1. Introduction

In recent decades, the international community has recognized the urgency of mitigating climate change effects and transitioning toward more sustainable economies. One of the fundamental pillars of this transition is the adoption of renewable energy sources, which offer a cleaner alternative to fossil fuels and promise to significantly reduce greenhouse gas emissions [1]. However, the implementation of renewable energies faces considerable challenges due to the variability of natural resources and the specific geographic and climatic conditions of each region. These factors directly influence the technical and economic feasibility of renewable technologies, complicating the efficient planning of energy matrices across diverse geographical areas [2].

Globally, countries like Germany, the United States, and China have made strides in diversifying their energy matrices by integrating a higher percentage of renewable energies such as solar, wind, and hydroelectric power. However, even in these nations, energy planning remains a complex task, requiring the analysis of multiple criteria to select the optimal energy sources based on regional characteristics and local energy needs [3]. Scientific literature has documented numerous approaches to support this decision-making process, with one of the most notable being the use of multicriteria decision-making methodologies, such as the analytic hierarchy process (AHP). These methodologies allow the decomposition of complex problems into hierarchical structures, facilitating the evaluation of alternatives based on technical, economic, social, and environmental criteria [4].

Colombia, a country characterized by its remarkable geographical and climatic diversity, faces a unique challenge in its transition toward a more sustainable energy system. Despite having abundant renewable resources, such as solar energy in the Caribbean and Orinoquía regions or wind potential in La Guajira, the diversification of its energy matrix has been limited [5]. The country’s historical reliance on hydropower, which accounts for ~70% of its installed capacity, has exposed Colombia to vulnerabilities from extreme weather events like El Niño, which directly impact the stability of its power supply. This underscores the urgent need to diversify energy sources, not only to reduce dependency on a single source but also to enhance the resilience of the national energy system [6].

Despite these challenges, energy planning in Colombia faces an additional obstacle: the variability of regional conditions. The country is divided into several geographical zones that differ significantly in terms of available energy resources, infrastructure, and energy demand needs [7]. Regions such as the Altiplano Cundiboyacense, the Eastern Plains, or the Pacific region present unique climatic and geographical conditions that make the application of a uniform national energy strategy unfeasible. In this context, it is imperative to develop an energy planning approach that adapts to the particularities of each region and allows the selection of the most appropriate energy sources based on a multicriteria analysis [8].

Previous literature has explored various approaches to address this problem. The AHP, developed by Thomas L. Saaty in the 1970s, has become a powerful tool for multicriteria decision-making in energy planning. This method allows complex problems to be broken down into a hierarchical structure that includes criteria and sub-criteria, facilitating the comparison of alternatives through both qualitative and quantitative evaluation [9]. AHP has been widely used in studies that address the selection of energy technologies in various contexts. However, one of the method’s main limitations lies in its inability to adequately handle the uncertainty and subjectivity inherent in human judgments, especially in situations where information is incomplete or imprecise. To address this issue, an extension of AHP that incorporates fuzzy logic has been proposed, known as the fuzzy AHP (FAHP). Fuzzy logic, introduced by Lotfi A. Zadeh in 1965, allows for the management of uncertainty and ambiguity by providing a mathematical framework to handle subjective judgments that cannot be expressed with numerical precision [10]. FAHP has proven particularly useful in applications where data is imprecise or incomplete, such as evaluating energy technologies in diverse geographical contexts with limited information. However, in Colombia, the application of FAHP in energy planning is still in its early stages, and research integrating this approach with the analysis of the country’s specific geographical and climatic conditions is scarce [11].

This study aims to close this gap in the literature by comparing the application of AHP and FAHP methodologies for selecting renewable energy sources across different regions of Colombia. The central hypothesis of this work is that the use of FAHP, in combination with traditional AHP, will provide a more robust decision-making tool, allowing for better management of uncertainty and subjectivity in judgments [12]. To this end, energy alternatives such as photovoltaic solar power, wind energy (onshore and offshore), biomass, geothermal energy, and hydroelectric power (large and small scale) will be evaluated. Each of these alternatives will be assessed based on a set of multidimensional criteria, including technical, economic, social, and environmental aspects. The implementation of these methodologies will not only allow for a more precise and adaptable evaluation of energy alternatives but also lead to more efficient and sustainable energy planning, tailored to the specific characteristics of each region. By integrating AHP and FAHP, this study will provide a practical guide for decision-making in Colombia’s energy planning, offering a methodological tool that can be replicated in other contexts with similar characteristics [13].

