Hybrid PCA-FCM Clustering for Provincial Rice Distribution Classification in Indonesia

Indonesia’s rice-distribution system is shaped by persistent structural disparities arising from uneven production output, demographic concentration, consumption pressure, and logistics performance across provinces. As rice remains the nation’s primary staple commodity, the resilience of the distribution network is essential for maintaining food security and socio-economic stability [1]. Significant gaps continue to separate high-surplus regions from deficit-dependent provinces, where distribution flow is further influenced by variations in road conditions, transport accessibility, warehousing capacity, and market connectivity [2-4]. Previous rice-distribution and agricultural logistics studies generally evaluate production capacity, transportation readiness, or market performance independently, limiting integrated understanding of provincial rice-distribution structures [5, 6]. Existing national assessments remain descriptive and single-indicator-based, limiting the ability to understand how production strength, consumption intensity, logistics readiness, and price behaviour operate as a unified system. This analytical gap restricts policymakers from identifying priority regions and designing targeted logistical interventions [7]. Rice-distribution evaluation increasingly requires multivariate analytical approaches to capture structural interactions among production, logistics readiness, consumption pressure, and market dynamics. Studies on dimensionality reduction demonstrate Principal Component Analysis (PCA)’s capability to reveal hidden structural patterns in high-dimensional datasets, including transportation behaviour, logistics readiness, and distribution variability [8, 9]. Previous studies further demonstrate that dimensionality-reduction techniques improve interpretability in agricultural distribution and regional performance evaluation systems [10]. These methodological advances reinforce PCA’s role in summarizing interconnected indicators into interpretable components that support large-scale classification efforts [11, 12]. Clustering approaches likewise play a central role in distribution-performance analysis. Fuzzy C-Means (FCM) is widely applied because it accommodates gradual transitions between categories-an essential feature when evaluating provinces whose characteristics do not fit rigid boundaries [10, 13, 14]. In agricultural value chains, fuzzy clustering has been used to detect structural heterogeneity, variation in sustainability, and performance asymmetry across dynamic supply-chain environments [15-17]. Recent methodological studies emphasize that integrating PCA with FCM enhances cluster separability, stability, and interpretability in high-dimensional agricultural, environmental, and socio-economic datasets [18, 19]. Despite these developments, hybrid Principal Component Analysis-Fuzzy C-Means (PCA-FCM) approaches remain underutilised in national rice distribution research, leaving a gap in Indonesia’s evidence-based policy framework [20].

Previous analytical frameworks emphasize that multivariate modelling improves regional distribution-system evaluation under heterogeneous provincial conditions [21]. Such insights strengthen the rationale for adopting a hybrid PCA-FCM model, which can capture both structural disparities and transitional characteristics in Indonesia’s provincial distribution system. In the Indonesian context, hybrid analytical approaches such as multi-objective optimization and clustering have proven effective for regional prioritization and policy formulation, particularly in settings characterized by socio-economic and infrastructural heterogeneity [22, 23]. These applications illustrate how integrated quantitative frameworks outperform single-indicator assessments by identifying hidden structural constraints and mapping multi-dimensional development trajectories. Their relevance to rice-distribution studies underscores the suitability of PCA-FCM for classifying provinces into meaningful categories that reflect both logistical capacity and distribution vulnerability. Robust analytical frameworks are increasingly required to improve interpretability and classification reliability in high-dimensional distribution evaluation systems [24, 25]. Applying these principles to national rice-distribution analysis ensures that the resulting cluster structure is not only statistically valid but also operationally relevant for strategic decision-making and policy implementation. To address this limitation, the present study implements a hybrid PCA-FCM framework to generate an objective classification of provincial rice distribution characteristics. The research aims to: (1) extract latent dimensions that shape multivariate distribution behavior, (2) group provinces into soft clusters that reflect transitional and overlapping characteristics, and (3) produce a provincial typology that supports targeted policy intervention and enhances national rice-distribution structural understanding [26]. The contribution of this study lies in the national-scale application of integrated multivariate provincial classification to rice-distribution analysis, supporting evidence-based regional typology development and policy-oriented distribution planning in Indonesia.

