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An efficient dynamic dispatching rule for scheduling in a job shop
Internationul Elsevier
Journal of Production
301
Economics, 32 (1993) 301- 313
An efficient dynamic shop
dispatching
rule for scheduling
in a job
T.S. Raghu and Chandrasekharan Rajendran Diaision qf Industrial Engineering and Management, Department qj’ Humanities and Sociul Sciences, Indian Institute of Technology, (Received
Mudras 600 036. lndiu
4 January
1993; Accepted
in revised form 28 May 1993)
Abstract In this paper we present a new dispatchmg rule which is dynamic and global in its structure. The rule adapts itself to the varration in the shop floor utilization level and assigns appropriate weights to the process time and due date information accordingly. Another important feature of the proposed rule is that the weights to the two components, viz. the process time and the due date components, differ from machine to machine. The results of a simulation study are presented to demonstrate the effectiveness of the rule with respect to the rules most commonly used in industry and the proven rules reported in the literature. The proposed rule is found to perform quite well with respect to mean flow time, mean tardiness and variation in tardiness.
1. Introduction Scheduling in a job shop is an important aspect of the shop floor management system, which can have a significant impact on the performance of the shop floor. The decision as to which job is to be loaded to a machine when it becomes free is normally made with the help of dispatching rules. Over the years many dispatching rules have been proposed by researchers. However, no rule has been found to perform well for important criteria such as the minimization of mean flow time and mean tardiness under different shop conditions. The choice of a rule depends on which criterion is intended to be improved upon. In general, it has been found that process-time-based rules fare better under tight load conditions, whereas due-date-based rules fare well under light Correspondence to: C. Rajendran, Division of Industrial Engineering and Management, Department of Humanities and Social Sciences, Indian Institute of Technology, Madras 600 036, India.
09255273/93/$06.00
(0 1993 Elsevier Science Publishers
load levels [l, 21. Another complication in the use of the dispatching rules is that some of them are sensitive to the shop floor configuration. A job shop can be classified as either an open or a closed shop, depending on the way in which the jobs are routed in the shop. In a closed shop the number of routings available to the job are fixed and an arriving job can follow one of the available routings. In an open shop there is no limitation on the routing of the jobs; thus each job could have a different routing. In this article we discuss a dynamic scheduling problem in an open shop configuration. A new dispatching rule, aimed at minimizing both flow time and tardiness, is proposed which is dynamic as well as global in its structure.
2. Terminology pii: processing time for operation j of job i, mi: number of operations on job i,
B.V. All rights reserved
Di: due date for job i, di.i: operation due date assigned to operation ,j of job i, t: time at which the dispatching decision is taken, W,: expected waiting time of job i for operatton q, Ci: completion time of job i, hi: time of arrival of job i, Zij: priority index given to operationj ofjob i (lowest Zij gets the highest priority).
3. Literature
survey
It is common practice to have the standard assumptions incorporated in the job shop model in the simulation studies [3]. Some of these assumptions, which have also been made in our simulation study, are listed below: 1. Each machine can perform only one operation at a time on a job. 2. A job once taken up for processing should be completed before another job is taken, i.e. job pre-emption is not allowed. 3. No two successive operations of a job can be performed on the same machine. 4. An operation cannot be taken up for processing until its previous operations are complete. 5. There are no limiting resources other than the machines. 6. There are no interruptions in the shop floor, e.g. no machine breakdowns. 8. There are no alternate routes. 9. The jobs are independent of each other, i.e. no assembly is involved. 10. There are no parallel machines. Dispatching rules can be classified in a number of ways [7]. One such classification is as follows: 1. processing-time-based rules, 2. due-date-based rules, 3. combination rules, 4. rules which arc neither processing-timebased nor due-date-based. The process-time-based rules totally ignore the due date information of the shop. The shortest processing time (SPT) is an example of
a process-time-based rule. The SPT rule has been found to minimize the flow time and a good due date performance is also observed under highly loaded conditions [ 1,2,4,5]. The due-date-based rules schedule the jobs based on the due date information of the job [6, 71. An example of a due-date-based rule is the earliest due date (EDD) rule, in which highest priority is given to a job which has the least due date. In general, the due-date-based rules give good results under light load conditions but the performance of these rules deteriorates under high load levels [S]. An obvious extension of these two types is a combination rule. Quite a number of combination rules have been tried in which priority of a job is determined based on both process time and due date [4,9, lo]. The rules which do not fall into any of the above categories load the jobs depending on the shop floor conditions rather than on the characteristics of the jobs. An example of this type of rule is WINQ (work in next queue), which sends the next job to the machine with the least queue length. The due-date-based rules and the combination rules can be categorized based on whether the priority value changes over time or not. A dispatching rule is called a dynamic rule if the priority value calculated at a particular instant differs from the value calculated at a later time. If the priority value once calculated remains the same throughout, such a rule is called a time-independent rule. The EDD rule falls into the latter category of a time-independent rule. On the other hand, the least slack rule, where the slack is defined as slack = due date - current time - total remaining process times, falls into the category of dynamic rules [l]. One more aspect in which the rules may differ from each other is the type of information they need. Local rules require information about jobs currently waiting for the machine under consideration whereas global rules make use of information about other machines
T.S. Raghu, C. RajmdranlAn
303
in the shop floor in addition to the machine under consideration. A detailed discussion of dispatching rules can be found in Blackstone et al. [6] and Haupt [7]. An obvious inference, which may not always be generalized, that can be made is that the more the information contained in the dispatching rule, the better its performance. Before the proposed rule is presented, some of the rules relevant to our study are discussed. The idea of an operational due date has been proposed by Conway et al. [S]. The operational due date is nothing but a deadline set for each operation of the job. The operation due dates help in closely monitoring the job’s due date, thus resulting in a better performance. The idea has been extended by Baker and Kanet [ 111, who proposed the modified operational due date (MOD) rule. In this rule the operational due date is used in deciding the priority of the job until it is operationally late, from which moment on the earliest finish time of the operation is treated as the due date. The rule can be expressed as follows:
where k is a look-ahead parameter which scales the slack according to the expected number of competing jobs and p denotes the expected processing time of an operation. The job with the maximum value of Zij is chosen for loading. The value of k used in their simulation study is 3. This look-ahead parameter can be adjusted depending on the load level of the shop. The suggested range of values for k is between 1.5 and 4.5. The waiting time for each operation, Wiqr is calculated as a multiple of the processing time of that operation. Anderson and Nyirenda [4] have come up with a modification of the MOD rule in which the operation due dates are set dynamically. They proposed two ways of setting the operational due dates which are described below. CR + SPT rule: This rule sets the operational date as a multiple of the processing time of the imminent operation and the critical ratio of that job, where the critical ratio is expressed as CR = allowance/total time
remaining
processing
Zij = max(&j, pij + t). The operation due dates, dij, are static and are calculated as follows:
dij
di,
=
j-l
+
tDi - Ti)Pij m,
-
3
C Piq q=l where die = Ti. Vepsalainen and Morton [12] proposed the apparent tardiness cost (ATC) rule, which uses an exponential function of operational slack. The rule can be expressed as follows:
Zij
= i
Pij “l,
Dtx
exp
c
16 ,o Ii
y=,+
(W,q+~,)-r-~,,
The operation due date, dij, for operation the job i is given as
j of
dij= CR pij.
