What economic production quantity (EPQ) is
The economic production quantity (EPQ, also called the economic manufacturing quantity or EOQ with gradual replenishment) is the run size that minimizes the sum of setup cost and inventory holding cost. Unlike the purchasing EOQ, it assumes the item is made in-house and joins stock gradually while demand is already drawing it down.
The idea goes back to E. W. Taft (1918), who adapted F. W. Harris's 1913 formula to manufacturing. If the production rate were infinite, the EPQ would collapse into the plain order quantity covered in inventory management with reorder point and economic order quantity.
Visual map of the economic production quantity
The map summarizes the formula, the inventory sawtooth and the worked example with minimum total cost.
EPQ formula
Q* = √[ 2 × D × S ÷ ( H × (1 − d ÷ p) ) ]
- D: annual demand (units/year).
- S: cost of one setup, per run.
- H: cost of holding one unit in stock for a year, per unit per year.
- d: daily demand (D ÷ working days).
- p: daily production rate, always greater than d.
The factor (1 − d ÷ p) lowers average inventory: during a run, stock builds only at the rate (p − d), so the peak is below the lot size. Maximum inventory: Imax = Q × (1 − d ÷ p). Average inventory: Imax ÷ 2.
Annual total cost = (D ÷ Q) × S + (Imax ÷ 2) × H
Worked example
Fictional example: a machined part has demand of 24,000 units/year over 240 working days. The machine makes 400 parts/day. Each setup costs $300 and holding one part costs $6 per year.
- Daily demand: d = 24,000 ÷ 240 = 100 parts/day.
- Factor: 1 − 100 ÷ 400 = 0.75.
- Lot: Q* = √(2 × 24,000 × 300 ÷ (6 × 0.75)) = √(14,400,000 ÷ 4.5) = √3,200,000 ≈ 1,789 parts.
- Maximum inventory: 1,789 × 0.75 ≈ 1,342 parts.
- Run time: 1,789 ÷ 400 ≈ 4.5 days. Full cycle: 1,789 ÷ 100 ≈ 17.9 days.
- Runs per year: 24,000 ÷ 1,789 ≈ 13.4.
At the optimal lot, setup cost and holding cost are equal:
| Lot Q | Runs/year | Setup ($) | Holding ($) | Total ($) |
|---|---|---|---|---|
| 1,000 | 24.0 | 7,200 | 2,250 | 9,450 |
| 1,500 | 16.0 | 4,800 | 3,375 | 8,175 |
| 1,789 | 13.4 | 4,025 | 4,025 | 8,050 |
| 2,000 | 12.0 | 3,600 | 4,500 | 8,100 |
| 2,500 | 9.6 | 2,880 | 5,625 | 8,505 |
Note how flat the curve is near the minimum: a lot of 2,000 costs only $50 more (0.6%). That gives you room to round to a practical quantity, such as a multiple of the container size or the shift output.
EPQ or EOQ: which one to use
With the purchasing formula (no 1 − d ÷ p factor), the same example gives Q = √(2 × 24,000 × 300 ÷ 6) ≈ 1,549 parts. Applied to a manufactured part, that lot leads to a cost of about $8,133, which is 1.0% above the optimum. The gap is small here because d ÷ p is 25%; it grows when production is slow relative to demand.
| Criterion | EOQ (purchasing) | EPQ (production) |
|---|---|---|
| Replenishment | Whole lot arrives at once | Enters gradually at rate p |
| Fixed cost per lot | Purchase order | Machine setup |
| Maximum inventory | Equal to lot Q | Q × (1 − d ÷ p) |
| Optimal lot | Smaller | Larger, due to production overlap |
Setup is the lever
Minimum cost grows with the square root of setup: Minimum cost = √(2 × D × S × H × (1 − d ÷ p)). In the example, if single-minute exchange of die (SMED) work cuts setup from $300 to $150, the optimal lot drops from 1,789 to about 1,265 parts and annual total cost falls from $8,050 to $5,692 (−29%). Smaller lots mean less inventory, a shorter lead time and more flexibility, which is the road to production leveling (heijunka).
When to use it and how to apply it on the shop floor
- Rank items with an ABC analysis; compute the EPQ for A and B items.
- Find the true S: idle machine hours, labor, adjustment scrap and test material. Do not use only the setter's wage.
- Estimate H: tied-up capital, storage, insurance, obsolescence. It is usually a percentage of unit cost per year.
- Confirm that p (real capacity, not nameplate) is greater than d. See production capacity.
- Compute Q*, round to a practical lot and review it whenever demand, cost or rate changes.
- Feed the result into MRP as the lot-sizing rule.
- Cover demand variability with safety stock, which the EPQ does not calculate.
Common mistakes and limitations
- Ignoring the true setup cost. Underestimating S gives lots that are too small; overestimating it builds needless inventory.
- Using nameplate capacity as p. With stoppages and losses the effective rate is lower; check the OEE.
- Treating the model as fixed truth. It assumes constant, known demand, a single item and linear costs.
- Forgetting the bottleneck. If several parts compete for the same machine, one part's lot affects the others. A per-item EPQ cannot see that.
- Stopping setup reduction. The optimal lot is a function of today's setup; in Lean the goal is to drive it down.
Frequently asked questions
What is the economic production quantity?
It is the run size that minimizes the sum of setup and holding costs, assuming the item is produced and enters stock gradually.
What is the EPQ formula?
Q* = √[2 × D × S ÷ (H × (1 − d ÷ p))], where D is annual demand, S the setup cost, H the annual holding cost per unit, d daily demand and p the daily production rate.
What is the difference between EPQ and EOQ?
EOQ assumes the lot arrives whole, as in a purchase. EPQ assumes gradual entry, so it includes the factor (1 − d ÷ p), which produces a slightly larger lot.
What lot does the example in this article give?
With D = 24,000, S = $300, H = $6, d = 100 and p = 400, the lot is about 1,789 parts with an annual total cost of $8,050.
What happens to the lot if setup time drops?
It shrinks with the square root of setup. Cutting S from $300 to $150 reduces the lot to about 1,265 parts and annual total cost by 29%.
Does the EPQ account for stockouts or demand variation?
No. The basic model assumes constant demand. Variation is covered separately with safety stock.
Sources
- TAFT, E. W. The most economical production lot. The Iron Age, 1918.
- HARRIS, F. W. How many parts to make at once. Factory, The Magazine of Management, 1913.
- NAHMIAS, S. Production and Operations Analysis. McGraw-Hill/Irwin.
- HOPP, W. J.; SPEARMAN, M. L. Factory Physics. Waveland Press.
- SILVER, E. A.; PYKE, D. F.; PETERSON, R. Inventory Management and Production Planning and Scheduling. Wiley.
Keep studying: trusted sources and links
- ASCM: Association for Supply Chain Management
- ILOS: Instituto de Logística e Supply Chain (in Portuguese)
- Wikipedia: Economic order quantity
- Wikipedia: Single-minute exchange of die
Save this address: Manufatura Enxuta is your school for lean manufacturing. Come back whenever you need to study, with a complete article, a visual map and trusted sources. Follow one of the learning paths or check the A–Z glossary.
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About the author
Lean Manufacturing & Behavioral Management Specialist
Over 20 years in the automotive and metalworking industries (GM and Dana), Lean Manufacturing practitioner since 2006. SENAI instructor and mentor in Brazil’s Brasil Mais Produtivo program, delivering consulting, training and audits for 50+ companies, combining quality, productivity and people development.
