Sustainable competitiveness evaluation of regional financial centers with fuzzy number intuitionistic fuzzy TODIM algorithms

Abstract

The competitiveness evaluation of regional financial centers is frequently looked as the multiple attribute group decision-making (MAGDM) problem. Based on the TODIM method and fuzzy number intuitionistic fuzzy sets (FNIFS), this paper proposes a new FNIF-TODIM method to evaluate the competitiveness of regional financial centers. First, some basic theories related to FNIFS are briefly introduced. In addition, the weights of the attributes are obtained objectively using the CRITIC weighting method. Then, the traditional TODIM method is extended to FNIFS to obtain the final order of alternatives. As a result, all alternatives can be ranked and the best one for the competitiveness assessment of regional financial centers can be identified. Finally, an example for competitiveness evaluation of regional financial centers and some decision comparative analysis is listed. The results show that the established algorithmic approach is useful. The main works of this work are: (1) the paper constructs the FNIF-TODIM method for the evaluation of the competitiveness of regional financial centers; (2) the established method is illustrated by a case study for competitiveness evaluation of regional financial centers; and (3) some comparisons prove the rationality and advantages.

Keywords

Multiple attribute group decision making (MAGDM)fuzzy number intuitionistic fuzzy sets (FNIFSs)TODIM method CRITIC method Competitiveness evaluation

1 Introduction

With the increasing complexity and ambiguity of MADM problems and the vagueness of subjective human perception, it is increasingly difficult for DMs to provide accurate judgments [1 –4]. Thereby deriving the optimal choice of a qualitative MADM that can be used to easily describe qualitative decision information [5, 6], Zadeh [7] produced the fuzzy sets (FSs) to deal with uncertain fuzzy domain [8 –10]. Then, the intuitionistic fuzzy sets (IFSs) [11, 12] was produced. Atanassov [13] produced the interval-valued IFSs (IVIFSs). Then, IFSs and IVIFSs have been widely used in different environments [14 –21]. Liu and Yuan [22] produced the fuzzy number IFSs (FNIFSs) based on TFSs [23]. Wang [24] produced the operational laws for FNIFSs. Li et al. [25] produced the entropy and similarity for FNIFSs.

With the increasing complexity of the MADM problems [26 –30], it is of great necessity to take the decision-makers (DMs)’ psychological factors into consideration [31 –33]. Tversky and Kahneman [34] produced the prospect theory (PT) for MADM problems under risk. Gomes and Lima [35] produced the TODIM to tackle the MADM issues where the DMs’ psychological behaviors may be covered under risk. Compared with other conventional method such as TOPSIS [36], VIKOR [37], SWARA [38], etc., the TODIM method is based on a two-by-two comparison of scenarios, where the dominance of each scenario over another is calculated under each attribute and thus ranked [39 –44].

As the role of finance in the development of modern economy is strengthened day by day, the theoretical research on regional financial competitiveness in the study of regional economic competitiveness is very important [45 –49]. For China at this stage, finance is in a period of rapid development, and the role of finance has penetrated into all aspects of social and economic life. Especially after China’s entry into WTO, the pace of opening up of the financial market has been further accelerated, comprehensively evaluating and analyzing financial competitiveness, fully understanding and accurately grasping China’s national conditions, national strength and international competitiveness, and promoting the long-term growth and development of China’s financial competitiveness [50 –53]. The sustainable development of the financial industry is of great significance. The competitiveness evaluation of regional financial centers is looked as the MAGDM [54 –58]. The main works and motivations of this paper are summarized as follows: (1) attribute weights were obtained by CRITIC; (2) the paper constructs the FNIF-TODIM method for the evaluation of the competitiveness of regional financial centers; (3) the established method is illustrated by a case study of the evaluation of the efficiency of the use of physical laboratory resources in universities; and (4) a comparative study of existing methods is conducted. In order to do so, this reminder of the paper continues. Section 2 presents the definition of FNIFS. The FNIFS is used to extend the MAGDM method of constructing TODIM in Section 3. An empirical application of regional financial center competitiveness assessment is given and some comparative analysis is performed to show the merits of the approach defined in Section 4. Finally, the conclusions is given in Section 5.

2 Preliminaries

Atanassov [11] defined the intuitionistic fuzzy sets (IFSs). Atanassov [13] further extended the IFSs and defined the IVIFSs. Liu and Yuan [22] produced the FNIFSs based on triangular fuzzy number (TFNs) [23].

Definition 1 [22]. The B is a FNIFS on Q is produced:

$B = {〈 q, {TT}_{B} (q), {FF}_{B} (q) com>rangle \in Q}$ (1)TT_B (q),FF_B (q) are TFNs, d TT_B (q) = (XX (q) , YY (q) , ZZ (q)), FF_B (q) = (AA (q) , SS (q) , DD (q)), 0 ⩽ ZZ (q) + DD (q) ⩽ 1,∀ q ∈ Q.

Let TT_B (q) = (XX (q) , YY (q) , ZZ (q)), FF_B (q) = (AA (q) , SS (q) , DD (q)), so VV (q) =〈 (XX (q) , YY (q) , ZZ (q)) , (AA (q) , SS (q) , DD (q)) 〉, VV (q) is FNIFN.

Wang [24] defined the basic operational laws for FNIFNs.