Thus, the CR + SPT rule can be expressed
as
Zij = max (CR pij; pij). S/RPT + SPT rule: Another rule proposed by Anderson and Nyirenda uses slack per remaining operation time as the multiplier of processing time in setting operational due dates. The rule can be expressed as S/RPT
= slack/total time.
remaining
processing
The operation due date, dij, for operation the job i is given as dij = (S/RPT)pij.
j of
Thus the S/RPT + SPT rule can be expressed as Zij = max(S/RPT
pij;Pij).
Both the rules have been found to have similar performance characteristics, with the S/RPT +SPT being more efficient in minimizing the number of tardy jobs and the CR + SPT rule being more efficient in minimizing mean tardiness. The rules MOD, ATC, CR + SPT and S/!RPT + SPT have one similarity in that they neglect the due date information soon after the job has become tardy and use only the processing time in assigning priorities. In situations where there is quite a variation in the processing times, this could make certain jobs very tardy just as the way the SPT rule tends to make some jobs very tardy.
4. Proposed rule The motivation of this study has been the results of the past simulation studies. It has generally been found that the SPT rule performs better under tight load conditions and the due-date-based rules are found to perform better under light load conditions [l, 2, 6-S]. Thus, if proper weights are given to the process time and due date components of the job depending on the utilization level in the shop, we could expect a good performance of such a rule under all utilization levels. We can therefore have a combination of processing time and slack per remaining processing time with different weights given to them, the weights being a function of the utilization level of the shop. We define the utilization level of a machine as ‘1 =
total total
service
service
time
time + total
free time‘
Obviously, we have 0 < q < 1. The weight factors are exp(q) for pij and exp( - 7) for (S/RPTpjj). Under tight load conditions, the process ttme component will be dominant, while the due date or slack compon-
ent will be dominant under light load conditions. The negative exponential term in the due date component also serves to reduce the magnitude of the slack and thus reduces the disparity in the magnitude of the process time and due date terms. In addition, the coefficients for the process time and due date components are influenced by the shop utilization level, rendering the coefficients dynamic in nature. Yet another indication of the utilization level of the shop is the waiting time at the subsequent machines. The expected waiting time beyond the next immediate machine cannot be estimated with a fair degree of accuracy due to the dynamic nature of the shop. It is obvious that we cannot judge the queue content of machines beyond the machine performing the next immediate operation. Past research has calculated the look-ahead parameter as either the total processing time of the jobs in the queue or the queue length at the next machine [l]. These approaches do not take into account the probable relative priority of the job when it enters the queue at the machine of the next operation. The look-ahead parameter used in our rule is different from these approaches in that it provides a more accurate estimate of waiting time for the job on the next machine. The look-ahead is calculated as follows: Assume that a job, i, is taken up for processing on the current machine at the time instant t, then it joins the queue on the next machine at t + pij. Let f’ be the time at which the next machine becomes free. The time at which the scheduling decision involving this job is made on the next machine is at 2’. The priority indices of all the jobs in the queue are calculated at this time t’. Thus. the expected waiting time of the job i at the next machine is the sum of the processing times of all the jobs which have a higher priority than i, and is denoted by fl,,,. The proposed rule’s priority index Zii is calculated as follows: Z,,; = S:RPTexp(
- ‘I)Pij + exp(pj)Pij + IV,,,.
The job with the least Zi,; is loaded machine.
on the
It is clear that three components, due date or slack ent of the waiting
the proposed rule consists of viz, process time component, component, and the compontime for the next operation.
5.2.3.
5.1. Dispatching
rules used in the simulation
The dispatching rules used in the simulation study for evaluation purposes are SPT, EDD, MOD, ATC, (S/RPT + SPT) and (CR + SPT). The SPT and EDD rules are chosen because they are simple to implement and are widely rules are used in practice. The remaining chosen because they have been found to perform the best.
Root mean square qf’ tardiness
Normally a shop floor manager would prefer many jobs to be a little tardy than to have a few jobs finishing very tardy. The root mean square tardiness is a good measure to evaluate this factor. Root mean square of tardiness, T rmS, is defined as (l/n) i i=
5.3. E.perimental 5.2. Pe~fkwmance measures The performance measures used in the simulation experiment to evaluate the dispatching rules are given below. 5.2.1. Flo\+’ time (Fi ) This refers to the time a job i spends in the shop floor or the resident time of a job i in the shop floor. This measure indicates the workin-process inventory. The term Fi is defined as Ci - Ti. Usually, the mean flow time is used as a performance criterion, which can be defined as F=
(l/n)
~ (Ci - pi), i=l
where n refers scheduled. 5.2.2.
to the total
of jobs
Tardiness
Mean tardiness ?:= (l/n)
number
i i=
can be defined as
max(O, Li), 1
where lateness,
Li, is defined as (Ci - Di).
of’ tardy ,johs
This gives the proportion of the total number of jobs which could not finish on time. 5.2.4.
5. Simulation experimentation
Proportion
[max(O, Li))’ 1
1’ ‘12.
conditions
The simulation experiment was conducted in an open shop configuration consisting of 12 machines. The routing for each order was different and generated randomly, with each machine having an equal probability of being chosen. The number of operations for each job was uniformly distributed between 4 and 12. The process times are drawn from rectangular distributions. Three process time distributions are used: 1. t-50, 2. l-100, 3. 25% of the orders with 50-100 and the rest with l--50. The last distribution is used to evaluate the rules in a shop where there is a large variability in process time distributions. The total work content (TWK) method of due date setting [6] is used in all the experiments with allowance factors of 4, 5 and 6. The order arrivals are generated using an exponential distribution. Two machine utilization levels are tested in the experiments, viz. 85% and 95%. Thus, there are three types of process time distributions, three different modes of due date setting and two different utilization levels, making a total number of 18 simulation experiment sets for every dispatching rule. Each simulation experiment consists of six different runs using antithetic random numbers for each
306
T.S. Rughu, C. RqjcwirranJ An efficient ciwmnic
number of replications is between 5 and 10 [14]. The method suggested by Fishman [15] has been followed in the present study to fix the total sample size. We have found that a sample size of 12 000 is adequate to yield the desired standard error, which limits the tolerance level of all performance criteria at less than 5% of the expected mean values at a 95% confidence level. The number of replications has been fixed in accordance with the suggestion made by Law and Kelton [14] and is fixed at 6, which in turn fixes the run length for a replication at 2000. The data from orders numbering from 501 to 2500 are collected for evaluation purposes, and the shop is continuously loaded with orders until the completion of the last of the 2000 orders. This helps in overcoming the problem of “censored data”
pair of runs. In each run the shop is continuously loaded with orders which are numbered on arrival. The orders are numbered as they arrive in the shop. 5.3.1.
Stea&-state
condition qf’ the shop
In order to ascertain when the system reaches steady state, we observed the shop parameters continuously. It was found using a graphical technique that the shop reaches a steady-state condition when 500 job orders are completed. Graphs of the utilization pattern of the shop for three dispatching rules, viz. SPT, S/RPT + SPT and NEW, are presented in Figs. l-3, respectively. The same phenomenon is also found to occur with other dispatching rules. 5.3.2.
disputching rule
Cll.
