## 3 Proposed Approach
### The Improved GEP
#### 3.2.2 Evolutionary Search Process  
_(1) Fitness evaluation:_ Fitness is crucial for GEP algorithms because it evaluates the quality of candidate solutions and directs the evolution process. The IGEP algorithm evolves a population of individuals and each one is decoded into a dispatching rule. For each specific objective, the average normalized objective value of a rule \(r\) through training scenarios \(S\) is taken as the rule fitness value \(fitness(r)\), as shown in Equation (7), where \(ref\) refers to a reference rule. The SPT/TWKR, PT + WINQ, and WATC rules are accepted as reference rules due to their remarkable performance in decreasing the MS, MFT, and MWT goals [14; 18; 49], respectively. Finally, Equation (8) is used to estimate the percentage change \(PC\) in the performance of a given method, with respect to a reference rule:  
\[fitness(r)=\frac{1}{|S|}\sum_{s=1}^{|S|}\frac{f(r,s)}{f_{ref}(s)} \tag{7}\]  
\[PC=100\cdot(1-fitness(r)) \tag{8}\]  
_(2) Evolution processes:_ The initial generation evolved in the GEP algorithm are a set of individuals represented as trees that can be decoded into the corresponding candidate dispatching rules. The GEP algorithm creates computer programs of varying lengths and sizes, each of which is recorded in a linear chromosome of a predetermined length. In addition, it is common for GEP chromosomes to consist of multiple genes that are of equivalent length. Functionally speaking, the GEP genes consist of two distinct regions, namely the head and tail. The interdependence between the head and tail, coupled with the limitation that the tail exclusively comprises terminal symbols, ensuring the generation \begin{table}
\begin{tabular}{c c} \hline \hline
**Node Name** & **Description** \\ \hline AT & The current time \\ PT & Processing time of the operation \\ NPT & Processing time of the next operation \\ OWT & The waiting time of the operation \\ NOPS & Number of operations of the job \\ WIQ & Work in the current queue \\ WINQ & Work in the next queue \\ NOIQ & Number of operations in the current queue \\ NOINQ & Number of operations in the next queue \\ DD & Due date of the job \\ MRT & Ready time of the machine \\ SL & Slack time of the job \\ WKR & Work remaining (including the current operation) \\ NOR & Number of operations remaining \\ FDD & Flow due date of the operation \\ W & Weight of the job \\ \hline Function set & +, -, \(\times\), /, max, min \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The terminal and function sets. of a valid tree expression for each gene. Figure 3 depicts a dispatching rule developed in accordance with the research. After the initial generation, a proportion of 10% of the most optimal individuals in the population is stored in the elite library. Then, the entire population (including 10% of individuals in the elite library) is equally divided into groups, and genetic operations (such as selection, replication, mutation, transportation, and recombination) are performed for each subpopulation after grouping. The selection process can preserve excellent individuals, allowing them to pass into the next generation. Roulette wheel sampling with elitism is used here to select the appropriate individuals in the subpopulation, based on their fitness for the next generation. The mutation is designed to generate perturbations for each individual to avoid premature and stagnant problems in evolution. To preserve the fundamental structure of chromosomes, it has been observed that symbols located in the head region have the ability to transform into any other function or terminal, whereas symbols situated in the tail region are only capable of transforming into terminals. The principle of transportation involves the stochastic selection of a chromosomal fragment, which is subsequently inserted into the head region. Three types of transport operators are applied for each part of the chromosome: IS transportation, RIS transportation, and gene transportation.