Application of Two-Step Entropy–TOPSIS Method and Complete Linkage Clustering for Water-Pumping Windmill Investment on Thailand Peninsula

Abstract

This study focuses on identifying suitable areas for the installation of water-pumping windmills in Thailand, which require wind speeds of at least 4 m/s to operate efficiently. A simple combined approach is introduced, integrating the Entropy–TOPSIS method complete linkage clustering to prioritize and categorize potential locations. Out of 271 initial areas, 28 have been selected based on their ability to meet the 4 m/s wind speed threshold. The Entropy–TOPSIS method first evaluates these areas based on monthly wind speed and agricultural area. The analysis reveals that regions with higher wind speeds generally score better for wind energy potential, while areas with larger agricultural spaces tend to score higher for farming suitability. The final integrated scores show that agricultural area is more significant, with a weight of 0.7788, compared to the wind speed weight of 0.2212. The areas are then ranked, and complete linkage clustering groups them into six categories, from the most to the least suitable for windmill installation. A sensitivity analysis confirms the robustness of the clustering method, as the group composition remains stable despite minor changes in weight adjustments. This approach simplifies decision-making for sustainable energy investments in Thailand agriculture sector.

1. Introduction

Thailand is well-known as a leading global agricultural producer, with almost 30% of its workforce engaged in the agricultural sector [1]. The country’s varied topography allows it to cultivate over 13 economic crops, exporting their products worldwide. Agriculture plays a crucial role in Thailand’s national income. However, Thai farmers face challenges such as low annual incomes and significant debt, largely due to high production costs, particularly the energy expenses associated with water-pumping systems. To manage these costs, many Thai farmers have traditionally replaced electric or gasoline/diesel water pumps with wind-powered water-pumping windmills. These windmills are usually 9 to 15 m tall, can operate efficiently at low wind speeds, starting from a cut-in speed of 2 m/s, and can provide water without incurring electricity or fuel costs. According to the Department of Alternative Energy Development and Efficiency (DEDE) [2], the average wind speed in Thailand ranges from approximately 3 to 5 m/s, which is classified as low to moderate. The southern regions of Thailand experience higher average wind speeds compared to other parts of the country. According to data from the Thailand Office of Agriculture Economics [3] and the Thailand Land Development Department [4], nearly 54% of the land in the western region of Southern Thailand is highly suitable for agriculture and five out of six provinces in this region are recognized for their high-quality agricultural production. This indicates strong potential for wind energy investment in these areas. Various researchers [5,6,7,8] evaluated wind potential in agricultural areas by collecting and analyzing wind data with precise instruments and mathematical methods. However, these methods are impractical for Thai farmers due to the lack of access to such instruments and budget constraints. Based on previous studies, it is recommended to prioritize wind quality in the western regions of Southern Thailand using monthly wind data and other agricultural data from secondary sources. Sakon et al. [9,10] utilized either the Analytical Hierarchy Process (AHP) or the complete linkage clustering method with secondary wind data. AHP successfully ranked areas based on wind energy potential, while the complete linkage clustering method categorized areas with similar wind quality. However, these methods were complex to apply to the wind dataset. Simpler approaches, such as the entropy weight method and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), have been explored for prioritizing alternatives based on final scores, aiding in rational decision-making. For instance, Jingwen [11] used a combination of entropy weight and TOPSIS to select an appropriate Management Information System (MIS) based on five purchasing criteria. Similarly, Xiangxin et al. [12] applied the entropy weight method to safety conditions in coal mines and used TOPSIS for evaluation. Jia et al. [13] determined the priority of 3 cold-chain logistics and distribution center locations by integrating 11 indexes, with weights assigned via the entropy weight method and then ranking them using TOPSIS. Fu et al. [14] and Salehi et al. [15] also employed the Entropy–TOPSIS method for selecting wastewater pollution control technologies and assessing management system quality, respectively. Yushan et al. [16] assessed and ranked the quality development levels of nine cities in the Pearl River Delta using a set of criteria across various dimensions: four criteria in economic development, three in innovative development, three in coordination and sharing, three in green development, and three in open development. The evaluation process began with determining the weights for all criteria. Subsequently, the cities’ quality development levels were ranked based on data from 2013 to 2020 using the TOPSIS method. Fu et al. [17] applied the entropy weight method to assign weights to 10 secondary indicators within four primary categories: access openness, transaction openness, exit openness, and transfer openness. They then used the TOPSIS method to evaluate and rank 22 digital platforms in China. Based on their analysis, the platforms were classified into three tiers: four in the top 20%, seven in the middle 20–50%, and eleven in the lower 50%. Wang et al. [18] used the entropy weight method and TOPSIS to choose suitable green retrofitting options for old multistory buildings in severe cold regions. They identified 16 evaluation indicators across 5 dimensions: energy savings, environmental benefits, economic benefits, thermal performance, and building fire protection. The entropy method was used to determine the weights for these indicators. Using these weights along with TOPSIS, they assessed different green retrofitting plans to significantly reduce energy consumption and pollutant emissions in these old multi-storey houses. For more complex multi-criteria decision-making methods, studies [19,20,21,22,23,24] introduced advanced techniques to help decision makers select the most appropriate alternatives. Andrii et al. [19] presented an extension of the Characteristic Objects Method (COMET) by integrating the Expected Solution Point (ESP) approach, resulting in the ESP-COMET method, which aims to improve individual decision-making. When compared to other advanced methods such as Stable Preference Ordering toward the Ideal Solution (SPOTIS) and Reference Ideal Method (RIM), the ESP-COMET method proved to be a simpler way to define customer preferences and better align decisions with individual needs and expectations. Jakub et al. [20] integrated Ranking Comparison (RANCOM) with Analytic Hierarchy Process (AHP) to assess the importance of criteria in housing location selection. This combined approach provided valuable insights into how these techniques perform in various decision-making contexts, contributing to a deeper understanding of their application in the field. The study emphasized RANCOM’s practical effectiveness in addressing complex decision-making problems, underscoring the importance of refining subjective weighting methods and developing robust strategies to handle uncertainties and inaccuracies in expert judgments. Jakub et al. [21] developed a hybrid model by combining Ranking Comparison (RANCOM) and Expected Solution Point Stable Preference Ordering Towards Ideal Solution (ESP-SPOTIS) methods, integrating personalization into multi-criteria decision-making. Validated through electric vehicle selection, the model provided more accurate and tailored recommendations than other multi-criteria methods. Jakub et al. [22] identified a gap in sensitivity analysis, particularly in assessing the effects of changes to the decision matrix. To fill this gap, they introduced the COMprehensive Sensitivity Analysis Method (COMSAM), which systematically adjusts multiple values within the matrix. COMSAM allowed for an in-depth exploration of the problem space, providing insights into how changes across different criteria influence the robustness of results. This study introduced a novel approach to sensitivity analysis, improving understanding of the impact of simultaneous changes on decision outcomes. Bartlomiej et al. [23] conducted a study to compare the Entropy and STD method with the SITW method for re-identifying the TRI medical function. The effectiveness of these methods was evaluated using Spearman’s weighted correlation coefficient across various scenarios and numbers of alternatives. The study concluded that the SITW method outperformed the Entropy and STD method in determining multi-criteria weights by leveraging previously assessed alternatives. Jingjing et al. [24] proposed an intelligent clustering routing approach (ICRA) for unmanned aerial vehicle networks (UANETs), consisting of clustering module, clustering strategy adjustment module and routing module. ICRA was designed to adapt to dynamic environmental conditions and application requirements and it was able to effectively determine the optimal clustering strategy. Extensive studies demonstrated ICRA’s robustness and superior performance over existing methods, with notable improvements in clustering efficiency, topology stability, energy efficiency and quality of service. Although studies [19,20,21,22,23,24] present more advanced, complex, and effective multi-criteria decision-making methods, studies [11,12,13,14,15,16,17,18] show that combining the entropy weight method with TOPSIS is also an effective, yet simpler approach to evaluating or ranking alternatives based on different criteria. Sakon et al. [25] employed a combined approach to prioritize wind quality across different regions in western Southern Thailand. In their study, they calculated entropy weight values based on monthly wind speed data and subsequently ranked the wind quality using the TOPSIS method. However, this approach focused exclusively on wind quality, potentially overlooking areas that, despite having favorable wind conditions, may not be suitable for agricultural purposes due to factors such as limited agricultural land or other constraints.