Run !ength and number qfreplications
Typically, the total sample size in simulation studies of job shop scheduling has been of the order of thousands of job completions [ 13,121. For the same total sample size it is preferable to have a smaller number of replications and a larger run length, and the recommended
6. Results and discussion The results of the simulation study are presented in Tables 1-6. The results shown are obtained by taking the mean of the mean
I
0
500
ORDER
Fig. 1.
1500
LOO0 llJW3tP
2000
2500
T.S. Raghu, C. Rqjendran!An
eficienf
ORDER
c(vnarnic dispatching rule
307
RlJHBtR
Fig. 2.
0 5 -
0 t
0
-
3 -
oil-
0
I
0
-
,.........,,1.......1.........,.........(.........,,
0
so0
1000 ORDER
1500
2000
7.500
IIUliEER
Fig. 3.
values of the six replications. All the performance criteria except for the percentage of tardy jobs are given relative to the best performing rule. “Percentage tardiness” indicates the percentage of jobs which are tardy. For
example, in Table 1 the flow time of the ATC rule is 8.4372% higher than the NEW rule and, likewise, the percentage number of tardy jobs for the SPT rule in Table 1 is 28. The relative percentage error values are calculated as
308 Table 1 Utilization
level of 85% and the process
time sampled
from a rectangular
( 1. SO)
T rm\
Percentage tardy jobs
86.8948 144.9596 38.1021 35.4636 20.4294 21.2010 0.0000
222.1262 67.92 11 152.4637 149.907 1 133.2953 98.7575 0.0000
28 81 50 40 37 62 52
0.0000
245.8172 243.1579 90.9972 89.0443 27.4654 27.9363 0.0000
407.5429 107.3889 226.1 114 231.5031 I SO.7978 106.8772 0.0000
20 62 31 25 1X 36 30
0.7227 31.1059 20.5982 16.9895 12.1477 I 1.2677 0.0000
727.0735 440.9233 227.5039 214.4366 27.1909 15.6886 0.0000
786.4308 159.2308 364.3282 358.7795 135.6615 71.7846 0.0000
15 43 20 15 7 15 12
Due date tightness
Rule
Flow time
Tardiness
4
SPT EDD MOD ATC SiRPT + SPT CR t SPT NEW*
1.9995 33.8338 10.7472 8.4372 7.2887 8.6131 0.0000
-
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW *Denotes
distribution
the proposed
1.7977 33.2096 15.9445 13.1420 10.0344
11.2902
rule
follows: If F,, denotes the mean flow time of the rule n, n = 1 indicates the SPT rule, n = 2 indicates the EDD rule, and so on. Relative percentage increase in flow time for rule y1with respect to the best performing rule is given by F, - min(F,,
Vn < 7)
min(F,,
7)
Vn d
-
of
x 100.
Relative percentage increase in tardiness and relative percentage increase in root mean square of tardiness are calculated in a similar manner. The results clearly show the superiority of the NEW rule with respect to flow time, tardiness and root mean square of tardiness, whereas the S/RPT + SPT rule and the SPT rule give a low value of the percentage
of tardy jobs. We now discuss detail.
the results
in
6. I. Flow time performance The NEW rule is seen to reduce the mean flow time in most of the cases, with SPT consistently giving the second best result. However, at 85% load level, the performances of the SPT rule and the NEW rule are comparable. The improvement in flow time performance of the NEW rule at high load levels can be attributed to the look-ahead feature incorporated in the NEW rule. Since the estimated waiting time on the next machine is also used in arriving at the priority of a job, the jobs which have less expected waiting times have a better chance of being selected. This ultimately helps in reducing the waiting time of the jobs; thus a reduction in flow time can be expected. The rules such as SPT, S/RPT + SPT, etc., do not incorporate such a look-ahead
T.S. Raghu, C. Rajwdran/An Table 2 Utilization
level of 95% and the process
time sampled
from a rectangular Tardiness
Due date tightness
Rule
Flow time
4
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
12.7611 3 I .0990 15.1008 14.9479 13.7099 13.5939 0.0000
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW SPT EDD MOD ATC S,‘RPT + SPT CR + SPT NEW
6
Table 3 Utilization
level of 85% and the process
309
crflikwt cfvnarnic dispatching rule
distribution
(1, 50)
T rnls
Percentage tardy jobs
47.9683 65.5118 32.1221 33.0388 29.1002 27.6019 0.0000
200.5432 30.3272 165.9366 166.6801 156.7419 131.1927 0.0000
40 97 75 66 67 92 86
16.7126 35.2069 22.5455 20.2993 16.2639 16.0689 0.0000
99.8229 100.2796 64.4060 60.7040 44.7732 42.1942 0.0000
289.348 1 48.4090 222.3287 217.1 173 177.8742 153.3769 0.0000
33 92 63 54 56 81 72
18.4309 36.2368 27.065 1 25.1079 16.9617 15.2724 0.0000
189.3485 145.9801 110.7025 107.6676 62.4279 53.0860 0.0000
429.5135 72.4707 296.6532 299.3521 223.7108 192.0309 0.0000
27 85 54 43 44 67 56
time sampled
from a rectangular
distribution
of
(1, 100)
Due date tightness
Rule
Flow time
Tardiness
T rms
Percentage tardy jobs
4
SPT EDD MOD ATC SJRPT + SPT CR + SPT NEW
4.0288 36.5182 12.0337 9.8324 8.705 1 10.1302 0.0000
96.0304 155.7973 44.7600 42.9279 28.3182 28.8844 0.0000
270.6791 73.1987 184.4453 190.0320 168.0830 134.1732 0.0000
28 85 50 41 40 66 56
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
3.9534 34.3271 16.9411 13.0661 10.8307 10.6296 0.0000
258.8882 245.559 1 99.2332 88.9137 35.0351 22.4473 0.0000
458.0680 100.3590 274.407 1 260.6600 192.2711 125.0260 0.0000
20 66 33 25 20 37 31
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
2.4000 3 1.8420 19.6040 15.6688 11.7946 11.1681 0.0000
639.8570 380.7938 186.9525 170.4684 21.1679 2.7676 0.0000
788.7714 137.8903 402.9274 387.2994 168.8643 84.2642 0.0000
15 46 18 15 7 15 15
of
Table 4 Utilization
level of 95% and the process
time sampled
from a rectangular Tardiness
distribution
(1, 100)
T rm\
Percentage tardy jobs
47.0532 61.8642 33.6945 32.5910 28.0518 23.7990 0.0000
213.7906 21.3361 184.1545 181.4566 166.3569 136.1070 0.0000
40 97 75 66 61 91 x7
16.2532 33.2861 21.7343 19.0402 14.3846 12.7156 0.0000
91.5708 88.5835 58.7976 53.8735 37.1804 31.0541 0.0000
304.194X 37.8069 236.8890 234.4065 189.5866 161.2468 0.0000
31 93 64 54 57 X2 15
17.7649 34.7261 25.6512 24.1722 13.7293 12.9730 0.0000
165.464 I 125.5363 94.5987 93.3008 44.5365 39.5098 0.0000
421.1002 53.7452 299.9797 300.9190 212.7873 188.6144 0.0000
25 86 56 46 44 68 61
Due date tightness
Rule
Flow time
4
SPT EDD MOD ATC S;RPT + SPT CR + SPT NEW
13.4241 30.6394 16.5362 15.4318 13.8251 12.2478 0.0000
SPT EDD MOD ATC S:‘RPT + SPT CR + SPT NEW SPT EDD MOD ATC .S/RPT + SPT CR + SPT NEW
Table 5 Utilization level of 85% and the process time sampled (50, 100) for 25% of jobs
from a rectangular
distribution
of
(1, 50) for 75% of jobs and from
Due date tightness
Rule
Flow time
Tardiness
T rm\
Percentage tardy jobs
4
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
0.0000 31.7145 9.4708 7.2456 5.3319 7.4174 0.5557
5X.8726 124.0100 24.2918 25.9650 9.8002 11.8837 0.0000
212.7690 41.5974 150.0647 162.3621 12 I .3298 96.8593 0.0000
17 19 44 28 28 58 50
