Two-stage portfolio optimization method achieved based on fuzzy preference and group consensus

Abstract

The invention discloses a two-stage investment portfolio optimization method achieved based on fuzzy preference and group consensus, which is more practical by providing an acceptable additive consistency framework of FPR, defining a consistency index and setting a threshold value and establishing a consistency evaluation standard conforming to an actual decision-making scene, and has the advantages that the method is more practical by constructing a two-stage FPR adjustment optimization framework, and the optimization efficiency is improved. In the first stage, the minimum adjustment distance is determined, and in the second stage, investment portfolio selection is optimized on the premise that the minimum information loss is guaranteed; according to the method, original preference information of experts is reserved to the maximum extent while mathematical preciseness is kept, and when the individual consensus level, the group consensus level and an acceptable group consensus threshold value are lower than the threshold value, a system automatically identifies the lowest expert and performs adjustment by applying a two-stage optimization model; according to the iteration process, individual consistency and investment portfolio performance constraints are ensured to be met in each iteration, and individual consistency and group consensus are effectively coordinated.

Description

Technical Field

[0001] This invention relates to the fields of financial engineering and investment decision-making technology, specifically a two-stage portfolio optimization method based on fuzzy preferences and group consensus. Background Technology

[0002] In the context of increasingly complex and volatile global financial markets, portfolio optimization plays a crucial role in securities trading. Since Markowitz proposed the groundbreaking mean-variance portfolio optimization model in 1952, this theory has been widely studied and applied in academic research and investment practice. For complex portfolio optimization problems, preference-based group decision-making methods are effective due to their ease of operation and flexibility. Fuzzy preference relations (FPRs), as an effective means of handling decision-makers' uncertain preferences, play an important role in the investment decision-making process of preference-based group decision-making. FPRs are often used as judgment matrices, employing a (0-1) scale to characterize experts' preferences for different alternatives. In the context of portfolio selection, consistency is considered the most important criterion for evaluating FPRs because it reflects the logic and rationality of experts' ranking of different securities. Inconsistent preferences may lead to contradictory investment recommendations and suboptimal portfolio performance. Assuming investment decisions include Given a pool of candidate securities, experts need to compare each security pairwise; a fuzzy preference relation is used. The matrix R is used to represent the matrix, where the elements are... Experts believe that securities Better than securities The degree of , ranging from 0 to 1; specifically, it means: if , indicating securities and No obvious advantages or disadvantages; if , indicating securities Better than securities The closer the value is to 1, the stronger the preference; if , indicating securities Better than securities This matrix has complementary properties, that is... This means that if securities are considered Superior The degree is 0.7, so it is considered that... Superior The degree is 0.3; Consistency is a key indicator for evaluating the quality of expert judgment. Intuitively, if an expert believes security A is superior to security B, and security B is superior to security C, then logically, A should be superior to C, and the degree of superiority should match the previous two judgments. Perfect consistency requires that all these judgments maintain a strict mathematical logical relationship. (Consistency index) It is used to quantify the degree of consistency in preference relationships, with values ​​between 0 and 1, where 0 represents perfect consistency and 1 represents complete inconsistency; However, in the practice of group decision-making (GDM), the consensus-reaching process (CRP) is often not adequately and effectively addressed. Because each expert has different professional backgrounds and investment experience, they may have significantly different evaluations and judgments of the same investment project. In this case, establishing an effective consensus-reaching mechanism becomes crucial. How to guide the expert group to reach a consensus on investment decisions through scientific methods while respecting the opinions of individual experts remains an important issue that needs to be explored in depth in current research. Among existing technologies, the closest similarity to this invention is a portfolio optimization model based on perfectly consistent FPR, proposed by Guo et al. The core of this approach lies in the assumption that FPR possesses perfect additive consistency; specifically, the FPR matrix... It is considered to have perfect additive consistency if and only if there exists a nonnegative vector. It satisfies the condition that the sum of the weights is 1, and also satisfies the following relation: This holds true for all i and j; Based on this, the scheme constructs two types of optimization models; the first type is the risk minimization model, given the target expected rate of return. Consistent with perfect additive property, the objective function of FPR is to minimize portfolio risk (i.e., minimize...). The mathematical model is as follows:

[0003] Wherein, constraint (a) is that the expected return is not lower than the target value. Constraint (b) is a perfect consistency constraint, constraint (c) is that the sum of all weights is 1, and constraint (d) is a non-negative weight constraint. The second type is the profit maximization model, which assumes a target risk tolerance. Consistent with perfect additive FPR, the objective function is to maximize the expected return (i.e., maximize the...). The mathematical model is as follows:

