Optimizing the operation of series and parallel reservoirs using the neutrosophic environment: A case study of the Karun watershed

The reservoir operating policies usually rely on the historical flow time series. The basis for using these series is the assumption that their statistics (mean, standard deviation, etc.) are constant over time, which means the time series is steady. Yet, the reservoir inflow during operation will differ from that of the past, even if the climate change is not a concern. For this reason, the reservoir operation policy may not be satisfactory and may not be appropriately applied. Ignoring these issues, known as hydrologic uncertainty, causes problems in water supply. Considering hydrological uncertainties in reservoir management and water releases from reservoirs will reduce difficulties such as water shortages during droughts, urban water rationing during dry seasons, instability of agricultural production downstream of dams, hydropower production problems, and environmental problems.

The chronological order of reservoir operations optimization techniques begins with customary methods involving linear programming (LP), nonlinear programming, dynamic programming, and stochastic dynamic programming (SDP). These techniques use mathematical models to mimic and optimize reservoir performance.

The evolution of artificial intelligence from the early 21st century has resulted in the development of problem-solving techniques and the emergence of metaheuristic optimization algorithms that have greatly improved the operational performance of reservoirs. Metaheuristics are search-guiding strategies. Metaheuristics sample a fraction of solutions that would otherwise be too large to enumerate or explore (Blum & Roli 2003). Metaheuristic algorithms started with evolutionary algorithms (EAs), which are based on mechanisms inspired by nature and simulate biological evolutionary behaviors of living creatures, including genetic algorithm (GA), genetic programming (GP), and differential evolution (DE). The nondominated sorting GA II, also called Pareto front, is the other popular modified GA method used widely in multiobjective optimization for reservoir operations (Liu et al. 2020). Then, swarm intelligence algorithms (SIs), which work based on the information transfer in the collective behavior of self-organizing societies in animals, such as ant colony optimization, bee optimization, grey wolf optimization, and particle swarm optimization (PSO), were introduced to optimize the operation of the reservoir. Solving complex optimization problems using metaheuristic algorithms had difficulties such as premature convergence (Lei et al. 2018), dimensionality (Jiang et al. 2018), massive computational burden (Wang et al. 2020), and parameter tuning (Niu & Feng 2021). Therefore, the hybrid EAs in swarm intelligence were introduced and widely used to solve the complexity of water resources management (Lai et al. 2022).

A combination of DE and PSO to optimize the operation of the multipurpose reservoirs has shown that it is easy to implement with appropriate performance. While PSO can find local optimal solutions, DE is highly dependent on its controlling parameters, and improper parameter settings can result in premature convergence (Sedki & Ouazar 2012).

Another popular technique for improving PSO was to combine PSO with a support vector machine (SVM). SVM has shown how machine learning simplifies complicated processes and nonlinear problems.

Lai et al. (2022) conducted a comprehensive review ranging from traditional models to metaheuristic algorithms in optimizing reservoir operation from 2011 to 2021. They reviewed recent research advances in reservoir operation optimization techniques and emphasized the importance of future climate variables in guiding operating policy. They mentioned that in the last few decades, artificial neural network (ANN) and GP have been the most used to model or simulate reservoir operating systems. These methods are known as regression or supervised machine learning methods, which are easy to parameterize. Combined algorithms also perform better than single metaheuristic algorithms and traditional methods.

Zadeh introduced the fuzzy set in 1965 (Goguen 1973) as a helpful tool for decision-making in water resources since it is often associated with inaccuracy and uncertainty (Kambalimath & Deka 2020). With fuzzy sets, the interpretation of the user regarding the characteristics of the model variables essentially determines the membership functions and their rule structure. However, it is impossible to identify the membership functions in some modeling circumstances by simply observing the data.

Rather than arbitrarily selecting the parameters associated with a particular membership function, the parameters can be optimized to fit the functions to the input and output data. In 1993, Jang (1996) presented the adaptive neuro-fuzzy inference system (ANFIS), in which fuzzy membership functions are optimized using adaptive neural learning techniques. ANFIS is the most widely used fuzzy inference model in water resources management studies (Deka & Chandramouli 2009).

Fuzzy sets use membership functions to model only one aspect of real-world problems. The degree of membership indicates the degree to which a subject belongs to fuzzy sets (FS), so the value of that must be less than or equal to 1. Intuitionistic fuzzy sets (IFS) theory by Atanasov (1986) deals with the problem of uncertainty in a global set by considering a nonmembership function in conjunction with a fuzzy membership function.

The degree of membership of a subject is complementary to the degree of nonmembership, which indicates the extent to which a subject is not a member of the IFS, so the total values of these two degrees must be less than or equal to one.

Neutrosophy by Smarandache (2020) is a new branch of philosophy with an ancient history that studies neutrality, its emergence, nature, scope, and its interactions with various ideologies. The word neutrosophy, with its Latin and Greek roots, means the knowledge of neutral thought. The fundamental thesis of neutrosophy is that any idea has not only a specific degree of truth, as is commonly assumed in multivalued logical contexts, but also degrees of falsity and indeterminacy that are independent.

