A robust multiple-objective decision-making paradigm based on the water–energy–food security nexus under changing climate uncertainties

From the perspective of the water–energy–food (WEF) security nexus, sustainable water-related infrastructure may hinge on multi-dimensional decision-making, which is subject to some level of uncertainties imposed by internal or external sources such as climate change. It is important to note that the impact of this phenomenon is not solely limited to the changing behavior patterns of hydro-climatic variables since it can also affect the other pillars of the WEF nexus both directly and indirectly. Failing to address these issues can be costly, especially for those projects with long-lasting economic lifetimes such as hydropower systems. Ideally, a robust plan can tolerate these projected changes in climatic behavior and their associated impacts on other sectors, while maintaining an acceptable performance concerning environmental, socio-economic, and technical factors. This study, thus, aims to develop a robust multiple-objective decision-support framework to address these concerns. In principle, while this framework is sensitive to the uncertainties associated with the climate change projections, it can account for the intricacies that are commonly associated with the WEF security network. To demonstrate the applicability of this new framework, the Karkheh River basin in Iran was selected as a case study due to its critical role in ensuring water, energy, and food security of the region. In addition to the status quo, a series of climate change projections (i.e., RCP 2.6, RCP 4.5, and RCP 8.5) were integrated into the proposed decision support framework as well. Resultantly, the mega decision matrix for this problem was composed of 56 evaluation criteria and 27 feasible alternatives. A TOPSIS/Entropy method was used to select the most robust renovation plan for a hydropower system in the basin by creating a robust and objective weighting mechanism to quantify the role of each sector in the decision-making process. Accordingly, in this case, the energy, food, and environment sectors are objectively more involved in the decision-making process. The results revealed that the role of the social aspect is practically negligible. The results also unveiled that while increasing the power plant capacity or the plant factor would be, seemingly, in favor of the energy sector, if all relevant factors are to be considered, the overall performance of the system might resultantly become sub-optimal, jeopardizing the security of other aspects of the water–energy–food nexus.

www.nature.com/scientificreports/ RCP 8.5) are also integrated into the proposed decision support framework. In essence, the main components of the proposed framework are simulation and operation of the water resources system based on the standard operation policy (SOP), evaluating the system's efficiency through a series of quantitative performance criteria, and finally, applying the MADM-based framework to opt for a robust system renovation setting.
Simulating the water resources system. SOP is a primitive, and perhaps the most-well-known realtime operation policy in water resources planning and management 14 . The core principle here is to minimize the prioritized water shortage at the current time step with no conservation policy (e.g., hedging rules) in place. SOP, as a standard rule curve (RC), determines how the operator should behave at any given state of a reservoir 15,16 . This rule curve is established as an attempt to balance various water demands including but not limited to flood control, hydropower, water supply, and recreation 17 . A SOP operating system attempts to release water to meet a water demand at the current time, with no regard to the future. In general, SOP can be mathematically expressed as 18 : where in which R t = amount of water supplied during the tth time step; D t = consumers' water demand during the tth time step; AW t = amount of available water during the tth time step; S t = amount of stored water during the tth time step; S min = dead storage of the reservoir; Q t = inflow during the tth time step; Loss t = net water loss (i.e., precipitation minus evaporation) of the reservoir during the tth time step; and T = total number of time steps in the operational horizon. In practice, however, a different type of water demand leads to a different interpretation of water shortage. There are cases in which the stakeholders' needs are represented by a set of volumetric demand targets, and the decision-makers' objective would be to minimize the water deficit based on a set of priorities for these demands. This is a typical case for agricultural, domestic, industrial, and environmental demands. For hydropower generation, however, a conventional interpretation of SOP would be to generate maximum electricity permitted by the power plant capacity (PPC) at each given time step 19 . For a hydropower system, the amount of water needed to reach a power plant capacity is given by 19 :  www.nature.com/scientificreports/ in which, γ w = water specific weight; g = gravitational acceleration; η = efficiency of the hydropower system; ΔH t = height difference between the reservoir water level and the tailwater level at time step t; Countday t = number of days within time step t; and PF = plant factor of the hydropower system. As stated earlier, applying an SOP-based plan requires a set of pre-defined priorities to advise decision-makers concerning the order, in which each of these demands is to be met. The major water demands include drinking, industry, environment, agriculture, and hydropower. Thus, according to the SOP's principle, the decision-makers, first, allocate the available water to meet the demand of the stakeholder with the highest priority (i.e., the domestic and industrial demand). After this first water demand is fully satisfied, the available water can be used for the next demand. Such an allocation process continues until no water is available. It should be noted, however, that if the released water in each stage passes through the penstock equipped with the turbines, electricity can be generated. The amount of energy generated in previous stages must be accounted for before computing the amount of water released for hydropower purposes.