As a final comment, it is important to note that Colombia’s geographic and climatic diversity, spanning mountainous regions, extensive coastal areas, and arid zones, presents a unique challenge for energy planning. This study focuses on optimizing the integration of renewable energy within this diverse context, drawing lessons from countries such as Germany, the United States, and China. For instance, Germany has demonstrated the potential of wind energy in regions with limited resources, while the United States has implemented strategies for integrating renewable sources into extensive power grids. China, on the other hand, has excelled in decentralized production, an approach particularly relevant for Colombia’s Non-Interconnected Zones. These international experiences are adapted in this study to address Colombia’s specific needs, exploring solutions that tackle energy inequalities and optimize the use of local resources, such as solar energy in La Guajira and wind energy along the Caribbean coast.

2. Material and Methodology

Today, the selection of suitable energy sources is a critical challenge faced by researchers, engineers, and decision-makers in the context of the transition to a more sustainable and efficient energy system. Growing awareness of environmental impacts, the depletion of fossil resources, and the demand for clean, renewable energy sources has driven the need to make informed decisions in the selection of energy sources. However, this decision-making process has become increasingly complex due to the multitude of variables and criteria that must be considered. In this context, metaheuristic methods have emerged as powerful and versatile tools to address the complexity of energy source selection. These advanced computational techniques allow researchers and practitioners to seek optimal solutions to highly complex and multidimensional decision problems [30]. Metaheuristic methods are based on heuristic search concepts, exploring and exploiting the solution space efficiently and effectively, Figure 1 masters the schematic diagram of the methodological process that is implemented in the AHP and FAHP methodologies in the present research [31].

The importance of metaheuristic methods lies in their ability to find approximate solutions to optimization problems in a reasonable time. In the context of energy source selection, this involves considering a wide range of factors, such as costs, environmental impact, resource availability, and energy demand, among others [31]. The complexity of these problems and the need to make decisions that balance multiple objectives make metaheuristic methods a natural choice. The implementation of metaheuristic methods in the selection of energy sources offers significant advantages. These techniques make it possible to evaluate the different alternatives in a comprehensive and efficient way, taking into account the constraints and preferences of the energy system in question. In addition, they provide an adaptable and scalable approach that can be applied in a variety of scenarios, from the selection of renewable energy technologies to large-scale energy system planning [32, 33].

Figure 2 shows the five main steps that are required to be carried out for the implementation of a metaheuristic method, the first step consists of the selection of criteria and sub-criteria that will serve for the evaluation of the group of experts, these criteria and sub-criteria must be selected very carefully, for this it is required to consult the specialized literature, consultation with personnel of mastery of the research topic, as a second step is the choice of the group of energy alternatives that these energy alternatives must be in accordance with the existing potential in the area of implementation of the research, as a third step is the elaboration of a survey where they are allowed to compare according to the Thomas L. Saaty scale [34], alternatives, criteria, and sub-criteria, the survey is sent to the group of experts where they from their experience and knowledge will give their opinion, when the survey of the experts is available, each matrix of responses is evaluated and determined from the radius and the consistency index in order to determine the validity of the answers given by each expert consulted, if these values are among those allowed according to the size of each matrix, the implementation step of the selected metaheuristic method will be carried out, which in this research were the AHP and the FAHP, finally, the results of the implemented methods will be obtained and the weighting and hierarchization of criteria, subcriteria, and evaluated alternatives will be carried out [35].

The AHP is a methodology developed by mathematician and economist Thomas L. Saaty in the 1970s. This technique has become a fundamental tool in multi-criteria decision-making and has found applications in a wide variety of fields, from investment selection to natural resource management and strategic planning. Thomas L. Saaty, a leading scholar in the field of game theory and operations research, began developing the ideas that would lead to AHP in the early 1970s. The motivation behind their work was to provide a framework that would allow people to make more informed and logical decisions in complex situations involving multiple criteria and alternatives [36]. Saaty formulated the mathematical foundations of the AHP, which are based on matrix theory and mathematical logic.