Neutrosophic sets (NS) are generalizations of fuzzy sets (Goguen 1973) and IFS (Atanasov 1986), which consider degrees of truth and falsity in addition to degrees of indeterminacy and incompatibility (Smarandache 1998). The truth membership function, the falsity membership function, and the indeterminacy membership function are features of a neutrosophic set. The indeterminacy of indefinite data introduced in the NS theory is crucial for appropriate decision-making, which is impossible with intuitionistic fuzzy set theory. Since indeterminacy usually occurs in the operation of the reservoirs at the inflow to the reservoir, the downstream demand, and the hydropower demand, the NS theory can analyze the various operating conditions well and make the modeling conditions closer to the reality.

Furthermore, some extensions of NS, such as interval NS (Ye 2014), bipolar NS (Said Broumi et al. 2016), single-valued NS (Ji et al. 2018), multivalued NS (Peng et al. 2018), and interval linguistic NS (Wang et al. 2018), are proposed and used to solve various problems (Broumi et al. 2019a, b, c, d, e; Edalatpanah 2018; Kumar et al. 2019a, b; Najafi & Edalatpanah 2013; Smarandache 2020).

In 2019, Kahraman and Otay classified the fields of neutrosophic application, with computer science, mathematics, and engineering accounting for the largest share of neutrosophic applications (Kahraman & Otay 2019). Edalatpanah and Smarandache conducted a data envelopment analysis for simple NS. They presented an input-oriented Data Envelopment Analysis (DEA) model with simplified neutrosophic numbers and proposed a new solution strategy (Edalatpanah & Smarandache 2019). Edalatpanah and Das introduced a new classification function of the triangular neutrosophic number and its implementation in integer programming (Das & Edalatpanah 2020).

Edalatpanah introduced a direct method for triangular neutrosophic LP (Edalatpanah 2020a). He presented a nonlinear approach (Edalatpanah 2019) to neutrosophic LP and introduced systems of neutrosophic linear (Edalatpanah 2020b) equations and the neutrosophic structured element (Edalatpanah 2020c).

The remarkable feature of the new model compared with the existing methods is that it can simply and efficiently handle triangulated neutrosophic information (Edalatpanah & Smarandache 2019). Edalatpanah et al. conducted neutrosophical studies on data envelopment analysis with undesirable outputs (Mao et al. 2020; Yang et al. 2020). Zhang et al. studied neutrosophic applications in civil engineering fields until 2022. The results of their research show that despite recent advances in civil engineering, decision-making in civil engineering has become more and more complex.

While the neutrosophic theory can reflect the uncertainty, ambiguity, and inadequacy of the data, the neutrosophic set has not yet found its proper place in the various branches of this engineering field. The neutrosophic theory has not yet explored many problems in civil engineering (Zhang & Ye 2023).

Li et al. (2019a) introduced the vector similarity metrics (indices) in the Neutrosophic Number System (NNS) environment to analyze the rock slope stability assessments.

They compared the estimated results with the measured values to validate the method's efficiency. Li et al. (2019b) developed tangent and arctangent similarity measures for NNs and then introduced a novel slope stability assessment method using tangent and polar similarity measures.

Qin et al. (2023) proposed a technique that combines Single-Valued Neutrosophic Numbers (SVNN) with Gaussian process regression to assess stability in open-pit slopes in indeterminate environments. Garai et al. (2023) presented a multicriterion decision-making approach based on fractional ranking to manage water resources in the bipolar neutrosophic fuzzy environments in Kolkata. First, they introduced a fractional ranking technique of single-valued triangular bipolar neutrosophic number based on grades and illegibility and then formulated a new Multi-Criteria Decision-Making (MCDM) method. By using the proposed MCDM procedure, they investigated a water resource management (WRM) problem in Kolkata in a bipolar neutrosophic environment.

Abdel-Monem et al. (2023) used the integrated neutrosophic regional management ranking technique to manage agricultural water. They integrated the proposed neutrosophic set decision-making approach with the MARCOS method to obtain optimal solutions. Kousar et al. (2023) employed appropriate membership functions and the multiobjective neutrosophic fuzzy linear programming model to solve the rice and wheat production model in Pakistan. Abouhawwash et al. presented a neutrosophic framework for evaluating distributed circular water to prepare for unexpected stressor events. They proposed a triangular neutrosophic set to compute the weights of the criteria (Abouhawwash et al. 2023).

There are many theoretical and practical advances to optimize the Dam Reservoir Operation. Despite the studies conducted on the optimal operation of reservoirs under conditions of uncertainty, no research has been reported on the performance of neutrosophic inference in reservoir operation and even in other areas of water resources management. The neutrosophic inference environment is very efficient due to its ability to use a large amount of erroneous information and to analyze dynamic systems, especially in cases where there is no correct understanding of the physical relationships between phenomena. Therefore, in this research, the neutrosophic inference environment is considered a suitable procedure to simulate and predict the behavior of optimal operation of the dam reservoir. The case study is the Karun River watershed in Iran. The study focuses on six major dams in the basin, five in series and one in parallel. The multiobjective function includes minimizing the shortages of downstream demand and maximizing the energy production of the entire system.

