**1. Introduction**

Since fuzzy set was proposed by Zadeh [1] in 1965, fuzzy theory is used to quantitatively depict the fuzziness of processes or attributes of things, especially for multiple attribute decision-making (MADM) in some real-life problems.

The applications of fuzzy theory and related techniques in the MADM area can be basically classified into two categories: (1) Applications based on modifications of fuzzy set theory and its extension theories. Fuzzy comprehensive evaluation (FCE) is one of the most classic applications of fuzzy set theory to MADM. One-level FCE can effectively deal with an evaluation problem with a small number of evaluation index parameters (also called attributes). Zhao [2] proposed an electrocardiogram signal quality evaluation method based on one-level FCE and simple heuristic fusion. Yang [3] proposed a method to evaluate exposure, sensitivity, and adaptive capacity based on one-level FCE for better flood vulnerability assessment. Multi-level FCE is preferred as the number of attributes is large. To map porphyry-copper prospectivity in the Gangdese district, Tibet, western China, Zuo [4] established a two-level binary geoscience FCE including favorable rocks, intrusive rocks, faults, and geochemical anomalies. Besides, FCE also has been successfully applied to other MADM problems, e.g., image analysis [5,6], risk assessment [7,8], energy managemen<sup>t</sup> [9,10], personnel selection [11], etc. As a successful extension of fuzzy set, intuitionistic fuzzy set (IFS) was initiated by Atanassov [12]. IFS

uses membership degree, non-membership degree, and hesitancy degree to deal with fuzziness and uncertainty information, which is very useful for the resolution of MADM problems with incomplete attribute weights and uncertain attribute information [13–17]. Xu [13] proposed the models for MADM with intuitionistic fuzzy information based on IFS theory. Wan [14] proposed a new risk attitudinal method for IFS and applied it to the MADM of the teacher selection problem. Furthermore, with the initiation of interval-valued intuitionistic fuzzy set (IVIFS) by Atanassov [18], MADM based on IVIFS becomes a hot topic for researchers [19–27]. The applications of type-2 fuzzy sets [28–30], hesitant fuzzy sets [31,32], and dual hesitant fuzzy linguistic term sets [33] were also reported recently. From these applications, we can find that the extension theories provided researchers with more and more profound theoretical models of fuzzy theory to depict and solve the complicated real-life MADM problems. (2) Applications based on the combination between fuzzy theory and other MADM methods. There are about 20 MADM methods in the literature [34]. The analytic hierarchy process (AHP) [35,36] and technique for order of preference by similarity to ideal solution (TOPSIS) [37–39] are two of the most popular methods combined with fuzzy theory in the area of supplier evaluation [40]. These applications are based on the methodology that one MADM method can be modified by the combination of the other MADM methods. It could be useful in most cases, but the disadvantage is that an increase in computational complexity is also obvious, which is rarely discussed by researchers.

The applications mentioned above focus on the representation and calculation of fuzziness and uncertainty of attributes in MADM problems, the attribute information of which is completely given. As the deepening of understanding of MADM problems grows, researchers began to study MADM problems with incomplete attribute information in recent years. Here, we call such property of attribute information as the incompleteness of attribute, which has two forms, i.e., incomplete weights or values of partial attributes. For the type of problems with incomplete attribute weights information, Park [41] provided mathematical tools for interactive MADM from the perspective of pairwise dominance. Xu [42] determined the attribute weights by the optimization model based on the maximizing deviation method. Wei [43] proposed a gray relationship analysis method to calculate the weights for IFS. Bao [44] proposed an intuitionistic fuzzy decision method based on prospect theory and the evidential reasoning approach. For the type of problems with incomplete attribute values and weights, Eum [45] established dominance and potential optimality to evaluate whether the alternative outperforms for a fixed feasible region denoted by the constraints. There also exists the third type of MADM problems with some partial attribute values and weights totally unknown, i.e., no constraints of incomplete attributes, which is the extreme type of the above two types of problem. Actually, this type of problem is common in the real world, e.g., decision maker could not provide some attribute values and weights because of discreet principles or cognitive impairment, or some attribute values could not be obtained because of the failure of the data acquisition system, or there exists some unclear or undefined attributes of new things or processes. Thus, it is necessary to study and find out a solution to these type of MADM problems.

