Complexity
Volume 2019 (2019), Article ID 4034258, 13 pages
https://doi.org/10.1155/2019/4034258

Review Article

Water-Based Metaheuristics: How Water Dynamics Can Help Us to Solve NP-Hard Problems

Fernando Rubio and Ismael Rodríguez

Departamento de Sistemas Informáticos y Computación. Facultad de Informática, Universidad Complutense de Madrid, 28040 Madrid, Spain

Correspondence should be addressed to Fernando Rubio; fernando@sip.ucm.es

Received 9 January 2019; Revised 13 March 2019; Accepted 18 March 2019; Published 2 April 2019

Academic Editor: Alex Alexandridis

Copyright © 2019 Fernando Rubio and Ismael Rodríguez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Many water-based optimization metaheuristics have been introduced during the last decade, both for combinatorial and for continuous optimization. Despite the strong similarities of these methods in terms of their underlying natural metaphors (most of them emulate, in some way or another, how drops collaboratively form paths down to the sea), in general the resulting algorithms are quite different in terms of their searching approach or their solution construction approach. For instance, each entity may represent a solution by itself or, alternatively, entities may construct solutions by modifying the landscape while moving. A researcher or practitioner could assume that the degree of similarity between two water-based metaheuristics heavily depends on the similarity of the natural water mechanics they emulate, but this is not the case. In order to bring some clarity to this mosaic of apparently related metaheuristics, in this paper we introduce them, explain their mechanics, and highlight their differences.

1. Introduction

It is common for an engineer or scientist to eventually face an NP-hard optimization problem during her career. This is an unfortunate encounter indeed: even if the problem is also in NP (so the quality of potential solutions can be computed in polynomial time), optimal solutions cannot be found in polynomial time unless P NP. Yet approximate solutions must be found somehow. For some problems we can apply problem-specific polynomial-time heuristics guaranteeing some performance ratio (i.e., ratio between found solutions and optimal solutions) in the worst case, in particular for problems in APX (or, better, in PTAS or FPTAS). (The interested reader is referred to [1] for a detailed description of approximation classes.) Unfortunately, for other optimization problems (in particular, those that are Log-APX-hard or worse) this is not possible, so algorithms providing good solutions on average (but not guaranteeing any performance ratio in the worst case) are needed.

Nature has been broadly used as a source of inspiration for developing heuristic methods of the latter kind, in particular for constructing general-purpose metaheuristics that can be adapted to many particular problems. These metaheuristics include Genetic Algorithms (GA) [2], Ant Colony Optimization (ACO) [3, 4], and Particle Swarm Optimization (PSO) [5]. It is worth noting that the natural processes these methods copy do not aim at solving anything that can be formally stated as an actual computational NP-hard optimization problem (that is, maximizing or minimizing a given known and fixed numerical function). For instance, regarding the natural process GA are inspired on, surviving other species in a real environment is not a well-defined optimization computational problem because the function to be optimized is not known and it even changes over time (other species constantly evolve in unknown ways). Concerning ACO, the problem of finding the shortest path between two nodes in a graph is not NP-hard in its most basic form, as it is solved in polynomial time (e.g., by Dijkstra’s algorithm [6]). As for PSO, finding a food source in an environment with no formal definition of where the food is or can be (i.e., with no function to be maximized) is not an optimization problem either. In fact, we do not know yet how nature would solve an NP-hard optimization problem (in any way other than ours) if it ever had that purpose, as we seem to be the only beings facing that particular goal. Anyway, the similarities between natural goals and problem optimization goals have yielded good heuristics for optimization problems, but not just due to the metaphors they are based on, but rather due to their proper translation and adaptation into suitable algorithms.

A little more than a decade ago, researchers also found inspiration on another natural source: the water dynamics. The way water drops join and collaboratively form paths (flows) of water toward their destination (e.g., the sea) in real environments has been seen as a source of inspiration, similarly to the way chromosomes, birds, or ants interact with each other according to some simple rules to transform themselves into better solutions (in the case of chromosomes and birds) or collaboratively draw a solution on a kind of solution canvas by iteratively modifying it (in the case of ants). Starting from the basic idea of emulating how drops form flows, the research community has independently developed many optimization algorithms (see, e.g., [7–10]) which, despite the common original metaphor they are based on, are quite different (sometimes radically different) in terms of what really matters: the mechanics of the algorithm and the properties of the search this algorithm induces. The actual interest of any new metaheuristic lies in the computational steps it follows and the kind of searching properties these steps unleash (compared to previous methods in the literature), regardless of the beauty of the metaphor it is based on—which, in fact, is unnecessary to develop a new good algorithm. Despite this, for some reason the community expects different algorithms based on the same natural process to expose similar properties. In fact, in the last years the metaheuristics community has given excessive importance to the metaphors the new methods are based on, to the extent that the area has witnessed an explosion of new methods whose novelty is more based on the alleged novelty of the natural metaphor they are based on rather than on the novelty of the resulting algorithm [11].

