A risk assessment method based on Pythagorean fuzzy set and artificial-neuron-like evaluation node

Abstract

Taking concepts from supply management, we developed a specification-assessment-compliance approach to obtain a transparent multi-criteria decision-making method. We designed an artificial-neuron-like node that allows the implementation of networks to represent hierarchies of evaluation criteria. A new graphical model based on functions in the unit segment uses the concept of Pythagorean fuzzy set (PFS). The specification PFSs’ entropies modulate the widths of one-sided triangular fuzzy numbers (TFNs) with positive slopes that become the evaluation nodes’ activation functions. All the specifications refer to the same point to facilitate the evaluation and ensure coherence. One-sided TFNs with negative slopes biunivocally represent the assessment PFSs at the input layer of the network. A risk case study on the options for the outsourcing of an information technology development project shows the proposed method’s implementation. We compare the results with those of the application of two other previous methods.

Keywords

Multi-criteria decision-making Pythagorean fuzzy set triangular fuzzy number artificial neuron information entropy

1 Introduction

Nowadays, organizations depend more and more on information technology (IT). Organizations launch IT projects across all departments to adapt to new required services, maintain competitiveness, meet emerging needs, and keep pace with technology in continuous evolution [1, 2]. IT projects are challenging to manage and show a higher failure rate than projects in other areas [3, 4]. The continuous evolution of technology generates cycles in which the rate of failure slowly drops thanks to the learning process, to rocket up again when new technological paradigms arise [5, 6]. Given that proper risk management is essential for projects of any type, this asseveration becomes more drastic in IT projects.

The motivation and the goal of this paper are to overcome the opacity of risk assessment methods. The research objective is to implement a flexible, transparent method that assesses the risk factors, analyzes their interactions, and evaluates the project’s overall risk.

In multi-criteria decision-making (MCDM), a set of evaluation criteria {C₁, C₂, … , C_M } determines how well the options (O₁, O₂, … , O_N) suit the desired solution. The MCDM implementation assigns different weights (W₁, W₂, … , W_M) to the criteria according to their importance and assesses each option concerning each evaluation criterion. The application of an MCDM model [7] permits implementing a risk evaluation method in which the risk factors become the evaluation criteria.

Risk characterization is uncertain by nature.

Fuzzy logic [8] uses fuzzy sets (FS) with membership functions. Many authors use triangular fuzzy numbers (TFN) [9] as membership functions.

FS and intuitionistic fuzzy sets (IFS) [10] help deal with uncertain values.

Pythagorean fuzzy sets (PFS) [11] overcome the limitations of classic IFS [12] and are better representing and handling uncertainties [13 –15].

The entropy of a fuzzy set provides information on the amount of knowledge it contains [14 –16].

Several authors have proposed fuzzy logic-based methods for project risk evaluation [17 –24] (Table 1).

Table 1
Fuzzy evaluation of project risk

Application Applied Paradigm Fixed reference Hierarchy or network rep-representation and calculation Specifications tolerances Individualized risk factor performance Analysis of interactions among risk factors Group decision

[17] IT project Fuzzy Analytic Hierarchy Process FS No Yes No No No No

[18] Outsourced Software project Risk functions Interval-valued IFS Yes No No Yes No Yes

[19] Outsourced Software project Risk functions Interval-valued IFS Yes No No Yes No Yes

[20] IT project Fuzzy Analytic Hierarchy Process, Fuzzy inference system FS No Yes No No No No

[21] Selection of risk management approach to an IT project Vectorial Analysis IFS No No No No No No

[22] Construction FMEA, VIKOR Hesitant FS No No No No No Yes

[23] Software effort estimation Fuzzy Similarity Measure, Fuzzy Reasoning Method FS No No No No No No

[24] Enterprise Resource Planning adoption Fuzzy Petri Net model FS No Yes No Yes No No

Proposed method Outsourced IT development project PFS/TFN, Artificial Neuron like evaluation nodes PFS Yes Yes Yes Yes Yes No

	Application	Applied	Paradigm	Fixed reference	Hierarchy or network rep-representation and calculation	Specifications tolerances	Individualized risk factor performance	Analysis of interactions among risk factors	Group decision
[17]	IT project	Fuzzy Analytic Hierarchy Process	FS	No	Yes	No	No	No	No
[18]	Outsourced Software project	Risk functions	Interval-valued IFS	Yes	No	No	Yes	No	Yes
[19]	Outsourced Software project	Risk functions	Interval-valued IFS	Yes	No	No	Yes	No	Yes
[20]	IT project	Fuzzy Analytic Hierarchy Process, Fuzzy inference system	FS	No	Yes	No	No	No	No
[21]	Selection of risk management approach to an IT project	Vectorial Analysis	IFS	No	No	No	No	No	No
[22]	Construction	FMEA, VIKOR	Hesitant FS	No	No	No	No	No	Yes
[23]	Software effort estimation	Fuzzy Similarity Measure, Fuzzy Reasoning Method	FS	No	No	No	No	No	No
[24]	Enterprise Resource Planning adoption	Fuzzy Petri Net model	FS	No	Yes	No	Yes	No	No
Proposed method	Outsourced IT development project	PFS/TFN, Artificial Neuron like evaluation nodes	PFS	Yes	Yes	Yes	Yes	Yes	No

As part of the case study, we compare the results obtained by the new method proposed here with the results obtained by applying a vectorial method [21]. In that method, IFS and TFN did not relate, and the latter was used only in calculating weights. The vectorial method used the classic representation of IFS. It calculated Euclidean distances from the points corresponding to the ideal virtual solution to the points that correspond to the option under evaluation. It is not possible to consider the tolerances with the vectorial method.

