Lesson · Inventory Models · Reorder points and review intervals

Periodic Review and the Base-Stock Level

Under periodic review, stock is inspected every R periods and topped up to a base-stock level S. The order placed now arrives after L periods; the next chance to order anything is R periods from now and it arrives L after that. So what is on hand and on order right now has to last R + L periods, and S is a quantile of the demand over that whole span. Covering only the lead time is the standard error, and on the worked instance it turns a 90% target into 34.38%.

Open the interactive lesson Practice

review every R periods, order up to the base-stock level S the order placed now lands after L; the NEXT one lands R + L from now so the exposure window is R + L periods, not L and not R S = the smallest level with P(demand over R + L periods ≤ S) ≥ α 0,1,2 at 1/4,1/2,1/4 with R = 2, L = 1: three periods, mean 3, S = 5 at α = 90% S = 5 actually achieves 63/64 = 98.44%; covering L alone gives S = 2 and 34.38%

One window, and the two shorter ones it is mistaken for

Core ideas

What this lesson establishes

Three ideas, and the first of them is the whole lesson: everything else is what follows once the window is right.

Think about when the next chance to act arrives, not when this order does

An order placed at this review lands L periods from now. If demand in the meantime turns out high, the earliest correction is the order placed at the next review, R periods from now, and that one lands R + L periods from now. So the stock in hand and on order at this moment is the last defence for the whole of R + L. The window is a statement about when you can next intervene, which is why it contains the review interval at all.

A base-stock level is a position, not an order quantity

The order placed is S minus whatever is already on hand and on order, so it varies from review to review while S does not. That is the difference between this and continuous review: there the quantity Q is fixed and the moment varies; here the moment is fixed and the quantity varies. Comparing the two policies by comparing S with Q compares two unlike things.

A discrete distribution overshoots a target, and that is not a rounding

The cumulative distribution jumps, so the smallest level reaching a target usually passes it. On the worked instance a 90% target is first cleared at S = 5, which delivers 63/64, or 98.44%. Reporting 90% there would understate the stock being held by a whole unit, and the gap is a property of the distribution rather than of any arithmetic done to it.

The material

Why the interval is in the formula

The policy, the argument that fixes the window, the quantile, and what covering the wrong window actually costs.

Definition · Periodic review, and the base-stock level

Stock is inspected every R periods — the review interval — and an order is placed bringing the total on hand plus on order up to the base-stock level S. The order arrives after a lead time of L periods.

S is chosen so that the probability of meeting all demand over the exposure window is at least the target α.

Everything turns on what the exposure window is, and the argument is worth doing slowly because the wrong answer is so plausible. Suppose a review happens now. The order placed now arrives at time L. Between now and then, nothing can be changed. At time R there is another review, and whatever is ordered there arrives at time R + L. So the first moment at which a decision made after now can affect the stock position is R + L, and the position taken now must carry the demand of the entire interval up to it.

now R L R + L |-----------|-----------------|------------------| | | | | review next review this order next order order to S arrives arrives the position taken NOW is the last one that can affect stock before R + L, so it must cover demand over R + L periods

Set L = 0 and the window is still R periods: the order arrives instantly, but the next one cannot be placed until the next review, so the stock has to last the interval. Set R = 1 and the window is 1 + L, which is the continuous-review answer with one period of granularity added. Both limits are worth checking on the panel, because they are the two cases in which a reader's intuition says the interval should disappear and it does not.