The research begins with optimizing the reservoir operation problem using conventional mathematical methods. In step 2, the relevant system is modeled in the ANFIS environment, and its behavior is derived based on fuzzy rules. One of the critical challenges in neutrosophic inference is to extract the functions ‘truth membership,’ ‘falsity membership,’ and ‘indeterminacy membership.’ The third step aims to generate these functions. The ‘truth membership’ functions are the same functions optimized in ANFIS, and the ‘falsity membership’ and ‘indeterminacy membership’ functions are created by setting the related relationship between the optimization results and ANFIS in such a way that these concepts are implicit in the mentioned relationships. The use of ANFIS to extract neutrosophic functions is one of the innovations of this research. As the fourth step, the corresponding problem is modeled and solved separately in the neutrosophic inference space. In the fifth step, the results of neutrosophic inference are compared with the optimization method, conventional operation (rule curve), and ANFIS.

Therefore, this study presents an innovative mathematical model for extracting optimal rules from a real dam reservoir operation problem in the neutrosophic inference environment.

Dealing with uncertainty in dam operations is crucial for optimal performance. Researchers have proposed various methods to address this.

Stochastic optimization methods use probabilistic models to account for uncertainty in input variables (e.g., input flows, demand). Techniques include SDP (Puterman 2014) and Monte Carlo simulation (Mooney 1997).

SDP uses probability distributions to model variability and uncertainty. SDP considers the future in terms of possible scenarios.

SDP computational demands rise exponentially with state variables, making it inappropriate for large or multireservoir systems. SDP often makes simple assumptions to reduce complexity. For example, SDP assumes that the probability distribution of the input flow and other variables is constant. In reality, environmental and climatic conditions may change, and these simplifying assumptions may reduce the accuracy and applicability of the results. Classical stochastic models like SDP require accurate and comprehensive data. However, many areas lack sufficient and accurate meteorological, hydrological, and water demand data. Therefore, these methods are less applicable in uncertain data conditions (Ross 2014). Wu et al. used SDP to operate a cascaded hydroelectric reservoir system with multiple local optima. They proved that multiple optima in nonconcave maximization models of reservoir exploitation significantly reduce the accuracy of the SDP solution. Therefore, this study used a two-step algorithm that combines navigation and search to achieve better solutions (Wu et al. 2018). Monte Carlo simulation uses stochastic methods and probability distributions to model and analyze the effects of random variables and uncertainties on reservoir performance. The accuracy of this method depends on the accuracy of the input data and probability distributions. Incomplete or inaccurate information can lead to unreliable results (Harrison 2010).

Optimum operation charts (OOCs) are graphical tools that set reservoir operation goals based on predicted and actual data. This chart gives operators more flexibility to respond to changing conditions. However, they can lead to inappropriate performance under extreme or unusual conditions (such as drought or flood) because they often present a simplified view of the reservoir system that assumes normal conditions. OOCs require accurate data to optimize reservoir performance, and incomplete or inaccurate data can lead to incorrect decisions since the accuracy and correctness of the optimal operation chart are highly dependent on the quality of the input data (Wen et al. 2021). Jiang et al. optimized the total output operation chart in cascade reservoirs and presented its application. They developed a new two-layer nested model based on the optimization model of the total output operation graph and the optimal output distribution model to maximize power generation. Two methods (stepwise regression and mean) were employed to extract the output distribution ratio.

Optimal operation functions (OOF) are mathematical models or algorithms that make optimal decisions based on input parameters (e.g., inflows, water levels, demand). While convenient, these features have shortcomings. Accurate data are required to extract reliable OOFs. These functions often rely on assumptions about system behavior or relationships between variables. These assumptions can oversimplify real-world dynamics and lead to inaccuracies in operational decisions (Saab et al. 2022).

Robust optimization finds feasible, high-performance solutions under uncertainty. This method does not require an exact probability distribution but considers a range of possible scenarios. It provides solutions that are less sensitive to model inaccuracy and variability. However, it can lead to overly conservative solutions that may not be optimal in some scenarios (Yu et al. 2023).

Scenario analysis evaluates different scenarios based on various assumptions about future conditions (e.g., climate change, fluctuations in input flows, and demand). This method helps understand possible outcomes and plans for different future conditions. However, it relies on selecting acceptable scenarios and may not include all possible uncertainties (Jiang et al. 2022).

The decision tree method uses tree-like models of decisions and their possible outcomes, including random events and uncertainty. This method provides a clear visual representation of decision-making under uncertainty and helps evaluate different decision paths. However, it can become complex with multiple branches and may not effectively manage ongoing uncertainty. In one study, decision trees, ANNs, and fuzzy logic algorithms were compared with the operating rules of a reservoir in North India. The results indicate that the fuzzy logic model performs better than other investigated computational techniques (Kumar et al. 2013).

Other methods, such as ANNs and stochastic dual dynamic programming (SDDP), will be discussed in the following sections. Existing methods of managing water resources and reservoirs, while still applicable, cannot fully address the complexities and uncertainties. Thus, they may be inadequate where multiple factors are involved in the uncertainty. Therefore, there is a need to develop more flexible and comprehensive methods for dealing with uncertainties in input data.

Fuzzy logic provides valuable tools for scenarios where uncertainties are qualitative or imprecise. This method uses fuzzy sets and rules to deal with inaccurate and uncertain information. Fuzzy logic systems can model fuzzy or qualitative aspects of uncertainty. This method is effective for linguistic and qualitative uncertainties but may not cover all uncertainties.

