Bidirectional projection method for multi-attribute group decision making under probabilistic uncertain linguistic environment

Abstract

The financial products selection in the financial services sector is a traditional multi-attribute group decision making (MAGDM) problem. Probabilistic uncertain linguistic sets (PULTSs) could be used to evaluate the financial products with uncertain linguistic terms and corresponding weights (probabilistic). The bidirectional projection (BP) method could take the bidirectional projection values into account. In this paper, we develop an integration model of information entropy and BP method under PULTSs. First of all, utilizing information entropy derives the priority weights of attributes. Next, utilizing the BP method of the PULTSs to obtain the final ranking of the alternatives. To depict the BP method, the formative vectors of two alternatives are defined, and a weighted vector model and inner product are improved under the PULTSs. In addition, through giving the case of financial products selection and some existing MAGDM methods for comparative analysis, it is proved that the method is practical and effective. The proposed approach also contributes to the effective selection of appropriate options in other decision-making matters.

Keywords

Multi-attribute group decision making (MAGDM)probabilistic uncertain linguistic sets (PULTSs)bidirectional projection (BP) method information entropy financial products selection

1 Introduction

Since objective things are intricacy, most decision-making problems are uncertain and vague [1 –5], and decision-making information is more appropriate to be expressed for linguistic term [6 –9]. Evaluation of qualitative variables by linguistic term [10] is a flexible method that makes human thinking and expression consistent. Herrera and Martinez [11] proposed the linguistic term sets (LTSs) in their developed computational technique. Then, scholars have come up with uncertain linguistic set [12, 13] and multi-granularity uncertain linguistic set [14]. By combining hesitant fuzzy sets and linguistic term sets, Rodriguez, Martinez [15] put forward hesitant fuzzy linguistic item sets (HFLTS) and provide several possible linguistic variables. When expressing preferences in a qualitative environment, several linguistic terms may have different weights (expressed in probability) and be considered at the same time. Thus, Pang, Wang [16] put forword the notion of probabilistic linguistic term sets (PLTS) in 2016, and proposed some PLTS basic laws of operation and aggregation operators. Based on which, an extended TOPSIS approch was gived and applied to practical cases. Liao, Jiang, Xu, Xu and Herrera [17] proposed linear programming method for solving the MADM with PLTSs. Lin, Chen, Liao and Xu [18] developed the probabilistic linguistic ELECTRE II method and applied it for edge computing. Lei, Wei, Lu, Wei and Wu [19] gave the GRA method for incomplete weight information MAGDM in the PLTS. Zhai, Xu and Liao [20] proposed the vector-term sets of probabilistic linguistic to be applied for MAGDM with multi-granular linguistic information. Feng, Liu and Wei [21] proposed a QUALIFLEX method with possibility degree comparison of probabilistic linguistic information. Wang, Wei, Wu, Wei and Guo [22] defined the probabilistic linguistic GRP method. Chen, Wang and Wang [23] applied the MULTIMOORA method for PLTSs to select cloud-based ERP. When uncertain linguistic terms are used to express evaluation information, these evaluation information is different, and the weights (probabilities) that occur for each evaluation are different. Based on the ideological inspiration of PLTS and uncertain linguistic terms,Lin, Xu [24] proposed the concept of probabilistic uncertain linguistic item sets (PULTS) and studied its basic laws of operation and aggregation operators,which is used for modeling and operating uncertain linguistic variable in decision-making. Probabilistic uncertain linguistic preference relationship was established by Xie, Ren [25], and they devised similarity degree measure and distance measure to improve consistency. Wei, He [26] proposed the probabilistic uncertain linguistic MABAC method to deal with MAGDM problem. Lei, Wei, Wu, Wei and Guo [27] defined the QUALIFLEX method for MAGDM with PULTSs. Wei, He, Lei, Wu, Wei and Guo [28] defined the uncertain probabilistic linguistic MABAC method. Wei, Lin, Lu, Wu and Wei [29] defined the generalized Dice similarity measures for probabilistic uncertain linguistic MAGDM.

There existed another similar concept called linguistic distribution for solving the corresponding problem. Zhang, Dong and Xu [30] presented the concept of distribution assessments in a linguistic term set. Zhang, Guo and Martinez [31] presented the linguistic hierarchies. Zhang, Yu, Martinez and Gao [32] presented the linguistic distribution to cope with the multigranular unbalanced HFLTSs in GDM. Wu, Dong, Qin and Pedrycz [33] presented the linguistic distribution and priority based approximation to flexible linguistic expressions. Zhang, Xiao, Palomares, Liang and Dong [34] presented the linguistic distribution for large-scale GDM with comparative linguistic information. Tang, Peng, Zhang, Pedrycz and Yang [35] presented consensus-driven models to personalize individual semantics with distribution linguistic preference relations. Dong, Wu, Zhang and Zhang [36] presented the linguistic distribution assessments with interval symbolic proportions.

The projection method is a very good method to deal with MAGDM problems [37, 38]. It can use the length and angle of the alternatives to calculate the projection value of the alternatives on the positive and negative ideal solutions, so as to sort the alternatives. But when the projection value is the same, the alternatives cannot be distinguished. So Ye [39] proposed a BP method to overcome this shortcoming, applied it to single-valued neutrosophic sets, and proved its superiority through studying instance. Because the BP method can consider bidirectional projection values, when the projection values on the ideal solution are the same, alternatives can also be distinguished. Therefore, compared with the projection method, the BP method is more comprehensive and more reasonable [40]. The BP method has been widely applied for decision making. A BP measure of interval numbers is proposed by Ye [41] and it is extended to the BP measure of neutrosophic numbers. Zang, Zhao [42] proposed a grey relational BP approach to solve IVHFLNs MCDM problems. The Power average operator is combined with the bidirectional projection, which is extended to the PLTS [40]. Liu and You [43] improved the BP method of linguistic neutrosophic numbers for MCGDM. Qu, An [44] established the bidirectional projection models to study dual hesitant fuzzy MADM problems. Wei, Zhang [45] established the picture fuzzy BP method to deal with MAGDM problem. Li, Huang [46] designed the grey relational BP method to solve MAGDM problem of the TrT2IFN. Yang, Wang [47] proposed the normal wiggly Pythagorean hesitant fuzzy BP approach for MADM problems. However, so far, there are still no studies on the BP method for MAGDM under PULTSs.

The PULTSs could well depict uncertain information with uncertain linguistic term and the probabilities of each uncertain linguistic term. and the BP method could not only consider the distance and angle of the alternatives, but also consider the bidirectional projection values. Combining the BP method and the PULTSs will better solve the problem of MAGDM.

In our study, the BP method of probability uncertain linguistic is used in the MAGDM with the completely unknown weight information. The innovation is as follows: (1) First use of BP method in probabilistic uncertain linguistic group decision; (2) A probabilistic uncertain linguistic BP group decision method is established, in which the entropy is used to determine the attribute priority weight; (3) Give an example studies on financial products selection to demonstrate the developed approach; (4) A number of comparative research methods, including PUL-TOPSIS operator, PULWA method, PUL-EDAS method and ULWA operator, are presented.

2 Preliminaries

Xu [48] proposed the additive LTSs and Gou, Xu and Liao [49] gave a mathematical transformation function between LTSs and [0, 1].

Definition 1 [48, 49]. Let L ={ γ_α|α = - θ, ⋯ , -1, 0, 1, ⋯ θ } be an LTS [48], the linguistic terms γ_α can be depicted as β which is derived by the mathematical function g [49]: $g : [γ_{- θ}, γ_{θ}] \to [0, 1], g (γ_{α}) = \frac{α + θ}{2 θ} = β$ (1)

Evidently, they are equivalent in mathematical information. Moreover, β could also be expressed as the linguistic term γ_α, which is derived by the mathematical function g^-1: $g^{- 1} : [0, 1] \to [γ_{- θ}, γ_{θ}], g^{- 1} (β) = γ_{(2 β - 1) θ} = γ_{α}$ (2)

Pang, Wang and Xu [50] proposed the concept of PLTSs.

