A Binary Black Widow Optimization Algorithm for Addressing the Cell Formation Problem Involving Alternative Routes and Machine Reliability

Abstract

The Cell Formation Problem (CFP) involves the clustering of machines to enhance productivity and capitalize on various benefits. This study addresses a variant of the problem where alternative routes and machine reliability are included, which we call a Generalized Cell Formation Problem with Machine Reliability (GCFP-MR). This problem is known to be NP-Hard, and finding efficient solutions is of utmost importance. Metaheuristics have been recognized as effective optimization techniques due to their adaptability and ability to generate high-quality solutions in a short time. Since BWO was originally designed for continuous optimization problems, its adaptation involves binarization. Accordingly, our proposal focuses on adapting the Black Widow Optimization (BWO) metaheuristic to tackle GCFP-MR, leading to a new approach named Binary Black Widow Optimization (B-BWO). We compare our proposal in two ways. Firstly, it is benchmarked against a previous Clonal Selection Algorithm approach. Secondly, we evaluate B-BWO with various parameter configurations. The experimental results indicate that the best configuration of parameters includes a population size $\left(Pop\right)$ set to 100, and the number of iterations $\left(Ma{x}_{i}ter\right)$ defined as 75. Procreating Rate $\left(PR\right)$ is set at 0.8, Cannibalism Rate $\left(CR\right)$ is set at 0.4, and the Mutation Rate $\left(PM\right)$ is also set at 0.4. Significantly, the proposed B-BWO outperforms the state-of-the-art literature’s best result, achieving a noteworthy improvement of 1.40%. This finding reveals the efficacy of B-BWO in solving GCFP-MR and its potential to produce superior solutions compared to alternative methods.

1. Introduction

Competitiveness in manufacturing is a complex and multifaceted issue. Several aspects must be considered, such as cost competitiveness, as well as adaptation to new technologies with newer and more sophisticated tools and strategies to improve efficiency. Nowadays, the demand for products with elevated quality standards, shorter lead times, and increased production capacities are on the rise. By effectively managing the layout and arrangement of machinery, companies can maximize their operational efficiency, flexibility, and productivity. Additionally, an optimal arrangement of machinery within a manufacturing facility plays a crucial role in promoting the long-term sustainability of a company striving to excel in various dimensions: social, environmental, and financial. (Triple Bottom Line (TBL), [1]).

The formation of manufacturing cells is implemented as a solution to correctly distribute machines in a plant and group them in that plant. A manufacturing cell is a group or cluster of machines, where the machines that are in each cell are selected based on production criteria. How to order the machines is a complex task, due to the number of combinations that can be generated. In the “Cell Formation Problem” (CFP), as mentioned in [2], the goal is to split a factory into a predetermined number of machine cells. A very relevant criterion is the “part families”. According to this criterion, machines within a cell should be engaged in manufacturing parts with similar requirements.

Among the advantages of manufacturing cell formation mentioned by Greene and Sadowski [3] are:

• Minimized material handling;
• Decreased tooling requirements;
• Reduced setup time;
• Reduced expediting efforts;
• Lowered work-in-process;
• Reduced part makespan;
• Enhanced human relations;
• Improved operator expertise.

The “Cell Formation Problem” (CFP) is an approach that seeks to group machines into a cluster to increase productivity and, at the same time, benefit from the aforementioned advantages. This allows achievement of an optimal distribution in the plant according to several criteria. According to [4], the similarity–difference criterion is the most commonly used criterion to address the CFP. This criterion relies on the concept of grouping machines with similar operations together within the same cell, while machines that have different operations should be grouped in different cells.

The CFP has often been considered unrealistic since important aspects that should be regarded are not used, like maintenance, setup, reconfiguration, alternative routes, breakdown, and others. Specifically, to increase the realism and applicability of CFP solutions, additional considerations and factors have been incorporated into the grouping problem.

The first consideration is that the parts that are produced may have more than a single production route. In other words, there are alternative ways and production sequences to manufacture a part. This consideration is known as “Alternative Routes”.

Another consideration included in the CFP indicates that a machine can present failures leading to additional costs and/or delays. In other words, the CFP includes “Machine Reliability” issues.

In the available literature, the Cell Formation Problem (CFP) is primarily described as a combinatorial optimization problem with the main objective of optimally organizing machines into groups. This process of machine grouping aims to minimize several factors, such as existing costs, the number of “special elements” located outside cells, and the number of “voids” present within cells, as discussed in [5]. By way of mentioning some aspects, we have the so-called “special elements” that appear when the part needs to be processed on a machine that is in a different cell than the one assigned for that part. On the other hand, the “voids” appear when the part does not need to be processed on a machine within the cell assigned for that part.

The high complexity of resolution for combinatorial-type problems (e.g., [6,7]), led researchers such as [8] to classify the CFP as an NP-Hard problem [9]. Talbi in [10] tells us that there are two methods to solve NP-Hard optimization problems: Exact Methods and Approximate Methods.

Exact Methods refer to algorithms that explore the entire search space, ensuring they can reach the global optimum. While these methods guarantee optimization, they may encounter significantly long resolution times when dealing with complex problems. On the other hand, Approximate Methods are algorithms that do not guarantee finding the global optimum, but they do produce high-quality solutions in a reasonable amount of time. This last characteristic results in the Approximate Methods being used to solve problems like the CFP.

Metaheuristics are a type of Approximate Methods algorithm widely used in the literature [11]. Metaheuristics are general-purpose algorithms that, with few modifications, can solve different optimization problems. As indicated in [12], metaheuristics perform the optimization process in two main phases: exploration and exploitation. Exploration consists of investigating the search space to detect promising regions where there may be good solutions. On the other hand, exploitation consists of intensifying the search in these regions detected in the exploration phase and trying to find better solutions. Considering this, achieving a proper equilibrium between exploration and exploitation is crucial for obtaining good solutions [13,14].

The Black Widow Optimization (BWO) algorithm, mentioned in [15], is one of the newest and most powerful metaheuristics to have been introduced so far. The algorithm has been subjected to comparisons with various optimization methods, such as the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Biogeography Based Optimization (BBO), and Artificial Bee Colony (ABC). Findings demonstrated that the BWO algorithm exhibits fast performance in solving the given objective function and converges efficiently toward the optimal values. However, according to [15], this does not imply that BWO is the most superior optimization algorithm ever created. Nevertheless, it can be viewed as a fitting and well-suited algorithm for various optimization problems.

This study aims to demonstrate the applicability of the BWO metaheuristic in addressing the CFP problem, considering alternative routes and machine reliability. The results are encouraging, and demonstrate that this metaheuristic is an effective and innovative tool for solving such problems.

This investigation has the following organization: Section 2 presents the theoretical background for analysis. Section 3 outlines the Cell Formation Problem with alternative routing and machine reliability. Section 4 defines the Black Widow Optimization algorithm. In Section 5, we introduce our proposal: Binary Black Widow Optimization for solving the CFP with alternative routing and machine reliability. The results, along with their analyses, are presented in Section 6. Finally, Section 7 provides the conclusions and discusses potential future trends for the CFP.

