The Single machine early/Tardy Problem* PENG si ow & thomas e. Morton

1. The Single machine early/Tardy Problem*PENG si ow & thomas e. Morton IE 573- Paper Presentation A. İrfan Mahmutoğulları *Ow, P. S., & Morton, T. E. (1989). The single machine early/tardy problem. Management Science, 35(2), 177-191.

2. Introduction

3. Introduction

4. Introduction • Heuristics to obtain good solutions to the problem • Dispatch method: Whenever a machine is free a priority function selects the next job • MRV (Morton, Rachamadugu and Vepsalainen 1984) • Earliest Due Date • LIN-ET • EXP-ET • Filtered Beam Search

5. Background • Sidney (1977): minimizing maximum job penalty (early or tardy) • Lakshminarayan et al. (1978) later provided an O(n log n) algorithm for this problem. • Seidmann et al. (1981) considered the problem of assigning individual job due dates andidentifying a sequence so as to minimize weighted earliness, tardiness and lead timescosts. All jobs had the same weights.

6. Background • Search Techniques: • Best-first search and depth-first search • Barr and Feigenbaum (1981) • Lawler and Woods (1966) • Nilsson(1980) • Baker (1974): Neighborhood search • Lowerre (1976): Beam search

7. Analysis of the Early/Tardy Problem

8. Analysis of the Early/Tardy Problem

9. Analysis of the Early/Tardy Problem

10. Analysis of the Early/Tardy Problem

11. Analysis of the Early/Tardy Problem • Special Cases of the Early/Tardy Problem:

12. Heuristics for the Early/Tardy Problem • Tardiness Heuristics • Morton et al. (1984) on the weighted tardinessproblem • A myopic heuristicthat attempts to achieve local optimality • Job i immediately precedes job j when • Pij(si) may be taken to be the priority of job i with respect to j at the earliest time themachine is free

13. Heuristics for the Early/Tardy Problem • A dispatch priority rule wasderived by comparing each job's priority to an average job with processing time

14. Heuristics for the Early/Tardy Problem • However, local optimality is far away from global optimality due to «clashes» between multiple jobs.

15. Heuristics for the Early/Tardy Problem • This insight led to the addition of a look ahead parameter, k to the priorityfunction. The resulting function is: • Morton et al. (1984)experimented with other functions to find a better approximation

16. Heuristics for the Early/Tardy Problem • Linear vs. Exponential priority rules for tardiness problem:

17. Heuristics for the Early/Tardy Problem • Early/Tardy Heuristics • Following Morton et al. (1984) • If (1) is divided by pipj

18. Heuristics for the Early/Tardy Problem • As in theweighted tardiness case, • A simple dispatch rule may be obtained by comparing each job'spriority to that of a job with average processing time and • Alookahead parametermay be used to attempt to extend the scope of optimalitybeyond two adjacent jobs. • Linear priority rule:

19. Heuristics for the Early/Tardy Problem • Exponential priority rule:

20. Heuristics for the Early/Tardy Problem • Linear vs. Exponential priority rules for early/tardy problem:

21. Heuristics for the Early/Tardy Problem • Choice of k: • k controls the time at which a job's prioritybegins to increase • Therefore, when job due dates are closetogether and the lead times of jobs are not very long, a large lookahead k should be used • A decision may then be made early enough to avoid the clash. In the case wheredue dates are evenly distributed, k should be small as few jobs will clash

22. Heuristics for the Early/Tardy Problem • Beam search methods • The goodness of each partial sequence is estimatedusing a function known as an «evaluation function» and the «best» two sequences areselected

23. Heuristics for the Early/Tardy Problem • Evaluation Function • Priority search Priority of last job added to the sequence is used • Probe search Schedule cost is estimated for each node • Filtered beam search Priority search + Probe search

24. Heuristics for the Early/Tardy Problem Filter width (α) = 3 Beam width (β) = 2 1 2 3 4 5 Evaluated by Priority search

25. Heuristics for the Early/Tardy Problem Filter width (α) = 3 Beam width (β) = 2 1 2 3 4 5 The best three are selected and Evaluated by Probe search

26. Heuristics for the Early/Tardy Problem Filter width (α) = 3 Beam width (β) = 2 1 2 3 4 5 The best two are selected

27. Heuristics for the Early/Tardy Problem Filter width (α) = 3 Beam width (β) = 2 1 2 3 4 5 5 2 4 1 1 5 2 3 Evaluated by Priority search

28. Heuristics for the Early/Tardy Problem Filter width (α) = 3 Beam width (β) = 2 1 2 3 4 5 5 2 4 1 1 5 2 3 The best three are selected for each parent - Evaluated by Probe search

29. Heuristics for the Early/Tardy Problem Filter width (α) = 3 Beam width (β) = 2 1 2 3 4 5 5 2 4 1 1 5 2 3

30. Computational Study • Design of the experiment • Tardiness factor (coarse measure of the proportion of the jobs that might be expected to be tardy in an arbitrary sequence) • Due date range (controls the range of the due date distribution)

31. Computational Study • Processing times and due dates: • A bivariate Normal distribution was used for processing times, due dates and the correlationbetween the processing times and due dates. • Numbers drawn were rounded to the nearestinteger. • Population mean for processing times was 15. • Coefficient of variation for the processing times, (std. dev./ mean), was 0.2. • Due dates range factor, R, was set at 0.4 and 1.0. • Correlation coefficient between processing times and due dates, ρ, was set at 0and 0.5. • Tardiness Factor,was set at 0.2 and 0.6.

32. Computational Study • Tardy cost rate: • w/p ~ uniform [0,5]. • wi= (w/p) xpi. • Early cost rate. • h / w was set at 25%, 10% and 5 • Number of jobs in each set of tests, n. 8, 15, and 25. • Twenty test problems were generated for each combination of test parameter settings,giving a total of 1440 test problems.

33. Computational Study • A preliminary study of the performances of the threeBeam Search methods discussed earlier was conducted using the 25-job problems withearly-to-tardy cost rate ratio of 25%. • The EXP-ET priority function was used for thepriority evaluation and to perform the probe in the cost evaluation. • Based on this study,Filtered beam search was determined to dominate the others in terms of search efficiencyand solution quality.

34. Computational Study • Performance = (Cost of Heu. – OPT or LB cost) / OPT or LB cost • Optimal solutions are obtained via Branch-and-Bound • 8 job and (some) 15 job instances • LBs are obtained by breaking each jobs that can be solved as assignment problem • (some)15 job and 25 job instances • Lower bounds were found quite tight

35. Computational Study • Effect of k parameter on EXP-ET • When the due daterange was wide, larger look aheads degraded performance • When the range was narrow and tardiness factor was high performance improved as k increased

36. Computational Study

37. Computational Study

38. Computational Study

39. Computational Study

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