1 Introduction

Three-way decision model is a granular computing methodology for solving decision-making problems with uncertainty (Yao 2010). In this model, a decision on an alternative is acceptance, abstaining, or rejection. This avoids hasty classification of the alternatives at the border of acceptance and rejection effectively and thus makes the model more flexible than traditional two-way decision model (Yao 2011). Because of this advantage, three-way decision model has received extensive attention during the past decade. A number of research topics related to three-way decision model, such as three-way approximations (Deng and Yao 2014), three-way decision model for incomplete information system (Liu et al. 2016), three-way concept analysis (Yao 2017), three-way clustering (Wang and Yao 2018), three-way conflict analysis (Yao 2019), sequential three-way decision model (Yang et al. 2019), three-way fuzzy partitions (Zhao and Yao 2019), application of three-way decision model (Zhang et al. 2019; Shen et al. 2020), and multi-criterion three-way decision-making (Jia and Liu 2019; Ye et al. 2020; Zhan et al. 2021a, b; Zhang and Dai 2022; Wang et al. 2022b), have been proposed in this period. In addition, three-way decision model has been extended to many fuzzy environments, such as intuitionistic fuzzy environment (Liu et al. 2020a; Gao et al. 2020; Jiang and Hu 2021; Wang et al. 2022c), linguistic intuitionistic fuzzy environment (Liu et al. 2022b), hesitant fuzzy environment (Liang et al. 2020; Wang et al. 2021a, b; Feng et al. 2022; Wang et al. 2022a), linguistic hesitant fuzzy environment (Lei et al. 2020), interval-valued intuitionistic fuzzy environment (Jia and Liu 2021; Ye et al. 2021; Liu et al. 2022a), interval type-2 fuzzy environment (Liang et al. 2019), Pythagorean fuzzy environment (Liang et al. 2018; Lang et al. 2019; Du et al. 2022), q-rung orthopair fuzzy environment (Zhang et al. 2021a), and interval-valued q-rung orthopair fuzzy environment (Liang et al. 2021), to solve corresponding multi-criterion decision-making (MCDM) problems. However, there is yet no evidence that three-way decision model has been extended to linguistic interval-valued intuitionistic fuzzy environment.

Linguistic interval-valued intuitionistic fuzzy set (LIVIFS) (Garg and Kumar 2019a, b), presented on the basis of interval-valued intuitionistic fuzzy set (Atanassov and Gargov 1989), linguistic term set (Herrera and Martínez 2001; Xu 2004), and linguistic intuitionistic fuzzy set (Zhang 2014), is one of the most important types of fuzzy sets for expressing the values of criteria in MCDM problems. An LIVIFS is defined by an element, a membership degree of the element to the LIVIFS, and a non-membership degree of the element to the LIVIFS. The membership degree and non-membership degree are respectively quantified by an interval of two linguistic terms. A pair composed of the two intervals is usually called a linguistic interval-valued intuitionistic fuzzy number (LIVIFN). Compared to interval-valued intuitionistic fuzzy numbers, linguistic terms, and linguistic intuitionistic fuzzy numbers, LIVIFNs provide stronger expressive capability and is more flexible for describing vague information. Because of these characteristics, LIVIFNs have been widely used to express criterion values in MCDM problems (Kumar and Garg 2018; Garg and Kumar 2019a, b; Liu and Qin 2019; Tang et al. 2019; Garg and Kumar 2020; Qin et al. 2020a, c; Zhu et al. 2020; Liu et al. 2020b; Garg 2020; Fahmi et al. 2020; Xu et al. 2020, 2021; Sajjad Ali Khan et al. 2021). To solve an MCDM problem whose criterion values are quantified by LIVIFNs, many researchers presented their respective methods: Kumar and Garg (2018) presented a method based on prioritised weighted averaging and prioritised weighted geometric operators; Garg and Kumar (2019a) presented an extended technique for order preference by similarity to ideal solution; Garg and Kumar (2019b) presented a method based on weighted averaging, weighted geometric, ordered weighted averaging, ordered weighted geometric, hybrid averaging, and hybird geometric operators; Liu and Qin (2019) presented a method based on weighted Maclaurin symmetric mean operator; Tang et al. (2019) presented a procedure that can cope with inconsistent and incomplete linguistic interval-valued intuitionistic fuzzy preference relations; Garg and Kumar (2020) presented a method based on possibility degree measure; Qin et al. (2020a) presented a method based on Archimedean power weighted Muirhead mean operators; Qin et al. (2020c) presented a method based on Archimedean prioritised ’and’ and Archimedean prioritised ’or’ operators; Zhu et al. (2020) presented a method based on Hamacher weighted averaging and Hamacher weighted geometric operators; Liu et al. (2020b) presented a method based on partitioned weighted Hamy mean operator; Garg (2020) presented a multi-criterion group decision-making method; Fahmi et al. (2020) presented a method based on Dombi hybrid weighted geometric operator; Xu et al. (2020) presented a method based on copula weighted Heronian mean operators; Xu et al. (2021) presented a method based on copula power aggregation operators; Sajjad Ali Khan et al. (2021) presented a technique for order preference by similarity to ideal solution with incomplete weights. Each of these methods can work well in specific situation. But they could generate undesirable decision-making results when the information for decision-making is insufficient or acquisition of the information requires a certain cost, as all of them are based on two-way decision model.

Two-way decision model is a commonly used granular computing methodology for solving decision-making problems with uncertainty. In this model, a decision on an alternative is either acceptance or rejection, which has certain advantage when the information for decision-making is sufficient or the cost of obtaining the information is small. However, it is usually difficult to make an acceptance decision or a rejection decision directly due to the inaccuracy and incompleteness of the information. In this case, three-way decision model is often used unconsciously. This model adds an abstaining decision for each alternative on the basis of two-way decision model. It is more flexible and advantageous than two-way decision model when the information is insufficient or acquisition of the information needs a certain cost, since it can avoid hasty classification of the alternatives at the border of acceptance and rejection (Yao 2010, 2011). Here is a real-life decision example for comparing the two decision models (Yao 2012). There were six eggs in the kitchen. A woman had beaten five eggs in a bowl, and her husband came to help beat the last one. He is a fan of decision theory. As he picked up the last egg to beat, two questions came to his mind: Was this egg a good egg? What decision model was most appropriate for the current situation? Traditional two-way decision model leads to two decision actions: beat the egg into the bowl and throw the egg away. The risk of the first action is: If the last egg is a bad egg, then the previous five eggs in the bowl will be wasted, while the risk of the second action is: If the last egg is a good egg, then this egg will be wasted. Table 1 lists the cost of taking each action. Naturally, if the probability that the last egg is a good egg (denoted as P) is relatively high (e.g. \(P \ge a\)), the first action is better than the second one. Otherwise (e.g. \(P \le b\)), the second action is better than the first one. However, if \(b< P < a\), both actions seem to be inappropriate. That is, two-way decision model seems to be inappropriate for the current situation. If three-way decision model is adopted, i.e. an additional decision action, use another bowl to beat the last egg, is added to the model, the added cost, as listed in Table 2, is just washing one more bowl. That is, if \(b< P < a\), the added cost is washing one more bowl. From this real-life decision example, it is not difficult to see the advantage of three-way decision model over two-way decision model.

Table 1 Decision costs for two-way decision model
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Table 2 Decision costs for three-way decision model
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Research gap is evident from the analysis and example above: The existing methods to solve MCDM problems with LIVIFNs could produce inappropriate decision-making results because all of them are based on conventional two-way decision model. Motivated by this research gap, three-way decision model is introduced to propose a new method for solving an MCDM problem with LIVIFNs in this paper. The objectives are to establish relative loss functions and corresponding three-way decision rules for an MCDM problem with LIVIFNs, to develop a three-way decision method for an MCDM problem with LIVIFNs, and to illustrate the application and demonstrate the effectiveness and advantage of the method. Compared to the existing methods, the proposed method is based on more flexible three-way decision model. It considers the characteristics of each alternative via calculating its relative losses on the basis of its evaluation values. This avoids possible issues caused by a common practice that different alternatives are assigned the same loss function based on experience. In addition, the method is more flexible and advantageous, especially when the information for decision-making is insufficient or acquisition of the information requires a certain cost.

