Inventory Control Guide Page 38
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404.7 Lead Time Variability
Up to now we always assumed that the demand during the lead time was known or that it
could be determined from the knowledge of the lead time itself and of the distribution of the
demand during a known period.
Here we consider the case where the lead time itself is also variable.
Lead Time parameters
lead time Lt is a random variable with:
mean:
E[Lt]
variance:
VAR[Lt]
In the previous pages, we assumed the lead time to be deterministic, that is with variance 0.
Now we assume it has some positive variance.
Demand Parameters
demand during t days is a random variable with:
t E[D
mean:
]
day
variance: t Var[D
]
day
This means the demand over t days is the sum of t daily demands.
It has an average equal to t times the average daily demand and a variance equal to t times
the variance of the daily demand.
Demand during lead time is a random variable with:
µ µ µ µ = E[Lt] E[D
mean:
]
day
2
2
variance: σ σ σ σ
=E[Lt]Var[D
] + E
[D
]Var[Lt]
day
day
Verification:
2=
if VAR[Lt] = 0, then σ
E[Lt] Var[D
]
day
(a sum of E[Lt] identical random variables)
This case is as before. The lead time is constant and the demand during the lead time is the
sum of daily demands over "lead time" days. The variance of a sum equals the sum of
variances.
2
2
] = 0, then σ
if VAR[D
=E
[D
] VAR[Lt]
day
day
the random variable
"demand during lead time" =
= random variable "lead time length" ×λ
This case is new. The daily demand is fixed. The lead time is variable. The distribution of the
demand during the lead time is equal to the distribution of the lead time after an appropriate
scaling. The days are replaced by demand units. You use here the rule :
var(k times a random variable X) = k*k var(X).
Prod 2100-2110
Inventory Control
37
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