Performance criteria. Performance criteria are, in essence, quantitative measures that can provide a practical insight for the decision-makers regarding the status of a system. This definition covers a broad spectrum of mathematical representations, which can range from simple mathematical formulas such as the average of a specific output to more complex and probability-based entities 20,21 . The most fundamental and universal probability-based performance criteria are reliability, resiliency, and vulnerability [22][23][24] . In essence, reliability is the probability of successful function of a system; resiliency measures the probability of successful functioning following a system failure; and vulnerability quantifies the severity of failure during an operation horizon 25 . It should be noted that these three criteria assess different aspects of a water resources system, and as such, they complement one another 26 . For more information regarding these probabilistic performance criteria, the readers can refer to Sandoval-Solis et al. 27 and Zolghadr-Asli et al. 20 .
In this study, the concept of levelized cost of energy (LCOE) is utilized for economic evaluation. The LCOE of a given hydropower system is the ratio of lifetime costs to lifetime electricity generation, both of which are discounted back to a common year using a discount rate that reflects the average cost of capital 28 . The LCOE of renewable energy systems depends on the technology, geographic criteria, capital and operating costs, and the efficiency of the system. The LCOE can be mathematically expressed as follows 29 : in which I t = investment expenditures in year t; M t = operation and maintenance expenditures in year t; F t = fuel expenditures in year t; E t = electricity generation in year t; r = discount rate; and n = economic life expectancy of the system.

MADM.
MADM is an umbrella term to describe a series of frameworks, which aim to help individuals or a group of individuals to prioritize a series of discretely defined alternatives with regard to a set of evaluation attributes 30,31 . MADM can provide the necessary means to conduct planning and management under changing circumstances such as those under climate change conditions 10,32 . According to one of the basic principles of MADM, the decision-maker can use the similarity of the feasible alternatives and the preferential result and/ or incongruity of the undesirable alternatives. The notion mentioned above is, chiefly, the core principle of the reference-dependent theory 33 . Accordingly, the reference-based branch of the MADM methods can, itself, be classified into two major groups: screening methods and ranking methods. Screening methods eliminate alternatives that cannot satisfy the pre-determined conditions for the desirable solution, while ranking methods order all the alternatives from the best to the worst 34 .
Pioneered by Hwang and Yoon 35 , the technique for order references by similarity to an ideal solution (TOPSIS) is a compensatory, objective MADM solving method rooted from the basic principles of the reference-dependent theory. The core idea is that the chosen alternative should have the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution 36 . The basic computation algorithm of TOPSIS can be summarized as follows 37,38 : Step I: Construct the original decision matrix (X), where m feasible alternatives are to be evaluated based on n evaluation criteria: in which x ij = the element of the ith alternative concerning the jth criterion.
Step II: Defining the reference alternatives [i.e., the ideal solution (s + ) and the negative-ideal solution (s − )]. To do so, first, the elements of the decision matrix that are associated with negative criteria must be redefined by using the following equation: The elements of the decision matrix that are associated with positive criteria would remain the same: www.nature.com/scientificreports/ The ideal alternative is an arbitrarily defined vector, which describes the aspired solution to the given problem, while the inferior alternative is an arbitrarily defined solution that represents the most undesirable option for the given MADM problem. Here, the ideal and negative-ideal solutions would be represented with two separate vectors where each pair of the corresponding elements in these vectors is, respectively, the maximum and minimum values of x * ij with regard to each of the evaluation criteria.
Step III: Each element of the decision matrix should be normalized by using the following equation: in which p ij = the normalized performance value for the ith alternative with respect to the jth criterion.
Step IV: The weighted normalized preference value (z ij ) can be computed as follows: in which w j = the weight (i.e., the importance value) of the jth criterion. The weights assigned to the evolution criteria reflect their relative importance to the decision-makers. The higher the weights are, the more crucial their roles would be in the selection process. Chiefly, these weighting mechanisms are either subjective in nature or follow an objective procedure. In the subjective approaches, the weights of the attributes are assigned based on the performance information given by the decision-maker, whereas in the objective approaches, the weights of the evaluation attributes would be obtained by using the objective information extracted from the decision matrix 39 . Shannon's Entropy method, used in this study as the weight assignment mechanism, is a well-known objective weighting technique 40 . This method tends to assign the highest weight to an evaluation attribute with the highest dispersity in its values. For more information on the computational framework of this method, the readers can refer to Lotfi and Fallahnejad 41 .
