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Distribution of Demand During Distribution of Demand During Lead TimeLead Time
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Sum of random variables
Let D1 and D2 be two independent random variables withexpected values µ1 and µ2 and standard deviations σ1 and σ2.ThThen,
( ) ( ) ( ) 22
221
22
21
221 σσ +=+=+ baDVarbDVarabDaDVar
( ) ( ) ( ) 212121 μμ baDbEDaEbDaDE +=+=+
( ) ( ) ( )( )( ) 2
22121
22
221
221
212121
σσ
σσ
+=+
+=+
DDstd
babDaDstd
Therefore,
( ) 2121 σσ ++ DDstd
( ) nnn nDDDE μμμμμμμ ====+++=+++ ..... if ,........... 212121
( ) nn nDDDstdn
σσσσσσσ ====+++=+++ ..... if ,........... 2122
22
21 1
2
Sum of random variables
Let D be the per unit demand with mean and standarddeviation . Let the demand be independent across periods.L t LT b th t t L d Ti Th L d Ti
DμDσ
Let LT be the constant Lead Time. Then, Lead TimeDemand has the following mean and standard deviation:
( ) LDDDE ++++++( ) DDDDLT LDDDE μμμμ =+++=+++ ...........21
( ) LTDDDstd σσσσ =+++=+++ 222( ) DLT LTDDDstdLT
σσσσ =+++=+++ 221 ...........1
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Distribution of Demand During Lead Time
TimeLeadofMean:Time Lead :
LT
LTμ
unit timeper Demand : Time Lead ofDeviation Standard :
TimeLead ofMean :
LT
LT
D σμ
Demand unit timeper ofDeviation Standard :unit timeper DemandMean :
D
D
σμ
TiL dd iD dfD i tiSt d dTime Lead during DemandMean :
TimeLeadduring Demand :
DDLT
DDLTμ
TimeLeadduringDemandofDeviation Standard :DDLTσ
Distribution of Demand During Lead Time
321 variablerandom a is ,....... L LDDDDDDLT +++= 321 ,
LTDDDLT
L
μμμ =From PS I: See Ross- Conditional
Variance Formula
{ } { }()()(
LTDDDLT
lLTDDLTEVarlLTDDLTVarEDDLTVar
μμμ
=+== { } { }
222
2
()()(
DD )Var(l)E(l
σμσμ
μσ
+=
+=
222
LTDDLTDDLT
LTDDLT
σμσμσ
σμσμ
+=
+=
Standard Normal Distribution
If D~ N(µ,σ) so that:⎤⎡ ⎞⎛
211Then, ⎥
⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −
−=2
21exp
21)(
σμ
σπxxf
( ) ( )[ ] [ ]
[ ]11
011
D
EDEDE
⎞⎛
=−=−=⎟⎠⎞
⎜⎝⎛ −
μ
μμσ
μσσ
μ
( ) ( )[ ] [ ]
( )10
1011 222
NDTh f
VarDVarDVar
⎟⎞
⎜⎛ −
=+=+=⎟⎠⎞
⎜⎝⎛ −
μ
σσ
μσσ
μ
( )1,0~, NTherefore ⎟⎠
⎜⎝ σ
μ
N(0,1) is called standard normal
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standard normal distribution
Loss Function (also called as Partial Expectation) in a Continuous Review (Q, R) system
When demand is normally distributed, the Expected stockouts percycle, E(n) is computed as: Loss Function (also
( ) ( ) ( )dxxfRxnE −= ∫∞
called PartialExpectation).
( ) ( )xtRzLetDDLT
DDLT
DDLT
DDLT
R
σμ
σμ −
=−
= ; Standardized loss function (also called
( ) ( ) ( ) ( )zEdttztnE DDLTz
DDLT σφσ =−= ∫∞ standardized Partial
Expectation).
where φ(t) is the standard normal density
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