Supply chains models

Transcription

1 Supply chains models Maria Ivanova Technische Universität München, Department of Mathematics, Haupt Seminar 2016 Garching, Germany July 3, 2016

2 Overview Introduction What is Supply chain? Traffic flow vs. production systems Basic model Notations Basic model Redefining the flux Constitutive relation Asymptotic validity of the conservation law Scaling Interpolation and weak formulation An exact solution for a single bottleneck Concept of virtual processors Numerical experiments Experiment 1: Verifying Theorem 2 Experiment 2: long supply chain with unstructured throughput times and capacities

3 What is Supply chain? 1. A network of suppliers that produce goods, both, for one another and for generic customers. Goods travel from origin-suppliers to destination-customers, possibly visiting intermediate suppliers and being altered or recombined in the process. Conservation rules at each supplier define its outputs as a function of its inputs. [C.Daganzo, A Theory of Supply Chains]

4 What is Supply chain? 2. A finite, connected directed, simple graph consisting of arcs J = {1,..., N} and vertices V = {1,..., N 1}. Each supplier j is modelled by an arc j, which is again parameterized by an interval [a j, b j ]. We use a 1 = and b N = + for the first respectively the last supplier in the supply chain. [S.Goettlich, M.Herty, A.Klar, Network Models for Supply Chains] We consider a chain of M suppliers or processors S 0,..., S M 1. Each supplier takes a certain good (measured in units or parts), processes it, and hands it over to the next supplier in the chain.

5 Traffic flow vs. production systems Conservation Laws: t ρ + x [φ(ρ)] = 0 ρ: Work in progress (WIP) density vs. density of vehicles; x: stage of the production process (degree of completion - DOC). x = 0 : unfinished product, x = X: finished product vs. traffic flow: road; φ: flux function.

6 Similarities between Traffic and Production Models Complexity and Topology: Complex re - entrant production systems vs. Networks of roads. Control: Policies for production systems vs. Traffic control mechanisms. Random behavior. First Principle Models: 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.

7 Differences between Traffic and Production Modeling Capacity Limits: Limited capacity of processors (machines) vs. limited space (capacity of the road) backward wave propagation in traffic flow φ(ρ) ρ > 0 and φ (ρ) < 0. Parameters and Control Variables: Traffic: randomly given influx and individual behavior, distribute road capacities. Production: randomly given demand, choose start rate and (to some extent) topology.

8 Notations n - part index; m - processor index; τ(m, n) - the time at which part number n passes from supplier number m 1 to supplier number m; U(t) - N-curve given by the number of parts which have passed from processor S m 1 to processor S m at time t, i.e., by U(m, t) = H(t τ(m, n)) n=0 F (m, t) - the flux from processor S m 1 into processor S m F (m, t) = d dt U(m, t) = δ(t τ(m, n)), m = 0,..., M; n=0

9 Notations W (m, t) - the work in progress (WIP) of processor S m at time t W (m, t) = U(m, t) U(m + 1, t) + K(m), m = 0,..., M 1. Combining the last two equations yields the conservation law d W (m, t) = F (m, t) F (m + 1, t) dt T (m) - minimal processing time.

10 Assumptions It holds: τ(m + 1, n) τ(m, n) + T (m) WIP W (m, t) can never become negative. Conservation law: t ρ + x min {µ, W T } = 0 x - artificial continuous variable which indexes the suppliers; µ - service rate; ρ(x, t) - product density over x, i.e., W = ρdx.

11 Basic model Transition times recursion: τ(m + 1, n) = max {τ(m, n) + T (m), τ(m + 1, n 1) + 1 µ(m,n 1) }, m = 0,..., M 1, n 1 We assume finite capacities of the processors, so that µ(m, n) C(m), m = 0,..., M 1, n 0 Initial and boundary conditions for the recursion: τ(0, n) = τ A (n), n 0, τ(m, 0) = τ I (m), m = 0,..., M, where τ A (n) - the arrival time of part n in the first processor; τ I (m) - the time the first part has arrived at supplier S m.

