## 4 Example of Formulating a Vector Criterion
The criteria for assessing control from an economic point of view include the following criteria: Time lost for anticollision maneuver is determined by the equation: \[F_{6}(x)=T_{t}=\frac{VS-V_{z}\text{\it{S}cos}(\psi_{z}-\psi)}{V_{z}V}\] (23) where \(V_{z},\psi_{z}\) are the set speed and course of the ship before a collision situation, \(V,\psi\) are the current speed and course of the ship, and \(S\) is the distance traveled; Fuel consumption per anticollision maneuver is determined by the equation: \[F_{7}(x)=\Delta Z_{p}=a\big{(}V_{z}^{3}-V^{3}\big{)}-b\big{(}V_{z}^{2}-V^{2} \big{)}+c(V_{z}-V)\] (24) where \(a,\ b,\ c\) are coefficients approximating the fuel consumption curve, depending on the type of propulsion system, ship's hull, and hydromechanical conditions of the basin; Loss of road for anticollision maneuver \[F_{8}(x)=\Delta S=\int_{t_{0}}^{t_{k}}[V_{z}-V\text{\it{cos}}(\psi_{z}-\psi)]dt\] (25) where \(t_{o},\ t_{k}\) are the start and end time of the anticollision maneuver, respectively, - Deviation from the set trajectory is determined by the equation: \[F_{9}(x)=\Delta y=\int_{t_{o}}^{t_{k}}V\text{\it{sin}}(\psi_{z}-\psi)dt\] (26) - Deviation from the set course: \[F_{10}(x)\ =\ \Delta\psi\ =\ \psi_{z}-\psi\] (27)  
For specified scalar quality criteria (18)-(27), subjective confidence levels \(\alpha_{n,1}\) were adopted, which are the arithmetic mean of confidence levels estimated by experts. However, the values of confidence levels \(\alpha_{n,2}\) resulting from the accuracy of determining partial criteria are calculated from the relationship:  
\[F_{2}(x)\ =T_{Dmin}^{j}=\frac{y_{j}V_{wy}^{j}+x_{j}V_{wx}^{j}}{\left(V_{w}^{j} \right)^{2}} \tag{19}\]0  
where \(\delta F_{n}(x)\) is the relative error value of the \(n\)-th partial criterion. Confidence levels \(\alpha_{n,3}\) resulting from restrictions imposed on partial criteria in the form of:  
\[F_{2}(x)\ =T_{Dmin}^{j}=\frac{y_{j}V_{wy}^{j}+x_{j}V_{wx}^{j}}{\left(V_{w}^{j} \right)^{2}} \tag{19}\]1  
are determined from the equation:  
\[F_{2}(x)\ =T_{Dmin}^{j}=\frac{y_{j}V_{wy}^{j}+x_{j}V_{wx}^{j}}{\left(V_{w}^{j} \right)^{2}} \tag{19}\]2  
Confidence levels \(\alpha_{n3}\) for the adopted partial criteria are determined with the following limitations:  
\[ \begin{array}{c}\min F_{1}(x)\geq D_{b},\min F_{2}(x)\geq T_{b},\;\min F_{3 }(x)\geq F_{3},\min F_{4}(x)\geq F_{4},\\ \min F_{5}(x)\geq F_{5},\max F_{6}(x)\leq 1.05T_{t},\;\max F_{7}(x)\leq 1.1Z_{ pt},\\ \min F_{8}(x)\geq F_{8},\min F_{9}(x)\geq F_{9},\;\min F_{10}(x)\geq F_{10} \end{array}  \tag{31}\]  
Subjective confidence levels determined by experts and other confidence levels determined based on the above relationships are presented in Table 1. Table 1 also presents the values of the total confidence levels of individual scalar criteria, which were determined from relationship (8). As a result of the analysis of the proposed partial criteria, a vector base criterion was formulated that includes two scalar criteria related to traffic safety and two scalar criteria assessing the economic aspect of control. The criterion is presented in the form of:  
\[F(x)=[F_{1}(x),\;F_{2}(x),\;F_{6}(x),\;F_{7}(x)] \tag{32}\]  
The total confidence level determined from (9) for the baseline criterion is \(Q(F)=0.9485\). Using Theorem 1, we check the possibility of extending the base vector criterion to the other defined scalar criteria. This condition is not met for criteria \(F_{3}(x)\), \(F_{4}(x)\), \(F_{5}(x)\), and \(F_{10}(x)\) because there are conditions such that \(Q(F)>\alpha_{3}=0.9113\), \(Q(F)>\alpha_{4}=0.8733\), \(Q(F)>\alpha_{5}=0.86\), and \(Q(F)>\alpha_{10}=0.9067\); therefore, these scalar criteria according to Theorem 1 cannot be entered into the vector criterion. For scalar criteria \(F_{8}(x)\) and \(F_{9}(x)\), there are conditions such that \(\alpha_{8}=0.9567>Q(F)\) and \(\alpha_{9}=0.9517>Q(F)\). In this case, we check by entering these scalar criteria into the vector criterion whether the conditions resulting from Theorem 3 are met.