Definition 2 [50]. L ={ γ_j|j = - θ, ⋯ , -1, 0, 1, ⋯ θ } is an LTS, a PLTS is designed as:

$\begin{matrix} L (p) = & {γ^{(τ)} (p^{(τ)}) | γ^{(τ)} \in L, p^{(τ)} ⩾ 0, \\ τ = 1, 2, \dots, # L (p), \sum_{τ = 1}^{# L (p)} p^{(τ)} ⩽ 1} \end{matrix}$ (3) where the τth probability linguistic term γ^(τ) (p^(τ)) contains both γ^(τ) and its probability p^(τ), and # L (p) is the number of elements in L (p). The probability linguistic term γ^(τ) (p^(τ)) of L (p) are arranged in ascending order.

Lin, Xu, Zhai and Yao [24] designed the probabilistic uncertain linguistic term sets (PULTS) based on ULTSs [51] and PLTS.

Definition 3 [24]. A PULTS is designed as follows:

$\begin{matrix} PUL (p) = & {[L^{(τ)}, U^{(τ)}] (p^{(τ)}) | p^{(τ)} ⩾ 0, τ = 1, 2, \\ \dots, # PUL (p), \sum_{τ = 1}^{# PUL (p)} p^{(τ)} ⩽ 1} \end{matrix}$ (4) where [L^(τ), U^(τ)] (p^(τ)) associate the ULTS [L^(τ), U^(τ)] with the probability p^(τ), L^(τ), U^(τ) are LTSs, L^(τ) ⩽ U^(τ), and # PUL (p) is the cardinality of PUL (p).

For ease of calculation, the PULTS PUL (p) is normalized Lin, Xu, Zhai and Yao [24] as $NPUL (p) = {[L^{(τ)}, U^{(τ)}] ({\tilde{p}}^{(τ)}) | {\tilde{p}}^{(τ)} ⩾ 0, τ = 1, 2, \dots, # NPUL (p), \sum_{τ = 1}^{# NPUL (p)} {\tilde{p}}^{(τ)} = 1}$ , where ${\tilde{p}}^{(τ)} = p^{(τ)} / \sum_{τ = 1}^{# PUL (p)} p^{(τ)}$ for all τ = 1, 2, ⋯ , # NPUL (p).

If there is overlap in a PULTS, the PULTS should be preprocessed by dividing the overlapped ULT into several non-overlapping ULTs which share the probabilities equally. For example, the PULTS {〈 [γ_-1, γ₁] , 0.6 〉 , 〈 [γ₀, γ₁] , 0.2 〉 , 〈 [γ₁, γ₂] , 0.2 〉} should be preprocessed into the PULTS {〈 [γ_-1, γ₀] , 0.3 〉 , 〈 [γ₀, γ₁] , 0.5 〉 , 〈 [γ₁, γ₂] , 0.2 〉}.

Definition 4 [24]. ${PUL}_{1} (p) = {[L_{1}^{(τ)}, U_{1}^{(τ)}] (p_{1}^{(τ)}) | τ = 1, 2, \dots, # {PUL}_{1} (p)}$ and ${PUL}_{2} (p) = {[L_{2}^{(τ)}, U_{2}^{(τ)}] (p_{2}^{(τ)}) | τ = 1, 2, \dots, # {PUL}_{2} (p)}$ are two PULTSs, where # PUL₁ (p) and # PUL₂ (p) are the numbers of uncertain linguistic terms in PUL₁ (p) and PUL₂ (p) respectively. If # PUL₁ (p) > # PUL₂ (p), then add # PUL₁ (p) - # PUL₂ (p) uncertain linguistic terms to PUL₂ (p), and the added uncertain linguistic terms should be the smallest linguistic term in PUL₂ (p) and the probabilities of added linguistic terms should be zero.

Definition 5 [52]. ${PUL}_{1} (p) = {[L_{1}^{(τ)}, U_{1}^{(τ)}] (p_{1}^{(τ)}) | τ = 1, 2, \dots, # {PUL}_{1} (p)}$ and ${PUL}_{2} (p) = {[L_{2}^{(τ)}, U_{2}^{(τ)}] (p_{2}^{(τ)}) | τ = 1, 2, \dots, # {PUL}_{2} (p)}$ are two PULTSs with # PUL₁ (p) = # PUL₂ (p) = # PUL (p), their Hamming distance HD (PUL₁ (p) , PUL₂ (p)) between PUL₁ (p) and PUL₂ (p) is defined below:

$\begin{matrix} HD ({PUL}_{1} (p), {PUL}_{2} (p)) \\ = \frac{\sum_{τ = 1}^{# PUL (p)} (| g (L_{1}^{(τ)}) p_{1}^{(τ)} - g (L_{2}^{(τ)}) p_{2}^{(τ)} | + | g (U_{1}^{(τ)}) p_{1}^{(τ)} - g (U_{2}^{(τ)}) p_{2}^{(τ)} |)}{2 # PUL (p)} \end{matrix}$ (5)

Definition 6 [24]. For the PUL (p) ={ [L^(τ), U^(τ)] (p^(τ)) |τ = 1, 2, ⋯ , # PUL (p) }, the expected value E (PUL (p)) and deviation degree D (PUL (p)) of PUL (p) are defined: $E (PUL (p)) = \frac{\sum_{τ = 1}^{# PUL (p)} (\frac{g (L^{(τ)}) p^{(τ)} + g (U^{(τ)}) p^{(τ)}}{2})}{\sum_{τ = 1}^{# PUL (p)} p^{(τ)}}$ (6)

$\begin{matrix} D (PUL (p)) \\ = \sqrt{\frac{\sum_{τ = 1}^{# PUL (p)} {p^{(τ)} (\frac{g (L^{(τ)}) + g (U^{(τ)})}{2} - E (PUL (p)))}^{2}}{\sum_{τ = 1}^{# PUL (p)} p^{(τ)}}} \end{matrix}$ (7)

According to the Equations (6)-(7), the size relation of two PULTSs is specified as follows: (1) if E (PUL₁ (p)) > E (PUL₂ (p)), then PUL₁ (p) > PUL₂ (p); (2) if E (PUL₁ (p)) = E (PUL₂ (p)), then if D (PUL₁ (p)) = D (PUL₂ (p)), then PUL₁ (p) = PUL₂ (p); if D (PUL₁ (p)) < D (PUL₂ (p)), then, PUL₁ (p) > PUL₂ (p).

3 Entropy-based BP method for MAGDM under the PULTS

In this section, the probabilistic uncertain linguistic Bidirectional Projection (PUL-BP) method is used to solve the MAGDM problems with completely unknown weight information. Utilizing information entropy to derive the priority weights of attributes, and utilizing the BP method of PULTSs to get the ranking of the alternatives. For the convenience of description, we use the following mathematical symbols to denote relevant content. Suppose there are s chosen alternatives, t qualitative attributes and q qualified experts, A ={ A₁, A₂, ⋯ , A_s } denotes the set of s chosen alternatives, C ={ C₁, C₂, ⋯ , C_t } represents the set of t qualitative attributes with weight vector ω = (ω₁, ω₂, ⋯ , ω_t), where ω_j ∈ [0, 1], j = 1, 2, ⋯ , t, $\sum_{j = 1}^{t} ω_{j} = 1$ , and the set of q experts is denoted as E ={ E₁, E₂, ⋯ , E_q }. Experts evaluate each alternative value under t qualitative attributes and denote them as uncertain linguistic term $[L_{ij}^{(k)}, U_{ij}^{(k)}] (i = 1, 2, \dots, s, j = 1, 2, \dots, t, k = 1, 2, \dots, q)$ .

The specific modeling steps of the PUL-BP method for MAGDM are presented below:

Step 1. Transform the cost attribute value [γ_α, γ_β] into the corresponding beneficial attribute value [γ_- β, γ_- α].

Step 2. Convert the ULTs $[L_{ij}^{(k)}, U_{ij}^{(k)}] (i = 1, 2, \dots, s, j = 1, 2, \dots, t, k = 1, 2, \dots, q)$ into probabilistic uncertain linguistic decision matrix PULM = (PULM_ij (p)) _s×t, ${PULM}_{ij} (p) = {[L_{ij}^{(τ)}, U_{ij}^{(τ)}] (p_{ij}^{(τ)}) | τ = 1, 2, \dots, # {PULM}_{ij} (p)} (i = 1, 2, \dots, s, j = 1, 2, \dots, t)$

Step 3. Preprocess and normalize all the PULTSs by using definitions 3 and 4, and then get the normalized probabilistic uncertain linguistic decision matrix NPULM = (NPULM_ij (p)) _s×t, ${NPULM}_{ij} (p) = {[L_{ij}^{(τ)}, U_{ij}^{(τ)}] (p_{ij}^{(τ)}) | τ = 1, 2, \dots, # {NPULM}_{ij} (p)} (i = 1, 2, \dots, s, j = 1, 2, \dots, t)$ .

