CYBERNETIC CONTROL IN A SUPPLY CHAIN: WAVE PROPAGATION AND RESONANCE

1
CYBERNETIC CONTROL IN A SUPPLY CHAIN WAVE
PROPAGATION AND RESONANCE

• Ken Dozier and David Chang
• USC Engineering Technology Transfer Center
• July 14, 2005

2
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

3
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

4
Quasistatic example reduction in unit cost of
productionPresented 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

5
Quasistatic example unit cost of production
Ln Output
High output N, High temperature 1/b
Costs down
High output N, Low temperature 1/b
Low output N, High temperature 1/b
Entropy up
Low output N, Low temperature 1/b
Unit costs
6
Semiconductor example Movement between 1992
and 1997 on Maxwell Boltzmann plot
Ln output
1997 High output N, Low temperature 1/b
Ln Output
1992 Low output N, High temperature 1/b
Unit costs
7
Heavy spring example Movement between 1992 and
1997 on Maxwell Boltzmann plot
Ln Output
1997 Low output N, High temperature 1/b
1992 Low output N, Low temperature 1/b
Unit costs
8
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

9
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 ß 0.167
per 106
10
Productivity Ratio (97/92) of the statistical
parameters

• Company size Large Intermediate
  Small
• IT rank 59 70 81
• 0.86 1.0
  0.90
• E(1000s) 0.78
  0.98 1.08
• /company 0.91 1.0 1.21
• Sh (million) 1.53 1.24
  1.42
• Sh/E (1000) 1.66 1.34
  1.35
• ß 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.

11
Time-dependent phenomena

• Cyclic phenomena in economics
• Ubiquitous
• Resource wasteful career disruptive
• Example oscillations in supply chain
  inventories

12
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

13
Observations of supply chain oscillations

• Prevalent inventory oscillations led to MITs
  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(i?t)
• 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(n1) (?T)2 y(n)
  0 1
• Normal mode form for N companies in chain
• y(pn) expi2?pn/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