Group decision-making method based on leader-follower dynamic game

Abstract

According to the group decision-making method based on the leader-follower dynamic game, a leader-follower hierarchical adjustment rule is designed to carry out iterative adjustment on an expert evaluation matrix, control and stability of a leader on a decision-making direction are guaranteed, bidirectional optimization is formed through feedback of the follower, and the decision-making efficiency is improved. The authority and the democratic are effectively coordinated, the group consensus degree is remarkably improved, the number of iterations is reduced, and the consensus achievement efficiency is improved. The method comprises the following steps: S1, collecting an evaluation matrix expressed by experts to hesitate a fuzzy language term set; s2, clustering experts through distance measurement and determining intra-class and clustering weights; s3, calculating multi-layer consensus degree to measure and evaluate consistency; s4, if the group consensus degree does not reach the preset threshold value, iteratively adjusting the expert evaluation matrix according to a hierarchical adjustment rule until the expert evaluation matrix reaches the standard; and S5, sorting the schemes by using an approximate positive and negative ideal solution.

Description

Technical Field

[0001] The present invention belongs to the technical field of group decision-making, and specifically relates to a group decision-making method based on leader-follower dynamic game applied in selecting environmental governance solutions. Background Art

[0002] Group decision-making refers to the process in which multiple members of a group or team discuss and decide on a complex issue. Sharing opinions, information, and knowledge among members is crucial in group decision-making, ultimately leading to a decision acceptable to all. Group decision-making is often used in areas such as optimizing pollution emission control plans, comprehensively evaluating ecological restoration projects, and product design. With the rapid changes in the social and economic environment, group decision-making is becoming increasingly widespread. To describe decision-making information, in 2012, Rodríguez proposed the concept of a hesitant fuzzy linguistic term set, building on the concepts of hesitant fuzzy sets and linguistic term sets. This concept can characterize situations in which decision-makers are hesitant and unable to choose between several consecutive linguistic terms. Based on human linguistic expressions, hesitant fuzzy linguistic term sets are more aligned with human thinking and cognition, can flexibly and comprehensively reflect decision-makers' true preferences, and are more conducive to experts expressing their evaluations of various attributes in a decision. Therefore, hesitant fuzzy term sets are widely used in multi-attribute group decision-making. Group consensus is a core concept in group decision-making, referring to a generally accepted, shared decision solution reached through discussion, negotiation, and adjustment among decision-making experts. The core of group consensus lies in forming a common decision-making plan through the coordination and adjustment of expert opinions. This requires not only the unification of opinions among decision-making experts, but also the elimination of differences of opinion and conflicts of preferences to a certain extent. However, traditional models often ignore the key role of leaders in reaching consensus in the group decision-making process, and lack a systematic identification and utilization mechanism for the professionalism, authority and representativeness of leaders, resulting in limited decision-making efficiency and quality. For example, Chinese patent application CN116187451A discloses a large-scale group decision-making VPC method based on consensus and priority coefficient. Although it takes into account the personal experience and knowledge of decision-makers as well as their subjective tendencies and preference information, it does not construct a hierarchical consensus framework. It is unable to use leaders to condense group preferences within clusters and reduce differences of opinion at the micro level. It is also difficult to quickly converge core contradictions across clusters through coordination between leaders at the macro level. Summary of the Invention

[0003] In response to the problem of insufficient interaction mechanism between leaders and followers in the process of reaching consensus in group decision-making, the present invention proposes a group decision-making method based on leader-follower dynamic game, and designs hierarchical adjustment rules for leaders and followers to iteratively adjust the expert evaluation matrix. This not only ensures the leader's control and stability over the decision-making direction, but also forms a two-way optimization through the feedback of followers, effectively coordinates authority and democracy, significantly improves the group consensus and reduces the number of iterations, thereby improving the efficiency of reaching consensus.

[0004] The technical solution adopted by the present invention to solve its technical problems is:

[0005] A group decision-making method based on a leader-follower dynamic game, comprising the following steps:

[0006] S1: Determine the evaluation attributes and schemes, and collect the evaluation matrix expressed by experts in the form of hesitant fuzzy language term sets;

[0007] Specifically: Suppose there are p experts forming a decision-making group E{e1,e2,...,e p}(p≥20), where e u (u∈1,2,...,p) represents the u-th expert; X={x1,x2,...,x a} represents a feasible alternative, where x i (i∈1,2,...,a) represents the i-th solution; C={c1,c2,...,c n} represents the attribute set of the scheme, where c j (j∈1,2,...,n) represents the jth attribute; W=(w1,w2,...,w n ) T Represents the weight vector of the attribute, w j ≥0 and represents the weight of the j-th attribute; Representative Experts u Provided evaluation matrix, where Indicates expert e u The evaluation of the jth attribute of the i-th solution; m and n represent the rows and columns of the evaluation matrix respectively;

[0008] The hesitant fuzzy language term set is expressed as:

[0009] in Represents the language term set, g represents the granularity of the language term set, if the language terms in S evenly cover [0,1] and satisfy the order, that is, if a<b, then Here a and b only represent the relationship between the magnitude of the quantity; δ l Indicates H S The level of granularity of the language terms in Represents the language term set in δ l Granular language expression; #H S The number of linguistic terms that represent the set of hesitant fuzzy linguistic terms.

