EPQ, quantity discounts and backorders

Economic production quantity with finite production rate, all-units quantity discounts, and the EOQ with planned backorders, with formulas and worked decisions.

Drafted with Aria, reviewed by the AiCanCode.org team. Spotted an error? Use Give Feedback at the bottom of the page.

Why it matters

The basic EOQ assumes the whole lot arrives at once, the price never changes and stock-outs are forbidden. Real plants make parts on their own machines while consuming them, suppliers offer lower prices for bigger lots, and some customers accept a short wait. The three extensions in this topic – EPQ, quantity discounts and planned backorders – handle exactly these cases and are favourite GATE numericals.

Key ideas

1. Economic production quantity (EPQ, finite replenishment rate). An item is produced at rate p while it is being consumed at rate d (p > d). During a production run stock builds up at the net rate (p − d); after the run it is drawn down at rate d.

  • Run (production) time t_p = Q/p; cycle time T = Q/d.
  • Maximum inventory I_max = (p − d)·t_p = Q(1 − d/p), not Q.
  • Average inventory = I_max/2, so holding cost is lower than in EOQ and the economic lot is larger.
  • The set-up cost S plays the role of the ordering cost.
  • As p → ∞ the EPQ reduces to the EOQ.
  • d and p must be in the same units (both per day or both per year).

2. Quantity discounts (all-units discount). The supplier charges a lower unit price C for every unit in the order once the order reaches a break quantity. Holding cost is usually H = i·C, so it also falls with price. Purchase cost D·C now depends on Q and must be included. Procedure:

  1. Starting from the lowest price, compute the EOQ with that price's H.
  2. If that EOQ lies in the price's quantity range, it is a candidate (and no higher-price range needs checking beyond it).
  3. If it is below the range, the candidate for that price is the lowest quantity of the range (the break point); if above, that price is not feasible at its EOQ – move to the next price.
  4. Compute total cost TC = D·C + (D/Q)·S + (Q/2)·H for each candidate and choose the lowest. (Incremental discounts, where only units above the break get the lower price, need a different procedure.)

3. Planned backorders (shortages allowed). If customers wait and the cost of a backorder is B per unit per year (time-weighted), it pays to let stock run negative for part of each cycle: each order first fills the waiting backorders.

  • Order quantity Q is larger than the EOQ; maximum inventory I_max = Q − b, where b is the maximum backorder.
  • Optimal fraction of each cycle spent in shortage = H/(H + B).
  • As B → ∞ (shortages very costly), the model returns to EOQ.

Formulas

EPQ Q* = √[2·D·S / (H·(1 − d/p))] I_max = Q·(1 − d/p) TC* (set-up + holding) = √[2·D·S·H·(1 − d/p)] t_p = Q/p, T = Q/d

  • D = annual demand (units/year), S = set-up cost (₹/set-up), H = holding cost (₹/unit·year), d = demand rate, p = production rate (same units, d < p), t_p = run time, T = cycle time.

Quantity discount (all-units) TC(Q) = D·C + (D/Q)·S + (Q/2)·i·C

  • C = unit price for the range containing Q (₹/unit), i = carrying rate (fraction/year).

Planned backorders (instantaneous replenishment) Q* = √(2·D·S/H) · √[(H + B)/B] b* = Q*·H/(H + B) I_max = Q* − b* = Q*·B/(H + B) TC* = √(2·D·S·H) · √[B/(H + B)]

  • B = backorder cost (₹/unit·year), b* = maximum backorder (units).

Worked examples

Example 1 (standard) – EPQ. A press shop uses 24,000 brackets/year (100 per day over 240 days) and can make them at 400 per day. Set-up cost S = ₹900, holding cost H = ₹12/unit·year.

