Multi-criterion decision BWM consistency threshold obtaining method based on preference of decision maker

Abstract

The invention discloses a decision maker preference-based multi-criterion BWM (BeastWorst Method, BWM) consistency threshold acquisition method, which is characterized in that specific thresholds suitable for various decision scenes are generated by utilizing Monte Carlo simulation according to different criterion quantities and evaluation scales, and a qualified basis is provided for the consistency of a decision maker evaluation process; based on error minimization constraint, ordinal number consistency and cardinal number consistency are comprehensively considered, so that criterion sorting logic conflicts and decision deviations are eliminated; by introducing an error term and an acceptance rate index, the preference of a decision maker is met, and the flexibility and the application range of the BWM method in practical application are further improved.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-criteria decision-making, and specifically to a method for obtaining a consistency threshold of a multi-criteria decision-making Best Worst Method (BWM) based on decision maker preferences. The method is particularly suitable for the scientific determination and dynamic adjustment of the consistency threshold in complex decision-making scenarios. Background Art

[0002] In recent years, multi-criteria decision-making methods have been increasingly applied in fields such as resource allocation, risk assessment, and healthcare. Among them, the Best-Worst Method (BWM) has gradually become a mainstream decision-making tool due to its high efficiency and minimal number of comparisons. Compared with the traditional Analytic Hierarchy Process (AHP), the BWM significantly reduces the complexity of subjective judgment by requiring decision makers to perform pairwise comparisons between the "best criterion" and the "worst criterion." Furthermore, it enhances the objectivity of weight calculation through a mathematical optimization model.

[0003] However, existing BWM methods still have key bottlenecks in practical applications:

[0004] (1) Existing BWM methods lack a clear consistency threshold, which makes it difficult for decision makers to assess in real time whether their evaluations meet the consistency requirements and when they need to be revised, thus affecting the credibility of decision results;

[0005] (2) In multi-criteria decision-making, it is often necessary to dynamically adjust the consistency threshold based on the decision-maker's intentions and scenario characteristics. For example, in some high-risk medical decisions, stricter thresholds are required to ensure the reliability of the results; whereas in daily consumer decisions, decision-makers may be more concerned with efficiency and accept looser thresholds. Existing technologies lack the ability to flexibly respond to such needs, which limits the applicability of the BWM method.

[0006] (3) Current BWM research focuses on cardinality consistency (i.e., whether the product relationship of the evaluation values satisfies a Bj ·a jW =a 2 BW ), but ignores the ordinal consistency (i.e. the logical consistency of the criterion importance ranking). The existing technology fails to establish a correlation model between ordinal consistency and cardinality threshold, which may cause logical contradictions and decision-making deviations. Summary of the Invention

[0007] To address the above problems, the present invention proposes a BWM consistency threshold acquisition method for multi-criteria decision-making based on decision maker preferences. The method aims to dynamically generate consistency thresholds that adapt to different scenarios and decision maker preferences through random simulation and error minimization strategy, so as to improve the applicability of BWM.

[0008] In order to solve the above technical problems, the present invention adopts the following technical solutions:

[0009] A method for obtaining a BWM consistency threshold for multi-criteria decision-making based on decision maker preference, characterized by comprising the following steps:

[0010] Step S1: Set the original parameters in the multi-criteria decision-making environment, including the number of criteria n, the evaluation scale a BW , simulation times m and variance σ, the evaluation scale a BW is the maximum value in the BWM scale.

[0011] Step S2: Generate random comparison samples, including:

[0012] S2.1) Generate the best criterion comparison array A Bj : Randomly generate m groups of integer arrays, each array contains n elements, two of which are 1 and a BW , the remaining n-2 elements are randomly generated from uniform distribution X~U(1,9);

[0013] S2.2) Generate the worst criterion comparison vector A jW : Randomly generate m groups of arrays, each array contains n elements, two of which are a BW and 1, the remaining n-2 elements are distributed by the normal distribution function X~N(a BW / a Bj ,σ) are generated one by one and rounded according to the following formula:

[0014]

[0015] Where f(·) is the rounding function; a Bj Compare array A for the best criterion Bj The ratio of the best criterion to criterion j; a jW Compare array A to the worst criterion jW The ratio of the middle criterion j to the worst criterion.

