Specially Structured Flow Shop Scheduling in Two Stage with Concept of Job Block and Transportation Time to Optimize Total Waiting Time of Jobs

Flow shop scheduling, where the machines are prearranged in order that the flow of each and every of the products that are processed in them is unidirectional. In the present paper Flow shop scheduling models in two stage are well thought of. The problem is specially structured with the idea that maximum of equivalent processing time on first machine remains always less than the minimum of equivalent processing time on second machine. The intention of the study is to find the schedule which lessens the total of the waiting time of all the jobs. The time to transport jobs from first machine to second machine is considered separately and two jobs considered as a group job. An Algorithm to optimize the desired function and the program in C++ language is proposed. Numerical example by applying the algorithm proposed is solved.

This paper is an addition in the study made by Gupta D. and et.al. [18] in the sense that we have taken into consideration the concept of making a job block for two jobs.
III. PRACTICAL SITUATION Industrialized units play an imperative role in the monetary growth of a country. Flow shop scheduling happens in banks, airports, service stations etc. Regular working in industries and factories has diverse jobs which are to be practiced on various machines. The idea of lessening the total of the waiting time for all the jobs may be a reasonable aspect from managers of Factory /Industry perspective when he has contract to made the work with less waiting with a viable party to finish the work.

IV. NOTATIONS
S j : Schedule of the jobs. m j 1 : Time taken by first machine to process j th job. m j 2 : Time taken by second machine to process j th job. ' X j : Equivalent processing time taken by machine X to process j th job.
' Y j : Equivalent Processing time taken by machine Y to process j th job. : Time taken to transport j th job from first machine to second machine. is the time consumed in transporting j th job from machine M 1 to machine M 2. The problem formulation in matrix form as defined in [6], [16] can be seen in TABLE I. Our goal is to come across a best possible sequence S j of jobs by considering two jobs (l, m) as a job block with the intention to optimize the total of the waiting time of all the jobs. The equivalent processing times ' X j and ' Y j of j th job on Fictitious machines X and Y as defined by Singh T.P. [6], Gupta D. and et.al. [16], [18] are given by

A. Assumptions
In the given flow shop scheduling the following assumptions are made: 1) Machines M 1 and M 2 are processing n jobs, firstly on machine M 1 then on machine M 2 and no passing is permissible. 2) At the same time no job will be processed by both of the machines.
3) The course of action of the machines can't be interrupted until a job which is in execution can't be completed. 4) Set-up time of machines, Break down interval of machines is negligible. 5) It is given two jobs l, m as a block with priority of processing job l over job m in the block (l, m). are the processing times of n jobs on machines X and Y correspondingly satisfying processing times structural relationship defined in equation (2) in that case for the n job sequence     n ,..., , : Proof. Using principle of Mathematical Induction on number of jobs: is true for n=1.
Assume the result holds for less than n jobs, Following the similar notations as used in B. Lemma, for n job sequence

D. Theorem
Following the similar notations as used in B. Lemma, for the n job sequence the total waiting time ) (W is given by

E. Equivalent Job Block Theorem
Assuming the two machines X and Y are processing n jobs in the sort XY.
on machine X and Y respectively.(l, m) is the group job or job block which can be made equivalent to the one job α (called equivalent job α). Job α has processing times ' X  and ' Y  on the machines X and Y and are given by The theorem is proved by Maggu P.L.and et.al. [4].
VI. ALGORITHM Step 1: Calculate the processing times for the fictitious machines X and Y denoted by ' X j and ' Y j defined as in equation (1 Step 2: Verify the processing time structural relationship ' ' Y Min X Max j j  as defined in equation (2).
Step 3: Take equivalent job α = (l, m) and calculate processing times using equations (7) and replace the pair of jobs (l, m) in this order by the single job α.
Step 4: Calculate the values for ' in the TABLE II.
Step 5: Assemble the jobs in ascending order of x j Assume the schedule thus found be ) ,....., , ( Step 6: Find the other schedules of jobs S 1 , S 2 ,….., S n-1 .Where S 1 is the schedule obtained in 5 th step, schedules S i , 1≤i≤n-1 can be obtained by taking i th job in the sequence S 1 to the 1 st position and considering respite of the schedule same. Step 7: Calculate the total waiting time W for all the sequences S 1 , S 2 ,….., S n-1 using the equations (6)   cout<<"Equivalent processing time of job block on machine X and Y\n"; cout<<"X = "<<xp<<"\n"<<"Y = "<<yp<<"\n"; cout<<"Fictitious Machine X\n";  TABLE I with n=5 can be seen in TABLE III. Our intention is to attain most favorable schedule of jobs lessening the total of the waiting time for all the jobs by considering 3, 5 in a block (3,5). Solution As per step 1-Calculate the processing time for the fictitious machines X and Y in TABLE IV as defined in [6], [16], [18] given by equations (1). As per step 2: , hence the processing time structural relationship defined in equation (2) is satisfied. As per step 3-Taking (3, 5) as a job block denoting this job block by α. The processing times on both of the machines X and Y for single job α are calculated using equations (7): for the given problem in TABLE V.
VII. CONCLUSION The idea of lessening the total of the waiting time for all the jobs in the flow shop scheduling is very significant in the case when the manufacturer or producer has a bond with the customers to made their work without waiting for too much time. Though it may raise the other costs such as penalty cost of the jobs or the total elapsed time etc. By taking into account the various parameters like time to set up the machines, time consumed in breakdown of machines etc. the study can be further generalized.