Acceptability analysis and priority weight elicitation for interval multiplicative comparison matrices

• Proposing acceptability of interval multiplicative comparison matrices (IMCMs).
• Deriving properties of consistent IMCMs and acceptable IMCMs.
• Conversion of interval multiplicative weights to acceptable IMCMs with consistency.
• Establishing an interval multiplicative weight derivation model for acceptable IMCMs.

Existing research on acceptability of pairwise interval comparison matrices focuses on acceptable consistency by controlling their inconsistency levels to within a certain threshold. However, a perfectly consistent but highly indeterminate interval comparison matrix can be unacceptable as it contains little (sometimes no) useful decision information. This paper first analyzes the current definition of acceptable consistency for interval multiplicative comparison matrices (IMCMs) and shows its technical deficiencies. We then introduce a new notion of acceptable IMCMs, considering both inconsistency and indeterminacy levels in IMCMs. A geometric-mean-based index is proposed to measure the indeterminacy ratio of an IMCM, and useful properties are derived for consistent IMCMs and acceptable IMCMs. An indeterminacy-ratio and geometric-mean-based transformation equation is subsequently put forward to convert normalized acceptable interval multiplicative weights into an acceptable IMCM with consistency. By introducing an auxiliary constraint, a logarithmic least square model is established to generate interval multiplicative weights from acceptable IMCMs. A geometric-mean-based possibility degree formula is designed to compare and rank normalized interval multiplicative weights. Two numerical examples are presented to illustrate how to utilize the proposed framework.