[0016] S2.3) Composing the original evaluation data: Compare the best criteria to the array A Bj Compare array A with the worst criterion jW The elements in A are combined in their corresponding order, that is, Bj The original 1 and a BW Respectively with A jW A BW Combined with 1, the remaining a Bj and a jW Combine them to obtain m groups of complete BWM evaluation raw data.

[0017] Step S3: Data grouping

[0018] The m sets of complete raw BWM assessment data were categorized into ordinal consistency and ordinal inconsistency groups. If the elements in an array satisfy any of the following equations at any i and j, the array is ordinal consistency and is classified into the ordinal consistency group; otherwise, it is classified into the ordinal inconsistency group.

[0019] (a Bi -a Bj )×(a jW -a iW )>0,

[0020]

[0021] Step S4: Cardinality consistency calculation

[0022] The following formula is used to calculate the data of each group in the ordinal consistent group and the ordinal inconsistent group respectively to obtain the cardinality consistency index CR of the m group evaluation.

[0023]

[0024] Among them, CR is the global cardinality consistency level, CR j Is with criterion c j The corresponding local cardinality consistency level.

[0025] Step S5: Set preferences and calculate thresholds

[0026] According to the decision maker's needs, the error terms α and β, and the acceptance rate ζ0 are constrained conditions, and the consistency threshold T that satisfies the minimum total error is obtained by solving the following formula d ;

[0027]

[0028] Among them, λ is the proportion of ordinal inconsistent groups in m groups of randomly generated arrays; α is the proportion of arrays with cardinality consistency index CR higher than the threshold in ordinal consistent groups; β is the proportion of arrays with cardinality consistency index CR lower than the threshold in ordinal inconsistent groups; ζ0 is the minimum acceptance rate acceptable to decision makers.

[0029] Preferably, the number of simulations m should be no less than 10,000 times to meet the accuracy requirements of multi-criteria decision-making.

[0030] Preferably, the variance σ ranges from [0, (a BW -1) / 6], the 3σ criterion based on the normal distribution is set to ensure that 99.7% of the generated data fall within the evaluation scale interval [1,a BW]; Specifically, according to the 3σ criterion, the probability that the value of the random number generated by the normal distribution X~N(μ,σ) falls within [μ-3σ,μ+3σ] is 99.73%. In order to satisfy the random generated value falling within the evaluation scale interval [1,a BW ], there is (μ+3σ)-(μ-3σ)=a BW -1, we get σ=(a BW -1) / 6, so the value range of σ is [0,(a BW -1) / 6]; the smaller the decision maker's cognitive uncertainty about the thing being decided, the smaller the value of σ, and the more concentrated the distribution of evaluation scores.

[0031] Preferably, in step S5, the decision maker preference is set in the following manner:

[0032] (a) If you want to completely eliminate ordinal inconsistency evaluation, set the constraint α = 0;

[0033] (b) If you need to fully accept the ordinal consistency evaluation, set the constraint β = 0;

[0034] (c) By setting the acceptance rate ζ0, the convenience of decision-making operations and the reliability of decision results are balanced. Generally speaking, the larger the value of ζ0, the fewer times the decision-making process requires modification and the better the convenience of use, but the more error terms are included in the decision and the less reliable the result.

[0035] Compared with the prior art, the present invention has the following beneficial effects:

[0036] The present invention proposes a multi-criteria BWM consistency threshold acquisition method based on decision maker preferences. This method combines parameters such as the decision maker's risk preference, the number of criteria, and the evaluation scale, and uses Monte Carlo simulation to generate specific thresholds applicable to various decision-making scenarios, providing a qualified basis for decision maker consistency assessment. Based on the error minimization constraint, by comprehensively considering ordinal consistency and cardinality consistency, the logical conflicts in criterion sorting and decision bias are eliminated. By introducing the acceptance rate indicator, the operational convenience of the decision process and the reliability of the results are balanced, thereby improving the flexibility and applicability of the BWM method in practical applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 This is a flow chart of the method of the present invention, showing the complete process of steps S1 to S5. DETAILED DESCRIPTION

[0038] The present invention will be described in further detail below with reference to the accompanying drawings.

