Evaluating Taiwan’s Geothermal Sites: A Bounded Rationality Data Envelopment Analysis Approach

Abstract

Amid rising global demand for renewable energy, geothermal power emerges as a vital, low-carbon solution to enhance energy security and sustainability. Taiwan, strategically located on the seismically active Pacific Ring of Fire, possesses an untapped geothermal potential that is underutilized due to complex site selection challenges. This study specifically addresses the need for a more precise and psychologically attuned site selection process, aiming to optimize the development of geothermal resources in regions with complex geological settings. Utilizing the Modified Bounded Rationality Data Envelopment Analysis (MB-DEA) model, this research integrates traditional DEA with bounded rationality to factor in the risk preferences of decision-makers, offering a novel approach that enhances accuracy in evaluating geothermal sites. This study addresses the critical challenge of accurately selecting geothermal energy sites in geologically complex regions like Taiwan, where traditional methods fall short, aiming to significantly boost the efficiency and effectiveness of geothermal energy exploitation as part of Taiwan’s transition to renewable energy sources. Applied to 30 potential sites across Taiwan, our model provides a detailed assessment based on technical, economic, and psychological criteria, revealing variations in site suitability influenced by stakeholder risk attitudes. Key locations such as Datun Mountain, Maoxing, and Taolin consistently rank highly, confirming their robust potential irrespective of risk preferences. At the same time, other sites show marked sensitivity to shifts in decision-making attitudes. This work significantly advances the methodology of renewable energy site selection by demonstrating the utility of incorporating psychological factors into analytical models, which not only refines decision-making processes but also aligns with Taiwan’s strategic energy planning goals. This study also underscores the importance of accurate geographical data in complex terrains, suggesting further refinement and dynamic integration of bounded rationality for future research.

1. Introduction

Renewable energy (RE) is increasingly crucial in the global transition toward sustainable, energy-secure futures. It plays a vital role in mitigating climate change by reducing carbon emissions and lessening reliance on fossil fuels. As of recent years, renewable sources have provided approximately 29% of global electricity, highlighting a shift toward more sustainable energy systems [1]. For Taiwan, this shift is not only a response to environmental concerns but also a strategic move to enhance energy security and economic stability [2].

Amidst a global pivot towards renewables, Taiwan has set ambitious targets to reshape its energy landscape. The government aims to increase renewable energy’s share to 320% of total power consumption by 2025, up from about 5% in recent years [2]. This initiative is part of a broader strategy to decrease dependency on imports, which constitute over 98% of Taiwan’s energy supply, and to phase out nuclear power by the same year. Taiwan’s commitment to environmental sustainability and reducing carbon footprints drives the urgency to diversify energy sources. Diversifying Taiwan’s energy portfolio is a strategic mission critical for achieving energy independence and sustainability. Current efforts focus on expanding capacity in established sectors like solar and wind, which already contribute significantly to the energy mix (over 4500 MW from solar and 800 MW from wind) [3]. However, Taiwan is also exploring less-utilized resources such as geothermal energy, leveraging its unique geological setting to tap into this reliable energy source.

Situated on the Pacific Ring of Fire, Taiwan boasts substantial geothermal potential, estimated to contribute up to 33.64 GW of power [4]. Despite these prospects, geothermal development has been minimal, currently accounting for less than 1% of the nation’s energy production [4]. The high geothermal gradients, averaging 100 °C per kilometer, present an opportunity to enhance Taiwan’s renewable energy output with a stable and sustainable source, providing a critical balance to the more variable solar and wind outputs [5].

The challenge of selecting optimal sites for geothermal development involves a complex interplay of geological, environmental, and socio-economic factors [6,7,8]. Effective site selection is critical, as it directly influences the feasibility and efficiency of energy production [9,10]. Strategic site selection not only optimizes resource extraction but also minimizes environmental impacts and maximizes social and economic benefits, ensuring the long-term sustainability of geothermal projects [11,12].

Data Envelopment Analysis (DEA) is a non-parametric method used in operations research and economics for the assessment of the production efficiency of decision-making units (DMUs) [13]. Unlike parametric techniques that require a predefined functional form relating inputs to outputs, DEA constructs an empirical production frontier that allows for the evaluation of relative efficiency among comparable entities based on multiple inputs and outputs [14]. This makes DEA particularly useful for assessing operational performance where multiple complex processes must be optimized simultaneously.

Bounded rationality is a concept introduced in decision-making processes that acknowledges the limitations of both the cognitive capabilities of individuals and the finite amount of information available to them [15,16]. Prospect theory, a cornerstone of bounded rationality, provides a framework for understanding how people make decisions in situations of uncertainty [17]. It contrasts with the traditional expected utility theory by suggesting that people value gains and losses differently, leading to decision-making that deviates from the classical rational agent model. This theory explains how decisions are influenced by a reference point, with losses typically felt more intensely than equivalent gains [18,19].

Integrating DEA with bounded rationality, specifically through frameworks like Prospect Theory, enhances traditional efficiency analyses by incorporating realistic psychological and behavioral insights into decision-making processes [20]. This integration allows for a more nuanced understanding of efficiency that considers not only the ‘what’ and ‘how’ but also the ‘why’ of decision-making. It acknowledges that decision-makers do not continuously pursue absolute efficiency but may be driven by other practical constraints and preferences, which are often overlooked in classical rationality models [19]. Integrating DEA with bounded rationality is particularly promising for geothermal energy site locations. This approach allows for evaluating sites not only based on their physical and economic outputs but also considering the decision-makers’ risk preferences and psychological biases. By applying this integrated model, planners can identify locations that are technically and economically feasible and align with the strategic goals and subjective preferences of stakeholders. This method addresses the complexity and uncertainty inherent in geothermal site selection, where multiple criteria, such as environmental impact, resource availability, and local community acceptance, must be balanced against economic outcomes.

As discussed in Section 2, there exists a research gap in comprehensively assessing the potential of geothermal energy sites, which are considered by the Taiwanese Government across Taiwan [4]. The primary objective of this research is to apply the Modified Bounded Rationality Data Envelopment Analysis (MB-DEA) model to select a geothermal energy site location (GESL) in Taiwan. This study aims to integrate classical efficiency analysis with concepts of bounded rationality to comprehensively assess potential geothermal energy locations. By doing so, it seeks to determine the most suitable locations for geothermal energy development by applying the MB-DEA model, which involves analyzing sites for their raw performance metrics and alignment with strategic, environmental, and social goals under the constraints of bounded rationality. Additionally, this research will explore the implications of integrating bounded rationality with DEA in environmental and resource management contexts, aiming to advance the theoretical framework of MB-DEA and demonstrate its applicability and value in complex decision-making scenarios typical of renewable energy management. This approach is expected to provide nuanced insights into site selection, balancing empirical efficiency with decision-making’s psychological and strategic dimensions.