The remainder of the paper is organised as follows: Sect. 2 gives a brief introduction of some prerequisites. Construction of the three-way decision model is described in Sect. 3. Section 4 documents development of the three-way decision method. Illustration of the application and demonstration of the effectiveness and advantage are reported in Sect. 5. Section 6 ends the paper with a conclusion.

2 Preliminaries

2.1 Three-way decision model

In rough set theory, an approximation space is defined as \(S = (U, R)\), where U is a finite non-empty set and R is an equivalence relation over U. Let [x] be the partition of U induced by R and X be a subset of U. According to the work of Yao (2010), the lower and upper approximations of X are respectively given by

$$\begin{aligned} {\underline{APR}}(X) = \{x \in U \mid P(X|[x]) \ge a\} \end{aligned}$$
(1)
$$\begin{aligned} {\overline{APR}}(X) = \{x \in U \mid P(X|[x]) > b\} \end{aligned}$$
(2)

where

$$\begin{aligned} P(X|[x]) = \frac{|[x] \cap X|}{|[x]|} \end{aligned}$$
(3)

denotes the conditional probability of \(x \in X\) when \(x \in [x]\), and a and b are two thresholds that satisfy \(0 \le b < a \le 1\).

According to Eqs. (1) and (2), the universe U is divided into a positive region, a boundary region, and a negative region, which respectively correspond to acceptance, abstaining, and rejection in three-way decisions (Yao 2010). The three regions can be defined by

$$\begin{aligned} POS(X) = \{x \in U \mid P(X|[x]) \ge a\} \end{aligned}$$
(4)
$$\begin{aligned} BND(X) = \{x \in U \mid b< P(X|[x]) < a\} \end{aligned}$$
(5)
$$\begin{aligned} NEG(X) = \{x \in U \mid P(X|[x]) \le b\} \end{aligned}$$
(6)

To provide semantic interpretations of the three regions, Yao (2010) introduced the Bayesian decision theory. Let X and \(\lnot X\) be two states that respectively denote \(x \in X\) and \(x \notin X\). Each of the two states corresponds to three decision actions: action \(A_P\) is deciding \(x \in POS(X)\); action \(A_B\) is deciding \(x \in BND(X)\); action \(A_N\) is deciding \(x \in NEG(X)\). A loss function regarding the cost of the three actions at the two states is defined by a \(3 \times 2\) matrix, as shown in Table 3. \(\lambda _{P,\in }\) denotes the cost of taking the decision action \(A_P\) when \(x \in X\). \(\lambda _{P,\notin }\) denotes the cost of taking the decision action \(A_P\) when \(x \notin X\). The meaning of \(\lambda _{B,\in }\), \(\lambda _{B,\notin }\), \(\lambda _{N,\in }\), and \(\lambda _{N,\notin }\) can be deduced by analogy.

Table 3 Loss function
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Let \(L(A_*|[x])\) \((* = P, B, N)\) be the expected loss when taking the decision action \(A_*\). According to Table 3, \(L(A_*|[x])\) can be calculated as

$$\begin{aligned} L(A_P|[x]) = \lambda _{P,\in }P(X|[x]) + \lambda _{P,\notin }P(\lnot X|[x]) \end{aligned}$$
(7)
$$\begin{aligned} L(A_B|[x]) = \lambda _{B,\in }P(X|[x]) + \lambda _{B,\notin }P(\lnot X|[x]) \end{aligned}$$
(8)
$$\begin{aligned} L(A_N|[x]) = \lambda _{N,\in }P(X|[x]) + \lambda _{N,\notin }P(\lnot X|[x]) \end{aligned}$$
(9)

According to the Bayesian decision theory, the best decision is the one with the minimum cost. Based on this, the following decision rules are obtained:

(\(R_P\)) If \(L(A_P|[x]) \le L(A_B|[x])\) and \(L(A_P|[x]) \le L(A_N|[x])\), then take the decision action \(A_P\);

(\(R_B\)) If \(L(A_B|[x]) \le L(A_P|[x])\) and \(L(A_B|[x]) \le L(A_N|[x])\), then take the decision action \(A_B\);

(\(R_N\)) If \(L(A_N|[x]) \le L(A_P|[x])\) and \(L(A_N|[x]) \le L(A_B|[x])\), then take the decision action \(A_N\).

If \(\lambda _{P,\in } \le \lambda _{B,\in } < \lambda _{N,\in }\), \(\lambda _{N,\notin } \le \lambda _{B,\notin } < \lambda _{P,\notin }\), and \(P(X|[x]) + P(\lnot X|[x]) = 1\), then the decision rules above can be simplified as follows:

(\(R_{P'}\)) If \(P(X|[x]) \ge a\) and \(P(X|[x]) \ge c\), then take the decision action \(A_P\);

(\(R_{B'}\)) If \(P(X|[x]) \le a\) and \(P(X|[x]) \ge b\), then take the decision action \(A_B\);

(\(R_{N'}\)) If \(P(X|[x]) \le b\) and \(P(X|[x]) \le c\), then take the decision action \(A_N\),

where

$$\begin{aligned} a = \frac{\lambda _{P,\notin } - \lambda _{B,\notin }}{(\lambda _{P,\notin } - \lambda _{B,\notin }) + (\lambda _{B,\in } - \lambda _{P,\in })} \end{aligned}$$
(10)
$$\begin{aligned} b = \frac{\lambda _{B,\notin } - \lambda _{N,\notin }}{(\lambda _{B,\notin } - \lambda _{N,\notin }) + (\lambda _{N,\in } - \lambda _{B,\in })} \end{aligned}$$
(11)
$$\begin{aligned} c = \frac{\lambda _{P,\notin } - \lambda _{N,\notin }}{(\lambda _{P,\notin } - \lambda _{N,\notin }) + (\lambda _{N,\in } - \lambda _{P,\in })} \end{aligned}$$
(12)

If \((\lambda _{P,\notin } - \lambda _{B,\notin })(\lambda _{N,\in } - \lambda _{B,\in }) > (\lambda _{B,\in } - \lambda _{P,\in })(\lambda _{B,\notin } - \lambda _{N,\notin })\), then \(0 \le b< c < a \le 1\) and the decision rules can be further simplified as follows:

(\(R_{P''}\)) If \(P(X|[x]) \ge a\), then take the decision action \(A_P\);

(\(R_{B''}\)) If \(b< P(X|[x]) < a\), then take the decision action \(A_B\);

(\(R_{N''}\)) If \(P(X|[x]) \le b\), then take the decision action \(A_N\).

To establish a loss function from the criterion value in an MCDM problem, Jia and Liu (2019) introduced relative loss function, as shown in Table 4, where \(\lambda '_{P,\notin } = \lambda _{P,\notin } - \lambda _{N,\notin }\), \(\lambda '_{B,\in } = \lambda _{B,\in } - \lambda _{P,\in }\), \(\lambda '_{B,\notin } = \lambda _{B,\notin } - \lambda _{N,\notin }\), and \(\lambda '_{N,\in } = \lambda _{N,\in } - \lambda _{P,\in }\).

Table 4 Relative loss function
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2.2 LIVIFS and related concepts

The definition of LIVIFS is as follows:

Definition 1

(Garg and Kumar 2019b) Let h be a positive integer and \(S_{[0, h]} = \{S_i \mid S_0 \le S_i \le S_h\}\) be a continuous linguistic term set such that for any \(S_j, S_k \in S_{[0, h]}\), \(S_j > S_k\) iff \(j > k\). An LIVIFS A over a finite universe of discourse U is

$$\begin{aligned} A = \big \{\big \langle x, \big [S_{p_A^\mathrm {L}(x)}, S_{p_A^\mathrm {U}(x)}\big ], \big [S_{q_A^\mathrm {L}(x)}, S_{q_A^\mathrm {U}(x)}\big ] \big \rangle \mid x \in U\big \} \end{aligned}$$
(13)

where \(\big [S_{p_A^\mathrm {L}(x)}, S_{p_A^\mathrm {U}(x)}\big ]\) and \(\big [S_{q_A^\mathrm {L}(x)}, S_{q_A^\mathrm {U}(x)}\big ]\) are respectively the linguistic membership and non-membership degrees of x to A, and \(p_A^\mathrm {U}(x) + q_A^\mathrm {U}(x) \le h\) for any \(x \in U\). The linguistic hesitancy degree of x to A is

\(\big [S_{h - p_A^\mathrm {U}(x) - q_A^\mathrm {U}(x)}, S_{h - p_A^\mathrm {L}(x) - q_A^\mathrm {L}(x)}\big ]\).

