Transcript Batching Policy in Kanban Systems U. S. Karmarkar, S. Kekre 20.03.2003

Batching Policy in Kanban
Systems
U. S. Karmarkar, S. Kekre
20.03.2003
Gökhan METAN
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Outline
Kanban System
 Markovian Models of
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Single-Card Kanban System
Dual-Card Kanban System
Two-Stage Kanban System
Numerical Example
 Conclusion

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Kanban System
The pull system means that materials are drawn or sent for by the users of
the material as needed.
[Hall]
The Kanban system is an information system that harmoniously controls
the production of the necessary products in the necessary quantities at the
necessary time in every process of a factory and also among companies,
which is known as the JIT production.
[Monden]
A Kanban is a tool to achieve JIT production. It is simply a card which is
usually put in a rectangular vinyl envelope. [Monden]
Two types of Kanban cards in general:
- Production-Ordering Kanban (or simply Production Kanban)
- Withdrawal Kanban (Conveyance or Transportation Kanban)
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Two Card Kanban System
Inbound
Stockpoint
Outbound
Stockpoint
Move Card
Production Card
Move
Cards
Production
Cards
Move
Cards
Production
Cards
Important properties of Kanban System
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Production is carried out in multiples of a minimum
quantity or batch.
The number of cards (or containers) in the system is fixed,
hence the total quantity of on-hand and on-order inventory
in the facility is also fixed (fixed-volume pull system).
Production is only initiated when finished inventory is
removed from the cell, thereby releasing a card (or
container).
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Analyzed System Configurations
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Single Card Kanban System:
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The production activity within the cell is controlled by the
Production Kanban cards, but transportation activity from the cell
is not controlled by the Withdrawal (transportation) Kanban cards.
Since the production within the cell is controlled by the Kanban
cards, there is an upper bound on the quantity in the cell.
Since the transportation is not controlled by the Kanban cards there
is no limit for the demand from the cell, which implies there is no
upper bound back orders (unfilled demand) that can accumulate.
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Analyzed System Configurations

Dual Card Kanban System:
–
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The only difference from the above is that there is an upper bound
on the back orders, which is limited by the number of transport
Kanbans, since they are controlled by the withdrawal Kanban
cards.
Two-stage Kanban System:
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Consists of two cells in series.
Simplest version of a multistage Kanban controlled process.
Interactions can be determined in order to get insights.
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Model & Basic Assumptions

Three system configurations are both analyzed by
Markovian models.
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The state of the system is represented by the number of Kanban
cards on order, the number of cards and batches in finished
inventory, and the number of batches on back order.
Models are used to link the system parameters (batch size & number
of cards) with the expected costs of operating the system.
The considered cost types are holding & back order or shortage
costs.
The inventory holding cost depends on the production lead times in
the cell.
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Model & Basic Assumptions

