Page 1 Stephen C. Graves Copyright 2010 All Rights Reserved Strategic Safety Stocks in Supply...

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

[email protected]://web.mit.edu/sgraves/www/

Joint work with Sean Willems, Boston University, Katerina Lesnaia, Oracle, Tor Schoenmeyr, FirstSolar

http://web.mit.edu/sgraves/www/


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!


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

http://www.sipmodel.com/


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


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”


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


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


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


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



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.



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)


KIMES 100


Supply Chain: Before


Supply Chain: Lead Times


Supply Chain: Costs


Supply Chain: Optimized


Supply Chain: Implemented


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


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


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


Battery Supply Chain


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


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


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


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


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)


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


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!


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


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


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


• Electronic test system manufactured by Teradyne, Inc.• 3,866 part/locations• Used real data on supply chain topology, lead times, costs of parts


• 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


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


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


• 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


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


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


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


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.


( ) 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


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?


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


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


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


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


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


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


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 !

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