In general,

\(\big (\big [S_{p_A^\mathrm {L}(x)}, S_{p_A^\mathrm {U}(x)}\big ], \big [S_{q_A^\mathrm {L}(x)}, S_{q_A^\mathrm {U}(x)}\big ]\big )\)

is called an LIVIFN, denoted as

\(\theta = \big (\big [S_{p_\theta ^\mathrm {L}}, S_{p_\theta ^\mathrm {U}}\big ], \big [S_{q_\theta ^\mathrm {L}}, S_{q_\theta ^\mathrm {U}}\big ]\big )\).

To compare two LIVIFNs, a score function is first needed:

Definition 2

(Garg and Kumar 2019b) Let

\(\theta = \big (\big [S_{p_\theta ^\mathrm {L}}, S_{p_\theta ^\mathrm {U}}\big ], \big [S_{q_\theta ^\mathrm {L}}, S_{q_\theta ^\mathrm {U}}\big ]\big )\)

be an arbitrary LIVIFN. The score value of \(\theta \) can be calculated by

$$\begin{aligned} S(\theta ) = \frac{2h + p_\theta ^\mathrm {L} + p_\theta ^\mathrm {U} - q_\theta ^\mathrm {L} - q_\theta ^\mathrm {U}}{4} \end{aligned}$$
(14)

Then two LIVIFNs can be compared via certain rules:

Definition 3

Let

\(\theta _1 = \big (\big [S_{p_{\theta _1}^\mathrm {L}}, S_{p_{\theta _1}^\mathrm {U}}\big ], \big [S_{q_{\theta _1}^\mathrm {L}}, S_{q_{\theta _1}^\mathrm {U}}\big ]\big )\)

\(\theta _2 = \big (\big [S_{p_{\theta _2}^\mathrm {L}}, S_{p_{\theta _2}^\mathrm {U}}\big ], \big [S_{q_{\theta _2}^\mathrm {L}}, S_{q_{\theta _2}^\mathrm {U}}\big ]\big )\)

be two arbitrary LIVIFNs. If \(S(\theta _1) < S(\theta _2)\), then \(\theta _1 \prec \theta _2\); If \(S(\theta _1) > S(\theta _2)\), then \(\theta _1 \succ \theta _2\); If \(S(\theta _1) = S(\theta _2)\), then \(\theta _1 = \theta _2\).

To obtain the difference of two LIVIFNs, a distance measure is usually required:

Definition 4

(Qin et al. 2020a) Let

\(\theta _1 = \big (\big [S_{p_{\theta _1}^\mathrm {L}}, S_{p_{\theta _1}^\mathrm {U}}\big ], \big [S_{q_{\theta _1}^\mathrm {L}}, S_{q_{\theta _1}^\mathrm {U}}\big ]\big )\)

\(\theta _2 = \big (\big [S_{p_{\theta _2}^\mathrm {L}}, S_{p_{\theta _2}^\mathrm {U}}\big ], \big [S_{q_{\theta _2}^\mathrm {L}}, S_{q_{\theta _2}^\mathrm {U}}\big ]\big )\)

be two arbitrary LIVIFNs. The distance between \(\theta _1\) and \(\theta _2\) can be calculated by

$$\begin{aligned} D(\theta _1, \theta _2) = \frac{\left|p_{\theta _1}^\mathrm {L} - p_{\theta _2}^\mathrm {L}\right| + \left|p_{\theta _1}^\mathrm {U} - p_{\theta _2}^\mathrm {U}\right| + \left|q_{\theta _1}^\mathrm {L} - q_{\theta _2}^\mathrm {L}\right| + \left|q_{\theta _1}^\mathrm {U} - q_{\theta _2}^\mathrm {U}\right|}{4h} \end{aligned}$$
(15)

To perform operations related to LIVIFNs, a set of operational laws are needed:

Definition 5

(Garg and Kumar 2019b) Let \(\theta = \big (\big [S_{p_\theta ^\mathrm {L}}, S_{p_\theta ^\mathrm {U}}\big ], \big [S_{q_\theta ^\mathrm {L}}, S_{q_\theta ^\mathrm {U}}\big ]\big )\) \(\theta _1 = \big (\big [S_{p_{\theta _1}^\mathrm {L}}, S_{p_{\theta _1}^\mathrm {U}}\big ], \big [S_{q_{\theta _1}^\mathrm {L}}, S_{q_{\theta _1}^\mathrm {U}}\big ]\big )\) \(\theta _2 = \big (\big [S_{p_{\theta _2}^\mathrm {L}}, S_{p_{\theta _2}^\mathrm {U}}\big ], \big [S_{q_{\theta _2}^\mathrm {L}}, S_{q_{\theta _2}^\mathrm {U}}\big ]\big )\) be three arbitrary LIVIFNs and t be an arbitrary positive number. The operations among \(\theta \), \(\theta _1\), \(\theta _2\), and t can be performed via

$$\begin{aligned}&{\overline{\theta }} = \big (\big [S_{q_\theta ^\mathrm {L}}, S_{q_\theta ^\mathrm {U}}\big ], \big [S_{p_\theta ^\mathrm {L}}, S_{p_\theta ^\mathrm {U}}\big ]\big ) \end{aligned}$$
(16)
$$\begin{aligned}&t\theta = \bigg (\bigg [S_{h\big (1 - (1 - p_\theta ^\mathrm {L} / h)^t\big )}, S_{h\big (1 - (1 - p_\theta ^\mathrm {U} / h)^t\big )}\bigg ],\nonumber \\&\bigg [S_{h(q_\theta ^\mathrm {L} / h)^t}, S_{h(q_\theta ^\mathrm {U} / h)^t}\bigg ]\bigg ) \end{aligned}$$
(17)
$$\begin{aligned}&\theta ^t = \bigg (\bigg [S_{h(p_\theta ^\mathrm {L} / h)^t}, S_{h(p_\theta ^\mathrm {U} / h)^t}\bigg ], \bigg [S_{h\big (1 - (1 - q_\theta ^\mathrm {L} / h)^t\big )},\nonumber \\&S_{h\big (1 - (1 - q_\theta ^\mathrm {U} / h)^t\big )}\bigg ]\bigg ) \end{aligned}$$
(18)
$$\begin{aligned}&\theta _1 \oplus \theta _2 = \bigg (\bigg [S_{p_{\theta _1}^\mathrm {L} + p_{\theta _2}^\mathrm {L} - p_{\theta _1}^\mathrm {L}p_{\theta _2}^\mathrm {L} / h}, S_{p_{\theta _1}^\mathrm {U} + p_{\theta _2}^\mathrm {U} - p_{\theta _1}^\mathrm {U}p_{\theta _2}^\mathrm {U} / h}\bigg ], \bigg [S_{q_{\theta _1}^\mathrm {L}q_{\theta _2}^\mathrm {L} / h},\nonumber \\&S_{q_{\theta _1}^\mathrm {U}q_{\theta _2}^\mathrm {U} / h}\bigg ]\bigg ) \end{aligned}$$
(19)
$$\begin{aligned}&\theta _1 \otimes \theta _2 = \bigg (\bigg [S_{p_{\theta _1}^\mathrm {L}p_{\theta _2}^\mathrm {L} / h}, S_{p_{\theta _1}^\mathrm {U}p_{\theta _2}^\mathrm {U} / h}\bigg ], \bigg [S_{q_{\theta _1}^\mathrm {L} + q_{\theta _2}^\mathrm {L} - q_{\theta _1}^\mathrm {L}q_{\theta _2}^\mathrm {L} / h},\nonumber \\&S_{q_{\theta _1}^\mathrm {U} + q_{\theta _2}^\mathrm {U} - q_{\theta _1}^\mathrm {U}q_{\theta _2}^\mathrm {U} / h}\bigg ]\bigg ) \end{aligned}$$
(20)

