A multi-attribute group decision-making method based on Z-number

Abstract

The present invention relates to a multi-attribute group decision-making method based on Z-number, comprising: determining a decision criterion; determining the weight of an attribute; suggesting a semantic variable table of an attribute; determining the evaluation of the attribute in group decision-making; converting the evaluation variable into a numerical value; constructing part A in Z-number; constructing part B in Z-number using information entropy; reconstructing Z-number based on the influence of part B on the attribute; fusing Z-numbers of different attributes to obtain Z-numbers of different options; sorting the options and making decisions to draw conclusions. Compared with the prior art, the present invention utilizes Z-number theory to directly generate part A and part B in Z-number from an original data set, and proposes a new Z-number generation method, which reconstructs Z-number by considering the influence of part B on the attribute, thereby solving the conflict problem existing in traditional Z-number fusion, making the fusion result in the final group decision more reasonable and more stable.

Description

Technical Field

[0001] The present invention relates to the technical field of uncertain information processing, and in particular to a multi-attribute group decision-making method based on Z-number. Background Art

[0002] With the vigorous development of the information age, the information that people can obtain is growing at an exponential rate, and all this information has a common characteristic - uncertainty. Therefore, identifying and studying uncertain information will play an indispensable role in human understanding and solving many problems in human society. In decision-making problems, the decision makers are often excellent experts or experienced operators in the field. When evaluating attributes, decision makers are often highly subjective. Decision makers have their own judgments on each evaluation criterion and each alternative plan based on their own preferences. This judgment may be due to the limitations of their own abilities or heavy reliance on past experience, which easily leads to uncertainty in the decision-making process.

[0003] To this end, existing technologies mainly use possibility theory, probability theory, evidence theory, fuzzy set theory and other methods to reduce the uncertainty in group decision-making. However, the above theories all express uncertain information in a qualitative or quantitative way, and do not give a complete description of uncertain information, resulting in low accuracy and poor stability of decision results. Summary of the invention

[0004] The purpose of the present invention is to overcome the defects of the above-mentioned prior art and provide a multi-attribute group decision-making method based on Z-number, which can improve the accuracy and stability of decision results.

[0005] The object of the present invention can be achieved by the following technical solution: A multi-attribute group decision-making method based on Z-number, comprising the following steps:

[0006] Step 1: Determine the decision criteria;

[0007] Step 2: Determine the weight of the attribute;

[0008] Step 3: Semantic variable table of proposed attributes;

[0009] Step 4: Determine the evaluation of attributes in group decision making;

[0010] Step 5: Convert the evaluation variables into numerical quantities;

[0011] Step 6: Construct the A part of Z-number;

[0012] Step 7: Use information entropy to construct the B part of Z-number;

[0013] Step 8: Reconstruct the Z-number based on the impact of part B on the attributes;

[0014] Step 9: Combine the Z-numbers of different attributes to obtain the Z-numbers of different options;

[0015] Step 10: Sort the options and reach a conclusion.

[0016] Furthermore, the specific process of step 1 is as follows:

[0017] Analyze specific group decision-making problems, obtain and classify the important elements in the decision-making problem: alternative options S k (k=1,2,…,t), attribute C i (i=1,2,…,m) and expert E j (j=1,2,…,n), and at the same time construct a logical framework for multi-attribute decision making.

[0018] Furthermore, the step 2 specifically includes the following steps:

[0019] (2.1) Determine the importance weights of different experts for their respective attributes for alternative options. Assume that the importance weights are represented by a trapezoidal fuzzy number It represents the weight value of the j-th expert for the i-th attribute, and the weight value is defuzzified:

[0020]

[0021] (2.2) The weight value corresponding to the i-th attribute can be obtained by the following calculation:

[0022]

[0023] (2.3) Since the above weight values ​​do not meet the normalization requirement, the obtained weight values ​​are further normalized:

[0024]

[0025] Furthermore, the step 3 specifically constructs a semantic variable table of attributes based on historical data or collective opinions of experts for reference in subsequent expert evaluation.

