Soft Computing Methodology to Optimize the Integrated Dynamic Models of Cellular Manufacturing Systems in a Robust Environment

Abstract

Machine learning, neural networks, and metaheuristic algorithms are relatively new subjects, closely related to each other: learning is somehow an intrinsic part of all of them. On the other hand, cell formation (CF) and facility layout design are the two fundamental steps in the CMS implementation. To get a successful CMS design, addressing the interrelated decisions simultaneously is important. In this article, a new nonlinear mixed-integer programming model is presented which comprehensively considers solving the integrated dynamic cell formation and inter/intracell layouts in continuous space. In the proposed model, cells are configured in flexible shapes during the planning horizon considering cell capacity in each period. This study considers the exact information about facility layout design and material handling cost. The proposed model is an NP-hard mixed-integer nonlinear programming model. To optimize the proposed problem, first, three metaheuristic algorithms, that is, Genetic Algorithm (GA), Keshtel Algorithm (KA), and Red Deer Algorithm (RDA), are employed. Then, to further improve the quality of the solutions, using machine learning approaches and combining the results of the aforementioned algorithms, a new metaheuristic algorithm is proposed. Numerical examples, sensitivity analyses, and comparisons of the performances of the algorithms are conducted.

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

1. Introduction

Facility layout (FL) considers the layout of machines within cells (Intracell layout) and the layout of cells (Intercell Layout) on the shop floor which can be counted as an essential element to plan a CMS. The cutdown in material handling cost, work-in-process, and throughput rate is the result of an efficient FL [1]. The system’s performance would have the capacity to be boosted by designing a capable layout and the expense of the production also would fall around 40% to 50% on average [2]. In the FL problem, the actual parameters in the flow of some objectives might be ignored. The objectives which might diminish the intercellular circulation but none of them would certainly cause the charge of material handling to be minimized could be the minimization of the number of exceptional elements (EEs) or another common one can be referred to as the cost reduction of intercell movement. Although layout design has been somehow neglected in the CMS as many of the related researches have only investigated the CF, the vital point in this design could be using the FL problem [3, 4]. As stated, the decisions that are made in the FL and CF problem are interrelated, and tellingly to have a favored CMS design inscribing these two concurrently is of high importance [5]. Nevertheless, it has been asserted a complicated issue to make any of these kinds of decisions [6, 7]. Therefore the simultaneous addressing of these decisions is a difficult issue.

In most studies, some of the mentioned decisions have been handled and sometimes all of them consecutively [8]. On the other hand, most approaches in the area of facility layout and CF problem, for simplicity, usually consider minimizing the number of intercell movements or intracell movements or both [9]. To make a minimum material handling cost in FL design, it is an important matter to inspect the exact details of this design like the concept of distance. On the other hand, these types of approaches might make some illusive presumptions being machine location or fixed cells in the FL problem which might lead to an incompetent outcome. Also, in earlier investigations, the machines existing in a cell space were placed only in one type of location which was a line-formed one. There is another way to locate the machines other than line-form as well which is a U-form but the problem would be the expenses add up to the system.

Roughly speaking, many models in the field of CMS can be classified as NP-hard problems. Thus, employing and proposing exact mathematical approaches to tackle the models of CMS is usually ineffective. To this end, different heuristic, metaheuristic, and machine learning approaches have been applied, which can effectively handle the NP-hard models of CMS.

Machine learning, neural networks, and metaheuristic algorithms are relatively new subjects employed broadly in different fields of industrial engineering and management studies. They are also closely related to each other: learning is somehow an intrinsic part of all of them [10, 11]. Computer science, probability and statistic rules, and information and decision theory play a major role in the development of machine learning and other algorithms of metaheuristics. Machine learning methods find applications in different fields of industrial engineering, vehicle routing problem, and lot sizing, and maintenance optimization models are among others [12].

As the proposed mixed-integer nonlinear model in this paper is NP-hard, four metaheuristic algorithms are employed to tackle the problem. In the first step, the Genetic Algorithm (GA), Keshtel Algorithm (KA), and Red Deer Algorithm (RDA) are designed to optimize the model. To further improve the solutions of the algorithms, using machine learning approaches, a novel metaheuristic algorithm, which benefits from the merits of the aforementioned technique, is proposed. Accordingly, using soft computing methodology to optimize the integrated dynamic cell formation problem in a robust environment is the main contribution of the current study.

