A New Priority Rule for Initial Ordering of Jobs in Permutation Flowshop Scheduling Problems

Scheduling in a permutation flowshop refers to processing of jobs in a set of available machines in the same order. Among the several possible performance characteristics of a flowshop, makespan remains one of the highest preferred metrics by researchers in the past six decades. The constructive heuristic proposed by Nawaz-Enscore-Ham (NEH) is one of the best for makespan minimization. The performance essentially depends on the initial ordering jobs according to a particular priority rule (PR). The popular priority rules are non-increasing order of the jobs' total processing time, the sum of average processing time and standard deviation and, the sum of average processing time, standard deviation and absolute skewness among others. The objective of this paper is to propose and analyse a new job priority rule for the permutation flowshop. The popular priority rules available in the literature are studied and, one of the best priority rules; the sum of average processing time and standard deviation is slightly modified, by replacing the standard deviation by mean absolute deviation (MAD). To assess the performance of the new rule, four benchmark datasets are used. The computational results and statistical analyses demonstrate the better performance of the new rule. Keywords—Priority rule; flowshop scheduling; makespan; NEH algorithm


I. INTRODUCTION
Permutation flowshop scheduling problems (PFSSP) remain one of the most studied domains in operations research in the past six decades. PFSSP refers to scheduling 'n' number of jobs for processing in 'm' number of machines in the same order. According to Rinnooy Kan [1], PFSSP is proved to be NP-hard when the number of machines is greater than three. Since the PFSSP is np-hard, the computation time grows exponentially for larger problems and hence, exact solution becomes impossible or expensive. As a result, several dispatching rules, heuristics and metaheuristics have been proposed over the decades. Efficient heuristics report solutions with acceptable accuracy levels in a reasonable time. Priority rules or dispatching rules are some form of heuristics and have been studied in both academia and industry domains extensively for decades (Tay and Ho 2008) [2]. A few simple dispatching rules that are being extensively used in jobs' scheduling are: Shortest Processing Time (SPT) rule, Longest Processing Time (LPT) rule, Earliest Due Date (EDD) rule, First Come First Serve (FCFS) rule.
The optimization parameters for the PFSSP are generally; flow time, idle time or makespan (Liu et al., 2016) [3] to satisfy different production line requirements. They find numerous real-time applications and could be combined with Internet of Things (IOT) for specific applications (Salis, 2021) [4].Among the many parameters that are being optimised, makespan minimization is widely considered by researchers over the years. Johnson's [5] algorithm proposed in 1954 yields an optimum solution for two machines and 'n' jobs PFSSPs which was extended to three machines cases.
As the problem is NP-hard, the exact solution becomes impossible for larger problems and the computation time grows exponentially with the problem size. Earlier approximate heuristics could not yield the expected accuracy and the breakthrough came in 1983 when Nawaz-Enscore-Ham (NEH) [6] algorithm was proposed. NEH algorithm which has a complexity of O(n 3 .m) uses the largest processing time (LPT) dispatching rule and is considered as one of the best constructive heuristics for makespan minimization even today. Many improvements and extensions have been proposed by many authors over the years. NEH essentially consists of three phases: • Pre-arranging the jobs according to the non-increasing order of their total processing times (Priority Rule).
• Selecting the first two jobs from the processed sequence as the initial partial sequence (Initial Sequence).
• Inserting other jobs one by one at a suitable place that minimises the partial makespan (Insertion Phase). Ribas et al. (2010) [7] tested four priority rules combined with the powerful insertion technique of NEH including rules from NEHKK1 (Kalczynski and Kamburowski 2008) [8], N&M (Nagano and Moccellin 2002) [9], LPT, and a random job sequence. Baskar and Xavior (2015) [10] analysed the job insertion technique for different initial sequences.
The authors improved the solution quality of NEH by suitably modifying the first and third phases. Several priority rules for NEH are proposed and available in the literature and the ones proposed by Dong et al. [11] and Liu et al. [12] yield better results.
Framinan et al. [13] analysed 177 initial sequences for the NEH heuristic and concluded that SUM PIJ / DECR (i.e. original NEH) is ranked 1 among all for the makespan minimisation.
During the insertion phase, we come across several occasions when the partial makespan remains the same for 408 | P a g e www.ijacsa.thesai.org more than one partial sequence. NEH breaks such ties randomly. Several effective tie-breaking rules are proposed and analysed for the solution quality. The tie-breaking rules proposed by Fernandez and Framinan [14], Lie et al. [12], Benavides [15] and the ones recently by Baskar and Xavior [16] are a few to mention. This paper considers the sum of the average processing times and means absolute deviation (MAD) for the initial ordering of jobs (Priority Rule) and analyses its impact using well-known benchmark datasets.
The structure of this paper is as follows: the new priority rule is presented in Section 2 followed by the benchmark and performance metrics used for the assessment in Section 3. The results and statistical analyses are detailed in Section 4 and Section 5 discusses about the conclusion, limitations and future work.