Definition 2 [24]. VV (q_i) =〈 (XX (q_i) , YY (q_i) , ZZ (q_i)) , (AA (q_i) , SS (q_i) , DD (q_i)) 〉 and VV (q_j) =〈 (XX (q_j) , YY (q_j) , ZZ (q_j)) , (AA (q_j) , SS (q_j) , DD (q_j)) 〉 are the FNIFNs, the basic operational laws for FNIFNs are defined: $\begin{matrix} VV (q_{i}) \oplus VV (q_{j}) = {(\begin{matrix} XX (q_{i}) + XX (q_{j}) - XX (q_{i}) XX (q_{j}), \\ YY (q_{i}) + YY (q_{j}) - YY (q_{i}) YY (q_{j}), \\ ZZ (q_{i}) + ZZ (q_{j}) - ZZ (q_{i}) ZZ (q_{j}) \end{matrix}), (\begin{matrix} AA (q_{i}) AA (q_{j}), \\ SS (q_{i}) SS (q_{j}), \\ DD (q_{i}) DD (q_{j}) \end{matrix})}; \end{matrix}$ $\begin{matrix} VV (q_{i}) \otimes VV (q_{j}) = {((\begin{matrix} XX (q_{i}) XX (q_{j}), \\ YY (q_{i}) YY (q_{j}), \\ ZZ (q_{i}) ZZ (q_{j}) \end{matrix})), ((\begin{matrix} AA (q_{i}) + AA (q_{j}) - AA (q_{i}) AA (q_{j}), \\ SS (q_{i}) + SS (q_{j}) - SS (q_{i}) SS (q_{j}), \\ DD (q_{i}) + DD (q_{j}) - DD (q_{i}) DD (q_{j}) \end{matrix}))}; \end{matrix}$ $\begin{matrix} λ VV (q_{i}) \\ = {(\begin{matrix} 1 - {(1 - XX (q_{i}))}^{λ}, \\ 1 - {(1 - YY (q_{i}))}^{λ}, \\ 1 - {(1 - ZZ (q_{i}))}^{λ} \end{matrix}), (\begin{matrix} {(AA (q_{i}))}^{λ}, \\ {(SS (q_{i}))}^{λ}, \\ {(DD (q_{i}))}^{λ} \end{matrix})}, \\ λ ⩾ 0; \end{matrix}$ $\begin{matrix} {(VV (q_{i}))}^{λ} \\ = {(\begin{matrix} {(XX (q_{i}))}^{λ}, \\ {(YY (q_{i}))}^{λ}, \\ {(ZZ (q_{i}))}^{λ} \end{matrix}), (\begin{matrix} 1 - {(1 - AA (q_{i}))}^{λ}, \\ 1 - {(1 - SS (q_{i}))}^{λ}, \\ 1 - {(1 - DD (q_{i}))}^{λ} \end{matrix})}, \\ λ ⩾ 0 . \end{matrix}$

Wang [24] defined score function and accuracy function for FNIFNs.

Definition 3 [24]. Let VV (q) =〈 (XX (q) , YY (q) , ZZ (q)) , (AA (q) , SS (q) , DD (q)) 〉, the score function is depicted:

$\begin{matrix} SF (VV (q)) = & \frac{XX (q) + 2 YY (q) + ZZ (q)}{4} \\ - \frac{AA (q) + 2 SS (q) + DD (q)}{4}, \\ SF (VV (q)) \in [- 1, 1] . \end{matrix}$ (2)

Definition 4 [24]. Let VV (q) =〈 (XX (q) , YY (q) , ZZ (q)) , (AA (q) , SS (q) , DD (q)) 〉, an accuracy function is defined:

$\begin{matrix} AH (VV (q)) = & \frac{XX (q) + 2 YY (q) + ZZ (q)}{4} \\ + \frac{AA (q) + 2 SS (q) + DD (q)}{4}, \\ AH (VV (q)) \in [- 1, 1] \end{matrix}$ (3)

Wang [24] defined the comparison for FNIFNs:

Definition 5 [24]. Let VV (q₁) and VV (q₂), then if SF (VV (q₁)) < SF (VV (q₂)), VV (q₁) < VV (q₂); if SF (VV (q₁)) = SF (VV (q₂)), then

(1) if AH (VV (q₁)) = AH (VV (q₂)), VV (q₁) = VV (q₂);

(2) if AH (VV (q₁)) < AH (VV (q₂)), VV (q₁) < VV (q₂).

Jiang and Liu [59] defined the Euclidean distance between two given FNIFNs.

Definition 6 [59]. Suppose VV (q_i) =〈 (XX (q_i) , YY (q_i) , ZZ (q_i)) , (AA (q_i) , SS (q_i) , DD (q_i)) 〉 and VV (q_j) =〈 (XX (q_j) , YY (q_j) , ZZ (q_j)) , (AA (q_j) , SS (q_j) , DD (q_j)) 〉, the Euclidean distance between two given FNIFNs is:

$\begin{matrix} FNIFNED (VV (q_{i}), VV (q_{j})) \\ = \sqrt{\frac{1}{6} [\begin{matrix} {(XX (q_{i}) - XX (q_{j}))}^{2} + {(YY (q_{i}) - YY (q_{j}))}^{2} + {(ZZ (q_{i}) - ZZ (q_{j}))}^{2} + \\ {(AA (q_{i}) - AA (q_{j}))}^{2} + {(SS (q_{i}) - SS (q_{j}))}^{2} + {(DD (q_{i}) - DD (q_{j}))}^{2} \end{matrix}]} \end{matrix}$ (4)

3 TODIM method for MAGDM with FNIFSs

Then, the FNIFN-TODIM for MAGDM is produced with FNIFSs. The calculating steps of produced model is listed. Let I ={ I₁, I₂, … I_n } be the decision attributes, ow = { ow₁, ow₂, …, ow_n } be weight I_j, where ${ow}_{j} \in [0, 1], \sum_{j = 1}^{n} {ow}_{j} = 1$ . Assume K = { K₁, K₂, … K_l } be DMs that have weight κ = { κ₁, κ₂, … κ_l }, where κ_o ∈ [0, 1] , $\sum_{o=1}^{l} κ^{o=1}$ . Let ℵ = { ℵ ₁, ℵ ₂, … ℵ _m } be alternatives. Subsequently, the given calculating steps are produced.

Step 1. Produce the DM’s IVIFN matrix $Ω^{o} = {(Ω_{ij}^{o})}_{m \times n}$ and overall IVIFN-matrix $O Ω = {(O Ω_{ij})}_{m \times n}$ is obtained.

