Table 16 Optimal matching scheme \(\Upsilon^{ * }\) based on weight vector \(\Omega = \left( {\omega_{1} ,\omega_{2} ,\omega_{3} } \right)\).

\(\Omega = \left( {{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3},{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3},{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}} \right)\)

\(\Upsilon_{1}^{ * } = \left\{ {\left( {\vartheta_{1} ,\chi_{2} } \right),\left( {\vartheta_{2} ,\chi_{2} } \right),\left( {\vartheta_{3} ,\chi_{1} } \right),\left( {\vartheta_{4} ,\chi_{2} } \right),\left( {\vartheta_{5} ,\chi_{3} } \right),\left( {\vartheta_{6} ,\chi_{4} } \right),\left( {\vartheta_{7} ,\chi_{1} } \right),\left( {\vartheta_{8} ,\chi_{4} } \right)} \right\}\)

\(\Omega = \left( {{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4},{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4},{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)\)

\(\Upsilon_{5}^{ * } = \left\{ {\left( {\vartheta_{1} ,\chi_{2} } \right),\left( {\vartheta_{2} ,\chi_{1} } \right),\left( {\vartheta_{3} ,\chi_{3} } \right),\left( {\vartheta_{4} ,\chi_{2} } \right),\left( {\vartheta_{5} ,\chi_{4} } \right),\left( {\vartheta_{6} ,\chi_{4} } \right),\left( {\vartheta_{7} ,\chi_{2} } \right),\left( {\vartheta_{8} ,\chi_{1} } \right)} \right\}\)

\(\Omega = \left( {{1 \mathord{\left/ {\vphantom {1 5}} \right. \kern-0pt} 5},{1 \mathord{\left/ {\vphantom {1 5}} \right. \kern-0pt} 5},{3 \mathord{\left/ {\vphantom {3 5}} \right. \kern-0pt} 5}} \right)\)

\(\Upsilon_{6}^{ * } = \left\{ {\left( {\vartheta_{1} ,\chi_{3} } \right),\left( {\vartheta_{2} ,\chi_{4} } \right),\left( {\vartheta_{3} ,\chi_{2} } \right),\left( {\vartheta_{4} ,\chi_{2} } \right),\left( {\vartheta_{5} ,\chi_{1} } \right),\left( {\vartheta_{6} ,\chi_{2} } \right),\left( {\vartheta_{7} ,\chi_{1} } \right),\left( {\vartheta_{8} ,\chi_{4} } \right)} \right\}\)

\(\Omega = \left( {{1 \mathord{\left/ {\vphantom {1 6}} \right. \kern-0pt} 6},{1 \mathord{\left/ {\vphantom {1 6}} \right. \kern-0pt} 6},{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}} \right)\)

\(\Upsilon_{7}^{ * } = \left\{ {\left( {\vartheta_{1} ,\chi_{3} } \right),\left( {\vartheta_{2} ,\chi_{4} } \right),\left( {\vartheta_{3} ,\chi_{2} } \right),\left( {\vartheta_{4} ,\chi_{4} } \right),\left( {\vartheta_{5} ,\chi_{2} } \right),\left( {\vartheta_{6} ,\chi_{2} } \right),\left( {\vartheta_{7} ,\chi_{1} } \right),\left( {\vartheta_{8} ,\chi_{1} } \right)} \right\}\)

\(\Omega = \left( {{3 \mathord{\left/ {\vphantom {3 7}} \right. \kern-0pt} 7},{3 \mathord{\left/ {\vphantom {3 7}} \right. \kern-0pt} 7},{1 \mathord{\left/ {\vphantom {1 7}} \right. \kern-0pt} 7}} \right)\)

\(\Upsilon_{1}^{ * } = \left\{ {\left( {\vartheta_{1} ,\chi_{2} } \right),\left( {\vartheta_{2} ,\chi_{2} } \right),\left( {\vartheta_{3} ,\chi_{1} } \right),\left( {\vartheta_{4} ,\chi_{2} } \right),\left( {\vartheta_{5} ,\chi_{3} } \right),\left( {\vartheta_{6} ,\chi_{4} } \right),\left( {\vartheta_{7} ,\chi_{1} } \right),\left( {\vartheta_{8} ,\chi_{4} } \right)} \right\}\)

\(\Omega = \left( {{3 \mathord{\left/ {\vphantom {3 8}} \right. \kern-0pt} 8},{3 \mathord{\left/ {\vphantom {3 8}} \right. \kern-0pt} 8},{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}} \right)\)

\(\Upsilon_{1}^{ * } = \left\{ {\left( {\vartheta_{1} ,\chi_{2} } \right),\left( {\vartheta_{2} ,\chi_{2} } \right),\left( {\vartheta_{3} ,\chi_{1} } \right),\left( {\vartheta_{4} ,\chi_{2} } \right),\left( {\vartheta_{5} ,\chi_{3} } \right),\left( {\vartheta_{6} ,\chi_{4} } \right),\left( {\vartheta_{7} ,\chi_{1} } \right),\left( {\vartheta_{8} ,\chi_{4} } \right)} \right\}\)

\(\Omega = \left( {{4 \mathord{\left/ {\vphantom {4 9}} \right. \kern-0pt} 9},{4 \mathord{\left/ {\vphantom {4 9}} \right. \kern-0pt} 9},{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-0pt} 9}} \right)\)

\(\Upsilon_{1}^{ * } = \left\{ {\left( {\vartheta_{1} ,\chi_{2} } \right),\left( {\vartheta_{2} ,\chi_{2} } \right),\left( {\vartheta_{3} ,\chi_{1} } \right),\left( {\vartheta_{4} ,\chi_{2} } \right),\left( {\vartheta_{5} ,\chi_{3} } \right),\left( {\vartheta_{6} ,\chi_{4} } \right),\left( {\vartheta_{7} ,\chi_{1} } \right),\left( {\vartheta_{8} ,\chi_{4} } \right)} \right\}\)