Transitivity and approximate consistency threshold determination for reciprocal preference relations in group decision making

In a group decision-making process, contradictory pairwise comparisons may exist in individual preferences or the group preferences even though the consensus level is reached. To avoid such a contradictory phenomenon, this study presents an approximate transitivity-based consistency threshold for reciprocal preference relations to ensure the reliability of the ranking of alternatives with reciprocal preference relations in group decision making. The natural inconsistency or intransitivity of reciprocal preference relations is analyzed and verified by numerical experiments. Then, using the results of numerical experiments, approximate transitivity-based consistency thresholds are introduced based on the objectives of minimising the Type I error, Type II error and total error in statistics. Moreover, a transitivity checking process regarding individual reciprocal preference relations and group reciprocal preference relations is incorporated in the group decision-making process. A transitivity-checking integrated group decision-making model is given for application. An example about the station selection for high-speed railway line is provided to show the necessity of the approximate transitivity-based consistency threshold in checking the transitivity of reciprocal preference relations for group decision making.