$Ω^{o} = {(Ω_{ij}^{o})}_{m \times n} = [\begin{matrix} Ω_{11}^{o} & Ω_{12}^{o} & \dots & Ω_{1 n}^{o} \\ Ω_{21}^{o} & Ω_{22}^{o} & \dots & Ω_{2 n}^{o} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ Ω_{m 1}^{o} & Ω_{m 2}^{o} & \dots & Ω_{mn}^{o} \end{matrix}]$ (5)

$O Ω = {[O Ω_{ij}]}_{m \times n} = [\begin{matrix} O Ω_{11} & O Ω_{12} & \dots & O Ω_{1 n} \\ O Ω_{21} & O Ω_{22} & \dots & O Ω_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ O Ω_{m 1} & O Ω_{m 2} & \dots & O Ω_{mn} \end{matrix}]$ (6)

$O Ω_{i j} = ([1 - \prod_{o = 1}^{l} {(1 - X X_{Ω_{i j}^{o}})}^{κ_{l}}, 1 - \prod_{o = 1}^{l} {(1 - Y Y_{Ω_{i j}^{o}})}^{κ_{l}}, 1 - \prod_{o = 1}^{l} {(1 - Z Z_{Ω_{i j}^{o}})}^{κ_{l}}], [\prod_{o = 1}^{l} {(A A_{Ω_{i j}^{o}})}^{κ_{l}}, \prod_{o = 1}^{l} {(S S_{Ω_{i j}^{o}})}^{κ_{l}}, \prod_{o = 1}^{l} {(D D_{Ω_{i j}^{o}})}^{κ_{l}}])$ (7)

Step 2. Normalize the $O Ω = {(O Ω_{ij})}_{m \times n}$ to $N Ω^{N} = {[N Ω_{ij}]}_{m \times n}$ .

$N Ω_{ij} = {\begin{matrix} (\begin{matrix} [1 - \prod_{o = 1}^{l} {(1 - {XX}_{Ω_{ij}^{o}})}^{κ_{l}}, 1 - \prod_{o = 1}^{l} {(1 - {YY}_{Ω_{ij}^{o}})}^{κ_{l}}, 1 - \prod_{o = 1}^{l} {(1 - {ZZ}_{Ω_{ij}^{o}})}^{κ_{l}}] \\ , [\prod_{o = 1}^{l} {({AA}_{Ω_{ij}^{o}})}^{κ_{l}}, \prod_{o = 1}^{l} {({SS}_{Ω_{ij}^{o}})}^{κ_{l}}, \prod_{o = 1}^{l} {({DD}_{Ω_{ij}^{o}})}^{κ_{l}}] \end{matrix}) \\ I_{j} is a benefit criterion; \\ (\begin{matrix} [\prod_{o = 1}^{l} {({AA}_{Ω_{ij}^{o}})}^{κ_{l}}, \prod_{o = 1}^{l} {({SS}_{Ω_{ij}^{o}})}^{κ_{l}}, \prod_{o = 1}^{l} {({DD}_{Ω_{ij}^{o}})}^{κ_{l}}], \\ [1 - \prod_{o = 1}^{l} {(1 - {XX}_{Ω_{ij}^{o}})}^{κ_{l}}, 1 - \prod_{o = 1}^{l} {(1 - {YY}_{Ω_{ij}^{o}})}^{κ_{l}}, 1 - \prod_{o = 1}^{l} {(1 - {ZZ}_{Ω_{ij}^{o}})}^{κ_{l}}] \end{matrix}) \\ I_{j} is a cost criterion \end{matrix}$ (8)

Step 3. Produce the decision weight.

The CRITIC method is proposed by Diakoulaki, Mavrotas and Papayannakis [60]. Peng and Garg [61] defined the intuitionistic fuzzy soft CoCoSo and CRITIC method. The CRITIC has been used in different setting and connected with methods[62 –69]. The compute procedures of the CRITIC method are designed. The objective weights’ methods are produced [59].

(1) Depending on matrix $N Ω = {[N Ω_{ij}]}_{m \times n}$ , the correlation coefficient is:

$\begin{matrix} {FNIFNCC}_{jr} = \\ \frac{\sum_{i = 1}^{m} (SF (N Ω_{ij}) - SF (N Ω_{j})) (SF (N Ω_{ir}) - SF (N Ω_{r}))}{\sqrt{\sum_{i = 1}^{m} {(SF (N Ω_{ij}) - SF (N Ω_{j}))}^{2}} \sqrt{\sum_{i = 1}^{m} {(SF (N Ω_{ir}) - SF (N Ω_{r}))}^{2}}}, j, r = 1, 2, \dots, n \end{matrix}$ (9)

where $S F (N Ω_{j}) = \frac{1}{m} \sum_{i = 1}^{m} S F (N Ω_{i j}) a n d S F (N Ω_{r}) = \frac{1}{m} \sum_{i = 1}^{m} S F (N Ω_{i r})$

(2) Produce FNIFN standard deviation.

$\begin{matrix} {FNIFNSD}_{j} \\ = \sqrt{\frac{1}{m - 1} \sum_{i = 1}^{m} {(SF (N Ω_{ij}) - SF (N Ω_{j}))}^{2}} \end{matrix}$ (10) with $SF (N Ω_{j}) = \frac{1}{m} \sum_{i = 1}^{m} SF (N Ω_{ij})$ .

(3) Produce the objective weights.

${ow}_{j} = \frac{{FNIFNSD}_{j} \sum_{j = 1}^{n} (1 - {FNIFNCC}_{jr})}{\sum_{j = 1}^{n} ({FNIFNSD}_{j} \sum_{j = 1}^{n} (1 - {FNIFNCC}_{jr}))}$ (11) where ow_j ∈ [0, 1], $\sum_{j = 1}^{n} {ow}_{j} = 1$ .

Step 4. Define relative weight:

${ow}_{{jr}_{0}} = {ow}_{j} / - {ow}_{r_{0}}, j = 1, 2, \dots, n .$ (12)ow_{r
₀} = max{ ow_j|j = 1, 2, ⋯ , n }, and 0 ⩽ ow_{jr
₀} ⩽ 1.

