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Article

Selection of Business Process Modeling Tool with the Application of Fuzzy DEMATEL and TOPSIS Method

1
School of Maritime Economics and Management, Dalian Maritime University, Dalian 116026, China
2
College of Engineering Science and Technology, Shanghai Ocean University, Shanghai 201306, China
*
Author to whom correspondence should be addressed.
Axioms 2022, 11(11), 601; https://doi.org/10.3390/axioms11110601
Submission received: 11 September 2022 / Revised: 24 October 2022 / Accepted: 25 October 2022 / Published: 28 October 2022
(This article belongs to the Special Issue Fuzzy Logic and Application in Multi-Criteria Decision-Making (MCDM))

Abstract

:
The business process modeling tool selection problem has a significant impact on the overall performance of enterprise business process modeling, which will directly affect the development of enterprise information systems. Apart from that, the process to select the business process modeling tool from all alternatives is a Multi-Criteria Decision Making (MCDM) problem. This paper develops a methodology based on the hybrid fuzzy Decision-Making Trial and Evaluation Laboratory (DEMATEL) and Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) method to help companies select the optimal business process modeling tool, where the business process modeling process is more efficient, economic and safe. The proposed method has the following state-of-the-art contributions and features: (1) the latest application of the MCDM methodology to the field of BPM tool selection, (2) addressing the direct and indirect impact between criteria in the selection of BPM tools, and (3) considering the hybrid fuzzy (uncertainty) decision-making issue in the BPM tool selection process. Meanwhile, the mathematical formula in TOPSIS can be regarded as a formula for solving a symmetric problem. The hybrid fuzzy DEMATEL method is used to obtain the weight for the criteria to be considered in the BPM tool selection process, and the TOPSIS method is used to obtain the final business process modeling tool.

1. Introduction

Business Process Modeling (BPM) is an effective activity of representing business processes for a company and is typically performed by business analysts who provide expertise in the modeling domain [1,2]. For enterprises, BPM is an essential proportion of the quality improvement of Information Systems (IS) [3]. BPM and management are becoming an essential part of today’s enterprises [4]. Meanwhile, Business Process Modeling (BPM) tools are considered a great way to connect BPM languages and modeling stakeholders. BPM tools can display the corresponding model to stakeholders, and stakeholders can also operate and control the model through BPM tools. For example, automobile manufacturing modelers can use BPM tools to build automobile manufacturing business process models.
Choosing a proper BPM tool can be seen as an essential problem for a company because poorly chosen BPM tools can have a serious and negative impact on a company’s BPM modeling process, which will directly affect the development of enterprise IS. Choosing an effective BPM tool for the company is a complicated process, and different criteria that affect BPM need to be considered [5]. Therefore, the process can be a difficult task for managers in the enterprise to carry out. Meanwhile, the selection of the BPM tool process can be seen as the Multi-Criteria Decision Making (MCDM) issue. The TOPSIS technique is a mainstream MCDM method that can approach the BPM tool selection problem. Additionally, to obtain the optimal BPM tool, the experts must analyze various information and factors in the company that affects BPM tool selection [6,7]. To select the optimal BPM tool, we have to consider different parameters, such as technical parameters (expressiveness, readability, usability, formability, etc.), economical parameters (application cost, operating cost, etc.), and so on. Meanwhile, there are direct and indirect affecting relationships between these parameters. Additionally, experts will have uncertainty when determining the influence between these parameters, which can be seen as a hybrid fuzzy problem. A fuzzy DEMATEL technique can approach hybrid fuzzy problems and the direct and indirect affecting relationships between the technical parameters.
Although BPM tool choice has a great impact on enterprise IS, there are very few related studies and references on this topic. Therefore, depending on the issues discussed above, the key target of the study is to propose a fuzzy DEMATEL and TOPSIS combination method to support companies in dealing with the BPM tool selection problem.
The main constitutions of this study are shown below:
  • Although the MCDM method has been applied in many fields, it has not been applied to the selection of BPM tools. Therefore, the first important constitution of this study is to propose criteria for the selection of BPM tools and an MCDM method for the selection of BPM tools.
  • In the BPM tool selection process for a company, there will be direct and indirect interdependence between all the criteria. Therefore, the second important conclusion of this paper is to use DEMATEL analysis to fix the direct and indirect influence problem between criteria in the BPM tool selection process.
  • When experts define the affecting rank between BPM tool selection criteria, there is uncertainty here because experts cannot clearly determine the impact of a specific scale value. Therefore, the third objective of this paper is to approach the hybrid fuzzy (uncertainty) decision-making issue.
The rest structure of the paper is shown below. The “Literature Review” section introduces business flow modeling tools and related methods for multicriteria decision-making. The section “Business Process Modeling Tool Selection Methodology” describes the proposed BPM tool selection methodology that integrates fuzzy DEMATEL and TOPSIS. Section “Results” depicts the simulation and experimental results of an example. A detailed discussion and future work will be presented in the “Discussion and Conclusion” section.

2. Literature Review

A business process is a continuous, gradual, and uncontrollable result that a series of intrinsically linked business activities or events produce [2]. It is very important for a company to effectively manage the business process. Apart from that, BPM [8] is one of the essential components of business process management, and BPM can describe the integration and relation of different enterprise activities [3]. The BPM is inseparable from a large number of BPM languages, and there are currently many BPM languages being advocated or practiced [9,10].
In the 1980s, since the first Framework Program of Research and Development (Esprit program), a large number of languages of BPM have appeared in North America and Europe [11]. These include MERISE [12], GRAI [13,14,15], NIAM [16], CIMOSA [17], IEM [18,19,20], UML [21], BPMN [22], EPC [23], Petri net [19], IDEFx [24], ARIS [25], 4EM [26], DEMO [27] and so on.
Here, all the business process modeling languages can be divided into the following three categories according to different modeling characterizations [20]:
  • Namely, early or activity-centered languages: NIAM, IDEFx, MERISE, IEM, GRAI, and so on.
  • Business process-centered languages: CIMOSA, ARIS, IEM, BPMN, EPC, Petrinet, and so on.
  • Enterprise knowledge-centered languages: 4EM, DEMO, etc.
In addition, various kinds of BPM tools can support all the BPM languages mentioned above. For example, GDToolkit [28], TimeNet [29], GreatSPN [30], JFern, JPetriNet and PIPE2 can support Petri nets [31]; ADONIS:CE, Bizagi Modeler, Cardanit, BPMN.io & family, Sparx Enterprise Architect, MagicDraw can support BPMN [32,33]; MO2GO can support IEM [20].
Here, some of the BPM tools are free of charge. For example, for the Petri net supporting tools, JFern, JPetriNet, and PIPE2 are free of charge, and GreatSPN, GDToolkit, and TimeNet are charged; for BPMN supporting tools, ADONIS: CE, Bizagi Modeler, Cardanit and BPMN.io & family are free of charge, and Sparx Enterprise Architect and MagicDraw are charged; for the IEM supporting tool, MO2GO is charged. Although some experts believe that more expensive BPMN tools have more comprehensive features and can handle more difficult business process modeling problems, they can also be more difficult to operate and control, and therefore, employee training can also be more difficult [33]. Meanwhile, some of the BPM languages and corresponding support tools are very easy to learn. For example, the IEM modeling language or MO2GO tool is very simple and straightforward to use, and beginners do not need much training when building a model [34]. Meanwhile, although some of the BPM tools are free of charge or cheap, the learning and expressive efficiency of these tools are not poor.
Therefore, it is very difficult for companies to select an optimal BPM tool, and when project managers select the BPM tools for their company, they must evaluate all candidate BPM tools depending on the different evaluation criteria, such as efficiency criteria, economic criteria, safety criteria, and so on.
Although it is very important for the company to select optimal BPM tools, which will directly affect the development of enterprise IS, there is very limited research and publications in this area. Many researchers and institutions are more focused on the analysis of BPM languages. Khouloud and Sonia proposed an approach for selecting a BPM language based on the requirements of the modeler and considered the different criteria for comparing modeling languages [3]. Vernadat reviewed and summarized important research works and contributions made to BPM over the last four decades, outlining major BPM constructs and their extensions as well as prominent modeling tools and methods [20]. Alotaibi analyzed BPM criteria, methods, and linguistics and divided them into different groups depending on their features [35]. Kožíšek and Vrana summarize the current knowledge of BPM languages, especially for UML, BPMN, and EPC, which are increasingly important in the agri-food industry, and they describe the history of BPM, currently mostly used alternatives [36]. Geiger et al., present an analysis of the current state and evolution of BPM Notation support and implementation [22].
However, these reference studies are related to the selection of BPM language or the review and introduction of different BPM languages, tools, or methods, and there is no specific method for BPM tool selection. Therefore, in this study, we will mainly focus on this issue.
The capability estimation and best choice of BPM tools are related to different levels and various criteria; therefore, the problem of BPM tool selection is the multiple criteria decision-making (MCDM) issue [37,38]. MCDM is a well-known branch of decision-making [39]. It is related to proposing and dealing with multistandard decision-making issues [40]. MCDM supports company BPM experts in quantifying special standards depending on their essentialness in different decision targets [41]. There are many different MCDM methodologies, and several mainstream MCDM methodologies are the Analytic Hierarchy Process (AHP) [42,43], Analytic Network Process (ANP) [44], Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) [45,46,47,48], classic MAUT [49], ELECTRE [50] and so on. Here, some methods are more adaptable than others in special decision issues.
The main objective of this research is to select the best BPM tool for companies where many parameters should be considered in decision-making.
These parameters are multidisciplinary (technical parameters, economical parameters, and time parameters), and we apply the TOPSIS method to obtain the final optimal BPM tool. The reasons for choosing the TOPSIS method are as follows [7]:
  • The pre- and post-steps of the TOPSIS method are logical and easy to comprehend.
  • The calculation steps at a glance.
  • The method can use straightforward mathematical criteria descriptions to find optimal candidate options.
  • The significance weights are considered in the decision-making process.
Meanwhile, the parameters that we considered in this paper are symmetry parameters. We have to consider not only benefit parameters (efficient parameters) but also cost parameters (economical parameters, time parameters). Therefore, we need to find a decision-making method that can handle positive and negative symmetrical parameters. TOPSIS is a well-known methodology as a symmetrical method used for solving MCMD problems [51,52]. To obtain the best options from all competitors, the TOPSIS methodology uses the key formula of the calculation result of the maximum distance from the Negative-Ideal (N-I) solution and the minimum distance from the Positive-Ideal (P-I) solution [51,52,53,54]. Therefore, it can be seen as a symmetric issue approaching the process in mathematical equations.
Apart from that, in the BPM tool selection process for a company, there will also be direct and indirect interdependence between all the criteria, such as the low readability of the BPM tool, which will directly affect and increase the difficulty of tool learning. Therefore, when we define the weight for the BPM tool selection criteria, we must consider direct and indirect interdependence. DEMATEL is an effective methodology to analyze the direct and indirect influence between criteria. There is an algorithm behind the DEMATEL analysis, which is always used to analyze and create the connections of causation between the assessment standards [55,56] or to derive interrelationships among factors [57]. Therefore, we can use DEMATEL analysis to fix the direct and indirect influence problem between criteria.
Additionally, in the DEMATEL analysis, when experts define the affecting rank between BPM tool selection criteria, there is uncertainty here because, in many cases, experts cannot clearly determine the impact of a specific scale value. This uncertain decision issue is the hybrid fuzzy decision-making issue. Many studies have dealt with hybrid fuzzy problems in decision-making. Mardani et al. [58] proposed a hybrid fuzzy AHP decision-making methodology to evaluate healthcare and medical problems. Akram et al. [59] proposed a fuzzy ELECTRE-II method for multicriteria decision-making problems. Celik et al. [60] used interval type-2 fuzzy AHP methods to approach decision-making problems in maritime transportation engineering. A Fuzzy ARAS method is also proposed for recycling facility location problems [61].
Although there are many methods of fuzzy application, there is no application for BPM tool selection. The above fuzzy methods do not consider the direct and indirect relationships between coefficients such as AHP and ELECTRE.
Hence, in this paper, the key objective of this study is to combine the hybrid fuzzy DEMATEL and TOPSIS methods to propose a BPM tool selection method considering the BPM process affecting parameters. This method can help company managers find the optimal BPM tools for their company.

