E. Gehrig & M. Kawski Optimal Control of Supply Chains with Variable Product Mixes INFORMS-03...

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E. Gehrig & M. Kawski Optimal Control of Supply Chains with Variable Product Mixes INFORMS-03

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Optimal Control of Supply Chains with Variable Product Mixes

Eric Gehrig* and Matthias Kawski*

Department of Mathematics and StatisticsArizona State University, Tempe, U.S.A.

___________Supported in part by Intel Corp. and by the NationalScience Foundation through the grant DMS 01-07666



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Outline

• Description of the physical system

• A mathematical model

• One control approach

• Simulations

• Theoretical challenges



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From AI to the customer

F / S A / TAI

GW

CW

GW

SW

D

D

D

D

D

D

D

F / S A / TAI CWSW

F / S A / TAI CWSW

“Global warehouse”“Silicon-warehouse”

“Components warehouse”“Assembly-inventory”

“Customer/demand”

“Fabrication / Sort” “Assembly-Test factory”



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Various supply-“mixes”Assembly inventory

A 1

A 2

B 1

B 2

Assembly / test factory

Supplies of wafers / dies from several different sources

with different grades / quality /….



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AssemblyAssembly inventory

A 1

A 2

B 1

B 2

Assembly / test factory

Simplified view of assembly:saw up wafers into dies and place them into

“packages” (combine with various components)

key choice: which source to use for which package.

laptop – A 1

laptop – A 2

desktop – B 1

desktop – B 2

laptop package

desktop package



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Testingassembled packages

laptop package

A 1

desktop package

B 1

laptop package

A 2

desktop package

B 2

laptop hi speed

laptop med speed

laptop lo speed

desktop hi speed

desktop med speed

desktop lo speed

assembly, …

“burn-in”, ...

basically the

same steps

for all pairs

no re-entry,

only small

variation of

the delay

Testing and binning

2 disjoint “complete”

directed digraphs

bins of “semi-finished” goods



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Delay finishing: “Vanilla boxes” †

Assembly and test factory W

D

D

D“Customer/demand”

SFGIFinish

& ship

D

D

D

Semi-finished goods inventory

AI

Assembly / testAI

† Swaminathan, Tayur: Managing broader prod lines through delayed

differentiation using vanilla boxes. Management Science. 1998

“Assembly inventory” “warehouse”



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Finish and ship

L 1

Semi-finished goods inventory

L 2

L 3

D 1

D 2

D 3

L 1

L 2

L 3

D 1

D 2

D 3

Customer /

“demand”configuring

here: 2 disjoint

directed graphs

shipping

Some semi-finished products may be “reconfigured” to meet more than one “demand”. Compare “outlet stores” (sporting goods), hotel and airline-seat pricing, ….



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Some data-points

• Large lead-time (approx 60 days) and large variabilityin supply from F/S (global network) buffered by AI

• Packages etc. also subject to lead-time, capacity constraints, …

• “Starts” from AI in “whole batches” of variable sizes

• A/T factory finite (variable) capacity (shared by all products)

• Stochastic and load dependent delay (about 2 weeks on average)

• Stochastic testing yields

• Demands vary stochastically on fast time-scale.

• Demands may be back-logged, but high “back-log” penalties.



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Fundamental modeling choices

• Material is continuous

• Time is discrete (unit = 1day, should be 1 shift = ½ d)– Due dates / shipments (truck/air): once or twice per day– Factory starts: traditionally assigned per shift (or week)

– Much easier math (finite dim stochastic models), can later always take averages and limits…..if desired



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Basic mathematical model• System of difference equations

Inventory level of product i=1..N at the end of day t = start of day (t+1)

Starts days ago for

j=1..M supply categories

Actual binning percentages of supply

category j=1..M into SFGI bins i=1..N

Percentages (of starts!) arriving early / on-time/ late

Total shipments (incl. configuring) from bin i=1..N in time period t

Starting inventory for each SFGI product i=1..N



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Some mathematical features• System of difference equations

Delays are stochastic and load-dependent. I.e. treat

i (percentage that exits i days after start )

as outputs of stochastic system driven by total load

Testing results are

MULTIPLICATIVE disturbances

Shipments follow external disturbances

Basically a simple cascade system



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Testing & binning (here M=4, N=6)

Percentage of supply category j=1..M that tests into SFGI bin i=1..N.