To solve an MCDM problem with LIVIFNs, it is often of necessity to group together multiple criterion values in the form of LIVIFNs to obtain a summary LIVIFN. This important task can be completed by an aggregation operator of LIVIFNs. So far, many aggregation operators of LIVIFNs have been presented. A representative example is the linguistic interval-valued intuitionistic fuzzy power weighted Muirhead mean (LIVIFPWMM) operator (Qin et al. 2020a), which has the capabilities in considering the interrelationships among multiple aggregated criteria and reducing the effect of extreme criterion values on the aggregation results. The definition of this aggregation operator is as follows:

Definition 6

(Qin et al. 2020a) Let

\(\theta _i = \big (\big [S_{p_{\theta _i}^\mathrm {L}}, S_{p_{\theta _i}^\mathrm {U}}\big ], \big [S_{q_{\theta _i}^\mathrm {L}}, S_{q_{\theta _i}^\mathrm {U}}\big ]\big )\) \((i = 1, 2,\ldots , n)\)

be n arbitrary LIVIFNs, \(r_i\) be n real numbers that respectively correspond to \(\theta _i\) such that all \(r_i \ge 0\) but not at the same time all \(r_i = 0\), P\(_n\) be the set of all permutations of (1, 2, ..., n), \(w_i\) be the weights of \(\theta _i\) such that \(0 \le w_i \le 1\) and \(\Sigma _{i=1}^n w_i = 1\), \(D(\theta _i, \theta _j)\) (\(j = 1, 2, ..., n\) and \(j \ne i\)) be the distance between \(\theta _i\) and \(\theta _j\), and \(S'(\theta _i, \theta _j) = 1 - D(\theta _i, \theta _j)\) be the degree of support for \(\theta _i\) from \(\theta _j\) that satisfies: \(0 \le S'(\theta _i, \theta _j) \le 1\); \(S'(\theta _i, \theta _j) = S'(\theta _j, \theta _i)\); if \(D(\theta _i, \theta _j) < D(\theta _{i'}, \theta _{j'})\) (\(i', j' = 1, 2, ..., n\) and \(j' \ne i'\)), then \(S'(\theta _i, \theta _j) \ge S'(\theta _{i'}, \theta _{j'})\). The LIVIFPWMM operator is given by

$$\begin{aligned} LIVIFPWMM(\theta _1, \theta _2, ..., \theta _n) = \big (\big [S_{p_\Theta ^\mathrm {L}}, S_{p_\Theta ^\mathrm {U}}\big ], \big [S_{q_\Theta ^\mathrm {L}}, S_{q_\Theta ^\mathrm {U}}\big ]\big ) \end{aligned}$$
(21)

where

$$\begin{aligned}&p_\Theta ^\mathrm {L} = h\bigg (1 - \prod \limits _{y \in \varvec{P}_n}^{}\big (1 - \prod \limits _{i = 1}^{n}\big (1 - (1 - p_{\theta _i}^\mathrm {L} / h)^{n\xi _{y(i)}}\big )^{r_i} \big )^{\frac{1}{n!}}\bigg )^{\frac{1}{\sum \limits _{i = 1}^{n}r_i}} \end{aligned}$$
(22)
$$\begin{aligned}&p_\Theta ^\mathrm {U} = h\bigg (1 - \prod \limits _{y \in \varvec{P}_n}^{}\big (1 - \prod \limits _{i = 1}^{n}\big (1 - (1 - p_{\theta _i}^\mathrm {U} / h)^{n\xi _{y(i)}}\big )^{r_i} \big )^{\frac{1}{n!}}\bigg )^{\frac{1}{\sum \limits _{i = 1}^{n}r_i}} \end{aligned}$$
(23)
$$\begin{aligned}&q_\Theta ^\mathrm {L} = h\bigg (1 - \big (1 - \prod \limits _{y \in \varvec{P}_n}^{}\big (1 - \prod \limits _{i = 1}^{n}\big (1 - (q_{\theta _i}^\mathrm {L} / h)^{n\xi _{y(i)}}\big )^{r_i}\big )^{\frac{1}{n!}}\big )^{\frac{1}{\sum \limits _{i = 1}^{n}r_i}}\bigg ) \end{aligned}$$
(24)
$$\begin{aligned}&q_\Theta ^\mathrm {U} = h\bigg (1 - \big (1 - \prod \limits _{y \in \varvec{P}_n}^{}\big (1 - \prod \limits _{i = 1}^{n}\big (1 - (q_{\theta _i}^\mathrm {U} / h)^{n\xi _{y(i)}}\big )^{r_i}\big )^{\frac{1}{n!}}\big )^{\frac{1}{\sum \limits _{i = 1}^{n}r_i}}\bigg ) \end{aligned}$$
(25)
$$\begin{aligned}&\xi _{y(i)} = \frac{w_{y(i)}\bigg (1 + \sum \limits _{k=1,k \ne y(i)}^{n}S'(\theta _{y(i)}, \theta _k)\bigg )}{\sum \limits _{j=1}^{n}w_j\bigg (1 + \sum \limits _{k=1,k \ne j}^{n}S'(\theta _{j}, \theta _{k})\bigg )} \end{aligned}$$
(26)

3 Three-way decision model for MCDM problem with LIVIFNs

Two critical tasks for constructing a three-way decision model for an MCDM problem with LIVIFNs are to establish a loss function for it and to develop corresponding decision rules. Let \(A_i\) (i is a number among 1, 2, ..., m) and \(C_j\) (j is a number among 1, 2, ..., n) be respectively an alternative and a criterion of an MCDM problem with m alternatives and n criteria,

\(\theta = \big (\big [S_{p_\theta ^\mathrm {L}}, S_{p_\theta ^\mathrm {U}}\big ], \big [S_{q_\theta ^\mathrm {L}}, S_{q_\theta ^\mathrm {U}}\big ]\big )\)

be an LIVIFN that quantifies the evaluation value of \(A_i\) under \(C_j\), and \(X_j\) and \(\lnot X_j\) be two states that respectively denote \(A_i\) satisfies \(C_j\) and \(A_i\) does not satisfy \(C_j\). Each of the two states corresponds to three decision actions: action \(A_P\) is acceptance; action \(A_B\) is abstaining; action \(A_N\) is rejection. Inspired by a relative loss function derived from an evaluation value in the form of intuitionistic fuzzy number (Liu et al. 2020a), a relative loss function derived from \(\theta \) is established in Table 5, where \(o = ([S_0, S_0], [S_h, S_h])\) and \(0 \le \delta \le 1\) is a risk avoidance coefficient.

Table 5 Relative loss function derived from an LIVIFN
Full size table

An explanation of the relative loss function in Table 5 is given as follows: Since the cost of making correct decisions is the lowest, the minimum LIVIFN \(([S_0, S_0], [S_h, S_h])\) is assigned to corresponding positions; When \(A_i\) satisfy \(C_j\), the cost of acceptance is \(([S_0, S_0], [S_h, S_h])\) and the cost of rejection is \(\theta \); When \(A_i\) does not satisfy \(C_j\), the cost of acceptance is \({\overline{\theta }}\) and the cost of rejection is \(([S_0, S_0], [S_h, S_h])\); Because of the conditions \(\lambda _{P,\in } \preceq \lambda _{B,\in } \prec \lambda _{N,\in }\) and \(\lambda _{N,\notin } \preceq \lambda _{B,\notin } \prec \lambda _{P,\notin }\), a risk avoidance coefficient \(0 \le \delta \le 1\) is leveraged to quantify the cost of abstaining.

If the operational laws of LIVIFNs in Eq. (16) and (17) are applied to the relative loss function in Table 5, a specific relative loss function derived from \(\theta \) is obtained, as shown in Table 6. According to the score function in Definition 2 and the comparison rules in Definition 3, it is easy to prove that the specific relative loss function in Table 6 satisfies the conditions \(\lambda _{P,\in } \preceq \lambda _{B,\in } \prec \lambda _{N,\in }\) and \(\lambda _{N,\notin } \preceq \lambda _{B,\notin } \prec \lambda _{P,\notin }\).

