One machine scheduling to minimize expected mean tardiness—Part II

Computers Ops Res. Vol. 21,No. 7,pp.787-195,1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved

Pergamon

0305-0548/9437.00+ 0.00

ONE MACHINE SCHEDULING TO MINIMIZE MEAN TARDINESS-PART II

EXPECTED

B. V. CADAMBI~‘$ Department of Statistics, University of Bombay, India (Received

December 1989; in revised form March 1991)

Scope and Purpose-This

paper presents a heuristic algorithm for the problem of scheduling a given set ofsimultaneously available stochastic jobs to minimize the expected mean or total tardiness. The processing times are assumed to follow the exponential distribution. The heuristic is based on the prinicples of Wilkerson-Irwin’s heuristic algorithm. For the deterministic version, the latter heuristic was found to give results which were good and within reasonable computational time. The proposed heuristic is evaluated. Finally, the solutions of the deterministic and stochastic versions are compared to examine the significance of difference between the two versions. Abstract-A heuristic algorithm is developed for the problem of scheduling a number of simultaneously available jobs with exponentially distributed processing times (with known means), on a single machine, to minimize the mean or total expected tardiness. The performance of the heuristic is evaluated by comparing the results with the optimal values or values of a proposed lower bound. The values of expected total tardiness of the sequences generated by the deterministic and stochastic versions, of the Shortest Processing times sequence and the Earliest Due Date Sequence are compared. Finally, the paper includes a discussion on the limitations of the proposed model and the issue of risk due to uncertainty in processing times.

INTRODUCTION

This paper considers the problem of scheduling several jobs, all available at the start and having exponentially distributed processing times (with known means), on a single machine, to mininize the mean or total expected tardiness. Efforts of several investigators to find a polynomial bounded algorithm for the deterministic version have failed although the problem has not been shown to be NP-complete [l]. Thus, it is probably impractical to use an algorithm which gives an optimal solution as discussed in [2] for problems of large sizes. A heuristic algorithm, analogous to the heuristic for the deterministic version proposed by Wilkerson and Irwin [lo] is developed in this paper. The theoretical developments are first outlined. The algorithm is then stated and illustrated using a numerical example. The heuristic is evaluated by comparing the solutions obtained with the optimal value or a proposed lower bound taking several test problems. For each test problem, the impact of using the Earliest Due Date (EDD) sequence, the Shortest Processing Time (SPT) sequence and the solution generated by the deterministic version for the stochastic case, are also examined. An approach to deal with problems involving unequal mean processing times is also discussed. The paper concludes with a discussion on the limitations of the proposed model and the issue of risk due to uncertainty in processing times. THEORETICAL

DEVELOPMENTS

The notations and assumptions are the same as in [2]. In [2], it is shown that if Q>z~+~ and &Gdlr+r then job k precedes job k+r in an optimal sequence. However, if rkcrk+, and dk
787

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B. V. CADAMBI

Theorem I: Suppose rk
Proof of this result is straightforward. Interpretation: If the ratio of the probability that job k is not completed on time when scheduled after 1, 2, . . ., k- 1, k+r to the probability that job k+ r is not completed on time when scheduled after 1, 2, . . ., k- 1, k is larger than the ratio of the mean processing times of job k to job k+r, then job k should precede job k+r. A heuristic, analogous to Wilkerson and Irwin’s heuristic for the deterministic version, has been developed. The principles are detailed below: The proposed heuristic, like the Wilkerson and Irwin’s algorithm, utilizes a pairwise comparison ofjobs in the construction of the desired sequence. The algorithm utilizes two ordered sets: (1) a set of unscheduled jobs denoted by US,; (2) a set of scheduled jobs denoted by SC. The scheduled set S, is a partially completed job sequence, which may be revised as the execution of the algorithm proceeds, The set US, contains the remaining jobs that appear in the non-decreasing order of due dates or EDD order. At each stage, the first job in US,, called the pivot job, is removed to implement the decision rule. Denoting a as the last job in S,, p as the pivot job and y as the first job in US, (after removing /?), the precedence of/l over job y for the given SCis examined. If, either r,> ry and dp
FOR

