Review of Decision Support Systems and Allocation Models for Integrated Water Resources Management Focusing on Joint Water Quantity-Quality

Integrated Water Resource Management, as defined by the GWP, introduces concepts that need to be considered together with analytical tools (allocation models and DSSs) that can effectively support the design and evaluation of demand management instruments. In this section, these concepts are described in the context of IWRM and then the theoretical foundation of the allocation models and systems is presented.

Economic optimization (EC) is the allocation of water resources among all uses/users, maximizing the net benefits (or minimizing scarcity costs) obtained from these uses. An optimal economic allocation effects the transfer of water until the net marginal benefits are equal across all uses (principle of microeconomic equimarginality). Uses with benefits that are difficult to measure, such as for recreational or leisure purposes, can also be considered, as there are methodologies for associating their economic benefits (Bekchanov et al. 2015; IWA 2019; Wang et al. 2019; Ward and Pulido-Velázquez 2008). Besides the economic issues, there are institutional restrictions that guarantee, for example, compliance with certain priorities or minimum values such as ecological flows, issues of equality, and sustainability.

When considering the economic objective as the criterion to be maximized, an economically efficient allocation is identified, which contributes to maximum benefit or well-being of all users of the available water resources. This allocation is considered optimal from an economic point of view, since aggregate gains offset losses, which means that the distributive aspects are not considered, that is, the issue of equality is not incorporated. Thus, a fair economic optimum implies the need for compensation from those who win to those who lose. Both equality (EQ) and sustainability (SU) can be incorporated into the optimization model through restrictions, which results in the so-called second-best economic optimum.

Equality concerns social justice or ensuring access for all users (mainly marginalized and poorer groups) to the quantity and quality of water needed to ensure dignity, human rights, and quality of life. We use the definition of equality of Peña (2011) as that policy environment which provides citizens with a sufficiently equal start in terms of goods, services, and conditions of life, and subsequently fair access to relevant resources that allow them to make use of their individual capacities. Equality can be measured through the Gini coefficient, for example (Hu et al. 2016). Sustainability requires that aquatic ecosystems be also recognized as users and that an appropriate allocation be made to sustain their natural functioning. Meeting this restriction also requires limitation or avoidance of land uses that can negatively impact these ecosystems (IWA 2019). Understanding the interactions between these factors is essential, especially regarding the negative impacts of human activities on the ecosystem. Thus, we consider the integration of quantity and environmental quality separately, with sustainability in this study associated with limits and allocations of the amount of water needed to maintain the natural functions of ecosystems rather than to water quantity-quality issues.

Another important socioeconomic aspect included in the concept of IWRM, in addition to economic optimization, is the maximization of social welfare (SOC). The social issue considered in the concept of equality, in general, is incorporated through restrictions on economic optimization. In practice, basic social needs in many countries are often met through institutional protection, which can also be modeled through constraints on optimization problems. The issue of maximizing social welfare involves elements that can be measured through impacts on the main social indicators, such as the number of jobs, income generation in different social strata, and by life expectancy, while equality is related to the distribution of resources that ensures that everyone’s basic needs are met (Peña 2011).

This assessment requires integration of network-based models with economy-wide models such as computable general equilibrium (CGE) models and input-output models (IOM). The integration of these two types of models expands the capacity for measuring the effects of different allocation policies, supporting their design and evaluation by considering important social welfare indicators.

Throughout the development of this review, it was found that some of the studies reviewed, although they did not consider economic optimization as such, presented measures of economic evaluation, including calculations of benefits and costs. Given the cohesion between economic and social well-being, we included works that presented economic measures but did not consider optimization. Any market context that does not include the criterion of allocative efficiency has a negative effect on the well-being of society (Callan and Thomas 2013).

Many authors have sought to insert environmental aspects in their water management models, whether by applying restrictions on maintaining the ecological flow (Chakroun et al. 2015); by exploitation sustainability constraints (Martinsen et al. 2019a); by simulating negative externalities due to water quality (Abbaspour et al. 2015); by assessing the costs of capturing, transporting, and treating water from available sources (Kahil et al. 2018); and by comparing scenarios of water scarcity and abundance, whether or not caused by climate change (Kahil et al. 2015, 2016a). These concepts are more related to sustainability and do not properly model quality aspects.