This research marks a significant advancement in the application of DEA by integrating it with bounded rationality principles, specifically tailored for geothermal energy site selection in Taiwan. While incorporating bounded rationality into DEA is established, its application to the geothermal energy sector represents a novel exploration. This approach is innovative in addressing the complex interplay of technical, environmental, and socio-economic factors crucial for sustainable energy development. By adjusting the DEA framework to incorporate decision-makers’ risk preferences and cognitive biases, this study pioneers a method that more accurately mirrors the real-world decision-making landscape, where such factors often overshadow purely empirical considerations. This research fills a critical gap by providing a more holistic evaluation tool for site selection. It enhances the robustness of decision-making processes in the context of renewable energy planning. Applying this MB-DEA model to Taiwan’s unique geothermal context is expected to yield insights that could guide policy and practice, setting a new standard for comprehensive site evaluation in the renewable energy domain.

This study introduces notable theoretical and practical novelties in geothermal energy site selection with its MB-DEA model. Theoretically, the MB-DEA enhances traditional DEA by introducing a decision variable to handle the “No Solution” problem and negative values in the objective function, allowing decision-makers to adapt the model based on their psychological and behavioral coefficients. This adjustment reflects a significant innovation, offering a method to capture the complexities of decision-making under risk and uncertainty. Practically, this study applies this novel model to comprehensively assess geothermal sites across Taiwan, which the Taiwanese Government identified.

This research is structured into five sections to provide a comprehensive analysis of the application of the MB-DEA model to geothermal energy site selection in Taiwan. Section 1, the Introduction, establishes the context and significance of the study, outlining the need for renewable energy diversification in Taiwan and introducing the innovative application of the MB-DEA model. Section 2, the Literature Review, examines previous studies, theoretical advancements related to DEA and bounded rationality, and previous applications within the renewable energy sector, highlighting gaps that this study aims to fill. Section 3, the Methodology, details the integration of DEA with bounded rationality principles and describes the approach and criteria used to evaluate potential geothermal sites. Section 4, Numerical Results, presents the findings from applying the MB-DEA model to various candidate sites and analyzing the data to discern the most viable locations for geothermal energy development. Finally, Section 5, the Conclusion, summarizes the study’s findings, discusses the implications for energy policy and site selection practice, and suggests directions for future research.

2. Literature Review

2.1. Overview of Renewable Energy in Taiwan

The global shift towards renewable energy is a response to urgent climate change challenges and the need for secure energy sources. Renewable energy, particularly geothermal power, is increasingly valued for its stability and continuous output. This section provides a focused exploration of the global renewable energy landscape, emphasizing geothermal energy’s significant role. Special attention is dedicated to Taiwan’s geothermal resources, which, due to its location on the seismically active Pacific Ring of Fire, present substantial untapped potential that could critically enhance the country’s energy mix.

Taiwan’s heavy reliance on imported energy catalyzed significant research into renewable resources, addressing energy security and environmental sustainability. A pivotal study by Chen et al. in 2010 reassessed Taiwan’s renewable reserves, including solar, wind, biomass, ocean, geothermal, and hydropower [21]. The findings highlighted the potential of renewables to produce up to 2.75 times the national power generation of 2008, with wind and solar identified as particularly promising due to their scalability and declining cost profiles. However, the challenges of low energy density and high generation costs, along with the instability of supply compared to fossil fuels, were noted as significant hurdles.

Following the Fukushima incident, Chen and Lee’s 2014 paper discussed Taiwan’s shift away from nuclear power, underscoring the necessity for diverse renewable sources [22]. This study reviewed the geographical and technological suitability of solar thermal, photovoltaic, and wind energy, advocating for enhanced government policies to support renewable development and ensure energy independence. By 2019, Chuang et al.‘s evaluation of Taiwan’s power supply stability predicted substantial contributions from offshore wind and solar energy by 2030, though they cautioned that peak summer demands could still pose risks of shortages [23].

In 2023, Tsai provided an extensive review of Taiwan’s renewable energy progress over the last two decades, highlighting the transformative impact of the Renewable Energy Development Act of 2009 [24]. The growth in solar PV from 2009 to 2021 exemplified significant advancements, with plans to expand renewable capacity to over 27 GW by 2025, aligning with global net-zero emission targets. Concurrently, Chen et al. explored the geothermal potential within the Tatun Volcanic Group, identifying promising sites despite environmental and regulatory challenges [25]. Their application of the play fairway methodology helped pinpoint feasible geothermal development sites, proposing the Matsao area as a prime candidate due to its considerable potential and minimal environmental impact.

These studies collectively chart Taiwan’s evolving energy landscape and highlight the crucial role of renewable energy in achieving sustainable development, energy independence, and reduced carbon emissions. The progression aligns with global shifts towards renewable energy, positioning Taiwan as a potential leader in renewable energy innovation and policy adaptation. This context sets the stage for the current study’s focus on enhancing geothermal site selection through advanced analytical methods that integrate psychological factors, offering a novel approach to address the complex challenges inherent in renewable energy development. Despite previous studies identifying and evaluating renewable energy potentials in Taiwan, there remains a significant research gap in comprehensively assessing geothermal sites across Taiwan, specifically those identified by the Taiwanese Government, using methods that integrate technical data and psychological factors influencing decision-makers preferences and risk attitudes. This gap highlights the need for advanced analytical approaches to provide a more holistic understanding of site feasibility and align geothermal development strategies with broader energy policy objectives.

2.2. Integration of DEA and Bounded Rationality

Integrating bounded rationality principles with DEA represents a significant advancement in efficiency analysis, particularly in addressing the cognitive limitations and heuristic biases of decision-makers that traditional DEA models tend to overlook. This subsection synthesizes the theoretical underpinnings and practical motivations for this integration. It highlights its capacity to yield more nuanced and realistic efficiency assessments, especially in environments marked by uncertainty and complex decision-making processes.

In recent studies, innovative methods have been proposed to refine DEA’s application by incorporating psychological factors. In 2023, Shi et al. proposed a method for cross-efficiency evaluation that aligns management objectives with Management by Objectives (MBO) theory, influenced by Prospect Theory [26]. The study emphasized the significant impact of ‘gain and loss’ psychology, which arises from choosing management objectives as reference points, potentially distorting decision-makers’ rationality. Their model, incorporating organizational, personal, and composite objectives, offers a more flexible approach to performance evaluation and has been validated through DEA ranking examples.

Similarly, Ning et al. introduced the EPCE model in the same year, an enhancement of traditional DEA cross-efficiency that accounts for decision-makers’ non-ideal rationality by integrating their risk attitudes [27]. This approach provides a more realistic framework for evaluating and ranking decision-making units, particularly in multi-criteria decision environments, as demonstrated through an empirical analysis of mutual fund investments in the Chinese market.