Neutrosophic logic extends fuzzy logic by incorporating degrees of uncertainty and inconsistency. Neutrosophic manages uncertainty by modeling truth, falsity, and indeterminacy and efficiently models uncertain environments. Neutrosophic logic is a crucial paradigm shift in the operation of dams because all existing methods, including fuzzy and nonfuzzy methods, manage uncertainty based on the available data. But neutrosophic offers a different logic. In this logic, truth membership functions cover available data spaces, and indeterminate functions cover possible and ambiguous spaces. False membership functions cover no-data spaces. Neutrosophic can, therefore, manage uncertainty much more comprehensively.

Every idea 〈A〉 tends to be neutralized, reduced, and balanced by non-A ideas as an equilibrium state (not anti-A, as Hegel claimed). Between 〈A〉 and anti-A, infinitely many non-A ideas exist that can balance 〈A〉 without necessarily being a version of anti-A. To neutralize an idea, one must discover all three aspects of it – truth, falsity, and indeterminacy – and then reverse/combine them. After that, the idea is considered neutral. In atoms, protons, electrons, and neutrons stay together. Every idea preserves the structure of the atom. The analysis of positive, negative, and neutral statements is the basis of this argument. It shall be called quantum philosophy.

Hegel's (dual) dialectic does not work. As a result, it should be made into a trialectic, and even more, to a pluralectic term because there are different degrees of positive, negative, and neutral, all interpenetrating. Going to a continuum power transalectic (∞-alectic), neutrosophy studies the possibility and the impossibility of an idea.

NS considers degrees of indeterminacy and inconsistency along with degrees of membership and nonmembership. The functions of truth, indeterminacy, and falsity membership are characteristic of a neutrosophic set.

This research defines the neutrosophic set (V) as the decision variables of the reservoir inflow, release, downstream demand, and storage.

Figures 8 and 9 show the flowchart of the steps in this research. As the flowchart shows, the following procedure will be performed step by step.

• 1. The first step is to group the data into two sets: training and testing. Primary data are the inflow to the reservoir (river flow rate) and downstream demands such as agriculture, industry, drinking water, hydropower supply, and reservoir topography.

• 2. The optimization problem is specified with the desired objectives, which include the conflicting purposes of simple operation to minimize the downstream shortage and the hydropower generation problem to reduce the energy shortfall. The corresponding optimization problem is solved in Lingo software using definite data and conventional mathematical methods.

• 3. The dam operation rule curve is derived using the results of solving the aforementioned optimization problem.

• 4. By setting up a reservoir operation simulation system and using training and test data separately, the operating policy is checked and controlled using the operation rule curve method with fuzzy performance criteria.

• 5. The ANFIS is modeled using the results of the aforementioned optimization problem. Inputs to the ANFIS system include reservoir inflow, demand (monthly statistical data), and storage (extracted from optimization). The reservoir release (derived from the optimization) is also considered as the output of the ANFIS system. In this way, ANFIS extracts the relationship between input and output data, and by giving the reservoir storage, monthly demand, and inflow, it can obtain the required release based on the ANFIS rules.

• 6. At this step, it is possible to simulate and solve the problem separately using training and test data and optimized membership functions in ANFIS.

• 7. In the reservoir operation simulation, the required releases are obtained from the ANFIS results, and the objective functions and fuzzy performance criteria (such as step 5) are checked and controlled separately for training and test data.

• 8. The ANFIS results are applied to extract the truth, indeterminacy, and falsity membership functions.

• 9.1. Truth membership functions:

The reservoir output in optimization (step 3) is the best solution to achieve the optimization goals. The rules and equations are extracted in ANFIS so that the optimal behavior of the reservoir can be modeled and formulated for further use in future operations. Therefore, with the results of ANFIS, the fuzzy membership functions optimized in ANFIS are considered truth membership functions in neutrosophic. To focus on presenting the fundamental principles of neutrosophic logic and to avoid unnecessary complexities, these membership functions are assumed to be triangular. Figures 4(a) and 7(a) show an example of truth membership functions in the Karun Dam for the reservoir input and output decision variables.

• 9.2. Falsity membership functions:

These functions are formed according to the membership function of truth and falsity and based on the concept of indeterminacy in such a way that every point whose degree of membership in the membership functions of truth and falsity is maximum and minimum (one and zero) has the minimum indeterminacy degree (zero). Because the membership degree in the truth and falsity functions is clearly defined at these points and leaves no doubt, the indeterminacy at these points is zero. On the other hand, in the points where the membership degree in the functions of truth and falsity is 0.5, the indeterminacy is maximal (one) because they have the most indeterminacy. For example, if the inflow to the reservoir is such that its membership in the truth function is 0.5 and its membership in the falsity function is also 0.5, then logically, the indeterminacy of this point should be at most (one). Figures 4(c) and 7(c) show an example of the indeterminate membership functions in the Karun-1 Dam for reservoir inflow and release decision variables.

• 10. One of the main challenges of neutrosophic inference is creating proper relationships between the system inputs and outputs. This step generates the relations between inputs and outputs in each of the truth, falsity, and indeterminacy membership functions according to neutrosophic rules.

