1. **Problem Statement:** A factory has an annual demand of 12,960 units, ordering cost of 750 per order, unit cost of 100, and inventory carrying rate of 10%. We need to find: (i) Economic Order Quantity (EOQ) (ii) Total cost at EOQ (iii) Optimal number of orders (iv) Optimal interval between orders (in days, assuming 360 days/year) (v) Discount rate to make buying 6480 units as attractive as EOQ purchase. 2. **Formula for EOQ:** $$EOQ = \sqrt{\frac{2DS}{H}}$$ where: - $D$ = annual demand = 12960 units - $S$ = ordering cost = 750 - $H$ = holding cost per unit per year = carrying rate \times unit cost = 0.10 \times 100 = 10 3. **Calculate EOQ:** $$EOQ = \sqrt{\frac{2 \times 12960 \times 750}{10}} = \sqrt{1944000} = 1394.64 \approx 1395 \text{ units}$$ 4. **Total cost at EOQ:** Total cost $= \text{Ordering cost} + \text{Holding cost} + \text{Purchase cost}$ Ordering cost $= \frac{D}{EOQ} \times S = \frac{12960}{1394.64} \times 750 = 9.29 \times 750 = 6967.5$ Holding cost $= \frac{EOQ}{2} \times H = \frac{1394.64}{2} \times 10 = 6973.2$ Purchase cost $= D \times \text{unit cost} = 12960 \times 100 = 1,296,000$ Total cost $= 6967.5 + 6973.2 + 1,296,000 = 1,309,940.7$ 5. **Optimal number of orders:** $$\text{Number of orders} = \frac{D}{EOQ} = \frac{12960}{1394.64} = 9.29 \approx 9 \text{ orders}$$ 6. **Optimal interval between orders:** $$\text{Interval} = \frac{\text{Number of working days}}{\text{Number of orders}} = \frac{360}{9.29} = 38.74 \approx 39 \text{ days}$$ 7. **Discount rate for 6480 units purchase:** We want purchase cost at 6480 units with discount rate $r$ to equal total cost at EOQ: $$6480 \times 100 \times (1 - r) = 1,309,940.7 - (\text{ordering cost} + \text{holding cost})$$ Ordering + holding cost at EOQ $= 6967.5 + 6973.2 = 13,940.7$ So purchase cost part $= 1,309,940.7 - 13,940.7 = 1,296,000$ Set: $$6480 \times 100 \times (1 - r) = 1,296,000$$ $$648,000 \times (1 - r) = 1,296,000$$ Divide both sides by 648,000: $$\cancel{648,000} \times (1 - r) = \frac{1,296,000}{\cancel{648,000}} = 2$$ This implies: $$1 - r = 2 \Rightarrow r = 1 - 2 = -1$$ A negative discount rate is impossible, meaning buying 6480 units at full price is not as attractive as EOQ purchase. To make it equally attractive, the supplier would have to pay the buyer, which is unrealistic. **Final answers:** - EOQ $= 1395$ units - Total cost at EOQ $= 1,309,940.7$ - Optimal number of orders $= 9$ - Optimal interval between orders $= 39$ days - Discount rate $= \text{not feasible (negative)}$