[0004] Wherein, constraint (a) is that the portfolio risk does not exceed the target value. Constraint (b) is a perfect consistency constraint, constraint (c) is that the sum of all weights is 1, and constraint (d) is a non-negative weight constraint. This scheme has the following obvious limitations; firstly, the perfect consistency assumption requires consistency, that is... This is almost impossible to achieve in practical applications due to the limitations of human cognition and subjective biases; secondly, the scheme does not provide any improvement or adjustment strategies for FPR that does not satisfy perfect consistency; thirdly, the scheme is only applicable to single-expert decision-making scenarios and does not consider the issue of multi-expert group decision-making and consensus-building; finally, from a numerical calculation perspective, perfect consistency can only asymptotically approach zero in optimization algorithms and cannot be precisely achieved, which brings difficulties to the actual solution. Some existing group portfolio selection methods employ a simple opinion aggregation approach. The basic steps are as follows: First, each expert independently provides their own FPR and investment preference judgment; second, the opinions of multiple experts are merged into a single group opinion using a certain aggregation operator (such as arithmetic mean, geometric mean, or weighted average); finally, the portfolio optimization problem is directly solved based on the aggregation result to obtain the final investment plan. The main drawback of this type of approach is the lack of a dynamic consensus-building process; it cannot effectively handle significant conflicts of opinion among experts, which may lead to the marginalization or neglect of the views of some experts; in addition, this static aggregation method fails to consider the coordination between individual consistency and group consensus, and does not provide experts with the opportunity to adjust and improve their judgments; therefore, although the method is simple, it is difficult to ensure that the decision results truly reflect collective wisdom and gain the approval of all experts. Based on the above-mentioned existing technology, its disadvantages are as follows: The assumption of perfect consistency is unrealistic: Existing portfolio selection methods based on fuzzy preference relations (FPR) assume that all FPRs are perfectly consistent. However, in actual decision-making, due to the limitations of human cognition and subjective biases, it is difficult for the preference relationships provided by experts to achieve perfect consistency; this idealized assumption limits the practical application value of the method. Lack of effective consistency improvement strategies: Existing research mainly focuses on checking the consistency of FPR, but there is a lack of effective adjustment and optimization strategies for FPR that has not reached an acceptable level of consistency; Traditional consistency adjustment methods only focus on mathematical consistency and do not adequately consider the impact on portfolio performance, which may lead to suboptimal investment results and excessive loss of original information; Ignoring the consensus mechanism of group decision-making: Most existing portfolio models are constructed from the perspective of a single expert, while actual major financial decisions (such as corporate investment committees and fund management teams) are essentially multi-expert collaborative processes. Although there are some group decision-making methods, these methods lack effective mechanisms to handle conflicts of opinion among experts and cannot achieve a dynamic consensus-building process. Directly applying traditional single-decision-maker models may not reflect collective wisdom or obtain group consensus. Summary of the Invention

[0005] In view of the problems existing in the prior art, the present invention discloses a two-stage portfolio optimization method based on fuzzy preference and group consensus. The technical solution adopted includes the following steps: Step A: Establish an acceptable consistency evaluation system for fuzzy preference relations (FPR): A1: Define the FPR matrix, let... The FPR matrix provided to experts, in which Ah, this indicates that experts believe securities Better than securities The degree of preference, ranging from 0 to 1; The number of securities to be selected. Let be the investment weight vector, satisfying and conditions; A2: For the FPR matrix R provided by the experts in A1, calculate its consistency index using the formula. The consistency index CI(R) ranges from [0,1], where 0 represents perfect consistency and 1 represents complete inconsistency. A3: Set an acceptable consistency threshold α, which is determined by the decision-maker based on actual needs; A4: Construct a minimum consistency index calculation model under portfolio performance constraints. The goal of this model is to ensure that the expected return is not lower than the target value. Or the expected risk does not exceed the target value. Under the constraints, calculate the minimum consistency index that the FPR can achieve. The calculated consistency index With threshold Comparison: If If the FPR is considered to have an acceptable level of consistency, it can be directly used for portfolio optimization; if If so, an optimization model needs to be built for further judgment or adjustment; Step B: Construct a two-stage FPR adjustment and portfolio optimization model: Phase 1: Determine the minimum adjustment distance; B1: Define the adjustment distance: original FPR matrix With the adjusted FPR matrix Adjust the distance between them: ; B2: Constructing a minimum adjustment distance optimization model: Determining the required minimum adjustment distance while satisfying acceptable consistency and investment constraints. ; respectively targeting expected returns Constraints and Expected Risks The objective is to minimize the adjustment distance under constraints; a model is constructed accordingly. B3: Solve the model established in step B2 using a mathematical optimization algorithm to obtain the minimum adjustment distance. ; Phase 2: Optimize the portfolio under the minimum adjustment distance constraint; B4: Construct a portfolio optimization model, ensuring that the adjustment distance does not exceed... Under the premise of optimizing the risk or return objectives of the investment portfolio; models are constructed for the objectives of minimizing risk and maximizing return respectively; B5: Solve and output the final result. Solve the model established in step B4 using a mathematical optimization algorithm to obtain the adjusted FPR matrix. Optimal investment weight vector The optimal portfolio risk or return.

