Optimization of Supply Chain Networks

Transcription

1 Optimization of Supply Chain Networks M. Herty TU Kaiserslautern September 2006 (2006) 1 / 41

2 Contents 1 Supply Chain Modeling 2 Networks 3 Optimization Continuous optimal control problem Discrete optimal control problem MIP 4 Numerical Results Simulation Results Optimization Results Scaling effects on MIP Braun example 5 Summary (2006) 2 / 41

3 Contents Supply Chain Modeling 1 Supply Chain Modeling 2 Networks 3 Optimization Continuous optimal control problem Discrete optimal control problem MIP 4 Numerical Results Simulation Results Optimization Results Scaling effects on MIP Braun example 5 Summary (2006) 3 / 41

4 Supply Chain Modeling Starting point PDE model of Armbruster, Degond and Ringhofer t ρ + x min{µ(x, t), v(x, t)ρ} = 0 Assumption: Chain of processors x with capacity µ ρ part density ρ may contain δ distributions where µ is discontinuous (2006) 4 / 41

5 Supply Chain Modeling Simplification Assume µ(x, t) = µ(x) and µ(x) is piecewise constant. Each constant µ j corresponds to one fixed processor j, say on interval x I j Chain of processors with different capacities µ j Assume the same for v(x, t) Introduce the part density ρ j in processor j (2006) 5 / 41

6 Simplification (cont d) Supply Chain Modeling Part density ρ j in processor j satisfies t ρ j + x min{µ j, v j ρ j } = 0, x I j, j = 1, 2,..., with µ j, v j now constant! Consecutive processors j, j + 1 have to be coupled Situation: PDE model of ADR: Flow conservation min{µ 1, v 1 ρ 1 (b 1, t)} = min{µ 2, v 2 ρ 2 (a 2, t)} lead to δ distributions (interpreted as buffers) Modification: Introduce a new buffer variable t q 2 (t) min{µ 1, v 1, ρ 1 (b 1, t)} = min{µ 2, v 2 ρ 2 (a 2, t)} + t q 2 (t) (2006) 6 / 41

7 Supply Chain Modeling Simplification (cont d) t ρ j + x min{µ j, v j ρ j } = 0, min{µ 1, v 1, ρ 1 (b 1, t)} min{µ 2, v 2 ρ 2 (a 2, t)} = t q Freedom in specifying the buffer dynamics: Set v 2 ρ 2 (a 2, t) := min{µ 2 ; min{µ 1, v 1 ρ 1 }} if q(t) = 0 Buffer is empty: Process at most µ 2 but no more than the incoming flux Set v 2 ρ 2 (a 2, t) := µ 2 if q(t) > 0 Buffer is full: Process the most possible: µ 2. Further: v j ρ j µ j and dynamics in the processor is just transport (2006) 7 / 41

8 Supply Chain Modeling Relation between both approaches ADR model for the N curve t ū = min{µ(x), x ū} = 0 (for v = 1, ρ = x ū) Network model: Set u as antiderivative of ρ j plus buffers u(x, t) = x x 0 ρ j (ξ, t)χ Ij (ξ)dξ j j Formally, t u satisfies the same equation as t ū. q j (t)h(x a j ) Result is independent of the specific buffer dynamics and uses only conservation property (2006) 8 / 41

9 Supply Chain Modeling Model from a different point of view Assume initial data such that v j ρ j (x, 0) < µ j Transport inside the processor: t ρ j + x v j ρ j = 0, j = 1, 2,..., Between processor a buffering which stores or releases goods: t q j (t) = v j ρ j min{µ j 1, v j 1 ρ j 1 } Some release rule for goods: ( ) min{µj ; min{µ v j ρ j (a j, t) = j 1 ; v j 1 ρ j 1 }} q j = 0 µ j q j > 0 Regularization of release rule (Ringhofer et. al.) v j ρ j (a j, t) = min{µ j ; q j }, ɛ << 1 ɛ (2006) 9 / 41

10 Contents Networks 1 Supply Chain Modeling 2 Networks 3 Optimization Continuous optimal control problem Discrete optimal control problem MIP 4 Numerical Results Simulation Results Optimization Results Scaling effects on MIP Braun example 5 Summary (2006) 10 / 41