Chen’s 2024 study further advanced DEA by integrating Prospect Theory to account for the non-rational subjective preferences of decision-makers [28]. This research developed a novel method of aggregating efficiency scores using a distance entropy function, which surpasses the limitations of traditional arithmetic averaging. Applied empirically to high-tech industries across 29 Chinese provinces, this model effectively provided a detailed and nuanced ranking of decision-making units.

Building on these advancements, Braaten and Tsai’s forthcoming study explores the implications of bounded rationality in legal settings, analyzing corporate fraud penalties with data from the Corporate Prosecution Registry [29]. Their findings underscore the influence of internal and external factors on prosecutorial decisions, such as the Department of Justice’s involvement and the corporation’s nationality.

Integrating DEA with bounded rationality principles is a critical evolution in efficiency analysis, effectively addressing decision-makers’ cognitive biases previously ignored by traditional DEA models. This progression is showcased across various sectors, from high-tech industries to the legal domain, enhancing the accuracy and realism of DEA evaluations under real-world constraints. Inspired by the pioneering work of Chen and colleagues in 2019 [30], who introduced the Behavior DEA (BDEA) model, this study seeks to address specific limitations observed in traditional DEA applications, particularly in scenarios where total gains are smaller than total losses, which can result in no solution due to negative objective function values. The current research builds on this foundation, aiming to refine and expand the application of DEA in geothermal site selection by incorporating a more comprehensive set of cognitive and behavioral dynamics.

2.3. Geothermal Energy Site Selection

Site selection is a pivotal process in geothermal energy development, involving complex evaluations that must consider geological, environmental, and socio-economic factors. This section delves into the methodologies and criteria traditionally employed in the site selection process for geothermal projects. It critically examines how these methods have evolved and discusses the incorporation of advanced analytical tools to improve accuracy and efficiency.

Mostafaeipour et al. evaluated the geothermal energy potential across 21 provinces in Afghanistan using multiple multi-criteria decision-making (MCDM) methods, including SWARA, ARAS, TOPSIS, VIKOR, and WASPAS, in 2020 [31]. The research highlights Afghanistan’s untapped geothermal resources and provides a systematic approach to identifying the most viable sites for geothermal development, with Ghazni Province emerging as the most suitable location. Sensitivity analysis further underscored the robustness of these methods, revealing that changes in criteria weights significantly impacted the rankings, except in the VIKOR method, where volcanic dome density was deemed the most critical criterion. In 2021, Uliasz-Misiak et al. explored the dual use of geological formations for carbon dioxide storage and geothermal energy production, addressing the need for criteria in selecting suitable sites [32]. The paper proposes a set of 12 geological criteria for this purpose, assessed through the Analytical Hierarchy Process (AHP) based on expert opinions. This approach facilitates a systematic evaluation and ranking of sites, contributing to more informed and effective decision-making in the integration of carbon capture and storage (CCS) with geothermal energy exploitation. Additionally, Meng et al. focus on accurately identifying geothermal resource locations in northeastern China’s Changbai Mountain region using five criteria: shallow ground temperature, fault lines, hot springs, basalt distribution, and iso gravity lines [33]. These criteria were mapped and weighted through the analytic hierarchy process and entropy weight method, resulting in a multi-criteria decision analysis using GIS. The geothermal potential areas were categorized into six grades, with three existing geothermal wells verifying the high accuracy of the identified favorable zones, demonstrating the effectiveness of the approach for efficient geothermal exploration. Currently, Yalcin et al. utilize the Maximum Entropy (MaxEnt) Method and Multi-criteria Decision Analysis (MCDA) to identify geothermal potential in the Büyük Menderes Graben (BMG) of Turkey’s Aegean Region [34]. Combining the GIS-based MaxEnt machine learning method and the AHP, the study produces geothermal favorability maps. These maps, categorized using the Natural Breaks Jenks Method, show high accuracy in identifying geothermal resources, with the MaxEnt method particularly sensitive to geological parameters like cap rock geology and faults. This method serves as a valuable guide for further geothermal exploration, enhancing potential applications in thermal tourism, housing heating, greenhouse operations, and renewable energy production in Turkey [34].

In renewable energy site selection, various methodologies such as SWARA, ARAS, TOPSIS, VIKOR, and WASPAS are utilized within multi-criteria decision-making frameworks to systematically assess potential sites, though their outcomes can be significantly impacted by changes in criteria weights. The AHP, often combined with GIS, structures decision-making and enhances geographic specificity but relies heavily on expert opinions and accurate geological data. Additionally, the MaxEnt method, used with MCDA, employs machine learning to create detailed geothermal favorability maps, yet its effectiveness is contingent on the quality of geological data. These methods, while innovative, face common challenges of data dependency and subjective criteria weighting, underscoring the need for methodological refinement and new approaches for more reliable site selection in renewable energy.

Despite significant advancements in the field of renewable energy, the integration and application of comprehensive methodologies for geothermal energy site selection remain underexplored. While global trends underscore the importance of renewable energy, particularly geothermal power, due to its stability and reliability, there is a lack of specific studies focusing on the potential within Taiwan, a region with considerable geothermal resources. Traditional site selection methodologies have been predominantly empirical, often neglecting the nuanced decision-making processes influenced by bounded rationality. Although the integration of DEA with bounded rationality principles has been theoretically proposed, its practical application, especially within the context of geothermal energy site selection, is limited. Existing approaches do not sufficiently address the complexity of site selection, which requires balancing technical efficiency with environmental and socio-economic considerations. This research aims to fill this gap by applying a bounded rationality DEA model to the geothermal energy sector in Taiwan, providing a more holistic and realistic framework for site selection that accounts for both empirical data and human decision-making behaviors.

3. Methodology

3.1. Preliminaries

3.1.1. Data Envelopment Analysis

In 1978, a landmark advancement in the field of operations research and efficiency assessment occurred when Charnes and his colleagues introduced the innovative DEA model, widely known as the CCR model. This model became instrumental in evaluating the technical efficiency of DMUs across various sectors. The core assumption of the CCR model was constant returns to scale, a critical concept in optimization [35]. However, as the model was applied in diverse practical contexts, it became clear that this assumption did not always hold true. This limitation led to further developments in DEA methodologies. Consequently, Banker and his research team developed the BCC model, which allowed for variable returns to scale, thereby providing greater flexibility and realism in the analysis [36]. The DEA framework, encompassing both the CCR and BCC models, serves as an essential tool for assessing the performance of a set of DMUs. Each DMU is represented as $i=1\dots I$ utilizes multiple inputs $(j=1\dots J)$ to produce various outputs $(t=1\dots T)$. To calculate the technical efficiency ($E_k$) for each $k$th DMU, one must solve the following mathematical model (1), which captures the complex relationships between inputs and outputs. This modeling approach is vital for not only measuring efficiency but also identifying opportunities for improvement and optimizing resource allocation within these DMUs. The DEA methodology continues to be a foundational approach to tackling efficiency and performance evaluation challenges across numerous industries and sectors, making it an ever-relevant and indispensable tool for decision-makers and analysts.