II. NEW PRIORITY RULE
The initial ordering of jobs or priority rule does affect the solution quality of the NEH heuristic. The priority rule considered by the original NEH is non-increasing order of the jobs' total processing times. Total processing time (TPT) for a particular job, j can be represented as, (1) m -Number of machines in the schedule. p ij denotes the processing time of job 'j' in the machine 'i'.
Dong et al. [11] added the standard deviation of the processing times with the average total processing time, AVG j for a job and reported improved results. Mathematically, standard deviation, Another priority rule proposed recently by Liu et al. [12] adds the absolute skewness with the priority rule of Dong et al. [11] and the rule improves the solution further. Skewness (SKE) for a job, j is defined as, In this paper, the standard deviation is replaced by another similar metric, the Mean Absolute Deviation (MAD) which is the average of the absolute deviations from the mean value.
For a job 'j', MAD was used as the variation measure in the tiebreaking rule of Liu et al. [12].Another metric, the median absolute deviation (MAD1) is also used instead of mean absolute deviation during the process; that is, descending order of average processing time + median absolute deviation as a priority rule. However, the results are not encouraging in this case.
For a job 'j', The benchmarks and analyses results using different combinations are presented and discussed in the coming sections.

III. BENCHMARK AND PERFROMANCE MATRIC
IV. COMPUTATIONAL RESULTS AND DISCUSSION Different priority rules (initial ordering of jobs) considered are summarized below: • PR-NEH: Descending order of their total processing times.
• PR-D: Descending order of the sum of their average processing time and standard deviation.
• PR-LJP: Descending order of the sum of their average processing time, standard deviation and absolute (skewness).
• PR-DB: Descending order of the sum of their average processing time and mean absolute deviation.
409 | P a g e www.ijacsa.thesai.org • "No PR" refers to no initial ordering of jobs; raw data are used as received.
Codes are generated for all priority rules in MATLAB R2012b and run in an i5 desktop PC with 4 GB RAM. Many heuristics initially order the jobs according to a priority rule and then refine the solution further using some strategy. The initial priority ordering may have significant impact on the final result. To start with, Taillard instances are initially ordered as per the priority rule (PR) considered and the makespans computed without any further processing. The average PRDs (APRDs) are presented in Table I. The bold digits indicate the best result for a particular problem set. There is not much difference in the results among the PRs considered. The best APRD of 21.40% is reported by PR-DB and the worst one by PR-D (21.64%); the difference being 1.12%. Even no "no priority rule" reports a better mean value of 21.61% and accounts for best results in 4/12 sets. Though PR-DB reports the lowest deviation, it reports best results in one case only, 10 machines and 100 jobs.
Subsequently, the impact of different initial sequences has been investigated. NEH considers the first two jobs of the processed sequence as the initial sequence. Baskar et al. [22] considered a few other sets of jobs as initial sequences and analysed (Table II) the impact using Taillard dataset. The performance was analysed by randomly selecting two jobs also from the processed sequence. The results show that the APRDs vary slightly with respect to the initial partial sequence. According to them, randomly selecting two jobs also results in a reasonably good APRD with 3.43% which may be due to the job insertion strategy that was originally used by the NEH algorithm. It is to be noted that many heuristics could not even better this mean value of 3.43% for the Taillard benchmark. Now, to assess the performance; the PR is applied to the classic NEH algorithm and the results are tabulated in Table  III to Table VI. The benchmark used are; Carlier's [17] proposed in 1978, Taillard's [19], in 1993, Demirkol's [18] in 1998 and the latest VFR benchmark proposed in 2015 by Vallada et al. [21]. The better APRDs are shown in bold and italics in all the tables.
For the Carlier 8 instances (Table III); Dong et al. priority rule, PR-D reports an APRD of 1.44%, an increase of 3.60% in the APRD over PR-NEH which is taken as the reference for the analysis in this paper. That is, for this dataset, the solution quality deteriorates when PR-D is used for initial ordering the jobs. The APRD of the new priority rule, PR-DB is 1.21%, a significant 12.95% improvement over the rule used by NEH.
