Normal wiggly hesitant fuzzy set (NWHFS) is a new fuzzy information form to help decision makers (DMs) express their evaluations, which can further dig the potential uncertain information hidden in the original data given by the DMs. Firstly, we define a new distance measure and new operational laws of NWHFSs. Then, for the situation where attribute weights are completely unknown, we propose an extended CCSD method to produce them objectively, which comprehensively uses standard deviation (SD) and correlation coefficient (CC). What's more, we introduce the MABAC (multiattributive border approximation area comparison) method, which takes the distance between alternatives and the border approximation area (BAA) into consideration for handling the complex and uncertain decision-making problems. Meanwhile, we combine the MABAC method with prospect theory (PT), which considers DMs' psychological behavior, and propose a new NWHF-CCSD-PT-MABAC method to cope with the multi-attribute decision making problems under normal wiggly hesitant fuzzy environment. Lastly, we illustrate the validity and advantages of the proposed method through an example of college book supplier selection.