Inventory models: EOQ, quantity discounts and safety stock

Inventory costs, the EOQ and EPQ models, all-units quantity discounts, reorder point and safety stock, and ABC/VED control, with EOQ, discount, EPQ and safety-stock numericals.

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Why it matters

A car plant buys thousands of different parts — fasteners, bearings, tyres, castings, electronic modules. Order too much and capital is tied up, storage space overflows and parts become obsolete when a model changes; order too little and the line stops. EOQ, quantity-discount and safety-stock calculations give the order size and reorder point for independent-demand items such as spare parts and consumables, and they are among the most frequently asked GATE numericals in industrial engineering.

Key ideas

Inventory costs.

  • Ordering (setup) cost S — cost per order: paperwork, transport, receiving inspection; for in-house production, the machine setup cost. Independent of order size.
  • Holding (carrying) cost H — cost of keeping one unit for a year: interest on capital, storage, insurance, obsolescence. Often written H = i·P, where i is the annual carrying rate and P the unit price.
  • Purchase cost D·P — matters only when price depends on order size (discounts).
  • Shortage cost — lost sales or line stoppage.

Classical EOQ model (Wilson). Assumptions: demand known and constant; lead time known and constant; whole order received at once; no shortages; costs constant. Inventory falls in a saw-tooth from Q to zero, so the average inventory is Q/2. Ordering cost per year falls with Q (D·S/Q); holding cost rises with Q (H·Q/2). Total cost is minimum where the two are equal — that Q is the EOQ. The total-cost curve is flat near the optimum: ordering 20 % more or less than EOQ raises cost by only about 2 %, so rounding to a pack size is safe.

Economic production quantity (EPQ). When a part is made in-house at rate p while being used at rate d (p > d), stock builds up only at p − d during production, so maximum inventory is Q(1 − d/p) and the optimal batch is larger than the EOQ.

Quantity discounts (all-units). The unit price drops when Q crosses a break point. Because H usually depends on price, the procedure is:

  1. Compute EOQ at each price, starting from the lowest price.
  2. If an EOQ is feasible (lies in its own price range), it is a candidate; for prices where the EOQ is below the range, take the lowest quantity of that range (the break point) as the candidate.
  3. Compute total annual cost (purchase + ordering + holding) for each candidate and pick the lowest.

Reorder point and safety stock. An order is placed when stock falls to the reorder point (ROP). With constant demand, ROP = demand during lead time. When demand varies, safety stock is added so that the probability of not running out during the lead time — the cycle service level — is high. With normally distributed daily demand, the standard deviation over the lead time grows with the square root of the lead time. The service factor z comes from the standard normal table (z ≈ 1.28 for 90 %, 1.645 for 95 %, 2.33 for 99 %).

Selective control. ABC analysis classifies items by annual consumption value (A: about 10–20 % of items, 70–80 % of value — tight control; C: many cheap items — simple rules). VED (vital, essential, desirable) is used for maintenance spares, where criticality matters more than value. FSN (fast, slow, non-moving) highlights obsolete stock.

Formulas

Q* = √(2·D·S / H)

  • Q* = EOQ (units); D = annual demand (units/year); S = ordering cost (₹/order); H = holding cost (₹/unit·year).

TC = D·P + (D/Q)·S + (Q/2)·H

  • Total annual cost (₹/year); P = unit price (₹/unit). At Q*, ordering cost = holding cost and TC* (excluding purchase) = √(2·D·S·H).

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

  • Number of orders per year and cycle time (years).

Q_p* = √( 2·D·S / (H·(1 − d/p)) ), I_max = Q·(1 − d/p)

  • EPQ (units) and maximum inventory (units); d = usage rate, p = production rate, in the same units (e.g. units/day).

ROP = d · L + SS

  • Reorder point (units); d = average demand per day; L = lead time (days); SS = safety stock (units).

SS = z · σ_d · √L

  • σ_d = standard deviation of daily demand (units/day), daily demands independent; z = standard normal factor for the service level. If σ is given for a week, L must be in weeks.

Worked examples

Example 1 (standard) — EOQ and reorder point. Given: a bearing used at D = 12,000 units/year (300 working days), S = ₹400 per order, P = ₹50, carrying rate 20 % per year, lead time 5 days, demand constant.

  1. H = i·P = 0.20 × 50 = ₹10 per unit-year
  2. Q* = √(2 × 12,000 × 400 / 10) = √960,000 = 979.8 ≈ 980 units
  3. TC* (ordering + holding) = √(2 × 12,000 × 400 × 10) = ₹9,798 per year
  4. N = 12,000 / 979.8 = 12.2 orders per year
  5. d = 12,000 / 300 = 40 per day; ROP = 40 × 5 = 200 units

Answer: order about 980 units whenever stock falls to 200; ordering + holding cost ≈ ₹9,798 per year.

Example 2 (GATE level) — all-units quantity discount. Given: same item; price ₹50 for Q < 1,500, ₹48 for 1,500 ≤ Q < 5,000, ₹47 for Q ≥ 5,000; H = 20 % of price.

  1. EOQ at ₹47: H = 9.4, Q = √(9,600,000 / 9.4) = 1,010.6 — below 5,000, so candidate Q = 5,000.
  2. EOQ at ₹48: H = 9.6, Q = √(9,600,000 / 9.6) = 1,000 — below 1,500, so candidate Q = 1,500.
  3. EOQ at ₹50: Q = 979.8 — feasible (< 1,500), candidate.
  4. Total costs TC = D·P + D·S/Q + H·Q/2:
    • Q = 979.8 at ₹50: 6,00,000 + 4,899 + 4,899 = ₹6,09,798
    • Q = 1,500 at ₹48: 5,76,000 + 3,200 + 7,200 = ₹5,86,400
    • Q = 5,000 at ₹47: 5,64,000 + 960 + 23,500 = ₹5,88,460

Answer: order 1,500 units (₹5,86,400 per year).

