Inventory models: EOQ, quantity discounts and safety stock

EOQ and its assumptions, reorder point, production-lot (EPQ) model, all-units quantity discounts and safety stock with worked numericals.

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Why it matters

Inventory ties up cash, floor space and handling effort, yet running out stops a line or loses a customer. EOQ-type models give the order size and reorder point that balance these costs, and they are the starting point for every MRP lot-sizing rule, purchase contract and safety-stock policy you will meet in a plant.

Key ideas

  • Costs that drive inventory decisions.
    • Ordering (set-up) cost S — the fixed cost of placing one order or one machine set-up (paperwork, transport, inspection, changeover). It does not depend on order size.
    • Holding (carrying) cost H — the cost of keeping one unit in stock for one year: interest on capital, storage, insurance, obsolescence. Often given as a fraction i of unit price: H = i·C.
    • Purchase cost C·D — constant in basic EOQ, but changes with Q when quantity discounts are offered.
    • Shortage cost — cost of a stockout or backorder (ignored in basic EOQ).
  • Basic EOQ (Wilson) model assumptions. Demand is known and constant; lead time is known and constant; the whole lot arrives at once (instantaneous replenishment); no shortages; S and H are constant; unit price does not depend on Q. The stock pattern is a saw-tooth that falls from Q to zero, so the average inventory is Q/2.
  • The trade-off. Annual ordering cost (D/Q)·S falls as Q rises; annual holding cost (Q/2)·H rises with Q. Their sum is minimum where the two are equal. The total-cost curve is flat near the optimum, so a Q 10–20% away from EOQ raises cost only slightly — EOQ is robust to estimation errors in D, S and H.
  • Reorder point (ROP). With constant demand and lead time, order when stock falls to the demand during lead time, ROP = d·L. The lead time does not change the EOQ; it only fixes when to order.
  • Production (EPQ / economic batch quantity) model. When a lot is produced in-house at rate p while being consumed at rate d (p > d), stock builds up gradually. Maximum inventory is Q(1 − d/p), not Q, so the optimum batch is larger than EOQ.
  • Quantity discounts (all-units). The supplier charges a lower price for every unit if Q crosses a break point. Because purchase cost now depends on Q, compare total annual cost TC = (D/Q)S + (Q/2)·i·C + C·D at (a) each price's EOQ if it is feasible in that price band, and (b) each break quantity where a lower price begins. Pick the lowest TC.
  • Safety stock (SS). Real demand and lead times vary. A buffer SS is added to the reorder point, ROP = d̄·L + SS. Assuming lead-time demand is normally distributed, SS = Z·σ_L, where Z is the standard normal value for the desired cycle service level (probability of no stockout in a replenishment cycle). Higher service level → larger Z → larger SS, and the cost rises steeply above about 95%.
  • Links. EOQ logic reappears as lot sizing in MRP; demand forecasts feed D and σ; ABC analysis decides which items deserve this careful control (A items: few items, most of the money value).

Formulas

  • Annual total cost (basic model): TC(Q) = (D/Q)·S + (Q/2)·H + C·D
  • Economic order quantity: Q* = √(2DS / H)
    • D = annual demand (units/year); S = ordering cost (₹/order); H = holding cost (₹/unit/year); C = unit price (₹/unit); Q = order quantity (units).
    • Applies when all basic EOQ assumptions hold.
  • Holding cost as a rate: H = i·C, i = carrying rate (per year, e.g. 0.2).
  • At the optimum: (D/Q*)·S = (Q*/2)·H and minimum variable cost TC* (excluding purchase) = √(2DSH)
  • Number of orders per year: N = D / Q*; cycle time: T = Q* / D (years)
  • Reorder point (deterministic): ROP = d·L — d = demand rate (units/day), L = lead time (days), same time unit for both.
  • Production (EPQ) model: Q* = √(2DS / (H·(1 − d/p))); maximum inventory I_max = Q*·(1 − d/p) — p = production rate, d = consumption rate, same units, p > d.
  • All-units discount: TC = (D/Q)·S + (Q/2)·i·C_j + C_j·D evaluated for each price C_j at its feasible EOQ or its break quantity.
  • Safety stock: SS = Z·σ_L; with constant lead time and daily demand standard deviation σ_d: σ_L = σ_d·√L
  • Reorder point with safety stock: ROP = d̄·L + Z·σ_L
    • Z = standard normal variate for the service level (1.645 for 95%, 2.33 for 99%; take other values from a normal table).

Worked examples

Example 1 (standard): EOQ, cost and reorder point Given: D = 12,000 units/year, S = ₹600/order, C = ₹50/unit, carrying rate i = 20% per year, 300 working days/year, lead time L = 5 days.

  1. Holding cost: H = i·C = 0.2 × 50 = ₹10/unit/year.
  2. EOQ: Q* = √(2DS/H) = √(2 × 12,000 × 600 / 10) = √1,440,000 = 1,200 units.
  3. Orders per year: N = D/Q* = 12,000/1,200 = 10; cycle time T = 300/10 = 30 working days.
  4. Annual ordering cost = (D/Q*)·S = 10 × 600 = ₹6,000; annual holding cost = (Q*/2)·H = 600 × 10 = ₹6,000 (equal, as expected).
  5. Daily demand d = 12,000/300 = 40 units/day; ROP = d·L = 40 × 5 = 200 units. Answer: Q = 1,200 units, variable cost ₹12,000/year, reorder at 200 units.*

Example 2 (GATE level): all-units quantity discount Given: D = 10,000 units/year, S = ₹300/order, i = 20% per year. Price: ₹50 for Q < 1,000; ₹48 for 1,000 ≤ Q < 2,500; ₹46 for Q ≥ 2,500. Find the best order quantity.

