Hyperbolic Models for Large Supply Chains Christian Ringhofer (Arizona State University) Hyperbolic Models for Large Supply Chains – p. 1/4
Introduction Topic: Overview of conservation law (traffic - like) models for large supply chains. Joint work with.... • S. G¨ ottlich, M. LaMarca, D. Marthaler, A. Unver • D. Armbruster (ASU), P . Degond (Toulouse), M. Herty (Aachen) • K. Kempf , J. Fowler (INTEL Corp.) Hyperbolic Models for Large Supply Chains – p. 2/4
Definition of a supply chain One supplier takes an item, processes it, and hands it over to the next supplier. Suppliers ( Items): ◮ Machines on a factory floor (product item), ◮ Agent (client), ◮ Factory, many items, ◮ Processors in a computing network (information), Hyperbolic Models for Large Supply Chains – p. 3/4
Example: Protocol for a Wafer in a Semiconductor Fab Hyperbolic Models for Large Supply Chains – p. 4/4
OUTLINE 03 ◮ Traffic flow like models ◮ Clearing functions: Quasi - steady state models - queueing theory. ◮ First principle models for non - equilibrium regimes (kinetic equations and hyperbolic conservation laws.) ◮ Stochasticity (transport in random medium). ◮ Policies (traffic rules). Hyperbolic Models for Large Supply Chains – p. 5/4
Traffic flow - like models ◮ Introduce the stage of the whole process as an artificial ’spatial’ variable. Items enter as raw product at x = 0 and leave as finished product at x = X . ◮ Define microscopic rules for the evolution of each item. • many body theory, large time averages ◮ • fluid dynamic models (conservation laws). ◮ Analogous to traffic flow models (items ↔ vehicles). Hyperbolic Models for Large Supply Chains – p. 6/4
Similarities between Traffic and Production Modeling 05 ◮ Complexity and Topology: Complex re - entrant production systems. Networks of roads. ◮ Many body problem: interaction not given by simple mean fields. ◮ Control: Policies for production systems. Traffic control mechanisms. ◮ Random behavior. ◮ Model Hierarchies: Discrete Event Simulation (DES), Multi - Agent Models (incorporate stochastic behavior) ⇒ kinetic equations for densities (mean field theories, large time asymptotics) ⇒ fluid dynamics ⇒ rate equations (fluid models). Simulation ⇒ optimization and control. Hyperbolic Models for Large Supply Chains – p. 7/4
Quasi - Steady State Models and Clearing functions 07 ◮ A clearing function relates the expectation of the throughput time in steady state of each item for a given supplier to the expectation of the load, the ’Work in Progress’. ◮ Derived from steady state queuing theory. ◮ Yields a formula for the velocity of an item through the stages (Graves ’96. Dai - Weiss ’99) and a conservation law of the form ∂ t ρ + ∂ x [ v ( x, ρ ) ρ ] = 0 ρ : item density per stage, x ∈ [0 , X ] : stage of the process. Hyperbolic Models for Large Supply Chains – p. 8/4
Example: M/M/ 1 queues and simple traffic flow models Arrivals and processing times governed by Markov processes: v ( x, ρ ) = c ( x ) 1 1+ ρ , c ( x ) = � processing times � c ( x ) : service rate or capacity of the processor at stage x . Simplest traffic flow model (Lighthill - Whitham - Richards) ρ v ( x, ρ ) = v 0 ( x )(1 − ρ jam ) ◮ In supply chain models the density ρ can become arbitrarily large, whereas in traffic the density is limited by the space on the road ρ jam . Hyperbolic Models for Large Supply Chains – p. 9/4
phase velocity: v phase = ∂ ∂ρ [ ρv ( x, ρ )] c ( x ) 2 ρ v phase = (1+ ρ 2 ) > 0 , v phase − traffic = v 0 ( x )[1 − ρ max ] ◮ In supply chain models the propagation of information (shock speeds) is strictly forward v phase > 0 , whereas in traffic flow models shock speeds can have both signs. ◮ Problem: Queuing theory models are based on quasi - steady state regime. Modern production systems are almost never in steady state. (short product cycles, just in time production). ◮ Goal: Derive non - equilibrium models from first principles (first for automata) and then including stochastic effects. Hyperbolic Models for Large Supply Chains – p. 10/4
First principle models for automata 12 ◮ Assume processors work deterministically like automata. A processor located in the infinitesimal stage interval of length ∆ x ∆ x needs a time τ ( x ) = v 0 ( x ) to process an item. ◮ It cannot accept more than c ( x )∆ t items per infinitesimal time interval ∆ t . Theorem (Armbruster, CR ’03): In the limit ∆ x → 0 , ∆ t T → 0 . this yields a conservation law for the density ρ of items per stage of the form ∂ t ρ + ∂ x F ( x, ρ ) = 0 , F ( x, ρ ) = min { c ( x ) , v 0 ( x ) ρ } Hyperbolic Models for Large Supply Chains – p. 11/4