3. Business Process Modeling Tool Selection Methodology

The BPM tool choice process for this research is shown in Figure 1.
First, it is necessary to define the benefit and cost standards that will be applied for the assessment of BPM tool alternatives. Here, we will also determine the corresponding rank value for all the criteria. Afterward, we can use the Fuzzy DEMATEL technique to define the weight for the standards. Then, it is possible to decide the candidate options assessment result according to the TOPSIS method. Finally, we can define the final BPM tool for the company.

3.1. Determining the Criteria to Be Used in Evaluation for BPM Tool Alternatives

The main objective of this method is to select the optimal BPM tool for the company that purchases or applies it so that the BPM tool can be used more efficiently, economically, quickly, and safely. The criteria to be considered in the selection of the BPM tool can be decided by the company experts based on the main objective. After the consideration, four important types of criteria to be used for BPM tool selection are proposed. Meanwhile, each criterion type contains several subcriteria. These standards are often what companies need and what company managers think. All these criteria allow the BPM process to be more efficient, economical, and time-saving. The four criteria and corresponding sub-criteria are shown in Table 1.
In Table 1, we find that there are four types of criteria (Efficiency parameters (C1), Economical parameters (C2), Time parameters (C3), and Safety parameters(C4)), and every type of criterion also has different corresponding sub-criteria. A detailed description of all these criteria can be seen as follows:
  • Efficiency parameters (C1) are related to the factors affecting BPM efficiency. Efficiency parameters include Expressiveness, Readability, Usability, Formality, and Ease of Learning.
    • Expressiveness (C11): This parameter checks whether the modeling tool can express various kinds of organizational environments on the basis of informational, structural, behavioral, and functional perspectives [62].
    • Readability (C12): This parameter checks whether the model is simple to comprehend for stakeholders.
    • Usability (C13): This parameter checks whether the modeling tool is easy to apply and install.
    • Formality (C14): This parameter checks whether the model has ambiguities and inaccuracies in model interpretation.
    • Ease of Learning (C15): This parameter checks whether the modeling tool and language are easy for the company modelers to learn.
  • Economical parameters (C2) related to the various expenses incurred when using the tool. Economical parameters include application and installment costs, operating costs, and training costs.
    • Application and installment cost (C21): The cost of parameter configuration, application, and installation when starting to use the modeling tool in the company.
    • Operating cost (C22): The rental cost of the modeling tool and the salary of the company modelers.
    • Training cost (C23): Training costs of modeling tools for company modelers.
  • Time parameters (C3) cluster includes application and installment time, operating time, and training time.
    • Application and installment time (C31): The time consumption of parameter configuration, application, and installation when starting to use the modeling tool in the company.
    • Operating time (C32): The time consumed by company modelers operating the modeling tool.
    • Training time (C33): The time consumed by company modelers to learn the modeling language and tool.
  • Safety parameters (C4) include all parameters that the BPM tool can affect the safety of the IS of the company. Safety parameters include Internal safety and External safety.
    • Internal safety (C41): The safety of BPM tools inside the company, such as the software freezes, disappearance, and error storing modeling data.
    • External safety (C42): The safety of the BPM tool outside of the company, for example, if the tool is vulnerable to network intrusion and whether the modeling data are easily leaked to the outside through the tool.
Then, we can define the evaluation rank for the sub-criteria. Here, we can use a 1–5 scale rank, as shown in Table 2.
Depending on Table 2, company experts can assess the BPM tool alternatives.

3.2. Determining the Fuzzy Weight for the Criteria

Although DEMATEL is an effective methodology to analyze the direct and indirect influence between criteria, it cannot deal with the problem of misjudgment with certainty, and the influence value between the criteria directly depends on the uncertainty decision result of experts on the actual case number. Therefore, for the purpose of approaching the uncertainty issue, Fuzzy DEMATEL extends DEMATEL.
Therefore, after we define the criteria and corresponding rank to be used in the evaluation of BPM tool alternatives, we can use the Fuzzy DEMATEL technique to determine the fuzzy weight for all the standards. The fuzzy set theory was proposed by Zadeh [63] to approach uncertain issues. Here, there is one variable value that must be known, and the variable is the Triangular Fuzzy Number (TFN). The fuzzy theory uses a fuzzy number to present outcome information that experts cannot determine or quantify in a decision model. Meanwhile, the TFN [63] is always applied to solve fuzzy problems in uncertain environments. TFNs are composed of triples (d, e, f). Here, the “d” and “f” values are the upper and lower limits of the fuzzy numbers, respectively, and “e” is the most likely number. The membership function fA(x) of TFN is expressed as Equation (1) and Figure 2.
f A ( y ) = { y d e d , ( d   y e ) f y f e , ( e   y f )   0 , ( y < d ,   y > f )
where fA(y) value will vary between 0 and 1.
After that, we can release the basic process of the Fuzzy DEMATEL method, which can be shown as follows:
  • Step 1: Collect the opinions of company experts for direct influence between criteria.
In this step, company experts are asked to define the rank of direct influence between criteria according to pairwise comparison. The fuzzy influence rank value and corresponding description are shown in Table 3. Direct fuzzy influence degree and corresponding triangular fuzzy scale.
In Table 3, we can find that the direct influence degree ranges from No impact to Very high impact, and the higher the degree, the greater the rank value. Various forms of scales, including sequential, exact, ratio, interval, or perhaps a combination of these, could have been included in this study. However, the sequential scales (linguistic variables) are more appropriate for expressing expert preferences, especially when the number of alternative and qualitative criteria is high. Meanwhile, in this research, the range of the Triangular Fuzzy Scale varies from 0 to 1, and there are 5 direct influence degrees (No impact to Very high impact in Table 3). Therefore, to ensure that there are five Triangular Fuzzy Scales and that the five direct influence degree points are at the center of the triangle (Figure 2), we need to define the Triangular Fuzzy Scale, as in Table 3. Such as Low impact has a triangular fuzzy scale (0, 0.25, 0.5), and the point at 0.25 is exactly at the center of the triangle.
BPM experts can use this degree to define the direct influence degree for every two criteria or sub-criteria, and we can collect opinions of all the experts and release value of TFN for every option. After that we have to defuzzify the TFN and obtain the crisp influence value between criterions. The mainstream and widely acknowledge defuzzification method is the CFCS [64], and the CFCS can obtain the optimal crisp value. If we assume that B ij = ( d ij q ,   e ij q ,   f ij q ) means the TFN for the criterion i influence criterion j in qth fuzzy survey, the detailed process for defuzzification of CFCS can be seen as following four steps:
(1)
Normalization:
nf ij q = ( f ij q min   d ij q ) / ( max f ij q min d ij q )
ne ij q = ( e ij q min   d ij q ) / ( max f ij q min d ij q )
nd ij q = ( d ij q min   d ij q ) / ( max f ij q min d ij q )
(2)
Calculate right and left normalized numbers:
rn ij q = nf ij q / ( 1 + nf ij q   ne ij q )
ln ij q = ne ij q / ( 1 + ne ij q   nd ij q )
(3)
Calculate total generalized crisp numbers:
tn ij q = [ ln ij q ( 1   ln ij q ) +   rn ij q × rn ij q ] / ( 1 ln ij q +   rn ij q )
(4)
Calculate crisp numbers:
p ij q = min d ij q +   tn ij q × ( max f ij q min d ij q )
After the defuzzification process through Equations (2)–(8), the TFN will be defuzzified to crisp numbers. After that, depending on the crisp numbers, we can create a nonnegative fuzzy matrix D = [pmn] (Equation (9)).
D = [   0           p 12           p 1 j           p 1 K     p 21       0               p 2 j           p 2 K                                                       p n 1   p n 2             0                 p nK                                                     p K 1   p K 2       p Kj           0           ]
where   p nj is the direct fuzzy influence rank to which the expert recognized standard n impacts on standard j, and K is the total number of criteria for BPM tool selection.
  • Step 2: Calculate average fuzzy matrix T = [tnj] (Equation (10)).
t nj = 1 TN k = 1 TN p nj k
where   p mn is the direct fuzzy influence rank to which the expert recognized standard n impacts on standard j, and TN is the total number of company experts.
  • Step 3: Calculate the normalized direct fuzzy impact matrix G.
The value in the normalized direct fuzzy influence matrix G ranges between [0, 1], and the calculation process of normalization is shown in Equations (11) and (12):
G = λ T ,
where
λ = Min [ 1 max 1 n J j = 1 J t nj ]
where G is the normalized direct fuzzy influence matrix, and J is the total number of factors to the fuzzy influence virtual team member selection.
In Equation (12), depending on Markov chain theory, the normalized direct influence matrix G after it has been multiplied by itself, all the values of the matrix will be close to 0, which is a zero matrix, and it means that lim o G o equals [ 0 ] n × n .
  • Step 4: Calculate the overall direct and indirect fuzzy impact matrix E.
The calculation process of the overall direct and indirect fuzzy impact matrix E can be seen as Equations (13) and (14).
H = lim q ( G + G 2 + +   G q ) = q = 1 G q  
where
q = 1 G q =   G 1 + G 2 + +   G q = G ( I + G 1 + G 2   +   G q 1 ) = G ( I G ) 1 ( I G ) ( I + G 1 + G 2   +   G q 1 ) = G ( I G ) 1 ( I G q ) H = G ( I G ) 1
  • Step 5: Calculate the sums of the rows and columns of matrix H.
The calculation process can be seen as Equations (15) and (16).
ro = [ ro n ] = [ j = 1 J h nj ]
co = [ co j ] = [ n = 1 J h nj ]
where ro is the sum of the nth row in matrix T, co is the sum of the jth column in matrix T, and [ ] denotes the matrix consisting of the resultant values.
In Equation (15), the ro n means the total given both direct and indirect fuzzy effects from the criteria n to the other criteria, and co j means the total received both direct and indirect fuzzy effects from other criteria to criteria j. Therefore, when n equals j, the value of ( ro n +   co j ) means the overall impacts both given and received by criteria j, and ( ro n +   co j ) means the centrality of the factors n in all factors. Centrality indicates the position of the factor in the evaluation index system and the magnitude of its effect. Therefore, the value of ( ro n +   co j ) means the direct and indirect fuzzy effects value of the factors j.
  • Step 6: Calculate the normalized ( ro n +   co j ) value.
    NRC l = ( ro l +   co l )   j = 1 J ( ro j +   co j )
    where l = 1 J NRC l = 1 and NRC l are normalized ( ro l +   co l ) values.
In Equation (17), we find that this is the easiest approach to reformulate the feature range from 0 to 1, and we can use the normalized ( ro n +   co j ) value (NRC) to define the fuzzy weight for the criteria to influence BPM tool selection.

