Meindl, P. (2015). *Supply Chain Management* (6th Ed.), Pearson. (Section: Quantity discounts and EOQ) ### **Example** Suppose a company faces a 2% defective rate and quantity discounts as above. If they order 6,001 units, they will receive 120 defective units per order (2% of 6,001), which will be disposed of immediately at a cost. The rest are usable, and the calculation above shows the cost-minimizing policy. --- **If you need a spreadsheet version or sensitivity analysis, let me know!**