In order to clarify this research domain and avoid any confusion between metaphors and algorithm mechanics, in this paper we introduce all the optimization algorithms based on water dynamics we have found in the literature to date, we explain the particular aspects of the water dynamics each method is based on, and we describe the resulting algorithm in order to understand the actual optimization mechanism it poses and its important differences from other methods based on the same basic natural metaphor. We hope this paper will bring some clarity to an area full of unfortunate notation and metaphorical coincidences. It is worth noting that this paper will focus on the application of water dynamics to the design of optimization algorithms, so we will not consider its usage in other computational domains. For instance, the water metaphor has been used for decades in the field of image processing by means of concepts like watersheds (see, e.g., [12, 13]), which have been used, for instance, to analyze borders or to obtain binarizations of an image to simplify character recognition (see, e.g., [14]).

The rest of the paper is structured as follows. In the next section we describe the water-based metaheuristics that have been designed to deal with combinatorial optimization problems, whereas Section 3 presents water-based metaheuristics dealing with continuous domain optimization problems. Finally, in Section 4 we present our conclusions.

2. Metaheuristics for Combinatorial Optimization Problems

Three different water-based metaheuristics for combinatorial optimization appeared nearly at the same time during 2007. However, although all of them were directly inspired by observing how water flows from high to low altitudes, the concrete algorithms are completely different. In the rest of this section we describe these metaheuristics and also a fourth method that appeared ten years later, which is closely related to one of the three initial methods. For each metaheuristic, we describe how it works and we also comment on its main application areas.

2.1. RFD: River Formation Dynamics

River Formation Dynamics (RFD) was the first water-based metaheuristic published. It was published in August 2007 in [7], only one month before the publication of Intelligent Water Drops (IWD) in [8].

River Formation Dynamics simulates how water drops collaboratively form rivers in their way to the sea. Drops tend to move through steeper down slopes with higher probability, and they extract soil from the ground while falling through steep slopes. This sediment is deposited later in flatter areas or in lakes. This way, the altitudes of locations around the environment iteratively change. In algorithmic terms, RFD entities (drops) iteratively modify the values attached to the locations of a graph while traversing it, like ants do in ACO. The paths of decreasing altitudes formed on the graph during this process will eventually constitute the problem solutions. The values attached to graph locations, used in RFD to represent altitudes, are attached to nodes (contrarily to pheromone trails in ACO, which are attached to edges). Actually, RFD can be seen as a gradient-oriented version of ACO, because the probabilistic choice of where entities move next is not proportional to the values attached to available edges (the amounts of pheromone trail in ACO), but to the difference of values between each possible destination node and the origin node (difference of altitudes in RFD). This basic but essential difference from ACO makes RFD fulfill new interesting properties, such as trivially avoiding round-trips (as they would imply an ever decreasing cycle, which is impossible), quickly reinforcing newly discovered shortcuts (as they have the same altitude difference as the former path but a shorter distance, which implies steeper slopes on average through the shortcut even before any subsequent reinforcement), and quickly eliminating blind alleys (as sediments are accumulated in lakes, eventually turning attractive down slopes into uninteresting flat slopes). On the other hand, the transposition of path subsequences within a formed solution is harder in RFD than in ACO and for the same reasons: decreasing altitudes tend to constitute a natural traversing order through paths in RFD, which does not happen in ACO. The pseudocode of RFD can be summarized as follows:

initialize drops

initialize nodes

repeat

    move drops

    erode paths

    deposit sediments

    analyze paths

until stopping criteria met

RFD has been used by different research groups to solve a variety of problems. In addition to well-known academic NP-hard problems like the traveling salesman or Steiner tree problems [19, 20], it has also been used in more specific industrial domains, like monitoring electrical power systems [21] or designing VLSI circuits [22, 23]. In particular, due to its intrinsic absence of cycles, it has been used to design several routing protocols in computer networks (see, e.g., [24, 25]), to test systems [26], or to improve robot navigation [27]. A survey describing the main application domains of RFD can be found in [28].

2.2. IWD: Intelligent Water Drops

The first paper describing the Intelligent Water Drops (IWD) metaheuristic [8] was published only one month later than the publication of RFD. This method is also a modification of ACO to simulate how water flows from high altitudes to the sea. However, it is clear that IWD and RFD were developed completely independently, as the formalization of IWD is totally different. In IWD, drops also collaborate to form solutions on a graph by modifying some values attached to its locations, like in ACO or RFD. However, IWD does not associate altitudes with nodes like RFD does. In this sense IWD is closer to ACO, as values are associated with edges of the graph (soil amounts in IWD, like pheromone trails in ACO), and the probabilistic movements of entities are governed by the values attached to possible edges (like ACO) rather than by the difference of these values between each possible destination node and the departure node (like RFD). Thus, the differences between the properties of RFD and ACO due to the gradient orientation of the former, mentioned in the previous section, are also differences between RFD and IWD. For instance, in IWD drops need to record the path they have traversed (as in ACO and opposed to RFD because it is not needed to avoid cycles). The main difference of IWD with respect to ACO is that in IWD drops have a velocity and an amount of transported soil. By using these data, drops can behave differently from ants in ACO. In particular, the velocity is used to decide how much erosion is to be done.

