SORTING APPROACHES IN THE HEURISTIC BASED ON REMAINING AVAILABLE AND PROCESSING PERIODS TO MINIMIZE TOTAL WEIGHTED TARDINESS IN PROGRESSIVE IDLING-FREE 1-MACHINE PREEMPTIVE SCHEDULING

Background. In preemptive job scheduling, total weighted tardiness minimization is commonly reduced to solving a combinatorial problem, which becomes practically intractable as the number of jobs and the numbers of their processing periods increase. To cope with this challenge, heuristics are used. A heuristic, in which the decisive ratio is the weighted reciprocal of the maximum of a pair of the remaining processing period and remaining available period, is closely the best one. However, the heuristic may produce schedules of a few jobs whose total weighted tardiness is enormously huge compared to the real minimum. Therefore, this heuristic needs further improvements, one of which already exists for jobs without priority weights with a sorting approach where remaining processing periods are minimized. Three other sorting approaches still can outperform it, but such exceptions are quite rare. Objective. The goal is to determine the influence of the four sorting approaches and try to select the best one in the case where jobs have their priority weights. The heuristic will be applied to tight-tardy progressive idling-free 1-machine preemptive scheduling, where the release dates are given in ascending order starting from 1 to the number of jobs, and the due dates are tightly set after the release dates. Methods. To achieve the said goal, a computational study is carried out with applying each of the four heuristic approaches to minimize total weighted tardiness. For this, two series of 4151500 scheduling problems are generated. In the solution of a scheduling problem, a sorting approach can “win” solely or “win” in a group of approaches, producing the heuristically minimal total weighted tardiness. In each series, the distributions of sole-and-group “wins” are ascertained. Results. The sole “wins” and non-whole-group “wins” are rare: the four sorting approaches produce schedules with the same total weighted tardiness in over 98.39 % of scheduling problems. Although the influence of these approaches is different, it is therefore not really significant. Each of the sorting approaches has heavy disadvantages leading sometimes to gigantic inaccuracies, although they occur rarely. When the inaccuracy occurs to be more than 30 %, this implies that 3 to 9 jobs are scheduled. Conclusions. Unlike the case when jobs do not have their priority weights, it is impossible to select the best sorting approach for the case with job priority weights. Instead, a hyper-heuristic comprising the sorting approaches (i. e., the whole group, where each sorting is applied) may be constructed. If a parallelization can be used to process two or even four sorting routines simultaneously, the computation time will not be significantly affected.


Introduction
Minimization of total weighted tardiness (TWT) in job schedules is an important task which aims at reducing costs of production delays [1]. This task is commonly reduced to solving a combinatorial problem which becomes practically intractable as volumes of jobs increase (i. e., as the number of jobs and/or the numbers of their processing periods increase) [2,3]. The tractability can be slightly stretched and strengthened by using an optimal substitute for infinity in respective integer linear programming models [4,5] and re-arranging jobs for either job ascending order input or job descending order input [6,7]. After solving long series of thousands of job scheduling problems, these methods can really decrease computation time on average, but the impact on the tractability is too tiny [8].
The tractability problem is resolved by applying heuristics which allow finding schedules whose TWT is approximately minimal [9,10]. However, the heuristically minimal TWT is not always the exact one. So, the tradeoff here is accuracy versus speed (i. e., computation time). A heuristic, in which the decisive ratio is the weighted reciprocal of the maxi mum of a pair of the remaining processing period (RPP) and remaining available period (RAP) [3], is closely the best one [11]. The accuracy of this heuristic (henceforward, let it be named the RPP-RAP heuristic) was studied in [3] on a pattern of tight-tardy progressive idling-free single machine preemptive (TPIF1MP) scheduling [8]. In general, the RPP-RAP heuristic produces about 92 % schedules [3] whose TWT is exactly minimal, i. e. an integer linear programming model in about 92 % of the cases is needless. Besides, the heuristic schedules 2 jobs always with the exactly minimal TWT. However, the RPP-RAP heuristic may produce schedules of a few jobs whose TWT is enormously huge compared to the real minimum of TWT. For instance, a problem of scheduling 4 jobs divided into 6, 4, 5, and 5 processing periods with their respective priority weights 10,4,9,8, whose due dates are 11, 5, 15, 18 by the progressive release dates 1, 2, 3, 4, respectively, has the TWT minimum of 16, whereas the RPP-RAP heuristic produces a schedule whose TWT is 36 (i. e., the relative gap [2,3,12] here is 125 %, which is obviously unacceptable). Therefore, this heuristic needs further improvements. One of such improvements is suggested in article [13] which considers minimization of total tardiness (i. e., without job priority weights) in TPIF1MP scheduling, where the release dates are given in ascending order starting from 1 to the number of jobs, and the due dates are tightly set after the release dates. Whereas the RPP-RAP heuristic sorts maximal decisive ratios by release dates, where the scheduling preference is given to the earliest job (this is the earliest-job sorting), three other sorting approaches are presented in [13]. In accordance with article [13], the earliest-job sorting ensures a heuristically minimal total tardiness in more than 97.6 % of scheduling problems, but it fails to minimize total tardiness in no less than 2.2 % of the cases. Nevertheless, a sorting approach with minimizing remaining processing periods produces a heuristically minimal total tardiness for almost any scheduling problem. If an exception occurs, this sorting approach "loses" to the other sorting approaches very little. Moreover, the exceptions are quite rare as just a one scheduling problem was registered (out of 31914 cases followed by a sole "win" of a heuristic version) whose minimal total tardiness was achieved by the earliest-job sorting.

