From the inferences gained by the review, certain research gaps are determined such as (i) stability as a performance metric is not well explored for CoCoSo ranking method; (ii) specifically, alternative and criteria based adequacy tests are rarely conducted to investigate the stability/rank reversal phenomenon of the approach; and (iii) inclusion of partial adequacy test result is lacking in the existing studies. The stability of the CoCoSo method is experimented by using the adequacy test that observes the effect on ranking order when alterations are made to alternatives and criteria. Junior et al. (2014) utilized this method in their study of comparing AHP and TOPSIS in the fuzzy context. This method evaluates the performance by inclusion or exclusion of alternatives and criteria. In the case of the alternatives, an additional alternative, a repetition of one of the existing alternatives, was inserted into the decision matrix. For testing criteria, an additional criterion, a repetition of one of the existing criteria, was inserted into the decision matrix. The CoCoSo method has been applied to this newly generated decision matrix for alternatives and criteria. The rank values of the results of the original decision matrix and the newly generated decision matrix were compared to analyse the stability of the CoCoSo method.

By gaining motivation from the inferences/gaps provided above, the authors attempt to demystify CoCoSo in terms of its stability to support researchers in MCDM with better understanding of the scope and utility of the popular method, which is considered as the main objective of this study. Specific contributions are presented, such as:

These contributions attempt to understand the stability and/or partial stability of the popular CoCoSo method in the fuzzy context, which is a novel proposition of this research and to the best of authors’ knowledge, such a study on the CoCoSo approach is conducted for the first time and this study aims to support/help researchers better understand the stability aspect (rank reversal behaviour) of the CoCoSo approach.

The paper is structured as follows. Section 2 explains the methodology of the CoCoSo method and the adequacy test. Section 3 describes the experiment done on the CoCoSo method. Section 4 provides the results obtained from this experiment and draws some discussion and insights. Section 5 concludes the paper and directs attention to future work.

The CoCoSo method was applied using the scalar and power approach. This method was applied with biased and unbiased weights for the criteria. Main reason for selecting CoCoSo method is that it is a fairly recent approach developed by Yazdani et al. (2019) and utilized by many reseachers for MCDM. Further, the method works on the principle of considering multiple compromise solutions in an integrated fashion, which provides certain level of rationale to experts in ranking alternatives. Thus, investigating the stability aspect of the method would help experts/researchers in their choice of choosing an appropriate approach for ranking alternatives. In the future, researchers can explore the stability of other approaches such as COPRAS (“complex proportional assessment”), MARCOS (“measurement alternatives and ranking according to compromise solution”), MABAC (“multi-attributive border approximation area comparison”), ARAS (“additive ratio assessment”), and VIKOR (“viekriterijumsko kompromisno rangiranje”) as well. The procedure for the CoCoSo method is shown in the following steps:

Step 1: Consider a decision matrix X i j defined in equation (1) having m alternatives and n criteria. The order of this matrix is m × n: (1) X i j = x 11 x 12 ⋯ x 1 n x 21 x 22 ⋯ x 2 n ⋯ ⋯ ⋯ ⋯ x m 1 x m 2 ⋯ x m n .

Step 2: Consider a weight vector w j defined in equation (2) which contains the weights for the n criteria. The order of this vector is 1 × n: (2) W j = [ w j ] , ∀ w j , where 0 < w j < 1 and ∑ j = 1 n w j = 1 .

Please note that if biased weights are used, the values of w j should be unequal, or if unbiased weights are used, the values of w j should be equal to the reciprocal of the number of criteria n defined in Step 1.

Step 3: The CoCoSo method can be applied using the scalar or power approach. For the matrix X i j defined in equation (1) using the weight vector w i j defined in equation (2), if the scalar approach is used, determine the weighted matrix W X i j using equation (3), or if you are using the power approach, determine the weighted matrix W X i j using equation (4). The order of this matrix is m × n. (3) W X i j = [ w x i j ] = [ x i j × w j ] , ∀ x i j ∈ X i j and w j ∈ W j , (4) W X i j = [ w x i j ] = [ x i j w j ] , ∀ x i j ∈ X i j and w j ∈ W j .

Step 4: Determine the multi-stage compromise solution for the matrix W X i j defined in equation (3) or equation (4) using equations (5)–(7). Each equation yields an m × 1 vector: (5) T 1 = ∑ i = 1 m w x i j ∑ j = 1 n x i j , (6) T 2 = ∑ i = 1 m w x i j min j ( w x i j ) , (7) T 3 = ∑ i = 1 m w x i j max j ( w x i j ) , where w x i j is the value of the matrix in the ith and jth position, and min ( . ) & max ( . ) are the minimum and maximum operators, respectively.

Step 5: Combine the compromised solution from equations (5)–(7) to obtain the net ranking vector of order m × 1 using equation (8). This step finally gives the rank value of the decision matrix X i j defined in equation (1) based on the approaches selected in Step 2 and Step 3. (8) T = T 1 × T 2 × T 3 3 + T 1 + T 2 + T 3 3 .

The adequacy test has been applied to both alternatives and criteria. The procedure for the adequacy tests is shown in the following steps:

Step 1: Construct children’s matrices for every matrix on which the CoCoSo method was applied. The children’s matrices should be created by repeating the alternatives or the criteria. •

For a matrix with dimension m × n, m children matrices will be generated by repeating the alternative. Every children matrix generated should have only one unique alternative repeated.