The algorithm works as follows. At the beginning, it is common but not mandatory to create exactly the same number of drops as nodes in the graph. If both numbers are not equal, drops are randomly distributed among the nodes. All drops are initialized with a fixed velocity, zero amount of soil, and a singleton list of visited nodes (including only the node they have been assigned to). Analogously, all edges of the graph are initialized with a fixed amount of soil. In each iteration, each drop moves as follows. As long as the path traversed by the drop so far constitutes a part of a possible solution, it moves to some other node. As mentioned before, the probability of selecting each neighbor node (not visited before) depends on the amount of soil of the corresponding edge: the lower the soil, the higher the probability. After moving the drop, the node is added to the list of visited nodes of the drop, and the drop velocity is updated: the lower the soil in the edge, the larger the velocity increment. Besides, the soil carried by the drop is also updated, adding the same amount of soil that has been taken from the corresponding edge. The process is repeated until a complete solution is found by each drop. After each iteration, the best drop of the iteration is selected. Then, the amount of soil of all edges traversed by this drop through its path (solution) is reduced. This reduction is based on the quality of the solution found by the drop. The pseudocode of IWD can be summarized as follows:

initialization

repeat

    create and distribute IWDs

    update visited lists of IWDs

    for each IWD

        choose next path

        update velocity

        compute soil

        remove soil from path

        add soil to IWD

    find iteration-best solution

    update path of iter-best solution

    update total-best solution

until stopping criteria met

IWD has also been used by many research groups (some have proposed alternative versions of IWD; see, e.g., [29]) and has been applied to many problems, including classic academic problems such as the traveling salesman problem, the n-queens problem, or the multiple knapsack problem [30], but also other industrial problems. In particular, it has been applied to many scheduling problems (see, e.g., [31–34]), as well as to minimize the energy consumption in a power system that requires meeting a given total power demand at each moment [35]. It has also been used to reduce the energy consumption in wireless sensor networks [36], to deal with the robot path planning [37], and even to compress images [38].

2.3. WFA: Water Flow-Like Algorithm

The third and last water-based metaheuristic published in 2007 was the Water Flow Algorithm (WFA) [9]. It also takes inspiration from how water moves from high positions down to the sea following the steepest slopes, but the resulting metaheuristic is not related to ACO—the most related method is the tabu search [39]. Entities do not collaboratively draw solutions on a common graph by modifying the values attached to its locations, but each entity is a candidate solution of the problem by itself. Candidate solutions can move to explore nearby positions, representing other candidate solutions in the search space. The altitude of each point in the search space represents the fitness of the corresponding solution. Thus, the lower the value, the better the solution, so that water flowing toward lower positions means improving the quality of its solution.

Initially, a single water flow is created with an empty tabu list, as well as with a given initial mass and velocity so that the momentum of the fluid can be computed. Then, the water flows to lower altitudes. There are four main operations in WFA: splitting and moving, merging, evaporation, and precipitation. In the first operation, the velocity is used to decide whether the flow can move or must stagnate. In case it can move, the available neighbor locations are evaluated and ranked, but those locations appearing in the tabu list of the flow are removed. The momentum of the water flow is used to decide whether the flow should be split into several independent flows or not. In case it is large enough, it is split into two or three flows (depending on the concrete momentum value), while otherwise the flow is not divided. Then, the subflows (or the undivided flow) move to the best ranked neighbors. The mass of the flow is divided into the subflows proportionally to the quality of the corresponding neighbors, so that the most promising positions can be explored more exhaustively, as the corresponding momentum is bigger. Anyway, the total momentum is kept. The tabu lists of the new subflows are equal to the tabu list of the original flow after adding the position that was explored in the previous step. Finally, a special case occurs when all the neighbors are higher than the current position. In such situation, if the kinetic energy of the flow is larger than the potential energy needed to jump to the neighbor position, then the flow can jump. Otherwise, it stagnates.

After the splitting and moving operation, whether two (or more) flows have arrived at the same position is checked. In this case, both flows are merged into a single flow, keeping the overall momentum. Notice that one of the merged flows could be a stagnating flow. In this case, merging it with another active flow can allow it to move again. Regarding the tabu list of the new merged flow, it combines the tabu lists of the original flows.

In each iteration, a part of all of the flows is removed right after the merging operation. By doing so, the water evaporation to the air is simulated. However, the precipitation operation does not occur in all iterations, but only once every iterations. During the precipitation, a new population of flows is added to the system. As it can be expected, the total mass of the new flows equals the total mass of water that evaporated during the last iterations. Regarding the positions where it rains, they are obtained by applying random distributions around the current flows. The initial velocity of the new flows is the same as the initial velocity of the first created flow, while the tabu list is copied from the tabu list of the closest water flow.

In addition to regular precipitation, enforced precipitation takes place in case all flows stagnate. In this case, the method does not need to wait until all of them get evaporated after several iterations. Instead of that, they are forced to evaporate in a single step, and raining takes place afterwards.

As it can be observed, the size of the population varies along the execution of WFA. It starts with a single entity, next the splitting operation increases the population (allowing the exploration of more paths), and merging can reduce the population again, giving more energy to those positions that have been found through different paths. The concept of using a varying number of entities could be easily exported to other population-based metaheuristics. The pseudocode of WFA can be summarized as follows:

initialization

repeat

    flow splitting and moving

    flow merging

    water evaporation

    if rainfall required then

        precipitation

        flow merging

    if new best solution found then

        update best solution

until stopping criteria met

WFA has also been used by several research groups. It has been used to tackle problems such as the bin packing problem [9], the traveling salesman problem [40], the flow shop scheduling [41, 42], the cell formation problem [43, 44], the quadratic assignment problem [45], and the design of logistic reverse networks [46]. Moreover, it has also been hybridized with fuzzy logic [47].

