*Article* **Fuzzy Attribute Expansion Method for Multiple Attribute Decision-Making with Partial Attribute Values and Weights Unknown and Its Applications**

#### **Jinbao Zhuo, Weifeng Shi \* and Ying Lan**

Logistics Engineering College, Shanghai Maritime University, Shanghai 201306, China; 201640211007@stushmtu.edu.cn (J.Z.); ylan@shmtu.edu.cn (Y.L.)

**\*** Correspondence: wfshi@shmtu.edu.cn; Tel.: +86-136-1192-3783

Received: 22 November 2018; Accepted: 30 November 2018; Published: 4 December 2018

**Abstract:** In the real world, there commonly exists types of multiple attribute decision-making (MADM) problems with partial attribute values and weights totally unknown. Symmetry among some attribute information that is already known and unknown, and symmetry between the pure attribute set and fuzzy attribute membership set can be a considerable way to solve this type of MADM problem. In this paper, a fuzzy attribute expansion method is proposed to solve this type of problem based on two key techniques: the spline interpolation technique and the attribute weight reconfiguration technique, which are respectively used for the determination of attribute values and the reconfiguration of attribute weights. The spline interpolation technique to expand attribute values can enhance the performance of some regression methods and clustering methods by the comparisons between the results of these methods dealing with practical cases with and without the application of the technique, which further illustrates the effectiveness of this technique. For MADM problems with partial attribute values and weights totally unknown, compared with traditional fuzzy comprehensive evaluation (FCE), FCE with the application of fuzzy attribute expansion method can obtain results more similar with the ones when all attribute values and weights are known, which is proved by the practical power quality evaluation example.

**Keywords:** fuzzy set; fuzzy attribute expansion; MADM