12 Redefining the flux Observe for an arbitrary test function ψ(t) τ I (m) ψ(t)f (x m, t) = n=0 ψ(τ(m, n)) Rewrite into a Riemann sum for an integral τ I (m) where ψ(t)f (x m, t) = ψ(τ(m, n)) n τ(m, n)f(x m, τ(m, n)), n=0 (a) n τ(m, n) := τ(m, n + 1) τ(m, n), (b) f(x m, τ(m, n)) := 1 nτ(m,n) On a time scale where n is small, it becomes τ I (m) ψ(t)f (x m, t) τ I (m) ψ(t)f(x m, t)

13 Redefining the flux Consider the case when the arrival times τ would be distributed continuously, i.e., if they were given as a function τ(x, y). Then we have (a) d dy U(x, τ(x, y)) = tu(x, τ) y τ = 1, (b) d dx U(x, τ(x, y)) = xu(x, τ) + t U(x, τ) x τ. So, for continuum τ(x, y) of arrival times, we set f(x, τ) = 1 yτ and ρ(x, τ) = xτ yτ ρ and f satisfy a conservation law t ρ + x f = 0. Approximate density ρ and flux f defined from the arrival times (a) f(x m, τ(m, n)) = 1, m = 0,..., M, n 0, nτ(m,n) mτ(m,n+1) h m nτ(m+1,n) (b) ρ(x m, τ(m + 1, n)) =, m = 0,..., M 1, n 0, (c) m τ(m, n) = τ(m + 1, n) τ(m, n), h m := x m+1 x m.

14 Constitutive relation THEOREM 1. Let the arrival times τ(m, n) satisfy the transition times recursion. Let the approximate density ρ and flux f be defined as above. Then the approximate flux can be written in terms of the approximate density via a constitutive relation of the form f(x m, τ(m, n)) = φ mn (ρ(x m 1, τ(m, n))), m = 1,..., M, n 0, with the flux function φ m given by φ mn (ρ) = min{µ(m 1, n), h m 1 ρ T (m 1) }.

15 Scaling We define by T 0 the average processing time, i.e., T 0 := 1 M 1 M m=0 T (m) Then the following holds for scaled variables τ(m, n) = MT 0 τ s (m, n), T (m) = T 0 T s (x m ), µ(m, n) = µs(xm,τs(m+1,n)) T 0. Consider a regime where M >> 1 holds and set ε = 1 M << 1. With this scaling the transition times recursion becomes (a) τ s (m + 1, n + 1) = max{τ s (m, n + 1) + εt s (x m ), τ s (m + 1, n) ε + µ s(x m,τ s(m+1,n)) }, m = 0,..., M 1, n 0, (b) τ s (0, n) = τs A (n), n 0, τ s (m, 0) = τs I (m), m = 0,..., M.

16 Scaling We set giving n τ(m, n) = τ(m, n + 1) τ(m, n) = T 0 ns τ s (m, n), m τ(m, n) = τ(m + 1, n) τ(m, n) = T 0 ms τ s (m, n), τ s (m + 1, n) = τ s (m, n) + ε ms τ s (m, n), τ s (m, n + 1) = τ s (m, n) + ε ns τ s (m, n). We scale the density ρ and the flux f by f(x, t) = 1 T 0 f s (x, t MT 0 ), ρ(x, t) = M X ρ s(x, t MT 0 ), where X is the length of the DOC interval. This gives us scaled flux and density (a) f s (x m, τ(m, n)) = 1 (b) ρ s (x m, τ s (m + 1, n)) = nsτ s(m,n) ε msτ(m,n+1) h m nsτ s(m+1,n), m = 0,..., M, n 0,, m = 0,..., M 1, n 0

17 Scaling Scaled version of the constitutive relation then reads (a) f s (x m, τ s (m, n)) = φ s (x m 1, τ s (m, n), ρ s (x m 1, τ s (m, n))), h (b) φ s (x m 1, t, ρ s ) = min{µ s (x m 1, t), m 1 ρ s εxt s(x m 1 ) }. (b) suggests a natural choice for the grid in x-direction, namely h m = εxt s (x m ) = XT (m) M 1 m = 0,..., M 1. m=0 T (m),

