The main themes of this encyclopaedia entry are an integrated resources management problem considering environmental and economic parameters, and its representation through different optimization types.

Water Resources Management (WRM) involves all measures and actions that we apply to water resources (surface, groundwater, freshwater, and seawater), to convert or improve their status and cover the multiple needs of societies and ecosystems [1,2]. As one can imagine, WRM is subject to multiple factors (e.g., meteorological, natural, ecological, and socio-economic), and should take into account various water sources and different users, with different characteristics [3]. There are also multiple direct and indirect implications of WRM in the short and long term, applicable to multiple sectors (such as ecology, biodiversity, hydrology, economy, energy, industry, urban planning, policy, etc.) [4]. Integrated Water Resources Management (IWRM) acknowledges the intricate nature of WRM, encompassing various factors, stakeholders, and short- and long-term implications across multiple sectors. Thus, WRM is often meant to be a holistic and integrated process, by nature [5,6]. Since water resources utilization is connected with multiple other uses and activities, WRM deals with complex problems that must take into account many different variables reflecting this complexity (different sources, different users, and gains from using water, as mentioned). These complex problems have theoretically infinite management solutions. The achievement of WRM objectives under the various restrictions posed by their natural, social, economic, and regulatory aspects can be closely related to an optimization process approach that aims to find the optimum solution(s) under specific constraints [7]. This logic has been a useful approach for several aspects of WRM research and practice, with various applications [8,9], including water resources allocation [10,11,12], water infrastructure, irrigation networks, dams and reservoirs, hydropower works, etc. [13,14,15], hydrology and hydraulics [16,17,18], disaster analysis and management [19,20,21], water quality management [22,23,24,25], transboundary water management [26,27,28,29], policy/governance/development [30,31,32,33,34,35], Water-Energy-Food Nexus [36,37,38], and other cross-disciplinary fields such as hydro-economics, socio-hydrology, ecohydrology, etc. [39,40,41,42,43,44,45,46].

Optimization is a mathematical representation of a problem that we want to solve in the best possible way, satisfying many (often conflicting) objectives. The solutions of such problems are not evident or clearly standing out, so optimization formulates the problems in a structured way (mathematically), helping us solve them while quantifying the impacts of these solutions, or their trade-offs with the constraints of the problem [13].

An objective (goal) is selected for our problem and the optimal solution to the problem will result from its minimization or maximization (e.g., maximum water supply, minimum costs, etc.). This is described mathematically by an objective function.

The objective function is subject to the constraints of the problem, which, as mentioned, express the physical, technical, economic, environmental, or regulatory restrictions of the problem. Each one of these constraints is expressed by a function, and all should be met. The variables in all these functions represent the decision parameters (decision variables) under our control to define them (i.e., the solutions of the optimization problem). The optimal solution values will provide the minimum or maximum result of the objective function, having met all the constraints of the problem [37].

For example, an objective function (Z) is set as a goal to maximize (or minimize), under constraints, which are all functions (h) of the decision variables 𝑥1, 𝑥2,𝑥3,…,𝑥𝑛

(1)

The objective function must satisfy a set (i) of constraints (functions y). These are subject to thresholds (e.g., 𝑎𝑖,

This describes the idea of the general (in this case linear) optimization logic. The different techniques are building on this logic, by following necessary modifications. Depending on the relations of the variables and constraints involved and the mathematical form of the functions used, there are many different optimization techniques, such as non-linear, fuzzy, dynamic, quadratic, or goal programming.

This entry presents a typical problem of integrated WRM, by providing its general formulation, and then its approach from different optimization types offering different representations and answers. The entry is expected to contribute as an educational resource, providing tangible and easily reproducible examples, along with the respective codes.