2. Theoretical Background

A cellular manufacturing system (CMS) [16] refers to the division of an entire manufacturing system into autonomous clusters of machines known as cells. Each cluster is dedicated to produce a specific subset of parts, constituting a family of parts. Proper planning and design of a CMS, as highlighted in [17], offers several advantages, including reduced setup time, work-in-process (WIP), lot sizes, tool requirements, required space, inventories, material handling efforts, flow time, and throughput time, as well as enhanced quality management.

One of the techniques that were initially used to address the CFP was group technology (GT) in [18], which is a method of organizing machines, tools, people, and other processing facilities belonging to a factory into groups. Each group has the task of manufacturing all parts in the processing phase in which it operates. The GT, as mentioned in [18], can present multiple advantages, including:

• Shorter production times;
• Better quality;
• Reduced material handling costs;
• Better accountability;
• Less indirect labor;
• Reduced set-up time;
• Increased capacity;
• Simplified automation;
• Easier internal promotion;
• Reduced material obsolescence;
• Job satisfaction.

Among the most common considerations in CFP research is the assumption that the parts to be produced can only be processed by a single production plan. This is not a realistic consideration, since there is no single way to manufacture parts, and it limits the field of exploration of possible combinations. It is for this reason that works such as [19] solved the CFP by considering several process plans (routes) for each part. That led to the creation of the generalized CFP (GCFP). This same research showed that GCFP improves the quality of part family groupings and machine cells. It should be emphasized that this is especially important in large instances resulting from large numbers of parts and machines.

CFP with alternative routes is not the only aspect that can be considered if one wants the CFP to be more accurate and realistic. There are several aspects to consider, but one of the most crucial considerations is the possibility of machines failing and experiencing breakdowns. Machine failures, according to [20], have the potential to influence the performance of the system and even stop the production of parts. Many CMS design investigations start from the assumption that machines cannot have failures, which can lead to a bad design for a production system, since a failure occurrence has the potential to adversely affect the entire production process. It is here where the concept of machine reliability emerges, as it helps to determine a more realistic GCFP solution.

2.1. Current Status of the Metaheuristics in the GCFP

This section will draw on documents that address the GCFP, a problem that is widely acknowledged to be of NP-Hard complexity [9], since it requires much more than traditional classical optimization due to the immense number of possible combinations. Metaheuristics [8] are a recommended and extensively employed approach for solving NP-Hard problems, including the GCFP in our case. There is a great variety of metaheuristics and, to show some of them, we will present

Table 1. Metaheuristics.

Reference	Metaheuristic	Year
[22]	Genetic Algorithm	1991
[23]	Particle Swarm Optimization	1995
[24]	Differential Evolution	1997
[25]	Photosynthetic Learning Algorithm	1998
[26]	Clonal Selection Algorithm	2000
[27]	Harmony Search	2001
[28]	Shuffled Frog Leaping Algorithm	2003
[29]	Beehive algorithm	2004
[30]	Wasp Swarm Optimization	2005
[31]	Small-World Optimization Algorithm	2006
[32]	Invasive Weed Optimization	2006
[33]	Artificial Bee Colony Algorithm	2007
[34]	Roach Infestation Optimization	2008
[35]	Biogeography Based Optimization	2008
[36]	Cuckoo Search	2009
[37]	Firefly Algorithm	2009
[38]	Gravitational Search Algorithm	2009
[39]	Bat Algorithm	2010
[40]	Teaching learning-based optimization	2011
[41]	Water Cycle Algorithm	2012
[42]	Krill Herd	2012
[43]	Social Spider Optimization	2013
[44]	Spider Monkey Optimization	2014
[45]	Grey Wolf Optimizer	2014
[46]	Water Wave Optimization	2015
[47]	Moth flame Optimization	2015
[48]	Whale Optimization Algorithm	2016
[49]	Dragonfly Algorithm	2016
[50]	Grasshopper Optimization Algorithm	2017
[51]	Coyote Optimization Algorithm	2018
[52]	Search and rescue optimization	2019
[53]	Bear smell search algorithm	2020
[54]	Mean-shift Algorithm	2021
[55]	Arithmetic Optimization Algorithm	2021
[56]	Golden Eagle Optimizer	2021
[57]	Coronavirus Optimization Algorithm	2022

adapted from [21].

One of the first metaheuristics applied to GCFP cases is the “Genetic Algorithm” (GA). Inspired by the evolution mechanism, it was developed by [58]. To apply a GA, the initial step involves creating a population of solutions represented by chromosomes. That initial population is usually obtained randomly. Subsequently, the solutions within the genetic algorithm are assessed and breeding probabilities are assigned in a way that chromosomes representing superior solutions for the GCFP (such as lower costs) have a higher likelihood of reproducing, compared to chromosomes that yield inferior solutions.

Over the years, new metaheuristics were applied to the GCFP. Among these, “Particle Swarm Optimization” (PSO) was proposed by [23] and applied to GCFP in [59]. PSO draws inspiration from the social behavior of foraging bird flocks. The algorithm begins with a population of birds (flock) randomly exploring the search space to discover an optimal position (food source). Each bird’s ability to determine its next position depends on three factors: inertial velocity, personal best position, and global best position. PSO retains information about previously visited best positions, allowing it to determine the subsequent positions.

Similar to PSO, the “Cuckoo Search” (CS) algorithm is designed for solving multimodal functions, aiming to replace a relatively inferior solution with a potentially better one. CS draws inspiration from bird behavior, and was applied to solve the GCFP in [60]. CS is inspired by the habits of a cuckoo bird. These birds lay their eggs in the nests of other birds (hosts). With this, they eliminate the eggs of the host, which increases the chances of their own eggs hatching successfully. If the host birds detect foreign eggs, they either discard them or abandon their nests and build new ones in a different place. One of the key advantages of the CS algorithm is the simplicity and the reduced number of parameters that need to be tuned compared to other metaheuristic algorithms.

There are two investigations mentioned in [60] that make use of another metaheuristic, not mentioned in Table 1, that was used to solve the GCFP. The first investigation is [61], and the metaheuristic is “Simulated Annealing” (SA). According to [61], SA is a versatile metaheuristic employed to overcome local minimums by permitting jumps to higher energy states. The analogy drawn here is to the annealing process utilized by a craftsman to craft a sword from an alloy. In the context of SA, the optimization of a solution of a problem is continually refined until the best solution is discovered within the search space. As an observation, according to [62], the main disadvantage of such a metaheuristic is that it can become trapped in a (locally optimal) cycle. To avoid being trapped in such a cycle, the SA enables the acceptance of certain similar solutions that may be worse than the incumbent solution. In order to decide whether to accept a worse solution than the incumbent one in order to break the cycle, a probability function of acceptance is performed on the postulating solution (worse solution), and the result of this function is evaluated based on a criterion that determines whether or not the replacement of the postulating solution by the solution that is generating the cycle will be accepted.

The second investigation is “Tabu Search” (TS) [63], which has demonstrated its success in solving a wide range of combinatorial problems. The main idea of TS is to avoid areas that were recently visited in the solution space and focus on reaching unexplored and promising areas. It permits non-improving moves to escape local optima, and it marks attributes of recently executed moves as tabu, or forbidden for a specified iteration number, to prevent revisiting the same solutions in cycles.

