Enhancing Small and Medium Enterprises: A Hybrid Clustering and AHP-TOPSIS Decision Support Framework

2.1 K-Means clustering

The fundamental clustering technique employs a leveraged approach by calculating the Euclidean distance between every pair of samples in the sample set matrix D, resulting in the distance matrix M. The first cluster center is the corresponding sample, and the maximum value is chosen based on the value distance (C_i = 1, 2, ..., i). The sample located the furthest from the first cluster center is selected as the subsequent initial cluster center. The comparison of the distances of each unselected sample and the first cluster center, with the most significantly separated from the two previously determined samples. Finally, until the number of clustering centers at the beginning matches the value of k ascertained [8].

Based on several related studies that have been described, the primary researchs used as a reference in this research are literatures [7, 10, 13]. The research [8] compared two clustering methods, K-Means and SOM, where the K-Means clustering method had lower accuracy than the SOM method. This research uses the SOM method with different data [24]; SME data is used in Madura. The algorithm step diagram is shown in the Figure 1.

Then, literatures [9, 10] evaluated the cluster results by assessing the Davies Bouldin Index (DBI) to get the optimal cluster on family welfare data. The clustering process is carried out by combining 3 clusters, learning rate, and epoch values to determine the results of the DBI value in determining optimal cluster results [7]. The usage of K-Means with Automatic Clustering using Differential Evolution (ACDE) for grouping SME innovation processes produces three optimal clusters. The usage SOM clustering on the grouping of potato chip SMEs is formed into 2 clusters: cluster 1 is a micro-scale, and cluster 2 is a small scale [4].

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Figure 1. Flow diagram of SOM algorithm

K-Means includes an algorithm for simple clustering and can group data according to similarities between data [25]. The steps for calculating the K-Means method are as follows [15].

(1) Determine the number of clusters (k) as a dynamic value for grouping similarity data. A rule of thumb approach based on n overall data that can follow Eq. (1):

$k=\sqrt{\frac{n}{2}}$   (1)

(2) Determine the initial centroid value (initial cluster center point). To determine the initial centroid value, generally use the random method or take values randomly.

(3) Calculate the distance of each data with the centroid. Euclidean Distance is used to calculate the distance for each data using Eq. (2):

$D_J=\sqrt{\sum_i^n\left(x_i-y_i\right)^2}$      (2)

(4) Grouping data based on the nearest cluster after going through the distance calculation process. It is necessary to pay attention to the cluster that has the closest distance to the data or centroid, then group the data according to the cluster.

The data has been grouped according to clusters, then calculate the centroid of the new cluster by calculating the average of the total data values in the new cluster. To get a new centroid value, use Eq. (3).

$C_i=\frac{\sum_{i=1}^n x_i \in S_i}{n}$    (3)

Repeat the third and fourth stages. The last step compares the new cluster obtained with the initial group; if there are still changes or the new cluster centroid is not the same as the initial cluster, iterates again, starting from determining the new cluster centroid. If they are the same, then the clustering process is complete.

2.2 Analytical hierarchy process

AHP is a Multi-Criteria Decision Making (MCDM) method introduced by Thomas L. Saaty in the 1970s. The AHP method is commonly used because it can determine the importance of many criteria by conducting a pairwise comparison analysis on each measure. In addition, AHP also considers the validity of the requirements and alternatives chosen by the decision maker to the tolerance limit to prevent inconsistent results. The following stages in the AHP method include [20]:

(1) Create a hierarchical structure starting from the goal to be achieved, followed by the criteria and alternatives you want to rank.

(2) Create a pairwise comparison matrix (Pairwise Comparisons).

(3) This comparison matrix describes the relative contribution or influence of each element on each objective or criterion level above it.

(4) Normalization of the pairwise comparison matrix. After compiling the criteria matrix values, the next step is to calculate the matrix normalization values by dividing the value of each cell by the number of each column with Eq. (4).

$X_{i j}=\frac{k_{i j}}{\sum_{i=1}^n k_{i j}}$        (4)

where, the comparison value of K_i to K_j is k_ij. The value of k is determined by the rule:

• If k_ij = x, then k_ji = $\frac{1}{x}$ and the value x $\neq$ 0 .

• Certain treatment occurs at k_ii = 1, applies to all i.

Pairwise comparisons were performed for the scoring criteria. The importance level scale value in the pairwise comparison matrix can be seen in Table 1 [21].

Table 1. AHP importance scale value

(5) Calculate the value of the priority weight (priority vector), then do a consistency test. If it is not consistent, taking the importance value must be repeated. Calculating priority weight values with Eq. (5).