Step 5. The dominance degree FNIFDD_j (ℵ _i, ℵ _t) of ℵ_i over ℵ_tfor I_jis derived with Equations (13–15):

$if SF (N Ω_{ij}) > SF (N Ω_{tj}),$ (13) then, ${FNIFDD}_{j} (ℵ_{i}, ℵ_{t}) = \sqrt{{ow}_{{jr}_{0}} FNIFNED (N Ω_{ij}, N Ω_{tj}) / \sum_{j = 1}^{n} {ow}_{{jr}_{0}}},$

$if SF (N Ω_{ij}) = SF (N Ω_{tj}),$ (14) then, DD_j (A_i, A_t) = 0,

$if SF (N Ω_{ij}) < SF (N Ω_{tj}),$ (15)

then, $F N I F D D_{j} (ℵ_{i}, ℵ_{t}) = - \frac{1}{θ} \sqrt{(\sum_{j = 1}^{n} o w_{j r_{0}}) F N I F N E D (N Ω_{i j}, N Ω_{t j}) / o w_{j r_{0}}}$

where the given parameterθ depicts the risk factor for the losses, and $FNIFNED (N Ω_{ij}, N Ω_{tj})$ is distances between $N Ω_{ij}$ and $N Ω_{tj}$ . If $SF (N Ω_{ij}) > SF (N Ω_{tj})$ then FNIFDD_j (ℵ _i, ℵ _t) signifies the gain; if $SF (N Ω_{ij}) < SF (N Ω_{tj})$ then FNIFDD_j (ℵ _i, ℵ _t) represents the loss.

Step 6. In order to illustrate FNIFDD_j (ℵ _i, ℵ _t) clearly, a dominance degree matrix FNIFDDM_j (ℵ _i, ℵ _t) for I_j is designed as:

$\begin{matrix} {FNIFDDM}_{j} = {[{FNIFDD}_{j} (ℵ_{i}, ℵ_{t})]}_{m \times m} \\ \begin{matrix} ℵ_{1} & ℵ_{2} & \dots & ℵ_{m} \end{matrix} \\ = \begin{matrix} ℵ_{1} \\ ℵ_{2} \\ ⋮ \\ ℵ_{m} \end{matrix} [\begin{matrix} 0 & {FNIFDD}_{j} (ℵ_{1}, ℵ_{2}) & \dots & {FNIFDD}_{j} (ℵ_{1}, ℵ_{m}) \\ {FNIFDD}_{j} (ℵ_{2}, ℵ_{1}) & 0 & \dots & {FNIFDD}_{j} (ℵ_{2}, ℵ_{m}) \\ ⋮ & ⋮ & \dots & ⋮ \\ {FNIFDD}_{j} (ℵ_{m}, ℵ_{1}) & {FNIFDD}_{j} (ℵ_{m}, ℵ_{2}) & \dots & 0 \end{matrix}] \end{matrix}$ (16)j = 1, 2, ⋯ , n

Step 7. Calculate the FNIF overall dominance degree FNIFODD (ℵ _i, ℵ _t) (i, t = 1, 2, ⋯ , m) of the ℵ_i over each offered alternative ℵ_t through Equation (17):

$\begin{matrix} FNIFODD (ℵ_{i}, ℵ_{t}) = & \sum_{j = 1}^{n} {FNIFDDM}_{j} (ℵ_{i}, ℵ_{t}), \\ (i, t = 1, 2, \dots, m) \end{matrix}$ (17)

Therefore, the FNIF overall dominance degree matrix (FNIFODDM) is obtained as:

$\begin{matrix} FNIFODDM = {[FNIFODD (ℵ_{i}, ℵ_{t})]}_{m \times m} \\ \begin{matrix} ℵ_{1} & ℵ_{2} & \dots & ℵ_{m} \end{matrix} \\ = \begin{matrix} ℵ_{1} \\ ℵ_{2} \\ ⋮ \\ ℵ_{m} \end{matrix} [\begin{matrix} 0 & FNIFODD (ℵ_{1}, ℵ_{2}) & \dots & FNIFODD (ℵ_{1}, ℵ_{m}) \\ FNIFODD (ℵ_{2}, ℵ_{1}) & 0 & \dots & FNIFODD (ℵ_{2}, ℵ_{m}) \\ ⋮ & ⋮ & \dots & ⋮ \\ FNIFODD (ℵ_{m}, ℵ_{1}) & FNIFODD (ℵ_{m}, ℵ_{2}) & \dots & 0 \end{matrix}] \end{matrix}$ (18)

Step 8. The FNIF overall dominance degree value FNIFODDV (ℵ _i) (i = 1, 2, ⋯ , m) of ℵ_i is obtained by Equation (19):

$\begin{matrix} FNIFODDV (ℵ_{i}) \\ = \frac{\sum_{t = 1}^{m} FNIFODD (ℵ_{i}, ℵ_{t}) - min_{i} {\sum_{t = 1}^{m} FNIFODD (ℵ_{i}, ℵ_{t})}}{max_{i} {\sum_{t = 1}^{m} FNIFODD (ℵ_{i}, ℵ_{t})} - min_{i} {\sum_{t = 1}^{m} FNIFODD (ℵ_{i}, ℵ_{t})}}, i = 1, 2, \dots, m . \end{matrix}$ (19)

And the order of these offered alternatives could be sorted with FNIFODDV (ℵ _i), that is, the greater FNIFODDV (ℵ _i), the better the alternative ℵ_iis.