3.3. Determine the TOPSIS Result Value for All the Candidate BPM Tool Alternatives

After we define the criteria and corresponding weight and rank to be used in the evaluation of BPM tool alternatives, we can use the TOPSIS method to obtain the final BPM tool for the company. The basic process of the TOPSIS method is shown as follows:
  • Step 7: Establish a decision matrix for alternatives (Equation (18)).
    D = [ A 1 A 2 A A I ] [   C 11 C 12 C 1 j C 1 J C 21 C 22 C 2 j C 2 J C i 1 C j 2 C i j C i J C I 1 C I 2 C I j C I J ]
    where D is the decision matrix, Ai is alternative to i, c ij is the jth standard number corresponding to the ith alternative (Ai), I is the number of alternatives, and J is the number of criteria.
  • Step 8: Get the normalized decision matrix Z(=zij) (Equation (19)).
    z ij = c ij i = 1 I c ij 2
  • Note. zij = Normalized number for jth standard corresponding to ith alternative. I = Sum of candidate options.
  • Step 9: Obtain the weighted normalized decision matrix X(=xij) (Equation (20)).
    x ij = w j · z ij
The fuzzy weighted normalized decision matrix value is obtained from the multiplication result between the matrix value “ z ij ” in Equation (19) and the corresponding fuzzy weights. In this research, the fuzzy weight can be defined by the normalized ( ro n +   co j ) value in Section 3.2, and the sum of fuzzy weights is 1 ( j = 1 n w j = 1 ).
  • Step 10: Decide the P-I and N-I solutions (Equations (21) and (22)).
    P-I   solution :   x j * = { max i x ij ,   i l min i x ij ,   i l
    where l′ is the value set associated with benefit criteria and l″ is the value set associated with cost criteria.
    N-I   solution :   x j 0 = { min i x ij ,   i l max i x ij ,   i l
    where l′ and l″ are the number set corresponding to benefit and cost standards, respectively.
  • Step 11: Calculate the n-dimensional Euclidean distance from each solution to the P-I solution and the N-I solution (Equations (22) and (23)).
    Distance   to   P-I   solution :   d i * = j = 1 J ( x ij   x j * ) 2
    Distance   to   N-I   solution :   d i 0 = j = 1 J ( x ij x j 0 ) 2
  • Step 12: Calculate the relative closeness to the idea solution (Equation (25)).
    H i * = d i 0 ( d i 0 + d i * )
    where the H i * index value lies between 0 and 1.
  • Step 13: According to the order of the Hi* number, determine the capability of the alternatives. A higher Hi* number indicates a better alternative capability. Then, we can rank the alternatives depending on the Hi* numbers for the purpose of showing the performance comparison results for all the alternatives.

4. Results

As stated earlier, choosing an effective BPM tool for the company is a complicated process, and different criteria that affect BPM need to be considered. Therefore, choosing an optimal BPM tool will be very difficult for enterprise managers to carry out. Therefore, selecting an optimal BPM tool is very important for the development of the company’s IS.
In this example, we consider 8 candidate BPM tools (JFern, JPetriNet, GreatSPN, GDToolkit, TimeNet, ADONIS: CE, Bizagi Modeler and Cardanit). Based on Table 1 and Table 2, we can create the rank table (Table 4) for all the sub-parameters to influence BPM tool selection. Here, the project manager can determine the ranking of all sub-parameters through expert surveys.
From Table 4, we can find that there are 8 candidate BPM tools with the different corresponding sub-criteria ranks. After we define all the rank values, we have to define the fuzzy weights of all the four main types of parameters (C1, C2, C3 and C4) and corresponding sub-parameters (C11, C12, C13, C14, C15, C21, C22, C23, C31, C32, C33, C41 and C42) in Table 1.
Therefore, to obtain fuzzy weights for all the criteria, 10 company experts are requested to define the rank of direct fuzzy impact between criteria according to pairwise comparison depending on the direct fuzzy influence level in Table 3. The direct fuzzy influence degrees for 10 company experts can be seen in Table A1, Table A2, Table A3, Table A4 and Table A5 (Appendix A). Depending on Equation (1), we can release the TFN for the degrees, as shown in Table A6, Table A7, Table A8, Table A9 and Table A10 (Appendix A). Then, we use Equations (2)–(8) to release the defuzzy crisp values as Table A11, Table A12, Table A13, Table A14 and Table A15 (Appendix A) for the triangle fuzzy triples.
Then, we can use the crisp values to create the average direct fuzzy influence matrix T (Equation (10)) for the BPM tool selection main criteria and corresponding sub-criterions like Table 5, Table 6, Table 7, Table 8 and Table 9.
In Table 5, Table 6, Table 7, Table 8 and Table 9, all the values are obtained from the average of the direct fuzzy influences, and all these direct influence values are defined by the 10 experts. For example, in Table 5, average direct fuzzy influence rank from C1 to C3 is 0.545 which is obtained from the average fuzzy influence value of 10 experts ( 0.73 + 0.5 + 0.27 + 0.27 + 0.25 + 0.73 + 0.5 + 0.5 + 0.73 + 0.97 10 = 0.545) (Equation (10)). After that, depending on the Equations (11) and (12), we can calculate the normalized direct influence matrix T (Table 10, Table 11, Table 12, Table 13 and Table 14) for all the criterion and sub-criterions.
In Table 10, for instance, the normalized direct influence value for C1 to C3 is 0.262, which is obtained from the multiplication between the direct influence rank from C1 to C3 (0.545 in Table 5) and the λ value ( 1 max 1 n 4   j = 1 4 t nj =   1 max ( j = 1 4 t 1 j ,       j = 1 4 t 2 j ,       j = 1 4 t 3 j ,       j = 1 4 t 4 j ) = 1 max ( 1.819 , 1.656 , 1.679 , 2.081 ) = 0.481 ) for the average direct influence matrix (Table 5). After that, depending on the Equations (13)–(16), we can obtain the total direct and indirect influence matrix E (Table 15, Table 16, Table 17, Table 18 and Table 19) and the corresponding sum of rows (or in Equation (15)) and columns (co in Equation (16)) of the E.
From the Table 15, Table 16, Table 17, Table 18 and Table 19, we can find the direct and indirect relationships between criteria or sub-criteria. The ron (highlighted in yellow) means the total given both direct and indirect effects from criteria n to the other criteria (for example, in Table 15, ro1 = 1.484 + 1.852 + 1.614 + 1.696 = 6.646), and coj (highlighted in pink) means the total received both direct and indirect effects from other criteria to criterion j (for example, in Table 15, co1 = 1.484 + 1.592 + 1.585 + 1.874 = 6.535). After, we obtain the overall direct and indirect impact matrix E with corresponding ron and coj values, when n equals j, we can release the (roj + coj) (highlighted in lime green) value, which means the centrality of criterion j, for all the criteria (Table 15, Table 16, Table 17, Table 18 and Table 19).
After that, we can calculate the normalized (ro + co) value for all the criterions and sub-criterions like Table 20.
In Table 20, for example, the NRC value for Efficiency parameters (C1) is 0.251, which is obtained by dividing ro1 + co1 (Efficiency parameters (C1) in Table 15 by the sum of column ron + coj in Table 15 ( 13.181 6.646 + 6.172 + 6.153 + 7.317 = 0.251 ). After that, the final weight of sub-parameter is the multiplication of the weights of the four main and corresponding sub-criteria.
After we define the weight for all the criteria and sub-criteria, we can use these weights (Table 20) and Equations (19) and (20), and Table 4 to obtain the weighted normalized values such as Table 21.
In Table 21, the abbreviation max denotes the benefit standard, and abbreviation min denotes the cost standard. The final weighted normalized value for alternative 1 (T1) corresponding to sub-criteria C11 is 0.009. The number is obtained by multiplying the final weight of sub-criteria C11 (0.251 × 0.194 = 0.049) and the normalized decision matrix value for C11 ( 0.190 = 2 i = 1 24 y i 1 2 ).
After that, depending on Table 21, Equations (21) and (22), the P-I and N-I solutions are determined. The P-I solution and N-I solution can be seen as Table 22.
Depending on the data from Table 21 and Table 22, Equations (23) and (24), the relative distances of each candidate option from the P-I and N-I solutions can be released. Finally, the relative distance of each candidate option to the P-I solution is released depending on Equation (25). The relative distances of each alternative from the P-I and N-I solutions and the result of the relative distance of each alternative from the P-I solution can be seen as Table 23.
Table 23 shows the evaluation result of the considered alternatives obtained by using TOPSIS technology. In Table 23, the Ci* value is obtained by the relative closeness to the ideal solution (Equation (25)). For example, the value for the Ci* value for T1 is 0.405 ( H i * = d i 0 ( d i 0 + d i * ) = 0.06 ( 0.06 + 0.088 ) = 0.405 ). Meanwhile, in Table 23, di* is the n-dimensional Euclidean distance from each solution in Table 21 to the P-I solution (T+ Table 22), and di0 is the n-dimensional Euclidean distance from each solution in Table 21 to the N-I solution (T Table 22). For example, the di* value for T1 is 0.88 ( j = 1 13 ( x ij x j * ) 2 =   ( 0.009 0.023 ) 2 + ( 0.015 0.025 ) 2 + ( 0.011 0.027 ) 2 + ( 0.019 0.025 ) 2 + ( 0.025 0.025 ) 2 + ( 0.012 0.012 ) 2 + ( 0.022 0.011 ) 2 + ( 0.021 0.011 ) 2 + ( 0.003 0.001 ) 2 + ( 0.027 0.009 ) 2 + ( 0.029 0.001 ) 2 + ( 0.014 0.071 ) 2 + + ( 0.013 0.063 ) 2   = 0.88). Meanwhile, the di0 value for T1 is 0.06 ( j = 1 13 ( x ij x j 0 ) 2 =   ( 0.009 0.009 ) 2 + ( 0.015 0.01 ) 2 + ( 0.011 0.05 ) 2 + ( 0.019 0.012 ) 2 + ( 0.025 0.001 ) 2 + ( 0.012 0.047 ) 2 + ( 0.022 0.045 ) 2 + ( 0.021 0.042 ) 2 + ( 0.003 0.05 ) 2 + ( 0.027 0.046 ) 2 + ( 0.029 0.048 ) 2 + ( 0.014 0.014 ) 2 + + ( 0.013 0.013 ) 2 = 0.06). From the Ci* value in Table 23, it is possible to find that the alternative T4 obtains the highest value (0.642) (highlighted in green). Here, when considering the four types of criteria (BPM efficiency (C1), various expenses incurred when using the BPM tool (C2), application, installment, operating and training time (C3), and security issues affected by BPM tools (C4)), alternative T4 is the best overall performing BPM tool. Therefore, managers can select BPM tool T4 for the company.

