1 scheduling problem—A heuristic approach

ADDRESSING THE N/l SCHEDULING PROBLEMA HEURISTIC APPROACH K. ROSCOEDAVIS* University of Georgia. Athens, Georgia, U.S.A. and JAMES E. WALTERS? Ball State University, Muncie, Indiana 47306. U.S.A. Scope and purpose-A problem of key significance in many manufacturing operations is the scheduling of n jobs through a single manufacturing facility (machine). Prior research conducted in this area has resulted in the development of scheduling algorithms. However there are some disadvantages in using an algorithm to solve the problem. Quite often lengthy computational times result. particularly when a large number of jobs are to be scheduled. And, in some cases computer core requirements for processing may be excessive. The purpose of this article is to demonstrate that a heuristic procedure can be employed for the n/l scheduling problem where the objective is to minimize maximum job lateness. A number of basic heuristic procedures are examined and tested with sample problems in order to develop a multiple-heuristic scheduling model. The model’s effectiveness is demonstrated for a range of problem sizes and due-dates. To support manual implementation of the procedure a detailed flowchart is given. Abstract-The problem of sequencing n-jobs on one machine (n/l) to minimize maximum job lateness has been the subject of much prior research. Most of this research has been directed at identifying optimal solutions to the problem via algorithmic search techniques. A weakness in employing an algorithm for solving the problem, however, is that lengthy computational times may result because of the necessity of searching n! sequences. By employing a multiple heuristic approach this limitation can be avoided. An optimal or near optimal schedule can be identified in a finite number of steps. This paper describes a multiple heuristic model that is effective more than eighty-ninety percent of the time in providing an optimal schedule for the N/I/L max scheduling program. Ten separate heuristics are described. and the results of testing the heuristics over fifteen hundred and sixty randomly generated problems is presented. Three of the heuristics are combined to form the heuristic-scheduling model. INTRODUCTION

Numerous procedures and techniques have been developed for solving the n job single resource scheduling problem. For example, Baker and Su[ I] in a recent article propose and describe a branch and bound algorithm for solving the problem where the objective is to minimize maximum job tardiness. McMahon and Florian[9] present a “blocking” algorithm for solving the problem where the objective is to minimize maximum job lateness. For the case where job preemption is allowed, Bratley et a/.[21 propose using a “labeling” algorithm. Three common elements exist among these as well as other approaches that have been proposed for solving the problem: (1) each employs an optimizing algorithm; (2) each employs a heuristic as an aid to the optimizing algorithm; (3) each contends to be the most efficient in solving the prob1em.S With exception of research conducted by Gavett[7] in early 1965 and some approaches employed in the early development of scheduling, limited research has been conducted in the use of heuristics for scheduling. *K. Roscoe Davis is an Associate Professor of Management Sciences at the University of Georgia. He received his B.S. and M.S. degrees in electrical engineering from Louisiana Polytechnic Institute and his Ph.D. in management science from North Texas State University. Before joining the faculty at Georgia he spent ten years with Texas Instruments and Honeywell. The author has publications in Production Inventory Management Journal, OMEGA-The International Journal of Management Science, Computers and Operations Research, Industrial Marketing Management, Managerial Planning, Jbumal of Cost and Management, Journal of fndustrial Engineering, Journal of Systems Management, Decision Sciences, and the Ioumal of Risk and insurance. tJames E. Walters is an Associate Professor of Finance and Management at Bail State University. He received his Bachelor of Science degree in electrical engineering from the University of Illinois, his MBA and DBA in decision sciences from Kent State University. Before joining the faculty at Ball State University he spent four years in industry with Louis Allis Company and Reliance Electric Company, and three years at Bowfing Green State University and the University of Georgia. SBaker and Sufll, McMahon and Florianl81,and others report actual running times and/or CPU times required to solve numerous example problems, in order to demonstrate the efficiency of their proposed algorithm. R9