4 Case study and comparative analysis

4.1 Case study

Since the 1990 s, with the establishment of the theme of peace and development, as well as the development of economic and financial globalization, the competition between countries has increasingly reflected the competition in the economic field. In this context, the competition between regions is essentially the competition of economic strength. Therefore, the research on regional economic competitiveness has been paid more and more given attention by people. As the role of finance in the development of modern economy is increasingly strengthened, the theoretical research on regional financial competitiveness in the study of regional economic competitiveness is also very important. A large number of studies have shown that there is a very high positive correlation between the regional financial competitiveness and the comprehensive competitiveness of the regional economy, and as the importance of finance increases, the correlation between them also shows an increasing trend. For China at this stage, finance is in a period of rapid development, and the role of finance has penetrated into all aspects of social and economic life. Especially since joining, the opening of the financial market has been further accelerated, and the financial competitiveness has been evaluated and analyzed in an all-round way. The sustainable development of the industry is of great significance. In modern society, finance has become an important driving force for economic development. It can not only bridge the broken buying and selling behavior and make social reproduction go smoothly, but also is an important driving force for social reproduction. In particular, the financial center, through the allocation of funds, promotes economic growth and at the same time enables people to grasp the beating of the economic pulse at any time through the transmission of financial information. Based on the above-mentioned reality, this paper tries to improve the evaluation index system of the competitiveness of regional financial centers by studying the competitiveness of cities that have been proposed to establish regional financial centers in the central region, introducing evaluation indicators, and using reasonable evaluation methods to establish a comprehensive evaluation function to evaluate the regional financial competitiveness. And analyze the specific situation of Changsha, and put forward policy suggestions to improve the competitiveness of Changsha, hoping to benefit the local economic developments. In this paper, the example about the competitiveness evaluation of regional financial centers under FNIFNs is utilized to illustrate the produced FNIF-TODIM methods. We shall give 5 regional financial centers ℵ_i (i = 1, 2, 3, 4, 5) to choose. The decision experts consider four decision attributes to assess these regional financial centers: ding172I₁ represents the financial contribution rate Loans indicating the ability of the region to actually grasp the financial resources); ding173I₂ means economic development(The development of regional finance is the financial development of a specific geographical space. Different economic development states provide completely different growth conditions and variable spaces for the development of finance, and the level and capability of regional financial development provide the internal support for the economic development of the region. Therefore, the development of regional economy provides an explanation for the development of regional finance in this region); ding174I₃ represents openness(The degree of openness is an important factor affecting the competitiveness of a region in the market, and the degree of openness of a place also has an important impact on the financial development of the region. Introducing capital, technology, labor force and advanced institutional culture and combining them effectively will enable capital to obtain higher utilization efficiency and output efficiency than under closed state, so that capital can play a greater role. This paper selects the actual utilization of foreign capital, total import and export, foreign investment contribution, actual utilization of foreign investment, and total economic opening to the outside world to evaluate the level of openness of a region); ding175I₄ means the city development(Faster urban development can bring potential profit opportunities and good expectations to the region, which will improve the cohesion and attractiveness of the region, form a virtuous circle, and further promote financial development). All attributes are beneficial. The competitiveness evaluation of regional financial centers is expressed with FNIFNs by using attributes (whose weighting vectorξ = (0.15, 0.40, 0.25, 0.20)), the FNIFN decision matrix as depicted in Tables 1– 3.

Table 1
The FNIFN matrix by K₁

I₁ I₂

ℵ₁ 〈 (0.3, 0.3, 0.4) , (0.3, 0.4, 0.5)〉〈 (0.1, 0.2, 0.3) , (0.5, 0.6, 0.6)〉

ℵ₂ 〈 (0.1, 0.4, 0.5) , (0.1, 0.2, 0.5)〉〈 (0.2, 0.3, 0.4) , (0.1, 0.3, 0.6)〉

ℵ₃ 〈 (0.4, 0.5, 0.7) , (0.1, 0.2, 0.3)〉〈 (0.2, 0.3, 0.8) , (0.1, 0.1, 0.2)〉

ℵ₄ 〈 (0.1, 0.3, 0.3) , (0.2, 0.3, 0.3)〉〈 (0.2, 0.3, 0.4) , (0.1, 0.2, 0.3)〉

ℵ₅ 〈 (0.4, 0.7, 0.7) , (0.1, 0.2, 0.3)〉〈 (0.2, 0.3, 0.4) , (0.2, 0.2, 0.5)〉

I₃ I₄

ℵ₁ 〈 (0.2, 0.3, 0.8) , (0.1, 0.1, 0.2)〉〈 (0.2, 0.5, 0.5) , (0.1, 0.4, 0.4)〉

ℵ₂ 〈 (0.1, 0.2, 0.5) , (0.2, 0.4, 0.4)〉〈 (0.2, 0.4, 0.6) , (0.1, 0.3, 0.5)〉

ℵ₃ 〈 (0.2, 0.4, 0.5) , (0.1, 0.1, 0.5)〉〈 (0.4, 0.5, 0.5) , (0.3, 0.3, 0.4)〉

ℵ₄ 〈 (0.1, 0.5, 0.7) , (0.1, 0.2, 0.3)〉〈 (0.6, 0.7, 0.9) , (0.1, 0.1, 0.1)〉

ℵ₅ 〈 (0.1, 0.2, 0.3) , (0.1, 0.2, 0.2)〉〈 (0.2, 0.3, 0.6) , (0.2, 0.3, 0.4)〉

	I₁	I₂
ℵ₁	〈 (0.3, 0.3, 0.4) , (0.3, 0.4, 0.5)〉	〈 (0.1, 0.2, 0.3) , (0.5, 0.6, 0.6)〉
ℵ₂	〈 (0.1, 0.4, 0.5) , (0.1, 0.2, 0.5)〉	〈 (0.2, 0.3, 0.4) , (0.1, 0.3, 0.6)〉
ℵ₃	〈 (0.4, 0.5, 0.7) , (0.1, 0.2, 0.3)〉	〈 (0.2, 0.3, 0.8) , (0.1, 0.1, 0.2)〉
ℵ₄	〈 (0.1, 0.3, 0.3) , (0.2, 0.3, 0.3)〉	〈 (0.2, 0.3, 0.4) , (0.1, 0.2, 0.3)〉
ℵ₅	〈 (0.4, 0.7, 0.7) , (0.1, 0.2, 0.3)〉	〈 (0.2, 0.3, 0.4) , (0.2, 0.2, 0.5)〉
	I₃	I₄
ℵ₁	〈 (0.2, 0.3, 0.8) , (0.1, 0.1, 0.2)〉	〈 (0.2, 0.5, 0.5) , (0.1, 0.4, 0.4)〉
ℵ₂	〈 (0.1, 0.2, 0.5) , (0.2, 0.4, 0.4)〉	〈 (0.2, 0.4, 0.6) , (0.1, 0.3, 0.5)〉
ℵ₃	〈 (0.2, 0.4, 0.5) , (0.1, 0.1, 0.5)〉	〈 (0.4, 0.5, 0.5) , (0.3, 0.3, 0.4)〉
ℵ₄	〈 (0.1, 0.5, 0.7) , (0.1, 0.2, 0.3)〉	〈 (0.6, 0.7, 0.9) , (0.1, 0.1, 0.1)〉
ℵ₅	〈 (0.1, 0.2, 0.3) , (0.1, 0.2, 0.2)〉	〈 (0.2, 0.3, 0.6) , (0.2, 0.3, 0.4)〉