11 Networks Network model extends to networks Each outgoing processor has its own queue Notation δ indices of incoming, δ + indices of outgoing processors at a fixed vertex Release rule for goods? Take all inflow and distribute among the outgoing processors with some rate (2006) 11 / 41

12 Networks Network model extends to networks (cont d) A j is the percentage of the total influx going to processor j t q j (t) = v j ρ j A j (t) v j ρ j (a j, t) = min{µ j ; q j ɛ } A j fulfill j A j = 1 for all t > intersection: A j 1 i δ min{µ i, v i ρ i }, (2006) 12 / 41

13 Networks One slide on theory for network model If TV (ρ j (x, 0)) C for some C > 0 and if the network does not contain closed loops, then there exists a weak solution (ρ j, q j ) j on the network such that ρ j C 0,1 (0, T ; L 1 (a j, b j )) and q j W 1,1 (0, T ). Proof relies on wave/front tracking Given piecewise constant initial data (i.e. traveling fronts) Estimate the number of interactions of fronts Bound the TV after interaction gives compactness for ρ j Bound TV ( t q j ; [t 0, t]) yields compactness for t q j (2006) 13 / 41

14 Contents Optimization 1 Supply Chain Modeling 2 Networks 3 Optimization Continuous optimal control problem Discrete optimal control problem MIP 4 Numerical Results Simulation Results Optimization Results Scaling effects on MIP Braun example 5 Summary (2006) 14 / 41

15 Optimization Distribution of goods among copies of same processor Sample production of Braun, Germany Network sketch Optimize for distribution rates A j (t) for j = 3,..., 8 for given inflow profile at j = 1 (2006) 15 / 41

16 Optimization Continuous optimal control problem Mathematical formulation as optimal control problem Notation: j e (edge) and v for vertices, f e (ρ e ) := v e ρ e min A v,e (t) e T b e 0 t q e = A v,e (t) ē a e f e (ρ e )dxdt + T 0 q e (t)dt subject to t ρ e + x f e (ρ e ) = 0, f ē(ρē(bē, t) f e (ρ e (a e, t)), v e ρ e (a e, t) = min{µ e ; qe (t) } ɛ Missing: Initial data (= 0), inflow data, e Av,e = 1 (2006) 16 / 41

17 Optimization Derivation of the optimality system Continuous optimal control problem Set: h e ( ρ e, A v ) := A v,e (t) ē f ē(ρē(bē, t) for e δ v + \{ẽ}, and hẽ( ρ e, A ( v ) := 1 ) e ẽ Av,e (t) f ē(ρē(bē, t)). ē Finally: min A v,e (t) e subject to T b e 0 a e v e ρ e dxdt + T 0 q e (t)dt t ρ e + x v e ρ e = 0, ρ e (x, 0) = 0, v e ρ e (a e, t) = min{µ e ; qe (t) }, ɛ t q e = h e ( ρ e, A v ) v e ρ e (a e, t), q e (0) = 0 (2006) 17 / 41

18 Optimization Continuous optimal control problem Optimality system is the stationary point of the Lagrangian Formally: L( ρ e, A v, q e, Λ e, P e ) = e A T e A 0 T b e 0 T b e e A 0 a e a e v e ρ e dxdt + T 0 q e dt Λ e t ρ e + Λ e v e x ρ e dxdt ( P e t q e h e ( ρ e, A ) v ) + min{µ e, q e /ɛ} dt Multipliers for part densities are Λ e Multipliers for the queues are P e Stationary point: ρ e L = A v L = Λ e L = P e L = 0 (2006) 18 / 41

19 Optimization Continuous optimal control problem Optimality system is a coupled system of pdes and odes t ρ e + v e x ρ e = 0, ρ e (x, 0) = 0, v e ρ e (a, t) = min{µ e, q e /ɛ}, t q e = h e ( ρ e, A v ) v e ρ e (a, t), q e (0) = 0, t Λ e v e x Λ e = v e, Λ e (x, T ) = 0, v e Λ e (b, t) = Pē(t) hē( ρ e, A v ), ρē ē δ v + s.t. e δv t P e = 1 (P e Λ e (a, t)) 1 ɛ H(µe q e /ɛ), P e (T ) = 0, e δ + v P e A v,ē he ( ρ e, A v ) = 0. (2006) 19 / 41