18 Interpolation and weak formulation Goal: initial boundary value problem for a conservation law t ρ + x f = 0, f = min{µ(x, t), ρ}, f(0, t) = f A (t) Complications: a domain bounded by t > τ I ; we cannot assume any kind of smooth relation between two consecutive processors; the influx function f can become discontinuous. Solution: Set ρ(x, t) = x u(x, t). This gives t u = min{µ(x, t), x u}, lim u(x, t) = x 0 ga (t), d dt ga (t) = f A (t)

19 Interpolation Interpolation in time direction: (a) f 1 (x m, t) = f(x m, τ(m, n)), τ(m, n) t < τ(m, n + 1); (b) ρ 1 (x m, t) = ρ(x m, τ(m + 1, n 1)), τ(m + 1, n 1) t < τ(m + 1, n); (c) f A (t) = 1 nτ A (n), τ A (n) t < τ A (n + 1). Define: u 1 (x m+1, t) = u 1 (x m, t) hm X ρ 1(x m, t), m = 0,..., M 1, u 1 (x 0, t) = t τ(0,0) f A (s)ds Functions of continuous space and time: (a) f 2 (x, t) = f 1 (x m+1, t), x m x < x m+1, m = 0,..., M 1, (b) τ I 2 (x) = τ I (m + 1), x m x < x m+1, m = 0,..., M 1, (c) u 2 (x, t) = u 1 (x m+1, t), x m x < x m+1, m = 0,..., M 1.

20 Weak formulation THEOREM 2. Given the scaled density and flux at the discrete points x m, τ(m, n), let the piecewise constant interpolant u 2 and f 2 be defined as above. Let the scaled throughput times T (x m ) stay uniformly bounded in m. Assume finitely many bottlenecks for a finite amount of time, i.e., let m τ(m, n) be bounded for ε 0 except for a certain number of nodes m and a finite number of parts n, which stays bounded as ε 0. Then, for ε 0 and max h m 0 the interpolated N-function and flux u 2, f 2 satisfy the initial boundary value problem (a) t u 2 = f 2, t > τ2 I (x), 0 < x < X, (b) u 2 (x, τ I (x)) = 0, lim u 2(x, t) = t x 0 τ 2 (0,0) f A (s)ds, in the limit ε 0, weakly in x and t.

21 An exact solution for a single bottleneck Suppose t ρ + x f = 0, f = min{µ(x, t), ρ}, f(0, t) = f A (t) is posed on x [0, 1], with a bottleneck at x = 1 2, f A - prescribed arrival rate { µ1 for 0 < x < 1 µ(x) = 2, 1 µ 2 for 2 < 0 < 1, µ 2 < f A (t) < µ 1.

22 An exact solution for a single bottleneck The solution is given by ρ(x, t) = ρ c (x, t) + q(t)δ(x 1 2 ), where ρ c satisfies t ρ c + x ρ c = 0, x (0, 1 2 ) ( 1 2, 1), ρ c (0, t) = f A (t), ρ c ( 1 2 +, t) = µ 2, whose solution is given by { f ρ c (x, t) = A (t x) for 0 < x < 1 2, 1 µ 2 for 2 < 0 < 1. In order the whole solution ρ(x, t) to be a spatially weak solution of the conservation law, we have to satisfy 1 0 φ(x) tρ(x, t) min{µ(x), ρ(x, t)} x φ(x)dx = = φ(0)f A (t) φ(1) min{µ 2, ρ(1, t)} for any arbitrary smooth test function φ(x).