The rest of the paper is organized as follows: In Section 2, the mathematical model for the problem is presented. Section 3 presents the metaheuristic algorithms for optimizing the model. Section 4 discusses how the algorithms are tuned and reports the results of computations and calculations. Finally, Section 5 concludes the paper.

2. Proposed Mathematical Model

2.1. Notations

2.1.1. Sets

  $i,i^{\prime }=\mathrm{1,2},\dots ,m$: machines group symbol

  $j=\mathrm{1,2},\dots ,n$: parts symbol

  $l,k,k^{\prime }=\mathrm{1,2},\dots ,c$: cells group symbol

  $h=\mathrm{1,2},\dots ,H$: periods group symbol

2.1.2. Parameters

  $D_{jh}$: parts demand for type $j$ in period $h$

  $B_j$: batch size for parts transportation type $j$

  $C_{\text{intra}}^j$: the expense of intracell material handling for the transportation of part $j$ per unit distance ($/unit)

  $C_{\text{inter}}^j$: the expense of intercell material handling for the transportation of part $j$ per unit distance ($/unit)

  $C_i$: the expense of relocating machine $i$ ($/unit)

  $R_{ij}$: the number of operations done on part $j$ using machine $i$

  $E$: shop floor horizontal length (the length of the shop floor)

  $F$: job shop vertical length (the width of the shop floor)

  $SP$: the pairs group $i,j$ such that $a_{ij}\ge 1$ (the group of elements that are not zero of a part-machine matrix)

  $NM$: every cell’s maximum figure of relocated machines per period

  ${\alpha }_j$: the expense coefficient (or penalty) caused by existence of any special part type $j$ per period

  $N$: an appropriate big positive figure

  $A_{kl},B_{kl}$,: a random variable and a zero

  $A_{i{i}^{\prime }h},B_{i{i}^{\prime }h}$: a random variable and a zero

  $f_{i{i}^{\prime }h}^j$: trips number for changing the location of part type j between machines $i$ and $i^{\prime }$ in period $h$:

$$\begin{array}{}\text{(1)}& f_{i{i}^{\prime }h}^j=\begin{array}{cc}\frac{D_{jh}}{B_j},& \text{if\hspace{0.17em}}{R}_{i^{\prime }j}-R_{ij}=1,\\ 0,& \text{if\hspace{0.17em}}{R}_{i^{\prime }j}-R_{ij}\ne 1.\end{array}\end{array}$$

2.1.3. Decision Variables

  $X_{ikh}=\begin{array}{cc}1,& \text{If\hspace{0.17em}machine}i\text{is\hspace{0.17em}allocated\hspace{0.17em}to\hspace{0.17em}cell}k\text{in\hspace{0.17em}period}h,\\ 0,& \text{Otherwise}\end{array}$

  $Y_{jkh}=\begin{array}{cc}1,& \text{If\hspace{0.17em}part}j\text{is\hspace{0.17em}allocated\hspace{0.17em}to\hspace{0.17em}cell}k\text{in\hspace{0.17em}period}h,\\ 0,& \text{Otherwise}\end{array}$

  $Z_{ih}=\begin{array}{cc}1,& \text{If\hspace{0.17em}machine}i\text{displaces\hspace{0.17em}through\hspace{0.17em}periods}h\text{\hspace{0.17em}and\hspace{0.17em}}h+1,\\ 0,& \text{Otherwise}\end{array}$

  $U_{ijkh}=\begin{array}{cc}1,& \text{if\hspace{0.17em}}{Y}_{jkh}=0\text{\hspace{0.17em}and\hspace{0.17em}}{X}_{ikh}=1,\\ 0,& \text{Otherwise}\end{array}$

  $V_{ijkh}=\begin{array}{cc}1,& \text{if\hspace{0.17em}}{Y}_{jkh}=1\text{\hspace{0.17em}and\hspace{0.17em}}{X}_{ikh}=0,\\ 0,& \text{Otherwise}\end{array}$

  $x_{ih}$: the center of machine $i$ in period $h$ in horizontal coordinate

  $y_{ih}$: the center of machine $i$ in period $h$ in vertical coordinate

  $p_{kh}^1$: the left side of cell $k$ in period $h$ in horizontal coordinate

  $p_{kh}^2$: the right side of cell $k$ in period $h$ in horizontal coordinate

  $q_{kh}^1$: the bottom side of cell $k$ in period $h$ in vertical coordinate

  $q_{kh}^2$: the top side of cell $k$ in period $h$ in vertical coordinate

Thus, the expense of displacing part j between machines i and i′ in period ℎ concerning the movement of the intercell or intracell could be determined as follows.