To enhance the assessment, we propose a more comprehensive analysis that includes not only monthly wind speed data [26] but also agricultural areas [27]. This can be achieved through a two-step approach that integrates the entropy weight method and TOPSIS (Entropy–TOPSIS) to rank potential sites for water-pumping windmill installations. Following this, we suggest applying the complete linkage method to group areas with similar characteristics, ensuring more accurate and meaningful clustering. This step is particularly useful, as studies [10,25,28,29,30,31] have shown that the performance scores of certain alternatives in TOPSIS are often very close to each other. The complete linkage method provides a straightforward and effective means to cluster these alternatives without introducing chain effects, which can sometimes distort the final results. This integrated approach would allow for a more holistic evaluation of suitable areas for the installation of water-pumping windmills, with a focus on both wind potential and agricultural suitability, thus facilitating investment decisions in the agricultural sector along the west coast of Southern Thailand. Additionally, a sensitivity analysis will be performed in the final stage of the study to verify the robustness of the clustering results. What makes this approach particularly appealing is its simplicity; it does not require complex mathematical processes, making it accessible for implementation using basic computational tools.

2. Materials and Methods

To effectively prioritize and group potential locations for installing water-pumping windmills using the dataset of monthly wind speed and agricultural areas [26,27], the process involves four primary steps:

• Initial evaluation of areas based on monthly wind speed data from dataset [26].
• Determining area priorities for each perspective of dataset [26,27] using the Entropy–TOPSIS method.
• Establishing area priorities considering a combination of all perspective using the Entropy–TOPSIS method.
• Clustering areas using the complete linkage method.
• Sensitivity analysis for Entropy–TOPSIS [32,33].

2.1. Initial Evaluation of Areas Based on Monthly Wind Speed Data from Dataset [26]

In this stage, the dataset comprising monthly wind speeds recorded at a height of 10 m across 271 areas along the west coast of southern Thailand [26]. (These monthly wind speed datasets at various heights across Thailand were recorded by Janjai et al. [26] and are served as secondary or open-source data by researchers in the field of renewable energy in Thailand.) Subsequently, all areas where the monthly wind speed of at least 4 m/s should be initially filtered out. This criterion is based on the optimal operational conditions for water-pumping windmills ranging from 9 to 15 m in height, which perform most efficiently when exposed to wind speeds faster than 4 m/s [8].

2.2. Determining Area Priorities for Each Perspective of Dataset Using the EntropyTOPSIS Method

At this stage, the dataset [26,27] consisting of monthly wind speed and agricultural areas is individually analyzed using Entropy–TOPSIS. (These agricultural area datasets across Thailand were recorded and updated by the Land Development Department of Thailand [27] and are served as secondary or open-source data for researchers in agriculture and renewable energy in Thailand.) Initially, data from two perspectives (monthly wind speed and agricultural areas) are organized into a tabular format, as illustrated in

Table 1. The initial table for each perspective of all m areas.

Area	Monthly Wind Speed/Agricultural Areas
	Attribute 1	Attribute 2	Attribute 3	…	Attribute n
A1	x1,1	x1,1	x1,1	…	x1,1
A2	x2,1	x2,2	x2,3	…	x2,n
A3	x3,1	x3,2	x3,3	…	x3,n
︙	︙	︙	︙	︙	︙
Am	xm,1	xm,2	xm,3	…	xm,n

, where “A_i” denotes area i and “x_i,j” signifies the value of area i for attribute j.

For the perspective of monthly wind speed, attributes include January, February, March, April, May, June, July, August, September, October, November, and December. The “x_i,j” values represent the average monthly wind speed (m/s) of each area A_i.

For the perspective of agricultural areas, attributes consist of the prime agricultural area and the agricultural area with high potential. The “x_i,j” values represent the agricultural area (rai) of each area A_i. (Note: 1 rai equals 1600 m².)

Next, all values of x_i,j in each column should be normalized with their own column sum (${\sum }_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{x}}_{\mathrm{i},\mathrm{j}}$) so that the normalized values of r_i,j are calculated from all x_i,j’s with Equation (1) and

Table 2. The normalized table for each perspective of all m areas.

Area	Monthly Wind Speed/Agricultural Areas
	Attribute 1	Attribute 2	Attribute 3	…	Attribute n
A1	r1,1	r1,2	r1,3	…	r1,12
A2	r2,1	r2,2	r2,3	…	r2,12
A3	r3,1	r3,2	r3,3	…	r3,12
︙	︙	︙	︙	︙	︙
Am	rm,1	rm,2	rm,3	…	rm,12

is the normalized table of all monthly wind speed data. Subsequently, all values of “x_i,j” in each column are normalized with their respective column sum (${\sum }_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{x}}_{\mathrm{i},\mathrm{j}}$). This normalization process computes the normalized values of “r_i,j” using Equation (1). Table 2 illustrates the normalized table containing all values of “x_i,j”.

$${\mathrm{r}}_{\mathrm{i},\mathrm{j}}={\displaystyle \frac{{\mathrm{x}}_{\mathrm{i},\mathrm{j}}}{{\sum }_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{x}}_{\mathrm{i},\mathrm{j}}}},\mathrm{i}=1 \mathrm{t}\mathrm{o} \mathrm{m} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{j}=1 \mathrm{t}\mathrm{o} \mathrm{n}$$

(1)

Now, the entropy values of “e_j” for each attribute are determined using Equation (2). Subsequently, attribute weight values of “w_j” are assigned using Equation (3) to quantify the attribute effect. These attribute weight values of “w_j” are incorporated into Table 1, resulting in the formation of the TOPSIS initial table for each perspective encompassing all m areas, with attribute weight as depicted in

Table 3. The initial TOPSIS table for each perspective of all m areas with the attributes’ weights.

Area	Monthly Wind Speed/Agricultural Areas
	Attribute Weight (wj)
	w1	w2	w3	…	wn
	Attribute 1	Attribute 2	Attribute 3	…	Attribute n
A1	x1,1	x1,1	x1,1	…	x1,1
A2	x2,1	x2,2	x2,3	…	x2,n
A3	x3,1	x3,2	x3,3	…	x3,n
︙	︙	︙	︙	︙	︙
Am	xm,1	xm,2	xm,3	…	xm,n

.

$${\mathrm{e}}_{\mathrm{j}}=-{\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{r}}_{\mathrm{i},\mathrm{j}}\mathrm{l}\mathrm{n}{(\mathrm{r}}_{\mathrm{i},\mathrm{j}})}{\mathrm{l}\mathrm{n}\left(\mathrm{m}\right)}},\mathrm{i}=1 \mathrm{t}\mathrm{o} \mathrm{m} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{j}=1 \mathrm{t}\mathrm{o} \mathrm{n}$$

(2)

$${\mathrm{w}}_{\mathrm{j}}={\displaystyle \frac{1-{\mathrm{e}}_{\mathrm{j}}}{\sum _{\mathrm{j}=1}^{\mathrm{n}}(1-{\mathrm{e}}_{\mathrm{j}})}},\mathrm{j}=1 \mathrm{t}\mathrm{o} \mathrm{n}$$

(3)

Next, the data from Table 3 are subjected to the TOPSIS analysis to compute priority values (P_k,i) for the given m areas. This process commences by deriving the b_i,j values from the x_i,j values in Table 3 using Equation (4), leading to the generation of the TOPSIS normalized tables for each perspective for all m areas, as shown in

Table 4. The normalized TOPSIS table for each perspective of all m areas.