For the comparatively larger dataset of Demirkol's which are 40 in numbers also, the proposed priority rule, PR-DB reports better APRD than the priority rules, PR-NEH and PR-D with an improvement of 4.70% over the reference PR-NEH. The improvement of the rule, PR-D over PR-NEH being 1.61%. Here, the lower bounds (LB) are considered instead of upper bounds (UB) for better comparison as the makespans better the UBs provided by Demirkol et al. in many cases. While using Taillard's dataset; three more priority rules are considered as detailed below: • PR-LJP1: Descending order of the sum of their average processing time, mean absolute deviation and absolute (skewness).
• PR-LJP2: Descending order of the sum of their average processing time, mean absolute deviation and skewness.
• PR-DB1: Descending order of the sum of their average processing time and median absolute deviation.  The PR-LJP proposed by Liu et al. [12] is proved to yield better results for the Taillard dataset. PR-LJP adds a third metric, absolute (skewness) to the priority rule, PR-D. However, when this absolute (skewness) is added to the sum of average processing time and mean absolute deviation (PR-LJP1); the APRD comes down to 3.16% from 3.06%. The APRD slightly improves to 3.12% when the absolute (skewness) is replaced by skewness (PR-LJP2). For this dataset, PR-D reports an APRD of 3.22% and PR-DB, 3.17%. The improvement of PR-DB is 4.52% over PR-NEH which is better than PR-D (3.01% improvement). When the median absolute deviation (MAD1) replaces the mean absolute deviation (PR-DB1), we get an APRD of 3.29% (Table V) for the Taillard instances.
The results of VFR (Small) and VFR (Large) datasets are given in Table VI. The performance is different here. For the VFR (Small) dataset, the performance of PR-DB worsens by 2.35% whereas; PR-D reports 0.78% improvement over the reference priority rule, PR-NEH. For the smaller problems, the priority rule, PR-D reports the best results. However, when the number of problem sets are considered, the reference rule, PR-NEH accounts for better results in 11/24 sets followed by PR-D, 8/24 sets and the new rule, PR-DB does well in only 5/24 cases.
However, for the VFR (Large) dataset, the performance of the newly proposed rule, PR-DB is better than PR-D and PR-NEH. The improvement over PR-NEH is 3.89% for this larger dataset. The new rule also reports best results in 13/24 problem sets. PR-D comes next with better results in 8/24 cases. The priority rule, PR-NEH which performs extremely well in smaller problems of VFR, could not match that performance and accounts for only 3/24 problem sets.
The summary of the results is presented in Table VII. For all the datasets except VFR (Small), the newly proposed priority rule, PR-DB outperforms PR-D and PR-NEH.
Paired t-tests are carried out at 95% confidence level using MINITAB17 and the results are presented in Table VIII. Since only PR-D and PR-DB are similar priority rules, they are compared with PR-NEH and listed in Table VII and  Table VIII. The analyses could prove the statistical significance in three cases with probability values of 0.001, 0.001 and 0.000 for the pairs; PR-NEH vs PR-DB (Demirkol), PR-NEH vs PR-D (VFR-Large) and PR-NEH vs PR-DB (VFR-Large) respectively.

V. CONCLUSION AND FUTURE WORK
This paper proposes a new priority rule, PR-DB for the permutation flowshop scheduling problems. The popular NEH algorithm considers the jobs according to the non-increasing order of their total processing times. Dong et al. added another metric, the standard deviation of the processing times of a particular job for initial ordering. In this work, the standard deviation used in the priority rule of Dong et al., PR-D is replaced by Mean Absolute Deviation. The PR is applied to the classic NEH heuristic for makespan minimisation. Computational results show that the performance is better than PR-D for the Carlier, Demirkol, Taillard and VFR (Large) datasets. The results do not improve in the case of VFR (Small) instances. The paired t-tests are carried out to assess the statistical significance. The main advantage of the new rule is that it is simple yet powerful. It can be easily applied in scheduling the jobs in any engineering industry. Similarly, it can be combined with any popular tie-breaking rule to improve the solution quality further. Also, the results obtained could be used as the seed solution for any metaheuristic for better solution and schedules. Future work includes the assessment of PR-DB for other potential benchmarks also.