Risk pooling: safety stock falls, the stock in transit does not

Scale

The four fixed panels share one scale, 0 to 3,600 units, and it does not move. The live chart above rescales with what you type.

Basis

Both columns carry the same demand and the same lead time. Only where the buffer is sized differs.

Assumptions

Demand adds normally; one correlation for every pair; the lead time is fixed; the units in transit are as typed.

Model version

1.0

The formula, in words

• Three blocks, and only one of them is in question here. Inventory sits in three kinds: what you hold because you buy in batches, what you hold because demand is uncertain, and what is in transit because a replenishment takes time. This page prices the second and draws the third and leaves the first out entirely. Any figure that quotes a change to inventory without saying which of the three moved is a figure with a definition missing, and separating them before anything is compared is the first thing the sources named below insist on.
• Spreads do not add. Squares of spreads do, once you account for how the demands move together. Add four uncertain demands and the spread of the total is the square root of a sum: each location's spread squared, plus the correlation multiplied by every cross term between one location and another. Four locations at 50 units a day each give 10,000 for the squares and 30,000 for the cross terms, so at a correlation of 0.25 the total spread is $\sqrt{17{,}500} = 132.29$, which is 132 rather than the 200 you get by adding the four straight. That gap between 132 and 200 is the whole of the arithmetic.
• The square root law is the special case, not the rule. Where the locations are the same size and nothing they have in common moves demand, the total spread is one location's spread times the square root of the count, against the count times that spread if each holds its own buffer, so the buffer falls by one over the square root of the count. That is a clean result and it is true only under both of those conditions. Break either one and it stops holding: at a correlation of 0.25 the four locations keep 66% of their buffer rather than 50%, and with one location carrying most of the demand they keep 76% of it on the same total spread. Equal locations that move independently produce the widest gap between the two columns, and every departure from either condition narrows it.
• The safety factor and the lead time scale both sides and cancel out of the percentage. How much of the buffer the arrangement removes depends on three things only: how many locations there are, how their sizes compare, and how much they move together. It does not depend on the safety factor or on the lead time at all, because both multiply the left column and the right column by the same amount. What those two decide is how many units that percentage is worth, which is what makes the figure material or trivial. The safety factor is also the door a service target comes in through: turning a target into a factor needs a distribution shape that nobody typed here, and the lesson on what the last percent of service costs is where that curve is drawn. This page takes the factor as your number and reports it back unchanged.
• The block that does not move is why the percentage is smaller than it looks. Because the stock in transit is identical on both sides, it cancels out of the difference: the two totals differ by exactly what the two buffers differ by, and not by a unit more. So the same arrangement that takes a third off the buffer takes a ninth off the inventory. And afterwards the untouched block is a larger share of what is left than it was before, not because it grew but because the total it is measured against shrank.

What this does not model

How far the stock then sits from whoever is waiting for it, and what the added response time is worth. Freight, handling and the extra moves a single location creates on the way out. Whether the same offsetting could be had without moving anything, through a rule that lets four locations draw on one another's stock and a record accurate enough to support it, which reaches most of the same arithmetic and none of the same buildings. Whether a buffer removed here reappears one level up, since moving a buffer is not the same as removing it and a total that improves locally can worsen across the chain. Cycle stock, which is not on this page in any form. Lead time variability, since the spread of demand over a lead time carries a second term for the lead time's own variance and this page holds the lead time as a fixed number. The square root of the lead time itself, which assumes demand in one period is independent of demand in the next; the fitted exponent is often not one half, and where it is not, the buffer scales differently with the lead time than this page draws. The distribution behind treating a factor times a spread as a buffer, which is a normal shape assumed rather than tested and which understates the buffer wherever the real demand has a fatter upper tail. What service level the factor buys, which needs that shape and is not printed here for that reason. That one correlation figure stands for every pair, where four real locations have six pair correlations and they will not agree. That the units in transit are taken as typed and are never checked against the lead time, though the two are linked by the same relation the sibling lesson on boundaries draws. Where the spread figures came from, which is the input carrying the most weight and the least evidence. And the largest omission of all: whether the demand at four locations in the period that produced those spreads is the demand they will face next.

A worked example

The four locations as shipped. Four spreads of 50 units a day each. Added straight they come to 200 units a day, and that is the figure four separate buffers are sized against. Squared and added they come to $4 \times 2{,}500 = 10{,}000$.

