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Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods

  • Zhiyuan Wang , Mahmut Baydaş EMAIL logo , Željko Stević EMAIL logo , Abdullah Özçil , Sayed Ameenuddin Irfan , Zhe Wu and Gade Pandu Rangaiah
Published/Copyright: November 24, 2023
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

The use of multi-criteria decision-making (MCDM) methods to select the most appropriate one from a range of alternatives considering multiple criteria is a suitable methodology for making informed decisions. When constructing a decision or objective matrix (DOM) for MCDM procedure, either crisp numerical values or fuzzy linguistic terms can be used. A review of relevant literature indicates that decision experts often prefer to give linguistic terms (instead of crisp numerical values) based on their domain knowledge, to establish a fuzzy DOM. However, previous research articles have not adequately studied the selection between fuzzy and crisp DOM in MCDM, especially under the context of assessing the financial performance (FP) of listed firms – a notably complex decision-making problem. As such, the primary motivation of this study is to bridge this research gap through comparative analyses of fuzzy and crisp DOM in MCDM. Along this path, and in order to handle fuzzy DOM, this work also proposes two new fuzzy MCDM methods: fuzzy preference ranking on the basis of ideal-average distance (PROBID) and fuzzy sPROBID (simpler PROBID), extending the applicability of the original crisp PROBID and sPROBID methods. Moreover, for the first time in the literature, this work compares the FP rankings obtained using fuzzy MCDM methods with an objective benchmark we have identified, i.e., the real-life stock return (SR)-based ranking. The case study of ranking the FP of 32 listed firms demonstrates that the fuzzy MCDM methods produce higher correlation results with the SR-based ranking. The results also suggest that the proposed fuzzy sPROBID method with triangular fuzzy DOM performs the best for assessing the FP of firms in terms of Spearman’s rank correlation coefficient with the SR-based ranking. Overall, the contributions of this work are three-fold: first, it proposes two new fuzzy MCDM methods (i.e., fuzzy PROBID and fuzzy sPROBID); second, it advances the application of fuzzy MCDM methods in assessing and ranking the FP of listed firms to make rational investment decisions in the financial market; third, it studies the selection between fuzzy and crisp DOM through comparisons with an objective benchmark.

Keywords: fuzzy decision matrix; crisp decision matrix; PROBID; sPROBID; MCDM
MSC 2010: 90C70; 90B50; 91G80

1 Introduction

Using multi-criteria decision-making (MCDM) methods is an effective strategy for making well-informed decisions when choosing one alternative out of many, considering multiple criteria. This is particularly useful in complex problems such as evaluating and ranking the financial performance (FP) of listed firms. MCDM methods allow for the evaluation of trade-offs among many criteria since a single criterion is usually inadequate to provide a comprehensive understanding of a firm’s performance. MCDM methods consider multiple criteria that reflect the various aspects of firms and combine them into a single measurement for a fair comparison. As a result, they are widely adopted for the ranking and selection of alternatives [1–13].

On the other hand, choosing the most suitable MCDM method for a particular application from the numerous MCDM methods available in the literature has become a challenging task [14–16]. Wang and Rangaiah [17] led the effort to recommend several MCDM methods based on the number of user inputs required from decision-makers, simplicity, and applicability of the methods. However, due to the lack of an objective benchmark, it remains difficult to ascertain the superiority of one MCDM method over the rest. In this study, we adopt the real-life ranking of 32 listed firms based on stock return (SR) as the objective benchmark, given that SR is a direct reflection of the capital gains and losses made by investors. SR can be calculated as the difference between the current stock price and the prior stock price divided by the prior stock price. Thereafter, the FP rankings generated by different MCDM methods, considering multiple FP criteria (i.e., FP indicators), are compared with the SR-based ranking using Spearman’s rank correlation coefficient. It is generally accepted that the MCDM method, which produces the highest correlation, most accurately represents the real-life situation, thereby asserting its superiority over other methods.

It is common to use fuzzy MCDM methods, especially for the ranking and selection problems that require expert opinions in situations characterized by uncertainty or incomplete information [18]. From the literature review, it is observed that crisp numerical values of the criteria are sometimes converted to fuzzy linguistic terms, and fuzzy MCDM methods are then used in selecting the best alternative. For instance, Lam et al. [19] suggested that, when crisp data are less suitable to model an event due to vagueness, interval judgment with linguistic terms can be used for initial evaluation; and they then proposed an entropy-fuzzy technique for order reference based on similarity to the ideal solution (TOPSIS) model to evaluate the FP of companies based on liquidity, solvency, efficiency, and profitability ratios for portfolio investment. Raheja and Jain [20] designed an intuitionistic fuzzy-based system using the concept of intuitionistic fuzzy set theory and converting crisp data into fuzzy data. Kumar and Kalpana [21] focused on the construction of fuzzy expert system, an artificial intelligence tool tailored for decision-making problems; their approach involved the use of fuzzification to convert crisp values into fuzzy values, a process that reportedly simplified decision-making for practitioners.

Another reason highlighted in the literature for the conversion from crisp values to fuzzy values is that some FP indicators or criteria (e.g., current ratio, cash ratio, equity/debt ratio) do not fit into the conventional categories of larger-the-better (i.e., maximization or benefit) or smaller-the-better (i.e., minimization or cost). For instance, the equity/debt ratio measures the financial leverage of a firm. A higher equity/debt ratio does not always indicate better performance, as it indicates a more conservative capital structure, which can lead to lower financial risk but may also result in a lower return on investment. Conversely, a lower equity/debt ratio indicates a more aggressive capital structure, with a larger proportion of debt financing, which can result in higher financial risk but also potentially higher returns on investment. The ideal equity/debt ratio may differ depending on the industry and specific business goals, making it difficult to consider this FP indicator in the classic architecture of MCDM methods [22]. Therefore, financial experts’ interpretation of FP indicators, using fuzzy linguistic terms based on their domain knowledge and industry experience, is necessary.