Inserted in the challenges of constructing an IWRM that implements sustainable economic development that ensures natural resources for future generations, water quality management integrated with quantity, or water quantity-quality (QQ), represents an important issue, given its complexity.

Network-based economic models, for example, hydro-economic models, use spatial representation to simulate demands from the various economic sectors and to represent offers in a hydrographic basin. This spatial representation of nodes is optimal for representing water quantity aspects; in general, however, these models consider only the quantitative balance (Souza da Silva and Alcoforado de Moraes 2021).

The incorporation of quality is required, however, because it affects water availability and can modify the optimal economic allocation.

The integration of quality into allocation models increases the number of equations and parameters that require calibration, demands sophisticated computational capacity, generally involves high degrees of nonlinearity, and consequently makes it difficult to present a solution. Nonetheless, this is not the most important issue, as mentioned in the Introduction section, because water quality equations incorporated into integrated models are not represented with the same level of detail that is common in traditional water quality simulation models. Distributed sewage return flow, for example, is usually spatially placed in the node closest to the user, which is not optimal for water quality modeling (Alcoforado de Moraes et al. 2010).

There are two main approaches used in modeling to support decision making in the allocation of water resources: (1) descriptive models, which show what will happen if a given strategy is adopted, and (2) prescriptive models, which determine the strategy that must be adopted to satisfy a given objective or decision criteria. It is desirable for water allocation support models to be as prescriptive as possible, but the complexities of operating rules for real systems often direct them toward a more descriptive direction.

Optimization models are inherently prescriptive because they determine values of decision variables that optimize an objective function. In fact, mathematical programming (optimization) techniques inevitably engender the development of more prescriptive models. Despite this, simulation and optimization models need not be rigidly categorized as either descriptive or prescriptive (Wurbs 2005).

Economic optimization modeling makes it possible to identify not only the economically optimum water allocations but also theassociated marginal values such as shadow prices and opportunity costs.

Optimization models also enable delimitation of a Pareto front through different objective functions. For example, in the case of two objective functions (OF1 and OF2), each of the points on the Pareto front represents the maximum OF1, given a certain level of OF2 and vice versa, which means that solutions on the curve form a set of nondominated solutions. Points under the curve, on the other hand, can improve both objectives. Even though these solutions on the Pareto front are nondominated, there are trade-offs among them (Alcoforado de Moraes et al. 2021). This gives policy makers the opportunity to select optimal solutions according to policy priorities.

Ideally, water allocation policy decisions should be based on results from both descriptive and prescriptive models. Table 1 summarizes the main characteristics of decision support models, categorized according to the proposed classification together with examples of applications found in the literature. The initials adopted in the column “types” will be the same as that adopted for the analysis that follows in this paper.

Regarding the model structure, prescriptive or descriptive models that use optimization algorithms can use linear programming where all the mathematical equations are either linear, nonlinear, or able to cope with some nonlinearities. Gradient-based nonlinear programming (NLP) algorithms are widely available and implementations such as CONOPT2 and MINOS have been used to solve complex problems. These are available in the general algebraic modeling system (GAMS), a modeling platform for programming mathematical problems which is especially useful for solving large, complex problems such as those involving a large number of variables and constraints or a high degree of nonlinearity. However, both speed and reliability of these problem-solvers decrease as problem size and complexity increase. Due to this limitation, other methods, such as evolutionary algorithms, have been applied to solve large nonlinear reservoir management models (Cai et al. 2001).

Even descriptive models that do not use any optimization algorithm are able to simulate relations among variables and parameters using linear or nonlinear functions. This is the meaning of linearity used in this review to classify the models in Table 3 (column 6). A model can also have deterministic or stochastic characteristics. In deterministic models, the output of the model is fully determined by the parameter values and the initial conditions with “perfect foresight.” Since water management evolves uncertainties, however, the consideration of stochastic models is important. These models possess some inherent randomness; thus, the same set of parameter values and initial conditions may lead to different outputs.

