Inventory and Variability
Stanley B. [email protected]
http://web.mit.edu/manuf-sys
Massachusetts Institute of Technology
December 13, 2010
Inventory and Variability 1/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
Inventory
◮ Inventory is a necessary evil, and we do our best to minimize it.
◮ Variability makes inventory necessary.
◮ Variability is the enemy of manufacturing.
Inventory and Variability 2/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
InventoryCosts
◮ Financial: raw materials have been paid for, but no revenue comes inuntil the finished product is sold.
◮ Demand risk: a product may not be sold due to (for example):◮ time value (newspaper)◮ obsolescence◮ going out of fashion◮ spoilage (food)
The longer an item is in storage, the more likely that one of these willoccur.
Inventory and Variability 3/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
InventoryCosts
◮ Holding cost (warehouse space)
◮ Damage/theft/loss
◮ Reduction of yield when inspection is not immediate.
◮ Concealment of causes of problems — the sources of variability.
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InventoryBenefits and Motivation
◮ The purpose of storage is to allow systems to survive even whenimportant events are unsynchronized.
◮ Storage improves system performance by decoupling parts of thesystem from one another.
◮ Storage reduces the propagation of disturbances, and thus reducesinstability and the fragility of complex, expensive systems.
◮ Storage allows production systems (of energy or goods) to be builtwith capacity less than the peak demand.
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InventoryBenefits and Motivation
◮ Reduces the propagation of disturbances (eg, machine failures).
◮ Allows economies of scale:◮ volume purchasing◮ set-ups
◮ Helps manage seasonality and limited capacity.
◮ Helps manage uncertainty:◮ Short term: random arrivals of customers or orders.◮ Long term: Total demand for a product next year.
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VariabilityCauses
◮ Random (uncontrollable)
◮ random demand◮ random supply◮ machine failures◮ human variability
◮ Deterministic (controllable)◮ setups◮ batches◮ order costs◮ MRP nervousness
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VariabilityHow variability causes inventory
B1
B2
largepipes
smallpipe
75
100
75
100
75
100
No variability:
◮ Assume the small pipe is thebottleneck.
◮ There is no accumulation of waterin either tank.
◮ The outflow rate in the third pipe isalways the same as the rate of flowinto the first pipe.
◮ Call this Case 1 .
Inventory and Variability 8/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
VariabilityHow variability causes inventory
M2
M1
M3
B1
B2
largepipes
all valves closed
0
0
0
Variable valves, Case 2
◮ In this figure, all valves are closed.
◮ While the system is in this state,the water level does not change inany tank, and there is no inflow oroutflow.
◮ But it does not stay in this stateforever.
Inventory and Variability 9/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
VariabilityHow variability causes inventory
M1
M 2
M3
B1
B2
0 100
0
0
Variable valves, Case 2
◮ Now the second valve is opened.
◮ While the system is in this state,the first tank loses water and thesecond tank gains water, and thereis no inflow or outflow.
◮ If it stays in this state long enough,the first tank becomes empty orthe second tank becomes full.
◮ But it does not stay in this stateforever either.
Inventory and Variability 10/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
VariabilityHow variability causes inventory
M1
M 2
M3
B1
B2
0 100
0
0 100
Variable valves, Case 2
◮ Now the second and third valvesare opened.
◮ While the system is in this state,the first tank loses water. The levelsecond tank stays constant. Thereis no inflow but there is outflow atmaximum rate ...
◮ until the first tank becomes empty,and then the second tank loseswater until it becomes empty.
◮ But it leaves this state eventuallyalso.
Inventory and Variability 11/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
VariabilityHow variability causes inventory
M2
M1
M3
B1
B2
0
0 100
0 100
Variable valves, Case 2
◮ There are a total of eight valve states.For each state, the flow rate in eachpipe is either 0 or 100.
◮ Think of an open valve as operationaland a closed valve as failed. Then thevalves are characterized by MTTF,MTTR.
◮ For some set of MTTFs and MTTRs,the average flow rate of Case 2 is thesame as the flow rate of Case 1...
◮ ... but Case 2 has inventory and Case1 does not.
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VariabilityHow variability causes inventory
Conclusions:
◮ Variability causes inventory.
◮ The tanks in Case 1 are always empty.
◮ The tanks in Case 2 gain or lose material due to the unsynchronizedvalve behavior.
◮ Variability makes inventory necessary.
◮ We would get the same flow rate in Case 1 if the tanks were not there.
