Solving the Manufacturing Cell Design Problem through an Autonomous Water Cycle Algorithm

Abstract

Metaheuristics are multi-purpose problem solvers devoted to particularly tackle large instances of complex optimization problems. However, in spite of the relevance of metaheuristics in the optimization world, their proper design and implementation to reach optimal solutions is not a simple task. Metaheuristics require an initial parameter configuration, which is dramatically relevant for the efficient exploration and exploitation of the search space, and therefore to the effective finding of high-quality solutions. In this paper, the authors propose a variation of the water cycle inspired metaheuristic capable of automatically adjusting its parameter by using the autonomous search paradigm. The goal of our proposal is to explore and to exploit promising regions of the search space to rapidly converge to optimal solutions. To validate the proposal, we tested 160 instances of the manufacturing cell design problem, which is a relevant problem for the industry, whose objective is to minimize the number of movements and exchanges of parts between organizational elements called cells. As a result of the experimental analysis, the authors checked that the proposal performs similarly to the default approach, but without being specifically configured for solving the problem.

1. Introduction

Worldwide manufacturing plants are usually structured into manufacturing entities, which are composed of machines processing a specific part for a product. This organization presents a problem for most companies because, generally, the output of a machine could be the input of another, which could be located in a faraway place, reducing the efficiency of the construction process because of the moving of parts. This relevant problem for the industry is a classical optimization problem defined by Flanders [1] and called the manufacturing cell design problem, whose objective is to minimize the number of movements and exchanges of parts between groups of machines, which are called cells.

During the last decade, the manufacturing cell design problem has been intensely addressed in the optimization field area. Exact techniques have been proposed for solving the problem, as in [2]. This type of exhaustive techniques is efficient in solving low-complexity instances. However, these techniques usually fail when larger instances are considered because of timing constraints [3]. On the other hand, approximate algorithms are capable of getting solutions for any instance size in reasonable computing time, but results could not be as high quality as expected because these methods are focused on providing near-optimal solutions [4,5].

Focusing on metaheuristics, many bio-inspired algorithms have successfully addressed complex optimization problems in the literature. These techniques are characterized by mimic the behavior of processes from nature, such as honey bees collect nectar from flowers, bats hunt their prey in darkness, the attraction between two objects by computing the gravitational force, among many other examples. Despite the great success of metaheuristics in different application areas, their proper design and implementation to reach optimal solutions is not a simple task. The operation of metaheuristics is governed by a set of parameters that demand a previous setting. Such a configuration is known to be a major task. In fact, selecting the appropriate parameter setting is crucial for the efficient exploration of the search space and therefore to the effective finding of high-quality solutions [6].

Under this line, the authors propose a variation of the water cycle algorithm technique inspired by the water cycle of the elements in the hydrosphere, which was originally proposed in [7,8]. The goal of this variation is that the algorithm has the capacity of updating the population size to manage the trade-off between exploration and exploitation, so that the process of getting good solutions to the optimization problem could be facilitated. To this end, the authors consider the autonomous search paradigm introduced in [9]. The idea of selecting the population size parameter to control this trade-off is not new and then, many works can be found in the literature following this trend [10]. As expected, other interesting approaches can be found for controlling this trade-off in the literature [10].

In this work, the authors compare the original and the autonomous approach of the metaheuristic while solving 160 known instances of the manufacturing cell design problem [11]. As a result, the authors checked that the autonomous water cycle algorithm performs similarly than the default approach, but without being specifically configured for solving a specific instance, which is valuable.

The manuscript is organized as follows. Section 2 discusses related works in the field. Section 3 defines the manufacturing cell design problem. Section 4 exposes the default water cycle algorithm. The proposed autonomous water cycle algorithm is presented in Section 5. Finally, experimental results and concluding remarks are shown in Section 6 and Section 7, respectively.

2. Related Work

Cellular manufacturing systems are widely considered in the industry due to the several benefits involved in economic terms; indeed, preliminary investigations in this context date back from 1960 with the work of Burbidge [12] about production flow analysis. From this point, mathematical programming was strongly involved in solving the problem. Several examples in this line can be found from classic linear programming to more advanced goal programming procedures [11,13,14,15,16,17,18]. Techniques derived from mathematical programming and artificial intelligence, such as constraint programming, were also reported [2,3]. More recently, the room for metaheuristics in this area has been growing since its use is more appropriate when problem instances are intractable with exact methods because of computing time restrictions. In this line, some authors applied classic metaheuristics as tabu search for solving different approaches of the problem [19,20,21,22,23]. The presence of genetic algorithms (GAs) is also large. For instance, a bi-criteria model for solving cell formation problems was presented in [24]. A similar approach was reported in [25], but involving three objective functions. Alternative routines for the part-flow were incorporated in [26,27]. In [28], another GA was proposed but mostly oriented to minimize the inter-cell flow cost rather than minimizing the number of inter-cell movements. Industrial cases from the automobile and steel industry were presented in [29,30] while applying GAs. Hybrid GAs, as well as variants such as the predator-prey GA, were also explored [31,32,33,34]. Classic simulated annealing, differential evolution, scatter search, and particle optimization algorithms also participate in the literature [35,36,37,38].

From the previous analysis, the authors reach that modern swarm intelligence metaheuristics are the most widely applied techniques for solving the problem in recent years. For instance, the authors may cite the following works, among many others. In [39], a migrating bird algorithm was applied, which was later parallelized in [40]. The artificial fish swarm algorithm and the shuffled frog leaping algorithm were applied in [41,42]. Bat algorithm and its autonomous version were applied in [43,44], respectively. The firefly algorithm was considered in [45,46], the cat swarm optimization in [47] and the flower pollination algorithm in [48]. Black hole algorithm has successfully been applied to fine tune a machine learning approach in [49]. Finally, an Egyptian vulture optimization algorithm was reported in [50].

As introduced before, although metaheuristics are widely considered in the literature for solving complex optimization problems, the usage of these methods is limited by the needed of configuring parameters, which definitively affect how the search is performed. This situation implies that it is needed to solve the fitting problem of the algorithm. However, it is unlikely to get the optimal solution to the problem because of the number of possible combinations. This fact means that the performance of the algorithm could be biased due to the usage of an inadequate configuration. A solution to this problem could be to consider metaheuristics with the less possible number of parameters. However, parameters usually provide metaheuristics the capacity of providing a good solution in a great range of instances and problems, i.e., adaptability. Another possibility is the one considered in this paper, where the authors study how to provide autonomous adaptability during the solving time. This is a novel concept, which has not been extensively considered in the literature. For instance, we may cite the following works defining an autonomous strategy for guiding specific metaheuristics [51,52,53]. On this basis, in this work, the authors study how to provide autonomous adaptability to the water cycle algorithm, which is a metaheuristic that was successfully applied to solve several optimization problems [54,55,56,57]. As far as the authors know, this is the first work in the field providing adaptability to the water cycle algorithm, while solving the cell design problem.

3. The Manufacturing Cell Design Problem

The cellular manufacturing strategy proposed by Flanders [1] promotes the separation of the machines involved in a production plant by following a specific strategy. The idea is to group parts of similar functions, geometry or fabrication process into families that are processed in the same section, and thus creating highly independent areas called cells. Some advantages of this production strategy include reduction of cost and material-handling time, labor, and paperwork, a decrease of in-process inventories, shortening delivery time, and an increase of machine utilization and production control. Further analysis of this technology may be found in [58,59,60].

