Training Industrial Engineering Inventory Models — EOQ & Reorder Point
3 / 5

Inventory Models — EOQ & Reorder Point

24 min Industrial Engineering

Inventory Models — EOQ & Reorder Point

Managing inventory involves balancing ordering costs (placing orders) and holding costs (storing items). The EOQ model finds the optimal order quantity.

Economic Order Quantity (EOQ)

$$Q^* = \sqrt{\frac{2DS}{H}}$$

where $D$ = annual demand, $S$ = ordering cost per order, $H$ = holding cost per unit per year.

Total Annual Cost

$$TC = \frac{D}{Q}S + \frac{Q}{2}H + DC$$

where $C$ = unit cost. At $Q^*$, ordering cost equals holding cost.

Reorder Point

$$ROP = d \times L$$

where $d$ = daily demand and $L$ = lead time in days. With safety stock: $ROP = dL + z\sigma_L$.

Example 1

$D = 10{,}000$ units/year, $S = \$50$/order, $H = \$2$/unit/year. Find EOQ.

$$Q^* = \sqrt{\frac{2(10{,}000)(50)}{2}} = \sqrt{500{,}000} = 707 \text{ units}$$

Example 2

Using Example 1, find the number of orders per year and total annual ordering + holding cost.

Orders/year: $D/Q^* = 10{,}000/707 = 14.1$ orders

Ordering cost: $14.1 \times 50 = \$707$

Holding cost: $(707/2) \times 2 = \$707$

Total: $\$1{,}414$ (at EOQ, they're equal!)

Example 3

Daily demand is 40 units, lead time is 5 days, safety stock is 50 units. Find the reorder point.

$$ROP = 40 \times 5 + 50 = 250 \text{ units}$$

When inventory drops to 250 units, place a new order.

Practice Problems

1. $D = 5{,}000$, $S = \$25$, $H = \$5$. Find EOQ.
2. For #1, find total cost (ordering + holding).
3. $D = 12{,}000$, $S = \$100$, $H = \$4$. Find EOQ and number of orders/year.
4. Daily demand $= 50$, lead time $= 7$ days, no safety stock. Find ROP.
5. Same as #4 but with safety stock for 95% service level ($z = 1.65$, $\sigma_d = 10$ units/day). Find safety stock and ROP.
6. If holding cost doubles (from $H$ to $2H$), how does EOQ change?
7. A supplier offers a 5% discount for orders ≥ 1{,}000 units. $D = 8{,}000$, $S = \$40$, $H = \$3$, $C = \$20$. Should you take the discount?
8. Production order quantity: $Q_p^* = \sqrt{\frac{2DS}{H(1-d/p)}}$ where $p$ = production rate, $d$ = demand rate. $D = 10{,}000$, $S = \$50$, $H = \$2$, $p = 50$/day, $d = 40$/day. Find $Q_p^*$.
9. Average inventory with EOQ $= Q^*/2$. If $Q^* = 500$, what is the average inventory?
10. Time between orders: $T = Q^*/D$. If $Q^* = 707$, $D = 10{,}000$ (250 working days). Find $T$ in days.
11. Annual holding cost is 25% of unit cost ($H = 0.25C$). If $C = \$40$, $D = 2{,}000$, $S = \$30$. Find EOQ.
12. ABC analysis: top 20% of items account for what % of total value?
Show Answer Key

1. $Q^* = \sqrt{2(5000)(25)/5} = \sqrt{50{,}000} = 224$ units

2. $TC = (5000/224)(25) + (224/2)(5) = 558 + 560 = \$1{,}118$

3. $Q^* = \sqrt{2(12000)(100)/4} = \sqrt{600{,}000} = 775$ units; $N = 12000/775 = 15.5$ orders

4. $ROP = 50 \times 7 = 350$ units

5. $\sigma_L = 10\sqrt{7} = 26.5$ units; SS $= 1.65 \times 26.5 = 43.7$; $ROP = 350 + 44 = 394$ units

6. $Q^* \propto 1/\sqrt{H}$; doubling $H$ → EOQ × $1/\sqrt{2} \approx 0.707$, reduced by ~29%

7. EOQ $= \sqrt{2(8000)(40)/3} = 461$. TC at 461: ordering+holding = $\$1{,}384$ + item cost $= \$161{,}384$. At 1000: TC $= 320 + 1500 + 152{,}000 = \$153{,}820$. Take the discount.

8. $Q_p^* = \sqrt{500{,}000/(2 \times 0.2)} = \sqrt{1{,}250{,}000} = 1{,}118$ units

9. 250 units

10. $T = 707/(10{,}000/250) = 707/40 = 17.7$ working days

11. $H = 10$; $Q^* = \sqrt{2(2000)(30)/10} = \sqrt{12{,}000} = 110$ units

12. Approximately 80% (the 80/20 rule or Pareto principle)

Statistical Process Control Queuing Theory & Work Measurement