1. MILP Approach to the Axxom Case Study Sebastian Panek

2. Introduction What is this talk about? MILP formulation for the scheduling problem provided by Axxom (lacquer production) What‘s new since our meeting in Sept. 02? Improved model and solution procedure, new results What about modeling TA as MILP? This work is still in progress...

3. Overview Short problem description MILP formulation Solution procedure Emprical studies Conclusions

4. Short problem description

5. Additional problem characteristics Additional restrictions for pairs of tasks: –start-start restrictions –end-start restrictions –end-end restrictions Parallel allocation of mixing vessels Machine allocation allowed interval

6. Problem simplifications Labs are non-bottleneck resources, no exclusive resource allocation is needed (provided by Axxom) Individual colors for lacquers => many different products –No batch merging is possible Only few jobs exceeding max. batch capacity –Batch splitting is not considered

7. General approaches For short-term scheduling problems in the processing industry [Kondili,Floudas, Pantelides, Grossmann,...]: –State Task Networks (STN) –Resource Task Networks (RTN) Early formulations: discrete time Recent work: continuous time Task 1 Task 2 State A State B State C 1 1 1 1

8. General approaches (2) Advantages: –Batch splitting/merging –Mass balances –Individual modeling of products –Restrictions on storages Disadvantages: –Continuous and discrete time models tend to require many points of time, number difficult to estimate –Very detailed view of the problem not always necessary Problem: large models, difficult to solve

9. Our approach: sequencing based continuous time model Continuous time Individual representation of time for machines Focused on tasks and machines Products (states) are not considered explicitly Fixed batch sizes (no merging and splitting of batches) Grows according to the number of tasks and not to the time horizon

10. MILP formulation of the continuous time model Real variables for starting and ending times of tasks Binary variables for the machine allocation –task i is processed on machine k : Binary variables for the sequencing of tasks –task i is processed before task h on machine k :

11. Starting and ending times for allocated machines Starting and ending dates for tasks i on machines k Extra linear equations are needed to express nonlinear products of binary and real variables

12. Restrictions on binary variables Each task must be processed on 1 machine If both tasks i and h are processed on machine k then either i is scheduled before h or vice versa

13. Sequencing restrictions Tasks i, h processed on the same machine k must not overlap each other Set iff task i is finished before task h

14. Objective function Minimize too late and too early job completions

15. Additional heuristics 1.Non-overtaking of non-overlapping jobs 2.Non-overtaking of equal-typed jobs (M. Bozga) 3.Earliest Due Date (EDD)

16. 2-step solution procedure 1.Apply heuristics 3 (EDD) by fixing some variables 2.Solve the problem 3.Relax and fix some variables according to heuristics 1+2 4.Solve the problem again reusing previous solution as initial integer solution

17. How is the model influenced by the heuristics? N: #Tasks, M: #Machines Most binary variables are variables. Worst case: # variables = O(N 2 M) (!!!) (i,h=1...N, k=1...M) # real variables = ~2NM But: When using heuristics, many binary variables are fixed and disappear from the model. We want to reduce O(N 2 M) to O(NM)! How that?

18. A little example: 1 machine, 4 jobs, 1 task/job Job#1234 Release0113 Deadline2345 Type1122 *** *** *** *** Matrix of variables i=1 2 3 4h=123 4

19. Heuristics #1 non-overlapping jobs Job#1234 Release0113 Deadline2345 Type1122 *1* *1* *** 00* Matrix of variables i=1 2 3 4h=123 4

20. Heuristics #2 equal-typed jobs Job#1234 Release0113 Deadline2345 Type1122 11* 01* **1 000 Matrix of variables i=1 2 3 4h=123 4

21. Heuristics #2 EDD Job#1234 Release0113 Deadline2345 Type1122 111 011 001 000 Matrix of variables i=1 2 3 4h=123 4

22. Empirical studies on the Axxom Case Study model scaled from 4 up to 29 jobs Jobs in job table sorted according to deadlines 2-stage solution procedure (heuristics 3, 1+2) CPU usage limited to 20+20 minutes Measurement of solution time, equations, real and binary variables, objective values and bounds Software: GAMS+Cplex Hardware: 1.5 GHz Athlon, 1 GB Ram

23. Objective values lower bounds integer solutions gap

24. Solution times 20 min. limit was active for >10 jobs

25. Variables and Equations equations total variables binary variables ~50% of all Variables!

26. Gantt chart: 29 jobs 2h of computation time, first integer solution after few min.(node 173)

27. 22 jobs, moving horizon procedure Horizon: 7 jobs, 16 steps a 25 minutes, 300 MHz machine

28. Conclusions from empirical studies EDD heuristics at 1. stage helps finding integer solutions quickly (even for large instances!) 2. stage usually cannot find better solutions (in short time)... but the number of binary variables is significantly reduced from O(N 2 M) to O(NM) without restricting the problem too much for <20 jobs very good gaps can be expected in short time first integer solutions within few minutes for the 29 jobs instance efficiency comparable to TA model from M. Bozga (VERIMAG)... but quantitative infos about integer solutions from the gaps A decomposition strategy helps improving the efficiency and the quality of results