3. When Should I Order?

Chapter 2 used EOQ to estimate a useful batch size. Now we add the other half of the decision: when should a replenishment order be triggered? The answer depends on how long supply takes and, for periodic policies, how often inventory is reviewed.

We begin with deterministic demand and fixed lead time so the mechanics stay visible. Real international lanes have variable customs, transport, and production delays; those uncertainties require additional protection beyond the baseline calculations here.

Notation and units

Symbol	Meaning
$d$	Demand per time period, such as units per week
$D$	Demand per year, when annual cost models are used
$L$	Supplier lead time, measured in the same period as $d$
$R$	Time between scheduled inventory reviews
$s$	Continuous-review reorder point
$S$	Periodic-review order-up-to level
$I_{transit}$	Average inventory in the supply pipeline

Always align time units before multiplying. Weekly demand needs lead time in weeks; annual demand needs lead time in years.

Continuous review: cover demand during lead time

Assume demand is constant at $d$ units per period and replenishment always arrives after a fixed lead time of $L$ periods. If there is no safety stock, order when inventory position reaches the amount expected to be used before the order arrives:

$$s = dL$$

For example, weekly demand of 3 units and a two-week lead time imply a reorder point of 6 units. Under a continuous-review $(s,Q)$ policy, reaching that threshold triggers the fixed order quantity $Q$.

In practice, calculate the trigger from the inventory position (on hand plus on order minus backorders), not from on-hand stock alone. This matters when several orders are already in transit. If demand or lead time is uncertain, the reorder point generally needs a safety-stock allowance; $dL$ is only the deterministic baseline.

Inventory in transit

Goods ordered but not yet received are pipeline, or in-transit, inventory. With stable flow, Little's Law says average pipeline inventory is throughput multiplied by time in the system:

$$I_{transit} = dL = DL$$

Use $d$ with $L$ in matching short-period units, or annual demand $D$ with lead time $L$ in years. For example, demand of 8 units per week and a three-week lead time imply an average of 24 units in transit.

This is an average-flow relationship, not an extra order quantity to add to every order. Making fewer, larger orders does not by itself reduce average pipeline stock when demand and lead time stay the same. To reduce it, reduce the lead time or the flow rate. The purchase contract also determines whether the buyer or supplier owns goods while they are in transit, and therefore who carries the associated cost and risk.

Periodic review: cover lead time and the next review interval

With periodic review, an order can only be placed at the next scheduled check. A deterministic order-up-to level must therefore cover demand over both the supplier lead time and the wait until the next review:

$$S = dL + dR = d(L+R)$$

If weekly demand is 3, lead time is 2 weeks, and reviews are 4 weeks apart, the baseline target is $S=3(2+4)=18$ units. At a review, order enough to raise the inventory position to $S$; the order quantity itself depends on the current position.

For steady deterministic demand, demand over one review interval is $dR$. That is the replenishment amount per interval in the stable case, and the cycle stock rises and falls across that interval. Its average is:

$$C_s = \frac{dR}{2}$$

The average pipeline amount remains $I_{transit}=dL$. Thus the order-up-to level is not the expected on-hand inventory. In a stable, no-shortage system, average net on-hand inventory is approximately $S-dL-dR/2$. With the baseline target $S=d(L+R)$, this reduces to the average cycle stock $dR/2$.

For example, with one unit of daily demand, a five-day lead time, one-day reviews, and an order-up-to level of 10, average net on-hand stock is about $10-5-0.5=4.5$ units, not 10. The target is an inventory-position target; pipeline units are part of that position.

Forecast-based targets

If demand is expected to change over the protection window, replace constant-rate products with forecasts for the actual periods:

$$s = F_L, \qquad S = F_L + F_R$$

Here $F_L$ is forecast demand during lead time and $F_R$ is forecast demand from now through the next review interval. These are sums over the relevant periods, not necessarily one average rate multiplied by a duration. Safety stock, if needed, is added separately.