◮ The flow rate in Case 2 is a function of the valve behavior (MTTF andMTTR) and the sizes of the tanks. To make the average flow rate ofCase 2 the same as the flow rate of Case 1, tanks may be necessary.
Inventory and Variability 13/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
What is an Operation?
OperationPart Part
Waste
MachineOperator
Consumable
Nothing happens until everything is present.
Inventory and Variability 14/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
What is an Operation?Waiting
Whatever does not arrive last must wait.
◮ Inventory: parts waiting.
◮ Under-utilization: machines waiting.
◮ Idle work force: operators waiting.
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What is an Operation?Waiting
OperationPart Part
Waste
MachineOperator
Consumable
◮ Reductions in the availability, or ...
◮ Variability in the availability ...
... of any one of these items causes waiting in the rest of them and reduces
performance of the system.
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Variability — Causes and CuresCreation
Causes
◮ random demand
◮ random supply
◮ machine failures
◮ human variability
◮ setups
◮ batches
◮ order costs
Cures
◮ manage demand,
◮ find a more reliable supplier
◮ improve machines
◮ improve human management
◮ reduce setup and order costs
◮ increase capacity and flexibility
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Variability — Causes and CuresPropagation
Causes
◮ blockage and starvation
Cures
◮ in-process inventory buffers
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Variability — Causes and CuresAmplification
Causes
◮ MRP nervousness – changes inschedule in the presence ofbatching
Cures
◮ reduce setup and order costs
◮ develop improved schedulingmethods
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Variability — Causes and CuresConclusion:
Therefore, factories should be
◮ designed and operated
to minimize the
◮ creation, propagation, and amplification
of uncertainty, variability, and randomness.
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How inventory and variability interactSimulations: Buffer Level vs. time
0
2
4
6
8
10
0 200 400 600 800 1000
n(t)
t
MTTRi = 10,MTTFi = 100, i = 1, 2;N = 10
Small buffer
0
20
40
60
80
100
0 2000 4000 6000 8000 10000n(
t)t
MTTRi = 10,MTTFi = 100, i = 1, 2;N = 100
Large buffer
Inventory and Variability 21/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
How inventory and variability interactSimulations: Buffer Level vs. time
0
20
40
60
80
100
0 2000 4000 6000 8000 10000
n(t)
t
MTTRi = 10, i = 1, 2,
MTTF1 = 50,
MTTF2 = 100;N = 100
First Machine is Bottleneck
0
20
40
60
80
100
0 2000 4000 6000 8000 10000
n(t)
t
MTTRi = 10, i = 1, 2,
MTTF1 = 100,
MTTF2 = 50;N = 100
Second Machine is Bottleneck
Inventory and Variability 22/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
How inventory and variability interactProduction Rate vs. Buffer Size
N
P
0.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0 20 40 60 80 100 120 140 160 180 200
Increasing a small buffer may pay off; increasing a large buffer will not.
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How inventory and variability interactAverage Inventory vs. Buffer Size
N
nBottleneck at first machine.
Bottleneck at second machine.
0
20
40
60
80
100
120
140
160
180
200
0 20 40 60 80 100 120 140 160 180 200
There will be less inventory if the bottleneck is upstream.
Inventory and Variability 24/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
WIP and Buffer Size in Long linesAll buffer sizes = 23, total buffer space = 437
0
5
10
15
20
25
0 2 4 6 8 10 12 14 16 18 20
aver
age
inve
ntor
y
Production Rate = 0.8806
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WIP and Buffer Size in Long linesTotal buffer space = 414
0
5
10
15
20
25
0 2 4 6 8 10 12 14 16 18 20
optim
al b
uffe
r si
ze
Production Rate = 0.8804
Inventory and Variability 26/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
WIP and Buffer Size in Long linesAll buffer sizes = 37, total buffer space = 703
0
10
20
30
40
50
0 2 4 6 8 10 12 14 16 18 20
aver
age
inve
ntor
y
Production Rate = 0.8804
Inventory and Variability 27/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
WIP and Buffer Size in Long linesTotal buffer space = 510
0
10
20
30
40
50
0 2 4 6 8 10 12 14 16 18 20
optim
al b
uffe
r si
ze
Production Rate = 0.8803
Inventory and Variability 28/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.
WIP and Buffer Size in Long linesConclusions
◮ Buffer space is not needed where inventory is greatest;
◮ Buffer space is needed where inventory variability is greatest.
Inventory and Variability 29/29 Copyright c©2010 Stanley B. Gershwin. All rights reserved.