1 Introduction

Nowadays, selection problems are becoming a more important problem type in multi-criteria decision-making (MCDM) environments due to the sharp competition in the different sectors. Selection problems are widely used in many areas, such as machine-tool selection, robot selection, manufacturing process selection, supplier selection, personnel selection, material selection, etc. Developments of new technologies impact the products and human capabilities to complete tasks more rapidly and efficiently. So, alternative capabilities are rapidly growing, and rating alternatives is a complex issue in the MCDM-based selection problems. We must assign criteria weights and determine alternatives’ ratings to set MCDM-based selection models. In these stages, expert opinions are very crucial to select the most appropriate alternative. In the criteria weight assignment and alternative ranking process, linguistic expressions of experts are very critical to assigning more accurate criteria weights and alternative rating scores.

The fuzzy set theory-based applications are incorporated into the MCDM models to cope with this complex issue. In the literature, TFN-based aggregation methods are widely used to assign experts’ linguistic expressions. For example, Memari et al. [18] proposed an intuitionistic fuzzy TOPSIS model to select the right sustainable supplier for automotive spare parts producer firms. Their presented model determined sustainable ranking scores of suppliers through a case study. Rahimdel and Karamoozian [21] used the technique for order preferences by similarity to the ideal solution (TOPSIS) model with fuzzy set theory to select the best primary crusher for the iron mine. Chu and Lin [8] presented a fuzzy TOPSIS approach to robot selection problems using TFN-based linguistic expressions. The membership function of each weight was determined by interval procedure for TFNs. Li et al. [14] presented an indicator system and a method for data integration by evaluating the specifications and role of third-party logistics, for 3PL provider selection. They established a comprehensive analysis approach for 3PL suppliers based on fuzzy sets. Lam et al. [13] proposed a fuzzy principal component analysis approach for solving the material supplier selection problem. They used TFNs to quantify the experts’ expressions. Then, they employed principal component analysis to compress the criteria data and eliminate the multi-collinearity among them. Li et al. [15] developed a fuzzy portfolio selection model with background risk based on the definitions of the possibilistic return and possibilistic risk. They used LR-type possibility distribution for the returns of assets. Amindoust et al. [1] determined the sustainable supplier selection criteria and sub-criteria and presented an approach to rank suppliers. They used the fuzzy inference system-based ranking model to handle the subjectivity of experts’ expressions for the selection problem. Liu [16] integrated fuzzy quality function deployment and the prototype product selection model to develop a product design and selection. They adopted the α-cut operation in the fuzzy quality function deployment model to determine the fuzzy set of each alternative. Keršulienė and Turskis [11] proposed a fuzzy MCDM model using the fuzzy information fusion principles, additive ratio assessment, and step-wise weight assessment ratio analysis models to select an architect. Their aggregation process was based on the unification of information using fuzzy sets on a basic linguistic term set. Mougouei and Powers [19] proposed a cost–value approach that considers the impacts of value-related requirement dependencies on the value of selected optimal requirements. They exploited the algebraic structure of fuzzy graphs for modeling value-related requirement dependencies and their strengths. Chan and Prakash [5] proposed a maintenance policy selection model at the level of the firm rather than the equipment level. Some selection criteria were crisp values, whereas others were obtained in linguistic terms. They presented a distance-based fuzzy MCDM model to select the appropriate maintenance policy. Their MCDM model was suitable for integrating data, in the form of linguistic variables, TFNs, and crisp numbers, into the analyzing study of maintenance policy alternatives. Chen and Lin [7] proposed a fuzzy geometric mean decomposition-based fuzzy MCDM method to enhance the flexibility of the fuzzy decision matrix. They used fuzzy sub-judgment matrices to diversify the original fuzzy judgment matrix. Their presented approach was used to select the smart technology applicant for supporting mobile health care during and after the COVID-19 pandemic. Chai et al. [4] developed a fuzzy MCDM model based on the cumulative prospect theory, interval-valued fuzzy sets, and a combination of intuitionistic fuzzy sets for supplier selection problems. Huang et al. [9] introduced the patent value evaluation model. They considered the fuzziness of decision makers’ expressions and the uncertainty of patent indicators and proposed a TFN-TOPSIS model based on the possibility degree relationship model. Chisale and Lee [3] prioritized barriers to renewable energy acceleration in Malawi using the analytic hierarchy process (AHP) and fuzzy TOPSIS combined model. They used TFNs to represent the expert’s subjective judgments. Zhang et al. [26] used the fuzzy-TOPSIS model to obtain the hexagonal close-packed metallic crystal best structure among all structure alternatives when investigated under more than one criterion evaluating the TFNs.

