Fuzzy complementary judgment matrix: a powerful tool for multi-attribute decision-making and Matlab practice
Introduction
In the real world, we often need to make multi-attribute decisions, and these decisions usually involve various uncertainties and ambiguities. To address these challenges, multi-attribute decision-making methods emerged. This article will conduct an in-depth study of a multi-attribute decision-making method, that is, a method based on fuzzy complementary judgment matrix, and how to apply this method in Matlab to solve complex decision-making problems.
What is multi-attribute decision-making?
Multi-attribute decision-making refers to the problem of selecting the optimal solution or decision when faced with multiple evaluation factors or attributes. These evaluation factors often involve different weights and priorities, and their relevance may also differ. The multi-attribute decision-making method aims to comprehensively consider these factors to help decision-makers make informed decisions.
Fuzzy complementary judgment matrix
Fuzzy Complementary Judgment Matrix (FCJM) is a multi-attribute decision-making method used to deal with fuzzy and uncertain information. It is based on complementary relationships and allows decision makers to make decisions under conditions of uncertainty and ambiguity.
Construct fuzzy complementary judgment matrix
The key steps to construct a fuzzy complementary judgment matrix include:
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Identify decision factors and attributes: First, the factors and attributes involved in the decision problem need to be clearly defined, as well as the relationships between them.
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Develop complementary relationships: Next, decision makers need to develop complementary relationships, that is, the relative importance and complementarity of each factor or attribute to other factors or attributes. These relationships are often expressed as fuzzy numbers, reflecting uncertainty.
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Constructing a fuzzy complementary judgment matrix: Using the formulated complementary relationship, a fuzzy complementary judgment matrix can be constructed. This matrix reflects the relative weight and complementarity between factors or attributes.
Mathematical representation of fuzzy complementary judgment matrix
Suppose we have
n
n
n factors or attributes, fuzzy complementary judgment matrix
C
C
The mathematical representation of C is as follows:
C
=
[
c
11
c
12
…
c
1
n
c
twenty one
c
twenty two
…
c
2
n
?
?
?
?
c
n
1
c
n
2
…
c
n
n
]
C = \begin{bmatrix} c_{11} & amp; c_{12} & amp; \ldots & amp; c_{1n} \ c_{21} & amp; c_{22} & amp; \ldots & amp; c_{2n} \ \vdots & amp; \vdots & amp; \ddots & amp; \vdots \ c_{n1} & amp; c_{n2} & amp; \ldots & amp; c_{nn} \ \end{bmatrix}
C=
in,
c
i
j
c_{ij}
cij? indicates factors
i
i
i relative to factors
j
j
j complementarity. These values are usually fuzzy numbers indicating the degree of complementarity.
Application of fuzzy complementary judgment matrix
Fuzzy complementary judgment matrices are often used in multi-attribute decision-making problems to help decision-makers determine the weight and relative importance of each factor or attribute. By analyzing and processing the fuzzy complementary judgment matrix, the final evaluation result can be obtained to support the decision-making process.
Application of fuzzy complementary judgment matrix in Matlab
Below, we will demonstrate how to apply fuzzy complementary judgment matrix in Matlab to solve a multi-attribute decision-making problem. Suppose we need to choose the best model of a new mobile phone, taking into account three evaluation factors: performance, price and appearance. We will use a fuzzy complementary judgment matrix to determine the relative weight of these factors.
Construct fuzzy complementary judgment matrix
First, we need to construct a fuzzy complementary judgment matrix, which contains the complementary relationships between performance, price and appearance. Suppose our complementary relationship is as follows:
- Complementarity of performance to price is moderate.
- Complementarity of performance to appearance is high.
- Price-to-performance complementarity is low.
- Complementarity of price to appearance is moderate.
- The complementarity of appearance to performance is high.
- Complementarity of appearance to price is moderate.
We can represent these relationships with fuzzy numbers and construct a fuzzy complementary judgment matrix.
% Construct fuzzy complementary judgment matrix
C = [0.5, 0.7, 0.5;
0.3, 0.5, 0.7;
0.7, 0.5, 0.5];
Calculate weight vector
Next, we will calculate the weight vector for each factor. By processing the fuzzy complementary judgment matrix, we can obtain the weight vector.
% Calculate weight vector n = size(C, 1); w = sum(C) / n;
The obtained weight vector w represents performance and price
and the relative weight of appearance.
Multi-attribute decision-making based on weight
Now, we can use the calculated weights to make multi-attribute decisions. Suppose we have rated each phone model on its performance, price, and appearance and received a corresponding score. We can multiply these scores with the weight vector and then take the weighted sum to determine the final review score.
% Assume that the performance, price, and appearance of the three mobile phones are perf_scores, price_scores, and appearance_scores respectively. perf_scores = [8, 7, 9]; price_scores = [6, 5, 7]; appearance_scores = [7, 8, 9]; % Calculate the weighted sum decision_scores = w(1) * perf_scores + w(2) * price_scores + w(3) * appearance_scores;
By comparing decision_scores of different phone models, we can choose the best phone model as the final decision.
Conclusion
Fuzzy complementary judgment matrix is a powerful multi-attribute decision-making method suitable for dealing with uncertainty and fuzzy information. This article introduces the principle and construction method of fuzzy complementary judgment matrix, and demonstrates how to apply this method in Matlab to solve multi-attribute decision-making problems. By constructing fuzzy complementary judgment matrices, calculating weight vectors, and making weight-based multi-attribute decisions, decision makers can better understand and deal with complex decision-making problems. If you need to face decision-making problems with multiple uncertainties and fuzziness factors, the fuzzy complementary judgment matrix method may be a useful tool worthy of in-depth study and application.