# An Integrated Fuzzy Structured Methodology for Performance Evaluation of High Schools in a Group Decision-Making Problem

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Proposed Methodology

#### 3.1. Delphi Technique

#### 3.2. AHP Technique

#### 3.3. TOPSIS Technique

_{ij}). As before, we use fuzzy numbers to settle the ambiguity in experts’ judgments. A seven-point scale to score the options based on each criterion is shown in Table 2.

^{+}) and FNIS (d

^{−}) should be calculated. If F

_{1}and F

_{2}are two fuzzy triangular numbers, then the distance between these two numbers will be calculated with the following formula:

## 4. Case Study

_{1}), we have the following 14 sub-criteria:

- S
_{11}: The proportion of teachers with a bachelor’s degree or higher to the total number of teachers; - S
_{12}: The proportion of teachers with a field of study related to the subject; - S
_{13}: The ratio of teaching to all teachers; - S
_{14}: The average age of teachers; - S
_{15}: The average service history of teachers; - S
_{16}: The average teaching hours of teachers per week; - S
_{17}: The training courses completed by teachers; - S
_{18}: The manager’s degree; - S
_{19}: The manager’s field; - S
_{110}: The amount of service history of the manager in management or deputy positions; - S
_{111}: The training courses (specialized) completed by the managers and deputies; - S
_{112}: The degree of deputies; - S
_{113}: The field of study of deputies; - S
_{114}: The average service history of deputies.

_{2}), we have the following 9 sub-criteria:

- S
_{21}: The per capita student; - S
_{22}: The ratio of income from extracurricular and public assistance, etc., to the student; - S
_{23}: The ratio of incurred expenses to approved expenses; - S
_{24}: The ratio of costs incurred to motivate teachers to the total budget; - S
_{25}: The ratio of educational expenses to total costs; - S
_{26}: The ratio of breeding expenses to total expenses; - S
_{27}: Cost per student; - S
_{28}: The proportion of expenditures with the approved budget; - S
_{29}: The quality of positive documents of costs.

_{3}), we have the following 12 sub-criteria:

- S
_{31}: Per capita student space; - S
_{32}: The ratio of the number of students to the toilets; - S
_{33}: The ratio of breeding space to the total area; - S
_{34}: The sports space per capita; - S
_{35}: The ratio of the number of students to the classroom space; - S
_{36}: The ratio of printers and photocopying machines to the needs of the school; - S
_{37}: The ratio of the number of computers to students; - S
_{38}: Suitability of educational tools and materials to students’ needs; - S
_{39}: The degree of up-to-date educational equipment and materials; - S
_{310}: Quality of teaching materials and tools; - S
_{311}: Variety of educational materials and tools; - S
_{312}: Suitability of tables, benches, and chairs to the needs of students.

_{4}), we have the following 5 sub-criteria:

- S
_{41}: The ratio of available books to students; - S
_{42}: The ratio of the number of CDs, educational videos, and tapes to students; - S
_{43}: The ratio of books, journals, and teaching guides to teachers; - S
_{44}: The average of teachers who use up-to-date books and publications; - S
_{45}: The average number of students using the updated library.

_{5}), we have the following 25 sub-criteria:

- S
_{51}: Number of training programs held for teachers; - S
_{52}: The ratio of the number of encouraged teachers to the total number of teachers; - S
_{53}: The ratio of the number of encouraged students to the total number of students; - S
_{54}: The quality of setting annual school programs; - S
_{55}: How to implement annual programs; - S
_{56}: The quality of compiling quarterly reports and sending them to the regional management; - S
_{57}: The reopening of the school on time and the preparation of teachers and students; - S
_{58}: Formation of school councils on time; - S
_{59}: The quality of council meetings; - S
_{510}: Registering and maintaining records and minutes of council meetings; - S
_{5111}: How to implement council approvals; - S
_{512}: The quality of actions performed in national and religious celebrations; - S
_{513}: How to perform the morning ceremony; - S
_{514}: The quality of congregational prayers; - S
_{515}: How to use leisure time; - S
_{516}: Actions were taken to identify the strengths and weaknesses of teachers; - S
_{517}: The number of training programs held for teachers; - S
_{518}: The ability of the manager to evaluate the performance of teachers; - S
_{519}: How to inform broadcast programs and instructions; - S
_{520}: The level of familiarity of the manager with the description of the duties of the employees; - S
_{521}: The manager’s familiarity with the principles and skills of educational management; - S
_{522}: The extent of the manager’s familiarity with the principles and philosophy of education; - S
_{523}: The manager’s familiarity with the principles of psychology; - S
_{524}: The degree of the manager’s familiarity with the laws and regulations of education; - S
_{525}: The quality of transportation service for students.