$P V_i=\frac{\sum_{j=1}^n X_{i j}}{n}$     (5)

(6) Calculate the eigenvector values of each pairwise comparison matrix by multiplying the pairwise comparison matrices by the priority criteria weight value for each row with Eq. (6).

$E V_{i j}=\left(k_{i j} \times P V_{j i}\right)$  (6)

(7) The AHP method is used to measure consistency based on the consistency ratio. The consistency test must be carried out in a decision to determine whether the value of importance given to each criterion has reached a consistent weight. The steps for calculating the consistency value are as follows:

(a) The results of the consistency test obtain a value by calculating the average value in the CM column using Eq. (7).

$C M_i=\frac{\sum_{i=1}\left(E V_{i j}\right)}{P V_i}$    (7)

(b) The results of the consistency test obtain a value by calculating the average value in the CM column using Eq. (8).

$\lambda \max =A V E R A G E(C M)$    (8)

(8) The Consistency Index (CI) value is calculated using Eq. (9).

$C I=\frac{(\lambda \max -n)}{(n-1)}$   (9)

(9) Evaluation of consistency in pairwise comparisons is done by calculating the CR using Eq. (10).

$C R=\frac{C I}{R I}$      (10)

The Ratio Index (RI) has been determined based on the matrix order or the number of criteria in the AHP method shown in Nilai RI.

Table 2. RI value

In Table 2, if CR≤0.1, the determined pairwise comparison value has reached a consistent deal because the error tolerance is 10%. Suppose CR>0.1, a pairwise comparison matrix is reassessed to obtain consistent results [22].

2.3 Technique for order preference by similarity to ideal solution

Yoon and Hwang presented the TOPSIS in 1981. The TOPSIS technique is a type of Multi-Criteria Decision Making (MCDM) approach that operates under the tenet that the optimal answer is the closest to the positive ideal solution and the furthest from the negative perfect solution [21]. AHP, ELECTRE III, PROMETHEE II, Gray Relational Analysis, Fuzzy TOPSIS, MOORA, WASPAS, VIKOR, and Fuzzy Analytical Hierarchy Process (FAHP) are some of the MCDM methodologies that are used as decision support systems. A clear rating of other options is provided by decision-making, which also handles a variety of criteria and alternatives, their benefits and drawbacks, and management constraints for complex data, including subjective assessment [26].

The usage of TOPSIS method has solved recommendations for SMEs that deserve coaching to help recommend alternatives by compiling all options for each cluster and calculating to get the highest preference value to obtain a ranking of alternative recommendations in each cluster, especially in recent years in the Asia Pacific region. The TOPSIS method can help calculate data as weight suitability of alternative approaches that will produce the best assessment, compute very efficiently, and measure relative performance levels and decision alternatives made in simple mathematics [22, 23].

The approach (FAHP-TOPSIS) has been utilized to prioritize impediments and methods for adopting manufacturing SMEs while giving strategic insights to guide decision-makers and practitioners of Indian manufacturing SMEs in implementing sustainable manufacturing. Furthermore, this study assists managers by offering a list of hierarchies' ranking systems based on their effectiveness, removing implementation constraints [27]. Figure 2 depicts the flow of the TOPSIS approach.

Many conflicting qualitative and quantitative criteria play a role in the success of high-tech SMEs. To overcome this problem, it is necessary to integrate the Analytic Hierarchy Process (AHP) considering the weight of the calculated criteria, which can increase the critical success factor (CSF) to be determined to produce a ranking used for recommendations [28].

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Figure 2. Process diagram of the TOPSIS method

4.1 Purpose of research

The system flow that will be carried out in the SME grouping research combines the SOM and AHP-TOPSIS mixed methods. The system flow design of the method proposed in the study is according to Figure 3.

The stages of the process in the system to be built first is the data preparation stage on the SME dataset, where this stage is carried out for data preparation, starting from transformation and normalization until the data is ready for processing. Then enter the clustering stage using the SOM method with input from the SME dataset that has undergone the data preparation process. For the optimal clustering results, a cluster evaluation is used using DBI. Then proceed to weighting the criteria for the weights used in making decisions using the AHP method. The alternative ranking stage in each cluster using the TOPSIS method. The input weight to the TOPSIS method was used the results of SME data that have gone through the clustering stage and priority weights obtained from the AHP method. The priority weights are used as criteria weighting in the TOPSIS method because the TOPSIS method does not have specific weight calculations, so the AHP method is needed to get more consistent weights.