Table 6 Specific relative loss function derived from an LIVIFN
Full size table

Let \(P(X_j|[A_i])\) be the conditional probability of \(A_i\) satisfy \(C_j\), \(P(\lnot X_j|[A_i])\) be the conditional probability of \(A_i\) does not satisfy \(C_j\), \(L(A_*|[A_i])\) \((* = P, B, N)\) be the expected loss when taking the decision action \(A_*\). According to Table 6, \(L(A_*|[A_i])\) can be calculated by

$$\begin{aligned}&L(A_P|[A_i]) = ([S_0, S_0], [S_h, S_h])P(X_j|[A_i]) \oplus \big (\big [S_{q_\theta ^\mathrm {L}}, S_{q_\theta ^\mathrm {U}}\big ], \big [S_{p_\theta ^\mathrm {L}},\nonumber \\&S_{p_\theta ^\mathrm {U}}\big ]\big )P(\lnot X_j|[A_i]) \end{aligned}$$
(27)
$$\begin{aligned}&L(A_B|[A_i]) = \bigg (\bigg [S_{h\big (1 - (1 - p_\theta ^\mathrm {L} / h)^\delta \big )}, S_{h\big (1 - (1 - p_\theta ^\mathrm {U} / h)^\delta \big )}\bigg ], \bigg [S_{h(q_\theta ^\mathrm {L} / h)^\delta },\nonumber \\&S_{h(q_\theta ^\mathrm {U} / h)^\delta }\bigg ]\bigg )P(X_j|[A_i]) \oplus \nonumber \\&\bigg (\bigg [S_{h\big (1 - (1 - q_\theta ^\mathrm {L} / h)^\delta \big )}, S_{h\big (1 - (1 - q_\theta ^\mathrm {U} / h)^\delta \big )}\bigg ], \bigg [S_{h(p_\theta ^\mathrm {L} / h)^\delta },\nonumber \\&S_{h(p_\theta ^\mathrm {U} / h)^\delta }\bigg ]\bigg )P(\lnot X_j|[A_i]) \end{aligned}$$
(28)
$$\begin{aligned}&L(A_N|[A_i]) = \big (\big [S_{p_\theta ^\mathrm {L}}, S_{p_\theta ^\mathrm {U}}\big ], \big [S_{q_\theta ^\mathrm {L}},\nonumber \\&S_{q_\theta ^\mathrm {U}}\big ]\big )P(X_j|[A_i]) \oplus ([S_0, S_0], [S_h, S_h])P(\lnot X_j|[A_i]) \end{aligned}$$
(29)

If \(P(X_j|[A_i]) + P(\lnot X_j|[A_i]) = 1\) and the operational laws of LIVIFNs in Eq. (17) and Eq. (19) are applied to the three equations above, \(L(A_*|[A_i])\) can be calculated by

$$\begin{aligned}&L(A_P|[A_i]) = \bigg (\bigg [S_{h\big (1 - (1 - q_\theta ^\mathrm {L} / h)^{1 - P_i}\big )}, S_{h\big (1 - (1 - q_\theta ^\mathrm {U} / h)^{1 - P_i}\big )}\bigg ],\nonumber \\&\bigg [S_{h(p_\theta ^\mathrm {L} / h)^{1 - P_i}}, S_{h(p_\theta ^\mathrm {U} / h)^{1 - P_i}}\bigg ]\bigg ) \end{aligned}$$
(30)
$$\begin{aligned}&L(A_B|[A_i]) = \bigg (\bigg [S_{h\big (1 - (1 - p_\theta ^\mathrm {L} / h)^{\delta P_i}(1 - q_\theta ^\mathrm {L} / h)^{\delta - \delta P_i}\big )},\nonumber \\&S_{h\big (1 - (1 - p_\theta ^\mathrm {U} / h)^{\delta P_i}(1 - q_\theta ^\mathrm {U} / h)^{\delta - \delta P_i}\big )}\bigg ], \nonumber \\&\bigg [S_{h(p_\theta ^\mathrm {L} / h)^{\delta - \delta P_i}(q_\theta ^\mathrm {L} / h)^{\delta P_i}},\nonumber \\&S_{h(p_\theta ^\mathrm {U} / h)^{\delta - \delta P_i}(q_\theta ^\mathrm {U} / h)^{\delta P_i}}\bigg ]\bigg ) \end{aligned}$$
(31)
$$\begin{aligned}&L(A_N|[A_i]) = \bigg (\bigg [S_{h\big (1 - (1 - p_\theta ^\mathrm {L} / h)^{P_i}\big )},\nonumber \\&S_{h\big (1 - (1 - p_\theta ^\mathrm {U} / h)^{P_i}\big )}\bigg ], \bigg [S_{h(q_\theta ^\mathrm {L} / h)^{P_i}}, S_{h(q_\theta ^\mathrm {U} / h)^{P_i}}\bigg ]\bigg ) \end{aligned}$$
(32)

where \(P_i = P(X_j|[A_i])\).

According to the Bayesian decision theory, the best decision is the one with the minimum cost. Based on this, the following decision rules are obtained:

(\(R'_P\)) If \(L(A_P|[A_i]) \preceq L(A_B|[A_i])\) and \(L(A_P|[A_i]) \preceq L(A_N|[A_i])\), then take the decision action \(A_P\);

(\(R'_B\)) If \(L(A_B|[A_i]) \preceq L(A_P|[A_i])\) and \(L(A_B|[A_i]) \preceq L(A_N|[A_i])\), then take the decision action \(A_B\);

(\(R' _N\)) If \(L(A_N|[A_i]) \preceq L(A_P|[A_i])\) and \(L(A_N|[A_i]) \preceq L(A_B|[A_i])\), then take the decision action \(A_N\).

According to the score function in Eq. (14), the score values of \(L(A_P|[A_i])\), \(L(A_B|[A_i])\), and \(L(A_N|[A_i])\) can be respectively calculated by

$$\begin{aligned}&S\big (L(A_P|[A_i])\big ) = h - \frac{h}{4}\big ((1 - q_\theta ^\mathrm {L} / h)^{1 - P_i} + (1 - q_\theta ^\mathrm {U} / h)^{1 - P_i} + (p_\theta ^\mathrm {L} / h)^{1 - P_i} +\nonumber \\&(p_\theta ^\mathrm {U} / h)^{1 - P_i}\big ) \end{aligned}$$
(33)
$$\begin{aligned}&S\big (L(A_B|[A_i])\big ) = h - \frac{h}{4}\big ((1 - p_\theta ^\mathrm {L} / h)^{\delta P_i}(1 - q_\theta ^\mathrm {L} / h)^{\delta - \delta P_i} + (p_\theta ^\mathrm {L} / h)^{\delta -}\nonumber \\&{\delta P_i}(q_\theta ^\mathrm {L} / h)^{\delta P_i} + \nonumber \\&(1 - p_\theta ^\mathrm {U} / h)^{\delta P_i}(1 - q_\theta ^\mathrm {U} / h)^{\delta - \delta P_i} + (p_\theta ^\mathrm {U} / h)^{\delta -}\nonumber \\&{\delta P_i}(q_\theta ^\mathrm {U} / h)^{\delta P_i}\big ) \end{aligned}$$
(34)
$$\begin{aligned}&S\big (L(A_N|[A_i])\big ) = h - \frac{h}{4}\big ((1 - p_\theta ^\mathrm {L} / h)^{P_i} + (1 - p_\theta ^\mathrm {U} / h)^{P_i}\nonumber \\&+ (q_\theta ^\mathrm {L} / h)^{P_i} + (q_\theta ^\mathrm {U} / h)^{P_i}\big ) \end{aligned}$$
(35)

Based on the score values, the decision rules can be rewritten as follows:

(\(R'_{P'}\)) If \(S(L(A_P|[A_i])) \le S(L(A_B|[A_i]))\) and \(S(L(A_P|[A_i])) \le S(L(A_N|[A_i]))\), then take the decision action \(A_P\);

(\(R'_{B'}\)) If \(S(L(A_B|[A_i])) \le S(L(A_P|[A_i]))\) and \(S(L(A_B|[A_i])) \le S(L(A_N|[A_i]))\), then take the decision action \(A_B\);

(\(R' _{N'}\)) If \(S(L(A_N|[A_i])) \le S(L(A_P|[A_i]))\) and \(S(L(A_N|[A_i])) \le S(L(A_B|[A_i]))\), then take the decision action \(A_N\).