_{6}), we have the following 3 sub-criteria:

- S
_{61}: The number of students examined in terms of health and treatment; - S
_{62}: The quality of Bogue food; - S
_{63}: Health quality of the school environment.

_{7}), we have the following 16 sub-criteria:

- S
_{71}: Average GPA of incoming students; - S
_{72}: The ratio of students to teachers; - S
_{73}: The ratio of students to classes; - S
_{74}: The ratio of students participating in camps and scientific trips to all students; - S
_{75}: The ratio of students participating in scientific, laboratory, Olympiads, artistic, and sports competitions to the total number of students; - S
_{76}: The ratio of students participating in extracurricular classes to total students; - S
_{77}: The number of exhibitions held of students’ scientific, cultural, and artistic activities; - S
_{78}: The rate of students who have completed their education within the official period; - S
_{79}: The average annual grade point average of students; - S
_{710}: The middle passing grade of each semester; - S
_{711}: Pass percentage of each semester; - S
_{712}: Annual acceptance rate; - S
_{713}: The amount of students’ participation in class management; - S
_{714}: The amount of student participation in group work; - S
_{715}: The level of student participation in decision making and planning; - S
_{716}: The condition of students’ appearance.

_{8}), we have the following 14 sub-criteria:

- S
_{81}: The amount of use of educational technology in the teaching process; - S
_{82}: Status of planning to improve educational quality; - S
_{83}: Analysis of the results of academic progress; - S
_{84}: Providing feedback on the results of academic achievement tests; - S
_{85}: The number of teachers using the plan; - S
_{86}: The amount of teachers’ use of educational tools and materials; - S
_{87}: The amount of teachers’ use of various teaching strategies; - S
_{88}: The extent to which teachers use multiple methods of evaluating academic progress; - S
_{89}: The level of familiarity of teachers with the goals and content of lessons; - S
_{810}: The level of collaboration and exchange of teachers’ experiences with each other; - S
_{811}: The level of teachers’ familiarity with educational goals, regulations, and guidelines; - S
_{812}: The level of teachers’ participation in decision making and planning; - S
_{813}: How to schedule teachers’ meetings with parents; - S
_{814}: The amount of teachers’ use of laboratories and workshops.

_{9}), we have the following 6 sub-criteria:

- S
_{91}: How to register students; - S
_{92}: The quality of student’s academic records; - S
_{93}: The quality of personnel and job files of employees; - S
_{94}: Quality office property; - S
_{95}: The quality of the examination book; - S
_{96}: The quality of the statistical office.

_{10}), we have the following 4 sub-criteria:

- S
_{101}: Cultural status of parents of students; - S
_{102}: Economic status of parents of students; - S
_{103}: Educational level of students’ parents; - S
_{104}: Parents’ satisfaction with the school.

_{10}) are shown in Table 4 and Table 5.

Criteria | ${\mathit{x}}_{\mathbf{max}}^{1}$ | ${\mathit{x}}_{\mathbf{max}}^{2}$ | ${\mathit{x}}_{\mathbf{max}}^{3}$ | De-Fuzzy | Normal W |
---|---|---|---|---|---|

S_{23} | 0.275 | 0.275 | 0.273 | 0.275 | 0.271 |

S_{25} | 0.492 | 0.492 | 0.487 | 0.492 | 0.484 |

S_{26} | 0.249 | 0.249 | 0.247 | 0.249 | 0.245 |

_{2}was found to be 0.049.

Criteria | ${\mathit{x}}_{\mathbf{max}}^{1}$ | ${\mathit{x}}_{\mathbf{max}}^{2}$ | ${\mathit{x}}_{\mathbf{max}}^{3}$ | De-Fuzzy | Normal W |
---|---|---|---|---|---|

S_{31} | 0.228 | 0.225 | 0.223 | 0.228 | 0.218 |

S_{32} | 0.357 | 0.354 | 0.351 | 0.357 | 0.342 |

S_{34} | 0.226 | 0.223 | 0.221 | 0.226 | 0.216 |

S_{36} | 0.183 | 0.181 | 0.178 | 0.183 | 0.175 |

S_{38} | 0.023 | 0.023 | 0.023 | 0.023 | 0.022 |

S_{310} | 0.028 | 0.028 | 0.028 | 0.028 | 0.027 |

_{3}was found to be 0.081.