Moreover, although each FP indicator is typically categorized as either cost- or benefit-oriented, real-life conditions do not always adhere to linearity. This nuanced situation, which presents a seeming contradiction to the conventional crisp MCDM methods, can be effectively addressed using fuzzy decision or objective matrix (DOM) made up of fuzzy linguistic terms. To handle such a DOM, this work proposes two new fuzzy MCDM methods: fuzzy preference ranking on the basis of ideal-average distance (PROBID) and fuzzy sPROBID (simpler PROBID), expanding upon the work of Wang et al. [23]. This expansion enhances the capabilities of the original PROBID and sPROBID methods, bringing their proven comprehensiveness, robustness, consistency, and stability [23–25] into the domain of fuzzy MCDM. Upon further examination of the literature, it is notable that there lacks a concrete recommendation regarding the choice of MCDM methods based on an objective benchmark [26–32]. Consequently, this work takes an unprecedented step in comparing the FP rankings obtained using fuzzy MCDM methods against an objective benchmark we have identified – the real-life SR-based ranking. The final outcomes of this study demonstrate that the fuzzy PROBID and fuzzy sPROBID methods lead to higher correlation results with the SR-based ranking than their crisp counterparts (i.e., the original PROBID and sPROBID methods).

Furthermore, crisp values of the FP indicators are typically derived from balance sheet and represent the performance of a firm at a specific time. However, the static values of FP indicators are insufficient to capture changes over time and address investor expectations. Therefore, dynamic FP indicators that evaluate the changes between two balance sheets across different times are necessary to provide a comprehensive view of a firm’s performance. For example, in investment analysis, it is essential to ensure that a good firm has a positive return on equity (ROE) for the current period. However, it may be more appealing if the ROE shows an increase compared to the base period. In this case, the dynamic ROE is as crucial as, or even more important than, the static ROE. In fact, as reported by Baydaş and Pamučar [15], dynamic FP indicators have been found to exhibit a stronger correlation with SR compared to static FP indicators. Thus, for interpreting with fuzzy linguistic terms, financial experts may consider both the values of the static and dynamic FP indicators.

Recently, Baydaş and Pamučar [15] and Baydaş et al. [33] studied the correlation between MCDM-based FP ranking with crisp data and the benchmark (i.e., SR-based ranking). This study aimed to push the boundary through benchmarking FP rankings by fuzzy MCDM methods (i.e., fuzzy PROBID and fuzzy sPROBID) coupled with two types of fuzzy linguistic terms (i.e., triangular and trapezoidal), against SR-based ranking. Besides, earlier scholarly publications have not adequately studied the selection between fuzzy and crisp DOM in MCDM. Thus, in this work, we not only make comparisons among the ranking results based on fuzzy DOM, but also cross-compare them with the ranking results derived from crisp DOM to identify the combination that best assesses the FP of firms. Overall, the primary motivation of this study is to bridge this research gap through comparative analyses of fuzzy and crisp DOM in MCDM. To facilitate this, fuzzy PROBID and fuzzy sPROBID methods are proposed to handle fuzzy DOM.

The contributions of this study are three-fold: first, it proposes two new fuzzy MCDM methods (i.e., fuzzy PROBID and fuzzy sPROBID); second, it advances the application of fuzzy MCDM methods in assessing and ranking the FP of listed firms to make rational investment decisions in the financial market; third, it studies the selection between fuzzy and crisp DOM through comparisons with an objective benchmark. The remainder of this article is organized as follows. Section 2 outlines the steps involved in the fuzzy PROBID and fuzzy sPROBID methods. Section 3 provides an illustration of the application of these methods in assessing the FP of 32 listed firms. Section 4 discusses the results of the comparative analyses and then presents the limitations of the current work. Finally, Section 5 summarizes the conclusions drawn from this study and suggests potential avenues for future research.

2 Fuzzy PROBID and fuzzy sPROBID methods

Wang et al. [23] proposed the original PROBID and sPROBID methods. This section builds upon these methods and advances toward the development of fuzzy PROBID and fuzzy sPROBID methods with triangular and trapezoidal fuzzy numbers. The methods take into consideration a DOM with m alternatives or solutions and n criteria or objectives. The n criteria may either be for maximization (i.e., a benefit criterion, larger-the-better) or be for minimization (i.e., a cost criterion, smaller-the-better). Some mathematical symbols that are used in the following descriptions are as follows: f ij refers to the original linguistic term of the jth criterion for the ith alternative in the DOM; a ij denotes the first fuzzy number of the jth criterion for the ith alternative after converting linguistic terms to fuzzy numbers; analogously, b ij denotes the second fuzzy number, c ij denotes the third fuzzy number, and so on; A ij , B ij , and C ij represent the value of a ij , b ij , and c ij after normalization, respectively; and w j is the weight of the jth criterion. In this paragraph and throughout this article, i ∈ { 1,2 , … , m } and j ∈ { 1,2 , … , n } .

2.1 Fuzzy PROBID with triangular fuzzy numbers

The eight steps of the fuzzy PROBID method, using triangular fuzzy numbers, are as follows.

Step 1. For the DOM with m alternatives and n criteria, first convert the original linguistic terms (f ij ) to triangular fuzzy numbers using Table 1. For instance, a linguistic term f ij = Low can be represented by three fuzzy numbers (1, 3, 5), in which a ij = 1, b ij = 3, and c ij = 5.

Table 1

Triangular fuzzy numbers for representing linguistic terms

Linguistic terms Triangular fuzzy numbers
a b c
Very low 1 1 3
Low 1 3 5
Average 3 5 7
High 5 7 9
Very high 7 9 9

Step 2. Construct the normalized DOM by applying the following normalization equations:

(1) F ij = ( A ij , B ij , C ij ) = a ij c j * , b ij c j * , c ij c j * for maximization criterion ,

(2) F ij = ( A ij , B ij , C ij ) = a j − c ij , a j − b ij , a j − a ij for minimization criterion ,

where c j * = max i { c ij } and a j − = min i { a ij } . Note that the symbol F ij is written in boldface to indicate that it represents a vector consisting of three scalar values ( A ij , B ij , C ij ) . The same convention is applied in the subsequent steps.

Step 3. Construct the weighted normalized DOM by multiplying the normalized values ( A ij , B ij , C ij ) with the assigned weight (w j ). Decision makers have the option of either assigning all numerical values to the weights (provided that the sum of all weights equals to 1) or assigning all the weights using the linguistic terms presented in Table 1, in which case the sum of all linguistic weights need not be unity.