4 Three-way decision method for MCDM problem with LIVIFNs

An MCDM problem with LIVIFNs is generally described via m alternatives \(A_i\) \((i = 1, 2, ..., m)\), n criteria \(C_j\) \((j = 1, 2, ..., n)\), a vector of risk avoidance coefficients of criteria \((\delta _1, \delta _2, ..., \delta _n)\) such that \(0 \le \delta _j \le 1\) is the risk avoidance coefficient of \(C_j\), a vector of weights of criteria \((w_1, w_2, ..., w_n)\) such that \(0 \le w_j \le 1\) is the weight of \(C_j\) and \(\Sigma _{j=1}^n w_j = 1\), \(h + 1\) linguistic terms \(S_k\) \((k = 0, 1, ..., h)\), and a decision matrix \(\varvec{M} = [\theta _{i,j}]_{m \times n} = \big [\big (\big [S_{p_{\theta _{i,j}}^\mathrm {L}}, S_{p_{\theta _{i,j}}^\mathrm {U}}\big ], \big [S_{q_{\theta _{i,j}}^\mathrm {L}}, S_{q_{\theta _{i,j}}^\mathrm {U}}\big ]\big )\big ]_{m \times n}\) such that each \(\theta _{i,j}\) is an LIVIFN that quantifies the evaluation value of \(C_j\) of \(A_i\). Based on the constructed three-way decision model, a three-way decision method to solve an MCDM problem with LIVIFNs is developed. This method consists of the following steps:

(1) Normalise the decision matrix \(\varvec{M}\) as the following matrix:

$$\begin{aligned} \begin{aligned} \varvec{M'} = [\theta '_{i,j}]_{m \times n} = \left\{ \begin{array}{ll} \big [\big (\big [S_{p_{\theta _{i,j}}^\mathrm {L}}, S_{p_{\theta _{i,j}}^\mathrm {U}}\big ], \big [S_{q_{\theta _{i,j}}^\mathrm {L}}, S_{q_{\theta _{i,j}}^\mathrm {U}}\big ]\big )\big ]_{m \times n} &{} \text{ if } C_j \text{ is } \text{ a } \text{ benefit } \text{ criterion } \\ \big [\big (\big [S_{q_{\theta _{i,j}}^\mathrm {L}}, S_{q_{\theta _{i,j}}^\mathrm {U}}\big ], \big [S_{p_{\theta _{i,j}}^\mathrm {L}}, S_{p_{\theta _{i,j}}^\mathrm {U}}\big ]\big )\big ]_{m \times n} &{} \text{ if } C_j \text{ is } \text{ a } \text{ cost } \text{ criterion } \end{array} \right. \end{aligned} \end{aligned}$$
(36)

(2) Convert each normalised evaluation value \(\theta '_{i,j}\) into a relative loss function according to Table 6:

$$\begin{aligned} \varvec{\lambda '}(\theta '_{i,j}) = \begin{bmatrix} ([S_0, S_0], [S_h, S_h]) &{} \lambda '_{P,\notin }(\theta '_{i,j}) \\ \lambda '_{B,\in }(\theta '_{i,j}) &{} \lambda '_{B,\notin }(\theta '_{i,j}) \\ \lambda '_{N,\in }(\theta '_{i,j}) &{} ([S_0, S_0], [S_h, S_h]) \\ \end{bmatrix} \end{aligned}$$
(37)

where

$$\begin{aligned}&\lambda '_{P,\notin }(\theta '_{i,j}) = \bigg (\bigg [S_{q_{\theta '_{i,j}}^\mathrm {L}}, S_{q_{\theta '_{i,j}}^\mathrm {U}}\bigg ],\nonumber \\&\bigg [S_{p_{\theta '_{i,j}}^\mathrm {L}}, S_{p_{\theta '_{i,j}}^\mathrm {U}}\bigg ]\bigg ) \end{aligned}$$
(38)
$$\begin{aligned}&\lambda '_{B,\in }(\theta '_{i,j}) = \bigg (\bigg [S_{h\big (1 - (1 - p_{\theta '_{i,j}}^\mathrm {L} / h)^{\delta _j}\big )}, S_{h\big (1 - (1 - p_{\theta '_{i,j}}^\mathrm {U} / h)^{\delta _j}\big )}\bigg ],\nonumber \\&\bigg [S_{h(q_{\theta '_{i,j}}^\mathrm {L} / h)^{\delta _j}}, S_{h(q_{\theta '_{i,j}}^\mathrm {U} / h)^{\delta _j}}\bigg ]\bigg ) \end{aligned}$$
(39)
$$\begin{aligned}&\lambda '_{B,\notin }(\theta '_{i,j}) = \bigg (\bigg [S_{h\big (1 - (1 - q_{\theta '_{i,j}}^\mathrm {L} / h)^{\delta _j}\big )}, S_{h\big (1 - (1 - q_{\theta '_{i,j}}^\mathrm {U} / h)^{\delta _j}\big )}\bigg ],\nonumber \\&\bigg [S_{h(p_{\theta '_{i,j}}^\mathrm {L} / h)^{\delta _j}}, S_{h(p_{\theta '_{i,j}}^\mathrm {U} / h)^{\delta _j}}\bigg ]\bigg ) \end{aligned}$$
(40)
$$\begin{aligned}&\lambda '_{N,\in }(\theta '_{i,j}) = \bigg (\bigg [S_{p_{\theta '_{i,j}}^\mathrm {L}}, S_{p_{\theta '_{i,j}}^\mathrm {U}}\bigg ],\nonumber \\&\bigg [S_{q_{\theta '_{i,j}}^\mathrm {L}}, S_{q_{\theta '_{i,j}}^\mathrm {U}}\bigg ]\bigg ) \end{aligned}$$
(41)

Then the normalised decision matrix \([\theta '_{i,j}]_{m \times n}\) can be converted into a relative loss function matrix \([\varvec{\lambda '}(\theta '_{i,j})]_{m \times n}\).

(3) Aggregate \(\varvec{\lambda '}(\theta '_{i,j})\) with respect to j using the LIVIFPWMM operator in Eq. (21) and a summary relative loss function for \(A_i\) is obtained as

$$\begin{aligned} \varvec{\lambda '}(\theta '_i) = \begin{bmatrix} ([S_0, S_0], [S_h, S_h]) &{} LIVIFPWMM\big (\lambda '_{P,\notin }(\theta '_{i,j})\big ) \\ LIVIFPWMM\big (\lambda '_{B,\in }(\theta '_{i,j})\big ) &{} LIVIFPWMM\big (\lambda '_{B,\notin }(\theta '_{i,j})\big ) \\ LIVIFPWMM\big (\lambda '_{N,\in }(\theta '_{i,j})\big ) &{} ([S_0, S_0], [S_h, S_h]) \\ \end{bmatrix} \end{aligned}$$
(42)

(4) Calculate the conditional probability of \(A_i\) using grey relational analysis. The conditional probability is one of the important components of a three-way decision model. It is assumed to be fixed in many cases. However, the conditional probabilities of alternatives in an MCDM problem are usually different. In addition, it is difficult to determine the conditional probability directly since one of the prerequisites for this is decision attribute, which is not included in an MCDM problem. Therefore, the conditional probabilities of alternatives are generally calculated indirectly (Liu et al. 2020a). In the three-way decision model of (Liang et al. 2018), the technique for order of preference by similarity to ideal solution is used to calculate the conditional probabilities of alternatives. The positive ideal solution and negative ideal solution are used to describe two states of decision attribute. The final relative closeness degrees imply the conditional probabilities. This approach, which takes distance as a scale, can only reflect the positional relation of data curves, and cannot embody the importance of alternatives via the trend difference of data sequence. Grey relational analysis, a technique for determining whether or not variables are correlated and the degree of their correlation, can provide a good measure for analysing the trend difference of data series and assessing the similarity between curve shapes (Liu et al. 2020a). To this end, grey relational analysis is leveraged to calculate the conditional probability of \(A_i\). Firstly, the score values of \(\theta '_{i,1}, \theta '_{i,2}, ..., \theta '_{i,n}\) are calculated using the score function in Eq. (14). According to the calculated score values and the comparison rules in Def. 3, the maximum and minimum criterion values of \(A_i\) can be found as

$$\begin{aligned} \theta '^+_i = \bigg (\bigg [S_{q_{\theta '^+_i}^\mathrm {L}}, S_{q_{\theta '^+_i}^\mathrm {U}}\bigg ], \bigg [S_{p_{\theta '^+_i}^\mathrm {L}}, S_{p_{\theta '^+_i}^\mathrm {U}}\bigg ]\bigg ) = \max \limits _{j=1}^{n}\big \{\theta '_{i,j}\big \} \end{aligned}$$
(43)
$$\begin{aligned} \theta '^-_i = \bigg (\bigg [S_{q_{\theta '^-_i}^\mathrm {L}}, S_{q_{\theta '^-_i}^\mathrm {U}}\bigg ], \bigg [S_{p_{\theta '^-_i}^\mathrm {L}}, S_{p_{\theta '^-_i}^\mathrm {U}}\bigg ]\bigg ) = \min \limits _{j=1}^{n}\big \{\theta '_{i,j}\big \} \end{aligned}$$
(44)

