## 2 Conflicting Objectives and Non-Dominated Solutions of EMS Design Problem
The following expression (4) can present the objective function describing the criterion. \[f^{2}(\mathbf{y})=\sum_{j=1}^{n}b_{j}\text{max}\big{\{}0,sign\big{(}\text{min}\big{\{} t_{ij}:i=1,\ldots,m;y_{i}=1\big{\}}-T\big{)}\big{\}} \tag{4}\]  
These two criteria stand in conflict, as shown in [18]. When one of the objective functions is minimized, it signifies that an increase in the other objective function compensates for the drop in the other. To make the final decision on suitable EMS design easier, we try to offer a significant spectrum of designs containing such solutions, which cannot be dominated by any other solution. We say that the solution \(\mathbf{y}\)\(\in\)\(Y\) is non-dominated if there is no such solution \(\mathbf{x}\)\(\in\)\(Y\) that \(f^{1,}(\mathbf{y})\)\(\
eq\)\(f^{1,}(\mathbf{x})\) or \(f^{2,}(\mathbf{y})\)\(\
eq\)\(f^{2,}(\mathbf{x})\) and \(f^{1,}(\mathbf{y})\)\(\geq\)\(f^{1,}(\mathbf{x})\) and \(f^{2,}(\mathbf{y})\)\(\geq\)\(f^{2,}(\mathbf{x})\). The maximal set of non-dominated solutions is called the Pareto front of solutions. As concerns the Pareto front identification, various methods were developed. Most of them are based on the usage of solving tools for Problems (5) and (6). \[\text{min}\Big{\{}f^{1}(\mathbf{y}):\mathbf{y}\in\mathbf{Y};f^{2}(\mathbf{y})\leq u\Big{\}} \tag{5}\]  
\[\text{min}\Big{\{}f^{2}(\mathbf{y}):\mathbf{y}\in\mathbf{Y};f^{1}(\mathbf{y})\leq u\Big{\}} \tag{6}\]  
To describe some of the methods, we introduce the following functions, which produce the optimal solutions of (5) and (6). The procedure producing the optimal solution of (5) will be denoted by \(GetSol1(u)\), and the procedure for Problem (6) will be denoted by \(GetSol2(u)\). Then, the leftmost component of the Pareto front for the given EMS design problem can be obtained by performing \(GetSol1(f^{2,}(GetSol2(+\infty)))\), and the rightmost member can be obtained by \(GetSol2(f^{1,}(GetSol1(+\infty)))\). We mention here two different approaches to obtaining the Pareto front or its very good approximation with \(\varepsilon\) accuracy. The first approach is based on the bisection process. It makes use of the proposition that if the inequality \(f^{1,}(GetSol1(u))>\)\(f^{1,}(GetSol1(v))\) holds for \(u<v\), then there must be at least one member \(\mathbf{y}\) of the Pareto front, for which the following inequalities are fulfilled: \(u<\)\(f^{2,}(\mathbf{y})\)\(\leq v\). Otherwise, there is no member of the Pareto front, for which the above inequalities hold. This proposition is used for a bisection process, which proved to be able to identify the complete Pareto front of the EMS designs [18]. Another approach [27; 28] alternatively changes the usage of \(GetSol1\) and \(GetSol2\) to determine the next member of the Pareto front with a lower value of \(f^{2}\). If \(\mathbf{x}\) is a member of the Pareto front, the next member with a better value of \(f^{2}\) can be obtained by the following two optimization processes with an accuracy of \(\varepsilon\):  
1. Perform \(\mathbf{z}=GetSol1(f^{2}(\mathbf{x})-\varepsilon)\). 2. Determine \(\mathbf{y}=GetSol2(f^{1,}(\mathbf{z}))\).