HEURISTIC APPROACH FOR N-JOB, 3-MACHINE FLOW SHOP SCHEDULING PROBLEM INVOLVING TRANSPORTATION TIME, BREAK DOWN TIME AND WEIGHTS OF JOBS

This paper provides a new simple heuristic algorithm for a ‘3-Machine, n-job’ flow-shop scheduling problem in which jobs are attached with weights to indicate their relative importance and the transportation time and break down intervals of machine are given. A heuristic approach method to find optimal or near optimal sequence minimizing the total weighted mean production flow time for the problem has been discussed.


INTRODUCTION
The practical importance of scheduling problem depends upon three factors i.e. job transportation time, relative importance of a job over another job and break down machine time.These three concepts were separately studied by various researchers as Johnson [1], Jackson [2], Bellman [3], Miyazaki and Nishiyama [4], Bansal [5].But in the present communiqué we have discussed all the three factors together which are more practical.The object of the paper pertains to determination of an optimal or near optimal sequence for n-job × 3-machine flow shop problem which associates 'weight' with a job in the sense of relative importance in the process and includes transportation time and break-down machine time minimizing the total weighted mean production flowtime.
Many applied and experimental situations exist in our day-to-day working in factories and industrial production concerns etc.The practical situation may be taken in a paper mill or sugar factory where various types of paper or sugar are produced with relative importance i.e. weight in jobs.The transportation time (which includes loading time, moving time and unloading time etc.) for the job and break-down of machine (due to failure of electric current or due to non-supply of raw material or other technical interruptions) which have been neglected in Johnson's [1] scheduling paper and which have a significant role in production concern.

FLOW-SHOP MODEL AND NOTATION
The flow-shop model can be stated as follows: (a)Let n-job be processed through three machines A, B and C in the order A BC. Let "i" denote the job in S where S is an arbitrary sequence.All jobs are available for processing at time zero.
(b)Let each job be completed through the same production stage, i.e.ABC.In other words, passing is not allowed in the flow shop.
(c)Let A i , B i , C i denote the processing time of job "i" on machine A, B, and C respectively t i and g i denote the transportation time of job "i" from A to B and from B to C respectively.(d)Let job "i" be assigned with a weight w i according to its relative importance for performance in the given sequence.
(e)The performance measure studied is weighted mean flow time defined by ,Where f i is flow time of i-th job.
(f)Let the break down interval (a,b) is already known to us, i.e. a deterministic nature.The break down interval length (b -a) which is known.
Then our aim is to find out optimal or near optimal sequence of jobs so as to minimize the total elapsed time.
The given problem in the tabular form may be stated as follows.
Job Machines with Processing time and Transportation time Weight of jobs 3. ALGORITHM Suppose that either one or both of the following structural conditions involving the processing time and transportation time of jobs hold.
Structural conditions : Then, the following steps of algorithm are : STEP I : Modify the problem into two machines flow-shop problem by introducing two fictitious machine G and H with processing time G i and H i respectively such that G i = A i + t i + B i + g i and H i = t i + B i + g i + C i The modified problem in the tabular form is ) has no effect on job i.

STEP VI :
Now repeat the procedure to get the optimal sequence.This sequence is either optimal or near optimal for the original problem.By this sequence we can determine the total elapsed time and weighted mean-flow time.
For clarification, the following numerical example is given below: Find optimal or near optimal sequence when the break-down interval is (a, b) = (18, 25).Also calculate the total elapsed time and weighted mean-flow time.

NUMERICAL EXAMPLE
SOLUTION : Now Min (A i + t i ) = 9 Max (t i + B i ) = 9 Hence, (I) structural condition is satisfied.Then, using the steps I to IV and applying Johnson technique for optimal and near optimal sequence we get the sequence (2, 3, 4, 1), Now to check the effect of break down interval (18, 25) on sequence (2, 3, 4, 1) is read as follows: Consider a four job and three machine problem with processing time, transportation time and the weight of jobs is given as: Now, repeating the procedure we get the sequence (2, 3, 4, 1) which is optimal or near optimal and the final table is: 7. R. G. Parker, Deterministic scheduling theory, Chapman and Hall, New York, 1995.