Bottlenecks ∂ t ρ + ∂ x F ( x, ρ ) = 0 , F ( x, ρ ) = min { c ( x ) , v 0 ( x ) ρ } ◮ No maximum principle (similar to pedestrian traffic with obstacles). ◮ The capacity c ( x ) is discontinuous if nodes in the chain form a bottleneck. ◮ Flux F discontinuous ⇒ density ρ distributional. (alternative model by Klar, Herty ’04). ◮ Random server shutdowns ⇒ bottlenecks shift stochastically. Hyperbolic Models for Large Supply Chains – p. 12/4
A bottleneck in a continuous supply chain Temporary overload of the bottleneck located at x = 1 . Hyperbolic Models for Large Supply Chains – p. 13/4
Stochasticity: Random breakdowns and random media 15 Hyperbolic Models for Large Supply Chains – p. 14/4
Availability Hyperbolic Models for Large Supply Chains – p. 15/4
Random capacities Random breakdowns modeled by a Markov process setting the capacity to zero in random intervals. ∂ t ρ + ∂ x [min { c ( x, t ) , v 0 ρ } ] = 0 sample capacity mu 3 2.5 2 1.5 1 0.5 0 −0.5 −1 0 0.5 1 1.5 2 2.5 3 3.5 t One realization of the capacity c ( x, t ) Hyperbolic Models for Large Supply Chains – p. 16/4
The Markov process: c ( x, t ) switches randomly between c = c up and c = 0 c ( x, t ) prob = 1 − ∆ tω ( x, c ) c ( x, t + ∆ t ) = c up ( x ) − c ( x, t ) prob = ∆ tω ( x, c ) ◮ Frequency ω ( x, c ) given by mean up and down times of the processors. ◮ Particle moves in random medium given by the capacities. Hyperbolic Models for Large Supply Chains – p. 17/4
One realization with flux F = min { c, vρ } using a stochastic c density rho 4 3.5 3 2.5 2 1.5 1 0.5 0 1 0.8 3.5 0.6 3 2.5 0.4 2 1.5 0.2 1 0.5 0 0 x t Goal: Derive equation for the evolution of the expectation � ρ � Hyperbolic Models for Large Supply Chains – p. 18/4
The many body problem 18 ◮ Formulate deterministic model in Lagrangian coordinates. ξ n ( t ) : position of part n at time t . ◮ A ’follow the leader’ model: d dt ξ n = min { c ( ξ n , t )[ ξ n − ξ n − 1 ] , v 0 ( ξ n ) } ◮ Particles move in a random medium, given by stochastic capacities c ( ξ n , t ) Hyperbolic Models for Large Supply Chains – p. 19/4
Kinetic equation for the many body probability density F ( t, x 1 , .., x N , y 1 , ., y K ) : probability that ξ 1 ( t ) = x 1 , .., ξ N ( t ) = x N and c 1 ( t ) = y 1 , .., c K ( t ) = y K . Satisfies a Boltzmann equation in high dimensional space. ∂ t F ( t, X, Y ) + ∇ X · [ V ( X, Y ) F ] = Q [ F ] K ( X, Y, Y ′ ) F ( t, X, Y ′ ) dY ′ − κ ( X, Y ) F � Q [ F ] = X = ( x 1 , .., x N ) : positions Y = ( y 1 , .., y K ) , Y ∈ { 0 , c up } K : kinetic variable ( discrete velocity model). Q ( F ) : interaction with random background. Hyperbolic Models for Large Supply Chains – p. 20/4
◮ Discrete event simulation corresponds to solving the kinetic many body equation by Monte Carlo. ◮ Use methodology for many particle systems. Mean field theory, long time averages, Chapman - Enskog. Hyperbolic Models for Large Supply Chains – p. 21/4
Theorem (Degond, CR ’06): On large time scales (compared to the mean up and down times of the processors ) the expectation � ρ � of the part - density satisfies F = � a � c up [1 − exp( − v 0 � ρ � � a � c up ) − εσ 2 ( a ) ∂ x � ρ � ] ∂ t � ρ � + ∂ x F = 0 , � T up � a : availability � a � = � T up � + � T down � , ε : ratio of � T up/down � to large time scale. Hyperbolic Models for Large Supply Chains – p. 22/4
One realization density rho 4 3.5 3 2.5 2 1.5 1 0.5 0 1 0.8 3.5 0.6 3 2.5 0.4 2 1.5 0.2 1 0.5 0 0 x t Hyperbolic Models for Large Supply Chains – p. 23/4
The steady state case density rho mean field rho 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 1 1 0.8 0.8 3.5 3.5 0.6 0.6 3 3 2.5 2.5 0.4 0.4 2 2 1.5 1.5 0.2 0.2 1 1 0.5 0.5 0 0 0 0 x x t t 60 stages, bottleneck processors for 0 . 4 < x < 0 . 6 Constant influx; F ( x = 0) = 0 . 5 × bottleneck capacity Left: DES (100 realizations), Right: mean field equations Hyperbolic Models for Large Supply Chains – p. 24/4
Verification of the transient case density rho mean field rho 5 5 4 4 3 3 2 2 1 1 0 0 1 1 0.8 0.8 3.5 3.5 0.6 0.6 3 3 2.5 2.5 0.4 0.4 2 2 1.5 1.5 0.2 0.2 1 1 0.5 0.5 0 0 0 0 x x t t Influx F ( x = 0) temporarily at 2 . 0 × the bottleneck capacity Left: 500 realizations, Right: mean field equations Hyperbolic Models for Large Supply Chains – p. 25/4
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