The supplier performance evaluation of sports event service under the COVID-19 outbreak: A novel hesitant fuzzy MARCOS method

3.1 Preliminaries

It is challenging to describe an FS’s membership function in multiple realistic conditions due to increasing complexity, temporal boundedness, the haziness of the human mind, and countless other variables. To solve this issue, Torra [7] proposed the Hesitant Fuzzy Set (HFS), which permits the membership degree of an element of a given set to have a few alternative values.

Definition 1. Given a fixed set X, an HFSs on X is a function that returns a subset of values in [0,1] when applied to X. For convenience, Xia et al. [40] completed the original HFs definition by including the HFS mathematical representation as follows: E = ( < x , h E ( x ) > | x ∈ X )(1)

Where h _E  (x) is a set of some values in [0,1] and denotes the possible membership degree of the element x ∈ X to the set E, h = h _E  (x) = H {γ¹, γ², ⋯ , γ^l(h)} (γ ^λ = (0, 1) , λ = 1, 2, ⋯ , l (h)) it is called a Hesitant Fuzzy Element (HFE).

Definition 2. Given three HFEs represented by h, h_1, and h₂, Torra [7], Xia et al. [40] defined some operations on them, which can be described as:

h ^c  = ∪  _γ∈h {1 - γ)

Definition 3. Given an HFE h, we define the intuitionistic fuzzy value Aenv(h) as the envelope of h, where Aenv(h) can be represented as (h^-, 1- h+), with h^- = min {γ|γ ∈ h} and h⁺ = max {γ|γ ∈ h}.

Definition 4. Let h be an HFE, with a score function of h defined by: s ( h ) = 1 l ( h ) ∑ γ ∈ h γ(2)

Where l(h) is the number of elements in h. Let h₁ and h₂ be two HFEs, then

if s (h₁) > s (h₂) then h₁ > h₂;

Definition 5. Xia et al. [41] Let h _j (j = 1, 2,  . . . , n) be a collection of HFEs. A hesitant fuzzy weighted averaging (HFWA) operator is a mapping H ⁿ  → H such that: A ( h 1 , h 2 , ⋯ , h n ) = ⊕ j = 1 n ( w j h j ) = ∪ γ 1 ∈ h 1 , γ 2 ∈ h 2 , ⋯ , γ j ∈ h j { 1 - ∏ j = 1 n ( 1 - γ j ) w j }(3)

where w = (w₁, w₂, ⋯ , w _n )  ^T is the weight vector of h _j (j = 1, 2,  . . . , n) with w _j =[0,1] and ∑ j = 1 n w j = 1 . Especially if w = (1/ n, 1/ n, ⋯ , 1/ n)  ^T , then, the HFWA operator reduces to the hesitant fuzzy averaging (HFA) operator: HFA ( h 1 , h 2 , ⋯ , h n ) = ⊕ j = 1 n ( 1 n h j ) = ∪ γ 1 ∈ h 1 , γ 2 ∈ h 2 , ⋯ , γ j ∈ h j { 1 - ∏ j = 1 n ( 1 - γ j ) 1 n }(4)

Definition 6. Xia et al. [41] Let h _j (j = 1, 2,  . . . , n) be a collection of HFEs and let HFWG: H ⁿ  → H, if HFWG ( h 1 , h 2 , ⋯ , h n ) = ⊗ j = 1 n ( h j w j ) = ∪ γ 1 ∈ h 1 , γ 2 ∈ h 2 , ⋯ , γ j ∈ h j { ∏ j = 1 n γ j w j }(5)

Then HFWG is called a hesitant fuzzy weighted geometric (HFWG) operator, where w = (w₁, w₂, ⋯ , w _n )  ^T is the weight vector of h _j  (j = 1, 2, ⋯ , n), with w _j  ∈ [0, 1], and ∑ j = 1 n w j = 1 . In the case where w=(1/n, 1/n,..., 1/n)^T, the HFWA operator reduces to the hesitant fuzzy geometric (HFG) operator: HFG ( h 1 , h 2 , ⋯ , h n ) = ⊗ j = 1 n ( h j 1 n ) = ∪ γ 1 ∈ h 1 , γ 2 ∈ h 2 , ⋯ , γ j ∈ h j { ∏ j = 1 n γ j 1 n }(6)