[0010] S2: Distance measurement is performed based on the evaluation matrix to calculate the evaluation matrix distance between different experts and cluster the experts. Then, based on the similarity of the evaluation matrices of each expert in the cluster, the weight of the expert cluster is determined. The more similar the evaluations of the experts in the cluster, the better the clustering effect. Finally, the corresponding cluster weight is assigned.

[0011] The distance measurement is performed based on the evaluation matrix to obtain the evaluation matrix distance between different experts, specifically:

[0012] like but and The distance measure between them is:

[0013]

[0014] in, and They are and Corresponding granularity level;

[0015] like Then and The set of less hesitant fuzzy language terms is expanded.

[0016] The clustering of experts is specifically as follows:

[0017] The present invention selects the K-medoids clustering algorithm when clustering. The clustering algorithm first randomly selects a representative object for each cluster, then assigns the remaining objects to the nearest cluster based on the distance between each object and the representative object, and finally repeatedly uses non-representative objects to replace the representative object to improve the quality of the clustering results.

[0018] The K-medoids clustering algorithm is used. The K-medoids clustering algorithm uses each expert as the cluster center of the next clustering, where the sample point of the cluster center is the leader and the experts in the class where the cluster center is located are the followers:

[0019] Let T = {t1, t2, ...t q} represents the clustering set of experts, where t k (k∈1,2,...,q) represents the kth cluster; where m k is cluster t k The number of experts in represents the rth expert in cluster k, represents the expert class weight vector in the k-th cluster, represents the intra-class weight of the r-th expert in the k-th cluster, and β=(γ1,γ2,...,γ q ) T represents the clustering weight vector, γ k ≥0 and L={E1,E2,...E q} represents the cluster set of leaders, where E k (k∈1,2,...,q) represents the leader of the k-th cluster.

[0020] The process of determining the weight within the expert class is as follows:

[0021] The high similarity of the evaluation matrix of each expert in the cluster indicates that the decision-making experts in the cluster are more likely to reach a consensus. The higher the weight, the lower the weight. In cluster k, the experts Evaluation Matrix V r The similarity metrics are as follows:

[0022]

[0023] in, Represents cluster t k Medium V r Comparison with other expert evaluation matrix V y The total difference, that is

[0024] expert The intra-class weights based on similarity metrics are as follows:

[0025]

[0026] The specific assignment of cluster weights is as follows: if the expert clustering satisfies the "high cohesion and low coupling" principle, the clustering effect will be better. That is, the more similar the expert evaluations within the cluster, the better the clustering effect, and the greater the weight assigned, and vice versa.

[0027] For experts in the cluster, the distance measurement formula is used to calculate the evaluation matrix distance between experts to obtain the cluster expert similarity matrix ESM k As shown below:

[0028]

[0029] where e r and e y Represents two experts in the cluster;

[0030] Calculate the internal similarity coefficient ISF(t k ):

[0031]

[0032] in, represents the number of elements in the expert similarity matrix;

[0033] Calculate the evaluation matrix G of each two different clusters c ,G d The distance between them is used to obtain the similarity matrix SM between clusters:

[0034] SM=[1-d(G c ,G d )](c,d∈k,c≠d)

[0035] Calculate the external similarity coefficient ESF(t k ):

[0036]

[0037] The larger the external similarity coefficient of a cluster is, the higher the similarity between the cluster and other clusters is, and the lower the weight is, and vice versa;

[0038] Get the normalized clustering weight γ k :

[0039]

[0040] S3: Calculate multi-layer consensus, including the consensus between followers and leaders in the cluster, the consensus of the cluster, the consensus between the cluster's evaluation matrix and the group decision matrix, and the group consensus, to measure the consistency of evaluations between different decision-making layers;

[0041] Specifically: According to the expert evaluation matrix V r and expert intra-class weights Determine cluster t k The evaluation matrix G k :

[0042]

[0043] The evaluation matrix G of each cluster is calculated using cluster weights k Perform fusion to obtain the final group decision matrix R:

[0044]

[0045] Calculate cluster t k Followers in With leader E kConsensus

[0046]

[0047] in represents the evaluation matrix of the follower; V(E k ) represents the leader’s evaluation matrix;

[0048] Calculate cluster t k Consensus CL k :