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
1.6386 30.3706 17.2665 13.8233 11.5322 11.1699 0.0000
199.2851 21 1.9181 80.5 102 80.7783 34.5601 23.438 I 0.0000
370.27 I 3 63.6063 2 19.4007 222.7526 149.5210 96.7980 0.0000
II 60 30 18 15 34 25
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
2.5865 24.9303 21.6656 19.0086 14.6390 13.4009 0.0000
674.6550 363.9649 201.2045 210.8156 30.0627 13.8519 0.0000
78 1.3986 102.8175 331.5508 336.7857 146.3879 83.0835 0.0000
9 37 18 11 5 14 10
of
T.S. Rqhu,
C. RujmdranlAn
Table 6 Utilization level of 95% and the process time sampled (50, 100) for 25% of jobs Due date tightness
Rule
Flow time
4
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
4.5261 32.9393 7.8406 6.8356 5.275 I 5.0869 0.0000
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
efficimt
d,mmic
from a rectangular
311
dispatching rule
distribution
and from
T TlllS
Percentage tardy jobs
30.4698 68.4007 15.5298 15.9419 10.9542 8.4829 0.0000
227.6415 26.1194 188.0045 192.9478 177.8967 148.3783 0.0000
21 95 62 40 43 81 80
I 1.5787 35.0905 19.7082 18.3125 11.7197 12.5400 0.0000
88.5973 102.0684 54.3595 55.2222 29.9789 29.9047 0.0000
339.1032 42.4643 271.1667 269.2080 215.8599 195.9052 0.0000
17 83 49 31 34 68 62
15.0032 37.6698 26.5705 25.7235 16.1041 13.3890 0.0000
179.6940 156.0178 101.5671 107.2468 52.3056 36.0592 0.0000
475.6203 60.9889 350.0465 354.5147 256.7947 196.0747 0.0000
14 74 40 25 25 54 44
factor. It is evident from the results that the improvement in flow time is more pronounced at high load levels than at low load levels. This is due to the fact that at high load levels the queue lengths at the machines are high, and hence the look-ahead feature, i.e. the expected waiting time at the next machine, becomes a very useful factor in reducing the overall waiting time of the jobs; thus an improvement in mean flow time is achieved. However, at lower levels the difference in performance between the NEW rule and the SPT rule becomes less pronounced due to relatively shorter queue lengths at the machines.
Tardiness
(1, 50) for 75% ofjobs
of
performance of the NEW rule and the next best performing rule is comparable since the look-ahead factor may become less effective under these conditions due to loose due date setting and relatively less waiting time. The SPT rule fares badly with the tardiness criterion as the due date tightness slackens. The better performance of the NEW rule with respect to the tardiness criterion can be attributed to the change in weights of the process time and slack components depending on the utilization level. 4.3. Root mean square value
qf tardiness
performance
6.2. Tardiness performance The NEW rule is consistently seen to give the best measures of mean tardiness compared to other rules, with S/RPT + SPT and CR + SPT yielding the next best results. However, sometimes at high due date allowances and low utilization levels it is found that the
A major difference between the proposed rule and the other rules tested, except for EDD, is the performance under root mean square value of tardiness. The proposed rule outperforms the rest of the rules by a large margin. This is not unexpected because the rules such as SPT, S/RPT + SPT and
CR + SPT ignore the due date information as soon as a job becomes tardy. This results in some jobs finishing very tardy. Although EDD is seen to do better under this criterion, its performance with respect to other criteria is not quite encouraging. It can therefore be inferred from the performance under this criterion that if minimization of variance in tardiness is the dominant criterion, it is not prudent to ignore the due date information altogether.
The SPT rule consistently gives a low number of tardy jobs. S/RPT + SPT is found to be the second best performing rule under this criterion. The use of SPT has always been found to result in a low number of tardy jobs. One plausible explanation for the better performance of S/RPT + SPT over CR + SPT is that the process time is used for priority calculation earlier in S/RPT + SPT. This result conforms with the findings of Anderson and Nyirenda [4] in their study. The NEW rule is found to perform better than CR + SPT and EDD with respect to this criterion.
coming very tardy. The simulation study shows that this is indeed the case. The proposed rule does not suffer from this drawback, as it retains the due date information even after a job becomes tardy. By giving appropriate weights to the process time and due date information depending on the utilization of the machines, the proposed rule tries to adapt to changes in the machine utilization levels of the shop floor. It should be noted that, unlike some of the combination rules, the proposed rule does not require a priori setting of weights for the different components. By estimating the waiting time at the next machine, the processing of jobs which have to wait for a long time on the next machine can be delayed. This can help in expediting some jobs which may otherwise be held up. The incorporation of such a look-ahead feature helps in reducing the expected waiting time of jobs. That this is indeed the case has been shown by the simulation results in which the proposed ruIe is found to give better flow time results that the SPT rule at high load levels.
Acknowledgement 7. Conclusions In this paper we have proposed a new rule which is dynamic and global in its structure. It consists of three components, viz. process time component, slack component and the component of waiting time for the next operation. We assign dynamic weights to the process time and the slack component. A simulation study conducted shows that the rule demonstrates the potential for improved performance in flow time, tardiness and root mean square tardiness. Some of the advantages of the proposed rule are listed below. 1. The rules such as CR + SPT, S/RPT + SPT, MOD and ATC, which ignore the due date information soon after a job becomes tardy, have the disadvantage that they result in a small number of jobs be-
We thank the referees for many constructive suggestions and comments which led to a revision of an earlier version of the paper.
References 111 Conway. R.W., 1965. Priority dispatching
and job lateness in a job shop. .I. Indust. Eng., 16: 123-130. [?I Rochette, R. and Sadowski, R.P., 1976. A statistical comparison of the performance of simple dispatching rules for a particular set of job shops. Int. J. Prod. Res.. 14: 63-75. to Sequencing and c31 Baker, K.R., 1974. Introduction Scheduling. Wiley, New York. c41 Anderson, E.J. and Nyirenda, J.C., 1990. Two new rules to minimize tardiness in a job shop. Int. J. Prod. Res., 28: 2277-2292. ISI Conway, R.W.. Maxwell, W.L. and Miller, L.W.. 1967. Theory of Scheduling. Addison-Wesley, Reading. MA.