The aggregation of the assessments by vector calculations did not permit the analysis of interactions among risk factors and was less intuitive in representing the decision-making problem.

Many intuitionistic evaluation methods measure the distance-based similarity. From the direct measurement of the Hamming distance [25] or the Euclidean distance [26] to more sophisticated approaches to the calculation of distance [21], different authors compare every parameter under evaluation with its respective reference. The similarity/dissimilarity method [27] relates knowledge to the distance to the reference. It improves coherence by fixing the reference at the most intuitionistic point. Some methods use a two-sided symmetric TFN’s width to measure the uncertainty or lack of confidence in an assigned value [28]. Other authors propose the entropy-based measuring of the knowledge contained in the IFSs to calculate the weights of criteria in decision-making problems [29].

There are geometric interpretations of IFS and PFS [30, 31] based on intuitionistic space points.

We can find two approaches to integrating IFS and artificial neurons. The most common approach consists of the combination of IFS and fuzzy inference networks. The fuzzy inference network includes an intuitionistic fuzzy inference layer in which each element represents a fuzzy inference rule [32, 33]. The least common approach adapts the classic artificial neural network to admit IFS or PFS as input, output, or weight [34]. A classic artificial neural network needs a previous training procedure and produces results through an opaque process.

To provide simplicity and transparency, permit partial evaluations, analyze inter-factors influence, and have flexibility, we do not design an inference network but an artificial-neuron-like-based network that implements a hierarchy of evaluation criteria representing the taxonomy of risk factors. This paper integrates the PFS and the TFN to obtain the artificial neuron-like evaluation node. As in the classic artificial neural network, not having an inference layer, it is not possible to use a PFS directly as an input of the activation function that, in our case, expresses the specification tolerance.

Several authors have demonstrated the use of TFN as input of artificial neural networks [35, 36], usually to integrate linguistic variables in their models [37, 38]. So that, since the PFS cannot directly feed the activation function, the main benefit of transforming the PFS into a one-sided TFN is coherently integrating its three values in a single function, permitting a PFS to become the input of an evaluation network. Thus, the PFS/TFN model, in association with the activation function, acts as a new adjustable defuzzification module. Furthermore, evaluators will find it easier to visualize than the representation as a point somewhere in the first quadrant of the unit circumference.

In this way, the new method benefits from both the hierarchical evaluation network capabilities and the PFS advantages to deal with uncertainty, thus implementing an understandable specification-assessment-compliance approach that facilitates the evaluator’s task.

Unlike previous proposals, we obtain an intuitive representation of the decision-making problem through an evaluation network. Furthermore, setting the reference at the maximum membership level is more intuitive and keeping it along the process facilitates the assessment’s coherence.

The method could also apply, with some minor modifications, to other evaluation or decision-making problems. Furthermore, we can explore the use of the PFS/TFN model as part of other specific fuzzy methods that apply to varied fields, from the web service selection for rejuvenation [39] or the fuzzy transportation problem [40] to the shortest path problem [41].

Table 1 compares the proposed method with other fuzzy logic-based proposals for the evaluation of risk in projects.

2 Preliminaries

2.1 Artificial neural network

In artificial neurons, each weight W_i multiplies an input x_i and the sum of all the weighted inputs (plus a usually necessary bias b) stimulates an activation function f (x).

The output y of each artificial neuron is [42]: $y = f (b + \sum_{i} W_{i} x_{i})$ (1)

Classic artificial neural networks arrange artificial neurons in a fully connected network that needs a training process to adjust its weights and biases.

2.2 Fuzzy set, intuitionistic fuzzy set, and Pythagorean fuzzy set

Given a universe of discourse X, a fuzzy set (FS) A is described by: $A = {(x, μ_{A} (x)) | x \in X}$ (2) where μ_A (x) denotes the level of membership of x to A, with 0 ⩽ μ_A (x) ⩽ 1. An intuitionistic fuzzy set (IFS) is described by [10]: $A = {(x, μ_{A} (x), ν_{A} (x)) | x \in X}$ (3) where μ_A (x) denotes the level of membership of x to A, and ν_A (x) denotes the level of non-membership of x to A, with the following restriction: $μ_{A} (x) + ν_{A} (x) ⩽ 1$ (4)

In an IFS, π_A (x) = 1 - μ_A (x) - ν_A (x) denotes the level of indeterminacy of x to A.

The Pythagorean fuzzy set (PFS) [11] is a generalization of the concept of IFS with the following restriction [12]: $μ_{A}^{2} (x) + ν_{A}^{2} (x) ⩽ 1$ (5)

In a PFS, $π_{A} (x) = \sqrt{1 - μ_{A}^{2} (x) - ν_{A}^{2} (x)}$ denotes the level of indeterminacy of x to A.