The central idea of the AHP is to break down a decision-making problem into a hierarchical structure of criteria and sub-criteria, and then use a comparison matrix to capture the preferences and relationships between these elements [34]. In 1980, Thomas L. Saaty published the book entitled “The Process of Analytic Hierarchy” (AHP), in which he exhaustively presented the AHP methodology and its applications in decision-making. This book became a fundamental reference in the field of multicriteria decision-making [37].

As the benefits of the AHP methodology spread, its adoption in various disciplines increased. It has been used in investment project selection, natural resource management, decision-making in strategic planning, choice of technologies, and more recently in artificial intelligence and machine learning applications [38]. Over the decades, Saaty and other researchers have continued to develop and refine the AHP, introducing variants and extensions of the methodology to address specific problems and improve its applicability in real-world situations [34].

The methodology section of this article deploys a meticulous and rigorous approach to addressing the particular challenges associated with the evaluation of renewable energy sources in Colombia. The implementation of metaheuristic methods, specifically TOPSIS and TOPSIS with fuzzy logic, constitutes the backbone of our methodological strategy. These approaches were carefully selected due to their demonstrated ability to deal with the complexity inherent in decision-making in a changing and diverse environment such as Colombia’s [39]. The choice of TOPSIS and TOPSIS with fuzzy logic as metaheuristic methods is the result of careful evaluation of their suitability to address the variables and uncertainty associated with the selection of renewable energy sources. TOPSIS, with its ability to handle multiple criteria and provide trade-offs, offers a robust basis for the weighting and prioritization of energy options [40]. On the other hand, the integration of fuzzy logic in TOPSIS is justified by its ability to model the imprecision inherent in the variability of renewable energy sources, ensuring a more realistic and adaptive approach [41].

The methodology developed takes into account the geographical diversity of Colombia, characterized by variations in altitude, climates, and topographies. The application of these methods is tailored to the particularities of each region, recognizing that there is no single approach that can encompass the complexity of the country’s geographical conditions. The adaptability of TOPSIS and TOPSIS with fuzzy logic is presented as an essential element in ensuring that renewable energy decisions are contextually relevant and strategically informed [30]. The methodology section details the key steps of implementation, from data collection to final evaluation [42]. Specific criteria, such as the availability of natural resources, environmental impact, and economic viability, are included to ensure a holistic assessment of each renewable energy source. In addition, sensitivity analyses are integrated to evaluate the robustness of the results in the face of possible changes in environmental or technological conditions [43].

The validation of the results obtained by applying AHP and AHP with fuzzy logic is done through comparisons with conventional approaches and the review of experts in the field [44]. This validation process guarantees the reliability and practical applicability of the results, strengthening the contribution of this research to the emerging field of efficient renewable energy management in Colombia [45], for this research, seven energy alternatives were used as evidenced in Table 1 and were implemented in five possible scenarios, according to the five geographical regions that comprise the study area, as evidenced in Tables 3 and 4.

As a synthesis of the methodology implemented in this research, it is expressed in Figure 1; however, in the following subsections a general explanation of each of the steps is made.

2.1. AHP Methodology

In summary, a series of explanatory steps for the implementation of the hierarchical analytical process methodology are presented, it is expected that this summary will be easy to understand:

2.1.1. Implementation of the AHP Methodology

2.1.1.1. Step 1: Modeling

In this phase, the hierarchical sequence of problems is implemented and the objectives, criteria, and alternatives to be implemented are defined according to the criteria of the experts. Next, the alternatives through which the criteria to be evaluated are established. These criteria should take into account the problem and identify attributes that help make good decisions [51]. These criteria should be measured at the level, which is the fundamental goal we must achieve to solve the problem, and at the second level, the criteria will be positioned according to a descending hierarchy of one or more variables for each criterion. The third and final layer will be the alternatives in decision-making [52].

2.1.1.2. Step 2: Reviews

Once the alternatives are understood and the criteria defined, each criterion is classified and weighted when selecting the alternatives. The intent of this process is to measure the importance that decision-makers assign to each criterion or alternative i, compared to each criterion or alternative j. A baseline scale of 1–9 was used to assess the relative preference of the items, Table 2 [53].

In this way, we proceed to construct the matrix of paired comparisons, where we obtain a square matrix Anxn = [aij], with 1 ≤ i, j≤ n.

