# Economic order quantity

EOQ as a bounded lot-sizing model: ordering/holding costs, consistent units, feasible pack sizes, sensitivity, reorder timing and model limits.

Reviewed: 2026-11-01

Tags: production, planning, inventory, lot-sizing

The illustrative EOQ model uses **12,000 units/year**, **75 currency/order** and **6 currency/(unit·year)** holding cost, with uniform demand, constant price and whole-lot receipt. The continuous optimum is **547.72 units**. When orders must be multiples of **100 units**, both **500** and **600** cost **3,300 currency/year** in ordering and cycle-stock holding. Compare feasible quantities and any other costs before selecting one.

EOQ balances recurring ordering and cycle-stock holding costs. MIT’s basic model assumes deterministic demand, no backlogs and whole-lot receipt. [Stanley B. Gershwin, MIT 2.854 Inventory, ©2016](https://www.ocw.mit.edu/courses/2-854-introduction-to-manufacturing-systems-fall-2016/6c8fec8f99bcf7059b73b82e96d43901_MIT2_854F16_Inventory.pdf)

## Define the model before using the formula

Use one item, uniform demand, constant costs and no price breaks for the basic model. Handle lead time separately from quantity. [SAGE, Operations Management, chapter 16](https://edge.sagepub.com/venkataraman/student-resources/chapter-16/learning-objectives)

Use a single time basis and one physical unit:

| Symbol | Meaning | Example unit |
| --- | --- | --- |
| D | Demand rate | units/year |
| S | Incremental fixed cost of one replenishment order | currency/order |
| H | Holding cost for one unit for the same time basis | currency/(unit·year) |
| Q | Quantity received per order | units/order |

A purchase order is not automatically the right cost event. Establish whether a setup, shipment, release or another activity creates the fixed cost being modeled. State which costs actually change when lot size changes. Allocated annual overhead should not silently become an incremental cost of each order.

## Cost expression and optimum

With inventory decreasing uniformly from Q to zero, average cycle stock is Q/2 and order frequency is D/Q. The relevant annual cost is:

`C(Q) = DS/Q + HQ/2`

For positive D, S and H, its continuous optimum is:

`Q* = sqrt(2DS/H)`

MIT's deterministic model supplies this tradeoff; the result assumes unrestricted lot size and the stated replenishment behavior. With constant unit purchase price, annual purchase cost is independent of Q and can be omitted when comparing these lot sizes. It must still be included when reporting total purchasing expenditure. [MIT inventory notes, EOQ section](https://www.ocw.mit.edu/courses/2-854-introduction-to-manufacturing-systems-fall-2016/6c8fec8f99bcf7059b73b82e96d43901_MIT2_854F16_Inventory.pdf)

At the unconstrained EOQ, annual ordering cost equals annual cycle-stock holding cost. Washington State University's teaching transcript also distinguishes per-unit holding cost from a carrying rate applied to item cost. That equality is a model check, not a target to enforce after constraints or quantity discounts change the problem. [Chuck Munson, WSU Management and Operations 340 transcript](https://apps.aoi.wsu.edu/onlineMedia/transcripts/mgtop340_04_v%28c132%29.pdf)

## Establish a credible holding-cost basis

MIT CTL presents holding cost as item value multiplied by a carrying charge. Identify the valuation and rate behind H and keep their currency and period consistent. A monthly rate combined with annual demand without conversion changes the result. [Jarrod Goentzel, MIT CTL inventory slides, ©2022](https://ctl.mit.edu/sites/ctl.mit.edu/files/attachments/S2%20-%20Aug%204%20-%20W2%20-%20Slides.pdf)

For an actual planning decision, document which financing, storage, insurance, deterioration or obsolescence elements are included and avoid double counting them. Specify whether the cost basis is historical purchase value, replacement value or another approved basis. EOQ does not determine an organization's accounting or investment policy.

Zero or missing inputs need an explicit decision. If ordering has no fixed cost in the chosen boundary, or H is not a meaningful positive estimate, the square-root expression is not a finite practical lot-size recommendation. Resolve the cost model and operating constraints rather than replacing missing values with convenient defaults.

