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Page 1 Stephen C. Graves Copyright 2010 All Rights Reserved
Strategic Safety Stocks in Supply Chains: Update on Recent Work
Stephen C. GravesMIT, E53-347
sgraves@mit.eduhttp://web.mit.edu/sgraves/www/
Joint work with Sean Willems, Boston University, Katerina Lesnaia, Oracle, Tor Schoenmeyr, FirstSolar
Page 2 Stephen C. Graves Copyright 2010 All Rights Reserved
Overview
• Motivation and assumptions for SIP model• Prior work – review of base model &
example (joint with Willems)• Recent work – extend to account for
capacity (joint with Schoenmeyr)• Recent work – extend to include evolving
forecasts (joint with Schoenmeyr)• Summary
Papers available on request!
Page 3 Stephen C. Graves Copyright 2010 All Rights Reserved
Strategic Safety Stock Model: Intent
• Tactical model to determine the amount and positioning of safety stocks in supply chains
• Tactical model to support supply chain improvement teams
• Simple model, easily accessible, runs on PC, understandable inputs/outputs; academic version available from www.sipmodel.com
• Commercialized by Optiant; applications support both tactical and operational decisions
Page 4 Stephen C. Graves Copyright 2010 All Rights Reserved
Assumptions
• Supply chain modeled by an acyclic graph
• Deterministic processing time for each stage
• No capacity constraints
• Deterministic yield
• Periodic review, base stock control for each stage; common review period and no lot sizing
Page 5 Stephen C. Graves Copyright 2010 All Rights Reserved
Assumptions
• Fixed service time between stages where service time is the decision variable
• Each stage quotes same service time to all adjacent downstream stages
• Stationary, bounded demand process for each end item
• Each stage provides 100% service: “Guaranteed service model”
Page 6 Stephen C. Graves Copyright 2010 All Rights Reserved
Stage k
Inventory
Processing
Orders d(t)Orders d(t)
kSService time
At time node k must deliver d(t) to the downstream node from its inventory kt S
At time , d(t) units are delivered as raw material from node k + 1, and at time the d(t) units are ready as inventory at node k
1kt S
1kS Service time Processing time kT
1k kt T S
Stage k must have a base stock level equal to max demand over the net replenishment time 1k k k kS T S
Review of guaranteed-service base-stock problem
Page 7 Stephen C. Graves Copyright 2010 All Rights Reserved
Base stock mechanicsB is base stock level. I (t) is inventory at end of time t.
Demand arrives:
d0 … dt-SI-T-1 dt-SI-T dt-S dt… …
shipped
received
Id0 … dt-SI-T
receivedd0 … dt-S
shipped
1;k kSI S S S
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Key results for guaranteed service, bounded demand
1 1
1 1
1 2
0 1
k k k k k k k k k k
k k k k k k k k k k k
I t B d t S T d t S T d t S
I t B Max d t S T d t S D S T S
1 1
k
k k k k k k k k
For D z
E I t B S T S z S T S
Page 9 Stephen C. Graves Copyright 2010 All Rights Reserved
Stage k
InventoryProcessing
Stage k+1
InventoryProcessing InventoryProcessing
Stage k-1
kT kS 1kS 1kS 1kT 1kT
Safety stock 2 1 1k k kz S T S 1k k kz S T S 1k k kz S T S
If we view the service times as decision variables we get a global optimization problem:
11
1 1
1
min
0
0
k
N
k k k kS
k
k
k k k
h z S T S
S k
S s
S T S k
Review of guaranteed-service base-stock problem
Page 10 Stephen C. Graves Copyright 2010 All Rights Reserved
11
1 1
1
min
0
0
k
N
k k k kSk
k
k k k
h S T S
S k
S s
S T S k
Simpson (1958): Solve serial system through enumeration. “All-or-nothing” property of optimal solution (i.e., either )
10 or k k k kS S S T
Graves and Willems (2000): Solve spanning tree system through polynomial-time dynamic programming (Lesnaia, 2004). Fast enough for large, real-life applications.
Review of guaranteed-service base-stock problem
Page 11 Stephen C. Graves Copyright 2010 All Rights Reserved
Algorithmic Results
• For serial systems, Simpson (1958) showed the all or nothing property for solution
• Graves and Willems (2000) developed a pseudo-polynomial DP for spanning trees; also Graves (1988), Inderfurth (1991) and Inderfurth and Minner (1998)
• Lesnaia (2004) provides polynomial DP for spanning tree and specialized algorithm for any two layer network
• General network is NP hard (Lesnaia, 2004); optimum occurs at an extreme point for concave bound function
• Several exact and heuristic algorithms for general networks: Humair and Willems (2006, 2008); Lesnaia (2004); Minner (2000); Magnanti et al. (2006)
Page 12 Stephen C. Graves Copyright 2010 All Rights Reserved
KIMES 100
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Supply Chain: Before
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Supply Chain: Lead Times
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Supply Chain: Costs
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Supply Chain: Optimized
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Supply Chain: Implemented
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Thermal Printer
Integrate &Test
Scanner
Kit OrderIDM Assembly
Software
Computer
Monitor
IDM MonitorCalibration
Cables, etc.