If all the assigned weights are numerical values, apply the following equation to obtain the weighted normalized values, v ij :

(3) v ij = ( A ij w , B ij w , C ij w ) = ( A ij × w j , B ij × w j , C ij × w j ) .

If all the assigned weights are linguistic terms, first convert them to triangular fuzzy numbers using Table 1, similar to Step 1; each linguistic w j will be converted to three fuzzy numbers ( a w j , b w j ,   c w j ). Next, apply the following equation to obtain the weighted normalized values, v ij :

(4) v ij = ( A ij w , B ij w , C ij w ) = ( A ij × a w j , B ij × b w j , C ij × c w j ) .

Step 4. Find the spectrum of fuzzy positive-ideal solutions (FPISs), which encompasses the most FPIS ( FPIS ( 1 ) ) , 2nd FPIS ( FPIS ( 2 ) ) , …, kth FPIS ( FPIS ( k ) ) , …, and mth FPIS ( FPIS ( m ) ) , using equation (5). Then, calculate the Euclidean distance from each alternative to each of the FPISs using equation (6):

(5) FPIS ( k ) = { v ( k ) 1 , v ( k ) 2 , … , v ( k ) j , … , v ( k ) n } j ∈ { 1 , 2 , … , n } ; k ∈ { 1 , 2 , … , m } ,

where v ( k ) j = ( A ξ k j w , B ξ k j w , C ξ k j w ) , ξ k is the row index (from 1 to m) of the kth largest value in C ij w ; in case of ties (i.e., multiple kth largest values in C ij w ), use B ij w to break the tie; if there is still a tie, use A ij w to aid. In the aforementioned equation, v ( k ) j represents the kth best value of the jth criterion. For example, v ( 1 ) 2 represents the best value of the second criterion and v ( 2 ) 3 represents the second best value of the third criterion.

Euclidean distance from the ith alternative to the kth FPIS is denoted as P S i ( k ) and calculated as follows:

(6) P S i ( k ) = ∑ j = 1 n 1 3 ‖ v ij − v ( k ) j ‖ 2 = ∑ j = 1 n 1 3 [ ( A ij w − A ξ k j w ) 2 + ( B ij w − B ξ k j w ) 2 + ( C ij w − C ξ k j w ) 2 ] ,

where i ∈ { 1 , 2 , … , m } , k ∈ { 1 , 2 , … , m } , v ij = ( A ij w , B ij w , C ij w ) as in Step 3, and v ( k ) j = ( A ξ k j w , B ξ k j w , C ξ k j w ) as in equation (5).

Step 5. Find the spectrum of fuzzy negative-ideal solutions (FNISs), which encompasses the most FNIS ( FNIS ( 1 ) ) , 2nd FNIS ( FNIS ( 2 ) ) , …, kth FNIS ( FNIS ( k ) ) , …, and mth FNIS ( FNIS ( m ) ) , using equation (7). Then, calculate the Euclidean distance from each alternative to each of the FNISs using equation (8):

(7) FNIS ( k ) = { v ( k ) 1 , v ( k ) 2 , … , v ( k ) j , … , v ( k ) n } j ∈ { 1 , 2 , … , n } ; k ∈ { 1 , 2 , … , m } ,

where v ( k ) j = ( A ξ k j w , B ξ k j w , C ξ k j w ) , ξ k is the row index (from 1 to m) of the kth smallest value in A ij w ; in case of ties (i.e., multiple kth smallest values in A ij w ), use B ij w to break the tie; if there is still a tie, use C ij w to aid. In the aforementioned equation, v ( k ) j represents the kth worst value of the jth criterion. For example, v ( 1 ) 2 represents the worst value of the second criterion and v ( 2 ) 3 represent the second worst value of the third criterion.

Euclidean distance from the ith alternative to the kth FNIS is denoted as N S i ( k ) and calculated as follows:

(8) N S i ( k ) = ∑ j = 1 n 1 3 ‖ v ij − v ( k ) j ‖ 2 = ∑ j = 1 n 1 3 [ ( A ij w − A ξ k j w ) 2 + ( B ij w − B ξ k j w ) 2 + ( C ij w − C ξ k j w ) 2 ] ,

where i ∈ { 1 , 2 , … , m } , k ∈ { 1 , 2 , … , m } , v ij = ( A ij w , B ij w , C ij w ) as in Step 3 and v ( k ) j = ( A ξ k j w , B ξ k j w , C ξ k j w ) as in equation (7).

Step 6. Find the average solution for each criterion using equation (9) and denote it as v ̅ j . Then, calculate the Euclidean distance from each alternative to the average solution using equation (10) and denote it as S i ( avg ) :

(9) v ̅ j = ( A ̅ j w , B ̅ j w , C ̅ j w ) = ∑ i = 1 m A ij w m , ∑ i = 1 m B ij w m , ∑ i = 1 m C ij w m for j ∈ { 1 , 2 , … , n } ,

(10) S i ( avg ) = ∑ j = 1 n 1 3 ‖ v ij − v ̅ j ‖ 2 = ∑ j = 1 n 1 3 [ ( A ij w − A ̅ j w ) 2 + ( B ij w − B ̅ j w ) 2 + ( C ij w − C ̅ j w ) 2 ] ,

where i ∈ { 1 , 2 , … , m } .

Step 7. Determine the overall positive-ideal distance using equation (11), which is essentially the weighted sum distance of one alternative to the first half of the FPISs:

(11) S i ( pos − ideal ) = ∑ k = 1 m + 1 2 1 k P S i ( k ) i ∈ { 1 , 2 , … , m } when m is an odd number ∑ k = 1 m 2 1 k P S i ( k ) i ∈ { 1 , 2 , … , m } when m is an even number .

Likewise, determine the overall negative-ideal distance using equation (12), which is essentially the weighted sum distance of one alternative to the first half of the FNISs:

(12) S i ( neg − ideal ) = ∑ k = 1 m + 1 2 1 k N S i ( k ) i ∈ { 1 , 2 , … , m } when m is an odd number ∑ k = 1 m 2 1 k N S i ( k ) i ∈ { 1 , 2 , … , m } when m is an even number .