2.3 Triangular fuzzy number

The triangular function $f_{\tilde{a}} (x)$ is denoted as a triangular fuzzy number (TFN) $\tilde{a} = (l, m, u)$ [9].

The function of a one-sided TFN with a positive slope (l, m, m) is [43]: $f_{\tilde{a} +} (x) = {\begin{matrix} \begin{matrix} 0; & x < l \end{matrix} \\ \begin{matrix} \frac{x - l}{m - l}; & l ⩽ x ⩽ m \end{matrix} \\ \begin{matrix} 0; & x > m \end{matrix} \end{matrix}$ (6) The function of a one-sided TFN with a negative slope (m, m, u) is: $f_{\tilde{a} -} (x) = {\begin{matrix} \begin{matrix} 0; & x < m \end{matrix} \\ \begin{matrix} \frac{u - x}{u - m}; & m ⩽ x ⩽ u \end{matrix} \\ \begin{matrix} 0; & x > u \end{matrix} \end{matrix}$ (7)

2.4 Entropy and lack of knowledge

The lack of knowledge of a PFS can be related to its entropy [16]. The following equation calculates the entropy of a PFS [15]: $E (x) = 1 - (μ_{A}^{2} (x) + ν_{A}^{2} (x)) | μ_{A}^{2} (x) - ν_{A}^{2} (x) |$ (8)

3 Methodology

The proposed method connects artificial neuron-like evaluation nodes to build a network representing the risk factors’ taxonomy.

This hierarchy of risk factors becomes the hierarchy of evaluation criteria and sub-criteria of the new MCDM method.

Figure 1 shows a schematic representation of the artificial neuron-like evaluation node model. With minor changes, the proposed method can use either IFS or PFS. We selected PFS because PFS has a lesser restricted range of membership and non-membership values than IFS, facilitating the evaluator’s task.

Fig. 1

Evaluation node.

At hierarchy levels other than the input layer, the crisp output values y that result from the previous layer are the inputs in_i to the evaluation nodes.

Each input to an evaluation node has an associated weight (W_i) related to the level of influence of the respective risk factor.

We can evaluate individual risk factors along the hierarchy and take the hierarchy output as the overall risk.

The evaluation method includes the following steps: Step 1. Evaluation hierarchy. Once we have identified the risk factors, we arrange them in a hierarchy of evaluation nodes.

Weights (W_i) set the importance of every evaluation criterion or sub-criterion. The calculation of weights for the evaluation criteria and sub-criteria is beyond the scope of this work.

The connection of the evaluation nodes implements the evaluation network.

Step 2. Specification of the desired characteristics and their tolerances. The evaluation reference O₀ is the ideal solution, consisting of a set of specifications.

For every evaluation sub-criterion of the input layer, we specify its desired characteristics employing a descriptive text, and we define the tolerance for every sub-criterion and criterion using a specification PFS (μ_s, ν_s, π_s): $μ_{s}^{2} + ν_{s}^{2} + π_{s}^{2} = 1; μ_{s} ⩾ 0, ν_{s} ⩾ 0, π_{s} ⩾ 0$ (9)

The membership value μ_s sets the level of confidence in the specification description or the level of requirement of its fulfillment. The non-membership value ν_s sets the lack of confidence in the specification description or the level of looseness in its requirement. The value π_s sets the level of hesitance or uncertainty concerning the given specification.

We use the entropy-based Equation (8) to calculate the width (0 ⩽ d = 1 - d_s ⩽ 1) of a specification ramp and so setting the activation function threshold, considering that it will not make sense to specify a requirement and be more confident in its inadequacy than in its suitability and that therefore, in practical specification cases, we will have that $μ_{s}^{2} ⩾ ν_{s}^{2}$ : $d_{s} = 1 - E_{s} = (μ_{s}^{2} + ν_{s}^{2}) | μ_{s}^{2} - ν_{s}^{2} |$ (10)

Figure 2 shows the threshold values d_s for ν_s = 0, and μ_s > π_s.

Fig. 2

d_s vs π_s; ν_s = 0.

Figure 3 shows the threshold values d_s for π_s = 0 and μ_s > ν_s.

Fig. 3

d_s vs ν_s; π_s = 0.

Figure 4 shows the specification function S (x): a one-sided TFN with a positive slope and maximum value at x = 1 that becomes a ramp-type activation function in the neuron-like evaluation node (Fig. 1).

Fig. 4

Activation function S (x).

For d_s < 1 : $S (x) = {\begin{matrix} \frac{x - d_{s}}{1 - d_{s}}; & d_{s} < x ⩽ 1 \\ 0; & x ⩽ d_{s} \end{matrix}$ (11) For d_s = 1 : $S (x) = {\begin{matrix} 1; & x = 1 \\ 0; & x < 1 \end{matrix}$ (12)

A greater width d (i.e., a smaller threshold value d_s) means a lesser restrictive specification. A null width d (i.e.,d_s = 1) represents the most restrictive condition.

