On consistency of the weighted arithmetical mean complex judgement matrix ^* * This project was supported by the National Natural Science Foundation of China (70525004; 70471035; 70121001).

The weighted arithmetical mean complex judgement matrix (WAMCJM) is the most common method for aggregating group opinions, but it has a shortcoming, namely the WAMCJM of the perfectly consistent judgement matrices given by experts canot guarantee its perfect consistency. An upper bound of the WAMCJM's consistency is presented. Simultaneously, a compatibility index of judging the aggregating extent of group opinions is also introduced. The WAMCJM is of acceptable consistency and is proved provided the compatibilities of all judgement matrices given by experts are smaller than the threshold value of acceptable consistency. These conclusions are important to group decision making.