Step 8. Calculate the positive-ideal/negative-ideal ratio (R i ) and then performance score (P i ) of each alternative as follows:

(13) R i = S i ( pos − ideal ) S i ( neg − ideal ) for i ∈ { 1 , 2 , … , m } ,

(14) P i = 1 1 + R i 2 + S i ( avg ) for i ∈ { 1 , 2 , … , m } .

The recommended alternative is that with the highest P i value.

2.2 Fuzzy sPROBID with triangular fuzzy numbers

As stated in Wang et al. [23], the sPROBID method is a simplified version of the PROBID method. In this study, we maintain the same principle and develop a simpler variant of the fuzzy PROBID method, referred to as the fuzzy sPROBID method. The first five steps of the fuzzy sPROBID are identical to Steps 1–5 outlined in Section 2.1, and Step 6 of finding the average solution for each criterion is omitted in the fuzzy sPROBID. The key difference lies in Steps 7 and 8, where instead of using the first half of the FPISs to determine S i ( pos − ideal ) and the first half of the FNISs to determine S i ( neg − ideal ) , the fuzzy sPROBID considers only the first quarters of the FPISs and FNISs in finding S i ( pos − ideal ) and S i ( neg − ideal ) , respectively.

Step 7. Determine the overall positive-ideal distance using equation (15), which is essentially the weighted sum distance of one alternative to the first quarter of the FPISs:

(15) S i ( pos − ideal ) = ∑ k = 1 m \ 4 1 k P S i ( k ) for i ∈ { 1 , 2 , … , m } when m ≥ 4 P S i ( 1 ) for i ∈ { 1 , 2 , … , m } when 0 < m < 4 .

Likewise, determine the overall negative-ideal distance using equation (16), which is essentially the weighted sum distance of one alternative to the first quarter of the FNISs:

(16) S i ( neg − ideal ) = ∑ k = 1 m \ 4 1 k N S i ( k ) for i ∈ { 1 , 2 , … , m } when m ≥ 4 N S i ( 1 ) for i ∈ { 1 , 2 , … , m } when 0 < m < 4 .

In both equations (15) and (16), m\ 4 is the integer quotient of m divided by 4, e.g., 10 \ 4 would yield 2.

Step 8. The performance score (P i ) is simplified to the ratio of the negative-ideal distance to the positive-ideal distance, expressed as follows:

(17) P i = S i ( neg − ideal ) S i ( pos − ideal ) for i ∈ { 1 , 2 , … , m } .

The recommended alternative is that with the highest P i value.

2.3 Fuzzy PROBID with trapezoidal fuzzy numbers

The eight steps of the fuzzy PROBID method, using trapezoidal fuzzy numbers, are as follows:

Step 1. For the DOM with m alternatives and n criteria, first convert the original linguistic terms (f ij ) to trapezoidal fuzzy numbers using Table 2. For instance, a linguistic term f ij = Average can be represented by four fuzzy numbers (4, 5, 5, 6), in which a ij = 4, b ij = 5, c ij = 5, and d ij = 6.

Table 2

Trapezoidal fuzzy numbers for representing linguistic terms

Linguistic terms Trapezoidal fuzzy numbers
a b c d
Very low 1 1 1 2
Low 1 2 2 3
Fairly low 2 3 4 5
Average 4 5 5 6
Fairly high 5 6 7 8
High 7 8 8 9
Very high 8 9 10 10

Step 2. Construct the normalized DOM by applying the following normalization equations:

(18) F ij = ( A ij , B ij , C ij , D ij ) = a ij d j * , b ij d j * , c ij d j * , d ij d j * for maxim ization c riterion ,

(19) F ij = ( A ij , B ij , C ij , D ij ) = a j − d ij , a j − c ij , a j − b ij , a j − a ij for minimization c riterion ,

where d j * = max i { d ij } and a j − = min i { a ij } . Note that the symbol F ij is written in boldface to indicate that it represents a vector consisting of four scalar values ( A ij , B ij , C ij , D ij ) . The same convention is applied in the subsequent steps.

Step 3. Construct the weighted normalized DOM by multiplying the normalized values ( A ij , B ij , C ij , D ij ) with the assigned weight (w j ). Decision makers have the option of either assigning all numerical values to the weights (provided that the sum of all weights equals to 1) or assigning all the weights using the linguistic terms presented in Table 2.

If all the assigned weights are numerical values, apply the following equation to obtain the weighted normalized values, v ij :

(20) v ij = ( A ij w , B ij w , C ij w , D ij w ) = ( A ij × w j , B ij × w j , C ij × w j , D ij × w j ) .

If all the assigned weights are linguistic terms, first convert them to trapezoidal fuzzy numbers using Table 2, similar to Step 1; each linguistic w j will be converted to four fuzzy numbers ( a w j , b w j , c w j , d w j ). Next, apply the following equation to obtain the weighted normalized values, v ij :

(21) v ij = ( A ij w , B ij w , C ij w , D ij w ) = ( A ij × a w j , B ij × b w j , C ij × c w j , D ij × d w j ) .

Step 4. Find the spectrum of FPISs, which encompasses the most FPIS ( FPIS ( 1 ) ) , 2nd FPIS ( FPIS ( 2 ) ) , …, kth FPIS ( FPIS ( k ) ) , …, and mth FPIS ( FPIS ( m ) ) , using equation (22). Then, calculate the Euclidean distance from each alternative to each of the FPISs using equation (23):

(22) FPIS ( k ) = { v ( k ) 1 , v ( k ) 2 , … , v ( k ) j , … , v ( k ) n } j ∈ { 1 , 2 , … , n } ; k ∈ { 1 , 2 , … , m } ,

where v ( k ) j = ( A ξ k j w , B ξ k j w , C ξ k j w , D ξ k j w ) , ξ k is the row index (from 1 to m) of the kth largest value in D ij w ; in case of ties (i.e., multiple kth largest values in D ij w ), use C ij w to break the tie; if there is still a tie, use B ij w to aid; if a tie persists, use A ij w . In the aforementioned equation, v ( k ) j represents the kth best value of the jth criterion. For example, v ( 1 ) 2 represents the best value of the second criterion; v ( 2 ) 3 represents the second best value of the third criterion.