5. Discussion and Conclusions

The business process modeling tool selection problem has a significant influence on the total performance of enterprise business process modeling, which will directly affect the development of the enterprise information system. Some candidate options have to be considered and assessed depending on the affecting parameters (efficiency parameters (C1), economical parameters (C2), time parameters (C3) and safety parameters (C4)) and corresponding sub-parameters (expressiveness (C11), readability (C12) usability (C13), formality (C14), ease of learning (C15), application and installation cost (C21), operating cost (C22), training cost (C23), application and installation time (C31), operating time (C32), training time (C33), internal safety (C41), and external safety (C42)) in a BPM tool choosing issue, causing various ambiguous information. Thus, an optimal assessment method is needed to make effective decisions. Therefore, the BPM tool selection methodology is proposed in this study with the consideration of different BPM tool selection influencing factors.
Apart from that, the process to select the BPM tool from all alternatives is a fuzzy multi-criteria decision-making (MCDM) issue. Thus, we use TOPSIS to obtain the priority of BPM tool alternatives. Here, TOPSIS is applied to determine the priorities of the BPM candidate tools. The proposed method has dramatically improved the usefulness and accuracy of decision-making in the BPM tool choice process. In addition, in the method, we find that the weights of the criteria and sub-criteria in the TOPSIS technique are determined by company expert options through the fuzzy DEMATEL method. After that, the final weight for sub-criteria is the multiplication of the defined two weights, and it is important and may change the ranking of the alternatives. This method can provide a reference for companies purchasing or applying BPM tools when selecting BPM tools so that the BPM tool can be used more efficiently, economically, quickly, and safely. Rather than providing assistance to the business process modeling tool manufacturer (Bonita [65], Camunda [66], etc. Of course, it is also possible for business process modeling tool manufacturers to use this method to evaluate the goodness of their own tools. However, the evaluation should involve the experts from the company interested in purchasing the tool.
Some scholars, such as Zhang and Su, 2019 [67], also proposed the fuzzy DEMATEL and TOPSIS combination method. However, the fuzzy part of this approach is based on a 2-tuple linguistic method and approaches the relationships between the attributes of the proof participants and determines their weights. The 2-tuple linguistic model is used to aggregate linguistic evaluation information and requires the interpretation of linguistic labels. Even though the model can increase the accuracy in the process of aggregation, it is mainly concerned with semantic ambiguity. However, the fuzzy part studied in this paper is the uncertainty of the decision data (not semantically ambiguous). The Triangular Fuzzy Number not only can be used not only to express the vagueness and uncertainty of decision data but also to represent fuzzy terms in decision data processing. Therefore, the application of triangular fuzzy numbers is more adaptable in our study.
Although the proposed method is developed for the BPM tool selection problem, it is also adaptable for other software tool selections with slight modifications. Such as ERP or office tool selection problems in manufacturing or trading companies. The main part that needs to be modified here is the criteria part. Companies can change the main criteria and sub-criteria (Table 1) according to the characteristics of the software tool they need to choose. For example, the ERP tool selection problem is more about data evaluation and control rather than graphical presentation, so the expressiveness (C11) and formality (C14) criteria can be removed and replaced by the data control or evaluation criteria. Meanwhile, although some criteria do not need to be changed, the contents inside the criteria need to be redefined.
Even though the method is proposed and applied for the BPM tool selection problem, and we can obtain the final BPM tool (T4 in Table 23), there are many study areas for expansion. Studies could help extend the approach by first determining the sub-criteria more rationally and precisely. Second, we find and establish a method of criteria and sub-criteria weight definition in a reasonable way. Meanwhile, as Zhang and Su, 2019 [67] consider the 2-tuple linguistic model in the fuzzy DEMATEL method, it is also very important for our research to consider semantic ambiguity. Therefore, further studies will focus on these study areas.
This study first describes the significance of BPM tool selection for information systems (IS) in companies. After that, the key target is expressed to approach the issue in BPM tool selection. Then, the method is proposed with the application of DEMATEL and the TOPSIS method to approach BPM tool selection with the consideration of different BPM tool evaluation criteria. Finally, depending on the method proposed in this paper, the company can choose the optimal and most suitable BPM tool. This method allows the BPM process in the company to be more economical, efficient, time-saving, and safe.

Author Contributions

Ideas, G.J. (Guangying Jin) and G.J. (GuangZhe Jin); Method, G.J. (Guangying Jin); Writing—Original Draft Preparation, G.J. (Guangying Jin); Writing—Review & Editing, G.J. (Guangying Jin); Supervision, H.H.; Funding Acquisition, G.J. (Guangying Jin). All authors have read and agreed to the published version of the manuscript.

Funding

The study was supported by Talent Research Start-up Funding of Dalian Maritime University, authorization code: 02502329.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