The specification PFS (μ_s, ν_s, π_s) modulates the width of the activation function S (x). Nevertheless, this TFN is not a representation of the PFS.

Step 3. Assessment. For a given option O_n, an assessment PFS (μ_a, ν_a, π_a) applies to each sub-criterion of the input layer, obtained by comparing the real option O_n with the ideal solution O₀.

For every evaluation sub-criterion of the input layer, the evaluator analyzes the extent to which the option under evaluation O_n matches the specification descriptive text.

The membership value μ_a sets the level of match between the specification and the option under evaluation. The non-membership value ν_a sets the level of incompatibility between the specification and the option under evaluation. The value π_a sets the level of hesitancy or uncertainty of the assessment. In this way, the assessment PFS (μ_a, ν_a, π_a) assesses the compliance with a specification by the option under evaluation.

We consider three extreme cases: When, for a given sub-criterion, the evaluated option O_n perfectly matches the specification, the assigned PFS is (1, 0, 0); if it does not match at all, the assigned PFS is (0, 1, 0); and if the evaluator is not able to produce any conclusion about the comparison between the option and the specification, the assigned PFS is (0, 0, 1).

Limited by these three extreme values, and considering that $μ_{a}^{2} + ν_{a}^{2} + π_{a}^{2} = 1$ , μ_a ⩾ 0, ν_a ⩾ 0, and π_A ⩾ 0, any other intermediate value is possible.

PFS/TFN

As shown in Fig. 5, a one-sided TFN A (x) with a negative slope biunivocally represents the assessment triplet (μ_a, ν_a, π_a): $A (x) = {\begin{matrix} 0; & x < μ_{a}^{2} \\ \frac{1 - ν_{a}^{2} - x}{1 - ν_{a}^{2} - μ_{a}^{2}}; & μ_{a}^{2} ⩽ x ⩽ μ_{a}^{2} + π_{a}^{2} \\ 0; & x > μ_{a}^{2} + π_{a}^{2} \end{matrix}$ (13)

Fig. 5

Assessment function A (x).

A (x) reaches its maximum value at the membership squared value of the assessment PFS: $A (μ_{a}^{2}) = 1$ .

The closer μ_a is to the point of the maximum level of membership (i.e., the closer μ_a is to x = 1), the better the option meets the specification.

The width of this TFN corresponds to the hesitance or uncertainty squared value $π_{a}^{2}$ and we can consider it as a measurement of the fuzziness of the TFN; the smaller π_a is (i.e., the narrower the TFN is), the more confident the evaluator is in the values assigned to μ_a and ν_a.

Step 4. Calculation of the compliance values v and the risk factors values r that derive from them. Figure 6 shows that the output v of an activation function at the hierarchy’s input layer is the value at the intersection between the assessment function A(x) and the specification function S(x). $v = {\begin{matrix} 0; & μ_{a}^{2} + π_{a}^{2} < ds \\ 0; & (μ_{a}^{2} + π_{a}^{2} = ds), (μ_{a} \neq 1) \\ \frac{μ_{a}^{2} + π_{a}^{2} - d_{s}}{1 + π_{a}^{2} - d_{s}}; & μ_{a}^{2} + π_{a}^{2} > ds \\ 1; & μ_{a} = ds = 1 \end{matrix}$ (14)

Fig. 6

Output v of an activation function S (x) in the input layer.

Figures 7 and 8 show examples of output values v of an activation function in the input layer with a threshold value d_s = 0.5.

Fig. 7

v vs. $μ_{a}^{2}$ for d_s = 0.5 and $π_{a}^{2} = 0.1$ .

Fig. 8

v vs. $π_{a}^{2}$ for d_s = 0.5 and $μ_{a}^{2} = 0.1$ .

Each risk factor value r is given by: $r = 1 - v$ (15)

At hierarchy levels other than the input layer, the previous layer’s crisp output values y are the inputs in to the evaluation nodes. Figure 9 shows the output v of the activation function S (in). $v = S (in) = {\begin{matrix} 0; & in < ds \\ 0; & in = ds \neq 1 \\ \frac{in - d_{s}}{1 - d_{s}}; & in > ds \\ 1; & in = ds = 1 \end{matrix}$ (16) The output y of an evaluation node (Fig. 1) is the weighted sum of the outputs v of all its activation functions: $y = \sum_{i = 1}^{i = I} w_{i} v_{i}$ (17)

Fig. 9

Output v of an activation function S(x) in layers other than the input one.

4 Case study

The study’s objective is to show the method application. The company performs development projects for its own sales network in Spain. In some cases, a third party will execute the project, and it is necessary to evaluate the risk of the different options before subcontracting. The study evaluates the risk in seven options for an outsourced IT development project related to electronic transactions. After interviewing some employees, an evaluator designed the hierarchy of risk factors, calculated the weights, described the evaluation reference, set the specifications tolerances, and assessed each option’s risk factors.

4.1 Evaluation process

Step 1. We arranged the hierarchy (Fig. 10) according to the following risk factors (evaluation criteria):

Technical

Infrastructure

Implementation of standards

Personnel education and experience

Management:

Financial health

Project management resources

Collaborative:

Previous collaborations

Country

Fig. 10

Case study. Evaluation hierarchy.