• 10.1. Relationships between inputs and outputs in truth membership functions:

The relations between the inputs and outputs of the truth membership functions are considered to be the same relations optimized in ANFIS. The ANFIS results concerning the relationships among membership functions are consistent with its concept. In response to how the system behaves, the study extracted the system behavior in the optimal state using the ANFIS method and answered the question.

• 10.2. Relationships between inputs and outputs in indeterminate membership functions:

The boundary points of the input neutrosophic membership functions (e.g., the boundary points of the functions shown in Figure 4) are placed in Equations 4, 6, and 10, and the boundary points of the output neutrosophic membership functions for the reservoir release are obtained.

Figure 7 shows the neutrosophic membership functions of the reservoir output in Karun-1 Dam. The inference of the output neutrosophic membership functions using the input neutrosophic membership functions, and the aforementioned method is one of the innovations of this research, which is for the first time.

• 12. The desired problem is simulated and solved separately with truth, indeterminacy, and falsity membership functions.

Due to the complexity of the Karun system, as a series of dams with the purpose of hydropower and agriculture hydropower supply, and also because there are many areas of demand in it, an integrated water resources management and modeling of the entire system is necessary. Therefore, the model was developed for the Karun-e Bozorg basin with series and parallel dams. Generally, dams built for power generation fall into two main categories. In the first category, which includes the Karun-1, 2, 3, and 4 and Godar-e Landar dams, the primary purpose of the project is energy production, and regardless of the other objectives of the project, the reservoir release relies on the demand for the electricity and maximizing hydropower production. The Karun-2 Dam is not included in the calculation because it has not yet been built. In the second category, which encompasses the Dez and Gotvand dams, water is released from the reservoir based on other downstream needs, regardless of the energy produced. The generated power is calculated based on the released water and heads of the turbines.

In series and parallel dam systems, changes in the inflow and storage of other reservoirs directly and indirectly affect the release of a particular reservoir. In series dams, changes in the inflow and storage in other reservoirs strongly influence management decisions on the release of water for each reservoir, because all of the reservoirs are directly interconnected, and the inflow of one reservoir is directly affected by the outflow of the upstream reservoirs. If upstream reservoirs release more water, the downstream reservoir must cope and may have to release more to prevent overflow. Similarly, if upstream reservoirs reduce water flow, the downstream reservoir inflow will be less, and the downstream reservoir may release less water. Water storage in upstream or downstream reservoirs also affects the release of a particular reservoir. For example, if the upstream reservoir stores a high amount of water and restricts the release, this can reduce the pressure on the downstream reservoir to release water. Also, if the downstream reservoirs have more storage capacity, the upstream reservoir may be able to release more safely because the downstream reservoirs can store that amount of water.

In parallel dams, changes in the inflow and storage of other reservoirs can indirectly affect the release of a particular reservoir. Compared to series dams, which are directly connected, parallel dams often operate separately. In the case of parallel dams, each reservoir usually has an independent inflow from a common water source (such as a large river or similar watershed). If the inflow to one reservoir changes (due to climate change, rainfall, or other factors), these changes may affect the need to release water from the other reservoir. If one of the reservoirs receives more inflow and has sufficient storage capacity, there may be less pressure on other parallel reservoirs to release water immediately. This can help other reservoirs in times of water scarcity. If one reservoir in the parallel system needs to store more water (e.g., due to a drought forecast), the release of water from other reservoirs can increase. Therefore, in a system of series and parallel dams, comprehensive and optimal management of water resources requires complex management models and precise coordination between reservoirs. Storage and release decisions in one reservoir should be based on the condition of other reservoirs to avoid the risk of flooding and overflow and to optimize water storage.

This research first optimizes the operation of the entire series and parallel dam system as an integrated system and naturally considers the effect of upstream reservoir release and storage on downstream reservoir inflow, storage, and release in the optimization. Then, according to the optimization results, the behavior of each dam is derived separately in ANFIS. ANFIS inputs are reservoir storage, demand, and inflow to the reservoir, and ANFIS outputs are reservoir releases. Reservoir volume and release result from the entire system's integrated optimization. The optimal release of upstream reservoirs also plays a role in the inflow to the reservoir. Therefore, the outflow and storage of other reservoirs play an indirect role in the ANFIS inference (Equation (4)). Neutrosophic inference follows the same logic (Equations (6) and (10)). If optimizing each dam were performed separately, the results of ANFIS and neutrosophic would not consider the effect of the reservoirs on each other. The system simulation sums the monthly release from ANFIS or neutrosophic inference in the upstream dam with the downstream dam inflow. Therefore, the downstream dam inflow corresponding equations consider the upstream dam influence.

An elevation volume curve is typically employed to define the relationship between reservoir elevation and volume. This curve illustrates how the stored volume changes with the increasing water elevation in the reservoir. The relationship depends on the geographical and geometric characteristics of the valley or area where the reservoir is. In practice, the closest function is fitted to the elevation volume curve, often using second-degree polynomials, to extract the elevation from the reservoir volume.