We can see from the literature that the aggregation procedures in the selection problems using MCDM models depend on the linguistic expressions of the expert opinions. The fuzzy extensions of the TOPSIS model are developed in the literature to overcome model uncertainty since human judgments in real-life studies. They used aggregation procedures to aggregate different expert opinions in a useful way. However, the presented procedures have some disadvantages. They used fuzzy number characteristics-based aggregation procedures and ignored the prefixed membership function extensions or modifiers that were “very” and “more or less”. In this work, we described an exparabolic and parabolic shape area calculation based on the TFN linguistic expression aggregating procedure for fuzzy TOPSIS models to fill this gap. Furthermore, this procedure provides the best alternative using extended linguistic hedges for the linguistic expressions, hence maintaining the descriptive capabilities of the fuzzy TFNs. The proposed aggregating procedure is very simple, reflects the classical TFN values in more extendable judgments, and provides a more tolerable way. Also, the provided aggregation procedure is capable of differentiating ranking results of alternatives that have more adjacent scores among them in the classical TFNs-based fuzzy TOPSIS models.

2 Methodology

2.1 Fuzzy TFN

In this study, triangular fuzzy numbers are used for linguistic expressions (see Fig. 1) [6, 22]. The fuzzy numbers are defined using the membership function (µ(x)) ranging from 0 to 1. A TFN is illustrated in Fig. 1, represented by (l, m, u), where l, m, and u are the smallest, medium, and largest possible values, respectively. The linear TFN can be defined with a membership function as follows:

$$ \mu \left( x \right) = \left\{ {\begin{array}{*{20}l} {1, \quad x = m} \hfill \\ {\frac{x - l}{{m - l}}, \quad l \le x \le m} \hfill \\ {\frac{u - x}{{u - m}},\quad m \le x \le u} \hfill \\ {0,\quad {\text{otherwise}}} \hfill \\ \end{array} .} \right. $$
(1)
Fig. 1
figure 1

Triangular fuzzy number (TFN)

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Fuzzy arithmetic for TFNs are as follows:

(i) TFN operations [6]:

Let \(\mathop A\limits^{\sim } =\) (l1, m1, u1), and \(\mathop B\limits^{\sim } =\) (l2, m2, u2) are positive TFN numbers, the arithmetic operations are as follows:

$$ \tilde{A} \oplus \tilde{B} = \left( {l_{1} + l_{2} ,\;m_{1} + m_{2} ,\;u_{1} + u_{2} } \right) $$
(2)
$$ \tilde{A} \otimes \tilde{B} = \left( {l_{1} \times l_{2} ,\;m_{1} \times m_{2} ,\;u_{1} \times u_{2} } \right) $$
(3)
$$ \tilde{A}\oslash\tilde{B} = \left( {l_{1} /u_{2} ,\;m_{1} /m_{2} , \, u_{1} /l_{2} } \right) $$
(4)

2.2 The new exparabolic and parabolic area calculation-based aggregation approach for TFNs

Reflecting the optimal meaning and appropriate system behavior for a given linguistic expression is difficult, even for problem-trained experts. This is a decisive factor in multi-criteria decision-making processes. The semantic representation of linguistic expressions creates a numerical value about the relevance of the concept that the expression represents. The well-known prefixed membership function modifiers are “very” and “more or less”. The first modifier causes a decrease in the membership degree of a value in the fuzzy set. The second modifier is a “more or less” fuzzy expansion operator because it increases the degree of membership [25]. These functions are

$$ \mu_{{{\text{very}}}} A_{i}^{k} \left( x \right) = \left( {\mu_{{A_{i}^{k} }} \left( x \right)} \right)^{2} $$
(5)
$$ \mu_{{{\text{more}} {\text{or}} {\text{less}}}} A_{i}^{k} \left( x \right) = \sqrt {\mu_{{A_{i}^{k} }} \left( x \right)} $$
(6)

where x,…,xN are the characteristics for the selection problem, \(A_{1}^{k}\),…,\(A_{N}^{k}\) are linguistic values used to discretize the continuous value of the criteria. The illustration of their effects on a normalized fuzzy set with TFN is shown in Fig. 2.