Then, the grey relational coefficient between \(A_i\) and \(\theta '^+_i\) and the grey relational coefficient between \(A_i\) and \(\theta '^-_i\) are respectively calculated as

$$\begin{aligned} \eta ^+_{i,j} = \frac{\min \limits _{i=1}^{m}\min \limits _{j=1}^{n}\big \{D(\theta '_{i,j}, \theta '^+_i)\big \} + \beta \max \limits _{i=1}^{m}\max \limits _{j=1}^{n}\big \{D(\theta '_{i,j}, \theta '^+_i)\big \}}{D(\theta '_{i,j}, \theta '^+_i) + \beta \max \limits _{i=1}^{m}\max \limits _{j=1}^{n}\big \{D(\theta '_{i,j}, \theta '^+_i)\big \}} \end{aligned}$$
(45)
$$\begin{aligned} \eta ^-_{i,j} = \frac{\min \limits _{i=1}^{m}\min \limits _{j=1}^{n}\big \{D(\theta '_{i,j}, \theta '^-_i)\big \} + \beta \max \limits _{i=1}^{m}\max \limits _{j=1}^{n}\big \{D(\theta '_{i,j}, \theta '^-_i)\big \}}{D(\theta '_{i,j}, \theta '^-_i) + \beta \max \limits _{i=1}^{m}\max \limits _{j=1}^{n}\big \{D(\theta '_{i,j}, \theta '^-_i)\big \}} \end{aligned}$$
(46)

where \(0< \beta < 1\) is the identification coefficient whose general value is 0.5, and \(D(\theta '_{i,j}, \theta '^+_i)\) and \(D(\theta '_{i,j}, \theta '^-_i)\) are respectively the distance between \(\theta '_{i,j}\) and \(\theta '^+_i\) and the distance between \(\theta '_{i,j}\) and \(\theta '^-_i\) which can be calculated using Eq. (15). Next, the grey relational degree of \(A_i\) with respect to \(\theta '^+_i\) and the grey relational degree of \(A_i\) with respect to \(\theta '^-_i\) are respectively calculated as

$$\begin{aligned} \eta ^+_i = \sum _{j=1}^{n}w_j\eta ^+_{i,j} \end{aligned}$$
(47)
$$\begin{aligned} \eta ^-_i = \sum _{j=1}^{n}w_j\eta ^-_{i,j} \end{aligned}$$
(48)

Lastly, the relative closeness of the grey relation of \(A_i\) is calculated as

$$\begin{aligned} \psi _i = \frac{\eta ^+_i}{\eta ^+_i + \eta ^-_i} \end{aligned}$$
(49)

This value is used to express the conditional probability of \(A_i\), i.e. \(P(X|[A_i]) = \psi _i\).

(5) Calculate the expected loss of \(A_i\) when taking the decision action \(A_*\) \((* = P, B, N)\). If \(P(X|[A_i]) + P(\lnot X|[A_i]) = 1\), \(L(A_*|[A_i])\) can be expressed as

$$\begin{aligned}&L(A_P|[A_i]) = \big (1 - P(X|[A_i])\big )LIVIFPWMM\big (\lambda '_{P,\notin }(\theta '_{i,j})\big ) \end{aligned}$$
(50)
$$\begin{aligned}&L(A_B|[A_i]) = P(X|[A_i])LIVIFPWMM\big (\lambda '_{B,\in }(\theta '_{i,j})\big ) \oplus \big (1 - P(X|[A_i])\big )LIVIFPWMM\big (\lambda '_{B,\notin }(\theta '_{i,j})\big ) \end{aligned}$$
(51)
$$\begin{aligned}&L(A_N|[A_i]) = P(X|[A_i])LIVIFPWMM\big (\lambda '_{N,\in }(\theta '_{i,j})\big ) \end{aligned}$$
(52)

which can be calculated using the operational laws of LIVIFNs in Eq. (17) and Eq. (19).

(6) Calculate the score values of \(L(A_P|[A_i])\), \(L(A_B|[A_i])\), and \(L(A_N|[A_i])\) using the score function in Eq. (14).

(7) Take decision actions according to the decision rules (\(R'_{P'}\)), (\(R'_{B'}\)), and (\(R'_{N'}\)).

(8) Make recommendations based on the results of three-way decisions.

5 Application and comparisons

5.1 Application of the proposed method

3D (three-dimensional) printing, formally known as additive manufacturing, is a set of technologies that create 3D objects via joining materials in a layer upon layer manner. At present, there are mainly seven categories of 3D printing technologies, which are vat photopolymerisation, material jetting, binder jetting, powder bed fusion, material extrusion, directed energy deposition, and sheet lamination. Based on these technologies, over 1,600 (data from Senvol database for industrial additive manufacturing machines and materials) industrial 3D printers have been manufactured and sold in the market so far. In practice, how to select a proper 3D printer from a set of alternatives to make a specific product is non-trivial, because 3D printer has a direct influence on the quality and cost of the product (Wang et al. 2017).

To assist selection of 3D printers, many types of methods have been developed within academia during the past two decades. One of the most popular and important types of methods is MCDM method. This type of method determines a proper 3D printers from a set of alternatives via weighing the values of multiple criteria of all alternatives obtained from evaluation of domain experts, estimation of theoretical models, or results of simulation experiments (Qin et al. 2020b). The following is an illustrative example about the application of the proposed multi-criterion three-way decision-making method in 3D printer selection.

In this example, a user needs to select a proper 3D printer from ten alternative 3D printers (denoted as \(A_1, A_2, ..., A_{10}\)) to print a 3D model using certain material. The user invited an experienced domain expert to evaluate the ten alternative 3D printers based on five criteria: predicted part error (\(C_1\)), predicted surface roughness (\(C_2\)), predicted tensile strength (\(C_3\)), predicted elongation (\(C_4\)), and predicted part cost (\(C_5\)). The weights and risk avoidance coefficients of the five criteria are given by (0.1, 0.1, 0.3, 0.3, 0.2) and (0.25, 0.25, 0.25, 0.25, 0.25), respectively. The domain expert was asked to use LIVIFNs to express the evaluation results. There are nine available linguistic terms: extremely low (\(S_0\)), very low (\(S_1\)), low (\(S_2\)), slightly low (\(S_3\)), medium (\(S_4\)), slightly high (\(S_5\)), high (\(S_6\)), very high (\(S_7\)), and extremely high (\(S_8\)). The evaluation results that form a decision matrix \(\varvec{M} = [\theta _{i,j}]_{10 \times 5}\) are listed in Table 7.

Table 7 Evaluation results of the domain expert
Full size table

Using the proposed three-way decision method, the 3D printer selection problem above can be solved via the following steps:

  1. (1)

    Since \(C_1\), \(C_2\), and \(C_5\) are cost criteria and \(C_3\) and \(C_4\) are benefit criteria, according to Eq. (36), \(\varvec{M}\) is normalised as \(\varvec{M'} = [\theta '_{i,j}]_{10 \times 5}\).

  2. (2)

    According to Eq. (37), the LIVIFNs in \(\varvec{M'}\) are converted into relative loss functions. Then \(\varvec{M'}\) is converted into a relative loss function matrix \([\varvec{\lambda '}(\theta '_{i,j})]_{10 \times 5}\).

  3. (3)

    According to Eq. (42) (When adapting the LIVIFPWMM operator, \(r_1 = 1\), \(r_2 = 2\), \(r_3 = 0\), \(r_4 = 0\), and \(r_5 = 0\)), \(\varvec{\lambda '}(\theta '_{i,j})\) is aggregated with respect to j and a summary relative loss function for \(A_i\) is respectively obtained, as shown in Table 8.

Table 8 Summary relative loss functions
Full size table

(4) According to Eqs. (43)–(49), the conditional probability of \(A_i\) is respectively calculated, as listed in Table 9.