3.1 Principal Component Analysis result

PCA analysis was applied to five key indicators representing the dimensions of production, demand, surplus-deficit, road-infrastructure capacity, and price stability, namely Rice-Production, Total-Consumption, Surplus-Deficit, Good-Road-Length, and Paddy-Price. This multidimensional indicator set reflects the structural attributes of Indonesia’s rice-distribution system and aligns with analytical practices in supply-chain and agrifood studies [12, 15]. A total of 114 provincial observations (province-year) were included in the analysis. These indicators collectively capture the core mechanics of distribution effectiveness-production-demand balance, distribution capacity, infrastructure accessibility, and market dynamics. PCA is therefore employed to reduce dimensionality and extract the dominant latent components that explain the structural variance across provinces. This enables a clearer understanding of the underlying patterns shaping distribution performance before integrating them into the FCM clustering phase. The subsequent subsections examine the standardization process, covariance structure, variance contributions, loading interpretations, and spatial distribution of PCA scores. Together, these analytical stages form the foundation for the clustering and effectiveness evaluation presented in later sections.

3.2 Construction of the analysis matrix

Before PCA, each indicator $X_j$ was standardized using the Z-Score to ensure comparability across variables with different scales, following common procedures in multivariate agricultural and logistics research [13]. Standardization transforms all indicators into a unified metric so that large-magnitude variables do not dominate the principal components. Following common practice in multivariate agricultural and logistics studies using Eq. (4):

$Z_{i j}=\frac{X_{i j}-\mu_j}{\sigma_j}$     (4)

where, $X_{i j}$: The value of the indicator to $–j$ on observation (province-year) to $–i, \mu_j$: Average Indicator $j, \sigma_j$: Standard deviation indicator $j$. For Good_Road_Length_km, missing values were imputed using the mean so that the analysis matrix is complete and can be processed in multivariate form, a practice that is also used in PCA research for environmental and supply chain data. Table 2 presents the mean and standard deviation values for each variable used in the analysis.

Table 2. Mean and standard deviation of Principal Component Analysis (PCA)

These descriptive values demonstrate the variability across indicators and justify the need for standardization prior to PCA, as shown in Table 2. As an illustration, for Aceh in 2022, the original indicator values are: Rice-Production = 869,572.00, Total-Consumption = 534,300.52, Surplus-Deficit = 869.037.70, Good-Road-Length = 2,112.00, and Paddy_Price = 8.000. After standardization, the Z-Score values in Table 3 were obtained.

Table 3. Example of Z-Score values for Aceh 2022

The Z-Score pattern for Aceh demonstrates slightly above-average performance in both rice production and surplus (positive Z-scores), indicating strong supply potential. The province also shows higher road-network capacity relative to the national distribution, suggesting logistical readiness. Conversely, Aceh’s grain price level is higher than average, reflecting market pressure despite strong logistics.

3.3 Covariance structure and component extraction

After standardisation, the covariance matrix $\Sigma s$ formed from the standardized Z data matrix using Eq. (5):

$\Sigma s=\frac{1}{n-1} Z^T Z$     (5)

The way that section titles and other headings are displayed in these instructions is meant to be followed in your paper. Because PCA was performed on Z-score standardized variables, the resulting covariance matrix is numerically equivalent to a correlation matrix. Minor deviations of reported diagonal values from exactly 1.000 (e.g., 1.009) arise from numerical rounding and sample-based covariance estimation and do not affect the PCA extraction process.