[0006] Step C: Constructing a consensus-building framework for group decision-making environments: It has Several experts participated in the decision-making process, and each expert... Provide FPR matrix Using the acceptable consistency FPR and corresponding investment weight vector of each expert obtained in step B Calculate the group priority weight vector: That is, the arithmetic mean of the weight vectors of each expert; based on this, the level of individual consensus is defined. and the level of group consensus : ; ; in, Measurement experts The degree of closeness between the FPR and the group consensus weight. Both measures the overall level of group consensus; their values ​​are both within... Between these values, the closer to 1, the higher the degree of consensus; a threshold for group consensus is set. ,when When it is considered that an acceptable group consensus has been reached; when If necessary, an iterative consensus adjustment will be performed.

[0007] As a preferred embodiment of the present invention, in step A2, the consistency index of the FPR matrix R provided by the expert is calculated using the following formula: .

[0008] As a preferred technical solution of the present invention, step A4 involves the FPR matrix provided by the expert. The minimum achievable consistency index is calculated using the following model: Expected returns Not lower than the target value Under the constraints and with the objective of minimizing the consistency exponent, the model is constructed as follows: ; Among them, constraint (a) is the expected return constraint; constraint (b) is the consistency index calculation; constraint (c) requires all weights to sum to 1; constraint (d) requires all weights to be non-negative. In anticipation of risks Not higher than the target value Under the constraints, and with the objective of minimizing the consistency exponent, the constraints are... Replace with risk constraints The specific model is as follows: .

[0009] As a preferred embodiment of the present invention, step A further includes step A5, solving the optimization model and determining the minimum consistency index value that the FPR can achieve under portfolio constraints, by solving the model constructed in step A4 using a mathematical optimization algorithm. Then, a judgment is made: like This indicates that the FPR meets the acceptable consistency requirement under the constraints and can be directly used for portfolio optimization; like This indicates that even under optimal conditions, the FPR cannot reach an acceptable level of consistency, and it is necessary to proceed to process two to adjust and improve the FPR.

[0010] As a preferred embodiment of the present invention, step B2 is aimed at the expected benefit. The objective of minimizing the adjustment distance under constraints is modeled as follows: ; Wherein, constraint (a) is the definition of the adjusted distance; constraint (b) is that the expected return is not less than Constraints (c) and (d) require that the adjusted FPR consistency index be below a threshold. Constraints (e) and (f) require all weights to be non-negative and sum to 1; constraints (g), (h), and (i) require the elements in the adjusted fuzzy preference relation to satisfy the basic definition of a fuzzy preference relation. Regarding anticipated risks The objective of minimizing the adjustment distance under constraints is to replace constraint (b) with a risk constraint. The model is constructed as follows: .

[0011] As a preferred embodiment of the present invention, step B4, targeting the risk minimization objective, constructs the following model: ; Among them, constraints (a) and (b) stipulate that the adjustment distance does not exceed the minimum value. The other constraints function the same as those in the previously given model; To maximize profit, the objective function is changed to And adjust the constraints accordingly, constructing the model as follows: .

[0012] As a preferred embodiment of the present invention, the complete algorithm flow for step B5 is as follows: Input: Original FPR matrix Consistency threshold Expected return target threshold or risk target threshold Optimize the type of objective, namely, minimize risk or maximize return; Execution steps: B51: Using steps A4 and A5, calculate the results under the constraints. Minimum Consistency Index ; B52: Determine if If so, then Since acceptable consistency has been achieved, we can directly construct a portfolio optimization model to solve it and output the optimal weight vector. Based on investment performance, the algorithm terminates; if Then proceed to the next step; B53: Calculate the minimum adjustment distance ; B54: Under the constraint of minimum adjustment distance The adjusted FPR matrix is ​​obtained. Optimal weight vector and investment performance; Output: The final, consistent FPR matrix after adjustment. Or not adjusted Optimal investment weight And the corresponding portfolio risks or returns.

