Existing studies mainly diagnose the degree of influence of important operation-related parameters of electricity enterprises on carbon emissions from a global perspective; however, there is a lack of a set of indicator systems to perceive the contribution of carbon emission reduction of RDNs, making it difficult to explain the support of urban green transformation (Yang Y. et al., 2023). In order to solve the above problems, the carbon emission reduction contribution evaluation indicator system of RDNs is constructed from four dimensions of economy, technology, environment, and management, as shown in Table 1.

A D = A D t | t = 1 , 2 , . . . , K is the set of semantic evaluations provided by the decision expert D t . A D t is converted to a spherical fuzzy number A ∼ S on the argument domain U , as shown in Equation 3. A ∼ S = u , μ A ∼ S u , υ A ∼ S u , π A ∼ S u | u ∈ U . (3) μ A ∼ S : U ∈ 0 , 1   υ A ∼ S : U ∈ 0 , 1   π A ∼ S : U ∈ 0 , 1 . (4)

The values μ A ∼ S , υ A ∼ S , and π A ∼ S , which are the degree of subordination, non-subordination, and hesitation of u to A ∼ S , respectively, are satisfied: 0 ≤ μ A ∼ S 2 u + υ A ∼ S 2 u + π A ∼ S 2 u ≤ 1 , ∀ u ∈ U . (5)

The basic operations of A ∼ S = μ A ∼ S u , υ A ∼ S u , π A ∼ S u and B ∼ S = μ A ∼ S u , υ A ∼ S u , π A ∼ S u for any two spherical fuzzy numbers A ∼ S and B ∼ S , where λ is real and λ ≥ 0 , are as follows: A ∼ S ⊕ B ∼ S = μ 2 A ∼ S + μ 2 B ∼ S − μ 2 A ∼ S μ 2 B ∼ S 1 / 2 , v A ∼ S v B ∼ S , 1 − μ 2 B ∼ S π 2 A ∼ S + 1 − μ 2 A ∼ S π 2 B ∼ S − π 2 A ∼ S π 2 B ∼ S 1 / 2 , (6) λ ⋅ A ∼ S = 1 − 1 − μ 2 A ∼ S λ , v A ∼ S λ ,   1 − μ 2 A ∼ S λ − 1 − μ 2 A ∼ S − π 2 A ∼ S λ , (7) where λ > 0 . The spherical distance between A ∼ S and B ∼ S is expressed as D A ∼ S , B ∼ S = arccos 1 − 1 2 μ A ∼ S 2 − μ B ∼ S 2 2   + v A ∼ S 2 − v B ∼ S 2 2 + π A ∼ S 2 − π B ∼ S 2 2 , (8) where the defining factor 2 π makes the spherical distance between the spherical fuzzy numbers to be 0 , 1 , instead of 0 , 2 π because μ A ∼ S 2 u + v A ∼ S 2 u + π A ∼ S 2 u = 1 , which is obtained by simplification: D A ∼ S , B ∼ S = 2 π ∑ i = 1 n arccos μ A ∼ S u i μ B ∼ S u i + v A ∼ S u i v B ∼ S u i + π A ∼ S u i π B ∼ S u i ] . (9)

The spherical weighted arithmetic mean operation of A ∼ S i i = 1 , 2 , . . . , n is as follows M w A S 1 , A S 2 , . . . , A S n = w 1 A S 1 + w 2 A S 2 + . . . + w n A S n = 1 − ∏ i = 1 n 1 − μ A S i 2 w i , ∏ i = 1 n v A S i w i , ∏ i = 1 n 1 − μ A S i 2 w i − ∏ i = 1 n 1 − μ A S i 2 − π A S i 2 w i , (10) where w i ∈ 0 , 1 , ∑ i = 1 n w i = 1 . The clear value of the spherical fuzzy number A ∼ S is calculated as follows: Def A ∼ S = μ A ∼ S − π A ∼ S 2 − v A ∼ S − π A ∼ S 2 . (11)

The weights play a crucial role in the model-solving process; however, the constructed indicator system contains a variety of uncertain information. Accordingly, accurate quantification of indicator weights will become a key problem in determining carbon emission reduction contributions.