We calculated the normalized weights (Table 2). For this purpose, we used FAHP, but the process for weights calculation is not within the scope of this paper, and any suitable method is valid.

Table 2

Normalized weights

W₁ = 0.4230	w₁₁ = 0.4966
	w₁₂ = 0.3368
	w₁₃ = 0.1666
W₂ = 0.3658	w₂₁ = 0.6667
	w₂₂ = 0.3333
W₃ = 0.2112	w₃₁ = 0.5000
	w₃₂ = 0.5000

Step 2. We specified the evaluation reference O₀ by mean of written descriptions of the risk factors (i.e., the evaluation sub-criteria) at the input layer of the network to obtain a set of specifications:

Technical:

Infrastructure: Availability of all the necessary equipment for the development and test of the required applications.

Implementation of standards: Skilled use of accepted engineering and quality standards.

Personnel education and experience: Sound experience in the development of similar applications.

Management:

Financial health: Favorable position concerning its assets and liabilities. Business continuity out of the question.

Project management resources: Use of project management methodologies. History of schedule and budget compliance.

Collaborative:

Previous collaborations: Existence of previous completely satisfactory experiences.

Country: Absolute legal certainty. Same time zone and common language.

We specified the tolerance for every sub-criterion and criterion. Table 3 shows the values of the specification PFS (μ_s, ν_s, π_s) for S_ij and S_i required for the evaluation sub-criteria (C_ij) and criteria (C_i) and the resulting threshold values d_s.

Table 3

Specifications tolerances

C_ij, C_i	S_ij, S_i	μ _s	ν _s	π _s	d_s
C₁₁	S₁₁	0.95	0.21	0.23	0.700
C₁₂	S₁₂	0.88	0.02	0.47	0.666
C₁₃	S₁₃	0.73	0.15	0.67	0.322
C₂₁	S₂₁	0.88	0.02	0.47	0.666
C₂₂	S₂₂	0.90	0.32	0.30	0.529
C₃₁	S₃₁	0.90	0.18	0.40	0.607
C₃₂	S₃₂	0.92	0.12	0.37	0.689
C₁	S₁	0.80	0.20	0.57	0.408
C₂	S₂	0.90	0.18	0.40	0.607
C₃	S₃	0.96	0.10	0.26	0.801
	S_out	$\sqrt{0.5}$	$\sqrt{0.5}$	0.00	0.000

Step 3. Table 4 shows the squared values of the assessment PFS of each sub-criterion C_ij for every option O_n.

Table 4

Squared values of the assessment

O_n	C _ij	$μ_{a}^{2}$	$ν_{a}^{2}$	$π_{a}^{2}$
O₁	C₁₁	0.4905	0.4905	0.0191
	C₁₂	0.8587	0.0194	0.1219
	C₁₃	0.4597	0.3109	0.2293
	C₂₁	0.9693	0.0193	0.0114
	C₂₂	0.9658	0.0215	0.0127
	C₃₁	0.4969	0.4969	0.0063
	C₃₂	0.9734	0.0157	0.0108
O₂	C₁₁	0.9875	0.0123	0.0002
	C₁₂	0.9866	0.0132	0.0002
	C₁₃	0.9866	0.0132	0.0002
	C₂₁	0.9497	0.0298	0.0205
	C₂₂	0.9658	0.0215	0.0127
	C₃₁	0.9212	0.0394	0.0394
	C₃₂	0.9875	0.0146	0.0100
O₃	C₁₁	0.4905	0.4905	0.0191
	C₁₂	0.9864	0.0133	0.0003
	C₁₃	0.9617	0.0304	0.0079
	C₂₁	0.9863	0.0133	0.0004
	C₂₂	0.3752	0.6014	0.0234
	C₃₁	0.9676	0.0108	0.0215
	C₃₂	0.4745	0.4745	0.0511
O₄	C₁₁	0.9899	0.0101	0.0000
	C₁₂	0.4449	0.4449	0.1102
	C₁₃	0.5884	0.3958	0.0158
	C₂₁	0.4969	0.4969	0.0063
	C₂₂	0.4973	0.4973	0.0055
	C₃₁	0.7673	0.2273	0.0064
	C₃₂	0.9780	0.0206	0.0014
O₅	C₁₁	0.9899	0.0101	0.0000
	C₁₂	0.4905	0.4905	0.0191
	C₁₃	0.4291	0.5560	0.0149
	C₂₁	0.1082	0.8886	0.0032
	C₂₂	0.9989	0.0011	0.0000
	C₃₁	0.3786	0.6051	0.0164
	C₃₂	0.4853	0.4853	0.0294
O₆	C₁₁	0.3752	0.6014	0.0234
	C₁₂	0.9947	0.0052	0.0001
	C₁₃	0.1964	0.7726	0.0311
	C₂₁	0.6764	0.3150	0.0086
	C₂₂	0.7663	0.2273	0.0064
	C₃₁	0.3786	0.6051	0.0164
	C₃₂	0.9780	0.0206	0.0014
O₇	C₁₁	0.9989	0.0011	0.0000
	C₁₂	0.9439	0.0281	0.0281
	C₁₃	0.3752	0.6014	0.0234
	C₂₁	0.4905	0.4905	0.0191
	C₂₂	0.9989	0.0011	0.0000
	C₃₁	0.6802	0.3078	0.0120
	C₃₂	0.9875	0.0123	0.0002

Step 4. Table 5 shows the output V and the partial results v_i of compliance with the specifications (the higher, the better) of criteria and sub-criteria for option O₁. We can also calculate the individual assessment of each risk factor (r = 1 - v).