[0039] A method for obtaining BWM consistency threshold of multi-criteria decision-making based on decision maker preference is as follows: Figure 1As shown in the figure, first, the original parameters are set according to the decision scenario, and random samples of decisions are generated based on the Monte Carlo method. Then, the samples are grouped and the consistency index of each group is calculated. Then, the relevant parameters are set according to the designer's preferences and the optimal solution is calculated. Finally, the consistency threshold result under the decision scenario is obtained.

[0040] The detailed steps of this method are as follows:

[0041] Step S1: Set the original parameters in the multi-criteria decision-making environment, including the number of criteria n, the evaluation scale a BW , simulation times m and variance σ, the evaluation scale a BW is the maximum value in the BWM scale.

[0042] In order to ensure the accuracy of decision making, the number of simulations m should satisfy m≥10000; the value range of the variance σ is [0, (a BW -1) / 6], based on the 3σ principle of normal distribution, to ensure that 99.7% of the generated data fall within the evaluation scale interval [1,a BW ]; Specifically, according to the 3σ criterion, the probability that the value of the random number generated by the normal distribution X~N(μ,σ) falls within [μ-3σ,μ+3σ] is 99.73%. In order to satisfy the random generated value falling within the evaluation scale interval [1,a BW ], there is (μ+3σ)-(μ-3σ)=a BW -1, we get σ=(a BW -1) / 6, so the value range of σ is [0,(a BW -1) / 6]; the smaller the decision maker's cognitive uncertainty about the thing being decided, the smaller the value of σ, and the more concentrated the distribution of evaluation scores.

[0043] Step S2: Generate a random comparison sample, including:

[0044] S2.1) Generate the best criterion comparison array A Bj : Randomly generate m groups of integer arrays, each array contains n elements, two of which are 1 and a BW , the remaining n-2 elements are randomly generated from uniform distribution X~U(1,9);

[0045] S2.2) Generate the worst criterion comparison vector A jW : Randomly generate m groups of arrays, each array contains n elements, two of which are a BW and 1, the remaining n-2 elements are distributed by the normal distribution function X~N(a BW / a Bj ,σ) are generated one by one and rounded according to the following formula:

[0046]

[0047] Where f(·) is the rounding function; a Bj Compare array A for the best criterion Bj The ratio of the best criterion to criterion j; a jW Compare array A to the worst criterion jW The ratio of the middle criterion j to the worst criterion.

[0048] S2.3) Composing the original evaluation data: Compare the best criteria to the array A Bj Compare array A with the worst criterion jW The elements in A are combined in their corresponding order, that is, Bj The original 1 and a BW Respectively with A jW A BW Combined with 1, the remaining a Bj and a jW Combine them to obtain m groups of complete BWM evaluation raw data.

[0049] Step S3: Data grouping

[0050] The m sets of complete raw BWM assessment data were categorized into ordinal consistency and ordinal inconsistency groups. If the elements in an array satisfy any of the following equations at any i and j, the array is ordinal consistency and is classified into the ordinal consistency group; otherwise, it is classified into the ordinal inconsistency group.

[0051] (a Bi -a Bj )×(a jW -a iW )>0,

[0052]

[0053] Step S4: Cardinality consistency calculation

[0054] The following formula is used to calculate the data of each group in the ordinal consistent group and the ordinal inconsistent group respectively to obtain the cardinality consistency index CR of the m group evaluation.

[0055]

[0056] Among them, CR is the global cardinality consistency level, CR j Is with criterion c j The corresponding local cardinality consistency level.