Euclidean distance from the ith alternative to the kth FPIS is denoted as P S i ( k ) and calculated as follows:

(23) P S i ( k ) = ∑ j = 1 n 1 4 ‖ v ij − v ( k ) j ‖ 2 = ∑ j = 1 n 1 4 [ ( A ij w − A ξ k j w ) 2 + ( B ij w − B ξ k j w ) 2 + ( C ij w − C ξ k j w ) 2 + ( D ij w − D ξ k j w ) 2 ] ,

where i ∈ { 1 , 2 , … , m } , k ∈ { 1 , 2 , … , m } , v ij = ( A ij w , B ij w , C ij w , D ij w ) as in Step 3, and v ( k ) j = ( A ξ k j w , B ξ k j w , C ξ k j w , D ξ k j w ) as in equation (22).

Step 5. Find the spectrum of FNISs, which encompasses the most FNIS ( FNIS ( 1 ) ) , second FNIS ( FNIS ( 2 ) ) , …, kth FNIS ( FNIS ( k ) ) , …, and mth FNIS ( FNIS ( m ) ) , using equation (24). Then, calculate the Euclidean distance from each alternative to each of the FNISs using equation (25):

(24) FNIS ( k ) = { v ( k ) 1 , v ( k ) 2 , … , v ( k ) j , … , v ( k ) n } j ∈ { 1 , 2 , … , n } ; k ∈ { 1 , 2 , … , m } ,

where v ( k ) j = ( A ξ k j w , B ξ k j w , C ξ k j w , D ξ k j w ) , ξ k is the row index (from 1 to m) of the kth smallest value in A ij w ; in case of ties (i.e., multiple kth smallest values in A ij w ), use B ij w to break the tie; if there is still a tie, use C ij w to aid. If a tie persists, use D ij w . In this step, v ( k ) j represents the kth worst value of the jth criterion. For example, v ( 1 ) 2 represents the worst value of the second criterion; v ( 2 ) 3 represents the second worst value of the third criterion.

Euclidean distance from the ith alternative to the kth FNIS is denoted as N S i ( k ) and calculated as follows:

(25) N S i ( k ) = ∑ j = 1 n 1 4 ‖ v ij − v ( k ) j ‖ 2 = ∑ j = 1 n 1 4 [ ( A ij w − A ξ k j w ) 2 + ( B ij w − B ξ k j w ) 2 + ( C ij w − C ξ k j w ) 2 + ( D ij w − D ξ k j w ) 2 ] ,

where i ∈ { 1 , 2 , … , m } , k ∈ { 1 , 2 , … , m } , v ij = ( A ij w , B ij w , C ij w , D ij w ) as in Step 3, and v ( k ) j = ( A ξ k j w , B ξ k j w , C ξ k j w , D ξ k j w ) as in equation (24).

Step 6. Find the average solution for each criterion using equation (26) and denote it as v ̅ j . Then, calculate the Euclidean distance from each alternative to the average solution using equation (27) and denote it as S i ( avg ) :

(26) v ̅ j = ( A ̅ j w , B ̅ j w , C ̅ j w , D j w ) = ∑ i = 1 m A ij w m , ∑ i = 1 m B ij w m , ∑ i = 1 m C ij w m , ∑ i = 1 m D ij w m for j ∈ { 1 , 2 , … , n } ,

(27) S i ( avg ) = ∑ j = 1 n 1 4 ‖ v ij − v ̅ j ‖ 2 = ∑ j = 1 n 1 4 [ ( A ij w − A ̅ j w ) 2 + ( B ij w − B ̅ j w ) 2 + ( C ij w − C ̅ j w ) 2 + ( D ij w − D ̅ j w ) 2 ] ,

where i ∈ { 1 , 2 , … , m } .

Step 7. Determine the overall positive-ideal distance using equation (28), which is essentially the weighted sum distance of one alternative to the first half of the FPISs:

(28) S i ( pos − ideal ) = ∑ k = 1 m + 1 2 1 k P S i ( k ) for i ∈ { 1 , 2 , … , m } when m is an odd number ∑ k = 1 m 2 1 k P S i ( k ) for i ∈ { 1 , 2 , … , m } when m is an even number .

Likewise, determine the overall negative-ideal distance using equation (29), which is essentially the weighted sum distance of one alternative to the first half of the FNISs:

(29) S i ( neg − ideal ) = ∑ k = 1 m + 1 2 1 k N S i ( k ) for i ∈ { 1 , 2 , … , m } when m is an odd number ∑ k = 1 m 2 1 k N S i ( k ) for i ∈ { 1 , 2 , … , m } when m is an even number .

Step 8. Calculate the positive-ideal/negative-ideal ratio (R i ) and then performance score (P i ) of each alternative as follows:

(30) R i = S i ( pos − ideal ) S i ( neg − ideal ) for i ∈ { 1 , 2 , … , m } ,

(31) P i = 1 1 + R i 2 + S i ( avg ) for i ∈ { 1 , 2 , … , m } .

The recommended alternative is that with the highest P i value.

2.4 Fuzzy sPROBID with trapezoidal fuzzy numbers

The first five steps of the fuzzy sPROBID are identical to Steps 1–5 outlined in Section 2.3, and Step 6 is omitted in the fuzzy sPROBID. Next, to determine the performance score (P i ) of each alternative based on P S i ( k ) and N S i ( k ) , Steps 7 and 8 in Section 2.2 can be directly applied, which are unchanged regardless of whether triangular or trapezoidal fuzzy numbers are used.

2.5 Program development and descriptions

The fuzzy PROBID and sPROBID methods discussed in the previous sections have been implemented in a program using visual basic application (VBA) in Microsoft Excel. The choice of Excel was made due to its widespread availability and popularity among academics and professionals alike. As a result, researchers and practitioners from any field with an interest in MCDM and familiarity with Microsoft Excel can use the program with ease.

The main interface of the program is shown in Figure 1. Decision makers simply need to follow the five steps outlined in the figure. The “Overview” sheet provides a brief guide on how to use the program, and the “Results from MOO” sheet is where decision makers should enter their DOM. This program for fuzzy PROBID and sPROBID methods (referred to as F-EMCDM) is built upon our well-received EMCDM program [17,34]. Readers who are interested in these programs (i.e., EMCDM for crisp DOM and F-EMCDM for fuzzy DOM), as well as other related programs that are listed on https://blog.nus.edu.sg/rangaiah/free-optimisation-software/, can obtain a copy by approaching the first author wangzhiyuan@u.nus.edu) or Professor Rangaiah chegpr@nus.edu.sg).