[0049]

[0050] Calculate cluster t k The evaluation matrix G k The consensus degree CCL(t k ):

[0051] CCL(t k )=1-d(G k ,R)

[0052] Finally, calculate the group consensus GCL:

[0053]

[0054] S4: If the group consensus is less than the preset consensus threshold, the expert evaluation matrix is iteratively adjusted according to the leader-follower hierarchical adjustment rule to obtain a new group decision matrix until the group consensus is greater than the preset consensus threshold. The group decision matrix at this time is the final evaluation matrix;

[0055] Ideally, consensus in group decision-making means that all experts reach a consensus on the alternative options. In actual decision-making, since each expert may have different views and perspectives on the problem, conflicts and disagreements often occur, resulting in an inability to reach a completely consistent consensus. Therefore, based on the judgment of consensus, if the group reaches an acceptable level of consensus, that is, the group consensus GCL is greater than or equal to the preset consensus threshold, it is considered that the group opinion has reached a consensus. In this case, the expert's evaluation information will be further adjusted to ensure that the group's consensus is gradually optimized and improved. The present invention adopts the proposed leader-follower hierarchical adjustment rule model to adjust the expert evaluation matrix, and then finally reach a consensus. If the group consensus GCL is less than the preset consensus threshold, the adjustment process is entered until the group consensus GCL is greater than the preset consensus threshold, and the process ends. At this time, the group decision matrix is the final evaluation matrix.

[0056] The leader-follower hierarchical adjustment rules are specifically as follows:

[0057] Assume that p experts are clustered into q cluster groups using K-mediods clustering. Each cluster group has 1 leader, i.e., there are q leaders and (pq) followers.

[0058] (1) The adjustment rules for the leadership decision-making layer are as follows:

[0059]

[0060] Among them, V(E k ) represents the leader’s evaluation matrix, G k Represents cluster t k The evaluation matrix, R represents the final group decision matrix, is the average value of cluster weight, RD represents random value;

[0061] If cluster t k The evaluation matrix G k The consensus degree with the final group decision matrix R is greater than the group consensus degree, that is, CCL(t k )>GCL, that is, leader E k When the consensus of the cluster group is greater than the group consensus, the leader's evaluation matrix V(E k ) is consistent with the preference of the group decision matrix R, and the leader E is selected. k And the evaluation matrix G of the cluster group k This means that in highly similar situations, the leader's preference can effectively guide the decision-making preferences of the cluster group.

[0062] If CCL(t k )<GCL, that is, leader E k When the consensus of the cluster group is less than the group consensus, the leader's preference is affected by the preference of the cluster group. That is, the leader E k When the cluster weight of the cluster group is greater than the average value, the group has a greater influence on the decision-making, and the leader's evaluation matrix V(E k ) and the final group decision matrix R; when That is, the leader E k When the cluster weight of the cluster group is less than the average value, the influence of this group on the decision is small, and the evaluation matrix G of the cluster group is selected. k The value between the final group decision matrix R;

[0063] (2) The follower decision-making layer adjustment rules are as follows:

[0064]

[0065] in, represents the evaluation matrix of the followers, is cluster t k The average value of internal expert weights;

[0066] If cluster t k Followers in With leader E k The consensus degree is greater than cluster t k The consensus degree, that is The preferences of followers and leaders tend to be consistent, which means that in highly similar situations, followers are confident enough in themselves to choose the leader in the follower's evaluation matrix. and the leader's evaluation matrix V(E k ) value;

[0067] If cluster t k Followers in With leader E k The consensus degree is less than cluster t k The consensus When the weight of the follower is large, the follower has more decision-making initiative and influence, so they can weigh their own preferences with the preferences of their cluster group and choose the follower's evaluation matrix. and the evaluation matrix G of the cluster group k When the follower's weight is small, it is more inclined to rely on the evaluation value of the leader of the cluster or the cluster group, and choose the leader's evaluation matrix V(E k ) and the evaluation matrix G of the cluster group k Take a value between .

[0068] (3) Consensus-reaching process

[0069] After completing the above steps, the group consensus is calculated again based on the final group decision evaluation matrix R. If the preset consensus threshold φ = 0.9 is reached, the iteration is stopped; if not, the iteration is returned to continue.

[0070] S5: Use the method of approximating positive and negative ideal solutions to sort the final solutions.

[0071] Specifically: According to the final group decision matrix R, define the positive ideal solution A + and negative ideal solution A - They are:

[0072] A + ={R 1+ ,R 2+ ,...,R n+}A + ={R 1- ,R 2- ,...,Rn-}

[0073] in,

[0074]

[0075] R ij+ and R ij- They represent the positive and negative evaluation values of the i-th solution under the j-th attribute respectively;

[0076] The distance between each solution of the final group decision matrix and the positive and negative ideal solutions:

[0077]

[0078] R ij represents the evaluation value of the i-th solution under the j-th attribute; represents a positive ideal solution; represents a negative ideal solution;

[0079] Calculate comprehensive evaluation value

[0080]

[0081] The plan with the highest comprehensive evaluation value is selected as the optimal plan, which is the decision result and the decision is completed.