Ship CustomerOrder
Palletize andMove to DC
Keyboard, etc.
Memory
Power Supply PlatformAssembly
Memory Chips
ROMs
DRAMs
Burn In
Kit Prep/CBA
OtherAssemblies
OtherComponents Ship to Bldg.
Supply-Chain
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KIMES 100
• Project results
–Sizing finished goods inventory
–Assess where to target lead-time reduction efforts
–Framework to work with suppliers on purchasing long lead-time parts
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Key Benefits & Learning• Shows value from “holistic” perspective
• Formalizes inventory-related supply chain costs, and provides an optimal benchmark
• Provides framework and standard terminology for cross-functional debate
• Shows the effectiveness of inventory, strategically positioned in a few places to de-couple the supply chain
• De-couple supply chain prior to a high-cost added stage; and prior to product explosion
• Most leverage from lead time reduction
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Battery Supply Chain
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HP Supplies Inventory Modeling ProjectRegionsFactoriesSuppliers Customers
Answer the bulk pen inventory question…
… in the context of what is best for the system
IPG Builds Roughly 1000 Network Models per Year
# Models Built
0
1000
2000
3000
4000
5000
6000
7000
8000
2001 2002 2003 2004 2005 2006 2007 2008 2009
standard training class and tier based support
Page 24 Stephen C. Graves Copyright 2010 All Rights Reserved
Key Limitations, circa 2000
• Stationary demand assumptions
• No capacity constraints
• DP algorithm for spanning tree only
• Deterministic lead times
• Common review period
• Common service time to all downstream customers
Page 25 Stephen C. Graves Copyright 2010 All Rights Reserved
Strategic safety stocks in supply chains with evolving forecasts
• Same assumptions as for base case, but now there is an evolving forecast for the end item demand
• Guaranteed service model – each stage commits to a guaranteed service but now for a bound on forecast errors
• Each stage uses a forecast-based ordering policy, rather than a base stock policy
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Forecast evolution model Graves et al. (1986), Heath and Jackson (1994)
( )tf t i is our forecast, in period t, for demand in period t + i
Each period the forecasts are revised
Assumptions (“rational forecasts”):(A1) are i.i.d. R.V.(A2) [ ( )] 0tE f j
( )tf t i
Unlike previous authors, we make no assumptions about ( )tf t H
We show an equivalence with this model and general, state-space models of demand (e.g., ARIMA)
The forecast is initialized as at the horizon( )tf t H
1( ) ( ) ( )t t tf t i f t i f t i
forecast revisionCurrent forecast is demand ( ) ( )tf t d t
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1
0Scheduled future demand
Downstream schedule changes
( ) ( ) ( ) .kL
k t k ti
P t f t L f t i
Forecast-based order policy:
For zero service times, this corresponds to orders in a simple (no lot sizing, etc) MRP system.
Both the forecast evolution model, and similar order mechanisms have been considered before; new contribution is to consider non-zero service times in a global optimization problem
1 1
k
k k jjL S T
cumulative lead timeOrder placed by stage k
For zero service times, the forecast-based orders have some local optimality properties (Aviv, 2003)
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01
1
( ) ( )k kt t L
k k k k ii t j i
I t S T I f j
We can use the equations for the evolving forecast and the order policy to derive the inventory
Safety stockconstant
Forecast revisions
1
( )k kt t L
ii t j i
f j t
If we can find a bound on the sum:
and set the safety stock level to0
1
max ( )k kt t L
k ii t j i
I f j
then the stage can guarantee service; i.e., 0kI t
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0 2 21k k kI z F L F L
1
( ) ( )t L
j tj t
F L D f j
For the cumulative forecast error:
Define
can be calculated from historical data on demand and forecasts
1
( )t L
j tj t
D f j
( )F L
we have a valid (probabilistic) bound. Loosely speaking, the stages will provide guaranteed service as long as the cumulative forecast errors are smaller than ( )zF L
( )k kt t L
2 2i k k 1
i t 1 j i
f j F L F L
We find that
which we propose as a bound. By setting the safety stock level
How might we set the bound?
D is demand RV; σ() is now a function too!