Another metaheuristic used to solve the GCFP is the “Clonal selection Algorithm” (CSA). The CSA, as mentioned in [64], is an algorithm that draws inspiration from the immune system. In this context, the CFP is represented as the antigen, and a potential solution to the problem is represented by the antibody. The CSA starts by generating feasible solutions (antibodies) that do not violate any constraint of the instance to be evaluated, and then calculates the fitness (e.g., cost) of each solution. Subsequently, the solutions of the clones (exact copies) with the best fitness found so far are generated. New solutions are created by mutating these clones and making slight adjustments. Next, a crossover is executed between the new solutions and, if they turn out to be infeasible, a repair process is applied. It is worth noting that repair can occur either after mutation or crossover. To enhance solution quality and avoid becoming stuck in local optima, a local search method is employed on two randomly chosen solutions. To conclude, the initial solutions are replaced by the new ones.

Finally, we will take into account the “Firefly optimization” (FF). This metaheuristic mimics the pairing behavior of fireflies, and was applied to the GCFP by [65]. As mentioned by [66], three main considerations were taken into account for this metaheuristic. The first consideration was to consider all fireflies without any specific gender so that any firefly can be paired with any other firefly regardless of its sex. The second consideration indicates that the attraction is proportional to the generated brightness (fitness), so that during any signal exchange between two fireflies, the less bright one will approach the brighter one. It is also mentioned that the attraction is proportional to the intensity of the brightness, resulting in the fact that when there is a greater the distance between fireflies, the brightness will be lower. In a case where the brightness is equal in both fireflies, one will be selected at random. The third and final consideration specifies that the objective function determines the brightness of a firefly.

2.2. Reliability in the GCFP

Reliability is the probability that a device will execute its designated function within a specified period and under the specified conditions, as stated in [67]. Taking advantage of the information provided by reliability, it is possible to apply functions where a low reliability is penalized, thus generating a pattern in the organization of machines according to their reliability to estimate the real economic convenience of a change in the way of managing these machines which, for the present research, derives from the creation of manufacturing cells. In the field of study of GCFP, there are different ways of incorporating reliability, but within the literature, we can identify the two most common distinct approaches: “System Failure Rate” (SFR) and “Breakdown Cost” (BC).

Research such as [8,68] make use of the SFR applied to the GCFP using a Markovian approach, and assume that the failures that machines may present have an exponential distribution. It should be noted that SFR is also known as the “Logarithmic Index of Reliability” (LIR). The SFR, or LIR, calculates the failure rate of a route that consists of multiple machines by adding up the individual failure rates of each machine. Considering the obtained failure rates, it will be possible to minimize the probability of failures by selecting the most appropriate processing route and assigning machines to cells.

The investigations that make use of BC [60,69,70] multiply the number of machine failures, which is quantified based on the penalty or breakdown cost associated with each failure per machine. To obtain the number of failures, the total production time is determined for each machine in a selected route and divided by its “Mean Time Between Failure” (MTBF). To determine the total Breakdown Cost, the Breakdown Costs for each selected route are added together. This will help in the selection of routes in a better way and assign machines to cells more efficiently.

Some variants incorporate other reliability concepts or adapt them. For instance, the penalty for non-production is an example of the aforementioned. Investigations such as [71,72] use BC, but also consider the time it will take to repair the machine after it fails. To calculate the time that a machine will be penalized for being repaired, “Mean Time To Repair” (MTTR) is used. This value is multiplied by each failure and then penalized.

2.3. Metaheuristics and Reliability in the GCFP

Currently, there are very few representatives that solve the GCFP with machine reliability, considering the use of metaheuristics. Among the investigations that were previously mentioned in the two previous subsections, only [60,69] made use of metaheuristics to solve the GCFP with a machine reliability consideration (“Tabu Search” and “Cuckoo Search”, respectively). It should be noted that both investigations used monetary penalties for occurring failure.

In some cases, reliability is considered, but not quantifiably, like in the research by [73] that made use of GA. In this specific case, a constraint is used to control the entry of existing machines per period. This constraint highlights that having extra machines in the system enhances reliability by offering alternative routes in the event of a machine failure. As can be seen, the GCFP with machine reliability was solved, but to date has not been solved by the “Black Widow Optimizer” (BWO) metaheuristic.

3. Cell Formation Problem with Alternative Routes and Machine Reliability Considerations (GCFP-MR)

The GCFP is now aiming to be more realistic, considering aspects relevant to grouping machines. One aspect to consider while creating cells is the existence of the fact there are processes that require machines located in different cells. This cell jumping is known as “intercellular movement”, and carries a cost of implementation. Another important aspect to consider is that we should take into account the idea that machines can break down at some point in time, stopping the processes carried out on the routes. To calculate the impact that the failure of a damaged machine would generate, a calculation of the probability of a failure occurring is performed, measured on a cost basis. These two aspects are considered in the model of Karoum and Elbenani [64], since it calculates the intercellular costs based on the quantity and price of the parts that carry out said movement, in addition to calculating the breakdown cost based on the calculation of the cost of the number of machine failures (Breakdown Cost). The model is presented in the form of a 0–1 integer optimization model.

3.1. Model Assumptions

GCFP works with the following premises:

• The total number of cells is defined;
• The lower limit of machines per cell is known;
• The upper limit of machines per cell is known;
• Each part has at least one process route, but only a single route can be selected;
• Each route has different operations with ordered sequences and these operations are performed by machines. The sequence must be taken into account, as it will help determine when a part passes from one cell to another, in addition to collaborating with the calculation of material handling costs;
• Each type of part has its time of operation (or processing) in each machine;
• Multiple identical machines were not considered;
• The production demand of each part is known and said demand is also deterministic;
• The Mean Time Between Failures (MTBF) was used for the machine reliability calculation.

3.2. Mathematical Model

(a) 

Parameters

The parameters considered are:

m:

Quantity of machines;

n:

Quantity of parts;

c:

Quantity of cells;

r:

Total quantity of routes;

P_i:

Part i production volume;

q_i:

Quantity of routes per part i;

L_i:

Lower limit of the quantity of machines in cell l;

U_i:

Upper limit of the quantity of machines in cell l;

K_ij:

Quantity of machines in route j of part i;

$u_{ij}^1$, $u_{ij}^2$,…, $u_{ij}^{k_{ij}}$: Machines index of route j of part i;

T_ik:

Duration required for processing a part i on machine k;

B_k:

Cost associated with machine k breakdown;

A_ij:

Cost associated with intercellular movement for part i in route j;

MTBF_k:

Mean time between failure of machine k.