Traditional DSSs aim to provide partial or complete support to unstructured decision-making activities using analytical models and techniques combined with traditional data access and recovery functions, all through a friendly interface (Paredes-Arquiola et al. 2010; Rousseau et al. 2000). While structured decisions are logical with well-defined factors and results, which are, therefore, programmable, unstructured ones involve a trial-and-error approach, heuristics, intuition, and common sense.

A DSS needs to have three sets of technical capabilities: a database management system (DBMS), a model base management system (MBMS), and a system that manages the interface between the user and the system, which has been named the Dialogue Management and Generation System (DGMS) [Sprague (1980) in Alcoforado de Moraes (1997)]. In the context of water resource management and planning, these systems have been used extensively in conjunction with a geographic information system (GIS) and given the name spatial decision support systems (SDSSs). Integration of a GIS with a DSS provides unique advantages for water management: representation of real world spatial relationships in a visual and analytical form; integration of socioeconomic, environmental, and physical components within a broad database; and the potential for modeling simulation and optimization techniques to support problem solving (McKinney et al. 1999). The possibility of connecting these models with geographic databases managed through a GIS is the main advantage of SDSSs in supporting the management of water resources in cases where both the spatial representation of the managing unit, or part of it, and the application of models for problem solving are required.

There are a number of strategies for coupling models with GIS (Watkins and McKinney 1997), ranging from loose coupling to tight coupling. A coupling is considered loose when the data are merely transferred between the models and the GIS based on separate systems and, generally, with individual data management. A tight coupling is characterized by integrated data management, where GIS and models share the same database. The strength of the coupling lies in the fact that the data and modeling share the same platform (McKinney et al. 1993; Watkins and McKinney 1997).

Making allocation models available and understandable to decision makers involves manipulating data entry and output as well as some graphical interfaces, thus making the input and output interface, or DGMS, in a DSS so important.

The ease and flexibility of defining, modifying, and visualizing any basin or section of it in a GIS interface also influences the modeling and analysis of results. Thus, it is possible to create a platform where specific applications (for example, optimization models with different criteria and restrictions) can be developed and modified to the direct specifications of the decision maker with little time and effort. At the same time, responses to changes (generating results from different models) required by the user are made available through the platform and can be easily analyzed and compared.

In addition to these technical characteristics, the interface of an SDSS must be able to include the participation of all the stakeholders involved. This interface promotes negotiation and cooperation through the provision of a common framework for modelers and decision makers, making economic decision tools and results available through, for example, participatory modeling. Participatory modeling can encompass the development and use of various computer-based models and analytical tools, communication and visualization tools, in addition to mental and cultural models (Mayer et al. 2017).

Another important aspect in the analysis of the results for decision making, in addition to the results of the modeling itself, are the exogenous variables or parameters and the functional forms adopted in the models.

As mathematical modeling makes an abstraction of the real world, it requires some exogenous variables (or parameters) to reduce its complexity. Equations can represent, through certain functional forms, the relationships between variables that constitute functions to be met or optimized, whether as restrictions or objective functions. When these parameters and equations are changed, the results will change, making it possible to analyze the robustness of a model based on the intensity of these changes. Sensitivity analysis refers to variations in model results due to changes in the parameters. On the other hand, the analysis of scenarios allows comparison of model results when changing equations and restrictions. DSSs are able to analyze scenarios and/or sensitivity to changes that represent specific conditions such as climate change, variation in demands, and different environmental and land use policies.

Articles and papers published between 2000 and 2021 were searched. We sought to review models and systems that could support water allocation policies applied to river basin levels using at least one of the concepts discussed in section “Allocation Models and Decision Support Systems for IWRM” (economic optimization; equality; sustainability; social welfare and economic assessment; and quality and quantity integration).

The search procedure, resulting in a selection of 68 publications, is described below:

Step 1. Search for publications related to models and DSSs to support IWRM at the basin level using the Web of Science database of scientific journals for articles that contained the following keywords with the conditional operator AND.

The result of the initial research came up with an almost equal distribution between topics related to Engineering (2,764) and to Economics (2,585), resulting in a total of 4,929 publications. Some of these articles were classified by the system under both topics.