[0026] Furthermore, step 4 is specifically based on the alternative option S determined in step 1. k , Attribute C i With expert E j And refer to the semantic variable table constructed in step 3 to obtain the evaluation variable matrix D = {D ij}(i=1,2,…,m,j=1,2,…,n).

[0027] Furthermore, the step 5 specifically includes the following steps:

[0028] (5.1) According to the evaluation variable matrix obtained in step 4, the semantic variables and fuzzy numbers are matched with each other;

[0029] (5.2) The fuzzy numbers are defuzzified to obtain an evaluation matrix consisting of exact numbers as the original data sample.

[0030] Furthermore, the specific process of step 6 is as follows:

[0031] According to the data clustering algorithm, the original data obtained in step 5 is calculated as follows to generate the A part of Z-number:

[0032]

[0033] Among them, for any j, To evaluate the variable D ij The minimum value of each row of elements in , is the row average, The maximum value of the row.

[0034] Furthermore, the step 7 specifically uses information entropy to characterize the consistency of evaluation values ​​of the same attribute by different experts, including the following steps:

[0035] (7.1) Calculate the probability of the evaluation value occurring:

[0036]

[0037] (7.2) Calculate the information entropy of a single attribute:

[0038]

[0039] (7.3) Normalize the information entropy and obtain the B part of Z-number:

[0040]

[0041] Furthermore, step 8 is specifically based on the influence of part B on the existence of the attribute, combined with the mathematical expectation theory, to reconstruct part A in the Z-number, and the calculation process is as follows:

[0042]

[0043]

[0044]

[0045] Among them, α is the mathematical expectation of part B, which is derived from the defuzzification step in step (2.1), A' is the reconstructed part A, B' is the reconstructed part B, λ is the adjustment coefficient, l is the number of fuzzy semantics, is the fuzzy number center value of the pth fuzzy semantics.

[0046] Furthermore, step 9 specifically combines the Z-number of each attribute after reconstruction obtained in step 8 with the attribute weight obtained in step 2, and fuses the two in a weighted combination form to obtain the Z-number expression of each backup option:

[0047]

[0048] Among them, Z" is the Z-number after fusion, Z i ' is the Z-number of the i-th attribute after reconstruction.

[0049] Furthermore, step 10 specifically uses part A of the Z-number as a sorting factor to sort the Z-number, and part B is used to assist in judging the effectiveness of the sorting. After that, the decision maker selects the optimal option that best meets the requirements of the group decision problem based on the sorting vector.

[0050] Compared with the prior art, the present invention uses Z-number theory to directly generate part A and part B in Z-number from the original data set, and proposes a new Z-number generation method. By considering the influence of part B on the attribute, the Z-number is reconstructed, and the central value of the unknown part is considered. Part A in the new Z-number is generated by weighted fusion to solve the conflict problem in the traditional fusion process and obtain a fusion result that is more in line with the situation, so that the fusion result in the final group decision is more reasonable and more stable. The fusion results are sorted and the best backup options are decided, which effectively improves the accuracy and stability of the decision results.

[0051] The present invention utilizes information entropy to describe the characteristics of the discrete degree of data well, and generates part B in Z-number. Part B generated according to information entropy can fully express the collective opinions of multiple experts on the evaluation of a certain attribute, thereby increasing the dimension of describing uncertain information - reliability, thereby ensuring that the uncertain information can be fully described. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is a schematic diagram of the method flow of the present invention;

[0053] Figure 2 A structural diagram of the edible oil supplier selected in the embodiment;

[0054] Figure 3 It is a comparative schematic diagram of the A part in the Z-number after fusion in the embodiment. DETAILED DESCRIPTION

[0055] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0056] Example

[0057] like Figure 1 As shown, a multi-attribute group decision-making method based on Z-number includes the following steps:

[0058] Step 1: Determine the decision criteria;

[0059] Step 2: Determine the weight of the attribute;

[0060] Step 3: Semantic variable table of proposed attributes;

[0061] Step 4: Determine the evaluation of attributes in group decision making;

[0062] Step 5: Convert the evaluation variables into numerical quantities;

[0063] Step 6: Construct the A part of Z-number;

[0064] Step 7: Use information entropy to construct the B part of Z-number;

[0065] Step 8: Reconstruct Z-number considering the impact of part B on the attributes;

[0066] Step 9: Merge the Z-numbers of different attributes;

[0067] Step 10: Sort the options and reach a conclusion.

