Implementation of Fuzzy C-Means and Topsis in College Rankings

Prior to now, the ranking of higher education institutions, particularly those at the Regional II Palembang Higher Education Service Institution, was based on one component of the work unit's criteria. This makes the university ranking results superior on one criterion but inferior on another. The number of instructors and the number of students at 100 universities in the South Sumatra region were split into two groups based on the outcome of the fuzzy c means algorithm grouping and regional criteria and calculated based on the resulting mean value. The grouping results using a topsis algorithm decision-making system with a weight determined by the number of lecturers with functional positions, college accreditation, number of certified lecturers, and percentage level of higher education database reports are used as a reference to rank universities. Based on the mean value of the fuzzy c means algorithm and the grouping results, seven colleges were chosen. Using the topsis method's way of making decisions, the final score for the highest-ranked college is 0.850.


INTRODUCTION
The ranking of higher education institutions needs to be done selectively according to the category used. Higher Education Service Institutions, the scale of tertiary institutions has been carried out using one class in the Region II Higher Education Service Institutions work group, which causes universities to excel in a tiny aspect and be weak in other parts. As a result of the ranking that has been carried out so far, of course, you will not get the best tertiary ranking results based on many components as the basis for grouping and evaluating tertiary institutions.
Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a decision-making method used in solving Multi-Criteria Decision Making (MCDM) problems or making decisions with many criteria. This method was introduced by Yoon and Hwang (1981). The main concept of the TOPSIS method is to find the best alternative solution that has the closest distance to the positive ideal solution and the farthest distance from the negative ideal solution. The TOPSIS method is a multi-criteria decision-making method that is simple, efficient in the calculation process, and can measure the relative performance of many alternatives [7].

METHODS
The research method uses research stages which are described in the form of research diagrams as follows.  Figure 1, the stages of the research can be explained as follows: 1. Collection and download of datasets (databases of higher education) that will be used for research, including data on tertiary institutions in the province of South Sumatra 2. Determination of criteria, sub-criteria and alternatives followed by cluster determination using the fuzzy c-means model. The Fuzzy C-Means cluster method aims to classify universities into groups based on the variables determined by the researcher. The first stage was carried out by grouping 16 tertiary institutions in the Bengkulu province area based on the university ranking indicator variables originating from the higher education database and determining the cluster centre, which would mark the average location for each cluster. By repairing the cluster centre and degree of membership of each data point repeatedly, the cluster centre will move towards the right location. The loop is based on minimizing the objective function, which describes the distance from a given data point to the cluster centre weighted by the degree of membership of the data point. The output of Fuzzy C-Means is a cluster centre series and several degrees of membership for each data. The software used as a tool in this research is Matlab 2021A. 3. Topsis for determining university rankings using MS. Excel and Matlab.
Testing is done to avoid errors from the system created. If an error occurs, the system will be repaired again until the process results are as expected. Determination of university ranking

TOPSIS
The research method used in this study uses data from the tertiary institutions in the higher education database with a dataset using 100 tertiary institutions in the South Sumatra region. The reason for selecting South Sumatra was due to geographical conditions, which reflect the area of LLDIKTI region II and the province with the highest number of tertiary institutions. Clustering using the Fuzzy C-Means Algorithm is as follows (Kusumadewi, 2010:80). 1. Determine the data be clustered X, in the form of a matrix of size nxm (n=number of data samples, m=attributes of each data). Xij=i-th sample data (i=1,2,...,n), jth attribute (j=1,2,...,m); 2. Determine the number of clusters (c), rank (w), maximum iteration (MaxIter), smallest expected error (ζ), the initial objective function (P0=0), initial iteration (t=1) 3. Generate random numbers µik, i=1,2,...,n; k=1,2,...,c; as elements of the initial partition matrix U. The partition matrix (U) in fuzzy grouping satisfies the following conditions. 5. Calculate the objective function in the t-th iteration, Pt; The objective function is used as a looping condition to get the right cluster center. So that the tendency of the data to enter which cluster is obtained in the final step. For the initial iteration the value of t = 1.
Calculate the change in the partition matrix; Check stop condition; a) |Pt -Pt-1| < ζ) or (t>MaxIter) then stop; b) If not, the iteration is increased t=t+1, repeat step 4 The grouping based on the initial dataset was divided into 2 clusters and based on three attribute data, namely 90 for the municipality of Palembang, 80 for municipalities other than Palembang and 70 for the district. The data on the number of lecturers and data on the number of students can be seen in Appendix 1.
For ranking using the TOPSIS method, there are several steps that must be passed to get the ideal solution. Here are the steps of the TOPSIS method.
1. Build a decision matrix. The X decision matrix refers to m alternatives that will be evaluated based on n criteria. An x decision matrix can be seen as follows:  D i + is the alternative distance from the positive ideal solution defined as: D i -is the alternative distance from the negative ideal solution defined as: Information : Di + is the distance of the I-th alternative from the positive ideal solution, Diis the distance of the I-th alternative from the negative ideal solution, Y ij is the element of the decision matrix that is weighted normalized Y, Yj + are elements of the positive ideal solution matrix, Yjare elements of the negative ideal solution matrix. 6. Calculates the closeness to the positive ideal solution.
The relative closeness of each alternative to the positive ideal solution can be calculated using the following equation: Vi + is the relative proximity of the I-th alternative to the positive ideal solution.
Di + is the I-th alternative distance from the positive ideal solution.
Diis the distance of the I-th alternative from the ideal solution negative. 7. Alternate ranking.
Alternatives are sorted from the largest C + value to the smallest value. The alternative with the largest C + value is the best solution.
The research results are seen in table 1, and the next step is to determine the initial parameters that will be used to solve the problem with the Fuzzy C-Means algorithm. These parameters are the number of clusters ( = 2), power ( = 2), maximum iteration (MaxIter = 100), smallest expected error ( = 0.01), the initial objective function ( 0 = 0), and the initial iteration ( = 1). The number of clusters specified is two. Using MatLab 2021A software, the results of calculating cluster centres, membership degrees or U matrices and the value of the objective function or object. Requires an initial iteration of 51 times before obtaining the optimal solution for the functional value Jw (U, V) of 3326327.260926. In the 51st iteration, the cluster center produced by MatLab software k = After obtaining the Centroid or centre point, the next step is to calculate the distance between the input criteria values from the user to each of the existing cluster centre points. The following is the Euclidean Distance formula: The following is the source code of the Euclidean Distance calculation used in this study.

Ranking Using Topsis
Based on the cluster data obtained after using the algorithm, fuzzy c means will be processed using the topsis method with four criteria which will then be  2. For the first cluster after using the fuzzy-c-means algorithm as follows: 3. Normalized matrix = root of the power of the value on each criterion (x=√C^2)

CONCLUSION
This study shows that the weaknesses in the ranking of tertiary institutions in Region II Higher Education Service Institutions, which rely on one category for each work section, can be overcome using the fuzzy c-means and topsis methods. The use of the Fuzzy C-Means method can divide college clusters by region, number of lecturers and number of students. By taking the mean value from the results of the calculation of the fuzzy c-means algorithm, two college clusters are created where there are seven universities with scores above the cluster mean value. Furthermore, from these clusters, the topsis method can be a solution for ranking the results of the clustering, where seven alternatives for the best higher education rankings are obtained.