Problem statement
In order to improve the RPP-RAP heuristic, the goal is to determine the influence of the four sorting approaches and try to select the best one. For this, a few series of TPIF1MP scheduling problems will be generated to minimize TWT, in which the distribution of sole "wins" is to be ascertained. The distribution of group "wins" is to be ascertained also. The possibility of the best sorting approach selection will be discussed and the corresponding conclusions on it will be made.

The earliest-job sorting and three other sorting approaches
Given N jobs to be scheduled, \ {1}, N ∈  with their respective processing periods [14,15] and due dates Then a subset * ( )  1.
Assignment (12) executed by condition (11) implies that the earliest job is preferred to be scheduled [18] when there are two or more maximal decisive ratios in subset (8 is an approximately minimal TWT corresponding to this schedule [3,8,18]. So, in the case when (11) is true, the RPP-RAP heuristic sorts maximal decisive ratios in subset (8) by release dates, where the scheduling preference is given to the earliest job (whose release date is the least). However, along with this earliest-job sorting, the other three approaches to sorting might be used: the RPP-or-due-date sorting, the min-RPP sorting, and random sorting [13]. The RPP-or-due-date sorting is executed as follows. If the RAP of job i * is * 0 Subsequently, a subset * * * * ** ** 1 ( ) is found and job ** 1 i is scheduled: 1.
For example, a TPIF1MP scheduling problem with 1 5 [ ] It is noteworthy that the RPP-RAP heuristic using the earliest-job sorting produces the same schedule (34), whereas the RPP-or-due-date sorting for job lengths (31), priority weights (32), and due dates (33) produces schedule 1 20 [ ] whose TWT is greater by 15 units (i. e., the in-heuristic gap here is 9.26 %): This is the example of that the min-RPP sorting can outperform the RPP-or-due-date sorting, and the latter, unlike the example with job lengths (20), priority weights (21), and due dates (22), can be outperformed by the earliest-job sorting. Finally, the random sorting consists in just a random selection of a job from subset (8). Thus, in the case when (11) is true, number m is randomly selected from subset {1, } L and job * m i is scheduled: However, if a schedule is built by using the random sorting by (38), (39), instead of (12) by (11), then, generally speaking, the heuristic produces random schedules and TWT. Nevertheless, the random sorting can outperform each of the three above-described sorting approaches. For example, a TPIF1MP scheduling problem with is solved by the random-sorting RPP-RAP heuristic producing two different schedules, one of which is The other schedule 1 16 [ ] whose probability is equal to the probability of schedule (43), is much worse because its TWT is 21 units greater: At the same time, the RPP-RAP heuristic using the earliest-job sorting and the min-RPP sorting also produces schedule (45). However, the RPP-or-duedate sorting produces schedule (43). So, the random sorting here outperforms (by 21 units of TWT) the earliest-job sorting and the min-RPP sorting at 50 % rate. In another example, where The other schedules by the random sorting have (7) 269.
The same TWT is produced by the earliest-job sorting, the RPP-or-due-date sorting, and the min-RPP sorting. Despite the random sorting produces more than two schedules, they have only two possible TWT (whose probabilities are equal): (7) 259 ϑ =  and (7) 269. ϑ =  Thus, the random sorting here outperforms (by 10 units of TWT) the other three sorting approaches at 50 % rate.

Generation of TPIF1MP scheduling problems
As there is no an obvious "leader" among the above-described four sorting approaches, the possibility of such a "leadership" (implying the capability to produce heuristically minimal TWT more often than others) should be ascertained via a statistical analysis of the performance. In TPIF1MP scheduling, the release dates for (3) can be given in ascending order as follows: The due dates for (4) are generated as [8,18] 1 n n n n by the respective random due date shift [7,8] ( ) with a pseudorandom number z drawn from the standard normal distribution (with zero mean and unit variance), and function y(x) returning the integer part of number x (e. g., see [7,8]). The job lengths for (1) are generated as [3] ( 2) with a pseudorandom number u [19,20] drawn from the standard uniform distribution on the open interval (0; 1). So, the job length is randomly generated between 2 and A -1 [3]. The job priority weights for (2) are generated similarly to (55): by a weight amplitude factor Once a vector of job lengths (1) is generated, due date shifts (54) are generated until and TWT is not 0 (i. e., the due dates are not very great, so at least one tardy job would exist).