[0068] Wherein, step 1 specifically includes the following steps:

[0069] Analyze specific group decision-making problems, obtain and classify the important elements in the decision-making problem: alternative options S k (k=1,2,…,t), attribute C i (i=1,2,…,m) and expert E j (j=1,2,…,n), and at the same time construct a logical framework for multi-attribute decision making.

[0070] Step 2 specifically includes the following steps:

[0071] (2.1) Determine the importance weights of different experts for their respective attributes for alternative options. Assume that the importance weights are represented by a trapezoidal fuzzy number It represents the weight value of the j-th expert for the i-th attribute, and the weight value is defuzzified:

[0072]

[0073] (2.2) The weight value corresponding to the i-th attribute can be obtained by the following calculation:

[0074]

[0075] (2.3) Since the above weight values ​​do not meet the normalization requirement, the obtained weight values ​​are normalized:

[0076]

[0077] Step 3 specifically includes the following steps:

[0078] Based on historical data or collective opinions of experts, a semantic variable table of attributes is constructed for reference in subsequent expert evaluation.

[0079] Step 4 specifically includes the following steps:

[0080] Based on the alternative S determined in step 1 k , Attribute C i With expert E j And referring to the semantic variable table constructed in step 3, further construct the evaluation variable matrix D = {D ij}(i=1,2,…,m,j=1,2,…,n).

[0081] Step 5 specifically includes the following steps:

[0082] The evaluation values ​​under the multi-expert system obtained in step 4 are used to correspond the semantic variables to the fuzzy numbers, and then the fuzzy numbers are defuzzified, and finally an evaluation matrix composed of exact numbers is obtained as the original data sample.

[0083] Step 6 specifically includes the following steps:

[0084] According to the data clustering algorithm, the original data obtained in step 5 is calculated as follows to generate the A part of Z-number:

[0085]

[0086] Among them, for any j, is the evaluation variable D ij The minimum value of each row of elements in , similarly, is its row average, is the row maximum value.

[0087] Step 7 specifically includes the following steps:

[0088] According to the evaluation variables of different attributes by multiple experts in multi-attribute group decision-making, information entropy is used to describe the characteristics of data dispersion. This technical solution uses information entropy to characterize the consistency of different experts' evaluation values ​​of the same attribute, so that part B generated according to information entropy can express the collective opinions of multiple experts on the evaluation of a certain attribute. The specific steps are as follows:

[0089] (7.1) Calculate the probability of the evaluation value occurring:

[0090]

[0091] (7.2) Calculate the information entropy of a single attribute:

[0092]

[0093] (7.3) Normalize the information entropy to obtain the B part of Z-number:

[0094]

[0095] Step 8 specifically includes the following steps:

[0096] Considering the influence of part B on the attributes, and based on the theory of mathematical expectation, part A in Z-number is reconstructed. The calculation process is as follows:

[0097]

[0098] Among them, α is the mathematical expectation of part B, which can be derived by the defuzzification step in step (2.1). λ is the adjustment coefficient, which can be the central value of the fuzzy number. The derivation process is as follows:

[0099]

[0100] l is the number of fuzzy semantics, is the fuzzy number center value of the pth fuzzy semantics, expressed as:

[0101]

[0102] Step 9 specifically includes the following steps:

[0103] According to the reconstructed Z-number of each attribute obtained in step 8, combined with the attribute weight obtained in step 2, the two are fused in the form of weighted combination to obtain the Z-number expression of each alternative option. The specific calculation process is as follows:

[0104]

[0105] Step 10 specifically includes the following steps:

[0106] According to the above steps, the Z-number values ​​of the alternative options composed of different attribute combinations under different expert evaluations are obtained. The Z-number is sorted based on the A part of the Z-number as the sorting factor, and the B part assists in judging the effectiveness of the sorting. At this point, the decision maker can refer to the sorting vector to select the optimal option that best meets the requirements of the group decision problem.