Criteria | ${\mathit{x}}_{\mathbf{max}}^{1}$ | ${\mathit{x}}_{\mathbf{max}}^{2}$ | ${\mathit{x}}_{\mathbf{max}}^{3}$ | De-Fuzzy | Normal W |
---|---|---|---|---|---|

S_{41} | 0.499 | 0.496 | 0.493 | 0.499 | 0.487 |

S_{44} | 0.399 | 0.396 | 0.394 | 0.399 | 0.389 |

S_{45} | 0.127 | 0.127 | 0.126 | 0.127 | 0.124 |

_{4}was found to be 0.077.

Criteria | ${\mathit{x}}_{\mathbf{max}}^{1}$ | ${\mathit{x}}_{\mathbf{max}}^{2}$ | ${\mathit{x}}_{\mathbf{max}}^{3}$ | De-Fuzzy | Normal W |
---|---|---|---|---|---|

S_{51} | 0.102 | 0.101 | 0.100 | 0.102 | 0.097 |

S_{55} | 0.067 | 0.066 | 0.065 | 0.067 | 0.064 |

S_{56} | 0.089 | 0.088 | 0.087 | 0.089 | 0.085 |

S_{58} | 0.074 | 0.073 | 0.072 | 0.074 | 0.070 |

S_{59} | 0.086 | 0.085 | 0.084 | 0.086 | 0.082 |

S_{514} | 0.068 | 0.067 | 0.066 | 0.068 | 0.065 |

S_{515} | 0.101 | 0.100 | 0.099 | 0.101 | 0.097 |

S_{517} | 0.105 | 0.104 | 0.103 | 0.105 | 0.101 |

S_{518} | 0.096 | 0.095 | 0.094 | 0.096 | 0.091 |

S_{519} | 0.097 | 0.096 | 0.95 | 0.097 | 0.093 |

S_{523} | 0.078 | 0.077 | 0.076 | 0.078 | 0.074 |

S_{524} | 0.085 | 0.085 | 0.085 | 0.085 | 0.081 |

_{5}was found to be 0.037.

Criteria | ${\mathit{x}}_{\mathbf{max}}^{1}$ | ${\mathit{x}}_{\mathbf{max}}^{2}$ | ${\mathit{x}}_{\mathbf{max}}^{3}$ | De-Fuzzy | Normal W |
---|---|---|---|---|---|

S_{61} | 0.357 | 0.348 | 0.339 | 0.357 | 0.350 |

S_{62} | 0.349 | 0.341 | 0.332 | 0.349 | 0.342 |

S_{63} | 0.296 | 0.313 | 0.330 | 0.313 | 0.307 |

_{6}was found to be 0.076.

Criteria | ${\mathit{x}}_{\mathbf{max}}^{1}$ | ${\mathit{x}}_{\mathbf{max}}^{2}$ | ${\mathit{x}}_{\mathbf{max}}^{3}$ | De-Fuzzy | Normal W |
---|---|---|---|---|---|

S_{72} | 0.148 | 0.147 | 0.145 | 0.148 | 0.142 |

S_{76} | 0.217 | 0.214 | 0.212 | 0.217 | 0.208 |

S_{77} | 0.160 | 0.158 | 0.156 | 0.160 | 0.153 |

S_{79} | 0.182 | 0.180 | 0.178 | 0.182 | 0.174 |

S_{712} | 0.172 | 0.170 | 0.169 | 0.172 | 0.165 |

S_{715} | 0.166 | 0.165 | 0.163 | 0.166 | 0.159 |

_{7}was found to be 0.021.