Implemented as i.i.d. random variables about (known, or in future: slowly-varyingaverages with “observer”). Typical “bin-splits” data:



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Basic control constraints

• Starts are constrained by total factory capacity and supply availability (assembly supply, j=1..4, and packages)



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Finish and ship

amounts taken from SFGI

bin j=1..N, and shipped as

product i=1..N, after possibly

being configured (if i > j )

The shipments are also constrained

by the current demands + backlogs

New / old backlog shipments

demand



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Small model: # of controls, constraints

• 4 start rates with polygonal constraint 16 inequalities: capacity, supplies, nonnegativity

expect sometimes interior, sometime on boundary • 12 finish/ship rates + 6 “scrap” rates

with polygonal constraint (30 inequalities: 6 inventory, 6 demand, 18 nonnegativity) expect almost always on boundary (on “face”, not “corner”):

• don’t create backlog unless absolutely unavoidable• “can’t ship product without a mailing label (address)”,

i.e. can never have negative backlog (shipping too early)



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Controllability

Red: (mean) demandBlue: starts (capacitated, fixed mix)

Scrap slow

Illustrations for

1 supply-mix (1 control)

and 2 SFGI bins (2 inventories)

Many more scenarios for

m controls, n products: n > m.

Simulations make sense only when

average demand, supply mix and

capacity provide reasonable con-

trollability of the averaged system.

Finish and ship

Can’t configure more than slow demand



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Many ways to be uncontrollable

Total demand df + ds > capacity Total demand df + ds << capacity !

but supply mix not “fa(s)t” enough:

df > (fraction of fast) * capacity



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Controls strategies – in words

• Currently:– Forecasts look ahead equal to total lead-time,

using only with aggregate plan for “configure-and-shipping”– Moderate LP (0.1 sec per time-unit of simulation) w/ many

extra var’s to accommodate pcw linear functions/cost-functions– First satisfy demand for most valuable product,

then consider configuring to satisfy other demands– Look ahead: “Don’t configure today if it compromises

tomorrow’s ability to fill hi-value demand”…..– Smoothing (“move suppression”) only (for) “total” start-rates –

but do NOT smooth the start rates for each individual product



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Simulations, sample data

Lots of other “stochasticity” data, objective function weights, …..

Simultaneous opposite steps in demand-mixes.



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Demands: Steps up/down in each “mix” Backlogs: rare, and almost never > 1 day



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Starts: “Undesired” oscillations between A1/A2, B1/B2, eventually don’t use one supply bin at all

(note: supply constraints not running here). Variability of starts variability of demands



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Stochasticity of testing / binning: Don’t get exactly what we had asked for (even w/o variable delay)



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Stochasticity of testing / binning: Don’t get exactly what we had looked for (even w/o variable delay)



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configuring from the SFGI-perspective and from the shipping-perspective



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Inventory levels (at noon and at midnite). Currently no scrapping, just let go up to sky.



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Almost EMPTY SFGInventories – yet hardly any backlogs, almost NEVER more than 1 day !!!



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Theoretical: Optimal inventory targets ?

• “stochastic dynamic programming problem”

- “controlled” stochastic dynamical system

- “cost” function (expected cost over N stages)



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Cost function

Minimize over all controls = re-order policies

Expected valueover all demands

Typical: piecewise linear stage cost

Final cost/value

Order cost Backlog penalty Inventory holding cost

2-step process: 1st find desired optimal target inventory ,

2nd find optimal feedback



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Known results• In case of no capacity constraints, “configuring”, “supply mix”,…:

Simple formula for constant optimal inventory target S, and “Order-up to S”- policy is well-known to be optimal

• General cases:computationally intractableexistence of optimal solutions (relying on convexity,contraction-mapping of value functions uses discount)

• Recent – incl. computationally feasible approximations Tayur (1993): Capacitated system using “infinitesimal perturbation analysis” Glasserman (1994): capacity and multiple products



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Capacitated systems, w/ or w/o configuring

• Order-up to” policies may still be optimal,but in general the optimal target inventory is no longer constant (but function of horizon!)

• “Configuring” (vanilla boxes) allows risk-pooling,generally lowering optimal inventory targets – butdetails depend delicately on (mis)match of supply-mix and demand-mix.



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Conclusion

• Established mathematical models

• Analyzed controllability properties

• Proposed simple feedback-control laws

• Simulations of closed loop system, includingsimulations with real factory and demand data

• Some analysis of optimal strategies- key issue is the “noise model” for demand.

E. Gehrig & M. Kawski Optimal Control of Supply Chains with Variable Product Mixes INFORMS-03...

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