[0082] The present invention also protects the application of the above-mentioned group decision-making method based on leader-follower dynamic game in selecting environmental governance solutions.

[0083] The beneficial effects of the present invention include:

[0084] (1) Through the hierarchical adjustment rules of leaders and followers, the group consensus is significantly improved and the number of iterations is reduced, making the efficiency of consensus reaching higher than that of traditional methods; (2) The dynamic game model based on the hierarchical adjustment rules of leaders and followers can accurately simulate the complex interactions between group members, enhance the adaptability of the decision-making process through strategy adjustment and information optimization, and show greater flexibility in large-scale groups with a large number of experts; (3) The hierarchical adjustment mechanism not only guarantees the leader's control and stability over the decision-making direction, but also forms a two-way optimization through the feedback of followers, effectively coordinates authority and democracy, can quickly respond to dynamic decision-making environments, and solve the variability problems that are difficult to deal with by traditional methods; (4) While improving the rationality of decisions, it provides a new paradigm for complex decisions that takes into account both efficiency and stability, reduces the practical cost of integrating large-scale expert opinions, and has significant engineering application value.

[0085] In summary, the proposed method can effectively adjust the expert group evaluation matrix, improve group consensus, and demonstrate greater adaptability and flexibility in large-scale expert groups. Compared with existing methods, the proposed method demonstrates greater efficiency and stability in optimizing group consensus in large groups, can more quickly and effectively cope with the large number of expert groups, and has important practical significance and application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0086] Figure 1 The figure is a flow chart of the overall method of the present invention. DETAILED DESCRIPTION

[0087] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the embodiments described are only some embodiments of the present invention, not all embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0088] In addition, the technical features involved in the different embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0089] This study found that leaders can significantly promote consensus formation by proactively adjusting opinions and guiding followers' behavior within a group. Followers' feedback and collaboration can also inversely optimize the leader's decision-making direction. This dynamic synergy improves the rationality and stability of decision-making. Based on this idea, the present invention provides a group decision-making method based on a leader-follower dynamic game, primarily designed to address the challenge of achieving group consensus in optimizing environmental governance solutions.

[0090] First, environmental governance experts formulate an evaluation matrix based on the evaluation attributes of different governance options (such as cost-effectiveness, ecological impact, and implementation cycle) using a hesitant fuzzy language term set. A granularity-based distance measure is used to calculate inter-expert evaluation differences. The expert group is then dynamically divided into clusters using the K-medoids clustering algorithm. Within each cluster, a representative leader and its follower group are automatically identified. Individual weights are then assigned based on the similarity of expert evaluations within the cluster, and cluster weights are calculated using the external similarity coefficients between clusters. A hierarchical consensus framework is constructed, comprising a leader decision-making layer, a follower decision-making layer, and a group decision-making layer. Multiple rounds of opinion adjustment are conducted through a two-way dynamic game mechanism. Leaders dynamically adjust their evaluations between their own preferences, cluster preferences, and the larger group's preferences based on the relative consensus between their cluster and the group's. Followers, based on their level of consensus with the leader and their own weights, strategically adjust their preferences between their individual preferences, the leader's opinions, and the group's preferences until the group's consensus reaches a preset consensus threshold. Finally, an improved TOPSIS method is used to calculate the closeness of each option to the positive and negative ideal solutions, ranking the options and ensuring that the group's preferences converge with the ideal solution.

[0091] This method expresses expert preferences by constructing a hesitant fuzzy terminology set, using K-medoids clustering to divide expert roles and establish a hierarchical consensus framework. A game mechanism is designed in which leaders dynamically adjust their preferences based on cluster consensus, and followers provide two-way feedback based on weights and consensus levels, achieving collaborative optimization of expert opinions. By calculating weights based on in-class and out-class similarities, iteratively evaluating multi-level consensus, and implementing hierarchical adjustment rules, a consensus-building path is formed that balances decision-making authority and group collaboration. Finally, a modified TOPSIS method is used to rank options. Through role division and a dynamic game mechanism, this method effectively improves the efficiency and stability of consensus among large-scale expert groups in complex environmental governance decision-making.

[0092] To verify its effectiveness, this method was applied to two real cases: the optimization of pollution control schemes in a river basin and the comparison of solid waste treatment technologies in an industrial park, verifying the consensus efficiency of group decision-making and the rationality of the schemes.

[0093] Example 1: Take the pollution control plan for a certain river basin as an example.

