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Heuristics for the Single Machine Early/Tardy Problem: Approaches and Analysis

This presentation by İrfan Mahmutoğulları and colleagues focuses on the single machine early/tardy scheduling problem, which aims to minimize penalties for early and tardy job completions. The authors review various heuristic approaches, including priority dispatch methods and search strategies like best-first and beam search. They analyze historical algorithms, evaluate new heuristics, and discuss the impact of due dates and processing times. This work provides insights into improving scheduling efficiency through informed decision-making processes in job sequencing.

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Heuristics for the Single Machine Early/Tardy Problem: Approaches and Analysis

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  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

  40. TeŞEKKÜRLER !

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