(b) 

Decision variables of the model

The decision variables are:

$$\begin{array}{c}{Z}_{ij}=\left\{\begin{array}{c}1,ifroutejofpartiisselected\\ 0,otherwise\end{array}\right.\hfill \\ Y_{kl}=\left\{\begin{array}{c}1,ifmachinekislocatedincelll\\ 0,otherwise\end{array}\right.\hfill \\ X_{ijklsl}=\left\{\begin{array}{c}1,ifroutejofpartiisselected,machinekislocatedin\\ celllandmachinesisnotlocatedincelll\\ 0,otherwise\end{array}\right.\hfill \end{array}$$

(c) 

Objective function

The mathematical model used by Karoum and Elbenani [64] is presented below:

$$MinTC=\sum _{i=1}^n\sum _{j=1}^{q_i}\sum _{k=1}^{K_{ij}-1}\sum _{l=1}^c{A}_{ij}{P}_i{X}_{ij\left(u_{ij}^{k_{ij}}\right)l\left(u_{ij}^{k_{ij}+1}\right)l}+\sum _{i=1}^n\sum _{j=1}^{q_i}\sum _{k=1}^{K_{ij}}Zij{\displaystyle \frac{P_i{T}_{i\left(u_{ij}^{k_{ij}}\right)}{B}_{\left(u_{ij}^{k_{ij}}\right)}}{MTB{F}_{\left(u_{ij}^{k_{ij}}\right)}}}$$

(1)

Subject to:

$$\sum _{j=1}^{q_i}{Z}_{ij}=1\forall i=1,2,\dots ,n$$

(2)

$$\sum _{l=1}^c{Y}_{kl}=1\forall k=1,2,\dots ,m$$

(3)

$$\sum _{k=1}^m{Y}_{kl}\le U_l\forall l=1,2,\dots ,c$$

(4)

$$\sum _{k=1}^m{Y}_{kl}⩾L_l\forall l=1,2,\dots ,c$$

(5)

$$X_{ijklsl}\le Z_{ij}\forall i=1,2,\dots ,n;\forall j=1,2,\dots ,q_i;\forall k,s=1,2,\dots ,m;\forall l=1,2,\dots ,c$$

(6)

$$X_{ijklsl}\le Y_{kl}\forall i=1,2,\dots ,n;\forall j=1,2,\dots ,q_i;\forall k,s=1,2,\dots ,m;\forall l=1,2,\dots ,c$$

(7)

$$X_{ijklsl}\le (1-Y_{sl})\forall i=1,2,\dots ,n;\forall j=1,2,\dots ,q_i;\forall k,s=1,2,\dots ,m;\forall l=1,2,\dots ,c$$

(8)

$$Z_{ij}+Y_{kl}+(1-Y_{sl})-X_{ijklsl}\le 2\forall i=1,2,\dots ,n;\forall j=1,2,\dots ,q_i;\forall k,s=1,2,\dots ,m;\forall l=1,2,\dots ,c$$

(9)

$$Z_{ij},Y_{kl},X_{ijklsl}\in \{0,1\}$$

(10)

The objective function (Equation (1)) has two sections. In the first section, the total cost of intercellular movements is calculated, while the second section computes the machine breakdown cost within the objective function (1).

In Equation (2), the constraint stipulates that only a single route per part can be chosen. In constraint (3), it mandates that each machine can be assigned to one and only one cell. The constraints presented in Equations (4) and (5) establish limits for the maximum and minimum number of machines allowed per cell. Equation (6) indicates that an intercellular move occurs from a selected path of a part. A similar case occurs with Equation (7), as it indicates that an intercellular movement can occur if and only if the cell and machine, where the intercellular movement originates, have already been selected. Furthermore, the target machine of the intercellular movement must not be in the same cell as the originating machine, according to Equation (8). Finally, Equation (9) indicates that if there is an intercellular movement, it must be from a route starting at a machine in a cell already selected and destined for a machine that is in a different cell. The variables are binary, according to Equation (10).

4. Black Widow Optimization Algorithm Methodology

The Black Widow Optimization algorithm, introduced in 2020 by Hayyolalam and Pourhajo Kazem [15], is a population-based metaheuristic. It takes inspiration from the mating behavior of black widow spiders in their natural habitat.

4.1. Inspiration: Black Widows in Mating Season

In the mating season of black widows, the female makes her web and marks certain areas in order to entice a male. After a male enters the web, it causes other males to lose interest in entering said web. Something unique about mating black widows is that the female might consume the male either during or after mating. After mating, the female proceeds to fill her egg sac with eggs. When the eggs hatch, cannibalism occurs between siblings, where the mother is sometimes also involved. This act of cannibalism causes only the strongest to survive [15].

4.2. Mathematical Modeling

This metaheuristic consists of five stages: (1) initialization, (2) procreation, (3) cannibalism, (4) mutation, and (5) termination criteria.

4.2.1. Initialization

Like all population-based metaheuristics, the initial population is generated as the first step. Black widows represent the solutions, where $Widow=[w_1,w_2,\dots ,w_{N_{var}}]$ and $N_{var}$ represent the counts of decision variables in the optimization problem. Each $Widow$ belongs to population W, and each decision variable $(w_1,w_2,\dots ,w_{N_{var}})$ is assigned a continuous random number that satisfies the constraints of the problem.

Therefore, we will have a matrix of $N_{pop}\times N_{var}$, where $N_{pop}$ corresponds to the number of Widows. The initial matrix is as follows:

$$W=\begin{array}{cc}Widow{s}_1& =\\ Widow{s}_2& =\\ ⋮& \\ Widow{s}_{N_{pop}}& =\end{array}\left[\begin{array}{ccccc}{w}_{1,1}& w_{1,2}& w_{1,3}& \cdots & w_{1,N_{var}}\\ w_{2,1}& w_{2,2}& w_{2,3}& \cdots & w_{2,N_{var}}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ w_{N_{pop},1}& w_{N_{pop},2}& w_{N_{pop},3}& \cdots & w_{N_{pop},N_{var}}\end{array}\right]$$

(11)

4.2.2. Procreate

At this stage, pairs of parents are randomly chosen to participate in the procreation process. In nature, black widows lay about 1000 eggs, of which only the strongest survive. To mimic this behavior, an array called alpha $\left(\alpha \right)$ is created whenever the widow array is composed of random numbers and, subsequently, offspring are generated using $\alpha$ through the following equation:

$$\left\{\begin{array}{c}C{H}_1=\alpha \times Pa{r}_1+(1-\alpha )\times Pa{r}_2\\ C{H}_2=\alpha \times Pa{r}_2+(1-\alpha )\times Pa{r}_1\end{array}\right.$$

(12)

In Equation (12), the $Pa{r}_1$ and $Pa{r}_2$ are the parents and $C{H}_1$ and $C{H}_2$ are the offspring. This operation is carried out $N_{var}/2$ times. It should be emphasized that $Pa{r}_1$ must be different from $Pa{r}_2$. Finally, both offspring and parents go through the cannibalization process, and the surviving individuals are included in a matrix and sorted based on their cost value.

4.2.3. Cannibalism

In nature, there are three types of cannibalism, which are the following:

• Sexual Cannibalism: similar to the female black widow consuming her mate during or after mating, the algorithm distinguishes between the female and male based on their fitness, where the one with the highest fitness is considered the female;
• Sibling Cannibalism: Just as strong black widow’s offspring feed on their weaker siblings, the algorithm determines the number of survivors based on the cannibalism rating (CR). The fitness of each individual is used to distinguish between the strong and weak black widows;
• Maternal Cannibalism: this type of cannibalism is often observed, and consists of the sons eating their mother if they have better fitness.

