Distribution of Demand During Distribution of Demand During Lead TimeLead Time

1

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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