The quantile, on the worked instance

Demand is 0, 1 or 2 a period with probabilities 1/4, 1/2, 1/4. Review every two periods, lead time one period, so the window is three periods and the distribution to quantile is the three-fold convolution: mean 3, support 0 to 6, and mass concentrated in the middle.

demand over R + L = 3 periods: total 0 1 2 3 4 5 6 prob 1/64 3/32 15/64 5/16 15/64 3/32 1/64 cumul 1/64 7/64 11/32 21/32 57/64 63/64 1 1.56% 10.94% 34.38% 65.63% 89.06% 98.44% 100.00% target 90%: the first cumulative at or above it is at 5 S = 5, achieving 63/64 = 98.44% target 95%: also S = 5. target 99%: S = 6, achieving 100%

S = 4 reaches 89.06%, just short of the 90% target, and S = 5 reaches 98.44%. There is nothing between them, so the policy that meets the target exceeds it by more than eight percentage points and holds a unit of stock that a continuous target would not have asked for. That is worth reporting rather than smoothing: the panel prints the achieved level beside the target for exactly this reason.

What covering the lead time alone costs

Example · The standard error, priced

Cover only the lead time. L = 1 period of demand, quantiled at 90%, gives S = 2 — a perfectly sensible-looking number reached by a perfectly sensible-looking calculation.

Over the real exposure window of three periods, a level of 2 delivers 11/32 of service: 34.38% against a target of 90%. The policy is not slightly under-stocked, it is wrong by a factor, and every figure used to produce it was computed correctly.

The panel shows both levels on the same table, the right one in amber and the wrong one in red, with the service each delivers over the real window. That is the design choice worth copying: a misconception that gets priced on the same page as the correct answer is much harder to keep than one that gets a warning in the margin.

Example · A longer interval, and where the stock goes

Raise the lead time to two periods, so the window becomes four. The mean rises to 4, and the base-stock level for a 90% target rises to 6, achieving 96.48%.

One extra period of lead time bought one extra unit of base-stock, which is less than the extra mean demand would suggest if the levels were driven by the mean alone — and more than nothing, which is what a reader who had fixed on the review interval might expect. The window is R + L and both letters do work.

Compare the two review policies on their exposure. Continuous review watches the shelf every moment, so it is exposed only for L; periodic review is exposed for R + L. The extra stock is what the reviewing discipline is not doing, and the trade is a real one: reviewing more often costs effort and holds less stock, reviewing less often costs stock and holds less attention. The model does not choose between them, and it does say exactly what the difference is.

Every figure on this page is exact. The convolution over R + L periods is a finite sum of products of rational probabilities, the cumulative column is a running sum of fractions, and the percentages are those fractions printed to two places rather than decimals standing in for them. 98.44% is 63/64, and 34.38% is 11/32; neither is rounded in the sense the earlier lessons of this course use the word, where an order quantity of 20√30 genuinely has no exact decimal form.

Interactive

Reviewing every R periods changes what the stock has to cover

R + L, and the service level a reader who used L alone would actually get

Set the review interval, the lead time and the target

The distribution quantiled here is the demand over the review interval PLUS the lead time, built by convolution on this page. The base-stock level is the first total whose cumulative probability reaches the target, and the achieved level is printed beside it because a discrete distribution overshoots. The comparison row prices the usual mistake — covering the lead time alone — over the real window, instead of warning about it.

Worked example

Demand in ONE period — “value probability”

Review interval R, in periods2

Lead time L, in periods1

Service target, in per cent90

Periods covered—

Mean over them—

Base-stock level S—

Achieved service—

S from L alone—

What that would cost—

Set the lead time to zero and the answer is still R periods of demand, not none: the order arrives instantly but the next one cannot be placed until the next review, so the stock has to last the interval. Set R to one and it becomes the continuous-review answer. The comparison row shows what covering only the lead time would leave exposed.

Method

Setting a base-stock level

Four steps. The first is the one this lesson exists for, and it takes one sentence to get right and one to get wrong.

Write down the window as R + L, in words, before computing anything

“The order I place now is the last one that can arrive before the one I place at the next review.” Say it about the actual situation, with the actual numbers in it. A reader who writes L here will get every subsequent step right and the answer wrong.

Convolve the per-period distribution R + L times

Not L times and not R times. The mean is (R + L) times one period's mean, and the spread grows more slowly than the mean does, which is why the safety component of S grows more slowly than the window.