Step 2. Filter the selection of articles based on the application of the terms ‘river basin’ AND ‘integrated modeling’ AND ‘planning water resources system’ throughout the text, which reduced the result to 820 publications.

Step 3. Analyze the titles and abstracts of the articles found in Step 2, removing those that did not consider any of the IWRM concepts described above, thus establishing the final sample at 68 publications, which we considered adequate for the purposes of the review proposed in this study.

Step 4. Analyze articles by reading all those selected in Step 3, tabulating their characteristics based on the IWRM concepts, the theoretical foundations of the models, and the DSSs as described in the section “Allocation Models and Decision Support Systems for IWRM”.

Step 5. Elaborate on and analyze the results.

Initially, in Step 4, an article’s main water management issue was identified based on the classification of tools to support the management of water resources presented by Harou et al. (2009), adapted as shown in Table 2.

Many of the studies reviewed here address more than one category of challenges. However, in the present work, only one category was assigned to each article cited.

Then, the models presented in the articles were evaluated in relation to their use of IWRM concepts, as discussed in the section “Allocation Models and Decision Support Systems for IWRM”: economic optimization, equality, sustainability, social welfare/economic assessment, and integration between water quality and quantity. The models were also classified according to their structure, as “linear” or “nonlinear” and “deterministic” or “stochastic” (Section “Theoretical Foundation of Water Allocation Models”), and based on their orientation, as being “descriptive” or “prescriptive” (see section “Theoretical Foundation of Water Allocation Models”).

The DSSs were classified according to specific aspects related to automated systems such as: coupling intensity (loose or tight), existence of a model or criteria base (availability of more than one model or criterion), analysis of scenarios or sensitivity (functionality present in the DSS), and existence of an interface developed for the user/stakeholder, following the concepts presented in section “Allocation Models and Decision Support Systems for IWRM”.

The analysis of the 68 selected articles was tabulated using these aspects, obtaining first a characterization of each allocation model and DSS described in the paper, related to the consideration (presence or not) of each of the IWRM concepts, its orientation, and structure as detailed in Tables 3 and S1. On the Table 3 it is indicated if the allocation models are available in a DSS, whose specific characteristics are presented in Table 4. A description of the main characteristics of the descriptive optimization-driven simulation (DODS) models is given a specific column (DODS APPROACHES MAIN OBJECTIVE in Table 3) since they do not fit into any other evaluation of the IWRM concepts.

The main characteristics of a DSS (see section “Theoretical Foundation of DSS Used for IWRM”) were observed in each of the studies, and then described and tabulated in Table 4. The presence or absence of a developed user interface and tight coupling are important features as these indicate maturity in the DSS structure, facilitating the insertion of data and the manipulation of simulations and results. Another important aspect is the existence of a model base able to use different criteria, allowing the adaptation of the DSS to the user’s purposes. Nonetheless, there are still many limitations on the interchange of models between platforms and systems while achieving the necessary flexibility.

Using the models and automated systems described in the papers analyzed (see Tables 3 and S1 for more detailed information), a quantitative analysis was conducted regarding the consideration of the IWRM concepts by the tools provided. Fig. 1 presents a representation of the publications mentioning the individual IWRM concepts.

As can be seen in Fig. 1, sustainability is the most used criterion (44). This can be explained by the global concern of countries and institutions about the need for measures to conserve ecosystems. Equality, on the other hand, was given few mentions within an integrated analysis, as were social welfare and economic assessment.

Another detail in the quantitative analysis was considered, dealing with the evolution and trend of the number of concepts treated simultaneously, regardless of which ones, and to what extent they have been incorporated over the years into the models/systems described in the publications analyzed. Recent publications show a tendency for more models to incorporate more IWRM concepts. Fig. 2 presents a weighted average and trend line of the number of concepts incorporated into the approaches reported annually, indicating an evolution in the tools available to support the integrated management of water resources.

Fig. 3 depicts an analysis of concepts included in the publications reviewed over the years selected for this study. The years are presented on the X axis, while the various combinations of concepts considered in the papers are indicated by their acronyms on the Y axis. The number of publications from a specific year using the combination of concepts specified on the Y axis determines the size of the circle with the number reported in its center. The circles are filled according to the number of concepts considered. For instance, circles that represent one criterion are filled with a light gray color. The year 2022 represents the total number of models and systems integrating each of the possible combinations of concepts, for the period 2000–2021.