3.1. Problem Statement

To find the optimal design of the production plant, where the inter-cell part exchange is minimized, the problem is schematized into the part-machine matrix, where each coordinate shows which machine processes a particular part. Through the reorganization of the rows and columns of the matrix different configurations can be tested. From this initial matrix, two others are derived: the machine-cell and the part-cell matrices, representing the cell that currently allocates the machines and parts, respectively.

The idea in this optimization problem is to minimize the so-called exceptional elements, which are parts that move from one cell to another to satisfy the production workflow [11]. The mathematical model representing this problem is described as follows:

• Modeling parameters:

  -

  M: the number of machines.

  -

  P: the number of parts.

  -

  C: the number of cells.

  -

  $M_{max}$: the maximum number of machines in a cell.

  -

  $x_{ij}$: the $(i,j)$ element in the machine-part incidence matrix, meaning

  $$x_{ij}=\left\{\begin{array}{cc}1& \mathrm{if}\mathrm{the}i-\mathrm{th}\mathrm{machine}\mathrm{processes}\mathrm{the}j-\mathrm{th}\mathrm{part},\hfill \\ 0& \mathrm{otherwise},\hfill \end{array}\right.$$

  (1)

  where $i\in \{1,\dots ,M$} is the machine number and $j\in \{1,\dots ,P\}$ is the part number.

• Decision variables:

  -

  $y_{ik}$: the $(i,k)$ element in the machine-cell incidence matrix, meaning

  $$y_{ik}=\left\{\begin{array}{cc}1& \mathrm{if}\mathrm{the}i-\mathrm{th}\mathrm{machine}\mathrm{is}\mathrm{located}\mathrm{in}\mathrm{the}k-\mathrm{th}\mathrm{cell},\hfill \\ 0& \mathrm{otherwise},\hfill \end{array}\right.$$

  (2)

  where $k\in \{1,\dots ,C$} is the cell number.

  -

  $z_{jk}$: the $(j,k)$ element in the part-cell incidence matrix, meaning

  $$z_{jk}=\left\{\begin{array}{cc}1& \mathrm{if}\mathrm{the}j-\mathrm{th}\mathrm{part}\mathrm{is}\mathrm{located}\mathrm{in}\mathrm{the}k-\mathrm{th}\mathrm{cell},\hfill \\ 0& \mathrm{otherwise}.\hfill \end{array}\right.$$

  (3)

The goal of the optimization problem is given by

$$\mathtt{minimize}\sum _{k=1}^C\sum _{i=1}^M\sum _{j=1}^P{x}_{ij}{z}_{jk}(1-y_{ik}),$$

(4)

subject to

$$\sum _{k=1}^C{y}_{ik}=1,\forall i,$$

(5)

$$\sum _{k=1}^C{z}_{jk}=1,\forall j,$$

(6)

$$\sum _{i=1}^M{y}_{ik}\le M_{max},\forall k.$$

(7)

3.2. Problem Example

An example of the manufacturing cell design problem is included in this section to clarify the previous definition. To this end, the authors consider the machine-part matrix in

Table 1. Problem example. Machine-part matrix.

		Machines
		A	B	C	D	E	F	G	H	I	J
	1		✓							✓
	2		✓	✓						✓
	3	✓				✓	✓		✓
	4				✓						✓
Parts	5				✓			✓			✓
	6		✓	✓
	7	✓				✓			✓
	8							✓			✓
	9			✓						✓
	10	✓					✓

, determining how machines and parts are related in a given industrial process. From this table, there are 10 machines ($M=10$) and 10 parts ($P=10$). Suppose that for the optimized design, there are 3 ($C=3$) available cells to organize the industrial process and there is a constraint in the maximum number of machines in a cell so that this maximum number equals 4 ($M_{max}$ = 4). With this information, the machine-cell and part-cell matrices are generated by randomly assigning machines and parts to cells. These two matrices are shown in

Table 2. Problem example. Randomly generated machine-cell and part-cell matrices.

		Cell					Cell
		1	2	3			1	2	3
	A	✓				1		✓
	B		✓			2		✓
	C		✓			3	✓
	D			✓		4			✓
Machines	E	✓			Parts	5			✓
	F	✓				6		✓
	G			✓		7	✓
	H	✓				8			✓
	I		✓			9		✓
	J			✓		10	✓

. From these two matrices, the objective of the optimization problem is to reorganize machines and parts so that inter-cell movements are minimized. A possible solution to this problem is shown in

Table 3. Problem example. Optimized machine-cell and part-cell matrices.

, where machines $\{A,E,F,H\}$ and parts $\{3,7,10\}$ are assigned to cell 1, machines $\{B,C,I\}$ and parts $\{1,2,6,9\}$ are assigned to cell 2, and machines $\{D,G,J\}$ and parts $\{4,5,8\}$ are assigned to cell 3. The cost of this solution is 0 because there are not inter-cell movements, meaning that this is an optimal solution to the problem.

4. The Water Cycle Inspired Solving-Method

As is well known, water exists in the earth in three different states: solid (ice, snow), liquid (water, sea, raindrops) and gaseous (vapor). Even though oceans, sea, rivers, streams, clouds, and rains are constantly changing, the total amount of water on the planet is not affected [61,62,63]. The interconnection of the three water states forms the water life cycle depicted in Figure 1. In this figure, raindrops travel in the mountains towards the sea, forming rivers or streams; however, there is also a possibility that some streams flow into rivers and not necessarily into the sea. The water life cycle is composed of three phases [64,65,66]. The first one occurs when the heat produced by the sun affects the water surface, producing that seawater begins to evaporate. This first phase is complemented by the photosynthesis of plants in a process known as transpiration. In the second phase, vapor rises to the atmosphere forming clouds which store plenty of evaporated water through condensation. The third phase, known as precipitation, occurs when stored vapor becomes liquid water because of clouds start to cool, for instance when clouds rise considerably.

The water cycle inspired algorithm is a population-based solving method. In the context of the algorithm, each individual in the population is a solution to the problem known as a raindrop. Thus, each raindrop has associated a machine-cell and a part-cell matrix, as well as a fitness value calculated as given by Equation (4). The population is composed of $N_{pop}$ raindrops.

The solutions in the population are organized on three levels: sea, rivers, and streams. To this end, the population is first sorted in decreased order of fitness quality. Then, the first solution is the sea. The following $N_{rivers}$ solutions are the rivers and the rest of the solutions are the streams. Thus, the sum of rivers and sea is given by

$$N_{sr}=N_{rivers}+1,$$

(8)

and the number of streams is given by

$$N_{streams}=N_{pop}-N_{sr}.$$

(9)

According to the flow magnitude (how good a solution is), each sea/river has associated a set of streams flowing to them. The cardinal of this set for a given sea/river is defined as

$$N{S}_n=round\left\{\left|\frac{c_n}{{\sum }_{i=1}^{N_{sr}}{c}_i}\right|\times N_{streams}\right\},n\in \{1,2,\dots ,N_{sr}\},$$

(10)

where $c_n$ is the fitness value of the n-th solution in the first $N_{sr}$ solutions in the population as given by Equation (4). From this expression, the sea will have a higher number of streams than a lower quality river. Following this idea, better solutions will have the capacity of attracting more water flows than worse ones, as shown in Figure 2. In this figure, rivers (stars) and streams (circles) modify their trajectory to follow stronger flows. When a stream flows into the sea (squares), it is taken as a solution. Additionally, the white color is used to detail advances in each iteration (a new position).