  1. EOQ at each price, Q = √(2DS/(i·C)):
    • C = 50: √(6,000,000/10) = 774.6 units → lies in its band (Q < 1,000), feasible.
    • C = 48: √(6,000,000/9.6) = 790.6 units → below 1,000, infeasible; candidate becomes the break quantity 1,000.
    • C = 46: √(6,000,000/9.2) = 807.6 units → below 2,500, infeasible; candidate becomes 2,500.
  2. Total annual cost TC = (D/Q)S + (Q/2)·i·C + C·D:
    • Q = 774.6, C = 50: 3,873 + 3,873 + 500,000 = ₹507,746.
    • Q = 1,000, C = 48: 3,000 + 500 × 9.6 + 480,000 = 3,000 + 4,800 + 480,000 = ₹487,800.
    • Q = 2,500, C = 46: 1,200 + 1,250 × 9.2 + 460,000 = 1,200 + 11,500 + 460,000 = ₹472,700.
  3. The lowest TC is at Q = 2,500. Answer: Order 2,500 units at ₹46; minimum total annual cost ₹472,700.

Example 3 (GATE level): safety stock and reorder point Given: mean demand d̄ = 40 units/day, standard deviation of daily demand σ_d = 8 units/day, constant lead time L = 9 days, desired service level 95% (Z = 1.645). Daily demands are independent.

  1. σ_L = σ_d·√L = 8 × √9 = 24 units.
  2. SS = Z·σ_L = 1.645 × 24 = 39.5 ≈ 40 units (round up to protect the service level).
  3. ROP = d̄·L + SS = 40 × 9 + 40 = 400 units. Answer: Safety stock ≈ 40 units; reorder point = 400 units.

Example 4 (short): production lot size D = 6,000 units/year, S = ₹500/set-up, H = ₹4/unit/year, p = 60 units/day, 250 days/year so d = 24 units/day. Q* = √(2 × 6,000 × 500 / (4 × (1 − 24/60))) = √(6,000,000/2.4) = √2,500,000 = 1,581 units; I_max = 1,581 × 0.6 = 949 units.

Common mistakes

  • Mixing time units: D per year with H per month, or daily demand with lead time in weeks.
  • Using H = i·C with the wrong price in discount problems — H changes with each price band.
  • Taking the EOQ of a lower price band without checking it falls inside that band; an infeasible EOQ must be moved to the break quantity.
  • Forgetting the purchase cost C·D when comparing discount options (it is the term that usually decides).
  • Writing ROP = d·L + SS but then adding SS to the EOQ as well — safety stock changes when you order, not how much.
  • Adding standard deviations instead of variances: σ_L = σ_d·√L, not σ_d·L.
  • Using the EOQ formula in the production model; it ignores consumption during the production run.

For GATE ME

Expect direct EOQ numericals (order quantity, number of orders, cycle time, minimum total cost), effect-of-change questions ("if demand doubles, EOQ rises by √2"), reorder point calculations, production-lot (EPQ) problems with I_max, and all-units discount problems where the trick is checking feasibility of each EOQ and including purchase cost. Safety-stock questions give Z or a normal-table value; practise computing σ_L from daily data. Practise doing these quickly with consistent units.

Quick check

  1. If annual demand becomes four times larger with all else equal, by what factor does EOQ change?
  2. At the EOQ, how do annual ordering cost and annual holding cost compare?
  3. In the EPQ model, why is the optimum batch larger than the EOQ?
  4. Daily demand has σ_d = 5 units and lead time is 16 days. What is σ_L?
  5. Does a longer lead time change the EOQ in the basic model?

Answers: 1. It doubles (EOQ ∝ √D). 2. They are equal. 3. Stock builds gradually because units are consumed during production, so average inventory per unit of Q is lower, I_max = Q(1 − d/p). 4. σ_L = 5 × √16 = 20 units. 5. No; it changes only the reorder point.

Interview questions

All Metrology, CIM and 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 purchased in bulk. Companies must analyze whether the savings from the discount outweigh the additional holding costs incurred by ordering 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 or supply chain disruptions. It acts as a buffer to ensure that a company can continue to meet customer demand even when unexpected events occur. Safety stock is crucial for maintaining service levels and avoiding lost sales.

  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 size 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.

  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 mitigate the risk of stockouts, which can lead to lost sales and dissatisfied customers. The cost of holding additional inventory is often justified by the potential revenue loss and damage to customer relationships that can occur if products are unavailable when needed. Safety stock ensures a higher level of service reliability.

  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 sizes.

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

    EOQ = √(2DS/H) = √(2 × 10,000 × 50 / 2) = √500,000 ≈ 707 units. That means about 14 orders a year, and at the optimum the annual ordering and holding costs are equal (≈ ₹707 each).

  8. 8.Explain how safety stock levels are determined in a company.Concept

    Safety stock levels are determined based on factors such as demand variability, lead time variability, desired service level, and the company's risk tolerance. Companies often use statistical methods to calculate safety stock, considering the standard deviation of demand and lead time, and applying a service level factor to ensure the desired probability of not running out of stock.

  9. 9.What are the potential drawbacks of holding too much safety stock?Application

    Holding 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 lead to waste if the stock becomes obsolete. It's important to balance the benefits of safety stock with the costs of holding it.

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