Johnsons Rule for Sequencing m jobs on n

Johnson’s Rule for Sequencing m jobs on n machines Let the machines be M 1, M 2, M 3, ……Mm One or both the following conditions must be satisfied, in order to use Johnson’s rule. 1. Minimum processing time on M 1 ≥ Maximum processing time on machines M 2, M 3… M(m-1) 2. Minimum processing time on Mm ≥ Maximum processing time on machines M 2, M 3… M(m-1)

Problem Six jobs J 1, J 2, … J 6, are required to be processed on four machines M 1, M 2, M 3 and M 4 in that order. Using the processing times given below, determine the optimal sequence(s) of job performance. Job Machine M 1 M 2 M 3 M 4 J 1 12 6 6 8 J 2 8 3 6 12 J 3 13 4 6 9 J 4 9 2 4 14 J 5 10 6 7 12 J 6 12 7 2 10

SOLUTION

Step 1 1. Minimum processing time on M 1 = 8 Maximum processing time on machines M 2, M 3 = 7 Minimum processing time on M 1 ≥ Maximum processing time on machines M 2 and M 3 2. Therefore the solution can be obtained using Johnson’s rule.

Step 2 Introduce two fictitious machines G and H such that time taken on G = time taken on M 1 + M 2 + M 3 and time taken on H = time taken on M 2 + M 3 + M 4 Job Machine G Machine H Machine M 1 M 2 M 3 M 4 J 1 12 6 6 8 J 2 8 3 6 12 J 3 13 4 6 9 J 4 9 2 4 14 J 5 10 6 7 12 J 6 12 7 2 10

Step 2 Introduce two fictitious machines G and H such that time taken on G = time taken on M 1 + M 2 + M 3 and time taken on H = time taken on M 2 + M 3 + M 4 Job Machine G H J 1 12 + 6 = 24 6 + 8 = 20 J 2 8 + 3 + 6 = 17 3 + 6 + 12 = 21 J 3 13 + 4 +6 = 23 4 + 6 + 9 = 19 J 4 9 + 2 + 4 = 15 2 + 4 + 14 = 20 J 5 10 + 6 + 7 = 23 6 + 7 + 12 = 25 J 6 12 + 7 + 2 = 21 7 + 2 + 10 = 19

Step 3 The converted problem looks like this: Job Machine G H J 1 24 20 J 2 17 21 J 3 23 19 J 4 15 20 J 5 23 25 J 6 21 19 Now find the optimal sequence(s) like 3 -machine problems and proceed to find the time taken.

Practice Problem The table below gives the processing time (in hours) of seven jobs to be processed on four machines M 1, M 2, M 3 and M 4, in the order M 1 M 2 M 3 M 4. Sequence the given jobs using Johnson’s method and find the overall processing time. Job Machine M 1 M 2 M 3 M 4 A 3 1 4 12 B 8 0 5 15 C 11 3 8 10 D 4 7 3 8 E 5 5 1 10 F 10 2 0 13 G 2 5 6 9 Optimal Sequence: A E F B G D C or A E F G B D C