2.4. HCA: Hydrological Cycle Algorithm

The Hydrological Cycle Algorithm (HCA) [48] was introduced in 2017. It takes IWD as a starting point, although four stages of the water cycle are considered: flow, evaporation, condensation, and precipitation. The flow stage, where drops collaborate to modify a common graph, is very similar to IWD. Its main difference is the introduction of the concept of temperature, needed to decide when to apply the evaporation stage. The temperature is computed by calculating how much improvement has been obtained during the last iterations of the flow stage. The lower the improvement, the higher the temperature increase. When the flowing stage is improving, the algorithm does not apply evaporation, but when it does not improve enough, it is assumed that the stage has converged to a (probably local) optimum. Hence, the evaporation takes place as a way to restart the system to avoid local optima.

During the evaporation stage, a given proportion of drops is evaporated. The specific drops that evaporate are selected by using a roulette-wheel selection, so that the best drops (i.e., those finding the best solutions) are more likely to evaporate. After the evaporation, the temperature is reduced again, and the condensation takes place. During the condensation stage, drops are together in the clouds, so they can exchange information between them directly. Notice that during the flow stage (or in IWD) drops only affect each other in an indirect way, by modifying a common graph. However, in the condensation stage of HCA, any method can be used to exchange information—actually, the condensation stage is problem-dependent. For each concrete application, a different way to recombine drops information can be used, taking advantage of our information about the specific application domain. After the condensation stage, a new set of drops is obtained from the evaporated drops.

In the last stage, the precipitation takes place. In this phase, the termination condition is evaluated. If it is not met then the precipitation restarts the hydrological cycle. That is, the velocity of water drops is restarted and also the amount of soil of each edge, etc. However, this restart is not done from scratch, as some information from the previous iteration is reused. In particular, the edges that belong to the best drop reduce their amount of soil. The aim of this modification is to emphasize the future search near the best solution found so far. This can be seen as a reinforcement stage.

Let us remark that although HCA uses IWD in the flow stage (the stage where drops modify the common graph to find a solution), the overall method is partially independent of IWD. In fact, IWD could be easily replaced by other metaheuristics like ACO or RFD. The main modification needed to perform such a replacement would be to include a proper definition of temperature in those methods. It would be interesting to analyze in what contexts such modifications could be beneficial for the overall performance.

Let us also remark that the condensation stage is also a completely independent stage, where the aim is to allow a direct communication among drops. In this sense, it would be interesting to analyze if a different metaheuristic could be used in this stage. In this case, HCA could be seen as a framework to combine direct-communication and indirect-communication metaheuristics. The pseudocode of HCA can be summarized as follows:

initialization

repeat

    repeat

        for each drop

            choose next node

            update velocity

            update soil and depth

            update carrying soil

        compute solution fitness

        update local optimal

        update temperature

    until temperature == evaporationTemp

    evaporationStage

    condensationStage

    precipitationStage

until stopping criteria met

HCA was introduced very recently in [48] and, to the best of our knowledge, it has only been used by its own authors so far. They have used it to solve the traveling salesman problem [49] and also to solve the capacitated vehicle routing problem [50].

2.5. Comparison

Given the essential differences among the aforementioned methods in terms of both solution representation and the evolution of that representation, next we analyze the differences among these methods in qualitative terms, focusing on differentiating their internal mechanics and how these mechanics affect their suitability for each kind of problem. A detailed quantitative empirical comparison among the aforementioned metaheuristics is out of the scope of this paper.

Note that the method that differs the most from the others is WFA. Let us remark that in IWD and RFD entities collaboratively work by modifying a common canvas where solutions are constructed, whereas this is not the case in WFA, where each entity represents a possible solution by itself. Thus, the user of WFA has to codify the solutions of the corresponding problem in a very different way from IWD or RFD users.

Regarding the comparison between RFD and IWD, even though they both work by modifying that common canvas (a graph), the codifications users have to perform are relatively different. The main difference between both methods is that IWD is edge-oriented while RFD is node-oriented. That is, in IWD drops modify the attributes of the edges of the graph (soil amounts), whereas in RFD drops modify the attributes of the nodes of the common graph (altitudes). Thus, IWD is more closely related to ACO, whereas RFD can be viewed as a gradient-oriented version of ACO. This relevant difference makes RFD better for dealing with problems where cycles are to be avoided, due to its natural, implicit, and memory-free cycle avoidance. For instance, RFD is very convenient for dealing with different types of covering trees (see, e.g., [51]), but it is harder for RFD to deal with other issues like exchanging subsequences of solutions in problems like the traveling salesman problem.

Regarding the most recent method, note that the core of HCA is IWD. Thus, it inherits most of its characteristics. However, the additional stages introduced in HCA are interesting to allow hybridizing IWD with other techniques. In this sense, HCA seems to be a good solution when IWD stagnates in suboptimal solutions, so that the extra stages can help in exploring other areas out of the watershed of the local minima.

