# Test Center Location Problem: A Bi-Objective Model and Algorithms
## 5 Simulation Results
### Results on TCLA  
We run TCLA on the model presented in Equation (5) to find Pareto optimal solutions. The code of the algorithm is implemented in the programming language Python 3.7 and runs on a standard PC (Intel(R) Core(TM) i7 (Santa Clara, CA, USA) and 32G RAM). To this end, we consider a rectangular environment with a size of \(1500\times 1000\) and generate instances with random locations for the demand points, \(P\), and potential centers, \(Q\). Also, we assign random weights for the demand points in \([10,100]\). Figure 1 shows the random instance with \(n=100\) weighted demand points and \(m=25\) locations at which test centers will be opened. We denote each random instance of the TCLP with a triplet \((n,m,k)\), where \(k\) is the number of opened centers. We run TCLA for \((100,25,k)\), where \(k\in\{5,8,12,15\}\). The combinatorial complexity of the search space of the TCLP is related to \(m\) and \(k\) such that the number of possible solutions is \((^{m}_{k})\). This implies that the worst case happens for \(k=\frac{m}{2}\). On the other hand, the complexity of TCLA, like all the population-based heuristics, is directly related to the size of the population, \(N\), and the number of generations, \(T\). We set the size of the population in TCLA to \(N=2\) cm and the number of generations to \(T=cN\), where \(c=min\{k,m-k\}\). We choose these values because they achieve an optimal balance between processing time and the quality of the obtained non-dominated solutions. This choice is informed by our analysis of the simulation results. Figures 2-5 show the results for the instance illustrated in Figure 1 for \(k=5\), \(k=8\), \(k=12\) and \(k=15\), respectively. In each figure, we select three solutions from the obtained non-dominated set, two extreme solutions and one middle solution. Indeed, we sort the obtained non-dominated solutions according to one of the objectives, i.e., \(F_{1}\), and choose the first (Figures 2-5a), last (Figures 2-5b) and middle (Figures 2-5c) solutions. Also, we depict all the obtained non-dominated solutions in the objective space (Figures 2-5a). The solid blue squares show the selected (opened) test center location in each solution, and for simplicity, we draw the Voronoi edges of them (the green lines). Consequently, the Voronoi region of each selected test center and the demand points that are allocated to each center can be recognized easily. Note that the demand points are weighted (see Figure 1). The running times of TCLA for \(k=5\), \(k=8\), \(k=12\), and \(k=15\) are approximately 7, 24, 41, and 33 s, respectively. TCLA finds 9, 13, 8, and 11 non-dominated solutions for \(k=5\), \(k=8\), \(k=12\), and \(k=15\), respectively. For \(k=5\), the range of \(F_{1}\) values spans from 127 to 829, and their corresponding \(F_{2}\) values vary between 23.1 and 21. The resulting solution set exhibits a good distribution pattern along the \(F_{1}\) axis. However, there exists a noticeable gap in the \(F_{2}\) values, from 23.1 to 26.6, where no solutions are found. In the cases of \(k=8\) and \(k=12\), the solution sets exhibit a well-distributed spread in both objective spaces. For \(k=8\), the \(F_{1}\) values range from 189 to 701, while the \(F_{2}\) values lie in the interval of (16.5, 18.6). Conversely, for \(k=12\), the \(F_{1}\) values span from 308 to 578, and the \(F_{2}\) values range from 14.3 to 13.5. Lastly, for \(k=15\), the obtained solution set covers a range of \(F_{1}\) values from 284 to 523 and \(F_{2}\) values between 12.3 and 14.3. Pareto-optimal solutions play a significant role in aiding decision-makers when selecting an efficient trade-off solution. It is essential to recognize that enhancing one objective often necessitates a trade-off with another objective. The degree of improvement and the associated trade-offs require careful examination.