Mastering Weight Generation and Multi‑Criteria Evaluation Methods for Modeling Competitions

In mathematical modeling contests, weight analysis and multi‑criteria evaluation are common topics. This article systematically reviews common weight‑generation and evaluation methods, helping participants handle such problems effectively.

1. Analytic Hierarchy Process (AHP)

1.1 Method Overview

AHP is a widely used tool for multi‑level, multi‑factor decision problems. It decomposes a complex decision into hierarchical levels and determines weights by pairwise comparisons of factors.

1.2 Mathematical Principle

AHP builds a hierarchical model with goal, criteria, and alternative layers. Pairwise comparison matrices express relative importance; the maximum eigenvalue and its eigenvector provide the weight vector. The calculation steps are:

Compute the maximum eigenvalue of the matrix;

Obtain the eigenvector;

Normalize the eigenvector to get final weights.

1.3 Advantages and Disadvantages

Advantages:

Simple and intuitive, easy to understand;

Suitable when decision makers have experience but cannot provide precise numbers.

Disadvantages:

Subject to personal bias;

For large problems the pairwise matrix grows quickly, increasing computational complexity.

AHP helps break down problems layer by layer, using simple pairwise comparisons to reveal true priorities.

2. Entropy Weight Method

2.1 Method Overview

The entropy weight method determines indicator weights based on information entropy, assigning larger weights to indicators with greater uncertainty.

2.2 Mathematical Principle

For each indicator, its entropy value is calculated from normalized data; then weights are derived from the entropy values.

2.3 Advantages and Disadvantages

Advantages:

Data‑driven, avoids subjective interference;

Suitable for abundant and relatively independent data.

Disadvantages:

Unstable with small sample sizes;

Sensitive to noise, requiring preprocessing.

The entropy method tells us that the more uncertain an indicator, the larger its weight, because it carries richer information.

3. TOPSIS

3.1 Method Overview

TOPSIS ranks alternatives by their distances to an ideal solution and a negative ideal solution, selecting the alternative closest to the ideal and farthest from the negative ideal.

3.2 Mathematical Principle

Normalize the decision matrix;

Compute the weighted normalized matrix;

Determine the positive and negative ideal solutions;

Calculate the distances of each alternative to both ideals;

Compute the relative closeness coefficient and rank the alternatives.

3.3 Advantages and Disadvantages

Advantages:

Provides a comprehensive evaluation result;

Applicable to multiple objectives.

Disadvantages:

Requires predefined ideal and negative‑ideal solutions;

Sensitive to outliers.

TOPSIS acts like a “quality detector,” always finding the solution nearest to the ideal.

4. Fuzzy Comprehensive Evaluation

4.1 Method Overview

The fuzzy comprehensive evaluation uses membership functions from fuzzy mathematics to assess systems with uncertainty or vagueness.

4.2 Mathematical Principle

Indicators are selected, fuzzy membership functions are constructed, and a weighted average yields the final evaluation.

4.3 Advantages and Disadvantages

Advantages:

Effectively handles uncertainty and fuzziness;

Suitable for highly subjective evaluation problems.

Disadvantages:

Results depend heavily on the choice of membership functions;

Requires extensive expert judgment.

Amid fuzziness, the method finds “clarity,” providing a soft landing for decision making.

5. CRITIC

5.1 Method Overview

CRITIC generates weights based on information content and inter‑criterion correlation, using each indicator’s standard deviation and correlation matrix.

5.2 Mathematical Principle

Calculate the standard deviation of each indicator;

Compute the correlation coefficient matrix;

Derive the information amount for each indicator;

Determine weights from the information amounts.

5.3 Advantages and Disadvantages

Advantages:

Considers both correlation and information content;

Suitable for problems with many conflicting criteria.

Disadvantages:

Requires high‑quality data;

Complex calculations, best for smaller datasets.

CRITIC reveals the most representative indicators by examining their information and correlation.

6. Principal Component Analysis (PCA)

6.1 Method Overview

PCA reduces dimensionality by extracting principal components that capture the most variance, providing objective weight vectors for evaluation.

6.2 Mathematical Principle

Standardize the data;

Compute the covariance matrix;

Perform eigenvalue decomposition to obtain eigenvalues and eigenvectors;

Select principal components based on eigenvalues;

Project the original data onto the selected components to obtain scores.

6.3 Advantages and Disadvantages

Advantages:

Reduces dimensionality while retaining important information;

Objectively determines indicator importance via eigenvalues;

Effectively integrates correlated indicators.

Disadvantages:

Sensitive to noise or outliers in the data;

Interpretation of components may be difficult;

Requires a sufficiently large sample size for stability.

PCA not only cuts dimensions but also preserves the most valuable information, making complex data concise and profound.

7. Factor Analysis

7.1 Method Overview

Factor analysis transforms many correlated variables into a few latent factors, revealing underlying structures and simplifying models.

7.2 Mathematical Principle

The model expresses observed variables as a linear combination of factor loadings and latent factors plus error terms.

7.3 Advantages and Disadvantages

Advantages:

Uncovers hidden structures among variables;

Reduces redundancy and simplifies problems.

Disadvantages:

Strict model assumptions;

Requires solid mathematical background.

Factor analysis extracts latent factors from complex variables, revealing the true underlying structure.

8. Data Envelopment Analysis (DEA)

8.1 Method Overview

DEA evaluates the relative efficiency of decision‑making units (DMUs) without preset weights, constructing an efficient frontier from input‑output data.

8.2 Mathematical Principle

DEA formulates a linear programming model (e.g., CCR and BCC models) to maximize efficiency scores subject to constraints that ensure all DMUs have efficiency ≤ 1 and that weights are non‑negative.

8.3 Advantages and Disadvantages

Advantages:

No need to pre‑assign weights; the model derives them from data;

Effective for comparing units with multiple inputs and outputs.

Disadvantages:

Computationally intensive for large datasets;

Highly dependent on data quality; errors can distort results.

DEA does not rely on a “standard” benchmark; it reveals the most efficient units through peer comparison.

Choosing the appropriate method and combining it with a suitable mathematical model enables optimal solutions for complex decision‑making and optimization problems.