For the construction of the matrix, the following axioms must be considered:

• Axiom of reciprocity: If A is a matrix of paired comparisons, then it is true that if aij = x then aji = 1/x with 1/9 ≤ x ≤ 9

For the property of reciprocity, only n (n−1)/2 comparisons are made:

• Axiom of homogeneity: The elements that are compared with each other will be of the same order of magnitude and hierarchy.

• Axiom of independence: When the decision-maker makes the comparisons, it is assumed that the criteria do not depend on the different alternatives.

By complying with the above axioms, it is possible to determine the desired comparison matrix, Table 5 and in Table 6, example of a comparison matrix.

Table 5. Implementation of the Saaty scale according to the degree of importance (Saaty, 1980).

Value	Definition
1	Equal importance
3	Moderate importance
5	Great importance
7	Very important
9	Extreme importance
2, 4, 6 and 8	Intermediate values

Table 6. Example of a comparison matrix (Own elaboration).

Scale according to the degree of importance	A1	A2	A3
A1	1	A12	A13
A2	A21	1	A23
A3	A31	A32	1

2.1.1.3. Step 3. Prioritization and Synthesis

After comparing the paired matrices, we proceed to calculate since this is a priority part to be made. This underscores the importance that decision-makers attach to each element.

The priority is represented as a vector or vectors, assuming that it is a matrix A (nxn), as obtained by making comparisons in pairs, we call the solution of the equation eigenvalues or eigenvectors of A (λ1, λ2,…,λn): det (A − λI) = 0. The principal eigenvalue (λmax) of the matrix is the maximum value of the eigenvalue obtained by performing the above formula [54].

The principal eigenvalue of {A} and {a} is the associated eigenvector. The eigenvectors associated with the value of probabilities are the weighting vectors that must be achieved [55].

Thus, the eigenvector achieved is that of the criteria matrix, which we call Vc, which represents the relative importance of each criterion selected in the joint evaluation of the alternatives on which we work [56]. When the eigenvector obtained is the eigenvector of the surrogate matrix for a given criterion, we call it Vai (column vector), which represents the relative importance of each surrogate matrix for criterion i where we obtain as many eigenvectors as standard. Something to keep in mind is that the final decision is the consistency of the decision-maker’s decision when completing the matching matrix [57].

In this way, the decision made by the insured is a personal judgment, which leads to an inconsistency that is evaluated to determine if the limits are below what corresponds.

2.1.1.4. Step 4. Consistency Analysis

This analysis takes into account the subjectivity of manufacturing decisions. When performing a process of comparing matrices in pairs, subjectivity is as real and objective as possible because the different elements of one matrix are in turn compared with another matrix [58].

On the other hand, there is a procedure to calculate it. If acceptable, the decision-making process can continue, but if not, more in-depth analysis is required, as it can review judgments about peer comparisons. Calculate the consistency ratio using Equation (1), see Figure 3 [60].

Standard Matrix A:

$$ (1)

The sum of the rows is obtained from Equation (2):

$$ (2)

The priority vector B that is formed is given by Equation (3):

$$ (3)

The product of the original matrix A and the priority vector B forms a matrix of column C, Equation (4):

$$ (4)

CI: Consistency Index

Consequently, the quotient between the column of matrix C and the priority vector B is calculated, obtaining another vector of column D, Equation (5):

$$ (5)

By adding and averaging its elements, the value of the consistency index (CI), Equation (6), is obtained:

$$ (6)

Subsequently, the CI obtained is compared with the random CI in Table 3:

The random consistency (CI) value as a function of the matrix size represents the CI value that would have been obtained if the numerical judgments of the scale had been entered into the comparison matrix in a completely random manner [34].

Therefore, the CI is divided by random consistency, thus obtaining the Inconsistency Index (IR), Equation (7):

$$ (7)

The calculation of the consistency coefficient should not be seen simply as a procedural step, but as a critical element to ensure that judges’ evaluations are fair and not arbitrary. A low consistency index (greater than or equal to 0.10) indicates a high level of agreement among the judges, which reinforces the validity and reliability of the results obtained from the analysis. Conversely, a higher proportion may suggest inconsistencies that need to be addressed to maintain the reliability of the findings. For those interested in understanding the precise methodology for calculating the consistency index, more details can be found in the articles mentioned above. These resources will guide you through the specific steps and formulas needed to accurately perform this important calculation, ensuring that the evaluation process remains robust and credible [34, 61]. Finally, a consistent matrix is considered when the stipulated values for the size of each matrix are not exceeded, Table 7. If a matrix exceeds the consistency coefficient, the assessments made are checked and modified. Figure 4 presents the methodological procedure addressed for each response of each expert in the calculation of the consistency index and radius of the present investigation.

