An heuristic approach for solving three machine FSSP with the objective of minimizing total completion time

Scheduling is the allocation of resources over a period of time to perform a collection of tasks. In this paper, we extended our earlier work [14] for three machine flow shop scheduling problem (FSSP) and investigated three heuristic algorithms for the objective of total completion time of all the jobs (TCT). Experimental results had shown that one of our algorithms gives better results than the other two when the machine order is reversed.

decision making process. Most of the researchers gave their attention to the practical importance and complexity of the problem. Scheduling flow shop will always been interesting to deal when more constraints added in its environment.
The main aim of FSSP is that to obtain a sequence of jobs to minimize certain performance measure such as maximum completion time of all the jobs or makespan and sum of the completion time of all the jobs or TCT in a schedule. In the production environment, resources play a vital role there and maximization of utilization of the resources ensures that minimization of the makespan. Also minimization of TCT guarantees the minimization of the work in process inventory [2] Framinan 2003.
Because of its combinatorial in nature, FSSP is very much essential in production planning and control process which sustains practical importance and offers a great theoretical challenge to the researchers. Moreover the importance of the FSSP has been strengthened by the emergence of flexible manufacturing system and manufacturing industries which are integrated by computer networking. In such areas, different manufacturing work stations have to produce similar parts and due to their similarity it might visit the machines in an identical order.
In general, shop scheduling problems contains m different machines and n different jobs each of which being processed on all the machines. So each job requires m different operations and in FSSP, the sequence of operations has been processed in the same technological order in all the machines, that is, a job which is processed in the first machine has to be processed in the second machine and so on. This scheme has to be followed for all the jobs in the sequence. The processing times of the jobs on the machines are known well in advance and it is non-negative. In a given time, only one job has to be processed by at most one machine and each machine may have a job for processing or it may wait for a job but any operation might not be stopped before its completion on its respective machine. All the jobs are available at time zero for its processing.
In general, heuristic algorithms ensures some reasonable near optimal solution but not global optimum always. Some of the composite heuristics and constructive heuristics with minimization of TCT objective are presented here. Liu and Reeves [8] 2001 developed some efficient heuristic algorithm which is the composite of constructive heuristics and local search. Allahverdi and Aldowaisan [1] 2002 produced a set of composite heuristics by many constructive heuristics such as Rajendran and Zeigler 1997 [11] and Woo and Yim 1998 [15] or a local search 2456-8686, 6(1), 2022: 067-077 https://doi.org/10.26524/cm121 Journal of Computational Mathematica Page 69 of 77 algorithm based on pair wise exchange. Framinan and Leisten [2] (2003) developed a heuristics for TCT objective and analyzed with those of Rajendran and Zeigler [11] (1997) and Woo and Yim [15](1998). Framinan  we analyzed three algorithms for two machine FSSP with TCT objective. In this paper, we extended the same for three machine FSSP with the objective TCT when the machine order is reversed.

Methodology
Johnson [6] provided a dominance condition for reducing three machine FSSP problem in to two machine FSSP. The same methodology is employed here. The condition for reducing three machine FSSP into two machine FSSP is same as that of Johnson's algorithm which is given below: For handling tie criteria, we proposed two priority rules as follows: Priority1: Select the job with smallest processing time on the other machines and process it first, if it belongs to machine A or last, if it belongs to machine B.
Priority2: Select the job with largest processing time on the other machines and process it first, if it belongs to machine A or last, if it belongs to machine B.
By adopting the above priority rules, two heuristic algorithms had been developed in [5,13]  Time (SMJ-LPT) algorithm for solving three machine FSSP which is given below:

SMJ-SPT Algorithm [5]
Step 1: Check the condition for reduction and reduce the problem into two machine problem if anyone or both the conditions are satisfied.
Step 2: Observe the processing times of the all the jobs and select a job with smallest one.
Step 3: If it belongs to first machine, then schedule the job first in the sequence. If it belongs to second machine, then schedule the job last in the sequence.
Step 4: In case of tie occurs on same machine, select the job with smallest index, In case of tie occurs on different machines ( ), apply the priority rule 1.
Step 5: Delete the corresponding job from the list then repeat the above steps until all the jobs are scheduled.

SMJ-LPT Algorithm [13]
Step 1: Check the condition for reduction and reduce the problem into two machine problem if anyone or both the conditions are satisfied.
Step 2: Observe the processing times of the all the jobs and select a job with smallest one.
Step 3: If it belongs to first machine, then schedule the job first in the sequence. If it belongs to second machine, then schedule the job last in the sequence.
Step 4: In case of tie occurs on same machine, select the job with smallest index, In case of tie occurs on different machines ( ), apply the priority rule 2.
Step 5: Delete the corresponding job from the list then repeat the above steps until all the jobs are scheduled.

Illustration
Consider a FSSP with 5 -jobs on 3 -machines.

SMJ-SPT algorithm
According to SMJ-LPT algorithm, to break the tie, priority should be given to a job with the longest processing time. So schedule job 2 in the beginning of the sequence as shown below: SMJ-LPT algorithm After breaking the tie situation, the remaining jobs are scheduled as per the respective algorithm. If the remaining jobs are allotted by applying Johnson's algorithm for this problem, we got the sequence as 4 -5 -3 -2 -1 which gives TCT as 413. We got the sequence as 5 -3 -2 -4 -1 if we apply SMJ-SPT algorithm which gives TCT as 389 where as by applying SMJ-LPT algorithm, we got the sequence as 2 -4 -5 -3 -1 which gives TCT as 379. From this we conclude that SMJ-LPT algorithm performs well than the other two algorithms for this problem.

Computational results
To test the efficiency of our proposed heuristic algorithms for the TCT objective, 70 problem instances were taken into consideration, each of 10 instances for 2machine with 4 jobs, 5jobs, 6jobs, 7jobs, 8jobs, 9jobs and 10jobs. The processing times are chosen arbitrarily from the numbers 1 to 99. Sum of the TCT values of 10 instances for each problem size is presented in Table 1. The data available in this table 1 had shown that that our proposed SMJ-LPT algorithm provides better results than Johnson's algorithm as well as SMJ-SPT algorithm for the TCT objective. It is further observed that for the TCT objective concern, SMJ-LPT algorithm outperforms well than Johnson's algorithm in 62 out of 70 problem instances were tested. Out of 70 problem instances were tested, SMJ-LPT algorithm provides best result than SMJ-SPT algorithm for 41 problem instances, equal values for 6 problem instances and worst result for 23 problem instances. Moreover SMJ-SPT algorithm outperforms well than Johnson's algorithm in 48 out of 70 problem instances were tested, equal value for 1 problem instance and worst result for 21 problem instances.   Table 2: TCT values for Johnson's, SMJ-SPT and SMJ-LPT algorithms

Conclusion
Two heuristic algorithms proposed by the authors based on Johnson's rule were tested for the TCT objective on three machine njobs FSSP. From the analysis, the heuristic SMJ-LPT algorithm is far superior to Johnson's rule as well as SMJ-SPT algorithm and hence we conclude that an algorithm with optimizes multi objectives is more essential in the field of production when the machine order is reversed. So one can utilize our proposed SMJ-LPT algorithm for the multi objective purpose.