Figure 1 Main interface of the F-EMCDM program implementing fuzzy PROBID and sPROBID methods.
Figure 1

Main interface of the F-EMCDM program implementing fuzzy PROBID and sPROBID methods.

3 Application to assess FP of firms

Inspired by the various numerical examples used in explaining mathematical steps [35], in this section, we first collect the crisp data on several FP indicators for 32 firms listed in the metal-goods sub-sector of the BIST (Borsa Istanbul, the main stock exchange in Turkey) manufacturing sector. Specifically, we gather the crisp data on ROE, Altman Z-score, Market Value Added (MVA) spread, Tobin’s Q, and SR from a commercial FINNET database. The crisp data for ROE, Altman Z-score, MVA spread, and Tobin’s Q are obtained for both December 2019 and December 2020, as shown in Table 3. On the other hand, SR represents the calculated SR for December 2020 based on the stock price in December 2019 and is included in the last column of Table 3. As studied and reported by Baydaş and Pamučar [15], the changes in the FP indicators (e.g., ΔROE, the difference in ROE between December 2019 and December 2020) are of more interest to investors and statistically have a stronger correlation with SR, compared to static values of the FP indicators (e.g., ROE in December 2020). Therefore, we proceed to calculate the changes (Δ) in ROE, Altman Z-score, MVA spread, and Tobin’s Q between December 2019 and December 2020, and they are presented in Table 4 and will be used as the crisp DOM.

Table 3

Collected crisp data on FP indicators (ROE, Altman Z-score, MVA spread, Tobin’s Q, and SR) for 32 listed firms

Firm December 2019 December 2020 SR
ROE Altman Z-score MVA spread Tobin’s Q ROE Altman Z-score MVA spread Tobin’s Q
ALCAR 0.041 4.002 0.567 1.797 0.136 10.897 7.586 6.113 5.082
ARCLK 0.097 1.657 −0.604 0.943 0.205 1.743 −0.155 0.827 0.459
ASUZU 0.039 1.515 0.536 0.748 0.024 1.388 1.067 1.013 0.479
AYES 0.051 3.518 −0.226 0.439 0.282 5.182 1.504 1.516 4.183
BALAT −0.142 0.48 0.619 0.702 0.147 3.132 14.894 4.456 5.645
BFREN 0.405 13.876 9.543 7.197 0.346 29.209 44.369 22.913 3.274
BNTAS 0.105 6.069 0.09 1.492 0.115 6.902 0.665 1.9 0.861
DITAS 0.041 2.429 1.844 1.466 0.175 2.396 2.798 1.375 0.363
EGEEN 0.292 6.851 1.062 2.161 0.371 11.532 6.632 5.653 1.976
EMKEL −0.026 0.48 −1.092 0.321 −0.015 0.953 0.676 0.997 2.723
FMIZP 0.529 46.975 10.912 10.914 0.533 80.338 29.931 25.599 1.492
FORMT −0.315 1.91 2.598 1.973 0.096 2.776 4.671 2.239 0.583
FROTO 0.42 4.163 3.666 1.796 0.596 4.373 4.592 2.224 0.778
GEREL −0.329 1.49 0.977 1.297 0.075 2.769 2.651 2.2 0.925
IHEVA 0.108 9.753 1.71 2.85 0.087 6.945 0.46 1.884 −0.399
JANTS 0.289 5.288 1.297 2.018 0.403 14.56 8.789 7.018 4.690
KARSN 0.031 1.124 0.395 0.727 0.028 1.72 1.803 1.564 1.057
KATMR −0.12 1.192 −3.311 1.544 −0.156 1.309 −0.524 1.661 −0.198
KLMSN 0.354 2.285 0.612 1.442 0.142 1.724 −0.632 1.293 −0.231
MAKTK 0.085 5.454 1.844 2.631 0.049 6.895 3.905 3.944 0.814
OTKAR 0.551 2.862 3.413 2.137 0.614 2.817 5.394 2.283 1.003
PARSN 0.013 0.965 −0.105 0.988 0.119 1.159 −0.031 0.997 0.434
PRKAB 0.097 2.82 0.475 0.953 0.105 4.51 5.443 2.728 3.654
SAFKR 0.091 3.441 0.241 1.22 0.169 7.41 4.142 4.007 3.991
SAYAS −0.031 2.878 1.923 1.686 0.531 3.024 4.903 1.818 2.799
SILVR −0.149 2.06 0.362 0.703 0.207 3 5.799 1.412 2.981
TMSN −0.047 1.234 −0.162 0.831 0.071 2.499 0.757 1.403 1.195
TOASO 0.342 2.88 1.559 1.298 0.399 2.319 1.872 1.183 0.263
TTRAK 0.146 2.6 1.487 1.665 0.539 4.274 4.985 2.726 2.298
ULUSE 0.232 12.249 4.429 4.738 0.019 11.09 9.229 6.71 0.930
VESBE 0.315 2.589 1.169 0.896 0.396 2.804 1.189 1.134 0.893
VESTL 0.093 0.787 −0.15 0.011 0.257 1.099 −0.389 0.231 0.669
Table 4

Crisp DOM: changes (Δ) in ROE, Altman Z-score, MVA spread, and Tobin’s Q between December 2019 and December 2020

Firm ΔROE ΔAltman Z-score ΔMVA spread ΔTobin’s Q
ALCAR 0.095 6.895 7.019 4.316
ARCLK 0.108 0.086 0.449 −0.116
ASUZU −0.015 −0.127 0.531 0.265
AYES 0.231 1.664 1.73 1.077
BALAT 0.289 2.652 14.275 3.754
BFREN −0.059 15.333 34.826 15.716
BNTAS 0.01 0.833 0.575 0.408
DITAS 0.134 −0.033 0.954 −0.091
EGEEN 0.079 4.681 5.57 3.492
EMKEL 0.011 0.473 1.768 0.676
FMIZP 0.004 33.363 19.019 14.685
FORMT 0.411 0.866 2.073 0.266
FROTO 0.176 0.21 0.926 0.428
GEREL 0.404 1.279 1.674 0.903
IHEVA −0.021 −2.808 −1.25 −0.966
JANTS 0.114 9.272 7.492 5
KARSN −0.003 0.596 1.408 0.837
KATMR −0.036 0.117 2.787 0.117
KLMSN −0.212 −0.561 −1.244 −0.149
MAKTK −0.036 1.441 2.061 1.313
OTKAR 0.063 −0.045 1.981 0.146
PARSN 0.106 0.194 0.074 0.009
PRKAB 0.008 1.69 4.968 1.775
SAFKR 0.078 3.969 3.901 2.787
SAYAS 0.562 0.146 2.98 0.132
SILVR 0.356 0.94 5.437 0.709
TMSN 0.118 1.265 0.919 0.572
TOASO 0.057 −0.561 0.313 −0.115
TTRAK 0.393 1.674 3.498 1.061
ULUSE −0.213 −1.159 4.8 1.972
VESBE 0.081 0.215 0.02 0.238
VESTL 0.164 0.312 −0.239 0.22