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This limited use of heuristics, the general practice of employing algorithms, and the push for improved processing efficiency has not occurred by chance. Where n jobs are to be processed by a single facility II! possible schedules exist by which the jobs can be completed. For even the finite case, where a small number of jobs are to be scheduled, an extremely large number of possible schedules exist (i.e. for n = 10 the number of schedules = 3.6 x 106). A complete enumerative search of n! schedules, with the objective of identifying the optimal schedule, is extremely time consuming, even with the use of a computer. But, by employing some form of “blocking,” “labeling,” or branch and bound algorithm the search problem can be reduced. Such algorithms subdivide the n! problem into smaller subset problems, eliminating the need for searching (testing) all n! schedules for optimality.* Research in scheduling thus has tended to be directed at developing improved “search” algorithms. But, all search algorithms are not necessarily efficient. As a matter of fact scheduling algorithms in general are not efficient, particularly for large problems. Baker and Su[l] found that for scheduling 30 jobs or less, a branch and bound procedure could require as much as a second of CPU time to determine the optimal schedule. Bratley et a/.[21 found that for scheduling 25 jobs or less, on the average a fraction of a second of CPU time was required for optimizing but often more than a second was required. When these latter authors examined the case where 100 jobs were to be scheduled, about one-third of the time extensive CPU times were encountered. In one case the authors found that more than 300 s were required to find the optimal schedule. These authors also found that in several cases their algorithm did not produce an optimal schedule even when the run time was extended beyond 300 s-the authors “stopped the run” when it was decided an excessive time level had been reached. Most authors would probably agree, as Bratley et al. [2] note, that when a large number of jobs are to be scheduled it is probably desirable to employ a scheduling heuristic. However, an argument against the use of heuristics is that they do not necessarily provide an optimal schedule. But this brings into focus the directive of this research and a question disregarded by prior researchers: “Is it really necessary to optimize ?” It may be desirable to suboptimize, assuming suboptimality occurs only a fractional part of the time. This would be particularly acceptable if computational times are extremely small. The authors contend that fractional suboptimality is a valid approach for the n job scheduling problem. The authors further contend this is true when a few or a large number of jobs are to be scheduled. To support these arguments a heuristic model is presented that provides optimal or near optimal solutions a large percentage of the time. The approach taken in developing the model was to examine and study a number of different scheduling heuristics, test the heuristics for optimality (each heuristic generated schedule was compared with the optimal schedule generated via an algorithm), and finally integrate the heuristics. The combined heuristics, i.e. the model, was then tested for optimality.

THE SCHEDULING

PROBLEM

Since numerous versions of the n/l scheduling problem have been examined in prior studies it is appropriate to define the particular problem addressed in this study. We consider the problem to be one of assigning n jobs to a single machine. Each job j of the II job set is described by the following characteristics: rj (release time): the time at which job j is released and processing thus may begin. pi (processing time): the time in which job j will occupy the machine. This time includes processing time and set-up time. dj (due-date): the deadline or completion date (time) for job j. The three characteristics ri, pj and dj are assumed to be known at time to, when the schedule is computed; however, rj, pj and dj may differ from job to job. That is, it is assumed the job j + 1 may have an entirely different release date (rj+l), due date (dj+,) and processing time @,+,) from job j + 2, or any of the remaining n - 1 jobs. As a result of scheduling, each job will be completed at a specific time (c). For job j this *Several authors have found that heuristics combined with a subdividing (branching) algorithm can also eliminate some schedules in the n ! search.

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91

time is defined as cfi Job j is considered to be late if ci > di. Job lateness &), thus is defined by Lj = (Cj- dj). If job lateness is computed for all n jobs, the maximum lateness (Lmax)of these jobs is a measure of the desirability of the schedule. The objective in the scheduling process is to minimize maximum job lateness. Since I.,,, is defined thus: L InsIX = max (L,); the objective is: min I.,,,,. Both preemptive (job splitting) and non-preemptive n/l scheduling have been considered by prior authors. However, for this study only the non-preemptive case is considered. Job splitting is not permitted. HEURISTICS

CONSIDERED

In approaching the n/l problem ten different heuristics were examined and studied under varying rj, di and pi conditions. The heuristics considered were as follows: (1) EDD rule. A schedule of jobs arranged in nondecreasing order of due dates, i.e.

this is traditionally known as “Jackson’s Theorem”181 for the static version of the n-job single machine problem. Jackson181 demonstrated that for the case where all ri’s are equal (release time is the same for all jobs), maximum lateness is minimized by processing the jobs in nondecreasing order of their due dates, Some authors have referred to this as the ELF rule[S], earliest latest finish rule. (2) ERD r&e. A schedule of jobs arranged in nondecreasing order of job release dates, i.e.

This is a modified version of the Jackson theorem where release date [r,] is employed for sequencing as opposed to due-dates [d,]. (3) ERAD rule. A SChe du1e of Jo ’ b s arranged in nondecreasing order of each job’s release date plus due date, i.e.