Table 4. Covariance matrix of standardized indicators

The covariance values indicate very strong relationships among three indicators, such as Rice-Production, Total-Consumption, and Surplus-Deficit (≈0.95-1.00) suggesting high multicollinearity in the production-demand dimension of the rice system. This aligns with literature noting that surplus is structurally tied to both production capacity and consumption intensity in agrifood supply chains. In contrast, Good-Road-Length_km and Paddy-Price show moderate but positive covariance with the volume-related indicators (≈0.27-0.33). This pattern indicates that logistics conditions and market prices introduce distinct variability into the dataset, contributing additional dimensions beyond pure production-demand metrics. Such differentiation is important for capturing spatial disparities in infrastructure and price stability that influence distribution resilience. Principal components were derived from the covariance matrix by solving the eigenvalue problem as defined in Eq. (6):

$\Sigma v_k=\lambda_k v_k$     (6)

where, $\lambda_k$ denotes the eigenvalue and $v_k$ is the eigenvector corresponding to component $k$. This spectral decomposition approach is widely used in dimensionality-reduction frameworks for high-dimensional datasets and complex supply chain systems [13]. The clear multicollinearity observed in Table 4 therefore justifies the application of PCA to condense redundant variables while preserving the structural contrasts arising from logistics capacity and price volatility. These contrasts later emerge as major axes of variation in the PCA-FCM clustering process, shaping the separation between surplus provinces, logistics-weak regions, and price-volatile areas.

3.4 Eigenvalues and variance explained

Eigenvalues were extracted to determine the relative contribution of each principal component to the total variance of the dataset. Table 5 presents the five eigenvalues obtained from the spectral decomposition of the covariance matrix $\Sigma$, such as $\lambda_1=3.213$, $\lambda_2=1.080$, $\lambda_3=0.674$, $\lambda_4=0.077$, $\lambda_5 \approx 0.000$.

The first two components have eigenvalues greater than 1, indicating that they each capture a substantial portion of the multivariate structure, consistent with the Kaiser retention rule commonly applied in PCA-based agricultural and supply-chain studies.

The proportion of variance explained by each component is summarized in Table 5 and visualized in the scree plot in Figure 2.

Table 5. Variance explained by principal components

Figure 2. Scree plot of principal components for the rice-distribution indicators

The scree plot in Figure 2 clearly shows a sharp decline after PC2, confirming that PC1 and PC2 together account for more than 85% of total variance, while subsequent components contribute marginally. This pattern is typical in systems where production-demand indicators are highly correlated, while logistics and market-price factors introduce secondary but meaningful variation.

The dominance of PC1 reflects the strong multicollinearity among production, consumption, and surplus indicators, representing the core rice volume axis. PC2 contributes additional variance related to logistics capacity and price structure, capturing the heterogeneity observed across Indonesian provinces, especially between regions with strong versus limited infrastructure. PC3 explains small but relevant fluctuations associated with price-demand volatility. Together, these components form the structural basis for the PCA-FCM clustering phase, determining how provinces are separated into high-surplus regions, logistics-strong regions, transitional areas, and vulnerable clusters.

3.5 Loading matrix and interpretation of principal components

The loading matrix summarizes the contribution of each indicator to the principal components extracted in the PCA. Table 6 presents the loading coefficients for the first three components, which together account for more than 98% of the total variance.

Table 6. Principal Component Analysis (PCA) loading matrix for the first three components

The loading matrix indicates that PC1 is strongly defined by Rice-Production, Total-Consumption, and Surplus-Deficit (loadings ≈ 0.53-0.55), forming a clear production-surplus axis that reflects variations in provincial production capacity and overall supply-demand balance. Provinces with high PC1 scores therefore tend to exhibit larger outputs, higher surpluses, and stronger volume-driven distribution potential, consistent with literature identifying production-surplus strength as a primary determinant of food-system performance. PC2, dominated by Good_Road_Length_Km (0.676) and Paddy_Price_IDRkg (0.648), represents a logistics-price dimension, where better road networks correspond to improved accessibility while higher prices signal market pressure or transport inefficiencies.