4.2.4. Mutation Process

The process of mutation begins with the selection of $nm$ individuals from the procreation population. The value of nm is determined by the mutation rate. The Widows of this subpopulation have a positional exchange between two of their decision variables, thus generating a new widow.

4.2.5. Termination Criteria

One of the most important sections of metaheuristic algorithms is the termination criterion. In this metaheuristic, three term criteria are used: (1) reaching a maximum number of iterations, (2) whether improvements are observed compared to the best fitness obtained within a specific number of iterations, and (3) achieving of the designated level of fitness. To assist in understanding, the Algorithm 1 (pseudocode) used in BWO is presented.

Algorithm 1 Black Widow Optimization Algorithm

Input: $Ma{x}_{iter}$, $N_{Pop}$, Procreating Rate (PR), Cannibalism Rate (CR) and Mutation Rate (PM) Output: The updated population $W^{\prime }=$$\{Wido{w}_1^{\prime },Wido{w}_2^{\prime },\dots ,Wido{w}_{N_{pop}}^{\prime }\}$ and $Best$ 1: Initialization of population W 2: repeat 3:     Determine the number of reproduction $\left(nr\right)$ based on PR ($nr$ = $N_{pop}$ × $PR$) 4:     Select the best $nr$ solutions in W and save them in $po{p}_1$ 5:     for $i=1$ to $nr$ do                                                           ▹ Procreating and cannibalism 6:           Choose two Widows randomly as parents $(Pa{r}_1,Pa{r}_2)$ from $Po{p}_1$ 7:           Generate $N_{var}$ children using Equation (12) 8:           Destroy father 9:           Eliminate some of the offspring based on CR. 10:         The surviving solutions are saved into $po{p}_2$. 11:     end for 12:     Determine the number of mutation $\left(nm\right)$ based on PM ($nm$ = $nr$ × $PM$) 13:     for $i=1$ to $nm$ do                                                                                          ▹ Mutation 14:         Select a solution from $po{p}_1$ 15:         Randomly mutate one decision variable of the solution and generate a new solution. 16:         Mutated solutions are saved into $po{p}_3$ 17:     end for 18:     Update $W=po{p}_2+po{p}_3$ 19: until the terminal criterion 20: return the best $Widow$ from W

5. Our Proposal: Binary Black Widow Optimization (B-BWO) for Solving GCFP-MR

In this section, the adaptation made to B-BWO so that it can solve the GCFP-MR will be exposed.

5.1. Solution Representation

The implementation of the BWO [15] in the GCFP-MR starts with the creation of solutions or Widows that are feasible and do not violate any of the restrictions of Section 3.2. Each $Widow$ is composed of two parts: (1) Routes $\left(Z_{ij}\right)$ and (2) $\left(Y_{kl}\right)$, as shown in Figure 1. Let us not forget that our problem is binary, therefore each decision variable is made up of ones or zeros.

As mentioned in Section 4.2.1, each $Widow$ belonging to the population W has a size of $N_{var}$ or Widow size. $N_{var}$ denotes the quantity of decision variables present in the problem. This value can be calculated with Equation (13).

$$N_{var}oWidowsize=r+\left(mc\right)$$

(13)

5.2. Binarization of Solutions

When we perform the procreation process as explained in Section 4.2.2, it results in continuous solutions within the range $[0,1]$. Since the problem to be solved is binary, we need to binarize these solutions. In the existing literature, there are diverse approaches to binarize solutions, as mentioned in [74,75].

Since the perturbed solutions in procreation are within the range $[0,1]$, in the present work we apply a simple binarization which is detailed in Equation (14):

$$C{H}_{new}=\left\{\begin{array}{cc}1& ifc{h}_i\ge 0.5\\ 0& otherwise\end{array}\right.$$

(14)

where $c{h}_i$ is the value obtained after procreation in the i-th dimension for the solution $CH$. When we perform the initial generation of solutions, and when we binarize the solutions after procreation, there is a possibility that we will have infeasible solutions. Subsequently, a feasibility test is conducted and, if the solutions are found to be infeasible, they undergo repair. The Feasibility Check is detailed in Section 5.3 and the Solution Repair Process is detailed in Section 5.4.

5.3. Feasibility Check

To demonstrate that the B-BWO obtained a feasible solution for the GCFP-MR, we will verify that the solution shown in Figure 2 does not violate any of the constraints shown in Section 3.2.

To be more precise and graphic, the widow found will be divided into two sections (routes and cells) that are represented by Figure 3 and Figure 4.

We will start with the first restriction, which is Equation (2), where it specifies that only one route j can be selected per part i. As we can see in Figure 3, the parts have been separated by colors (red) according to the number of routes they have, to facilitate the detection that only one route j has been selected per part i (i.e., there is only one box with a value of 1 in each part).

Moving on to the next restriction, Equation (3) indicates that each machine k can only be in one cell l. Here we will use Figure 4 with the difference that the color, in this case, is different (blue). As we can see, the machines k have been separated by colors, and since there are two existing cells, each machine has two boxes. As in the above case, we can see that each machine k has selected only one cell l, so this restriction was not violated either.

Continuing with the restrictions of upper and lower limits of machines per cell, we have Equations (4) and (5), respectively. We will use Figure 4 again and say that there are three machines in the first cell (l = 1) and six machines in the second cell (l = 2). We conclude that these restrictions are also not violated, considering that the upper limit is six machines per cell and the lower limit is two machines.

In the case of other restrictions, feasibility is in the process and algorithm of the BWO.

• Once the eight routes j have been selected for the eight parts i, the other 12 routes, out of the initial 20, are eliminated, so in case where is an intercellular jump $X_{ijklsl}$, it belongs to its respective route j selected by the variable $Z_{ij}$, thus complying with the fifth restriction (Equation (6));
• For the case of Equations (7) and (8), the following was applied. To each machine k that participated in the production of a selected route j, a vector of five elements or data was assigned (Figure 5), where the fifth element ($E_5$) is the cell in which said machine is located. To verify compliance with the restrictions, it is first verified that both $E_1$ and $E_2$ of the two machines we are evaluating are equal, to ensure that we are on the appropriate route j from the same part i. Subsequently, if $X_{ijklsl}$ of this route j is equal to 1, this indicates that there is an intercellular jump from machine k to s, the two participating machines (k and s) are selected, and it is verified that their elements $E_4$ are consecutive numbers and that their elements $E_5$ are different, to guarantee that the intercellular jump initiated in machine k is in cell l (checking compliance with Equation (7)), and that the intercellular jump completed in machine s is not in the same cell l (checking compliance with Equation (8)).

Finally, Equation (9) is an equation generated by the linearization of the model, so, by complying with the other restrictions, the last restriction is not violated automatically.

5.4. Solution Repair Process

It should be understood that in both the offspring generation process and the mutation process, there is the possibility of generating $Widows$ that become infeasible and violate one or more constraints. For this purpose, a solution repairer was created, which makes changes to the infeasible widow to make it feasible. The present mechanism grabs an unfeasible $Widow$ and alters it so that only one route j is selected for each part i (Equation (2)), only one cell l is selected for each machine k (Equation (3)), and the solution repair ensures that the upper and lower limits for the number of machines in cells are adhered to (Equations (4) and (5)).