Step 4. Calculate the weight values for attributes with information entropy[53].

Information entropy can reflect the average of uncertainty information amount contained in each attribute. In a given attribute, the greater value of entropy illustrates the greater similarity. The greater similarity illustrates that this attribute provides less information. So, this attribute should have a smaller weight value [54 –56].

First, the normative decision matrix NPULNM_ij (p) is obtained by the following expression:

$\begin{matrix} {NPULNM}_{ij} (p) = \frac{\sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{g (L_{ij}^{(τ)}) p_{ij}^{(τ)} + g (U_{ij}^{(τ)}) p_{ij}^{(τ)}}{2}}{\sum_{i = 1}^{s} \sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{g (L_{ij}^{(τ)}) p_{ij}^{(τ)} + g (U_{ij}^{(τ)}) p_{ij}^{(τ)}}{2}}, \\ j = 1, 2, \dots, t \end{matrix}$ (8)

Then, the information entropy E_j (j = 1, 2, ⋯ t) is calculated by the following: $E_{j} = - \frac{1}{ln s} \sum_{i = 1}^{s} {NPULNM}_{ij} (p) ln {NPULNM}_{ij} (p)$ (9) if NPULNM_ij (p) =0, NPULNM_ij (p) ln NPULNM_ij (p) is specified as 0.

Finally, the attribute weights vector ω = (ω₁, ω₂, ⋯ , ω_t) is calculated: $ω_{j} = \frac{1 - E_{j}}{\sum_{j = 1}^{t} (1 - E_{j})}, j = 1, 2, \dots, t$ (10) where ω_j ∈ [0, 1] and $\sum_{j = 1}^{t} ω_{j} = 1$ .

Step 5. Determine the ideal alternatives A⁺ and A^-.

NPULM = (NPULM_ij (p)) _s×t (i = 1, 2, ⋯ , s ; j = 1, 2, ⋯ , t) is a normalized PUL matrix, the positive ideal alternative is denoted as: $A^{+} = (A_{1}^{+}, A_{2}^{+} \dots, A_{t}^{+})$ (11) where $\begin{matrix} A_{j}^{+} = {max_{i} {[L_{ij}^{(τ)}, U_{ij}^{(τ)}] (p_{ij}^{(τ)})} | τ = 1, 2, \dots, \\ # {NPULM}_{ij} (p)}, (i = 1, 2 \dots, s; j = 1, 2, \dots, t) \end{matrix}$

Analogously, the negative ideal alternative is denoted as: $A^{-} = (A_{1}^{-}, A_{2}^{-} \dots, A_{t}^{-})$ (12) where $\begin{matrix} A_{j}^{-} = {min_{i} {[L_{ij}^{(τ)}, U_{ij}^{(τ)}] (p_{ij}^{(τ)})} | τ = 1, 2, \dots, \\ # {NPULM}_{ij} (p)} (i = 1, 2 \dots, s; j = 1, 2, \dots, t) \end{matrix}$

Step 6. Defined the formative vector A^-A⁺, A^-A_i, A_iA⁺ as follows: $A^{-} A^{+} = (A_{1}^{-} A_{1}^{+}, A_{2}^{-} A_{2}^{+}, \dots, A_{t}^{-} A_{t}^{+})$ (13) $A^{-} A_{i} = (A_{1}^{-} A_{i 1}, A_{2}^{-} A_{i 2}, \dots, A_{t}^{-} A_{it})$ (14) $A_{i} A^{+} = (A_{i 1} A_{1}^{+}, A_{i 2} A_{2}^{+}, \dots, A_{it} A_{t}^{+})$ (15) where

$A_{j}^{-} A_{j}^{+} = {[α_{j}^{* (τ)}, β_{j}^{* (τ)}] | τ = 1, 2, \dots, # {NPULM}_{ij} (p)}$ $A_{j}^{-} A_{ij} = {[α_{j}^{- (τ)}, β_{j}^{- (τ)}] | τ = 1, 2, \dots, # {NPULM}_{ij} (p)}$ $A_{ij} A_{j}^{+} = {[α_{j}^{+ (τ)}, β_{j}^{+ (τ)}] | τ = 1, 2, \dots, # {NPULM}_{ij} (p)}$

where $\begin{matrix} α_{j}^{* (τ)} = & min ((max_{i} {g (L_{ij}^{(τ)}) p_{ij}^{(τ)}} - min_{i} {g (L_{ij}^{(τ)}) p_{ij}^{(τ)}}), \\ (max_{i} {g (U_{ij}^{(τ)}) p_{ij}^{(τ)}} - min_{i} {g (U_{ij}^{(τ)}) p_{ij}^{(τ)}})) \\ β_{j}^{* (τ)} = & max ((max_{i} {g (L_{ij}^{(τ)}) p_{ij}^{(τ)}} - min_{i} {g (L_{ij}^{(τ)}) p_{ij}^{(τ)}}), \\ (max_{i} {g (U_{ij}^{(τ)}) p_{ij}^{(τ)}} - min_{i} {g (U_{ij}^{(τ)}) p_{ij}^{(τ)}})) \\ α_{j}^{- (τ)} = & min ((g (L_{ij}^{(τ)}) p_{ij}^{(τ)} - min_{i} {g (L_{ij}^{(τ)}) p_{ij}^{(τ)}}), \\ (g (U_{ij}^{(τ)}) p_{ij}^{(τ)} - min_{i} {g (U_{ij}^{(τ)}) p_{ij}^{(τ)}})) \\ β_{j}^{- (τ)} = & max ((g (L_{ij}^{(τ)}) p_{ij}^{(τ)} - min_{i} {g (L_{ij}^{(τ)}) p_{ij}^{(τ)}}), \\ (g (U_{ij}^{(τ)}) p_{ij}^{(τ)} - min_{i} {g (U_{ij}^{(τ)}) p_{ij}^{(τ)}})) \\ α_{j}^{+ (τ)} = & min ((max_{i} {g (L_{ij}^{(τ)}) p_{ij}^{(τ)}} - g (L_{ij}^{(τ)}) p_{ij}^{(τ)}), \\ (max_{i} {g (U_{ij}^{(τ)}) p_{ij}^{(τ)}} - g (U_{ij}^{(τ)}) p_{ij}^{(τ)})) \\ β_{j}^{+ (τ)} = & max ((max_{i} {g (L_{ij}^{(τ)}) p_{ij}^{(τ)}} - g (L_{ij}^{(τ)}) p_{ij}^{(τ)}), \\ (max_{i} {g (U_{ij}^{(τ)}) p_{ij}^{(τ)}} - g (U_{ij}^{(τ)}) p_{ij}^{(τ)})) \end{matrix}$

Step 7. Compute the projection values of A^-A_i on A^-A⁺ and A^-A⁺ on A_iA⁺.

The weight vector of each alternative is ω = (ω₁, ω₂, ⋯ , ω_t), ω_j ∈ [0, 1], the module of A^-A⁺ is computed below: $| A^{-} A^{+} | = \sum_{j = 1}^{t} ω_{j} \sqrt{\sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{(α_{j}^{* (τ)})^{2} + (β_{j}^{* (τ)})^{2}}{2}}$ (16)

Analogously, the modules of A^-A_i and A_iA⁺ are computed below: $| A^{-} A_{i} | = \sum_{j = 1}^{t} ω_{j} \sqrt{\sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{(α_{j}^{- (τ)})^{2} + (β_{j}^{- (τ)})^{2}}{2}}$ (17) $| A_{i} A^{+} | = \sum_{j = 1}^{t} ω_{j} \sqrt{\sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{(α_{j}^{+ (τ)})^{2} + (β_{j}^{+ (τ)})^{2}}{2}}$ (18)

The inner product of A^-A_i and A^-A⁺ is computed below:

$\begin{matrix} A^{-} A_{i} \cdot A^{-} A^{+} \\ = {(\sum_{j = 1}^{t} ω_{j} \sqrt{\sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{α_{j}^{- (τ)} α_{j}^{* (τ)} + β_{j}^{- (τ)} β_{j}^{* (τ)}}{2}})}^{2} \end{matrix}$ (19)

The inner product of A^-A⁺ and A_iA⁺ is computed below:

$\begin{matrix} A^{-} A^{+} \cdot A_{i} A^{+} \\ = {(\sum_{j = 1}^{t} ω_{j} \sqrt{\sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{α_{j}^{* (τ)} α_{j}^{+ (τ)} + β_{j}^{* (τ)} β_{j}^{+ (τ)}}{2}})}^{2} \end{matrix}$ (20)

The cosine of A^-A_i and A^-A⁺ is computed below:

$\begin{matrix} \begin{matrix} \end{matrix} cos (A^{-} A_{i}, A^{-} A^{+}) = \frac{A^{-} A_{i} \cdot A^{-} A^{+}}{| A^{-} A_{i} | | A^{-} A^{+} |} = \\ \frac{{(\sum_{j = 1}^{t} ω_{j} \sqrt{\sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{α_{j}^{- (τ)} α_{j}^{* (τ)} + β_{j}^{- (τ)} β_{j}^{* (τ)}}{2}})}^{2}}{(\sum_{j = 1}^{t} ω_{j} \sqrt{\sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{(α_{j}^{- (τ)})^{2} + (β_{j}^{- (τ)})^{2}}{2}}) (\sum_{j = 1}^{t} ω_{j} \sqrt{\sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{(α_{j}^{* (τ)})^{2} + (β_{j}^{* (τ)})^{2}}{2}})} \end{matrix}$ (21)

The projection value of A^-A_i on A^-A⁺ is calculated as: $\begin{matrix} {prj}_{A^{-} A^{+}} (A^{-} A_{i}) = | A^{-} A_{i} | cos (A^{-} A_{i}, A^{-} A^{+}) \\ = \frac{A^{-} A_{i} \cdot A^{-} A^{+}}{| A^{-} A^{+} |} \\ = \frac{{(\sum_{j = 1}^{t} ω_{j} \sqrt{\sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{α_{j}^{- (τ)} α_{j}^{* (τ)} + β_{j}^{- (τ)} β_{j}^{* (τ)}}{2}})}^{2}}{\sum_{j = 1}^{t} ω_{j} \sqrt{\sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{(α_{j}^{* (τ)})^{2} + (β_{j}^{* (τ)})^{2}}{2}}} \end{matrix}$ (22)

Analogously, the projection value of A^-A⁺ on A_iA⁺ is calculated as: $\begin{matrix} {prj}_{A_{i} A^{+}} (A^{-} A^{+}) = | A^{-} A^{+} | cos (A^{-} A^{+}, A_{i} A^{+}) \\ = \frac{A^{-} A^{+} \cdot A_{i} A^{+}}{| A_{i} A^{+} |} \\ = \frac{{(\sum_{j = 1}^{t} ω_{j} \sqrt{\sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{α_{j}^{* (τ)} α_{j}^{+ (τ)} + β_{j}^{* (τ)} β_{j}^{+ (τ)}}{2}})}^{2}}{\sum_{j = 1}^{t} ω_{j} \sqrt{\sum_{τ = 1}^{# {NPULM}_{ij} (p)} \frac{(α_{j}^{+ (τ)})^{2} + (β_{j}^{+ (τ)})^{2}}{2}}} \end{matrix}$ (23)

Step 8. Compute the closeness degree of A_i to ideal alternatives. $C (A_{i}) = \frac{{prj}_{A^{-} A^{+}} (A^{-} A_{i})}{{prj}_{A^{-} A^{+}} (A^{-} A_{i}) + {prj}_{A_{i} A^{+}} (A^{-} A^{+})}$ (24)

Step 9. According to the C (A_i) (i = 1, 2, ⋯ , s), give the ranking order of all alternatives.

As shown in Fig. 1, the larger the value of prj_{A
^-
A
⁺} (A^-A_i), the closer the alternative A_i is to the positive ideal alternative A⁺. Similarly, the smaller the value of prj_{A
_i
A
⁺} (A^-A⁺), the farther the alternative A_i will be to negative ideal alternative A^-. Therefore, the alternative with the largest C (A_i) (i = 1, 2, ⋯ , s) value is the optimal alternative. Thus, the optimal alternative not only is closest to the positive ideal alternative but also should be far away from the negative ideal alternative.

Fig. 1

The graphical representation of prj_{A
^-
A
⁺} (A^-A_i) and prj_{A
_i
A
⁺} (A^-A⁺).

When the distance of the two alternatives from the positive and negative ideal alternatives are same, the optimal alternative can still be obtained. As shown in Fig. 2, we can see |A^-A_i| = |A_iA⁺|, |A^-A_k| = |A_kA⁺|, prj_{A
^-
A
⁺} (A^-A_i) = prj_{A
^-
A
⁺} (A^-A_k), prj_{A
_i
A
⁺} (A^-A⁺) < prj_{A
_k
A
⁺} (A^-A⁺), C (A_i) > C (A_k) and A_i > A_k. The reason is that the bidirectional projection method not only considers the relationship between the alternatives and the positive and negative ideal alternatives, but also considers the relative importance between the positive ideal alternative and the negative ideal alternative. Therefore, the research of the PUL-BP method is very meaningful and worth studying. It can solve the problem effectively.

Fig. 2

Schematic diagram of Bi-directional projection.

4 The numerical case and comparative analysis

4.1 The numerical case

Financial markets are booming in today’s society, and the financial services sector provide many financial products for investors. It is very important to choose the optimal financial products because of the vital interests of investors [57, 58]. The financial products selection in the financial service sectors is a traditional MAGDM problem. Fuzzy linguistic is often used to describe people’s preference of financial products, so some scholars have studied the selection methods of financial products with fuzzy system approaches. Merigo and Lafuente [59] developed a method that the ordered weighted averaging operator was used to select financial products. Liu and Wei [58] developed a weighted averaging operator of dynamic 2-tuple linguistic for the financial products investment. An incomplete additive probabilistic linguistic preference relation was proposed and used to select financial products[60]. When probabilistic uncertain linguistic term is used to be expressed decision information by decision maker, we proposed the BP method of PULTSs to select the optimal financial product.

Thus, we give a numerical case of financial products selection to demonstrate the practicability of the improved method. Five possible financial products A_i (i = 1, 2, ⋯ , 5) under four attributes are provided to select for five experts. The four attributes are as: ① C₁ is manager; ② C₂ is starting point; ③ C₃ is seven-day annualized interest rate (%); ④ C₄ is Term. The starting point (C₂) is cost attribute, the other three attributes are beneficial. The evaluation values of the five potential financial products employ the linguistic term in set:

$\begin{matrix} L = {l_{- 3} = extremelypoor (EP), l_{- 2} = verypoor (VP), \\ l_{- 1} = poor (P), l_{0} = ordinary (O), l_{1} = good (G), \\ l_{2} = verygood (VG), l_{3} = extremelygood (EG)} \end{matrix}$

Under the above four attributes, the evaluation results of five experts are listed in the Tables 1 –5.

Table 1
Decision information of the first decision maker

Alternatives C₁ C₂ C₃ C₄

A₁ [P, O] [VP, O] [VP, P] [P, O]

A₂ [VP, P] [P, O] [P, O] [O, G]

A₃ [O, G] [EP, P] [O, G] [EP, P]

A₄ [VP, O] [VP, P] [O, G] [O, G]

A₅ [EP, P] [O, G] [VG, EG] [VP, P]

Alternatives	C₁	C₂	C₃	C₄
A₁	[P, O]	[VP, O]	[VP, P]	[P, O]
A₂	[VP, P]	[P, O]	[P, O]	[O, G]
A₃	[O, G]	[EP, P]	[O, G]	[EP, P]
A₄	[VP, O]	[VP, P]	[O, G]	[O, G]
A₅	[EP, P]	[O, G]	[VG, EG]	[VP, P]

Table 2

Decision information of the second decision maker

Alternatives	C₁	C₂	C₃	C₄
A₁	[O, G]	[O, G]	[O, G]	[O, VG]
A₂	[O, G]	[O, VG]	[EP, VP]	[P, O]
A₃	[G, VG]	[P, O]	[P, G]	[O, G]
A₄	[O, G]	[P, O]	[G, VG]	[P, G]
A₅	[P, O]	[G, VG]	[P, O]	[O, G]

Table 3

Decision information of the third decision maker

Alternatives	C₁	C₂	C₃	C₄
A₁	[P, O]	[G, VG]	[G, VG]	[O, VG]
A₂	[VP, P]	[P, O]	[O, G]	[P, O]
A₃	[O, G]	[O, G]	[G, VG]	[P, O]
A₄	[G, VG]	[VP, P]	[O, G]	[G, VG]
A₅	[VP, P]	[O, G]	[P, O]	[VP, P]

Table 4

Decision information of the fourth decision maker

Alternatives	C₁	C₂	C₃	C₄
A₁	[G, VG]	[O, G]	[VP, P]	[P, O]
A₂	[P, O]	[O, VG]	[O, G]	[O, G]
A₃	[G, VG]	[O, G]	[O, G]	[EP, P]
A₄	[O, G]	[O, G]	[G, VG]	[G, VG]
A₅	[VP, P]	[G, VG]	[O, VG]	[O, G]

Table 5

Decision information of the fifth decision maker

Alternatives	C₁	C₂	C₃	C₄
A₁	[G, VG]	[O, G]	[G, VG]	[O, VG]
A₂	[O, G]	[O, VG]	[O, G]	[G, EG]
A₃	[P, G]	[P, O]	[O, G]	[P, O]
A₄	[G, VG]	[P, O]	[G, VG]	[O, G]
A₅	[P, O]	[O, G]	[O, VG]	[G, VG]

Then, the PUL-BP method is used to select the financial products.