Table A1. Degree of direct fuzzy influence between criterions.
Table A1. Degree of direct fuzzy influence between criterions.
E1C1
(DFN)
C2
(DFN)
C3
(DFN)
C4
(DFN)
E2C1
(DFN)
C2
(DFN)
C3
(DFN)
C4
(DFN)
E3C1
(DFN)
C2
(DFN)
C3
(DFN)
C4
(DFN)
E4C1
(DFN)
C2
(DFN)
C3
(DFN)
C4
(DFN)
E5C1
(DFN)
C2
(DFN)
C3
(DFN)
C4
(DFN)
Efficiency (C1)0233C10224C10312C10313C10111
Economical (C2)3022C22031C23023C21022C22012
Time (C3)1201C33402C33301C31102C33102
Safety (C4)1420C41230C43420C41320C42220
E6C1
(DFN)
C2
(DFN)
C3
(DFN)
C4
(DFN)
E7C1
(DFN)
C2
(DFN)
C3
(DFN)
C4
(DFN)
E8C1
(DFN)
C2
(DFN)
C3
(DFN)
C4
(DFN)
E9C1
(DFN)
C2
(DFN)
C3
(DFN)
C4
(DFN)
E10C1
(DFN)
C2
(DFN)
C3
(DFN)
C4
(DFN)
C10134C10323C10323C10333C10342
C22043C23013C23013C23013C21023
C33401C32403C32302C32201C33402
C44330C44340C44340C44330C43420
Note. DFI = Direct Fuzzy Influence. C = Criteria. E = Expert.
Table A2. Degree of direct fuzzy influence between Efficiency (C1) sub-criterions.
Table A2. Degree of direct fuzzy influence between Efficiency (C1) sub-criterions.
E1C11
(DFN)
C12
(DFN)
C13
(DFN)
C14
(DFN)
C15
(DFN)
E2C11
(DFN)
C12
(DFN)
C13
(DFN)
C14
(DFN)
C15
(DFN)
E3C11
(DFN)
C12
(DFN)
C13
(DFN)
C14
(DFN)
C15
(DFN)
E4C11
(DFN)
C12
(DFN)
C13
(DFN)
C14
(DFN)
C15
(DFN)
E5C11
(DFN)
C12
(DFN)
C13
(DFN)
C14
(DFN)
C15
(DFN)
EX (C11)01112C1102313C1101233C1102121C1100302
RE (C12)20242C1210131C1220142C1230232C1220142
US (C13)11033C1322033C1312033C1331043C1302033
FO (C14)23204C1413204C1412003C1443203C1423101
EL (C15)41020C1532200C1531030C1541020C1542200
E6C11
(DFN)
C12
(DFN)
C13
(DFN)
C14
(DFN)
C15
(DFN)
E7C11
(DFN)
C12
(DFN)
C13
(DFN)
C14
(DFN)
C15
(DFN)
E8C11
(DFN)
C12
(DFN)
C13
(DFN)
C14
(DFN)
C15
(DFN)
E9C11
(DFN)
C12
(DFN)
C13
(DFN)
C14
(DFN)
C15
(DFN)
E10C11
(DFN)
C12
(DFN)
C13
(DFN)
C14
(DFN)
C15
(DFN)
C110232 C1101312C1102132C1101222C1103213
C122024 C1220142C1220242C1210243C1210323
C131103 C1321033C1311033C1333033C1331032
C142320 C1433104C1412203C1413203C1433204
C153221 C1541010C1531010C1531020C1532020
Note. E = Expert. DFI = Direct Fuzzy Influence. C = Criteria. EX = Expressiveness. RE = Readability. US = Usability. FO = Formality. EL = Ease of Leaning.
Table A3. Degree of direct fuzzy influence between Economical (C2) sub-criterions.
Table A3. Degree of direct fuzzy influence between Economical (C2) sub-criterions.
E1C21
(DFN)
C22
(DFN)
C23
(DFN)
E2C21
(DFN)
C22
(DFN)
C23
(DFN)
E3C21
(DFN)
C22
(DFN)
C23
(DFN)
E4C21
(DFN)
C22
(DFN)
C23
(DFN)
E5C21
(DFN)
C22
(DFN)
C23
(DFN)
Application and Installment cost (C21)023C21012C21022C21013C21032
Operating cost (C22)404C22301C22203C22203C22303
Training cost (C23)330C23220C23120C23130C23230
E6C21
(DFN)
C22
(DFN)
C23
(DFN)
E7C21
(DFN)
C22
(DFN)
C23
(DFN)
E8C21
(DFN)
C22
(DFN)
C23
(DFN)
E9C21
(DFN)
C22
(DFN)
C23
(DFN)
E10C21
(DFN)
C22
(DFN)
C23
(DFN)
C21012C21034C21032C21034C21034
C22303C22302C22303C22303C22303
C23240C23320C23440C23420C23240
Note. E = Expert. DFI = Direct Fuzzy Influence. C = Criteria.
Table A4. Degree of direct fuzzy influence between Time (C3) sub-criterions.
Table A4. Degree of direct fuzzy influence between Time (C3) sub-criterions.
E1C31
(DFN)
C32
(DFN)
C33
(DFN)
E2C31
(DFN)
C32
(DFN)
C33
(DFN)
E3C31
(DFN)
C32
(DFN)
C33
(DFN)
E4C31
(DFN)
C32
(DFN)
C33
(DFN)
E5C31
(DFN)
C32
(DFN)
C33
(DFN)
Application and Installment time(C31)012C31001C31011C31012C31002
Operating time (C32)003C32002C32002C32003C32003
Training time (C33)020C33030C33010C33020C33020
E6C31
(DFN)
C32
(DFN)
C33
(DFN)
E7C31
(DFN)
C32
(DFN)
C33
(DFN)
E8C31
(DFN)
C32
(DFN)
C33
(DFN)
E9C31
(DFN)
C32
(DFN)
C33
(DFN)
E10C31
(DFN)
C32
(DFN)
C33
(DFN)
C31012C31002C31002C31012C31003
C32004C32003C32002C32003C32003
C33020C33020C33030C33030C33030
Table A5. Degree of direct fuzzy influence between Safety (C4) sub-criterions.
Table A5. Degree of direct fuzzy influence between Safety (C4) sub-criterions.
E1C41
(DFN)
C42
(DFN)
E2C41
(DFN)
C42
(DFN)
E3C41
(DFN)
C42
(DFN)
E4C41
(DFN)
C42
(DFN)
E5C41
(DFN)
C42
(DFN)
Internal safety(C41)01C4102C4101C4102C4103
External safety (C42)30C4220C4230C4230C4230
E6C41
(DFN)
C42
(DFN)
E7C41
(DFN)
C42
(DFN)
E8C41
(DFN)
C42
(DFN)
E9C41
(DFN)
C42
(DFN)
E10C41
(DFN)
C42
(DFN)
C4102C4101C4102C4102C4102
C4220C4240C4220C4230C4230
Note. E = Expert. DFI = Direct Fuzzy Influence. C = Criteria.
Table A6. The corresponding triangle fuzzy numbers in Table A1.
Table A6. The corresponding triangle fuzzy numbers in Table A1.
E1C1 (TFN)C2 (TFN)C3 (TFN)C4 (TFN)E2C1 (TFN)C2 (TFN)C3 (TFN)C4 (TFN)
Efficiency (C1)0(0.25, 0.5, 0.75)(0.5, 0.75, 1)(0.5, 0.75, 1)C10(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)(0.75, 1, 1)
Economical (C2)(0.5, 0.75, 1)0(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)C2(0.25, 0.5, 0.75)0(0.5, 0.75, 1)(0, 0.25, 0.5)
Time (C3)(0, 0.25, 0.5)(0.25, 0.5, 0.75)0(0, 0.25, 0.5)C3(0.5, 0.75, 1)(0.75, 1, 1)0(0.25, 0.5, 0.75)
Safety (C4)(0, 0.25, 0.5)(0.75, 1, 1)(0.25, 0.5, 0.75)0C4(0, 0.25, 0.5)(0.25, 0.5, 0.75)(0.5, 0.75, 1)0
E3C1 (TFN)C2 (TFN)C3 (TFN)C4 (TFN)E4C1 (TFN)C2 (TFN)C3 (TFN)C4 (TFN)
C10(0.5, 0.75, 1)(0, 0.25, 0.5)(0.25, 0.5, 0.75)C10(0.5, 0.75, 1)(0, 0.25, 0.5)(0.5, 0.75, 1)
C2(0.5, 0.75, 1)0(0.25, 0.5, 0.75)(0.5, 0.75, 1)C2(0, 0.25, 0.5)0(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)
C3(0.5, 0.75, 1)(0.5, 0.75, 1)0(0, 0.25, 0.5)C3(0, 0.25, 0.5)(0, 0.25, 0.5)0(0.25, 0.5, 0.75)
C4(0.5, 0.75, 1)(0.75, 1, 1)(0.25, 0.5, 0.75)0C4(0, 0.25, 0.5)(0.5, 0.75, 1)(0.25, 0.5, 0.75)0
E5C1 (TFN)C2 (TFN)C3 (TFN)C4 (TFN)E6E6C1 (TFN)C2 (TFN)C3 (TFN)
C10(0, 0.25, 0.5)(0, 0.25, 0.5)(0, 0.25, 0.5)C1C10(0, 0.25, 0.5)(0.5, 0.75, 1)
C2(0.25, 0.5, 0.75)0(0, 0.25, 0.5)(0.25, 0.5, 0.75)C2C2(0.25, 0.5, 0.75)0(0.75, 1, 1)
C3(0.5, 0.75, 1)(0, 0.25, 0.5)0(0.25, 0.5, 0.75)C3C3(0.5, 0.75, 1)(0.75, 1, 1)0
C4(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)0E6C4(0.75, 1, 1)(0.5, 0.75, 1)(0.5, 0.75, 1)
E7C1 (TFN)C2 (TFN)C3 (TFN)C4 (TFN)E8C1 (TFN)C2 (TFN)C3 (TFN)C4 (TFN)
C10(0.5, 0.75, 1)(0.25, 0.5, 0.75)(0.5, 0.75, 1)C10(0.5, 0.75, 1)(0.25, 0.5, 0.75)(0.5, 0.75, 1)
C2(0.5, 0.75, 1)0(0, 0.25, 0.5)(0.5, 0.75, 1)C2(0.5, 0.75, 1)0(0, 0.25, 0.5)(0.5, 0.75, 1)
C3(0.25, 0.5, 0.75)(0.75, 1, 1)0(0.5, 0.75, 1)C3(0.25, 0.5, 0.75)(0.5, 0.75, 1)0(0.25, 0.5, 0.75)
C4(0.75, 1, 1)(0.5, 0.75, 1)(0.75, 1, 1)0C4(0.75, 1, 1)(0.5, 0.75, 1)(0.75, 1, 1)0
E9C1 (TFN)C2 (TFN)C3 (TFN)C4 (TFN)E10C1 (TFN)C2 (TFN)C3 (TFN)C4 (TFN)
C10(0.5, 0.75, 1)(0.5, 0.75, 1)(0.5, 0.75, 1)C10(0.5, 0.75, 1)(0.75, 1, 1)(0.25, 0.5, 0.75)
C2(0.5, 0.75, 1)0(0, 0.25, 0.5)(0.5, 0.75, 1)C2(0, 0.25, 0.5)0(0.25, 0.5, 0.75)(0.5, 0.75, 1)
C3(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)0(0, 0.25, 0.5)C3(0.5, 0.75, 1)(0.75, 1, 1)0(0.25, 0.5, 0.75)
C4(0.75, 1, 1)(0.5, 0.75, 1)(0.5, 0.75, 1)0C4(0.5, 0.75, 1)(0.75, 1, 1)(0.25, 0.5, 0.75)0
Note. TFN = Triangle Fuzzy Number. C = Criteria.
Table A7. The corresponding triangle fuzzy numbers in Table A2.
