1. 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. 2 Outline Background –Application of statistical physics to economic phenomena 3 –Quasistatic examples 4-10 –Time-dependent phenomena11 Implications of supply chain oscillations for cybernetic control 12 –Inventory oscillation observations13 –Simple model of supply chain oscillations14 –Normal mode equations15 –Implications16 Conclusions 17

3. 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. 4 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

5. 5 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

6. 6 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

7. 7 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/ 

8. 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. 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 β = 0.167 per $10 6

10. 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 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.

11. 11 Time-dependent phenomena Cyclic phenomena in economics –Ubiquitous –Resource wasteful & career disruptive Example: oscillations in supply chain inventories

12. 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. 13 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

14. 14 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

15. 15 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[i2  pn/N] [2] Normal mode dispersion relation  =  (2/T) sin(  p/N) where p is any integer [3]

16. 16 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

17. 17 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