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Figure 3. System flow diagram

4.2 Clustering SME

The stage of presenting the results of the trial scenario to answer research questions, namely the first, is in the form of the effect of learning rate and epoch values on clustering results with the DBI cluster evaluation value. The second is ranking accuracy results from the clustering results of the SOM method and the combined AHP and TOPSIS methods. Test scenario one aims to obtain the optimal cluster results based on scala 0.01 of learning rate and epoch in Table 1 dan Table 2. Tests were carried out on 800 SME data used in this study; then, the cluster evaluation stage will be carried out using the DBI method to determine how optimal the cluster results obtained for each combination. The overall results of the learning rate and epoch combination experiments, along with the DBI values in test scenario 1, are summarized in Table 3 and Table 4.

Table 3. The results of SOM test

Table 4. The results of the K-Means test

Table 3 displays SOM that there is a significant change in the DBI when k varies from 1 to 12. Additionally, the rate at which the DBI changes with k slows down when k > 4. K-Means experienced reduced errors in clusters 4 to 8, and the optimal number of clusters must be 4 to 6. As k increases, the DBI value rises steadily; the more significant the grouping effect, the smaller the DBI value. SOM can be seen that the most optimal cluster of each experiment is found in combination 9 with a learning rate parameter value of 0.05 and epoch 1000 and a DBI value of 0.74785 is obtained with the total distribution of data in each cluster namely cluster 1 totaling 256 data, cluster 2 destroying 102 data, and cluster 3 totaling 442 data. Otherwise, Table 4 shows K-Means produces a higher DBI than the result in SOM. K-Means depends on the value of k and the centroid update contain in 0.7709, so That requires more iterations.

Figure 4 shows the test results based on changes in the number of clusters k, which vary from 1 to 12, based on the SSE value. The results show that in the SOM and K-Means methods, after the number of clusters reaches 4, the SSE has experienced a sharp decline, meaning that the higher the cluster value, the lower the SSE value.

The SSE value for each of the best performances is k = 12, the SOM algorithm is 1.988, and the K-Means is 1.545, both of which have experienced a decline. A comparison of the two reveals that the results of cluster analysis have more remarkable similarities within clusters and more noticeable differences between clusters.

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Figure 4. Clustering performance comparison with SSE

4.3 Discussion

The most optimal combination among the other combinations because it produces the lowest DBI value, where the lower the DBI value, the better the clusters obtained from grouping. The test results on the selected combination are good because it has a significant epoch value, where the greater the number of epochs, the training takes a long time but will produce good enough cluster data. Still, conversely, the smaller the number of epochs, the training process requires a relatively short time but will produce cluster data that has yet to be grouped optimally. Then the smallest learning rate value is selected, where the more significant the learning rate, the faster the training process becomes unstable, and the cluster results must be better grouped. The optimal cluster results obtained are evaluated based on SSE, which is used to assess the accuracy of predictions made by the model. So, the difference between SOM and K-Means is based on cluster differences, as shown in Figure 5.

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Figure 5. Accuracy value of the ranking of each cluster

Figure 5 explain experiments on clusters 1, 2, and 3 of test scenario three were carried out using different priority weights in each cluster with six criteria: the number of workers, average turnover, average production costs, type of business difficulty, getting credit, and needing external loans. These weights are obtained from the results of the consistency test on the AHP method, where the results of these weights get different CI and CR values. The details of the results of the AHP priority weights are shown in Table 5.

Experiments on clusters 1, 2, and 3 of test scenario two were carried out using the same priority weight with six criteria, namely number of workers = 0.031, average turnover = 0.38, average production costs = 0.255, type of business difficulty = 0.192, get credit = 0.08, and need external loans = 0.063. These weights were obtained from the consistency test on the AHP method, where the weight results received a CI value of 0.09 and a CR of 0.072. The usage AHP priority weight results were obtained with different CR values for each cluster originating from the questionnaire containing priority weights, as shown in Table 5. Even though different CR values were obtained, namely cluster 1 = 0.072, cluster 2 = 0.026, and cluster 3 = 0.052, all of these CR values indicate that the resulting priority weights are consistent because the CR value is ≤ 0.1 or ≤ 10%. The process of ranking the data by the system shows the results with the highest preference value in each cluster are influenced by the decision value factor at the time of calculation, resulting in the data distance from the near-positive ideal solution. Therefore, the data gets a high preference value.

(1) Most of cluster 1 shown SME are with individual businesses, the business field sector in other service activities, with an average number of workers, turnover, and high production costs; all companies in this cluster cannot get credit and do not need external.

(2) Most of cluster 2 shown SME are in Sampang with UD businesses, the business field sector in trade, with an average number of workers, turnover, and moderate production costs. Companies in this cluster get credit but do not need external loans.

(3) The majority in cluster 3 of SME are in Pamekasan with individual businesses and UD, the business field sector in other service activities and food and beverage accommodation, with an average number of workers, turnover, and production costs which are relatively low, in this cluster the dominant businesses do not get credit, but all companies need outside loans.

Table 5. AHP priority weight for SOM and K-Means