# Test Center Location Problem: A Bi-Objective Model and Algorithms
## 5 Simulation Results
### Comparison Results  
We employ the _Set Coverage Metric_ (SCM) to assess the Pareto-optimality of the ultimate solutions acquired [36]. In the context of two solution sets denoted as \(A\) and \(B\), the SCM (denoted as \(scm(A,B)\)) is defined as follows. \[scm(A,B)=\frac{|\{b\in B|\exists\;a\in A:a\;dominates\;b\}|}{|B|}. \tag{7}\]  
Here, we utilize the notation \(|.|\) to represent the cardinality (size) of a set. The metric value \(scm(A,B)=1\) signifies that any solutions within set \(B\) are dominated by at least one solution in set \(A\). Conversely, when \(scm(A,B)=0\), it implies that no solution in \(B\) is dominated by any solutions in \(A\). Consequently, when \(scm(A,B)\) approaches 1 while \(scm(B,A)\) approaches 0, it indicates that solution set \(A\) outperforms solution set \(B\) in terms of Pareto optimality. In the scenario where the set \(A\) comprises Pareto-optimal solutions, it is evident that \(scm(B,A)=0\) holds true for any set \(B\); however, \(scm(A,B)\) serves as a gauge of set \(B\)'s effectiveness in achieving Pareto optimality, measuring its efficiency in this regard. In this part, we employ the Gurobi 5.6.3 optimization solver [33] with a specified parameter of \(MIPGap=1\times 10^{-3}\). Our goal is to identify a collection of solutions that are either Pareto-optimal or very close to being Pareto-optimal. These solutions will subsequently be used as one of the sets in the calculation of \(scm(A,B)\). Similarly, we employ TCLA and identify the resulting non-dominated solutions, which will serve as the other set in the calculation of \(scm(A,B)\). It is important to note that the solutions obtained through Gurobi may not necessarily be Pareto-optimal due to the presence of an optimality gap. This gap signifies the difference between the best-known integer solution and the current best solution discovered during Gurobi's branch-and-bound process. The optimality gap is expressed as a percentage of the objective function value and plays a crucial role in balancing the trade-off between solution quality and solution computation time. The metric \(scm(A,B)\) calculates the number of solutions within a set \(B\) that are dominated by at least one solution in set \(A\). However, it lacks the ability to quantify \"to what extent\" a solution \(b\in B\) may be dominated by a solution \(a\in A\). To address this concern, we extend the concept of the _approximation factor_ from single-objective optimization to bi-objective optimization as follows: If we represent the objective values of a solution \(b\) from the set \(B\) as \(F_{1}(b)\) and \(F_{2}(b)\), when solution \(a\) dominates solution \(b\), we have  
\[[F_{1}(a)<F_{1}(b)\;\;and\;\;F_{2}(a)\leq F_{2}(b)]\;\;\;or\;\;\;[F_{1}(a)\leq F _{1}(b)\;\;and\;\;F_{2}(a)<F_{2}(b)] \tag{8}\]  
Now, let \(\alpha\) and \(\beta\) be the smallest values that satisfy the following equations. \[F_{1}(a)\geq(1-\alpha)F_{1}(b)\;\;and\;\;F_{2}(a)\geq(1-\beta)F_{2}(b) \tag{9}\]  
Indeed, parameter \(\alpha\) (\(\beta\)) signifies the percentage by which solution \(b\) must enhance its performance to surpass solution \(a\) in objective \(F_{1}\) (\(F_{2}\)). Consequently, decreasing \(\alpha\) percent in the direction of \(F_{1}\), or \(\beta\) percent in the direction of \(F_{2}\), will render solution \(b\) no longer dominated by solution \(a\). Furthermore, augmenting in both directions simultaneously will yield a stronger solution. Now, identifying the minimum pair of \((\alpha,\beta)\) that satisfies equation Equation (9) for all solutions \(a\in A\) reflects the quality of the solution \(b\). Furthermore, extending this fact to all solutions \(b\in B\) and choosing the maximum \(\alpha\) and \(\beta\) values among them will represent the quality of solution set \(B\) in comparison to solution set \(A\). Let us denote this metric by \(\alpha\beta(A,B)\). To utilize the Gurobi solver, we implement the \(\epsilon\)-constraint methodology, as elucidated in Section 4.1.