# Effects of Decision Variables Selection on Sewer Optimization Problem
## 2 Methods
### Artificial Protozoa Optimizer (APO)
The autotrophic mode weight factor \(w_{a}\) helps in updating the movement of the protozoan as a function of the neighbors' positions, such as \(X_{k-}\) and \(X_{k+}\). The factor \(f\) represents the foraging factor, which decides upon the step size, and \(M_{f}\) indicates mapping vector to decide which dimension of the protozoan position needs to be modified. \[X_{i}^{new}=X_{i}+f\cdot\left(X_{near}-X_{i}+\frac{1}{np}\cdot\sum_{k=1}^{np}w_ {h}(X_{i-k}-X_{i+k})\right)\bigodot M_{f}, \tag{12}\]  
In this equation, \(X_{near}\) is a position of a nearby nutrient-rich region and \(w_{h}\) is the weight factor in heterotrophic mode. The protozoan locates towards \(X_{near}\) for exploiting nutrient sources with \(f\) controlling the magnitude of the move while \(M_{f}\) gives the updated dimensions. \[X_{i}^{new}=X_{min}+rand\bigodot(X_{max}-X_{min}), \tag{13}\]  
Here, \(X_{min}\) and \(X_{max}\) represent the lower and upper bounds of the search space, respectively, while \(rand\) is a random vector with elements in the range [0, 1]. \[X_{i}^{new}=X_{i}\pm rand\bigodot(X_{min}+rand\bigodot(X_{max}-X_{min})\big) \bigodot M_{r}, \tag{14}\]  
In this equation, \(X_{i}\) is the current position of the protozoan, and \(X_{i}^{new}\) represents the new position after reproduction. The scalar \(rand\) introduces randomness, and \(M_{r}\) is the mapping vector for reproduction. Few critical parameters form the basis upon which the artificial protozoa optimizer algorithm can regulate the optimization process so as to ensure effectiveness: the number of protozoa, a candidate solution from the population of size specified by \(ps\), such that the larger the value, the more the number of protozoa, the greater the diversity, at the cost of computational cost; \(dim\) specifies the dimensionality within the optimization problem, i.e., the number of decision variables, structuring the search space that protozoa will traverse. The number of neighbor pairs (\(np\)) is considered during a foraging to align the positions of protozoans based on the influences from neighbors, thus helping in exploration and exploitation. _MaxFEs_ bound the algorithm, defining the stopping criterion. Proportion fraction of dormancy and reproduction (\(pf\)) and its upper limit (\(pf_{max}\)) control the balance between introducing new solutions and exploiting those of high quality. An important factor, the foraging factor (\(f\)), controls the adaptive step size of the foraging process to balance exploration and exploitation. In this formulation, the autotrophic (\(w_{a}\)) and heterotrophic (\(w_{h}\)) weight factors modulate the dynamics of the movement towards the light source and nutrient-rich areas, respectively. Lastly, an adaptive setting of the probability of autotrophic and heterotrophic behavior (\(p_{ab}\)) determines the propensity to exhibit either foraging mode, gradually shifting the movement from a state of exploration to one of exploitation over iterations. The trade-off between entering the dormancy stage and reproduction is controlled by the \(p_{dr}\). The parameter values of the APO algorithm were chosen based on the recommendations of the initial study by [PERSON] et al. [34]. The flowchart of the APO algorithm is given in Figure 2.