Step 1. Convert cost attribute C₂ into beneficial attribute. The specific way is converting the cost attribute value [γ_α, γ_β] (- 3 ⩽ α, β ⩽ 3) into the corresponding beneficial attribute value [γ_-β, γ_-α] (See Tables 6 –10).

Table 6

Decision information of the first decision maker

Alternatives	C₁	C₂	C₃	C₄
A₁	[P, O]	[O, VG]	[VP, P]	[P, O]
A₂	[VP,P]	[O, G]	[P, O]	[O, G]
A₃	[O, G]	[G, EG]	[O, G]	[EP, P]
A₄	[VP, O]	[G, VG]	[O, G]	[O, G]
A₅	[EP, P]	[P, O]	[VG, EG]	[VP, P]

Table 7

Decision information of the second decision maker

Alternatives	C₁	C₂	C₃	C₄
A₁	[O, G]	[P, O]	[O, G]	[O, VG]
A₂	[O, G]	[VP, O]	[EP, VP]	[P, O]
A₃	[G, VG]	[O, G]	[P, G]	[O, G]
A₄	[O, G]	[O, G]	[G, VG]	[P, G]
A₅	[P, O]	[VP, P]	[P, O]	[O, G]

Table 8

Decision information of the third decision maker

Alternatives	C₁	C₂	C₃	C₄
A₁	[P, O]	[VP, P]	[G, VG]	[O, VG]
A₂	[VP, P]	[O, G]	[O, G]	[P, O]
A₃	[O, G]	[P, O]	[G, VG]	[P, O]
A₄	[G, VG]	[G, VG]	[O, G]	[G, VG]
A₅	[VP, P]	[P, O]	[P, O]	[VP, P]

Table 9

Decision information of the fourth decision maker

Alternatives	C₁	C₂	C₃	C₄
A₁	[G, VG]	[P, O]	[VP, P]	[P, O]
A₂	[P, O]	[VP, O]	[O, G]	[O, G]
A₃	[G, VG]	[P, O]	[O, G]	[EP, P]
A₄	[O, G]	[P, O]	[G, VG]	[G, VG]
A₅	[VP, P]	[VP, P]	[O, VG]	[O, G]

Table 10

Decision information of the fifth decision maker

Alternatives	C₁	C₂	C₃	C₄
A₁	[G, VG]	[P, O]	[G, VG]	[O, VG]
A₂	[O, G]	[VP, O]	[O, G]	[G, EG]
A₃	[P, G]	[O, G]	[O, G]	[P, O]
A₄	[G, VG]	[O, G]	[G, VG]	[O, G]
A₅	[P, O]	[P, O]	[O, VG]	[G, VG]

Step 2. Integrate the above decision information into PULTSs decision matrix (Table 11).

Table 11

Probabilistic uncertain linguistic decision matrix

Alternatives	C₁	C₂
A₁	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.4 〉, 〈 [γ_{0}, γ_{1}], 0.2 〉, \\ 〈 [γ_{1}, γ_{2}], 0.4 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 2}, γ_{0}], 0.2 〉, 〈 [γ_{0}, γ_{1}], 0.6 〉, \\ 〈 [γ_{1}, γ_{2}], 0.2 〉} \end{matrix}$
A₂	$\begin{matrix} {〈 [γ_{- 2}, γ_{- 1}], 0.4 〉, 〈 [γ_{- 1}, γ_{0}], 0.2 〉, \\ 〈 [γ_{0}, γ_{1}], 0.4 〉} \end{matrix}$	{〈 [γ_-1, γ₀] , 0.4 〉 , 〈 [γ₀, γ₂] , 0.6 〉}
A₃	$\begin{matrix} {〈 [γ_{- 1}, γ_{1}], 0.2 〉, 〈 [γ_{0}, γ_{1}], 0.4 〉, \\ 〈 [γ_{1}, γ_{2}], 0.4 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 3}, γ_{- 1}], 0.2 〉, 〈 [γ_{- 1}, γ_{0}], 0.4 〉, \\ 〈 [γ_{0}, γ_{1}], 0.4 〉} \end{matrix}$
A₄	$\begin{matrix} {〈 [γ_{- 2}, γ_{0}], 0.2 〉, 〈 [γ_{0}, γ_{1}], 0.4 〉, \\ 〈 [γ_{1}, γ_{2}], 0.4 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 2}, γ_{- 1}], 0.4 〉, 〈 [γ_{- 1}, γ_{0}], 0.4 〉, \\ 〈 [γ_{0}, γ_{1}], 0.2 〉} \end{matrix}$
A₅	$\begin{matrix} {〈 [γ_{- 3}, γ_{- 1}], 0.2 〉, 〈 [γ_{- 2}, γ_{- 1}], 0.4 〉, \\ 〈 [γ_{- 1}, γ_{0}], 0.4 〉} \end{matrix}$	{〈 [γ₀, γ₁] , 0.6 〉 , 〈 [γ₁, γ₂] , 0.4 〉}
Alternatives	C₃	C₄
A₁	$\begin{matrix} {〈 [γ_{- 2}, γ_{- 1}], 0.4 〉, 〈 [γ_{0}, γ_{1}], 0.2 〉, \\ 〈 [γ_{1}, γ_{2}], 0.4 〉} \end{matrix}$	{〈 [γ_-1, γ₀] , 0.4 〉 , 〈 [γ₀, γ₂] , 0.6 〉}
A₂	$\begin{matrix} {〈 [γ_{- 3}, γ_{- 2}], 0.2 〉, 〈 [γ_{- 1}, γ_{0}], 0.2 〉, \\ 〈 [γ_{0}, γ_{1}], 0.6 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.4 〉, 〈 [γ_{0}, γ_{1}], 0.4 〉, \\ 〈 [γ_{1}, γ_{3}], 0.2 〉} \end{matrix}$
A₃	$\begin{matrix} {〈 [γ_{- 1}, γ_{1}], 0.2 〉, 〈 [γ_{0}, γ_{1}], 0.6 〉, \\ 〈 [γ_{1}, γ_{2}], 0.2 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 3}, γ_{- 1}], 0.4 〉, 〈 [γ_{- 1}, γ_{0}], 0.4 〉, \\ 〈 [γ_{0}, γ_{1}], 0.2 〉} \end{matrix}$
A₄	{〈 [γ₀, γ₁] , 0.4 〉 , 〈 [γ₁, γ₂] , 0.6 〉}	$\begin{matrix} {〈 [γ_{- 1}, γ_{1}], 0.2 〉, 〈 [γ_{0}, γ_{1}], 0.4 〉, \\ 〈 [γ_{1}, γ_{2}], 0.4 〉} \end{matrix}$
A₅	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.4 〉, 〈 [γ_{0}, γ_{2}], 0.4 〉, \\ 〈 [γ_{2}, γ_{3}], 0.2 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 2}, γ_{- 1}], 0.4 〉, 〈 [γ_{0}, γ_{1}], 0.4 〉, \\ 〈 [γ_{1}, γ_{2}], 0.2 〉} \end{matrix}$

Step 3. Process the above evaluation matrix into the normalized PULTS decision matrix (Table 12).