[0057] Step S5: Set preferences and calculate thresholds

[0058] According to the decision maker's needs, the error terms α and β, and the acceptance rate ζ0 are constrained conditions, and the consistency threshold T that satisfies the minimum total error is obtained by solving the following formula d ;

[0059]

[0060] Among them, λ is the proportion of ordinal inconsistent groups in m groups of randomly generated arrays; α is the proportion of arrays with cardinality consistency index CR higher than the threshold in ordinal consistent groups; β is the proportion of arrays with cardinality consistency index CR lower than the threshold in ordinal inconsistent groups; ζ0 is the minimum acceptance rate acceptable to decision makers.

[0061] Decision makers can completely exclude ordinal inconsistency evaluations or fully accept ordinal consistency evaluations by setting α = 0 or β = 0, respectively. Furthermore, decision makers can also balance the convenience of decision-making operations with the reliability of decision results by setting the acceptance rate ζ0. Generally speaking, a larger ζ0 value means fewer revisions during the decision-making process and greater ease of use, but also more error terms are included in the decision and less reliable the results.

[0062] The present invention will be further described in detail below with reference to specific embodiments.

[0063] Xiao Ming needs to choose one of three top-tier hospitals in his city, A, B, and C, for a physical examination. He needs to consider each hospital's performance in five areas: reputation, technology, equipment, service, and cost. The BWM standard nine-point scale is used to evaluate these five criteria and determine their relative weights.

[0064] according to Figure 1 The process shown in the figure is to obtain the BWM consistency threshold of the multi-criteria decision-making in this scenario, including the following steps:

[0065] (a) According to the decision scenario, set the number of criteria n = 5 (corresponding to reputation, technology, equipment, service and cost), the evaluation scale a BW =9, number of simulations m = 10000, variance σ = 0.67;

[0066] (b) Generate the optimal criterion comparison array A Bj : Randomly generate 10,000 integer arrays. In addition to 1 and 9, each array contains 3 more elements using the uniform distribution X~U(1,9), with a total of 5 elements in each array, as shown in Table 1.

[0067] Table 1 Randomly generated optimal criterion comparison array A Bj

[0068]

[0069] (c) Generate the worst criterion comparison array A jW : Randomly generate 10,000 arrays. In addition to 9 and 1, each element in the array is distributed by the normal distribution function X~N(a BW / a Bj ,σ) and perform rounding operation according to the following formula to obtain the data shown in Table 2. 3W =1 is generated by the normal distribution function X~N(9 / 7,0.67) and rounded, and a in the second group 5W =2 is generated by the normal distribution function X~N(9 / 6,0.67) and rounded. The rounding operation is as follows:

[0070]

[0071] Where f(·) is the rounding function.

[0072] Table 2 Randomly generated worst criterion comparison array A jW

[0073]

[0074] (d) Compare the best criteria to array A Bj Compare array A with the worst criterion jW The elements in are combined in their corresponding order to obtain 10,000 sets of complete BWM evaluation raw data, as shown in the following table.

[0075]

[0076] (e) Classify 10,000 groups of original data into ordinal consistent groups and ordinal inconsistent groups. When all elements in a group satisfy any one of the following two formulas, the data in this group are ordinal consistent and are classified into the ordinal consistent group; otherwise, they are classified into the ordinal inconsistent group.

[0077] (a Bi -a Bj )×(a jW -a iW )>0,

[0078]

[0079] (f) Use the following formula to calculate the data of each group in the ordinal consistent group and the ordinal inconsistent group respectively to obtain the cardinality consistency index CR of 10,000 evaluation groups.

[0080]

[0081] Among them, CR is the global cardinality consistency level, CR j Is with criterion cj The corresponding local cardinality consistency level.

[0082] (g) In order to balance operability and the credibility of the results, Xiao Ming sets the acceptance rate ζ0 = 0.7 and calculates the final consistency threshold T by solving the following formula d =0.069.

[0083]

[0084] Among them, λ is the proportion of ordinal inconsistent groups in m groups of randomly generated arrays; α is the proportion of arrays with cardinality consistency index CR higher than the threshold in ordinal consistent groups; β is the proportion of arrays with cardinality consistency index CR lower than the threshold in ordinal inconsistent groups; ζ0 is the minimum acceptance rate acceptable to decision makers.