As shown in Figure 8, the optimization is performed using the training data. The optimum output and the monthly minimum and maximum volumes are then obtained, based on which the reservoir operating curve is obtainable. The operation policy performance (Equations (20) to (23)) is investigated by simulation using the test data. Several methods, described in Section 1, can be used for optimization. The aforementioned optimization problem with 30 years of monthly data has 34,219 decision variables and 29,452 constraints, of which 3,865 are nonlinear. The aforementioned problem was solved using Lingo version 20 and then simulated and analyzed using MATLAB software. By employing advanced and optimized algorithms, Lingo can solve large and complicated issues in a short time. This feature is crucial for industrial and operational issues requiring rapid, large-scale problem-solving. Lingo has other advantages, including support for nonlinear problems, multiobjective models, sensitivity analysis, and postprocessing. In new versions of Lingo, the developers have tried to reduce the problems and limitations associated with nonlinear problems and local optima traps by using more advanced algorithms, hybrid methods, metaheuristics, support for multistart techniques, and global search techniques. For solving nonlinear problems, Lingo utilizes gradient-based methods when the objective function is smooth and differentiable. However, discontinuities in the objective function or constraints may make them inefficient (Schrage 2006). In recent versions of Lingo, the developers have incorporated more advanced algorithms, hybrid methods, metaheuristics, multistart techniques, and global search techniques to mitigate problems related to nonlinear problems and the risk of getting trapped in local optima. These improvements have allowed Lingo to solve complex nonlinear issues more accurately and efficiently, increasing the probability of reaching global optima (Men & Yin 2018; Gupta & Ali 2021).

The results of the optimization (reservoir storage and release) are applied to establish the relationship between the inputs and outputs of the ANFIS model. Several key factors affect reservoir releases, the most important of which are the inflow to the reservoir, the amount of water stored in the reservoir, and downstream demand. The inflow to the reservoir is one of the most critical determinants of output. Precipitation, snowmelt, surface, and river flow determine it. The storage volume of the reservoir also plays a significant role in specifying the outflow; for instance, when the water volume in the reservoir is high, it is possible to align the release more reliably with the demand. The needs of downstream reservoir users, such as cities, industry, agriculture, and hydroelectric power plants, must be considered managing releases. All of these factors are critical in determining the optimal performance of the dams and need to be coped with simultaneously. In this research, fuzzy and neutrosophic input data, including reservoir inflow, storage, downstream demand, and output data corresponding to the reservoir release, have been considered. The flowchart of the proposed neutrosophic inference environment is shown in the figure 9.

The rule curve in reservoir dams is a graph that shows the relationship between the volume of water in the reservoir and time. This curve serves as a guideline or structured protocol for managing and regulating the reservoir to ensure optimal achievement of various objectives. After optimization of the lower and upper rule curves, the development proceeds for 12 months (Macian-Sorribes & Pulido-Velazquez 2020). To implement the rule curve, the reservoir storage at the beginning of each month is calculated using Equation (14) in the simulation, after which the release policy is applied based on Equations (20)–(23). These equations are presented by Hyung-Il Eum et al. in the integrated management system for the Nakdong River reservoir in South Korea (Eum & Simonovic 2010).These equations divide the reservoir into four zones:

As shown in Figure 3, in the Karun-e Bozorg area, the Karun-4, Karun-3, Karun-1, and Godar-e Landar dams with the purpose of hydropower and the Gotvand dam with the primary purpose of meeting the downstream needs and the secondary purpose of hydropower are located in series. Each of the aforementioned dams also has interbasin inlets to the reservoir. In Godar-e Landar Regulatory Dam, the reservoir inflow is equal to its release, so its operation depends on the performance of the upstream dams. The Dez Dam was built parallel to the aforementioned dams with the primary purpose of supplying downstream demands and the secondary purpose of hydropower generation. Because there is no dam in Band-e Qir, so it is modeled as a node where the Gotvand and Dez releases meet and merge after satisfying the Band-e Qir demand. After the downstream demands of the Band-e Qir are satisfied, the water flows to Ahvaz and the Persian Gulf. Therefore, the extent to which Band-e Qir's downstream demands are satisfied also depends on the operating policies of the Dez and Gotvand dams. The first 15 years of data are the train data, and the next 15 years are the test data. The minimum environmental requirement at Dez Dam is 120 m³/s. The minimum environmental demand at Gotvand and Band-e Qir is 150 m³/s. According to the statistical data, the average monthly demand downstream of Band-e Qir is about 1.3 times higher than the average inflow from the Dez and Gotvand dams.

Considering the different and conflicting objectives and the mentioned series and parallel arrangement of the dams, there is a big problem with many complexities and constraints for the optimal operation of the dams in the Karun area.

The fuzzy performance criteria and objective functions for the train and test data in every dam are shown separately in Tables 2 and 4 and for the entire system in Tables 3 and 5. The comparison of the objective functions shows the extent to which each method has succeeded in meeting the criteria of the model (Equation (13)). Therefore, the closer the objective function obtained in the rule curve, ANFIS, and neutrosophic methods is to the optimal objective function, the better that method has inferred the relationships between the decision variables in the optimal operating mode.

The objective function of the corresponding problem is minimizing the total shortage of downstream needs (hydropower, agricultural and urban, etc.), thus avoiding significant vulnerabilities.

