1. **Problem Statement:** ABC Machine Manufacturing uses 500,000 pounds of steel annually at a cost of 2.20 per pound. The inventory carrying cost is 20% of the purchase price, and the ordering cost is 1,000 per order. Safety stock is 20,000 pounds, delivery time is 10 days, and the company operates 365 days a year. We need to find: - Economic Order Quantity (EOQ) - Number of orders per year - Average inventory level - Inventory reorder point - Total annual inventory cost 2. **Formulas and Important Rules:** - EOQ formula: $$EOQ = \sqrt{\frac{2DS}{H}}$$ where: - $D$ = annual demand - $S$ = ordering cost per order - $H$ = holding cost per unit per year - Average inventory = $$\frac{EOQ}{2} + \text{safety stock}$$ - Number of orders per year = $$\frac{D}{EOQ}$$ - Reorder point = $$d \times L + \text{safety stock}$$ where: - $d$ = daily demand - $L$ = lead time in days - Total annual inventory cost = $$\text{Ordering cost} + \text{Holding cost} + \text{Purchase cost}$$ 3. **Calculations:** - Given: - $D = 500,000$ pounds/year - $S = 1,000$ per order - Purchase price per pound = 2.20 - Holding cost rate = 20% of purchase price - Safety stock = 20,000 pounds - Lead time $L = 10$ days - Operating days = 365 - Holding cost per unit per year: $$H = 0.20 \times 2.20 = 0.44$$ - Calculate EOQ: $$EOQ = \sqrt{\frac{2 \times 500,000 \times 1,000}{0.44}} = \sqrt{\frac{1,000,000,000}{0.44}} = \sqrt{2,272,727,272.73} \approx 47,667.63$$ pounds - Number of orders per year: $$\frac{D}{EOQ} = \frac{500,000}{47,667.63} \approx 10.49 \approx 10 \text{ orders (rounded down)}$$ - Average inventory level: $$\frac{EOQ}{2} + \text{safety stock} = \frac{47,667.63}{2} + 20,000 = 23,833.82 + 20,000 = 43,833.82$$ pounds - Daily demand $d$: $$d = \frac{D}{365} = \frac{500,000}{365} \approx 1369.86$$ pounds/day - Reorder point: $$d \times L + \text{safety stock} = 1369.86 \times 10 + 20,000 = 13,698.63 + 20,000 = 33,698.63$$ pounds - Total annual inventory cost: - Ordering cost: $$\text{Number of orders} \times S = 10.49 \times 1,000 = 10,490$$ - Holding cost: $$H \times \text{average inventory} = 0.44 \times 43,833.82 = 19,288.88$$ - Purchase cost: $$D \times \text{purchase price} = 500,000 \times 2.20 = 1,100,000$$ - Total cost: $$10,490 + 19,288.88 + 1,100,000 = 1,129,778.88$$ 4. **Final Answers:** - Economic Order Quantity (EOQ): **47,668 pounds** - Number of orders per year: **10 orders** - Average inventory level: **43,834 pounds** - Inventory reorder point: **33,699 pounds** - Total annual inventory cost: **1,129,779** (rounded)