# Effects of Decision Variables Selection on Sewer Optimization Problem
## 1 Introduction
Optimization algorithms are used to solve mathematical models after the mathematical model of the problem is obtained. Optimization algorithms are designed to find the best combinations of decision variables' values for one or more objective functions. These algorithms are utilized in all disciplines, especially economics, basic sciences, or engineering. The effectiveness of optimization algorithms of real-life problems has been proved in many studies [21; 22; 23; 24]. The objective function is considered as the cost of the sewer network; therefore, it consists of the pipe cost, the manhole cost, and the excavation cost. Pipe diameters and pipe slopes that are unknown are sought to minimize the objective function. These unknowns can be directly selected as decision variables, or different variables can be selected as decision variables. In the literature, three different cases are encountered in this matter. In the first case, the pipe diameter, in the second case, the pipe slopes, and in the third case,the pipe diameter and pipe slopes can be selected as decision variables. Constraints consist of lower and upper bounds of velocities and relative flow depths in pipes, lower and upper bounds of slopes, and cover depths. [25]  
If the pipe diameter is chosen as the decision variable, the pipe slope and therefore the cover depths can be calculated hydraulically. If the pipe slope or cover depths are chosen as decision variables, the pipe diameter can also be calculated hydraulically. If the pipe diameter and the pipe slope or cover depths are chosen as decision variables, there is no need for hydraulic calculations. So, choosing decision variables also determines the number of decision variables [25]. Different methods such as linear programming, nonlinear programming, and recently meta-heuristic algorithms are used to solve the sewer optimization problem. When the studies in which different decision variables were made in the sewer optimization problem using meta-heuristic methods are examined, the following studies can be summarized:  
[PERSON] et al. [7] used the genetic algorithm and tabu search algorithm in solving the hydraulic optimization problem of the sewer network. Pipe diameter and pipe slope were selected as decision variables. They were applied to the network of The East-I Wastewater Collection Engineering project built in Taiwan. [PERSON] et al. [9] used a genetic algorithm in the solution of the hydraulic optimization problem of the sewer network in their study. Two different alternatives were preferred for decision variable selection. In the first alternative, nodal elevation and pipe diameter were selected as decision variables. In the second alternative, only nodal elevation was selected. The application was made on the network proposed by [PERSON] and [PERSON] [26]. [PERSON] et al. [27] used the particle swarm algorithm in solving the hydraulic optimization problem of the network in their study. Pipe diameter and pipe slope were selected as decision variables. The application was made on a small network. [PERSON] [25] used the rebirthing genetic algorithm in solving the hydraulic optimization problem of the sewer network. A pipe slope was selected as the decision variable. The application was made on the network proposed by [PERSON] and [PERSON] [26]. [PERSON] and [PERSON] [28] used an adaptive genetic algorithm in solving the hydraulic optimization problem of the network in their study. Pipe diameter, pipe slope, and pump location were selected as decision variables. The application was made on the network proposed by [PERSON] and [PERSON] [29]. [PERSON] and [PERSON] [30] used the ant colony algorithm in their study to solve the hydraulic optimization problem of the sewer network. Nodal elevations were selected as decision variables. The application was made on three different sized networks proposed hypothetically. [PERSON] and [PERSON] [11] used the simulated annealing algorithm in their study to solve the hydraulic optimization problem of the network.