This is also a modified Jackson theorem. Baker and SufIl, employ this heuristic in the branch and bound procedure to discard sequences to be searched in the n ! search process. (4) SLACK rule. A schedule of jobs resulting by sequential selection, based upon slack time (di -pi). Whenever the machine completes the processing of a job, the next job to be processed is that job, in the available set of jobs, having the smallest slack time. If the machine is idle, then at the time the next job is to be released, the job having the smallest slack time, of the available set of jobs, is scheduled. The schedule is complete when all jobs have been scheduled. This is a dynamic version of the Jackson theorem, where slack time (di -pi) is employed. (5) SPT rule. A schedule of jobs arranged in nondecreasing order of processing time, i.e.

Conway, Maxwell and Miller[3], refer to this as the “shortest processing time” rule. This rule is traditionally used in cases where due dates are not imposed on the completion of jobs. C,A 0 R.. Vol. 4, No. ?-B

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K. ROSCOE DAVISand JAMESE. WALTERS

(6) FIFO rule. A schedule of jobs arranged in the sequence in which the jobs are received, not necessarily in order of release date. This is the rule traditionally used in practice where no scheduling system, algorithm, or technique exists. An example of this rule is the processing of jobs at a computer center where job priorities are not allowed, i.e. jobs are processed on first come-first serve basis. (7) DERD rule. A schedule of jobs resulting by sequential selection. Whenever the machine completes the processing of a job the next job to be processed is that job, in the available set of jobs, having the earliest due date. If the machine is idle, then at the time the next job is to be released, the job having the earliest due date, of the available set of jobs, is scheduled. The schedule is complete when all jobs are scheduled. This is a modified ERD rule developed by Schrage[ll]. The structure of the rule is defined as follows: (a) Let 3 be the set of unscheduled jobs. (b) Set t = 0. ~-(c) If there exists at least one job j E 3 such that r; 5 t, then go to step (e). (d) Set t = min (ri).

le.3 (e) Among all jobs j E 3 such that ri 5 t choose job j that has the smallest due date di.

Break ties on the due date by selecting the job with the largest processing time pi, (f) Schedule the chosen job and update the time monitor, i.e. t = t +pj. (g) If .?# C$go to step (c); otherwise schedule is complete. (8) DEDD rule. A schedule of jobs resulting by sequential selection. Whenever the machine completes the processing of a job or whenever a job is released the next job to be processed is that job having the earliest due date of the set of jobs available for processing. The schedule is complete when all jobs are scheduled. This is a modified version of the DERD rule where job preemption is allowed. In using this rule the sequential scheduling process is updated each time a job is released or completed, but the set available to be scheduled, s, includes both released jobs as well as those being processed. This rule was included in the set of heuristics in order to study the impact of job preemption. This heuristic does not necessarily result in a preemptive schedule, therefore it was valid to include the rule in the ten heuristic set. (9) FDEDD rule. A schedule of jobs that results by accepting the next job to be scheduled as that job occurring first in the DEDD schedule. (10) RDP rule. A schedule of jobs arranged in a modified nondecreasing order of release date. A schedule results by first arranging jobs in nondecreasing order of release date; when ties result jobs are arranged in nondecreasing order of processing times. This is a modified version of the ERD rule. This is also a modified version of the Jackson theorem. TESTING PROCEDURES

AND RESULTS

The heuristics set, i.e. the ten different heuristics, was tested over a range of test problems in order to identify that group of heuristics which would provide the “best” solution (i.e. a solution value for L,,, that was equal or close to the optimal a large percentage of the time). The set was tested on 1560 different problems, designed specifically to explore performance variation for changes in n (number of jobs to be scheduled) and tightness of due-dates (difference between due-date and release date).* The release time (rj), process time (pj) and due-date (dj) for each test problem was generated by repeatedly sampling from a uniform distribution and applying the randomly generated numbers according to the following equations: Release date (rj) = 100x (random number,) Processing time (pj) = [200 x (random numberz)]/n + 1 Due-date (dj) = rj + 100 x (random number3 x M, where M is the due-date tightness coefficient. Parametric problem sets were generated for six different problem sizes: n = 5, 10, 15, 20, 25 and 30, and for thirteen different due-date tightness coefficients: M = 0.1, 0.25, 0.5, 0.75, 1.0, *The procedure for generating problems is consistent with Baker and Su[2] and others. The procedure was used in order to allow comparison of heuristics with prior analyses.