Sustainability with integration of quality and quantity were presented in 4 publications. Other publications also considered these two concepts integrated with other components [$\mathrm{SU}+\mathrm{SOC}+\mathrm{QQ}$ (3); $\mathrm{EC}+\mathrm{SU}+\mathrm{QQ}$ (1); $\mathrm{EQ}+\mathrm{SU}+\mathrm{QQ}$ (1); $\mathrm{EC}+\mathrm{EQ}+\mathrm{SU}+\mathrm{QQ}$ (2)], which means that sustainability and integration of quality and quantity were present in 11 publications out of the 68 analyzed. They were also considered individually, in 9 and 7 publications, respectively. In summary, sustainability and integration of quality and quantity were included in 57 of the publications studied, individually or in addition to other aspects.

Section “Models and Decision Support Systems for IWRM Considering Integration of Water Quantity and Quality” will discuss in more detail what was found in the publications reviewed concerning the characteristics of the models and systems that integrated the quantity and quality of water, both those which used this concept in isolation and those which included sustainability (most frequently) and other IWRM concepts.

The economic optimization criterion was used as the sole criterion in 5 of the publications studied. Its frequency increases significantly when integrated with other concepts [$\mathrm{EC}+\mathrm{SU}$ (7) - the most frequent combination of two concepts; $\mathrm{EC}+\mathrm{EQ}$ (1); $\mathrm{EC}+\mathrm{SOC}$ (1); $\mathrm{EC}+\mathrm{QQ}$ (4); $\mathrm{EC}+\mathrm{EQ}+\mathrm{SU}$ (5); $\mathrm{EC}+\mathrm{SU}+\mathrm{QQ}$ (1); $\mathrm{EC}+\mathrm{EQ}+\mathrm{SU}+\mathrm{QQ}$ (2)], being referred to in 27 of the analyzed publications.

The economic optimization models, in general, incorporated equality and/or sustainability issues through constraints. Identification of the economic optimum helps in establishing incentives that lead to economic efficiency. This enables decision makers to incorporate, measure, and evaluate economic institutional, environmental, and access restriction issues, facilitating the design of instruments based on economic theory, making the benefits and economic costs associated with the different strategies of government regulation and intervention transparent for decision makers and for society. Overall, sustainability in these economic optimization models was represented in the reviewed publications by ecological use, established in terms of minimum flows in allocation, which, in general, did not significantly increase the complexity of the model. For example, the economic optimization models considered ecological flow as a quantitative restriction for incorporating the sustainability criterion (Souza da Silva and Alcoforado de Moraes 2018).

Social welfare/economic assessment appeared as the sole criterion in 3 models and combined with others [$\mathrm{EC}+\mathrm{SOC}$ (1); $\mathrm{SOC}+\mathrm{QQ}$ (1); $\mathrm{SU}+\mathrm{SOC}$ (5); $\mathrm{EQ}+\mathrm{SOC}$ (1); $\mathrm{SU}+\mathrm{SOC}+\mathrm{QQ}$ (3); $\mathrm{EQ}+\mathrm{SU}+\mathrm{SOC}$ (6)] in approximately 30% of the publications studied. Most of the models/systems in this category carried out economic assessment, but only 8 included social welfare direct measures. The economically driven simulation model developed by Marques et al. (2006) is a good example of the use of an IWRM tool with economic assessment. This model provides estimates of economic and operational impacts of alternative policies for the modeled system, represents demands based on users’ willingness to pay for water, and improves economic approaches to water management. Bekchanov et al. (2017) have reported the difficulty of combining network-based economic models with models such as IOM and CGE. This can be associated with the difficulties of incorporating the criterion of social welfare, which is measured through social indicators such as employment and income. Meanwhile, the importance of integrating this aspect is increasingly recognized. Alcoforado de Moraes et al. (2021) integrated a network-based optimization with an input-output model, made available through an SDSS (the HEAL System), to support the design and evaluation of water allocation policies. In a case study using four interlinked hydrographic basins in northeastern (NE) Brazil, they calculated the ratio between the proportionate economic impact, assessed through changes in social (employment) and economic [gross domestic product (GDP)] indicators, in response to the proportionate water impact (changes in the water allocations) for each economic sector. The HEAL System is also able to identify and support water management in another northeastern basin at Brazil (São Francisco river basin–SFRB), as it recently (Souza da Silva and Alcoforado de Moraes 2021) has incorporated the hydro-economic model, which identifies an optimal economic allocation of water resources in that basin (Souza da Silva and Alcoforado de Moraes 2018). Habibi Davijani et al. (2016) presented social welfare as the objective function in the optimal allocation of water resources. Their results showed an increase of 13% in the number of jobs created in the industrial and agricultural sectors when compared to the previous water use situation.

