Multiplicative consistency of linguistic preference relations with weakened hedges

Preference relations are often used to derive the priority of attributes and/or alternatives. Linguistic term with weakened hedges (LTWH) as a type of complex linguistic expressions can more straightforwardly describe the linguistic information provided by DMs when evaluating under uncertainties. The preference relations represented by LTWHs are an effective tool to model linguistic information. The concept and properties of additive consistency have been proposed before. This paper aims to study the multiplicative consistency of preference relations expressed by LWTHs. This paper constructs the principle of inspection for multiplicative consistency. Especially, the theories and algorithms for consistency checking and improving are proposed. We develop an automatic approach to improve a LWHPR that is not multiplicatively consistent. Finally, we demonstrate the practicality of the proposed method through a case study of evaluating the attributes in the prevention of haze pollution in China.