T.S. Raghu, C.. RqjendronlAn [6] Blackstone, J.H., Phillips, D.T. and Hogg, G.L., 1982. A state-of-the-art survey of dispatching rules for manufacturing job shop operations. Int. J. Prod. Res., 20: 2745. [7] Haupt, R., 1989. A survey of priority rule-based scheduling. OR Spektrum. 11: 3--16. [8] Ramasesh, R., 1990. Dynamic job shop scheduling: A survey of simulation research. OMEGA. 16: 43357. [9] Eilon. S. and Cotterill. D.J., 1968. A modified SI rule in job shop scheduling. Int. J. Prod. Res., 7: 135-145. [lo] Oral, M. and Malouin. J.L., 1973. Evaluation of shortest processing time scheduling rule with truncation process. AIIE Trans., 5: 357- 365.
t$ficirnt ~Jwamic~ disputching ruk
313
[I 11 Baker, K.R. and Kanet, J.J., 1982. Job shop scheduling with modified due dates. J. Oper. Mgmt.. 4: 11-22. 1123 Vepsalainen. A.P.J. and Morton, T.E., 1987. Priority rules for job shops with weighted tardiness costs. Mgmt. Sci., 33: 103551047. [ 131 Conway, R.W., Johnson, B.M. and Maxwell, W.L., 1960. An experimental investigation of priority dispatching. J. Indust. Eng., 11: 221 230. [ 141 Law, A.M. and Kelton, W.D., 1984. Confidence intervals for steady state-simulation: I. A survey of fixed sample size procedures. Oper. Res.. 32: 1221~ 1239. [IS] Fishman. G.S.. 1971. Estimating sample size in computing simulation experiments Mgmt. Sci., 18: 21l38.
Journal of Production
301
Economics, 32 (1993) 301- 313
An efficient dynamic shop
dispatching
rule for scheduling
in a job
T.S. Raghu and Chandrasekharan Rajendran Diaision qf Industrial Engineering and Management, Department qj’ Humanities and Sociul Sciences, Indian Institute of Technology, (Received
Mudras 600 036. lndiu
4 January
1993; Accepted
in revised form 28 May 1993)
Abstract In this paper we present a new dispatchmg rule which is dynamic and global in its structure. The rule adapts itself to the varration in the shop floor utilization level and assigns appropriate weights to the process time and due date information accordingly. Another important feature of the proposed rule is that the weights to the two components, viz. the process time and the due date components, differ from machine to machine. The results of a simulation study are presented to demonstrate the effectiveness of the rule with respect to the rules most commonly used in industry and the proven rules reported in the literature. The proposed rule is found to perform quite well with respect to mean flow time, mean tardiness and variation in tardiness.
1. Introduction Scheduling in a job shop is an important aspect of the shop floor management system, which can have a significant impact on the performance of the shop floor. The decision as to which job is to be loaded to a machine when it becomes free is normally made with the help of dispatching rules. Over the years many dispatching rules have been proposed by researchers. However, no rule has been found to perform well for important criteria such as the minimization of mean flow time and mean tardiness under different shop conditions. The choice of a rule depends on which criterion is intended to be improved upon. In general, it has been found that process-time-based rules fare better under tight load conditions, whereas due-date-based rules fare well under light Correspondence to: C. Rajendran, Division of Industrial Engineering and Management, Department of Humanities and Social Sciences, Indian Institute of Technology, Madras 600 036, India.
09255273/93/$06.00
(0 1993 Elsevier Science Publishers
load levels [l, 21. Another complication in the use of the dispatching rules is that some of them are sensitive to the shop floor configuration. A job shop can be classified as either an open or a closed shop, depending on the way in which the jobs are routed in the shop. In a closed shop the number of routings available to the job are fixed and an arriving job can follow one of the available routings. In an open shop there is no limitation on the routing of the jobs; thus each job could have a different routing. In this article we discuss a dynamic scheduling problem in an open shop configuration. A new dispatching rule, aimed at minimizing both flow time and tardiness, is proposed which is dynamic as well as global in its structure.
2. Terminology pii: processing time for operation j of job i, mi: number of operations on job i,
B.V. All rights reserved
Di: due date for job i, di.i: operation due date assigned to operation ,j of job i, t: time at which the dispatching decision is taken, W,: expected waiting time of job i for operatton q, Ci: completion time of job i, hi: time of arrival of job i, Zij: priority index given to operationj ofjob i (lowest Zij gets the highest priority).
3. Literature
survey
It is common practice to have the standard assumptions incorporated in the job shop model in the simulation studies [3]. Some of these assumptions, which have also been made in our simulation study, are listed below: 1. Each machine can perform only one operation at a time on a job. 2. A job once taken up for processing should be completed before another job is taken, i.e. job pre-emption is not allowed. 3. No two successive operations of a job can be performed on the same machine. 4. An operation cannot be taken up for processing until its previous operations are complete. 5. There are no limiting resources other than the machines. 6. There are no interruptions in the shop floor, e.g. no machine breakdowns. 8. There are no alternate routes. 9. The jobs are independent of each other, i.e. no assembly is involved. 10. There are no parallel machines. Dispatching rules can be classified in a number of ways [7]. One such classification is as follows: 1. processing-time-based rules, 2. due-date-based rules, 3. combination rules, 4. rules which arc neither processing-timebased nor due-date-based. The process-time-based rules totally ignore the due date information of the shop. The shortest processing time (SPT) is an example of
a process-time-based rule. The SPT rule has been found to minimize the flow time and a good due date performance is also observed under highly loaded conditions [ 1,2,4,5]. The due-date-based rules schedule the jobs based on the due date information of the job [6, 71. An example of a due-date-based rule is the earliest due date (EDD) rule, in which highest priority is given to a job which has the least due date. In general, the due-date-based rules give good results under light load conditions but the performance of these rules deteriorates under high load levels [S]. An obvious extension of these two types is a combination rule. Quite a number of combination rules have been tried in which priority of a job is determined based on both process time and due date [4,9, lo]. The rules which do not fall into any of the above categories load the jobs depending on the shop floor conditions rather than on the characteristics of the jobs. An example of this type of rule is WINQ (work in next queue), which sends the next job to the machine with the least queue length. The due-date-based rules and the combination rules can be categorized based on whether the priority value changes over time or not. A dispatching rule is called a dynamic rule if the priority value calculated at a particular instant differs from the value calculated at a later time. If the priority value once calculated remains the same throughout, such a rule is called a time-independent rule. The EDD rule falls into the latter category of a time-independent rule. On the other hand, the least slack rule, where the slack is defined as slack = due date - current time - total remaining process times, falls into the category of dynamic rules [l]. One more aspect in which the rules may differ from each other is the type of information they need. Local rules require information about jobs currently waiting for the machine under consideration whereas global rules make use of information about other machines
T.S. Raghu, C. RajmdranlAn
303
in the shop floor in addition to the machine under consideration. A detailed discussion of dispatching rules can be found in Blackstone et al. [6] and Haupt [7]. An obvious inference, which may not always be generalized, that can be made is that the more the information contained in the dispatching rule, the better its performance. Before the proposed rule is presented, some of the rules relevant to our study are discussed. The idea of an operational due date has been proposed by Conway et al. [S]. The operational due date is nothing but a deadline set for each operation of the job. The operation due dates help in closely monitoring the job’s due date, thus resulting in a better performance. The idea has been extended by Baker and Kanet [ 111, who proposed the modified operational due date (MOD) rule. In this rule the operational due date is used in deciding the priority of the job until it is operationally late, from which moment on the earliest finish time of the operation is treated as the due date. The rule can be expressed as follows:
where k is a look-ahead parameter which scales the slack according to the expected number of competing jobs and p denotes the expected processing time of an operation. The job with the maximum value of Zij is chosen for loading. The value of k used in their simulation study is 3. This look-ahead parameter can be adjusted depending on the load level of the shop. The suggested range of values for k is between 1.5 and 4.5. The waiting time for each operation, Wiqr is calculated as a multiple of the processing time of that operation. Anderson and Nyirenda [4] have come up with a modification of the MOD rule in which the operation due dates are set dynamically. They proposed two ways of setting the operational due dates which are described below. CR + SPT rule: This rule sets the operational date as a multiple of the processing time of the imminent operation and the critical ratio of that job, where the critical ratio is expressed as CR = allowance/total time
remaining
processing
Zij = max(&j, pij + t). The operation due dates, dij, are static and are calculated as follows:
dij
di,
=
j-l
+
tDi - Ti)Pij m,
-
3
C Piq q=l where die = Ti. Vepsalainen and Morton [12] proposed the apparent tardiness cost (ATC) rule, which uses an exponential function of operational slack. The rule can be expressed as follows:
Zij
= i
Pij “l,
Dtx
exp
c
16 ,o Ii
y=,+
(W,q+~,)-r-~,,
The operation due date, dij, for operation the job i is given as
j of
dij= CR pij.
