Sequencing Problem. Dr.S.S.Kulkarni

Transcription

1 Sequencing Problem The selection of an appropriate order for a series of jobs to be done on a finite number of service facilities, in some preassigned order, is called sequencing. The general sequencing problem may be defined as; Let there be n jobs to be processed one of a time on each of m machines. The sequence (order) of the machines in which each job should be performed as given. The actual as expected time requires by the job on each machine is also given. The general sequencing problem, therefore, is to find the sequence out of (n!) m possible sequence which minimize the total elapsed time between the stat of the job in the first machine and the completion of the last job on the machine.

2 Following are the assumptions undulying a sequencing problem. 1. Each job, once started on a machine, is to be performed up to completion on that machine 2. The processing time on each machine is known. Such a time is independent of the order of the jobs in which they are processed. 3. The tine taken by each job in changing order from one machine to another is negligible. 4. A job starts on the machine as soon as the job and the machine both are idle and job is next to the machine and machine is also next to the job. 5. No machine may process one job simultaneously. 6. The order of completion of job has no. significance i.e. no job is to be given priority. The order of completion of jobs is independent of sequence of jobs.

3 Basic terms used in sequencing 1.Number of machines If means the service facilities through which a job must pass before it is completed. 2.Processing order It refers to the order in which various machines are required for completing the job. 3.Processing time It indicates the time required by a job on each machine. 4.Total elapsed time It is the time for which a machine does not have a job to process. i.e. time from the end of job (i-1) the start of job.

4 5.Total elapsed time This is the time between starting the first job and completing the last job including the idle time (if any) in a particular order by the given set of machines. It is denoted by T. 6.No passing rules. It refers to the rule f maintaining the order in which jobs are to be proceed on given machines.

5 Processing of n jobs through two machines The problem can be described as follows 1. Only two machines A & B are available 2. The processing order of the job is processing i.e. AB 3. Passing is not allowed 4. Processing times either exact or expected A 1, A 2.. A n & B 1, B 2.. B n are known. The problem is to sequence (order) the jobs so as to minimize the total elapses time T.

6 The solution procedure adopted by Johnsom (1954) is given below. Step I List the jobs along with their processing times in a table as shown below. Job n Processing time on machine A A 1 A 2 A 3 A n B B 1 B 2 B 3 B n Step II Find the smallest processing time in each machine i.e. find out min (A j, B j ) for all j

7 Step III If the smallest processing time is for the first machine A, then place the corresponding job in the first available position in the sequence. Step IV If there is a tie in selecting the mimimum of all the processing times, then three situations may arise. Minimum among all processing time is same for the machine, i.e. min (A j, B j )=A k = B r then k th job first and the r th job last. If the tie for the minimum occurs among processing times A j on machine A only, then select the job to process that corresponds to the minimum of B j s and process it first of all. If the tie for the minimum occurs among the B j s, select the job corresponding to the minimum of A i s and process it in the last go to next step.

8 Step V Cross out the jobs already assigned and repeat the step II through IV until all the jobs have been assigned. Processing n jobs through 3 machines I. The problem can be described as II. Only three machines A 1 B and C are involved. III. Each job is processed in the prescribed order ABC IV. Transfer of jobs is not permitted Exact or expected processing times are given as Job Machine A B C 1 A 1 B 1 C 1 2 A 2 B 2 C 2 3 A 3 B 3 C 3 n A n B n C n

9 There is no general method available by which we can obtain optimal sequence in problem involving processing of n jobs on 3 machines. The method adopted by Johnson can be extended to corner the special cases where either one or both of the following conditions hold. The minimum time on machine A the maximum time on machine B The minimum time on machine C the maximum time on machine B. The procedure explained here is to replace the problem with an equivalent problem, involving n jobs and two fictitious machines denoted by G & H and corresponding time G i & H i are defined by G i = A i + B i H i = B i + C i

10 If this problem with prescribed order GH is solved, then the resulting optimal sequence will also be optional for the original problem. Processing n jobs through m Machines Consider the situation when there are n jobs each of which to be processed on m machine M 1, M 2...M n in order M 1, M 2...M n and T ij denote the time taken by procedure for obtaining an optimal sequence is as follows Step I First find Min T ij j Min T mj j Max (T 2j T 3j.. T m-ij ) for j=1,2, n. J

11 Step II Check whether Min T ij Max T iji = 2,3,. m-1 j j Min T ij Max T ij i = 1, 2,. m-1 Step III If inequalities of step II are not satisfied then method fails and problem is not solvable. Step IV Convert the m machine problem into 2 machine problem considering two fictitious machines G & H so that T Gj = T ij +T 2j + T (m-1)j and T Hj = T 2j + T 3j +.. T mj

12 Now determine the optimal sequence of n jobs through 2 machines wing the optimal sequencing algorithm Step V In addition to conditions given in step IV, if T 2j + T 3j T (m-1)j = c (a fixed positive constant) j = 1, 2,., n. Then determine the optimal sequence for n jobs and two machines M 1 & M m in the order M 1 M 2 by using the optimal sequencing algorithm.

13 Notes: 1. If in addition to the condition given in the step IV, 2. If T ij = T mj and T Gj = T Hj for j = 1,2,. A then n! number of optimal sequences will exist This procedure for sequencing n jobs through m machines is not a general procedure. The method is applicable to only such sequencing problem in which minimum time of processing the jobs through h first and/ or last machine is greater than or equal to the time of processing the jobs through medio machine.