If $X_{ikh},X_{i^{\prime }kh}>0$ this cost is equal to the following equation:$$\begin{array}{}\text{(2)}& C_{i{i}^{\prime }h}^j=x_{ih}-x_{i^{\prime }h}+y_{ih}-y_{i^{\prime }h}{C}_{\text{intra}}^j.\end{array}$$

If $X_{ikh}{X}_{i^{\prime }kh}=0\text{\hspace{0.17em}and\hspace{0.17em}}{X}_{ikh}{X}_{i^{\prime }{k}^{\prime }h}>0$ this cost is equal to the following equation:$$\begin{array}{}\text{(3)}& C_{i{i}^{\prime }h}^j=x_{ih}-x_{i^{\prime }h}+y_{ih}-y_{i^{\prime }h}{C}_{\text{inter}}^j.\end{array}$$

The way cells are configured, the machines plot inside them, and also their layout on the shop floor are the goals of this model to be defined simultaneously in kinetic situations somehow that some expenses such as cells redesigning, parts total transportation cost, and EES number are reduced. In the given model, the job shop figure is considered for the intracellular layout. Some mixed-integer nonlinear programming models are discussed below along with multiple presumptions, parameters, and decision variables.

2.2. Model Assumption

The following presumptions are taken into account to simulate the model:

(i) In every period, the flow between machines is defined. The demand for parts, operational paths of parts, and also the parts transportation batch size are how this figure is gained from.

(ii) If each product’s batch size is in all periods constant and determined, the parts can move within the batches. For both inter- and intracell displacements, the parts batch largeness is considered the same.

(iii) By applying rectilinear distance, the expense of material handling is assessed regarding the center-to-center distance between machines.

(iv) The expense of material handling for inter- and intracell relocations for machines and parts pertains to the distance traveled.

(v) The unit expense of intercell and intracell relocations for every part type is predestined and stays constant in the planning horizon.

(vi) During the periods, the unit expense of machine displacement remains the same and is predestined for every kind of machine. The expense contains the opening, transferring, and resetting of the machine.

(vii) In every period, by using the expected workload in every cell, the number of cells to be shaped can be ascertained beforehand. Nevertheless, during the planning horizon, since the cells are malleable, they can be shaped easily; hence, their configuration would not be predetermined.

(viii) Only one number of each machine kind exists.

(ix) During the planning horizon, the maximum capacity of cells is determined and stays constant.

(x) The machines are supposed to have a unit dimension, since they are squares of equal area. No extra inventory should exist between the periods; each demand must be supplied in the related period and no delayed orders are permitted.

(xi) 100% is assumed for machines and production efficiency.

2.3. Mathematical Formulation

Concerning input parameters and variables, the presented nonlinear model for this problem is as follows:$$\begin{array}{}\text{(4)}& \text{minimize}{\displaystyle \sum _{h=1}^H{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{i^{\prime }=1}^m{f}_{i{i}^{\prime }h}^j{C}_{i{i}^{\prime }h}^j}}}}+{\displaystyle \sum _{h=2}^H{\displaystyle \sum _{i=1}^m{C}_i{Z}_{ih}}}+{\displaystyle \sum _{h=1}^H{\displaystyle \sum _{k=1}^C{\displaystyle \sum _{i,j\in sp}{\alpha }_j.\frac{U_{ijkh}+V_{ijkh}}{2}}}}.\end{array}$$