$$\begin{gathered} maximize E_k=\varphi +\sum _{t=1}^T{u}_t{m}_{tk} \\ subject to \\ \sum _{j=1}^J{v}_j{n}_{jk}=1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \\ \rho +\sum _{t=1}^T{u}_t{m}_{ti}-\sum _{j=1}^J{v}_j{n}_{ji}\le 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}i=1, \dots ,I \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{u}_t, v_j\ge 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}j=1, \dots ,J ; t=1,\dots ,T \\ \rho \mathrm{is} \mathrm{free}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \end{gathered}$$

(1)

In the model (1), $u_t$ and $v_j$ signify the weights assigned to the $t$th and the $j$th input, respectively. These weights are essential as they indicate the relative significance of each input and output in the efficiency evaluation process. Additionally, the values $n_{ji} \mathrm{and} m_{ti}$ are important, where $n_{ji}$ refers to the value of the $j$th input, and $m_{ti}$ refers to the value of the $t$th output for the $i$th DMU. These values provide the actual data for the inputs and outputs used in the efficiency calculation, serving as the basis upon which DEA assesses the performance of DMUs. The primary objective of DEA is to determine the efficiency of each DMU. A DMU is deemed efficient when its technical efficiency $\left(E_k\right)$ equals 1. In this framework, technical efficiency means that the DMU is operating at its optimal level, utilizing its inputs fully to produce the desired outputs without any inefficiencies. Achieving a technical efficiency score of 1 signifies outstanding performance and indicates that the DMU is operating on the frontier of its production possibilities, setting a benchmark for other units to follow.

3.1.2. Bounded Rationality and Prospect Theory

Bounded rationality, introduced by Herbert A. Simon in the 1950s, posits that individuals are limited in their cognitive capabilities, information availability, and time when making decisions [37]. Unlike classical economic theory, which assumes that individuals have unlimited cognitive resources and access to all relevant information, bounded rationality acknowledges the constraints and imperfections in human decision-making processes. According to bounded rationality, individuals use heuristics, or mental shortcuts, to simplify complex decision-making tasks. These heuristics are not always optimal but are satisfactory and practical given the limitations faced by individuals. Bounded rationality emphasizes the idea of “satisficing”—seeking a solution that is good enough rather than optimal. This approach better reflects real-world decision-making, where perfect rationality is unattainable.

Prospect theory, developed by Daniel Kahneman and Amos Tversky in 1979, offers an alternative to the expected utility theory, which traditionally explained decision-making under risk [17]. Prospect theory provides a more accurate description of how people perceive and evaluate potential losses and gains. The theory is based on several key principles:

• Reference Dependence: Individuals evaluate outcomes relative to a reference point, typically their current state, rather than considering absolute outcomes. Gains and losses are perceived relative to this reference point.
• Probability Weighting: People tend to overweigh small probabilities and underweight large probabilities. This leads to non-linear probability weighting, where unlikely events are given more weight than they statistically deserve and more likely events are underweighted.
• Diminishing Sensitivity: The value function in prospect theory is concave for gains and convex for losses, indicating diminishing sensitivity to changes in wealth. This means that the subjective difference between gaining USD 100 and USD 200 is greater than between gaining USD 1100 and USD 1200, and similarly for losses.

Loss Aversion: Losses loom larger than gains. People tend to experience the pain of a loss more intensely than the pleasure of an equivalent gain. This asymmetry explains why individuals are often risk-averse when it comes to gains but risk-seeking when trying to avoid losses.

Prospect theory has been influential in explaining various economic behaviors and anomalies, such as why people buy insurance or lottery tickets, and it has broad applications in finance, marketing, and public policy [38,39]. These principles collectively shape the construction of the prospective value function, which describes how individuals perceive and evaluate outcomes. This function is typically depicted as an asymmetrical S-shaped curve, visually representing the interaction of these principles, as illustrated in Figure 1. Reference dependence divides the prospect value function into two distinct regions: one for gains and one for losses. Loss aversion is evident in the differing slopes of these regions, with the loss side being steeper due to individuals’ heightened sensitivity to losses. The principle of diminishing sensitivity is reflected in the function’s convex shape for gains and concave shape for losses. Mathematically, the prospect value function is represented by Equation (2), where $\mathsf{\Delta }t$ indicates the amount of loss or gain relative to the reference point. The parameters $\gamma$, $\delta$, and $\theta$ correspond to the decision-makers risk attitude towards gains, risk attitude toward losses, and degree of loss aversion, respectively. These parameters create a quantitative model that elucidates how individuals’ decisions are shaped by prospect theory, making it a powerful tool for understanding and predicting human behavior in various decision-making scenarios.

$$f\left(\Delta t\right)=\left\{\begin{array}{cc}{\left(\Delta t\right)}^{\gamma }& ,\forall \Delta t\ge 0;0<\gamma <1\\ -{\theta \left(-\Delta t\right)}^{\delta }& ,\forall \Delta t<0;0<\delta <1\end{array}\right.$$

(2)

3.2. Bounded Rationality Data Envelopment Analysis Model

Building on the core concepts of prospect theory, Chen et al. have developed an innovative approach in the context of DEA, introducing a novel perspective on behavioral decision-making and its implications for evaluating efficiency when considering risk [30]. To overcome the “No Solution” problem and give decision-makers freedom in determining psychological and behavioral coefficients, the model in this study has been modified, and a decision variable has been added. Its purpose is to handle the problem of the model’s objective function receiving negative values. This MB-DEA model is structured through a series of distinct steps, each tailored to reflect the cognitive dynamics exhibited by individuals when making decisions under conditions of risk and uncertainty.

Step 1: Normalization of Inputs and Outputs

The initial step of this approach involves the normalization of inputs ($x_{ji}$) and outputs ($y_{ij}$), as outlined in Equations (3) and (4). Normalization is essential, as it ensures a fair and balanced comparison across different decision-making units while also addressing the biases and subjective elements inherent in human decision-making.

$$x_{ji}={\displaystyle \frac{n_{ji}^{max}-n_{ji}}{n_{ji}^{max}-n_{ji}^{min}}}\hspace{1em}\hspace{1em}\hspace{1em}i=1,\dots ,I;j=1,\dots ,J$$

(3)

$$y_{ti}={\displaystyle \frac{m_{ti}-m_{ti}^{min}}{m_{ti}^{max}-m_{ti}^{min}}}\hspace{1em}\hspace{1em}\hspace{1em}i=1,\dots ,I;t=1,\dots ,T$$

(4)

Step 2: Identification of Reference Points

Incorporating the psychological insights of prospect theory, the next phase of the model involves establishing both positive and negative reference points. These are detailed in Equations (5) and (6) and play a crucial role in shaping how gains and losses are perceived by individuals. These reference points act as benchmarks, allowing for the evaluation of performance in terms of gains and losses, and are a direct application of the reference dependence principle from prospect theory.