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1 2 3 4 5 6 7 8 9 10
Time into the futureF
ore
ca
st
err
or
j
1L 3L2L
2 21
1
1 1
1 1
1
min
0
0
k
N
k k kS
k
k
k k jj
k
k k k
h F L F L
L S T k
S k
S s
S T S k
Optimization problem: how do we find the least cost safety stock configuration that maintains guaranteed service for any forecast/demand realization within the bounds
Page 31 Stephen C. Graves Copyright 2010 All Rights Reserved
2 21
1
1 1
1 1
1
min
0
0
k
N
k k kS
k
k
k k jj
k
k k k
h F L F L
L S T k
S k
S s
S T S k
Forecast problem
11
1 1
1
min
0
0
k
N
k k k kS
k
k
k k k
h z S T S
S k
S s
S T S k
Base stock problem
The problem is very similar to the base stock problem solved by Simpson (1958), and extended by Graves and Willems (2000) and others.
Under some mild assumptions about the forecasts, we show that the all-or-nothing property holds
We can use existing, effective algorithms to find optimal service times, after modifying the bound function
Page 32 Stephen C. Graves Copyright 2010 All Rights Reserved
Managerial insight nugget
• Assuming forecast revisions at time t are independent and with proposed bound on forecast errors:
1
0 21
1
;k
k
j L
k k k k tj L
I z S T S D t j f t j
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• Electronic test system manufactured by Teradyne, Inc.• 3,866 part/locations• Used real data on supply chain topology, lead times, costs of parts
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• Schedule contained booked and “preliminary” orders, and got increasingly locked down as the date of delivery approach• The schedule was effectively a forecast, and we used data on past schedule changes to calculate F(L)• As a forecast of actual demand, it was fairly accurate in the short term but useless >10 weeks out
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10
Weeks into the future
Co
rre
lati
on
fo
rec
as
t -
de
ma
nd
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Current Optimal base stock Optimal forecast/MRP
25.5% improvement
Total cost??? ?
Difficult to compare with current situation because no consistent optimization procedure/ service level used
?
• In the forecasted case, most savings were far downstream, where forecasts were accurate • Optimization time ~1 minute on a laptop computer
• Schoenmeyr thesis discusses generalizations for multi-product networks
Page 36 Stephen C. Graves Copyright 2010 All Rights Reserved
0
1
2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Total lead time (weeks)
To
tal h
old
ing
co
sts
Base stock policy (Graves-Willems)5 week forecast
10 week forecast (current)
20 week forecast
Page 37 Stephen C. Graves Copyright 2010 All Rights Reserved
• We have shown how to map the optimization method used for base stock systems, so that it can be used for forecast-driven (push) systems
• This approach enables optimization of large system with
• Evolving schedule in make-to-order context• Evolving demand forecast in make-to-stock context
• Benefit relative to base stock case depends on forecast quality; in one case study it was ~25%
Summary of results
Page 38 Stephen C. Graves Copyright 2010 All Rights Reserved
Strategic safety stocks in supply chains with capacity constraints
• Same assumptions as before, but now there can be a capacity constraint at each stage
• Guaranteed service model – each stage commits to a guaranteed service for bounded demand
• Deterministic production lead time T • Each stage follows a base stock policy, subject to
capacity constraint
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Open problem: what if there are capacity constraints?
1k k kz S T S Now may not be enough safety stock, because any units that get “stuck” will be delayed.
Q: How much extra inventory do we need?Q: How do we optimize a supply chain with one or more capacity constraints?Q: Do the structural results from before (“all or nothing”) hold up?
Stage k
InventoryProcessing
Orders d(t)Orders d(t)
kSService time1kS Service time Processing time kT
/ periodkc“stuck” units
Page 40 Stephen C. Graves Copyright 2010 All Rights Reserved
Q: How much extra inventory do we need?
A: The base stock level for the node with capacity constraint can be calculated with a functional transformation of the unconstrained base stock level.
Original base stock level/order bound: 1( ) ( ), k k kB D for S T S
Base stock with capacity constraint: 0
( ) ( )( ) max ( ) kn
B D D n c n
In general
Page 41 Stephen C. Graves Copyright 2010 All Rights Reserved
Q: How much extra inventory do we need?
A: The base stock level for the node with capacity constraint can be calculated with a functional transformation of the unconstrained base stock level.
Original base stock level/order bound:
2 2
2
2
2
for ( ) ( )
( ) ( )( )
for ( )
kk k
k
cc c
B D
Dc
Common example
( ) ( ) 2B D
Base stock with capacity constraint:
The base stock grows hyperbolically as we decrease capacity. On the other hand, if the capacity constraint is large enough it becomes irrelevant.
