Inventory costs and the EOQ model

Inventory cost components, the basic EOQ model with its assumptions and derivation, total cost, cycle time, the cost penalty of non-optimal lots, and reorder point with long lead times.

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Why it matters

Inventory typically ties up 20–40 % of a manufacturing firm's working capital. Order too often and the purchase department drowns in paperwork and freight charges; order too much at once and money sits on shelves, rusts or becomes obsolete. The economic order quantity (EOQ) model finds the balance and is the base on which every other inventory model (EPQ, discounts, safety stock) is built.

Key ideas

Why hold inventory. To decouple stages of production, to buy or make in economical lots (cycle stock), to protect against uncertainty (safety stock), to meet seasonal peaks (anticipation stock) and to fill the pipeline (in-transit stock).

Inventory-related costs.

  • Purchase (item) cost – unit price × annual demand. It matters only when the price depends on lot size (quantity discounts).
  • Ordering cost S (set-up cost for manufactured items) – cost per order independent of quantity: preparing the order, follow-up, receiving, inspection, invoice processing; for production, machine set-up and first-piece inspection.
  • Holding (carrying) cost H – cost of keeping one unit in stock for a year: interest or opportunity cost of capital, storage space, handling, insurance, taxes, obsolescence, deterioration. Often expressed as a fraction i of unit cost: H = i·C (typically 15–30 % per year).
  • Shortage (stock-out) cost – lost sales, backorder handling, idle production, goodwill. Not part of the basic EOQ model.

Basic EOQ (Wilson/Harris) model – assumptions.

  • Demand is known and constant at D units per year.
  • The whole order arrives at once (instantaneous replenishment).
  • Lead time is known and constant.
  • Ordering cost per order and holding cost per unit-year are constant; no quantity discounts.
  • No shortages are allowed.
  • One item, no interaction with others.

The saw-tooth. Stock rises from 0 to Q at each delivery and falls linearly to 0, so average inventory is Q/2. Annual ordering cost = (D/Q)·S falls as Q rises; annual holding cost = (Q/2)·H rises with Q. The total is minimum where the derivative is zero, which is also the point where the two costs are equal.

Robustness. The total-cost curve is flat near the optimum. Ordering Q instead of Q* raises cost only by the factor ½(Q/Q* + Q*/Q): a 25 % error in Q costs only about 2.5 % extra. So rounding Q to a pack size or using rough cost estimates is acceptable. Because EOQ depends on the square root, doubling demand raises EOQ by only √2 ≈ 1.41.

Reorder point (ROP). With constant demand and lead time, place the next order when stock on hand falls to the demand during lead time. If the lead time is longer than the cycle time (L > T), more than one order is outstanding at a time; the on-hand reorder level is then lead-time demand minus Q times the number of orders already in the pipeline (equivalently, track inventory position = on hand + on order).

Formulas

TC(Q) = (D/Q)·S + (Q/2)·H + D·C

  • TC = total annual cost (₹/year), D = annual demand (units/year), Q = order quantity (units), S = ordering cost (₹/order), H = holding cost (₹/unit·year), C = unit price (₹/unit). Drop D·C when price is constant.

Q* = √(2·D·S / H)

  • Economic order quantity (units). D and H must use the same time base.

H = i · C

  • i = annual carrying rate (fraction per year).

TC* (ordering + holding) = √(2·D·S·H) = H·Q*

  • At Q*, ordering cost = holding cost = TC*/2.

N = D / Q*, T = Q* / D

  • N = orders per year, T = cycle time (years; × working days per year to get days).

TC(Q) / TC* = ½ (Q/Q* + Q*/Q)

  • Cost penalty for ordering a non-optimal quantity (ordering + holding only).

ROP = d · L (if L ≤ T); ROP = d·L − n·Q (if L > T, n = whole number of cycles in L)

  • d = demand rate (units/day), L = lead time (days), ROP in units on hand.

Worked examples

Example 1 (standard) – EOQ, cost and cycle. D = 12,000 units/year, S = ₹500/order, unit price C = ₹200, carrying rate i = 15 % per year, 250 working days per year.

  1. H = i·C = 0.15 × 200 = ₹30/unit·year.
  2. Q* = √(2 × 12,000 × 500/30) = √4,00,000 = 632.5 units.
  3. Orders per year N = 12,000/632.5 = 18.97 (about 19).
  4. Cycle time T = 632.5/12,000 × 250 = 13.2 working days.
  5. Ordering cost = 18.97 × 500 = ₹9487; holding cost = (632.5/2) × 30 = ₹9487 – equal, as expected.
  6. TC* (ordering + holding) = √(2 × 12,000 × 500 × 30) = ₹18,974 per year; with purchase cost (12,000 × 200 = ₹24,00,000) total = ₹24,18,974.

Example 2 (GATE level) – forced lot size and reorder point with long lead time. D = 4800 units/year, S = ₹600/order, H = ₹16/unit·year, 240 working days per year. The supplier ships only in pallets of 800 units.

  1. Q* = √(2 × 4800 × 600/16) = √3,60,000 = 600 units; TC* = √(2 × 4800 × 600 × 16) = ₹9600/year.
  2. With Q = 800: ordering = (4800/800) × 600 = ₹3600; holding = (800/2) × 16 = ₹6400; TC = ₹10,000/year.
  3. Check by the penalty formula: ½(800/600 + 600/800) = ½(1.3333 + 0.75) = 1.0417; 9600 × 1.0417 = ₹10,000. A 33 % larger lot costs only 4.2 % more.
  4. Demand rate d = 4800/240 = 20 units/day; cycle time with Q = 800 is 800/20 = 40 days.
  5. Lead time 20 days (≤ 40): ROP = 20 × 20 = 400 units.
  6. Lead time 50 days (> 40): lead-time demand = 20 × 50 = 1000 units; one order of 800 is always in transit, so the on-hand ROP = 1000 − 800 = 200 units (inventory position at reorder = 1000).