Step V: In this step, every given alternative is compared to the reference points, namely, the ideal and inferior alternatives. The described procedure, which is known as the separation measurement in TOPSIS, can be mathematically expressed as follows 35 : And in which D + j and D − j = separation measurements of the jth criterion with respect to the ideal and inferior alternatives, respectively.
Step VI: The relative closeness to the ideal solution (χ i ), which can be used to rank the desirability of the feasible alternative, can be computed as follows 35 : The further this distance (i.e., larger values of χ i ), the more desirable the alternative would be.
Robust multi-attribute framework. As stated, each climate change scenario depicts a unique future with regard to the changing climate, which in turn introduces an element of uncertainty to the projected performance of water resources systems during their operation horizon. Furthermore, downscaling methods, which link these projected changes in the global climatic pattern to a local or regional scale, can be another source of uncertainty. Naturally, for long-lasting water infrastructure such as a hydropower system, addressing these uncertainties in a proper and timely manner can be one of the key components of a robust project. Thus, this study aims to not only evaluate the system's performance under the status quo but also assess the credibility of the system under the projected climate change conditions.
The other characteristic one might expect from a robust project is its ability to take into account the multidimensionality nature of water-related infrastructure. Most notably, addressing the WEF security nexus must be a priority in water resources planning and management. Resultantly, any robust decision-making paradigm for water resources planning and management purposes should also account for the other pillars of the WEF nexus (i.e., energy and food sectors), as they would be consequentially affected by such decisions. It is also important to note that these sectors could be affected by the climate change phenomenon. The other crucial feature of a robust decision-making paradigm is that it should be able to account for the socio-economic, environmental, and technical factors that determine the overall quality of the project. Such a decision-making paradigm is depicted in Fig. 2. This notion in practice, however, can typically lead to a mega decision matrix composed of numerous criteria and alternatives that can be overwhelming if the subjective MADM methods are to be www.nature.com/scientificreports/ employed. This study, thus, employs an objective MADM framework (i.e., TOPSIS/Entropy) to help overcome the above-described problem. The basic idea is to promote a universal and practical decision support framework that enables the water resources planners and managers to account for the intricacies of the WEF security nexus while simultaneously taking the uncertainties of climate change projections into account. Figure 3 illustrates the flowchart of the proposed decision support framework.

Case study
The Karkheh River basin, with an area of 51,000 km 2 , is located in the southwest of Iran. The basin's strategic role in water and food security in the country has earned it the nickname: Iran's food basket. Furthermore, the basin's immense potential for hydropower has encouraged the decision-maker to actively pursue and promote new projects to harness these renewable energy resources. Thus, any practical long-term plans for the water resources of the Karkheh River basin must, in a way, address the WEF security nexus. Typically, the Karkheh River basin experiences high spatiotemporal variations in precipitation and air temperature 42 . Accordingly, the southern part of the basin receives an average annual precipitation of approximately 150 mm, while the average annual precipitation in the northern and northeastern parts is reported to be as high as 1000 mm. The air temperature in the basin ranges from − 25 to 50 °C. Overall, the southern area of the catchment can be classified as semi-arid with mild winters and long hot summers, while the northern part and  www.nature.com/scientificreports/ the alpine regions have cold winters and mild summers. As a result, both rain-fed and irrigated agriculture are practiced in the northern part of the basin, while irrigated agriculture is solely practiced in the southern part of the basin due to its arid climate 42,43 . The Karkheh River, which is originated from the Zagros Mountains in Western Iran, is approximately 900 km long and discharges into the Hoor-Al-Azim swamp at the Iran-Iraq border. Based on the available datasets, the average annual discharge of the Karkheh River under the baseline condition is about 5916 million m 3 (MCM). In addition to its permanent role in the energy and agriculture sectors, the Karkheh River also plays a crucial part in establishing the downstream ecosystem and supplying water for the domestic needs of the nearby communities 43 .
Due to the considerable potential of the basin in terms of renewable energy resources, many new hydropower projects have been implemented to actively expand the basin's capacity to generate electricity 19 . The Karkheh Dam, with an installation capacity of 400 MW, is among one of the major infrastructures that has been in an operational stage since 1999 to help meet the drinking, industrial, agricultural, and energy generation demands of the region. Figure 4 illustrates the Karkheh River basin and the location of the dam. The characteristics of the Karkheh Dam are shown in Table 1.