The assumptions are:
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Demand (D) ~ Poisson Process
Free cards enter the process queue
Production Process (P) ~ Exponential Distribution
Cell producing a single-item class
As batch sizes change  demand arrival & production rates are
adjusted accordingly
The inputs to the production process (raw material or labor) are
always available
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Single-Card Kanban System
N = Total number of cards and batches (in inventory and in process)
I = Average number of batches in inventory
Q = Batch Size
D = Total demand / production (units / year)
t = Setup time
P = Processing rate
1
Q
x =
= Average time to produce a batch = t +
m
P
D
l = Average arrival rate of batched demand =
Q
l
Dt
D
r=
=
+
m
Q
P
D
u=
(utilization rate)
P
p (i ) : probability that system is in state i (i = ..., -2, -1, 0,1, 2,..., N )
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Single-Card Kanban System
Semi-infinite Birth/Death Process
µ
i
µ
-2
λ
µ
-1
λ
µ
0
λ
µ
1
λ
# of Batches of Backlogged Demand
µ
2
λ
N
λ
# of Batches of Inventory
Markovian Model of a Single-Card Cell with N Cards
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Single-Card Kanban System
The steady state balance equations:
m p (i ) = l p (i + 1)
å
" i£ N- 1
p (i ) = 1
i
Solving these equations yields:
p (i ) = r N - i (1 - r )
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Single-Card Kanban System
b = Cost of back order ($ / unit time)
h = Cost of inventory ($ / unit time)
B = Average number of batches back ordered
The cost function:
C ( N , Q) = bBQ + hNQ
¥
where
B=
å
ir N + i (1 - r )
i= 1
The analysis based on N :
é æDt öæN (1 - r )+ 1öù
¶C
÷
֍
ú+ hN
÷
= bB êê1 - çç ÷ç
÷
÷
ú
ç
ç
÷
¶N
øú
êë è Q øçè r (1 - r ) ÷
û
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Single-Card Kanban System
The Analysis based on Q:
é Dt
ù
ê
ú Þ B - without bound
Q¯ê
ú
êë1- (D / P)ú
û
éu N + 1 ù
ú & C becomes asymptotically linear increasing in Q
Q- Þ B- ê
ê1 - u ú
ë
û
Dimensionless lot size:
q = (Q / Dt )
Expected dimensionless cost function:
C ( N , Q)
c( N , q) =
= (b / h) Bq + Nq
hDt
Service level factor that captures the weight given
to back orders relative to inventory holding
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Dual-Card Kanban System
Truncated Birth/Death Process
µ
µ
-M
-2
λ
µ
-1
λ
µ
0
λ
# of Batches of Backlogged Demand
µ
1
λ
µ
2
λ
N
λ
# of Batches of Inventory
Markovian Model of a Dual-Card Cell with N
Production Cards & M Withdrawal Kanban Cards
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Dual-Card Kanban System
The steady state equations:
p (i ) = (l / m) p(i + 1) , i = - M , - M + 1, ..., 0, 1, 2, ..., ( N -1)
N
å
p (i ) = 1
i= - M
The solution of these equations yields:
1-r
p(N) =
r éêë1-r N+M ù
ú
û
r N - i (1 - r )
p (i ) =
i = - M , - M + 1, ..., 0, 1, ..., ( N -1)
N+ M
1- r
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Dual-Card Kanban System
Average Back order:
r N - i (1 - r )
1- r
B = å ip (i ) + N p( N ) = å i
+
N
1- r N + M
r éêë1- r N + M ù
i= - M
i= - M
ú
û
Average Inventory:
N- 1
N -1
N
I=
å
ip (i )
i= 1
Cost function:
C ( N , M , Q) = bBQ + hNQ
--OR-- (If it is assumed that in process inventory is valued at less than finished inventories):
C ( N , M , Q) = bBQ + hIQ + h ¢( N - I )Q
where h ¢ is the average cost of work-in-process per unit per time.
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Two-Stage Kanban System
STAGE-2
STAGE-1
σ
μ
E2
F2
E1
Stage-2 Container
Stage-2 Output
Stage-1 Container
Stage-1 Output
F2
λ
Two-Stage Kanban System
l = Demand rate for Stage-1 output (containers/unit time)
m = Processing rate at Stage-1
s = Processing rate at Stage-2
I = Number of containers in circulation at Stage-1
J = Number of containers in circulation at Stage-2
i = Index for full stage-1 containers at F1, i = 0, 1, 2, ..., I
j = Index for full stage-2 containers at F 2, j = 0, 1, 2, ..., J
h1 = Inventory holding cost for stage-1
h2 = Inventory holding cost for stage-2
b = Shortage cost (only incurred for stage-1)
p (i, j ) = Steady state probability that there are i full containers at Stage-1
and j full containers at Stage-2
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Two-Stage Kanban System
λ
J
# of Full Stage-2
Containers
σ
σ
λ
j
σ
σ
0
1
σ
λ
σ
μ
σ
λ
σ
μ
σ
σ
λ
σ
λ
λ
σ
μ
σ
σ
λ
0
σ
λ
λ
σ
μ
σ
σ
σ
σ
λ
λ
λ
1
μ
σ
σ
σ
λ
λ
λ
2
Stage-1
Processing
Blocked
λ
σ
λ
2
i
I
# of Full Stage-1 Containers
State Transition Diagram for Markovian Model of Two-Stage Kanban System
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Two-Stage Kanban System
The Expected Cost Function:
I
C ( I , J , l , m, s ) =
J
å å
cij p(i, j )
i= 0 j = 0
where cij is the associated cost of holding i and j numbers of containers
at Stage 1 and 2 respectively, and given by:
(iQh1 + jQh2 )
for i ¹ 0
cij =
(iQh1 + jQh2 + b)
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Two-Stage Kanban System
To incorporate the effects of lot sizing, relate the state transition
rates with container size and define:
D = External demand rate for Stage-1 finished goods.
Q = Batch Size (Container Size)
P1 = Processing rate at Stage-1
P2 = Processing rate at Stage-2
t 1 = Setup time at Stage-1
t 2 = Setup time at Stage-2
Then:
D
Q
1 Q
=
+ t1
m P1
l=
1
Q
=
+ t2
s
P2
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Two-Stage Kanban System
Analysis of Results:
- For large Q, the inventory holding costs for both stage 1 & 2
grow asymptotically linearly with Q. This is because λ, μ, σ stay
in the same relative position as Q increases and the transition
probabilities stabilize.
- When Q decreases, shortage costs rise for any choice of
Kanban card numbers. This is because of the fact that the
production lead times increases in the sense of Setup Times.
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Conclusion

The batch size associated with each card has a significant
effect on the performance of the Kanban system.
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The effect of the number of Kanban cards in the system is
also significant.
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In fact, since the batch size and lead times are correlated,
its effect is much more complex than the number of
containers in the system.
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Conclusion

In multistage Kanban system, the parameters at one stage
affect the performance at other stages. Increasing the
number of cards at one stage leads to an increase in
inventory levels at a succeeding stage, and reduces the
inventory levels at a preceding stage.
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References
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Y. Monden. Toyota Production System, Industrial Engineering and
Management Press, Norcross, Georgia, 1983.
W.R. Hall. Zero Inventories, Dow Jones, Irwin Illinois, 1983.
U.S. Karmarkar, S. Kekre. “Batching Policy in Kanban Systems”,
Journal of Manufacturing Systems, Vol. 8, No. 4.
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