Equality was the least incorporated criterion and did not appear individually in any model studied, but was associated with other concepts in 18 publications [$\mathrm{EC}+\mathrm{EQ}$ (1); $\mathrm{EQ}+\mathrm{SU}$ (1); $\mathrm{EQ}+\mathrm{SOC}$ (1); $\mathrm{EC}+\mathrm{EQ}+\mathrm{SU}$ (5); $\mathrm{EQ}+\mathrm{SU}+\mathrm{SOC}$ (6); $\mathrm{EQ}+\mathrm{SU}+\mathrm{QQ}$ (1); $\mathrm{EC}+\mathrm{EQ}+\mathrm{QQ}$ (1); $\mathrm{EC}+\mathrm{EQ}+\mathrm{SU}+\mathrm{QQ}$ (2)]. Among the few studies that considered equality, most did this through definitions of priority for human use. In particular, de Azevedo et al. (2000) and Cai et al. (2002) incorporated temporal equality by introducing restrictions to ensure that the benefits of water use would not diminish over the years; they and incorporated spatial equality by ensuring that users in different locations in the basin would have equitable access to water.

Four aspects of the IWRM definition [$\mathrm{EC}+\mathrm{EQ}+\mathrm{SU}+\mathrm{QQ}$ (2)] were considered by Zhou et al. (2015) and Cai et al. (2002). To address these aspects simultaneously, Zhou et al. (2015) developed an integrated optimal allocation model (IOAM) for a complex system of water resources management combined with adaptive allocation, dynamic allocation, and multi-objective optimization.

Another important aspect to consider is the development of integration between concepts categorized as either economic or physical (engineering). To evaluate this integration, a matrix was built (Table 5), recording the frequency with which the paired concepts are present in the analyzed models. The matrix can be analyzed by criterion both along the rows and down the columns. Among the economic concepts, economic optimization is the one that appears most integrated with the engineering concepts (23 times), followed by equality (19) and social welfare/economic assessment (18). Among the economic concepts, the combinations occur less frequently, with the occurrence of the integration of economic optimization and equality (9) and between equality and social welfare/economic assessment (7). Analysis of the engineering concepts shows that environmental sustainability appears integrated with economic concepts more often (44 times) than the criterion of water quantity-quality (16).

Even though QQ is a quite common criterion among the revised models, quantity-quality integration appears less frequently combined with economic optimization concepts. This occurs in only 7 models: [$\mathrm{EC}+\mathrm{QQ}$ (4); $\mathrm{EC}+\mathrm{SU}+\mathrm{QQ}$ (1); $\mathrm{EC}+\mathrm{EQ}+\mathrm{SU}+\mathrm{QQ}$ (2)].

In fact, economic optimization integrated with quantity-quality confers many important benefits to IWRM, indicating the importance of developing more of these models. Thus, the introduction of water quality restrictions for ecological purposes can be priced out, enabling measurement of the costs of environmental restrictions of different intensities. These expenses can be discussed with society and compared with estimates of prevention and correction costs. Compensation schemes and shadow prices can be used to deal with externalities.

Regarding the structure of the models, there was a very balanced division between descriptive (28) and prescriptive (40), with most of the analyzed models being prescriptive. Since both descriptive and prescriptive models generally employ equations and nonlinear functions, this demonstrates the complexity of the mathematical modeling required by IWRM.