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1.25, 1.50, 1.75,2.0,2.25, 2.50,2.75 and 3.00. Twenty different test problems were generated for each problem-size, and due-date tightness coefficient combination. To test the optimality generating ability of each heuristic, the L,,, generated by the respective heuristic was compared with the optimal I,,,,. The optimal L,., was generated by employing a branch and bound algorithm.* Table 1 shows the optimality generating ability of each heuristic. The ERAD rule provided the optimal schedule for 979 of the 1560 problems, (i.e. the rule provided the optimal scheduled 62.7% of the time), while the SPT (shortest processing time) rule produced an optimal for only 9 problems, (i.e. the rule provided an optimal schedule less than 1% of the time.) From examining Table 1 it was concluded that a single heuristic could not be employed since the best heuristic provided the optimal schedule less than 63% of the time. Noting however that the non-optimal solution set (those problems where the optimal schedule was not generated) for each heuristic was not a mutually exclusive set when compared with the other heuristics, further study was warranted. When due-date tightness is small, i.e. when the release date and the due-date are very close, one could logically reason that a heuristic rule based upon release date might provide a “good” (i.e. near optimal) schedule. Likewise, when due-date tightness is large, a good schedule would probably result by employing a heuristic based upon due-date. When the heuristics in this study were examined for varying due date tightness this is precisely what resulted. Figure 1 shows the number of optimal schedules generated for the respective heuristic when M was varied from 0.1 to 3.0. For each due-date tightness level 120 test problems were generated, (i.e. for each tightness level 20 problems were generated for n = 5, 10, 15, 20. 25 and 30). Table 1. Optimal schedule generating ability of heuristics (comparison of L,., (heuristic) vs L,., (optimal)) for 1560 problems Rule

No. of Optimal Solutions

Percent of Total

847 279 979 573 009 015 722 1% 628 301

54.29 17.88

1. EDD

2. 3. 4. 5. 6. 7. 8. 9. 10.

ERD ERAD SLACK SPT FIFO DERD DEDD FDEDD RDP

62.76 36.73 0.57 0.96 46.28 12.56 40.26 19.29

100

z

--DEIX

: z

0 o oFDEDD -ERO

; i: 8 G

-----ERAD 50

X-xEDD -SLACK

& E 3

0,

025

05

075

10

125

Due

date

,5

175

20

225

25

275

30

tightness

Fig. 1. Optimal schedule generating ability of heuristic for varying levels to due date tightness (M). *The branch and bound algorithm employed for generating the optimal L,.,, was a modified version of the Baker and Su algorithm. The algorithm was developed by one of the authors and has been reported previously[l3].

K. ROSCOEDAVISand JAMES E. WALTERS

94

From Fig. 1 it can be noted that both the DERD and the FDEDD rules provided the optimal solution for a large number of problems when due-date tightness was less than 0.5 but performed poorly when the tightness level was 0.75 or greater. On the other hand, the EDD and the ERAD rules provided the opposite results: when the tightness level was small the optimality generating ability of the rule was poor; when the tightness level was greater than 1.25 the number of optimal solutions generated increased significantly. These data suggested that a combined heuristic model resulting from integrating the EDD and the DERD rules, the DERD and the ERAD rules, or some other combination of rules would provide a heuristic with high optimality generating ability. But, before studying the heuristics in a combined form, the impact of job size (n) was examined. This did not involve testing additional problems; a reordering of the data were required. The data were arranged in job size (number of jobs to be scheduled) versus optimal schedule attainment for due date tightness levels from 0.1 to 3.0. Two hundred and sixty problems were examined for each job size (i.e. 20 problems had been generated for the 13 different tightness levels). Figure 2 is the results of reordering the data. On examining Fig. 2 it can be noted that while all of the rules tended to be less effective with increasing job size, the DERD, ERAD, EDD and the FDEDD again were the most effective rules. These results supported the conclusion regarding rule effectiveness for due date tightness (Fig. 1). Based upon the results shown in Figs. 1 and 2, five heuristic sets were examined for optimality effectiveness: (1) all heuristics, (2) the DERD, ERAD, EDD and SLACK rules, (3) set without the SLACK rule (DERD, ERAD and EDD), (4) the DERD and ERAD rules, and finally (5) the DERD and EDD rules. The SLACK rule was included in set 2 in lieu of the FDEDD rule because it exhibited characteristics in Fig, 1 that were not common to the other rules. Tables 2 and 3 are the summary results of the five sets. Table 2 is a display of heuristic set effectiveness for varying job sizes (n) and Table 3 is a display of the same data for varying levels of due-date tightness @I).* The data in Tables 2 and 3 indicated that the overall effectiveness of sets 1, 2 and 3 were nearly equal. Each set produced approximately 1350 optimal solutions for the 1560 problems tested. Since set 3 employed a smaller number of heuristics the results would suggest adoption of this set as the heuristic model. But, before adopting this group the data were further analyzed. Since all three sets produced nearly the same number of optimal solutions it appeared logical to ask: “For those solutions which were not optimal was there as large difference between the heuristic generated I,,,,,, and the optimal I+,,,,?” Table 4 is a summary of this analysis.