Subsequently, we seek the expertise and domain knowledge of a financial expert to create two separate sets of fuzzy DOMs using triangular and trapezoidal linguistic terms, respectively, based on the crisp data pertaining to ROE, Altman Z-score, MVA spread, and Tobin’s Q from Tables 3 and 4. The created triangular and trapezoidal fuzzy DOMs are shown in Table 5.

Table 5

Fuzzy DOMs: created by financial expert using triangular and trapezoidal linguistic terms, respectively

Firm Triangular fuzzy DOM Trapezoidal fuzzy DOM
ROE Altman Z-score MVA spread Tobin’s Q ROE Altman Z-score MVA spread Tobin’s Q
ALCAR Average Very high Very high Very high Average Very high Very high Very high
ARCLK Average Low Average Low Average Fairly low Fairly low Fairly low
ASUZU Low Low Average Low fairly low Fairly low Average Fairly low
AYES Very high Average Average Average Very high Fairly low Fairly high Average
BALAT Very high High Very high Very high Very high High Very high Very high
BFREN Very high Very high Very high Very high Very high Very high Very high Very high
BNTAS Average Average Low Average Average Fairly high Fairly Low Average
DITAS Average Average Average Average Fairly high Average Fairly low Average
EGEEN Very high High High Very high Very high High High Very high
EMKEL Low Low Average Average Fairly low Fairly low Fairly low Average
FMIZP Very high Very high Very high Very high Very high Very high Very high Very high
FORMT High Average Average Average High Average Average Average
FROTO Very high Average Average Average Very high Average Average Average
GEREL High Average Average Average High Average Average Average
IHEVA Low Average Low Low Low Fairly Low Fairly Low Fairly Low
JANTS Very high Very high High Very high Very high Very high High Very high
KARSN Average Average Average Average Average Average Average Average
KATMR Very Low Average Low Average Very low Average Fairly low Average
KLMSN Low Low Low Average fairly low Fairly low Fairly low Average
MAKTK Average Average Average High Average Fairly high Average Fairly high
OTKAR Very high Average Average Average Very high Average Fairly high Average
PARSN Average Low Low Average Fairly low Average fairly low Average
PRKAB Average High High Average Average Fairly high High Fairly high
SAFKR Average Very high High High Fairly high Very high High High
SAYAS Very high High High Average Very high High High Average
SILVR High High High Average High High High Average
TMSN Average Average Average Average Fairly high Average Average Average
TOASO Very High Average Average Average Very high Average Average Average
TTRAK Very high High High Average Very high Fairly high High Average
ULUSE Low High High High Fairly low High High High
VESBE Very high Average Average Average Very high Average Average Average
VESTL Average Average Low Low Fairly high Average Fairly low Fairly low

Next, MCDM analyses are carried out using our EMCDM program for the crisp DOM (Table 3) and F-EMCDM program for the fuzzy DOMs (Table 5). In this process, the four benefit (i.e., maximization) criteria, namely, ROE, Altman Z-score, MVA spread, and Tobin’s Q, are assigned equal weights (i.e., 0.25 for each) to assess the MCDM-based FP of the 32 firms and rank them accordingly. SR serves as the benchmark, and the FP ranking based solely on SR is used as the ground truth. Finally, we calculate Spearman’s rank correlation coefficient (ρ) to evaluate the correlation between each MCDM-based FP ranking and the SR-based FP ranking. ρ ranges from −1 to +1, where −1 indicates a perfect negative correlation, +1 represents a perfect positive correlation, and 0 suggests no correlation between the two. Figure 2 effectively summarizes the abovementioned steps. Spearman’s rank correlation coefficient of +1 is generally desirable.

Figure 2 Flowchart illustrating the systematic approach to determine MCDM-based FP rankings using crisp and fuzzy DOMs, and find their Spearman’s rank correlation coefficient (ρ) with SR-based FP ranking.
Figure 2

Flowchart illustrating the systematic approach to determine MCDM-based FP rankings using crisp and fuzzy DOMs, and find their Spearman’s rank correlation coefficient (ρ) with SR-based FP ranking.

4 Discussions and limitations

The calculated Spearman’s rank correlation coefficients between each MCDM-based FP ranking and the SR-based FP ranking are presented in a descending order in Figure 3. It is worth mentioning that the p-values of all Spearman’s rank correlation coefficients are less than 0.001, indicating high statistical significance and providing strong evidence to support the calculated correlation coefficients. These results reveal important insights regarding the performance of MCDM methods tested and different types of DOMs in this application. Specifically, first, when using the crisp DOM, sPROBID outperforms PROBID by leading to a higher correlation coefficient with the SR-based FP ranking (i.e., 0.655 vs 0.569). Second, fuzzy sPROBID performs better than fuzzy PROBID for both triangular and trapezoidal fuzzy DOMs. Third, the correlation results based on the fuzzy DOMs are consistently higher than those based on the crisp DOM. Fourth, fuzzy sPROBID provides slightly better correlation result with the triangular fuzzy DOM than with the trapezoidal fuzzy DOM (i.e., 0.770 vs 0.769), whereas fuzzy PROBID gives better correlation result with the trapezoidal fuzzy DOM than with the triangular fuzzy DOM (i.e., 0.704 vs 0.674). Finally and notably, fuzzy sPROBID with triangular fuzzy DOM performs the best overall, yielding a high Spearman’s rank correlation coefficient of 0.770 with the SR-based FP ranking.