To explain the operation of the repairer more clearly, it should be noted that three repair phases were considered:

(a) 

Only one route per part (Phase 1): The repairer starts by selecting the first “r” elements of the Widow, we will call this selection $W_Z$. Within $W_Z$ it will be evaluated that the sum of the elements (routes) of each part i equals 1. In case this equality is not fulfilled, there are two alternatives:

• The first alternative of error in $W_Z$ would occur if there is more than one element with a value equal to 1, which would indicate that more than one route is being selected for the part i. To solve this error, a route j is randomly selected from among the routes that already have a value equal to 1 in that part, this route j will keep its value equal to 1, but all the other routes will be filled with a value equal to 0;
• The second alternative of error in $W_Z$ would occur if no route is selected or, in other words, the sum of the routes of part i is 0. To solve this error, a route j is randomly selected and assigned the value 1, the other routes will keep their value equal to 0.

(b) 

Only one cell per machine (Phase 2): To continue with Phase 2, the remaining elements of the $Widow$ are selected $N_{var}$; we will call this selection $W_Y$. With $W_Y$ already selected, we will evaluate that the sum of the selected cells of each machine k is equal to 1. In case this equality is not fulfilled, there are two alternatives:

• The first alternative of error in $W_Y$ would occur if there is more than one cell with a value equal to 1, which would indicate that more than one cell per machine k is being selected. To solve this error, a cell l is randomly selected from those already selected with a value equal to 1, this cell l will keep its value equal to 1, but all the other cells will be filled with a value equal to 0;
• The second alternative error in $W_Y$ would occur if no cell is selected for machine k, resulting in the sum of cells of machine k being equal to 0. To solve this error, a cell l is randomly selected and assigned the value 1, the other cells will keep their value equal to 0.

(c) 

Respecting cell size limits (Phase 3): Finally, the amount of machines per cell is repaired if the upper or lower limit is not respected. For this phase, we will use the $W_Y$ repaired and obtained after Phase 2.

• Lower Limit: In case the lower limit is not respected by a cell ($l_{LB}$), we proceed to randomly select another cell ($l_{Ale}$) different from $l_{LB}$. Then, a machine k belonging to cell $l_{Ale}$ is randomly selected and moved to the cell that does not respect the lower limit. In other words, a machine k is shifted from cell $l_{Ale}$ to cell $l_{LB}$.
• Upper Limit: If the upper limit is not respected by a cell ($l_{UB}$), a machine k belonging to the same cell $l_{UB}$ is randomly selected. This machine k will be designated with a value equal to 0 in cell $l_{UB}$, and with a value equal to 1 in all the other cells. This is performed with the intention that machine k is changed to any other cell except $l_{UB}$. As we know, we cannot have more than two cells selected per machine k, so if this situation were to arise, this new cell assignment would lead to an infeasible but repairable result if we go through Phase 2 again.

All three phases are re-evaluated until no constraints are violated and the solution becomes feasible. This is easily accomplished with a “while” loop in the coding.

5.5. Complexity Analysis

An important aspect of software development is its computational cost. The Big-O asymptotic notation allows to quantify the algorithm complexity in terms of the worst case [76]. Our proposal was analyzed in three different sections: solution initialization, solution perturbation, and problem aspects such as solution repair and fitness calculation.

5.5.1. Initialization of Solution

In our proposal, we use a random generator of feasible solutions. This generator has a complexity of $O(Z_i·Z_j+Y_k·Y_l+Y_l)$, where $Z_i$ corresponds to the number of parts, $Z_j$ corresponds to the number of routes of each part, $Y_k$ corresponds to the number of machines, and $Y_l$ corresponds to the number of cells in each machine.

5.5.2. Solution Perturbation

For each iteration, the solutions are perturbed in two phases. The first of them is related to the crossover phase, which has a complexity of $O(nr+nr·\frac{N_{var}}{2}+nr·cr)$, where $nr$ corresponds to the number of reproduction, $N_{var}$ corresponds to the number of decision variables, and $cr$ corresponds to the number of cannibalism.

On the other hand, we have the mutation stage. This has a complexity of $O(nm+N_{pop})$, where $nm$ corresponds to the number of mutation and $N_{pop}$ corresponds to the population size.

Thus, the complexity of the solution perturbation process is $O(T·(nr+nr·\frac{N_{var}}{2}+nr·cr+nm+N_{pop}))$, where T corresponds to the number of iterations.

5.5.3. Problem Aspects

As for the problem aspects, we have two processes involved: solution repair and fitness calculation. On the solution repair side, we have a complexity of $O(Z_i·Z_j+Y_k·Y_l+Y_l)$. On the fitness calculation side, we have a complexity of $O(A_j·Y_k·Y_l+Z_j·Y_k·Z_i)$, where $A_j$ corresponds to the cost of movement between the l-cells. In conclusion, the aspects of the problem have a complexity of $O(T·N_{pop}·(Z_i·Z_j+Y_k·Y_l+Y_l)+(A_j·Y_k·Y_l+Z_j·Y_k·Z_i))$.

5.5.4. Algorithmic Complexity of Proposal

Adding the algorithmic complexities of the three processes of the proposal, we obtain the following complexity of the proposal:

$$O(T·((nr+nr·\frac{N_{var}}{2}+nr·cr)+(nm+N_{pop})+(N_{pop}·(Z_i·Z_j+Y_k·Y_l+Y_l)+(A_j·Y_k·Y_l+Z_j·Y_k·Z_i))))$$

(15)

6. Experimental Results

To illustrate the efficacy of our method in addressing the GCFP-MR, we compare ourselves with the results obtained by Karoum and Elbenani [64]. The mentioned case has a total of 20 existing routes between eight parts, in addition to having nine machines that can be divided into two cells. The intercellular movement costs, as well as the production sequences and times, are in

Table 2. Case of [64].

Sequence/ Processing Time (min)	Parts (Volume)
	P1(75)			P2(130)			P3(110)		P4(145)		P5(110)		P6(105)		P7(140)				P8(115)
Machines	R1	R2	R3	R1	R2	R3	R1	R2	R1	R2	R1	R2	R1	R2	R1	R2	R3	R4	R1	R2
M1	1(5)			1(4)			1(4)	1(4)	1(5)	1(5)		1(4)		1(4)	1(5)			1(5)	1(4)
M2		1(5)	1(5)		1(5)	1(5)					1(3)		1(5)			1(3)	1(3)	2(3)		1(4)
M3							2(3)	2(3)	2(3)	2(3)
M4	2(3)			2(4)				3(3)	3(5)	3(5)		2(4)			2(4)		2(4)
M5	3(4)		2(4)	3(3)	2(3)	2(3)	3(4)			4(4)					3(3)		3(3)
M6		2(5)														2(4)		3(4)	2(3)	2(4)
M7											2(5)	3(5)	2(5)	2(5)
M8		3(4)		4(4)		3(4)	4(3)	4(3)	4(3)	5(4)			3(5)	3(5)			4(5)
M9	4(5)		3(5)		3(3)						3(5)	4(5)			4(5)	3(5)		4(5)
Intercellular cost	375	375	0	1300	0	650	1100	0	0	1450	1100	550	525	0	700	0	700	700	575	0

, and the MTBFs with Breakdown Costs (BC) are in

Table 3. MTBF and BC from [64].