The way in which a stream modify its trajectory is depicted in Figure 3, where X is the new position of the stream as a random displacement in the interval $[0,\alpha ·d]$, d is the distance between the river and the stream (usually in terms of fitness value), and $\alpha \in (1,2)$.

According to the attraction capacity of water flows, the algorithm applies three phases to transform a solution during one iteration of the optimization process. First, it moves streams forward to the rivers as given by

$$X_{stream}(t+1)=X_{stream}\left(t\right)+rand·\alpha ·(X_{river}\left(t\right)-X_{stream}\left(t\right)).$$

(11)

where rand is a random number in the interval $[0,1]$. Second, streams are directed to the sea as given by

$$X_{stream}(t+1)=X_{stream}\left(t\right)+rand·\alpha ·(X_{sea}\left(t\right)-X_{stream}\left(t\right)).$$

(12)

Third, rivers flow towards the sea as given by

$$X_{river}(t+1)=X_{river}\left(t\right)+rand·\alpha ·(X_{sea}\left(t\right)-X_{river}\left(t\right)).$$

(13)

Next, the algorithm evaluates if the newly generated solution is better than its connection, i.e., if the stream provides a better fitness value than the river. In such a case, roles are exchanges as depicted in Figure 4.

The algorithm also includes a mechanism to avoid premature convergence based on the evaluation of the evaporation condition. This process evaluates how close are rivers and streams to the sea so that evaporation is produced. Thus, the evaporation condition for rivers is evaluated as

$$\parallel X_{sea}-X_{river}\parallel <d_{max}orrand<0.1,$$

(14)

and for streams is evaluated as

$$\parallel X_{sea}-X_{stream}\parallel <d_{max}orrand<0.1,$$

(15)

where $d_{max}$ is updated over iterations as given by

$$d_{max}^{t+1}=d_{max}^{t=0}-\frac{d_{max}^t}{\max \mathrm{iterations}}.$$

(16)

Note that the $d_{max}$ is reduced over iterations, meaning that the evaporation condition is more complicated to meet so as execution progresses. If the evaporation condition is met, a rain process occurs, meaning that the water cycle starts again and then, a new population is generated replacing the previous one. That is,

$$X_{\mathrm{new}\mathrm{population}}=LB+rand·(UB-LB),$$

(17)

where $LB$ and $UB$ represent the lower and upper bound values, respectively. Based on the above description, the main steps in the algorithm are described in Algorithm 1.

Algorithm 1: Water cycle inspired algorithm.

5. The Proposed Autonomous Water Cycle Algorithm

The integration of autonomous search into the water cycle algorithm will be responsible for varying the population size ($N_{pop}$), while maintaining the same proportion of rivers/sea ($N_{sr}$) and streams ($N_{streams}$). This proportion was experimentally defined as $30\%$ for $N_{sr}$ and 70% for $N_{streams}$. The calculation of $N_{pop}$ is inspired by one of the most important expressions in the water cycle algorithm, Equation (10), defining the trade-off between exploitation and exploration in the metaheuristic. Thus, the population size is calculated according to the relationship between the worst and best solution in the current population. That is:

$$N_{pop}=round\left\{\left|\frac{c_{best}^{N_{pop}}-c_{wrong}^{N_{pop}}}{{\sum }_{i=1}^{N_{pop}}cos{t}_i}\right|\times 100\right\},$$

(18)

where $c_{best}^{N_{pop}}$ and $c_{worst}^{N_{pop}}$ are the best and worst fitness value in the current population. In this expression, population size will be larger as the difference between the best and the worst fitness value is increased. Otherwise, population size will be smaller.

The criterion for updating the population size is defined according to the differences observed, in percentage, between the best and worst solution in $N_{sr}$ in two different time intervals. Thus, let $dif{f}_{N_{sr}}^{t1}$ and $dif{f}_{N_{sr}}^{t2}$ the differences observed between the best and worst solution in $N_{sr}$ in time $t1$ and $t2$, $t1<t2$, respectively. Then, if $dif{f}_{N_{sr}}^{t2}-dif{f}_{N_{sr}}^{t1}<0$, it means that the algorithm could be trapped in a local minimum and then, the population size should be increased in a number of elements as given by Equation (18). Otherwise, if $dif{f}_{N_{sr}}^{t2}-dif{f}_{N_{sr}}^{t1}>dif{f}_{th}$, it means that the differences observed could be very large and then, the algorithm should focus the search on the most promising areas, so reducing the population size in terms of Equation (18). Note that $dif{f}_{th}$ was experimentally defined as $3.00\%$. The previously discussed hybridization of autonomous search and the water cycle algorithm can be find in Algorithm 2.

Algorithm 2: Hybridization of autonomous search and the water cycle algorithm.

When proposing a hybrid approach is also important to analyze the time complexity of the proposal in comparison to the default approach. If we focus on the default water cycle algorithm, the time complexity has a linear relationship between the maximum number of iterations $\left(T\right)$ and the population size. Hence, the time complexity of the algorithm is bounded by $O(k·n)$, where k is related to the number of iterations and n with the population size. Analyzing the hybrid approach, we also reach that time complexity is also bounded by a linear relationship between the number of iterations, which remains constant in both approaches, and the population size, which varies over the execution of the algorithm. Hence, theoretical complexity is not increased. As expected, execution time could differ for both approaches because of the size of the population, as occurs for two runs of the default algorithm with two different values for the population parameter. If we focus on the memory footprint, we reach that it depends on the size of the population. Hence, the usage of the main memory will be similar for both approaches, while considering the same population size.

6. Experimental Analysis

This section discusses the experimental methodology followed by conducting the experimentation, as well as a discussion about the results obtained while comparing both default and autonomous approaches.

6.1. Experimental Methodology

Two sets of instances are considered. The first set includes the usual 90 Boctor’s instances, where the usage of 16 machines and 30 parts are optimized (see

Table 4. Boctor’s instances.

Instance	C	M${}_{\mathit{max}}$	Best	Instance	C	M${}_{\mathit{max}}$	Best	Instance	C	M${}_{\mathit{max}}$	Best
ID			Known	ID			Known	ID			Known
BP01	2	8	11	BP11	2	9	11	BP21	2	10	11
BP02	2	8	7	BP12	2	9	6	BP22	2	10	4
BP03	2	8	4	BP13	2	9	4	BP23	2	10	4
BP04	2	8	14	BP14	2	9	13	BP24	2	10	13
BP05	2	8	9	BP15	2	9	6	BP25	2	10	6
BP06	2	8	5	BP16	2	9	3	BP26	2	10	3
BP07	2	8	7	BP17	2	9	4	BP27	2	10	4
BP08	2	8	13	BP18	2	9	10	BP28	2	10	8
BP09	2	8	8	BP19	2	9	8	BP29	2	10	8
BP10	2	8	8	BP20	2	9	5	BP30	2	10	5
BP31	2	11	11	BP41	2	12	11	BP51	3	6	27
BP32	2	11	3	BP42	2	12	3	BP52	3	6	7
BP33	2	11	3	BP43	2	12	1	BP53	3	6	9
BP34	2	11	13	BP44	2	12	13	BP54	3	6	27
BP35	2	11	5	BP45	2	12	4	BP55	3	6	11
BP36	2	11	3	BP46	2	12	2	BP56	3	6	6
BP37	2	11	4	BP47	2	12	4	BP57	3	6	11
BP38	2	11	5	BP48	2	12	5	BP58	3	6	14
BP39	2	11	5	BP49	2	12	5	BP59	3	6	12
BP40	2	11	5	BP50	2	12	5	BP60	3	6	10
BP61	3	7	18	BP71	3	8	11	BP81	3	9	11
BP62	3	7	6	BP72	3	8	6	BP82	3	9	6
BP63	3	7	4	BP73	3	8	4	BP83	3	9	4
BP64	3	7	18	BP74	3	8	14	BP84	3	9	13
BP65	3	7	8	BP75	3	8	8	BP85	3	9	6
BP66	3	7	4	BP76	3	8	4	BP86	3	9	3
BP67	3	7	5	BP77	3	8	5	BP87	3	9	4
BP68	3	7	11	BP78	3	8	11	BP88	3	9	10
BP69	3	7	12	BP79	3	8	8	BP89	3	9	8
BP70	3	7	8	BP80	3	8	8	BP90	3	9	5

). The first 50 Boctor’s instances consider a value of C equalling 2, i.e., two cells, and a value of $M_{max}$ in the range 8 to 12, i.e., a maximum number of machines per cell from 8 to 12. The next 40 Boctor’s instances consider a value of C equalling 3 and a value of $M_{max}$ in the range 6 to 9.

