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 and identifying a sequence so as to minimize weighted earliness, tardiness and lead times costs. 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

11. Special Cases of the Early/Tardy Problem:

12. Heuristics for the Early/Tardy Problem Tardiness Heuristics Morton et al. (1984) on the weighted tardiness problem A myopic heuristic that attempts to achieve local optimality Job i immediately precedes job j when P ij (s i ) may be taken to be the priority of job i with respect to j at the earliest time the machine is free

13. Heuristics for the Early/Tardy Problem A dispatch priority rule was derived 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 priority function. 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 p i p j

18. Heuristics for the Early/Tardy Problem As in the weighted tardiness case, A simple dispatch rule may be obtained by comparing each job's priority to that of a job with average processing time and A look ahead parameter may be used to attempt to extend the scope of optimality beyond 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. Choice of k: k controls the time at which a job's priority begins to increase Therefore, when job due dates are close together and the lead times of jobs are not very long, a large look ahead k should be used A decision may then be made early enough to avoid the clash. In the case where due dates are evenly distributed, k should be small as few jobs will clash Heuristics for the Early/Tardy Problem

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

23. 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 Heuristics for the Early/Tardy Problem

24. 12345 Filter width (α) = 3 Beam width (β) = 2 Evaluated by Priority search

25. Heuristics for the Early/Tardy Problem 12345 Filter width (α) = 3 Beam width (β) = 2 The best three are selected and Evaluated by Probe search

26. Heuristics for the Early/Tardy Problem 12345 Filter width (α) = 3 Beam width (β) = 2 The best two are selected

27. Heuristics for the Early/Tardy Problem 12345 Filter width (α) = 3 Beam width (β) = 2 1 24 5 1 23 5 Evaluated by Priority search

28. Heuristics for the Early/Tardy Problem 12345 Filter width (α) = 3 Beam width (β) = 2 1 24 5 1 23 5 The best three are selected for each parent - Evaluated by Probe search

29. Heuristics for the Early/Tardy Problem 12345 Filter width (α) = 3 Beam width (β) = 2 1 24 5 1 23 5

30. 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) Computational Study

31. Processing times and due dates: A bivariate Normal distribution was used for processing times, due dates and the correlation between the processing times and due dates. Numbers drawn were rounded to the nearest integer. 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 0 and 0.5. Tardiness Factor, was set at 0.2 and 0.6. Computational Study

32. Tardy cost rate: w/p ~ uniform [0,5]. w i = (w/p) x p i. 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. Computational Study

33. A preliminary study of the performances of the three Beam Search methods discussed earlier was conducted using the 25-job problems with early-to-tardy cost rate ratio of 25%. The EXP-ET priority function was used for the priority 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 efficiency and solution quality. Computational Study

34. 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 Computational Study

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

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