Table 7. Comparison between the obtained CI and the random CI [53].

Die size (n)	1	2	3	4	5	6	7	8	9	10
Random consistency	0.0	0.0	0.52	0.89	1.11	1.25	1.35	1.40	1.45	1.49

2.2. FAHP Methodology

The AHP with FAHP methodology is an extension of the AHP that incorporates fuzzy logic to address uncertainty and ambiguity in multi-criteria decision-making. Although the exact history of its development is not as widely documented as that of the AHP. Fuzzy logic, developed by He, Lu, and Li [62] in the 1960s, provides a framework for representing and managing uncertainty in data and judgments [62]. In an effort to combine the advantages of AHP with fuzzy logic’s ability to deal with uncertainty, researchers began exploring the extension of AHP with fuzzy logic [61]. Throughout the 1990s and 2000s, there was significant growth in research and literature related to FAHP. Researchers from various disciplines, such as engineering, management, and artificial intelligence, contributed to the development and adaptation of FAHP methods for specific applications. As the FAHP methodology gained popularity, specific variants and extensions were developed to address problems [62]. These variants are tailored to the needs of a variety of applications and industries, including project management, engineering decision-making, and strategic planning. Although there is no specific event or prominent figure behind the development of the FAHP as in the case of the AHP, its evolution is a testament to the adaptability and versatility of multicriteria decision-making methods in response to decision-making needs in uncertain environments. The FAHP methodology has proven to be a valuable approach to addressing complex problems in which uncertainty and ambiguity are key elements [63].

In the same way that the steps for the AHP methodology were explained, the steps to achieve the implementation of the fuzzy hierarchical analytical process FAHP are now summarized, see Figure 5 [55]:

This section presents the fundamental concepts that guided this study. The intention is not to cover all topics, but to provide essential support information to understand in a very good way the objective of the research, the context and the results, in this research initially a characterization of a specific population, I seek to evidence the social, economic, environmental, and energy reality of this. To carry out this characterization, the different existing government and research databases were consulted.

It happens that the characterizations of the population nature manage to obtain as results information on the structure and large number of identity attributes of various groups of people with evolutionary continuity over time, which according to their differences configure particular ways of being and being in a territory, in this research it was taken to the department of La Guajira in the country of Colombia, because this area is a territory very rich in natural resources, which makes it a potential in the implementation of renewable energy sources; however, despite this natural disposition, this area of that country is one where its inhabitants suffer from a poor quality of life [65].

These studies that characterize a population in relation to its energy potential also allow us to focus attention on guaranteeing or restoring the effective enjoyment of the rights of population groups, the recognition of their diversity and multiculturalism as social wealth, the particularities and inequalities that hinder or enable their access to the dynamics and benefits of social and territorial development [66]. The importance of correctly directing the appropriate forms of population characterization exercises lies in the fact that they allow the design, adjustment, and implementation of public policies to be based with a view to transforming situations considered problematic and offering goods and services that satisfactorily respond to the needs and interests of population groups [67].

On the other hand, the hierarchical analytical method with fuzzy logic was implemented with the aim of prioritizing the barriers that may arise in the implementation of energies with renewable sources, in the aforementioned study area [68] for the explanation of this methodology was separated into sections for a better understanding.