The evaluation indicator system constructed in this study has many indicators. The commonly used indicator weight determination methods (for example, hierarchical analysis) are cumbersome and complex to calculate, so this article adopts the BWM to prioritize the priority ranking of the indicator assignment weights to obtain consistent results through less pairwise comparison information (Xiao et al., 2023). In addition, the “best-worst” comparison can more directly reflect the relative importance of each criterion, which can help researchers identify the key influencing factors more quickly and formulate targeted emission reduction measures (Jia et al., 2019). The BWM method ensures the scientificity and rationality of the decision-making results through steps such as consistency checking, which helps reduce the influence of subjective judgments on the decision-making results and improves the objectivity and accuracy of the study.

Therefore, the BWM is expanded into the SFS to develop a new method of determining indicator weights. The specific steps are as follows:

a. For the constructed evaluation indicator system A i , determine the best and the worst indicators, denoted as A B and A W , respectively.

The preference comparison shown in Figure 3 is carried out and converted into the corresponding spherical fuzzy numbers, as shown in Table 2.

O ∼ B t and O ∼ W t separately represent the vector of fuzzy preferences of the best indicator A B over the other indicators provided, and the other indicators over the worst indicator A W provided by the expert D t . The form of O ∼ B t is as follows: O ∼ B t = O ∼ B 1 t , O ∼ B 2 t , . . . , O ∼ B n t , (13) where O ∼ B j t = μ B j t , v B j t , π B j t . Integrate the K fuzzy preference vectors O ∼ B t t = 1 , 2 , . . . , k provided by experts in the matrix O ∼ B 0 . It is as follows: O ∼ B 0 = O ∼ B 1 1 O ∼ B 2 1 ⋯ O ∼ B n 1 O ∼ B 1 2 O ∼ B 2 2 ⋯ O ∼ B n 2 ⋮ ⋮ ⋱ ⋮ O ∼ B 1 K O ∼ B 2 K ⋯ O ∼ B n K . (14)

Similarly, the fuzzy preference vectors of other indicators are compared to the worst indicator O ∼ W t t = 1 , 2 , . . . , k in matrix O ∼ W 0 . It is as follows: O ∼ W 0 = O ∼ 1 W 1 O ∼ 2 W 1 ⋯ O ∼ n W 1 O ∼ 1 W 2 O ∼ 2 W 2 ⋯ O ∼ n W 2 ⋮ ⋮ ⋱ ⋮ O ∼ 1 W K O ∼ 2 W K ⋯ O ∼ n W K . (15)

c. Integrate the preference vector matrices O ∼ B and O ∼ W .

According to Equations 3–7, integrate matrix O ∼ B 0 and matrix O ∼ W 0 into matrix O ∼ B and matrix O ∼ W as follows: O ∼ B = O ∼ B 1 , O ∼ B 2 , . . . , O ∼ B n , (16) O ∼ W = O ∼ 1 W , O ∼ 2 W , . . . , O ∼ n W , (17) where O ∼ B B = 0.5 , 0.5 , 0.5 , O ∼ W W = 0.5 , 0.5 , 0.5 , O ∼ B j , and O ∼ j W are calculated as follows: O ∼ B j = 1 k ∑ t = 1 k O ∼ B j t = 1 − 1 − μ B j 2 1 k , v B j 1 k , 1 − μ B j 2 1 k − 1 − μ B j 2 − π B j 2 1 k . (18) O ∼ j W = 1 k ∑ t = 1 k O ∼ j W t = 1 − 1 − μ j W 2 1 k , v j W 1 k , 1 − μ j W 2 1 k − 1 − μ j W 2 − π j W 2 1 k . (19)

d. Determination of indicator weights.