Step 3: Compare the rank values of the original matrix and the children’s matrices by sorting the values in descending order. The comparison can be made in the following two methods:

The PASS and FAIL percentages can be calculated with equations (9) and (10). The PASS COUNT and FAIL COUNT are numerical values denoting the number of matrices that have passed and failed the adequacy test, respectively. (9) PASS % = PASS COUNT PASS COUNT + FAIL COUNT × 100 , (10) FAIL % = FAIL COUNT PASS COUNT + FAIL COUNT × 100.

The PASS percentage, PARTIAL PASS percentage, and FAIL percentage can be calculated with equations (11)–(13). The PASS COUNT, PARTIAL PASS COUNT, and FAIL COUNT are numerical values denoting the number of matrices that have passed, partially passed, and failed the adequacy test, respectively. (11) PASS % = PASS COUNT PASS COUNT + PARTIAL PASS COUNT + FAIL COUNT × 100 , (12) PARTIAL PASS % = PARTIAL PASS COUNT PASS COUNT + PARTIAL PASS COUNT + FAIL COUNT × 100 , (13) FAIL % = FAIL COUNT PASS COUNT + PARTIAL PASS COUNT + FAIL COUNT × 100.

Figures 1 and 2 summarize the results of applying the CoCoSo method using the scalar approach with unbiased weights. The criteria for unbiased weights and the scalar approach of the CoCoSo method can be referred from Step 2 and Step 3 of Section 2.1, respectively.

Figures 1 and 2 present the pass, fail, and partial pass percentages for various simulated matrices of different sizes. This affords a comprehensive understanding of the CoCoSo method, significantly illuminating its workings for its more efficient application by future researchers. Key insights can be confidently drawn: (i) CoCoSo exhibits a commendable level of stability when alternative adjustments are contrasted with criteria changes; (ii) with the expansion of matrix dimensions, the pass rate experiences variability in response to both alternative and criteria modifications; (iii) however, fluctuations in the matrix sets do not yield decisive conclusions about the pass and fail rates; (iv) notably, the incorporation of partial pass in the process of criteria alteration substantially diminishes the fail rate. This confirms that there are situations where the top-ranking alternative sustains its position, regardless of significant alterations made to alternatives and criteria.

Figures 3 and 4 summarize the results of applying the CoCoSo method using the power approach with unbiased weights. The criteria for unbiased weights and the power approach of the CoCoSo method can be referred to from Step 2 and Step 3 of Section 2.1, respectively.

Figures 3 and 4 offer the stability outcomes associated with the CoCoSo method when employing the power approach and unbiased weight condition. It is unequivocally evident that modifications regarding alternatives have a diminished impact on the method’s stability, resulting in an increased stability value and a decrease in failure instances. These results demonstrate that when a partial pass is factored in, the failure rate plummets dramatically, signalling an elevated level of stability. We can confidently deduce the following: (i) substantial adjustments to the criteria have a profound impact on the stability of the CoCoSo method, triggering a rank reversal phenomenon, and (ii) the partial pass score is notably high relative to substantial changes to criteria, affirming that in a majority of cases, the top-ranked alternative retains its standing even after significant alterations to the criteria are executed.

Figures 5 and 6 summarize the results of applying the CoCoSo method using the scalar approach with biased weights. The criteria for biased weights and the scalar approach of the CoCoSo method can be referred to from Step 2 and Step 3 of Section 2.1, respectively.

Figures 5 and 6 outline the percentages for pass, fail, and partial pass across different sizes of simulated matrix sets, offering a comprehensive perspective of the CoCoSo method using a scalar approach and biased weights. It is worth noting that the findings presented here strongly resemble those detailed in Fig. 1 and Fig. 2. From this set of results, we can observe that (i) the CoCoSo method maintains a satisfactory stability level when alternative changes are compared with criteria modifications; (ii) the pass percentage exhibits volatility with the increase in matrix dimensions regarding both alternative and criteria changes; (iii) changes in matrix sets do not provide an apparent effect on the percentages of pass and fail; (iv) however, incorporating a partial pass in the criteria modification process significantly cuts down the failure rate, suggesting instances where the highest-rated alternative retains its rank even after substantial changes to alternatives and criteria are implemented.

Figures 7 and 8 summarize the results of applying the CoCoSo method using the power approach with biased weights. The criteria for biased weights and the power approach of the CoCoSo method can be referred to from Step 2 and Step 3 of Section 2.1, respectively.

Figures 7 and 8 present the stability outcomes of the CoCoSo method when applied with the power approach and biased weight condition. From this, we can deduce that alterations related to alternatives have a minimal impact on the method’s stability, leading to a higher stability value and fewer failures. The data suggest that including partial pass results in a substantial drop in failure rate, signifying enhanced stability. It is essential to point out that the findings here exhibit a weak similarity to those in Fig. 3 and Fig. 4. The following points can be drawn: (i) substantial modifications to the criteria impact the CoCoSo method’s stability, resulting in the phenomenon of rank reversal, and (ii) the partial pass score is noticeably high in relation to substantial criteria modifications, indicating that in numerous cases, the top-rated alternative maintains its rank even after extensive criteria changes are implemented.

A feature based comparison of proposed work with earlier works is also performed to clearly understand the unique proposition of the developed work. For this purpose, works such as Rani and Mishra (2022), Stanujkic et al. (2021), Liu and Han (2022), and Mathew et al. (2020) are compared with the proposed framework to understand the efficacy of the model in Table 2.

From Table 2, it must be noted that these models follow compromise strategy and/or considers functions that provides relative benefits based on diverse ranking approaches and they are actively utilized for ranking in the decision process. Some key features are considered based on which the comparison is provided and from the comparison it is clear that the proposed work is innovative and supports value addition to the ranking process in decision-making by demystifying the properties of the ranking scheme.