[0013] As a preferred embodiment of the present invention, the iterative consensus adjustment mechanism in step C is as follows: C1: Experts with the lowest Individual Consensus Level (ICL) ; C2: FPR of expert k The adjustments were made, and the revised FPR takes the following form: ; in, To adjust the coefficient, and Balancing the preservation of original preference information with convergence towards group consensus; when At the same time, the original preferences are completely preserved. At that time, the consensus of the group was fully adopted; C3: Construct an optimization model to determine the optimal adjustment for adjustments that have reached the consensus threshold. Let the set of all its experts be denoted as . The requirement is that during the adjustment process, The consensus among experts within the group will not decrease to The value must be below the threshold; during the adjustment process, the consensus index of the experts being adjusted must not be lower than the threshold. Adjusted individual consensus index of experts The objective function is to minimize the adjustment distance. In expected returns Under these conditions, its mathematical model is as follows: ; Wherein, constraints (a) and (b) represent the adjusted FPR; constraints (c)-(j) have the same function as the constraints of the first-stage model in step B; constraints (k) and (l) are used to calculate the adjusted group priority weights. and the adjusted Constraints (m) ensure the adjusted Reaching the set threshold Constraints (n) and (o) are used to ensure the set The FPRs of other experts who have reached an acceptable consensus in China will remain unchanged, while ensuring that the adjusted FPRs of these experts are maintained. It can still reach the set threshold; For the goal of maximizing returns, constraint (d) is replaced with a risk constraint. The model is as follows: ; C4: Under the minimum adjustment distance constraint determined in step C3, a step similar to consistency adjustment makes the first... The adjustment distance of each expert does not exceed To further optimize the portfolio objective, namely minimizing risk or maximizing return, we obtain the adjusted FPR. And weights; in expected returns Under these conditions, a model is constructed, and for the goal of maximizing returns, constraint (d) is replaced with a risk constraint. Build a model; C5: Update group priority weights Group consensus level (GCL); C6: Determine if If the maximum number of iterations is reached, output the final result and the algorithm terminates; otherwise, return to step C1 to continue adjusting.

[0014] As a preferred embodiment of the present invention, step C4 is expected to yield... Under these conditions, the risk minimization model is constructed as follows: ; For the goal of maximizing returns, constraint (d) is replaced with a risk constraint. The profit maximization model is constructed as follows: .

[0015] As a preferred embodiment of the present invention, the complete algorithm flow for step C is as follows: enter: The original FPR matrix set of experts Consistency threshold Group consensus threshold Expected return target or risk target Maximum number of iterations ; Preprocessing: Apply step B to each expert to obtain an acceptablely consistent FPR matrix. and weight vector ; Iterative process: Initialization: Let , , ; when hour: 1. Calculate the current and ; 2. If If a consensus is reached, the result is output and the process terminates. 3. Identify experts with the lowest ICL (Internal Classification) ; 4. For experts Applying a two-stage adjustment model, we obtain and ; 5. Update: Other experts remain unchanged: ; 6. ; Output: The final set of FPR matrices Investment weight vector set ICL and GCL values.

[0016] The beneficial effects of this invention are: 1. This invention introduces a consistency threshold. ,allow This means that the requirements are met, making the method practically operable. Secondly, the proposed two-stage optimization method provides a systematic improvement strategy for FPR that does not meet the consistency requirements. The first stage determines the minimum adjustment distance, and the second stage optimizes the investment objective under this constraint. This design ensures that while improving consistency, the original preference information is preserved to the greatest extent and investment performance is optimized. 2. This invention establishes a dynamic, iterative consensus-building mechanism. By identifying the expert with the lowest ICL and making targeted adjustments, it enables experts to gradually reach a genuine consensus based on a full understanding of the group's opinions, avoiding the marginalization of some expert viewpoints that may occur with simple aggregation methods. Secondly, the ICL and GCL quantification indicators proposed in this invention provide a scientific basis for evaluating the consensus state; by setting a group consensus threshold... By continuously monitoring the GCL value, we can clearly determine whether an acceptable level of consensus has been reached and provide an objective standard for whether further adjustments are needed. 3. This invention simultaneously ensures individual consistency and portfolio performance constraints during the consensus-building process. In each iterative adjustment, this invention ensures that the adjusted FPR meets the requirements. (Individual consistency) (Individual consensus) (Group consensus) and expected returns or risk control requirements; the coordination and optimization of this triple constraint ensures that the final decision has both a good group consensus basis and guarantees the logical rationality of individual judgment and the actual feasibility of investment. 4. In summary, this two-stage portfolio optimization method based on fuzzy preferences and group consensus proposes an acceptable additive consensus framework for FPR, defines a consensus index, and sets a threshold. A consensus assessment standard that aligns with real-world decision-making scenarios was established, making the method more practical. A two-stage FPR adjustment and optimization framework was constructed: the first stage determines the minimum adjustment distance, and the second stage optimizes portfolio selection while minimizing information loss. This method maintains mathematical rigor while preserving the original preferences of experts to the greatest extent possible by defining individual consensus level (ICL), group consensus level (GCL), and an acceptable group consensus threshold. When GCL falls below the threshold, the system automatically identifies the expert with the lowest ICL and adjusts it using a two-stage optimization model. This iterative process ensures that individual consistency and portfolio performance constraints are met in each iteration, effectively coordinating individual consistency and group consensus. Attached Figure Description

[0017] Figure 1 This is a flowchart of the present invention. Detailed Implementation