SOLUTION

For convenience, the following notation is used for job sets G4B9 Y)= SC,,

Let

Zb(y/SC)is proportional predecessors SC.

to the increment in the tardiness of job y when B is added to the existing

Step 1: Set SC= 4, US,= N. If 71 >t2 or 1,(2/9)< Z,(l/@ then assign job 1 to the first position in SC.Otherwise assign job 2 to the first position in SC.In either case, the assigned job becomes 01and the other job becomes /3,the pivot job. Step 2: The next job in US, becomes y. If TV > 7,, or Z,(y/S,) < 1,(B/S,) then /3 is added on to S,. Job /3 becomes a, job y becomes /I (it is removed from US,) and the next job in US, becomes y. Step 2 is repeated unless US, is empty (after removing #?)in which case job /3is added to the scheduled list and the iterations stop. If, however, neither of the above inequalities hold, return fl to US, and let job y become job 8. Proceed to Step 3. Step 3: If

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then /? is added to S, and placed after a. Z?becomes a, the first job in US, becomes /3 (the pivot job) and the next job in US, becomes job y. Go to Step 2. If, however, the conditions do not hold, a jump condition results. Go to Step 4. Step 4: (Jump condition). Remove job a from S, and add it to US, in the EDD order. If jobs remain in S,, the last job becomes a. Return to Step 3. If SCbecomes empty, job /? is assigned to the first position in SCand becomes job a. The second job in US, becomes /I and the third job becomes y. Go to Step 2. NUMERICAL

EXAMPLE

The same example used in [Z] is considered again. Table 1. Data used Job j Mean processing

time l/rj

Due date d,

1

2

3

4

5

5

7 8

6 9

3 10

8 12

6

Stage 1: S,=C$, US,=N

B=l y=2 Since t1 >r2 job 1 is added to S,.

Stage 2: S,=(l) x=1

U&=(2, /?=2

3, 4, S} y=3 Z&y/S,) = 1055.985 < Z@/S,) = 1143.177

Hence job 2 is returned to US, and job 3 becomes ~3. Stage 3:

us, = (2, 3, 4, 5}

&=W a=1

/?=3

Since r1 >r3 job 3 is added to S, us, = {2,4,5}

Stage 4: S,={l,3}

x=3

/?=2

y=4 zB(y/s,) = 1703.394
Hence job 2 is returned to US, and job 4 becomes /?. Stage 5: S,={l, 3) a=3

us, = {2,4,5}

/Z=4 Z&3/S, - {a}) = 185.22
Hence a jump condition case arises. Job 3 is returned to US, and job 1 becomes a. Stage 6: SC= (1) a=1

us,=

(2, 3, 5}

/?=4 Z&?/S, - {a}) = 24.75
Hence job 4 is added to SC

B. V. CADAMBI

790

Stage 7: S, = { 1, 4) Lx=4

us, = (2, 3, S}

/I=2 I&?/S, - {a}) = 485.2365 > Z&a/S, - {a}) = 244.0935

Hence job 4 is retained in S, Stage 8: S,={l, a=4

4)

US,={2, 3, 5}

/I=2

y=3 Z&y/S,) = 3406.788 < I#/&)

= 3732.939

Hence job 2 is returned to US,. Stage 9: S,={l,

4)

U&=(2,

a=4

3, 5)

p=3 Job 4 is shown to precede job 3 when 1 alone is scheduled in Stage 5. Hence job 3 is added to S,.

Stage 10: S,={l, Lx=3

4, 3)

US,={2, 5}

/I=2

y=5

Since rz > rg, job 2 is added to S,. Stage 11: S, = { 1, 4, 3, 2)

us,=

(5)

Hence job 5 is added to S,. The required sequence is therefore 1+4+3+2-+5.

