Optimization Techniques for Supply Network Models

1. Optimization Techniques for Supply Network Models • Simone Göttlich • Joint Work with: • M. Herty, A. Klar (TU Kaiserslautern) • A. Fügenschuh, A. Martin (TU Darmstadt) Workshop Math. Modelle in der Transport- und Produktionslogistik Bremen, 11. Januar 2008

2. Motivation Simulation and Optimization of a Production Network Typical questions: • Investment costs and rentability • Topology of the network • Production mix and policy strategies Simone Göttlich

3. Network - Model Modeling Optimization Extensions Existence Discrete Continuous Simone Göttlich

4. Main Assumptions No goods are gained or lost during the production process. The production process is dynamic. MODEL The output of one supplier is fed into the next supplier. Each supplier has fixed features. Simone Göttlich

5. Network Model Fixed parameters • Idea: • Each processor is described by one arc • Describe dynamics inside the processor • Add equations for queues in front of the processor queues Simone Göttlich

6. Network Model See Göttlich, Herty, Klar (2005):Network models for supply chains Definition A production network is a finite directed graph (E,V) where each arc corresponds to a processor on the intervall Each processor has an associated queue in front. Processor: Queue: Simone Göttlich

7. Network Coupling Definition We define time-dependent distribution rates for each vertex with multiple outgoing arcs. The functions are required to satisfy and Inflow into queue of arc 2 Example: Inflow into queue of arc 3 will be obtained as solutions of the optimization problem Simone Göttlich

8. Network - Model Modeling Optimization Extensions Existence Discrete Continuous Simone Göttlich

9. Optimization Problem Controls Positive weights Cost functional Constraints Processor Queue Initial conditions Simone Göttlich

10. Solution Techniques Optimal control problem with PDE/ODE as constraints! How to solve ? Mixed Integer Programming (MIP) Adjoint Calculus MIP LP Simone Göttlich

11. Network - Model Modeling Optimization Extensions Existence Discrete Continuous Simone Göttlich

12. Mixed Integer Program (MIP) See Fügenschuh, Göttlich, Herty, Klar, Martin (2006): A Discrete Optimization Approach to Large Scale Networks based on PDEs A mixed-integer program is the minimization/maximization of a linear function subject to linear constraints. Problem: Modeling of the queue-outflow in a discete framework Solution: Reduce complexity (as less as possible binary variables) Relaxed queue-outflow1 1See Armbruster et al. (2006): Autonomous Control of Production Networks using a Pheromone Approach Simone Göttlich

13. Derivation MIP Problem: Suitable discretization of the -nonlinearity Idea: Introduce binary variables (decision variables) Outflow of the queue = Inflow to a processor Example: implies and Simone Göttlich

14. Mixed Integer Program Two-point Upwind discretization (PDE) and explicit Euler discretization(ODE) leads to maximize outflux processor queue queue outflow initial conditions Simone Göttlich

15. Advantage MIP Constraints can be easily added to the MIP model: • Bounded queues: • Optimal inflow profile: In other words: Find a maximum possible inflow to the network such that the queue-limits are not exceeded. Simone Göttlich

16. Disadvantage MIP The problem has binary variables! • Goal • Reduce computational complexity • Avoid rounding errors in the CPLEX solver • Accelerate the solution procedure • How to do that? • Create a new preprocessing where PDE-knowledge is included • Compute lower und upper bounds by bounds strengthening Simone Göttlich

17. Network - Model Modeling Optimization Extensions Existence Discrete Continuous Simone Göttlich

18. Adjoint Calculus See Göttlich, Herty, Kirchner, Klar (2006): Optimal Control for Continuous Supply Network Models Adjoint calculus is used to solve PDE and ODE constrained optimization problems. Following steps have to be performed: 1. Define the Lagrange – functional: Cost functional PDE processor ODE queue with Lagrange multipliersand Simone Göttlich

19. Adjoint Calculus 2. Derive the first order optimality system (KKT-system): Forward (state) equations: Backward (adjoint) equations: Gradient equation: Simone Göttlich

20. Network - Model Modeling Optimization Extensions Existence Discrete Continuous Simone Göttlich

21. Linear Program (LP) Idea: Use adjoint equations to prove the reformulation of the MIP as a LP MIP LP Outflow of the queue Remove the complementarity condition! Remark: The remaining equations remain unchanged! Simone Göttlich

22. Numerical Results Example Solved by ILOG CPLEX 10.0 interior network points Simone Göttlich

23. Thank you! Simone Göttlich