[0018] Example 1 like Figure 1 As shown, this invention discloses a two-stage portfolio optimization method based on fuzzy preferences and group consensus. The technical solution adopted includes the following steps: Step A: Establish an acceptable consistency evaluation system for fuzzy preference relations (FPR): A1: Define the FPR matrix, let... The FPR matrix provided to experts, in which Experts believe that securities Better than securities The degree of preference, with a value range of ; The number of securities to be selected. Let be the investment weight vector, satisfying and conditions; A2: Regarding the FPR matrix provided by experts in A1 The consistency index is calculated using the following formula: The consistency index The range of values ​​is Where 0 represents perfect consistency and 1 represents complete inconsistency; this index is calculated based on the actual preference relationships given by experts. The level of consistency is quantified by the deviation from the preference relationship that should be theoretically perfectly consistent. A3: Set an acceptable consistency threshold This threshold is determined by decision-makers based on actual needs, and can be set to a value that is appropriate for each individual situation. This threshold serves as a reasonable and acceptable level; it is used to subsequently determine whether the FPR meets the acceptable consistency requirements. A4: Construct a minimum consistency index calculation model under portfolio performance constraints. The goal of this model is to ensure that the expected return is not lower than the target value. Or the expected risk does not exceed the target value. Under the constraints, calculate the minimum consistency index that the FPR can achieve. .

[0019] In expected returns Not lower than the target value Under the constraints and with the objective of minimizing the consistency exponent, the model is constructed as follows: ; Among them, constraint (a) is the expected return constraint; constraint (b) is the consistency index calculation; constraint (c) requires that the sum of all weights be 1; constraint (d) requires that all weights be non-negative. In anticipation of risks Not higher than the target value Under the constraints, and with the objective of minimizing the consistency exponent, the constraints are... Replace with risk constraints The specific model is as follows: ; A5: Solve the optimization model and determine the minimum consistency index value that the FPR can achieve under portfolio constraints. Using mathematical optimization algorithms such as linear programming and quadratic programming, solve the model constructed in step A4 to obtain the minimum consistency index value that the FPR can achieve. Then, a judgment is made: like This indicates that the FPR meets the acceptable consistency requirement under the constraints and can be directly used for portfolio optimization; like This indicates that even under optimal conditions, the FPR cannot reach an acceptable level of consistency, and it is necessary to proceed to process two to adjust and improve the FPR; Step B: Construct a two-stage FPR adjustment and portfolio optimization model: When the FPR provided by the experts cannot reach an acceptable level of consistency under the constraints, it needs to be adjusted. The two-stage optimization method proposed in step B ensures that the adjusted FPR meets the consistency requirements, retains the original preference information of the experts to the greatest extent, and optimizes the portfolio performance. Phase 1: Determine the minimum adjustment distance; B1: Define the adjustment distance: Define the original FPR matrix to quantize the adjustment magnitude. With the adjusted FPR matrix Adjust the distance between them: The smaller the distance, the better the original preference information is preserved. B2: Constructing a minimum adjustment distance optimization model: Determining the required minimum adjustment distance while satisfying acceptable consistency and investment constraints. ; respectively targeting expected returns Constraints and Expected Risks The objective is to minimize the adjustment distance under constraints; a model is constructed accordingly. Regarding expected returns The objective of minimizing the adjustment distance under constraints is modeled as follows: ; Wherein, constraint (a) is the definition of the adjusted distance; constraint (b) is that the expected return is not less than Constraints (c) and (d) require that the adjusted FPR consistency index be below a threshold. Constraints (e) and (f) require all weights to be non-negative and sum to 1; constraints (g), (h), and (i) require the elements in the adjusted fuzzy preference relation to satisfy the basic definition of a fuzzy preference relation. Regarding anticipated risks The objective of minimizing the adjustment distance under constraints is to replace constraint (b) with a risk constraint. The model is constructed as follows:

[0020] B3: Solve the model established in step B2 using a mathematical optimization algorithm to obtain the minimum adjustment distance. ; Phase 2: Optimize the portfolio under the minimum adjustment distance constraint; B4: Construct a portfolio optimization model, ensuring that the adjustment distance does not exceed... Under the premise of optimizing the risk or return objectives of the investment portfolio; models are constructed for the objectives of minimizing risk and maximizing return respectively; To minimize risk, the model is constructed as follows: ; Among them, constraints (a) and (b) stipulate that the adjustment distance does not exceed the minimum value. The other constraints function the same as those in the previously given model. This is because the adjustment model is divided into two stages. The first stage solves for the minimum adjustment distance, but the priority weights of the portfolio obtained at this stage may not represent the minimum risk or the maximum return. In the second stage, based on this minimum adjustment distance, the focus shifts to the portfolio's performance to solve for the corresponding minimum risk or maximum return. Therefore, the minimum adjustment distance constraint is added to the constraints, while the other constraints do not need to be changed. Thus, the function of the other constraints is the same as in the first stage of the model. To maximize profit, the objective function is changed to And adjust the constraints accordingly, constructing the model as follows: ; B5: Solve and output the final result. Solve the model established in step B4 using a mathematical optimization algorithm to obtain the adjusted FPR matrix. Optimal investment weight vector The optimal portfolio risk or return; The complete algorithm flow is as follows: Input: Original FPR matrix Consistency threshold Expected return target threshold or risk target threshold Optimize the type of objective, namely, minimize risk or maximize return; Execution steps: B51: Using steps A4 and A5, calculate the results under the constraints. Minimum Consistency Index ; B52: Judgment If so, then Since acceptable consistency has been achieved, we can directly construct a portfolio optimization model to solve it and output the optimal weight vector. Based on investment performance, the algorithm terminates; if Then proceed to the next step; B53: Calculate the minimum adjustment distance ; B54: Under the constraint of minimum adjustment distance The adjusted FPR matrix is ​​obtained. Optimal weight vector and investment performance; Output: The final, consistent FPR matrix after adjustment. Or not adjusted Optimal investment weight And the corresponding portfolio risks or returns; Step C: Constructing a consensus-building framework for group decision-making environments: In multi-expert investment decision-making scenarios, it is necessary to coordinate the opinions of different experts to reach a group consensus. The method of step B is extended to the group decision-making environment, and an iterative consensus-reaching framework is proposed. It has Several experts participated in the decision-making process, and each expert... Provide FPR matrix Using the acceptable consistency FPR and corresponding investment weight vector of each expert obtained in step B Calculate the group priority weight vector: That is, the arithmetic mean of the weight vectors of each expert; based on this, the level of individual consensus is defined. and the level of group consensus : ; ; in, Measurement experts The degree of closeness between the FPR and the group consensus weight. Both measures the overall level of group consensus; their values ​​are both within... Between these values, the closer to 1, the higher the degree of consensus; a threshold for group consensus is set. ,when When it is considered that an acceptable group consensus has been reached; when When this happens, iterative consensus adjustments are made. The iterative consensus adjustment mechanism is as follows: C1: Experts who identify the lowest ICL That is, the expert's FPR deviates the most from the group consensus; C2: FPR of expert k The adjustments were made, and the revised FPR takes the following form: ; in, To adjust the coefficient, and Balancing the preservation of original preference information with convergence towards group consensus; when At the same time, the original preferences are completely preserved. At that time, the consensus of the group was fully adopted; C3: Construct an optimization model to determine the optimal adjustment for adjustments that have reached the consensus threshold. Let the set of all its experts be denoted as . The requirement is that during the adjustment process, The consensus among experts within the group will not decrease to The value must be below the threshold; during the adjustment process, the consensus index of the experts being adjusted must not be lower than the threshold. Adjusted individual consensus index of experts The objective function is to minimize the adjustment distance. In expected returns Under these conditions, its mathematical model is as follows: ; Wherein, constraints (a) and (b) represent the adjusted FPR; constraints (c)-(j) have the same function as the constraints of the first-stage model in step B; constraints (k) and (l) are used to calculate the adjusted group priority weights. and the adjusted Constraints (m) ensure the adjusted Reaching the set threshold Constraints (n) and (o) are used to ensure the set The FPRs of other experts who have reached an acceptable consensus in China will remain unchanged, while ensuring that the adjusted FPRs of these experts are maintained. It can still reach the set threshold; For the goal of maximizing returns, constraint (d) is replaced with a risk constraint. The model is as follows: ; C4: Under the minimum adjustment distance constraint determined in step C3, a step similar to consistency adjustment makes the first... The expert's adjustment distance does not exceed To further optimize the portfolio objective, namely minimizing risk or maximizing return, we obtain the adjusted FPR. And weights; in expected returns Under these conditions, the risk minimization model is constructed as follows: ; For the goal of maximizing returns, constraint (d) is replaced with a risk constraint. The profit maximization model is constructed as follows: ; C5: Update group priority weights and GCL; C6: Determine if If the maximum number of iterations is reached, output the final result and the algorithm terminates; otherwise, return to step C1 to continue adjusting. The complete algorithm flow for step C is as follows: enter: The original FPR matrix set of experts Consistency threshold Group consensus threshold Expected return target or risk target Maximum number of iterations ; Preprocessing: Apply step B to each expert to obtain an acceptablely consistent FPR matrix. and weight vector ; Iterative process: Initialization: Let , , ; when hour: 1. Calculate the current and ; 2. If If a consensus is reached, the result is output and the process terminates. 3. Identify experts with the lowest ICL (Internal Classification) ; 4. For experts Applying a two-stage adjustment model, we obtain and ; 5. Update: Other experts remain unchanged: ; 6. ; Output: The final set of FPR matrices Investment weight vector set ICL and GCL values.

[0021] Components not described in detail in this article are existing technologies.

[0022] While the specific embodiments of the present invention have been described in detail above, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention, and modifications or variations without creative effort are still within the protection scope of the present invention.

