Analysis of Algorithms for Solving Two Machine FSSP

This paper studies a scheduling problem of two-machine flow shop (FSSP). Minimization of maximum completion time always leads to the improvement of all the relevant measures of flow lines. Minimization of total completion time also much important in the production environment. In our earlier work we had developed two new heuristics based on Johnson’s rule for solving FSSP with make span objective which reduces the total completion time criteria also. In this paper, we have extended the investigation of our algorithms to a objective which combines both the objectives. Through case studies, we showed that one of our algorithms performed well than the other two.


Introduction
In production scheduling problem, FSSP is one of the most important field on which many of the objectives has been investigated by many researchers for decades. In flow shop environment, all the jobs are to be executed by a series of machines in the same technological order. Since Johnson [7] compiled a famous heuristic rule for makespan objective, a large number of researchers developed heuristic and meta heuristic to enhance the quality or performance analysis of their algorithms for the different objectives in variety of situations.
Indeed, many of the papers resulting the same general performance but produces different to resolve this inconsistency, they introduced a new objective for the n jobs m machine FSSP. This new approach includes the overall performance of min and min ∑ on all the machines which is better than individual objectives. We have taken into consideration of this objective for the two machine case in this paper when the machine order is reversed.
The remainder of our paper is organized as follows: In section 2, we have given a brief literature review. Main assumptions relating to our problem have been modeled in section 3.
The implementing technique was given in section 4 followed by an illustration of analysis of the algorithms in section 5. Experimental results and conclusions were given in section 6 and 7.

Literature review
A large amount of research has been done on FSSP which is formulated initially by Johnson [7] (1) to analyze our developed algorithms along with Johnson's rule.
In this paper, we have been discussed about the performance of the objective in (1) for the two machine n jobs FSSP.

Statement of the problem
In a FSSP, a set of n -jobs has to be processed on 2 -different machines in the same order. Each job , = 1, 2, … , must process on machines A and B with the non negative processing times 1 , 2 , … , 1 , 2 , … , . Each machine can processes at most one job and each job can be handled at most one machine at any given time. Once the job started its processing, it should be completed without any interruption. The machines process the jobs in a first come first served manner. The jobs are processed on machine B first followed by machine A (i.e., in the order B -A). Johnson algorithm breaks the tie situation occur on same machines by process the job with smallest index. In case of tie occur in between , , it can be broken by process the job which falls on machine while in CDS method, the tie situation was broken by giving priority to the job in the same position of the previous stage.
For handling this tie criterion, we presented two priority rules as follows:

Methodology
Based on Johnson's rule and the above said priority rules, Sathiya Shanthi et. al [4,12] developed two heuristic algorithms which were given below.

SMJ-SPT Algorithm [4]
Step 1: Observe the processing times of the all the jobs and select a job with smallest one.
Step 2: If it is for the machine A, then schedule the job first in the sequence. If it is for the machine B, then schedule the job last in the sequence.
Step 3: 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 4: Delete the corresponding job from the list and repeat the above steps until all the jobs are scheduled.

SMJ-LPT Algorithm [12]
Step 1: Observe the processing times of the all the jobs and select a job with smallest one.
Step 2: If it is for the machine A, then schedule the job first in the sequence. If it is for the machine B, then schedule the job last in the sequence.
Step 3: 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 4: Delete the corresponding job from the list and repeat the above steps until all the jobs are scheduled.
For a given two machine FSSP, we obtained three different sequences by applying three algorithms such as Johnson rule and our proposed above said algorithms. The illustration of the various sequence obtaining method are given in the next section in detail. Johnson's rule If we apply SMJ-SPT algorithm for the above problem, the first tie situation can be handled by applying priority rule 1 and after observing the processing times of job 2 and job 4 on the respective other machines, we found that the smallest processing time is 31 which fall in machine A for job 4 and so we schedule job 4 first in the sequence and job 2 after job 4. Similarly in the second tie situation, the smallest processing time is 21 which fall in machine B for job 1 and so we schedule job 1 last in the sequence. After all the jobs are scheduled, the sequence for the SMJ-SPT algorithm is as follows: If we apply SMJ-LPT algorithm for the above problem, the first tie situation can be handled by applying priority rule 2 and after observing the processing times of job 2 and job 4 on the respective other machines, we found that the largest processing time is 51 which fall in machine B for job 2 and so we schedule job 2 last in the sequence and job 4 before job 2. Similarly in the second tie situation, the largest processing time is 32 which fall in machine A for job 3 and so we schedule job 3 first in the sequence. After all the jobs are scheduled, the sequence for the SMJ-LPT algorithm is as follows:  Table 1 showed that the completion times of the sequence obtained by SMJ-LPT algorithm. To test the performance of the equation (1) for our proposed heuristic algorithms in two machine cases, the job number is chosen from 4,5,6,7,8,9,10. Consequently there are 7 scales of instances and there are 10 instances for each scale and 70 problem instances in total.

Jobs
The processing times are chosen arbitrarily from the numbers 1 to 99. Sum of the values of equation (1) obtained for 10 instances for each problem size is presented in Table 2. From that, it is observed that our proposed SMJ-LPT algorithm provides better results than Johnson's rule as well as SMJ-SPT algorithm. It is further observed that for the performance of equation (1) 2  4  8977  8664  7814  2  5  15314  15054  14061  2  6  13886  12463  11751  2  7  32042  31203  26218  2  8  27687  26630  23129  2  9  29688  29024  27902  2  10  51088  50166  47623   TOTAL  178682 173204 158498

Conclusion
In manufacturing systems, minimization of production line utilization related to makespan which is more important objective of FSSP. Moreover minimization of work in process inventories is also related to TCT. These two common objectives of FSSP are the main concerns of many researchers over the years. In this paper, we analyzed three algorithms for the performance of the objective in equation (1) and results shows that our SMJ-LPT algorithm performs well for the two machine case. Many real life disturbances in the production line control with various constraint environments may be addressed for this objective to achieve adoptive production control.