Table 12

The normalized probabilistic uncertain linguistic decision matrix

Alternatives	C₁	C₂
A₁	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.4 〉, 〈 [γ_{0}, γ_{1}], 0.2 〉, \\ 〈 [γ_{1}, γ_{2}], 0.4 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 2}, γ_{- 1}], 0.2 〉, 〈 [γ_{- 1}, γ_{0}], 0.6 〉, \\ 〈 [γ_{0}, γ_{2}], 0.2 〉} \end{matrix}$
A₂	$\begin{matrix} {〈 [γ_{- 2}, γ_{- 1}], 0.4 〉, 〈 [γ_{- 1}, γ_{0}], 0.2 〉, \\ 〈 [γ_{0}, γ_{1}], 0.4 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 2}, γ_{0}], 0 〉, 〈 [γ_{- 2}, γ_{0}], 0.6 〉, \\ 〈 [γ_{0}, γ_{1}], 0.4 〉} \end{matrix}$
A₃	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.1 〉, 〈 [γ_{0}, γ_{1}], 0.5 〉, \\ 〈 [γ_{1}, γ_{2}], 0.4 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.4 〉, 〈 [γ_{0}, γ_{1}], 0.4 〉, \\ 〈 [γ_{1}, γ_{3}], 0.2 〉} \end{matrix}$
A₄	$\begin{matrix} {〈 [γ_{- 2}, γ_{0}], 0.2 〉, 〈 [γ_{0}, γ_{1}], 0.4 〉, \\ 〈 [γ_{1}, γ_{2}], 0.4 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.2 〉, 〈 [γ_{0}, γ_{1}], 0.4 〉, \\ 〈 [γ_{1}, γ_{2}], 0.4 〉} \end{matrix}$
A₅	$\begin{matrix} {〈 [γ_{- 3}, γ_{- 2}], 0.1 〉, 〈 [γ_{- 2}, γ_{- 1}], 0.5 〉, \\ 〈 [γ_{- 1}, γ_{0}], 0.4 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 2}, γ_{- 1}], 0 〉, 〈 [γ_{- 2}, γ_{- 1}], 0.4 〉, \\ 〈 [γ_{- 1}, γ_{0}], 0.6 〉} \end{matrix}$
Alternatives	C₃	C₄
A₁	$\begin{matrix} {〈 [γ_{- 2}, γ_{- 1}], 0.4 〉, 〈 [γ_{0}, γ_{1}], 0.2 〉, \\ 〈 [γ_{1}, γ_{2}], 0.4 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0 〉, 〈 [γ_{- 1}, γ_{0}], 0.4 〉, \\ 〈 [γ_{0}, γ_{2}], 0.6 〉} \end{matrix}$
A₂	$\begin{matrix} {〈 [γ_{- 3}, γ_{- 2}], 0.2 〉, 〈 [γ_{- 1}, γ_{0}], 0.2 〉, \\ 〈 [γ_{0}, γ_{1}], 0.6 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.4 〉, 〈 [γ_{0}, γ_{1}], 0.4 〉, \\ 〈 [γ_{1}, γ_{3}], 0.2 〉} \end{matrix}$
A₃	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.1 〉, 〈 [γ_{0}, γ_{1}], 0.7 〉, \\ 〈 [γ_{1}, γ_{2}], 0.2 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 3}, γ_{- 1}], 0.4 〉, 〈 [γ_{- 1}, γ_{0}], 0.4 〉, \\ 〈 [γ_{0}, γ_{1}], 0.2 〉} \end{matrix}$
A₄	$\begin{matrix} {〈 [γ_{0}, γ_{1}], 0 〉, 〈 [γ_{0}, γ_{1}], 0.4 〉, \\ 〈 [γ_{1}, γ_{2}], 0.6 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.1 〉, 〈 [γ_{0}, γ_{1}], 0.5 〉, \\ 〈 [γ_{1}, γ_{2}], 0.4 〉} \end{matrix}$
A₅	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.4 〉, 〈 [γ_{0}, γ_{2}], 0.4 〉, \\ 〈 [γ_{2}, γ_{3}], 0.2 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 2}, γ_{- 1}], 0.4 〉, 〈 [γ_{0}, γ_{1}], 0.4 〉, \\ 〈 [γ_{1}, γ_{2}], 0.2 〉} \end{matrix}$

Step 4. Compute the attributes priority weight from Equations (8)–(10): ω₁ = 0.4079, ω₂ = 0.2464, ω₃ = 0.1228, ω₄ = 0.2229.

Step 5. Obtain the positive and negative ideal alternatives (Table 13) by Equations (11)-(12):

Table 13

The ideal alternatives

C₁	C₂
A⁺	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.4 〉, 〈 [γ_{0}, γ_{1}], 0.5 〉, \\ 〈 [γ_{1}, γ_{2}], 0.4 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0 . 4 〉, 〈 [γ_{- 1}, γ_{0}], 0 . 6 〉, \\ 〈 [γ_{1}, γ_{2}], 0.4 〉} \end{matrix}$
A^–	$\begin{matrix} {〈 [γ_{- 3}, γ_{- 2}], 0.1 〉, 〈 [γ_{- 1}, γ_{0}], 0.2 〉, \\ 〈 [γ_{- 1}, γ_{0}], 0.4 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 2}, γ_{- 1}], 0 〉, 〈 [γ_{- 2}, γ_{- 1}], 0.4 〉, \\ 〈 [γ_{0}, γ_{2}], 0.2 〉} \end{matrix}$
C₃	C₄
A⁺	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.4 〉, 〈 [γ_{0}, γ_{1}], 0.7 〉, \\ 〈 [γ_{1}, γ_{2}], 0.6 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0.4 〉, 〈 [γ_{0}, γ_{1}], 0.5 〉, \\ 〈 [γ_{0}, γ_{2}], 0.6 〉} \end{matrix}$
A^-	$\begin{matrix} {〈 [γ_{0}, γ_{1}], 0 〉, 〈 [γ_{- 1}, γ_{0}], 0.2 〉, \\ 〈 [γ_{1}, γ_{2}], 0.2 〉} \end{matrix}$	$\begin{matrix} {〈 [γ_{- 1}, γ_{0}], 0 〉, 〈 [γ_{- 1}, γ_{0}], 0.4 〉, \\ 〈 [γ_{0}, γ_{1}], 0.2 〉} \end{matrix}$

Step 6. Compute the formative vector A^-A⁺, A^-A_i and A_iA⁺ (i = 1, 2, ⋯ , 5) (Table 14) from Equations (13)–(15):

Table 14

The formative vectors A^-A⁺ and A^-A_i

	C₁	C₂
A^–A⁺	[0.133,0.183], [0.183,0.233], [0.133,0.133]	[0.133,0.200], [0.133,0.167], [0.167,0.167]
A^–A₁	[0.133,0.183], [0.033,0.033], [0.133,0.133]	[0.033,0.067], [0.133,0.167], [0.000,0.000]
A^–A₂	[0.067,0.117], [0.000,0.000], [0.067,0.067]	[0.000,0.000], [0.033,0.167], [0.100,0.100]
A^–A₃	[0.033,0.033], [0.183,0.233], [0.133,0.133]	[0.133,0.200], [0.133,0.133], [0.033,0.033]
A^–A₄	[0.033,0.083], [0.133,0.167], [0.133,0.133]	[0.067,0.100], [0.133,0.133], [0.167,0.167]
A^–A₅	[0.000,0.000], [0.017,0.067], [0.000,0.000]	[0.000,0.000], [0.000,0.000], [0.100,0.133]
C₃	C₄
A^–A⁺	[0.133,0.200], [0.283,0.367], [0.267,0.333]	[0.133,0.200], [0.117,0.133], [0.200,0.367]
A^–A₁	[0.067,0.133], [0.033,0.033], [0.133,0.167]	[0.000,0.000], [0.000,0. 000], [0.200,0.367]
A^-A₂	[0.000,0.033], [0.000,0.000], [0.167,0.233]	[0.133,0.200], [0.067,0.067], [0.033,0.067]
A^-A₃	[0.033,0.050], [0.283,0.367], [0.000,0.000]	[0.000,0.133], [0.000,0.000], [0.000,0.000]
A^-A₄	[0.000,0.000], [0.133,0.167], [0.267,0.333]	[0.033,0.050], [0.117,0.133], [0.167,0.200]
A^-A₅	[0.133,0.200], [0.133,0.233], [0.033,0.033]	[0.067,0.133], [0.067,0.067], [0.033,0.033]