200 I-

‘i

----DERD - -DEDD 0 0 OFDEDD -ERD *---ERAD x-xEDD &--&SLACK wRDP a--oSPT . . . . ..F.FD

Number

Fig.

2.

of JObS scheduled,

n

Optimal schedule generating ability of heuristics for varying number of job sizes (II).

*The reader should recognize that Table 2 is a tabular display of the type analysis shown graphically in Fig. 2 and likewise Table 3 is a tabular represen~tion of the type analysis shown ~aphic~y in Fig. 1.

95

Addressing the N/l scheduling problem-a heuristic approach Table 2. Optimal schedule generating ability of heuristic sets for varying number of job sizes (n) Set 1

Number of jobs scheduled ( n )

(All)

Set 2 DERD ERAD EDD SLACK

Set 3

Set4

Set 5

DERD ERAD EDD

DERD ERAD

DERD EDD

5

250

250

250

10 15 20 25 30 Total number of optimal solutions generated:

232 231 214 221 205

231 231 214 220 205

231 231 214 217 x4

24? 213 217 203 195 I86

234 204 201 182 192 I82

1353

1351

1347

12%

1195

Table 3. Optimal schedule generating ability of heuristic sets for varying levels of due-date tightness (M) Set I Due-date tightness 00

Set 2 DERD ERAD EDD SLACK

Set 3

Set 4

Set 5

DERD ERAD EDD

ERD BRAD

DERD EDD

120 119 95 82 77 82 107 108 106 112 117 117 Ill

120 119 95 82 75 82 107 108 lo6 112 117 117 Ill

120 119 95 81 74 81 107 106 106 112 117 117 III

119 119 94 18 65 78 90 88 99 105 110 ill 107

120 118 90 69 64 69 86 88 88 96 104 I05 103

1353

1351

1347

1256

1195

W)

0.10 0.25 0.50 0.75 1.00 1.25 1.50 1.15

2.00 2.25 2.50 2.15 3.00 Total number of optimal solutions generated:

Analysis of the data in Table 4 supports the adoption of set 3 as the heuristic model. These data show that for deviations of 2 or less, set 3 produced an optimal solution for 93.25% of the problems while sets 1 and 2 produced optimal solutions 93.20% of the time.* Having adopted set 3 as the heuristic model an attempt was made to evaluate its computational efficiency. Table 5 is a display of the average CPU-run times for the 1.560problems run with set 3. Since limited data exist in the literature where heuristics have been tested, it is difficult to compare the results with other heuristic models. However, the results can be compared with algorithmic based models. Baker and Su [ l]t report similar statistics for their branch and bound algorithm. Compared with the Baker and Su algorithm the heuristic model was able to generate a solution in a fraction of the time. This result is not su~rising since an algorithm would require a lengthy search process, whereas the time consuming factor in the heuristic was a multiple sort procedure. THE HEURISTIC

MODEL: DESCRIPTION,

IMPLEMENTATION,

AND USE

Based upon the results from Tables 4 and 5 one would be inclined to accept the adopted model as a valid n/l scheduling device. However, a key question related to use of the model has *The deviations that resulted were very small compared with those reported by other authors (see Baker and Su[l]). These results strongly support the use of these particular heuristics for the particular problem studied. Wee Table 1 on p. 174 in Baker and Su[l].

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Table 4. Performance of top heuristic sets-optimal and near-optimalsolutions Deviation L,., (heuristic) L,., (optimal)

0

1 2 3 4 5 6 I 8 9 10 I1 12 13 14 15

Set 1

Set 2 DERD, ERAD, EDD and SLACK rules

AU heuristics (Number) 1353 55 46 33 32 14 13 03 04 02 01 01 01 01 00 01

(Proportion) 0.8673 0.0352 0.0295 0.0212 0.0205 0.0090 0.0083 0.0019 0.0026 0.0013 0.0006 0.9006 O.CCO6 0.0006 0.0000 0.0006

(Number) 1351 55 47 33 32 14 13 04 04 02 01 01 01 01 00 01

(Proportion) 0.8666 0.0353 0.0301 0.0212 0.0205 0.0090 0.0083 0.0026 0.0026 0.0013 0.0906 0.0006 0.0006 O.ooo6 O.OtHO O&306

Set 3 DERD, ERAD and EDD rules (Number) 1347 56 47 34 33 14 14 04 04 02 01 01 01 01 00 01