Criteria | ${\mathit{x}}_{\mathbf{max}}^{1}$ | ${\mathit{x}}_{\mathbf{max}}^{2}$ | ${\mathit{x}}_{\mathbf{max}}^{3}$ | De-Fuzzy | Normal W |
---|---|---|---|---|---|

S_{81} | 0.149 | 0.148 | 0.146 | 0.149 | 0.143 |

S_{82} | 0.212 | 0.210 | 0.208 | 0.212 | 0.203 |

S_{83} | 0.159 | 0.157 | 0.155 | 0.159 | 0.152 |

S_{86} | 0.180 | 0.178 | 0.177 | 0.180 | 0.172 |

S_{89} | 0.198 | 0.196 | 0.194 | 0.198 | 0.190 |

S_{814} | 0.146 | 0.145 | 0.143 | 0.146 | 0.140 |

_{8}was found to be 0.097.

Criteria | ${\mathit{x}}_{\mathbf{max}}^{1}$ | ${\mathit{x}}_{\mathbf{max}}^{2}$ | ${\mathit{x}}_{\mathbf{max}}^{3}$ | De-Fuzzy | Normal W |
---|---|---|---|---|---|

S_{92} | 0.187 | 0.186 | 0.184 | 0.187 | 0.181 |

S_{93} | 0.274 | 0.271 | 0.269 | 0.274 | 0.265 |

S_{94} | 0.284 | 0.282 | 0.280 | 0.284 | 0.275 |

S_{95} | 0.150 | 0.148 | 0.146 | 0.150 | 0.145 |

S_{96} | 0.139 | 0.138 | 0.137 | 0.139 | 0.134 |

_{9}was found to be 0.075.

Criteria | ${\mathit{x}}_{\mathbf{max}}^{1}$ | ${\mathit{x}}_{\mathbf{max}}^{2}$ | ${\mathit{x}}_{\mathbf{max}}^{3}$ | De-Fuzzy | Normal W |
---|---|---|---|---|---|

S_{102} | 0.290 | 0.288 | 0.287 | 0.290 | 0.282 |

S_{103} | 0.737 | 0.732 | 0.723 | 0.737 | 0.718 |

_{10}was found to be 0.084.

_{2}) by the weight of the main criteria (W

_{1}). For example, for C

_{1}and the related sub-criteria, we have the final weight in Table 20.

#### Sensitivity Analysis

_{1}) in combination (T) and in the view of administrators (A) ranks 1st; in contrast, from the parents’ point of view (P), this criterion ranks fifth in importance. The three most important criteria from the principals’ point of view are management staff (C

_{1}), administrative affairs (C

_{9}), and library (C

_{4}), respectively; this ranking shows that they consider most of the system’s internal factors, especially those directly related to themselves, to be important. On the other hand, the three most important criteria from the point of view of parents are educational equipment (C

_{3}), social environment (C

_{10}), and credits and costs (C

_{2}), respectively. Considering that the fourth most important criterion from the parent’s point of view is health (C

_{6}), it is clear that they consider a combination of internal and external factors of the system to be important in their analysis.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Table 1.**Changing the LVs into fuzzy numbers [56].

Linguistic Variables | Fuzzy Numbers (FNs) | Inverse of FNs |
---|---|---|

Equally preferred | $(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,1,\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)$ | $(\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.,1,2)$ |

Moderately preferred | $(1,\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,2)$ | $(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.,1)$ |

Strongly preferred | $(\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,2,\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)$ | $(\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.)$ |

Very strongly preferred | $(2,\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,3)$ | $(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.,\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)$ |

Extremely preferred | $(\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,3,\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.)$ | $(\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$7$}\right.,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.,\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.)$ |

The main diagonal of the matrix | $(1,1,1)$ | $(1,1,1)$ |

Linguistic Variables | Fuzzy Numbers |
---|---|

Very poor | (0, 0, 1) |

Poor | (0, 1, 3) |

Medium poor | (1, 3, 5) |

Fair | (3, 5, 7) |

Medium good | (5, 7, 9) |

Good | (7, 9, 10) |

Very good | (9, 10, 10) |

Criteria | Symbol | Source |
---|---|---|

Management staff | C_{1} | Literature |

Credits and costs | C_{2} | Literature |

Educational equipment | C_{3} | Literature |

Library | C_{4} | Literature |

Educational leadership | C_{5} | Literature |

Health | C_{6} | Recommended |

Students | C_{7} | Recommended |

Teaching and learning process | C_{8} | Literature |

Administrative affairs | C_{9} | Literature |

Social environment | C_{10} | Recommended |

Criteria | Sub-C | Expert 1 | Expert 2 | Expert 3 | Expert 4 | Expert 5 | Expert 6 | Expert 7 | Expert 8 | Expert 9 | Expert 10 | Average |
---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{10} | S_{101} | 7 | 5 | 7 | 9 | 8 | 9 | 7 | 7 | 7 | 8 | 7.4 |