## Original example: pack sizes can change the decision

Assume an invented item with **D = 12,000 units/year**, **S = 75 currency/order**, and **H = 6 currency/(unit·year)**. Price is constant, demand is uniform and replenishment arrives as a whole lot:

`Q* = sqrt(2 × 12,000 × 75 / 6) = 547.72 units`

At that continuous optimum, each annual cost term is approximately **1,643.17**, totaling **3,286.34 currency/year**. Average cycle stock is **273.86 units**. These are model outputs, not a statement about actual demand or costs.

Now suppose the supplier only permits positive multiples of **100 units**:

| Feasible Q | Orders/year D/Q | Ordering cost DS/Q | Holding cost HQ/2 | Sum/year |
| --- | --- | --- | --- | --- |
| 500 | 24 | 1,800 | 1,500 | 3,300 |
| 600 | 20 | 1,500 | 1,800 | 3,300 |
| 700 | 17.14 | 1,285.71 | 2,100 | 3,385.71 |

The cost function decreases up to Q* and increases afterward, so the adjacent feasible sizes **500 and 600** are sufficient candidates for the pack-constrained minimum in this example. Both cost **3,300**, about **0.42%** above the continuous minimum. They have different order frequencies and average cycle stocks, which can inform a further decision under explicitly added constraints. Rounding only upward would conceal the tie.

The example excludes safety stock, discounts, freight tiers, capacity and storage restrictions, shortages, uncertainty and finite-horizon effects. Fractional annual order counts are long-run rates, not a literal schedule of partial orders within one calendar year.

## Quantity is separate from reorder timing

SAGE distinguishes the fixed order quantity from the reorder point, which depends on demand, lead time and desired service level. EOQ therefore does not itself calculate safety stock or protect against an uncertain delivery. Select a replenishment policy with the appropriate uncertainty model. [SAGE, reorder-point and review-system distinctions](https://edge.sagepub.com/venkataraman/student-resources/chapter-16/learning-objectives)

Identify whether a reorder decision uses on-hand stock or inventory position, including open orders and committed demand under the chosen policy. Keep pipeline stock and safety stock separate from the Q/2 cycle-stock quantity. A recurring production release driven by a dated component schedule may need time-phased lot sizing rather than a constant-demand approximation.

## Recognize when the basic model is insufficient

Quantity discounts require comparing annual purchase cost as well as ordering and holding cost. Gradual production replenishment calls for a production-quantity model; seasonal or one-time demand has a different overstock/shortage tradeoff. These are distinct models in the SAGE teaching resource, not adjustments obtained by changing EOQ's label. [SAGE, EPQ, discounts and single-period systems](https://edge.sagepub.com/venkataraman/student-resources/chapter-16/learning-objectives)

Also check minimum orders, pallet/container multiples, shelf life, storage space, cash limits, supplier capacity, shared setups and transport consolidation. Adding such constraints can couple items or make a nominal EOQ infeasible. Evaluate the complete constrained cost and service problem; a mathematically correct square root is not evidence that a purchase release is executable.

## Planning handoff and review

Record item/revision, demand estimate and period, cost assumptions, quantity unit, eligible pack sizes, supplier terms, capacity/shelf-life limits and the proposed replenishment policy. Compare feasible alternatives with the same cost boundary and retain the reason for choosing among ties or near-ties. Revisit the model when demand, setup costs, terms or material life changes.

Report the calculated continuous EOQ, feasible selected quantity and actual order schedule as separate outputs. This prevents a benchmark from being mistaken for an authorized order or a stock reservation. The chosen organization-specific policy and approval responsibilities are inputs, not rules invented by this reference.

## Source versions

Reviewed 2026-11-01. Gershwin's MIT notes carry **©2016**; Goentzel's MIT CTL slides carry **©2022**. No separately verified publication date was established for the linked WSU transcript or SAGE chapter resource. Hosting/crawl dates are not treated as editions. The numerical example, input checks and handoff recommendations are original and bounded to the stated model.

## Related reference

[Production planning guides](index.md)