Area	Monthly Wind Speed/Agricultural Areas
	Attribute Weight (wj)
	w1	w2	w3	…	wn
	Attribute 1	Attribute 2	Attribute 3	…	Attribute n
A1	b1,1	b1,2	b1,3	…	b1,n
A2	b2,1	b2,2	b2,3	…	b2,n
A3	b3,1	b3,2	b3,3	…	b3,n
︙	︙	︙	︙	︙	︙
Am	bm,1	bm,2	bm,3	…	bm,n

.

$${\mathrm{b}}_{\mathrm{i},\mathrm{j}}={\displaystyle \frac{{\mathrm{x}}_{\mathrm{i},\mathrm{j}}}{\sqrt{{\sum }_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{x}}_{\mathrm{i},\mathrm{j}}^2}}},\mathrm{i}=1 \mathrm{t}\mathrm{o} \mathrm{m} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{j}=1 \mathrm{t}\mathrm{o} \mathrm{n}$$

(4)

Note: Since the data for monthly wind speed and agricultural areas in Table 3 represent the values of the benefit-type criteria, all x_i,j values in Table 3 should be normalized using the vector normalization formula, as shown in Equation (4), rather than the min–max normalization method.

Subsequently, all b_i,j values in Table 4 are multiplied by their corresponding w_j to compute the weighted normalized values of V_i,j as in Equation (5). Consequently, the TOPSIS weighted normalized table for each perspective for all m areas is generated, as shown in

Table 5. The weighted normalized TOPSIS table for each perspective of all m areas.

Area	Monthly Wind Speed/Agricultural Areas
	Attribute Weight (wj)
	w1	w2	w3	…	wn
	Attribute 1	Attribute 2	Attribute 3	…	Attribute n
A1	V1,1	V1,2	V1,3	…	Vn,12
A2	V2,1	V2,2	V2,3	…	Vn,12
A3	V3,1	V3,2	V3,3	…	Vn,12
︙	︙	︙	︙	︙	︙
Am	Vm,1	Vm,2	Vm,3	…	Vn,12
Vj+	V1+	V2+	V3+	…	Vn+
Vj−	V1−	V2−	V3−	…	Vn−

. In Table 5, the V_j+ represents the ideal best value selected from the V_i,j values in the jth column, while the V_j- represents the ideal worst value, also chosen from the V_i,j values in the jth column, following Equations (6) and (7).

V_i,j = e_j × b_i,j

(5)

V_j⁺ = max or min (V_1,j, V_2,j, V_3,j, … , V_n,j)

(6)

V_j⁻ = min or max (V_1,j, V_2,j, V_3,j, … , V_n,j)

(7)

At this point, the Euclidean distance from the ideal best values of each area (S_i⁺) and the Euclidean distance from the ideal worst values of each area (S_i⁻) are calculated using Equations (8) and (9) respectively. Finally, the performance scores (P_k,i) of all areas are determined by Equation (10), with higher P_k,i values indicating greater suitability for the corresponding perspective and lower P_k,i values indicating lesser suitability.

$${\mathrm{S}}_{\mathrm{i}}^{+}=\sqrt{{\sum }_{\mathbf{j}=1}^{\mathit{n}}{({\mathbf{V}}_{\mathbf{i},\mathbf{j}}-{\mathbf{V}}_{\mathbf{j}}^{+})}^2}$$

(8)

$${\mathrm{S}}_{\mathrm{i}}^{-}=\sqrt{{\sum }_{\mathbf{j}=1}^{\mathit{n}}{({\mathbf{V}}_{\mathbf{i},\mathbf{j}}-{\mathbf{V}}_{\mathbf{j}}^{-})}^2}$$

(9)

$${\mathrm{P}}_{\mathrm{k},\mathrm{i}}={\displaystyle \frac{{\mathbf{S}}_{\mathbf{i}}^{-}}{({\mathbf{S}}_{\mathbf{i}}^{+}+{\mathbf{S}}_{\mathbf{i}}^{-})}},\mathrm{i}=1 \mathrm{t}\mathrm{o} \mathrm{m} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{k}=1 \mathrm{t}\mathrm{o} 2$$

(10)

Note: (1) k = 1 corresponds to the perspective of monthly wind speed. (2) k = 2 corresponds to the perspective of agricultural areas.

Finally, all P_k,i values are applicable for generating the performance score table across these two perspectives: monthly wind speed and agricultural areas, as depicted in

Table 6. The performance scores (P_k,i) table for two perspectives of all m areas.

Area	Monthly Wind Speed	Agricultural Areas
A1	P1,1	P2,1
A2	P1,2	P2,2
A3	P1,3	P2,3
︙	︙	︙
Am	P1,m	P2,m

.

2.3. Establishing Area Priorities Considering a Combination of All Perspective Using the EntropyTOPSIS Method

Currently, the two perspectives of monthly wind speed and agricultural areas are being together considered to prioritize all m areas using the Entropy–TOPSIS method. The process begins by normalizing all P_k,i values in Table 6 using Equation (11). Subsequently, the TOPSIS normalized tables for all m areas are generated, as illustrated in

Table 7. The normalized table for two perspectives of all m areas.

Area	Monthly Wind Speed	Agricultural Areas
A1	C1,1	C2,1
A2	C1,2	C2,2
A3	C1,3	C2,3
︙	︙	︙
Am	C1,m	C2,m

. Next, the perspective weight values denoted as “wp_k” are determined using Equation (12) to quantify the perspective effect. These perspective weight values are then integrated into Table 7, resulting in the generating of the normalized table of two perspectives for all m areas represented in

Table 8. The initial TOPSIS table for two perspectives of all m areas.

Area	Perspective Weight (wpk)
	wp1	wp2
	Monthly Wind Speed	Agricultural Areas
A1	C1,1	C2,1
A2	C1,2	C2,2
A3	C1,3	C2,3
︙	︙	︙
Am	C1,m	C2,m

.

$${\mathrm{C}}_{\mathrm{k},\mathrm{i}}={\displaystyle \frac{{\mathrm{P}}_{\mathrm{k},\mathrm{i}}}{{\sum }_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{P}}_{\mathrm{k},\mathrm{i}}}},\mathrm{k}=1 \mathrm{t}\mathrm{o} 2 \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{i}=1 \mathrm{t}\mathrm{o} \mathrm{m}$$

(11)

$${\mathrm{w}\mathrm{p}}_{\mathrm{k}}={\displaystyle \frac{1+\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{C}}_{\mathrm{k},\mathrm{i}}\mathrm{l}\mathrm{n}{(\mathrm{C}}_{\mathrm{k},\mathrm{i}})}{\mathrm{l}\mathrm{n}\left(\mathrm{m}\right)}}{\sum _{\mathrm{j}=1}^4(1+\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{C}}_{\mathrm{k},\mathrm{i}}\mathrm{l}\mathrm{n}{(\mathrm{C}}_{\mathrm{k},\mathrm{i}})}{\mathrm{l}\mathrm{n}\left(\mathrm{m}\right)})}},\mathrm{k}=1 \mathrm{t}\mathrm{o} 2$$

(12)

$${\mathrm{P}}_{\mathrm{i}}={\displaystyle \frac{{\mathrm{S}}_{\mathrm{i}}^{-}}{({\mathrm{S}}_{\mathrm{i}}^{+}+{\mathrm{S}}_{\mathrm{i}}^{-})}}, \mathrm{i}=1 \mathrm{t}\mathrm{o} \mathrm{m}$$

(13)

Note: (1) k = 1 corresponds to the perspective of monthly wind speed. (2) k = 2 corresponds to the perspective of agricultural areas.