20 Optimization Continuous optimal control problem Remarks on the optimality system Equation for Λ e is backwards in time and space and network State equation: h 2 ( ρ e, A v ) = A v,2 (t)v 1 ρ 1 (b, t), h 3 ( ρ e, A v ) = (1 A v,2 (t))v 1 ρ 1 (b, t) Adjoint equation: v 1 Λ 1 (b, t) = A v,2 p 2 (t)v 2 + (1 A v,2 )p 3 (t)v 3. Gradient equation: ( p 2 (t) p 3 (t) ) v 1 ρ 1 (b, t) = 0 (2006) 20 / 41

21 Optimization Continuous optimal control problem Solving the optimality system Discretize then optimize here: Discretization can be chosen such that the problem is a mixed integer problem Optimize then discretize Several choices possible: Iterative solution of the optimality system, Newton type method for the full system, gradient methods for reduced cost functional,... (2006) 21 / 41

22 Optimization Discrete optimal control problem MIP Basic requirements for a mixed integer model The cost functional has to be linear in the unknowns All appearing equations have to be linear and will give equality constraints to a linear program Only box constraints are admissible Nonlinearities appearing have to linearized using binary variables Products of unknowns are not allowed (2006) 22 / 41

23 Optimization The details of the discretization Discrete optimal control problem MIP Recall the control problem min A v,e (t) e T b e 0 a e v e ρ e dxdt + T 0 q e (t)dt, subject to t ρ e + x v e ρ e = 0, ρ e (x, 0) = 0, v e ρ e (a e, t) = min{µ e ; qe (t) }, ɛ t q e = h e ( ρ e, A v ) v e ρ e (a e, t), q e (0) = 0 Trapezoidal rule for cost functional Two point Upwind discretization of the pde Explicit Euler discretization of the ode Reformulation of the min using binary variables Reformulation of function h e which contains products A v,ē e v e ρ e (2006) 23 / 41

24 Optimization Details of the discretization Discrete optimal control problem MIP Variables ρ e a,t, ρ e b,t, qe t and ht e (explained next) ) ρ e b,t+1 = ρe b,t + t b e a v (ρ e e e a,t ρ e b,t Alternatively: Just delay relation ρ e b,t = ρe a,t (b a)/v q e t+1 = qe t + t(h e t v e ρ e a,t) Box constraints 0 ρ e a/b,t µe and 0 q e t t has to satisfy CFL condition and stiffness of ode: t = min{ɛ; (b e a e )/v e : e} Implicit discretization also possible (2006) 24 / 41

25 Optimization Reformulation of the function h e Discrete optimal control problem MIP Continuous formulation: t q e = h e ( ρ e, A v ) v e ρ e (a e, t) Discretized formulation: q e t+1 = qe t + t(h e t v e ρ e a,t) h e ( ρ e, A v ) = A v,e (t) ē f ē(ρē(bē, t)) for e δ v + \{ē} and hẽ( ρ e, A ( v ) := 1 ) e ẽ Av,e (t) f ē(ρē(bē, t)). ē Replaced by the under determined linear equations for all t and e : ht e = v e ρ e b,t e δ + v e δ v The optimization determines the exact values of h e t (2006) 25 / 41

26 Discretization of the min Optimization Discrete optimal control problem MIP Continuous formulation: v e ρ e (a e, t) = min{µ e ; qe (t) ɛ } Discrete formulation for M sufficiently large and binary variables ξ e t {0; 1} µ e ξ e t v e ρ e a,t µ e, q e t ɛ Mξe t v e ρ e a,t qe t ɛ, µ e ξ e t q e t ɛ ξ e t determined by q e t /ɛ and µ e µ e (1 ξ e t ) + Mξ e t If q e t /ɛ > µ e then ξ e t = 1 (otherwise right part of last inequality violated) If ξ e t = 1, then v e ρ e a,t = µ e by first inequality For the case q e t /ɛ < µ e we obtain ξ e t = 0 (2006) 26 / 41

