Lot-sizing and scheduling with energy constraints and costs

1. Lot-sizing and scheduling with energy constraints and costs Journée P2LS "Lot-sizing dans l'industrie" LPI6 Paris 20 Juin 2014 Grigori German, Claude Lepape, Chloé Desdouits

2. Agenda Dealing with energy constraints and costs Scheduling versus lot-sizing A case study of manufacturing scheduling with energy costs Lot-sizing perspectives

3. Energy constraints and costs

4. Introduction Test Data Test Data Cost Time Day Night

5. Objectives • Determine whether it is worth considering energy costs in the planning and scheduling of a given factory or workshop • Determine what kinds of tradeoffs are worth considering between energy and: • Intermediate or final product inventory • Work shift organization • Other production costs • Tardiness risks • … • Determine what kinds of models and techniques can be used to answer the questions above • Process simulation • Scheduling with energy costs • Scheduling with energy (power) constraints, i.e., do not exceed a given power limit • Lot-sizing • … • Determine how generic can such models and techniques be?

6. Several questions Production planning and scheduling taking into account given energy tariffs • Reducing energy-intensive production during high-cost days and hours • Can mean different things: producing less, producing less energy-intensive products, avoiding energy-intensive steps, during the high-cost days and hours • Often impacting indirect CO2 emissions • Selecting or negotiating a better contract based on the energy-aware planning and scheduling capability • In particular concerning power subscription levels and penalties • Identifying demand-response opportunities • Maintaining a higher stock level to be able to reduce power consumption under rather short notice • When demand-response is “likely”

7. An example: the Sarel plant

8. Measuring chain Energy sensor • Self powered • Wireless communication • Non intrusive installation Accumulator • Provide the Energy value Collector transmitter • Send historical data periodically to the time series repository

9. SimulationOptimization Input Optimization Constraints • Based on a commercial production flow simulator (Rockwell Arena ) Cost Objectives Time Day Night

10. Scheduling versus lot sizing

11. Scheduling versus lot-sizing: differentiating questions What are the time scales? • Duration for the execution of a recipe or of its critical activities • Versus the frequency of tariff changes • What is the relationship between the critical resources time-wise and the critical resources energy-wise? • Do I have batch sizing flexibility and can it impact energy consumption? • Ovens, etc. • Energy-consuming setups / cleaning steps Cost Time Cost Time M1 M2 M1 M2

12. Three motivations for lot-sizing • To exploit batch sizing flexibility • As an abstraction of the scheduling problem • Less variables • Easiest constraints • … • As a tool to decompose the scheduling problem • Depending on the plant, coupled lot-sizing and scheduling can be the best solution

13. A case study of manufacturing scheduling with energy costs

14. Overview of the scheduling problem

15. Adding the energy dimension capacity calendar interval capr act2 cmax capacity cost interval Resr act1 act3 cmin et st time calendar

16. Optimization Input Optimization Constraints Cost Objectives Time Day Night

17. Method 1: Constraint Programming Simple, classical formulation Branching strategy: Earliest Due Date No simple formulation for computing the energy cost • Time-based formulation • Perspective: global constraint Generates a good first solution

18. Method 2: MIP How to express the energy cost? Overlap Variables Taille du bucket act dépasse à gauche Durée de act act dépasse à droite act et le bucket sont disjoints

19. Method 2: MIP How to express the energy cost? Constraints

20. Method 2: MIP Other constraints and variables • Disjunctive constraints: Applegate and Cook (1991) formulation Relaxed MIP • Too many variables and constraints (e.g., 700k+ variables and 1.2M+ constraints with 200 activities and a 400 days horizon) • Energy binary variables continuous in [0,1] • Stills leads CPLEX towards a good solution Perspectives • Explore different strategies (e.g., branch on all the variables before the energy variables) • Other formulations with precomputed intervals

21. Method 3: Hybrid local search Algorithm Perspectives • Adapted time windows size • Sliding time windows • Intensification • Local search • While there is still time • Find a time window F • Set all the variables outside F • Keep the best between S and S’ Constraint Programming S Optimize F with MIP S’ S

22. Comparison of the 3 methods without the energy Adapted benchmark instances from the literature CP, MIP & LS versus best known results

23. And with energy ? MIP versus LS Local search with and without energy

24. Scheduling perspectives Application to the SAREL use case Multi-objectives: Pareto-optimal schedules Piecewise linear energy costs

25. Lot-sizing perspectives

26. Models with lot-sizing periods corresponding to tariff intervals (buckets) • Makes sense only if recipe or critical activity execution duration is smaller than tariff intervals duration • Recipe-based model • Quantity of recipe r executed in period p for each period p and recipe r • Linked to the energy consumption in period p and hence to the energy cost (with a linear or piecewise linear relation between consumption and cost – could be subtle in some cases, e.g., if several resources in parallel consume and there is a penalty for exceeding a given amount of power …) • Linked to quantity of materials produced (or consumed) in period p • Linked to customer demands in different ways: either (i) no tardiness authorized with the risk that there is no solution, or (ii) delivering the demand when ready, either early or late, or (iii) delivering either just in time or late … • As a result linked to an evaluation of storage and tardiness costs • Activity-based model • For relevant activities of given batches, deciding in which period they execute • Variation (relaxation) of the model used in our scheduling study • With subtleties to look at when there are multiple energy-relevant activities or if the energy-relevant activity is not the bottleneck time-wise …

27. Models with lot-sizing periods exceeding or not consistent with tariff intervals Assuming recipe or critical activity execution duration is smaller than the lot-sizing period An open question is how to approximate the energy cost • An optimistic viewpoint assumes that inside each period we will be able to exploit intervals with the lowest tariffs, up to some given maximal power • Can we use historical data to better evaluate an expected cost? • Shall we do this through some smart coupling of lot-sizing and scheduling? Cost Cost 1 week period Max power = 10kW 88 hours at 0.05€/kWh 80 hours at 0.10€/kWh 1 week period Max power = 10kW 88 hours at 0.05€/kWh 80 hours at 0.10€/kWh (1680, 124) (1680, 124) (880, 44) (880, 44) (0, 0) (0, 0) Energy Energy

28. Conclusion Energy cost reduction is a growing concern • But usually one among multiple optimization criteria Multiple technical approaches and models can be considered Lot-sizing is one of them • Depending on time scales, relationships between the critical resources time-wise and the critical resources energy-wise, and on batch sizing flexibility • Sometimes (often) to be coupled with detailed scheduling • A very open topic at this point

29. Thank you for your attention!