Thus, the CR + SPT rule can be expressed
as
Zij = max (CR pij; pij). S/RPT + SPT rule: Another rule proposed by Anderson and Nyirenda uses slack per remaining operation time as the multiplier of processing time in setting operational due dates. The rule can be expressed as S/RPT
= slack/total time.
remaining
processing
The operation due date, dij, for operation the job i is given as dij = (S/RPT)pij.
j of
Thus the S/RPT + SPT rule can be expressed as Zij = max(S/RPT
pij;Pij).
Both the rules have been found to have similar performance characteristics, with the S/RPT +SPT being more efficient in minimizing the number of tardy jobs and the CR + SPT rule being more efficient in minimizing mean tardiness. The rules MOD, ATC, CR + SPT and S/!RPT + SPT have one similarity in that they neglect the due date information soon after the job has become tardy and use only the processing time in assigning priorities. In situations where there is quite a variation in the processing times, this could make certain jobs very tardy just as the way the SPT rule tends to make some jobs very tardy.
4. Proposed rule The motivation of this study has been the results of the past simulation studies. It has generally been found that the SPT rule performs better under tight load conditions and the due-date-based rules are found to perform better under light load conditions [l, 2, 6-S]. Thus, if proper weights are given to the process time and due date components of the job depending on the utilization level in the shop, we could expect a good performance of such a rule under all utilization levels. We can therefore have a combination of processing time and slack per remaining processing time with different weights given to them, the weights being a function of the utilization level of the shop. We define the utilization level of a machine as ‘1 =
total total
service
service
time
time + total
free time‘
Obviously, we have 0 < q < 1. The weight factors are exp(q) for pij and exp( - 7) for (S/RPTpjj). Under tight load conditions, the process ttme component will be dominant, while the due date or slack compon-
ent will be dominant under light load conditions. The negative exponential term in the due date component also serves to reduce the magnitude of the slack and thus reduces the disparity in the magnitude of the process time and due date terms. In addition, the coefficients for the process time and due date components are influenced by the shop utilization level, rendering the coefficients dynamic in nature. Yet another indication of the utilization level of the shop is the waiting time at the subsequent machines. The expected waiting time beyond the next immediate machine cannot be estimated with a fair degree of accuracy due to the dynamic nature of the shop. It is obvious that we cannot judge the queue content of machines beyond the machine performing the next immediate operation. Past research has calculated the look-ahead parameter as either the total processing time of the jobs in the queue or the queue length at the next machine [l]. These approaches do not take into account the probable relative priority of the job when it enters the queue at the machine of the next operation. The look-ahead parameter used in our rule is different from these approaches in that it provides a more accurate estimate of waiting time for the job on the next machine. The look-ahead is calculated as follows: Assume that a job, i, is taken up for processing on the current machine at the time instant t, then it joins the queue on the next machine at t + pij. Let f’ be the time at which the next machine becomes free. The time at which the scheduling decision involving this job is made on the next machine is at 2’. The priority indices of all the jobs in the queue are calculated at this time t’. Thus. the expected waiting time of the job i at the next machine is the sum of the processing times of all the jobs which have a higher priority than i, and is denoted by fl,,,. The proposed rule’s priority index Zii is calculated as follows: Z,,; = S:RPTexp(
- ‘I)Pij + exp(pj)Pij + IV,,,.
The job with the least Zi,; is loaded machine.
on the
It is clear that three components, due date or slack ent of the waiting
the proposed rule consists of viz, process time component, component, and the compontime for the next operation.
5.2.3.
5.1. Dispatching
rules used in the simulation
The dispatching rules used in the simulation study for evaluation purposes are SPT, EDD, MOD, ATC, (S/RPT + SPT) and (CR + SPT). The SPT and EDD rules are chosen because they are simple to implement and are widely rules are used in practice. The remaining chosen because they have been found to perform the best.
Root mean square qf’ tardiness
Normally a shop floor manager would prefer many jobs to be a little tardy than to have a few jobs finishing very tardy. The root mean square tardiness is a good measure to evaluate this factor. Root mean square of tardiness, T rmS, is defined as (l/n) i i=
5.3. E.perimental 5.2. Pe~fkwmance measures The performance measures used in the simulation experiment to evaluate the dispatching rules are given below. 5.2.1. Flo\+’ time (Fi ) This refers to the time a job i spends in the shop floor or the resident time of a job i in the shop floor. This measure indicates the workin-process inventory. The term Fi is defined as Ci - Ti. Usually, the mean flow time is used as a performance criterion, which can be defined as F=
(l/n)
~ (Ci - pi), i=l
where n refers scheduled. 5.2.2.
to the total
of jobs
Tardiness
Mean tardiness ?:= (l/n)
number
i i=
can be defined as
max(O, Li), 1
where lateness,
Li, is defined as (Ci - Di).
of’ tardy ,johs
This gives the proportion of the total number of jobs which could not finish on time. 5.2.4.
5. Simulation experimentation
Proportion
[max(O, Li))’ 1
1’ ‘12.
conditions
The simulation experiment was conducted in an open shop configuration consisting of 12 machines. The routing for each order was different and generated randomly, with each machine having an equal probability of being chosen. The number of operations for each job was uniformly distributed between 4 and 12. The process times are drawn from rectangular distributions. Three process time distributions are used: 1. t-50, 2. l-100, 3. 25% of the orders with 50-100 and the rest with l--50. The last distribution is used to evaluate the rules in a shop where there is a large variability in process time distributions. The total work content (TWK) method of due date setting [6] is used in all the experiments with allowance factors of 4, 5 and 6. The order arrivals are generated using an exponential distribution. Two machine utilization levels are tested in the experiments, viz. 85% and 95%. Thus, there are three types of process time distributions, three different modes of due date setting and two different utilization levels, making a total number of 18 simulation experiment sets for every dispatching rule. Each simulation experiment consists of six different runs using antithetic random numbers for each
306
T.S. Rughu, C. RqjcwirranJ An efficient ciwmnic
number of replications is between 5 and 10 [14]. The method suggested by Fishman [15] has been followed in the present study to fix the total sample size. We have found that a sample size of 12 000 is adequate to yield the desired standard error, which limits the tolerance level of all performance criteria at less than 5% of the expected mean values at a 95% confidence level. The number of replications has been fixed in accordance with the suggestion made by Law and Kelton [14] and is fixed at 6, which in turn fixes the run length for a replication at 2000. The data from orders numbering from 501 to 2500 are collected for evaluation purposes, and the shop is continuously loaded with orders until the completion of the last of the 2000 orders. This helps in overcoming the problem of “censored data”
pair of runs. In each run the shop is continuously loaded with orders which are numbered on arrival. The orders are numbered as they arrive in the shop. 5.3.1.
