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

Background. In preemptive job scheduling, which is a part of the flow-shop sequencing tasks, one of the most crucial goals is to obtain a schedule whose total tardiness would be minimal. Total 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 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 tardiness is 25 % greater than the minimum or even worse. Therefore, this heuristic needs a corrective branch which would further try to minimize total tardiness under certain conditions. Objective. The goal is to ascertain what is to be corrected in the heuristic so that the total tardiness value could be obtained lesser. 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. In this scheduling problem, the inaccuracy of finding the minimal total tardiness has the strongest negative impact, so this is almost the worst case, which defines the accuracy limit of the heuristic and positively serves just as the principle of minimax guaranteeing decreasing losses in the worst conditions. Methods. The heuristic sorts maximal decisive ratios by release dates, where the scheduling preference is given to the earliest job. To achieve the said goal, three other sorting approaches are presented and a computational study is carried out with applying each of the four heuristic approaches to minimize total tardiness. For this, two series of 266000 and 1064000 scheduling problems are generated. Results. The earliest-job sorting ensures a heuristically minimal total tardiness value 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 it has been registered just a one scheduling problem (out of 31914 cases followed by a sole “win” of a heuristic version) whose minimal total tardiness is achieved by the earliest-job sorting. Conclusions. The best heuristic version is that one which uses the sorting approach with minimizing remaining processing periods. This, however, is confirmed only for the case where jobs do not have any priorities. The case when jobs have their priority weights is to be yet analyzed.


Introduction
Flow-shop sequencing optimization is a very important problem whose solutions determine high economic and human-resource performance in multistep processes of assembling, building, dispatching, manufacturing, etc. [1,2]. In preemptive job scheduling, which is a part of the flow-shop sequencing tasks, one of the most crucial goals is to obtain a schedule whose total tardiness would be minimal [3,4]. Total tardiness minimization is commonly reduced to solving a combinatorial problem. In particular, the well-known Boolean linear programming model [5,6] is used for that. Nevertheless, struggling to compute a schedule with the exactly minimal total tardiness may become very resource-consuming (implying processor clock speed, memory space, and time of computations) even for a few jobs divided into more than 10 processing periods [6]. Thus, as the number of jobs and the numbers of their processing periods increase, the exact solution of the total tardiness minimization problem may become practically intractable [5,7,8]. The problem of tractability can be slightly stretched and strengthened by using an optimal substitute for infinity in the Boolean linear programming model [9] and re-arranging jobs for either job ascending order input or job descending order input [6,10,11]. However, these methods allow decreasing computation time only on average, after solving long series of thousands of job scheduling problems.
Along with obtaining an exact solution, there are a lot of heuristics allowing to find an approximate solution, which often coincides with the exact one and ensures thus the minimal total tardiness also [3,8,11,12]. In general, the heuristics operate with the remaining available period [8,12], remaining slack [3,8,11], and remaining processing period [5,8,11,12]. A heuristic, in which the decisive ratio is the reciprocal of the maximum of a pair of the remaining processing period (RPP) and remaining available period (RAP) [8], is closely the best one [12,13]. The accuracy of this heuristic (henceforward, let it be named the RPP-RAP heuristic) was studied in [8] on a pattern of tight-tardy progressive idling-free single machine preemptive (TPIF1MP) scheduling [11]. In the TPIF1MP scheduling, where the release dates are given in ascending order starting from 1 to the number of jobs, the inaccuracy of finding the minimal total tardiness has the strongest negative impact. Therefore, this is almost the worst case, which defines the accuracy limit of the RPP-RAP heuristic (and other heuristics as well) and positively serves just as the principle of minimax guaranteeing decreasing losses in the worst conditions (the maximum of unfavorable states) [8,13,14].
Article [8] ascertained that the RPP-RAP heuristic schedules 2 jobs always with the minimal total tardiness. In scheduling 3 to 7 jobs whose lengths (processing periods) are randomly generated within an integer interval from 2 to 5, the risk of missing the minimal total tardiness is just 1.5 % to 3.2 %. Moreover, it is expected that scheduling 12 and more jobs (divided into 2 to 5 processing periods) has at the most the same risk or even lower. In general, the RPP-RAP heuristic produces about 92 % schedules [8] whose total tardiness is exactly minimal, i.e. the Boolean linear programming model in about 92 % of the cases is needless.
In spite of such a promising performance, the RPP-RAP heuristic may produce schedules of a few jobs whose total tardiness is 25 % greater than the minimum or even worse. For instance, a problem of scheduling 4 jobs divided into 7, 7, 11, and 7 processing periods, respectively, whose due dates are 14, 20, 4, 13 by the progressive release dates 1, 2, 3, 4, has the total tardiness minimum of 29, whereas the RPP-RAP heuristic produces a schedule whose total tardiness is 37 (i.e., the relative gap [7,8,15] here is 27.5862 %, which is obviously unacceptable). Therefore, this heuristic needs a corrective branch which would further try to minimize total tardiness under certain conditions.

