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CYBERNETIC CONTROL IN A SUPPLY CHAIN: WAVE PROPAGATION AND RESONANCE

Ken Dozier and David Chang

USC Engineering Technology Transfer Center

July 14, 2005

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Outline

• Background– Application of statistical physics to economic phenomena 3– Quasistatic examples 4-10– Time-dependent phenomena 11

• Implications of supply chain oscillations for cybernetic control 12– Inventory oscillation observations 13– Simple model of supply chain oscillations 14– Normal mode equations

15– Implications 16

• Conclusions 17

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Applications of statistical physics to economics

• Quasistatic phenomena

– Approach: Constrained maximization of microstates corresponding to a macrostate

– Applications to date: unit cost of production & productivity

• Time-dependent phenomena

– Approach: normal mode analysis – Current application: supply chain oscillations

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Quasistatic example: reduction in unit cost of production

[Presented at 2004 T2S meeting in Albany, N.Y.]

• Background question– What is required for technology transfer to reduce

production costs throughout an industrial sector?

• Approach– Apply statistical physics to develop a “first law of

thermodynamics” for technology transfer, where “energy” is replaced by “unit cost of production”

• Result & significance– Find that technology transfer impact can be

increased if “entropy” term and “work” term act synergistically rather than antagonistically

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Quasistatic example: unit cost of production

Ln Output

Unit costs

High output N,High “temperature” 1/

High output N,Low “temperature” 1/

Low output N,High “temperature” 1/

Low output N,Low “temperature” 1/

Costs down

Entropy up

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Semiconductor example: Movement between 1992 and 1997 on Maxwell Boltzmann plot

Ln Output

Unit costs

1997:High output N,Low “temperature” 1/

1992:Low output N,High “temperature” 1/

Ln output

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Heavy spring example: Movement between 1992 and 1997 on Maxwell Boltzmann plot

Ln Output

Unit costs

1997:Low output N,High “temperature” 1/

1992: Low output N,Low “temperature” 1/

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Quasistatic example: Improve productivity [CITSA ’04 conference (July, 2004); Paper submitted to JITTA for

publication (March, 2005) ]

• Background – Information paradox: Value of technology transfer – and more

generally, of information – on productivity has been called into question

• Approach– Apply statistical physics approach to show how productivity

is distributed across an industry sector– Compare evolution of distributions for information-rich and

information-poor sectors [US economic census data for LA]

• Results & significance– Find that productivity decreases but output increases in small

company sectors that invest in information, while productivity increases in information-rich large company sectors

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Productivity: Comparison of U.S. economic census cumulative number of companies vs shipments/company (diamond points) in LACMSA in 1992 and the statistical physics cumulative distribution curve (square points) with β = 0.167 per $106

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500

1000

1500

2000

2500

3000

3500

4000

0 10 20 30 40 50 60

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Productivity: Ratio (‘97/’92) of the statistical parameters Company size: Large Intermediate Small

IT rank 59 70 81# 0.86 1.0 0.90E(1000s) 0.78 0.98 1.08#/company 0.91 1.0 1.21Sh ($million) 1.53 1.24 1.42Sh/E ($1000) 1.66 1.34 1.35 β 1.11 0.90 0.99

Findings:

Sectors with large companies spend a larger percentage on IT.Largest % increases in shipments are in large & small company sectors.Small companies increased in size while large companies decreased.Number of large and small companies decreased by 10%.Employment decreased 20% in large companies, but increased 8% in small companies.Largest productivity occurred in large companies.

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Time-dependent phenomena

• Cyclic phenomena in economics– Ubiquitous– Resource wasteful & career disruptive

• Example: oscillations in supply chain inventories

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Implications of supply chain oscillations for cybernetic control

• Approach

– Develop a simple model of important interactions between supply chain companies that give rise to oscillations

– Determine structure of normal mode oscillations– Find governing dispersion relation for supply chain normal

modes

• Results & significance

– Identify opportunities for resonant, adiabatic, and short-time technology transfer efforts

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Observations of supply chain oscillations

• Prevalent inventory oscillations led to MIT’s “Beer game” simulation

• Simulations and observations both show– Oscillations– Phase dependence of oscillations on position

in supply chain– Instabilities

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Development of a model for normal modes in a supply chain

• Assume oscillations in supply chain inventories of the form exp(it)

• Obtain a simple form for normal modes by any of three approaches– Inventory dependent on nearest neighbor inventories– Conservation equations for inventory and sales– Fluid flow model of a supply chain

• Derive dispersion relation giving dependence of oscillation frequency on form of normal mode

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Resulting normal modes in a supply chain with uniform processing times

• Supply chain normal mode equation

y(n-1) – 2y(n) + y(n+1) +(T)2 y(n) = 0 [1]

• Normal mode form for N companies in chain

y(p:n) = exp[i2pn/N] [2]

• Normal mode dispersion relation

= (2/T) sin(p/N) where p is any integer [3]

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Implications of normal modes

• Supply chains naturally oscillate at frequencies below and up to inverse of processing times– In agreement with observations

• Disturbances in inventories propagate through supply chain at different velocities– Phase velocities increase to saturation as disturbance

wavelength decreases– Group velocities decrease as disturbance wavelength decreases

• Maximum control exerted by resonant interactions (Landau damping) with propagating waves– Control by surfing

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Conclusions

• Normal mode analysis provides a good framework for optimizing cybernetic control of undesirable oscillations in supply chains

• Optimization of cybernetic control will involve development of quasilinear equations for calculating the impact of resonant interactions