Step 7. Compute the projection values of A^-A_i on A^-A⁺ and A^-A⁺ on A_iA⁺ from Equations (16)–(23) (i = 1, 2, ⋯ , 5): $\begin{matrix} {prj}_{A^{-} A^{+}} (A^{-} A_{1}) = 0.1671, {prj}_{A^{-}} A^{+} (A^{-} A_{2}) = 0.1098, {prj}_{A^{-}} A^{+} (A^{-} A_{3}) = 0.1611 \end{matrix}$ $\begin{array}{l} p r j_{A^{-} A^{+}} (A^{-} A_{4}) = 0.2185, \\ p r j_{A^{-} A^{+}} (A^{-} A_{5}) = 0.0726 \end{array}$ $\begin{matrix} {prj}_{A_{1} A^{+}} (A^{-} A^{+}) = 0.2454, \\ {prj}_{A_{2} A^{+}} (A^{-} A^{+}) = 0.2973, \\ {prj}_{A_{3} A^{+}} (A^{-} A^{+}) = 0.2321 \end{matrix}$ $\begin{matrix} {prj}_{A_{4} A^{+}} (A^{-} A^{+}) = 0.2599, \\ {prj}_{A_{5} A^{+}} (A^{-} A^{+}) = 0.3109 \end{matrix}$

Step 8. Compute the closeness degree of A_i to the ideal alternatives from Equation (24): $\begin{matrix} C (A_{1}) = 0.4051, C (A_{2}) = 0.2697, C (A_{3}) = 0.4098, \\ C (A_{4}) = 0.4568, C (A_{5}) = 0.1893 \end{matrix}$

Step 9. According to the C (A_i) (i = 1, 2, ⋯ , 5), all the financial products is ranked as A₄ ≻ A₃ ≻ A₁ ≻ A₂ ≻ A₅, and the optimal financial product among five alternatives is A₄.

4.2 Comparation with some existing methods

The existing methods PUL-TOPSIS method [24], PULWA operator [24], PUL-EDAS method [53] and ULWA operator [51] are provided to compare with the PUL-BP method in this section.

4.2.1 PUL-TOPSIS method

PUL-TOPSIS method [24] is used to compare with PUL-BP method and the results of computing and sorting are listed in Table 16. Thus, the optimal financial product is A₄. It can be seen that the PUL-TOPSIS method and the PUL-BP method obtain the same ranking of the alternatives and the same optimal alternative.

Table 15
The formative vector A_iA⁺

C₁ C₂

A₁A⁺ [0.000,0.000], [0.150,0.200], [0.000,0.000] [0.100,0.133], [0.000,0.000], [0.167,0.167]

A₂A⁺ [0.067,0.067], [0.183,0.233], [0.067,0.067] [0.133,0.200], [0.000,0.100], [0.067,0.067]

A₃A⁺ [0.100,0.150], [0.000,0.000], [0.000,0.000] [0.000,0.000], [0.000,0.033], [0.133,0.133]

A₄A⁺ [0.100,0.100], [0.050,0.067], [0.000,0.000] [0.067,0.100], [0.000,0.033], [0.000,0.000]

A₅A⁺ [0.133,0.183], [0.167,0.167], [0.133,0.133] [0.133,0.200], [0.133,0.167], [0.033,0.067]

C₃ C₄

A₁A⁺ [0.067,0.067], [0.250,0.333], [0.133,0.167] [0.133,0.200], [0.117,0.133], [0.000,0.000]

A₂A⁺ [0.133,0.167], [0.283,0.367], [0.100,0.100] [0.000,0.000], [0.050,0.067], [0.167,0.300]

A₃A⁺ [0.100,0.150], [0.000,0.000], [0.267,0.333] [0.067,0.133], [0.117,0.133], [0.200,0.367]

A₄A⁺ [0.133,0.200], [0.150,0.200], [0.000,0.000] [0.100,0.150], [0.000,0.000], [0.033,0.167]

A₅A⁺ [0.000,0.000], [0.133,0.150], [0.233,0.300] [0.067,0.067], [0.050,0.067], [0.167,0.333]

	C₁	C₂
A₁A⁺	[0.000,0.000], [0.150,0.200], [0.000,0.000]	[0.100,0.133], [0.000,0.000], [0.167,0.167]
A₂A⁺	[0.067,0.067], [0.183,0.233], [0.067,0.067]	[0.133,0.200], [0.000,0.100], [0.067,0.067]
A₃A⁺	[0.100,0.150], [0.000,0.000], [0.000,0.000]	[0.000,0.000], [0.000,0.033], [0.133,0.133]
A₄A⁺	[0.100,0.100], [0.050,0.067], [0.000,0.000]	[0.067,0.100], [0.000,0.033], [0.000,0.000]
A₅A⁺	[0.133,0.183], [0.167,0.167], [0.133,0.133]	[0.133,0.200], [0.133,0.167], [0.033,0.067]
C₃	C₄
A₁A⁺	[0.067,0.067], [0.250,0.333], [0.133,0.167]	[0.133,0.200], [0.117,0.133], [0.000,0.000]
A₂A⁺	[0.133,0.167], [0.283,0.367], [0.100,0.100]	[0.000,0.000], [0.050,0.067], [0.167,0.300]
A₃A⁺	[0.100,0.150], [0.000,0.000], [0.267,0.333]	[0.067,0.133], [0.117,0.133], [0.200,0.367]
A₄A⁺	[0.133,0.200], [0.150,0.200], [0.000,0.000]	[0.100,0.150], [0.000,0.000], [0.033,0.167]
A₅A⁺	[0.000,0.000], [0.133,0.150], [0.233,0.300]	[0.067,0.067], [0.050,0.067], [0.167,0.333]

Table 16

PUL-TOPSIS method calculating and sorting results

TOPSIS method	Results
The distances from PULPIS	$d_{1}^{+} = 0.3004, d_{2}^{+} = 0.4645 {, d}_{3}^{+} = 0.2567, d_{4}^{+} = 0.1303 {, d}_{5}^{+} = 0.6257$
The distances from PULNIS	$d_{1}^{-} = 0.5737, d_{2}^{-} = 0.3963 {, d}_{3}^{-} = 0.5632, d_{4}^{-} = 0.7125 {, d}_{5}^{-} = 0.2730$
Closeness coefficients	$\begin{matrix} {CI}_{1} = - 1.4994, {CI}_{2} = - 3.0073, {CI}_{3} = - 1.1787, \\ {CI}_{4} = 0.0000, {CI}_{5} = - 4.4174 \end{matrix}$
Ordering	A₄ ≻ A₃ ≻ A₁ ≻ A₂ ≻ A₅

4.2.2 PULWA operator

Then, PULWA operator [24] is used to compare with the PUL-BP method, the weight of attributes is obtained as: ω₁ = 0.4079, ω₂ = 0.2464, ω₃ = 0.1228, ω₄ = 0.2229, then the Z_i (ω) (i = 1, 2, ⋯ , 5) is obtained by using PULWA operator. $\begin{matrix} Z_{1} (ω) = {[γ_{- 0.3600}, γ_{- 0.0984}], [γ_{- 0.2370}, γ_{0.1061}], \\ [γ_{0.2123}, γ_{0.7906}]} \end{matrix}$ $\begin{matrix} Z_{2} (ω) = {[γ_{- 0.4891}, γ_{- 0.2123}], [γ_{- 0.4019}, γ_{0.0891}], \\ [γ_{0.0446}, γ_{0.4691}]} Z_{3} (ω) = {[γ_{- 0.4191}, γ_{- 0.0891}], \\ [γ_{- 0.0891}, γ_{0.3885}], [γ_{0.2370}, γ_{0.5679}]} \end{matrix}$ $\begin{matrix} Z_{4} (ω) = {[γ_{- 0.2347}, γ_{0.0000}], [γ_{0.0000}, γ_{0.4223}], \\ [γ_{0.4246}, γ_{0.8491}]} \end{matrix}$ $\begin{matrix} Z_{5} (ω) = {[γ_{- 0.3498}, γ_{- 0.1707}], [γ_{- 0.6051}, γ_{- 0.1152}], \\ [γ_{- 0.2173}, γ_{0.1628}]} \end{matrix}$

Next, the score of these alternatives Z_i (ω) (i = 1, 2, ⋯ , 5) are derived by Definition 9 [24]: $\begin{matrix} E (Z_{1} (ω)) = γ_{0.0689}, E (Z_{2} (ω)) = γ_{- 0.0834}, \\ E (Z_{3} (ω)) = γ_{0.0993} E (Z_{4} (ω)) = γ_{0.2435}, \\ E (Z_{5} (ω)) = γ_{- 0.2159} \end{matrix}$

Finally, the order could be derived: A₄ ≻ A₃ ≻ A₁ ≻ A₂ ≻ A₅. Thus, the optimal financial product is A₄. It can be seen that the PULWA operator and the PUL-BP method obtain the decision results.