S_{102} | 7 | 9 | 9 | 6 | 4 | 3 | 5 | 7 | 6 | 7 | 6.3 | |

S_{103} | 5 | 6 | 8 | 6 | 9 | 9 | 6 | 7 | 9 | 8 | 7.3 | |

S_{104} | 5 | 6 | 4 | 3 | 4 | 3 | 6 | 7 | 5 | 4 | 4.7 |

Criteria | Sub-Criteria |
---|---|

C_{1} | 3, 4, 8, 9, 12, 13, 14 |

C_{2} | 1, 2, 4, 7, 8, 9 |

C_{3} | 3, 5, 7, 9, 11, 12 |

C_{4} | 2, 3 |

C_{5} | 2, 3, 4, 7, 10, 11, 12, 13, 16, 20, 21, 22, 25 |

C_{6} | - |

C_{7} | 1, 3, 4, 5, 8, 10, 11, 13, 14, 16 |

C_{8} | 4, 5, 7, 8, 10, 11, 12, 13 |

C_{9} | 1 |

C_{10} | 2, 4 |

Criteria | Sub-C | Expert 1 | Expert 2 | Expert 3 | Expert 4 | Expert 5 | Expert 6 | Expert 7 | Expert 8 | Expert 9 | Expert 10 | Average |
---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{10} | S_{101} | 8 | 8 | 9 | 9 | 8 | 8 | 8 | 8 | 8 | 8 | 8.2 |

Removed | - | - | - | - | - | - | - | - | - | - | - | |

S_{103} | 7 | 8 | 7 | 7 | 9 | 7 | 9 | 9 | 7 | 9 | 7.9 | |

Removed | - | - | - | - | - | - | - | - | - | - | - |

Rounds | Items | Experts | Kendall’s C | D.F. | Sig. |
---|---|---|---|---|---|

The first | 108 | 10 | 0.333 | 107 | 0.000 |

The second | 53 | 10 | 0.402 | 52 | 0.003 |

Criteria | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | |
---|---|---|---|---|---|---|---|---|---|---|---|