Calculate the clear values of matrix O ∼ B and matrix O ∼ W as follows: R O ∼ B = R O ∼ B 1 , R O ∼ B 2 , . . . , R O ∼ B n , (20) R O ∼ W = R O ∼ 1 W , R O ∼ 2 W , . . . , R O ∼ n W , (21) where w ∼ B is the weight of the best indicator A B , w ∼ W is the weight of the worst indicator A W , and w ∼ j is the weight of the indicator A i . R(⋅) is obtained from Equation 11 w ∼ j is satisfied: min w ∼ B w ∼ j − R O ∼ B j . (22) min w ∼ j w ∼ W − R O ∼ j W . (23)

It implies that the optimal weights of the indicators are consistent between the optimal indicators compared to the other indicators and between the other indicators compared to the worst indicators as follows: w ∼ B w ∼ j = R O ∼ B j . (24) w ∼ j w ∼ W = R O ∼ j W . (25)

Let the objective value be ξ ∼ and construct an optimization model to determine the best weights of the indicators. It is as follows: min ξ ∼ s . t . w B / w j − R O ∼ B j ≤ ξ ∼ w j / w W − R O ∼ j W ≤ ξ . ∼ ∑ t = 1 k w ∼ j = 1 0 ≤ w ∼ j ≤ 1 j = 1 , 2 , . . . , n (26)

Solve the model by Lingo software to determine the indicator weight set w ∼ 1 , . . . , w ∼ j , . . . , w ∼ n .

e. Consistency test.

The consistency ratio ( C R ) is a key indicator to test the consistency of pairwise comparisons, and the consistency indices ( C I s ) under different semantic evaluations have been given, as shown in Table 2. The consistency ratio C R is determined from the C I s and the target value ξ * . The formula is given below: C R = ξ * C I , (27) where C R is closer to 0, indicating higher consistency, and C R is closer to 1, indicating lower consistency, where C R ∈ 0 , 1 .

Based on multi-dimensional indicator weights, the multi-attribute decision-making method is extended into the spherical fuzzy environment to calculate the carbon emission reduction contribution of RDNs. Considering that the decision-making process needs to deal with a large and highly uncertain amount of information, this research applies the MARCOS method. It is compared with common multi-attribute decision-making methods such as the multi-attributive border approximation area comparison (MABAC), the fuzzy comprehensive evaluation (FCE), the complex proportional assessment (COPRAS), and the preference ranking organization method for enrichment evaluations (PROMETHEE)-Ι and PROMETHEE-Ⅱ (Jia et al., 2019; Zhang et al., 2024; Rani et al., 2020; Seikh and Mandal, 2023; Yu et al., 2023).

MARCOS considers the inverse ideal solution and the ideal solution in the formation of the initial matrix, and at the same time, covers many indicators and decision scenarios and maintains stability (Thangaraj et al., 2023). Therefore, this article extends MARCOS into an SFS environment to determine the contribution of RDNs to carbon emission reduction. The specific steps are as follows:

a. Constructing a group decision matrix x ∼ i j m × n .

The ideal solution A I and anti-ideal solution A A I are introduced into the group decision matrix x ∼ i j m × n to obtain the extended group decision matrix: X ¯ m + 2 × n = A A I A 1 A 2 ⋮ A m A I C 1 C 2 ⋯ C n x ∼ a a 1 x ∼ 11 x ∼ a a 2 x ∼ 12 ⋯ x ∼ a a n x ∼ 1 n x ∼ 21 x ∼ 22 ⋯ x ∼ 2 n ⋮ ⋮ ⋱ ⋮ x ∼ m 1 x ∼ a 1 x ∼ m 2 x ∼ a 2 ⋯ x ∼ m n x ∼ a n , (30) where A I represents the ideal solution of indicator A j , and A A I represents the anti-ideal solution of indicator A j . x ∼ a j and x ∼ a a j are calculated as follows: x ∼ a j = μ a j , v a j , π a j = min ⁡ μ i j , ⁡ max ⁡ v i j , ⁡ min ⁡ π i j , i f j ∈ C , max ⁡ μ i j , ⁡ min ⁡ v i j , ⁡ min ⁡ π i j , i f j ∈ D (31) x ∼ a a j = μ a a j , v a a j , π a a j = max ⁡ μ i j , ⁡ min ⁡ v i j , ⁡ min ⁡ π i j , i f j ∈ C min ⁡ μ i j , ⁡ max ⁡ v i j , ⁡ min ⁡ π i j , i f j ∈ D , (32) where D and C denote the benefit and cost indicators, respectively.

c. Calculate the weighting matrix S ∼ j .