4.2.3 PUL-EDAS method

PUL-EDAS method [53] is used to compare with the PUL-BP method and the results of computing and sorting could be obtained (Table 17). Thus, the optimal financial product is A₄. The same ranking of the alternatives and the same optimal alternative are obtained by these two methods.

Table 17
PUL-EDAS method calculating and sorting results

Alternatives A₁ A₂ A₃ A₄ A₅

PULNSP 0.3880 0.0715 0.7094 1.0000 0.0256

PULNSN 0.8204 0.4892 0.7047 1.0000 0.0000

PULAS 0.6042 0.2804 0.7071 1.0000 0.0128

Ordering A₄ ≻ A₃ ≻ A₁ ≻ A₂ ≻ A₅

Alternatives	A₁	A₂	A₃	A₄	A₅
PULNSP	0.3880	0.0715	0.7094	1.0000	0.0256
PULNSN	0.8204	0.4892	0.7047	1.0000	0.0000
PULAS	0.6042	0.2804	0.7071	1.0000	0.0128
Ordering	A₄ ≻ A₃ ≻ A₁ ≻ A₂ ≻ A₅

4.2.4 ULWA operator

In this section, the ULWA operator [51] with same weight information is used to aggregate these ULTSs into a group matrix with ULTSs (See Table 18).

Table 18
Uncertain linguistic group decision matrix

Alternatives C₁ C₂ C₃ C₄

A₁ [γ₀, γ₁] [γ_-1, γ_0.2] [γ_-0.4, γ_0.6] [γ_-0.4, γ_1.2]

A₂ [γ_-1, γ₀] [γ_-1.2, γ_0.4] [γ_-0.8, γ_0.2] [γ_-0.2, γ₁]

A₃ [γ_0.2, γ_1.4] [γ_-0.2, γ₁] [γ₀, γ_1.2] [γ_-1.6, γ_-0.2]

A₄ [γ₀, γ_1.2] [γ_0.2, γ_1.2] [γ_0.6, γ_1.6] [γ_0.2, γ_1.4]

A₅ [γ_-1.8, γ_-0.6] [γ_-1.4, γ_-0.4] [γ₀, γ_1.4] [γ_-0.6, γ_0.4]

Alternatives	C₁	C₂	C₃	C₄
A₁	[γ₀, γ₁]	[γ_-1, γ_0.2]	[γ_-0.4, γ_0.6]	[γ_-0.4, γ_1.2]
A₂	[γ_-1, γ₀]	[γ_-1.2, γ_0.4]	[γ_-0.8, γ_0.2]	[γ_-0.2, γ₁]
A₃	[γ_0.2, γ_1.4]	[γ_-0.2, γ₁]	[γ₀, γ_1.2]	[γ_-1.6, γ_-0.2]
A₄	[γ₀, γ_1.2]	[γ_0.2, γ_1.2]	[γ_0.6, γ_1.6]	[γ_0.2, γ_1.4]
A₅	[γ_-1.8, γ_-0.6]	[γ_-1.4, γ_-0.4]	[γ₀, γ_1.4]	[γ_-0.6, γ_0.4]

The attributes weight is obtained as: ω₁ = 0.4079, ω₂ = 0.2464, ω₃ = 0.1228, ω₄ = 0.2229, then the Z_i (ω) (i = 1, 2, ⋯ , 5) is derived by using ULWA operator [51].

$\begin{matrix} Z_{1} (ω) = [γ_{- 0.3847}, γ_{0.7983}], Z_{2} (ω) = [γ_{- 0.8464}, γ_{0.3460}], \\ Z_{3} (ω) = [γ_{- 0.3243}, γ_{0.9203}] Z_{4} (ω) = [γ_{0.1675}, γ_{1.2937}], \\ Z_{5} (ω) = [γ_{- 1.2130}, γ_{- 0.0823}] \end{matrix}$

Next, the score of Z_i (ω) (i = 1, 2, ⋯ , 5) are derived by Definition 9 [24]: $\begin{matrix} E (Z_{1} (ω)) = γ_{0.2068}, E (Z_{2} (ω)) = γ_{- 0.2502}, \\ E (Z_{3} (ω)) = γ_{0.2980} \end{matrix}$ $E (Z_{4} (ω)) = γ_{0.7306}, E (Z_{5} (ω)) = γ_{- 0.6476}$

Finally, the order could be obtained: A₄ ≻ A₃ ≻ A₁ ≻ A₂ ≻ A₅. Thus, the optimal financial product is A₄. It can be seen that the decision results derived from the ULWA operator and the PUL-BP method are also the same.

5 Conclusion

In this paper, the BP method is improved to probability uncertain linguistic information for dealing with MAGDM problem with completely unknown weight. First of all, the necessary theories and PULTSs’ Hamming distance are simply reviewed. Secondly, the vector of information entropy based on the PULTS is computed to obtain the attribute priority weights. Then, the formative vector of two alternatives is defined and the weighted module of the formative vector and the weighted inner product of two formative vectors are improved under the PULTS. Next, the closeness degree of each alternative to ideal alternatives is derived by computing the bidirectional projection values. Finally, we give a practical numerical example about financial products selection in financial service sectors to elaborate the proposed method and use some existing methods for comparison to proof its applicability in realistic MAGDM problems. The results of the research show that the developed approach is effective and practical for solving uncertain and fuzzy decision-making problems. The PUL-BP method applies the BP method to probability uncertain linguistic environment. It not only considers the distance and angle of the alternatives, but also considers bidirectional projection values. So, it is more comprehensive and reasonable. Future research can consider the use of aggregation operators [61 –63] to model the relationship between attributes to improve effectiveness. Similarly, future research can also extend the models and methods designed in this study and apply them to other uncertain and vague decision making [64 –67] and other uncertain and vague environment [68 –70].

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D.F.

and Fei

, A novel method for heterogeneous multi-attribute group decision making with preference deviation, & Industrial Engineering 124 (2018), 58–64.

Bidirectional projection method for multi-attribute group decision making under probabilistic uncertain linguistic environment

Abstract

Keywords

1 Introduction

2 Preliminaries

4.1 The numerical case

Table 1 Decision information of the first decision maker Alternatives C1 C2 C3 C4 A1 [P, O] [VP, O] [VP, P] [P, O] A2 [VP, P] [P, O] [P, O] [O, G] A3 [O, G] [EP, P] [O, G] [EP, P] A4 [VP, O] [VP, P] [O, G] [O, G] A5 [EP, P] [O, G] [VG, EG] [VP, P]

4.2.1 PUL-TOPSIS method

4.2.3 PUL-EDAS method

Table 17 PUL-EDAS method calculating and sorting results Alternatives A1 A2 A3 A4 A5 PULNSP 0.3880 0.0715 0.7094 1.0000 0.0256 PULNSN 0.8204 0.4892 0.7047 1.0000 0.0000 PULAS 0.6042 0.2804 0.7071 1.0000 0.0128 Ordering A4 ≻ A3 ≻ A1 ≻ A2 ≻ A5

References

Table 1
Decision information of the first decision maker

Alternatives C₁ C₂ C₃ C₄

A₁ [P, O] [VP, O] [VP, P] [P, O]

A₂ [VP, P] [P, O] [P, O] [O, G]

A₃ [O, G] [EP, P] [O, G] [EP, P]

A₄ [VP, O] [VP, P] [O, G] [O, G]

A₅ [EP, P] [O, G] [VG, EG] [VP, P]

Table 17
PUL-EDAS method calculating and sorting results

Alternatives A₁ A₂ A₃ A₄ A₅

PULNSP 0.3880 0.0715 0.7094 1.0000 0.0256

PULNSN 0.8204 0.4892 0.7047 1.0000 0.0000

PULAS 0.6042 0.2804 0.7071 1.0000 0.0128

Ordering A₄ ≻ A₃ ≻ A₁ ≻ A₂ ≻ A₅