The positive reference points ($n_j^{+} \mathrm{a}\mathrm{n}\mathrm{d} m_t^{+}$):

$$n_j^{+}=\underset{i}{\min }\left(x_{ji}\right);m_t^{+}=\underset{i}{\max }\left(y_{ti}\right)$$

(5)

The negative reference points ($n_j^{-} \mathrm{a}\mathrm{n}\mathrm{d} m_t^{-}$):

$$n_j^{-}=\underset{i}{\max }\left(x_{ji}\right);m_t^{-}=\underset{i}{\min }\left(y_{ti}\right)$$

(6)

Step 3: Application of MB-DEA Model

The third and crucial final step involves constructing the MB-DEA model, detailed in Equation (7). This model effectively integrates the normalized values of inputs and outputs, the established reference points, and the principle of diminishing sensitivity that dictates responses to gains and losses. Within this framework, the parameter $\lambda$ plays a critical role, representing the relative emphasis placed on gains versus losses. When $\lambda$ is set at 0.5, it indicates a balanced approach, suggesting that decision-makers weigh gains and losses equally.

$$\begin{gathered} maximize Z=\beta +\lambda \left(\rho +\sum _{t=1}^T{u}_{tk}{\left(y_{tk}-m_t^{-}\right)}^{\gamma }+\sum _{j=1}^J{v}_{jk}{\left(n_j^{-}-x_{jk}\right)}^{\gamma }\right) \\ -\left(1-\lambda \right)\left(\rho +\sum _{t=1}^T{u}_{tk}\theta {\left(m_t^{+}-y_{tk}\right)}^{\delta }+\sum _{j=1}^J{v}_{jk}\theta {\left(x_{jk}-n_j^{+}\right)}^{\delta }\right) \\ subject to\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\sum _{j=1}^J{v}_{jk}{n}_{jk}=1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \\ \rho +\sum _{t=1}^T{u}_{ti}{m}_{ti}-\sum _{j=1}^J{v}_{ji}{n}_{ji}\le 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}i=1, \dots ,I \\ {u}_{ti}, v_{ji}\ge 0,\rho \mathrm{is} \mathrm{free}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}j=1, \dots ,J;t=1,\dots ,T;i=1, \dots ,I \\ \left(1-\lambda \right)\left(\rho +\sum _{t=1}^T{u}_{tk}\theta {\left(m_t^{+}-y_{tk}\right)}^{\delta }+\sum _{j=1}^J{v}_{jk}\theta {\left(x_{jk}-n_j^{+}\right)}^{\delta }\right) \\ -\lambda \left(\rho +\sum _{t=1}^T{u}_{tk}{\left(y_{tk}-m_t^{-}\right)}^{\gamma }+\sum _{j=1}^J{v}_{jk}{\left(n_j^{-}-x_{jk}\right)}^{\gamma }\right)\le \beta \end{gathered}$$

(7)

4. Numerical Results

4.1. Problem Descriptions

This section presents the numerical results obtained from the application of the MB-DEA model to the evaluation of potential geothermal energy sites in Taiwan. In this study, thirty potential geothermal energy site locations (GESL) across Taiwan are considered, as identified in the current research supported by the Taiwanese Government [4]. These sites have been selected based on their geological and geothermal properties, which suggest a high potential for sustainable energy production. Detailed information about each of these locations is systematically presented in

Table 1. The potential geothermal energy site locations in Taiwan.

Province	GESL Name	GESL Codename	Latitude	Longitude
New Taipei City	Datun Mountain	GESL1	25.1714752	121.5199143
	Wulai	GESL2	24.8919809	121.5094713
Yilan County	Jiaoxi	GESL3	24.8507003	121.7399571
	Tuchang	GESL4	24.7488416	121.7102775
	Renze	GESL5	24.5633874	121.5148056
	Wujie Township	GESL6	24.4261040	121.4812896
	District four	GESL7	24.5077571	121.8076125
	Smelly site	GESL8	24.3302224	121.6699538
	Maoxing	GESL9	24.3898246	121.5628072
	Cingshui	GESL10	24.3842906	121.7517199
Taichung	Malling	GESL11	24.2318907	121.0588558
	Guguan	GESL12	24.2000738	121.0050623
Hualien County	Fuyuan	GESL13	23.5885555	121.3683908
	Ruisui	GESL14	23.5077679	121.3276944
	Antong	GESL15	23.2967448	121.3438667
Chiayi County	Jhonglun	GESL16	23.3759671	120.5640324
Nantou County	Knaziy	GESL17	24.1641297	121.1755126
	Lushan	GESL18	24.0319737	121.2114060
	LeLe	GESL19	23.5536214	120.9609119
	Dongpu	GESL20	23.5610973	120.9327165
Tainan	Guanziling	GESL21	23.3356312	120.4962705
Kaohsiung city	Baolai	GESL22	23.1072742	120.7057937
Pingtung County	Sichongxi	GESL23	22.0914702	120.7502172
Taitung County	Wulu	GESL24	23.1692704	121.0397300
	Vakangan	GESL25	22.8941100	121.0682539
	Taolin	GESL26	22.8850218	120.9350856
	Zhiben	GESL27	22.6923428	121.0186029
	Jinfeng Jalan	GESL28	22.5927120	120.9387033
	Jinlu	GESL29	22.5275494	120.9477393
	Green Island	GESL30	22.6701884	121.4789464

and marked in Figure 2.

In the pursuit of identifying optimal GESL in Taiwan, our research employs the MB-DEA model, incorporating a thoughtfully selected array of eight inputs and two outputs, as shown in Figure 3. This selection is deeply rooted in established research methodologies and reference studies within the field of geothermal energy, ensuring a thorough and comprehensive assessment of each site’s potential. The inputs, carefully chosen based on their relevance to geothermal site evaluations, include both geophysical indicators and logistical considerations: Bouguer Gravity Anomalies (I1) and Free Air Gravity Anomalies (I2) in mGal, which are critical for detecting subsurface density variations and geological formations linked to geothermal activity [10,12]. These anomalies are complemented by logistical inputs such as the distances to the nearest electrical grid (I3), substations (I4), accessible roads (I5), and railways (I6) [32,40]. These measures assess the feasibility and potential costs associated with site development and energy distribution. Additionally, the distances to the closest population centers (I7) and essential water supplies (I8) are considered to evaluate the impact on local communities and the sustainability of site operations in terms of resource availability [6].

On the output side, Heat Flow in mW/m² (O1) and Population Density in People/km² (O2) serve as primary indicators of a site’s efficacy and its integration within the local context [13,41]. Heat Flow directly measures the thermal energy emanating from the Earth, which is pivotal in gauging the site’s energy generation capacity. Meanwhile, Population Density provides insight into the local demand for energy and potential market size, which are crucial for assessing the socio-economic viability of developing a geothermal plant at each location. The rationale behind the selection of these particular inputs and outputs is to construct a robust framework that not only examines the physical and economic aspects of site development but also incorporates considerations of environmental impact and community engagement. This approach aligns with the broader objectives of sustainable development and is supported by a rich body of literature that emphasizes the importance of integrating diverse factors in the evaluation of renewable energy projects.