Fig. 4 details this division in the inner two rings. On the third ring, the studies are divided according to the absence or presence of stochasticity. Finally, the IWRM concepts are shown on the outside ring as they are incorporated into the models and systems. The smallest division shown in the figure represents one unit, that is, one study, which enables visualization of the ratio between shapes on the graph and the number of studies.

Stochastic models appear in a much higher proportion among the prescriptive models than in the descriptive ones. As prescriptive models maximize or minimize a specific objective function to identify a plan that can be adopted that satisfies the decision criteria, they need to provide values for decision variables simultaneously, in all time periods, as well as consider inflows related to any future time points and scenarios under analysis. Thus, in general, prescriptive models are associated with stochastic optimization techniques and solvers. These tools are becoming more accessible and are able to deal with more complexity, making them well suited to IWRM and its uncertainties.

Most of the prescriptive and stochastic studies include concepts of sustainability or the integration of quality and quantity within economic optimization, reflecting the need to consider the question of uncertainty in this integration. On the other hand, both linear and nonlinear descriptive models are mostly deterministic and usually consider only the ecological aspects, more specifically, questions of sustainability and integration of quantity and quality, whether combined or isolated. The economic aspects of equality and social welfare/economic assessment appear much less frequently, either isolated or combined with only one of the ecological criteria. There is only one occurrence of the aggregation between equality, social welfare/economic assessment, and an ecological criterion (sustainability). This analysis is important to guide future studies, as well as to develop methodologies that will enable effective support for real integration of the various criteria in management decisions.

It is possible to verify through Table 4 and Fig. 5 that 11 of the 16 DSSs that have scenarios/sensitivity analyses also have friendly-user interfaces, which means they facilitate the design of scenarios and the comparison of the results obtained. 76.5% presented multicriteria analyses and 64.7% included a tight coupling, offering common geodatabases that allow linkages of model input and output to the associated simulation models.

An important observation regarding the DSS found in this review is that few studies provided allocation models that considered any economic criterion. For the most part, these automated systems introduce models that consider only engineering concepts: sustainability and quantity-quality.

As already noted, the integration of quantity and quality combined with sustainability criteria was very frequent in the water allocation models and in those that are made available through a DSS, which increases the potential of applying these tools.

To a lesser extent, there were models which considered this combination together with economic optimization [$\mathrm{EC}+\mathrm{SU}+\mathrm{QQ}$ (1); $\mathrm{EC}+\mathrm{EQ}+\mathrm{SU}+\mathrm{QQ}$ (2)], or other economic aspects [$\mathrm{SU}+\mathrm{SOC}+\mathrm{QQ}$ (3); $\mathrm{EQ}+\mathrm{SU}+\mathrm{QQ}$ (1)]. Of the latter, with some economic component integrated to the two engineering/sustainability concepts (SU and QQ), almost none were available through a DSS. In isolation, the quantity-quality criterion only appears less frequently than models that consider the aspect of sustainability alone, which does not include water quality.

In general, the consideration of the quantity-quality criterion in management models, in isolation or added to other concepts, was frequent in the studies (24) and demonstrated the ability to deal with greater complexity (17 of these studies are nonlinear models). This integration and sophistication of tools thus represents a recognition of the challenges associated with water quality management and water availability. The details of the quality modeling, integrated with the allocation models described in Table 4, are given in Table 6. It is important to highlight the existence of other important papers that present the integration of quantity-quality such as (Xu et al. 2019). This paper addressed reservoir operation by running a quasi-three-dimensional water quality model to simulate reservoir hydrodynamic conditions, nutrient cycles, water-sediment exchanges, and algal dynamics under various water supply schedules.

There is a lack of hydro-economic models with water quality modeling associated with water availability limitations. Few models were found (Alcoforado de Moraes et al. 2010; de Azevedo et al. 2000; Cai et al. 2003; Davidsen et al. 2015; Gunawardena et al. 2018; Zeng et al. 2016; Zhou et al. 2015), and progress in the development of these kinds of integrated models remains limited and slow with the need for improvements in the links between water quality (pollution and impacts) and economic benefits (Bekchanov et al. 2017). This is attributed to the complexity underlying the dynamics of water quality (von Braun 2016; Momblanch et al. 2016).