EVALUATION

OF THE

HEURISTIC

ALGORITHM

A. Objectives

(i) To evaluate the performance of the heuristic algorithm: the solutions obtained by the heuristic will be compared with the optimal value or a proposed lower bound. The branch and bound algorithm is used to determine the optimal value. The lower bound to be used will be discussed in C. (ii) To examine the effect of utilizing the solution obtained by the equivalent deterministic version for the stochastic version: since the stochastic version is fairly complicated, it is natural to consider the scope for using the deterministic solution (obtained by assuming that the mean processing time as the actual processing time), in the stochastic version. The deterministic version is solved by Wilkerson Irwin’s approach and the expected total tardiness of the sequence obtained is calculated. The value is compared with the expected total tardiness of the sequence obtained by the proposed heuristic. (iii) To examine the performance of the EDD and SPT sequences to determine whether sequencing decisions are significant. B. Choice of test problems

The test problems are chosen, based on the following considerations, anologous to that discussed by Srinivasan [4].

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(4 Problem Size: To ensure that the performance of the heuristic is size independent, the problem size has to be varied. Hence problems of size 9, 12, and 15, are considered.

V-4Correlation between processing times and due dates: When the due dates are constant or when the correlation is high, Theorem 1 of [2] clearly shows that the heuristic will work well. On the other hand, negative correlations are of less practical significance, since in practice, due dates are either random or to some extent, based on processing times. Hence it is sufficient to consider zero correlations and low positive correlations. Hence, we take p = 0 and p = 0.3. 14 Level of congestion: I: for the deterministic case for extreme levels of congestion, the solutions are trivial. Hence, there is a strong possibility of a good heuristic solution. Hence the level of congestion has to be varied and extreme values can be ignored. We take I=O.25,0.5 and 0.75. When i is the level of congestion, (1 - r) is the fraction of total work that could be completed on time, so that, with an average pl, the amount of work finished by pd is n(l - r)it,.

(4 The dispersions in the execution times and due dates: when the dispersions of both are low, the jobs tend to become homogeneous so that the sequencing decisions do not affect performance. When one of the two parameters exhibits low dispersion, the Shortest Processing Time Sequence (SPT) or EDD sequence is likely to perform well. Since both parameters are positive, it is infeasible to have high dispersions. To control dispersion, we consider the two coefficients of variation pd

&

We assign values 0.3, 0.5 and 0.8 and consider combinations. Since consideration of all possible combinations of the factors discussed above will result in a large number of test problems, the number of combinations will be reduced as follows: For problems of size 12, the two coefficients of variation are allowed to take values 0.3 and 0.8, while the coefficient of correlation is fixed as 0.3. Ail three values of I are considered. Hence there will be 12 problems. For problems of size 9 and 15, the coefficients of variation will be fixed as 0.5. All three values of level of congestion are considered. When I = 0.5, the correlation coefficient will be chosen to take both values. For other values of I, it will be fixed as 0.3. Hence, there will be four problems of each of these two values. Table 2. Parameters of the

Problem size (*I 9 9 9 9 12 12 12 12 12 12 12 12 12 12 12 12 15 1s 15 1s

Problem No. 1 2 3 4

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4

test problems

Level of congestion

Average of processing times

Standard deviation of processing

Average of Average of due dates

Standard deviation of due dates

Uf

@J

(0,)

(a)

(Cd)

0.25 0.5 0.5 0.75 0.25 0.25 0.25 0.25 0.5 0.5 0.5 0.5 0.75 0.75 0.75 0.75 0.25 0.5 0.5 0.75