27 Optimization Discrete optimal control problem MIP Discretized mixed integer problem (MIP) min ρ e a/b,t, qe t, h e t, ξ e t v e ( ρ e a,t + ρ e b,t) subject to, e t µ e ξ e t v e ρ e a,t µ e, q e t ɛ Mξe t v e ρ e a,t qe t ɛ, µ e ξt e qe t ɛ µe (1 ξt e ) + Mξt e, ht e = v e ρ e b,t, e δ + v e δ v q e t+1 = q e t + t(h e t v e ρ e a,t), 0 q e t, 0 ρ e a/b,t µe (2006) 27 / 41

28 Optimization Discrete optimal control problem MIP Problems and expectations Number of real variables is 4 N t N P, N t N P binary variables for N T time steps and N P arcs Usually, branch and bound methods used to solve MIP Sophisticated branch and bound methods for MIP exist however usually no information on the structure is used Drawback of MIP: No theory on convergence rates and complexity for general problems In particular: Dependence on data like µ e, v e is not predictable (2006) 28 / 41

29 Optimization Discrete optimal control problem MIP Related approaches SLP methods for finite dimensional constrained optimization problems Predecessor of the SQP methods commonly used today because of quasi linear convergence Optimal routing in congested traffic networks, c.f. Möhring et. al., TU Berlin used in DaimlerChrysler routing systems Optimization of simplified gas dynamics, c.f. Martin et. al., TU Darmstadt, still work in progress Water contamination detection and traffic flow (2006) 29 / 41

30 Optimization Discrete optimal control problem MIP Model extensions in the Mixed Integer formulation Bounded queues: q e t M Maintenance shutdown for N consecutive times t: φ e t {0; 1}, t φe t = 1 and ht+l e M(1 φ e t ) l = 0,..., N 1 Optimize for the inflow profile for given total number of goods: t f 0 t = N (2006) 30 / 41

31 Contents Numerical Results 1 Supply Chain Modeling 2 Networks 3 Optimization Continuous optimal control problem Discrete optimal control problem MIP 4 Numerical Results Simulation Results Optimization Results Scaling effects on MIP Braun example 5 Summary (2006) 31 / 41

32 Numerical Results Simulation Results Simulation results for network model Density and queues for chain of processors with transient inflow temporarily exceeding µ 1, µ 1 > µ 2 (2006) 32 / 41

33 Numerical Results Simulation Results Computing times for network model and discrete event simulator DES: (1.2): generating initial conditions; (1.3) computing recursion τ(m, n); (1.4) evaluation of Newell curves K is the number of consecutive processors, n number of parts (2006) 33 / 41

34 Numerical Results Optimization Results Dependence of cost functional on distribution rates Shown is e v e ρ e dxdt + q e dt for constant distribution rates α i (2006) 34 / 41

35 Numerical Results Scaling effects on MIP CPU times for MIP solution for chain of processors Chain of N P processors; discretization with N T time steps; sinusoidal inflow prescribed MIP contains N T N P binary and 3N T N P real variables (2006) 35 / 41

36 Numerical Results Scaling effects on MIP CPU times for MIP solution for highly connected networks X nodes in x direction, N T = 100 in first and X Y = 4 3 in second table (2006) 36 / 41

37 Numerical Results Braun example Effect of maintenance of processor three in Braun example Time evolution of buffers 2, 3 and 4 is shown (2006) 37 / 41

38 Numerical Results Braun example Effect of maintenance of processor three in Braun example Optimal time dependent distribution rate 2 3 with maintenance in at least 20 time intervals (2006) 38 / 41

39 Contents Summary 1 Supply Chain Modeling 2 Networks 3 Optimization Continuous optimal control problem Discrete optimal control problem MIP 4 Numerical Results Simulation Results Optimization Results Scaling effects on MIP Braun example 5 Summary (2006) 39 / 41

40 Summary Summary & Outlook Supply chain model for piecewise constant capacities Extension to networks Optimization by coarse grid discretization and reformulation as MIP Extension of optimization to priority models Currently: Optimization by black box MIP solver Multiple product flows on networks Stochastic effects (2006) 40 / 41

41 Summary Presented results are joint work with A. Fügenschuh, S. Göttlich, A. Klar and A. Martin. Thank you for your attention. (2006) 41 / 41