In the formula, S ∼ j is in the form of μ i , v i , π i . w i x i j is calculated by Equation 10 as follows: w i x i j = 1 − 1 − μ 2 x ∼ i j w i , v x ∼ i j w i , 1 − μ x ∼ i j 2 w i − 1 − μ x ∼ i j 2 − π x ∼ i j 2 w i . (34)

d. The sum S ∼ A A I , S ∼ A I of ideal and non-ideal solutions is calculated:

In the formula, K i + and K i − represent the utility degree of indicator A i associated with ideal solution A I and anti-ideal solution A A I , respectively, R S ∼ i represents the clear value of S ∼ i .

f. Determine the utility function value f K i .

Finally, based on Equations 12–41, the utility function values of the calculated solutions, f K i are used to rank the perceived carbon emission reduction contributions of the RDNs.

Shanghai has promoted renewable energy and a circular economy in Chongming Island, where distributed power supply, new energy generation, and intelligent operation and maintenance have achieved certain successes, driving the city’s economy toward low-carbon development. However, the significant contribution of regional carbon emission reduction is not currently perceived in depth, and the role of corresponding mitigation initiatives to support the green transformation of cities is not clearly defined. Therefore, this article selects 11 RDNs in Shanghai for analysis.

The evaluation system contains 12 indicators. A 1 , A 4 – A 11 are revenue-based indicators, and A 2 , A 3 , and A 12 are cost-based indicators. The decision-making expert group consists of the director of the finance department of the Chongming area, the deputy dean of the School of Economics and Management of Shanghai University of Electric Power, the head of the low-carbon evaluation team of the Economic Research Institute, and the personnel of the operation and maintenance center of Shibei Area, who provide pairwise comparative information on the indicators and information on the evaluation of the alternatives, as shown in Figures 5, 6., respectively. After their discussion, it was determined that the best indicator was A 6 and the worst indicator was A 11 . The case study adopts the SFS-BWM and SFS-MARCOS modeling proposed in this article to quantify the carbon emission reduction contribution of RDNs and analyze the results, and Figure 4 shows the area of each RDN.

a. Conversion of pairwise comparison preference information and alternative evaluation information into spherical fuzzy numbers.

The pairwise comparison preference information and alternative evaluation information provided by the experts, as shown in Figures 5, 6, respectively, are converted into spherical fuzzy numbers, as shown in Tables 3–5. O ∼ 7 W 1 is denoted as (0.9.0.1, 0.1), and x ∼ 11 1 is denoted as (0.6, 0.4, 0.4).

b. Integration of pairwise comparative preference information and alternative indicator performance evaluation information provided by experts.

The pairwise comparative preferences and indicator performance evaluations provided by the four experts are integrated to generate the group preference information of the best indicators relative to other indicators O ∼ B i , the group spherical fuzzy preference information of other indicators for the worst indicators O ∼ i W , and the group evaluation information of the RDN x ∼ i j .

c. Indicator weight calculation.

The clear values of R O ∼ B i and R O ∼ i W for the vectors O ∼ B i and O ∼ i W , are generated, and the weights of the indicators w ∼ j are determined by Lingo calculation, as shown in Figure 7.

d. Consistency test.

The target value ξ ∼ is obtained as 0.18 by using Equation 26, and the consistency ratio C R is calculated as 0.038 by using Equation 27, which is much smaller than 1 and close to 0. This low value indicates that BWM shows strong consistency in determining the weights of indicators.

e. Determination of ideal solution A I and anti-ideal solution A A I .