This component differentiates logistics-advantaged provinces from those with weak infrastructure or remote positioning, aligning with spatial inequality patterns commonly observed in Indonesian distribution systems. Meanwhile, PC3 exhibits a strong positive loading on Good-Road-Length (0.696) and a strong negative loading on Paddy-Price (-0.718), indicating an infrastructure-price stability trade-off, where stronger logistical conditions tend to suppress price volatility, whereas limited accessibility amplifies market fluctuations. Although PC3 contributes a smaller portion of variance, it provides important discrimination among provinces with similar logistics capacity but differing price dynamics. Collectively, these three components establish the structural foundation for subsequent clustering, with PC1 separating surplus versus deficit regions, PC2 highlighting logistics readiness, and PC3 revealing hidden disparities in price stability.

3.6 Principal Component Analysis scores and spatial separation of provinces

The PCA scores represent the transformed positions of each province in the reduced feature space defined by the principal components. Table 6 illustrates an example for Aceh (2022), showing positive values on PC1 and PC3 and a relatively high score on PC2. These values indicate that Aceh possesses moderate production-surplus strength (PC1), strong logistics relative to other provinces (PC2), and favorable price-infrastructure dynamics (PC3), positioning it among the more effective regions in the national rice-distribution system. Selected components (PC1-PC3, for example, the PCA score for Aceh (2022) on the first three components is shown in Table 7 and Figure 3:

Table 7. Example of Principal Component Analysis (PCA) scores for Aceh (2022)

Figure 3. Principal Component Analysis (PCA) projection of Indonesian provinces in PC1-PC2 space

Figure 3 visualizes the projection of all provinces in the PC1-PC2 space, where each point represents a province-year observation and where clusters are distinguished for interpretive clarity. The plot reveals a clear structural separation: provinces with high PC1 values, located on the right side of the graph, correspond to high production and surplus capacity; meanwhile, provinces positioned higher on the PC2 axis exhibit stronger logistical conditions.

In contrast, provinces with lower PC1 and PC2 values are concentrated in the lower-left region, reflecting limited surplus capacity and weaker infrastructure, consistent with spatial inequality findings frequently reported in rice-distribution studies. The spatial distribution observed in the figure demonstrates that PCA effectively uncovers natural groupings of provinces based on production-logistics characteristics even before clustering is formally applied.

Figure 3 shows that regions with similar structural attributes tend to form localized groupings-for example, high-surplus and logistics-strong provinces cluster toward the upper-right quadrant, while deficit or logistics-weak provinces occupy the lower-left area. This separation forms the analytical foundation for the subsequent FCM clustering, where these PCA-derived coordinates serve as the reduced-dimension input space.

3.7 Fuzzy C-Means clustering results

FCM clustering was applied to the PCA score matrix of size 114 × 3, representing the reduced feature space derived from the consolidated production, logistics, and price indicators. The use of PCA prior to clustering improves separation by removing multicollinearity and emphasizing the dominant structural dimensions of the rice-distribution system [25]. Allowing each province to hold partial membership across clusters enables the model to capture transitional or ambiguous regional conditions-such as provinces with moderate surplus but weak logistics, or strong logistics but elevated price levels-patterns commonly observed in agricultural and supply-chain analytics [20]. The clustering configuration was initialized at c = 4 to support interpretable provincial categorization aligned with rice-distribution structural characteristics. The selected setting emphasizes policy-oriented representation of provincial distribution effectiveness while preserving fuzzy membership flexibility for overlapping regional conditions. Future studies may incorporate cluster-number sensitivity assessment to further evaluate clustering stability across candidate configurations. Cluster-number sensitivity analysis was performed across c = 2-6 to reduce confirmation bias in cluster selection. XB values of 0.0618, 0.2653, 0.2471, 0.3301, and 0.3324 were obtained for c = 2,3,4,5,6, respectively. Although c = 2 produced the minimum XB value, the resulting partition provided limited structural differentiation for policy-oriented provincial categorization. The four-cluster configuration was retained because it preserved interpretability and fuzzy representation of overlapping provincial rice-distribution characteristics while maintaining acceptable clustering compactness. The fuzziness coefficient was fixed at m = 2.0, a widely adopted setting in logistics and agri-food clustering applications to ensure smooth membership transitions while preserving meaningful cluster separation. During each FCM iteration, the centroid for cluster $j$ is computed using a fuzzy-weighted average of all observations, as shown in Eq. (7):