Problem statement
As the RPP-RAP heuristic has "weak" places, the goal is to ascertain which they are and try to correct them. For this, a few series of TPIF1MP scheduling problems will be generated, in which the "weak" places are to be analyzed. The "weakness" will be treated with respect to using RPP and RAP in an alternative way allowing to obtain a total tardiness lesser than that by the RPP-RAP heuristic. Eventually, the corrected and updated RPP-RAP heuristic (henceforward, let it be named the CU-RPP-RAP heuristic) will be discussed and the corresponding conclusions on it will be made.

The RPP-RAP heuristic and its corrections
Given N jobs to be scheduled, N ∈  \ { } 1 , with their respective processing periods release dates and due dates the RPP-RAP heuristic builds stepwise a schedule , as time t progresses [7,8,11]. Before the start, RPPs are For every set of available jobs Then a subset is determined. If | | A t * ( ) = 1, then Assignment (11) executed by condition (10) implies that the earliest job is preferred to be scheduled [16] when there are two or more maximal decisive ratios in subset (7). Thus, job n is completed after mo- is an approximately minimal total tardiness that corresponds to this schedule [8,11,16]. Thus, in the case when (10) is true, the RPP-RAP heuristic sorts maximal decisive ratios in subset (7) 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, other approaches to sorting might be used. They consist in sorting by RPPs and due dates. Consider them as follows.
Along with that, one should remember that the random-sorting RPP-RAP heuristic may be outperformed as well in the same scheduling problem. So, stating that the random sorting produces a schedule or schedules with a definite total tardiness must be followed with a respective (approximate) probability.

Generation of TPIF1MP scheduling problems
In TPIF1MP scheduling, the release dates can be given in ascending order as follows: The due dates are generated as [11,16] d r H b n n n n by the respective random due date shift [10,11] b H n n with a pseudorandom number ζ drawn from the standard normal distribution (with zero mean and unit variance), and function ψ ξ ( ) returning the integer part of number ξ (e.g., see [10,11]). The job lengths are generated as [8] with a pseudorandom number u [17] drawn from the standard uniform distribution on the open interval (0; 1). So, the job length is randomly generated between 2 and A -1 [8]. Once a vector of job lengths (1) is generated, due date shifts (42) are generated until and the total tardiness value is not 0 (i.e., the due dates are not very great, so at least one tardy job would exist).
The 6487 sole "wins" in the first series constitute just 2.4387 % of the series volume. The remaining 97.5613 % are group "wins", where two to four sorting approaches have produced the same (minimal) total tardiness. Table 2 presents the distribution of the number of group "wins", where seven "winning" groups have been disclosed. The highest and dominating percentage (93.4836 %) of group "wins" has been revealed to be for the group consisting of all four sorting approaches. Moreover, every group contains the min-RPP sorting which is the incontestable "winner" in the con sideration of sole "wins" (see Table 1). This implies that the min-RPP sorting itself is capable of producing a total tardiness value, which either is minimal among the total tardiness values by the other three sorting approaches or is not greater (this is the 97.5613 % of the 266000 cases in the first series) than these values.
To get convinced that the obtained results are statistically reliable, a second series of TPIF1MP scheduling problems is analogously generated by (40) -(44), where 4000 scheduling problems are generated for every N = 2 15 , and A = 2 20 , . Table  3 presents the number of generated scheduling problems (among those 1064000 generated scheduling problems) whose total tardiness has been revealed to be minimal for the given sorting approach, whereas the other three sorting approaches have produced greater total tardiness values. As in the case of the first series, here the number of sole "wins" constitute just a small percentage (which is 2.3901 %) of the series volume, whereas the min-RPP sorting "has won" almost everywhere, expect for three scheduling problems. Now the random sorting "has won" two times for scheduling problems with  It is worth noting that the min-RPP sorting for job lengths (49) and due dates (50) produces a schedule whose total tardiness is  ϑ( ) 13 601 = , and  ϑ( ) 14 438 = for job lengths (51) and due dates (52), i.e. the min-RPP sorting "loses" to the random sorting just 0.67 % and 0.2288 %, respectively. Moreover, for job lengths (53) and due dates (54), the min-RPP sorting produces a schedule whose total tardiness is  ϑ( ) 14 454 = "losing" thus 0.2208 % to the earliest-job sorting.  Table 4 presents the distribution of the number of group "wins", where the same seven "winning" groups have been disclosed in the second series. The group "wins" do not much differ from those in Table 2 (for a working accuracy, see the bottom row of Table 4). This confirms the statistical reliability of the results in Tables 1 and 2. Therefore, the min-RPP sorting is really the incontestable "winner" in minimizing total tardiness of TPIF1MP scheduling problems generated by (40) -(44).
A special case when all the jobs have identical volumes is to be studied also. For this, a series of TPIF1MP scheduling problems of N jobs, each of which is of A processing periods, is generated by (40) -(42) and (44), where 1000 scheduling problems are generated for every N = 3 15 , and A = 2 10 , . So, altogether 117000 scheduling problems are generated in this series. There are no sole "wins", and the group "wins" are distributed only among four groups (Table 5). Once again, the highest and dominating percentage (96.2197 %) of group "wins" has been revealed to be for the group consisting of all four sorting approaches. As for the first and second series of non-constant job length scheduling problems, every "winning" group in the series of constant job length scheduling problems contains the min-RPP sorting. This additionally confirms the min-RPP sorting is very efficient.