Additionally, the authors apply the algorithms to a set of more complex problems from several authors, resulting in 70 instances to be optimized. In this set of instances, the number of machines is from 5 to 40, the number of parts is from 7 to 100, $M_{max}$ is from 2 to 20, and C is from 2 to 3. Additional information about these instances is found in

Table 5. A set of complex instances from several authors.

Instance	M	P	ID	M${}_{\mathit{max}}$	C	Best	ID	M${}_{\mathit{max}}$	C	Best
(Author)						Know				Know
King	5	7	CFP01	3	2	0	CFP02	2	3	2
Waghodekar	5	7	CFP03	3	2	5	CFP04	2	3	8
Seifoddini	5	18	CFP05	3	2	5	CFP06	2	3	11
Kusiak	6	8	CFP07	3	2	2	CFP08	2	3	7
Kusiak	7	11	CFP09	4	2	3	CFP10	3	3	5
Boctor	7	11	CFP11	4	2	2	CFP12	3	3	2
Seifoddini	8	12	CFP13	4	2	6	CFP14	3	3	7
Chandrasekharan	8	20	CFP15	4	2	7	CFP16	3	3	14
Chandrasekharan	8	20	CFP17	4	2	28	CFP18	3	3	39
Mosier	10	10	CFP19	5	2	1	CFP20	4	3	0
Chan	10	15	CFP21	5	2	4	CFP22	4	3	0
Askin	14	24	CFP23	7	2	1	CFP24	5	3	2
Stanfel	14	24	CFP25	7	2	2	CFP26	5	3	22
McCormick	16	24	CFP27	8	2	16	CFP28	6	3	17
Srinivasan	16	30	CFP29	8	2	12	CFP30	6	3	–
King	16	43	CFP31	8	2	15	CFP32	6	3	–
Carrie	18	24	CFP33	9	2	13	CFP34	6	3	–
Mosier	20	20	CFP35	10	2	27	CFP36	7	3	–
Kumar	20	23	CFP37	10	2	25	CFP38	7	3	–
Carrie	20	35	CFP39	10	2	1	CFP40	7	3	–
Boe	20	35	CFP41	10	2	–	CFP42	7	3	–
Chandrasekharan	24	40	CFP43	12	2	–	CFP44	8	3	–
Chandrasekharan	24	40	CFP45	12	2	–	CFP46	8	3	–
Chandrasekharan	24	40	CFP47	12	2	–	CFP48	8	3	–
Chandrasekharan	24	40	CFP49	12	2	–	CFP50	8	3	–
Chandrasekharan	24	40	CFP51	12	2	–	CFP52	8	3	–
Chandrasekharan	24	40	CFP53	12	2	–	CFP54	8	3	–
McCormick	27	27	CFP55	14	2	–	CFP56	9	3	–
Carrie	28	46	CFP57	14	2	–	CFP58	10	3	–
Kumar	30	41	CFP59	15	2	–	CFP60	10	3	–
Stanfel	30	50	CFP61	15	2	–	CFP62	10	3	–
Stanfel	30	50	CFP63	15	2	–	CFP64	10	3	–
King	30	90	CFP65	18	2	–	CFP66	12	3	–
McCormick	37	53	CFP67	19	2	–	CFP68	13	3	–
Chandrasekharan	40	100	CFP69	20	2	–	CFP70	14	3	–

.

The two approaches, the default water cycle algorithm and the autonomous water cycle algorithm, are applied to solve both sets of problems. The static approach is executed for several population sizes (30, 50, 80, and 100), while maintaining $N_{rivers}=2$, $T=1200$ (number of generations), and $d_{max}=3$, all them experimentally fixed. The autonomous approach considers the same values for T and $d_{max}$, while the population size is dynamically updated in the range 10 to 100. Note that both algorithms were implemented using Java SE 7 and all the experiments were performed on a 2.3 GHz Intel Core i3 laptop with 4 GB RAM running Windows 7.

6.2. Discussion of the Experimental Results

Table 6. Results solving the Boctor’s instances (1/3).