[0094] To address the complex intertwined issues of industrial wastewater, agricultural non-point source pollution, and domestic sewage within the river basin, the environmental protection department proposed three remediation plans: Plan A (a combined system of ecological wetland restoration and artificial enhanced treatment), Plan B (full coverage of industrial sewage interception pipe networks and clean production transformation), and Plan C (a riparian vegetation buffer zone and biofilm synergistic purification project). The remediation plans were evaluated based on four core indicators: cost-effectiveness (pollution reduction per unit of investment), ecological impact (biodiversity recovery rate and sustained water quality improvement cycle), technical feasibility (project implementation complexity and maturity), and social acceptance (public support and impact on residents' lives). These indicators were weighted equally based on expert evaluation. Eighteen experts in environmental engineering, ecology, and sociology were invited to conduct a multi-dimensional evaluation of the four attributes of the three plans using a hesitant fuzzy language terminology set.

[0095] The following is a detailed introduction to the decision-making process:

[0096] Step S1: For the determined evaluation attributes and solutions, the experts provide an evaluation matrix in the form of a hesitant fuzzy language term set:

[0097]

[0098] Step S2: Calculate the distance between different experts by performing distance measurement based on the evaluation matrix given by the experts; perform clustering and determine the weights within the expert cluster based on the similarity of the expert evaluation matrix; the more similar the expert evaluations within the cluster, the better the clustering effect, and finally assign corresponding clustering weights.

[0099] Table 1 shows the clustering weight information, which is obtained by using the K-mediods clustering algorithm and combining it with step S2.

[0100] Table 1 Clustering weight information

[0101]

[0102] Step S3: Calculate the consensus of each layer based on the distance between leaders, followers, and clusters in large groups.

[0103] Table 2 shows the cluster consensus information, which can be obtained by calculating in step S3.

[0104] Table 2 Cluster consensus information

[0105]

[0106]

[0107] Step S4: Perform expert opinion adjustment according to the leader-follower hierarchical adjustment rule proposed by the present invention.

[0108] The consensus iteration process is explained using the leader E6 and follower e3 as an example:

[0109] Because the group consensus GCL < φ, the iteration continues.

[0110] CCL(t1)<GCL, Therefore, V(E1), that is, the t1 group where E6 is located, has little influence on decision-making. The leader is more inclined to choose a value between the evaluation matrix G1 of the cluster where he is located and the evaluation value of the group decision matrix R. The same can be achieved for other leaders according to the adjustment mechanism.

[0111] so That is, e3 is more inclined to rely on the evaluation matrix V(E1) of the leader of the group t1, namely E6, or the evaluation matrix G1 of the cluster, and chooses a value between the evaluation matrix of the leader of the group t1 and the evaluation matrix of the cluster group. The other followers can be obtained in the same way according to the adjustment mechanism.

[0112] The expert evaluation information obtained by the above adjustment mechanism is used to recalculate the group consensus until the group consensus GCL>φ is stopped. (In this invention, φ=0.9)

[0113] The method is compared with other different opinion adjustment principle methods in other literatures. Table 3 is a comparison of the opinion adjustment principle methods in different literatures. It can be seen that the opinion adjustment principle of method [1] iterates through the particle swarm optimization algorithm to find the group consensus decision matrix with the highest consensus and the smallest deviation opinion as the final result, and pays more attention to the global optimization of group consensus. However, due to its global search characteristics in the optimization process, when dealing with a group with high heterogeneity, it may be necessary to further investigate the expert opinions and refine the plan before evaluation; the opinion adjustment principle of method [2] mainly finds the element with the lowest consensus and adjusts the lowest consensus element of the expert at that position. This method focuses on local adjustment, that is, by focusing on the experts or evaluation elements with low consensus in the group, and improving local problems to promote the overall improvement of group consensus. However, the local adjustment strategy may result in limited adjustment effects, especially in a complex large group decision environment, and may not be able to quickly and effectively resolve decision differences; the opinion adjustment principle of method [3] finds the subgroup with a consensus less than the consensus threshold, and then finds the experts in the subgroup with a consensus less than the consensus threshold, and adjusts their evaluation matrix. The proposed method combines the leader-follower decision model from a dynamic game and optimizes consensus by adjusting the group's overall evaluation matrix. By simulating the interaction between leaders and followers and adjusting each expert's evaluation information, the optimal group decision is ultimately achieved.

[0114] Table 3 Comparison of different opinion adjustment principles and methods

[0115]

[0116] Method [1] Literature information: Yang Yanpu. Product styling design perceptual evaluation method based on hesitant fuzzy language term set and particle swarm optimization algorithm [J]. Journal of Graphics, 2021, 42(04): 680-687.