C_{1} | U | 1 | 0.58 | 0.41 | 0.44 | 0.45 | 0.46 | 0.68 | 0.54 | 0.49 | 1.7 |

M | 1 | 0.45 | 0.32 | 0.32 | 0.36 | 0.36 | 0.54 | 0.44 | 0.39 | 1.35 | |

L | 1 | 0.37 | 0.26 | 0.25 | 0.35 | 0.31 | 0.45 | 0.38 | 0.33 | 1.03 | |

C_{2} | U | 2.67 | 1 | 1.00 | 1.20 | 1.88 | 1.35 | 2.01 | 2.11 | 1.31 | 1.1 |

M | 2.22 | 1 | 0.78 | 0.96 | 1.52 | 1.13 | 1.71 | 1.75 | 1.11 | 0.94 | |

L | 1.71 | 1 | 0.62 | 0.79 | 1.28 | 0.96 | 1.41 | 1.36 | 0.95 | 0.82 | |

C_{3} | U | 3.79 | 1.61 | 1 | 1.34 | 1.29 | 1.30 | 1.81 | 0.55 | 1.00 | 1.40 |

M | 3.15 | 1.28 | 1 | 1.05 | 1.09 | 1.00 | 1.54 | 0.42 | 0.78 | 1.02 | |

L | 2.41 | 1.00 | 1 | 0.82 | 0.89 | 0.82 | 1.28 | 0.35 | 0.55 | 0.78 | |

C_{4} | U | 3.97 | 1.26 | 1.22 | 1 | 0.77 | 1.64 | 1.71 | 1.52 | 1.90 | 1.56 |

M | 3.17 | 1.04 | 0.95 | 1 | 0.62 | 1.26 | 1.32 | 1.22 | 1.59 | 1.31 | |

L | 2.29 | 0.84 | 0.75 | 1 | 0.52 | 0.98 | 1.08 | 0.98 | 1.30 | 1.04 | |

C_{5} | U | 2.82 | 0.78 | 1.12 | 1.94 | 1 | 3.07 | 3.82 | 0.68 | 1.04 | 0.51 |

M | 2.74 | 0.66 | 0.92 | 1.62 | 1 | 2.49 | 3.16 | 0.54 | 0.87 | 0.43 | |

L | 2.24 | 0.53 | 0.78 | 1.30 | 1 | 1.94 | 2.59 | 0.46 | 0.72 | 0.37 | |

C_{6} | U | 3.23 | 1.05 | 1.22 | 1.02 | 0.52 | 1 | 0.40 | 1.52 | 1.90 | 1.56 |

M | 2.74 | 0.89 | 1 | 0.79 | 0.40 | 1 | 0.32 | 1.22 | 1.59 | 1.31 | |

L | 2.16 | 0.74 | 0.77 | 0.61 | 0.33 | 1 | 0.26 | 0.98 | 1.30 | 1.04 | |

C_{7} | U | 2.23 | 0.71 | 0.78 | 0.93 | 0.39 | 3.79 | 1 | 3.07 | 3.82 | 0.68 |

M | 1.84 | 0.59 | 0.65 | 0.76 | 0.32 | 3.15 | 1 | 2.49 | 3.16 | 0.54 | |

L | 1.47 | 0.50 | 0.55 | 0.58 | 0.26 | 2.50 | 1 | 1.94 | 2.59 | 0.46 | |

C_{8} | U | 2.64 | 0.74 | 2.86 | 1.02 | 2.19 | 1.02 | 0.52 | 1 | 1.04 | 0.51 |

M | 2.28 | 0.57 | 2.38 | 0.82 | 1.85 | 0.82 | 0.40 | 1 | 0.87 | 0.43 | |

L | 1.86 | 0.47 | 1.83 | 0.66 | 1.48 | 0.66 | 0.33 | 1 | 0.72 | 0.37 | |

C_{9} | U | 3.05 | 1.05 | 1.28 | 0.77 | 1.39 | 0.77 | 0.39 | 1.39 | 1 | 0.40 |

M | 2.58 | 0.90 | 1.00 | 0.63 | 1.15 | 0.63 | 0.32 | 1.15 | 1 | 0.32 | |

L | 2.06 | 0.76 | 0.79 | 0.53 | 0.96 | 0.53 | 0.26 | 0.96 | 1 | 0.26 | |

C_{10} | U | 0.97 | 1.22 | 1.32 | 0.96 | 2.67 | 0.96 | 2.19 | 2.67 | 3.79 | 1 |

M | 0.74 | 1.06 | 0.99 | 0.76 | 2.32 | 0.76 | 1.85 | 2.32 | 3.15 | 1 | |

L | 0.59 | 0.91 | 0.72 | 0.64 | 1.98 | 0.64 | 1.48 | 1.98 | 2.50 | 1 |

Criteria | ${\mathit{x}}_{\mathbf{max}}^{1}$ | ${\mathit{x}}_{\mathbf{max}}^{2}$ | ${\mathit{x}}_{\mathbf{max}}^{3}$ | De-Fuzzy | Normal W |
---|---|---|---|---|---|

C_{1} | 0.192 | 0.190 | 0.189 | 0.192 | 0.184 |

C_{2} | 0.073 | 0.073 | 0.072 | 0.073 | 0.070 |

C_{3} | 0.087 | 0.086 | 0.085 | 0.087 | 0.083 |

C_{4} | 0.076 | 0.075 | 0.074 | 0.076 | 0.073 |

C_{5} | 0.092 | 0.091 | 0.090 | 0.092 | 0.088 |

C_{6} | 0.110 | 0.109 | 0.107 | 0.110 | 0.105 |

C_{7} | 0.106 | 0.105 | 0.103 | 0.106 | 0.101 |

C_{8} | 0.109 | 0.108 | 0.107 | 0.109 | 0.104 |

C_{9} | 0.126 | 0.124 | 0.123 | 0.126 | 0.120 |

C_{10} | 0.075 | 0.074 | 0.074 | 0.075 | 0.072 |

Criteria | ${\mathit{x}}_{\mathbf{max}}^{1}$ | ${\mathit{x}}_{\mathbf{max}}^{2}$ | ${\mathit{x}}_{\mathbf{max}}^{3}$ | De-Fuzzy | Normal W |
---|---|---|---|---|---|