	Water Cycle Algorithm
Instance	Autonomous Approach			Default Approach
ID	${\mathit{N}}_{\mathbf{pop}}\in \{10,100\}$			${\mathit{N}}_{\mathbf{pop}}=30$			${\mathit{N}}_{\mathbf{pop}}=50$			${\mathit{N}}_{\mathbf{pop}}=80$			${\mathit{N}}_{\mathbf{pop}}=100$
	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$
BP01	11,631	196.71	11.13	9372	157.52	11.81	11,572	115.16	11.26	13,097	169.13	11.00	13,163	149.52	11.39
BP02	11,427	172.71	11.00	9081	137.71	11.26	11,345	80.29	11.06	12,921	71.58	11.00	13,247	46.39	11.00
BP03	11,572	222.68	11.00	9021	178.55	11.13	10,329	126.68	11.00	12,642	72.00	11.00	13,281	90.23	11.00
BP04	11,594	128.81	11.16	8964	156.00	11.23	11,378	158.52	11.00	12,863	118.97	11.00	12,903	61.13	11.06
BP05	11,311	96.58	11.15	9102	146.90	11.65	10,363	211.39	11.06	12,759	154.00	11.10	13,347	136.61	11.19
BP06	11,601	370.87	7.03	8969	135.03	7.26	11,729	241.48	7.29	12,865	264.16	7.29	12,829	252.94	7.16
BP07	11,872	141.26	6.15	9093	111.71	6.23	11,377	96.13	6.29	13,042	82.84	6.45	13,171	65.48	6.45
BP08	11,646	199.06	5.10	8972	136.48	5.35	11,404	178.87	4.94	12,825	144.29	4.77	13,067	142.13	4.90
BP09	12,054	33.35	4.26	8836	86.35	4.74	11,581	74.97	3.71	12,548	35.77	4.19	13,066	70.52	4.00
BP10	11,942	81.61	3.37	9177	106.03	3.74	11,602	47.55	3.68	12,539	114.10	3.29	13,147	72.10	3.48
BP11	11,723	5.97	5.38	9128	8.77	4.77	11,900	3.55	5.52	12,743	1.16	5.00	13,292	0.71	5.48
BP12	11,606	73.65	4.39	9038	52.65	4.65	11,840	40.55	4.52	12,614	46.23	4.52	13,347	55.19	4.65
BP13	11,576	95.94	4.13	8941	178.19	4.45	11,707	114.23	4.32	12,596	65.13	4.13	13,016	69.84	4.00
BP14	11,384	35.26	3.15	9019	37.77	3.55	10,789	16.94	3.58	12,839	26.84	3.45	13,431	11.65	3.65
BP15	11,365	77.19	2.15	9062	36.19	2.13	11,062	31.90	2.26	12,514	7.19	2.45	13,168	49.26	2.06
BP16	11,449	121.23	13.15	8907	199.42	14.45	11,851	251.45	14.26	12,709	203.16	14.23	13,245	226.06	14.32
BP17	11,478	71.32	13.00	8996	53.32	13.00	10,805	48.23	13.00	12,801	23.13	13.00	13,104	20.71	13.00
BP18	11,272	110.13	13.00	8947	68.87	13.00	10,695	96.48	13.00	12,711	63.32	13.00	13,219	58.71	13.00
BP19	11,334	186.81	13.19	9028	127.32	13.06	11,087	140.52	13.13	12,679	169.61	13.10	13,037	85.10	13.00
BP20	11,416	269.32	13.10	9022	167.45	13.45	11,509	192.06	13.13	12,919	186.65	13.13	13,075	142.06	13.06
BP21	11,444	19.61	9.14	9040	3.58	9.97	11,886	0.81	9.77	12,992	4.03	9.65	12,990	1.97	9.84
BP22	11,413	30.97	7.03	9079	52.19	7.58	11,608	23.97	7.61	12,772	9.35	7.13	13,208	11.00	7.45
BP23	11,118	99.03	7.02	9063	65.39	7.19	10,607	23.42	7.13	12,784	85.42	7.06	12,870	119.84	7.13
BP24	11,397	25.55	6.27	9134	99.32	7.10	11,718	76.16	6.19	12,778	61.77	5.68	13,112	59.16	6.00
BP25	11,925	164.10	5.16	9103	76.39	4.87	10,513	69.39	4.90	13,099	140.42	4.65	13,255	119.94	4.55
BP26	11,724	112.58	5.16	9071	189.94	5.26	11,784	139.71	5.39	13,111	176.39	5.06	13,023	218.58	5.10
BP27	11,566	189.65	3.81	9100	115.06	4.29	11,956	83.00	3.71	12,857	71.61	4.23	13,149	65.55	3.48
BP28	11,605	95.65	3.81	9008	31.48	4.23	11,347	67.23	3.94	13,074	118.10	3.61	13,024	21.16	3.81
BP29	11,481	124.29	3.06	9042	193.13	3.13	10,895	68.65	3.13	13,116	48.35	3.06	13,241	153.68	3.39
BP30	11,459	3.39	3.10	9078	69.71	2.68	11,750	67.26	2.65	12,876	137.97	2.61	13,217	91.58	2.68

,

Table 7. Results solving the Boctor’s instances (2/3).

	Water Cycle Algorithm
Instance	Autonomous Approach			Default Approach
ID	${\mathit{N}}_{\mathbf{pop}}\in \{10,100\}$			${\mathit{N}}_{\mathbf{pop}}=30$			${\mathit{N}}_{\mathbf{pop}}=50$			${\mathit{N}}_{\mathbf{pop}}=80$			${\mathit{N}}_{\mathbf{pop}}=100$
	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$
BP31	11,191	262.52	7.09	8989	188.87	7.29	11,711	196.19	7.29	12,982	173.74	7.32	13,080	152.23	7.13
BP32	11,224	46.35	5.19	9063	80.13	4.77	10,666	77.32	4.94	12,875	67.61	4.58	13,378	45.16	4.58
BP33	11,792	52.06	4.58	8958	66.55	4.58	11,547	24.58	4.19	12,882	74.39	4.45	13,397	53.06	4.19
BP34	11,492	57.84	4.12	9071	54.32	5.03	10,804	72.81	4.65	13,077	34.03	4.58	13,438	42.13	4.39
BP35	11,456	53.19	4.11	9005	88.13	4.74	11,588	16.94	4.74	12,972	47.87	4.77	13,230	53.74	4.58
BP36	11,251	427.90	13.16	9071	336.52	13.48	10,919	185.87	13.29	12,983	187.45	13.23	13,262	275.68	13.06
BP37	11,321	51.35	10.07	9082	98.32	10.94	11,770	35.55	10.97	12,519	76.81	11.03	13,192	79.58	10.77
BP38	11,318	155.97	9.16	8933	81.16	9.00	10,856	124.61	9.68	12,861	221.61	8.77	13,195	117.52	8.81
BP39	11,571	126.26	7.52	8985	174.61	6.13	10,582	123.16	5.52	12,842	119.45	5.39	13,405	145.16	5.74
BP40	11,763	116.68	6.75	9115	142.13	6.26	11,869	112.06	5.97	13,096	55.77	5.58	13,025	75.97	5.52
BP41	11,275	174.68	9.38	9100	144.42	10.29	11,951	86.90	9.55	12,762	99.55	9.45	13,404	29.68	9.61
BP42	11,458	150.06	8.00	8975	100.55	8.29	11,709	180.94	8.39	12,570	106.16	8.10	12,975	113.13	8.00
BP43	11,436	77.35	8.27	9051	133.90	8.65	11,838	49.97	9.06	12,875	168.19	8.87	13,407	128.10	8.58
BP44	11,222	79.26	6.24	9051	151.81	6.87	11,795	127.74	6.42	12,710	136.87	6.71	13,171	77.26	6.06
BP45	11,304	205.94	6.15	9148	78.19	7.32	11,609	95.97	6.81	13,193	120.45	6.26	13,356	223.10	5.77
BP46	11,363	243.42	8.02	9061	55.45	9.32	11,550	90.23	8.84	12,923	196.61	8.65	12,970	106.48	8.65
BP47	11,168	235.16	6.16	9034	161.23	6.32	10,677	99.29	5.58	12,712	165.58	5.87	13,188	149.35	5.84
BP48	11,692	109.35	5.75	9079	139.94	5.81	10,213	131.94	5.61	12,891	60.26	5.61	13,326	36.03	5.71
BP49	11,552	121.35	5.47	9038	96.77	7.06	10,171	196.90	6.39	12,762	123.42	5.77	12,924	173.68	5.39
BP50	11,471	97.00	5.59	9111	73.61	6.42	11,449	139.06	5.71	12,618	76.00	5.84	13,044	82.23	6.10
BP51	11,771	39.16	28.00	11,966	21.03	31.26	10,418	44.68	30.52	13,079	39.19	30.87	13,604	17.84	23.71
BP52	11,506	0.23	22.18	11,516	0.98	23.32	10,305	2.74	22.94	13,249	56.39	22.74	13,531	0.03	22.10
BP53	12,134	239.48	15.12	11,932	185.84	14.00	10,860	236.35	14.61	13,095	233.81	13.84	13,483	226.90	13.81
BP54	11,653	165.26	14.12	11,880	168.32	13.00	11,000	209.32	13.26	13,482	158.58	13.45	13,677	159.45	15.26
BP55	12,182	57.52	14.29	11,780	92.74	14.10	10,664	96.06	12.97	13,224	29.71	11.84	13,883	22.58	11.32
BP56	11,777	64.48	12.10	11,683	27.32	13.81	11,313	43.58	9.13	13,085	132.39	10.55	13,605	72.84	11.48
BP57	12,008	13.23	11.23	11,634	30.55	11.61	11,126	0.10	9.32	13,129	57.81	10.16	13,547	0.00	10.26
BP58	12,096	161.23	8.14	11,741	114.45	9.90	11,314	48.03	8.29	13,197	75.71	8.35	13,777	163.19	8.90
BP59	11,767	11.45	13.07	12,015	41.19	12.65	11,006	5.90	12.16	13,566	134.52	11.87	13,846	22.16	9.29
BP60	11,662	9.42	10.26	11,772	65.58	9.58	10,997	28.42	9.06	13,290	3.87	9.19	13,670	1.58	9.58

,

Table 8. Results solving the Boctor’s instances (3/3).