Table A7. The corresponding triangle fuzzy numbers in Table A2.
E1C11 (TFN)C12 (TFN)C13 (TFN)C14 (TFN)C15 (TFN)E2C11 (TFN)C12 (TFN)C13 (TFN)C14 (TFN)C15 (TFN)
EX (C11)0(0, 0.25, 0.5)(0, 0.25, 0.5)(0, 0.25, 0.5)(0.25, 0.5, 0.75)C110(0.25, 0.5, 0.75)(0.5, 0.75, 1)(0, 0.25, 0.5)(0.5, 0.75, 1)
RE (C12)(0.25, 0.5, 0.75)0(0.25, 0.5, 0.75)(0.75, 1, 1)(0.25, 0.5, 0.75)C12(0, 0.25, 0.5)0(0, 0.25, 0.5)(0.5, 0.75, 1)(0, 0.25, 0.5)
US (C13)(0, 0.25, 0.5)(0, 0.25, 0.5)0(0.5, 0.75, 1)(0.5, 0.75, 1)C13(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)0(0.5, 0.75, 1)(0.5, 0.75, 1)
FO (C14)(0.25, 0.5, 0.75)(0.5, 0.75, 1)(0.25, 0.5, 0.75)0(0.75, 1, 1)C14(0, 0.25, 0.5)(0.5, 0.75, 1)(0.25, 0.5, 0.75)0(0.75, 1, 1)
EL (C15)(0.75, 1, 1)(0, 0.25, 0.5)(0, 0, 0.25)(0.25, 0.5, 0.75)0C15(0.5, 0.75, 1)(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)(0, 0, 0.25)0
E3C11 (TFN)C12 (TFN)C13 (TFN)C14 (TFN)C15 (TFN)E4C11 (TFN)C12 (TFN)C13 (TFN)C14 (TFN)C15 (TFN)
C110(0, 0.25, 0.5)(0.25, 0.5, 0.75)(0.5, 0.75, 1)(0.5, 0.75, 1)C110(0.25, 0.5, 0.75)(0, 0.25, 0.5)(0.25, 0.5, 0.75)(0, 0.25, 0.5)
C12(0.25, 0.5, 0.75)0(0, 0.25, 0.5)(0.75, 1, 1)(0.25, 0.5, 0.75)C12(0.5, 0.75, 1)0(0.25, 0.5, 0.75)(0.5, 0.75, 1)(0.25, 0.5, 0.75)
C13(0, 0.25, 0.5)(0.25, 0.5, 0.75)0(0.5, 0.75, 1)(0.5, 0.75, 1)C13(0.5, 0.75, 1)(0, 0.25, 0.5)0(0.75, 1, 1)(0.5, 0.75, 1)
C14(0, 0.25, 0.5)(0.25, 0.5, 0.75)(0, 0, 0.25)0(0.5, 0.75, 1)C14(0.75, 1, 1)(0.5, 0.75, 1)(0.25, 0.5, 0.75)0(0.5, 0.75, 1)
C15(0.5, 0.75, 1)(0, 0.25, 0.5)(0, 0, 0.25)(0.5, 0.75, 1)0C15(0.75, 1, 1)(0, 0.25, 0.5)(0, 0, 0.25)(0.25, 0.5, 0.75)0
E5C11 (TFN)C12 (TFN)C13 (TFN)C14 (TFN)C15 (TFN)E6C11 (TFN)C12 (TFN)C13 (TFN)C14 (TFN)C15 (TFN)
C110(0, 0, 0.25)(0.5, 0.75, 1)(0, 0, 0.25)(0.25, 0.5, 0.75)C110(0.25, 0.5, 0.75)(0.5, 0.75, 1)(0.25, 0.5, 0.75)(0.75, 1, 1)
C12(0.25, 0.5, 0.75)0(0, 0.25, 0.5)(0.75, 1, 1)(0.25, 0.5, 0.75)C12(0.25, 0.5, 0.75)0(0.25, 0.5, 0.75)(0.75, 1, 1)(0.5, 0.75, 1)
C130(0.25, 0.5, 0.75)0(0.5, 0.75, 1)(0.5, 0.75, 1)C13(0, 0.25, 0.5)(0, 0.25, 0.5)0(0.5, 0.75, 1)(0.25, 0.5, 0.75)
C14(0.25, 0.5, 0.75)(0.5, 0.75, 1)(0, 0.25, 0.5)0(0, 0.25, 0.5)C14(0.25, 0.5, 0.75)(0.5, 0.75, 1)(0.25, 0.5, 0.75)0(0.5, 0.75, 1)
C15(0.75, 1, 1)(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)(0, 0, 0.25)0C15(0.5, 0.75, 1)(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)(0, 0.25, 0.5)0
E7C11 (TFN)C12 (TFN)C13 (TFN)C14 (TFN)C15 (TFN)E8C11 (TFN)C12 (TFN)C13 (TFN)C14 (TFN)C15 (TFN)
C110(0, 0.25, 0.5)(0.5, 0.75, 1)(0, 0.25, 0.5)(0.25, 0.5, 0.75)C110(0.25, 0.5, 0.75)(0, 0.25, 0.5)(0.5, 0.75, 1)(0.25, 0.5, 0.75)
C12(0.25, 0.5, 0.75)0(0, 0.25, 0.5)(0.75, 1, 1)(0.25, 0.5, 0.75)C12(0.25, 0.5, 0.75)0(0.25, 0.5, 0.75)(0.75, 1, 1)(0.25, 0.5, 0.75)
C13(0.25, 0.5, 0.75)(0, 0.25, 0.5)0(0.5, 0.75, 1)(0.5, 0.75, 1)C13(0, 0.25, 0.5)(0, 0.25, 0.5)0(0.5, 0.75, 1)(0.5, 0.75, 1)
C14(0.5, 0.75, 1)(0.5, 0.75, 1)(0, 0.25, 0.5)0(0.75, 1, 1)C14(0, 0.25, 0.5)(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)0(0.5, 0.75, 1)
C15(0.75, 1, 1)(0, 0.25, 0.5)0(0, 0.25, 0.5)0C15(0.5, 0.75, 1)(0, 0.25, 0.5)(0, 0, 0.25)(0, 0.25, 0.5)0
E9C11 (TFN)C12 (TFN)C13 (TFN)C14 (TFN)C15 (TFN)E10C11 (TFN)C12 (TFN)C13 (TFN)C14 (TFN)C15 (TFN)
C110(0, 0.25, 0.5)(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)C110(0.5, 0.75, 1)(0.25, 0.5, 0.75)(0, 0.25, 0.5)(0.5, 0.75, 1)
C12(0, 0.25, 0.5)0(0.25, 0.5, 0.75)(0.75, 1, 1)(0.5, 0.75, 1)C12(0, 0.25, 0.5)0(0.5, 0.75, 1)(0.25, 0.5, 0.75)(0.5, 0.75, 1)
C13(0.5, 0.75, 1)(0.5, 0.75, 1)0(0.5, 0.75, 1)(0.5, 0.75, 1)C13(0.5, 0.75, 1)(0, 0.25, 0.5)0(0.5, 0.75, 1)(0.25, 0.5, 0.75)
C14(0, 0.25, 0.5)(0.5, 0.75, 1)(0.25, 0.5, 0.75)0(0.5, 0.75, 1)C14(0.5, 0.75, 1)(0.5, 0.75, 1)(0.25, 0.5, 0.75)0(0.75, 1, 1)
C15(0.5, 0.75, 1)(0, 0.25, 0.5)(0, 0, 0.25)(0.25, 0.5, 0.75)0C15(0.5, 0.75, 1)(0.25, 0.5, 0.75)(0, 0, 0.25)(0.25, 0.5, 0.75)0
Note. TFN = Triangle Fuzzy Number. C = Criteria.
Table A8. The corresponding triangle fuzzy numbers in Table A3.
Table A8. The corresponding triangle fuzzy numbers in Table A3.
E1C21 (TFN)C22 (TFN)C23 (TFN)E2C21 (TFN)C22 (TFN)C23 (TFN)
Application and Installment cost (C21)0(0.25, 0.5, 0.75)(0.5, 0.75, 1)C210(0, 0.25, 0.5)(0.25, 0.5, 0.75)
Operating cost (C22)(0.75, 1, 1)0(0.75, 1, 1)C22(0.5, 0.75, 1)0(0, 0.25, 0.5)
Training cost (C23)(0.5, 0.75, 1)(0.5, 0.75, 1)0C23(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)0
E3C21 (TFN)C22(TFN)C23 (TFN)E4C21 (TFN)C22(TFN)C23 (TFN)
C210(0.25, 0.5, 0.75)(0.25, 0.5, 0.75)C210(0, 0.25, 0.5)(0.5, 0.75, 1)
C22(0.25, 0.5, 0.75)0(0.5, 0.75, 1)C22(0.25, 0.5, 0.75)0(0.5, 0.75, 1)
C23(0, 0.25, 0.5)(0.25, 0.5, 0.75)0C23(0, 0.25, 0.5)(0.5, 0.75, 1)0
E5C21 (TFN)C22(TFN)C23 (TFN)E6C21 (TFN)C22(TFN)C23 (TFN)
C210(0.5, 0.75, 1)(0.25, 0.5, 0.75)C210(0, 0.25, 0.5)(0.25, 0.5, 0.75)
C22(0.5, 0.75, 1)0(0.5, 0.75, 1)C22(0.5, 0.75, 1)0(0.5, 0.75, 1)
C23(0.25, 0.5, 0.75)(0.5, 0.75, 1)0C23(0.25, 0.5, 0.75)(0.75, 1, 1)0
E7C21 (TFN)C22(TFN)C23 (TFN)E8C21 (TFN)C22(TFN)C23 (TFN)
C210(0.5, 0.75, 1)(0.75, 1, 1)C210(0.5, 0.75, 1)(0.25, 0.5, 0.75)
C22(0.5, 0.75, 1)0(0.25, 0.5, 0.75)C22(0.5, 0.75, 1)0(0.5, 0.75, 1)
C23(0.5, 0.75, 1)(0.25, 0.5, 0.75)0C23(0.75, 1, 1)(0.75, 1, 1)0
E9C21 (TFN)C22(TFN)C23 (TFN)E10C21 (TFN)C22(TFN)C23 (TFN)
C210(0.5, 0.75, 1)(0.75, 1, 1)C210(0.5, 0.75, 1)(0.75, 1, 1)
C22(0.5, 0.75, 1)0(0.5, 0.75, 1)C22(0.5, 0.75, 1)0(0.5, 0.75, 1)
C23(0.75, 1, 1)(0.25, 0.5, 0.75)0C23(0.25, 0.5, 0.75)(0.75, 1, 1)0
Note. TFN = Triangle Fuzzy Number. C = Criteria.
Table A9. The corresponding triangle fuzzy numbers in Table A4.
Table A9. The corresponding triangle fuzzy numbers in Table A4.
E1C31 (TFN)C32 (TFN)C33 (TFN)E2C31 (TFN)C32 (TFN)C33 (TFN)
Application and Installment time (C31)0(0, 0.25, 0.5)(0.25, 0.5, 0.75)C310(0, 0, 0.25)(0, 0.25, 0.5)
Operating time (C32)(0, 0, 0.25)0(0.5, 0.75, 1)C32(0, 0, 0.25)0(0.25, 0.5, 0.75)
Training time (C33)(0, 0, 0.25)(0.25, 0.5, 0.75)0C33(0, 0, 0.25)(0.5, 0.75, 1)0
E3C31 (TFN)C32 (TFN)C33 (TFN)E4C31 (TFN)C32 (TFN)C33 (TFN)
C310(0, 0.25, 0.5)(0, 0.25, 0.5)C310(0, 0.25, 0.5)(0.25, 0.5, 0.75)
C32(0, 0, 0.25)0(0.25, 0.5, 0.75)C32(0, 0, 0.25)0(0.5, 0.75, 1)
C33(0, 0, 0.25)(0, 0.25, 0.5)0C33(0, 0, 0.25)(0.25, 0.5, 0.75)0
E5C31 (TFN)C32 (TFN)C33 (TFN)E6C31 (TFN)C32 (TFN)C33 (TFN)
C310(0, 0, 0.25)(0.25, 0.5, 0.75)C310(0, 0.25, 0.5)(0.25, 0.5, 0.75)
C32(0, 0, 0.25)0(0.5, 0.75, 1)C32(0, 0, 0.25)0(0.75, 1, 1)
C33(0, 0, 0.25)(0.25, 0.5, 0.75)0C33(0, 0, 0.25)(0.25, 0.5, 0.75)0
E7C31 (TFN)C32 (TFN)C33 (TFN)E8C31 (TFN)C32 (TFN)C33 (TFN)
C310(0, 0, 0.25)(0.25, 0.5, 0.75)C310(0, 0, 0.25)(0.25, 0.5, 0.75)
C32(0, 0, 0.25)0(0.5, 0.75, 1)C32(0, 0, 0.25)0(0.25, 0.5, 0.75)
C33(0, 0, 0.25)(0.25, 0.5, 0.75)0C33(0, 0, 0.25)(0.5, 0.75, 1)0
E9C31 (TFN)C32 (TFN)C33 (TFN)E10C31 (TFN)C32 (TFN)C33 (TFN)
C310(0, 0.25, 0.5)(0.25, 0.5, 0.75)C310(0, 0, 0.25)(0.5, 0.75, 1)
C32(0, 0, 0.25)0(0.5, 0.75, 1)C32(0, 0, 0.25)0(0.5, 0.75, 1)