As mentioned earlier, any infrastructure with a long-life expectancy, such as hydropower systems, must take into account the potential impacts of a changing climate. In this study, three representative concentration pathway (RCP)-based climate change projections, including RCP 2.6, RCP 4.5, and RCP 8.5, are considered. These scenarios are identified by their approximate radiative forcing (RF) values. Emission scenario RCP 2.6 peaks at 3.0 W/m 2 and then declines to 2.6 W/m 2 in the year of 2100; RCP 4.5 stabilizes after 2100 at 4.2 W/m 2 , and RCP 8.5 reaches 8.3 W/m 2 in 2100 on a rising trajectory 44 . The primary objective of these scenarios is to provide all www.nature.com/scientificreports/ input variables necessary to run comprehensive climate models in order to reach a target RF 45 . Furthermore, to generate a local climate change trajectory, one needs to downscale the projections of a global or regional circulation model. Indeed, the model selection and downscaling technique may have a tremendous effect on the outcome of these regional climate change trajectories 19 . To carefully address this issue, this study employs three different approaches to obtain the downscaled regional climate change projections. For the first and second trajectories, the outcomes of the second generation Canadian Earth System Model (CanESM2), a global circulation model (GCM), were respectively downscaled by using the change factor technique and the statistical downscaling model (SDSM). For the third trajectory, the results from the Coordinated Regional Downscaling Experiment (COR-DEX), a regional climate model (RCM), were bias-corrected by using the quantile mapping (QM) method. Then, the results were averaged out for each of the selected climate change scenarios (i.e., RCP 2.6, RCP 4.5, and RCP 8.5). For more information on the process of generating these regional climate change projections, readers can refer to Zolghadr-Asli et al. 19 and Enayati et al. 46 . The observed and predicted hydrographs of the Karkheh River under the baseline condition (1982-2011) and the three climate change conditions (2020-2070) are respectively shown in Figs. 5 and 6. Given the nature of this study, a fair estimation of the environmental demand is necessary to adequately address the multi-dimensionality of water resources planning and management. To that end, the fair principle of the Montana method 47 was employed to estimate the environmental demands under the three different climate change conditions (Fig. 7). According to this principle, 10% of the mean annual flow would be set as the environmental demand in wet seasons, while in dry seasons it would increase to 30% of the mean annual flow. Accordingly, this method can provide reasonable estimations for environmental demands [48][49][50] . Figure 8 illustrates the remaining downstream demands including the domestic and industrial demands, and the environmental and agricultural demands under the baseline condition, and the agricultural demands based on the future development projects 18 . At this juncture, it is necessary to recap the importance of projecting the impacts of climate change on all the pillars of the WEF nexus. As demonstrated herein, in addition to the changing patterns in hydro-climatic variables that could in turn influence the status of the involved parties, climate change could also directly have an impact on these sectors. As such, a robust decision-making paradigm for water resources planning and management should be flexible enough to account for all these changes.

Results and discussion
Inspired by the decision hierarchy depicted in Fig. 2 the main components of the current MADM problem were selected. Resultantly, the mega decision matrix of this problem was composed of 56 evaluation criteria and 27 feasible alternatives (Table 2). In this study, the environmental attributes measured how the system was able to meet the environmental water demands using a performance criterion. Similarly, the social, food, and energy attributes showed how successful the system satisfied the water demands for the municipal, agricultural, and hydropower sectors, in terms of the performance criteria. Lastly, the cost-effect measures were used to access the quantifiable economic attribute.
The majority of these criteria are the probability-based performance criteria (e.g., C1-C13), providing a glance at the technical aspect of these feasible renovation plans. The other notable technical criterion is the average    www.nature.com/scientificreports/ annual generated electricity (GW.H) (e.g., C14). It should be noted that, besides the status quo, all these criteria are measured and considered under the three different climate change conditions (i.e., RCP 2.6, RCP 4.5, and RCP 8.5). While most of these criteria are positive, some criteria that are used to measure vulnerability (e.g., C9, C10, C11, and C12) are technically referred to as negative criteria. As for the alternatives, there are 27 feasible options, each of which represents a different renovation setting for the hydropower system. The first alternative   www.nature.com/scientificreports/ (A1), for instance, depicts a situation in which the power plant's installation capacity (PPC) would be 400 MW while the plant factor (PF) would be set to 0.17. After a series of extensive simulations, all the evaluation criteria for the 27 feasible alternatives were measured. Next, using Shanon's Entropy, a set of weights were determined and assigned to the evaluation criteria (Fig. 9), which were obtained by an objective approach such as Shannon's Entropy to reflect their relative importance to the decision-maker. The higher the weights, the more crucial their roles in the selection process. The analysis of the assigned weights unveiled a consistent pattern for all of the climate change conditions (i.e., RCP 2.6, RCP 4.5, and RCP 8.5). Presumably, the weights assigned to the social probabilistic performance criteria (i.e., reliability, resiliency, and vulnerability) were zeros, indicating that, objectively, these criteria had the lowest priority in the decision-making process. The status quo, however, portrayed a different situation. With respect to the RCP 2.6 and RCP 8.5 conditions, and the status quo, the top-priority criterion was the vulnerability of the energy sector; and the computed weights for this top criterion under these three conditions were, respectively, 0.110, 0.131, and 0.027. As for RCP 4.5, the criterion with the highest priority was the resiliency of the energy sector, with the assigned weight of 0.070.  www.nature.com/scientificreports/ Figure 10 shows the average weights assigned to the evaluation criteria based on different aspects of the project (i.e., economy, energy, environment, social, and food). Accordingly, the energy, food, and environment sectors, with the average weights of 0.034, 0.022, and 0.016, respectively, are objectively more involved in the decision-making process. The results revealed that the role of the social aspect of this project, on the other hand, is practically negligible.