It is subject to$$\begin{array}{}\text{(5)}& {\displaystyle \sum _{k=1}^C{X}_{ikh}}=1,\hspace{1em}i=\mathrm{1,2},\dots ,m,\forall h,\text{(6)}& {\displaystyle \sum _{k=1}^C{Y}_{jkh}}=1,j=\mathrm{1,2},\dots ,n,\forall h,\text{(7)}& 1\le {\displaystyle \sum _{i=1}^m{X}_{ikh}}\le NM,k=\mathrm{1,2},\dots ,C,\forall h,\text{(8)}& N{Z}_{ih}\ge x_{ih}-x_{ih+1}+y_{ih}-y_{ih+1},\hspace{1em}\forall i,h<H,\text{(9)}& x_{ih}-x_{i^{\prime }h}+y_{ih}-y_{i^{\prime }h}\ge 1,\text{(10)}& \begin{array}{c}{x}_{ih}\ge p_{kh}^1-N1-X_{ikh},\\ x_{ih}\le p_{kh}^2+N1-X_{ikh},\\ y_{ih}\ge q_{kh}^1-N1-X_{ikh},\\ y_{ih}\le q_{kh}^2+N1-X_{ikh}\end{array},\hspace{1em}\forall i,k,h,\text{(11)}& \begin{array}{c}{p}_{kh}^1\ge 0,\\ q_{kh}^1\ge 0,\\ p_{kh}^2\le E,\\ q_{kh}^2\le F\end{array},\hspace{1em}\forall k,h,\text{(12)}& \begin{array}{c}{p}_{kh}^1-p_{lh}^2+N{A}_{kl}+N{B}_{kl}\ge 0,\\ p_{kh}^2-p_{lh}^1-N{A}_{kl}-N1-B_{kl}\le 0,\\ q_{kh}^1-q_{lh}^2+N1-A_{kl}+N{B}_{kl}\ge 0,\\ q_{kh}^2-q_{lh}^1-N1-A_{kl}-N1-B_{kl}\le 0,\\ 0\le k<l\le C,\end{array}\text{(13)}& \begin{array}{c}{x}_{ih}-x_{i^{\prime }h}+N{A}_{i{i}^{\prime }h}+N{B}_{i{i}^{\prime }h}\ge 1,\\ x_{i^{\prime }h}-x_{ih}-N{A}_{i{i}^{\prime }h}-N1-B_{i{i}^{\prime }h}\ge 1,\\ y_{ih}-y_{i^{\prime }h}+N1-A_{i{i}^{\prime }h}+N{B}_{i{i}^{\prime }h}\ge 1,\\ y_{i^{\prime }h}-y_{ih}-N1-A_{i{i}^{\prime }h}-N1-B_{i{i}^{\prime }h}\ge 1\end{array},\hspace{1em}\forall 1\le i<i^{\prime }\le M.\end{array}$$

The intracellular and intercellular material transferring costs are represented by the first term of the objective function. The expense of cell reshaping which might alter from period to period is defined in the following term. The decreasing number of exceptional parts is in connection with the third term. The double calculation of decision variables when they are 1 in this relationship is the reason behind the coefficient of $1/2$. The first set of constraints (Equation (5)) guarantees that every machine is allocated to just one cell. Every part that has been allocated to one part family only is ensured by the second constraint (Equation (6)). Constraint (7) is the limitation for the number of machines in one cell. Variable Z_ih equals 1 when machine type $i$ is displaced during periods h and (1 + h), which is ensured by the fourth constraint (Equation (8)). The fifth constraint (Equation (9)) which is replaced with Equation (13) prevents machines from being overlapped. As mentioned, the machines are considered squares with a unit dimension. Each machine that has to displace in space of its corresponding cell is indicated in the set of relationships (10). The next constraint (Equation (11)) is developed to control the cells which are in space of the job shop. Preventing cells from being overlapped is shown in the set of relationships in (12).

2.4. Proposed Robust Model

Robust optimization is used when there is uncertainty in the parameters. In this case, with a slight change in the value of one of the parameters, the optimality and justification of the answer may be compromised. Therefore, to control the situation, it is necessary to use an optimization model that, considering the existing uncertainties, obtains an optimal answer that remains an optimal and justified answer to the problem under investigation in the face of changes in uncertain parameters.

In this paper, to face the uncertainty of the parameters, set-induced robust optimization is used. In this optimization, it is assumed that uncertain data belongs to an uncertainty set, and the aim is to choose the best solution among those “immunized” against data uncertainty, that is, candidate solutions that remain feasible for all realizations of the data from the uncertainty set.

According to the above, a symmetric interval for the range of parameter changes is considered as follows.