Stević et al. [18] developed the MARCOS approach based on the evaluation of alternatives and ranking them as a compromise. Although the MARCOS approach is a powerful tool for making decisions, it cannot convey information about fuzziness and ambiguity. In order to improve the technique’s capabilities, a fuzzy extension is essential. The following is a definition of the HF-MARCOS approach.

e = 1, 2,  . . . , f is the set of decision experts (DEs), and their weights are φ _e  = [φ₁, φ₂, ⋯ , φ _f ] and ∑ e = 1 f φ e = 1 . φ e = ∑ γ e λ ∈ h e ( x ) γ e λ ∑ e = 1 f ∑ γ e λ ∈ h e ( x ) γ e λ , e = 1 , 2 , ⋯ , f ; λ = 1 , 2 , ⋯ , l ( h )(7)

Step 1. Assessment of criteria by experts

The linguistic assessments for the criteria and DMs are realized using Table 2. The linguistic variables are converted into hesitant fuzzy numbers (HFNs).

For example, let the ratings for three DMs (DM1, M2, DM3) be EI, I, and M, respectively. Then, the weight of the first DM is computed by using Equation (7) as follows: φ DE 1 = 0.800 + 0.875 + 0.950 ( 0.800 + 0.875 + 0.950 ) + ( 0.650 + 0.725 + 0.800 ) + ( 0.550 + 0.600 + 0.650 ) = 0.417

We, therefore, obtain the weights of DMs as 0.417, 0.345, and 0.238, respectively.

Step 2. Construct HF decision matrix of alternatives

Let X ^e is the HF decision matrix of the eth decision-makers (DM _e ); here, a ij e indicates the assessment of ith alternatives about the jth criteria by eth expert, the linguistic evaluations for the alternatives are carried out utilizing Table 3. a ij e is utilized by hesitant fuzzy number (HFN).

X e = { C 1 C 2 ⋯ C n A 1 a 11 e a 12 e ⋯ a 1 n e A 2 a 21 e a 22 e ⋯ a 2 n e ⋮ ⋮ ⋮ ⋱ ⋮ A m a m 1 e a m 2 e ⋯ a mn e }(8)

Step 3. Construct aggregated HF decision matrix

The aggregated HF decision matrix is presented as R = [ a ij ] m × n , a ij = ( γ ij 1 , γ ij 2 , ⋯ , γ ij l ( h ) ) a ij = HFWA ( a ij 1 , a ij 2 , ⋯ , a ij f ) = ⊕ e = 1 f ( φ e a ij e ) = ∪ γ ij 11 ∈ a ij 1 , γ ij 21 ∈ a ij 2 , ⋯ , γ ij f 1 ∈ a ij f { 1 - ∏ e = 1 f ( 1 - γ ij f ) φ e } R = { C 1 C 2 ⋯ C n A 1 a 11 a 12 ⋯ a 1 n A 2 a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋮ ⋱ ⋮ A m a m 1 a m 2 ⋯ a mn }(9)

Step 4. Determine the HF ideal solutions

Let τ⁺ is the HF positive ideal solution (HFPIS) and τ^- is the HF negative ideal solution (HFNIS), then it can be determined respectively by the following equation:

τ + = { a j , max i = 1 m 〈 γ ij λ 〉 | j = 1 , 2 , ⋯ , n ; λ = 1 , 2 , ⋯ , l ( h ) } = { < a 1 , H { ( γ 1 1 ) + , ( γ 1 2 ) + , ⋯ , ( γ 1 l ( h ) ) + > , < a 2 , H { ( γ 2 1 ) + , ( γ 2 2 ) + , ⋯ , ( γ 2 l ( h ) ) + > , ⋯ , < a n , H { ( γ n 1 ) + , ( γ n 2 ) + , ⋯ , ( γ n l ( h ) ) + > }(10) τ - = { a j - , min i = 1 m 〈 γ ij λ 〉 | j = 1 , 2 , ⋯ , n ; λ = 1 , 2 , ⋯ , l ( h ) } = { < a 1 - , H { ( γ 1 1 ) - , ( γ 1 2 ) - , ⋯ , ( γ 1 l ( h ) ) - > , < a 2 - , H { ( γ 2 1 ) - , ( γ 2 2 ) - , ⋯ , ( γ 2 l ( h ) ) - > , ⋯ , < a n - , H { ( γ n 1 ) - , ( γ n 2 ) - , ⋯ , ( γ n l ( h ) ) - > }(11)

Step 5. Calculate the distance measures

The distance measurement is calculated using a fuzzy normalized Euclidean distance equation. The equations below use δ ij + and δ ij - illustrate positive and negative distance measures, respectively. δ ij + = d ( a ij , a j + ) = 1 l ( h ) ∑ λ = 1 l ( h ) ( γ ij λ - ( γ j λ ) + ) 2(12) δ ij - = d ( a ij , a j - ) = 1 l ( h ) ∑ λ = 1 l ( h ) ( γ ij λ - ( γ j λ ) - ) 2(13)

Step 6. Determine the Closeness Coefficient (CC) values

CW _ij is the CC of the jth criterion, and it is defined utilizing HFPIS δ ij + and HFNIS δ ij - as follows: CW ij = δ ij - δ ij + + δ ij -(14)

Step 7. Calculate the weights of the criteria

The CC values are used to determine each criterion’s significance. It is mentioned that the final weights are calculated using normalization, and the total weights should equal 1 [29].

Step 8. Generate an initial CW value decision matrix

After Step 6, we get a decision matrix D ˜ = [ x ij ] m × n x ˜ ij = CW ij = δ ij - δ ij + + δ ij - . D ˜ = C 1 C 2 ⋯ C n A 1 x ˜ 11 x ˜ 12 ⋯ x ˜ 1 n A 2 x ˜ 21 x ˜ 22 ⋯ x ˜ 2 n ⋯ ⋯ ⋯ ⋱ ⋯ A m x ˜ m 1 x ˜ m 2 ⋯ x ˜ mn(15)

Step 9. Construct an extended decision matrix

The anti-ideal AI and ideal ID solutions are computed, and an extended decision matrix E ˜ is created. E ˜ = C 1 C 2 ⋯ C n AI x ˜ ai 1 x ˜ ai 2 ⋯ x ˜ ain A 1 x ˜ 11 x ˜ 12 ⋯ x ˜ 1 n A 2 x ˜ 21 x ˜ 22 ⋯ x ˜ 2 n ⋯ ⋯ ⋯ ⋱ ⋯ A m x ˜ m 1 x ˜ m 2 ⋯ x ˜ mn ID x ˜ id 1 x ˜ id 2 ⋯ x ˜ idn(16)

The ID is the alternative with the most acceptable performance, while the AI is the worst. Applying Equations (17) result in their computation. AI = min i x ˜ ij , ifj ∈ B and max i x ˜ ij , ifj ∈ C _(17) ID = max i x ˜ ij , ifj ∈ B and min i x ˜ ij , ifj ∈ C(18)

Where B is the benefit criteria, and C represents the cost criteria.

Step 10 . Build the normalized decision matrix

By using the following equations, the variables of the normalized matrix N ˜ = [ n ˜ ij ] m × n are obtained: n ˜ ij = x ˜ idj x ˜ ij , ifj ∈ C(19) n ˜ ij = x ˜ ij x ˜ idj , ifj ∈ B(20)

where elements x ˜ ij and x ˜ idj represent the elements of the matrix X.

