*By Joannes Vermorel, January 2012*

EOQ is the purchase order quantity for replenishment that minimizes total inventory costs. The purchase order is triggered when the inventory level hits the reorder point. The EOQ is calculated in order to **minimize a combination of costs** such as the purchase cost (which may include volume discounts), the inventory holding cost, the ordering cost, etc. The order quantity optimization is complementary to the safety stock optimization that focuses on finding the optimal *threshold* to trigger the reorder.

*Update July 2020*: The EOQ is a fairly dated perspective, deeply rooted in

*stationary*perspective where future demand is not only perfectly known, but constant as well. Those formulas given in the following might “textbook” supply chain, but the underlying perspective is broken by design. Probabilistic demand forecasts associated with the assessment of the marginal economic contribution of every single extra unit of stock is a much better approach. We do not recommend to use the EOQ concept anymore.

## Model and formula

The *classical* EOQ formula (see the Wilson Formula below) is essentially a trade-off between the ordering cost, assumed to be a flat fee per order, and inventory holding cost. Although this formula dating for 1913 is extremely well-known, **we advise against using such a formula in any modern supply chain environment**. The underlying mathematical assumptions behind this formula are simply incorrect nowadays.

The historical formula assumes that the cost of the *act of ordering* is the one key business driver. It certainly was an important factor back in 1913 when an army of clerks was required to manually keep track of the books, but with inventory management software and possibly EDI, this factor is usually insignificant. As a result, the “optimization” performed by the formula makes little sense, and completely ignores any price break that can be available when larger quantities are ordered.

**Download Excel sheet:** eoq-calculator.xlsm (illustrated calculation)

Thus, we propose here an EOQ formula variant that **optimizes the trade-off of carrying costs vs volume discounts**. Let’s introduce the variables:

- $${Z}$$ be the lead demand.
- $${H}$$ be the
*carrying cost*per unit for the duration of the lead time (1). - $$\delta$$ be the delta inventory quantity needed to reach the reorder point (2).
- $${P}$$ be the per unit purchase price, a function that depends on the order quantity q

(1) The **time scope** considered here is the lead-time. Hence, instead of considering the more usual *annual* carrying cost $$H_y$$, we are considering $$H = \frac{d}{365}H_y$$ assuming that $$d$$ is the lead time expressed in days.

(2) The delta quantity needs to take into account both the stock on hand $$q_{hand}$$ and the stock on order $$q_{order}$$, which gives the relationship $$\delta = R - q_{hand} - q_{order}$$ where $$R$$ is the reorder point. Intuitively, $$\delta+1$$ is the minimal quantity to be ordered in order to maintain the desired service level.

Despite it’s seemingly complicated look, this function can be easily computed with Microsoft Excel, as illustrated by the sheet provided here above.

### What about the order cost?

At first glance, it might look as if we are assuming a zero ordering cost, but not quite so. Indeed, the framework we introduce here is relatively flexible and the order cost (if any) can be embedded into the price function $$\mathcal{P}$$.

## Cost function

In order to model the cost function for the order quantity which takes into account volume discounts, let’s introduce $${R}$$ the reorder point. The inventory cost is the sum of the inventory *carrying* cost plus the *purchase cost*, that is

Indeed, taking an amortized viewpoint over the lead time period, the total quantity to be ordered will be $${Z}$$ the lead demand.

Then, the inventory level is varying all the time, but if we consider strict minimal reorders (i.e. $${q=δ+1}$$) then, the average stock level over time is equal to $${R}$$ the reorder point. Then, since we are precisely considering order quantity greater than $${δ+1}$$, those extra ordered quantities are shifting upward the average inventory level (and also postponing the time when the next reorder point will be hit).

The $${(q−δ−1)/2}$$ represents the inventory shift caused by the reorder assuming that the lead demand is evenly distributed for the duration of the lead time. The factor $${1/2}$$ is justified because an increased order quantity of $${N}$$ is only increasing the average inventory level of $${N/2}$$.

## Minimization of the cost function

In order to minimize $${C(q)}$$, we can start by isolating the part that does not depends of $${q}$$ with:

Since $${RH}$$ does not depend on $${q}$$, optimizing $${C(q)}$$ is the same as optimizing $${C∗(q)}$$ where:

Then, in this context, since the volume discount function $$\mathcal{P}$$ is an arbitrary function, there is **no direct algebraic solution** to minimize this formula. Yet, it does not imply that this minimization is hard to solve either.

A simple minimization for $${C^∗(q)}$$ consists of a (naive) **extensive numerical exploration**, that is computing the function for a large range of $${q}$$ values. Indeed, virtually no business is needing order quantities greater than 1,000,000 units, and letting a computer explore all costs values for $${q=1..1,000,000}$$ takes less 1 second even if the calculation is done within Excel on a regular desktop computer.

However, in practice, this computation can be vastly accelerated if we assume that $$\mathcal{P}(q)$$ is a **strictly decreasing function**, that is to say that the price *per unit* strictly decreases when the order quantity increases. Indeed, if $$\mathcal{P}(q)$$ decreases, then we can start the value exploration at $${q=δ+1}$$, iterates, and finally stop whenever the situation $${C^∗(q+1)>C^∗(q)}$$ gets encountered.

In practice, unit price rarely increases with quantities, yet, some local *bumps* in the curve may be observed if shipments are optimized for pallets, or any other container that favors certain package sizes.

*EoqVD()*to revert back to a naive range exploration.

## Wilson Formula

The most well-known EOQ formula is the Wilson Formula developed in 1913. This formula relies on the following assumptions:

- The ordering cost is flat.
- The rate of demand is known, and spread evenly throughout the year.
- The lead time is fixed.
- The purchase unit price is constant i.e. no discount is available.

Let’s introduce the follow variables:

- $${D_y}$$ be the annual demand quantity
- $${S}$$ be the fixed
*flat*cost per order (not a*per unit*cost, but the cost associated to the operation of ordering and shipping). - $${H_y}$$ the
*annual*holding cost

Under those assumptions, the Wilson optimal EOQ is:

In practice, we suggest to use a more **locally adjusted** variant (time-wise) of this formula where $${D_y}$$ is replaced by $${D}$$ the forecast demand rate for the duration of the lead time (aka the lead demand $${Z}$$ divided by the lead time), and where $${H_y}$$ is replaced by $${H}$$, the carrying cost for the duration of the lead time.

### Comparison of the two EOQ formula

For retail or wholesale, we believe that our *ad-hoc* EOQ formula presented at the top of this page, that emphasizes volume discounts is better suited, hence more profitable, than the Wilson formula. For manufacturers, it depends. In particular, if the order triggers a new production, then indeed, there might be a significant ordering cost (production setup) and little or no benefits in marginal unit cost afterward. In such a situation, the Wilson Formula is more appropriate.