$c_j=\frac{\sum_{i=1}^N u_{i j}^m y_i}{\sum_{i=1}^N u_{i j}^m}$     (7)

The membership degree $u_{i j}$ is then updated based on the relative distance between observation $x_i$ and each centroid, following Eq. (8):

$u_{i j}=\frac{1}{\sum_{k=1}^c\left(\frac{\left\|y_i-c_j\right\|}{\left\|y_i-c_k\right\|}\right)^{\frac{2}{m-1}}}$     (8)

The iterative process continues until the maximum change in the membership matrix falls below a predefined threshold, indicating that the cluster structure has converged and the final centroids are stable. This procedure follows the classical FCM formulation extensively used in warehousing performance analysis, distribution planning, and multidimensional agricultural efficiency studies.

3.8 Membership degree distribution across provinces

The FCM procedure produces a membership matrix $U \in$ $\mathbb{R}^{114 \times 4}$, in which each province-year observation is assigned four membership values $u_{i 1}, u_{i 2}, u_{i 3}, u_{i 4}$ representing its proximity to the four identified distribution-effectiveness clusters. The resulting membership values range from approximately 0.37 to 0.99, with an average dominant membership of about 0.70. This indicates that most provinces display a clearly dominant cluster affiliation while still retaining partial association with other clusters-an expected characteristic of soft clustering methods like FCM, which capture transitional or mixed structural behaviors in regional distribution systems [18].

This heterogeneous membership pattern reflects substantial variability across Indonesian provinces in terms of production capacity, logistical readiness, and price stability. Provinces with very high membership in a single cluster demonstrate strong alignment with a specific distribution profile, whereas others occupy more intermediate or hybrid states. Table 8 and Figure 4 present selected examples of provincial membership distributions (2022-2023), derived from the PCA-reduced feature space (PC1-PC3).

Figure 4. Fuzzy C-Means (FCM) membership degrees for selected provinces

Table 8. Final Fuzzy C-Means (FCM) membership degrees for selected provinces

The results shown in Table 8 and Figure 4 reveal several notable patterns. Aceh demonstrates a clearly dominant alignment with Cluster 1 (0.884), consistent with its profile as a highly effective region supported by strong production and logistics capacity. In contrast, provinces such as Sumatera Barat and Jambi display more moderate membership levels in Cluster 1 -0.57 and 0.29, respectively indicating transitional or mixed structural characteristics. Meanwhile, Papua Barat Daya and Papua Barat exhibit strong membership in Cluster 3 (0.757 and 0.677), reflecting surplus conditions constrained by logistical bottlenecks, a pattern commonly observed in geographically remote or infrastructure-limited regions. Riau, with a very high membership value in Cluster 4 (0.794), aligns with deficit-prone and logistically vulnerable provinces.

3.9 Characteristics of cluster centroids

The structural characteristics of each cluster can be interpreted from the centroid coordinates in PCA space, where each centroid represents the fuzzy-weighted average of provincial scores. These centroid values capture the dominant production, logistics, and price-volatility features that distinguish the four rice-distribution effectiveness clusters.

Table 9. Fuzzy C-Means (FCM) cluster centroids in Principal Component Analysis (PCA) space (PC1-PC3)

The centroid patterns in Table 9 show clear structural separation among clusters. Cluster 2 displays an exceptionally high PC1 value, indicating provinces with very strong production-surplus capacity and stable distribution conditions. Cluster 3 shows strongly negative PC1 and PC2 values, reflecting very low production output and poor logistics infrastructure, characteristic of structurally vulnerable regions. Cluster 1 combines moderately positive PC1 with the highest PC2 value, identifying provinces with balanced surplus capacity but superior logistics readiness. Cluster 4 presents moderately negative PC1 and small positive PC2, together with the most negative PC3, which signals deficit-prone provinces with weak transport accessibility and volatile price dynamics.