Considering the influence of the indicators on the contribution of areas to carbon emission reduction, the ideal solution A I and the anti-ideal solution A A I of the indicator system are determined.

f. Calculate the ideal solution S A I and anti-ideal solution S A A I .

The indicator weights obtained in step c are brought into the operation of a spherical fuzzy number to obtain the ideal solution S A I and the anti-ideal solution S A A I of each indicator. The clear values of the ideal solution S A I and the anti-ideal solution S A A I are obtained through Equation 6.

g. Calculate and rank the utility function value f K i .

Equations 37, 38 are used to obtain the utility degree K i + of alternative A i with the ideal solution A I and the utility degree K i − of alternative A i with the anti-ideal solution A A I . In addition, Equations 40 and 41 are used to obtain the utility function f K i + is 1.03 and f K i − is −0.03. Based on this, Equation 39 is used to obtain the utility function value f K i of the carbon emission reduction contribution of the RDNs, as shown in Table 6.

The indicator weight parameters of RDNs in terms of their perceived carbon emission reduction contributions are shown in Figure 7. The weight parameters of the indicators obtained by the SFS-BWM indicate that A 4 , A 6 , A 12 , and A 7 are four relatively important indicators. After the investigation, big data and artificial intelligence can predict energy demand more accurately, fault handling technology effectively processes and transforms energy, and energy storage technology can store and utilize energy to solve the contradiction between supply and demand in the power system so the evaluation of indicator weights is consistent with the actual situation, and the assignment of indicators is reasonable. However, the performance of indicators A 6 , A 8 , and A 12 is unevenly distributed among RDNs that must make targeted improvements according to their own situation. Meanwhile, the weight of A 4 is generally low, and the application of digitization technology in areas must still be strengthened. In the future, in order to improve their carbon emission reduction contribution, the RDNs will still face many challenges in low-carbon development. Therefore, they must continue to improve their equipment investment, technological innovation, environmental protection, and comprehensive management.

Under the influence of multi-dimensional indicators, the SFS-MARCOS method is used to calculate the carbon emission reduction contribution of RDNs, as shown in Figure 8. The results show that C 4 has the highest performance and C 10 has the lowest. Chongming has sufficient photovoltaic power and wind power generation sources and is now connected to the new energy installed capacity of 35 kV above, totaling 529 MW. The second-ranked Pudong region has created several green factories, green supply chains, green products, and green parks. Shanghai’s first green finance regulations have been implemented in the Pudong New Area. Changxing County has established a regulatory forcing mechanism to guide enterprises to increase investment in green manufacturing. In addition, the low-carbon agricultural development practice area of Langxia Town, Jinshan District, has realized the reduction, low-carbonization, and resource utilization of agricultural waste through projects such as the resource utilization of livestock and poultry manure.

The economic development in the southern region of the city heavily relies on high-energy-consuming and high-emission industries, with the electricity sector being the primary contributor to this challenge, so the total amount and intensity of carbon emissions from the south grid are high. For areas with low contribution to carbon emission reduction, a variety of measures should be taken according to local conditions to improve the effect of carbon emission reduction. Therefore, the case results are in line with the actual situation, which verifies the reliability of the modeling method and shows that the evaluation is effective.

In order to analyze the influence of indicator weights on the rankings of carbon emission reduction contribution of areas, sensitivity analysis based on weight fluctuation was implemented. For this analysis, all indicator weights were reduced or increased by 10% from the original, as shown in Figure 9.

Assuming that the original weight of A i becomes w i ′ = α w i , the weights of other indicators will change accordingly w o ′ = β w o i ≠ o . The modified weights satisfy α w i + ∑ o = 1 , o ≠ i 12 β w o = 1 , where α takes the values of 90% and 110%.

