Lecture. Simulation and optimization

Transcription

1 Course Simulation Lecture Simulation and optimization 1 4/3/2015 Simulation and optimization

2 Platform busses at Schiphol Optimization: Find a feasible assignment of bus trips to bus shifts (driver and bus) such that our predefined robustness measure is maximal. Simulation: Evaluate the day of operation for a given planning with stochastic disturbances 2 4/3/2015 Simulation and optimization

3 Optimization vs Simulation Optimization: Decision variables Objective function Constraints (usually deterministic data) Simulation: Scenarios, decision parameters Performance measures States, events, event-handling, uncertainty 3 4/3/2015 Simulation and optimization

4 Orthopedia Capacity: B beds nurses on day time in operation theatre on day t 15 types of identical patients. Patient of type i has Known operation time Known required nursing pattern Needs a bed during stay in hospital Find: Cyclic patient admission profile over the week Number of patients of type per week between and Objective: Equally divide load over time 4 4/3/2015 Simulation and optimization

5 Optimization vs Simulation Optimization find optimal or very good solution considers large number of alternatives high level model with aggregated data. Simulation: evaluation a few alternatives more details, more modelling flexibility uncertainty included more easily 5 4/3/2015 Simulation and optimization

6 Combining simulation and optimization 1. Evaluate optimization solution (or similar solutions) by simulation 2. Iterative approach 1. Evaluate optimization solution (or similar solutions) by simulation 2. Use simulation result to change objective function or constraints, go to 1 3. Combined optimization and simulation problem 6 4/3/2015 Simulation and optimization

7 Inventory system Single product Time between demands: exponential with mean 0.1 month Demand: 1 w.p. 1/6 2 w.p. 1/3, 3 w.p. 1/3, 4 w.p. 1/6 Lead time Uniform[0.5 month, 1 month] 7 4/3/2015 Simulation and optimization

8 Inventory system (2) Cost: Ordering cost for quantity Z: Z Holding cost: 1 EURO per item per month Backlogging: shortage cost 5 EURO per item per month (r,q) strategy: order q as soon as I r 8 4/3/2015 Simulation and optimization

9 Inventory system (3) Decision variables: (r,q) Objective: total cost are minimal Constraints: 0 r, q r+q warehouse capacity Supplier has production/transportation capacity to deliver q Total cost have to be computed by simulation 9 4/3/2015 Simulation and optimization

10 10 4/3/2015 Simulation and optimization Combined simulation and optimization problem Decision variables: input factors x 1, x 2,,x k for simulation Objective function (simulation result) Min f(x)=e(r(x 1,,x k )) where R(x 1,,x k ) simulation performance measure Constraints e.g.: p k pk p k k k k k c x a x a c x a x a u x l u x l L M L K ,,

11 Continuous variables Stochastic approximation (SA, gradient search): x = Π( x α ˆ R( x ( n+ 1) ( n) ( n) n Response surface = meta-model (inventory example r 0.22 q) Metaheuristics (local search) )) 11 4/3/2015 Simulation and optimization

12 Gradient search Contour plot 12 4/3/2015 Simulation and optimization

13 Discrete variables Small number: statistical selection Metaheuristics (local search methods) Nested Partition 13 4/3/2015 Simulation and optimization

14 Local search: iterative improvement 1. Determine starting solution x(start) 2. Set current solution x = x(start) 3. Determine new solution neighbour(x) 4. If neighbour(x) is better than x, set x = neighbour(x) and go to step 3 5. Else STOP 14 4/3/2015 Simulation and optimization

15 Local search: simulated annealing P 1. Determine starting solution x(start) and starting temperature T(start), set k=0 2. Set current solution x = x(start) 3. If k = k(temp-decrease), decrease T to αt and set k=0, otherwise k=k+1, 4. Determine new solution neighbour(x) 5. Set x = neighbour(x) with probability e ( n ) ( accept neighbour ( x)) = R( neighbour ( x)) R( x ) T 1 if R( neighbour ( x)) otherwise R( x) 6. If x is better than the best solution so far x best, set x best = x 6. Go to Step3, unless stopcriterium is met 15 4/3/2015 Simulation and optimization

16 Local search: tabu search Maintain tabu-list, e.g. the last 7 accepted solutions x is your best neighbor, accept x unless it is tabu. If tabu, try something else. 16 4/3/2015 Simulation and optimization

17 Local search: genetic algorithm Population of solutions Iteration 1 Cross-over Mutation Iteration 2 Do many iterations: Remember the best New population of Solutions (same size) 17 4/3/2015 Simulation and optimization

18 NP+SSM+HC (Pichitlamken and Nelson 2003) Initialisation: Initial solution Most Promising Region is all feasible solutions Iteration: 1. Partition Most Promising Region 2. Find random solutions in regions: MIX-D and MIX-DS 3. Sequential Selection with Memory 4. Hill Climbing 5. If best solution in MPR go to 1, otherwise backtrack 18 4/3/2015 Simulation and optimization

19 NP (nested partition) 19 4/3/2015 Simulation and optimization

20 MIX-D: find random solutions in given region 20 4/3/2015 Simulation and optimization

21 NP+SSM+HC Outperforms simulated annealing in terms of solution quality NP: convergence SSM: prevent selection error HC: strengthens searching 21 4/3/2015 Simulation and optimization