[0107] This embodiment applies the above technical solution to select the best supplier from multiple oil suppliers. In this embodiment, the supplier problem involves a case company that needs to purchase a series of edible oil products in the food industry, namely olive oil, palm oil, sunflower seed oil, and soybean oil. Among the companies participating in the competition, there are 4 olive oil suppliers, 3 palm oil suppliers, and 3 suppliers that provide both sunflower seed oil and soybean oil. Since the purchasing company adheres to the concept of green environmental protection and based on the data in procurement and strategic planning, four evaluation indicators were designed in the evaluation process: service level, quality, price and green standards, and a total of three experts participated in the evaluation process.

[0108] The specific process is as follows:

[0109] (1) Determine the decision-making criteria.

[0110] Analyze the group decision-making problem in the embodiment, and obtain three alternative options: olive oil supplier set {O1, O2, O3, O4}, palm oil set {P1, P2, P3}, sunflower oil and soybean oil supplier set {SS1, SS2, SS3}; attribute C i (i=1,2,3,4) respectively represent four evaluation indicators, E j (j=1,2,3) represents 3 experts and establishes Figure 2 Diagram showing the structure of the edible oil supplier selection.

[0111] (2) Determine the weight of the attribute.

[0112] After the experts’ multi-dimensional evaluation of the attributes, the importance weights of the four attributes by three experts are shown in Table 1.

[0113] Table 1

[0114]

[0115] Firstly, the trapezoidal fuzzy numbers are defuzzified to obtain clear numbers, and the weight values ​​of different experts under the same attribute conditions are accumulated. After normalization, the weight values ​​of the attributes are calculated to be 0.2971, 0.2794, 0.2471, and 0.1764, respectively.

[0116] (3) Semantic variable table of suggested attributes.

[0117] According to the actual situation, a semantic variable table of the attribute is suggested for expert decision making. This embodiment adopts the following semantic variable table.

[0118] Table 2

[0119] importance symbol Fuzzy Number Special Very Poor(VP) (0.0,0.0,0.1,0.2) Difference Poor(P) (0.1,0.2,0.2,0.3) Slightly worse Medium Poor(MP) (0.2,0.3,0.4,0.5) neutral Fair(F) (0.4,0.5,0.5,0.6) Slightly better Medium Good(MD) (0.5,0.6,0.7,0.8) good Good(G) (0.7,0.8,0.8,0.9) Very good Very Good(VG) (0.8,0.9,1.0,1.0)

[0120] (4) Determine the evaluation of attributes in group decision making.

[0121] Three experienced experts selected three different edible oil suppliers in the embodiment and constructed evaluation matrices as shown in Tables 3-1 to 3-3 with reference to the semantic variable table in step 3.

[0122] Table 3-1

[0123]

[0124] Table 3-2

[0125]

[0126] Table 3-3

[0127]

[0128] (5) Convert the evaluation variables into numerical quantities.

[0129] Use the evaluation values ​​obtained under the multi-expert system in step 4. Match the semantic variables with the fuzzy numbers, defuzzify the fuzzy numbers, and finally obtain the evaluation matrix composed of exact numbers as shown in Table 4-1 to Table 4-3.

[0130] Table 4-1

[0131]

[0132] Table 4-2

[0133]

[0134] Table 4-3

[0135]

[0136] (6) Construct the A part of Z-number.