The data used for the inputs and outputs in this study were meticulously gathered from public and trustworthy databases provided by prominent international organizations, namely the International Renewable Energy Agency (IRENA) and the International Energy Agency (IEA), as shown in

Table 2. The potential geothermal energy site location data.

GESL	I1	I2	I3	I4	I5	I6	I7	I8	O1	O2
GESL1	151.92	20.25	2.51	242.34	0.99	4.14	5.21	6.22	49.65	3.22
GESL2	141.7	33.77	0.27	266.72	1.83	5.52	5.82	3.2	49.65	5.51
GESL3	143.61	30.19	0.86	284.37	2.14	4.51	4.27	6.95	52.33	3.54
GESL4	143.61	30.19	0.05	291.53	0.07	4.7	1.64	12.19	52.33	1.22
GESL5	129.21	48.36	12.06	297.63	1.16	7.94	18.06	32.5	49.65	17.43
GESL6	120.73	58.99	22.21	309.09	8.36	7.83	18.06	30.8	51.47	31.25
GESL7	139.5	26.76	1.35	318.97	0.39	2.13	4.61	22.89	51.78	4.27
GESL8	149.25	16.28	0.82	326.97	7.47	9.61	19.39	9.6	51.78	19.04
GESL9	149.25	16.28	13.01	316.66	11.7	10.11	19.39	21.81	51.78	25.41
GESL10	139.5	26.76	1.24	327.17	0.42	2.99	10.45	8.75	51.78	10.09
GESL11	101.18	84.12	0.75	310.69	0.49	19.35	22.35	11.19	51.47	21.82
GESL12	101.18	84.12	0.15	312.15	0.32	13.34	16.71	17.42	51.47	16.6
GESL13	74.52	119.68	1.01	388.48	0.22	1.38	28.93	34.02	47.18	27.8
GESL14	174.52	119.68	3.88	395.41	1.72	4.55	28.93	40	47.18	18.42
GESL15	204.98	117.07	5.06	417.99	0.91	3.33	4.92	16.95	47.18	4.19
GESL16	130.06	57.66	0.67	390.2	0.12	13.6	13.44	8.15	45.16	12.46
GESL17	106.28	89.8	9.51	321.96	1.97	30.71	30.21	9.82	51.47	28.62
GESL18	117.43	102.92	0.28	336.91	0.03	35.38	25.75	7.83	51.47	24.24
GESL19	136.43	126.66	25.3	379.28	3.57	14.55	30.36	31.31	47.18	29.9
GESL20	136.43	126.66	25.8	377.74	0.6	12.05	28.64	30.2	47.18	28.2
GESL21	131.34	61.91	2.35	393.64	0.54	12.46	8.47	8.52	45.16	8.02
GESL22	131.34	61.91	11.86	421.92	0.43	42.15	8.47	15.41	45.16	23.45
GESL23	189.85	10.31	0.12	533.71	0.09	20.32	23.74	3.24	46.46	3091
GESL24	155.22	124.05	15.07	422.49	0.33	15.69	18.41	37.97	40.73	17.95
GESL25	169.88	107.59	2.2	452.75	0.24	7.2	12.25	15.75	40.73	11.94
GESL26	169.88	107.59	15.86	450.62	12.45	20.12	19.95	27.47	40.73	19.49
GESL27	180.67	79.96	1.98	473.31	0.04	2.79	3.87	19.29	40.73	3.54
GESL28	180.67	79.96	2.95	482.37	0.07	4.92	3.87	32.41	42.8	6.81
GESL29	189.77	49.28	0.24	489.63	0.45	1.76	7.13	36.85	42.8	10.59
GESL30	316.32	49.28	35.62	488.19	0.47	38.94	11.04	0.98	40.73	33.83

[3,42,43,44]. These databases are renowned for their reliability and comprehensive coverage of energy statistics, which ensure the integrity and credibility of the data utilized in our analysis. Utilizing these sources allows for a standardized and validated set of data that supports the robust assessment of geothermal energy site locations. The quality and transparency of the data from IRENA and IEA are critical, as they underpin the empirical foundation of our MB-DEA model, ensuring that our findings are both accurate and replicable in the broader context of renewable energy research.

4.2. Geothermal Energy Site Location Prioritization by MB-DEA

In this section, the MB-DEA model was applied to collected data to evaluate the efficiency of 30 identified potential geothermal energy site locations in Taiwan. Initially, the data corresponding to the inputs and outputs were normalized using the procedures outlined in Equations (3) and (4). As shown in

Table 3. The normalized data.

GESL	I1	I2	I3	I4	I5	I6	I7	I8	O1	O2
GESL1	0.680	0.915	0.931	1.000	0.923	0.932	0.876	0.134	0.769	0.001
GESL2	0.722	0.798	0.994	0.916	0.855	0.898	0.854	0.943	0.769	0.001
GESL3	0.714	0.829	0.977	0.856	0.830	0.923	0.908	0.847	1.000	0.001
GESL4	0.714	0.829	1.000	0.831	0.997	0.919	1.000	0.713	1.000	0.000
GESL5	0.774	0.673	0.662	0.810	0.909	0.839	0.428	0.192	0.769	0.005
GESL6	0.809	0.582	0.377	0.771	0.329	0.842	0.428	0.236	0.926	0.010
GESL7	0.731	0.859	0.963	0.737	0.971	0.982	0.897	0.438	0.953	0.001
GESL8	0.691	0.949	0.978	0.710	0.401	0.798	0.382	0.779	0.953	0.006
GESL9	0.691	0.949	0.636	0.745	0.060	0.786	0.382	0.466	0.953	0.008
GESL10	0.731	0.859	0.967	0.709	0.969	0.961	0.693	0.801	0.953	0.003
GESL11	0.890	0.366	0.980	0.765	0.963	0.559	0.279	0.738	0.926	0.007
GESL12	0.890	0.366	0.997	0.760	0.977	0.707	0.475	0.579	0.926	0.005
GESL13	1.000	0.060	0.973	0.498	0.985	1.000	0.050	0.153	0.556	0.009
GESL14	0.586	0.060	0.892	0.475	0.864	0.922	0.050	0.000	0.556	0.006
GESL15	0.460	0.082	0.859	0.397	0.929	0.952	0.886	0.591	0.556	0.001
GESL16	0.770	0.593	0.983	0.493	0.993	0.700	0.589	0.816	0.382	0.004
GESL17	0.869	0.317	0.734	0.727	0.844	0.281	0.005	0.773	0.926	0.009
GESL18	0.823	0.204	0.994	0.675	1.000	0.166	0.161	0.824	0.926	0.007
GESL19	0.744	0.000	0.290	0.530	0.715	0.677	0.000	0.223	0.556	0.009
GESL20	0.744	0.000	0.276	0.535	0.954	0.738	0.060	0.251	0.556	0.009
GESL21	0.765	0.557	0.935	0.481	0.959	0.728	0.762	0.807	0.382	0.002
GESL22	0.765	0.557	0.668	0.384	0.968	0.000	0.762	0.630	0.382	0.007
GESL23	0.523	1.000	0.998	0.000	0.995	0.535	0.231	0.942	0.494	1.000
GESL24	0.666	0.022	0.578	0.382	0.976	0.649	0.416	0.052	0.000	0.005
GESL25	0.606	0.164	0.940	0.278	0.983	0.857	0.631	0.621	0.000	0.003
GESL26	0.606	0.164	0.556	0.285	0.000	0.540	0.362	0.321	0.000	0.006
GESL27	0.561	0.401	0.946	0.207	0.999	0.965	0.922	0.531	0.000	0.001
GESL28	0.561	0.401	0.918	0.176	0.997	0.913	0.922	0.195	0.178	0.002
GESL29	0.523	0.665	0.995	0.151	0.966	0.991	0.809	0.081	0.178	0.003
GESL30	0.000	0.665	0.000	0.156	0.965	0.079	0.673	1.000	0.000	0.011

