Fuzzy Comprehensive Evaluation

Fuzzy Comprehensive Evaluation is a decision-making method that integrates fuzzy mathematics with comprehensive evaluation, often used for multi-criteria assessments. It is located in SPSSAU -> Comprehensive Evaluation -> Fuzzy Comprehensive Evaluation.

SPSSAU Operations

After dragging the analysis items to the right panel, click "Start". If evaluation index weights are available, they should be moved to the corresponding panel. The fuzzy operator parameters include four types as below:

No.	Fuzzy Operator Type	Description
1	Weighted Average: M(., +)	Utilizes both A matrix and R matrix information; recommended.
2	Dominant Factor Emphasis: M(Λ, V)	Uses minimal A matrix and R matrix information; not recommended.
3	Dominant Factor Emphasis: M(., V)	Uses minimal A matrix and R matrix information; not recommended.
4	Weighted Average：M(Λ, +)	Uses more R matrix information; recommended.

SPSSAU Data Format

One column should contain one evaluation item (e.g., "Dissatisfied," "Somewhat Dissatisfied," "Satisfied," "Very Satisfied"), represented as either selection proportions or counts.

If index items have weights, a separate column should be used to specify "Index Weight Values." This is optional; if no weights are provided, SPSSAU will assume equal weighting for all indices.

The index column does not need to be included in the analysis panel; it is only for researchers' reference.

The numbers in each evaluation item column represent selection percentages. For example, for Index Item 1, if the selection proportion for Evaluation Item 1 is 0.2 (20%), and for Evaluation Item 2, it is 0.5 (50%), the system will automatically normalize the values so that they sum to 1 within each index item.

Algorithm

1.Establish the Evaluation Index Set

Determine the evaluation indicators to form an index set: U = {u₁,u₂,...,u_n}.

2.Establish the Evaluation Set

Define possible evaluation levels to form an evaluation set V = {v₁,v₂,...,v_m} (e.g., "Dissatisfied/Satisfied").

3.Determine Weights

If weights are available, provide A = {a₁,a₂,...,a_n} and ensure they sum to 1. If the sum does not equal 1, SPSSAU will automatically normalize them. If no weight column is provided, the weight of all indicators is 1/n.

4.Construct the Fuzzy Evaluation Matrix

Using expert ratings or other methods, construct the fuzzy evaluation matrix R:

$$R=\left[\begin{array}{cccc}{r}_{11}& r_{12}& \cdots & r_{1m}\\ r_{21}& r_{22}& \cdots & r_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ r_{n1}& r_{n2}& \cdots & r_{nm}\end{array}\right]$$

r_ij represents the membership degree of index i under evaluation j.

5.Fuzzy Comprehensive Evaluation Results

The fuzzy comprehensive evaluation result is calculated as: Y = A · R, where A is the weight vector, and R is the fuzzy evaluation matrix. SPSSAU provides four types of fuzzy operators, with the default being M(·,+). The formulas for the four fuzzy operators are as follows:

$$M(\cdot ,+):Y_k=min(1,\sum _{j=1}^m{a}_j{r}_{jk}),k=1,2,\cdots ,n$$ $$M(\wedge ,\vee ):Y_k=\underset{1\le j\le m}{max}\{min(a_j,r_{jk})\},k=1,2,\cdots ,n$$ $$M(\cdot ,\vee ):Y_k=\underset{1\le j\le m}{max}\{a_j\cdot r_{jk}\},k=1,2,\cdots ,n$$ $$M(\wedge ,+):Y_k=min\{1,\sum _{j=1}^{m}min(a_j\cdot r_{jk})\},k=1,2,\cdots ,n$$

6.Membership Degree Normalization

The membership degrees obtained in the previous step will be normalized. SPSSAU also provides defuzzification results using the maximum membership degree principle to obtain the final evaluation result.

7.Composite Score Calculation

The composite score is obtained by multiplying the membership degrees with their corresponding evaluation scores and summing the results:

$$\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}.\mathrm{S}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}=\sum _{j=1}^m(b_j\times v_j)$$

b_j is the membership degree of evaluation j (default uses normalized values).

v_j is the score corresponding to evaluation j (e.g., "Very Welcoming" = 4, "Welcoming" = 3, "Neutral" = 2, "Not Welcoming" = 1).

For example, if an evaluation’s membership degrees are: Very Welcoming: 0.205, Welcoming: 0.320, Neutral: 0.390, Not Welcoming: 0.085 and the corresponding scores are 4, 3, 2, 1, then the composite score is calculated as: Comp.Score = (0.205 × 4) + (0.320 × 3) + (0.390 × 2) + (0.085 × 1) = 2.645. The final composite score provides an overall description of the evaluation and facilitates comparisons between different evaluation systems.