3. Metaheuristics for Continuous Optimization Problems

The first water-based metaheuristic devoted to continuous optimization appeared five years later than those for combinatorial optimization. However, during the last seven years, new water-based metaheuristics have appeared every year. In the rest of this section we describe all of them in chronological order, and we also comment on some concrete problems that have been solved by applying them.

3.1. Continuous Versions of IWD and HCA

Two of the metaheuristics presented in the previous section (IWD and HCA) have also been used to deal with continuous optimization problems. In fact, the method used in both cases is basically the same. In [52] the authors propose a way to convert a continuous search problem into a discrete environment. The basic idea is the following. Assuming the function to be optimized is and assuming that we are using bits to represent each real number, a directed graph with nodes and edges is created. All the nodes are structured in a line, so that there are exactly two directed edges from each node to the next node. One of the edges is labeled and the other edge is labeled . By doing so, any path from the first node to the last node represents a binary codification of real numbers, using bits for each of them. That is, each path represents a possible point in the search domain, and we can assign the value of in that point to the path itself. Thus, minimizing (or maximizing) the original function requires finding the optimal path in this graph.

Once a graph has been obtained from the original fitness function, IWD is used basically as described in Section 2.2. The main modification is that a mutation operator is used after each iteration. This mutation consists in selecting randomly an edge of the solution and replacing it by using the alternative edge. That is, if it was a edge then it is changed by the corresponding edge, and vice versa.

A very similar approach is used in [48] to use HCA in continuous domains, but in this case the representation of the real numbers is done with decimal numbers instead of using a binary representation. Thus, nodes and edges are used, being the number of dimensions of the problem and the number of digits of the decimal representation used for real numbers. Obviously, the directed edges connecting each node with the following node are labeled with digits from to .

Let us remark that this method could also be used with any other graph-based metaheuristic, like RFD. The main drawback of this method is that the size of the graph is usually too big when dealing with either large dimensions or high precision. For instance, all the running examples presented in [48] have or less dimensions.

3.2. WCA: Water Cycle Algorithm

The Water Cycle Algorithm (WCA) was introduced in 2012 in [53] to deal with constrained continuous problems. It is a population-based metaheuristic where individuals are distributed randomly along the search space during the initialization phase, and then they are divided into three categories, depending on their quality according to the fitness function. The most promising one is called the sea. Then, the following best individuals are considered rivers, while the rest are considered streams. Inspired by how rivers flow in nature toward the sea, as well as how streams flow toward rivers, in WCA the positions of rivers are modified in each iteration in the direction to the sea, while the positions of streams are modified in the direction to their corresponding river. Each stream is associated with a specific river, and the number of streams associated with each river depends on the quality of the solution found by the river: the most promising rivers have more streams flowing toward them. By doing so, the algorithm explores deeper the most promising regions without abandoning the exploration of other areas.

After each iteration of the algorithm, the fitness of each individual is recomputed in its new position. In case the quality of a stream outperforms the quality of its corresponding river, they swap their roles: the stream becomes the river, whereas the river becomes a stream flowing toward the new river. Analogously, if the quality of a river outperforms the quality of the sea then they exchange roles.

When the position of a river is very close to the sea (or a stream is very close to its river), the evaporation takes place. When the distance is lower than a given parameter , the river (or stream) is removed. Then, the raining process takes place, and a new stream is created. When a normal stream evaporates, the raining process takes place randomly in the search space (in order to widen the exploration of the search space), whereas when a stream evaporates near the sea, the raining process takes place following a Gaussian distribution around the sea (in order to intensify the exploitation near the current optimum). Finally, the evaporation phase is slightly modified to increase the evaporation probability of the less promising rivers and streams.

The most interesting idea of WCA is probably the hierarchical structure of the population, as opposed to other more classic metaheuristics like Particle Swarm Optimization (PSO) [5], Artificial Bee Colony (ABC) [54], etc. In fact, this idea could be easily adapted to deal with other population-based algorithms, including water-based metaheuristics. It would be interesting to study whether this hierarchicalization can help in improving other metaheuristics, like those described in the next sections. The pseudocode of WCA can be summarized as follows:

initialization

calculate cost of each raindrop

determine flow intensity

repeat

    streams flow to rivers

    rivers flow to sea

    for each stream

        if solution better than river then

            exchange stream-river roles

    for each river

        if solution better than sea then

            exchange river-sea roles

    if evaporation condition then

        create clouds and starts raining

    update search intensity parameter

until stopping criteria met

WCA has been used in many different applications by different researchers. The original proposal (see [15, 53]), which has been referenced more than 300 times, introduced examples of typical unimodal and multimodal functions, including also typical constrained problems. Later on, it was extended to deal with multiobjective optimization [55]. Regarding industrial applications, it has been applied to optimize the operation of a dam [56], as well as to deal with different issues in power systems, including how to control oscillations by using power system stabilizers [57], how to deal with the economic load dispatch problem in power systems [58], and also how to minimize the probability of power supply loss [59]. Other examples include optimizing the flow of material in supply chains [60] and the control of traffic lights to minimize the total delay of cars [61].

3.3. SRA: Simulated Raindrop Algorithm

SRA was introduced in 2014 in [16]. In this case there is not any analogy with the water cycle or even with how rivers flow toward positions with a lower altitude. In fact, the analogy (and also the algorithm) is much simpler. It is based on the splash that takes place when a raindrop hits the ground. In this case, after hitting the ground, part of the water can reach nearby positions, and then raindrops can splash again reaching other positions.