After applying Equations (4)–(9) and (13) to data in Table 8, the performance scores (P_i) for all m areas are determined as shown in

Table 9. The TOPSIS performance scores (P_i) of all m areas (combining two perspectives).

Area	A1	A2	A3	…	Am
Pi	P1	P2	P3	…	Pm

. The higher P_i value for an area signifies greater suitability for water-pumping windmill installation, whereas the lower P_i value indicates lesser suitability.

2.4. Clustering Areas Using the Complete Linkage Method

In this step, the pairwise distances are calculated based on all performance scores (P_i) for all m areas in Table 9 using the Euclidean distance defined in Equation (14) to obtain their pairwise distance value (D_ab or D_ba) between areas a and b. Subsequently, the initial m x m distance matrix depicted in Figure 1 is constructed using these D_ab or D_ba values. The next phase involves refining this initial distance matrix using the complete linkage method to cluster similar areas together.

$${\mathrm{D}}_{\mathrm{a}\mathrm{b}}={\mathrm{D}}_{\mathrm{b}\mathrm{a}}=\{\begin{array}{cc}\sqrt{{\left({\mathrm{P}}_{\mathrm{a}}-{\mathrm{P}}_{\mathrm{b}}\right)}^2}=\mid {\mathrm{P}}_{\mathrm{a}}-{\mathrm{P}}_{\mathrm{b}}\mid & ,\mathrm{a}\ne \mathrm{b}\\ 0& ,\mathrm{a}=\mathrm{b}\end{array}$$

(14)

In the subsequent step, the smallest value among all cells in the initial distance matrix (Figure 1) is identified. For example, if the lowest value of D_ab in Figure 1 is D₂₃ or D₃₂ (where D₂₃ = D₃₂), this value is designated as the cluster distance (C.D.) for the n^th round of improvement. If multiple instances of the lowest D_ab values exist, any of these equal values can be selected as the cluster distance (C.D.). Once C.D. = D₂₃ or D₃₂ is determined, areas A₂ and A₃ are merged into a single group, resulting in the updated distance matrix shown in Figure 2. All values of E_a,(b,c) = E_(b,c),a are computed using Equation (15) based on the maximum distance between neighboring groups. These steps are repeated iteratively to group similar areas together until the desired number of clusters is achieved.

E_a,(b,c) = E_(b,c),a = max {D_ab, D_ac}

(15)

2.5. Sensitivity Analysis for EntropyTOPSIS [32,33]

In the final step, the perspective weights for monthly wind speed (wp₁) and agricultural areas (wp₂) in Table 8 are adjusted. This adjustment, by its nature, will influence the TOPSIS performance scores (Pi) for all m areas, as it integrates both perspectives into the overall evaluation. To ensure that the clustering results for all m areas remain consistent when using the complete linkage method in Section 2.4, it is necessary to vary the values of wp₁ and wp₂ within a suitable range. This range can be determined by setting wp₁ and wp₂ based on the relationships specified in Equations (16) and (17). By adjusting these weights within the prescribed range, the integrity of the clustering outcomes from Section 2.4 is preserved, while still allowing for flexibility in how the two perspectives are weighted in the final analysis.

wp₁ + wp₂ = 1

(16)

where 0 ≤ wp₁ ≤ 1 and 0 ≤ wp₂ ≤ 1

wp_{1, new} = σ wp₁ and wp_1,new + wp_2,new = 1

(17)

where (1) σ is the changing parameter. (2) The value of σ can be varied from 0 to ${\displaystyle \frac{1}{{\mathrm{w}\mathrm{p}}_1}}$. (3) The value of wp_1,new is the new perspective weight of monthly wind speed. (4) The value of wp_2,new is the new perspective weight of agricultural areas.

It is important to highlight that the calculation methods described in Section 2.1, Section 2.2, Section 2.3, Section 2.4 and Section 2.5 are relatively simple and do not involve complex mathematical processes. As a result, they can be easily performed using basic computational tools. Any standard spreadsheet software that supports functions for creating, editing and analyzing datasets can be effectively employed for conducting these calculations. This accessibility ensures that the study’s analytical tasks can be carried out with minimal technical requirements, making the methodology user-friendly and adaptable for a wide range of users without the need for specialized software or advanced computational expertise. This flexibility in tool selection also facilitates efficient data manipulation, analysis and visualization, which are integral components of the research process outlined in this study.

3. Results

3.1. Initial Evaluation of Areas Based on Monthly Wind Speed Data from Dataset [26]

Most agricultural farms in Thailand use water-pumping windmills that are 9 to 15 m in height, as they operate effectively even at cut-in wind speeds as low as 3 m/s. However, research [8] suggests that for optimal performance and efficiency, the cut-in wind speed should ideally exceed 4 m/s. Based on this criterion, 28 out of 271 sub-districts on the western side of southern Thailand [26] have a minimum monthly wind speed at a height of 10 m that exceeds 4 m/s.

Table 10. The latitude and longitude coordinates of 28 areas (The monthly wind speed of at least 4 m/s) [26].

Area	A1	A2	A3	A4	A5	A6	A7	A8	A9	A10
Lat. (°N)	8.19	8.38	8.52	8.31	8.03	7.47	7.76	7.81	7.73	7.85
Long. (°E)	99.25	98.72	98.71	98.84	99.37	99.79	99.46	99.69	99.72	99.66
Area	A11	A12	A13	A14	A15	A16	A17	A18	A19	A20
Lat. (°N)	7.87	7.81	7.54	7.60	8.42	8.61	8.82	8.55	10.05	10.13
Long. (°E)	99.73	99.46	99.79	99.75	98.57	98.55	98.50	98.58	98.81	98.73
Area	A21	A22	A23	A24	A25	A26	A27	A28
Lat. (°N)	10.18	10.03	10.03	9.62	9.56	10.64	10.30	6.96
Long. (°E)	98.78	98.85	98.88	98.62	98.68	98.85	98.85	99.99

shows the locations of these 28 areas along with their monthly wind speeds at a height of 10 m.

3.2. Determining Area Priorities for Each Perspective of Dataset Using the EntropyTOPSIS Method

Next, the dataset [26,27], which includes monthly wind speeds (

Table 11. The monthly wind speeds at a height of 10 m of 28 areas (The monthly wind speed of at least 4 m/s) [26].

