# Test Center Location Problem: A Bi-Objective Model and Algorithms
## 4 Solution Approach for Test Center Location Problem
### An \(\epsilon\)-Constraint Method for the TCLP  
The \(\epsilon\)-constraint method is a popular, simple and flexible method for multi-objective optimization, but it typically has limited ability to provide in-depth insights into Pareto-optimal solutions [28; 32]. In fact, the \(\epsilon\)-constraint method requires the designation of one objective as primary and the others as constraints. This categorization of primary and secondary of course can be subjective and may lead to biased results. So, the decision maker needs to have extra knowledge and perform additional analyses. One significant challenge of \(\epsilon\)-constraint lies in determining the optimal value of \(\epsilon\), as selecting values that are either too small or too large can lead to narrow or overly expansive sets of feasible solutions, respectively. Additionally, the method's sensitivity to changes in \(\epsilon\) can result in instability and inconsistency in the obtained Pareto front. Furthermore, the approach may fail to adequately explore complex or non-convex solution spaces, potentially overlooking valuable solutions lying outside the \(\epsilon\)-defined range. While the \(\epsilon\)-constraint method can identify non-dominated solutions, it may struggle to capture the trade-offs between con flicting objectives, leading to suboptimal or incomplete representations of decision-maker preferences. Finally, computational complexity can pose a significant challenge, particularly for large or complex problems, where the computational burden associated with generating and evaluating candidate solutions may become prohibitive. Despite these limitations, the \(\epsilon\)-constraint method remains a useful, easy and popular approach, especially in cases where computational resources are limited or where simplicity and interpretability are prioritized.  
In the implementation of the \(\epsilon\)-constraint method, we select \(F_{2}\) and represent it as a constraint within the TCLP model, Equation (5). To facilitate decision-maker comprehension and practicality, we establish both a lower and an upper bound for \(\epsilon\) values. This ensures that \(F_{2}\) objectives remain within this predefined range. Specifically, for this purpose, we set \(k=m\) and compute \(F_{2}\) values represented as \(\frac{1}{\sum_{i=1}^{m}w_{i}}\Sigma_{i=1}^{n}(w_{i}\min\limits_{1\leq j\leq m }d_{ij})\). This corresponds to solutions where each demand point is allocated to its closest potential facility center. Conversely, by configuring \(k=1\), where all the demand points are allocated to the same center and \(F_{1}\) is no longer important, the upper bound for the \(F_{2}\) value can be established within polynomial time. Finally, the decision maker's preferences for the number of desired Pareto optimal solutions sets the number of \(\epsilon\) values, which are then uniformly selected from this defined range and set in the following constraint.  
\[\frac{1}{\Sigma_{i=1}^{n}w_{i}}\Sigma_{i=1}^{n}w_{i}(\Sigma_{j=1}^{m}x_{ij}d_{ ij})\leq\epsilon, \tag{6}\]  
Upon relocating the previously mentioned equation to the constraint section, we transform the problem into a single-objective optimization model, i.e., as an ILP. This model can be efficiently solved using widely available commercial solvers like Gurobi [33], yielding a single Pareto-optimal solution in each run. By introducing variations in the \(\epsilon\) values and iteratively executing the process, we can systematically generate a diverse set of Pareto-optimal solutions.