Service level: what the last percent of it costs

Moving the target from 90% to 95% adds one step of buffer. Moving it from 99% to 99.9% adds more than twice that step, for a tenth of the risk, and most policies were set without anybody drawing the curve.

Live result

The next point of service costs $517 a year and removes $730 of expected stockout cost.

service level you set95.0%
where the two costs meet96.7%
safety stock at that point1,116 units
days of demand it covers11.2 days

Below that point every added point of service still removes more than it costs.

Try a scenario

Service level you are aiming at95.0%

Units a dayunits/day

How steady demand is

Lead timedays

How much the lead time movesdays

Unit cost

Shown in any single currency; the arithmetic is identical.

Holding rate a year%

What one stockout costs you

At the balance service level: 96.733%

Safety stock	1,116 units, 11.2 days
Holding it a year	$9,822
Expected stockout cost	$2,385
Both together	$12,206
One more point costs	$730
One more point saves	$730

Field record

The policy curve is cycle service. Demand over the protection period is assumed close to normal, and one stockout has one entered cost.

What the buffer costs

Four measures, one run

at 95.0% policy

95.0%

numerator: · cycles that end without a stockout
denominator: · 12 replenishment cycles
window: · a year

Units filled

99.6%

numerator: · units shipped from stock
denominator: · 3,000 units demanded in a cycle
window: · one cycle

Filled within 3 days

98.4%

numerator: · units served inside the threshold
denominator: · 3,000 units demanded in a cycle
window: · one cycle

Units still short

13 units · 154 a year

numerator: · units not shipped on the day asked for
denominator: · not a percentage, a count
window: · one cycle, and a year

At a 95.0% policy, 95.0% of cycles end without a stockout, 99.6% of units ship from stock, 98.4% are filled within 3 days, and about 13 units of the 3,000 in a cycle are still short.

A needs a lot size and this page models none, so the denominator here is one cycle of demand, 3,000 units. That is stated rather than assumed, because the whole argument of this page is that an undeclared denominator is not a measurement.

What one more point costs and saves

What the balance point is not

It is not a target and it is not a recommendation. It is the place where two of your own numbers cross, and one of them, what a stockout costs you, is usually an estimate somebody made once. Move it and the crossing moves with it, which is the most useful thing this page can show you: the argument about service level is almost always an argument about that figure, held indirectly.

It also assumes a stockout costs the same whether it lasts an hour or a fortnight, that demand and lead time are independent, and that missing a cycle costs you the same as missing the one before. None of those are true, and all of them are why the number below the curve is a starting point for a conversation rather than the end of one.

The cost of one stockout

From

In steps of

It does not check the values in between.

Across the values sampled, where your service level sits stays below the balance point from $5,000 to $13,000, the highest value sampled. The nearest value where it changes is $4,000, where it becomes above the balance point.

The answer	below the balance point
Values sampled	$3,000 to $13,000, 11 values, step $1,000
Holds from	$5,000
Holds to	$13,000, highest sampled
Nearest value that changes it	$4,000
The answer there	above the balance point

What an unchanged region does and does not prove is set out in a short lesson.

The sweep samples only the values you entered. Between samples the answer is not checked, so read the boundary as observed, not proven.

Here is the maths

The late unit threshold stays fixed at 3 days in this example.

The formula, in words

Safety stock is one number times a lookup. Combine the two sources of uncertainty into one figure, then multiply it by how many of them you want to cover; the second part is the whole of the service level decision.
The lookup is why the curve bends. Going from 90% to 95% is 0.36 of those figures. Going from 99% to 99.9% is 0.76 of them, twice as much buffer for a tenth of the risk, and nothing in the policy document says so.
A stockout has a price and almost nobody writes it down.Multiply the cycles in a year by the chance of missing one by what missing one costs.
Where the two slopes meet. One more point of service always saves the same amount and always costs more than the point before it, so they cross once, and the crossing moves when any input does.

\sigma_D=D\times v

where

\sigma_D

is daily demand standard deviation,

D

demand, and

v

the variability factor.

\sigma_L=\frac{r}{2}

where

\sigma_L

is lead time standard deviation, and

r

the lead time spread.

\sigma=\sqrt{L\times\sigma_D^2+D^2\times\sigma_L^2}

where

\sigma

is combined standard deviation, and

L

lead time.

S=z\times\sigma=\Phi^{-1}\!\left(\frac{s}{100}\right)\times\sigma

where

S

is safety stock,

z

the normal score for the service level,

\Phi

the normal cumulative distribution, and

s

the service level in percent.

H=S\times c\times h

where

H

is holding cost,

c

unit cost, and

h

the annual holding rate.

O=n\times p\times b

where

O

is stockout cost,

n

cycles a year,

p

stockout probability, and

b

the cost of one stockout.

\Delta H=\frac{\sigma\times c\times h}{\phi(z)}\times0.01

where

\Delta H

is the cost of one more point, and

\phi(z)

normal density.

\Delta O=n\times b\times0.01

where

\Delta O

is the stockout saving from one more point.

\ell=\phi(z)-z\times(1-\Phi(z))

where

\ell

is normal loss.

Q=\ell\times\sigma

where

Q

is units short.

F=\left(1-\frac{Q}{B}\right)\times100

where

F

is fill rate in percent, and

B

cycle demand.

