Traditional model limitations
-
Upload
khaneducation -
Category
Technology
-
view
1.172 -
download
0
description
Transcript of Traditional model limitations
![Page 1: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/1.jpg)
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
![Page 2: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/2.jpg)
WORKING AND SAFETY STOCK
Safety Stock
QU
AN
TI T
Y
TIME
B
Q + S
S
Working Stock
Working Stock
![Page 3: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/3.jpg)
IDEAL INVENTORY MODEL
B
Q + S
SQU
AN
TIT
Y
Order Lot Order LotPlaced Received Placed Received
SafetyStock
Reorder Point
LeadTime
TIME
![Page 4: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/4.jpg)
Q + S
S
LeadTime
LeadTime
LeadTime
REALISTIC INVENTORY MODEL
TIME
B
QU
AN
TIT
Y
Stockout
![Page 5: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/5.jpg)
SAFETY STOCK VERSUS SERVICE LEVEL
.50 1.00
high
SA
FE
TY
S
TO
CK
low
SERVICE LEVEL (Probability of no stockouts)
![Page 6: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/6.jpg)
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
![Page 7: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/7.jpg)
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
![Page 8: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/8.jpg)
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
![Page 9: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/9.jpg)
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
![Page 10: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/10.jpg)
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
![Page 11: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/11.jpg)
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
![Page 12: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/12.jpg)
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
![Page 13: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/13.jpg)
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)
![Page 14: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/14.jpg)
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
![Page 15: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/15.jpg)
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
![Page 16: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/16.jpg)
SAFETY STOCK : BACKORDERING
MBS
MdMfMMdMfB
MdMfMBS
-=
)()()()(
)()()(
00
0
![Page 17: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/17.jpg)
BACKORDERING
CostStockoutCostHoldingTCS+=
BMPQ
ARH
dBdTCS 0)(
BMEQ
ARHMB )()(
MdMfBMQ
ARSH )()()(
B
AR
HRsPBMP )()(
![Page 18: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/18.jpg)
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
![Page 19: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/19.jpg)
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.
![Page 20: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/20.jpg)
SAFETY STOCK : LOST SALES
)()(0
MdMfMBSB
)( BMEMBS >+-=
-=
)()( MdMfBMMBB
-+-=
)()()()(0
Md MfMBMdMfMBB
---=
![Page 21: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/21.jpg)
LOST SALES
CostStockoutHolding CostTCS =
HQARHQsPBMP== )()(
BMPHQ
ARH
dB
dTCS=
= 0)(
BMEQARHBMEMB = )()(
MdMfBMQ
ARSHB
-+=
)()(
BMEHQ
ARHMB
= )()(
![Page 22: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/22.jpg)
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
![Page 23: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/23.jpg)
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
![Page 24: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/24.jpg)
VARIABLE DEMAND / VARIABLE LEAD TIME
LD DL 2222
Independent Distributions
LDM
L DD DL
LDM
22222
Dependent Distributions
L
![Page 25: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/25.jpg)
SERVICE PER ORDER CYCLE
c
c
SLBMP
BMP
cyclesorderofnototalstockoutawithcyclesofno
SL
=
>=
=
1)(
)(1
..
1
![Page 26: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/26.jpg)
IMPUTED STOCKOUT COSTS
)(
)(
/cost
BMPRHQ
A
ARHQ
BMP
unitBackorder
)(
)(1
)(
/
BMPR
BMPHQA
HQARHQ
BMP
unitsales costLost
![Page 27: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/27.jpg)
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)
![Page 28: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/28.jpg)
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
![Page 29: Traditional model limitations](https://reader033.fdocuments.us/reader033/viewer/2022061111/545610b8af795998788b4be7/html5/thumbnails/29.jpg)
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