2.2.1. Fuzzy Assembly Operations [69]

• •

  The complementary set of a fuzzy set $$ One is one whose characteristic function is defined by [70]:

$$ (8)

• •

  The union of two fuzzy sets A and B is a fuzzy set A ∪ B in U whose membership function is:

$$ (9)

• •

  The intersection of two fuzzy sets A and B is a fuzzy set A° B in U

With feature function:

$$ (10)

The main operators that meet the conditions for t-conorms are the maximum operator and the algebraic sum

$$ (11)

And the main operators that meet the conditions to be t-standards are the minimum operator and the algebraic product

$$ (12)

2.2.2. Fuzzy Relationships [71]

$$ (13)

Suppose R(x, y) and S(x, y)

They are relationships in the same product space UxV. The intersection or union between R and S, which are compositions between the two relations, is defined as [61]:

$$ (14)

$$ (15)

If R° S, R, and S belong to discrete universes of discourse. It is defined as a fuzzy relationship in U x W whose membership function is given by

$$ (16)

where the SUP operator is the maximum and the operator ^∗ can be any T-standard. Depending on the t-standard chosen, we can obtain different compositions: The two most commonly used compositions are the maximum-minimum composition and the maximum composition of the product:

• •

  The max-min composition of the fuzzy relations R (U, V) and S (V, W) is a fuzzy relationship R ° S in U x W defined by the membership function

$$ (17)

where (x, z)ɛUxW

• •

  The composition of the maximum product of the fuzzy relations R (U, V) and S (V, W) is a fuzzy relation to R ° S

in U x W defined by characteristic function

$$ (18)

where (x, z)ɛUxW

2.2.3. Fuzzy Implication

As we have already seen, in terms of the theory of fuzzy logic, the proposition “if you is A, then v is B”, where u ε you and v ε V, has an associated characteristic function that takes values in the interval. Examples of possible associated characteristic functions, drawn from the application of the analogies between operators and the aforementioned tautology, are m_A→B = (x, y) [72]:

$$ (19)

$$ (20)

$$ (21)

In fuzzy logic, the Modus Ponens extends to what is called Generalized Modus Ponens and can be summarized as follows:

Premise 1: “u is A^∗”

Premise 2: “if you are A THEN v is B”

Consequence: “v is B^∗”

Where the fuzzy set $$ does not necessarily have to be the same as the fuzzy set A of the rule’s antecedent and the fuzzy set $$ does not necessarily have to be the same as the fuzzy set B that appears in the rule’s consequence.

Thus, the generalized Modus Ponens is a fuzzy composition in which the first fuzzy relation is the fuzzy set $$ and which can be expressed:

$$ (22)

Taking into account that, in the applications of fuzzy logic to engineering, the characteristic function of the implication is built with the minimum and product operators, which in addition to being the simplest preserve the cause–effect relationship, we will have two options to choose from:

$$ (23)

$$ (24)

2.2.4. Methods of Organizing

• •

  Maximum Method: The output variable for which the characteristic function of the fuzzy output set is maximum is chosen. In general, it is not an optimal method, as this maximum value can be reached by several outputs.

• •

  Centroid method: Uses the center of gravity of the output characteristic function as the output of the system. Mathematically [73]:

$$ (25)

It is the most widely used method in applications ranging from fuzzy logic to engineering as a single solution is obtained, although it is sometimes difficult to calculate.

• •

  Height method: The center of gravity of the diffuse output set Bm is calculated for each ruler and then the system output is calculated as the weighted average:

$$ (26)

2.2.5. Fuzzy Logic Analysis

If it is a set of objects and a set of objectives, according to Chang’s (1996) extended analysis method, the extended analysis is developed for each of the values of the objects; In this way, they can be obtained for each objective X = {x₁, x₂, x₃, …x_n}U = {u₁, u₂, u₃, …u_n} [68] Therefore, the extended analysis values of m can be obtained with the following notation [74]:

$$ (27)

where everything is fuzzy triangular numbers. $$

Key steps of the model proposed by Chang (1996) [32]

Step 1: The value of the object-th of the extended analysis is defined as i

$$ (28)

To obtain, the fuzzy addition operation of m values of the extended analysis is performed for a particular matrix, such that: $$

$$ (29)

To get, perform the fuzzy sum of values operation, so that: $$

$$ (30)

Then the inverse vector of the equation is calculated as follows:

$$ (31)

Step 2: The degree of chance of it occurring is defined as M₂ = (l₂, m₂, u₂) ≥ M₁ = (l₁, m₁, u₁)

$$ (32)

And it can be expressed equivalently as follows:

$$

Yes m₂ ≥ m₁

$$ (33)

where d is the ordinate of the highest point of intersection D between y To compare and the values of y are required μ_M1μ_M2M₁M₂V(M₂ ≥ M₁)V(M₁ ≥ M₂)

Step 3: The degree of chance that a fuzzy convex number is greater than k

Convex numbers are defined as

$$

Therefore, assuming that:

$$ (34)

The weight of the vector is given by $$

Where are n elements? A_i(i = 1, 2, 3, …, n)

Step 4: Vector normalization is presented as follows:

$$ (35)

where W is not a fuzzy number, but the set of weights of each matrix.