Fig. 2
figure 2

Linguistic hedges

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We can offer a new expression for the aggregation procedure for “more or less” or “very” hedges using some geometric properties of area elements (Fig. 3). These properties use aggregation operations to convert linguistic terms as crisp equivalents. Exparabolic and parabolic area calculations are a suitable way to convert linguistic terms to a crisp value.

Fig. 3
figure 3

Triangular (blue line) parabolic (green line) and exparabolic (red line) shapes

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Let \(\widetilde{A}\)=(a, b, c), a fuzzy triangular number, the aggregated equivalent of exparabolic shape (Fig. 4) can be expressed as follows:

Fig. 4
figure 4

Exparabolic shape area

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Total area for exparabolic is

$$ A = \frac{b - a}{3} + \frac{c - b}{3} = \frac{c - a}{3} $$
(7)

To aggregation process, we can divide the medium number of the TFN to this area:

$$ {\text{Aggregate value for exparabolic area}} = \;b/A = b/\left( {\frac{c - a}{3}} \right) $$
(8)

To calculate cross lined segment area (Fig. 5), we express following equations:

$$ B = \left[ {\left( {b - a} \right) \times \left( 1 \right)} \right] - \left[ {\frac{b - a}{3} + \frac{b - a}{3}} \right] $$
(9)
$$ B = \left[ {\left( {b - a} \right) \times \left( 1 \right)} \right] - \left( {\frac{2b - 2a}{3}} \right) = \frac{5}{3}\left( {b - a} \right) $$
(10)
Fig. 5
figure 5

Special segment area

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Semi-area (Fig. 6) for aggregation can be calculated for more or less is

$$ C = \frac{5}{3}\left( {b - a} \right) + \frac{b - a}{3} = 2\left( {b - a} \right) $$
(11)
Fig. 6
figure 6

Semi-parabolic area segment

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Finally, we can calculate total area for more or less total area (Fig. 7) is as follows:

$$ D = 2\left( {b - a} \right) + 2\left( {c - b} \right) = 2\left( {c - a} \right). $$
(12)
Fig. 7
figure 7

More or less segment total area

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To aggregation process, we can divide the medium number of the TFN to this area:

$$ {\text{Aggregate value for exparabolic area}} = b/\left[ {2\left( {c - a} \right)} \right]. $$
(13)

The same procedure can be developed for triangular shape:

$$ {\text{Aggregate value for triangular area}} = b/\left[ {\left( {c - a} \right)/2} \right]. $$
(14)

Now, we can develop the new fuzzy TOPSIS application steps:

Step 1. Determine fuzzy decision matrix (\(\tilde{D}\))

$$ \tilde{D} = \left[ {\begin{array}{*{20}c} {\tilde{x}_{11} } & {\tilde{x}_{12} } & \cdots & {\tilde{x}_{1n} } \\ {\tilde{x}_{21} } & {\tilde{x}_{22} } & \cdots & {\tilde{x}_{2n} } \\ {\tilde{x}_{k1} } & {\tilde{x}_{k2} } & \cdots & {\tilde{x}_{kn} } \\ \end{array} } \right] $$
(15)

where the element \({\widetilde{x}}_{ij}\) represents the triangular fuzzy number equivalent of the linguistic performance value of the alternative. Here, m is expressed as (i = 1,2,…,k alternatives) and j = 1,2,…., n, criteria [2, 6, 10, 23, 24].

Step 2. Obtain aggregate TFN.

For triangular area:

$$ {\text{Aggregate value for triangular area}} = x_{ij} = b/A = b/\left( {\frac{c - a}{2}} \right). $$
(16)

For exparabolic area:

$$ {\text{Aggregate value for exparabolic area}} = x_{ij} = b/A = b/\left( {\frac{c - a}{3}} \right). $$
(17)

For parabolic area:

$$ {\text{Aggregate value for exparabolic area}} = x_{ij} = b/\left[ {2\left( {c - a} \right)} \right] = b/\left[ {2\left( {c - a} \right)} \right]. $$
(18)

These aggregation operations provide better identification of TFN-based expert opinions. Dividing the medium value of the TFN (b) to the geometric shape area indicates not only the importance of the criteria or alternative rating but also the uncertainty degree of the linguistic expression. The larger value is the better type of evaluation suitable for assigning the criteria weights and alternative rantings.

