Economic production quantity

The worked EPQ model assumes uniform production and demand of 48,000 and 12,000 units/year, 90 currency/run setup and 8 currency/(unit·year) holding cost. Output is usable as produced. The 600-unit lot reaches 450 units peak cycle stock and 225 units average. On a 250-working-day calendar, the run lasts 3.125 days and stock drains for 9.375 days. The calculation assumes continuous demand and availability within that calendar.

EPQ balances setup and holding costs when output becomes available gradually. MIT shows that the EOQ result is recovered as production rate increases without bound. MIT CTL, SC1x Key Concepts, v6.2 Fall 2016, printed p. 47.

Check the calculated lot against the actual schedule and capacity.

Define rates, costs and availability

Use one physical unit and a consistent time basis:

The basic no-shortage model requires P > D, positive setup and holding costs, constant rates and costs, and usable output entering inventory as it is produced. Rossetti’s inventory textbook derives EPQ as the finite-production model with backorders prohibited; this is a different assumption from allowing shortages at zero cost. Manuel D. Rossetti, Analysis of Inventory Systems, section 3.5.2.

If inspection holds every unit until the complete batch is released, the gradual-availability assumption may be wrong even though the machine physically produces pieces one by one. Identify the stock boundary: good units available for consumption, not merely machine starts. Do not mix an annual demand rate with an hourly production rate without conversion, or confuse the production rate during a run with annual output after all idle time and setups.

The textbook model excludes demand uncertainty, scrap/rework, breakdowns, shortages and interactions with other products unless explicitly extended. Constant unit production cost adds an annual term independent of Q; quantity-dependent costs, discounts or overtime can change the comparison. SAGE’s teaching resource distinguishes EPQ from quantity-discount and other inventory models. Venkataraman and Pinto, chapter 16 learning objectives.

Inventory geometry and cost

During production, units arrive at rate P and leave at rate D. Therefore net inventory buildup is P − D, not P. Once the run stops, remaining stock declines at rate D. The educational chapter on non-instantaneous replenishment describes this simultaneous production-and-consumption pattern. INFLIBNET, Operations Management, chapter 33.

For the no-shortage cycle beginning with zero stock:

production duration = Q/P

cycle duration = Q/D

peak cycle stock = Q(1 − D/P)

average cycle stock = Q(1 − D/P)/2.

Average stock is half the peak, not Q/2. Rossetti derives the resulting holding-cost term and optimal lot size from that inventory geometry. Rossetti, section 3.5.2.

Using the notation above, relevant cost per year and the continuous optimum are:

C(Q) = DS/Q + HQ(1 − D/P)/2

Q* = sqrt[2DS / {H(1 − D/P)}].

MIT gives this finite-replenishment tradeoff, including the correction relative to EOQ. At the same positive D, S and H, finite P produces a larger optimum lot than EOQ because some production is consumed before the run finishes. That does not mean EPQ necessarily holds more average inventory. MIT CTL, printed p. 47.

Original worked example: reconcile one cycle

Assume an invented item with D = 12,000 units/year, P = 48,000 units/year, S = 90 currency/run, and H = 8 currency/(unit·year). Both rates use 250 working days/year; production and consumption are uniform during that working-time calendar. Output is immediately usable, all pieces are good, shortages are forbidden and unit cost is constant.

The replenishment factor is 1 − 12,000/48,000 = 0.75. Thus:

Q* = sqrt[(2 × 12,000 × 90)/(8 × 0.75)] = 600 units.

Annual setup cost is 12,000/600 × 90 = 1,800. Average cycle stock is 600 × 0.75 / 2 = 225 units, giving annual holding cost 225 × 8 = 1,800. Relevant annual cost is 3,600 currency/year. The equal cost terms check this unconstrained optimum; they are not a rule to impose after operating constraints change the chosen quantity.

Daily demand is 48 units and production rate is 192 units per working day. One run therefore lasts 600/192 = 3.125 working days. Demand consumes 150 units during production, leaving 450 units at the peak. Production then stops for 450/48 = 9.375 working days while stock drains. The complete cycle is 12.5 working days, consistent with 600/48; there are 20 runs per working year.

After 150 units are consumed during the run, 450 remain for the drain phase. The 225-unit average is finished cycle stock only.

For comparison, the same inputs under instantaneous-receipt EOQ give 519.62 units and average cycle stock 259.81 units. The EPQ lot is larger, but its average finished cycle stock is lower because production and consumption overlap. This comparison changes only the replenishment assumption, not the costs or demand.

Check the schedule beyond the square root

Continue the example with an additional, independent assumption: each setup occupies the machine for two hours, and each working day has eight available hours. Annual processing occupies 12,000/192 = 62.5 days, or 500 hours. Twenty setups occupy another 40 hours. The item therefore uses 540 of 2,000 hours/year, or 27%, before other losses or products.

Setup cost S prices a run; setup time consumes capacity. Including setup cost in EPQ does not reserve those 40 hours. In this example, setup can be placed within the nonproduction interval while stock covers demand. A real plan must also fit other products, maintenance and material readiness. An annual capacity percentage alone does not prove that every required run fits before its due date.

For multiple stages, work-in-process holding and transfer batches can matter. Szendrovits’s original multistage EPQ paper explicitly includes overlapping operations and both WIP and finished-goods holding costs; the single-stock formula above does not represent that complete problem. Szendrovits, Management Science 22(3), November 1975, pp. 298–308.

Select a feasible quantity and review the assumptions

Evaluate permitted batch multiples, minimum runs, storage and shelf-life limits using the actual feasible quantities. Keep the same cost boundary for every alternative. If larger lots alter yield, overtime, material price or handling, revise the model rather than treating those effects as free.

When P approaches D from above, the EPQ expression becomes very large; this is a warning about the assumptions and lack of replenishment slack, not an instruction to make an enormous batch. At P = D there is no net buildup to cover a later production stoppage; at P < D production cannot sustain the stated demand. Investigate capacity, calendars and the replenishment policy instead of forcing a finite answer.

A planning handoff should retain demand and rate units, cost inputs, the inventory-release boundary, continuous optimum, feasible selected quantity, run/cycle/peak-stock results and capacity reconciliation. Determine when to launch each run through a replenishment and scheduling policy; EPQ alone does not calculate uncertainty protection or a customer delivery commitment.

Source versions

MIT’s PDF explicitly carries v6.2 Fall 2016. The historical multistage paper is dated November 1, 1975 by its publisher. No independently verified publication date is assigned here to the linked Rossetti, INFLIBNET or SAGE web chapters. The numerical inputs and calculations are original assumptions, not factory performance data.