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

5 5 5 5 3 7 3 7 3 7 3 7 3 7 3 7 5 5 5 5

67.5 45 45 22.5 90 90 90 90 60 60 60 60 30 30 30 30 112.5 15 75 31.5

33.75 22.5 22.5 Il.25 27 21 63 63 18 18 42 42 9 9 21 21 56.25 37.5 37.5 18.75

Correlation 0.3 0.3 0 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0 0.3

792

B. V. CADAMBI

Clearly, by fixing pl. the other parameters can be determined. & is fixed as 10 for all problems. The parameters are as indicated in Table 2. C. Lower bound on the optimal value Since most branch and bound procedures fail to generate optimal solutions within a reasonable time, a lower bound is proposed which may be helpful to get some idea about the performance of the heuristic. The proposed bound is a composite bound based on three bounds. The value of PB obtained during the branch and bound algorithm [Z], based on obvious precedences, is a lower bound and is denoted by LBl. Clearly, extension of due dates of one or more jobs reduces the optimal value. Hence, a second lower bound is obtained by considering a perturbed problem, which is as follows: Increase the due date of each job to the minimal extent but so as to satisfy the condition + pair (i, j)qdj’ where dj’ and dj are the altered due dates. For the perturbed problem, the optimal sequence is clearly the Shortest Processing Time Sequence. The total tardiness of this sequence is another lower bound and is denoted by tB2. (It is infeasible to obtain a lower bound by reducing the mean processing times in a manner analogous to due dates, because the mathematical model has been developed for the case when the mean processing times are distinct.) A third lower bound will be developed based on the following theorem. Theorem 2: For a given sequence S, the expected tardiness of job k is at least as large as its tardiness on the equivalent deterministic problem. ProoE Without any loss of generality assume that S is 1 --f 2 -_)3 --, , . . -+ n. E(T,)=E[MaxfX,+X,+++++X,-d,,O}] >[MaxE(X,+X,+...+X,--$},O)]

Table 3. Percentage deviation of the heuristic solution value from tbe lower bound Lower bound PB based on obvious precedences (J-El)

Lower bound by extending due dates (LB2)

Lower bound based on dete~i~st~ optimal value (LB3)

Problem size

Problem No.

Heuristic solution vaiue (H)

9 9 9 9 12 12 12 12 12 12 12 12

1 2 3 4 i 2 3 4 5 6 7 8

34.6807 52.7427 19.2918 308.7709 56.6451 61.3118 1.992 65.6987 192.4658 233.9088 61.5070 320.4822

15.10 25.40 14.10 165.0 2.5.50 23.20 1.17 16.90 51.50 142.0 26.80 34.1

27.42 1.334 32.44 227.0 28.50 19.40 1.242 15.67 123.10 164.3 18.29 120.2

0 5.8159 29.3 1 237.8508 28.SO 0 0 0.1775 145.6678 187.4073 0 mi.i712

12 12 12 12 15 15 15

9 10 11 12 1 2 3

459.3698 554.2602 410.1272 489.9968 62.1838 t

216.0 251.0 260.0 156.0 17.50 21.1

316.7 381.1 309.0 204.2 21.50 29.2

15

4

24.2 262.0

33.6 395.6

t 640.773

tTbe heuristic algorithm failed to give a solution within reasonable time.

Overall bound (LB)= Max (LBl, LB2, LB3)

Percentage deviation AE!!x*OO LB

27.42 25.40 32.44 237.8508 28.50 23.m 1.243 16.90 145.6678 187.4073 26.80

26.4196 107.6484 144.4260 29.8171 98.7S47 164.2750 60.2570 288.7497 32.1265 24.8131 129.SO37

391.3871 455.446 354.1690 346.6545 0 37.3556

201.1712 391.3871 445.446 354.1690 346.6545 21.5 37.355

59.3082 17.3697 21.6961 is.7999 41.3502 189.227

30.04So 531.8439

33.6 531.8439

20.49

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Table 4. Percentage deviation of the heuristic value from the optimal value Percentage deviation Problem size 9 9 9 9 12 12 12 12 12 12

No.

Heuristicsolution value(H) 34.6807 52.7427 87.0799 308.7709 56.6451 61.3118 1.992 65.6987 61.5070 320.4822

Optimal solution (0)

,H_0XlOO 0

34.6807 50.81 t 258.5076 t 56.95 1.901 65.6987 60.5807 291.0940

0

3.8038 t 19.4436 t 7.6589 4.7870 0 1.5290 10.0958

tThe branch and bound algorithm failed to give a solution within the execution time. The following tables gives the expected total tardiness of the deterministic solution and the stochastic solution to examine the significance ofthedifference between the two versions. The values for the EDD Sequence and SPT Sequence are also included.