Step 11 . Build the weighted decision matrix

The normalized matrix N ˜ is multiplied by the weight coefficients of the criterion w _j to get the weighted matrix V ˜ = [ v ˜ ij ] m × n . v ˜ ij = n ˜ ij × w j(21)

Step 12 . Calculation of the utility degree of alternatives K _i

Calculating the utility degrees of an alternative to the ideal and anti-ideal solutions requires the application of Equations (22). K i - = s i s ai(22) K i + = s i s id(23)

where s _i  (i = 1, 2, ⋯ , m) represents the sum of the elements of the weighted matrix V ˜ , equation (10). s i = ∑ j = 1 n v ˜ ij(24)

Step 13 . Determination of the utility function of alternatives f (K _i )

Using the following expression, the utility function is defined for each alternative. f ( K i ) = K i + + K i - 1 + 1 - f ( K i + ) f ( K i + ) + 1 - f ( K i - ) f ( K i - )(25)

Where f ( K i + ) represents the utility function of the ideal solution, whereas f ( K i - ) represents the utility function as per the anti-ideal solution. f ( K i + ) and f ( K i - ) are calculated using Equations (26). f ( K i + ) = K i - K i - + K i +(26) f ( K i - ) = K i + K i - + K i +(27)

Step 14 . Rank the alternatives

The final phase involves ranking the suggestions based on their ultimate utility function values. In other words, the alternative with the highest utility value is the one that is most chosen.

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Fig. 4

Influence of change of weight coefficients DM₁ on change of utility functions.

The discussion of how subjectively determined input parameters affect the model outputs and what outcomes can be attained by using alternative multi-criteria models arises after the novel MARCOS model has produced its initial results [42]. Checking the results’ robustness and examining how sensitive they are to changes in the input parameters of the MCDM model is thus a crucial stage in multi-criteria decision-making [42]. Taking into consideration the literature’s suggestions [43–45], the sensitivity analysis and validation of the results were carried out through three stages in the following section. The first part examined the effects of the decision-makers weight coefficients changing. The impact of changing the most important criterion on the ranking results was examined in the second section, and in the third section, the robustness of the obtained solution was determined through comparison with other MCMD methods.

4.1.1 Influence of the change of weight coefficients of decision-maker on ranking results

The weighting coefficients of the decision-maker, which were established based on subjective evaluations, were utilized to determine the weighting coefficients of the criteria. It is vital to analyze the effects of changes in the decision-makers (DMs) weights on the final ranking conclusions since the DMs’ weights have an immediate impact on the values of the criteria weights. The upcoming section describes an experiment in which alterations are made to the weight coefficients of the DM that exert the most significant impact on the decision-making process. Specifically, changes will be simulated for those weight coefficients that hold the highest values. The importance of the DM was defined earlier as follows: w_DM1 = 0.417, w_DM1 = 0.345, w_DM1 = 0.238. From the given values, we can observe that DM1 holds the highest sway on the outcomes of the multi-criteria model as it possesses the largest weight coefficients.

The change in the weight coefficients’ values was performed through 21 scenarios, wherein in each scenario, the importance of the weight coefficients DM₁ was reduced by 2%, mimicking the shift in the weighting coefficients range of w_DM1 ∈ [0.017, 0.417], The scenarios’ values of the other criteria were correspondingly changed to satisfy the requirement ∑ j = 1 n w j = 1 . As a result, novel vectors of weight coefficients for the criteria were obtained (Fig. 3), and Fig. 4 examined their impact on how utility functions changed. According to the analysis that has been presented in Fig. 5, it is clear that the ranking is not changed. Alternatives A₃ and A₄ continue to occupy the first and second positions in all alternatives, which demonstrated the stability and reliability of the MARCOS method. However, As the weight values of DM₁ decrease gradually, there is a simultaneous reduction in the utility function values of alternatives A₁, A₃, and A₄. On the other hand, there is a slight uptick observed in options A₂ and A₅ function values. Thus, when the weightings assigned by decision-makers are altered, it can lead to a variation in the function values of alternative solutions and hence have an impact on the final decision results.

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Fig. 3

DM’s weights through 21 scenarios.

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Fig. 5

Criteria weights through 22 scenarios.