Claims

1. A two-stage portfolio optimization method based on fuzzy preferences and group consensus, characterized in that, Includes the following steps: Step A: Establish an acceptable consistency evaluation system for fuzzy preference relations (FPR): A1: Define the FPR matrix, let... The FPR matrix provided to experts, in which Experts believe that securities Better than securities The degree of preference, with a value range of ; The number of securities to be selected. Let be the investment weight vector, satisfying and conditions; A2: Regarding the FPR matrix provided by experts in A1 The consistency index is calculated using a formula. The range of values ​​is , where 0 represents perfect consistency and 1 represents complete inconsistency; A3: Set an acceptable consistency threshold This threshold is determined by decision-makers based on actual needs; A4: Construct a minimum consistency index calculation model under portfolio performance constraints. The goal of this model is to ensure that the expected return is not lower than the target value. Or the expected risk does not exceed the target value. Under the constraints, calculate the minimum consistency index that the FPR can achieve. The calculated consistency index With threshold Comparison: If If the FPR is considered to have an acceptable level of consistency, it can be directly used for portfolio optimization; if If so, an optimization model needs to be built for further judgment or adjustment; Step B: Construct a two-stage FPR adjustment and portfolio optimization model: Phase 1: Determine the minimum adjustment distance; B1: Define the adjustment distance: original FPR matrix With the adjusted FPR matrix Adjust the distance between them: ; B2: Constructing a minimum adjustment distance optimization model: Determining the required minimum adjustment distance while satisfying acceptable consistency and investment constraints. ; respectively targeting expected returns Constraints and Expected Risks The objective is to minimize the adjustment distance under constraints; a model is constructed accordingly. B3: Solve the model established in step B2 using a mathematical optimization algorithm to obtain the minimum adjustment distance. ; Phase 2: Optimize the portfolio under the minimum adjustment distance constraint; B4: Construct a portfolio optimization model, ensuring that the adjustment distance does not exceed... Under the premise of optimizing the risk or return objectives of the investment portfolio; models are constructed for the objectives of minimizing risk and maximizing return respectively; B5: Solve and output the final result. Solve the model established in step B4 using a mathematical optimization algorithm to obtain the adjusted FPR matrix. Optimal investment weight vector The optimal portfolio risk or return; Step C: Constructing a consensus-building framework for group decision-making environments: It has Several experts participated in the decision-making process, and each expert... Provide FPR matrix Using the acceptable consistency FPR and corresponding investment weight vector of each expert obtained in step B Calculate the group priority weight vector: That is, the arithmetic mean of the weight vectors of each expert; based on this, the level of individual consensus is defined. and the level of group consensus : ； ； in, Measurement experts The degree of closeness between the FPR and the group consensus weight. Both measures the overall level of group consensus; their values ​​are both within... Between these values, the closer to 1, the higher the degree of consensus; a threshold for group consensus is set. ,when When it is considered that an acceptable group consensus has been reached; when If necessary, an iterative consensus adjustment will be performed.

2. The two-stage portfolio optimization method based on fuzzy preference and group consensus as described in claim 1, characterized in that: In step A2, the FPR matrix provided by the expert is... The consistency index is calculated using the following formula: .

3. The two-stage portfolio optimization method based on fuzzy preference and group consensus as described in claim 1, characterized in that: In step A4, the FPR matrix provided by the expert is... The minimum achievable consistency index is calculated using the following model: Expected returns Not lower than the target value Under the constraints and with the objective of minimizing the consistency exponent, the model is constructed as follows: ； Among them, constraint (a) requires the expected return. Constraint (b) is the consistency index of the objective function. The calculation; constraint (c) requires that the sum of all weights be 1; constraint (d) requires that all weights be non-negative; In anticipation of risks Not higher than the target value Under the constraints, and with the objective of minimizing the consistency exponent, the constraints are... Replace with risk constraints The specific model is as follows: 。 4. The two-stage portfolio optimization method based on fuzzy preference and group consensus as described in claim 3, characterized in that: Step A further includes step A5, which involves solving the optimization model and determining the minimum consistency index value that the FPR can achieve under portfolio constraints. This is done by solving the model constructed in step A4 using a mathematical optimization algorithm. Then, a judgment is made: like This indicates that the FPR meets the acceptable consistency requirement under the constraints and can be directly used for portfolio optimization; like This indicates that even under optimal conditions, the FPR cannot reach an acceptable level of consistency, and it is necessary to proceed to process two to adjust and improve the FPR.