	Water Cycle Algorithm
Instance	Autonomous Approach			Default Approach
ID	${\mathit{N}}_{\mathbf{pop}}\in \{10,100\}$			${\mathit{N}}_{\mathbf{pop}}=30$			${\mathit{N}}_{\mathbf{pop}}=50$			${\mathit{N}}_{\mathbf{pop}}=80$			${\mathit{N}}_{\mathbf{pop}}=100$
	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$
BP61	11,506	17.77	7.19	11,474	0.27	7.81	10,972	80.61	6.13	13,279	97.03	5.81	13,815	17.13	6.29
BP62	11,726	44.61	5.31	11,784	85.97	7.06	10,870	78.74	5.00	13,466	31.55	5.81	13,957	60.52	6.00
BP63	11,636	152.97	27.34	11,661	148.00	27.90	11,335	9.61	27.58	13,338	113.97	27.58	13,835	85.55	21.52
BP64	11,752	96.84	14.23	11,425	135.97	20.68	11,282	141.26	20.45	13,343	53.71	20.58	13,856	115.90	21.29
BP65	11,670	70.65	14.84	11,657	40.74	18.03	11,050	67.68	16.35	13,074	98.87	15.81	13,787	81.74	17.45
BP66	11,539	206.23	14.34	11,679	180.13	16.65	11,192	197.90	14.65	13,373	265.32	14.23	13,781	150.58	14.58
BP67	11,872	162.65	11.21	12,170	132.87	14.10	10,912	71.58	13.32	13,983	155.06	12.87	13,475	141.35	12.74
BP68	11,871	42.45	12.00	11,712	9.32	14.81	11,054	60.97	12.03	13,314	106.90	10.35	13,763	59.97	12.03
BP69	11,827	35.13	11.45	11,852	11.77	11.16	10,909	53.42	9.71	13,184	17.55	10.06	13,925	57.26	11.06
BP70	11,901	30.65	10.77	12,290	18.65	11.84	11,455	76.35	9.52	13,020	62.65	9.55	13,821	48.97	9.26
BP71	11,989	72.90	9.12	11,907	80.71	9.61	10,874	51.94	8.87	13,427	77.52	9.35	13,609	90.61	7.19
BP72	11,639	32.29	9.01	11,968	8.35	8.94	10,678	19.52	8.45	13,562	41.71	8.74	13,774	50.06	7.52
BP73	11,596	39.52	7.13	12,019	7.65	9.35	10,812	0.52	7.45	13,443	32.68	8.00	13,793	7.48	7.06
BP74	11,509	34.65	7.87	11,793	56.74	6.97	10,951	6.71	6.52	13,496	23.42	6.23	13,652	45.97	6.94
BP75	11,827	0.46	10.10	11,648	55.00	17.81	11,041	65.06	15.13	13,420	84.13	14.26	13,600	28.87	10.48
BP76	11,796	8.68	8.34	11,476	7.61	12.42	11,097	5.94	9.81	13,104	58.45	9.42	13,537	27.45	11.87
BP77	11,860	15.71	10.35	11,527	1.39	10.68	10,473	0.29	9.06	13,456	22.48	9.06	13,591	42.71	9.74
BP78	11,747	128.97	7.47	11,564	22.32	9.52	11,200	25.29	7.61	13,210	20.19	5.65	13,690	78.48	8.35
BP79	11,810	183.87	13.65	11,531	91.84	20.29	10,865	3.68	18.35	13,131	129.42	17.90	13,919	121.03	16.32
BP80	11,778	45.77	16.37	11,592	2.94	18.23	11,170	39.84	16.29	13,220	42.32	17.06	13,919	48.97	16.90
BP81	11,493	101.68	14.04	11,600	64.87	14.19	11,368	55.00	14.65	13,353	85.00	14.71	13,568	73.81	15.61
BP82	11,569	92.13	14.65	11,539	68.97	13.45	11,236	56.52	12.94	13,231	57.87	13.68	13,859	42.35	14.58
BP83	11,564	104.06	11.26	11,432	38.19	19.61	10,682	60.87	16.19	13,383	74.19	16.77	13,746	110.13	15.03
BP84	11,235	44.87	13.17	11,542	6.97	18.13	10,737	44.58	16.03	13,639	54.48	15.26	13,623	16.26	16.06
BP85	11,469	29.19	11.18	11,673	60.29	14.45	10,784	21.65	10.52	13,262	36.65	11.65	13,756	128.87	12.87
BP86	11,654	110.94	12.32	11,560	67.19	13.45	11,154	97.52	11.23	13,619	128.48	9.35	13,749	69.77	11.10
BP87	11,354	9.35	14.35	11,845	68.26	16.84	10,779	5.19	15.45	13,316	63.61	15.03	13,899	23.45	12.35
BP88	11,278	16.06	15.26	11,487	31.13	16.13	10,887	16.35	12.77	13,153	26.48	12.81	13,227	27.23	13.19
BP89	11,455	35.87	11.87	11,888	13.71	12.81	11,057	83.87	10.68	13,220	58.65	11.13	13,803	89.71	10.45
BP90	11,394	54.77	8.24	11,714	45.39	10.45	10,895	89.32	10.32	13,028	149.81	8.26	13,914	75.42	9.71

,

Table 9. Results solving complex instances (1/3).

