Inventory Models: Economic Order Quantity

Order too often and a business drowns in delivery charges and paperwork; order too much at once and cash sits idle on shelves, gathering storage costs and risk. The Economic Order Quantity (EOQ) model finds the order size that balances these two opposing costs perfectly. It is one of the oldest and most useful results in operations management, and every BBA, MBA and supply-chain student should be able to derive and apply it.

The Two Costs That Pull in Opposite Directions

The basic EOQ model looks at two annual costs:

• Ordering cost. Every order incurs a fixed cost \( S \) (processing, transport, inspection) regardless of size. If annual demand is \( D \) units and each order is \( Q \) units, there are \( D/Q \) orders a year, so annual ordering cost is \( (D/Q)S \). Bigger orders mean fewer of them.
• Holding cost. Each unit kept in stock costs \( H \) per year (storage, insurance, spoilage, and the opportunity cost of tied-up cash). Stock falls steadily from \( Q \) to 0 and is then replenished, so the average inventory is \( Q/2 \) and the annual holding cost is \( (Q/2)H \). Bigger orders mean more stock.

The total relevant cost is

$$ TC(Q) = \frac{D}{Q}S + \frac{Q}{2}H. $$

Deriving the EOQ

To minimise, differentiate with respect to \( Q \) and set the result to zero:

$$ \frac{dTC}{dQ} = -\frac{DS}{Q^2} + \frac{H}{2} = 0. $$

Solving,

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

The second derivative \( 2DS/Q^3 \) is positive, confirming a minimum. A neat property follows: at \( Q^* \), annual ordering cost equals annual holding cost. If your calculation does not show this equality, something has gone wrong.

Assumptions

The classic model assumes demand is known and constant, lead time is fixed, the whole order arrives at once, there are no stock-outs, and the unit price does not depend on order size. Real life bends these assumptions, but EOQ is surprisingly forgiving, as we will see.

Worked Example 1: A Lahore Electronics Distributor

A distributor sells 12,000 phone chargers a year. Each order costs PKR 2,500 to place and receive, and holding one charger in stock for a year costs PKR 60. The business works 300 days a year and the supplier's lead time is 5 days.

Step 1: EOQ.

$$ Q^* = \sqrt{\frac{2 \times 12{,}000 \times 2{,}500}{60}} = \sqrt{\frac{60{,}000{,}000}{60}} = \sqrt{1{,}000{,}000} = 1{,}000 \text{ units} $$

Step 2: Number of orders per year.

$$ \frac{D}{Q^*} = \frac{12{,}000}{1{,}000} = 12 \text{ orders} $$

Step 3: Annual costs.

$$ \text{Ordering} = 12 \times 2{,}500 = \text{PKR } 30{,}000 $$

$$ \text{Holding} = \frac{1{,}000}{2} \times 60 = \text{PKR } 30{,}000 $$

Total relevant cost is PKR 60,000. Ordering and holding costs are equal, as the theory predicts.

Step 4: Time between orders. With 300 working days and 12 orders, the cycle is \( 300/12 = 25 \) working days.

Step 5: Reorder point. Daily demand is \( 12{,}000/300 = 40 \) units. During the 5-day lead time the business uses \( 40 \times 5 = 200 \) units, so

$$ \text{ROP} = d \times L = 40 \times 5 = 200 \text{ units}. $$

When stock falls to 200 chargers, the next order of 1,000 is placed. Many firms add safety stock to the ROP to protect against uncertain demand; for example, a safety stock of 100 would raise the ROP to 300.

How Sensitive Is the EOQ?

Suppose the manager orders 1,200 instead of 1,000 because cartons come in that size. The cost becomes

$$ TC(1200) = \frac{12{,}000}{1{,}200}(2{,}500) + \frac{1{,}200}{2}(60) = 25{,}000 + 36{,}000 = 61{,}000. $$

A 20% deviation in quantity raised cost by only PKR 1,000, about 1.7%. In general,

$$ \frac{TC(Q)}{TC(Q^*)} = \frac{1}{2}\left(\frac{Q}{Q^*} + \frac{Q^*}{Q}\right), $$

and with \( Q/Q^* = 1.2 \) this gives \( 0.5(1.2 + 0.8333) \approx 1.0167 \). The total-cost curve is flat near its minimum, so rounding EOQ to a convenient pack size costs very little. This robustness is why the model remains so popular.

