### Meta-heuristic algorithms and their applications in hydrological science

A meta-heuristic is a higher level procedure that can be used to find a near global optimal solution to optimization problems that include incomplete or imperfect information or limited computation capacity. Meta-heuristics often make few assumptions about the optimization problem being solved, which makes them suitable for a wide variety of problems. Moreover, because they can search through large sets of feasible solutions, meta-heuristics can often find good solutions with less computational effort than traditional iterative methods or simple heuristics [3, 4].

In hydrological science, meta-heuristics such as ACO, GAs, PSO, SA, and TS are used in various sub-disciplines. GAs (which are search algorithms based on natural evolutionary mechanisms) are the most widely known meta-heuristic algorithms and were initially proposed by Holland [5]; they are also utilized more widely in hydrological science than any other algorithm. Figure 1 describes historical applications of GAs in various sub-disciplines of hydrological science. First, Savic and Walters [6] applied a GA to minimize the cost of a water distribution system, illustrating the potential of GAs as tools for planning of water distribution networks. Subsequently, Wang and Zheng [7] coupled a GA and SA to obtain the optimal pumping rate for a groundwater system, finding that their GA-based model was able to obtain almost identical solutions to other programming methods (or even offer improvements). Similarly, Samuel and Jha [8] developed computer code to optimize various aquifer parameters under different hydrogeological conditions using the GA technique. GAs have also been applied in water resources management: Sharif and Wardlaw [9] developed an optimal multi-reservoir system using a GA; Wardlaw and Bhaktikul [10] investigated problems with water allocation in an irrigation system; and Chen et al. [11] forecasted stream flow to help ensure more effective use of water resources. Recently, a methodology for determining the optimum reuse discharges of agriculture drainage water was proposed by Fleifle et al. [12]. Water-related disasters such as drought [13] and flood [14] have also been solved using GAs; for example, Sen and Oztopal [15] adopted a GA to achieve the optimum classification of rainy and non-rainy day occurrences based on selected hydrological data. In the field of rainfall–runoff modeling, Nasseri et al. [16] forecasted rainfall using a GA coupled with artificial neural networks, and Ragab et al. [17] used a GA to conduct parameter estimation for rainfall–runoff models. Recently, optimal control of sediment has become a key issue in watershed management. Accordingly, some studies have investigated the use of GAs as an optimization tool [18–20]. At present, water quality issues (from the environmental perspective) are some of the most pressing optimization problems that can be solved by meta-heuristic algorithms. For instance, Maringanti et al. [21] utilized a GA to investigate best management practices (BMPs) in a watershed and select optimal locations for nonpoint-source pollution control. Additionally, GAs have been used to solve a chlorine injection optimization problem in a water distribution system [22] and to predict the distribution of suitable habitats for selected species in deep water [23]. Based on the various applications described above, it is clear that GA optimization techniques are effective tools for the development of hydrological models and management of water resources. Other meta-heuristic algorithms (including ACO, PSO, SA, and TA) have also been applied in hydrological sciences, although their applications are much more limited than those of GAs. Since the 2000s, the application of meta-heuristic algorithms has focused on a few key fields, particularly reservoir operation, groundwater management, and water distribution network design. Nevertheless, other meta-heuristic techniques have the potential to exhibit good performance in various hydrological sectors based on the positive results achieved for GAs.

### Harmony search algorithms as a tool in hydrological science

As discussed above, HS algorithms can be used to achieve better harmony in the process of musical performance, unlike other meta-heuristic algorithms. Musical performances seek fantastic harmony, which is typically determined by aesthetic estimation, whereas optimization algorithms seek a global optimum that can be determined from the value of an objective function. HS algorithms preserve the history of past vectors and can vary the adaptation rate during computation. Moreover, as with GAs, HS algorithms manage several vectors simultaneously, although HS algorithms are different from GAs in many respects. HS algorithms offer two primary advantages: 1) they produce a new vector after considering all existing vectors, rather than considering only two parent vectors (as in GAs); and 2) they do not require the initial values of decision variables to be specified. Thus, it can be generally said HS algorithms offer more flexibility and better solutions than GAs.

Since the HS algorithm was first developed in 2001, it has been applied to various engineering problems and has garnered increasing attention globally. According to Google Scholar, the study that introduced the concept of the HS algorithm has been cited over 1,200 times (prior to August 2013). In case of major literature, the number of citation is increased up to 541 times in Web of science (Science Citation Index Expanded (SCIE), Social Science Citation Index (SSCI), Arts & Humanities Citation Index (A&HCI)). Approximately 50% of the existing literature pertaining to HS algorithms relates to the fields of civil engineering and mathematics/algorithms (Figure 2); thus, it is clear that HS algorithms have been utilized in various ways, both applied (e.g., engineering) and theoretical.

Applications of HS algorithms in hydrological science can be divided by discipline, as with other meta-heuristic algorithms, and the first hydrological applications of the HS technique were described in 2001. HS algorithms were applied to minimize the cost of a pipe network during the design process and to estimate parameters for the non-linear Muskingum routing model [1, 2]. Subsequently, Paik et al. [24] developed a parameter calibration method for rainfall–runoff models and demonstrated how adopting powerful meta-heuristic optimization algorithms such as HS algorithms can enable researchers to focus on aspects of the rainfall–runoff model other than parameter calibration. Ayvaz [25] first introduced HS algorithms to aquifer modeling, using the technique to determine aquifer parameters and the zonal structure of these parameters based on a given set of observations of piezometric head. More recently, Kougias and Theodossiou [26] solved problems related to dam operation using HS techniques; in particular, they investigated the optimum operation of a four-reservoir system over one day, with the aim of maximizing the daily benefits obtained from the reservoir system. The studies cited here represent only a small portion of the existing literature in this field, yet their results demonstrate the potential for use of HS algorithms in solving complex optimization problems successfully and efficiently.