3.10 Effectiveness evaluation

The effectiveness of the PCA-FCM clustering model was assessed using the XB index, a widely used cluster-validity metric capable of evaluating cluster compactness and centroid separation simultaneously. This evaluation approach is consistent with fuzzy clustering studies that emphasize the need for both structural separation and interpretability in multidimensional agrifood data [13]. Table 10 provides an example of the numerator components ($u_{i j}^2 d_{i j}^2$) for five selected provinces. These values represent each observation’s weighted contribution to its nearest centroid.

Table 10. Numerator component of cluster contribution

The total numerator for the PCA-FCM-based XB computation is obtained through four steps:

Step 1: Each province contributes a value of $u_{i j}^2 d_{i j}^2$ based on its membership degree and squared distance to each centroid. For example, for the province of Aceh which is seen as follows:

$0.884^2 d_1^2+0.007^2 d_2^2+0.024^2 d_3^2+0.085^2 d_4^2=0.693$

Step 2: Summing the total contribution of the entire province. After the individual contributions are calculated, all of those values are added together to form the total FCM numerator as seen:

Total numerator $=\sum\left(u_{i j}^2 d_{i j}^2\right)=55.812$

Step 3: Calculating the minimum separation between centroids is obtained by calculating the smallest Euclidean distance between different centroid pairs. The calculation shows the extent of the distance between the clusters:

$\min _{j \neq k}\left\|c_j-c_k\right\|^2=2.419$

Step 4: Substituting numerator and separation values into the XB index formula to obtain the clustering validity value:

$X B=\frac{55.812}{114 \times 2.419}=0.489$

PCA-FCM was selected because provincial rice-distribution patterns exhibit overlapping structural conditions requiring fuzzy membership representation and soft cluster boundaries [34]. A comparison of XB values for benchmarking the PCA-FCM model against two baselines (FCM without PCA and PCA-based K-Means) is shown in Table 11.

Table 11. Comparison of Xie-Beni (XB)

The results demonstrate that PCA-FCM provides a more informative clustering representation than conventional FCM without dimensionality reduction, which remains susceptible to redundancy among the five indicators. Although PCA-KMeans yields the lowest XB index, indicating superior compactness and separation under hard-clustering assumptions, it cannot represent the graded memberships needed to characterize provinces with transitional or overlapping distribution patterns.

By contrast, PCA-FCM achieves a suitable balance between cluster quality and fuzzy interpretability. This capability is particularly important for agrifood distribution systems, in which provinces may exhibit mixed characteristics that cannot be adequately represented by mutually exclusive clusters. As illustrated in Figure 5, the proposed PCA-FCM model therefore offers an effective compromise between structural cluster quality and policy-relevant interpretability for classifying provincial rice-distribution profiles.

Model robustness was examined by varying the fuzzification parameter . This sensitivity analysis ensures that cluster performance is not dependent on a single parameter setting, consistent with fuzzy-clustering evaluation practices in environmental and food-system contexts.

Figure 5. Xie-Beni (XB) comparison chart

Figure 6. Principal Component Analysis-Fuzzy C-Means (PCA-FCM) stability analysis graph

The stability values are summarized in Table 12. Key Findings: At m = 1.8, XB values remain low and consistent, indicating good stability. At m = 2.0, the model achieves optimal performance, with balanced compactness and separation. At m = 2.3, XB increases noticeably, indicating reduced centroid separation and lower structural quality. Figure 6 shows that the increase in the value of m leads to a decrease in cluster separation, reflected in the increase in the value of XB. To complement XB evaluation, additional validation metrics including Silhouette Score and Davies-Bouldin Index were examined to assess cluster cohesion and separation robustness. These complementary metrics strengthen validation reliability beyond fuzzy compactness assessment alone.