Table 2

The FNIFN matrix by K₂

	I₁	I₂
ℵ₁	〈 (0.2, 0.3, 0.3) , (0.3, 0.4, 0.5)〉	〈 (0.2, 0.3, 0.3) , (0.3, 0.4, 0.5)〉
ℵ₂	〈 (0.2, 0.2, 0.5) , (0.2, 0.3, 0.4)〉	〈 (0.2, 0.5, 0.5) , (0.3, 0.4, 0.5)〉
ℵ₃	〈 (0.1, 0.5, 0.7) , (0.1, 0.1, 0.2)〉	〈 (0.2, 0.3, 0.6) , (0.1, 0.2, 0.2)〉
ℵ₄	〈 (0.1, 0.2, 0.3) , (0.2, 0.3, 0.4)〉	〈 (0.2, 0.3, 0.4) , (0.1, 0.3, 0.5)〉
ℵ₅	〈 (0.2, 0.3, 0.4) , (0.2, 0.2, 0.3)〉	〈 (0.2, 0.3, 0.5) , (0.2, 0.3, 0.4)〉
	I₃	I₄
ℵ₁	〈 (0.2, 0.3, 0.4) , (0.4, 0.5, 0.6)〉	〈 (0.1, 0.2, 0.4) , (0.2, 0.3, 0.5)〉
ℵ₂	〈 (0.4, 0.6, 0.7) , (0.3, 0.3, 0.3)〉	〈 (0.2, 0.2, 0.4) , (0.3, 0.3, 0.5)〉
ℵ₃	〈 (0.3, 0.5, 0.7) , (0.3, 0.3, 0.4)〉	〈 (0.3, 0.4, 0.6) , (0.2, 0.3, 0.4)〉
ℵ₄	〈 (0.1, 0.2, 0.3) , (0.3, 0.4, 0.5)〉	〈 (0.4, 0.6, 0.6) , (0.1, 0.2, 0.4)〉
ℵ₅	〈 (0.1, 0.1, 0.4) , (0.1, 0.2, 0.3)〉	〈 (0.2, 0.4, 0.5) , (0.2, 0.4, 0.5)〉

Table 3

The FNIFN matrix by K₃

	I₁	I₂
ℵ₁	〈 (0.1, 0.3, 0.4) , (0.1, 0.3, 0.5)〉	〈 (0.1, 0.2, 0.2) , (0.1, 0.5, 0.6)〉
ℵ₂	〈 (0.1, 0.1, 0.1) , (0.1, 0.1, 0.3)〉	〈 (0.3, 0.5, 0.7) , (0.1, 0.1, 0.3)〉
ℵ₃	〈 (0.3, 0.4, 0.7) , (0.1, 0.2, 0.3)〉	〈 (0.1, 0.3, 0.4) , (0.2, 0.5, 0.6)〉
ℵ₄	〈 (0.3, 0.4, 0.5) , (0.2, 0.3, 0.4)〉	〈 (0.2, 0.4, 0.5) , (0.2, 0.3, 0.5)〉
ℵ₅	〈 (0.4, 0.5, 0.8) , (0.1, 0.2, 0.2)〉	〈 (0.1, 0.3, 0.7) , (0.2, 0.2, 0.3)〉
	I₃	I₄
ℵ₁	〈 (0.2, 0.2, 0.5) , (0.1, 0.2, 0.4)〉	〈 (0.2, 0.3, 0.5) , (0.1, 0.4, 0.5)〉
ℵ₂	〈 (0.4, 0.5, 0.8) , (0.1, 0.2, 0.2)〉	〈 (0.1, 0.3, 0.6) , (0.1, 0.2, 0.4)〉
ℵ₃	〈 (0.3, 0.3, 0.5) , (0.1, 0.1, 0.2)〉	〈 (0.1, 0.1, 0.3) , (0.2, 0.5, 0.7)〉
ℵ₄	〈 (0.1, 0.1, 0.5) , (0.2, 0.2, 0.5)〉	〈 (0.1, 0.2, 0.3) , (0.2, 0.3, 0.7)〉
ℵ₅	〈 (0.2, 0.3, 0.6) , (0.2, 0.4, 0.4)〉	〈 (0.1, 0.3, 0.5) , (0.2, 0.3, 0.5)〉

Then, the FNIF-TODIM method is produced for competitiveness evaluation of regional financial centers.

Step 1. According to Equation (7), the overall FNIFN-matrix is obtained (See Table 4).