In view of theorem 2, we have Optimal Value of the deterministic problem < Value of the total tardiness of the stochastic optimal sequence for the deterministic problem < Optimal value for the stochastic problem Hence the optimal value of the deterministic problem is a third bound and is denoted by LB3. An overall bound is LB = Max(LB1, LB2, LB3). D. Analysis of test data

Since it is difficult to obtain the branch and bound solution within a reasonable time, the following procedure is adopted to evaluate the heuristic algorithm: The percentage deviation of the heuristic solution value from the lower bound is calculated for all problems of each size. The branch and bound procedure is then applied to all problems of size 9 and those problems of size 12, where the percentage deviation from the lower bound exceeds 50%. The branch and bound is not applied to problems of size 15. For each problem, for which the branch and bound is applied, the percentage deviation of the heuristic value from the optimal value is calculated. The next two tables give the percentage deviation of the heuristic value from the lower bound and the optimal value. Observations

1. The lower bounds are in general a gross underestimate of the optimal value. This is because the test problems were chosen so that the correlation between processing times and due dates is low. Hence, the lower bounds are by themselves not an indicator of the performance. 2. Whenever the deviation of the heuristic solution value from the optimal value was high, the deterministic sequence gave better results. (Size 9, Problem 4 and Size 12, Problem 8.) For other problems the deviation is quite low. 3. Whenever the stochastic sequence gave a better solution than the deterministic sequence, the deviation was not too high. 4. Comparison with the Earliest Due Date and Shortest Processing Time Sequence indicate that the algorithms are probably appropriate when the dispersions of both the processing times and due dates are at least moderate and/or when the correlations are low. (Size 9, Problem 3; Size 12, Problem 8 and Size 12, Problem 12.) 5. In view of (2) and (3) the deterministic sequence is recommended for extensive experimental investigation for its appropriateness to the stochastic version, since the stochastic version is complicated from the computational standpoint.

794

B. V. CADAMBI NON-DISTINCT

MEAN

PROCESSING

TIMES

It is difficult to develop expressions for the probability density function of the completion times of jobs when the assumption that no two jobs have identical mean processing times does not hold. Hence it is difficult to develop an anologous algorithm. An alternative approach utilizing a decomposition principle anlogous to that of Potts and Van Wassenhove [S] is proposed. For simplicity, we assume that jobs k and k + 1 are the only pair with Zk=Tk+l.

Jobs k+ 1 is temporarily removed from the set of jobs. The algorithm is applied to the other jobs. The sets of jobs are then partitioned into two-the first set a, containing job k and its predecessors in the sequence and the other set g2 containing the remaining jobs. Next, the algorithm is applied to the jobs in a2 and k+ 1, but with the due dates of jobs reduced by a value equal to the sum of the mean processing times of jobs in 0,. The final sequence is obtained by taking jobs in a, to precede jobs in lJ,U{k + l} and each subsequence in the order identified by the algorithm. The above approach may be considered for extensive experimentation. However, if experimentation with problems involving unequal mean processing times indicates that Wilkerson Irwin’s algorithm performs well for the stochastic version also, the algorithm can be utilized even when the mean processing times are not necessarily unequal. DISCUSSION

The proposed model is applicable in scheduling situations where meeting deadlines is crucial. Usually in such cases, time dependent penalties are levied for jobs that are delivered after the deadlines, but no benefits are obtained for jobs that are completed early [6]. The proposed model has the following limitations.

(4 Absence of weights: the relative importance of jobs, in practice, have to be distinguished by assigning weights. The deterministic version of the total weighted tardiness problem has been studied by Picard and Queyranne [7] and Rinnoykan et al. [8]. The weighted tardiness criterion has been reported to be very difficult. Thus, a separate treatment of the stochastic version is likely to be discouraging. (b) Non-linear penalty functions: the model is inapplicable when the penalty functions are non-linear. Emmons [9] and Picard and Queyranne [7] have explored the deterministic version. Here too, the stochastic version would be too complicated.

Table 5. Expected total tardiness

of the deterministic

Expected total tardiness Problem NO.