Read S off the cumulative distribution at the target

The smallest total whose cumulative probability reaches α. The cumulative column is monotone, so the first one that clears is the answer and there is nothing to search.

Report the achieved service, not the target

A discrete distribution jumps past the target. On the worked instance a 90% target is met at S = 5 and achieves 98.44%, which is a unit of stock more than the target strictly required. Both numbers belong in the answer, and the gap between them is the cost of a discrete world.

Windows, positions and overshoot

Score 0 / 0

Choose the answer you can justify.

Three periods of exposure, and the level that covers them

Demand 0, 1, 2 a period at 1/4, 1/2, 1/4. Review every two periods, lead time one, target 90%. Every figure is the panel's own and all of them are exact fractions.

window: R + L = 2 + 1 = 3 periods mean over it: 3 total 0 1 2 3 4 5 6 prob 1/64 3/32 15/64 5/16 15/64 3/32 1/64 cumul 1/64 7/64 11/32 21/32 57/64 63/64 1 1.56% 10.94% 34.38% 65.63% 89.06% 98.44% 100.00% first cumulative at or above 90%: S = 5, achieving 63/64 = 98.44% S = 4 would give 57/64 = 89.06%, short of the target by under 1% covering the LEAD TIME alone: one period of demand, quantiled at 90% ⟹ S = 2 what S = 2 delivers over the real three periods: 11/32 = 34.38% changing the window: R = 1, L = 0 ⟹ 1 period, S = 2, 100.00% R = 2, L = 0 ⟹ 2 periods, S = 3, 93.75% R = 2, L = 1 ⟹ 3 periods, S = 5, 98.44% R = 2, L = 2 ⟹ 4 periods, S = 6, 96.48%

The two middle blocks are the same distribution read twice. S = 2 is the right answer to the wrong question — it is exactly the 90% quantile of one period's demand — and it delivers 34.38% against the target. Nothing in the calculation that produced it is wrong, which is what makes the error durable: it survives checking, and it can only be caught by asking what window the number is supposed to cover.

The bottom block shows both letters doing work. With L = 0 the answer is still S = 2 over one period, because the order arrives instantly but the next one cannot be placed until the next review. Adding a period of lead time to a two-period review takes S from 3 to 5; adding another takes it to 6. The level is not proportional to the window, because the safety component grows like the spread rather than like the mean.

For a rehearsal, keep the distribution and set the target to 95%. The supplied first move is that the cumulative column does not change at all — only the threshold does. Predict the base-stock level and its achieved service before you move the slider, and then say what that result implies about how precisely a service target can be specified on a discrete distribution.

Common mistakes

Covering the lead time and forgetting the intervalThe order placed now is not the last one before the stock runs down; it is the last one before the next review's order arrives. Omitting R turns a 90% target into 34.38% on the worked instance, and every figure in the wrong calculation is correct, so nothing in the arithmetic will reveal it.

Treating the base-stock level as an order quantityS is a target position, and the order placed is S minus what is already on hand and on order. Ordering S units at every review ignores everything in transit and over-orders by exactly the amount already coming, which is the classic way a periodic-review system oscillates.

Reporting the target instead of the achievementA discrete distribution overshoots, so “we hold a 90% service level” is usually false and usually false in the expensive direction: here the policy achieves 98.44% and holds the stock to match. The pair of numbers is the honest report, and the gap between them is a real cost that a single figure hides.

Check your understanding

✓

Finish when you can say why the review interval belongs in the window without looking it up.

You should be able to derive R + L from the timing of the next review and its arrival, build the demand distribution over that window by convolution, read the base-stock level off the cumulative column at a target, report the service it actually achieves, and state what covering the lead time alone would deliver on the same data.

Note: That completes the course: a quantity from a discriminant, two corrections to the cycle, and three models in which demand is a distribution rather than a rate. What none of them can do is check the distribution against the world it came from — the demand model is an input here, and Simulation and Variance Reduction is where a system too tangled for any of these formulas gets run instead of solved.