A Makespan Optimization Scheme for NP-Hard Gari Processing Job Scheduling Using Improved Genetic Algorithm

Algorithm 3

Johnson’s 𝑚-machine scheduling algorithm.

Stage 1:  (Conversion of 𝑚 machines (or processes to 2-Machine/process equivalent)

Step  0. Set  𝑖=𝑗=𝑘=0

Step  1. Set  𝑘=𝑘+1

Step  2. Set  𝑗=𝑗+1

Step  3. Set  𝑖=𝑖+1

Step  4. Read  𝑃_𝑖𝑗 (the  processing  time  for  the  𝑖th  job on  the  𝑗th  machine/process) and corresponding job number 𝐽_𝑖

1≤𝑗≤𝑚 for  each  job 𝑖 (1≤𝑖≤𝑛)

Step  5. 𝑄_𝑖𝑘=𝑄_𝑖𝑘+𝑃_𝑖𝑗  (update  of  the  cumulative  job  time  for  the  𝑖th  job  on  machine 𝑘=1,2)

Step  6. If   𝑖<𝑛GO  TO Step  3

Else Set 𝑖=0 and GO  TO  Step  7

Step  7. 𝐈𝐟𝑗<𝑚/2GO  TO  Step  3

    If  𝑗=𝑚/2GO  TO  Step  1

    𝐈𝐟𝑚/2<𝑗<𝑚GO  TO  Step  2

    𝐄𝐥𝐬𝐞𝐢𝐟𝑗=𝑚, STOP

Stage 2: 2-machine (process) equivalent Scheduling Algorithm

Step  8. 𝐑𝐞𝐚𝐝𝑆=𝑄_𝑖𝑘forall𝑖,𝑘;1≤𝑖≤𝑛,𝑘=1,2

Step  9. Set  𝑟=0, 𝑠=𝑛+1

Step  10. 𝐑𝐞𝐚𝐝𝑃_𝑖𝑘=𝑄Min𝑖,𝑘^𝑖𝑘 for  all𝑖,𝑘𝜖𝑆

   If  𝑘=1, set 𝑟=𝑟+1

   Set job 𝐽_𝑖 with time 𝑃_𝑖𝑘as  Scheduled  job𝐽_𝑟

Else Set 𝑠=𝑠−1

   Set job 𝐽_𝑖 with time 𝑃_𝑖𝑘as Scheduled  job𝐽_𝑠

Step  11. Set 𝑆=𝑆−𝑃_𝑖1−𝑃_𝑖2

Step  12. If 𝑆=∅ (empty) STOP

      Else GO TO 𝑆𝑡𝑒𝑝10