Worked Example 2: A Pharmacy and a Quantity Discount

A pharmacy chain sells 4,800 boxes of a medicine per year. Each order costs PKR 1,800, the price is PKR 500 per box, and annual holding cost is 20% of the price. The supplier offers a 2% discount (price PKR 490) on orders of 800 boxes or more. Should the pharmacy take it?

When discounts exist, purchase cost \( DP \) must be included in total cost:

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

Step 1: EOQ at the normal price. Here \( H = 0.20 \times 500 = 100 \).

$$ Q^* = \sqrt{\frac{2 \times 4{,}800 \times 1{,}800}{100}} = \sqrt{172{,}800} \approx 415.7, \text{ so about 416 boxes.} $$

Step 2: Total cost at 416 boxes.

• Purchase: \( 4{,}800 \times 500 = 2{,}400{,}000 \)
• Ordering: \( (4{,}800/416) \times 1{,}800 \approx 20{,}769 \)
• Holding: \( (416/2) \times 100 = 20{,}800 \)

Total ≈ PKR 2,441,569.

Step 3: Total cost at the discount break point, 800 boxes. Now \( H = 0.20 \times 490 = 98 \). The EOQ at this price, \( \sqrt{2 \times 4{,}800 \times 1{,}800 / 98} \approx 420 \), is below 800, so the cheapest allowed quantity in this price band is exactly 800.

• Purchase: \( 4{,}800 \times 490 = 2{,}352{,}000 \)
• Ordering: \( (4{,}800/800) \times 1{,}800 = 6 \times 1{,}800 = 10{,}800 \)
• Holding: \( (800/2) \times 98 = 39{,}200 \)

Total = PKR 2,402,000.

Step 4: Compare. Ordering 800 boxes saves about \( 2{,}441{,}569 - 2{,}402{,}000 = \text{PKR } 39{,}569 \) a year. Holding cost almost doubles, but the PKR 48,000 saving on purchase price outweighs it. The pharmacy should take the discount, provided the medicine's shelf life comfortably exceeds the two-month cycle (\( 800/4{,}800 \) of a year).

Sanity check: the discount saves 2% on PKR 2.4 million, i.e. PKR 48,000, and the extra holding-plus-ordering cost is \( (10{,}800 + 39{,}200) - (20{,}769 + 20{,}800) \approx 8{,}431 \). Net saving \( 48{,}000 - 8{,}431 = 39{,}569 \), matching Step 4.

Extensions You Will Meet

1. Economic Production Quantity (EPQ): stock is produced gradually at rate \( p \), so \( Q^* = \sqrt{2DS / (H(1 - d/p))} \).
2. Planned shortages: allowing back-orders at a cost changes the optimum.
3. Probabilistic models: safety stock is set using the standard deviation of lead-time demand and a target service level.
4. ABC analysis: focus EOQ effort on the few items that make up most of the inventory value.

Did You Know?

• The EOQ formula was published by Ford W. Harris in 1913, making it more than a century old.
• It is sometimes called the "Wilson formula" after R. H. Wilson, a consultant who popularised it in the 1930s.
• Because EOQ grows with the square root of demand, doubling sales raises optimal order size by only about 41%.

Key Takeaways

• EOQ balances ordering cost \( (D/Q)S \) against holding cost \( (Q/2)H \), giving \( Q^* = \sqrt{2DS/H} \).
• At the EOQ, annual ordering and holding costs are equal, which is a quick check.
• The reorder point is daily demand times lead time, plus any safety stock.
• Total cost is flat near the optimum, so small deviations cost little.
• With quantity discounts, compare total costs including purchase price at each break point.

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