In this case, it must be taken into account that W are no longer fuzzy numbers, but vectors with the final weights. Table 8 shows the values used in the conversion of the linguistic syntax used by the experts and their respective assessment into fuzzy triangular and triangular numbers. Finally, Figure 6 presents a diagram that explains how the hierarchical analytical process can be implemented or developed with fuzzy logic, with this scheme it is intended to provide an explanatory tool of this process for future research [14, 76].

Table 8. Fuzzy triangular conversion scale [75].

Linguistic scale	Triangular diffuse triangular
Just the same	(0,0,0)
Degradably important	(0,1,3)
Important	(1,3,5)
Much more important	(3,5,7)
Very strongly more important	(5,7,9)
Absolutely more important	(7,9,9)

3. Results and Discussions

The results of this research were grouped by each of the geographical regions, as well as according to the methodology implemented.

4. Conclusion and Recommendations

This research has produced significant insights into the evaluation and selection of renewable energy sources across different regions of Colombia, using the AHP and FAHP methodologies. In examining various energy alternatives, solar PV consistently emerged as a promising option in most regions due to its feasibility and potential for deployment. However, it is crucial to recognize that each region has unique characteristics and resources that influence the weighting and ranking of energy sources. The differences in results between AHP and FAHP highlight the importance of incorporating uncertainty and subjectivity into energy projects. Both methodologies provide valuable, complementary perspectives and should be evaluated based on the specific circumstances and objectives of each region.

The application of AHP and FAHP revealed key differences in their mathematical foundations. AHP is based on linear algebra, while FAHP utilizes fuzzy logic, which addresses uncertainty. A major distinction lies in how these methodologies represent uncertainty—AHP overlooks it, while FAHP incorporates fuzzy logic to handle it. Additionally, AHP relies on subjective judgments, whereas FAHP minimizes subjectivity by applying de-fuzzification techniques.

In the first scenario (SC1), AHP assigns considerably higher weights to alternatives A1 and A2, with 14.35% and 16.22%, respectively. In contrast, FAHP displays more variability, with A6 reaching 35.22%, suggesting that FAHP may better capture fluctuations in preferences. The technical criterion holds significant weight in both methodologies, although FAHP assigns more relevance to environmental aspects, indicating a shift toward more sustainable decisions. FAHP’s greater range of variation in the sub-criteria suggests heightened sensitivity to changes in the evaluated criteria.

The second scenario (SC2) reflects a similar trend, with AHP assigning higher weights to A1 and A2. However, FAHP shows a more balanced weight distribution. The environmental criterion remains important, with AHP allocating 14.16% to A1, while FAHP assigns 21.18% to the same criterion. This highlights the growing importance of sustainability in decision-making, particularly in the context of environmental considerations. In the sub-criteria, FAHP demonstrates more variability, suggesting it is better suited to capturing decision complexities.

In the third scenario (SC3), AHP assigns substantial weight to A1 (34.00%), while FAHP presents a similar value of 32.98%. The environmental criterion remains dominant, with AHP awarding 22.73% to A1. The variability in sub-criteria is notable, with FAHP showing a broader range of weight assignment, indicating its sensitivity to differences in the assessed sub-criteria.

In the fourth scenario (SC4), both AHP and FAHP follow similar patterns, though FAHP places greater emphasis on environmental criteria. A1 and A4 are the leading alternatives, with AHP assigning 11.12% and 34.87%, respectively. The environmental criterion remains crucial, with AHP allocating 23.15% to A1. Variability in sub-criteria is evident, with FAHP exhibiting more dispersion, suggesting that it may more effectively capture decision-making complexity.

Finally, in the fifth scenario (SC5), AHP assigns a weight of 8.52% to A1, while FAHP assigns 10.73%. The environmental criterion retains its significance, with AHP giving 14.68% to A1. The variability in sub-criteria is marked, with FAHP displaying a broader range in weight assignments, suggesting its greater sensitivity to changes in the sub-criteria.