Step 3. Set the normalized decision matrix using vector normalization method [23]:

$$ r_{ij} = \frac{{x_{ij} }}{{\mathop \sum \nolimits_{i = 1}^{k} x_{ij} }} $$
(19)

Step 4. Set the weighted normalized decision matrix (vij):

$$ V_{ij} = w_{j} \times r_{ij} $$
(20)

where wj is the aggregate criteria weight calculated from triangular, exparabolic, or parabolic area calculation procedures in Eqs. (15–17).

Step 5. Calculate positive and negative ideal solutions:

$$ A^{*} = \left\{ {\mathop {(\max }\limits_{i} v_{ij} \left| {j \in I),(\mathop {\min }\limits_{i} v_{ij} \left| {j \in I^{\prime } } \right.} \right.} \right\} $$
(21)
$$ A^{ - } = \left\{ {\mathop {(\min }\limits_{i} v_{ij} \left| {j \in I),\left( {\mathop {\max }\limits_{ij} v_{ij} \left| {j \in I^{\prime } } \right.} \right.} \right.} \right\} $$
(22)

where I is a benefit type measure and I′ is the cost type measure.

Step 6. Calculate the separation measures:

$$ D_{{}}^{ + } = \sqrt {\sum\limits_{j = 1}^{{\mathop n\nolimits_{{}} }} {(v_{i} - v_{i}^{*} )^{2} } } $$
(23)
$$ D_{{}}^{ - } = \sqrt {\sum\limits_{j = 1}^{{\mathop n\nolimits_{{}} }} {(v_{i} - v_{i}^{ - } )^{2} } } $$
(24)

Step 7. Calculate the ranking scores:

$$ C_{i}^{*} = \frac{{D_{i}^{ - } }}{{D_{i}^{ - } + D_{i}^{ + } }}\quad i = 1, \ldots ,k $$
(25)

3 Examples

We applied the proposed methodology to the two selection problems using the fuzzy TOPSIS method in the literature. The first example is related to robot selection [8], and the second example is related to software programmer selection [17]. Firstly, we applied the proposed methodology to these problems and discussed the suitability and advantages of the proposed methodology using a comparative analysis in the following sections.

3.1 Example 1: Robot selection problem

The robot selection problem from Chu and Lin [8] is applied to show the suitability of the proposed procedure. Chu and Lin [8] assumed that a manufacturing firm requires a robot to perform a material-handling task. Three candidates, a1, a2, and a3, are selected for the study. The four experts are set to evaluate the criteria weights and alternative rating scores. The triangular, exparabolic aggregation procedure-based solution and parabolic aggregation procedure-based solution results are presented in Tables 1, 2 and 3, respectively.

Table 1 Triangular area-based fuzzy TOPSIS model’s result for example 1
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Table 2 Exparabolic area-based fuzzy TOPSIS model’s result for example 1
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Table 3 Parabolic area-based fuzzy TOPSIS model’s result for example 1
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3.2 Example 2: Software programmer selection problem

Mahdavi et al. [17] proposed a TOPSIS approach using TFNs. They applied the measurement approach using fuzzy distance values with a lower bound of alternatives. They supposed that a software company desires to select a programmer. Three alternatives, a1, a2, and a3, are determined, and three experts are assigned for the evaluation process. Mahdavi et al. [17] used five benefit criteria that are considered emotional steadiness, oral communication skills, personality, past experience, and self-confidence in their study. The triangular, exparabolic aggregation procedure-based solution and parabolic aggregation procedure-based solution results are presented in Tables 4, 5, and 6, respectively.

Table 4 Triangular area-based fuzzy TOPSIS model’s result for example 2
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Table 5 Exparabolic area-based fuzzy TOPSIS model’s result for example 2
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Table 6 Parabolic area-based fuzzy TOPSIS model’s result for example 2
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3.3 Discussion of the examples’ results

We list the results in Table 7 and apply Spearman’s rank correlation test [24] to analyze ranking differentiations. The Spearman’s rank correlation calculation equation is as follows:

$$ r_{{\text{s}}} = 1 - \frac{{6\sum {d_{i}^{2} } }}{{t\left( {t^{2} - 1} \right)}} $$
(26)

where di is the difference between the two ranks of each observation, t is the number of observations.