When indicator A 8 is reduced by 10%, the ranking of carbon emission reduction contribution of the original RDNs changes. C 3 changes from ninth to tenth place, and C 10 changes from tenth to ninth. In addition, when indicator A 3 was increased by 10%, the carbon reduction contribution of C 3 moved forward by one place from ninth to eighth, while C 1 changed from eighth to ninth, exchanging rankings with each other. Changes in the weights of the remaining indicators did not bring about a change in the ranking of the carbon emission reduction contributions of RDNs.

This indicates that the environmental conditions of the region where areas C 3 and C 10 are located have a significant impact on the carbon emission reduction contribution of areas. When the regional environmental condition deteriorates ( A 8 indicator decreases), the carbon emission reduction contribution of some areas may fall in the ranking, while other areas may benefit from it and rise. This reflects the more prominent influence of environmental policies, regional resource utilization efficiency, and other factors on the carbon emission reduction contribution ability of RDNs in C 3 and C 10 . Indicator A 3 , changes in the cost of purchasing carbon allowances, can also affect the ranking of RDNs in terms of their contribution to carbon emission reductions. Increased costs may prompt areas to optimize equipment operations and improve energy efficiency, thereby increasing carbon emission reduction contributions. Conversely, changes in costs may also affect areas’ emission reduction efforts or willingness to invest. It is worth noting that this change is bidirectional, that is, cost increases are positive for some areas (e.g., C 3 ) and negative for others (e.g., C 1 ). This further emphasizes the importance of cost control and operational efficiency in the reduction of carbon emissions by RDNs.

In order to verify the feasibility and superiority of the SFS-MARCOS method in evaluating the carbon emission reduction contribution of RDNs, MABAC, FCE, COPRAS, PROMETHEE-Ι, and PROMETHEE-II are used to replace the MARCOS method. The application of MABAC in sorting problems is based on the distances of the alternatives from the border approximation areas. The FCE calculates the comprehensive evaluation value based on the set of weights and the single-factor judgement matrix, thus obtaining the overall evaluation of the evaluation object. The COPRAS considers the relative importance of the options and takes into account the actual validity of the options when dealing with the alternatives to ensure the comprehensiveness and practicability of the assessment results. The PROMETHEE combines the concepts of ideal and anti-ideal solutions and evaluates the solutions by constructing preference and aversion functions. The different results obtained by the different methods are analyzed, and the following rankings of utility function value of RDNs’ carbon emission reduction contribution are obtained, as shown in Figure 10.

As can be seen in Figure 10, the carbon reduction contribution of C 4 is the highest, and this result is largely consistent across the various decision-making methods, except for SFS-COPRAS and SFS-PROMETHEE-Ι. C 4 finishes in the second-best place in the SFS-PROMETHEE-Ι. All the methods unanimously identify that the carbon reduction contribution of C 6 is lower than that of other RDNs. Except for the SFS-MABAC calculations, which yielded the second-lowest C6 ranking. In addition, it can be clearly seen that the carbon emission reduction contribution of C 5 and C 11 is in the top three of all methods, C 6 , C 10 , and C 3 are in the bottom three of all methods, and the remaining C 1 , C 2 , C 7 , and C 8 are evenly ranked in the middle four. The difference in the rankings of the largest and smallest carbon reduction contribution of each RDN under all decision-making methods does not differ by more than two places. The only exception is the COPRAS method in C 10 , which ranks seventh and is due to the fact that COPRAS focuses more on a combined assessment of guideline scores and weights, with each area having a score under each guideline, which contributes to the differences in ranking results. The slight difference in the comparison results is within the acceptable range due to the variability of the method characteristics. All in all, the robustness and effectiveness of the ranking results by the suggested framework are verified.

Under the premise of ensuring the completeness of the decision-making evaluation information, it is compared with the SFS-MABAC, SFS-FCE, SFS-COPRAS, SFS-PROMETHEE-Ι, and SFS-PROMETHEE-Ⅱ methods to make the evaluation result objective and authentic and to verify that the SFS-MARCOS method is feasible, effective, and meets the needs of decision-making in practice.