Areas	Monthly Wind Speed (m/s) at the Height of 10 m												Average Wind Speed (m/s)
	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
A1	4.63	4.56	4.33	4.29	4.69	4.75	4.78	4.95	4.93	4.39	4.05	4.36	4.56
A2	4.01	4.12	4.02	4.27	4.34	4.38	4.36	4.61	4.70	4.43	4.10	4.13	4.29
A3	4.20	4.18	3.98	4.02	4.17	4.23	4.25	4.36	4.36	4.06	4.01	4.39	4.18
A4	4.00	4.07	3.99	4.23	4.35	4.35	4.35	4.55	4.64	4.36	4.05	4.13	4.25
A5	5.17	5.04	4.74	4.26	4.36	4.40	4.44	4.59	4.58	4.19	4.24	4.85	4.57
A6	5.38	5.06	4.71	4.08	4.21	4.21	4.24	4.42	4.39	4.03	4.34	5.40	4.54
A7	5.02	4.84	4.45	4.00	4.01	3.99	4.05	4.23	4.29	4.00	4.12	4.89	4.32
A8	5.12	5.01	4.62	4.30	4.52	4.55	4.57	4.79	4.78	4.26	4.35	5.03	4.66
A9	5.58	5.41	4.92	4.23	4.31	4.38	4.36	4.67	4.64	4.08	4.48	5.39	4.70
A10	4.72	4.65	4.29	4.13	4.38	4.40	4.42	4.59	4.58	4.16	4.08	4.63	4.42
A11	6.42	6.23	5.67	5.02	5.32	5.39	5.4	5.69	5.67	4.97	5.3	6.33	5.62
A12	6.20	5.92	5.40	4.57	4.51	4.52	4.54	4.84	4.94	4.53	4.92	6.06	5.08
A13	6.77	6.47	5.90	4.82	4.80	4.79	4.80	5.08	5.09	4.61	5.28	6.64	5.42
A14	5.71	5.56	5.16	4.39	4.17	4.10	4.11	4.32	4.35	4.00	4.50	5.45	4.65
A15	4.57	4.55	4.25	4.19	4.18	4.19	4.26	4.48	4.54	4.54	4.51	4.75	4.42
A16	5.28	5.08	4.51	4.22	4.44	4.54	4.56	4.90	4.93	4.41	4.52	5.52	4.74
A17	4.91	4.77	4.33	4.03	4.00	4.00	4.04	4.16	4.14	3.96	4.16	4.85	4.27
A18	4.67	4.68	4.73	4.61	4.53	4.59	4.74	5.01	4.79	4.74	4.75	5.04	4.74
A19	6.15	6.03	5.48	4.67	4.51	4.61	4.58	4.91	4.89	4.21	4.81	5.83	5.06
A20	5.29	5.29	4.83	4.35	4.41	4.46	4.48	4.69	4.71	4.09	4.29	4.97	4.66
A21	6.10	6.04	5.51	4.70	4.31	4.25	4.26	4.57	4.69	4.17	4.80	5.70	4.92
A22	6.88	6.74	6.19	5.15	4.67	4.77	4.74	5.16	5.16	4.45	5.26	6.37	5.46
A23	6.56	6.43	5.96	5.08	4.69	4.79	4.75	5.21	5.28	4.58	5.20	6.15	5.39
A24	5.96	5.63	5.39	4.83	4.91	4.79	4.88	4.82	4.62	4.25	4.62	5.60	5.03
A25	5.52	5.29	4.84	4.14	4.09	4.19	4.18	4.44	4.47	3.96	4.39	5.33	4.57
A26	5.72	5.65	5.18	4.43	4.03	4.02	4.00	4.35	4.47	4.07	4.53	5.30	4.65
A27	5.35	5.36	5.00	4.45	4.14	4.10	4.09	4.42	4.52	4.15	4.47	5.01	4.59
A28	6.46	6.18	5.53	4.58	4.04	4.00	3.99	4.29	4.45	4.26	5.12	6.35	4.94

) and agricultural areas (

Table 12. The agricultural areas of 28 areas [27].

Areas	Agricultural Areas (rai)		Areas	Agricultural Areas (rai)
	Prime Agricultural Area	Agricultural Area with High Potential		Prime Agricultural Area	Agricultural Area with High Potential
A1	22,786	52,158	A15	3633	2222
A2	9906	10,782	A16	15	2
A3	32,089	7840	A17	9653	6562
A4	26,450	19,311	A18	2469	1202
A5	9845	9232	A19	1770	35
A6	13,087	4966	A20	1728	12
A7	5753	9226	A21	3698	727
A8	7154	6343	A22	2992	483
A9	9613	4012	A23	964	1
A10	11,971	1536	A24	7448	10,896
A11	385	1335	A25	2786	774
A12	8949	7428	A26	19,890	6616
A13	3155	563	A27	5497	495
A14	3802	2371	A28	18,934	19,704

(Note: 1 rai equals 1600 m²).

), undergoes individual analysis using Entropy–TOPSIS. The analysis begins by applying Equations (1)–(3) to the data in Table 11 and Table 12. This process assigns attribute weight values for each perspective using the Entropy Weight method, as illustrated in

Table 13. The monthly weight values (w_1,j) of monthly wind speeds at a height of 10 m.

The Weight Values (w1,j) for Data in Table 10
Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
W1,1	W1,2	W1,3	W1,4	W1,5	W1,6	W1,7	W1,8	W1,9	W1,10	W1,11	W1,12
0.1920	0.1658	0.1353	0.0458	0.0406	0.0444	0.0448	0.0462	0.0409	0.0299	0.0675	0.1465
Summer Season				Rainy Season						Summer Season

and

Table 14. The weight values (w_2,j) of agricultural areas.

The Weight Values (w2,j) for Data in Table 11
Prime Agricultural Area	Agricultural Area with High Potential
W2,1	W2,2
0.3364	0.6636

. Then, the attribute weight values for each perspective are combined with the data from Table 11 and Table 12 to produce the initial TOPSIS tables for each perspective across all 28 areas, as presented in

Table 15. The initial TOPSIS table for monthly wind speeds at a height of 10 m of 28 areas (the monthly wind speed of at least 4 m/s) with monthly weight values.

Areas	Monthly Wind Speed (m/s) at the Height of 10 m
	W1,1	W1,2	W1,3	W1,4	W1,5	W1,6	W1,7	W1,8	W1,9	W1,10	W1,11	W1,12
	0.1920	0.1658	0.1353	0.0458	0.0406	0.0444	0.0448	0.0462	0.0409	0.0299	0.0675	0.1465
	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
A1	4.63	4.56	4.33	4.29	4.69	4.75	4.78	4.95	4.93	4.39	4.05	4.36
A2	4.01	4.12	4.02	4.27	4.34	4.38	4.36	4.61	4.70	4.43	4.10	4.13
A3	4.20	4.18	3.98	4.02	4.17	4.23	4.25	4.36	4.36	4.06	4.01	4.39
A4	4.00	4.07	3.99	4.23	4.35	4.35	4.35	4.55	4.64	4.36	4.05	4.13
A5	5.17	5.04	4.74	4.26	4.36	4.40	4.44	4.59	4.58	4.19	4.24	4.85
A6	5.38	5.06	4.71	4.08	4.21	4.21	4.24	4.42	4.39	4.03	4.34	5.40
A7	5.02	4.84	4.45	4.00	4.01	3.99	4.05	4.23	4.29	4.00	4.12	4.89
A8	5.12	5.01	4.62	4.30	4.52	4.55	4.57	4.79	4.78	4.26	4.35	5.03
A9	5.58	5.41	4.92	4.23	4.31	4.38	4.36	4.67	4.64	4.08	4.48	5.39
A10	4.72	4.65	4.29	4.13	4.38	4.40	4.42	4.59	4.58	4.16	4.08	4.63
A11	6.42	6.23	5.67	5.02	5.32	5.39	5.4	5.69	5.67	4.97	5.3	6.33
A12	6.20	5.92	5.40	4.57	4.51	4.52	4.54	4.84	4.94	4.53	4.92	6.06
A13	6.77	6.47	5.90	4.82	4.80	4.79	4.80	5.08	5.09	4.61	5.28	6.64
A14	5.71	5.56	5.16	4.39	4.17	4.10	4.11	4.32	4.35	4.00	4.50	5.45
A15	4.57	4.55	4.25	4.19	4.18	4.19	4.26	4.48	4.54	4.54	4.51	4.75
A16	5.28	5.08	4.51	4.22	4.44	4.54	4.56	4.90	4.93	4.41	4.52	5.52
A17	4.91	4.77	4.33	4.03	4.00	4.00	4.04	4.16	4.14	3.96	4.16	4.85
A18	4.67	4.68	4.73	4.61	4.53	4.59	4.74	5.01	4.79	4.74	4.75	5.04
A19	6.15	6.03	5.48	4.67	4.51	4.61	4.58	4.91	4.89	4.21	4.81	5.83
A20	5.29	5.29	4.83	4.35	4.41	4.46	4.48	4.69	4.71	4.09	4.29	4.97
A21	6.10	6.04	5.51	4.70	4.31	4.25	4.26	4.57	4.69	4.17	4.80	5.70
A22	6.88	6.74	6.19	5.15	4.67	4.77	4.74	5.16	5.16	4.45	5.26	6.37
A23	6.56	6.43	5.96	5.08	4.69	4.79	4.75	5.21	5.28	4.58	5.20	6.15
A24	5.96	5.63	5.39	4.83	4.91	4.79	4.88	4.82	4.62	4.25	4.62	5.60
A25	5.52	5.29	4.84	4.14	4.09	4.19	4.18	4.44	4.47	3.96	4.39	5.33
A26	5.72	5.65	5.18	4.43	4.03	4.02	4.00	4.35	4.47	4.07	4.53	5.30
A27	5.35	5.36	5.00	4.45	4.14	4.10	4.09	4.42	4.52	4.15	4.47	5.01
A28	6.46	6.18	5.53	4.58	4.04	4.00	3.99	4.29	4.45	4.26	5.12	6.35

and

Table 16. The initial TOPSIS table for agricultural areas of 28 areas with area weight values.