Machine	MTBF 1 (h)	BC 2
M1	90	900
M2	51	2000
M3	73	2000
M4	60	1600
M5	76	1500
M6	62	1800
M7	71	1400
M8	58	1700
M9	65	1500

¹ Mean time between failure; ² Breakdown cost.

.

The best result obtained by the researchers in [64] was USD 4737.54, and the solution vector is as follows:

$$W_{CSA}=[0,0,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,0]$$

(16)

Equation (16) shows the solution vector in the format exposed in Section 5.1. To help with the understanding of the solution representation, Figure 6 is used. As can be seen in Figure 6, the red section is the set of existing r routes (20) in the case of [64], while all the options (18) of l cells that can be chosen by the k machines are in the blue section. Applying the calculation of Widow size, we have a total of 38 elements (variables) that are all the necessary information to be able to calculate the cost of the case presented in [64], and that gives a total cost of USD 4737.54.

6.1. Experiment Configuration

As evidenced in Section 4, Black Widow Optimization has five key parameters: (1) Population Size $\left(Pop\right)$, (2) Maximum of Iterations $(Max\_iter)$, (3) Procreating Rate $\left(PR\right)$, (4) Cannibalism Rate $\left(CR\right)$, and (5) Mutation Rate $\left(PM\right)$.

Assigning the correct value to each parameter is key, since a bad configuration can lead to bad results. In the present work, different values were used for each parameter, as can be seen in

Table 4. Parameter combinations.

Pop 1	25	50	75	100
$Max\_iter$ 2	25	50	75	100
$PR$ 3	0.2	0.4	0.6	0.8
$CR$ 4	0.2	0.4	0.6	0.8
$PM$ 5	0.2	0.4	0.6	0.8

¹ Population size; ² Number of iterations; ³ Procreation Rate; ⁴ Cannibalism Rate; ⁵ Mutation Rate.

.

Thus, for this experiment, we used $4\times 4\times 4\times 4\times 4=1024$ combinations, where a combination takes a value from Table 4 to each parameter. For example, a combination is the following:

$$Combinatio{n}_1\left\{\begin{array}{c}Pop=25\\ Max\_iter=25\\ PR=0.2\\ CR=0.2\\ PM=0.2\end{array}\right.$$

(17)

To determine the best combination, each combination was evaluated 30 times, and an average of the 30 final costs obtained for each combination was calculated. Once we have the averages calculated, we will use the Relative Percentage Deviation (RPD) [77] to determine the extent of deviation from the optimum or best solution found. The RPD is calculated as follows:

$$RPD=\frac{{\overline{Cost}}_{comb}-Z_{opt}}{Z_{opt}}·100$$

(18)

where ${\overline{Cost}}_{comb}$ corresponds to the average cost obtained with the combination ($comb$), and $Z_{opt}$ is the best fitness.

Thus, we performed $4\times 4\times 4\times 4\times 4\times 30$ = 30,720 experiments for the present work.

6.2. Analysis of the Results

As previously mentioned, our target to at least reach is the one reported by [64], whose best-reported result is USD 4737.54. Once the metaheuristic was applied, our best-obtained solution is observed in Equation (19), and obtained a cost equal to USD 4671.34. As can be seen, the cost found is lower than that presented in [64] (1.40% lower), which leads to the determination that a better solution has been found.

$$W_{BWO}=[1,0,0,0,1,0,0,1,1,0,0,1,0,1,1,0,0,0,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0,1]$$

(19)

To help with the understanding of Equation (19), we have

Table 5. Route selected per part.

Part (i)	1	2	3	4	5	6	7	8
Route (j)	1	2	2	1	2	2	1	1

and

Table 6. Cell selected per machine.

Machine (k)	1	2	3	4	5	6	7	8	9
Cell (l)	2	1	1	2	2	2	2	1	2

, which show the routes selected for each part and the cell selected for each machine, respectively.

To verify that the value found by the metaheuristic is the optimum, an exhaustive search was conducted, determining that there are 483,840 feasible solutions for this instance, and the optimal result is USD 4671.34, thus confirming that the BWO found the optimum value. In other words, we have found a new best solution to the problem instance.

6.3. Black Widow Optimization Configuration Analysis

Once the 1024 averaged costs ($\overline{Cost}$) are determined, we examine whether the number of iterations (Max_iter) and the population size (Pop) influence the attainment of $RPD$. The heat map in

Table 7. Pop and Max_iter (heat map).

$\mathit{R}\mathit{P}\mathit{D}$	Max_iter				Pop Overall Average
Pop	25	50	75	100
25	23.64%	26.15%	23.81%	16.89%	22.62%
50	1.98%	1.24%	1.29%	1.35%	1.46%
75	0.94%	0.92%	0.94%	1.67%	1.12%
100	0.75%	0.79%	0.79%	0.76%	0.77%
Max_iter overall average	6.83%	7.28%	6.71%	5.17%	6.49%

illustrates the influence of these parameters. The color scale determines whether the result is good (green) or bad (red). As can be observed, it is evident that a larger population size (Pop) leads to a smaller $RPD$. On the other hand, the value of Max_iter does not have a significant influence on the size of $RPD$, so we can disregard it as a parameter of great influence.

Since we have determined that Pop has a significant influence, let us see which other parameter also affects the obtaining of a lower $RPD$. Among the remaining three parameters, we have:

• Procreation rate (PR);
• Cannibalism rate (CR);
• Mutation rate (PM).

Each of these three parameters was evaluated with Pop to determine their degree of influence on minimizing $RPD$, and the findings are presented in

Table 8. Pop and PR (heat map).

$\mathit{R}\mathit{P}\mathit{D}$	PR				Pop Overall Average
Pop	0.2	0.4	0.6	0.8
25	74.16%	11.72%	3.10%	1.51%	22.62%
50	2.59%	1.35%	1.06%	0.86%	1.46%
75	2.13%	0.98%	0.73%	0.63%	1.12%
100	1.27%	0.74%	0.60%	0.49%	0.77%
PR overall average	20.04%	3.70%	1.37%	0.87%	6.49%

,

Table 9. Pop and CR (heat map).

$\mathit{R}\mathit{P}\mathit{D}$	CR				Pop Overall Average
Pop	0.2	0.4	0.6	0.8
25	34.50%	30.14%	16.90%	8.96%	22.62%
50	2.08%	1.28%	1.28%	1.21%	1.46%
75	1.66%	0.97%	0.95%	0.90%	1.12%
100	0.82%	0.77%	0.76%	0.75%	0.77%
CR overall average	9.76%	8.29%	4.97%	2.95%	6.49%

and

Table 10. Pop and PM (heat map).