23 An exact solution for a single bottleneck Integrating by parts separately on the intervals (0, 1 2 ) and ( 1 2, 1) gives 1 0 φ(x) tρ c (x, t)dx + q (t)φ( 1 2 ) + φ(x) x min{µ(x), ρ c (x, t)}dx min{µ 1, ρ c ( 1 2, t)}φ( 1 2 ) + min{µ 1, ρ c (0, t)}φ(0) + min{µ 1, ρ c (0, t)}φ(0) + min{µ 2, ρ c ( 1 2 +, t)}φ( 1 2 ) = φ(0)f A (t) Since ρ c (x, t) < µ(x) everywhere and ρ c (0, t) = f A (t), this reduces to q (t) = f A (t 1 2 ) µ 2 = 0

24 Concept of virtual processors Basic idea: One processor with a processing time T and a service rate µ can be replaced by K virtual processors with the same service rate µ and processing times T K. THEOREM 3. Let the first processor S 0 in the chain be governed by the transition times recursion. If we replace the single processor by K virtual processors with the same processing rates and the same total throughput time, i.e., by ˆτ(m + 1, n + 1) = max{ˆτ(m, n + 1) + T (0) 1 K, ˆτ(m + 1, n) + µ(m,n) }, m = 0,..., K 1, ˆτ(0, n) = τ(0, n), ˆτ(m, 0) = τ(1, 0) (1 m K )T (0), m = 1,..., K, then we obtain the same outflux, i.e., holds. ˆτ(K, n) = τ(1, n), n 0,

25 Experiment 1: Verifying Theorem 2 Algorithm: Solve the transition times recursion; Compute the WIP W, the N-curve and the flux F ; Compare it to ρ, u and f computed from the solution of the hyperbolic equation. The hyperbolic problem for the approximate N-curve u is solved via a standard finite difference scheme of the form (a) u(x m, t n+1 ) = u(x m, t n ) + tf(x m, t n ), m = 1,..., M, t = t n+1 t n, X min{µ(x m 1, t), MT 0 x m 1 [u(x m, t) u(x m 1, t)]}, (b) f(x m, t) = m = 1,..., M, f A (t n ), m = 0

26 Experiment 1: Verifying Theorem 2 Chain of 3 suppliers with respective nodes S 0, S 1, S 2 ; Throughput times: T (0) = 1, T (1) = 3, T (2) = 1; Capacities C(0) = 15, C(1) = 10, C(2) = 15 parts per time unit; Create the regime of Theorem 2 by introducing virtual processors: split the nodes S 0 and S 2 into 10 virtual nodes and node S 1 into 30 virtual nodes. Set the service rates µ equal to the capacities, giving 15 for 0 < x < 0.2 µ(x, t) = 10 for 0.2 < x < for 0.8 < x < 1

27 Experiment 1: Verifying Theorem 2 We first solve the recursion for the transition times τ, starting with all empty queues, i.e., holds, and set τ I (M) = 0. τ I (m + 1) τ I (m) = T (m) = 0.1 We compute the arrival times randomly according to τ A (n + 1) τ A (n) = 1 f A (τ A (n)), τ A (0) = τ I (0) where f A (t) is a prescribed influx rate.

28 Experiment 1: Verifying Theorem 2

29 Experiment 1: Verifying Theorem 2

30 Experiment 1: Verifying Theorem 2

31 Experiment 1: Verifying Theorem 2

32 Experiment 1: Verifying Theorem 2

33 Experiment 1: Verifying Theorem 2

34 Experiment 2: long supply chain with unstructured throughput times and capacities

35 Experiment 2: long supply chain with unstructured throughput times and capacities

36 Experiment 2: long supply chain with unstructured throughput times and capacities

37 Experiment 2: long supply chain with unstructured throughput times and capacities

38 Experiment 2: long supply chain with unstructured throughput times and capacities

39 Experiment 2: long supply chain with unstructured throughput times and capacities

40 References [D. Armbruster, P. Degond and C. Ringhofer]: A model for the dynamics of large queuing networks and supply chains. SIAM J. Applied Mathematics, 66 (2006), pp [S. Goettlich, M. Herty, A. Klar]: Network models for supply chains. Communication in Mathematical Sciences, 3 (2005), pp [C. F. Daganzo]: A theory of supply chains. Lecture Notes in Economics and Mathematical Systems, 526. Berlin: Springer. viii, 123 p, [C. Ringhofer]: Kinetic and fluid dynamic models for complex supply network. chris/cime09/cime09.htm.