Deterministic solution (T,)

Stochastic solution (r,)

9 9 9 9 12 I2 12 12 12 12 12 12 12

I 2 3 4 1 2 3 4 5 6 7 8 9

35.5956 56.4437 87.0799 258.8163 56.6451 69.9544 1.9907 65.6987 190.2191 236.7274 64.4880 291.0940 412.3700

34.6807 52.7427 79.2918 308.7709 56.6451 61.3118 1.9920 65.6987 192.4568 233.9088 61.5070 320.4822 459.3698

12 12 12 15 15

10 11 12 1 4

482.5130 366.8709 396.1127 69.0974

554.2602 410.1272 489.9968 62.1838

559.6139

640.7773

Problem size

tThe value was not obtained

within reasonable

time.

and stochastic Percentage deviation

sequences Expected total tardiness EDD sequence

SPT sequence

2.6381 7.0171 9.8221 -16.1785 0 14.9061 -0.0653 0 - 1.1673 1.2050 4.8466 - 9.1700 - 10.2314

35.5956 56.4437 91.1603 329.3691 56.6451 69.9544 1.9907 65.6987 t 230.4087 68.1006 341.6798 429.0609

44.0082 135.0024 106.0321 262.8326 106.2624 69.5817 3.6242 123.5036 t 246.7679 t 318.566 969.1206

- 12.9447 - 10.5470 - 19.1601 11.1180 - 12.6664

474.8955 t 506.5347 67.1788 t

t t 402.2249 187.2456 t

T’-CxlOO r,

One machine scheduling-Part

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(6 Cases of controllable processing times: the model does not consider situations of controllable processing times as discussed by Vickson [S]. In practice, jobs may be expedited using overtime or through subcontracting. In such cases, the proposed model serves as a guideline to evaluate the workload prior to planning overtime/subcontracting rather than formal scheduling. The planning and scheduling functions in an organisation, as stated by Baker [6], frequently interact to arrive at a suitable plan. Hence such a model is useful in facilitating the interaction. Stochastic sequencing problems, like other decision making problems involving uncertainty should incorporate a measure of risk. Hence, it is desirable to compute variance of total tardiness as follows: \ ‘I

The values of variance of total tardiness besides expectation of total tardiness of the deterministic algorithm may be compared with the corresponding values for the EDD sequence, SPT sequence, the sequence obtained by ordering jobs in the increasing order of the average of mean processing time and due date and the optimal sequence using a large number of suitably chosen test problems. This will help to study the quality of solutions generated by the algorithm and clearly pinpoint the scenario where the algorithm can be useful. Acknowledgements-The

author is grateful to Dr Y. S. Sathe, Head, Department of Statistics, University of Bombay for his valuable guidance and to MS W. D’Souza for the neat typing of the manuscript. The author also expresses his sincere thanks to the two anonymous reviewers for their recommendations which helped to improve the paper. Editors note-Part I of this paper appeared in Vol. 18, No. 8 of this Journal in 1991. Part II should have appeared soon thereafter. It did not due to an oversight on my part.

REFERENCES 1. M. R. Garey and D. S. Johnson, Computers and Interactability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979). 2. B. V. Cadambi, One maching scheduling to minimise expected mean tardiness-Part I. Computers Ops Res. l&787-796 (1991). 3. R. G. Vickson, Choosing the job sequence and processing times to minimise total processing plus flow cost on a single machine. Ops Res. 28, 1155-1167 (1980). 4. V. Srinivasan, A hybrid algorithm for one machine sequencing problem to minimise total tardiness. Nav. Res. Logist Quart. 18, 317-327 (1971). 5. C. N. Potts and L. N. Van Wassenhove, A decomposition algorithm for the single machine total tardiness problem. Ops Res. Letters 1, 177-181 (1982). 6. K. R. Baker, Introduction to Sequencing and Scheduling. Wiley, New York (1974). 7. J. C. Picard and M. Queyranne, The time-dependent traveling salesman problem and applications to the tardiness problem in one-machine scheduling. Ops Res. 26, 86-110 (1978). 8. A. H. G. Rinnoykan, B. J. Lageweg and J. K. Lenstra, Minimising total costs in one machine scheduling. Ops Res. 23, 908-927 (1975). 9. H. Emmons, One machine sequencing problem to minimisecertain function ofjob tardiness. OpsRes.17,701-715 (1969). 10. L. J. Wilkerson and J. D. Irwin, An improved method for scheduling independent tasks. A.I.I.E. Trans. 3,239-245(1971),