Table 9 Calculated conditional probabilities
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(5) According to Eqs. (50)–(52) and the operational laws of LIVIFNs in Eq. (17) and Eq. (19), the expected loss of \(A_i\) when taking the decision action \(A_*\) \((* = P, B, N)\) is respectively calculated, as listed in Table 10.

Table 10 Calculated expected losses
Full size table

(6) Using the score function in Eq. (14), the score value of the expected loss of \(A_i\) when taking the decision action \(A_*\) \((* = P, B, N)\) is respectively calculated, as listed in Table 11.

Table 11 Calculated score values of the expected losses
Full size table

(7) According to the decision rules (\(R'_{P'}\)), (\(R'_{B'}\)), and (\(R'_{N'}\)), a certain decision action is respectively taken for \(A_i\) and results of three-way decisions are obtained as \(POS(X) = \{A_6, A_7\}\), \(BND(X) = \{A_1, A_2, A_3, A_4, A_5, A_8\}\), and \(NEG(X) = \{A_9, A_{10}\}\).

(8) Based on the results of three-way decisions, \(A_6\) and \(A_7\) are recommended to the user, \(A_9\) and \(A_{10}\) are not recommended, and the remaining alternative 3D printers need more information to make further decisions.

5.2 Comparison with existing methods

To demonstrate the effectiveness of the proposed three-way decision method, a comparison of the decision-making results of the method and the MCDM methods based on aggregation operators of LIVIFNs presented by Kumar and Garg (2018), Qin et al. (2020c), Garg and Kumar (2019b), Zhu et al. (2020), Garg (2020), Liu and Qin (2019), and Qin et al. (2020a) is carried out. In this comparison, each method is respectively applied to solve the practical MCDM problem with LIVIFNs above: The linguistic interval-valued intuitionistic fuzzy prioritised weighted averaging (LIVIFPWA) operator and the linguistic interval-valued intuitionistic fuzzy prioritised weighted geometric (LIVIFPWG) operator are respectively the key technique in the method of Kumar and Garg (2018); The linguistic interval-valued intuitionistic fuzzy prioritised ‘and’ (LIVIFPA) operator and the linguistic interval-valued intuitionistic fuzzy prioritised ‘or’ (LIVIFPO) operator are respectively the key technique in the method of Qin et al. (2020c); The linguistic interval-valued intuitionistic fuzzy weighted averaging (LIVIFWA) operator and the linguistic interval-valued intuitionistic fuzzy weighted geometric (LIVIFWG) operator are respectively the key technique in the method of Garg and Kumar (2019b); The linguistic interval-valued intuitionistic fuzzy Hamacher weighted averaging (LIVIFHWA) operator and the linguistic interval-valued intuitionistic fuzzy Hamacher weighted geometric (LIVIFHWG) operator are respectively the key technique in the method of Zhu et al. (2020); The linguistic interval-valued Pythagorean fuzzy weighted averaging (LIVPFWA) operator and the linguistic interval-valued Pythagorean fuzzy weighted geometric (LIVPFWG) operator are respectively the key technique in the method of Garg (2020); The linguistic interval-valued intuitionistic fuzzy weighted Maclaurin symmetric mean (LIVIFWMSM) operator is the key technique in the method of Liu and Qin (2019); The LIVIFPWMM operator is the key technique in the method of Qin et al. (2020a). For the LIVIFPWA (LIVIFPA) and LIVIFPWG (LIVIFPO) operators, the weights of criteria given in the MCDM problem are directly taken as their priority weights. Thus they reduce to the LIVIFWA and LIVIFWG operators, respectively. When adapting the LIVIFHWA and LIVIFHWG operators, 1 is assigned to the controlling parameter of Hamacher t-norm and t-conorm. When adapting the LIVIFWMSM operator, \(\lfloor n/2 \rfloor = \lfloor 5/2 \rfloor = 2\) (\(\lfloor \rfloor \) is the round down function) is assigned to the controlling parameter of Maclaurin symmetric mean. When adapting the LIVIFPWMM operator, (1, 2, 0, 0, 0) is assigned to the vector of the controlling parameters of Muirhead mean. To facilitate comparison, all methods based on aggregation operators of LIVIFNs use the score function in Def. 2 and the comparison rules in Def. 3 in generation of orders of alternative 3D printers. The results of the comparison are given in Table 12.

Table 12 Results of the comparison with existing methods
Full size table

As can be seen from Table 12, \(A_6\) and \(A_7\) are the alternative 3D printers in the first two positions of each order, \(A_9\) and \(A_{10}\) or \(A_{10}\) and \(A_9\) are the alternative 3D printers in the last two positions of each order, and the remaining alternative 3D printers are in the middle of each order. Such results are completely consistent with the decision-making results of the proposed method: \(A_6\) and \(A_7\) are recommended, \(A_9\) and \(A_{10}\) are not recommended, and the remaining alternative 3D printers need more information to make further decisions. This demonstrates the feasibility and effectiveness of the proposed method in solving practical MCDM problems with LIVIFNs.

The comparison above is based on the perspective of order of alternatives. It has been verified that the proposed three-way decision method is as effective as some existing methods based on aggregation operators of LIVIFNs. Since the proposed method comes from a combination of three-way decision model and MCDM under linguistic interval-valued intuitionistic fuzzy environment, it has a significant advantage over the existing methods. The proposed method is based on more flexible three-way decision model, while all existing methods are based on traditional two-way decision model. In two-way decision model, the decision on an alternative is either acceptance or rejection. This has certain advantage when the information is sufficient or the cost of obtaining the information is small. But it might be oversimplified and could provide undesirable and costly decision-making results when the information is insufficient or acquisition of the information requires a certain cost. In this situation, three-way decision model is more flexible and advantageous, as it adds an abstaining decision for each alternative. This effectively avoids hasty classification of the alternatives at the border of acceptance and rejection.

6 Conclusion

In this paper, a multi-criterion three-way decision-making method is presented to solve an MCDM problem with LIVIFNs. Firstly, a three-way decision model for an MCDM problem with LIVIFNs is constructed. This model consists of a specific relative loss function derived from an LIVIFN and corresonding three-way decision rules. Based on the constructed model, a three-way decision method for solving an MCDM problem with LIVIFNs is then proposed. In this method, the operational laws of LIVIFNs based on algebraic t-norm and t-conorm are used to calculate the relative losses and expected loss of each alternative, the LIVIFPWMM operator is applied to aggregate the relative loss function matrix, and grey relational analysis is adopted to compute the conditional probability of each alternative. After that, a practical MCDM problem with LIVIFNs is presented to illustrate the application of the proposed method. Finally, an experimental comparison with some existing methods is carried out to demonstrate the effectiveness and advantage of the method.

The major contribution of the paper is the development of a multi-criterion three-way decision-making method under linguistic interval-valued intuitionistic fuzzy environment. Compared to the existing methods, the developed method considers the characteristics of each alternative in an MCDM problem with LIVIFNs via calculating its relative losses according to its evaluation values, which avoids possible issues in some practical MCDM problems that different alternatives are assigned the same loss function according to experience. Further, the method is more flexible and advantageous for solving an MCDM problem with LIVIFNs when the information for decision-making is insufficient or acquisition of the information needs a certain cost.

Future work will aim especially at improving the proposed three-way decision method to overcome a main limitation: Inappropriate risk avoidance coefficients may cause no or many alternatives to be classified into the positive region, thus making further decisions difficult (Liu et al. 2020a). In the proposed method, the risk avoidance coefficients are manually assigned according to experience. This does not present such limitation for the 3D printer selection problem. However, the limitation could appear for other MCDM problems with LIVIFNs. To this end, an approach for calculating appropriate risk avoidance coefficients for different MCDM problems with LIVIFNs would be developed to improve the proposed method. Further, extensions of three-way decision model to solve trapezoidal fuzzy decision-making problem (Uluçay et al. 2018; Uluçay 2020), intuitionistic trapezoidal fuzzy decision-making problem (Uluçay et al. 2019; Bakbak and Uluçay 2019; Bakbak et al. 2019), consensus based group decision-making problem (Zhang et al. 2021d; Zhang and Li 2021; Gao and Zhang 2021), and two sided matching decision-making problem (Zhang et al. 2021b, c) would also be studied.