Discussion
One of the "weak" places in the RPP-RAP heuristic by (4) -(12) is the earliest-job sorting. Despite it ensures a heuristically minimal total tardiness value in more than 97.6 % of TPIF1MP scheduling problems (which is derived from the respective group "wins" in Tables 2 and 4), the earliest-job sorting still fails to minimize total tardiness in no less than 2.2 % of the cases (this is a lower bound of such an estimate). On the other hand, the min-RPP sorting produces a heuristically minimal total tardiness for almost any TPIF1MP scheduling problem, where exceptions like (45) -(54) are quite rare (technically, this has been just 4 occurrences out of 31914 cases followed by sole "wins", i.e. the exception is only 0.0125 % probable). All the more that the min-RPP sorting "loses" to the other sorting approaches very littleit has not been revealed a difference greater than 0.75 %, although there have been registered only 5 such occurrences including the case with job lengths (34) and dues dates (35). Moreover, if to consider non-random sorting approaches, it has been registered just a one TPIF1MP scheduling problem whose minimal total tardiness is achieved by the earliest-job sorting.
Apart from the min-RPP sorting, the other "weak" places in the RPP-RAP heuristic are tied to the RPP-or-due-date sorting and the random sorting. The RPP-or-due-date sorting cannot "win" solely, and "wins" of the random sorting are very rare and unreliable. Consequently, these sorting approaches cannot be used in minimizing total tardiness of TPIF1MP scheduling problems. On the contrary, the min-RPP sorting successfully corrects almost any fails of the RPP-RAP heuristic. There-fore, the CU-RPP-RAP heuristic can be defined as the RPP-RAP heuristic by (4) -(10) using the min-RPP sorting approach as (25) with (17) by (18).
Despite the CU-RPP-RAP heuristic is not perfect against the RPP-RAP heuristic (using the earliest-job sorting), it is impossible to figure out which processing periods (1) and due dates (3) should be to have the least total tardiness value produced solely by the earliest-job sorting approach. The task of searching for such a pattern was simplified with setting release dates (2) as (40), but the pattern is not clear. This pattern does possibly exist but it is too tricky if even the simplification has not given a primitive sketch of it. Nevertheless, this does not really matter owing to the highly efficient min-RPP sorting.

Conclusions
To minimize total tardiness in TPIF1MP scheduling by the heuristic based on RAPs and RPPs, it is the best to use the min-RPP sorting instead of the earliest-job sorting to schedule jobs when two or more maximal decisive ratios become equal. It is plausible that the respective CU-RPP-RAP heuristic is not the best itself, but it generally outperforms or equal to the heuristic if the earliest-job sorting, the RPP-or-due-date sorting, and the random sorting are used. Therefore, the CU-RPP-RAP heuristic using the min-RPP sorting is the best considering the RPP-RAP heuristic with the other three sorting approaches in TPIF1MP scheduling. However, the research on the RPP-RAP heuristic should be furthered for the case when jobs have their priority weights, where the sorting improvement is to be yet analyzed.