The performance criteria of water resource systems are one of the most critical components in assessing the sustainability of water resources. Reliability, resilience, and vulnerability are widely used criteria for evaluating the performance of water resource systems. Many studies have used them over the past three decades. Loucks (1997) first developed the sustainability index (SI) to quantify the sustainability of water resources systems using the three mentioned criteria (Loucks 1997). After that, many studies used the proposed index (Safavi et al. 2013; Sarang et al. 2008; Sandoval-Solis et al. 2011). Golmohammadi and Safavi explained the underlying problems in formulating performance criteria from the point of view of classical logic. They employed the concept of membership functions in fuzzy systems to modify their related relationships so that the output of these relationships is more consistent with the tangible reality of water resource systems' performance (Safavi & Gol Mohammadi 2016). The fuzzy performance criteria used in this research (Safavi & Gol Mohammadi 2016) provide decision-makers with a more appropriate decision space than the classical performance criteria.

According to Table 2, the objective functions in all dams except Karun-1 and Gotvand are closer to the optimal state in the neutrosophic system than in ANFIS and the rule curve. The SI in train data in all dams except Gotvand in the neutrosophic system is more suitable than all methods, even the optimum operation method. The objective functions of the whole system in training data are 6.84 in the optimum method, 8.13 in the rule curve, 7.27 in ANFIS, and 7.1 in neutrosophic. Therefore, the objective functions in the neutrosophic mode improved by 80 and 40%, respectively, compared to the rule curve and ANFIS in the train data.

The data in Tables 2 and 3 show that neutrosophic outperformed all competing methods in the training data for the fuzzy vulnerability, fuzzy resilience, and fuzzy reliability criteria in almost all dams. In the whole system, neutrosophic improved fuzzy reliability by 9, 9, and 11%, and fuzzy vulnerability by 3, 5, and 3% on the training data compared to optimum methods, rule curve, and ANFIS, respectively. Neutrosophic has improved fuzzy resilience by 15 and 8% compared to optimum and rule curve, respectively, and reduced it by 7% compared with ANFIS. Considering the data in Table 3, neutrosophic has improved the SI of the whole system in the training data by 14, 7, and 11%, respectively, compared to the optimum methods, rule curve, and ANFIS.

As shown in Flowchart 8, rule curve, ANFIS, and neutrosophic inference are obtained using training data, so it is reasonable that simulation with training data leads to better answers than test data. Because the test data are not used in the inference stage, the simulation and comparison results of the relevant methods with the test data are more important for estimating the efficiency of the compared methodologies. In neutrosophic, the SI in the test data in all dams except Gotvand is more suitable than ANFIS and the rule curve. The value of the SI in the Dez Dam was obtained as 0.51, which is a significant improvement compared to all methods, even optimal operation (Table 4).

The objective functions of the whole system in the test data are 7.31, 8.93, 8.15, and 7.85. As a result, the objective functions in neutrosophic optimization were improved by 36 and 67%, respectively, compared to the rule curve and ANFIS in the test data. In the whole system, in the test data, neutrosophic has improved the fuzzy reliability criterion by 4, 10, and 6%, the fuzzy vulnerability criterion by 4, 7, and 4%, and the fuzzy resilience criterion by 12, 8, and 10%, respectively, compared to the optimum method, rule curve, and ANFIS (Table 5). Table 5 shows that neutrosophic improved the SI on the test data by 11, 16, and 11%, respectively, compared to the optimum method, rule curve, and ANFIS.

In the optimization problems of hydropower plant operation, the reservoir release and the storage decision variables are contradictory. The greater the reservoir storage (water level in the reservoir) and the greater the outflow from the reservoir, the greater the power or energy produced (equations 17 through 19). However, an increase in reservoir storage requires a decrease in release, and on the other hand, an increase in release reduces reservoir storage and water level. Therefore, maximizing production capacity is achieved by balancing these two conflicting variables.

The storage of the different operation methods in the Dez Dam test data in Figure 10(b) shows that the storage in the neutrosophic is the closest to the optimum storage. This figure shows that the rule curve method has more fluctuations than other methods because it divides the reservoir into regions and releases water according to a specific equation. On the other hand, the reservoir release is the only storage function. However, in the ANFIS and neutrosophic inference methods, the release is not subject to these crispy divisions, and the reservoir release is also a function of the inflow to the reservoir at the beginning of the period.

Considering that the performance criterion of vulnerability in Dez Dam in neutrosophic is 0.71, while in the optimal method, rule curve, and ANFIS, the values are 0.9, 0.9, and 0.86, respectively, and shortage in neutrosophic is less than competing methods. The vulnerability index is higher in rule curve and ANFIS than in neutrosophic.

As seen in all of the aforementioned figures, release, shortage, power, and storage in neutrosophic are closer to optimal, and the results in the rule curve and sometimes in the ANFIS have high fluctuations. Therefore, the conclusion can be that neutrosophic infers the system's behavior better than other methods.