Computational study
A first series of TPIF1MP scheduling problems is generated by (52)-(58), where 500 scheduling problems are generated for every 3, 25 N = (owing to that the schedules of 2 jobs by the RPP-RAP heuristic always have the exactly minimal TWT) and 2, 20 A = and (57). So, altogether 4151500 scheduling problems are generated in the first series. Each scheduling problem is solved by the RPP-RAP heuristic using the four sorting approaches: the earliest-job sorting (which initially constitutes the RPP-RAP heuristic itself) by (11)-(13), the RPPor-due-date sorting by (15)-(19), the min-RPP sorting by (27) with (18) by (19) or (28)-(30), and the random sorting by (38), (39). Table 1 presents the number and percentage of generated scheduling problems whose TWT has been revealed to be minimal for the given sorting approach, whereas the other three sorting approaches have produced greater values of TWT (i. e., the heuristic with the given sorting approach "has won"). The most "winning" sorting approach is the min-RPP sorting covered nearly a half of all sole "wins". Despite its rather probabilistic nature, the random sorting has had almost a quarter of all sole "wins". In the case of TPIF1MP scheduling problems without priority weights, sole "wins" constitute about 2.39 % to 2.44 % of the volume of generated problems [13]. Here, in the first series, the percentage is just 0.5814 % (for 24136 sole "wins" out of 4151500 generated scheduling problems). Nevertheless, this is the average value. The sole "wins" percentage decreases as the weight amplitude factor increases (Fig. 1). Meanwhile, the interrelationship among the percentages of "wins" given in Table 1 remains almost the same (Fig. 2). Obviously, group "wins", where two to four sorting approaches have produced the same (minimal) TWT, dominate. They constitute 99.4186 % (for 4127364 group "wins" out of 4151500 generated scheduling problems) of the volume of generated problems, where 11 "winning" groups have been revealed (in fact, this is the total set of groups; there cannot be other groups). The distribution of the number of group "wins" is presented in Table 2 where the group membership is highlighted with a gray color. The highest and dominating percentage (98.397 %) of group "wins" has been revealed to be for the group consisting of all four sorting approaches. As the weight amplitude factor increases, this whole-group "wins" percentage increases (Fig. 3, where dotted markers are used), whereas the 10 numbers of non-whole-group "wins" decrease (Fig. 4). The polyline of the interrelationship among the percentages of nonwhole-group "wins" almost vanishes as the weight amplitude factor becomes sufficiently great (Fig. 5).  Unlike the case of TPIF1MP scheduling problems without priority weights [13], here the min-RPP sorting is not the incontestable "winner" because it does not belong to every non-whole group (i. e., to groups ##1-10). However, it is worth to note that it has had 51443 non-wholegroup "wins" in the first series, whereas the earliest-job sorting, the RPP-or-due-date sorting, and the random sorting have had 38418, 36917, and 35421 non-whole-group "wins", respectively. So, the min-RPP sorting does have nonetheless some advantage, which might be taken into account when a preference to a single sorting approach is to be given (because applying simultaneously two or more sorting approaches to solving a scheduling problem will slow down the process of obtaining a solution, whichever short computation time is). To get convinced that the obtained results are statistically reliable and repeatable, a second series of TPIF1MP scheduling problems is analogously generated by (52)-(58). Table 3 presents the number of sole "wins" in the second series along with the relative difference from the first series. Sole "wins" constitute 0.5836 % of the volume of generated problems in the second series. The difference between the series does not exceed 7.5 %. This is why the barred plot of the number of sole "wins" versus the weight amplitude factor (Fig. 6) is very similar to that for the first series (Fig. 1). The difference between the barred plots in Figs. 1 and 6 is tiny, indeed. Besides, the interrelationship among the percentages of "wins" given in Table 3 remains almost the same (Fig. 7) by just nearly repeating the plot in Fig. 2. The difference between the polylines in Figs. 2 and 7 is not that tiny, but the shape of those two polyline bunches is the same.  Group "wins" constitute 99.4164 % (for 4127270 group "wins" out of 4151500 generated scheduling problems) of the volume of generated problems in the second series. The distribution of the number of group "wins" is presented in Table 4 along with the relative difference from the first series. The difference between the series does not exceed 2.5 %, where the difference for whole-group "wins" is just 0.01 %. As the weight amplitude factor increases, this whole-group "wins" percentage increases (Fig. 8) similarly to that in the first series, as well as the 10 numbers of non-whole-group "wins" decrease ( Fig. 9). The difference between the barred plots in Figs. 4 and 9 is tiny, indeed. The polyline of the interrelationship among the percentages of non-whole-group "wins" similarly vanishes as the weight amplitude factor becomes sufficiently great (Fig. 10). Even the peaks and cavities in Figs. 5 and 10 are very resembling that additionally confirms the repeatability of the first series.  Finally, it is worth to note that, in the second series, the earliest-job sorting, the RPP-or-due-date sorting, the min-RPP sorting, and the random sorting have had 38313, 36903, 51728, and 35499 non-wholegroup "wins", respectively. These numbers differ from those in the first series by no more than 0.56 %, so it is valid (along with the above-mentioned similarities between the barred plots and polyline bunches) to affirm the series repeatable and statistically reliable. The repeatability, however, does not imply itself that it will be so easy to select the best sorting approach in order to make the RPP-RAP heuristic more accurate (or, in other words, to improve the RPP-RAP heuristic).