  1. d/p = 100/400 = 0.25, so 1 − d/p = 0.75.
  2. Q* = √(2 × 24,000 × 900/(12 × 0.75)) = √(4,32,00,000/9) = √48,00,000 = 2191 units.
  3. Run time t_p = 2191/400 = 5.48 days; cycle time T = 2191/100 = 21.9 days.
  4. I_max = 2191 × 0.75 = 1643 units (not 2191).
  5. TC* = √(2 × 24,000 × 900 × 12 × 0.75) = ₹19,718/year (set-up + holding).

Example 2 (GATE level) – all-units quantity discount. D = 5000 units/year, S = ₹490/order, carrying rate i = 20 %/year. Prices: ₹50 for Q < 1000; ₹48 for 1000 ≤ Q < 2500; ₹47 for Q ≥ 2500.

  1. Price ₹47: H = 9.40; EOQ = √(2 × 5000 × 490/9.40) = 722 < 2500, infeasible; candidate Q = 2500.
  2. Price ₹48: H = 9.60; EOQ = 714 < 1000, infeasible; candidate Q = 1000.
  3. Price ₹50: H = 10.00; EOQ = √4,90,000 = 700, feasible (< 1000); candidate Q = 700.
  4. TC(700) = 5000 × 50 + (5000/700) × 490 + 350 × 10 = 2,50,000 + 3500 + 3500 = ₹2,57,000.
  5. TC(1000) = 5000 × 48 + 5 × 490 + 500 × 9.60 = 2,40,000 + 2450 + 4800 = ₹2,47,250.
  6. TC(2500) = 5000 × 47 + 2 × 490 + 1250 × 9.40 = 2,35,000 + 980 + 11,750 = ₹2,47,730.
  7. Order 1000 units at ₹48; minimum total cost = ₹2,47,250/year. The ₹47 price saves ₹5000 on purchases but the extra holding cost outweighs it.

Example 3 – planned backorders. D = 10,000 units/year, S = ₹200, H = ₹4/unit·year, B = ₹12/unit·year.

  1. EOQ part = √(2 × 10,000 × 200/4) = 1000; factor √(16/12) = 1.1547.
  2. Q* = 1155 units; b* = 1155 × 4/16 = 289 units; I_max = 866 units.
  3. TC* = √(2 × 10,000 × 200 × 4) × √(12/16) = 4000 × 0.866 = ₹3464/year, lower than ₹4000 without backorders.

Common mistakes

  • Using Q instead of Q(1 − d/p) for maximum inventory in the EPQ model.
  • Mixing a daily production rate with an annual demand rate in d/p.
  • In discount problems, forgetting to recompute H = i·C for each price, or leaving out the purchase cost D·C.
  • Accepting an EOQ that lies outside its price range, or not testing the break quantities.
  • Inverting the backorder factor: Q* is multiplied by √((H + B)/B), which is greater than 1.
  • Treating B as a one-time cost per unit short when the model needs a cost per unit per year.

For GATE PI

Expect EPQ calculations (lot size, maximum inventory, run time, number of runs), all-units discount decisions with two or three price breaks, and EOQ with planned shortages (order quantity, maximum shortage, total cost). Practise the discount procedure in a table and always check feasibility of each EOQ against its range.

Quick check

  1. D = 3600/year, p = 14,400/year, S = ₹400, H = ₹6/unit·year. Find EPQ.
  2. In Q1, what is the maximum inventory?
  3. With planned backorders, H = ₹5 and B = ₹15 (both per unit·year). What fraction of each cycle is spent in shortage?
  4. An EOQ at the discounted price falls below the price break. Which quantity do you test for that price?

Answers: 1. √(2 × 3600 × 400/(6 × 0.75)) = √6,40,000 = 800 units. 2. 800 × 0.75 = 600 units. 3. 5/20 = 0.25. 4. The break quantity (the lowest quantity that earns the price).

Interview questions

All Production Planning and Operations Management interview questions

Try answering each one aloud before you open it.