A future research method comparable to neutrosophic is SDDP. This method is an efficient approach to reservoir exploitation problems with stochastic and accurate data that manage uncertainty through different scenarios (Shapiro 2011).This method suits multistage problems, such as reservoir operation, which require optimal decisions over time under changing conditions. It can adapt to various factors, including changes in reservoir inflows, fluctuations in water demand, and other stochastic parameters, making it easier to solve large-scale problems. However, SDDP has shortcomings. It relies on approximations to solve complex problems, which can reduce the final accuracy of the solution in some cases. SDDP requires rigorous scenario planning, and the results highly depend on uncertainty scenarios. If scenarios are not appropriately defined or key uncertainties are neglected, the results may not be optimal. In this method, uncertainty modeling is limited to scenarios that may not represent all possible situations.

Despite the reduction in computational cost compared to other methods, SDDP can still be very time consuming for large and complex problems. Rouge et al. applied SDDP to issues with multiple near-optimal solutions. He demonstrated that SDDP can exhibit unstable and unpredictable behavior if the problem is not adequately parameterized (Rougé & Tilmant 2016). Zephyr et al. proposed a hybrid SDP method for multireservoir optimization. They used the convexity properties of the value function to sample the reservoir surface space based on the local curvature of the value function estimated by the difference between a lower bound and an upper bound (error bound) (Zephyr et al. 2024). Neutrosophic manages uncertainty, contradiction, and indeterminacy simultaneously, modeling uncertainty as a continuum of truth (T), falsity (F), and indeterminacy (I) rather than relying on specific scenarios. This feature allows the system to analyze complex and uncertain situations and manage conflicting data, while SDDP does not. In general, the neutrosophic method may be more appropriate in situations with uncertainty, inconsistency, and instability in the data, and SDDP is appropriate in situations with more accurate data.

ANNs are another method that may be comparable to neutrosophic in future research. Khad et al. used an optimization-simulation framework based on implicit stochastic optimization, genetic algorithms (GAs), hidden Markov model, and recurrent neural network to extract and simulate operation rules of multiobjective reservoirs. The proposed method was applied in one of the largest reservoirs in the Ruhr River basin in Germany. Artificial flow scenarios were produced using the Thomas–Feering model with statistical characteristics very close to historical datasets (Khadr & Schlenkhoff 2021). One of the advantages of ANN is that there is no need to specify precise and explicit mathematical models; the neural network can automatically learn patterns in the data. Neural networks require large and diverse data for optimal performance. If sufficient and high-quality data are not available, the performance of the model will be affected. Neural networks can become overly dependent on training data and perform poorly with new data and uncertainty. This phenomenon, called overfitting, is one of the common problems of neural networks (Zhang et al. 2018). Although both ANNs and neutrosophic systems are in the field of artificial intelligence, there are many differences in the type of application and their principal purpose. Neural networks are highly effective at learning from large amounts of data and building complex patterns, while neutrosophic systems are highly effective at modeling uncertainty and inconsistency. This research used ANFIS to generate truth membership functions because ANFIS is an efficient method to combine ANNs and fuzzy logic.

Neutrosophic is a new theory that defines indeterminacy differently. Previous studies have used specific statistical data, usually in time series, to model the reservoir. Although data values are typically associated with uncertainty, inaccuracy, vagueness, inconsistency, and incomplete information, fuzzy modeling considers pieces of reality as membership functions. In this research, by using neutrosophic inference and creating ‘truth membership,’ ‘indeterminacy membership,’ and ‘falsity membership’ functions, the statistical information and problem conditions, including uncertainty in reservoir inflow, downstream demand, and hydropower need, have been brought closer to reality and achieve better results than other methods. The results showed that the proposed neutrosophic model has high potential to infer the behavior of complex operating systems in series and parallel reservoirs, and has led to improved system performance compared to conventional methods. Therefore, it can be a reliable method for extracting rule curves. The results showed that neutrosophic is a valuable method for decision-making because it quantifies and considers uncertainties in the data, including uncertainties in inflow, downstream and hydropower demand, etc., as well as the uncertainties that usually occur in the boundary values between the ‘true membership’ and ‘false membership’ functions in modeling problems, and thus well covers the prediction error.

Therefore, the suggestion is to use neutrosophic inference in WRM problems where data contain uncertainty, ambiguity, and error. It is also proposed that neutrosophic optimization be used in future research to define the problem of optimal operation of dams and other optimization problems in water resources management. Considering that the neutrosophic environment provides the possibility for each decision maker to use the ‘truth membership,’ ‘falsity membership,’ and ‘indeterminacy membership’ functions for each option as a set of several possible values between 0 and 1 to agree, disagree, or abstain, respectively, the suggestion is to use this model in WRM issues that require expert opinions and information.

Future investigations can develop truth, indeterminate, and false membership functions using Gaussian, trapezoidal, and other functions. While this may complicate the relationship between influencing factors and decision variables, it will likely improve results. It is possible to consider the regression relations between ANFIS results and optimal values as nonlinear and to study their effect on improving the results compared to the linear relations.

The neutrosophic method is more conceptually complex than classical methods, and it requires more expertise for implementation due to its triple nature and the need to analyze each data with three components. Neutrosophic logic is new to engineering. Therefore, extracting indeterminacy and falsity membership functions is still a challenge.

New uncertainty research methods can enhance the engineering applications of neutrosophic theory. Combining neutrosophic theory with other uncertainty theories is an emerging research direction.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.