	Water Cycle Algorithm
Instance	Autonomous Approach			Default Approach
ID	${\mathit{N}}_{\mathbf{pop}}\in \{10,100\}$			${\mathit{N}}_{\mathbf{pop}}=30$			${\mathit{N}}_{\mathbf{pop}}=50$			${\mathit{N}}_{\mathbf{pop}}=80$			${\mathit{N}}_{\mathbf{pop}}=100$
	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$
CF01	4463	20.09	0.00	4890	30.20	0.00	5583	41.42	0.00	5210	22.02	0.00	5564	41.23	0.00
CF02	4100	10.03	5.00	4982	40.12	5.00	4677	34.56	5.00	5267	34.89	5.00	6045	34.23	5.00
CF03	4562	1.05	5.00	4729	31.55	5.00	5129	102.29	5.00	5241	45.99	5.00	6680	45.45	5.00
CF04	4140	19.08	2.00	4999	129.09	2.00	5577	234.55	2.30	5578	182.33	2.45	6137	192.44	2.00
CF05	4321	7.08	2.00	4762	29.42	3.55	4579	56.89	3.49	5023	197.38	3.00	5805	43.23	3.56
CF06	4749	20.50	3.12	4534	12.23	2.32	5124	12.45	2.45	5310	38.03	2.50	6136	87.93	2.00
CF07	4200	58.98	6.07	4700	79.20	6.33	4894	0.02	6.09	5272	56.10	6.33	6179	31.56	6.34
CF08	4890	63.88	9.65	4124	145.67	8.23	5200	56.12	7.02	5161	52.48	8.23	4271	21.68	8.34
CF09	4387	1.80	29.68	4019	88.76	30.34	4679	87.92	29.20	5124	4.19	33.12	6012	56.23	29.23
CF10	4662	107.07	1.87	4589	56.78	2.44	5579	23.12	1.28	5812	46.79	3.66	5967	87.34	1.98
CF11	4136	72.00	6.71	4692	46.89	4.00	5489	98.12	4.12	5681	17.23	4.77	6126	69.34	4.12
CF12	4123	34.70	2.01	4532	109.23	1.92	5313	76.12	4.21	5125	168.78	2.45	6175	35.30	4.12
CF13	4980	186.07	2.67	4523	37.34	3.44	5092	51.22	3.10	5239	78.99	2.00	6376	45.23	3.12
CF14	4215	123.90	16.16	4423	35.46	18.23	5333	32.56	16.22	4569	35.67	17.34	6125	76.35	17.33
CF15	4578	31.85	12.07	4278	78.54	16.23	4578	87.99	12.45	4172	92.34	12.56	5569	53.34	12.46
CF16	4567	123.98	15.12	4982	77.77	17.44	4572	182.23	15.29	5374	45.67	19.34	6162	90.63	15.89
CF17	4991	12.45	14.67	4729	67.97	13.45	4793	124.56	17.23	5517	35.46	13.45	4579	87.34	14.22
CF18	4235	128.09	27.99	4934	56.78	29.03	4346	98.23	31.23	5771	68.89	29.34	5764	54.23	28.45
CF19	4215	201.12	25.91	4579	76.23	28.33	5141	41.23	26.34	5279	23.78	26.34	6579	51.24	26.34
CF20	4721	69.08	2.02	4689	34.87	2.34	5025	22.45	3.12	6011	94.23	2.56	6189	74.67	2.34
CF21	4676	41.00	22.42	4689	43.12	25.00	4578	77.24	22.24	5102	144.22	21.34	5834	13.68	22.24
CF22	4666	67.67	1.77	5325	43.65	2.32	4557	98.23	0.23	5129	46.38	4.12	6122	56.34	1.02
CF23	4733	38.92	6.84	4835	53.75	6.99	5034	129.18	9.22	5808	67.45	7.23	6123	124.68	7.34
CF24	4935	125.88	10.94	4912	45.78	11.29	5178	34.22	10.22	5198	66.94	9.34	6632	83.49	9.84
CF25	4235	56.72	19.86	4523	12.97	23.44	4689	56.89	21.32	6986	87.23	19.34	4829	34.67	22.23
CF26	4568	80.91	22.21	4867	67.68	23.44	4333	81.28	24.55	6567	64.23	23.68	5976	22.67	23.24
CF27	4377	132.01	22.42	4667	78.23	22.09	5129	86.23	26.23	6118	51.01	22.45	6136	163.49	23.56
CF28	4213	100.82	32.05	4899	45.23	32.99	5199	23.24	33.22	5292	30.45	34.56	6378	42.56	33.34
CF29	4912	31.50	37.35	4567	97.23	38.12	5275	45.23	37.23	5123	50.20	39.34	4562	78.93	37.34
CF30	4762	66.78	5.94	4987	156.34	6.01	5331	121.47	5.12	6399	34.85	5.68	6565	30.01	6.23

,

Table 10. Results solving complex instances (2/3).

	Water Cycle Algorithm
Instance	Autonomous Approach			Default Approach
ID	${\mathit{N}}_{\mathbf{pop}}\in \{10,100\}$			${\mathit{N}}_{\mathbf{pop}}=30$			${\mathit{N}}_{\mathbf{pop}}=50$			${\mathit{N}}_{\mathbf{pop}}=80$			${\mathit{N}}_{\mathbf{pop}}=100$
	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$
CF31	4589	81.44	7.48	5744	232.67	7.09	4123	87.27	8.12	5509	156.56	8.99	6195	63.56	7.97
CF32	4412	146.34	25.32	4935	57.68	26.97	5124	56.92	25.22	5982	49.03	25.23	6512	78.46	24.21
CF33	4567	98.92	40.77	5567	23.32	40.01	5794	72.08	40.23	5612	78.02	41.57	6391	56.23	42.23
CF34	4567	86.12	218.04	4924	12.34	232.12	4568	82.96	231.59	5857	57.89	217.67	5893	98.32	221.15
CF35	4912	40.71	20.55	4678	122.23	14.23	4512	12.39	13.55	5662	83.46	12.56	5704	112.34	14.45
CF36	4809	50.34	2.00	4579	61.92	2.00	5123	53.23	2.00	5823	45.23	2.00	6723	67.23	2.00
CF37	4579	132.45	8.00	4986	48.13	8.34	4982	34.23	8.00	5902	53.34	8.00	5723	87.12	8.00
CF38	4790	98.34	11.23	5230	51.09	11.23	5234	192.43	11.00	6623	98.12	11.00	6234	32.47	11.00
CF39	4902	55.10	7.00	4123	69.23	7.00	5912	30.34	7.00	5091	48.23	7.00	6871	54.12	7.00
CF40	4891	41.34	5.00	4926	66.34	5.00	4827	56.43	5.00	6123	77.34	5.00	6123	76.12	5.00
CF41	4891	64.12	2.00	5209	82.12	2.00	5782	58.23	2.23	5092	79.57	2.00	5983	12.68	2.00
CF42	4981	77.23	7.00	5500	56.19	7.00	5123	73.23	7.00	5397	52.12	7.00	6612	65.69	7.00
CF43	4520	56.56	14.00	4987	57.23	15.23	4982	80.23	14.00	6660	31.56	16.20	5981	77.23	14.00
CF44	4532	52.45	40.23	4562	55.23	42.23	4672	51.23	41.70	5234	12.57	45.34	6981	76.35	41.23
CF45	4873	64.98	2.15	4982	124.59	2.23	5088	22.45	1.23	5098	18.47	4.35	6912	91.34	2.34
CF46	4762	91.23	0.23	4309	77.34	0.23	5091	65.78	1.10	5230	57.21	1.23	6125	88.31	2.34
CF47	4623	102.34	3.69	4986	76.23	3.13	5982	74.23	2.20	5809	87.34	4.12	6898	10.34	3.34
CF48	4981	73.34	3.65	5210	54.33	2.34	4562	128.23	3.12	5729	73.23	2.34	5354	34.57	3.12
CF49	4871	57.09	25.69	4975	58.12	22.35	4123	12.34	23.40	6678	10.23	24.34	4981	64.46	22.34
CF50	4812	53.89	17.28	4987	61.23	20.23	5722	29.23	18.23	5982	45.24	17.35	6198	264.12	18.23
CF51	4718	93.56	22.23	4198	57.12	24.56	4982	77.34	23.40	5091	91.23	23.46	6987	136.34	22.35
CF52	4412	53.23	19.23	5012	105.23	18.23	5987	52.34	20.23	5890	31.45	20.43	5871	38.23	20.34
CF53	4092	67.75	41.89	4876	72.23	46.12	5512	77.35	41.23	5789	67.23	45.35	5123	65.25	42.34
CF54	4902	73.98	34.47	4764	52.31	35.67	5423	93.34	36.98	6123	54.12	36.34	6712	59.21	36.45
CF55	4782	53.46	18.98	5562	76.23	19.32	5982	192.34	17.98	5092	69.23	16.87	6258	54.13	15.56
CF56	4782	94.12	35.79	4357	98.23	36.89	5917	77.23	36.98	5689	77.20	35.12	6289	72.46	36.45
CF57	4757	42.10	4.56	4189	57.23	5.23	4728	88.34	6.98	6234	82.23	4.12	6012	38.57	4.48
CF58	4578	21.85	8.69	5762	133.23	9.23	5982	79.23	9.54	5819	78.35	8.23	5982	48.23	9.86
CF59	4684	12.87	14.58	4527	72.34	14.55	5123	67.34	14.87	5123	31.34	15.12	6230	69.12	14.97
CF60	4898	45.82	33.87	5342	45.12	35.43	5335	45.01	33.52	6123	23.12	32.24	6498	61.68	33.56

and

Table 11. Results solving complex instances (3/3).

