In a recently published paper by Liu et al. [Liu, F., Zhang, W.G., Wang, Z.X. (2012). A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making. European Journal of Operational Research 218, 747–754], two equations are introduced to define consistency of incomplete interval multiplicative preference relations (IMPRs) and employed to develop a goal programming model for estimating missing values. This note illustrates that such consistency definition and estimation model are technically incorrect. New transitivity conditions are proposed to define consistent IMPRs, and a two-stage goal programming approach is devised to estimate missing values for incomplete IMPRs. 
A note on “A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making”