[0117] Method [2] Literature information: Wei Cuiping, Ma Jing. Consensus model for hesitant fuzzy language group decision making [J]. Control and Decision, 2018, 33(02): 275-281.

[0118] Methods[3] Literature information: Liang Xia, Guo Jie, Liu Peide, et al. Dynamic large group emergency decision-making method based on group consensus[J]. Systems Science and Mathematics, 1-17.

[0119] Step S5: Use the method of approximating positive and negative ideal solutions to sort the final solutions.

[0120] Table 4 is a comparison of the results of different literature methods, specifically the final evaluation value obtained according to step S5, and the final ranking results compared with other literature.

[0121] As can be seen from Table 4, method [1] uses the particle swarm optimization algorithm to perform two iterative adjustments on the entire evaluation matrix, and the final group consensus is 0.9066. Method [2] finds the element with the lowest consensus and performs 31 iterative adjustments on it, and the final group consensus is 0.9001. This method promotes consensus through local adjustment, but it has a large number of iterations, especially in large groups, which requires a large number of iterations to complete the adjustment. Method [3] adjusts the sub-groups and performs one iterative adjustment, and the final group consensus is 0.9272. This method promotes consensus through sub-group division and threshold selection. Although the method of the present invention also adopts sub-group division, it is different from method [3] in that the present invention introduces the leader-follower adjustment mechanism in the dynamic game, so that members in the group can interact flexibly. The two-way interaction between the leader and the follower enables each expert's feedback to dynamically adjust the decision-making strategy, thereby avoiding the limitations that may be brought about by single sub-group division. The method of the present invention only requires one iteration when the number of iterations is the same as that of the literature

[54] , and the final group consensus is 0.9372, which can obtain a better consensus and better cope with changes among group members. The result sorting schemes in different literatures are consistent, which shows that the present invention has certain feasibility and effectiveness while ensuring a higher consensus. In summary, the method of the present invention is superior to other methods in optimizing group consensus by introducing the leader-follower feedback adjustment mechanism in the dynamic game, further improving the rationality and reliability of the decision-making results.

[0122] Table 4 Comparison of results of different literature methods

[0123]

[0124] Example 2: Take the comparison of solid waste treatment technologies in industrial parks as an example.

[0125] To address the need for coordinated disposal of multiple hazardous wastes within the park, including heavy metal sludge, organic waste residue, and waste acid and alkali, four treatment solutions were designed: Solution D (a combination of high-temperature plasma melting and heavy metal solidification technology), Solution E (an anaerobic fermentation-aerobic composting bio-synergistic conversion system), Solution F (a multi-stage pyrolysis and gasification process with residual energy recovery), and Solution G (a combined chemical stabilization-chelation precipitation-safe landfill process). The technical solutions were evaluated based on four core indicators: treatment efficiency (solid waste volume reduction rate per unit time), cost-effectiveness (comprehensive treatment cost per ton and resource utilization benefits), environmental risk (dioxin emission concentration and leachate leakage probability), and technical applicability (adaptability to multiple waste types and process stability). Twenty-five experts in hazardous waste disposal, chemical processing, and environmental risk assessment were invited to conduct a multi-attribute evaluation of the four solutions using a hesitant fuzzy language terminology set.

[0126] The following is a detailed introduction to the decision-making process:

[0127] Step S1: For the determined evaluation attributes and solutions, the experts provide an evaluation matrix in the form of a hesitant fuzzy language term set:

[0128]

[0129] Step S2: Calculate the distance between different experts by performing distance measurement based on the evaluation matrix given by the experts; perform clustering and determine the weights within the expert cluster based on the similarity of the expert evaluation matrix; the more similar the expert evaluations within the cluster, the better the clustering effect, and finally assign corresponding clustering weights.

[0130] Table 5 shows the clustering weight information, which is obtained by using the K-mediods clustering algorithm and combining it with step S2.

[0131] Table 5 Clustering related information

[0132]

[0133]

[0134] Step S3: Calculate the consensus of each layer based on the distance between leaders, followers, and clusters in large groups.

[0135] Table 6 shows the cluster consensus information, which can be obtained by calculating in step S3.

[0136] Table 6 Cluster consensus information

[0137]

[0138] Step S4-5: The decision-making process is the same as that in Example 1 and will not be described in detail.

[0139] Table 7 compares the results of different literature methods. The final evaluation value is obtained by the TOPSIS method based on formula (4.16)-formula (4.18), and is compared with the final ranking results of other literature.

[0140] Table 7 Comparison of results of different literature methods

[0141]

[0142] As can be seen from Table 7, method [2] finds the element with the lowest consensus and adjusts it, and finally iterates 89 times to reach a group consensus of 0.9001. Compared with the 18 experts in Example 1, this example has 25 experts. It can be seen that when the group is larger, method [2] requires more iterations to complete the adjustment. Method [3] adjusts the sub-group and reaches a consensus of 0.9175 through 2 iterations. The method of the present invention also requires 2 iterations to reach a consensus of 0.9201. When the number of iterations is the same as that of method [3], the method of the present invention can obtain a better consensus. The result sorting schemes in different documents are consistent, which shows that the present invention has both feasibility and effectiveness while ensuring a higher consensus.

