Post on 01-Nov-2014
description
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