$\mathit{R}\mathit{P}\mathit{D}$	PM				Pop Overall Average
Pop	0.2	0.4	0.6	0.8
25	38.78%	19.33%	16.99%	15.39%	22.62%
50	1.37%	1.35%	1.93%	1.20%	1.46%
75	1.67%	0.95%	0.95%	0.91%	1.12%
100	0.84%	0.73%	0.74%	0.78%	0.77%
PM overall average	10.67%	5.59%	5.15%	4.57%	6.49%

. As indicated in Table 8, it is evident that with the increase in the PR value, similar to Pop, the $RPD$ value decreases, indicating that PR has a direct influence on $RPD$. On the other hand, in the case of CR, as shown in Table 9, it has an influence on $RPD$ when Pop values are small. However, as Pop increases, it can be observed that CR does not have a significant influence on $RPD$. Finally, by examining Table 10, we can determine that PM has a slight influence, as no significant change in $RPD$ is observed.

From the provided data, we can ascertain that the most influential parameters for this case are the population size (Pop) and the procreation rate (PR). To further support this, we provide

Table 11. Top ten best parameter combinations.

Position	PR	CR	PM	Pop	Max_iter	$\mathit{R}\mathit{P}\mathit{D}$
1	0.8	0.4	0.4	100	75	0.24%
2	0.8	0.8	0.4	100	50	0.28%
3	0.6	0.4	0.4	100	100	0.33%
4	0.8	0.4	0.4	100	25	0.33%
5	0.8	0.6	0.4	100	100	0.33%
6	0.8	0.6	0.8	100	50	0.33%
7	0.8	0.8	0.4	75	100	0.33%
8	0.8	0.4	0.4	75	25	0.33%
9	0.6	0.4	0.8	100	25	0.38%
10	0.6	0.6	0.4	100	25	0.38%

and

Table 12. Top ten worst parameter combinations.

Position	PR	CR	PM	Pop	Max_iter	$\mathit{R}\mathit{P}\mathit{D}$
1015	0.2	0.2	0.6	25	100	153.35%
1016	0.2	0.2	0.4	25	50	160.64%
1017	0.2	0.2	0.8	25	50	175.45%
1018	0.2	0.2	0.2	25	75	176.65%
1019	0.2	0.6	0.2	25	75	202.28%
1020	0.2	0.2	0.2	25	50	213.79%
1021	0.2	0.4	0.2	25	100	239.78%
1022	0.2	0.4	0.2	25	25	287.10%
1023	0.2	0.4	0.6	25	50	305.24%
1024	0.2	0.4	0.2	25	75	395.50%

, which display the top 10 best and top 10 worst combinations, respectively, according to their $RPD$.

As can be seen in Table 11, at least one of the parameters Pop or PR utilizes high values, resulting in the best combinations. It is noteworthy that the best combination utilizes the highest values for both parameters (PR = 0.8, Pop = 100) and, among the top ten solutions, the averages show a tendency for PR to have a value of 0.8 (≈0.74) and Pop to have a value of 100 (≈95). Conversely, Table 12 only uses low values for Pop and PR, resulting in the worst combinations.

According to the computed $RPD$, we can conclude that the most favorable combination of $PR, CR, PM, Pop$, and $Max\_iter$ is 0.8, 0.4, 0.4, 0.4, 100, 75 (combination 1), while the least effective combination is 0.2, 0.4, 0.2, 0.2, 25, 75 (combination 1024). Nevertheless, to further validate our findings rigorously, we intend to conduct a statistical test in Section 6.4.

6.4. Statistical Test

Before performing the statistical test, it is necessary to know what type of test is to be performed: whether a parametric test or a non-parametric test.

A parametric statistical test is performed when the data under analysis meet a normal distribution, i.e., come from nature. Since the data being analyzed come from machines and not from nature, the statistical analysis to be performed is a non-parametric test. In addition, since our samples are independent of each other, we must apply the Wilcoxon–Mann–Whitney test [78,79].

From the scipy python library, we can apply this statistical test where the python function is called scipy.stats.mannwhitneyu. One parameter of the above function is “alternative”, which we define as “less”. We evaluate and contrast two distinct combinations ($Comb$), as previously stated in Table 4. As this has a p-value of less than 0.05, we can say that sample $CombA$ is statistically smaller than sample $CombB$. Thus, we can state the following hypotheses:

$$H_0=CombA\ge CombB$$

$$H_1=CombA<CombB$$

If the result of the statistical test is obtained a p-value < 0.05, we cannot assume that $Comb$ A has worse performance than $Comb$ B, rejecting $H_0$. This comparison is made because our problem is a minimization problem.

To verify the findings presented in Table 11 and Table 12, we conducted the Wilcoxon–Mann–Whitney test on

Table 13. Wilcoxin top 5 vs. bottom 5.

Comb	1 1	2 1	3 1	4 1	5 1	1020 2	1021 2	1022 2	1023 2	1024 2
1 1		≥0.05	≥0.05	≥0.05	≥0.05	<0.05	<0.05	<0.05	<0.05	<0.05
2 1	≥0.05		≥0.05	≥0.05	≥0.05	<0.05	<0.05	<0.05	<0.05	<0.05
3 1	≥0.05	≥0.05		≥0.05	≥0.05	<0.05	<0.05	<0.05	<0.05	<0.05
4 1	≥0.05	≥0.05	≥0.05		≥0.05	<0.05	<0.05	<0.05	<0.05	<0.05
5 1	≥0.05	≥0.05	≥0.05	≥0.05		<0.05	<0.05	<0.05	<0.05	<0.05
1020 2	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05		≥0.05	<0.05	≥0.05	≥0.05
1021 2	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05		≥0.05	≥0.05	≥0.05
1022 2	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05		≥0.05	≥0.05
1023 2	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05	<0.05		≥0.05
1024 2	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05	≥0.05

¹ Top 5 combinations; ² bottom 5 combinations

, using the five best results (1–5) and the five worst results (1020–1024) obtained from the various combinations listed in Table 4.

In Table 13, the combinations listed in the first column ($CombA$) are compared against the other combinations in the subsequent columns ($CombB$) to determine the corresponding p-value.

As observed, the top five combinations ($Comb$ 1–5) clearly outperform the worst five combinations ($Comb$ 1020–1024), providing additional support for the results obtained in Table 11 and Table 12. Interestingly, $Comb$ 1022 exhibits a tendency to be one of the worst combinations. This observation arises from its statistical inferiority not only compared to the five best combinations ($Comb$ 1–5) but also to $Comb$ 1020 and 1023, both of which are part of the worst five combinations ($Comb$ 1020–1024).

7. Conclusions

In this article, a metaheuristic never before implemented in solving the GCFP-MR was carried out. The goal was to generate feasible solutions and then determine the best options for minimizing costs in the GCFP-MR. To accomplish this, a case of the GCFP-MR was solved using the “Clonal Selection” metaheuristic in [64] to compare the results of this metaheuristic with the results obtained by the BWO. As observed in the results section, the BWO can successfully solve GCFP-MR instances, surpassing the performance of other metaheuristics and achieving the optimal value. We were able to confirm that the BWO found a cost 1.40% lower than [64]; in other words, we found a better solution. To consolidate the feasibility of the results, it was demonstrated that none of the constraints shown for the evaluated case were violated, and better parameters necessary to obtain good results with lower computational cost were determined. In future work, the application of the BWO to more complex models of the GCFP-MR is proposed, as well as its application to other types of combinatorial problems that are NP-Hard and, therefore, challenging to solve due to the required time and computational resources.