(Proportion) 0.8634 0.0390 0.0301 0.0218 0.0212 O.OO9il O.OQ90 0.0026 0.0026 0.0013 0.0006 0.0006 O&56 O.tXN% 0.0000 OX@06

Table 5.* Average CPU-run times (in seconds) for 1560parametric problem sets using heuristic set 3 m

5

10

15

20

25

30

Overall average

0.10 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.2s 2.50 2.15 3.00

0.00539 0.00561 0.00563 0.00532 0.~5~ 0.00547 0.00561 0.00566 0.08707 0.00915 0.00832 0.00998 0.00873

0.00738 0.00756 0.00787 0.~729 0.~787 0.00770 0.00816 0.00772 0.01206 0.01331 0.01373 0.01331 0.01331

0.01051 0.01061 0.01097 0.01039 0.01104 0.01107 0.01086 0.01044 0.02246 0.02412 0.02163 0.02288 0.02080

0.01499 0.01499 0.01539 0.01482 0.01558 0.01510 0.01485 0.01434 0.02745 0.02579 0.03203 0.02745 0.02870

0.01999 0.02011 0.01996 0.02067 0.02117 0.02108 0.02008 0.01927 0.03577 0.0403s 0.04118 0.03577 0.04243

0.02686 0.02685 0.02615 0.02731 0.02744 0.02758 0.02712 0.02517 0.05699 0.05616 0.05616 0.05657 0.05366

0.01245 0.01428 0.01432 0.01430 0.01483 0.01466 0.01444 0.01376 0.02696 0.02814 0.02884 0.02766 0.02793

Overall average maximum

0.00675 0.91103

0.00979 0.01460

0.0152f 0.02671

0.02011 0.03421

0.02753 0.04617

0.03800 0.06f84

*The test data was for the heuristic model written in PLI and run on the University of iBh4 370 Model 158 computer system.

Georgia

not been addressed, that is, “how does one determine or know whether a generated schedule is optimal, near optimal, or far removed from optimal ?” Fortunately, because of prior research conducted by McMahon and Florian [8], as well as Baker and Su [ 11,and others, we can answer this question. A heuristic procedure can be used to compute a lower bound (LB) on the minimum maximum lateness of a schedule. By comparing the L,,, of the heuristic generated schedule with the lower bound one can determine the degree of optimality of the schedule: (1) if I.,,, for the schedule equals the lower bound then the schedule is optimal; (2) if L,,, is greater than the lower bound (LB), the schedule may be optimal, but we have no way to determine exactly; (3) if L,,, is greater than the lower bound (LB) we know that the schedule is either optimal or lies somewhere between L ,,,_ and LB. This is a measure of the maximum error in the model. As noted above, McMahon and Florian[8], and Baker and Su[l] employ bounding procedures in their algorithms. This bounding procedures are heuristic based and are used to establish a boundary for the branch and bound process. One might logically ask, “if the lower bound procedure is heuristic based why not employ the concept to produce a scheduie?” The answer to this is that the LB process does not always result in the generation of a usable schedule. Tire LB procedure of Baker and Su[l], the procedure used in this study, is

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91

based, thus a schedule resulting from the procedure may allow job preemption. Because the LB procedure is preemption based however, does not prevent us from employing its L,,, as the bench mark for optimality. By comparing columns 6 and 7 from Table 4 with Table 6 one can determine the reliability of using LB as a bench mark for measuring degree of optima&y. The data show that the model did not duplicate exactly the resufts of the he~~stics-algorithm tests (Table 4), however the data show similar results: (1) the model identified 1236of 1347optimal solutions it had generated; (2) the model identified 91.28% of the 1560schedules with an L,,, difference of 2 or less (from Table 4, the data show the heuristics had the capability of generating 93.25% of the schedules with differences of 2 or less); (3) the maximum error in the model was 16; this was slightly greater than expected. Overall the model performance was acceptable. To further test the model and demonstrate its effectiveness in handling large problem sets, test problems were generated where the number of jobs to be scheduled, n, was 50, 100,150 and 200, and the due date tightness, M, was 0.05, 1.0, 1.5, 2,O and 3.0. Thirty problems were generated for each. parametric-set combination. A total of 600 problems were thus employed. The performance of the model for generating optimal or near-optimal solutions is displayed in Tabfe 7. The computational efficiency, expressed in terms of CPU-run times is displayed in Table 8. Based upon these data, the optimal schedule generating ability of the model was not significantly affected by increasing job size. For the 600 problems examined, the model was able to generate 491 optimal schedules and in 74 cases the L,,, of the generated scheduled differed by 1 from the optimal. The model thus generated an optimal or near-optimal schedule in 94.16% of the cases.