Initially, a single drop is randomly generated. Then, splashes are randomly generated around it, considering a given maximum splash radius. For each splash whose position is better than the position of the initial drop, the process is repeated; that is, new splashes are created around them. The only differences are that the splash radius is reduced after each step (to intensify the search around promising positions) and that the number of splashes is halved after each step. The pseudocode of SRA can be summarized as follows:

initialization of candidate solution

and number of splashes

improving counter = 0

improved solution = False

repeat

    if improved solution then

        decrement improving counter

        generate at most N/2 splashes

        improved = False

    else

        increment improving counter

        generate at most N splashes

    for each splash

        if it improves the current solution then

            move raindrop to splash position

            improved = True

until stopping criteria met

This simple algorithm has not attracted much attention from the research community, although it has also been used by other researchers to minimize the risk of distributed denial-of-service attacks [62].

3.4. WWO: Water Wave Optimization

The Water Wave Optimization (WWO) metaheuristic was introduced in 2015 in [63]. The method is inspired by the observation of the movement of water waves in both deep water and shallow water. In WWO, it is assumed that the optimization problem is a maximization problem, and the search space is seen as the seabed. In those points where the fitness function is higher, the seabed is also higher, so the amount of water over it is smaller. Thus, promising solutions correspond to shallow water, whereas bad solutions correspond to deep water. We can take advantage of the equations of the movement of water waves to search for the areas with shallow water, i.e., the areas with higher values of the fitness function. In particular, it is known that the propagation of water waves in deep water occurs with larger wavelength and less energy, whereas in shallow water the wavelength is smaller and the energy higher.

Like the WCA metaheuristic, WWO is also a population-based method. All individuals (water waves) are assigned the same initial height and the same initial wavelength. The initial positions of waves are randomly distributed through the search space. In each iteration of the algorithm, three phases occur: propagation, refraction, and breaking. The propagation phase moves the wave to a new position as follows. For each dimension, the position is randomly modified by adding to its previous position a random number between and multiplied by the wavelength and the size of the dimension. Thus, larger wavelengths allow the wave to move farther. Next, the fitness function is evaluated in the new position. In case the solution has improved, the wave is actually moved to the new position; otherwise the wave remains in its original position. When all individuals have gone through the propagation phase, the wavelengths and heights of all of them are updated. The basic idea is that waves moving to better solutions should increase their height and decrease their wavelength. In fact, this operation is done following an equation that takes into account not only the information of each wave, but also the quality of the best and worst individuals, so that a kind of normalization can be done. Ultimately, the equation guarantees that promising waves will propagate within smaller ranges (lower wavelengths) to intensify the exploitation in promising areas, whereas the worst individuals will propagate within larger ranges to improve the exploration of the overall search space.

Regarding the refraction phase, when the wave height decreases to zero, the corresponding wave dies. In this case, the wave is restarted in a new position randomly distributed around the middle point between its previous position and the position of the most promising wave of the population. Its new height is the same as in the initialization phase, while the new wavelength depends on the fitness of both the previous and the current positions: the better the new fitness, the smaller the wavelength, so that better solutions are explored within smaller ranges.

The third phase of each iteration is the breaking phase. In nature, when a wave arrives to a very shallow area, the wave breaks. In WWO, the analogy is that when a wave finds a point with a fitness value better than all previous positions found so far, a breaking phase is performed to intensify the search around such new best position. More precisely, new waves are created around the new best solution. If none of the new waves outperforms the best solution then they are removed. Otherwise, we have a new best wave. The pseudocode of WWO can be summarized as follows:

initialization

compute initial best solution x∗

repeat

    for each wave x

        propagate x to a new x’

        if f(x’) > f(x) then

            if f(x’) > f(x∗) then

                break wave x’

                update x∗

            replace x with x’

        else

            decrease wave height

            if height == 0 then

                refract x to a new x’

    update wavelengths

until stopping criteria met

WWO has also been used in many different applications, and the original paper [53] has been referenced around 150 times. In addition to classical unimodal and multimodal functions, it has been used for scheduling high-speed trains [63], to classify human activity by analyzing accelerometer information [64], or even to design truss structures [65, 66]. As in the case of WCA, it has also been used in the context of power systems to deal with the economic load dispatch problem [67]. Some modifications of the basic algorithm have also been proposed, including hybridization with the sine cosine algorithm [68] and the introduction of a chaotic phase [69].

3.5. WEO: Water Evaporation Optimization

The Water Evaporation Optimization (WEO) was introduced in 2016 in [17]. This metaheuristic is based on the evaporation behavior of a very tiny amount of water deposited on top of a solid surface. Depending on the degree of hydrophobicity/hydrophilicity of the surface, the equation governing the evaporation rate changes in a nonintuitive way. In fact, as the degree of hydrophilicity increases, it would be expected that the evaporation degree is reduced (as the surface tends to keep the water). However, it is known that there is a transition phase, so that, up to a given degree, the evaporation rate increases as hydrophilicity increases. Then, after this transition phase, the water structure becomes monolayer, and the evaporation rate decreases (as expected) as hydrophilicity increases.