Table 12. Additional cluster validation metrics

Table 12 provides a consolidated summary of the defining structural attributes for each cluster, integrating both quantitative PCA-FCM outputs and qualitative distributional indicators. Silhouette Score (0.509), Davies-Bouldin Index (0.581), and XB value (0.489) collectively indicate satisfactory cluster separation and compactness, supporting the robustness and reliability of PCA-FCM for provincial rice-distribution classification. DBSCAN, SOM, Hierarchical Clustering, and GMM were evaluated during model selection. PCA-FCM was adopted because provincial rice-distribution patterns exhibit overlapping characteristics requiring soft membership representation, whereas alternative approaches provide limited support for transitional regional structures.

3.11 Cluster interpretation

The PCA-FCM hybrid model identified four structurally distinct provincial clusters that capture variations in production capacity, logistics performance, price stability, and distribution resilience within Indonesia’s rice-distribution system. Cluster interpretation was performed by combining the indicator-level characteristics, PCA component contributions, and fuzzy membership patterns derived in earlier sections [35]. The PCA-FCM hybrid model identifies four provincial clusters that reflect structural differences in production capacity, logistics readiness, and price stability within Indonesia’s rice-distribution system [36]. Cluster interpretation was based on the average indicator values presented in Table 13, the final FCM centroid positions in the PCA space reported in Table 9, and the corresponding membership degrees. Collectively, these measures provide a concise yet comprehensive explanation of differences in distribution effectiveness across provinces.

Table 13. Stability analysis of Principal Component Analysis-Fuzzy C-Means (PCA-FCM)

Table 14. Indicator characteristics defining each cluster

As summarized in Table 14, Cluster 1 represents provinces with balanced production-consumption conditions, strong logistics readiness, and stable grain prices-characteristics consistent with moderately effective distribution characteristics. Cluster 2 represents provinces with very high production capacity, strong surplus generation, and stable logistics conditions, indicating highly effective structural distribution performance. Cluster 3 reflects provinces facing structural limitations in production and logistics conditions, indicating distribution vulnerability and constrained system resilience. Cluster 4 represents provinces characterized by structural production deficits and logistical limitations, indicating persistent distribution vulnerability associated with limited infrastructure readiness and unstable market conditions. These differences demonstrate how production conditions, logistics readiness, and market stability collectively shape provincial rice-distribution characteristics.

Table 15. Membership comparison across selected provinces

The membership values in Table 15 illustrate variations in cluster dominance across provinces. Aceh and North Sumatra show strong alignment with Cluster 1, indicating relatively balanced production-logistics characteristics. West Java and Riau show prominent memberships in Cluster 4, reflecting surplus conditions constrained by logistical limitations. Papua Barat and Papua Barat Daya strongly align with Cluster 3, indicating structurally vulnerable distribution conditions associated with weaker production-logistics performance. These patterns highlight the usefulness of fuzzy clustering in capturing transitional or mixed characteristics that may not be represented under rigid classification approaches [23].

Cluster assignments should be interpreted as province-year observations rather than fixed provincial identities. Provincial cluster positions may vary across annual observation periods due to changes in production conditions, logistics readiness, market dynamics, and supply-demand balance. This temporal variability reflects dynamic rice-distribution conditions captured by the analytical framework and supports year-sensitive interpretation of provincial clustering patterns.

3.12 Provincial mapping

The PCA-FCM classification results were projected onto a provincial mapping framework to visualize the spatial distribution of rice-distribution effectiveness across Indonesia. This mapping procedure followed established spatial-analysis practices in agrifood logistics and regional accessibility studies [24]. Table 16 summarizes the distribution of provinces assigned to each cluster category.

Table 16. Distribution of provinces by cluster category

The distribution in Table 16 indicates that Cluster 2 (Highly Effective) contains the smallest number of province-year observations, whereas Cluster 4 (Deficit & Logistically Vulnerable) represents the largest group. This pattern reflects national imbalance in production capacity, logistics readiness, and distribution resilience captured across annual provincial observations. The dominance of Cluster 3 suggests that production capacity alone does not guarantee distribution effectiveness when logistical limitations remain present. A detailed listing of the PCA-FCM classifications for all provinces and years is provided in Table 17.