preemptive

Table 6. Performance of heuristic model in generating op timal and near-optimal solutions Number of schedules 0

I 2 3 4 5 6 7 8 9 10 11

12 13 14 15 16

Proportion of to&i schedules

1236 92 76 37 45 30 15 It 5 3 2

0.8051 0.0590 0.0487 0.0237 0.0288 0.0192 o.Ou96 0.007f o.OQ32 O”QO19 0.0013

1 2

O.ooo6

2 t 1

0.0013 o&-l13 o.OoM 0.0006

1

0.0006

In terms of processing efficiency the overall average CPU-run times were small compared to times required by algorithmic models. Bratley et al.@] found that for job size of 50, the minimum time for generating an optimal schedule was 0.152 seconds, with the maximum time being 0,868 seconds. From Table 8, the average time for generating a solution for n = 50 using the heuristic model was 0.14269 s. The average time required to schedule IO0 jobs using the he~~st~~s was 0.54705 s, whereas Bratfey et al. found the minimum time for handling this number of jobs to be 0.994 s. These authors reported the maximum time to be 18.250s for scheduling 100 jobs. Since processing-running times are not reported in the literature beyond job sizes of 100 we have no basis for comparing the heuristics for the cases where n was 150 and 200. However, the data in Table 7 fully demonstrates that excessive run times do not result in using the model, even fur the cases with large job size.

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98

Table 7. Model performance--optimal and near-optimal solutions for large problem sizes (n = 50-200) Deviations (I,,., -LB) n

50

M 0.5

1.0

1.15

2.0

3.0

Subtotal

Prounion

Number of schedules

100

c-21 1-5 2-2 3-2 O-12 l-8 2-l 3-3 O-10 I-10 2-7 3-4 O-13 l-6 2-7 3-4 O-28 1-2

O-84 1-31 2-23 3-12 O-O.%00 I-O.2066 2-0.1533 3-0.0800

150

200

n-30

O-30

O-30

O-12 I-10

O-24 l-6

O-30

O-22 1-8

c-30

O-30

n-15 1-15

O-26 14

O-30

O-30

O-30

O-30

O-117 l-33 2 3 ~.78~ l-0.2200 2 3

O-140 l-10 2 3 O-o.9333 l-O.0667 2 3

O-150 1-O 2 3 ~1.~

Subtotal O-111 1-5 2-2 3-2 O-84 l-24 2-l 3-3 O-92 1-18 2-7 3-4 O-84 l-25 2-7 3-4 O-118 1-2 2 b491 l-74 2-23 3-12 f

Proportion

O-a.9250 l-0.0416 2-0.0167 3-0.0167 O-O.7166 1-0.2OOll 2-0.0583 3-0.0250 O-o.7666 1-0.1500 2-0.0583 3-0.0250 O-0.7000 l-0.2083 2-0.0583 3-0.0250 O-O.9833 l-0.0167 2 3 O-0.8183 t-0.1233 2-0.0383 3-0.0200 Total--.-%

1 2 3

Table 8. Average CPU-run times (in seconds) for large problem sizes (n = 50-200) n m

0.5 1.0 1.5 2.0 3.0 Overall average Maximum

50

100

150

200

Overall average

0.12723 0.11887 0.11766 0.24017 0.12753

0.47010 0.44733 0.43552 0.91861 0.46372

1.02573 1.01525 0.98954 2.10940 1.03598

1.85181 1.82519 1.75423 3.85882 1.80111

0.86871 0.85166 0.82423 1.78175 0.85706

0.14269

0.54705

1.23520

2.21823

0.27164

0.94731

2.41371

3.97131

CONCLUSIONS

The results of this study demonstrate that heuristics procedures can be employed to effectively solve the n/l scheduling problem, particularly for the case where the objective is minimizing maximum job lateness. A user’s willingness to adopt and employ a heuristic approach can be sign~cantly increased by providing a means for identifying those cases where an optimal schedule has been produced. For this study it has been demonstrated that a lower bounding heuristic can be used to “test” a schedule for optimality. The bounding heuristic can also provide a numeric measure of the maximum degree of error in the schedule. This error would be the maximum improvement that would result if an algorithmic procedure were used to generate a schedule. The compu~tional results of the study indicate that problems involving up to 200 jobs can be solved quite rapidly a high percentage of the time. One can expect the heuristics to provide an optimal or near-optimal schedule almost ninety percent of the time with minimal com-

