TRADITIONAL MODEL LIMITATIONS

• CERTAINTY EXISTS

- demand is known, uniform, and continuous

- lead time is known and constant

- stockouts are backordered or not permitted

• COST DATA ARE AVAILABLE

- order/setup cost known and constant

- holding cost is known, constant, and linear

• NO RESOURCE LIMITATIONS

- no inventory dollar limits

- storage space is available

WORKING AND SAFETY STOCK

Safety Stock

QU

AN

TI T

Y

TIME

B

Q + S

S

Working Stock

Working Stock

IDEAL INVENTORY MODEL

B

Q + S

SQU

AN

TIT

Y

Order Lot Order LotPlaced Received Placed Received

SafetyStock

Reorder Point

LeadTime

TIME

Q + S

S

LeadTime

LeadTime

LeadTime

REALISTIC INVENTORY MODEL

TIME

B

QU

AN

TIT

Y

Stockout

SAFETY STOCK VERSUS SERVICE LEVEL

.50 1.00

high

SA

FE

TY

S

TO

CK

low

SERVICE LEVEL (Probability of no stockouts)

STATISTICAL CONSIDERATIONS

maxM

0M) M(M P

0Md)M(M f

CONTINUOUS DISCRETEVARIABLE DISTRIBUTIONS DISTRIBUTIONS

M

maxM

1BM)M(P)BM(

BMd)M(f)BM(QuantityStockoutExpected

maxM

1BM)M(P

BMd)M(f

maxM

0M)M(P2)MM(

0Md)M(f2)MM(VarianceDemandTimeLead

2

E(M > B)

P(M > B)

B = reorder point in units. M = lead time demand in units (a random variable). f(M) = probability density function of lead time demand.P(M) = probability of a lead time demand of M units. = standard deviation of lead time demand

Demand Time Lead Mean

Probability of a Stockout

PROBABILISTIC LEAD TIME DEMAND

DEMAND DURING LEAD TIME (M)

PROBABILITY OF A STOCKOUT, P(M>B)

SAFETY STOCK

REORDER POINT

PR

OB

AB

ILIT

Y

P(M

)

0 M B

NORMAL PROBABILITY DENSITY FUNCTION

stockoutaofprobabilityBMPBF

functiondistributioncumulativeMdMfBF

functiondensityprobabilityMfB

=>=-

==

=

)()(1

)()(

)(

2)(

22/2)( MMeMf

Lead Time Demand (M)

M

= 1 - F(B) = P(M >B)

f(M)

f(B)

B

Area

P(M) =M M e- M

M!

POISSON DISTRIBUTION

LEAD TIME DEMAND (M)

PR

OB

AB

ILIT

Y

P(M

)

0.00

0.10

0.20

0.30

0.40

0 4 8 12 16 20 24

M=2

M=4M=6

M=8

M=10

M=1

NEGATIVE EXPONENTIAL DISTRIBUTION

LEAD TIME DEMAND (M)

PR

OB

AB

ILIT

Y D

EN

SIT

Y F

(M)

0

1/M f(M) = eM/M

M

NEGATIVE EXPONENTIAL DISTRIBUTION

0.0

0.5

1.0

1.5

2.0

2.5

0 2 4 6 8 10 12

LEAD TIME DEMAND (M)

PR

OB

AB

ILIT

Y D

EN

SIT

Y

f(M

)

M=1

M=2M=3

M=0.5

M=5

f(M) = eM/M

M

INDEPENDENT DEMAND : PROBABILISTIC MODELS

LOT SIZE : 2CR / H

REORDER POINT : B = M + S

I. KNOWN STOCKOUT COST

A. Obtain Lead Time Demand Distribution constant demand, constant lead time

variable demand, constant lead time

constant demand, variable lead time

variable demand, variable lead time

B. Stockout Cost

backorder cost / unit

lost sale cost / unit

II. SERVICE LEVEL

A. Service per Order Cycle

Demand Probability Demand Probability Lead time Probability

first week second week demand (col. 2)(col. 4)

(D) P(D) (D) P(D) (M) P(M)

1 0.60 1 0.60 2 0.36

3 0.30 4 0.18

4 0.10 5 0.06

3 0.30 1 0.60 4 0.18

3 0.30 6 0.09

4 0.10 7 0.03

4 0.10 1 0.60 5 0.06

3 0.30 7 0.03

4 0.10 8 0.01

CONVOLUTIONS(variable demand/week and constant lead time of 2 weeks)

Lead time demand (M) Probability P(M)

0 0

1 0

2 0.36

3 0

4 0.36

5 0.12

6 0.09

7 0.06

8 0.01

1.00

INVENTORY RISK( VARIABLE DEMAND, CONSTANT LEAD TIME )

J

S0

W

Q + S

-W

B

TIME

QU

AN

TIT

Y

L

P(M>B)

