# Crossover Operator Inspired by the Selection Operator
## 2 Modified Evolutionary Algorithm for Task Sequencing
### New Crossover Operator  
During the evolution process, individuals in the new population undergo modifications using genetic operators. Typically, crossover and mutation operators are employed, but in this study, individuals will be modified only using the crossover operator. As previously mentioned, the exclusion of the mutation operator is dictated by the need to exclude its influence on the analyzed results. Figure 2: The chromosome encodes the task sequence. Each task is described by a set of parameters, \(\overline{\mathbf{p}}\). Each node is connected to two neighbors by a connection, the length of which represents the cost \(C\) of the transition between tasks and is a function of the parameters of the adjacent tasks. During optimization, the chromosome forms a cyclic graph with no indication of the start of the sequence. The authors proposed a new, in contrast to the existing crossover methods (PMX, OX, Single-Point Crossover, and SCX), partially deterministic method, preferring the exchange of sequence fragments with a lower cost of external connections. The new crossover method consists of the following stages:  
1. Two individuals are selected randomly from the population for the crossover operation. One acts as a donor of a sequence fragment, and the other acts as a recipient of a sequence fragment. 2. For the donor individual in the randomly selected pair, all fragments of the task sequence are evaluated with a length constituting a metaparameter of the algorithm. The evaluation is based on the total cost of connections between tasks within a sequence fragment: \[C=\sum_{i=1}^{M-1}c(\overline{P}_{i},\overline{P}_{i+1})\] (2) where \(M\)--number of tasks in the analyzed fragment of the sequence (fragment length), \(\overline{P}_{i}\)--parameters of the \(i\)-th task in the fragment of the sequence, \(c()\)--the cost of transitioning between two tasks. 3. Having the cost of each fragment of the sequence, we can select the fragment transferred from the donor to the recipient using a method analogous to the proportional selection operator: \[P_{i}^{F}=\left(1-\frac{C_{i}-C_{min}}{\sum_{j=1}^{N}(C_{i}-C_{min})}\right)^{a}\] (3) where \(P_{i}^{F}\)--probability of selecting the sequence fragment \(i\); \(C_{i}\)--cost of internal transitions for the \(i\)-th fragment, \(C_{min}\)--internal cost of the cheapest fragment, \(N\) is the number of fragments (consistent with the number of tasks in the sequence), \(a\)--exponent controlling the strength of fragment selection. The coefficient \(a\) allows you to control the strength of selection. The value 0 means a completely random selection of a fragment of the sequence, which is analogous to solutions known in the literature [15; 22; 23]. Increasing the value of the coefficient \(a\) increases the deterministic impact of the internal cost of a sequence fragment. Further increasing the value of the coefficient \(a\) towards infinity brings us to a completely deterministic selection of the best fragment (with the lowest cost). The coefficient \(a\) is the next metaparameter of the discussed algorithm. The probabilities determined according to the Equation (1) are finally scaled to a sum equal to 1. 4. On the recipient side, for the same length of a sequence fragment, the cost of internal connections in each fragment is determined based on Equation (2). Figure 3: Example of a chromosome in the form of a ring (cycle) graph for 12 tasks. Then, a fragment is cut out from the recipient's chromosome (sequence), and a fragment taken from the donor will be inserted in its place. The probability of selecting such a fragment can be determined by the proportional (4) method. As you can see, the determined probabilities are complements to probabilities (3). \[P_{i}^{F}=\left(\frac{C_{i}-C_{min}}{\sum_{j=1}^{N}(C_{i}-C_{min})}\right)^{a}\] (4)
6. Division of genes (tasks) deleted from parent 2 (recipient) into two groups (Figure 4).