Stea&-state
condition qf’ the shop
In order to ascertain when the system reaches steady state, we observed the shop parameters continuously. It was found using a graphical technique that the shop reaches a steady-state condition when 500 job orders are completed. Graphs of the utilization pattern of the shop for three dispatching rules, viz. SPT, S/RPT + SPT and NEW, are presented in Figs. l-3, respectively. The same phenomenon is also found to occur with other dispatching rules. 5.3.2.
disputching rule
Cll.
Run !ength and number qfreplications
Typically, the total sample size in simulation studies of job shop scheduling has been of the order of thousands of job completions [ 13,121. For the same total sample size it is preferable to have a smaller number of replications and a larger run length, and the recommended
6. Results and discussion The results of the simulation study are presented in Tables 1-6. The results shown are obtained by taking the mean of the mean
I
0
500
ORDER
Fig. 1.
1500
LOO0 llJW3tP
2000
2500
T.S. Raghu, C. Rqjendran!An
eficienf
ORDER
c(vnarnic dispatching rule
307
RlJHBtR
Fig. 2.
0 5 -
0 t
0
-
3 -
oil-
0
I
0
-
,.........,,1.......1.........,.........(.........,,
0
so0
1000 ORDER
1500
2000
7.500
IIUliEER
Fig. 3.
values of the six replications. All the performance criteria except for the percentage of tardy jobs are given relative to the best performing rule. “Percentage tardiness” indicates the percentage of jobs which are tardy. For
example, in Table 1 the flow time of the ATC rule is 8.4372% higher than the NEW rule and, likewise, the percentage number of tardy jobs for the SPT rule in Table 1 is 28. The relative percentage error values are calculated as
308 Table 1 Utilization
level of 85% and the process
time sampled
from a rectangular
( 1. SO)
T rm\
Percentage tardy jobs
86.8948 144.9596 38.1021 35.4636 20.4294 21.2010 0.0000
222.1262 67.92 11 152.4637 149.907 1 133.2953 98.7575 0.0000
28 81 50 40 37 62 52
0.0000
245.8172 243.1579 90.9972 89.0443 27.4654 27.9363 0.0000
407.5429 107.3889 226.1 114 231.5031 I SO.7978 106.8772 0.0000
20 62 31 25 1X 36 30
0.7227 31.1059 20.5982 16.9895 12.1477 I 1.2677 0.0000
727.0735 440.9233 227.5039 214.4366 27.1909 15.6886 0.0000
786.4308 159.2308 364.3282 358.7795 135.6615 71.7846 0.0000
15 43 20 15 7 15 12
Due date tightness
Rule
Flow time
Tardiness
4
SPT EDD MOD ATC SiRPT + SPT CR t SPT NEW*
1.9995 33.8338 10.7472 8.4372 7.2887 8.6131 0.0000
-
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW *Denotes
distribution
the proposed
1.7977 33.2096 15.9445 13.1420 10.0344
11.2902
rule
follows: If F,, denotes the mean flow time of the rule n, n = 1 indicates the SPT rule, n = 2 indicates the EDD rule, and so on. Relative percentage increase in flow time for rule y1with respect to the best performing rule is given by F, - min(F,,
Vn < 7)
min(F,,
7)
Vn d
-
of
x 100.
Relative percentage increase in tardiness and relative percentage increase in root mean square of tardiness are calculated in a similar manner. The results clearly show the superiority of the NEW rule with respect to flow time, tardiness and root mean square of tardiness, whereas the S/RPT + SPT rule and the SPT rule give a low value of the percentage
of tardy jobs. We now discuss detail.
the results
in
6. I. Flow time performance The NEW rule is seen to reduce the mean flow time in most of the cases, with SPT consistently giving the second best result. However, at 85% load level, the performances of the SPT rule and the NEW rule are comparable. The improvement in flow time performance of the NEW rule at high load levels can be attributed to the look-ahead feature incorporated in the NEW rule. Since the estimated waiting time on the next machine is also used in arriving at the priority of a job, the jobs which have less expected waiting times have a better chance of being selected. This ultimately helps in reducing the waiting time of the jobs; thus a reduction in flow time can be expected. The rules such as SPT, S/RPT + SPT, etc., do not incorporate such a look-ahead
T.S. Raghu, C. Rajwdran/An Table 2 Utilization
level of 95% and the process
time sampled
from a rectangular Tardiness
Due date tightness
Rule
Flow time
4
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
12.7611 3 I .0990 15.1008 14.9479 13.7099 13.5939 0.0000
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW SPT EDD MOD ATC S,‘RPT + SPT CR + SPT NEW
6
Table 3 Utilization
level of 85% and the process
309
crflikwt cfvnarnic dispatching rule
distribution
(1, 50)
T rnls
Percentage tardy jobs
47.9683 65.5118 32.1221 33.0388 29.1002 27.6019 0.0000
200.5432 30.3272 165.9366 166.6801 156.7419 131.1927 0.0000
40 97 75 66 67 92 86
16.7126 35.2069 22.5455 20.2993 16.2639 16.0689 0.0000
99.8229 100.2796 64.4060 60.7040 44.7732 42.1942 0.0000
289.348 1 48.4090 222.3287 217.1 173 177.8742 153.3769 0.0000
33 92 63 54 56 81 72
18.4309 36.2368 27.065 1 25.1079 16.9617 15.2724 0.0000
189.3485 145.9801 110.7025 107.6676 62.4279 53.0860 0.0000
429.5135 72.4707 296.6532 299.3521 223.7108 192.0309 0.0000
27 85 54 43 44 67 56
time sampled
from a rectangular
distribution
of
(1, 100)
Due date tightness
Rule
Flow time
Tardiness
T rms
Percentage tardy jobs
4
SPT EDD MOD ATC SJRPT + SPT CR + SPT NEW
4.0288 36.5182 12.0337 9.8324 8.705 1 10.1302 0.0000
96.0304 155.7973 44.7600 42.9279 28.3182 28.8844 0.0000
270.6791 73.1987 184.4453 190.0320 168.0830 134.1732 0.0000
28 85 50 41 40 66 56
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
3.9534 34.3271 16.9411 13.0661 10.8307 10.6296 0.0000
258.8882 245.559 1 99.2332 88.9137 35.0351 22.4473 0.0000
458.0680 100.3590 274.407 1 260.6600 192.2711 125.0260 0.0000
20 66 33 25 20 37 31
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
2.4000 3 1.8420 19.6040 15.6688 11.7946 11.1681 0.0000
639.8570 380.7938 186.9525 170.4684 21.1679 2.7676 0.0000
788.7714 137.8903 402.9274 387.2994 168.8643 84.2642 0.0000
15 46 18 15 7 15 15
of
Table 4 Utilization
level of 95% and the process
time sampled
from a rectangular Tardiness
distribution
(1, 100)
T rm\
Percentage tardy jobs
47.0532 61.8642 33.6945 32.5910 28.0518 23.7990 0.0000
213.7906 21.3361 184.1545 181.4566 166.3569 136.1070 0.0000
40 97 75 66 61 91 x7
16.2532 33.2861 21.7343 19.0402 14.3846 12.7156 0.0000
91.5708 88.5835 58.7976 53.8735 37.1804 31.0541 0.0000
304.194X 37.8069 236.8890 234.4065 189.5866 161.2468 0.0000
31 93 64 54 57 X2 15
17.7649 34.7261 25.6512 24.1722 13.7293 12.9730 0.0000
165.464 I 125.5363 94.5987 93.3008 44.5365 39.5098 0.0000
421.1002 53.7452 299.9797 300.9190 212.7873 188.6144 0.0000
25 86 56 46 44 68 61
Due date tightness
Rule
Flow time
4
SPT EDD MOD ATC S;RPT + SPT CR + SPT NEW
13.4241 30.6394 16.5362 15.4318 13.8251 12.2478 0.0000
SPT EDD MOD ATC S:‘RPT + SPT CR + SPT NEW SPT EDD MOD ATC .S/RPT + SPT CR + SPT NEW
