the single machine early/tardy problem* peng si ow & thomas e. morton

THE SİNGLE MACHİNE EARLY/TARDY PROBLEM*

PENG Sİ 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.

Introduction

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

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.

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

Analysis of the Early/Tardy Problem

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

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

• Pij(si) may be taken to be the priority of job i with respect to j at the earliest time the machine is free


• A dispatch priority rule was derived by comparing each job's priority to an average job with processing time


• However, local optimality is far away from global optimality due to «clashes» between multiple jobs.


• 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


• Linear vs. Exponential priority rules for tardiness problem:


• Early/Tardy Heuristics• Following Morton et al. (1984)

• If (1) is divided by pipj


• 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:


• Exponential priority rule:


• Linear vs. Exponential priority rules for early/tardy problem:

• 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


• 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


• 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



1 2 3 4 5

Filter width (α) = 3 Beam width (β) = 2

Evaluated by Priority search


1 2 3 4 5


The best three are selected and Evaluated by Probe search


1 2 3 4 5


The best two are selected


1 2 3 4 5


1 2 4 5 1 2 3 5

Evaluated by Priority search


1 2 3 4 5


1 2 4 5 1 2 3 5

The best three are selected for each parent - Evaluated by Probe search


1 2 3 4 5


1 2 4 5 1 2 3 5

• 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

• 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

• Tardy cost rate: • w/p ~ uniform [0,5]. • wi = (w/p) x pi.

• 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

• 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

• 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

• 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

Computational Study

TEŞEKKÜRLER !

the single machine early/tardy problem* peng si ow & thomas e. morton

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