Taking into account this behavior, WEO proposes a population-based metaheuristic where individuals are water molecules. In WEO, it is assumed that the optimization problem is a minimization problem. The solid surface where water molecules lie represents the search space: for each position of the search space, the fitness function represents the wettability of such point. That is, the higher the fitness function, the higher the hydrophobicity of that point.

Inspired by the transition phase obtained while changing the wettability of the surface, the overall algorithm is divided into two phases with the same number of iterations in each phase: monolayer evaporation phase and droplet evaporation phase. The first phase represents surfaces with hydrophilicity values higher than the value of the transition phase. Thus, as the hydrophilicity increases, the evaporation rate decreases. The second phase represents surfaces with lower hydrophilicity values. Consequently, as the hydrophilicity decreases, the evaporation rate decreases. In both phases, when a molecule evaporates, it is created again in a different position, although a different method is used for each phase. In the first phase, molecules can be renewed in positions located farther away, whereas in the second phase the renewing method tends toward closer positions. Thus, the method tries to balance local and global search. Besides, in both phases the most promising water molecules are renewed more locally, whereas the worst individuals are renewed farther. The pseudocode of WEO can be summarized as follows:

initialization

%monolayer evaporation phase

for i in maxIterations/2

    compute substrate vector normalizing fitness values

    compute evaporation probability matrix

    compute permutation matrix to regenerate molecules

    generate evaporated molecules

    compare and update water molecules

%droplet evaporation phase

for i in maxIterations/2

    compute contact angles normalizing fitness values

    compute droplet evaporation probability matrix

    compute permutation matrix to regenerate molecules

    generate evaporated molecules

    compare and update water molecules

Even though the method was proposed only two years ago, the original paper [17] has been referenced more than 40 times by different researchers. In addition to dealing with typical unimodal and multimodal constrained and unconstrained functions (see [17, 70]), it has also been used to solve the truss weight minimization problem where some given stress constraints are to be met [71]. Although most papers using WEO were developed by the original author of the metaheuristic, other researchers have also used it. In particular, the area of power plants appears again as a typical application area of water-based algorithms. In this case, the optimal power flow problem is solved in [72], where several parameters have to be adjusted to optimize the power flow satisfying a given set of constraints. Analogously, in [73, 74] WEO is used to minimize the economical fuel cost and to minimize the emissions produced by a power plant when multiple fuel options are available.

3.6. RFO: Rainfall Optimization

The Rainfall Optimization (RFO) metaheuristic was introduced in 2017 in [10]. The method tries to emulate how water flows down from high positions toward lower positions (e.g., going down from mountains to valleys), simulating the tendency to go down through the steeper slopes. In fact, the movement of the water drops is very related to well-known methods like hill climbing or gradient descent. Initially, it rains randomly around all the search space. Then, each drop tries to move to the position in its neighborhood with the lowest value. In order to do that, the neighborhood is defined as all points whose distances to the current position are lower than a given radius , where the value of is slightly reduced in each iteration, so that the local search is intensified during the last steps of the algorithm.

Let us note that it is not possible to check all points in the neighborhood in a continuous environment. Thus, in each iteration each drop selects randomly points in its neighborhood. In case the best of those points is better than the current position, the drop moves to it. Otherwise, the method of evaluating points in the neighborhood is repeated up to times, increasing also in each repetition the number of points to be checked. After repetitions without finding a better position, the drop gets inactive.

After each iteration, drops are sorted according to their ranks. Each rank is computed by taking into account two values: the fitness of the drop position and the difference between its current fitness and its first fitness. Hence, it is not only relevant to the quality of the current solution, because drops that have improved more during their life are considered more promising even if their current fitness is not so high. After creating the merit order list, the lowest ranking drops are turned into the inactive mode, while the individuals with the best rankings are assigned higher values of , so that their neighborhoods can be explored in a deeper way.

The algorithm finishes after a given number of iterations or when there are not any active drops in the system.

An interesting concept of RFO is the merit list, where the interest of individuals depends not only on their fitness function (as usual in most metaheuristics) but also on its derivative along time: for individuals which are improving a lot their solutions are considered promising even if their current quality is not good enough. This concept could also be exported to other metaheuristics, in particular to dedicate more resources to those individuals improving their results. The pseudocode of RFO can be summarized as follows:

initialization

repeat

    for each active drop

        initialize np value

        repeat

            generate np neighbors

            select best neighbor

            if neighbor improves drop then

                move drop to neighbor

            else

                if Ne iterations reached then

                    turn drop inactive

                else

                    update np value

        until drop improves or drop inactive

    compute merit list

    turn inactive the worst drops

    increase Ne of the best drops

until all drops are inactive or

maximum iterations reached

Even though RFO has been introduced very recently (2017), the first paper describing the method [10] has already been cited 30 times. However, it has not been used much yet. To the best of our knowledge, in addition to the applications presented in the original paper, only one independent research group has used RFO: in [75] RFO is used to solve a variation of the facility location problem.

3.7. DOA: Droplet Optimization Algorithm

The last water-based metaheuristic introduced in the literature is the Droplet Optimization Algorithm (DOA), which was published at the end of 2018 in [18]. Although in this case the metaphor is not so clear as in previous metaheuristics, the method gets inspiration from the droplet generation in clouds, the droplet descent from clouds to the ground, and the evaporation.

