Inventory Control

Economic order quantity is the order size that minimises the total annual inventory cost (ordering plus holding).

Assumptions

• Demand $D$ per year is known and constant.
• Lead time is constant and no shortage is allowed.
• Whole order of $Q$ units arrives at once.
• Ordering cost $S$ per order and holding cost $H$ per unit per year are constant.
• Price per unit does not depend on the quantity (no discount).

Derivation

Inventory falls uniformly from $Q$ to 0, so the average inventory is $Q/2$.

$$\text{Annual ordering cost} = \frac{D}{Q} S, \qquad \text{Annual holding cost} = \frac{Q}{2} H$$ $$TC = \frac{D}{Q}S + \frac{Q}{2}H$$

(The purchase cost $DC$ is constant and is left out of the minimisation.) For minimum cost, differentiate with respect to $Q$:

$$\frac{dTC}{dQ} = -\frac{DS}{Q^2} + \frac{H}{2} = 0 \;\Rightarrow\; Q^2 = \frac{2DS}{H}$$ $$\boxed{Q^* = \sqrt{\frac{2DS}{H}}}$$

The second derivative $2DS/Q^3$ is positive, so this is a minimum.

Ordering cost equals holding cost at EOQ

From $Q^{*2} = 2DS/H$, we get $\dfrac{D}{Q^*}S = \dfrac{Q^*}{2}H$ because both equal $\sqrt{DSH/2}$. Hence

$$TC_{min} = 2\sqrt{\frac{DSH}{2}} = \sqrt{2DSH}$$

Related results

Number of orders per year $N = D/Q^*$; time between orders $T = Q^*/D$; reorder point $= d \times L$ (daily demand times lead time). The total cost curve is flat near the EOQ, so small changes in $Q$ change cost very little.

Given

$D = 12000$ units/year, $S =$ Rs 300, $H = 0.20 \times 150 =$ Rs 30 per unit per year, 300 working days, $L = 10$ days.

(a) EOQ

$$Q^* = \sqrt{\frac{2DS}{H}} = \sqrt{\frac{2 \times 12000 \times 300}{30}} = \sqrt{240000} = 489.9 \approx 490\ \text{units}$$

(b) Orders and cycle time

$$N = \frac{D}{Q^*} = \frac{12000}{489.9} = 24.5\ \text{orders/year}$$ $$T = \frac{300}{24.5} = 12.2\ \text{working days}$$

(c) Annual ordering plus holding cost

Ordering cost $= 24.5 \times 300 =$ Rs 7,348; holding cost $= (489.9/2) \times 30 =$ Rs 7,348.

$$TC = 7348 + 7348 = \text{Rs } 14{,}697$$

(Purchase cost is $12000 \times 150 =$ Rs 1,800,000, so total annual cost including purchase is Rs 1,814,697.)

(d) Reorder point

Daily demand $d = 12000/300 = 40$ units/day.

$$ROP = d \times L = 40 \times 10 = 400\ \text{units}$$

An order of 490 units is placed when the stock falls to 400 units.

(e) Order quantity of 600

$$TC_{600} = \frac{12000}{600}(300) + \frac{600}{2}(30) = 6000 + 9000 = \text{Rs } 15{,}000$$

Extra cost $= 15000 - 14697 =$ Rs 303 per year (about 2.1% higher).

Answer: EOQ = 490 units; 24.5 orders/year, every 12.2 working days; TC = Rs 14,697; ROP = 400 units; ordering 600 costs Rs 303 more per year.

Method

ABC analysis ranks items by annual consumption value (usage x unit cost) and divides them into classes; a few items carry most of the money value.

Step 1: annual value

Total value $=$ Rs 155,700. Items are sorted in descending order of value.

Step 2: classes

Control policy

• A items: tight control; accurate records, frequent review, small lots and careful forecasting, strict approval, low safety stock, close supplier follow-up.
• B items: moderate control; periodic review, normal records, reasonable safety stock.
• C items: loose control; simple records, bulk or two-bin system, large safety stock, infrequent review.

Answer: A = I2, I4, I3 (70.1% of value); B = I1, I8, I6 (18.3%); C = I10, I9, I5, I7 (11.6%).

(a) Definitions

• Lead time: the time between placing an order and receiving the goods in stock.
• Reorder level (point): the stock level at which a new order is placed; $ROP = \text{demand during lead time} + \text{safety stock}$.
• Safety stock (buffer stock): extra stock kept to protect against variation in demand and lead time. It is kept because demand and supply are uncertain, and a stock-out causes lost sales or production stoppage.

(b) Constant lead time

Demand during lead time $= d \times L = 50 \times 9 = 450$ units. Standard deviation of demand during lead time:

$$\sigma_L = \sigma_d \sqrt{L} = 8\sqrt{9} = 24\ \text{units}$$ $$SS = z\,\sigma_L = 1.645 \times 24 = 39.5 \approx 40\ \text{units}$$ $$ROP = 450 + 39.5 = 489.5 \approx 490\ \text{units}$$

(c) Variable lead time ($\sigma_{LT} = 2$ days)

$$\sigma_{dL} = \sqrt{L\sigma_d^2 + d^2\sigma_{LT}^2} = \sqrt{9(64) + 2500(4)} = \sqrt{10576} = 102.8\ \text{units}$$ $$SS = 1.645 \times 102.8 = 169.2 \approx 169\ \text{units}$$ $$ROP = 450 + 169.2 = 619.2 \approx 619\ \text{units}$$

Variation in lead time raises the required safety stock more than four times, showing the value of reliable suppliers.

Answer: (b) Safety stock = 40 units, ROP = 490 units; (c) safety stock = 169 units, ROP = 619 units.