Projection models for multiple attribute decision making with picture fuzzy information

In this paper, we investigate the picture fuzzy multiple attribute decision making problems where the attribute values are expressed in picture fuzzy numbers. We introduce some notions, such as picture fuzzy ideal point, the modules of picture fuzzy numbers. We also introduce the cosine of the included angle between the attribute value vectors of each alternative and the picture fuzzy ideal point. Then we establish the projection model to measure the similarity degrees between each alternative and the picture fuzzy ideal point. Based on the projection models, we can rank the given alternatives and then select the most desirable one. Finally, we illustrate the developed projection models with a numerical example for potential evaluation of emerging technology commercialization.