	Water Cycle Algorithm
Instance	Autonomous Approach			Default Approach
ID	${\mathit{N}}_{\mathbf{pop}}\in \{10,100\}$			${\mathit{N}}_{\mathbf{pop}}=30$			${\mathit{N}}_{\mathbf{pop}}=50$			${\mathit{N}}_{\mathbf{pop}}=80$			${\mathit{N}}_{\mathbf{pop}}=100$
	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$	$\mathbf{Stime}$	$\mathbf{It}$	$\mathbf{Opt}$
CF61	4787	67.34	35.01	4581	98.23	34.56	5512	53.23	38.34	6987	19.23	36.12	6213	45.12	35.34
CF62	4899	122.56	32.58	4198	34.23	32.45	5982	31.24	37.23	6349	149.23	32.89	6098	59.23	33.56
CF63	4700	83.57	71.69	5929	67.60	72.35	4627	45.12	78.23	5234	98.34	71.23	6928	49.12	71.35
CF64	4698	75.29	53.68	4721	56.78	55.32	4982	56.23	59.23	4012	71.23	54.12	6340	111.35	54.34
CF65	4677	87.31	11.67	5892	77.89	11.01	4635	85.30	17.23	5821	161.24	10.23	6329	51.57	10.23
CF66	4986	52.58	15.56	4912	56.78	15.57	5872	34.67	18.33	5898	104.23	15.23	6213	71.23	14.23
CF67	4872	81.23	43.99	4719	43.32	47.45	4983	25.67	49.12	6318	76.34	57.23	6391	33.47	46.89
CF68	4798	75.23	46.46	4689	76.34	57.34	4982	92.34	51.23	5981	207.23	50.47	6305	55.58	48.23
CF69	4821	65.12	318.08	4986	87.34	331.32	5093	56.45	367.23	6102	87.23	320.23	6982	97.24	318.54
CF70	4986	54.67	43.65	4700	120.30	47.77	5928	134.12	46.23	6124	41.23	41.23	6469	87.23	41.67

show the results obtained by solving the Boctor’s instances and the set of more complex instances through both approaches, respectively. In these tables, it is shown the average execution time ($Stime$), in seconds, and the average number of iterations needed ($It$) to reach the best solution found, whose fitness value is shown in the $Opt$ field. Note that the best fitness value found for each instance of the problem appears in bold font to facilitate the understanding of the results.

Analyzing Table 6 for instances BP01 to BP30, the authors check that the best configuration for the default approach was $N_{pop}=80$, getting the best solution in 13 cases, while the autonomous approach reached the best solution in 12 cases. The rest of configurations for the default approach performs significantly worse than $N_{pop}=80$. That means that the autonomous approach could be working in a similar way than the default one with $N_{pop}=80$, but without being specifically configured. This fact is exemplified in Figure 5, which shows the evolution of the population size over iterations for the autonomous approach. In this figure, it is observed that the algorithm has the capacity of reducing or increasing the population according to the needed of exploitation or exploration, respectively. Additionally, it is also observed that the algorithm tends to consider large populations, matching with the best configuration found for the default approach. Regarding $Stime$, the authors check that this value is significantly lower than when considering the autonomous approach, which could be due to the improvement in the searchability of the algorithm. Note that the linear trend observed in Figure 5 is just something anecdotal and then, it does not represent the usual behavior of the population size. That is, the algorithm adapts the population size according to the needed for exploration or exploitation.

Focusing on the results in Table 7 for instances BP31 to BP60, the authors check that the best configuration for the default approach was $N_{pop}=100$, getting the best solution in 12 cases, while the autonomous approach reached the best solution in 10 cases. As before, the rest of configurations for the default approach performs worse than $N_{pop}=100$ and the autonomous approach tends to provide a similar behavior than the best static configuration, but without being specifically configured. Note that $Stime$ is also better in the autonomous approach.

Regarding the results in Table 8 for instances BP61 to BP90, the authors check that there are three configurations for the default approach providing a good performance ($N_{pop}=50$, $N_{pop}=80$, and $N_{pop}=100$), getting the best solutions in 7 cases. On the other hand, the autonomous approach provides the best solutions in 10 $cases$. Thus, the autonomous approach tends to perform in a similar way than the best corresponding default approach, but without being specifically configured. This situation implies that for a real case where the best configuration could not be obtained for a problem before the execution, the usage of an autonomous approach could increase the probabilities of getting a good solution.

If we focus on the results solving more complex instances in Table 9, the authors check that the best configuration for the default approach was $N_{pop}=30$, getting the best solution in 6 cases, followed by $N_{pop}=50$ with 5 cases, while the autonomous approach obtained the best solution in 12 cases. As before, there is no general configuration for the default approach which performs well in most of the instances, and then the autonomous approach seems to be a promising way to address the problem.

Focusing on the results in Table 10, the authors check that there is a configuration for the default approach that performs well in most of the instances. This configuration is $N_{pop}=80$, getting the best solution in 11 cases, while the autonomous approach provided the best solution in 8 cases, which is better than the rest of configurations for the default approach. That means that the autonomous approach tended to work with a similar configuration as $N_{pop}=80$, but without being specifically programmed. Finally, in the last set of instances in Table 11, the authors check that there is no general configuration for the default approach performing well in most of the instances. As expected from the previous analysis, the autonomous approach performs better in this case.

From this analysis, the authors conclude that the performance of the autonomous water cycle algorithm is similar than the default approach when the latter was successfully configured. This fact is especially valuable because it implies that the autonomous approach could replace the original one in cases when it is not possible to identify an adequate configuration before running the algorithm. Additionally, the autonomous approach has shown a lesser execution time finding a good solution because of the improvement in the search capacity due to the usage of a dynamic population.

7. Conclusions

In this research, the authors have demonstrated a study of an alternative approach to the application of the water cycle metaheuristic algorithm. Specifically, the authors considered concepts from the autonomous search paradigm, which is a particular case of adaptive systems, for updating the population parameter of the bio-inspired algorithm. Thus, the proposed metaheuristic can control this relevant parameter during the search process.

To know the performance of the proposal, the default water cycle algorithm and the autonomous water cycle algorithm were considered for solving the manufacturing cell design problem, which is a relevant optimization problem in the industry. Thus, two sets of instances were considered, i.e., 90 classic instances proposed by Boctor and 70 more complex instances proposed by several authors. As a result of solving these instances, the authors concluded that the performance of the autonomous approach tends to be similar to the default approach when the latter was successfully configured. This conclusion is relevant because the autonomous approach could perform in a similar way than a well-configured default approach, but without being specifically configured. That means that the autonomous approach could be considered for solving optimization problems, where configurations are unknown.

As future works, it could be interesting to improve the autonomy of the algorithm by including the capacity of dynamically updating the rest of parameters in the metaheuristic. It could also be interesting to study how to provide autonomy to other metaheuristics in the literature. Additionally, the authors plan to explore the Learnheuristic approach [67], which considers concepts from the machine learning field, to improve the search capacity of the metaheuristic.