[0137] Using the data clustering algorithm, the original data obtained in step 5 is calculated to generate the A part of the Z-number.

[0138] (7) Use information entropy to construct part B in Z-number.

[0139] Using the information entropy algorithm, the original data in Table 4-1 to Table 4-3 are calculated to generate the B part of the Z-number, and finally the Z-number of different attributes in each alternative option is generated as shown in Table 5.

[0140] Table 5

[0141]

[0142] (8) Reconstruct the Z-number considering the impact of part B on the attributes.

[0143] Considering the influence of part B on the attributes, and based on the theory of mathematical expectation, part A in Z-number is reconstructed to obtain Z-number as shown in Table 6.

[0144] Table 6

[0145]

[0146] (9) Fusion of Z-numbers with different attributes.

[0147] According to the reconstructed Z-number of each attribute obtained in step 8, combined with the attribute weight obtained in step 2, the two are fused in the form of weighted combination to obtain the Z-number of each backup option as shown in Table 7.

[0148] Table 7

[0149]

[0150] (10) Sort the options and make decisions to draw conclusions.

[0151] According to the above steps, the Z-number values ​​of the backup options composed of different attribute combinations under the final evaluation of different experts are obtained, and the Z-number is sorted to obtain the sorting vector, and the following is obtained: Figure 3 The comparison of part A in the fused Z-number is shown in Figure 8. At this point, the decision maker can refer to the sorting vector and select the optimal option that best meets the requirements of the group decision problem as shown in Table 8.

[0152] Table 8

[0153] Cooking Oil Best Supplier olive oil <![CDATA[O3]]> Palm Oil Suppliers <![CDATA[P1]]> Sunflower Oil and Soybean Oil Suppliers <![CDATA[SS1]]>

[0154] In summary, this technical solution innovatively solves the multi-attribute group decision-making problem based on the Z-number theory. Z-number is a new mathematical model proposed by Zadeh in 2011. Z-number includes possibility theory, probability theory, evidence theory, and fuzzy set theory, and adds a new dimension to describe uncertainty, namely reliability. Using Z-number theory to process, model, fuse, and analyze raw data can provide a new development direction for mathematical models that describe uncertain information. Based on this, in order to solve the uncertainty problem in multi-attribute decision-making, this technical solution is based on Z-number theory. Compared with previous mathematical models for uncertain information, Z-number is more complete in expressing uncertainty. This technical solution also reconstructs Z-number during calculation, thereby solving the conflict problem of fusion results in the Z-number fusion process, thereby effectively improving the accuracy and stability of decision results.

Claims

1. A multi-attribute group decision-making method based on Z-number, characterized in that: The following steps are involved: Step 1: Determine the decision criteria; Step 2: Determine the weight of the attribute; Step 3: Semantic variable table of proposed attributes; Step 4: Determine the evaluation of attributes in group decision making; Step 5: Convert the evaluation variables into numerical quantities; Step 6: Construct the A part of Z-number; Step 7: Use information entropy to construct the B part of Z-number; Step 8: Reconstruct the Z-number based on the impact of part B on the attributes; Step 9: Combine the Z-numbers of different attributes to obtain the Z-numbers of different options; Step 10: Sort options and make decisions to draw conclusions; Among them, the three alternative options obtained in step 1 are: olive oil supplier set {O1, O2, O3, O4}, palm oil set {P1, P2, P3}, sunflower oil and soybean oil supplier set {SS1, SS2, SS3}; attribute C i (i=1,2,3,4) respectively represent four evaluation indicators, E j (j=1,2,3) represents 3 experts; In step 2, after the experts’ multi-dimensional evaluation of the attributes, the importance weights of the four attributes for the three experts were obtained; In step 3, a semantic variable table of attributes is suggested based on the actual situation for expert decision making; In step 4, three experts selected three different edible oil suppliers and constructed an evaluation matrix based on the semantic variable table in step 3; In step 5, the evaluation values ​​obtained in the multi-expert system in step 4 are used to correspond the semantic variables to the fuzzy numbers, and then the fuzzy numbers are defuzzified to finally obtain an evaluation matrix composed of exact numbers.