The use of metaheuristic methods in decision-making for renewable energy selection is crucial, especially in a geographically diverse country like Colombia. In such a context, conventional approaches may fall short. Metaheuristic methods like AHP and FAHP can manage the complexity of evaluating various energy alternatives and multidimensional criteria. Their ability to handle uncertainty and subjectivity, particularly in dynamic environments, provides valuable perspectives in decision-making. By overcoming the limitations of traditional methods, these techniques highlight the importance of accounting for the variability of energy potential and resource availability in each region. In Colombia, where geographic diversity means different regions may be more suited to specific renewable energy sources, metaheuristics offer more accurate, informed assessments and contribute to strategic planning for a sustainable, cleaner energy system.

The findings of this study are highly relevant not only for Colombia but also for the broader Latin American region and beyond. In Colombia, the results provide policymakers, energy planners, and industry stakeholders with a systematic, adaptable framework for selecting optimal renewable energy sources. This approach helps diversify the country’s energy matrix and enhance its resilience. The methodology accounts for uncertainty in regional energy planning, contributing to more sustainable and robust energy solutions, particularly in areas with limited data and high variability in natural resources. This approach can also be extended to other Latin American countries, many of which share similar geographical, economic, and environmental challenges. Nations in LATAM, with their own diverse energy potentials, can benefit from a replicable model that enables region-specific planning. On a global scale, this research contributes to the field of energy planning by demonstrating how combining AHP and FAHP can navigate the complexities of renewable energy transitions in countries with diverse geographic landscapes and uncertain resource availability. It serves as a model for other developing economies, supporting global efforts to combat climate change and promote energy sustainability.

Key recommendations include tailoring decisions to the specific characteristics of each region, considering the divergent results of methodologies, and promoting the use of FAHP or other metaheuristic techniques to better address uncertainty in future projects.

Disclosure

It is declared that this document has a forecast which has as cite: Christian Manuel Moreno Rocha, Daina Arenas Buelvas. Comparison between AHP and FAHP for the best Selection of Renewable Energies in Colombia, a country with diversity in its geographical regions: Optimization and Sustainability, February 01, 2024, PREPRINT (Version 1) available at Research Square [https://doi.org/10.21203/rs.3.rs-3910690/v1].

Conflicts of Interest

The authors declare no conflicts of interest.

Author Contributions

I, Christian Manuel Moreno Rocha, am writing to you with the purpose of presenting our article entitled “A Comprehensive Evaluation of Renewable Energy Sources in Colombian Regions Using AHP and FAHP Methodologies” for your consideration and possible publication. Our study focuses on an exhaustive analysis of renewable energy sources in several regions of Colombia, using the AHP and Fuzzy AHP methodologies. Adapting the selection of alternatives, including solar photovoltaic, onshore/offshore wind, biomass, geothermal and hydropower on a large and small scale, aims to provide sustainable solutions for the country’s growing energy demand. In our approach, we have considered multidimensional criteria covering social, economic, technical and environmental aspects, broken down into sub-criteria for meticulous evaluation. The results reveal different weightings and hierarchies for each region, influenced by the variability in energy potential and resource availability. We emphasize the importance of introducing innovative metaheuristic methods, especially AHP and FAHP, in the Colombian energy sector. In a geographically diverse country like Colombia, where regional variability demands flexible approaches, these methodologies emerge as crucial tools for decision-making. Our study not only offers a valuable guide for the effective implementation of renewable energies in Colombia, but also underlines the importance of accommodating regional variability in the transition toward a cleaner and more sustainable energy system. The application of metaheuristic methods contributes to a more effective strategic planning, promoting environmental sustainability within the Colombian context. We believe that this work contributes significantly to the existing knowledge in the field of renewable energies and decision-making in the energy sector. We hope you will consider our article for review and possible inclusion in the next issues. Finally, it is important to highlight the participation of each of the co-authors of this research, to thank Dr. Daina Arena, Mr. Itzjak Vega Machado and Mr. Juan Pacheco Peña, for accepting the invitation to conduct this research and their significant contributions. Thank you in advance for your time and consideration.

Funding

This research was funded with the authors’ resources, however the APC positions will be assumed by the University of Santiago de Compostela, Spain.