Page 42 Stephen C. Graves Copyright 2010 All Rights Reserved
( ) 2D
2 2
2
2
2
for ( ) ( )
( )( )
for ( )
kk k
k
cc c
D
Dc
For sufficiently large net replenishment times the capacity constraint does not matter
A stage with a capacity constraint needs safety stock even at zero net replenishment time
With a capacity constraint, we permit negative net replenishment time
Page 43 Stephen C. Graves Copyright 2010 All Rights Reserved
Q: How do we optimize a supply chain with one or more capacity constraints?
A: We have shown how to calculate new base stock levels for a single stage. Other stages are not affected (orders placed/delivered as before). Hence after transforming affected bounds, we can use existing optimization procedures.
A: The functional transformation
preserves concavity, and hence the “all or nothing” property holds.
0
( )( ) max ( ) kn
D D n c n
Q: Do the structural results from before (“all or nothing”) hold up?
Page 44 Stephen C. Graves Copyright 2010 All Rights Reserved
Stage k
InventoryProcessing
OrdersOrders
kSService time1kS Service time Processing time kT
/ periodkc“stuck” units
But can we do better?
“Why should we ever order units if we cannot process them when they arrive?”
kc
1( ) min( ( ) ( 1), )k k k kd t d t BL t c
1( ) max ( 1) ( ) ,0k k k kBL t BL t d t c
Place the censored order
( )d t ( )d t1( )kd t( )kd tCensored orders
( )kBL t
where we keep a backlog of delayed orders
Page 45 Stephen C. Graves Copyright 2010 All Rights Reserved
Q: But how much inventory do we need/ how do we optimize supply chain?
A: We find that base stock transformation remains the same but we need another functional transformation to obtain a new bound for orders (demand) placed by a censoring node
1 ( ) min( , ( ))k k k kD c D
Q: Do the structural results from before (“all or nothing”) hold up?A: Yes. (Φ also preserves concavity)
1 1 2( ) ( )k k kD D Order bound
Base stock
1kc kc1kc
1 1 2( ) ( )k k kB D
1( ) ( )k k kD D 1 1( ) ( )k k kD D
1( ) ( )k k kB D 1 1( ) ( )k k kB D
Page 46 Stephen C. Graves Copyright 2010 All Rights Reserved
Average Inventory8 7 6 5 4 3 2 1
Total cost
No capacity constraint
0 0 0 17.9 0 0 0 17.9 2,377
Capacity constraint, no censorship
0 0 15.5 0 0 15.5 0 12.6 2,433
Capacity constraint, censorship
5 5 5 5 5 6.5 0 12.6 2,233
Serial system with 8 nodes and capacity constraint at node 3. Assumed processing time 5 at each node; holding costs increase with 40% per stage.
• Censorship reduces cost impact of constraint• Censorship cost is sometimes even lower than uncapacitated problem!• “Paradox”: Under censorship, add constraint → better solution• Explanation 1: Censorship smoothes demand and reduces safety stocks upstream• Explanation 2: The (uncensored) local base stock policy is not optimal in a multi-stage system with guaranteed service• It may be of interest to censor even in the absence of actual capacity constraints
Page 47 Stephen C. Graves Copyright 2010 All Rights Reserved
Summary of results for capacity constraints
• We can generalize the base stock model to incorporate capacity constraints.
• For serial systems, we find exact analytical transformations, under which existing algorithms can be used with small modifications
• Known structural results (“all-or-nothing”) hold.
• These results also hold if we censor orders with the capacity. The necessary safety stocks are reduced.
• Censored orders sometimes lead to costs that are even lower than for the same problem without capacity constraints (in many examples 30-40% reductions by censoring the right amount at the right location)
• Development is for serial systems, and extends immediately to assembly structures; more general networks require a calculus to combine bounds
Page 48 Stephen C. Graves Copyright 2010 All Rights Reserved
Overall Summary• Motivation, assumptions and review of guaranteed service supply
chain model
• Extension for capacity– Requires transformation of base stock and of demand bound– Structural results and algorithms extend directly– Capacity constraint can lead to lower cost solution– Multi-item supply chains requires more work
• Extensions for evolving forecast– Requires forecast-based ordering and bound on forecast errors– Structural results and algorithms extend directly– Incorporating forecast can lead to lower costs– Multi-item supply chains requires more work
Page 49 Stephen C. Graves Copyright 2010 All Rights Reserved
Example demand data
0
100
200
300
400
500
6000 2 4 6 8
10
12
14
16
18
20
Net Replenishment Time
Un
its Maximum Demand
Expected Demand
Safety Stock
Page 50 Stephen C. Graves Copyright 2010 All Rights Reserved
We then set the safety stock level
0 2 21 1
1
k k k k k
k k k
I F L F L z L L
z S T S
1
( ) ( )t L
j tj t
F L z D f j z L
When we have no forecast
to get the base-stock model !