C33(0, 0, 0.25)(0.5, 0.75, 1)0C33(0, 0, 0.25)(0.5, 0.75, 1)0
Table A10. The corresponding triangle fuzzy numbers in Table A5.
Table A10. The corresponding triangle fuzzy numbers in Table A5.
E1C41 (TFN)C42 (TFN)E2C41 (TFN)C42 (TFN)
Internal safety (C41)0(0, 0.25, 0.5)C410(0.25, 0.5, 0.75)
External safety (C42)(0.5, 0.75, 1)0C42(0.25, 0.5, 0.75)0
E3C41 (TFN)C42 (TFN)E4C41 (TFN)C42 (TFN)
C410(0, 0.25, 0.5)C410(0.25, 0.5, 0.75)
C42(0.5, 0.75, 1)0C42(0.5, 0.75, 1)0
E5C41 (TFN)C42 (TFN)E6C41 (TFN)C42 (TFN)
C410(0.5, 0.75, 1)C410(0.25, 0.5, 0.75)
C42(0.5, 0.75, 1)0C42(0.25, 0.5, 0.75)0
E7C41 (TFN)C42 (TFN)E8C41 (TFN)C42 (TFN)
C410(0, 0.25, 0.5)C410(0.25, 0.5, 0.75)
C42(0.75, 1, 1)0C42(0.25, 0.5, 0.75)0
E9C41 (TFN)C42 (TFN)E10C41 (TFN)C42 (TFN)
C410(0.25, 0.5, 0.75)C410(0.25, 0.5, 0.75)
C42(0.5, 0.75, 1)0C42(0.5, 0.75, 1)0
Table A11. The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A6.
Table A11. The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A6.
E1C1
(DCV)
C2
(DCV)
C3
(DCV)
C4
(DCV)
E2C1
(DCV)
C2
(DCV)
C3
(DCV)
C4
(DCV)
E3C1
(DCV)
C2
(DCV)
C3
(DCV)
C4
(DCV)
E4C1
(DCV)
C2
(DCV)
C3
(DCV)
C4
(DCV)
E5C1
(DCV)
C2
(DCV)
C3
(DCV)
C4
(DCV)
Efficiency(C1)0.000.500.730.73C10.000.500.500.97C10.000.730.270.50C10.000.730.270.73C10.000.250.250.25
Economical(C2)0.730.000.500.50C20.500.000.730.27C20.730.000.500.73C20.260.000.490.49C20.490.000.260.49
Time(C3)0.260.490.000.26C30.730.970.000.50C30.730.730.000.27C30.260.260.000.49C30.730.270.000.50
Safety(C4)0.270.970.500.00C40.270.500.730.00C40.730.970.500.00C40.270.730.500.00C40.490.490.490.00
E6C1
(DCV)
C2
(DCV)
C3
(DCV)
C4
(DCV)
E7C1
(DCV)
C2
(DCV)
C3
(DCV)
C4
(DCV)
E8C1
(DCV)
C2
(DCV)
C3
(DCV)
C4
(DCV)
E9C1
(DCV)
C2
(DCV)
C3
(DCV)
C4
(DCV)
E10C1
(DCV)
C2
(DCV)
C3
(DCV)
C4
(DCV)
C10.000.270.730.97C10.000.730.500.73C10.000.730.500.73C10.000.730.730.73C10.000.730.970.50
C20.500.000.970.73C20.730.000.270.73C20.730.000.270.73C20.730.000.270.73C20.270.000.500.73
C30.730.970.000.27C30.500.970.000.73C30.500.730.000.50C30.490.490.000.26C30.730.970.000.50
C40.970.730.730.00C40.970.730.970.00C40.970.730.970.00C40.970.730.730.00C40.730.970.500.00
Note. DCV = De-fuzzy Crisp Number. C = Criteria.
Table A12. The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A7.
Table A12. The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A7.
E1C11
(DVC)
C12
(DVC)
C13
(DVC)
C14
(DVC)
C15
(DVC)
E2C11
(DVC)
C12
(DVC)
C13
(DVC)
C14
(DVC)
C15
(DVC)
E3C11
(DVC)
C12
(DVC)
C13
(DVC)
C14
(DVC)
C15
(DVC)
E4C11
(DVC)
C12
(DVC)
C13
(DVC)
C14
(DVC)
C15
(DVC)
E5C11
(DVC)
C12
(DVC)
C13
(DVC)
C14
(DVC)
C15
(DVC)
EX (C11)0.000.400.400.400.92C110.000.600.950.280.95C110.000.280.600.950.95C110.000.920.400.920.40C110.000.050.950.050.60
RE (C12)0.450.000.450.970.45C120.280.000.280.950.28C120.450.000.210.970.45C120.950.000.600.950.60C120.450.000.210.970.45
US (C13)0.280.280.000.950.95C130.600.600.000.950.95C130.280.600.000.950.95C130.700.210.000.970.70C130.050.600.000.950.95
FO (C14)0.450.700.450.000.97C140.210.700.450.000.97C140.280.600.050.000.95C140.970.700.450.000.70C140.600.950.280.000.28
EL (C15)0.540.540.380.080.00C150.410.950.600.050.00C150.410.500.050.170.00C150.970.540.380.080.00C150.540.640.270.030.00
E6C11
(DVC)
C12
(DVC)
C13
(DVC)
C14
(DVC)
C15
(DVC)
E7C11
(DVC)
C12
(DVC)
C13
(DVC)
C14
(DVC)
C15
(DVC)
E8C11
(DVC)
C12
(DVC)
C13
(DVC)
C14
(DVC)
C15
(DVC)
E9C11
(DVC)
C12
(DVC)
C13
(DVC)
C14
(DVC)
C15
(DVC)
E10C11
(DVC)
C12
(DVC)
C13
(DVC)
C14
(DVC)
C15
(DVC)
C110.000.450.700.450.97C110.000.280.950.280.60C110.000.600.280.950.60C110.000.400.920.920.92C110.000.950.600.280.95
C120.450.000.450.970.70C120.450.000.210.970.45C120.450.000.450.970.45C120.210.000.450.970.70C120.280.000.950.600.95
C130.280.280.000.950.60C130.600.280.000.950.95C130.280.280.000.950.95C130.950.950.000.950.95C130.950.280.000.950.60
C140.600.950.600.000.95C140.700.700.210.000.97C140.280.600.600.000.95C140.280.950.600.000.95C140.700.700.450.000.97
C150.650.950.600.050.00C150.730.540.210.030.00C150.410.500.500.050.00C150.410.950.500.110.00C150.950.950.500.110.00
Note. DCV = De-fuzzy Crisp Number. C = Criteria.
Table A13. The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A8.
Table A13. The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A8.
E1C21
(DVC)
C22
(DVC)
C23
(DVC)
E2C21
(DVC)
C22
(DVC)
C23
(DVC)
E3C21
(DVC)
C22
(DVC)
C23
(DVC)
E4C21
(DVC)
C22
(DVC)
C23
(DVC)
E5C21
(DVC)
C22
(DVC)
C23
(DVC)
Application and Installment cost (C21)0.000.500.73C210.000.270.50C210.000.500.50C210.000.270.73C210.000.730.50
Operating cost (C22)0.970.000.97C220.730.000.27C220.500.000.73C220.500.000.73C220.730.000.73
Training cost (C23)0.730.730.00C230.500.500.00C230.270.500.00C230.270.730.00C230.500.730.00
E6C21
(DVC)
C22
(DVC)
C23
(DVC)
E7C21
(DVC)
C22
(DVC)
C23
(DVC)
E8C21
(DVC)
C22
(DVC)
C23
(DVC)
E9C21
(DVC)
C22
(DVC)
C23
(DVC)
E10C21
(DVC)
C22
(DVC)
C23
(DVC)
C210.000.270.50C210.000.730.97C210.000.730.50C210.000.730.97C210.000.730.97
C220.730.000.73C220.730.000.50C220.730.000.73C220.730.000.73C220.730.000.73
C230.500.970.00C230.730.500.00C230.970.970.00C230.970.500.00C230.500.970.00
Note. DCV = De-fuzzy Crisp Number. C = Criteria.
Table A14. The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A9.
Table A14. The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A9.
E1C31
(DVC)
C32
(DVC)
C33
(DVC)
E2C31
(DVC)
C32
(DVC)
C33
(DVC)
E3C31
(DVC)
C32
(DVC)
C33
(DVC)
E4C31
(DVC)
C32
(DVC)
C33
(DVC)
E5C31
(DVC)
C32
(DVC)
C33
(DVC)
Application and Installment time(C31)0.000.350.65C310.000.080.50C310.000.500.50C310.000.350.65C310.000.050.65
Operating time (C32)0.030.000.73C320.050.000.65C320.050.000.65C320.030.000.73C320.030.000.73
Training time (C33)0.050.650.00C330.030.730.00C330.080.500.00C330.050.650.00C330.050.650.00
E6C31
(DVC)
C32
(DVC)
C33
(DVC)
E7C31
(DVC)
C32
(DVC)
C33
(DVC)
E8C31
(DVC)
C32
(DVC)
C33
(DVC)
E9C31
(DVC)
C32
(DVC)
C33
(DVC)
E10C31
(DVC)
C32
(DVC)
C33
(DVC)
C310.000.350.65C310.000.050.65C310.000.050.65C310.000.350.65C310.000.030.73
C320.030.000.97C320.030.000.73C320.050.000.65C320.030.000.73C320.030.000.73
C330.050.650.00C330.050.650.00C330.030.730.00C330.030.730.00C330.030.730.00
Note. DCV = De-fuzzy Crisp Number. C = Criteria.
Table A15. The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A10.
Table A15. The de-fuzzy crisp values for Triangle Fuzzy Numbers in Table A10.
E1C41
(DVC)
C42
(DVC)
E2C41
(DVC)
C42
(DVC)
E3C41
(DVC)
C42
(DVC)
E4C41
(DVC)
C42
(DVC)
E5C41
(DVC)
C42
(DVC)
Internal safety (C41)0.000.27C410.000.50C410.000.27C410.000.50C410.000.73
External safety (C42)0.730.00C420.500.00C420.730.00C420.730.00C420.730.00
E6C41
(DVC)
C42
(DVC)
E7C41
(DVC)
C42
(DVC)
E8C41
(DVC)
C42
(DVC)
E9C41
(DVC)
C42
(DVC)
E10C41
(DVC)
C42
(DVC)
C410.000.50C410.000.27C410.000.50C410.000.50C410.000.50
C420.500.00C420.970.00C420.500.00C420.730.00C420.730.00
Note. DCV = De-fuzzy Crisp Number. C = Criteria.