Example 3 (GATE level) — in-house production batch. Given: D = 12,000 units/year, production rate p = 48,000 units/year, setup cost S = ₹400, H = ₹10 per unit-year.

  1. d/p = 12,000 / 48,000 = 0.25
  2. Q_p* = √(2 × 12,000 × 400 / (10 × 0.75)) = √1,280,000 = 1,131.4 units
  3. I_max = 1,131.4 × 0.75 = 848.5 units

Answer: batch ≈ 1,131 units; maximum stock ≈ 849 units.

Example 4 — safety stock. Given: average demand 40 units/day, σ_d = 8 units/day, lead time 9 days, 95 % cycle service level (z = 1.645).

  1. σ_LT = 8 × √9 = 24 units
  2. SS = 1.645 × 24 = 39.5 ≈ 40 units
  3. ROP = 40 × 9 + 40 = 400 units

Common mistakes

  • Mixing time units: annual demand with a monthly holding cost, or weekly σ with a lead time in days.
  • Taking H as the unit price instead of i × P.
  • In discount problems, comparing only ordering + holding cost and forgetting the purchase cost D·P, which usually decides the answer.
  • Checking only the EOQ at the lowest price; every price break must be costed.
  • Multiplying σ by L instead of √L for independent daily demands.
  • Using EOQ for dependent-demand line parts that should be planned by MRP.

For GATE ME

Expect EOQ, number of orders, cycle time and total cost; EPQ with finite production rate; all-units discount decisions; and reorder point with safety stock. Conceptual questions ask about the effect of doubling D or S on EOQ (it rises by √2), the equality of ordering and holding costs at EOQ, and ABC classification. Practise keeping all quantities on one time basis.

Quick check

  1. If annual demand is quadrupled, by what factor does EOQ change?
  2. At EOQ, how do annual ordering cost and annual holding cost compare?
  3. D = 2,400 per year, S = ₹100, H = ₹12. Find the EOQ.
  4. Daily demand 50, lead time 4 days, safety stock 30. Reorder point?
  5. In ABC analysis, which class has few items but most of the value?

Answers: 1. It doubles. 2. They are equal. 3. √(480,000/12) = 200 units. 4. 230 units. 5. Class A.

Interview questions

All Production, Maintenance & Industrial Engineering interview questions

Try answering each one aloud before you open it.

  1. 1.What is the Economic Order Quantity (EOQ) model in inventory management?Concept

    The Economic Order Quantity (EOQ) model is a formula used in inventory management to determine the optimal order quantity that minimizes the total cost of inventory. This includes the costs of ordering and holding inventory. The EOQ formula helps businesses decide the most cost-effective quantity to order, balancing the trade-off between ordering costs and holding costs.

  2. 2.Explain the concept of quantity discounts in inventory management.Concept

    Quantity discounts are price reductions offered by suppliers to encourage buyers to purchase larger quantities. In inventory management, this concept is used to reduce the per-unit cost of items when buying in bulk. Companies must analyze whether the savings from the discount outweigh the additional holding costs incurred by purchasing larger quantities.

  3. 3.What is safety stock and why is it important in inventory management?Concept

    Safety stock is an additional quantity of inventory kept on hand to prevent stockouts caused by uncertainties in demand and supply. It acts as a buffer against unexpected fluctuations in demand or delays in supply. Safety stock is important because it helps maintain service levels and ensures that customer demand can be met even when there are disruptions in the supply chain.

  4. 4.How does the EOQ model help in reducing inventory costs?Application

    The EOQ model helps reduce inventory costs by determining the optimal order quantity that minimizes the total cost of ordering and holding inventory. By calculating the EOQ, businesses can avoid overstocking, which increases holding costs, and understocking, which can lead to stockouts and lost sales. This balance ensures efficient inventory management and cost savings.

  5. 5.Why might a company choose to hold safety stock even if it increases holding costs?Application

    A company might choose to hold safety stock to ensure a high level of customer service and avoid stockouts, which can lead to lost sales and customer dissatisfaction. The cost of holding safety stock is often justified by the potential revenue loss and damage to reputation that can occur if customer demand cannot be met. Safety stock acts as a buffer against uncertainties in demand and supply chain disruptions.

  6. 6.What happens if a company ignores quantity discounts when ordering inventory?Application

    If a company ignores quantity discounts, it may end up paying a higher price per unit than necessary, leading to increased purchasing costs. By not taking advantage of discounts, the company might miss opportunities to reduce overall inventory costs. However, it's important to balance the benefits of discounts with the potential increase in holding costs due to larger order quantities.

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

    To calculate the EOQ, use the formula: EOQ = √((2 * D * S) / H), where D is the annual demand, S is the ordering cost, and H is the holding cost. Plugging in the values: EOQ = √((2 * 10,000 * 50) / 2) = √(1,000,000) = 1,000 units.

  8. 8.Explain how safety stock levels are determined in inventory management.Concept

    Safety stock levels are determined based on factors such as demand variability, lead time variability, desired service level, and the standard deviation of demand during lead time. Companies use statistical methods to calculate the appropriate safety stock level that will provide a buffer against uncertainties while minimizing holding costs. The goal is to maintain a balance between service level and cost efficiency.

  9. 9.What are the potential risks of maintaining too much safety stock?Application

    Maintaining too much safety stock can lead to increased holding costs, including storage, insurance, and obsolescence costs. Excessive safety stock ties up capital that could be used elsewhere in the business and may result in waste if the stock becomes obsolete or expires. It is important to find the right balance to avoid these risks while still protecting against stockouts.

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