Table 4

The overall FNIFN matrix

	I₁	I₂
ℵ₁	$〈 \begin{matrix} (0.2770, 0.3021, 0.3048), \\ (0.3351, 0.3537, 0.4407) \end{matrix} 〉$	$〈 \begin{matrix} (0.1680, 0.2042, 0.2681), \\ (0.3537, 0.5013, 0.5620) \end{matrix} 〉$
ℵ₂	$〈 \begin{matrix} (0.2424, 0.2747, 0.3918), \\ (0.1347, 0.2410, 0.6729) \end{matrix} 〉$	$〈 \begin{matrix} (0.4056, 0.4231, 0.5282), \\ (0.2388, 0.2542, 0.4000) \end{matrix} 〉$
ℵ₃	$〈 \begin{matrix} (0.4056, 0.5068, 0.5673), \\ (0.2681, 0.2932, 0.3212) \end{matrix} 〉$	$〈 \begin{matrix} (0.2042, 0.2681, 0.6366), \\ (0.1680, 0.2886, 0.3396) \end{matrix} 〉$
ℵ₄	$〈 \begin{matrix} (0.2681, 0.3351, 0.3760), \\ (0.2348, 0.2650, 0.3000) \end{matrix} 〉$	$〈 \begin{matrix} (0.3358, 0.3448, 0.3596), \\ (0.2042, 0.2388, 0.4152) \end{matrix} 〉$
ℵ₅	$〈 \begin{matrix} (0.3786, 0.5238, 0.6524), \\ (0.1680, 0.2042, 0.2348) \end{matrix} 〉$	$〈 \begin{matrix} (0.3743, 0.3842, 0.4759), \\ (0.2332, 0.2758, 0.3448) \end{matrix} 〉$
	I₃	I₄
ℵ₁	$〈 \begin{matrix} (0.2348, 0.3786, 0.5180), \\ (0.2440, 0.2601, 0.4231) \end{matrix} 〉$	$〈 \begin{matrix} (0.3048, 0.3286, 0.4152), \\ (0.2146, 0.3786, 0.3937) \end{matrix} 〉$
ℵ₂	$〈 \begin{matrix} (0.4354, 0.4759, 0.6085), \\ (0.2745, 0.2942, 0.3048) \end{matrix} 〉$	$〈 \begin{matrix} (0.2348, 0.2440, 0.5421), \\ (0.2770, 0.2948, 0.4152) \end{matrix} 〉$
ℵ₃	$〈 \begin{matrix} (0.3351, 0.3458, 0.5783), \\ (0.1680, 0.1723, 0.3196) \end{matrix} 〉$	$〈 \begin{matrix} (0.3537, 0.3851, 0.4354), \\ (0.3743, 0.3396, 0.4482) \end{matrix} 〉$
ℵ₄	$〈 \begin{matrix} (0.1032, 0.2886, 0.5282), \\ (0.2348, 0.2546, 0.3537) \end{matrix} 〉$	$〈 \begin{matrix} (0.4759, 0.4879, 0.6963), \\ (0.1347, 0.2042, 0.4549) \end{matrix} 〉$
ℵ₅	$〈 \begin{matrix} (0.2348, 0.3132, 0.3348), \\ (0.1680, 0.2770, 0.2932) \end{matrix} 〉$	$〈 \begin{matrix} (0.2681, 0.2940, 0.5358), \\ (0.2681, 0.3396, 0.4152) \end{matrix} 〉$

Step 2. the attributes weight can be obtained:ow = (0 . 22, 0 . 25, 0 . 18, 0 . 35) ^T.

Step 3. Computing the relative weights, ow₄ = max{ ow₁, ow₂, ow₃, ow₄ }, then reference weight is ow_r = 0.35. Therefore, the defined relative weights are: ow_1r = 0.6286, ow_2r = 0.7143, ow_3r = 0.5143,ow_4r = 1.0000.

Step 4. The FNIFDD_j (ℵ _i, ℵ _t) of ℵ_iover ℵ_t for I_j is derived based on the Equations (13–15). Letθ = 2.5, FNIFDD_j (ℵ _i, ℵ _t) could be obtained in Table 5–8.

Table 5

The FNIFDD₁ (ℵ _i, ℵ _t)

Alternatives	ℵ₁	ℵ₂	ℵ₃	ℵ₄	ℵ₅
ℵ₁	0.0000	–0.2981	0.1308	–0.2843	–0.2202
ℵ₂	0.1640	0.0000	0.1483	0.0494	0.1563
ℵ₃	–0.2378	–0.2697	0.0000	–0.2843	–0.3241
ℵ₄	0.1563	–0.0899	0.1563	0.0000	0.1483
ℵ₅	0.1211	–0.2843	0.1783	–0.2697	0.0000

Table 6

The FNIFDD₂ (ℵ _i, ℵ _t)

Alternatives	ℵ₁	ℵ₂	ℵ₃	ℵ₄	ℵ₅
ℵ₁	0.0000	–0.2667	0.1491	–0.2530	0.1491
ℵ₂	0.1667	0.0000	0.1581	0.1667	0.1667
ℵ₃	–0.2385	–0.2530	0.0000	–0.2797	0.1054
ℵ₄	0.1581	–0.2667	0.1748	0.0000	0.1826
ℵ₅	–0.2385	–0.2667	–0.1687	–0.2921	0.0000

Table 7

The FNIFDD₃ (ℵ _i, ℵ _t)

Alternatives	ℵ₁	ℵ₂	ℵ₃	ℵ₄	ℵ₅
ℵ₁	0.0000	0.0707	0.1095	–0.3220	0.1612
ℵ₂	–0.1571	0.0000	0.1095	–0.3583	0.1612
ℵ₃	–0.2202	–0.2202	0.0000	–0.3114	0.1308
ℵ₄	0.1449	0.1612	0.1549	0.0000	0.1549
ℵ₅	–0.3583	–0.3583	–0.2629	–0.3443	0.0000

Table 8

TheFNIFDD₄ (ℵ _i, ℵ _t)

Alternatives	ℵ₁	ℵ₂	ℵ₃	ℵ₄	ℵ₅
ℵ₁	0.0000	–0.3266	0.2333	–0.2364	–0.1886
ℵ₂	0.2858	0.0000	0.1871	–0.3107	0.2333
ℵ₃	–0.2667	–0.2138	0.0000	–0.2851	–0.2364
ℵ₄	0.2068	0.2718	0.2494	0.0000	0.1462
ℵ₅	0.1650	–0.2667	0.2068	–0.1671	0.0000

Step 5. Calculate the FNIFODD (ℵ _i, ℵ _t) by Equation (18) (See Table 9).

