# Test Center Location Problem: A Bi-Objective Model and Algorithms
## 5 Simulation Results
### Comparison Results
* A trade-off exists between the running times of both algorithms and the quality of the non-dominated solutions they produce. As previously mentioned, the running time of TCLA is directly influenced by the size of potential center locations, denoted as \(m\), and the number of selected centers, denoted as \(k\). To simplify its application, we set the population size to \(N=2\) cm and the number of generations to \(T=cN\), where \(c\) is the minimum of either \(k\) or \(m-k\). On the other hand, in the case of Gurobi, since the model is an integer linear program, its solution quality is influenced by the parameter \(MIPGap\). Smaller values of \(MIPGap\) result in higher solution quality but longer running times. In general, increasing the number of demand points has a noticeable impact on the running time of both Gurobi and TCLA. However, the impact is less pronounced in the case of TCLA. For larger instances, Gurobi may take more than 3 h to complete. * Another important consideration is that the running time of Gurobi is closely tied to the spatial distribution of points and the shape of the search space. In addition to instance size, the specific locations of the points play a crucial role in Gurobi'sperformance. However, this factor has a less significant impact on the running time of TCLA. * The average \(scm\) metric for \(scm(TCLA,Gurobi)\) and \(scm(Gurobi,TCLA)\) are 0.098 and 0.081, respectively. So, TCLA finds solutions slightly close to real Pareto-optimal solutions compared to Gurobi's solutions. However, because of the small size of the obtained non-dominated solutions, it appears that the \(\alpha\beta\) metric provides a better assessment of solution quality. * According to \(\alpha\beta\) metric, aside from two instances, namely (100,40,15) and (100,40,20), where the difference between the obtained solutions is approximately 2.23% and 3.67%, respectively, the efficiency of both approaches is similar. * While the Pareto-optimal queue of the TCLP is generally non-convex, the \(\epsilon\)-constraint approach demonstrates an ability to discover a diverse array of solutions. * The reported time for the Gurobi is the total time to run the Gurobi for different \(\epsilon\) values in the \(\epsilon\)-constraint approach. So, if a decision maker is interested in finding just one single Pareto solution with a preferable level of work balance or average travel distance, the Gurobi will run faster than TCLA on small and medium-sized instances. Generally, the comparison results indicate that both TCLA and the Gurobi-based \(\epsilon\)-constraint approach show significant promise in effectively tackling the test center location problem. The choice between these approaches may depend on various factors such as problem size and computational resources (time and space), with each approach demonstrating its advantages. TCLA excels in providing swift and reasonably high-quality solution sets, making it particularly suitable for scenarios where quick decision-making is essential. On the other hand, the \(\epsilon\)-constraint approach with the Gurobi solver offers a quicker solution for small and medium-sized instances of the TCLP, especially when the objective is to identify a single optimal solution. This advantage stems from Gurobi's proficiency in handling integer linear programming models, while TCLA proves its adaptability in situations where linearity is not a critical constraint. As a result, TCLA holds the potential for broader applicability and extension to various problem variations, especially those demanding nonlinear modeling, such as scenarios where distance calculations, such as Euclidean distance, should be integrated directly into the model. Finally, it becomes evident that Gurobi struggles to handle large-size instances of the