# Effects of Decision Variables Selection on Sewer Optimization Problem
## 1 Introduction  
Optimization problems consist of objective function(s), decision variables, and constraints. The most important step of optimization is to model the problem mathematically. Determination of decision variables also plays a key role in creating the mathematical model. In some optimization problems, the same problem can be solved with different numbers of decision variables. This situation depends on the decision variables derived from each other. The Sewer optimization problem is one of these types of problems. In the literature, Sewer system optimization has been traditionally divided into three broad groups: layout optimization, hydraulic optimization, and combined layout and hydraulicoptimization. In layout optimization, optimal positions for system elements, such as manholes, pipelines, and pumping stations, are determined to provide efficient flow paths. For instance, [PERSON] and [PERSON] [1] made a comparison between deterministic models and stochastic models for layout optimization of sewers. [PERSON] [2] introduced a loop-by-loop cut algorithm to develop efficient drainage layouts. [PERSON] et al. [3] created an optimization strategy utilizing a depth-first search to develop optimal layouts for gravity pipe networks. [PERSON] et al. [4] utilized graph theory to develop feasible sanitary sewer networks. A hybrid genetic algorithm has been used by [PERSON] et al. [5] to develop optimal sewer network layout designs. Hydraulic optimization is concerned with the best values of the pipe diameters, slopes, and depths, assuming a pre-defined network layout. The earliest cost-based models for branched sewer system optimal design have been proposed by [PERSON] and [PERSON] [6]. [PERSON] et al. [7] presented a hybrid metaheuristic approach by combining genetic algorithms and tabu search to reach more efficient and reliable sewer system hydraulic optimization. [PERSON] [8] used a genetic algorithm to optimize the storm sewer network, demonstrating the algorithmic usability to treat difficult topologies and constraints. [PERSON] et al. [9] formulated a hydrograph-based optimization approach using genetic algorithms, which was effective for the simulation of unsteady stormwater flow. [PERSON] [10] formulated a parameter-free continuous ant colony optimization algorithm for storm sewer systems, both for constrained and unconstrained formulations. [PERSON] and [PERSON] [11] utilized simulated annealing to solve sewer system optimization problems by using Excel. [PERSON] et al. [12] formulated an optimized procedure for a sewer series of pipes by integrating analytical calculations with hydraulic constraints to reach cost-efficient and technically feasible designs. The usability of genetic algorithms has been improved by [PERSON] and [PERSON] [13] by adapting the mutation rate dynamically. [PERSON] et al. [14] proposed a heuristic algorithm to aid efficient hydraulic optimization for sewer network planning. A two-phase simulation-optimization strategy combining cellular automata with evolutionary algorithms has been proposed by [PERSON] et al. [15] for large-scale sewer systems. A number of studies investigated combined layout optimization and hydraulic optimization. [PERSON] and [PERSON] [16] used a combined GA-QP approach to simultaneously solve layout and hydraulic optimization problems. [PERSON] and [PERSON] [17] utilized particle swarm optimization (PSO) to optimize both layout and pipe sizes simultaneously. [PERSON] and [PERSON] [18] considered an arc-based ant colony optimization method to perform combined gravity-driven network design. [PERSON] et al. [19] modeled the sewer layout problem as a mixed-integer linear program, inspired by rectilinear Steiner tree algorithms, to reduce the cost of pipes as well as excavation via the utilization of alternative connection nodes. [PERSON] et al. [20] provided an inclusive mathematical framework that combines layout choice with the hydraulic optimization problem with realistic constraints.