MIP Models and Hybrid Algorithms for Simultaneous Job Splitting and Scheduling on Unrelated Parallel Machines

Algorithm 1

Local search.

For all generations 𝐺

For the population 𝑃

For all chromosomes 𝐶

𝑖=0 to 𝑔

𝑗=𝑖+1 to 𝑔

If Gene(i)_Machine == Gene(j)_Machine → (inter-machine interchange)

 If 𝑗=𝑖+1→ (adjacent jobs) (Figure 5)

   𝑎=0; 𝑏=0; 𝑐=0;

   If 𝑖>0 & Gene(i)_Machine == Gene(𝑖−1)_Machine

      𝑎=ST_{Mach(𝑖)Job(𝑖−1,𝑗)}−ST_{Mach(𝑖)Job(𝑖−1,𝑖)}+ST_{Mach(𝑖)Job(𝑗,𝑖)}−ST_{Mach(𝑖)Job(𝑖,𝑗)} → (“a” on Figure 5)

   If 𝑖==0∥ Gene(i)_Machine != Gene(𝑖−1)_Machine → (Gene(i)_Job is the first job on Gene(i)_Machine)

      𝑏=ST_{Mach(𝑖)Job(𝑗,𝑗)}−ST_{Mach(𝑖)Job(𝑖,𝑖)}+ST_{Mach(𝑖)Job(𝑗,𝑖)}−ST_{Mach(𝑖)Job(𝑖,𝑗)} → (“b” on Figure 5)

   If 𝑗<𝑔−1 & Gene(j)_Machine == Gene(𝑗+1)_Machine

      𝑐=ST_{Mach(𝑖)Job(𝑖,𝑗+1)}−ST_{Mach(𝑖)Job(𝑗,𝑗+1)} → (“c” on Figure 5)

   If 𝑎+𝑏+𝑐<0

      Let to interchange Gene(i)_Job and Gene(j)_Job on the schedule

      Calibrate random key numbers → (Section 4.2.2, item “i”)

 If 𝑗>𝑖+1 → (non-adjacent jobs) (Figure 6)

   𝑎=0;𝑏=0;𝑐=0;

   If 𝑖>0 & Gene(i)_Machine == Gene(𝑖−1)_Machine

      𝑎=ST_{Mach(𝑖)Job(𝑖−1,𝑗)}−ST_{Mach(𝑖)Job(𝑖−1,𝑖)}+ST_{Mach(𝑖)Job(𝑗,𝑖+1)}−ST_{Mach(𝑖)Job(𝑖,𝑖+1)}

         +ST_{Mach(𝑖)Job(𝑗−1,𝑖)}−ST_{Mach(𝑖)Job(𝑗−1,𝑗)}

   If 𝑖==0∥ Gene(i)_Machine != Gene(𝑖−1)_Machine → (Gene(i)_Job is the first job on Gene(i)_Machine)

      𝑏=ST_{Mach(𝑖)Job(𝑗,𝑗)}−ST_{Mach(𝑖)Job(𝑖,𝑖)}+ST_{Mach(𝑖)Job(𝑗,𝑖+1)}−ST_{Mach(𝑖)Job(𝑖,𝑖+1)}

         +ST_{Mach(𝑖)Job(𝑗−1,𝑖)}−ST_{Mach(𝑖)Job(𝑗−1,𝑗)}

   If 𝑗<𝑔−1 & Gene(j)_Machine == Gene(𝑗+1)_Machine

      𝑐=ST_{Mach(𝑖)Job(𝑖,𝑗+1)}−ST_{Mach(𝑖)Job(𝑗,𝑗+1)}

   If 𝑎+𝑏+𝑐<0

      Let to interchange Gene(i)_Job and Gene(j)_Job on the schedule

      Calibrate random key numbers → (Section 4.2.2, item “i”)

If Gene(i)_Machine != Gene(j)_Machine → (intra-machine exchanging) (Figure 7)

   𝑎=0;𝑎₁=0;𝑎₂=0;𝑏=0;𝑏₁=0;𝑏₂=0; Makespan_𝐷1 = 0; Makespan_𝐷2 = 0;

   a = CT_Gene(i)_Machine + PT_{Mach(𝑖)Job(𝑗)}−PT_{Mach(𝑖)Job(𝑖)}

   If 𝑖>0 & Gene(i)_Machine == Gene(𝑖−1)_Machine

      𝑎₁=ST_{Mach(𝑖)Job(𝑖−1,𝑗)}−ST_{Mach(𝑖)Job(𝑖−1,𝑖)}

   If 𝑖==0∥ Gene(i)_Machine != Gene(𝑖−1)_Machine → (Gene(𝑖)_Job is the first job on Gene(i)_Machine)

      𝑎₁=ST_{Mach(𝑖)Job(𝑗,𝑗)}−ST_{Mach(𝑖)Job(𝑖,𝑖)}

   If Gene(i)_Machine == Gene(𝑖+1)_Machine

      𝑎₂=ST_{Mach(𝑖)Job(𝑗,𝑖+1)} − ST_{Mach(𝑖)Job(𝑖,𝑖+1)}

   b = CT_Gene(j)_Machine + PT_{Mach(𝑗)Job(𝑖)}−PT_{Mach(𝑗)Job(𝑗)}

   If Gene(j)_Machine == Gene(𝑗−1)_Machine

      𝑏₁=ST_{Mach(𝑗)Job(𝑗−1,𝑖)}−ST_{Mach(𝑗)Job(𝑗−1,𝑗)}

   If Gene(j)_Machine ! = Gene(𝑗−1)_Machine

      𝑏₁=ST_{Mach(𝑗)Job(𝑖,𝑖)}−ST_{Mach(𝑗)Job(𝑗,𝑗)} → (Gene(j)_Job is the first job on Gene(j)_Machine)

   If 𝑗<𝑔−1 & Gene(j)_Machine == Gene(𝑗+1)_Machine

      𝑏₂ = (ST_{Mach(𝑗)Job(𝑖,𝑗+1)} − ST_{Mach(𝑗)Job(𝑗,𝑗+1)}

   𝑎=𝑎+𝑎₁+𝑎₂

   𝑏=𝑏+𝑏₁+𝑏₂

   Makespan_𝐷1 = Max(CT_Gene(i)_Machine; CT_Gene(j)_Machine)

   Makespan_𝐷2 =Max(𝑎;𝑏)

   If Makespan_𝐷2 < Makespan_𝐷1

      Let to exchange Gene(i)_Job and Gene(j)_Job on the schedule

      Calibrate random key numbers → (Section 4.2.2, item “ii”)