The cross terms. Adding the four spreads and squaring gives $200^2 = 40{,}000$. Take the squares away and $40{,}000 - 10{,}000 = 30{,}000$ is left, and that is the part of the total that depends on how the four move together. Nothing else in the arithmetic does.

The total spread at the typed correlation. $10{,}000 + 0.25 \times 30{,}000 = 10{,}000 + 7{,}500 = 17{,}500$, and $\sqrt{17{,}500} = 132.29$ units a day. Four locations that move together a little behave like one location with a spread of 132, not 200.

The multiplier both sides share. A safety factor of 2.00 across a lead time of 9 days is $2.00 \times \sqrt{9} = 2.00 \times 3 = 6$. It multiplies the left column and the right column identically, which is why it will not appear in any percentage below.

The two buffers. Four separate buffers: $6 \times 200 = 1{,}200$ units. One buffer against the four demands added first: $6 \times 132.29 = 793.73$ units, shown as 794. The difference is $1{,}200 - 793.73 = 406.27$ units, shown as 406, or 33.9% of the buffer.

The block that does not move. 2,400 units are in transit, and the arrangement does not change how much is flowing or how long it is in flight, so 2,400 units are in transit on both sides. It is the same number in the left column and the right column and it is drawn as the same rectangle in both.

Everything held. Four locations: $2{,}400 + 1{,}200 = 3{,}600$ units. One location: $2{,}400 + 793.73 = 3{,}193.73$ units, shown as 3,194. At $85 a unit that is $306,000 against $271,467, and the difference in dollars is $406.27 \times 85 = 34{,}533$.

The two percentages, and the distance between them. The buffer fell by 33.9%. Everything held fell by $\frac{34{,}533}{306{,}000} = 11.3\%$. Both are true and they are three times apart, and the whole of the distance between them is the 2,400 units that were never in the comparison.

The share that rises without anything moving. Before, the stock in transit was $\frac{2{,}400}{3{,}600} = 66.7\%$, or 66.7% of everything held. After, it is $\frac{2{,}400}{3{,}193.73} = 75.1\%$, or 75.1%. Not one unit of it moved.

The four panels, one parameter, four values. Hold everything and walk the correlation across the whole range four locations can occupy.

Read the first row and the last together. At the furthest apart four locations can move, the whole buffer goes and $204,000 is still standing. At demand that peaks in the same week everywhere, nothing goes at all. Between those two the curve is steep early and flat late: half of what independence promises has already gone by a correlation of 0.42, which is not a high figure for four locations sharing a season, a working week and a set of customers.

The same total spread, arranged differently. Take 125, 25, 25 and 25 units a day. They add to the same 200, so four separate buffers are still $6 \times 200 = 1{,}200$ units. But the squares now come to $15{,}625 + 625 + 625 + 625 = 17{,}500$, and the cross terms to $40{,}000 - 17{,}500 = 22{,}500$. At the same correlation of 0.25 the total is $17{,}500 + 5{,}625 = 23{,}125$, and $\sqrt{23{,}125} = 152.07$. One buffer is now 912 units rather than 794, the difference is $1{,}200 - 912.41 = 287.59$ units or $24,445, and the buffer falls by 24.0% rather than 33.9%.

Nothing on the page changed except how one number was split among four locations, and $10,089 of the case went with it. Where one location carries most of the demand there is less for the others to offset, and the square root law, which is quoted as though it were a property of counting warehouses, turns out to be a property of counting warehouses that are the same size.

What the arithmetic will not do. It will not say whether $34,533 is worth a building. It will not say what the stock costs to move once it is in one place, or what the extra day to the customer is worth, or whether the six real pair correlations behind that single 0.25 look anything like it. It says that on these numbers the buffer falls by a third and the inventory by a ninth, and that the distance between those two fractions is a block of 2,400 units that was never listening.

Check it yourself

The four spreads become 125, 25, 25 and 25 units a day, which add up to 200, exactly as the four fifties did. The correlation does not move, the lead time does not move, the safety factor does not move, and the units in transit do not move. The buffer one location would hold goes from 794 units to 912, the difference falls from $34,533 to $24,445, and $10,089 of the case disappeared without a single total changing. Two thirds of it was never in the argument.

The sources named for this page are Cachon and Terwiesch, Matching Supply with Demand, fourth edition; Hopp and Spearman, Factory Physics, third edition; John D. Sterman, Business Dynamics; Silver, Pyke and Thomas, Inventory Management and Production Planning and Scheduling, fourth edition; Sunil Chopra, Supply Chain Management, seventh global edition; and Savage, The Flaw of Averages.