1.3. Rule based

1.3.1. COMET

COMET is designed to evaluate decision alternatives according to the following steps:

Step 1. Definition of the space of the problem - the expert determines the dimensionality of the problem by selecting \(r\) criteria, \(C_{1}, C_{2}, \ldots, C_{r}\). Then, a set of fuzzy numbers is selected for each criterion \(C_{i}\), e.g., \(\{\tilde{C}_{i1}, \tilde{C}_{i2}, \ldots, \tilde{C}_{ic_{i}}\}\) (1.93):

(1.93)\[\begin{split}\begin{equation} \label{equ:criteria} \begin{gathered} C_{1}=\left\{\tilde{C}_{11}, \tilde{C}_{12}, \ldots, \tilde{C}_{1 c_{1}}\right\} \\ C_{2}=\left\{\tilde{C}_{21}, \tilde{C}_{22}, \ldots, \tilde{C}_{2 c_{2}}\right\} \\ \cdots \\ C_{r}=\left\{\tilde{C}_{r 1}, \tilde{C}_{r 2}, \ldots, \tilde{C}_{r c_{r}}\right\} \end{gathered} \end{equation}\end{split}\]

where \(C_{1}, C_{2}, \ldots, C_{r}\) are the ordinals of the fuzzy numbers for all criteria.

Step 2. Generation of the characteristic objects - the characteristic objects (\(CO\)) are obtained with the usage of the Cartesian product of the fuzzy numbers’ cores of all the criteria (1.94):

(1.94)\[\begin{equation} CO = \langle C\left(C_{1}\right) \times C\left(C_{2}\right) \times \cdots \times C\left(C_{r}\right) \rangle \end{equation}\]

As a result, an ordered set of all \(CO\) is obtained (1.95):

(1.95)\[\begin{split}\begin{equation} \begin{gathered} CO_{1} = \langle C(\tilde{C}_{11}), C(\tilde{C}_{21}),\ldots,C(\tilde{C}_{r1}) \rangle \\ CO_{2} = \langle C(\tilde{C}_{11}), C(\tilde{C}_{21}),\ldots,C(\tilde{C}_{r1}) \rangle \\ \cdots \\ CO_{t} = \langle C(\tilde{C}_{1c_{1}}), C(\tilde{C}_{2c_{2}}),\ldots,C(\tilde{C}_{rc_{r}}) \rangle \end{gathered} \end{equation}\end{split}\]

where \(t\) is the count of \(CO\)):

(1.96)\[\begin{equation} t=\prod_{i=1}^{r} c_{i} \end{equation}\]

Step 3. Evaluation of the characteristic objects - the expert determines the Matrix of Expert Judgment (\(MEJ\)) by comparing the \(CO\)):

(1.97)\[\begin{split}\begin{equation} M E J=\left(\begin{array}{cccc} \alpha_{11} & \alpha_{12} & \cdots & \alpha_{1 t} \\ \alpha_{21} & \alpha_{22} & \cdots & \alpha_{2 t} \\ \cdots & \cdots & \cdots & \cdots \\ \alpha_{t 1} & \alpha_{t 2} & \cdots & \alpha_{t t} \end{array}\right)\end{equation}\end{split}\]

where \(\alpha_{ij}\) is the result of comparing \(CO_{i}\) and \(CO_{j}\) by the expert. The function \(f_{exp}\) denotes the mental judgement function of the expert. It depends solely on the knowledge of the expert. The expert’s preferences can be presented as (1.98):

(1.98)\[\begin{split}\begin{equation} \alpha_{i j}=\left\{\begin{array}{l} 0.0, f_{\exp }\left(C O_{i}\right)<f_{\exp }\left(C O_{j}\right) \\ 0.5, f_{\exp }\left(C O_{i}\right)=f_{\exp }\left(C O_{j}\right) \\ 1.0, f_{\exp }\left(C O_{i}\right)>f_{e x p}\left(C O_{j}\right) \end{array}\right.\end{equation}\end{split}\]

After the MEJ matrix is prepared, a vertical vector of the Summed Judgments (\(SJ\)) is obtained as follows (1.99):

(1.99)\[\begin{equation} S J_{i}=\sum_{j=1}^{t} \alpha_{i j}\end{equation}\]

Eventually, the values of preference are approximated for each characteristic object. As a result, a vertical vector \(P\) is obtained, where the \(i-th\) row contains the approximate value of preference for \(CO_{i}\).

Step 4. The rule base – each characteristic object and its value of preference is converted to a fuzzy rule as (1.100):

(1.100)\[\begin{equation} IF ~~ C\left(\tilde{C}_{1 i}\right) ~~AND~~ C\left(\tilde{C}_{2 i}\right) ~~AND~~ \ldots ~~THEN~~ P_{i} \end{equation}\]

In this way, a complete fuzzy rule base is obtained.

Step 5. Inference and the final ranking - each alternative is presented as a set of crisp numbers, e.g. \(A_{i} = \{\alpha_{i1},\alpha_{i2},\alpha_{ri}\}\). This set corresponds to the criteria \(C_{1}, C_{2}, \ldots, C_{r}\). Mamdani’s fuzzy inference method is used to compute the preference of the \(i - th\) alternative. The rule base guarantees that the obtained results are unequivocal. The bijection makes the COMET completely rank reversal free.