Common mistakes

  • Using monthly demand with an annual holding cost (or vice versa).
  • Taking average inventory as Q instead of Q/2 in the holding cost.
  • Including the purchase cost when comparing quantities at a fixed price (it does not change Q*) – but it must be included when prices change with quantity.
  • Rounding Q* too early and then computing a "minimum" cost that is not equal for ordering and holding.
  • Setting ROP = d·L when the lead time exceeds the cycle time without accounting for orders in transit.
  • Forgetting that H = i × C changes when the unit price changes.

For GATE PI

EOQ appears almost every year in some form: compute Q*, total cost, number of orders, cycle time or reorder point; find the effect of changes in D, S or H (square-root relations); or the cost penalty of a non-optimal lot. Practise doing the arithmetic cleanly with consistent time units and recognising when lead time exceeds cycle time.

Quick check

  1. D = 2000 units/year, S = ₹100, H = ₹10/unit·year. Find Q*.
  2. If annual demand becomes four times larger, by what factor does EOQ change?
  3. At the EOQ, how do annual ordering and holding costs compare?
  4. Q* = 400 units and a buyer orders 800 units. By what factor does (ordering + holding) cost rise?

Answers: 1. √(2 × 2000 × 100/10) = 200 units. 2. It doubles (√4 = 2). 3. They are equal. 4. ½(2 + 0.5) = 1.25, i.e. 25 % higher.

Interview questions

All Production Planning and Operations Management interview questions

Try answering each one aloud before you open it.

  1. 1.What is Economic Order Quantity (EOQ) and why is it important in inventory management?Concept

    Economic Order Quantity (EOQ) is a formula used to determine the optimal order quantity that minimizes the total inventory costs, which include ordering costs and holding costs. It is important because it helps businesses reduce costs associated with ordering and storing inventory, ensuring that they have the right amount of stock at the right time.

  2. 2.Explain the components of inventory costs.Concept

    Inventory costs typically consist of three main components: ordering costs, holding costs, and shortage costs. Ordering costs are the expenses related to placing and receiving orders. Holding costs, also known as carrying costs, include storage, insurance, and opportunity costs of capital tied up in inventory. Shortage costs arise when demand cannot be met due to insufficient inventory, leading to lost sales or customer dissatisfaction.

  3. 3.How does the EOQ model help in balancing ordering and holding costs?Application

    The EOQ model helps in balancing ordering and holding costs by calculating the order quantity that minimizes the sum of these costs. By determining the optimal order size, businesses can reduce the frequency of orders (thus lowering ordering costs) while also minimizing the amount of inventory held (thus lowering holding costs). This balance ensures cost efficiency in inventory management.

  4. 4.What assumptions does the EOQ model make?Concept

    The EOQ model makes several assumptions: demand is constant and known, lead time is fixed, ordering and holding costs are constant, and there are no stockouts or shortages. Additionally, it assumes that each order is delivered in full and that there are no quantity discounts.

  5. 5.Why might a company choose not to use the EOQ model?Application

    A company might choose not to use the EOQ model if the assumptions of the model do not hold true for their operations. For example, if demand is highly variable, lead times fluctuate, or if there are significant quantity discounts available, the EOQ model may not provide an accurate or cost-effective solution. Additionally, if the costs of implementing and maintaining the EOQ system outweigh the benefits, a company might opt for a different inventory management approach.

  6. 6.What happens if a company orders more than the EOQ?Application

    If a company orders more than the EOQ, it may incur higher holding costs due to excess inventory. This can lead to increased storage costs, higher risk of obsolescence, and tied-up capital that could be used elsewhere. While ordering more might reduce ordering frequency, the additional holding costs could outweigh these savings, leading to overall higher inventory costs.

  7. 7.Calculate the EOQ for a company with an annual demand of 10,000 units, an ordering cost of ₹50 per order and a holding cost of ₹2 per unit per year.Numerical

    EOQ = √(2DS/H) = √(2 × 10,000 × 50 / 2) = √5,00,000 = 707.1 units. That gives about 14 orders a year, and annual ordering and holding costs of about ₹707 each (₹1414 in total). Note the input must use the same time base: annual demand with annual holding cost.

  8. 8.How does lead time affect inventory management and the EOQ model?Application

    Lead time affects inventory management by determining how quickly an order can be fulfilled. In the EOQ model, lead time is assumed to be constant, which simplifies the calculation of reorder points. However, if lead time is variable, it can complicate inventory management by increasing the risk of stockouts or excess inventory, requiring safety stock or adjustments to reorder points.

  9. 9.What is the impact of quantity discounts on the EOQ model?Application

    Quantity discounts can impact the EOQ model by altering the cost structure. While the EOQ model assumes constant costs, quantity discounts reduce the per-unit cost as order size increases. This can make larger orders more cost-effective, potentially leading to a different optimal order quantity than the EOQ would suggest. Companies must weigh the savings from discounts against the increased holding costs.

  10. 10.A company has a holding cost of ₹3 per unit per year and an ordering cost of ₹60 per order. If the EOQ is 500 units, what is the annual demand?Numerical

    From EOQ = √(2DS/H), D = EOQ² × H / (2S) = 500² × 3 / (2 × 60) = 7,50,000/120 = 6250 units per year. Check: √(2 × 6250 × 60/3) = √2,50,000 = 500 units.

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