Addressing the N/l scheduling problem-a heuristic approach

99

costs. One of the significant advantages that the model has over algorithmic based models is that it provides a schedule and requires limited computational time. This is particularly significant if we note that when Bratley et al. examined the 100 job scheduling problem about one third of the problems were not completed and run times greater than 300s were encountered. The heuristic model proposed here should never terminate with this dilemma.

putational

REFERENCES 1. K. R. Baker and Zaw-Sing Su, Sequencing with due dates and early start times to minimize maximum tardiness, Naval Res. Logistics Quart. 21, 171(1974). _ 2. P. M. Bratley, M. Florian and P. Robillard, Scheduling with earliest start and due date constraints, Naval Res. Logistics Quart. 18, 511 (1971). 3 R. W. Conway, W. L. Maxwell and L. W. Miller. Theory of Scheduling p. 20. Addison-Wesley, Reading, Ma (1967). 4. G. B. Dantzig, A machine job scheduling model, Mgmt Sci. 6. 191 (1969). 5. M. I. Dessouky and C. R. Margenthaler, The one-machine sequencing problem with early start and due dates, AIDE Trans. 4. 214 (1972). 6. S. E. Elmaghraby, The sequencing of related jobs, Naval Res. Logistics Quart. 15, 23 (1%8). I. 1. W. Gavett. Three heuristic rules for sequencing jobs to a single production facility, Mgmt Sci. 11,B166 (l%S). 8. J. R. Jackson, Scheduling a production line to minimize maximum tardiness, Research Rep. No. 43, Management Science Research Project, UCLA, January (l%S). 9. G. McMahon and M. Florian, On scheduling with ready times and due dates to minimize maximum lateness, Ops Res. 23, 475 (1975). 10. W. Pounds, The scheduling environment, Industrial Scheduling (Eds. W. Muth and J. Thompson), p. 5. Prentice Hall, Englewood Cliffs, NJ (1%3). 11. L. S&rage. Obtaining optimal solutions to resource constrained network scheduling problems, unpublished manuscript, March (1971). 12. J. Shwimer, On the N-Job, one-machine, sequence independence scheduling problem with tardiness penalties: a branch and bound solution, Mgmr Sci. 18, B301 (1972). 13. J. E. Walters, One machine deterministic job shop scheduling with variable release dates to minimize certain nondecreasing lateness penalty functions, Unpublished DBA Dissertation, Kent State University (1974). 14. .I. E. Walters and L. F. Simmons, Sequencing with due dates and early start times to minimize maximum lateness, J+oc. Eleventh Annual Meeting of the Southeastern Chapter of the Institute of Management Sciences, Charleston, South Carolina, October (1975). 15 J. E. Walters and J. Randall Brown, Blocking algorithm to minimize maximum lateness in job shop scheduling, presented at the ORSA-TIMS Special Interest Conference on the Theory and Applications of Scheduling, Orlando, Florida, February (1976).

(Received 27 October 1976)

K. ROSCOE DAVISand JMM~SE. WALTERS

100

iieurist*c -

LB Model

for

the

N,l,LMx

Problem

Let Set

of

Ci

= “AX(CI_i

ri)

+ Pi

where

i = 1, 2,

.,.,

N and Co = 0

Lmax = MAX(Ci - di) IES

,

ri)

+ Pi

where

i

= 1.

2,

by sequencing the DERB schedule(S) the available non-finished fobs having due date at the followin$ tines: a) at the fime that the current Job is

Form

a~

the

STEP 0 STEP 1

cime

that

the

next

fob

is

.,.)

N and

co =

to

that job of the earliest completed be released

,.a the unscheduled set of jabs be 5 SAL the current time (t) equal to the date of the earliest job in s t - “in ri IES

release

STEP 2

J = i where STEP 3

f E A and

i c 3 if

ci $, t

of the available set of jobs (A), schedule the JOb (j) having the earliest due date next in the schedule (S) where J = MIN di, 3’S iCA

STEP 4

STEP 5

3

Let

the

best

schedule

(B) and xts

unscheduled set of iobs be 3 ~“rrent time (t) e&al fo the job in s. t = MIN ri irS

earliest

J =i

where

j E A and

i P 5 if

ri

release

0

maximum lateness

(L)

Schedule

(9) of schedule is optimal if

date

2 t

Of the available set of jabs (A), schedule the (3) having the earliest due date next in the schedule (S) ohvre j - MIN di, J E S IFA

and 9.

h)

rhe the

the

and print restits L = LB. Q * I, - LB,

job

8,

L,