S_{11} | 0.135 | 0.133 | 0.132 | 0.135 | 0.129 |

S_{12} | 0.170 | 0.168 | 0.166 | 0.170 | 0.163 |

S_{15} | 0.111 | 0.110 | 0.109 | 0.111 | 0.106 |

S_{16} | 0.149 | 0.147 | 0.146 | 0.149 | 0.143 |

S_{17} | 0.161 | 0.160 | 0.158 | 0.161 | 0.154 |

S_{110} | 0.149 | 0.147 | 0.146 | 0.149 | 0.143 |

S_{111} | 0.168 | 0.167 | 0.165 | 0.168 | 0.161 |

Criteria | Weight | Sub-C | Weight | Final W |
---|---|---|---|---|

C_{1} | 0.197 | S_{11} | 0.129 | 0.025 |

S_{12} | 0.163 | 0.032 | ||

S_{15} | 0.106 | 0.021 | ||

S_{16} | 0.143 | 0.028 | ||

S_{17} | 0.154 | 0.030 | ||

S_{110} | 0.143 | 0.028 | ||

S_{111} | 0.161 | 0.032 |

School | d^{+} | d^{−} | CL | Rank |
---|---|---|---|---|

1 | 0.130 | 0.145 | 0.527 | 11 |

2 | 0.092 | 0.165 | 0.641 | 1 |

3 | 0.124 | 0.148 | 0.544 | 9 |

4 | 0.129 | 0.127 | 0.497 | 13 |

5 | 0.112 | 0.152 | 0.575 | 6 |

6 | 0.106 | 0.163 | 0.607 | 4 |

7 | 0.119 | 0.134 | 0.529 | 10 |

8 | 0.161 | 0.097 | 0.376 | 14 |

9 | 0.095 | 0.166 | 0.635 | 2 |

10 | 0.122 | 0.156 | 0.562 | 7 |

11 | 0.097 | 0.150 | 0.606 | 5 |

12 | 0.102 | 0.158 | 0.607 | 3 |

13 | 0.120 | 0.147 | 0.551 | 8 |

14 | 0.120 | 0.127 | 0.504 | 12 |

15 | 0.158 | 0.095 | 0.375 | 15 |

Criteria | W (P) | Rank | W (A) | Rank | W (T) | Rank |
---|---|---|---|---|---|---|

C_{1} | 0.110 | 5 | 0.258 | 1 | 0.184 | 1 |

C_{2} | 0.125 | 3 | 0.015 | 9 | 0.070 | 10 |

C_{3} | 0.143 | 1 | 0.023 | 8 | 0.083 | 7 |

C_{4} | 0.016 | 10 | 0.130 | 3 | 0.073 | 8 |

C_{5} | 0.077 | 8 | 0.099 | 5 | 0.088 | 6 |

C_{6} | 0.122 | 4 | 0.088 | 7 | 0.105 | 3 |

C_{7} | 0.099 | 7 | 0.103 | 4 | 0.101 | 5 |

C_{8} | 0.110 | 5 | 0.098 | 6 | 0.104 | 4 |

C_{9} | 0.062 | 9 | 0.178 | 2 | 0.120 | 2 |

C_{10} | 0.135 | 2 | 0.009 | 10 | 0.072 | 9 |

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**MDPI and ACS Style**

Li, P.; Edalatpanah, S.A.; Sorourkhah, A.; Yaman, S.; Kausar, N.
An Integrated Fuzzy Structured Methodology for Performance Evaluation of High Schools in a Group Decision-Making Problem. *Systems* **2023**, *11*, 159.
https://doi.org/10.3390/systems11030159

**AMA Style**

Li P, Edalatpanah SA, Sorourkhah A, Yaman S, Kausar N.
An Integrated Fuzzy Structured Methodology for Performance Evaluation of High Schools in a Group Decision-Making Problem. *Systems*. 2023; 11(3):159.
https://doi.org/10.3390/systems11030159

**Chicago/Turabian Style**

Li, Pengfei, Seyyed Ahmad Edalatpanah, Ali Sorourkhah, Saziye Yaman, and Nasreen Kausar.
2023. "An Integrated Fuzzy Structured Methodology for Performance Evaluation of High Schools in a Group Decision-Making Problem" *Systems* 11, no. 3: 159.
https://doi.org/10.3390/systems11030159