Decision making with incomplete interval multiplicative preference relations based on stochastic program and interval category

• The risk attitude of DM is introduced to define a new consistency index of IMPR. • The most pessimistic and optimistic acceptably consistent IMPRs are obtained. • A stochastic programming model is built to generate the priority weights. • A 0–1 mixed integer program is established to determine the DMs’ weights. • Derive the adjusted DMs’ weights with the group consensus and category information. Interval multiplicative preference relations (IMPRs) have been widely used in decision making for their ability to efficiently express the uncertainty of information. This paper investigates the decision making with incomplete IMPRs. First, a new consistency index of IMPR is defined. By minimizing the consistency index, the missing values in an incomplete IMPR can be estimated. Subsequently, considering the risk attitude of decision-makers (DMs), two optimization models are constructed to obtain the most pessimistic and optimistic acceptably multiplicative consistent IMPRs, respectively. Based on the triangular distribution of intervals, a stochastic programming model is built to derive the priority weights from an acceptably multiplicative consistent IMPR. Thus, a new method is put forward for individual decision making with an incomplete IMPR. To reach maximum group support degree, a 0–1 mixed integer programming model is established to determine DMs’ weights for group decision making with IMPRs. Considering the category information of the individual intervals and collective intervals, the adjusted DMs’ weights are defined. The individual IMPRs are integrated into the collective IMPR. The ranking of alternatives is generated by the collective IMPR. Two real-life examples are demonstrated to validate the proposed methods. A decision support system based on the proposed methods is designed.