5. The two-stage portfolio optimization method based on fuzzy preference and group consensus as described in claim 1, characterized in that: Step B2 is for the expected return. The objective of minimizing the adjustment distance under constraints is modeled as follows: ； Wherein, constraint (a) is the definition of the adjusted distance; constraint (b) is that the expected return is not less than Constraints (c) and (d) require that the adjusted FPR consistency index be below a threshold. Constraints (e) and (f) require all weights to be non-negative and sum to 1; constraints (g), (h), and (i) require the elements in the adjusted fuzzy preference relation to satisfy the basic definition of a fuzzy preference relation. Regarding anticipated risks The objective of minimizing the adjustment distance under constraints is to replace constraint (b) with a risk constraint. The model is constructed as follows: 。 6. The two-stage portfolio optimization method based on fuzzy preference and group consensus as described in claim 1, characterized in that: Step B4, targeting the risk minimization objective, constructs the following model: ； Among them, constraints (a) and (b) stipulate that the adjustment distance does not exceed the minimum value. The other constraints function the same as those in the previously given model; To maximize profit, the objective function is modified as follows: And adjust the constraints accordingly, constructing the model as follows: 。 7. The two-stage portfolio optimization method based on fuzzy preference and group consensus as described in claim 1, characterized in that: The complete algorithm flow for step B5 is as follows: Input: Original FPR matrix Consistency threshold Expected return target threshold or risk target threshold Optimize the type of objective, namely, minimize risk or maximize return; Execution steps: B51: Using steps A4 and A5, calculate the results under the constraints. Minimum Consistency Index ; B52: Determine if If so, then Since acceptable consistency has been achieved, we can directly construct a portfolio optimization model to solve it and output the optimal weight vector. Based on investment performance, the algorithm terminates; if Then proceed to the next step; B53: Calculate the minimum adjustment distance ; B54: Under the constraint of minimum adjustment distance The adjusted FPR matrix is ​​obtained. Optimal weight vector and investment performance; Output: The final, consistent FPR matrix after adjustment. Or not adjusted Optimal investment weight And the corresponding portfolio risks or returns.

8. The two-stage portfolio optimization method based on fuzzy preference and group consensus as described in claim 1, characterized in that: The iterative consensus adjustment mechanism in step C is as follows: C1: Recognition Lowest expert ; C2: For experts FPR The adjustments were made, and the revised FPR takes the following form: ； in, To adjust the coefficient, and Balancing the preservation of original preference information with convergence towards group consensus; when At the same time, the original preferences are completely preserved. At that time, the consensus of the group was fully adopted; C3: Construct an optimization model to determine the optimal adjustment for adjustments that have reached the consensus threshold. Let the set of all its experts be denoted as . The requirement is that during the adjustment process, The consensus among experts within the group will not decrease to The value must be below the threshold; during the adjustment process, the consensus index of the experts being adjusted must not be lower than the threshold. Adjusted individual consensus index of experts The objective function is to minimize the adjustment distance. In expected returns Under these conditions, its mathematical model is as follows: ； Wherein, constraints (a) and (b) represent the adjusted FPR; constraints (c)-(j) have the same function as the constraints of the first-stage model in step B; constraints (k) and (l) are used to calculate the adjusted group priority weights. and the adjusted Constraints (m) ensure the adjusted Reaching the set threshold Constraints (n) and (o) are used to ensure the set The FPRs of other experts who have reached an acceptable consensus in China will remain unchanged, while ensuring that the adjusted FPRs of these experts are maintained. It can still reach the set threshold; For the goal of maximizing returns, constraint (d) is replaced with a risk constraint. The model is as follows: ； C4: Under the minimum adjustment distance constraint determined in step C3, a step similar to consistency adjustment makes the first... The adjustment distance of each expert does not exceed To further optimize the portfolio objective, namely minimizing risk or maximizing return, we obtain the adjusted FPR. And weights; in expected returns Under these conditions, a model is constructed, and for the goal of maximizing returns, constraint (d) is replaced with a risk constraint. Build a model; C5: Update group priority weights and GCL; C6: Determine if If the maximum number of iterations is reached, output the final result and the algorithm terminates; otherwise, return to step C1 to continue adjusting.

9. The two-stage portfolio optimization method based on fuzzy preference and group consensus as described in claim 8, characterized in that: Step C4 is in the expected return Under these conditions, the risk minimization model is constructed as follows: ； For the goal of maximizing returns, constraint (d) is replaced with a risk constraint. The profit maximization model is constructed as follows: 。 10. The two-stage portfolio optimization method based on fuzzy preference and group consensus as described in claim 8, characterized in that: The complete algorithm flow for step C is as follows: enter: The original FPR matrix set of experts Consistency threshold Group consensus threshold Expected return target or risk target Maximum number of iterations ; Preprocessing: Apply step B to each expert to obtain an acceptablely consistent FPR matrix. and weight vector ; Iterative process: Initialization: Let , , ; when hour:

1. Calculate the current and ; 2. If If a consensus is reached, the result is output and the process terminates.

3. Identify experts with the lowest ICL (Internal Classification) ; 4. For experts Applying a two-stage adjustment model, we obtain and ; 5. Update: Other experts remain unchanged: ; 6. ； Output: The final set of FPR matrices Investment weight vector set , and value.

Images