Table 5

Output V and partial results v for option O₁

V = 0.2888	v₁ = 0.0000	v₁₁ = 0.0000
		v₁₂ = 0.6898
		v₁₃ = 0.4045
	v₂ =0.2888	v₂₁ = 0.9112
		v₂₂ = 0.9292
	v₃ =0.0000	v₃₁ = 0.0000
		v₃₂ = 0.9176

Table 6 shows the outputs V_nof the hierarchy (the higher, the better), the results for the overall risk R_n = 1 - V_n(the higher, the worse), and the normalized scores S_n = V_n/Max (V) obtained for every option O_n.

Table 6

Outputs V_n, overall risk R_na nd normalized score S_n

O_n	V_n	R_n	S_n
O₁	0.2888	0.7112	0.3979
O₂	0.7259	0.2741	1.0000
O₃	0.0822	0.9178	0.1132
O₄	0.0997	0.9003	0.1374
O₅	0.0723	0.9277	0.0996
O₆	0.0000	1.0000	0.0000
O₇	0.2783	0.7217	0.3834

The results were coherent: criteria and sub-criteria with higher weights presented a more remarkable influence on the assessment, and options assessed as closer to the specification produced outputs V_n with higher values.

4.2 Analysis of risk dependencies

Let us suppose that the risk factor C₃₁ influences risk factor C₁₃. We add a new level in the hierarchy and perform a backward connection (Fig. 11). For simplicity, both inputs to S₁₃ are equally weighted (w₁₃₁ = w₁₃₂ = 0.5), and we set the two new activation functions for a specification PFS $(\sqrt{0.5}, \sqrt{0.5}, 0)$ (i.e., d_s = 0). The rest of the activation functions and weights maintain their original values.

Fig. 11

Analysis of risk dependencies.

Table 7 shows the output V_n, the risk R_n and the normalized score S_n for every option O_n considering the influence of C₃₁ on C₁₃. Some values show slight differences from those in Table 6.

Table 7

Influence of C₃₁ on C₁₃

O_n	V_n	R_n	S_n
O₁	0.2888	0.7112	0.4032
O₂	0.7162	0.2838	1.0000
O₃	0.0809	0.9191	0.1129
O₄	0.1003	0.8997	0.1400
O₅	0.0618	0.9382	0.0863
O₆	0.0000	1.0000	0.0000
O₇	0.2844	0.7156	0.3970

4.3 Comparison of the results with those obtained with other methods

We applied two other methods to the case study: the vectorial method [21] and the Mean-Variance model. Table 8 shows the results. The best option was always O₂, but the ranking in the proposed method varies from those in the other two methods.

Table 8
Results obtained with three different methods

Proposed method Vectorial Mean-Variance

Tolerances set Tolerances unset method

O_n S_n S_n S_n E_n σ _n

O₁ 0.3979 0.8049 0.5152 0.6950 0.0516

O₂ 1.0000 1.0000 1.0000 0.8459 0.0344

O₃ 0.1132 0.7816 0.5945 0.7015 0.0305

O₄ 0.1374 0.7112 0.4299 0.6011 0.0349

O₅ 0.0996 0.5711 0.3882 0.5773 0.0227

O₆ 0.0000 0.6640 0.4184 0.6095 0.0296

O₇ 0.3834 0.8186 0.6234 0.7441 0.0304

Proposed method	Vectorial	Mean-Variance
O_n	S_n	S_n	S_n	E_n	σ _n
O₁	0.3979	0.8049	0.5152	0.6950	0.0516
O₂	1.0000	1.0000	1.0000	0.8459	0.0344
O₃	0.1132	0.7816	0.5945	0.7015	0.0305
O₄	0.1374	0.7112	0.4299	0.6011	0.0349
O₅	0.0996	0.5711	0.3882	0.5773	0.0227
O₆	0.0000	0.6640	0.4184	0.6095	0.0296
O₇	0.3834	0.8186	0.6234	0.7441	0.0304

If we disable the tolerances, setting all the specification PFSs to $(\sqrt{0.5}, \sqrt{0.5}, 0)$ (i.e., d_s = 0), the ranking becomes more similar to the other two. Figure 12 shows the relevance of tolerances in the assessment.

Fig. 12

Options ranking.

5 Discussion and conclusions

IT projects are particularly prone to failure, making risk evaluation essential for their success. Evaluation methods are usually opaque, presenting difficulties in explaining the results and not analyzing the possible interrelationships among risk factors.

This study’s main result is developing a transparent and flexible MCDM method that integrates purposely designed fuzzy decision nodes to implement evaluation networks, facilitating the analysis of risk factors and their interaction.

