## 4 Genetic Algorithm
### Genetic Operators
#### 4.5.4 Tournament Selection
In Step-1, several genes are arbitrarily selected (each one with 50% chance). In Step-2, the contents of Parent-1 are copied to Child-1 without the machine assignment information of the arbitrarily chosen genes. In the last step, the missing machine assignment information is copied from Parent-2. Figure 5: Illustration of the crossover operators SSC-1 and SSC-2. Figure 6: Illustration of the JLGSC crossover operator (an asterisk * in Step-1 denotes the location of an arbitrarily selected gene). Child-2 is created in a similar manner by starting Step-1 from Parent-2 for the same locations of the arbitrarily selected genes in creating Child-1 (i.e., locations 6, 10, 14, 18, 20, 23, 25, 27, and 31). Here, it is essential to mention that, although there are many crossover operators, whenever crossover is to happen between a pair of parent chromosomes, only one crossover operator will be arbitrarily selected and applied with a probability (\(p_{\textit{cross}}\)) to create two offspring. If the selected crossover operator is not applied (by chance), the parent chromosomes will move to the next generation (with or without mutation operators applied, again by chance). #### 4.5.6 Mutation Operators

Mutation operators are applied with a small probability on newly generated offspring to alter the genetic material. This category of operators used in this paper is listed below. They are adapted from [2], and the need to divide the genetic search into two stages is taken into consideration. The first two mutation operators, SGVM and SGSM, are applicable in both Stage-1 and -2 of the GA search. Whereas OGSM is applicable only in Stage-1, and

Figure 8: Illustration of the MAC crossover operator (the asterisks * denote the locations of arbitrarily selected genes). Figure 7: Illustration of the JLOSC crossover operator (an asterisk * in Step-1 denotes the location of an arbitrarily selected gene). OSSM, ROAM, and IOAM are applicable only in Stage-2. Each one of the six mutation operators listed below is applied with a probability \(p_{mut}\) on a newly generated offspring as long as it is eligible for the stage of the search of the GA.

1. [label=()]
2. Sublot Gene Value Mutation (SGVM). 3. Sublot Gene Swap Mutation (SGSM). 4. Operation Gene Shift Mutation (OCSM). 5. Operations Sequence Shift Mutation (OSSM). 6. Random Operation Assignment Mutation (ROAM). 7. Intelligent Operations Assignment Mutation (IOAM). SGVM is used to alter the value of a gene \(\alpha_{j,s}\) in Segment-1 of a newly born offspring chromosome. When this operator is applied, a single gene is arbitrarily selected, and its value is either increased or decreased (50% by chance) with a small quantity according to Equations (65) or (66), respectively. In these two equations, rand() is a function that returns a random number in the interval [0, 1], and \(\delta\) is the maximum increment or decrement quantity, which can be regarded as the GA's parameter that needs to be set. In this research, we found that a value of \(\delta\) between 0.1 and 0.2 is preferable. The second mutation operator, SGSM, arbitrarily selects a job \(j\) in Segment-1 and swaps the values of two arbitrarily selected genes \(\alpha_{j,s}\) and \(\alpha_{j,s^{\prime}}\) corresponding to subplots \(s\) and \(s^{\prime}\) (\(s\
eq s^{\prime}\)). \[\alpha_{j,s} =\min\{1,\alpha_{j,s}+\text{rand(}0\times\delta\} \tag{65}\] \[\alpha_{j,s} =\max\{1,\alpha_{j,s}-\text{rand(}0\times\delta\} \tag{66}\]

Operation gene shift mutation (OGSM) is applied on form-1 of Segment-2 of the solution representation (Figure 1c) during Stage-1 of the GA search. When this operator is applied, it first arbitrarily selects a gene \((j,s,o)\), and then relocates it to an arbitrarily location after and before the locations of the genes \((j,s,o-1)\) and \((j,s,o+1)\), respectively.