, this standardization step is crucial to ensure consistency and comparability across all data points, facilitating accurate and meaningful analysis in the subsequent stages of the model application. Subsequently, the MB-DEA model (7) was solved, employing default values for the psychological behavior coefficients, which include the decision-maker’s risk attitude towards gains $(\gamma =0.85)$, risk attitude towards losses $(\delta =0.92)$, degree of loss aversion $(\theta =2.25)$, and relative emphasis on gains $(\lambda =0.5)$ [17,18,39]. This setup provides a foundational standard for analyzing how these specific aspects of bounded rationality influence the decision-making process, offering a baseline from which further nuanced adjustments can be explored. The efficiency results for the GESLs have been comprehensively tabulated and are presented in

Table 4. The BR-Efficiency of GESL in Taiwan with default coefficients.

GESL	BR-Efficiency	GESL	BR-Efficiency	GESL	BR-Efficiency
GESL1	0.056	GESL11	0.016	GESL21	0.009
GESL2	0.008	GESL12	0.013	GESL22	0.022
GESL3	0.012	GESL13	0.024	GESL23	0.015
GESL4	0.012	GESL14	0.024	GESL24	0.021
GESL5	0.017	GESL15	0.013	GESL25	0.012
GESL6	0.017	GESL16	0.009	GESL26	0.051
GESL7	0.012	GESL17	0.030	GESL27	0.010
GESL8	0.012	GESL18	0.023	GESL28	0.016
GESL9	0.048	GESL19	0.027	GESL29	0.020
GESL10	0.012	GESL20	0.024	GESL30	0.024

. This table provides a detailed overview of how each location fares in terms of operational and potential efficiency, allowing for a clear and structured comparison across all evaluated sites.

In Table 4 and Figure 4 of our study, significant variation in the BR-Efficiency scores among the evaluated GESLs across Taiwan is demonstrated, providing important insights into regional geothermal potential and associated challenges. Datun Mountain (GS1) in New Taipei City, identified with the highest efficiency score of 0.056, is highlighted as a standout location. This high score indicates favorable logistics and infrastructural readiness, suggesting that optimal utilization of this site could be achieved with relatively lower investments compared to other sites, making it a strategic choice for initiating or expanding geothermal energy projects.

Conversely, adverse geological conditions or logistical challenges are suggested by the notably lower scores of 0.008 and 0.009 at sites like Wulai (GS2) and Jhonglun (GS16), respectively. These challenges could deter development unless significant investments are made to overcome them, emphasizing the need for a nuanced understanding of each site’s unique context and constraints to inform investment decisions and development strategies.

A potential regional pattern influenced by common geological or infrastructural characteristics is indicated by the clustering of similar efficiency scores among sites in Yilan County, such as Jiaoxi (GS3), Tuchang (GS4), and District Four (GS7). This observation is crucial for regional energy planning, suggesting areas where collective improvements in infrastructure or technology could elevate the overall efficiency of multiple sites simultaneously.

Moreover, the distribution of scores across different regions, such as the relatively higher scores observed in Nantou County at sites like Knaziy (GS17) and LeLe (GS19), points to the presence of several promising sites within a single region. The potential for a regional development strategy that leverages multiple sites to enhance the area’s contribution to national geothermal production capacity is suggested, which would not only enhance energy security but also distribute economic benefits more broadly across the region.

These results underscore the importance of integrating detailed, location-specific data into the site selection process. A more accurate picture of each site’s potential, tailored to the complex dynamics of risk preferences and logistical variables, is captured by the application of the MB-DEA model. This approach refines the decision-making process for geothermal development significantly, aligning it more closely with Taiwan’s strategic energy goals and the practical realities of geothermal exploitation.

In order to comprehensively assess the impact of the relative emphasis on gains within the decision-making process, the parameter $\lambda$, which represents this specific aspect, was varied across a spectrum of values. This methodological approach was implemented to determine how changes in $\lambda$ influence the overall evaluation outcomes of the study. The varying values of $\lambda$ were systematically applied within our model to generate a range of scenarios, with the resulting data captured and analyzed. The outcomes of these variations are meticulously documented and presented in Figure 5, providing a visual representation of how shifts in the relative emphasis on gains can significantly alter the results of the evaluation. This analysis is crucial for understanding the sensitivity of the model to changes in decision-makers’ risk preferences, particularly towards gains, and offers valuable insights into optimizing decision-making processes in geothermal site evaluations.

According to Figure 5, the analysis of the rankings for GESLs under different values of $\lambda$, which adjusts the relative emphasis on gains, highlights distinct variations in how decision-maker preferences influence site evaluations. GESL1 stands out for maintaining the top position across all values of $\lambda$, indicating a robust choice that performs well under various decision-making scenarios, suggesting strong inherent qualities that likely offer substantial benefits universally. Conversely, sites like GESL2 and GESL21 display notable fluctuations in rankings as $\lambda$ varies, with GESL2 oscillating between the lowest and somewhat higher positions. This variability highlights their sensitivity to changes in decision-making frameworks, especially those that alter the weighting of potential gains. These fluctuations suggest that the attractiveness and feasibility of such sites depend heavily on specific stakeholder risk attitudes, necessitating comprehensive risk management strategies to mitigate potential downsides. Sites such as GESL6 and GESL8 exhibit marked rank improvements as the emphasis on gains increases, suggesting that these sites become particularly favorable under optimistic decision-making scenarios that heavily favor potential rewards. This trend indicates that these sites may possess characteristics that are particularly advantageous under certain conditions, which might be missed under more conservative evaluation models.