[0143] Obviously, the above embodiments are merely examples for clarity of explanation and are not intended to limit the implementation methods. Those skilled in the art will readily appreciate that other variations or modifications based on the above descriptions are possible. It is not necessary and impossible to enumerate all implementation methods here. Obvious variations or modifications arising therefrom remain within the scope of protection of the present invention.

Claims

1. A group decision-making method based on leader-follower dynamic game, its characteristic steps include: S1: Determine the evaluation attributes and schemes, and collect the evaluation matrix expressed by experts in the form of hesitant fuzzy language term sets; S2: Perform distance measurement based on the evaluation matrix to find the evaluation matrix distance between different experts and cluster the experts; then determine the weight within the expert class based on the similarity of the evaluation matrix of each expert in the cluster; finally, assign the corresponding cluster weight; S3: Calculate multi-layer consensus, including the consensus between followers and leaders in the cluster, the consensus of the cluster, the consensus between the cluster's evaluation matrix and the group decision matrix, and the group consensus, to measure the consistency of evaluations between different decision-making layers; S4: If the group consensus is less than the preset consensus threshold, the expert evaluation matrix is iteratively adjusted according to the leader-follower hierarchical adjustment rule to obtain a new group decision matrix until the group consensus is greater than the preset consensus threshold. The group decision matrix at this time is the final evaluation matrix; S5: Use the method of approximating positive and negative ideal solutions to sort the final solutions.

2. A group decision-making method based on leader-follower dynamic game according to claim 1, characterized in that: The evaluation matrix expressed by the experts in step S1 in the form of a hesitant fuzzy language term set is specifically: Suppose there are p experts forming a decision-making group E{e1,e2,...,e p }(p≥20), where e u (u∈1,2,...,p) represents the u-th expert; X={x1,x2,...,x a } represents a feasible alternative, where x i (i∈1,2,...,a) represents the i-th solution; C={c1,c2,...,c n } represents the attribute set of the scheme, where c j (j∈1,2,...,n) represents the jth attribute; W=(w1,w2,...,w n ) T Represents the weight vector of the attribute, w j ≥0 and represents the weight of the j-th attribute; Representative Experts u Provided evaluation matrix, where Indicates expert e u The evaluation of the jth attribute of the i-th solution; m and n represent the rows and columns of the evaluation matrix respectively; The hesitant fuzzy language term set is expressed as: δ l ∈{a,...,b}, in Represents the language term set, g represents the granularity of the language term set, if the language terms in S evenly cover [0,1] and satisfy the order, that is, if a<b, then Here a and b only represent the relationship between the magnitude of the quantity; δ l Indicates H S The level of granularity of the language terms in Represents the language term set in δ l Granular language expression; #H S The number of linguistic terms that represent the set of hesitant fuzzy linguistic terms.

3. A group decision-making method based on leader-follower dynamic game according to claim 2, characterized in that: In step S2, distance measurement is performed based on the evaluation matrix to obtain the evaluation matrix distance between different experts, specifically: like but and The distance measure between them is: in, and They are and Corresponding granularity level; like Then and The set of less hesitant fuzzy language terms is expanded.

4. A group decision-making method based on leader-follower dynamic game according to claim 3, characterized in that: The clustering of experts in step S2 is specifically as follows: The K-medoids clustering algorithm is used. The K-medoids clustering algorithm uses each expert as the cluster center of the next clustering, where the sample point of the cluster center is the leader and the experts in the class where the cluster center is located are the followers: Let T = {t1, t2, ...t q } represents the clustering set of experts, where t k (k∈1,2,...,q) represents the kth cluster; where m k is cluster t k The number of experts in represents the rth expert in cluster k, represents the expert class weight vector in the k-th cluster, represents the intra-class weight of the r-th expert in the k-th cluster, and β=(γ1,γ2,...,γ q ) T represents the clustering weight vector, γ k ≥0 and L={E1,E2,...E q } represents the cluster set of leaders, where E k (k∈1,2,...,q) represents the leader of the k-th cluster.