Table 5 Utilization level of 85% and the process time sampled (50, 100) for 25% of jobs
from a rectangular
distribution
of
(1, 50) for 75% of jobs and from
Due date tightness
Rule
Flow time
Tardiness
T rm\
Percentage tardy jobs
4
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
0.0000 31.7145 9.4708 7.2456 5.3319 7.4174 0.5557
5X.8726 124.0100 24.2918 25.9650 9.8002 11.8837 0.0000
212.7690 41.5974 150.0647 162.3621 12 I .3298 96.8593 0.0000
17 19 44 28 28 58 50
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
1.6386 30.3706 17.2665 13.8233 11.5322 11.1699 0.0000
199.2851 21 1.9181 80.5 102 80.7783 34.5601 23.438 I 0.0000
370.27 I 3 63.6063 2 19.4007 222.7526 149.5210 96.7980 0.0000
II 60 30 18 15 34 25
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
2.5865 24.9303 21.6656 19.0086 14.6390 13.4009 0.0000
674.6550 363.9649 201.2045 210.8156 30.0627 13.8519 0.0000
78 1.3986 102.8175 331.5508 336.7857 146.3879 83.0835 0.0000
9 37 18 11 5 14 10
of
T.S. Rqhu,
C. RujmdranlAn
Table 6 Utilization level of 95% and the process time sampled (50, 100) for 25% of jobs Due date tightness
Rule
Flow time
4
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
4.5261 32.9393 7.8406 6.8356 5.275 I 5.0869 0.0000
SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW SPT EDD MOD ATC S/RPT + SPT CR + SPT NEW
efficimt
d,mmic
from a rectangular
311
dispatching rule
distribution
and from
T TlllS
Percentage tardy jobs
30.4698 68.4007 15.5298 15.9419 10.9542 8.4829 0.0000
227.6415 26.1194 188.0045 192.9478 177.8967 148.3783 0.0000
21 95 62 40 43 81 80
I 1.5787 35.0905 19.7082 18.3125 11.7197 12.5400 0.0000
88.5973 102.0684 54.3595 55.2222 29.9789 29.9047 0.0000
339.1032 42.4643 271.1667 269.2080 215.8599 195.9052 0.0000
17 83 49 31 34 68 62
15.0032 37.6698 26.5705 25.7235 16.1041 13.3890 0.0000
179.6940 156.0178 101.5671 107.2468 52.3056 36.0592 0.0000
475.6203 60.9889 350.0465 354.5147 256.7947 196.0747 0.0000
14 74 40 25 25 54 44
factor. It is evident from the results that the improvement in flow time is more pronounced at high load levels than at low load levels. This is due to the fact that at high load levels the queue lengths at the machines are high, and hence the look-ahead feature, i.e. the expected waiting time at the next machine, becomes a very useful factor in reducing the overall waiting time of the jobs; thus an improvement in mean flow time is achieved. However, at lower levels the difference in performance between the NEW rule and the SPT rule becomes less pronounced due to relatively shorter queue lengths at the machines.
Tardiness
(1, 50) for 75% ofjobs
of
performance of the NEW rule and the next best performing rule is comparable since the look-ahead factor may become less effective under these conditions due to loose due date setting and relatively less waiting time. The SPT rule fares badly with the tardiness criterion as the due date tightness slackens. The better performance of the NEW rule with respect to the tardiness criterion can be attributed to the change in weights of the process time and slack components depending on the utilization level. 4.3. Root mean square value
qf tardiness
performance
6.2. Tardiness performance The NEW rule is consistently seen to give the best measures of mean tardiness compared to other rules, with S/RPT + SPT and CR + SPT yielding the next best results. However, sometimes at high due date allowances and low utilization levels it is found that the
A major difference between the proposed rule and the other rules tested, except for EDD, is the performance under root mean square value of tardiness. The proposed rule outperforms the rest of the rules by a large margin. This is not unexpected because the rules such as SPT, S/RPT + SPT and
CR + SPT ignore the due date information as soon as a job becomes tardy. This results in some jobs finishing very tardy. Although EDD is seen to do better under this criterion, its performance with respect to other criteria is not quite encouraging. It can therefore be inferred from the performance under this criterion that if minimization of variance in tardiness is the dominant criterion, it is not prudent to ignore the due date information altogether.
The SPT rule consistently gives a low number of tardy jobs. S/RPT + SPT is found to be the second best performing rule under this criterion. The use of SPT has always been found to result in a low number of tardy jobs. One plausible explanation for the better performance of S/RPT + SPT over CR + SPT is that the process time is used for priority calculation earlier in S/RPT + SPT. This result conforms with the findings of Anderson and Nyirenda [4] in their study. The NEW rule is found to perform better than CR + SPT and EDD with respect to this criterion.
coming very tardy. The simulation study shows that this is indeed the case. The proposed rule does not suffer from this drawback, as it retains the due date information even after a job becomes tardy. By giving appropriate weights to the process time and due date information depending on the utilization of the machines, the proposed rule tries to adapt to changes in the machine utilization levels of the shop floor. It should be noted that, unlike some of the combination rules, the proposed rule does not require a priori setting of weights for the different components. By estimating the waiting time at the next machine, the processing of jobs which have to wait for a long time on the next machine can be delayed. This can help in expediting some jobs which may otherwise be held up. The incorporation of such a look-ahead feature helps in reducing the expected waiting time of jobs. That this is indeed the case has been shown by the simulation results in which the proposed ruIe is found to give better flow time results that the SPT rule at high load levels.
Acknowledgement 7. Conclusions In this paper we have proposed a new rule which is dynamic and global in its structure. It consists of three components, viz. process time component, slack component and the component of waiting time for the next operation. We assign dynamic weights to the process time and the slack component. A simulation study conducted shows that the rule demonstrates the potential for improved performance in flow time, tardiness and root mean square tardiness. Some of the advantages of the proposed rule are listed below. 1. The rules such as CR + SPT, S/RPT + SPT, MOD and ATC, which ignore the due date information soon after a job becomes tardy, have the disadvantage that they result in a small number of jobs be-
We thank the referees for many constructive suggestions and comments which led to a revision of an earlier version of the paper.
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