Discussion
The min-RPP sorting appears to have an advantage but it is hard to select this approach as the best. Indeed, there are examples of the scheduling problem in which the min-RPP sorting fails to be as accurate as other sorting approaches are. Moreover, the inaccuracy (with respect to the approach producing a lesser value of TWT) can be just gigantic. For instance, a TPIF1MP scheduling problem with 1 3 [ ] [9 16 3], The in-heuristic gap here is more than 923 % (!), which defies the min-RPP sorting best-selection outright. Such shocking counterexamples exist for the earliest-job sorting and the random sorting also. The RPP-or-due-date sorting, however, produce less shocking counterexamples. Table 5 contains a numerical description of the shocking counterexamples along with ordinary counterexamples in the first series, in which the inaccuracy of the sorting approach (with respect to the approach producing a lesser value of TWT) is that gigantic. The minimal and maximal number of jobs, at which the instances with TWT exceeding the in-heuristic gap have been registered, are given also. The similar results in the second series are given in Table 6. It is worth noting that the shocking counterexample with (59)-(65) falls beyond the statistics of Tables 5 and 6. Overall, Tables 5 and 6 allow seeing that the shocking counterexamples occur when 3 to 9 jobs are scheduled (when the in-heuristic gap is more than 30 %). To exclude the occurrence of gigantic inaccuracies, the respective integer linear programming model to exactly minimize TWT should be used instead of the heuristic. This model will be applicable owing to that scheduling up to 10 jobs is practically tractable [2,3,15,21,22].
It is also seen that the RPP-or-due-date sorting can produce an in-heuristic gap in more than 20 % by scheduling up to 11 jobs. This upper number is 9 for the earliest-job sorting and the random sorting, and is 10 for the min-RPP sorting. A pretty huge in-heuristic gap in more than 10 % can be produced by the min-RPP sorting when up to 20 jobs are scheduled (see Table 6, in which all the maximal numbers of jobs in the first row are greater than those in Table 5). This upper number is 21 for the earliest-job sorting, but it is 25 for the RPP-ordue-date sorting and the random sorting. Scheduling such numbers of jobs is unlikely to be tractable by the respective integer linear programming model. Therefore, each of the sorting approaches has heavy disadvantages, although they occur rarely. As sole "wins" and non-whole-group "wins" are rare also, it is impossible to select the best sorting approach. Instead, a hyper-heuristic comprising the sorting approaches (i. e., the whole group, where each sorting is applied) may be constructed (although the process of obtaining a solution will be slowed down, unless a parallelization is used) [23,24].

Conclusions
Pairwise comparison of Tables 1 and 2 to Tables 3 and 4 allows affirming that it is sufficient to generate 500 scheduling problems for obtaining statistically reliable results. This is confirmed by comparing Figs. 1-5 to Figs. 6-10, which are a deeper dissection of the tables. Figs. 1, 4, 6, 9 also confirm that the four sorting approaches become more indistinguishable as the weight amplitude factor increases.
In minimizing TWT by the RPP-RAP heuristic, the earliest-job sorting, the RPP-or-due-date sorting, the min-RPP sorting, and the random sorting approaches produce schedules with the same TWT in over 98.39 % of TPIF1MP scheduling problems.
Although the influence of these approaches is different, it is therefore not really significant. The RPP-RAP heuristic is truly improved only by applying all the four sorting approaches to solving a scheduling problem. If a parallelization can be used to process two or even four sorting routines simultaneously, the computation time will not be significantly affected.
Based on Tables 5 and 6 with the shocking counterexamples, it is recommended to use the RPP-or-due-date sorting for scheduling up to a few tens of jobs. To reduce the likelihood of gigantic inaccuracy, the volume of a scheduling problem should be as great as possible. An open question is how to recognize a shocking counterexample in a scheduling problem with (1)-(4) beforehand.