Areas	Agricultural Areas (rai)		Areas	Agricultural Areas (rai)
	W3,1	W3,2		W3,1	W3,2
	0.3364	0.6636		0.3364	0.6636
	Prime Agricultural Area	Agricultural Area with High Potential		Prime Agricultural Area	Agricultural Area with High Potential
A1	22,786	52,158	A15	3633	2222
A2	9906	10,782	A16	15	2
A3	32,089	7840	A17	9653	6562
A4	26,450	19,311	A18	2469	1202
A5	9845	9232	A19	1770	35
A6	13,087	4966	A20	1728	12
A7	5753	9226	A21	3698	727
A8	7154	6343	A22	2992	483
A9	9613	4012	A23	964	1
A10	11,971	1536	A24	7448	10,896
A11	385	1335	A25	2786	774
A12	8949	7428	A26	19,890	6616
A13	3155	563	A27	5497	495
A14	3802	2371	A28	18,934	19,704

(Note: 1 rai equals 1600 m²).

. Next, Equations (4)–(10) of the TOPSIS method is applied to the data in Table 15 and Table 16. This process calculates the performance scores (P_k,i) of 28 areas for each perspective, which are then summarized in

Table 17. The performance scores (P_k,i) for two perspectives of 28 areas.

Areas	The Performance Scores (Pk,i) for Two Perspectives		Areas	The Performance Scores (Pk,i) for Two Perspectives
	Monthly Wind Speed	Agricultural Areas		Monthly Wind Speed	Agricultural Areas
A1	0.2099	0.9161	A15	0.2016	0.0532
A2	0.0716	0.2178	A16	0.4166	0.0000
A3	0.0741	0.2960	A17	0.2622	0.1498
A4	0.0635	0.4193	A18	0.3018	0.0321
A5	0.3583	0.1922	A19	0.6983	0.0167
A6	0.4160	0.1497	A20	0.4138	0.0163
A7	0.2930	0.1771	A21	0.6712	0.0372
A8	0.3591	0.1340	A22	0.9009	0.0295
A9	0.4927	0.1152	A23	0.8484	0.0091
A10	0.2195	0.1114	A24	0.6209	0.2111
A11	0.8330	0.0246	A25	0.4566	0.0299
A12	0.7040	0.1594	A26	0.5328	0.2122
A13	0.8857	0.0315	A27	0.4341	0.0519
A14	0.5332	0.0563	A28	0.7385	0.3995

. Therefore, Table 17 should serve as the primary dataset for the subsequent analysis using the Entropy–TOPSIS method in the next step. For clearer comparison, Figure 3 displays the area prioritization results based on average wind speed, monthly wind speed and agricultural areas.

3.3. Establishing Area Priorities Considering a Combination of All Perspective Using the EntropyTOPSIS Method

In this process, data from Table 17 undergo an initial analysis using the entropy weight method, as described by Equations (11) and (12), to determine weighting values for two perspectives: Wp₁ = 0.2212 for monthly wind speeds and Wp₂ = 0.7788 for agricultural areas. Subsequently, these weights are integrated with the data from Table 17 to construct the initial TOPSIS table covering these two perspectives across all 28 areas, shown in

Table 18. The initial TOPSIS table for 2 perspectives of 28 areas.

Areas	The performance Scores (Pk,i) for Two Perspectives		Areas	The Performance Scores (Pk,i) for Two Perspectives
	Wp1	Wp2		Wp1	Wp2
	0.2212	0.7788		0.2212	0.7788
	Monthly Wind Speed	Agricultural Areas		Monthly Wind Speed	Agricultural Areas
A1	0.2099	0.9161	A15	0.2016	0.0532
A2	0.0716	0.2178	A16	0.4166	0.0000
A3	0.0741	0.2960	A17	0.2622	0.1498
A4	0.0635	0.4193	A18	0.3018	0.0321
A5	0.3583	0.1922	A19	0.6983	0.0167
A6	0.4160	0.1497	A20	0.4138	0.0163
A7	0.2930	0.1771	A21	0.6712	0.0372
A8	0.3591	0.1340	A22	0.9009	0.0295
A9	0.4927	0.1152	A23	0.8484	0.0091
A10	0.2195	0.1114	A24	0.6209	0.2111
A11	0.8330	0.0246	A25	0.4566	0.0299
A12	0.7040	0.1594	A26	0.5328	0.2122
A13	0.8857	0.0315	A27	0.4341	0.0519
A14	0.5332	0.0563	A28	0.7385	0.3995

. Following the application of Equations (4)–(9) and Equation (13) to the data in Table 18, performance scores (P_i) for all 28 areas are calculated and sorted from highest to lowest, as presented in

Table 19. The area performance scores (P_i) of 28 areas.

Area	A1	A4	A28	A3	A24	A26	A2	A5	A7	A12
Pi	0.9114	0.4519	0.4416	0.3199	0.2401	0.2382	0.2357	0.2122	0.1945	0.1917
Area	A6	A17	A8	A9	A10	A22	A13	A11	A23	A21
Pi	0.1690	0.1647	0.1507	0.1373	0.1227	0.1121	0.1111	0.1029	0.1007	0.0898
Area	A14	A19	A27	A25	A15	A20	A18	A16
Pi	0.0876	0.0849	0.0753	0.0621	0.0607	0.0505	0.0476	0.0472

. In Table 19, the higher performance scores indicate that an area is more suitable for installing the water-pumping windmill system compared to an area with lower scores.

3.4. Clustering Areas Using the Complete Linkage Method

In this step, all performance scores (P_i) for all 28 areas listed in Table 18 are clustered using the complete linkage method, following Equations (14) and (15). Initially, the 28 × 28 distance matrix based on these performance scores is generated, depicted in Figure 4. After 23 iterations, the 28 areas are clustered into 6 groups with a cluster distance (C.D.) of 0.1218, illustrated in Figure 5. Subsequently, Figure 6 is created to clearly display the 6 groups: Group I, II, III, IV, V and VI. Specifically, Group I represents the most suitable areas for installing the water-pumping windmill system, characterized by very high performance scores, whereas Group VI comprises the least suitable areas with lower performance scores.

3.5. Sensitivity Analysis for EntropyTOPSIS [32,33]

By adjusting the weights of wp₁ = 0.2212 and wp₂ = 0.7788 in Table 18 according to Equations (16) and (17), the calculation results indicate that the new weights, wp₁ and wp₂ are related as follows: wp_1,new = σ⋅wp₁ and wp_2,new = 1−wp_1,new where σ is a variable that can range from 0 to 4.5208. It is important to note that when σ = 1, the values of wp₁ and wp₂ remain at their initial values of 0.2212 and 0.7788, respectively. Under these conditions, the weights wp₁ and wp₂ can vary within a range such that their sum always equals 1. Through sensitivity analysis, it shows that to maintain the clustering results from the complete linkage method where the 28 areas are grouped into 6 clusters with the same group members as shown in Figure 6, the value of wp₁ should be adjusted within the range of 0.1749 to 0.2973. Correspondingly, wp₂ should be adjusted within the range of 0.7027 to 0.8251. This adjustment ensures that the clustering results remain consistent, provided that σ is within the range of 0.7904 to 1.3438.