With specification functions configured by PFSs and assessment functions that biunivocally represent PFSs, both in the unit segment, a new graphical model implements the proposed specification-assessment-compliance approach.

The transformation of PFS into TFN allows the PFS as input to the evaluation network, which otherwise had only been possible by adding an inference layer that would have reduced the simplicity and flexibility of the method. Furthermore, the consistent output of the activation functions at the input layer of the evaluation network reinforces the demonstration of the coherence of the PFS/TFN model. It also shows its capability to act in association with the activation function as a new adjustable defuzzification module.

The case study results demonstrate the influence of tolerances on evaluating the options, the new method’s coherence, its easy implementation, and transparency. In addition, the results obtained in the analysis of risk dependencies demonstrate the model’s versatility that permits the hierarchy’s reconfiguration, allowing even backward connections.

However, the lack of a procedure to implement group assessment and the lack of a method to establish the relationship between the risk factors and their Pythagorean assessment can produce biases. Further research should tackle these limitations.

The work described in this article opens the following prospects for future work: Maintaining the specification-oriented approach, we propose the search for other specification and assessment functions and the research on the suitability of the different possibilities, their possible link with each specific evaluation problem, and the suitability of using the intuitionistic procedures to configure the new functions. We should also define a procedure and a nomenclature to analyze more complex cross-relationships between risk factors. We could investigate the substitution of the evaluation hierarchy by a classic full-connected artificial neural network to implement a different approach requiring a previous training process. Such a training process might lead to conclusions about the interaction among evaluation criteria. We can also investigate the possibilities of using the proposed specification-oriented node as an artificial neuron for its integration in artificial neural networks. We should also test the method to verify its suitability for the application to other managerial decision-making problems. Furthermore, we should explore using the PFS/TFN model as part of other specific fuzzy methods. Finally, we can also use the one-sided TFN for the biunivocal representation of classic IFSs using intuitionistic values instead of its squared values.

References

Bayrak

, Evaluating large-scale IT investment decisions, Technology in Society 54 (2018), 128–138.

Idler

and Spang

, Balancing Facts and Feelings: Gaining Transparency on IT Project Decisions and Determinants, Proceedings of the 2018 International Conference on Software Engineering and Information Management, 2018, 92–97.

Kivijärvi

, Theorizing IT Project Success: Direct and Indirect Effects in a Hierarchical Framework, International Journal of Information Technology Project Management 11(1) (2020), 71–98.

Ebad

S.A.

, An exploratory study of ICT projects failure in emerging markets, Journal of Global Information Technology Management 21(2) (2018), 139–160.

Pasha

, Qaiser

and Pasha

, A critical analysis of software risk management techniques in large scale systems, IEEE Access 6 (2018), 12412–12424.

Öbrand

, Augustsson

N.P.

, Mathiassen

and Holmström

, The interstitiality of IT risk: An inquiry into information systems development practices, Information Systems Journal 29(1) (2019), 97–118.

-Durić

, Mitrović

Č.

, Komatina

, Tadić

and Vorotović

, The hybrid MCDM model with the interval Type-2 fuzzy sets for the software failure analysis, Journal of Intelligent & Fuzzy Systems 37(6) (2019), 7747–7759.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

Chang

D.Y.

, Applications of the extent analysis method on fuzzy AHP, European Journal of Operational Research 95(3) (1996), 649–655.

10.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1) (1986), 87–96.

11.

Yager

R.R.

, Pythagorean fuzzy subsets, Proceedings of Joint IFSA World Congress and NAFIPS Annual Meeting, 2013, 57–61.

12.

Gou

, Xu

and Ren

, The properties of continuous Pythagorean fuzzy information, International Journal of Intelligent Systems 31(5) (2016), 401–424.

13.

Guleria

and Bajaj

R.K.

, Pythagorean fuzzy (R, S)-norm discriminant measure in various decision making processes, Journal of Intelligent & Fuzzy Systems 38(1) (2020), 761–777.

14.

Thao

NX.

and Smarandache

, A new fuzzy entropy on Pythagorean fuzzy sets, Journal of Intelligent & Fuzzy Systems 37(1) (2019), 1065–1074.

15.

Xue

, Xu

, Zhang

and Tian

, Pythagorean fuzzy LINMAP method based on the entropy theory for railway project investment decision making, , International Journal of Intelligent Systems 33(1) (2018), 93–125.

16.

Szmidt

, Kacprzyk

and Bujnowski

, How to measure the amount of knowledge conveyed by Atanassov’s intuitionistic fuzzy sets, Information Sciences, 257 (2014), 276–285.

17.

Tüysüz

and Kahraman

, Project risk evaluation using a fuzzy analytic hierarchy process: an application to information technology projects, International Journal of Intelligent Systems 21(6) (2006), 559–584.

18.

Zhang

, Hu

, Ma

, Xu

, Yuan

and Chen

, Some risk functions of IVIFS applied to outsourced software project, Journal of Intelligent & Fuzzy Systems 31(6) (2016), 3103–3111.

19.