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Figure 1. Whole process of the BPM tool selection method.
Figure 1. Whole process of the BPM tool selection method.
Axioms 11 00601 g001
Figure 2. The membership function of the Triangular Fuzzy Number.
Figure 2. The membership function of the Triangular Fuzzy Number.
Axioms 11 00601 g002
Table 1. The criteria and sub-criteria for BPM tool selection.
Table 1. The criteria and sub-criteria for BPM tool selection.
CriteriaSub-Criteria
Efficiency parameters (C1)Expressiveness (C11)
Readability (C12)
Usability (C13)
Formality (C14)
Ease of Learning (C15)
Economical parameters (C2)Application and Installment cost (C21)
Operating cost (C22)
Training cost (C23)
Time parameters (C3)Application and Installment time (C31)
Operating time (C32)
Training time (C33)
Safety parameters (C4)Internal safety (C41)
External safety (C42)
Note. C = Criteria.
Table 2. Evaluation rank used for ranking BPM tools.
Table 2. Evaluation rank used for ranking BPM tools.
RankExplanation
1Very bad
2Good
3Normal
4Bad
5Very good
Table 3. Direct fuzzy influence degree and corresponding triangular fuzzy scale.
Table 3. Direct fuzzy influence degree and corresponding triangular fuzzy scale.
Direct Influence DegreeFuzzy Rank ValueTriangular Fuzzy Scale
No impact0000.25
Low impact100.250.5
Medium impact20.250.50.75
High impact30.50.751
Very high impact40.7511
Table 4. Evaluation rank value for all the business process modelling tool selection sub-criterions in Table 1.
Table 4. Evaluation rank value for all the business process modelling tool selection sub-criterions in Table 1.
AlternativeEfficiency (C1)Economical (C2)Time (C3)Safety (C4)
C12C12C13C14C15C21C22C23C31C32C33C41C42
TL12323512233311
TL23323512222211
TL34432334331145
TL44543432323355
TL54234343244222
TL63212211155221
TL74543213412145
TL85453334422545
Note. C = Criteria. TL = Tool.
Table 5. Average direct fuzzy influence matrix T between BPM tool selection main criterions.
Table 5. Average direct fuzzy influence matrix T between BPM tool selection main criterions.
TC1 (ADFI)C2 (ADFI)C3 (ADFI)C4 (ADFI)
Efficiency (C1)00.590.5450.684
Economical (C2)0.56700.4760.613
Time (C3)0.5660.68500.428
Safety (C4)0.6640.7550.6620
Note. ADFI = Average Direct Fuzzy Influence value.
Table 6. Average direct fuzzy influence matrix T between Efficiency (C1) sub-criterions.
Table 6. Average direct fuzzy influence matrix T between Efficiency (C1) sub-criterions.
T (C1)C11 (ADFI)C12 (ADFI)C13 (ADFI)C14 (ADFI)C15 (ADFI)
Expressiveness (C11)00.4930.6750.5480.786
Readability (C12)0.44200.4260.9290.548
Usability (C13)0.4970.43600.9520.855
Formality (C14)0.5070.7550.41400.866
Ease of Learning (C15)0.6020.7060.3990.0760
Note. ADFI = Average Direct Fuzzy Influence value.
Table 7. Average direct fuzzy influence matrix T between Economical (C2) sub-criterions.
Table 7. Average direct fuzzy influence matrix T between Economical (C2) sub-criterions.
T (C2)C21 (ADFI)C22 (ADFI)C23 (ADFI)
Application and Installment cost (C21)00.5460.687
Operating cost (C22)0.70800.685
Training cost (C23)0.5940.710
Note. ADFI = Average Direct Fuzzy Influence value.
Table 8. Average direct fuzzy influence matrix T between Time (C3) sub-criterions.
Table 8. Average direct fuzzy influence matrix T between Time (C3) sub-criterions.
T (C3)C31 (ADFI)C32 (ADFI)C33 (ADFI)
Application and Installment time (C31)00.2160.628
Operating time (C32)0.03600.73
Training time (C33)0.0450.6670
Note. ADFI = Average Direct Fuzzy Influence value.
Table 9. Average direct fuzzy influence matrix T between Safety (C4) sub-criterions.
Table 9. Average direct fuzzy influence matrix T between Safety (C4) sub-criterions.
T (C4)C41 (ADFI)C42 (ADFI)
Internal safety (C41)00.454
External safety (C42)0.6850
Note. ADFI = Average Direct Fuzzy Influence value.
Table 10. Normalized direct influence matrix G between BPM tool selection main criteria.
Table 10. Normalized direct influence matrix G between BPM tool selection main criteria.
GEfficiency (C1)Economical (C2)Time (C3)Safety (C4)
Efficiency (C1)00.2840.2620.329
Economical (C2)0.27200.2290.295
Time (C3)0.2720.32900.206
Safety (C4)0.3190.3630.3180
Table 11. Normalized direct influence matrix T between Efficiency (C1) sub-criteria.
Table 11. Normalized direct influence matrix T between Efficiency (C1) sub-criteria.
G (C1)C11C12C13C14C15
Expressiveness (C11)00.180.250.20.29
Readability (C12)0.1600.160.340.2
Usability (C13)0.180.1600.350.31
Formality (C14)0.190.280.1500.32
Ease of Learning (C15)0.220.260.150.030
Table 12. Normalized direct influence matrix T between Economical (C2) sub-criteria.
Table 12. Normalized direct influence matrix T between Economical (C2) sub-criteria.
G (C2)C21C22C23
Application and Installment cost (C21)00.390.49
Operating cost (C22)0.5100.49
Training cost (C23)0.430.510
Table 13. Normalized direct influence matrix T between Time (C3) sub-criteria.
Table 13. Normalized direct influence matrix T between Time (C3) sub-criteria.
G (C3)C31C32C33
Application and Installment time (C31)00.260.74
Operating time (C32)0.0400.86
Training time (C33)0.050.790
Table 14. Normalized direct influence matrix T between Safety (C4) sub-criteria.
Table 14. Normalized direct influence matrix T between Safety (C4) sub-criteria.
G (C4)C41C42
Internal safety (C41)00.66
External safety (C42)10
Table 15. Total direct and indirect influence matrix E between BPM tool selection main criteria.
Table 15. Total direct and indirect influence matrix E between BPM tool selection main criteria.
EEfficiency (C1)Economical (C2)Time (C3)Safety (C4)ronron + coj
Efficiency (C1)1.4841.8521.6141.6966.64613.181
Economical (C2)1.5921.5141.4931.5736.17213.356
Time (C3)1.5851.7561.2991.5136.15312.347
Safety (C4)1.8742.0621.7881.5937.31713.692
coj6.5357.1846.1946.375
The ron (highlighted in yellow) means the total given both direct and indirect effects from criteria n to the other criteria, and coj (highlighted in pink) means the total received both direct and indirect effects from other criterions to criterion j. The overall direct and indirect impact matrix E with corresponding ron and coj values, when n equals j, (roj + coj) (highlighted in lime green) value, represents the centrality of criterion j.
Table 16. Total direct influence matrix T between Efficiency (C1) sub-criteria.
Table 16. Total direct influence matrix T between Efficiency (C1) sub-criteria.
E (C1)C11C12C13C14C15ronron + coj
Expressiveness (C11)1.051.361.181.341.666.5912.25
Readability (C12)1.141.161.071.381.546.2912.81
Usability (C13)1.281.441.041.521.787.0612.34
Formality (C14)1.211.441.111.171.686.6112.97
Ease of Learning (C15)0.981.120.880.951.085.0112.75
coj5.666.525.286.367.74
Table 17. Total direct influence matrix T between Economical (C2) sub-criteria.
Table 17. Total direct influence matrix T between Economical (C2) sub-criteria.
E (C2)C21C22C23ronron + coj
Application and Installment cost (C21)4.734.895.214.8214.82
Operating cost (C22)5.515.035.6516.1916.19
Training cost (C23)5.275.185.1215.5715.57
coj15.5115.115.97
Table 18. Total direct influence matrix T between Time (C3) sub-criteria.
Table 18. Total direct influence matrix T between Time (C3) sub-criteria.
E (C3)C31C32C33ronron + coj
Application and Installment time (C31)0.343.544.047.927.92
Operating time (C32)0.353.043.737.127.12
Training time (C33)0.343.363.156.856.85
coj1.039.9410.92
Table 19. Total direct influence matrix T between Safety (C4) sub-criteria.
Table 19. Total direct influence matrix T between Safety (C4) sub-criteria.
E (C4)C41C42ronron + coj
Internal safety (C41)1.941.943.888.76
External safety (C42)2.941.944.888.76
coj4.883.88
Table 20. Normalized (ro + co) value (weight) for all the all the business process modelling tool selection criteria and sub-criteria.
Table 20. Normalized (ro + co) value (weight) for all the all the business process modelling tool selection criteria and sub-criteria.
Direct Influence DegreeNRC (W)Sub-CriteriaNRC (W)
Efficiency parameters (C1)0.251Expressiveness (C11)0.194
Readability (C12)0.203
Usability (C13)0.196
Formality (C14)0.205
Ease of Learning (C15)0.202
Economical parameters (C2)0.254Application and installment cost (C21)0.32
Operating cost (C22)0.35
Training cost (C23)0.33
Time parameters (C3)0.235Application and installment time (C31)0.36
Operating time (C32)0.33
Training time (C33)0.31
Safety parameters(C4)0.26Internal safety (C41)0.5
External safety (C42)0.5
Note. NRC = Normalized (ro + co) value. W = Weight.
Table 21. The weighted normalized value of all the business process modelling tool alternatives in Table 4.
Table 21. The weighted normalized value of all the business process modelling tool alternatives in Table 4.
AlternativeEfficiency (C1)Economical (C2)Time (C3)Safety (C4)
C11
Max
C12
Max
C13
Max
C14
Max
C15
Max
C21
Min
C22
Min
C23
Min
C31
Min
C32
Min
C33
Min
C41
Max
C42
Max
T10.0090.0150.0110.0190.0250.0120.0220.0210.030.0270.0290.0140.013
T20.0140.0150.0110.0190.0250.0120.0220.0210.020.0180.0190.0140.013
T30.0180.020.0160.0120.0150.0360.0450.0320.030.0090.010.0570.063
T40.0180.0250.0210.0190.020.0360.0220.0320.020.0270.0290.0570.063
T50.0180.010.0160.0250.0150.0470.0340.0210.040.0370.0190.0710.025
T60.0140.010.0050.0120.010.0120.0110.0110.050.0460.0190.0290.013
T70.0180.0250.0210.0190.010.0120.0340.0420.010.0180.010.0290.063
T80.0230.020.0270.0190.0150.0360.0450.0420.020.0180.0480.0570.063
Note. C = Criteria. T = Tool.
Table 22. The P-I and N-I solution of the considered BPM tool alternatives.
Table 22. The P-I and N-I solution of the considered BPM tool alternatives.
AlternativeEfficiency (C1)Economical (C2)Time (C3)Safety (C4)
C12
Max
C12
Max
C13
Max
C14
Max
C15
Max
C21
Min
C22
Min
C23
Min
C31
Min
C32
Min
C33
Min
C41
Max
C42
Max
T+0.0230.0250.0270.0250.0250.0120.0110.0110.010.0090.010.0710.063
T0.0090.010.0050.0120.010.0470.0450.0420.050.0460.0480.0140.013
Note. C = Criteria. T = Tool.
Table 23. TOPSIS result for all 8 candidate Business Process Modelling tools.
Table 23. TOPSIS result for all 8 candidate Business Process Modelling tools.
Alternativedi*di0Ci*
T10.0880.060.405
T20.0820.0710.464
T30.0560.090.616
T40.0480.0860.642
T50.0740.0730.497
T60.0920.0670.421
T70.0610.0920.601
T80.0690.0830.546
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Jin, G.; Jin, G.; Huo, H. Selection of Business Process Modeling Tool with the Application of Fuzzy DEMATEL and TOPSIS Method. Axioms 2022, 11, 601. https://doi.org/10.3390/axioms11110601

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Jin G, Jin G, Huo H. Selection of Business Process Modeling Tool with the Application of Fuzzy DEMATEL and TOPSIS Method. Axioms. 2022; 11(11):601. https://doi.org/10.3390/axioms11110601

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Jin, Guangying, Guangzhe Jin, and Haibo Huo. 2022. "Selection of Business Process Modeling Tool with the Application of Fuzzy DEMATEL and TOPSIS Method" Axioms 11, no. 11: 601. https://doi.org/10.3390/axioms11110601

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