slides 46 13 inv stoch

8/8/2019 Slides 46 13 Inv Stoch

1/14

Assumptions for the ( Q, R) policy

Probabilistic demand Demand is NOT deterministicbut probability distribution is known

Lead time MIGHT NOT BE deterministicShortages MAY OCCURAll ordered units arrive at oncePurchasing cost is independent of the order quantity


2/14

A Probabilistic Inventory Model.

Assumptions:Probabilistic lead-time demand ( D L)

mean Q D L standard deviation W D

L probability distribution fcn. P ( D L) / density fcn. f ( D L)

cumulative distribution fcn. F ( D L)= P (lead-time demand

Continuous review ( Q, R) system ( ( s, Q) system) fixed order size Q

order point R (or s), i.e., variable order period

Demand during stock-out periods is backlogged

L De


3/14

Determination of theorder quantity Q ?

H euristic approach:

S imply use the EOQ(with or without quantity discounts)or the Economic Batch Size model

=> Robust and simple model


4/14

Inventory levels:

Order point: R = E [ D L] + SS = Q D L + SS S afety stock: SS = R - Q D L or: SS = Z W D L

Average inventory (approximations): Expected max. inventory on hand: SS + Q

Expected min. inventory on hand: SS

Average inventory on hand: SS + Q/2 = R - Q D L

+ Q/2


5/14

Determination of the order point R ?3 alternative models:

1. S pecified probability of no stockout during lead time

Fservice level:

2. S pecified proportion of demand satisfied from inventoryon hand

P service level:

3 . Cost minimizationc s shortage cost

Fu!e R F R D P L

? A? A

P ud eman d

shortunitso#1


6/14

0.0000.050

0.1000.1500.2000.2500.3000.350

10 11 12 13 14 15 16 17 18 19

Lead time demand

P r o b a b i l i t y

Discrete probability distribution

Lead timedemand Probability Cumulativeprobability10 0.025 0.02511 0.050 0.07512 0.150 0.22513 0.050 0.27514 0.150 0.425

15 0.300 0.72516 0.150 0.87517 0.050 0.92518 0.050 0.97519 0.025 1.000

17

9.0

9.0

!ue

!

R

R D P L

F

F service level


7/14

0 .0000 .0500 . 000 . 500 . 00

0 . 500 . 000 . 50

10 11 12 13 14 15 16 17 18 19

L d ti d nd

o b b

i l i t

T he expected number of shortagesduring lead time for a given order point R?

R = 16: E (# shortages | R =16) == (17-16) P (17) + (18-16) P (18) + (19-16) P (19)

= (1)0.05 + (2)0.05 + (3

)0.025 = 0.225

P service level


8/14

Expected number of shortages duringlead time E (S ):

? A

? A

g

g

!

!

!

R L L L

R D L L

dD D f R DS E

D P R DS E L

:ondistributiContinuous

:ondistributiDiscrete

)1(

? A

L L

L

L L

D D

D

D D L

R Z

Z N

Z N S E

N D

W Q

W

W Q

!

!

}

and

function'lossnormalUnit'

theiswhere

,

,

:demandtimelead

ddistribute Normally

P service level


9/14


10/14

A maximal expected number of shortages? A

? A

? A ? A ? A

? A

? A ? A

ee

e

ee

e

P P

QS P QS

QS P

D

S Q DS

P D

Q DS

P

E E

P

1:salesLost1

1

1:salesLost1

year per d eman dyear per shortunitso#

1

P service level


11/14

0.0000.050

0.1000.1500.2000.2500.3000.350

10 11 12 13 14 15 16 17 18 19

Lead time demand

P r o b a b i l i t y

Discrete probability distribution

Lead timedemand Probability Cumulativeprobability10 0.025 0.02511 0.050 0.07512 0.150 0.22513 0.050 0.27514 0.150 0.425

15 0.300 0.72516 0.150 0.87517 0.050 0.92518 0.050 0.97519 0.025 1.000

Q = 100 P = 0.999=>

E (S ) < 100(1-0.999) = 0.1

E (S |R=19) = 0 E (S |R=18) = (1)0.025 = 0.025 E (S |R=17) = (1)0.05 + (2)0.025 = 0.1 E (S |R=16) = (1)0.05 + (2)0.05 + ( 3 )0.025 = 0.225

=> R = 17

P service level


12/14

Order point R?3 . Cost minimization: A marginal cost approach

Given a fixed order quantity Q,suppose R is increased to R + 1.

Then:T he safety stock SS +1 = R+1 - Q D L increases by one (1) unit,causing an increase of the annual inventory holding cost by ch

T he expected number of shortages E [S ] during the lead timedecreases by:

causing a decrease of the

annual shortage cost by:

R F R D P D P L R D L L !e! " 11

? A R F Q

Dc s 1


13/14

W hy increasing R decreases E (S ):

? A ? A ? A

R D P D P D P R P

D P D P R D

D P R D R P R R

D P R D D P R D RS E RS E

L

R D

L

R D

L

R D

L

R D

L L

R D

L L

R D

L L

R D

L L

L L

L L

L

L L

e

-

-

""

""

"

""

11

11

11

1

11

1

1


14/14

T he marginal cost approach

T he order point R is increased as long asthe shortage cost decrease exceeds the holding cost increase:

? A

? A

DcQc DcQc

R F

Dc

Qc R F

c R F Q

Dc

sh s

h

s

h

h s

ee

u

u

i1

1

1

E

E

slides 46 13 inv stoch

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