models of heijunka-levelled kanban- systems kai furmans fifth international conference on ``analysis...

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


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


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

Setup

Expected

downtim

es

Product A


Setup

Setup

Expected

downtim

es

EPEI


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


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:


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


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:


9

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)


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%


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:


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

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


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

models of heijunka-levelled kanban- systems kai furmans fifth international conference on ``analysis...

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