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Models of Heijunka-levelled Kanban-Systems
Kai Furmans
Fifth International Conference on
``Analysis of Manufacturing Systems – Production Management’’
13.01.2005 © Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH)
2
IFL
A Kanban System
Withdrawal Kanban
OrMarket Demand
Production Kanban
Finished GoodsWarehouse Production
Raw MaterialsWarehouse Transport Customer
13.01.2005 © Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH)
3
IFL
C C B B A A A C C B B A A A
A Kanban System with Levelling
Withdrawal Kanban
OrMarket Demand
Production Kanban
Finished GoodsWarehouse
Raw MaterialsWarehouse Transport Customer
TypeABC
13
2
2
3
2
43
5
2
Sequence
5
2
ON622
Next day
Overflow
3
Production
13.01.2005 © Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH)
4
IFL
EPEI – „Every Part Every Interval“ Calculation
EPEI
Product A
Setup Product B Product C
Setup
Setup
Expected
downtim
es
EPEI
Product A
Setup Product B Product C
Setup
Setup
Expected
downtim
es
Product A
Setup Product B Product C
Setup
Setup
Expected
downtim
es
EPEI
13.01.2005 © Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH)
5
IFL
No Capacity Limits, iid Demands, no Levelling
Assumptions:• Periodic demand and review
• Fixed and constant lead times tr• Demand is iid distributed , a demand of j items is described by the probability dj
• Stock development (up or down)
• Distribution of stocks in replenishment period
13.01.2005 © Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH)
6
IFL
Basic Model with Unlimited Capacity: Example
Probability distribution of the stock level within the replenishment period
0,00
0,02
0,04
0,06
0,08
0,10
0,12
-16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16
Service level >= 95%
Necessary base stock level: 6 units
demand distribution
0,00
0,10
0,20
0,30
0,40
0,50
1 2 3 4 5 6 7 8
am ount
pro
bab
ility
Replenishment time:
3 time units
Input parameter:
13.01.2005 © Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH)
7
IFL
Basic Model with Unlimited Capacity
Necessary base stock level to satisfy an service level of 0,95
0
2
4
6
8
10
12
14
16
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7
coefficient of variation
bas
e st
ock
leve
l
The necessary base stock level increases linear with coefficient of variation of demand
13.01.2005 © Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH)
8
IFL
Capacity Limits, iid Demands, with Levelling
• The assembly line has a capacity of c units per EPEI
• In period n Qn units are sold
• Kanbans, that exceed capacity c in one period, are collected in the overflow box• If in one period, less than c units are sold, Kanbans from the overflow box are moved to the
board (if available) Daily production is never more than c units, but can be less
. Difference between Capacity and Demand:
Number of waiting Kanbans in period n + 1
• If replenishment interval equals 1, (for assembly systems often the case), thenthe number of not yet replaced items in stock is:
13.01.2005 © Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH)
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IFL
Duality: Heijunka Model as GI/D/1-System or D/GI/1-System
• Stochastic demand – deterministic capacity: GI/D/1-System? • Discrete Time GI/D/1-Queueing-System:
• Distance between requests vary, requests for capacity are homogenous• Discrete Time D/GI/1-System Queueing-System:
• Distance between requests is identical (one period), workload is varying
• Lindleys Equation in discrete time:
• Distribution of probabilities can be computed with existing algorithms(i.e. Grassmann / Jain using Wiener-Hopf-Factorization)
13.01.2005 © Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH)
10
IFL
Limited Capacity - Heijunka Controlled: Example
Necessary base stock level: 17 units
0,00
0,02
0,04
0,06
0,08
0,10
0,12
0,14
0 5 10 15 20 25 30
amount
pro
ba
bil
ity
Service level >= 95%
13.01.2005 © Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH)
11
IFL
Reduction of Variability of Demand
• yi describes the distribution of the idle capacity per time interval for arriving Kanbans.
• zi describes the distribution of the idle capacity per time interval.
• The difference between the available capacity c and the idle capacity zi is the demand of parts which is requested from the supplier to replenish the raw materials.
• Thus:
• The requested replenishment quantity at the supplier now is:
13.01.2005 © Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH)
12
IFL
Limited Capacity - Heijunka Controlled: Example
The coefficient of variation of supplier demand decreases:
v customer demand = 0,27 v demand to supplier = 0,17
Reduction of the bullwhip effect
number of kanbans in the overflow
0
0,1
0,2
0,3
0,4
0,5
0,6
0 1 2 3 4 5 6 7 8 9
amount
pro
bab
ility
comparison: customer demand vs demand to supplier
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
5 6 7 8 9 10 11 12 13 14 15
amount
pro
ba
bili
ty
demand customer demand supply
Capacity: 10
If supplier has a capacity of 10
all demands will be fulfilled immediately
13.01.2005 © Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH)
13
IFL
Multiple Product Case, iid Demands, Limited Capacity
• The Multiple product case can be treated exactly as the single product case,if the capacity c is preallocated on the different products
– The calculations can be done for each product separately.• If all requests are handled by the same assembly unit, then the single product case has to
be applied– With a subsequent stock sizing using the waiting time distribution of Kanbans for
the determination of the respective stock sizes.
• Question: On which level should Levelling be performed?
Multiple Product Case, Demands generated from Kanban Loop, Limited Capacity
• Ongoing work
13.01.2005 © Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH)
14
IFL
Conclusions
We have not succeeded in answering all of your problems. The answers we
have found only serve to raise a whole set of new questions. In some ways
we feel we are as confused as ever, but we believe we are confused on a
higher level and about more important things Sign in a computer shop