On the other hand, GESL9 remains consistently high in the rankings regardless of $\lambda$ values, reinforcing its status as an efficient and stable investment choice, perceived favorably under a variety of risk preferences. This stability makes it a potentially safer and more reliable option for development. The analysis also shows that GESL26 decreases rank with higher $\lambda$ values, implying that it becomes less attractive as more emphasis is placed on gains, possibly due to lower potential or greater associated risks compared to other locations.

The comparison of geothermal site selection across previous studies—Chyi Wang et al. (2021) employing Play Fairway Analysis (PFA), Chao-Shing Lee et al. with a focus on geological and tectonic evaluations, and our own application of the MB-DEA—highlights significant overlaps and distinctions that are crucial for informed geothermal development strategy in Taiwan [5,45]. Chyi Wang et al. provide a broad, visually engaging overview identifying high-potential regions like the east coast mountain foothills and areas like the Tatun Volcano Group and Chingshui/Tuchang, which are recognized for their geothermal potential due to favorable geological features [45]. On the other hand, Lee et al. detail specific areas such as the Central Range and volcanic zones like the Tatun Volcano Group, highlighting their geological suitability due to active tectonic processes and high geothermal gradients [5].

Our study complements this geographical and geological focus by evaluating the operational efficiency of specific sites, such as Datun Mountain, which our MB-DEA model ranks as highly efficient. This detailed operational insight is invaluable, pinpointing exact locations where investments could yield optimal results. When both studies point to areas like the Tatun Volcano Group as having high potential, it underscores a strong consensus on their viability, suggesting these areas should be prioritized for development.

On the other hand, any discrepancies in site recommendations between the studies call for additional analysis to align geological potential with practical economic and infrastructural considerations. By integrating broad geological assessments with targeted operational evaluations from our MB-DEA model, stakeholders gain a robust, multi-dimensional perspective that supports strategic decision-making, ensuring the effective utilization of Taiwan’s rich geothermal resources.

When considering real-world applications, a synthesized approach that incorporates the broad geological insights of Lee et al., the geographic accessibility and initial screening of Wang et al.’s PFA, combined with the detailed operational and economic assessment provided by our MB-DEA model, would likely yield the most comprehensive and actionable results. Such an integrated approach would allow for both macro- and micro-level planning and development, harnessing Taiwan’s geothermal potential effectively while addressing the specific needs and constraints of each location.

4.3. Managerial Implications

The findings of this study, which assesses GESLs using the MB-DEA model under various risk preference scenarios, offer significant managerial implications that can shape strategic decision-making in the geothermal energy sector. Firstly, the importance of aligning site selection with the risk preferences of decision-makers and stakeholders is evident, suggesting that investment strategies should be tailored to match the psychological and economic profiles involved. This ensures financial viability and broad-based support for projects. Additionally, the variability in site rankings based on risk attitudes underscores the need for robust risk management strategies, including contingency planning and phased investments, to mitigate potential risks associated with variable stakeholder preferences.

Effective stakeholder engagement also emerges as a crucial factor, enhancing project acceptability and fostering a supportive development environment, especially given the community impacts of geothermal projects. Moreover, the consistent ranking of certain sites regardless of risk preference variations indicates that these locations are potentially safer investments, suggesting a strategic focus on these sites could be beneficial.

Policymakers, too, can leverage these insights to develop supportive policies that encourage investment in both higher-risk and lower-risk geothermal projects tailored to attract a diverse range of investors. Additionally, the integration of bounded rationality into traditional DEA models highlights the value of using advanced analytical tools in evaluating complex investment decisions. Such tools can capture the nuances of human decision-making, improving the relevance and accuracy of strategic planning and project evaluations in the renewable energy sector.

5. Conclusions

This study was initiated against the backdrop of increasing global interest in renewable energy sources, with a specific focus on geothermal energy due to its potential for providing reliable, low-carbon power. Taiwan, with its significant geothermal resources, particularly along the Pacific Ring of Fire, presents a compelling case for the exploration and development of geothermal energy. The motivation for this research stemmed from the need to apply sophisticated analytical methods to enhance the site selection process, thereby optimizing resource allocation and maximizing the potential benefits of geothermal development.

To address this challenge, the MB-DEA model was employed, integrating traditional DEA with elements of bounded rationality to account for the risk preferences of decision-makers. Moreover, this model is revised to make sure that this objective function is positive in all cases. This approach allowed for a nuanced analysis of potential geothermal sites, considering not only the physical and economic data but also the psychological factors that influence decision-making. The model was applied to 30 potential geothermal site locations across Taiwan, using multiple criteria for inputs and outputs to comprehensively assess each site’s efficiency and suitability.

The findings revealed significant variability in site rankings based on the decision-maker’s emphasis on gains, illustrating the profound impact of psychological factors on the evaluation of geothermal sites. Sites like GESL1 in Datun Mountain (New Taipei City), GESL9 in Maoxing (Yilan County), and GESL26 in Taolin (Taitung County) demonstrated robustness, consistently ranking high across different values of λ, indicating their strong potential irrespective of risk preference variations. In contrast, other sites displayed sensitivity to these changes, suggesting that their perceived value could significantly fluctuate with differing stakeholder risk attitudes.

This study provides practical contributions to Taiwan’s geothermal development by enhancing the site selection process with a sophisticated analytical model that incorporates both traditional and psychological factors. The integration of MB-DEA offers a significant advancement over traditional site evaluation methods, which typically focus solely on empirical data. By including decision-makers risk preferences and psychological biases, this model allows for a more comprehensive evaluation of potential sites, identifying those that are not only technically and economically feasible but also most likely to be supported by local stakeholders and policymakers. The application of this model to Taiwan’s diverse geothermal landscape has led to the identification of optimal locations that could significantly boost the country’s renewable energy output. This is particularly crucial as Taiwan aims to increase its renewable energy consumption amidst rising energy demands and environmental concerns. This study’s findings can help prioritize investment and development in regions that offer the highest returns and stability, thus supporting Taiwan’s strategic energy policies and its goals for sustainability and energy independence. Furthermore, the insights provided by this study can aid in resource allocation by highlighting areas that require more detailed geological and feasibility studies, thereby reducing unnecessary expenditure and focusing efforts where they are most likely to succeed. This targeted approach can accelerate the deployment of geothermal energy projects, contributing to Taiwan’s economic growth and environmental protection objectives. This contribution is not only strategic but also timely, as it aligns with global trends toward sustainable energy solutions and provides a model that can be adapted for other renewable energy initiatives both within and beyond Taiwan.

One notable limitation of this study is the potential inaccuracy in the geographical coordinates of some locations, which may vary by a few kilometers. Such discrepancies could impact the precision of the site evaluations, especially in geologically complex terrains where small shifts in location might lead to significant differences in geothermal potential. Future research should aim to refine the geographical data used in the model and explore the impact of such variations more thoroughly. Additionally, further studies could expand the model to include more dynamic elements of bounded rationality and explore the integration of other decision-making theories to enrich the analytical framework used in this and similar contexts.