1. Efficient Approaches to Scheduling
for Unrelated Parallel Machines with Release Dates
Tatiana Avdeenko, Yury Mezentsev Economic Informatics Department
Novosibirsk State Technical University
Novosibirsk, Russia
8th IFAC Conference on Manufacturing, Modelling, Management, and Control (MIM 2016), Troyes, France

2. Mathematical Formulation of the Problem
Contents
State of Art
Mathematical Formulation of the Problem
Approaches to solving the problem
Algorithm of Dynamical Programming with Sifting Variants
Example
Comparison of the approaches
Conclusion
8th IFAC Conference on Manufacturing, Modelling, Management, and Control (MIM 2016), Troyes, France

3. identical (with the same speed pj of processing for each job j),
State of Art
In parallel scheduling problem a set J of n jobs has to be scheduled on m parallel machines.
The machines can be
identical (with the same speed pj of processing for each job j),
uniform (with different speeds vi)
unrelated (with different time pij of processing the job j on the machine i).
Although there is extensive literature on parallel machine scheduling problem, overwhelmingly majority of papers is limited to situations in which processing times or speed rates are the same across all machines.
Kashan, A. H. and Karimi, B. (2009). A discrete particle swarm optimization algorithm for scheduling parallel machines, Computers and Industrial Engineering, vol. 56, no. 1, pp. 216–223.
Pfund, M., Fowler J.W., and Gupta J.N.D. (2004). A survey of algorithm for single and multi-objective unrelated parallel-machine deterministic scheduling problems. Journal of the Chinese Institute of Industrial Engineers. Vol. 21, pp. 230–241
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4. State of Art
We consider the most difficult statement of unrelated parallel machines problem without preemptions and with release dates when the processing of job j cannot be started before its deterministic release date rj. As a performance measure we consider a makespan Cmax=max{C1,C2,...,Cn}, where Cj is the completion time of job j. Minimizing makespan not only completes all jobs as quickly as possible but also allow to maximize the utilization of machines. Corresponding to classification of (2008), the problem is referred to as R| rj | Cmax that is NP-hard in general.
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5. To the best of our knowledge we know only few papers devoted to
State of Art
To the best of our knowledge we know only few papers devoted to
R | rj | Cmax , such as
Lin, Y.K. , Hsieh, H.T. and Hsieh, F.Y. (2012). Unrelated Parallel Machine Scheduling Problem Using an Ant Colony Optimization Approach. World Academy of Science, Engineering and Technology, vol. 6, pp. 1798–1803.
Lin Y. K. (2013). Particle Swarm Optimization Algorithm for Unrelated Parallel Machine Scheduling with Release Dates. Mathematical Problems in Engineering, Volume 2013, Article ID , 9 p.
Bank, J., and Verner F. (2001).Heuristic algorithms for unrelated parallel machine scheduling with a common due date, release dates, and linear earliness and tardiness penalties. Mathematical and Computer Modelling, Vol. 33(4-5), pp
Gharbi A. and Haouari M. (2002). Minimizing Makespan on Parallel Machines Subject to Release Dates and Delivery Times. Journal of Scheduling, Vol, 5(4), pp
8th IFAC Conference on Manufacturing, Modelling, Management, and Control (MIM 2016), Troyes, France

6. State of Art
In Lin (2013) a heuristic and a very effective particle swarm optimization (PSO) algorithm have been proposed to tackle the problem. Proposed heuristic was compared with the heuristic initially offered for identical parallel machines. The performance of PSO algorithm has been analyzed for the case small problem instances (18 jobs on 4 machines) and for the case of large problem instances (100 jobs on 10 machines).
In our we proposed efficient approaches that has better performance for large dimensions of R | rj | Cmax problem, that outperforms the PSO algorithm in Lin (2013) for the case of large problem instances. We have obtained promising results for the case J=100 and I=5 ,10 and 30.
8th IFAC Conference on Manufacturing, Modelling, Management, and Control (MIM 2016), Troyes, France

7. Mathematical formulation of the problem,
Let be the initial job schedule
(job schedule in the input of the system).
Let be Boolean variables if the job j is assigned to the machine i;
be the time of processing the job j by the machine i ( )
be the final schedule.
Now we can formulate the following mathematical model for the problem: ,
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8. 8th IFAC Conference on Manufacturing, Modelling, Management, and Control (MIM 2016), Troyes, France

9. 8th IFAC Conference on Manufacturing, Modelling, Management, and Control (MIM 2016), Troyes, France

10. Basic difficulty in solving the problem (1)-(8) is presence of Boolean variables, recursive functions and knapsack restrictions, defining the problem as NP-hard.
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11. Transformation (1)-(8) to the mixed integer linear programming through expanding recursive functions
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12. continuous variables for compensation of negative delays.
where
continuous variables for compensation of negative delays.
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13. Approach based on solving relaxed two-criterion problem
Another possible approach consists in simplifying the conditions of the initial problem (1)-(8). Instead of the conditions (4)‑(7) containing recursive functions, two criteria are introduced in the following way:
This assumes compromise solution both on the “pure” speed without accounting delays and on the uniform distribution of the delays between the machines. Then Pareto-optimal solution of (1)-(3), (18)-(21) is determined, all components of which are admissible for (1)-(8) and (9)-(17). The best schedule is separated from the Pareto-optimal solution.
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14. Dynamical programming method with sifting variants
Since the number of variants of intermediate schedules doubles at each stage, we propose sifting a half of locally worth variants at each stage starting with th.
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15. Algorithm A.
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16. Illustrative example TABLE I. Initial data Number of job (j)
Processing time
Delay of entering the job ( )
machine number(i) 1 2 4 3 5 6 7
TABLE II Results of the first stage
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17. TABLE III. Results of the second stage
TABLE IV Results of the third stage and sifting of variants
TABLE IV (continuation) 1
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18. TABLE V. Results of the fourth stage and sift of variants
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19. TABLE VI. Results of the fifth stage and sift of variants
TABLE VI (continuation)
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20. 1 TABLE VII. Results of the sixth stage and sift of variants
TABLE VII (continuation) 1
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21. TABLE VШ. Results and selection of the best variant
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22. Estimates of complexity of algorithm A and dynamical programming for the considered example are
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23. Some performance results of the approaches
makespan for the compromise solution (1)-(3),(18)-(21), obtained by special decomposition algorithm
and for algorithm A of dynamic programming with sifting variants
(in sec) is computation time for the decomposition algorithm, and for algorithm A
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24. Conclusion
Thus, the main result of the present paper is an efficient hybrid polynomial algorithm for optimal scheduling of unrelated parallel machines and give preliminary test of its effectiveness. The algorithm is based on the scheme of dynamical programming with sifting of locally worse variants at each stage.
By empirical way we have obtained the best number of iteration on which we begin to sift variants. There is still a lot of research, but it is already that the method shows promising results. We have the best time of computation and good closeness to optimal solution for large scale problems.
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25. Future work
One of the approaches to flexible job shop problem is decomposition of general task and separating local subproblems determining the schedule for appropriate parallel subsystem.
It can be shown that if we know schedules for all the subsystems of the whole system at each time moment, then it is possible to solve general flexible job shop problem by successive approximations, solving coordinating task at each stage. That is why synthesis of schedules for unrelated parallel systems with release dates is a key subproblem in solving general flexible job shop problem.
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26. Thank you for your attention !
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