Since the value of σ can be adjusted within the range of 0.7904 to 1.3438 without causing any changes to the clustering results from the complete linkage method, where the 28 areas are grouped into 6 clusters with the group members as shown in Figure 6 (with σ = 1). To provide a clearer understanding of how the clustering results remain stable across this range of σ, the following sensitivity analysis is presented for two specific values of σ: σ = 0.7904 and σ = 1.3438.

(1)

At σ = 1 (For reference), wp₁ = 0.2212 and wp₂ = 0.7788, the grouping is as follows:

Group I: A₁                      Group II: A₄, A₂₈

Group III: A₃                   Group IV: A₂₄, A₂₆, A₂, A₅, A₇, A₁₂

Group V: A₆, A₁₇, A₈, A₉, A₁₀, A₂₂, A₁₃, A₁₁, A₂₃

Group VI: A₂₁, A₁₄, A₁₉, A₂₇, A₂₅, A₁₅, A₂₀, A₁₈, A₁₆

(2)

At σ = 0.7904, wp₁ = 0.1749 and wp₂ = 0.8251, the grouping is as follows:

Group I: A₁                      Group II: A₄, A₂₈

Group III: A₃                   Group IV: A₂, A₂₄, A₂₆, A₅, A₇, A₁₂

Group V: A₆, A₁₇, A₈, A₉, A₁₀, A₂₂, A₁₃, A₁₁, A₂₃

Group VI: A₁₄, A₂₁, A₂₇, A₁₉, A₁₅, A₂₅, A₁₈, A₂₀, A₁₆

(3)

At σ = 1.3438, wp₁ = 0.2973 and wp₂ = 0.7027, the grouping is as follows:

Group I: A₁                      Group II: A₂₈, A₄

Group III: A₃                   Group IV: A₂₄, A₂₆, A₂, A₅, A₁₂, A₇

Group V: A₆, A₁₇, A₈, A₂₂, A₁₃, A₉, A₁₁, A₂₃, A₁₀

Group VI: A₂₁, A₁₉, A₁₄, A₂₇, A₂₅, A₂₀, A₁₆, A₁₅, A₁₈

Table 20. Sensitivity analysis to preserve the clustering results, where all 28 areas are clustered into 6 groups with the same group members as shown in Figure 6.

σ	Wp1	Wp2	Pairwise Reordering Within the Same Group
			Group I	Group II	Group III	Group IV	Group V	Group VI
0.7904	0.1749	0.8251	-	-	-	A2 ⟷ A26 A2 ⟷ A24	-	A15 ⟷ A25 A14 ⟷ A21 A18 ⟷ A20 A19 ⟷ A27
0.8238	0.1822	0.8178	-	-	-	A2 ⟷ A26 A2 ⟷ A24	-	A15 ⟷ A25 A14 ⟷ A21 A18 ⟷ A20 A19 ⟷ A27
0.8239	0.1823	0.8177	-	-	-	A2 ⟷ A26	-	A15 ⟷ A25 A14 ⟷ A21 A18 ⟷ A20 A19 ⟷ A27
0.8691	0.1923	0.8077	-	-	-	A2 ⟷ A26	-	A15 ⟷ A25 A14 ⟷ A21 A18 ⟷ A20
0.8695	0.1927	0.8076	-	-	-	-	-	A15 ⟷ A25 A14 ⟷ A21 A18 ⟷ A20
0.9281	0.2053	0.7947	-	-	-	-	-	A15 ⟷ A25 A14 ⟷ A21
0.9698	0.2146	0.7854	-	-	-	-	-	A15 ⟷ A25
1	0.2212	0.7788	See Figure 6 for reference
1.0133	0.2242	0.7758	-	-	-	-	-	A16 ⟷ A18
1.0664	0.2359	0.7641	-	-	-	-	-	A16 ⟷ A18 A14 ⟷ A19
1.0706	0.2369	0.7631	-	-	-	A7 ⟷ A12	-	A16 ⟷ A18 A14 ⟷ A19
1.0896	0.2433	0.7567	-	-	-	A7 ⟷ A12	A10 ⟷ A22	A16 ⟷ A18 A14 ⟷ A19
1.0999	0.2433	0.7566	-	-	-	A7 ⟷ A12	A10 ⟷ A22 A10 ⟷ A13	A16 ⟷ A18 A14 ⟷ A19
1.1767	0.2603	0.7369	-	-	-	A7 ⟷ A12	A10 ⟷ A22 A10 ⟷ A13 A10 ⟷ A11	A16 ⟷ A18 A14 ⟷ A19
1.1897	0.2632	0.7368	-	-	-	A7 ⟷ A12	A10 ⟷ A22 A10 ⟷ A13 A10 ⟷ A11 A10 ⟷ A23	A16 ⟷ A18 A14 ⟷ A19
1.2047	0.2665	0.7335	-	-	-	A7 ⟷ A12	A10 ⟷ A22 A10 ⟷ A13 A10 ⟷ A11 A10 ⟷ A23	A16 ⟷ A18 A14 ⟷ A19 A15 ⟷ A20
1.2549	0.2776	0.7224	-	-	-	A7 ⟷ A12	A10 ⟷ A22 A10 ⟷ A13 A10 ⟷ A11 A10 ⟷ A23	A16 ⟷ A18 A14 ⟷ A19 A15 ⟷ A20 A15 ⟷ A16
1.2771	0.2825	0.7175	-	-	-	A7 ⟷ A12	A10 ⟷ A22 A10 ⟷ A13 A10 ⟷ A11 A10 ⟷ A23 A9 ⟷ A22	A16 ⟷ A18 A14 ⟷ A19 A15 ⟷ A20 A15 ⟷ A16
1.2875	0.2848	0.7152	-	A4 ⟷ A28	-	A7 ⟷ A12	A10 ⟷ A22 A10 ⟷ A13 A10 ⟷ A11 A10 ⟷ A23 A9 ⟷ A22	A16 ⟷ A18 A14 ⟷ A19 A15 ⟷ A20 A15 ⟷ A16
1.2953	0.2866	0.7134	-	A4 ⟷ A28	-	A7 ⟷ A12	A10 ⟷ A22 A10 ⟷ A13 A10 ⟷ A11 A10 ⟷ A23 A9 ⟷ A22 A9 ⟷ A13	A16 ⟷ A18 A14 ⟷ A19 A15 ⟷ A20 A15 ⟷ A16
1.3438	0.2973	0.7027	-	A4 ⟷ A28	-	A7 ⟷ A12	A10 ⟷ A22 A10 ⟷ A13 A10 ⟷ A11 A10 ⟷ A23 A9 ⟷ A22 A9 ⟷ A13	A16 ⟷ A18 A14 ⟷ A19 A15 ⟷ A20 A15 ⟷ A16

presents the results of the sensitivity analysis, which show that pairwise reordering of two members within the same group occurs gradually as the value of σ adjusts (See Figure 6 for reference). Following are examples:

(1)

When σ = 0.9698, wp₁ = 0.2146 and wp₂ = 0.7854, the order of areas A₁₅ and A₂₅ in Group VI changes. Specifically, the order of area members in Group VI shifts from A₂₁, A₁₄, A₁₉, A₂₇, A₂₅, A₁₅, A₂₀, A₁₈, A₁₆ to A₂₁, A₁₄, A₁₉, A₂₇, A₁₅, A₂₅, A₂₀, A₁₈, A₁₆.