Table 9

The FNIFODD (ℵ _i, ℵ _t)

Alternatives	ℵ₁	ℵ₂	ℵ₃	ℵ₄	ℵ₅
ℵ₁	0.0000	–0.8207	0.6228	–1.0957	–0.0984
ℵ₂	0.4593	0.0000	0.6031	–0.4529	0.7176
ℵ₃	–0.9632	–0.9567	0.0000	–1.1604	–0.3243
ℵ₄	0.6662	0.0765	0.7355	0.0000	0.6321
ℵ₅	–0.3107	–1.1759	–0.0465	–1.0732	0.0000

Step 6. Obtain the FNIFODDV (ℵ _i) of ℵ_i through Equation. (19). $\begin{matrix} \begin{matrix} FNIFODDV (ℵ_{1}) = 0.3649, \\ FNIFODDV (ℵ_{2}) = 0.8580 \\ FNIFODDV (ℵ_{3}) = 0.0000, \\ FNIFODDV (ℵ_{4}) = 1.0000 \\ FNIFODDV (ℵ_{5}) = 0.1447 \end{matrix} \end{matrix}$

Step 7. Finally, the order of all the offered alternatives with FNIFODDV (ℵ _i) (i = 1, 2, 3, 4, 5) could be obtained: ℵ₄ ≻ ℵ ₂ ≻ ℵ ₁ ≻ ℵ ₅ ≻ ℵ ₃, and thus the optimal regional financial center is ℵ₄.

4.2 Comparative analysis

Then, we can compare the developed model with other models, and the conclusions drawn are shown in Table 10.

Table 10
Comparative analysis

Order

FNIFWA [24] ℵ₄ ≻ ℵ ₂ ≻ ℵ ₁ ≻ ℵ ₅ ≻ ℵ ₃

FNIFWG [70] ℵ₄ ≻ ℵ ₂ ≻ ℵ ₁ ≻ ℵ ₅ ≻ ℵ ₃

FNIFWHM [71] ℵ₄ ≻ ℵ ₂ ≻ ℵ ₅ ≻ ℵ ₁ ≻ ℵ ₃

FNIFHPWG [72] ℵ₄ ≻ ℵ ₂ ≻ ℵ ₁ ≻ ℵ ₅ ≻ ℵ ₃

IFNIFHCA [73] ℵ₄ ≻ ℵ ₂ ≻ ℵ ₅ ≻ ℵ ₁ ≻ ℵ ₃

FNIF-TODIM ℵ₄ ≻ ℵ ₂ ≻ ℵ ₁ ≻ ℵ ₅ ≻ ℵ ₃

	Order
FNIFWA [24]	ℵ₄ ≻ ℵ ₂ ≻ ℵ ₁ ≻ ℵ ₅ ≻ ℵ ₃
FNIFWG [70]	ℵ₄ ≻ ℵ ₂ ≻ ℵ ₁ ≻ ℵ ₅ ≻ ℵ ₃
FNIFWHM [71]	ℵ₄ ≻ ℵ ₂ ≻ ℵ ₅ ≻ ℵ ₁ ≻ ℵ ₃
FNIFHPWG [72]	ℵ₄ ≻ ℵ ₂ ≻ ℵ ₁ ≻ ℵ ₅ ≻ ℵ ₃
IFNIFHCA [73]	ℵ₄ ≻ ℵ ₂ ≻ ℵ ₅ ≻ ℵ ₁ ≻ ℵ ₃
FNIF-TODIM	ℵ₄ ≻ ℵ ₂ ≻ ℵ ₁ ≻ ℵ ₅ ≻ ℵ ₃

According to the above analysis in Table 10, the constructed FNIF-TODIM are compared with the results of FNIFWA and FNIFWG, which have slightly different rankings of the given solutions and no differences in the best alternatives. Compared with some existing methods, the FNIF-TODIM method has the following defined advantages: (1) based on the prospect theory, FNIF-TODIM method proposed an interactive MAGDM method with fewer parameters and simple computation, etc; (2) The FNIF-TODIM method can better reflect the subjective risk preferences of decision makers and does not require prior information on decision reference points.

5 Conclusion

At present, Chinese reform and opening up and socialist modernization are in a period of important strategic opportunities. With the further enhancement of comprehensive national strength, the further expansion of opening to the outside world, and the further improvement of the market economic system, the financial reform and development has entered an important turning point and development stage. In the new stage of development, some cities in our country take the construction of financial centers as a strategic measure to speed up economic development. Due to the unbalanced development of Chinese economy and the need for strong support points for the development of the financial industry, the formation of several regional financial centers in Chinese has its objective inevitability. As the strategic hinterland of Chinese economy, the central provinces and regions lack the driving and support of a strong regional growth center. On the whole and in all provinces and regions, there have been “small horse-drawn carts” to varying degrees. The rise of the central region objectively requires the implementation of the driving and support of the regional growth center. Under this background, in order to comply with the situation of regional economic development in the central region and the requirements of regional financial development, we should seize the opportunity to cultivate regional financial centers in the central region. It is not only the needs of the development of the central region, but also the inherent requirement of promoting the regional economic development of the entire central region. How to build a regional financial center in the central region is an important subject, and there is a lot of work to be done, and improving the regional competitiveness of the central region is the most important step to realize this subject. Therefore, the research on the competitiveness of financial centers in the central region has important theoretical significance and far-reaching practical significance. Thus, the competitiveness evaluation of regional financial centers is regarded as the MAGDM. Based on the TODIM method and FNIFSs, this paper proposes a new FNIF-TODIM method is built for MAGDM. Finally, an example for competitiveness evaluation of regional financial centers and some decision comparative analysis is listed. The results show that the established approach is useful. For our future work, comparing the constructed FNIF-TODIM method with other MAGDM methods under FNIFS (e.g., GRA method, TOPSIS method, PROMETHEE method, and ELECTRE method, FUCOM, BWM, MAIRCA, MABAC, VIKOR) deserves further research and exploration. It is also interesting to apply the constructed FNIF-TODIM method to other MAGDM problems (e.g., investment decision problem and supplier selection problem) [74 –76].

Footnotes

Acknowledgments

This work was supported by the launching scientific research funds for doctors from the Henan University of Animal Husbandry and Economy(2020HNUAHEDF035), Soft Science Research Project in Henan Province (222400410608), Humanities and Social Science Research Project of Universities in Henan Province(2021-ZZJH-148).

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