  1. 1.What is the Economic Production Quantity (EPQ) model?Concept

    The Economic Production Quantity (EPQ) model is an inventory management framework used to determine the optimal production quantity that minimizes total inventory costs, including setup and holding costs. Unlike the Economic Order Quantity (EOQ) model, EPQ considers production rates and allows for gradual inventory buildup over time. It is particularly useful in manufacturing settings where production and consumption occur simultaneously.

  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 of a product. In inventory management, these discounts can influence the order quantity decision, as purchasing in bulk may reduce the per-unit cost. However, it also requires balancing the cost savings against increased holding costs and potential risks of overstocking.

  3. 3.What are backorders, and how do they affect production planning?Concept

    Backorders occur when customer orders cannot be fulfilled from current inventory due to stockouts, leading to delayed delivery. In production planning, backorders can affect customer satisfaction and increase costs due to expedited shipping or overtime production. Effective management involves balancing inventory levels to minimize backorders while avoiding excessive holding costs.

  4. 4.Why is the EPQ model preferred over the EOQ model in certain manufacturing environments?Application

    The EPQ model is preferred over the EOQ model in manufacturing environments where production and consumption occur simultaneously. This is because EPQ accounts for the production rate and allows for inventory to build up gradually, which is more realistic in a production setting. It helps in optimizing production runs and reducing setup costs while maintaining adequate inventory levels.

  5. 5.What happens if a company ignores quantity discounts when planning inventory?Application

    If a company ignores quantity discounts, it may miss out on potential cost savings from reduced per-unit prices. This oversight can lead to higher overall purchasing costs and reduced competitiveness. However, it is also important to consider the trade-off with increased holding costs and the risk of overstocking, which can negate the benefits of the discount.

  6. 6.How can backorders be minimized in a production system?Application

    Backorders can be minimized by improving demand forecasting accuracy, maintaining safety stock, and optimizing production schedules to align with demand patterns. Implementing a responsive supply chain and enhancing communication with suppliers can also help in quickly addressing potential stockouts. Additionally, using inventory management software can provide real-time insights to prevent backorders.

  7. 7.Calculate the EPQ for a company with an annual demand of 10,000 units, a set-up cost of ₹500 per production run, a holding cost of ₹2 per unit per year, and a production rate of 50,000 units per year.Numerical

    EPQ = √[2DS / (H(1 − d/p))]. Here d/p = 10,000/50,000 = 0.2, so EPQ = √(2 × 10,000 × 500 / (2 × 0.8)) = √62,50,000 = 2500 units. The maximum inventory is only 2500 × 0.8 = 2000 units because the item is consumed while it is being produced.

  8. 8.A supplier offers a 5 % discount on orders of 1,000 units or more. If the regular price is ₹10 per unit, what is the discounted price, and how would you decide whether to take it?Numerical

    Discounted price = 10 × (1 − 0.05) = ₹9.50 per unit. To decide, compute total annual cost (purchase + ordering + holding, with holding = carrying rate × price) at the EOQ for the regular price and at Q = 1000 for the discounted price. Take the discount only if the purchase saving plus the lower ordering cost exceed the extra holding cost of the larger lot.

  9. 9.Explain how safety stock can be used to manage backorders.Application

    Safety stock acts as a buffer to protect against uncertainties in demand and supply, reducing the likelihood of stockouts and backorders. By maintaining an additional inventory level beyond the expected demand, companies can continue to fulfill customer orders even when unexpected demand spikes or supply chain disruptions occur. This helps in maintaining service levels and customer satisfaction.

  10. 10.What are the potential drawbacks of using the EPQ model?Application

    The potential drawbacks of using the EPQ model include its assumptions of constant demand and production rates, which may not hold true in dynamic environments. It also assumes that production and consumption occur simultaneously without any delays, which might not be realistic. Additionally, the model may not account for variations in setup costs or holding costs over time, leading to suboptimal decisions.

Finished this topic? Mark it so your progress, study plan and readiness keep up.

Stuck on something here?

PreviousInventory costs and the EOQ modelNext Safety stock and reorder point under uncertainty