4.1.2 Influence of the change of weight coefficients of criteria on ranking results OPEN IN VIEWER

Fig. 6

Influence of change of criterion weights on change of utility functions.

An experiment was conducted to analyze how the variation of weight coefficients impacts ranking results. The experiment consisted of generating 22 different scenarios. To simulate changes in the weighting factor’s interval w3 ∈ [0.00–0.22], the value of the most influential criterion’s weight (C3) was reduced by 1% through each scenario. The remaining criteria in the scenarios had their values adjusted proportionally to fulfill the requirement ∑ j = 1 n w j = 1 . Therefore, obtained new weight coefficient vectors for the criteria (Fig. 5) and analyzed their impact on the change of utility functions as demonstrated in Fig. 6.

Based on the analysis presented in Fig. 6, it is apparent that alterations in the weight coefficient for criterion C3 tend to impact alternatives A1, A2, A4, A5, and A6 more significantly compared to other alternative A3. The data illustrates that schemes A1, A2, and A4 have a notable reduction in function values, whereas schemes A5 and A6 have observed a remarkable increase in function utility values. This criteria weight variation has been found to have a more significant influence compared to the alterations in the DM weight value. However, Alternative A3 showed no significant changes, yet maintained its top-ranking position with a stable utility value, reinforcing the practicability and dependability of the Alternative A3.

4.1.3 Comparison with hesitant fuzzy MCDM methodologies

Three HF multi-criteria techniques were selected to compare the results because hesitant fuzzy sets were used in this paper to deal with uncertainty and inaccuracy: hesitant fuzzy TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) [46] and hesitant fuzzy VIKOR (VLseKriterijumaska Optimizacija I Kompromisno Resenje) technique [14], and HF-SCORE [47]. Fig. 7 provides a comparison of the use of various hesitant fuzzy MCMD approaches. The ranking results obtained by the proposed method are consistent with those obtained by the HF-SCORE method, indicating the reliability of the proposed method. The rankings produced by the suggested method and the conventional HF-TOPSIS multi-criteria techniques are comparable, according to the results. The ranking conclusions of the HF-MARCOS approach also convey that the distinctions between Alternatives A₁ and A₂ are more prominent. The HF-MARCOS and HF-TOPSIS rank is A_3  > A_4  > A_2  > A_1  > A_6  > A₅, which differs from the ranking result of the HF-VIKOR approach, which is A_2  > A_3  > A_1  > A_4  > A_5  > A₆. The primary cause of this difference is that HF-VIKOR considers decision-maker’s psychological preferences. The top four alternatives can still be found in A₃, A₄, A₂, and A₁. The proposed analysis demonstrates the value of the approach suggested in this paper, and the choice of the suggested alternative A₃ is believable.

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Fig. 7

Ranks of the alternatives based on different HF methodologies.

In this study, the HF-MARCOS multiple-criteria group decision-making approach is proposed. Its goal is to select the most suitable sports suppliers in terms of sports event services among multiple alternatives while taking into account various conflicting criteria. The suggested framework gives decision-makers the ability to convey their thoughts and judgments using linguistic variables and enables a variety of decision-makers to collaborate using a robust method to reach a final decision. Since MARCOS provides a compromise solution as per ideal and non-ideal solutions, satisfactory performance is achieved in a hesitant fuzzy environment. The main contributions of the work are summarized below:

The proposed approach ensures the selected sports suppliers of decision-makers.

The study suggested a novel MCGDM framework for modeling and analyzing evaluation information with a high degree of ambiguity.

Due to its hesitant basis, the proposed model can better capture the inconsistency, deficiency, and uncertainty of human judgment.

Although this study contributes to the literature, it also has some limitations. The limitations of this work are emphasized below:

One of the limitations of that approach may be the difficulty in obtaining the data and information required to implement it.

In this paper, various MCDM approaches have been used in HF contexts simply to validate the ranking order of sports suppliers. There might be certain limitations to the lack of comparison with other fuzzy set extensions (intuitionistic, spherical, neutrosophic, image, etc.).