Next, the TOPSIS method was used to rank the feasible alternatives to determine their relative desirability based on the weighted evaluation criteria. The results of this objective evaluation are displayed in Fig. 11. The first alternative (A1) emerged as the most robust solution, implying that if all crucial factors in the WEF security nexus are evaluated under the uncertain climate change projections, the first alternative would be selected, objectively, as the most desirable solution. From a statistical point of view, however, it should be noted that the relative closeness to the ideal solution (χ i ), which is used to rank the alternatives, indicates that A1 (χ 1 = 0.96), A2 (χ 2 = 0.96), and A3 (χ 3 = 0.96) may not be distinctively different, and thus these three solutions can all be labeled as robust ones. While these three alternatives can dominate other plausible solutions in terms of robustness, further in-depth assessments may be required to determine the ultimate unique order of the undominated alternatives.
It also should be noted that the desirability of the alternatives is sensitive to the variations in the plant factor (PF) variable (Fig. 11). The first, sixteenth, and twenty-third alternatives (i.e., A1, A16, and A23) are all set to have a PPC value of 400 MW, while their PF values are 0.17, 0.20, and 0.23, respectively. The overall ranks of these three alternatives are 1st, 9th, and 23rd, respectively, which suggests that the ranking is highly sensitive to the PF variable and less sensitive to the PPC variable. This notion indicates that increasing the PF of the system, while other factors remain the same, may decrease the overall performance of the system while it might seemingly be in favor of the energy sector.

Concluding remarks
From the perspective of water resources management, a sound, sustainable, and long-term plan should reflect the notion that water is an integrated part of the WEF security nexus. The strategic role of water in a modern society imposes complex challenges to the decision-makers. According to the requirements of the WEF security nexus, most, if not all, major decisions concerning water resources planning and management need to not only account for all the pillars of the WEF nexus, but also address the environmental, socioeconomic, and technical aspects. Meanwhile, the integrity of decisions made within that platform hinges on the hydro-climatic drivers, which are, by nature, exposed to some degree of uncertainty. Climate change is one of the sources for this type of uncertainty, which projects different plausible futures for hydro-climatic drivers, and in turn, the associated hydrological responses. Thus, climate change can alter the perceived historical patterns of hydro-climatic variables. More importantly, this phenomenon can also have major direct or indirect impacts on energy and food-two other pillars of the WEF nexus. As such, for water-related infrastructures, especially those with longlasting economic lifetimes such as hydropower systems, their performance can be significantly affected by these projected changes. A robust project, in terms of water resources planning and management, can be interpreted as a sustainably developed infrastructure that would not only take the multi-dimensionality of water resources into account but also can maintain a certain degree of acceptable performance under the projected fluctuations in the system inputs.
This study used this interpretation of robustness to promote an objective decision support framework (i.e., a fusion of TOPSIS and Shannon's Entropy) to explore further the notion of a robust decision-making paradigm for water resources planning and management within the context of WEF security nexus, which also accounts for the uncertainties imposed by the climate change phenomenon. Derived from an objective point-of-view, this framework provides a pragmatic perspective to opt for a robust solution that considers the environmental, socioeconomic, and technical factors; and more importantly, with minor computational adjustment it can assess even more evaluation criteria and/or feasible alternatives. To demonstrate the applicability of this framework, the Karkheh River basin in Iran was selected as a case study. Using a set of environmental, socioeconomic, and technical attributes, the most robust renovation plan for the Karkheh Dam was identified for both status quo and climate change conditions. As the robust solution, the system has a power plant capacity of 400 MW and a plant