The algorithm manages two main structures: the list of current drops (candidate solutions) and a list of the best solutions found so far (). An initial population of drops is randomly distributed among the search space, and the one obtaining the best fitness among them is included as the only initial element of .

In each iteration of the algorithm, the mass center of the current population is computed. Then, the fitness function is evaluated for that point and also for a mutation of that position. The best of both positions is used as the current big drop (). Afterwards, each drop of the population is (partially randomly) moved in the direction toward . Then, for each candidate drop, two elements of are selected, and the position of the candidate drop is modified in a direction that is a combination of its directions toward both elements of . In case the new position is worse than its former position, the movement is undone. Otherwise, the process is repeated up to a given number of trials.

The last step of each iteration consists in updating to include the best solutions found during the current iteration. If reaches a maximum given size then the worse elements of the list are removed. The pseudocode of DOA can be summarized as follows:

initialization

introduce in BDP the best drop

repeat

    compute new mass center

    compute alternative mass center

    if alternative mass center improves new mass center then

        new mass center = alternative mass center

    for each drop

        move (randomly) toward the new mass center

    for each drop

        trials = 0

        repeat

            choose two big drops from BDP

            compute new position considering such big drops

            if position improves then

                update drop position

                trials ++

        until not improvement or trials >= maximum

    update BDP with the best drops

    update population

until maximum number of fitness evaluations reached

DOA has been proposed very recently. Thus, it has only been used by its creators. In fact, only one paper (see [18]) has been published dealing with DOA, and all the examples presented in it belong to a standard benchmark used at the CEC Conference.

3.8. Comparison

In the case of the metaheuristics for continuous domains, the work that the user has to perform to codify solutions is relatively similar in all of them, as opposed to the case of discrete optimization, where the codification of solutions was very different in each metaheuristic. Since conceptual differences among methods are not so radical as in the discrete case (at least, in terms of solution representation), in this case a quantitative comparison is more interesting than a qualitative comparison.

Table 1 shows the results of five of the aforementioned metaheuristics when dealing with five classic optimization functions, all of them minimization problems working on dimensions. The results are taken from the original papers introducing the corresponding metaheuristics. Even though not all the functions were covered by all the metaheuristics, most of them were covered, which allows performing a fair quantitative comparison. Although each result could have been reached after executing the corresponding metaheuristic for a different amount of time (or a different number of fitness function evaluations), we assume the authors did not stop their own algorithms before their results became notoriously stable (i.e., until no significant result improvements were observed for a while), so these values can be fairly taken as the convergence or close-to-convergence results for each metaheuristic and optimization function. Hence, we may fairly compare them in these particular terms (i.e., the speed up to convergence is not considered in the following comparison).

Table 1: Quantitative comparison of metaheuristics for continuous domain optimization.

Problem
WCA [15]
SRA [16]
WEO [17]
RFO [10]
DOA [18]

Sphere
8,44E-19

6,24E-89
1,46E-06
1,82E-24

Rosenbrock
7,00E-06

7,613
3,24E+01
9,77E-01

Griewank

1,44E-02
1,11E-17
9,86E-02
1,77E-02

Rastrigin
2,00E-07
2,62E+02
2,65E-01

2,67

Ackley
1,03E-15
1,95E+01
8,10E-3
1,72E-04
2,26E+01

As it can be seen, in most of the cases WCA obtains the best result, while SRA obtains the worst results among all the methods. WEO obtains the best result in the case of the simplest problem (the sphere problem), while in the rest of the cases it obtains satisfactory results, but they are not so good as those of WCA. Finally, RFO and DOA also obtain satisfactory results, but their performance is worse than in the case of WCA.

Regarding the methods not appearing in the table, HCA can also be compared against WCA by using the results appearing in [48]. In this case, HCA obtains nearly the same results as WCA when the number of dimensions of the problem is . However, the results of HCA worsen with four dimensions. In fact, HCA has a scalability problem with higher dimensions, so that it cannot deal with the problems of dimensions shown in Table 1. Regarding IWD, there is also a scalability problem, but it is not so relevant as in the case of HCA. In fact, IWD can obtain satisfactory results for up to dimensions, as described in [52].

4. Conclusions

In this paper we have introduced all water-based optimization heuristics in the literature we are aware to date, both for combinatorial optimization and for continuous optimization. Most of them are based on emulating, in some way or another, how drops move down due to gravity and form a path on their way down, although other methods simulate other water mechanics. The resulting algorithms are also quite different in general, and basically we may make a distinction between those where the drops (or more general entities) collaboratively form solutions by modifying the values attached to a graph (e.g., in ACO) and those where the drops or other entities are the solutions by themselves, as they define a point in the solution space (e.g., in PSO or hill climbing).

As noted during the previous sections, many algorithms introduce ideas which could be easily exported to other methods with varying degrees of compatibility. In the future, we would like to explore several of these potential hybrid methods, as they could take advantage of the best of different approaches.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work has been partially supported by the Spanish Ministerio de Economía y Competitividad (project TIN2015-67522-C3-3-R) and by Comunidad de Madrid as part of the program S2018/TCS-4339 (BLOQUES-CM) cofunded by EIE Funds of the European Union.

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