Acceptable consistency analysis of interval reciprocal comparison matrices

When a decision maker expresses his/her opinions by means of an interval reciprocal comparison matrix, the study of consistency becomes a very important aspect in decision making in order to avoid a misleading solution. In the present paper, an acceptably consistent interval reciprocal comparison matrix is defined, which can be reduced to an acceptably consistent crisp reciprocal comparison matrix when the intervals become exact numbers. An interval reciprocal comparison matrix with unacceptable consistency can be easily adjusted such that the revised matrix possesses acceptable consistency. Utilizing a convex combination method, a family of crisp reciprocal comparison matrices with acceptable consistency can be obtained, whose weights are further found to exhibit a style of convex combination, and aggregated to obtain interval weights from an acceptably consistent interval reciprocal comparison matrix. A novel, simple yet effective formula of possibility degree is presented to rank interval weights. Numerical results are calculated to show the quality and quantity of the proposed approaches and compare with other existing procedures.