Q = order quantityB = reorder pointL = lead timeS = safety stock

B - S = expected lead time demand B - J = minimum lead time demand B + W = maximum lead time demand P(M>B) = probability of a stockout

J

SAFETY STOCK : BACKORDERING

MBS

MdMfMMdMfB

MdMfMBS

-=

)()()()(

)()()(

00

0

BACKORDERING

CostStockoutCostHoldingTCS+=

BMPQ

ARH

dBdTCS 0)(

BMEQ

ARHMB )()(

MdMfBMQ

ARSH )()()(

B

AR

HRsPBMP )()(

TCs = (B - M)H + E(M > B) =

B = 67 E(M > B) =

= (68- 67).08 + (69- 67).03 + (70- 67).01 = .17 units

TCs = (67- 65)(2)(.30) + = 1.20 + 2.04

= $3.24

B = 68 E(M > B) =

= (69- 68).03 + (70- 68).01 = .05 units

TCs = (68- 65)(2)(.30) + = 1.80 + 0.60

= $2.40

AR E(M>B)

Q

2(3600)(.05)

600

2(3600)(.17)

600

+=

-70

168)()68(

MMPM

max

1

)()(M

BMMPBM

+=

-70

167

)()67(M

MPM

B = 69 E(M > B) =

= (70- 69).01 = .01 units

TCs = (69- 65)(2)(.30) + = 2.40 + 0.12

= $2.52

+=

-70

169)()69(

MMPM

2(3600)(.01)

600

Therefore, the lowest cost reorder point is 68 units with an expected annual cost of safety stock of $2.40.

SAFETY STOCK : LOST SALES

)()(0

MdMfMBSB

)( BMEMBS >+-=

-=

)()( MdMfBMMBB

-+-=

)()()()(0

Md MfMBMdMfMBB

---=

LOST SALES

CostStockoutHolding CostTCS =

HQARHQsPBMP== )()(

BMPHQ

ARH

dB

dTCS=

= 0)(

BMEQARHBMEMB = )()(

MdMfBMQ

ARSHB

-+=

)()(

BMEHQ

ARHMB

= )()(

INVENTORY RISK(CONSTANT DEMAND, VARIABLE LEAD TIME)

Q + S

S

B

Lm

L

QU

AN

TIT

Y

TIMEP(M > B)

L = expected lead timeP(M > B) = probability of a stockout

B - S = expected lead time demand

Q = order quantity B = reorder point S = safety stock Lm = maximum lead time

0

J

S0

Q + S

- W

B

QU

AN

TIT

Y

Lm

INVENTORY RISK(VARIABLE DEMAND, VARIABLE LEAD TIME)

L

TIME

P(M >B)

P(M > B) = probability of a stockout B - S = expected lead time demand

B + W = maximum lead time demand

Q = order quantity B = reorder point S = safety stock L = expected lead time Lm = maximum lead time

B - J = minimum lead time demand

VARIABLE DEMAND / VARIABLE LEAD TIME

LD DL 2222

Independent Distributions

LDM

L DD DL

LDM

22222

Dependent Distributions

L

SERVICE PER ORDER CYCLE

c

c

SLBMP

BMP

cyclesorderofnototalstockoutawithcyclesofno

SL

=

>=

=

1)(

)(1

..

1

IMPUTED STOCKOUT COSTS

)(

)(

/cost

BMPRHQ

A

ARHQ

BMP

unitBackorder

)(

)(1

)(

/

BMPR

BMPHQA

HQARHQ

BMP

unitsales costLost

SAFETY STOCK : 1 WEEK TIME SUPPLY(Normal Distribution : Lead Time = 4 weeks)

Weekly Demand Safety Stock

D D

1000 100 1000 5.00 0

1000 200 1000 2.50 0.0062

1000 300 1000 1.67 0.0480

1000 400 1000 1.25 0.1057

1000 500 1000 1.00 0.1587

4

1000

D

SZ

SP(M>B)

PROBABILISTIC LOGIC

Service Levels

Service/units demanded, E(M>B) = Q(1 - SLU) E(M>B) = E(Z)

Convolution over lead time

Multiply dist. by demand, M = DL, = DL

Analytical Combination /Monte Carlo simulation

Service/cycle,

P(M>B) = 1 - SLc

Variable demand,variable lead time

Variable demand,constant lead time

Constant demand,variable lead time

Lost Sale, P(M>B) = HQ/(AR+HQ)

Backordering, P(M>B) = HQ/AR

Lead time demand distribution ?

Known stockoutcosts ?

No

Yes

Yes

No

Start

RISK : FIXED ORDER SIZE SYSTEMS

FOSS

Order

Quantity (Q) Set by Management

EOQ

EPQ

Reorder Point (B)

Service

Level

Per Cycle

Per Units Demanded

Known

Stockout Cost

Lost Sale

BackorderPer Outage

Per Unit

Per Outage

Per Unit