[0085] Therefore, when Xiao Ming's evaluation cardinality consistency index CR>0.069 during the decision-making process, it means that there is a serious inconsistency in the evaluation, and Xiao Ming needs to re-evaluate the evaluation until CR≤0.069 is met.

[0086] Finally, it should be noted that the above-described embodiments of the present invention are merely examples for illustrating the present invention and are not intended to limit the embodiments of the present invention. Although the applicant has described the present invention in detail with reference to preferred embodiments, a person skilled in the art will be able to make other variations and modifications based on the above description. It is not possible to enumerate all embodiments here. Any obvious changes or modifications derived from the technical solution of the present invention remain within the scope of protection of the present invention.

Claims

1. A method for obtaining BWM consistency threshold for multi-criteria decision-making based on decision maker preference, characterized by: The following steps are involved: Step S1: Set the original parameters in the multi-criteria decision-making environment, including the number of criteria n, the evaluation scale a BW , number of simulations m and variance σ; Step S2: Generate random comparison samples, including: S2.1) Generate the best criterion comparison array A Bj : Randomly generate m groups of integer arrays, each array contains n elements, two of which are 1 and a BW , the remaining n-2 elements are randomly generated from uniform distribution X~U(1,9); S2.2) Generate the worst criterion comparison vector A jW : Randomly generate m groups of arrays, each array contains n elements, two of which are a BW and 1, the remaining n-2 elements are distributed by the normal distribution function X~N(a BW / a Bj ,σ) are generated one by one and rounded according to the following formula: Where f(·) is the rounding function; a Bj Compare array A for the best criterion Bj The ratio of the best criterion to criterion j; a jW Compare array A to the worst criterion jW The ratio of the middle criterion j to the worst criterion; S2.3) Composing the original evaluation data: Compare the best criteria to the array A Bj Compare array A with the worst criterion jW The elements in A are combined in their corresponding order, that is, Bj The original 1 and a BW Respectively with A jW A BW Combined with 1, the remaining a Bj and a jW Combine them to obtain m groups of complete BWM evaluation raw data; Step S3: Data grouping The m groups of complete BWM evaluation raw data are classified into ordinal consistent groups and ordinal inconsistent groups. When the elements in an array satisfy any one of the following equations at any i and j, the array is ordinal consistent and is classified into the ordinal consistent group; otherwise, it is classified into the ordinal inconsistent group; (a Bi -a Bj )×(a jW -a iW )>0, Step S4: Cardinality consistency calculation The following formula is used to calculate the data of each group in the ordinal consistent group and the ordinal inconsistent group respectively to obtain the cardinality consistency index CR of the m group evaluation; Among them, CR is the global cardinality consistency level, CR j Is with criterion c j the corresponding local cardinality consistency level; Step S5: Set preferences and calculate thresholds According to the decision maker's needs, the error terms α and β, and the acceptance rate ζ0 are constrained conditions, and the consistency threshold T that satisfies the minimum total error is obtained by solving the following formula d ; Among them, λ is the proportion of ordinal inconsistent groups in m groups of randomly generated arrays; α is the proportion of arrays with cardinality consistency index CR higher than the threshold in ordinal consistent groups; β is the proportion of arrays with cardinality consistency index CR lower than the threshold in ordinal inconsistent groups; ζ0 is the minimum acceptance rate acceptable to decision makers.

2. The method according to claim 1, wherein: The value range of the variance σ is [0, (a BW -1) / 6], based on the 3σ principle of normal distribution, to ensure that 99.7% of the generated data fall within the evaluation scale interval [1,a BW ]Inside.

3. The method according to claim 1, wherein: In step S5, the decision maker preferences are set in the following manner: (a) If you want to completely eliminate ordinal inconsistency evaluation, set the constraint α = 0; (b) If you need to fully accept the ordinal consistency evaluation, set the constraint β = 0; (c) By setting the acceptance rate ζ0, the convenience of decision-making operation and the reliability of decision results are balanced.

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