(2)

When σ = 1.0706, wp₁ = 0.2369 and wp₂ = 0.7631, the order of areas A₇ and A₁₂ in Group IV changes. The order in Group IV adjusts from A₂₄, A₂₆, A₂, A₅, A₇, A₁₂ to A₂₄, A₂₆, A₂, A₅, A₁₂, A_7.

Similarly, in Group VI, the order of areas A₁₆ and A₁₈ changes first, followed by the reordering of areas A₁₄ and A₁₉. The sequence of area members in Group VI is adjusted as follows:

-

Initially, the order changes from A₂₁, A₁₄, A₁₉, A₂₇, A₂₅, A₁₅, A₂₀, A₁₈, A₁₆ to A₂₁, A₁₄, A₁₉, A₂₇, A₂₅, A₁₅, A₂₀, A₁₆, A₁₈.

-

Subsequently, the order of areas changes again from A₂₁, A₁₄, A₁₉, A₂₇, A₂₅, A₁₅, A₂₀, A₁₆, A₁₈ to A₂₁, A₁₉, A₁₄, A₂₇, A₂₅, A₁₅, A₂₀, A₁₆, A₁₈.

4. Discussion

Generally, Thai farmers have relied on the average wind velocity data from Table 11 to make investment decisions for water-pumping windmill systems because this approach is straightforward and cost-effective. However, this method overlooks to consider the significant impact of seasonal variations in wind velocity. In this study, the Entropy–TOPSIS method is applied to monthly wind speed data from Table 11. The results indicate that wind speeds during the summer are notably higher than those in the rainy season, with summer wind speeds generally receiving higher weight values compared to the rainy season as shown in Table 13. Additionally, the Entropy–TOPSIS analysis of agricultural areas revealed that the quantity of agricultural area with high potential among these 28 areas is a crucial factor for area prioritization, as demonstrated by the value W_2,2 = 0.6636 in Table 14. In Figure 3, prioritizing all 28 areas based on performance scores derived from monthly wind speeds, compared to using only the average wind velocity data, reveals that seasonal factors significantly affect the results. As a result, the rankings of the areas differ slightly when using monthly wind speeds versus average wind speeds. For instance, area A₁₁ ranks first based on average wind speed data but drops to fourth when considering monthly wind speeds with seasonal variations. Conversely, area A₂₂ ranks 23rd based on the quantity of agricultural area (Total of 3475 rai as shown in Table 12), yet it is ranked 2nd based on average wind speed and 1st based on monthly wind speed. This example suggests that areas with high wind speeds but limited agricultural potential may not be ideal for investment in water-pumping windmill systems. Therefore, it is important to analyze both monthly wind speeds and agricultural land availability together for more accurate area prioritization. To analyze both monthly wind speeds and agricultural land availability together, the Entropy–TOPSIS method is applied once more, incorporating both monthly wind speed performance scores and agricultural area performance scores. The results, presented in Table 18, reveal that agricultural areas hold significantly more weight in area prioritization compared to monthly wind speed, with a weight value of Wp₂ = 0.7788. This influence affects the area prioritization, making the results in Table 19, which combine both monthly wind speed and agricultural area data, somewhat similar to the area prioritization based solely on agricultural areas shown in Figure 3.

To make the results of area prioritization in Table 19 more practical, the complete linkage clustering method is applied. Here is a summary of the process:

(1)

Generate dissimilarity matrix: the 28 × 28 dissimilarity matrix called the 28 × 28 initial matrix is created shown in Figure 4.

(2)

Apply complete linkage clustering: the clustering method is applied iteratively for 23 rounds with C.D. = 0.1218, resulting in the final clustering matrix as depicted in Figure 5.

(3)

Form clusters: the 28 areas were grouped into 6 clusters, as shown in Figure 6:

Group I: Area A₁ (the performance score is 0.9114).

Group II: Areas A₄ and A₂₈ (the performance scores are from 0.4426 to 0.4519).

Group III: Area A₃ (the performance score is 0.3199).

Group IV: Areas A₂, A₅, A₇, A₁₂, A₂₄ and A₂₆ (the performance scores are from 0.1917 to 0.2401).

Group V: Areas A₆, A₈, A₉, A₁₀, A₁₁, A₁₃, A₁₇, A₂₂ and A₂₃ (the performance scores are from 0.1007 to 0.1690).

Group VI: Areas A₁₄, A₁₅, A₁₆, A₁₈, A₁₉, A₂₀, A₂₁, A₂₅ and A₂₇ (the performance scores are from 0.0472 to 0.0898).

Considering both monthly wind speeds and agricultural areas, Group I is identified as the most suitable for investment in water-pumping windmill systems due to its ample agricultural land and adequate wind speeds. In contrast, Group VI is the least suitable, as these areas have limited agricultural land, despite having sufficient wind speeds.

To assess the robustness of the clustering results, the weights wp₁ = 0.2212 and wp₂ = 0.7788 are adjusted across a range of σ values from 0 to 4.5208. Within the range 0.7907 ≤ σ ≤ 1.3438, the corresponding weights 0.1749 ≤ wp₁ ≤ 0.2973 and 0.7027 ≤ wp₂ ≤ 0.8251 are found to maintain the clustering results from the complete linkage method, where all 28 areas are grouped into 6 clusters (Group I to Group VI) with the same group members as shown in Figure 6. However, within this range of σ, some area members may reorder within their respective groups without changing the overall cluster composition. In contrast, when σ falls below 0.7907 (With wp₁ < 0.2212 and wp₂ > 0.7027) or above 1.3438 (With wp₁ > 0.2973 and wp₂ > 0.8251), the clustering results from the complete linkage method no longer hold. In such cases, the clustering outcomes would differ from the results shown in Figure 6, indicating that the stability of the clustering structure is sensitive to values outside this range.

5. Conclusions

This study introduces a two-step Entropy–TOPSIS method in combination with the complete linkage method to prioritize and cluster areas along the west coast of Southern Thailand, based on an analysis of monthly wind speed and agricultural data. Initially, 28 out of 271 areas are selected using the criterion that the minimum monthly wind speed at a height of 10 m exceeds 4 m/s. This threshold is crucial, as water-pumping windmills, which perform most efficiently when installed at heights between 9 and 15 m, require wind speeds above 4 m/s for optimal operation. The Entropy–TOPSIS method is then applied to evaluate these 28 areas based on both monthly wind speed and agricultural land area. First, the areas are ranked individually according to their performance scores for wind speed and agricultural area. Afterward, the Entropy–TOPSIS method combines these individual performance scores into a comprehensive evaluation. The final performance scores for all 28 areas are calculated and ranked using the two-step Entropy–TOPSIS approach. In the subsequent step, the complete linkage method is used to cluster the 28 areas into 6 groups (Group I through Group VI) based on variations in wind speed quality and agricultural area quality. Areas in Group I are identified as the most suitable locations for installing water-pumping windmill systems, while those in Group VI are the least favorable. To ensure the robustness and stability of the clustering results, a sensitivity analysis is conducted. The results show that, although some area members may reorder within their respective groups, the overall composition of the six clusters remains unchanged. This demonstrates that the clustering structure is stable within the given parameters. Finally, this combined method demonstrates its practical value, primarily due to its simplicity, ease of use, and ability to consider multiple criteria (both qualitative and quantitative) simultaneously. The procedure requires no complex calculations, and only standard spreadsheet software with basic functions for data creation, editing and analysis is needed to perform the calculations. This makes the method accessible to decision makers who may not have expertise in computational analysis, while still enabling them to obtain meaningful and reliable results. Furthermore, this approach can be adapted for other decision-making processes related to sustainable energy investments in Thailand’s agriculture sector, offering a versatile tool for similar projects in the future.