C=\Phi\!\left(z+\frac{D\times\tau}{\sigma}\right)\times100

where

C

is the share filled within the threshold in percent, and

\tau

threshold days.

A worked example

Shipped defaults: 100 units a day, normal demand, lead time 30 days, spread 12, unit cost $40, holding 22%, one stockout $6,000.

\sigma_D=100 \times 0.15 = 15

\sigma_L=12 \div 2 = 6

. The summed variance is

(30\times225+10{,}000\times36)

, with value

366{,}750

\sigma=\sqrt{30\times225+10{,}000\times36}=\sqrt{366{,}750}=605.60

units. The cost of one unit of z is

605.60 \times 40 \times 0.22 = 5{,}329.27

dollars, and that single number carries the whole curve. Cycles a year are

365 \div 30 = 12.1667

, so a year of certain stockouts would cost

12.1667 \times 6{,}000 = 73{,}000

dollars and one point of service saves a hundredth of it,

$730, at every point on the axis.

Service level	z	Holding the buffer	Stockouts	Both	One more point costs
90%	1.28155	$6,830	$7,300	$14,130	$304
95%	1.64485	$8,766	$3,650	$12,416	$517
96.73%	1.84299	$9,821	$2,385	$12,206	$730
99%	2.32635	$12,398	$730	$13,128	$2,000
99.9%	3.09025	$16,469	$73	$16,542	$15,829

The balance point, two ways, and they must agree. Closed form: $5{,}329.27 \div 73{,}000 = 0.073004$ , so $e^{-z^2/2}=\frac{0.073004}{0.398942}=0.182995$ , where $z$ is the normal score. That gives $z^{2} = -2 \times \ln(0.182995) = 3.39661$ and then $z = 1.84299$ . Finally $\Phi(1.84299)=0.96733$ , so 96.73%. Scan: the 100 point grid puts its lowest total at 96.7, at $12,206.45, and the totals either side of it sit above it. Both routes land on the same tenth, and that agreement is the test.

The two sentences the page exists for. The step from 90% to 95% adds $8{,}766 - 6{,}830 = 1{,}936$ dollars of buffer. The step from 99% to 99.9% adds $16{,}469 - 12{,}398 = 4{,}071$ dollars, 2.10 times as much for a tenth of the risk. A policy sitting at 99.9% carries $16{,}542 - 12{,}206 = 4{,}336$ dollars a year more than the balance point, 35.5% more, to move the chance of a stockout in any one cycle by 3.2 points. The last column is the sharper version of the same fact: a point of service costs $304 at 90% and $15,829 at 99.9%, and the saving it buys is $730 at both ends.

At 95%, normal density is 0.103145 and the normal loss is 0.020893. Multiplying by 605.5989 gives 12.65 units short in one cycle. One cycle asks for

100 \times 30 = 3{,}000

units, so the fill rate is 99.6%. Allowing a late unit 3 days shifts z by

100 \times 3 \div 605.5989 = 0.49538

, taking the reading to 98.4%. Across

365 \div 30 = 12.1667

cycles,

12.65 \times 12.1667

gives 153.9 units still short in a year.

What this does not model

Order quantity and its cost, review period, lost sales against back orders, partial stockouts, a stockout cost that grows with its length, correlated demand and lead time, supplier allocation in a shortage, obsolescence, and the fact that a service level measured on cycles is not the fill rate most people mean when they say service level.

The four readings are not combined and there is no fifth number under them. Cycle service and fill rate answer different questions, and averaging two answers to two questions produces an answer to neither.

Equation trace

Combine daily demand spread and lead time spread into one protection period deviation.
Convert the named cycle service level into its normal distribution value.
Multiply that value by the combined deviation to obtain safety stock.
Calculate annual holding cost and expected stockout cost on the same inputs.
Compare the cost and saving of the next service point.

Boundary

The calculation treats service as cycle service for the policy curve, assumes demand over the protection period is close to normal, and assigns one fixed cost to a stockout. It does not set a target or equate cycle service with fill rate.

Source anchors

Foundations of Inventory Management. The published book provides context for service measures, safety stock, and continuous review inventory policy.

Calculation version

1.0.0

The whole record prints. Your browser can save it as a PDF instead of printing it.

It draws the service curve and marks where two marginal costs meet on the figures you type. It describes the arithmetic and does not set inventory policy.

Service level field record

Service level: what the last percent of it costs

Accepted inputs

Named service level: 95.0%
Daily demand: 100 units
Lead time: 30 days
Lead time spread: 12 days
Unit cost: $40
One stockout: $6,000

At the balance service level: 96.733%

Safety stock: 1,116 units
Holding it a year: $9,822
Expected stockout cost: $2,385
Both together: $12,206
One more point costs: $730
One more point saves: $730

Field record

Equation trace: Combine uncertainty. Convert service to z. Calculate safety stock. Price holding and stockouts. Compare the next point.
Boundary: Cycle service with protection period demand close to normal and one fixed stockout cost. No target is recommended.
Source: Foundations of Inventory Management
Calculation version: 1.0.0

At the balance service level: 96.733%

Field record

What the buffer costs

Four measures, one run

What one more point costs and saves

What the balance point is not

The cost of one stockout

Here is the maths

The formula, in words

A worked example

What this does not model

Equation trace

Boundary

Source anchors

Calculation version

Related

Accepted inputs

At the balance service level: 96.733%

Field record