Table 7 Spearman’s rank correlation test results
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Spearman’s correlation coefficient increases in impact as two rankings become closer to being perfectly monotone functions of each other. When two rankings are perfectly matched, Spearman’s correlation coefficient will equal 1.

The ranking scores from the parabolic area calculation-based TOPSIS model are different from triangular and exparabolic area calculation-based TOPSIS results in Chu and Lin’s [8] case problem. So, it is difficult to say which alternative is exactly more suitable than the other. However, a2 is suitable for Chu and Lin’s [8] case problem due to ranking first or second for all approaches. If we can see the decision matrix of example 1, the expert evaluations are very close to each other. So, the proposed aggregation procedures provide two different results based on the “very” or “more or less” hedges for TFNs, and these new procedures are capable of the two-way perspectives for alternatives to select the more suitable one for the defined expectations.

On the other hand, we obtain similar results in example 2. Mahdavi et al. [17] ranking scores results are very close to each other (rs = 0.976). The only differentiation is in a1 and a3 ranking scores. Mahdavi et al. [17] study depends on the small ranking score as a better type of methodology. Also, their ranking scores are very close to alternatives. But the proposed aggregation procedures based on TOPSIS ranking scores are differentiated. The Mahdavi et al. [17] rankings present a smooth graph, but the proposed procedures-based result has a fluctuated shape (Fig. 8). The main advantage of the proposed aggregation procedure is the alternative separation capability analyzed with their linguistic diversification. They have modeling capability for the selection problems that alternative ratings are very closed. They used a geometrical area calculation procedure for the aggregation operator. So, the linguistic differences are much better modeled using developed procedures.

Fig. 8
figure 8

Comparison graphics for Mahdavi et al. [17] study

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According to the comparative analysis results, we reach the following outcomes about the advantages of the proposed methodology:

  1. i.

    If the expert’s linguistic evaluations are very close to each other, the proposed procedure uses two-way perspectives (“very” or “more or less” hedges) for alternatives to select the more suitable one for the defined expectations.

  2. ii.

    The fuzzy TOPSIS model’s ranking scores are very close for exemplified three alternatives (0.5308, 0.451, 0.455) in the Chu and Lin [8] approach. However, the proposed methodology’s result has differentiated between each alternative for the same example (0.346, 0.669, 0.567). In Mahdavi et al.’s [17] ranking, scores are 0.5304, 0.4626, and 0.494, respectively, in example 2. The proposed methodology’s results, for example 2, are 0.208, 0.991, and 0.148, respectively. So, the main advantage of the proposed aggregation procedure is the alternative ranking scores separation capability modeled with their linguistic diversification.

  3. iii.

    The modeling capability of the proposed methodology has an advantage when alternative ratings are very close. The proposed methodology uses a simple geometrical area calculation procedure for the aggregation operation. So, it has much better modeling capability when the expert’s linguistic expressions are very close.

4 Conclusions

In this work, we described an exparabolic and parabolic shape area calculation-based TFN linguistic expression aggregating procedure for fuzzy TOPSIS models. Furthermore, this procedure provides the best alternative using extended linguistic hedges for the linguistic expressions, hence maintaining the descriptive capabilities of the fuzzy TFNs.

We evaluated the results of different types of fuzzy TFNs-based TOPSIS model in selecting the best alternative. Results from the examples provide only the best alternative, so the presented new procedures consider the extended expressions as hedges and capture the information provided by overlapped expert opinions. The use of the proposed aggregating procedures that consider the information given by all the experts in the TFN expression process increases the generalization ability of the aggregating expressions. Nevertheless, it cannot determine a unique aggregate value as the best suitable for any type of problem, so it will be necessary to set aggregated TFNs for the new problems. This specification is an advantage for the modeling stage of the special TFNs for a specific selection problem. In future work, we intend to extend the application area to design a new kind of TFNs for the different MCDM methods such as GRA, VIKOR, PROMETHEE, and MOORA. On the other hand, the proposed model can be easily applied in different application areas requiring fuzzy multi-criteria decision-making processes, such as product or process quality improvement [12, 20], via evaluating the performance of artificial intelligence model-based predicted parameter values on the experimental or real quality characteristics’ values.