2. The multi-attribute group decision-making method based on Z-number according to claim 1, characterized in that: The specific process of step 1 is as follows: Analyze specific group decision-making problems, obtain and classify the important elements in the decision-making problem: alternative options S k (k=1,2,…,t), attribute C i (i=1,2,…,m) and expert E j (j=1,2,…,n), and at the same time construct a logical framework for multi-attribute decision making.

3. The multi-attribute group decision-making method based on Z-number according to claim 2 is characterized in that: The step 2 specifically includes the following steps: (2.1) Determine the importance weights of different experts for their respective attributes for alternative options. Assume that the importance weights are composed of a trapezoidal fuzzy number It represents the weight value of the j-th expert for the i-th attribute, and the weight value is defuzzified: (2.2) The weight value corresponding to the i-th attribute can be obtained by the following calculation: (2.3) Since the above weight values ​​do not meet the normalization requirement, the obtained weight values ​​are further normalized:

4. The multi-attribute group decision-making method based on Z-number according to claim 1, characterized in that: The step 3 specifically constructs a semantic variable table of attributes based on historical data or collective opinions of experts for reference in subsequent expert evaluation.

5. The multi-attribute group decision-making method based on Z-number according to claim 3 is characterized in that: The step 4 is specifically based on the alternative option S determined in step 1. k , Attribute C i With expert E j And refer to the semantic variable table constructed in step 3 to obtain the evaluation variable matrix D = {D ij }(i=1,2,…,m,j=1,2,…,n).

6. The multi-attribute group decision-making method based on Z-number according to claim 5, characterized in that: The step 5 specifically comprises the following steps: (5.1) According to the evaluation variable matrix obtained in step 4, the semantic variables and fuzzy numbers are matched with each other; (5.2) The fuzzy numbers are defuzzified to obtain an evaluation matrix consisting of exact numbers as the original data sample.

7. The multi-attribute group decision-making method based on Z-number according to claim 6, characterized in that: The specific process of step 6 is as follows: According to the data clustering algorithm, the original data obtained in step 5 is calculated as follows to generate the A part of Z-number: Among them, for any j, To evaluate the variable D ij The minimum value of each row of elements in , is the row average, The maximum value of the row.

8. The multi-attribute group decision-making method based on Z-number according to claim 7, characterized in that: The step 7 specifically uses information entropy to characterize the consistency of evaluation values ​​of the same attribute by different experts, including the following steps: (7.1) Calculate the probability of the evaluation value occurring: (7.2) Calculate the information entropy of a single attribute: (7.3) Normalize the information entropy and obtain the B part of Z-number:

9. The multi-attribute group decision-making method based on Z-number according to claim 8, characterized in that: The step 8 is specifically based on the influence of part B on the existence of the attribute, combined with the mathematical expectation theory, to reconstruct part A in the Z-number, and the calculation process is as follows: Among them, α is the mathematical expectation of part B, which is derived from the defuzzification step in step (2.1), A' is the reconstructed part A, B' is the reconstructed part B, λ is the adjustment coefficient, l is the number of fuzzy semantics, is the fuzzy number center value of the pth fuzzy semantics; Specifically, step 9 is to combine the Z-number of each attribute after reconstruction obtained in step 8 with the attribute weight obtained in step 2, and fuse the two in a weighted combination form to obtain the Z-number expression of each backup option: Among them, Z" is the Z-number after fusion, Z i ' is the Z-number of the i-th attribute after reconstruction.

10. The multi-attribute group decision-making method based on Z-number according to claim 1, characterized in that: The step 10 specifically uses part A of the Z-number as a sorting factor to sort the Z-number, and part B is used to assist in judging the effectiveness of the sorting. Afterwards, the decision maker selects the optimal option that best meets the requirements of the group decision problem based on the sorting vector.

Images