# Test Center Location Problem: A Bi-Objective Model and Algorithms
## 5 Simulation Results
### Comparison Results
We establish a range encompassing the lower and upper bounds for the feasible values of \(F_{2}\), denoting the average travel distance between individuals and their nearest open center. Subsequently, based on the desired quantity of Pareto-optimal solutions, we evenly select various \(\epsilon\) values from this specified interval. For example, for an interval \([left,right]\), if we are interested in finding at most \(h\) Pareto-optimal solutions, we run the Gurobi solver for \(\epsilon=left+i(\frac{right-left-left}{h-1})\), for \(i=0,1,2,...,h-1\). It's worth notingthat for certain \(\epsilon\) values, particularly those close to the lower boundary of the interval, no feasible solutions may be attainable. Additionally, some of these \(\epsilon\) values may yield identical optimal solutions, indicating that in the bi-objective space, the solutions obtained from larger \(\epsilon\) values are dominated by those obtained from smaller ones. Consequently, we run the Gurobi solver for all \(h\) sampled \(\epsilon\) values independently and subsequently report solely the Pareto-optimal solutions that have been ultimately obtained. We generate random instances of varying sizes for the test center location problem and, for each instance, employ both TCLA and Gurobi solver using the described \(\epsilon\)-constraint approach. For each run, we provide the number of non-dominated solutions obtained by TCLA, the number of Pareto optimal solutions obtained by Gurobi, and the set coverage metric \(scm(A,B)\), where \(A\) and \(B\) are the obtained solutions by Gurobi and TCLA, respectively. In addition, we report the running time (in seconds) for both approaches. It's worth noting that TCLA can find a non-dominated set in a single run, whereas Gurobi executes separately for each \(\epsilon\) value, resulting in one Pareto-optimal solution per run. Therefore, for Gurobi runs, we report two types of running times: the total running time for finding all Pareto-optimal solutions, and the average running time to discover a single Pareto-optimal solution, excluding runs that yield no feasible solutions. The comparison results are presented in Table 1, and the corresponding obtained non-dominated solutions in the objective space are depicted in Figure 6. We ran the algorithms and asked to find at most 10 non-dominated solutions. The table represents the number of non-dominated solutions by each of the approaches (TCLA is denoted by \(T\) and Gurobi is denoted by \(G\)) as well as \(scm(.,. )\) metric, \(\alpha\beta(.,. )\) metric and the running time (in seconds) for all 18 different instances of the problem. The first 8 instances are generated randomly in a 150 \(\times\) 100 rectangular shape, while for the larger instances with 200, 500, and 1000 weighted demand points, a bigger rectangle with a size of 1500 \(\times\) 1000 is used. The weights are also assigned randomly in the interval \([10,100]\). The final two runs, pertaining to the instances (500,100,50) and (1000,100,50), represent exceedingly large cases of the TCLP. In these scenarios, there exists a staggering number of possible center combinations, approximately on the order of \(5.39\times 10^{23}\). To conduct TCLA runs for these instances, we configured the population size to \(N=1000\) and the number of generations to T = 15,000. The computational time for TCLA under these settings amounted to 1061 s for the (500,100,50) instance and 1937 s for the (1000,100,50) instance. In contrast, Gurobi encounters substantial challenges when dealing with such formidable instances. For the former case, it necessitates a staggering 12,841 s (over 3 h and 30 min), while for the latter, we were compelled to terminate the program after 6 h due to a lack of any discernible outcome. It's worth noting that TCLA exhibited a relatively modest memory usage of approximately 200 MB. In contrast, when using Gurobi, the memory consumption significantly surpassed this, exceeding 3800 MB. Moreover, the ensuing results have been derived from the comparison between TCLA and Gurobi.