Table 17. Results of Principal Component Analysis-Fuzzy C-Means (PCA-FCM) clusters by province and year

To support spatial visualization, a reproducible pseudocode snippet was generated to link provincial-level shapefiles with cluster labels for the creation of a choropleth map.

These spatial patterns further affirm that the mapping workflow illustrated in Pseudocode 1 reliably represents the regional distribution-effectiveness structure. The spatial patterns revealed by the provincial map align with the structural interpretation of clusters. Western Indonesia-especially Java and northern Sumatra-shows dominance in C2 Highly Effective, supported by strong road infrastructure and production centers. Central regions exhibit mixed performance (C2-C3), while eastern and island provinces predominantly fall under C3 Surplus but Logistically Weak or C4 Deficit & Logistically Vulnerable, reflecting transport disadvantages and price volatility [37]. These spatial outcomes are formalized into a regional typology in Table 18, which links cluster membership with operational policy implications. This table functions as a bridge between statistical results and actionable planning, enabling policymakers to identify region-specific intervention priorities.

As shown in Table 18, each effectiveness category is defined using representative indicators and its associated policy relevance.

Table 18. Regional typology by effectiveness category

This study applied a hybrid PCA-FCM framework to evaluate provincial rice-distribution characteristics in Indonesia by integrating production capacity, consumption balance, logistics readiness, and price stability into a unified assessment model. PCA successfully reduced multicollinearity among rice-volume indicators and revealed three dominant structural dimensions-surplus capacity (PC1), logistics accessibility (PC2), and price-demand volatility (PC3). These components provided a clear foundation for cluster formation and improved interpretability of provincial disparities. The FCM clustering process identified four distinct distribution-effectiveness profiles: (1) Highly Effective regions with strong surplus generation and robust logistics, (2) Moderately Effective regions characterized by balanced production-consumption and stable markets, (3) Surplus but Logistically Weak regions with high production but infrastructure bottlenecks, and (4) Deficit & Logistically Vulnerable regions facing structural constraints in both production and price stability. Spatial and structural analysis further showed that Western Indonesian provinces tended to be concentrated in stronger distribution clusters, while central, eastern, and archipelagic regions tended to fall into weaker clusters due to limited road networks, long distribution distances, and fragmented transport systems. These disparities highlight the persistent geographic inequalities shaping national distribution characteristics. The evaluation using the XB index indicated that the PCA-FCM hybrid model provides an effective balance between cluster compactness and fuzzy interpretability for multidimensional agrifood analysis and policy-oriented classification. This study has several limitations. The analytical framework primarily relies on structural provincial indicators and does not explicitly incorporate operational logistics variables such as interprovincial rice flow, warehousing capacity, transport cost, delivery performance, or market accessibility. Additionally, infrastructure-related indicators contained incomplete observations, particularly concentrated in the 2024 observation period. To preserve empirical authenticity and avoid artificial spatial distortion, original statistical records were retained without synthetic value generation. Future studies may incorporate updated infrastructure statistics and additional operational logistics indicators to strengthen temporal representation and policy-oriented interpretation. Furthermore, partial structural dependency exists between production-related indicators and surplus-deficit measurements because surplus-deficit is mathematically derived from production-demand balance. The surplus-deficit indicator was retained because it directly reflects provincial supply-demand equilibrium conditions that remain operationally relevant for rice-distribution structural profiling. Future studies may further evaluate alternative feature configurations to assess clustering robustness under different indicator compositions. Overall, this research demonstrates that rice-distribution characteristics in Indonesia are influenced not solely by production levels but by the combined influence of logistics capacity and market stability. The resulting cluster typology offers a practical evidence-based tool for policymakers to identify priority regions, strengthen supply corridors, address infrastructure deficits, and enhance the resilience of the national rice-distribution system. The contribution of this study lies in the national-scale implementation of integrated multivariate provincial classification to support evidence-based rice-distribution profiling and policy-oriented regional prioritization.