Choosing a review interval with EOQ

For a periodic policy under constant demand, orders in steady state cover demand since the previous review. If $R$ is measured in years, the average cycle stock is $DR/2$ and the number of orders per year is $1/R$. The annual relevant cost is:

$$C(R) = \frac{K}{R} + h\frac{DR}{2}$$

This is the EOQ cost model written in review-period form. Using $Q=DR$ gives the optimal review interval:

$$R^* = \frac{Q^*}{D} = \sqrt{\frac{2K}{hD}}$$

With annual $D$, annual per-unit holding cost $h$, and fixed cost $K$ per order, $R^*$ is in years. A fixed lead time adds pipeline inventory; if its carrying cost is included, that term is constant with respect to $R$ and does not change this baseline optimum. Constraints such as supplier calendars or order consolidation can still make a different interval more practical.

Turn an optimum into a workable schedule

An optimal interval is a model output, not automatically a sensible meeting on a calendar. If a single item has an ideal review every 1.23 days, an organization may choose one day or another workable multiple. The sensitivity curve is fairly flat near the optimum, so modest rounding may have little modeled cost impact.

To coordinate many products, a company can choose a common base period $R_B$ and schedule reviews at powers of two:

$$R_k = 2^k R_B, \qquad k=0,1,2,\ldots$$

This groups products into a small number of recurring schedules, such as every week, two weeks, four weeks, or eight weeks. Choose among the feasible periods by comparing their cost ratios to the ideal interval:

$$\frac{C(R)}{C(R^*)} = \frac{1}{2}\left(\frac{R}{R^*}+\frac{R^*}{R}\right)$$

When adjacent power-of-two intervals bracket $R^*$, choosing the less costly neighboring option limits this model's cost premium to about 6.1%. That bound depends on having feasible schedule options on both sides of the optimum; a base period or calendar constraint can remove that guarantee.

As an example, if $R^*=29.12$ days and the base period is one day, the neighboring choices are 16 and 32 days. The 32-day schedule has a cost ratio of about $0.5(32/29.12+29.12/32)=1.0045$, or roughly a 0.45% modeled premium. With a five-day base period, the neighboring powers are 20 and 40 days; compare those against the same formula before choosing.

Practice: calculate and schedule

Concept check

Quiz

Lead time and review periods

1 / 6

Demand is 3 units per week and supplier lead time is 2 weeks. With no safety stock, what is the continuous-review reorder point?

Pick an answer to see if you are right.

Rank the periodic-review procedure

Put the decision steps in order for a deterministic $(R,S)$ review.

Quiz

Run a periodic replenishment review

1 / 2

The order-up-to target S has already been calculated. At a scheduled review, determine and release the order.

OrderDrag the handle to rearrange — top should be first.

1. ⋮⋮1Calculate inventory position from those quantities.
2. ⋮⋮2Read the current on-hand, on-order, and backorder quantities.
3. ⋮⋮3Release that quantity if it is positive; otherwise record that no order is needed.
4. ⋮⋮4Calculate the shortfall max(0, S - inventory position).

Drag items into the correct order (or use ↑↓), then check.

Recap

Decision	Deterministic baseline	What it represents
Continuous reorder point	$s=dL$	Demand expected during supplier lead time
Average pipeline stock	$I_{transit}=dL$	Average units traveling through the supply lead time
Periodic order-up-to level	$S=d(L+R)$	Demand over lead time plus time until the next review
Periodic average cycle stock	$C_s=dR/2$	Average on-hand buffer across a review cycle, excluding safety stock
Optimal review interval	$R^*=\sqrt{2K/(hD)}$ years	EOQ expressed as time between replenishments

These equations assume constant demand and fixed lead time. Forecasts can replace constant-rate estimates, and safety stock is needed when uncertainty or service requirements demand it.

Previous: How Much Should I Order?.

Next: Safety Stocks — quantify a demand buffer for a chosen cycle service target.