The Impact of Information on Supply Chain Oscillations

1. The Impact of Information on Supply Chain Oscillations Ken Dozier & David Chang Western Research Application Center IRMA International , Inc Washington D.C. May 23, 2006

2. Bio

3. A System of Forces in Organization Direction Cooperation Efficiency Proficiency Competition Concentrat\ion Innovation Source: “The Effective Organization: Forces and Form”, Sloan Management Review, Henry Mintzberg, McGill University 1991

4. Make & Sell vs Sense & Respond Chart Source:“Corporate Information Systems and Management”, Applegate, 2000

5. Market Redefinition Supply-chain Expansion Supply-chain Discovery Business Model Redefinition Business Model Refinement Business Process Redesign Business Process Improvement Theoretical Environment Seven Organizational Change Propositions Framework, “Framing the Domains of IT Management” Zmud 2002

6. Supply Chain (Firm) Source: Gus Koehler, University of Southern California Department of Policy and Planning, 2002

7. Supply Chain (Government) Source: Gus Koehler, University of Southern California Department of Policy and Planning, 2002

8. Supply Chain (Framework) Source: Gus Koehler, University of Southern California Department of Policy and Planning, 2002

9. Supply Chain (Interactions) Source: Gus Koehler, University of Southern California Department of Policy and Planning, 2002

10. Why statistical physics? • Proven formalism for “seeing the forest past the trees” • Well established in physical and chemical sciences • Our recent verification with data in economic realm • Simple procedure for focusing on macro-parameters • Most likely distributions obtained by maximizing the number of micro-states corresponding to a measurable macro-state • Straightforward extension from original focus on energy to economic quantities • Unit cost of production • Productivity • R&D costs • Self-consistency check provided by distribution functions

11. Plasma theories • Advanced plasma theories are extremely important when one tries to explain, for example, the various waves and instabilities found in the plasma environment. Since plasma consist of a very large number of interacting particles, in order to provide a macroscopic description of plasma phenomena it is appropriate to adopt a statistical approach. This leads to a great reduction in the amount of information to be handled. In the kinetic theory it is necessary to know only the distribution function for the system of particles. Source: University of Oulu, FInland

12. 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

13. Quasi-static Comparison of Statistical Formalism in Physics and Economics VariablePhysicsEconomics State (i) Hamiltonian eigenfunction Production site Energy Hamiltonian eigenvalue Ei Unit prod. cost Ci Occupation number Number in state Ni Output Ni = exp[-βCi+βF] Partition function Z ∑exp[-(1/kBT)Ei] ∑exp[-βCi] Free energy F kBT lnZ (1/β) lnZ Generalized force fξ ∂F/∂ξ ∂F/∂ξ Example Pressure Technology Example Electric field x charge Knowledge Entropy (randomness) - ∂F / ∂T kBβ2∂F/∂b

14. 4000 3500 3000 2500 2000 1500 1000 500 0 0 10 20 30 40 50 60 Quasi-static Comparison of U.S. economic census cumulative number of companies vs shipments/company (blue diamond points) in LACMSA in 1992 and the statistical physics cumulative distribution curve (square pink points) with β = 0.167 per $106

15. 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.

16. Oscillations in Supply Chains • Observations • Cyclic phenomena in economics ubiquitous & disruptive • Example: Wild oscillations In supply chain inventories • MIT “beer game” simulation • Supply chain of only 4 companies for beer production, distribution, and sales • Results of observations and simulations • Oscillations • Phase dependence of oscillations on position in chain • Spatial instability

17. IRMA 2006 Objectives • To show with a simple product-flow model of a supply chain that universal information exchange • Changes the character of oscillations from those of nearest neighbor information exchange • Causes an increase in the damping of oscillations

