## 4 Illustrative Examples  
In this section, examples are given to illustrate the optimal completion of the _incomplete_ PCMs using the _Nelder-Mead algorithm_. We are interested in matrices of order 4 and above, because matrices of order 3 have an analytic formula [58], and hence the optimal completion will be trivial. **Example 1**. : _Consider the \(4\times 4\) incomplete pairwise comparison matrix \(\mathbf{A}:\)_  
\[\mathbf{A}= \begin{pmatrix}1&*&1/3&1\\ *&1&1/9&*\\ 3&9&1&3\\ 1&*&1/3&1\end{pmatrix} \]  
_where \(*\) represents the missing comparisons. Clearly, by using the consistency condition, one obtains \(a_{12}=a_{13}a_{32}=3\) and \(a_{24}=a_{23}a_{34}=1/3\). Equivalently, the above pairwise comparison matrix with unknown variables \(x_{1}\) and \(x_{2}\) can be rewritten as_  
\[\mathbf{A}(\mathbf{x})= \begin{pmatrix}1&x_{1}&1/3&1\\ 1/x_{1}&1&1/9&x_{2}\\ 3&9&1&3\\ 1&1/x_{2}&1/3&1\end{pmatrix} \]  
_where \(\mathbf{x}=(x_{1},x_{2})\)._  
As mentioned before, it could be worthwhile that the expert expresses their preferences in the form of intervals. Therefore, we can formulate, as examples, two instances of the eigenvalue minimization problem with interval constraints:  
\[ \begin{array}{ll}\min&\lambda_{max}(\mathbf{A}(\mathbf{x}))\\ \text{s.t.}&1/9\leq x_{1}\leq 9\\ &1/9\leq x_{2}\leq 9\end{array}  \tag{11}\]  
and:  
\[ \begin{array}{ll}\min&\lambda_{max}(\mathbf{A}(\mathbf{x}))\\ \text{s.t.}&5\leq x_{1}\leq 7\\ &1/9\leq x_{2}\leq 9.\end{array}  \tag{12}\]  
With initial value \(\mathbf{x}_{0}=(1,1)\), applying the proposed algorithm on minimization (11), the optimal solution is \(\mathbf{x}^{*}=(3,1/3)\), and hence \(\lambda_{max}(\mathbf{A}(\mathbf{x}^{*}))=4\). In other words, it rebuilds a consistent matrix with entries in the Saaty's discrete scale. Moreover, the iteration and change of function values are reported in Table 2 and in Figure 1. The red point on the contour plot depicted in Figure 1 indicates the constrained minimum \((3,1/3)\). In this example corresponding to problem (11), we used the values for termination: \(\texttt{TolX}=10^{-4}\), \(\texttt{TolFun}=10^{-4}\), \(\texttt{MaxIter}=18\) and \(\texttt{MaxFunEvals}=35\). There is no variation in the \(\lambda_{max}\) values after the 18 th iteration. Again, solving the optimization problem (12) with initial value \(\mathbf{x}_{0}=(6,1)\), the algorithm reaches the solution \(x_{1}=5\) and \(x_{2}=0.2582\) with \(\lambda_{max}=4.0246\). Here, we omitted reporting the table and figure of this problem, as the simplex procedure was similar to that in Table 2. Note that, due to the constraint on \(x_{1}\), it is no longer possible to obtain a consistent matrix. Again, if we change the constraint \(1/7\leq x_{1}\leq 1/5\) on the same problem, the solution becomes \(\mathbf{x}^{*}=(0.2000,1.2910)\) with \(CR=0.2887\). Applying the _method of cyclic coordinates_[15] on minimization (12), the optimal solution is \(\mathbf{x}^{*}=(5.0003,0.2582)\). The solution is very similar to the previous optimal solution except for \(x_{1}=5.0003\). MATLAB's built-in function fminbnd in the _method of cyclic coordinates_ actually returns the optimal point in the interior of the interval \((5,7)\), even though theexact solution lies on the boundary. This is due to the slow convergence when the optimal solution saturates some constraints. Conversely, in the case of our algorithm (using the coordinate transformation technique), the optimal point returned by the algorithm can be a boundary point. Furthermore, the search performed by the _cyclic coordinate method_ is \"blind\", whereas the _Nelder-Mead_ does a better job in interpreting the topology of the function, even in the absence of information on the derivatives. **Example 2**. : _Consider the \(7\times 7\) incomplete pairwise comparison matrix \(\mathbf{A}\) similar to a matrix from an application to leakage control in the water supply [59], except for the six missing comparisons:_  
\[\mathbf{A}=\left(