Zhang

, Hu

, Ma

, Xu

, Yuan

and Chen

, Incentive-punitive risk function with interval valued intuitionistic fuzzy information for outsourced software project risk assessment, Journal of Intelligent & Fuzzy Systems 32(5) (2017), 3749–3760.

20.

Rodríguez

, Ortega

and Concepción

, A method for theevaluation of risk in IT projects, Expert Systems withApplications 45 (2016), 273–285.

21.

Rodríguez

, Ortega

and Concepción

, Anintuitionistic method for the selection of a risk managementapproach to information technology projects, InformationSciences 375 (2017), 202–218.

22.

Zolfaghari

and Mousavi

, Construction-project risk assessment by a new decision model based on De-Novo multi-approaches analysis and hesitant fuzzy sets under uncertainty, Journal of Intelligent & Fuzzy Systems 35(1) (2018), 639–649.

23.

Ezghari

and Zahi

, Uncertainty management in software effort estimation using a consistent fuzzy analogy-based method, Applied Soft Computing 67 (2018), 540–557.

24.

Bharathi

S.V.

, Pramod

and Ramakrishnan

, Risks assessment using Fuzzy Petri Nets for ERP Extension in Small and Medium Enterprises, In Start-Ups and SMEs: Concepts, Methodologies, Tools, and Applications, IGI Global, 2020, 564–588.

25.

Joshi

, Kumar

and Kumar

, Application of Intuitionistic fuzzy and Hamming Distance Measure in Multiple Attribute Decision Making (MADM), International Journal of Applied Science-Research and Review 3(1) (2016), 085–090.

26.

Ejegwa

P.A.

, Akubo

A.J.

and Joshua

O.M.

, Intuitionistic fuzzy set and its application in career determination via normalized Euclidean distance method, European Scientific Journal 10(15) (2014).

27.

Nguyen

, A novel similarity/dissimilarity measure for intuitionistic fuzzy sets and its application in pattern recognition, Expert Systems with Applications 45 (2016), 97–107.

28.

Rodríguez

, Ortega

and Concepción

, A method for theselection of customized equipment suppliers, Expert Systemswith Applications 40(4) (2013), 1170–1176.

29.

Das

, Dutta

and Guha

, Weight computation of criteria in a decision-making problem by knowledge measure with intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set, Soft Computing 20(9) (2016), 3421–3442.

30.

Naz

, Ashraf

and Akram

, A novel approach to decision-making with Pythagorean fuzzy information, Mathematics 6(6) (2018), 95.

31.

Kahraman

, Onar

S.C.

and Oztaysi

, A comparison of wind energy investment alternatives using interval-valued intuitionistic fuzzy benefit/cost analysis, Sustainability 8(2) (2016), 118.

32.

Kuo

R.J.

and Cheng

W.C.

, An intuitionistic fuzzy neural network with gaussian membership function, Journal of Intelligent & Fuzzy Systems 36(6) (2019), 6731–6741.

33.

Rani

and Elangovan

, An emerging intuitionistic fuzzy based groundwater level prediction, Indian Journal of Geo-Marine Sciences 46(06) (2017), 1213–1219.

34.

Chen

, Zhang

, Wang

L.L.

and Han

, Prediction of cloud resources demand based on hierarchical Pythagorean fuzzy deep neural network, IEEE Transactions on Services Computing (2019).

35.

Liu

H.T.

, Wang

, He

Y.L.

and Ashfaq

R.A.R.

, Extreme learning machine with fuzzy input and fuzzy output for fuzzy regression, Neural Computing and Applications 28(11) (2017), 3465–3476.

36.

Jeswal

S. K.

and Chakraverty

, Connectionist model for solving static structural problems with fuzzy parameters, Applied Soft Computing 78 (2019), 221–229.

37.

Vimal

K.E.K.

and Vinodh

, Application of artificial neural network for fuzzy logic based leanness assessment, Journal of Manufacturing Technology Management 24(2) (2013).

38.

Qiao

, Liu

, Ma

and Liu

, A methodology to evaluate human factors contributed to maritime accident by mapping fuzzy FT into ANN based on HFACS, Ocean Engineering 197 (2020), 106892.

39.

Rezaei Kalantari

, Ebrahimnejad

and Motameni

, Presenting a new fuzzy system for web service selection aimed at dynamic software rejuvenation, Complex & Intelligent Systems 6 (2020), 697–710.

40.

Ebrahimnejad

, New method for solving Fuzzy transportation problems with LR flat fuzzy numbers, Information Sciences 357 (2016), 108–124.

41.

Abbaszadeh Sori

, Ebrahimnejad

and H. Motameni,

Homayun

, The fuzzy inference approach to solve multi-objective constrained shortest path problem, Journal of Intelligent & Fuzzy Systems 38(4) (2020), 4711–4720.

42.

Lin

J.W.

, Artificial Neural Network Related to Biological Neuron Network: A Review, Advanced Studies in Medical Sciences 5(1) (2017), 55–62.

43.

Gasilov

N.A.

, Amrahov

S.E.

and Fatullayev

A.G.

, On a solution of the fuzzy Dirichlet problem for the heat equation, International Journal of Thermal Sciences 103 (2016), 67–76.