5. A group decision-making method based on leader-follower dynamic game according to claim 4, characterized in that: The process of determining the weights within the expert class in step S2 is as follows: The high similarity of the evaluation matrix of each expert in the cluster indicates that the decision-making experts in the cluster are more likely to reach a consensus. The higher the weight, the lower the weight. In cluster k, the experts Evaluation Matrix V r The similarity metrics are as follows: in, Represents cluster t k Medium V r Comparison with other expert evaluation matrix V y The total difference, that is expert The intra-class weights based on similarity metrics are as follows:

6. A group decision-making method based on leader-follower dynamic game according to claim 5, characterized in that: The step S2 assigns corresponding cluster weights, specifically: the more similar the expert evaluations within the cluster are, the better the clustering effect is, and the greater the weight is, and vice versa; For experts in the cluster, the distance measurement formula is used to calculate the evaluation matrix distance between experts to obtain the cluster expert similarity matrix ESM k As shown below: where e r and e y Represents two experts in the cluster; Calculate the internal similarity coefficient ISF(t k ): in, represents the number of elements in the expert similarity matrix; Calculate the evaluation matrix G of each two different clusters c ,G d The distance between them is used to obtain the similarity matrix SM between clusters: SM=[1-d(G c ,G d )](c,d∈k,c≠d) Calculate the external similarity coefficient ESF(t k ): The larger the external similarity coefficient of a cluster is, the higher the similarity between the cluster and other clusters is, and the lower the weight is, and vice versa; Get the normalized clustering weight γ k :

7. A group decision-making method based on leader-follower dynamic game according to claim 6, characterized in that: Step S3 is specifically as follows: According to the expert evaluation matrix V r and expert intra-class weights Determine cluster t k The evaluation matrix G k : The evaluation matrix G of each cluster is calculated using cluster weights k Perform fusion to obtain the final group decision matrix R: Calculate cluster t k Followers in With leader E k Consensus in represents the evaluation matrix of the follower; V(E k ) represents the leader’s evaluation matrix; Calculate cluster t k Consensus CL k : Calculate cluster t k The evaluation matrix G k The consensus degree CCL(t k ): CCL(t k )=1-d(G k ,R) Finally, calculate the group consensus GCL:

8. A group decision-making method based on leader-follower dynamic game according to claim 7, characterized in that: The leader-follower hierarchical adjustment rules in step S4 are specifically as follows: Assume that p experts are clustered into q cluster groups using K-mediods clustering. Each cluster group has 1 leader, i.e., there are q leaders and (pq) followers. The adjustment rules for the leader decision-making layer are as follows: Among them, V(E k ) represents the leader’s evaluation matrix, G k Represents cluster t k The evaluation matrix, R represents the final group decision matrix, is the average value of cluster weight, RD represents random value; If cluster t k The evaluation matrix G k The consensus degree with the final group decision matrix R is greater than the group consensus degree, that is, CCL(t k )>GCL, that is, leader E k When the consensus of the cluster group is greater than the group consensus, the leader's evaluation matrix V(E k ) is consistent with the preference of the group decision matrix R, and the leader E is selected. k And the evaluation matrix G of the cluster group k Take values between If CCL(t k )<GCL, that is, leader E k When the consensus of the cluster group is less than the group consensus, the leader's preference is affected by the preference of the cluster group. That is, the leader E k When the cluster weight of the cluster group is greater than the average value, the leader's evaluation matrix V(E k ) and the final group decision matrix R; when That is, the leader E k When the cluster weight of the cluster group is less than the average value, select the evaluation matrix G of the cluster group k The value between the final group decision matrix R; The follower decision-making layer adjustment rules are as follows: in, represents the evaluation matrix of the followers, is cluster t k The average value of internal expert weights; If cluster t k Followers in With leader E k The consensus degree is greater than cluster t k The consensus degree, that is The preferences of followers and leaders tend to be consistent, and the choice is in the follower's evaluation matrix and the leader's evaluation matrix V(E k ) value; If cluster t k Followers in With leader E k The consensus degree is less than cluster t k The consensus When the follower's weight is large, choose and the evaluation matrix G of the cluster group k When the follower's weight is small, the leader's evaluation matrix V(E k ) and the evaluation matrix G of the cluster group k Take a value between .

9. A group decision-making method based on leader-follower dynamic game according to claim 8, characterized in that: Step S5 is specifically as follows: Based on the final group decision matrix R, define the positive ideal solution A + and negative ideal solution A - They are: A + ={R 1+ ,R 2+ ,...,R n+ }A + ={R 1- ,R 2- ,...,R n- } in, R ij+ and R ij- They represent the positive and negative evaluation values of the i-th solution under the j-th attribute respectively; The distance between each solution of the final group decision matrix and the positive and negative ideal solutions: R ij represents the evaluation value of the i-th solution under the j-th attribute; represents a positive ideal solution; represents a negative ideal solution; Calculate comprehensive evaluation value The plan with the highest comprehensive evaluation value is selected as the optimal plan, which is the decision result and the decision is completed.

10. Application of a group decision-making method based on a leader-follower dynamic game as claimed in any one of claims 1 to 9 in selecting an environmental governance solution.

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