Figure 3 Descending order of Spearman’s rank correlation coefficient (ρ) between each MCDM-based FP ranking and SR-based FP ranking.
Figure 3

Descending order of Spearman’s rank correlation coefficient (ρ) between each MCDM-based FP ranking and SR-based FP ranking.

In addition, Sałabun and Urbaniak [36] recently proposed a new coefficient, termed the WS coefficient, to measure the similarity between two rankings. The key innovation of this coefficient is that positions at the top of the ranking have a more significant impact on the similarity than those at the bottom of the ranking. For instance, a reversal of alternatives between the first and second positions would be regarded as more impactful than a reversal between the fourth and fifth positions, during the similarity measurement process. On the other hand, Spearman’s rank correlation coefficient is indifferent to whether the reversals occur at the top or bottom of the ranking, treating each position with equal impact.

In this study, WS coefficient is also adopted to supplement the findings derived from the more commonly used Spearman’s rank correlation coefficient. Note that, in the present application, the WS coefficient values obtained are significantly higher than the Spearman’s rank correlation coefficient values, and the range of the former is less than that of the latter. This is because the top-ranked alternatives do not fluctuate significantly among the different rankings; these alternatives weigh more heavily when calculating the WS coefficient and lead to the overall higher WS coefficient values. Additionally, as seen from Figure 4, even when we account for the varied impacts of different positions using WS coefficient, the principal observation underscored by Spearman’s rank correlation coefficient from Figure 3 remains intact, i.e., the correlations (or similarities) based on the fuzzy DOMs are still consistently higher than those based on the crisp DOM. The remaining findings are largely congruent with those from Figure 3. For instance, when using crisp DOM, sPROBID outperforms PROBID by generating a slightly higher WS coefficient (i.e., 0.8552 vs 0.8543). Fuzzy sPROBID also performs slightly better than fuzzy PROBID for both triangular and trapezoidal fuzzy DOMs (i.e., 0.8823 vs 0.8821, and 0.8869 vs 0.8864).

Figure 4 WS coefficient between each MCDM-based FP ranking and SR-based FP ranking.
Figure 4

WS coefficient between each MCDM-based FP ranking and SR-based FP ranking.

The findings demonstrate the effectiveness of fuzzy DOM, not only under conditions of fuzzy information, but also in contexts where crisp numerical information is present. Indeed, financial experts interpret numerical information from balance sheets using their domain knowledge to form linguistic assessments. For instance, the financial expert consulted in this study prioritizes dynamic, time-varying ratios over static ones. Furthermore, financial experts are also acutely aware of the implications and significance of any increases or decreases in the FP criteria. The use of crisp DOM, however, does not always successfully navigate these complex and nonlinear cases. Implementing fuzzy techniques leads to a more flexible linguistic interpretation that incorporates the insights of decision makers/experts, thus facilitating the assessment of nonlinear situations. This flexibility is observed from the results obtained in the present study and is a key factor contributing to the superior performance of fuzzy MCDM methods coupled with fuzzy DOM in better representing real-world financial market situations.

Regarding limitations, this study is grounded in some static dimensions for financial data, including the use of MCDM methods, normalization technique, assigned weights, time range, criteria, and alternatives; any change in these may affect the final results to some extent. However, the focus of this study revolves around the proposed exploration of the selection between fuzzy and crisp DOM through comparisons with an identified objective benchmark, and the introduction of the new fuzzy PROBID and fuzzy sPROBID methods for fuzzy MCDM. The comparison between crisp and fuzzy DOM can be expanded to more MCDM methods, more objective benchmarks, and more problem scenarios.

5 Conclusions

In conclusion, accurate assessment of FP is crucial for informed decision-making in the financial market. MCDM methods are useful in evaluating trade-offs among multiple criteria and making informed decisions. Nonetheless, selecting the most appropriate MCDM method for a specific application remains a challenge. In this study, we took real-life SR-based ranking as the objective benchmark to compare FP rankings generated by different MCDM methods. We observed from the literature review that decision makers/experts, based on their domain knowledge, prefer to give fuzzy linguistic terms (instead of crisp numerical values) of criteria, to establish a fuzzy DOM. Fuzzy MCDM methods are then used to rank and select the best alternative. Moreover, previous publications have not adequately studied the selection between fuzzy and crisp DOM in MCDM. This work bridged the research gap through comparative analyses of fuzzy and crisp DOM in MCDM and also proposed two new fuzzy MCDM methods, namely, fuzzy PROBID and fuzzy sPROBID. These fuzzy methods coupled with fuzzy DOM produced higher correlation results with the SR-based FP ranking than their crisp counterparts. Fuzzy sPROBID with triangular fuzzy DOM produces a high Spearman’s rank correlation coefficient of 0.770 with the SR-based FP ranking. Our study advanced the investigation into the application of fuzzy MCDM methods and provided insights into the performance of MCDM methods tested with different types of DOMs (triangular fuzzy, trapezoidal fuzzy, and crisp). For future research, the proposed methodology in this work can be expanded to more MCDM methods, more objective benchmarks, and more problem scenarios.

  1. Conflict of interest: Dr. Željko Stević is the Guest Editor of the “Special Issue on Development of Fuzzy Sets and Their Extensions”, but was not involved in the review process of this article.

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Received: 2023-03-24
Revised: 2023-07-21
Accepted: 2023-09-21
Published Online: 2023-11-24

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
  26. Feature fusion-based text information mining method for natural scenes
  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
  34. Small perturbations of critical nonlocal equations with variable exponents
  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
  81. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives
  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
  88. Petri net analysis of a queueing inventory system with orbital search by the server
  89. Asymptotic stability of an epidemiological fractional reaction-diffusion model
  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
https://doi.org/10.1515/dema-2023-0117
Keywords for this article
fuzzy decision matrix; crisp decision matrix; PROBID; sPROBID; MCDM
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
  26. Feature fusion-based text information mining method for natural scenes
  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
  34. Small perturbations of critical nonlocal equations with variable exponents
  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
  81. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives
  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
  88. Petri net analysis of a queueing inventory system with orbital search by the server
  89. Asymptotic stability of an epidemiological fractional reaction-diffusion model
  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
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