In the real world, attributes are partially correlated and continuous in the attribute space. Hence, if the fuzzy mapping is linear, the fuzzy values and weights of these attributes are also continuous, which means that for the correlated attributes of things or processes, fuzzy values and weights of some unknown attributes can be approximately estimated by the ones of some known attributes. Thus, based on the above new cognition of the mapping relationship between the objective world and fuzzy attribute set, we propose a fuzzy attribute expansion method, which consisted of the spline interpolation technique and attribute weight reconfiguration technique, to deal with the MADM problems with some partial attribute values and weights totally unknown.

The rest of the paper is organized as follows: Section 2 provides some basic definitions about the fuzzy set and some related sets. In Section 3, the formulaic expression of the third type of MADM problem is given based on the definitions in Section 2. The geometric analysis of the pure attribute set (PAS), the measurable attribute set (MAS), and the fuzzy attribute membership set (FAMS) of the problem is conducted in Section 4, which is the theoretic basis of the fuzzy attribute expansion method proposed in Section 5. Applications in regression, clustering, and power quality evaluation are presented in Section 6. In Section 7, the conclusions of the paper are given.

## **2. Basic Definitions**

In this section, some basic concepts related to the method proposed in this paper are introduced and defined.

**Definition 1.** ([1]) *A fuzzy set A in the universe of discourse X* = {*<sup>x</sup>*1, *x*2, ··· , *xn*} is defined as follows:

$$A = \{ \langle \mathbf{x}, \mu\_A(\mathbf{x}) \rangle | \mathbf{x} \in X \},\tag{1}$$

*where μA*(*x*) : *X* → [0, 1] *is the membership function.*

**Definition 2.** *Assume that x has m kind of striking attributes (denoted as axj* (*j* = 1, ··· , *m*) *), if a set Ux*p *of x* satisfying the following conditions:


*where hi is relationship function, then Ux*p *is defined as the pure attribute set (PAS) of x*.

*If t* (*t m*) *kind of attributes of x are only known, then define these t kind of attributes as knowable fuzzy attributes (KFA) and the other* (*m* − *t*) *kind of attributes as unknowable fuzzy attributes (UFA).*

**Definition 3.** *If a set Ux*m *of x satisfying the following conditions:*

*(a)* <sup>∀</sup>*axj*(*j* = 1, ··· , *m*) ∈ *Ux*p,

$$(b) \quad \forall a\_j^{\ge} =\_j \left(a\_j^{\ge}\right) (j = 1, \cdot, \cdot, m) \in \mathcal{U}\_{\mathbf{m}^{\nu}}^{\mathbf{x}}\left(a\_j^{\ge}\right) : a\_j^{\ge} \to \mathbb{R}^{1 \times 1},$$

*where j is an unknown function to measure the attribute j in the real world, axj is the projection of attribute vector axj in some kind of space and expressed in numerical form, then Ux*m *is defined as the measurable attribute set (MAS) of x*.

**Definition 4.** *If a set Ux*f*of x satisfying the following conditions:*

*(a)* <sup>∀</sup>*axj*(*j* = 1, ··· , *m*) ∈ *Ux*p,

*(b)* <sup>∀</sup>*axj*=*j axj*(*j* = 1, ··· , *m*) ∈ *Ux*m,

$$(\mathbf{c}) \quad \forall \tilde{a}\_{\mathbf{j}}^{\mathbf{x}} = \nu\_{\mathbf{j}}(a\_{\mathbf{j}}^{\mathbf{x}}) (j = 1, \dots, m) \in \mathcal{U}\_{\mathbf{f}}^{\mathbf{x}}, \ \nu\_{\mathbf{j}}(a\_{\mathbf{j}}^{\mathbf{x}}) : a\_{\mathbf{j}}^{\mathbf{x}} \to [0, 1], \nu\_{\mathbf{f}}$$

*where <sup>ν</sup>jaxj* : *axj* → [0, 1] *is the attribute membership function, axj is fuzzy membership grade for attribute j of x, then Ux*f*is defined as the fuzzy attribute membership set (FAMS) of x*.

*For example, if axj is the volume attribute of a box, axj could be the length of the box and the length unit is meter, vj is the length attribute membership of satisfaction for customer, then axj is the fuzzy membership grade of length of the box.*
