Multiple Criteria Decision Making using Parametric Graded Mean Integration representation with preference index on ranking interval type 2 fuzzy sets

Multiple Criteria Decision Making (MCDM) is an important and effective methodology for dealing with the decision making. The interval type 2 fuzzy sets (IT2FS) are the best quantities to characterize the linguistic terms in connection with the uncertain and incomplete judgments. The final and critical stage for the MCDM task is the ranking of the alternatives. In this article, ranking methodology with parametric graded mean integration representation (PGMIR) with preference index of the IT2FS is extended. Application of the average PGMIR with preference index ranking approach to the MCDM is introduced. The preference index is the optimism attitude of the decision makers. The overall IT2FS rates for the alternatives are defuzzified and then ranked by sorting the average PGMIR. The average PGMIR ranking method is superior to the IT2FS centroid ranking in that the rigorous mathematical ordering properties are satisfied. First, the embedded type 1 fuzzy sets(T1FS) of an IT2FS are constructed. Secondly, extending from the GMIR of T1FS, the parametric PGMIR with preference index for each embedded T1FS is calculated as a function of the parameter. Finally, the average PGMIR is then obtained by taking the function average of the PGMIR. The ranking metric is defined based on the average PGMIR of the IT2FS. The average PGMIR is a multi-linear function of the endpoints of IT2FS. The rigorous mathematical ordering properties are satisfied. Therefore, it can be served as a well-defined ordering metric. The presented approach is illustrated by a supplier selection decision making problem.