18. Local Exchange of Information • Instead of designating each level in the chain by a discrete label n • the position in a chain was designated by a continuum variable x. • Flow of production units through each position in the chain was designated by a velocity variable v. • A differential distribution function f(x,v,t)dxdv denotes the number of production units in the intervals dx and dv at x and v at time t. • ∂f/ ∂t + ∂[fdx/dt]/ ∂x + ∂[fdv/dt]/ ∂v = 0 [1] • A thermodynamic force F that gives the rate at which v changes in time, this equation can be rewritten • ∂f/ ∂t + ∂[fv]/∂x +[∂fF]/ ∂v = 0 [2]

19. Nearest neighbor information exchange This becomes Vlasov-like equation for f(x,v,t) ∂f/∂t + v∂f/∂x + F∂f/∂v = 0 [5] This is the equation for collisionless plasmas When the inventory of the level below the level of interest is less than normal, the production rate (v) will be diminished because of the smaller number of production units being introduced to that level. At the same time, when the inventory of the level above the level of interest is larger than normal, the production rate will also be diminished because the upper level will demand less input so that it can “catch up” in its production through-put. Both effects give production rate changes proportional to the gradient of n. It is resonable also that the fractional changes are related rather than the changes themselves, since deviations are always made from the inventories at hand. ∂f/∂t + v∂f/∂x - 2ξv2(1/n)(dn/dx) ∂f/∂v = 0 [13]

20. Nearest neighbor dispersion relation Perturbed distribution f(x,v,t) = f0(v) + f1(v) exp[-i(t – kx)] [15] -i(-kv)f1 - ik 2ξv2(1/no)n1∂fo/∂v = 0 [16b] f1 = -2ξk(1/no) ∫dv’f1(v’) v2∂fo/∂v(-kv)-1 [17] This leads to the dispersion relation between  and k 1+ 2ξk (1/no) ∫dvv2∂fo/∂v(-kv)-1 =0 [18] Principal and imaginary parts ∫dvv2∂fo/∂v(-kv)-1 = PP∫dvv2∂fo/∂v (-kv)-1 - iπ(/k)2(1/k)∂fo(/k) /∂v [19]

21. Nearest neighbor dispersion relation (cont) Solving for   = 4ξkVo [1+ (1/n0)iπ(4ξVo )2∂fo(4ξVo ) /∂v] [23] Significance f0(v) peaked around V0, ∂f0(4ξV0 ) / ∂v <0. Oscillation resembles a sound-like wave Oscillation exhibits small Landau damping that is because of distance of phase velocity from Vo

22. Universal information exchange Introduce an information exchange potential Φ ∂2Φ/∂x2 = - [C/no]∫dv f(x,v,t) [24] from which the thermodynamic force F is obtained F = - ∂Φ/∂x [25] This reduces to the former results for nearest neighbor interactions when we choose C = ξVo2 / l2 [29]

23. Universal information exchange dynamic equations Introduction of potential into Vlasov equation ∂f/∂t + v∂f/∂x - ∂Φ/∂x ∂f/∂v = 0 [31] Perturbation in distribution function caused by Φ f1 = -kΦ1∂fo /∂v (-kv) -1 [33] Self-consistency condition Φ1 = (1/k2) [ξVo2 /nol2] ∫dv f1(v) [34]

24. Dispersion relation for universal information exchange ≈ kVo + ξ1/2(Vo/l) [1 + i {πξVo2/(2k2l2no)}∂fo/∂v ] [42] where ∂f0/∂v is evaluated at v = /k ≈ Vo + (ξ1/2Vo/kl) [43] Significance Oscillations resemble plasma oscillations Oscillations always exhibit Landau damping. This changes the form of the supply chain oscillation and in suppression of the resulting oscillation

25. Conclusions Washington DC • Supply chain oscillations can be described by a simple flow model of product through chain • Flow model shows that • Character of oscillation changes from sound-like to plasma-like when information exchange becomes universal rather than just between nearest neighbors • Damping of oscillation can be large when information exchange becomes universal

26. Future Work • Create a simulation that allows the study of various IT architectures on the optimization issues of supply chain management • kdozier@usc.edu • Visit the Learning Center • http:wesrac.usc.edu • Google wesrac • Google Ken Dozier