How much should you stock?

One order, before demand is known. Too few and you miss sales, too many and you eat the leftovers. The optimal order balances the two ā€” and is usually not the average demand.

The trade
Each unit sells forā‚¬, costsā‚¬to stock, and salvagesWhat you recover per unsold unit ā€” clearance price, return credit, or zero if it's a total loss. forā‚¬if unsold.
Demand averagesunits, give or take.
Running out costs ā‚¬30/unit Ā· overstocking costs ā‚¬10/unit ā†’ stock to the 75% critical fractileThe service level where the expected cost of stocking one more unit equals its expected benefit. At this fraction, demand is satisfied ā€” the next unit's gain just breaks even with its risk of going unsold.
Optimal order
240 units

Covers demand about 75% of the time (zĀ = +0.67). Stocking to the average (200) would leave you 40 units over.

240Optimal order Q*
75%Service level
ā‚¬5,2kExpected profit
Expected profit by order quantity
ā‚¬5,6k
20100300avg 200Q* 240380
Stock to each service level
Service levelOrder qtyExpected profit
50%200
ā‚¬5k
75%optimal240
ā‚¬5,2k
90%277
ā‚¬5,1k
95%299
ā‚¬5k
99%340
ā‚¬4,6k

When stockouts hurt more than leftovers (Cu > Co) the optimal order sits above average demand ā€” you deliberately over-order to guard against missed sales. When leftovers hurt more, it sits below. The optimum is demand at the critical fractile, not the mean.

the newsvendor model Ā· Q* at the critical fractile