The objective coefficients and the constraint coefficients possibly change with an unknown distribution but they are symmetrical and independent in the following intervals:$$\begin{array}{c}\text{(14)}& {\tilde{C}}_{\text{intra}}^j\in C_{\text{intra}}^j-{\widehat{C}}_{\text{intra}}^j,C_{\text{intra}}^j+{\widehat{C}}_{\text{intra}}^j,{\tilde{C}}_{\text{inter}}^j\in C_{\text{inter}}^j-{\widehat{C}}_{\text{inter}}^j,C_{\text{inter}}^j+{\widehat{C}}_{\text{inter}}^j,\\ {\tilde{C}}_i\in C_i-{\widehat{C}}_i,C_i+{\widehat{C}}_i,\\ {\tilde{D}}_{jh}\in D_{jh}-{\widehat{D}}_{jh},D_{jh}+{\widehat{D}}_{jh},\\ {\tilde{C}}_{i{i}^{\prime }h}^j\in x_{ih}-x_{i^{\prime }h}+y_{ih}-y_{i^{\prime }h}{C}_{\text{intra}}^j-{\widehat{C}}_{\text{intra}}^j{x}_{ih}-x_{i^{\prime }h}+y_{ih}-y_{i^{\prime }h}{C}_{\text{intra}}^j+{\widehat{C}}_{\text{intra}}^j,\\ {\tilde{C}}_{i{i}^{\prime }h}^j\in x_{ih}-x_{i^{\prime }h}+y_{ih}-y_{i^{\prime }h}{C}_{\text{inter}}^j-{\widehat{C}}_{\text{inter}}^j{x}_{ih}-x_{i^{\prime }h}+y_{ih}-y_{i^{\prime }h}{C}_{\text{inter}}^j+{\widehat{C}}_{\text{inter}}^j,\\ {\tilde{f}}_{i{i}^{\prime }h}^j\in \begin{array}{cc}\frac{D_{jh}-{\widehat{D}}_{jh}}{B_j},\frac{D_{jh}+{\widehat{D}}_{jh}}{B_j},& \text{if\hspace{0.17em}}{R}_{i^{\prime }j}-R_{ij}=1,\\ 0,& \text{if\hspace{0.17em}}{R}_{i^{\prime }j}-R_{ij}\ne 1.\end{array}\end{array}$$

For each parameter ${\tilde{a}}_{ij}$ which is subject to uncertainty, $a_{ij}$ represents the nominal value of the parameters and ${\widehat{a}}_{ij}$ represents constant perturbation (which are positive).

In this paper, a robust approach developed by Bertsimas and Sim is used to face the uncertainty of the parameters. Therefore, the robust counterpart of the original problem is obtained by replacing the original constraint with its robust counterpart constraint:$$\begin{array}{c}\text{(15)}& F\ge {\displaystyle \sum _{h=1}^H{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{i^{\prime }=1}^m{f}_{i{i}^{\prime }h}^j{C}_{i{i}^{\prime }h}^j}}}}+{\displaystyle \sum _{h=2}^H{\displaystyle \sum _{i=1}^m{C}_i{Z}_{ih}+{\Gamma }_0{Z}_0}}+{\displaystyle \sum _{h=1}^H{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{i^{\prime }=1}^m{q}_{i{i}^{\prime }h}^j}}}}+{\displaystyle \sum _{h=2}^H{\displaystyle \sum _{i=1}^m{s}_{ih}}}+{\displaystyle \sum _{h=1}^H{\displaystyle \sum _{k=1}^C{\displaystyle \sum _{i,j\in sp}{\alpha }_j.\frac{U_{ijkh}+V_{ijkh}}{2}}}},\\ Z_0+q_{i{i}^{\prime }h}^j\ge \frac{{\widehat{D}}_{jh}}{B_j}x{x}_{ih}-x{x}_{i^{\prime }h}+y{y}_{ih}-y{y}_{i^{\prime }h}{\widehat{C}}_{\text{intra}}^j,\hspace{1em}\forall i,h,j,i^{\prime },\\ Z_0+q_{i{i}^{\prime }h}^j\ge \frac{{\widehat{D}}_{jh}}{B_j}x{x}_{ih}-x{x}_{i^{\prime }h}+y{y}_{ih}-y{y}_{i^{\prime }h}{\widehat{C}}_{\text{inter}}^j,\hspace{1em}\forall i,h,j,i^{\prime },\\ Z_0+s_{ih}\ge {\widehat{C}}_{i}Z{Z}_{ih},\hspace{1em}\forall i,h,\\ -x{x}_{ih}\le x_{ih}\le x{x}_{ih},\hspace{1em}\forall i,h,\\ -x{x}_{i^{\prime }h}\le x_{i^{\prime }h}\le x{x}_{i^{\prime }h},\hspace{1em}\forall i^{\prime },h,\\ -y{y}_{ih}\le y_{ih}\le y{y}_{ih},\hspace{1em}\forall i,h,\\ -y{y}_{i^{\prime }h}\le y_{i^{\prime }h}\le y{y}_{i^{\prime }h},\hspace{1em}\forall i^{\prime },h,\\ -Z{Z}_{ih}\le Z_{ih}\le Z{Z}_{ih},\hspace{1em}\forall i,h,\\ {\displaystyle \sum _{k=1}^C{X}_{ikh}}=1,\hspace{1em}i=\mathrm{1,2},\dots ,m,\forall h,\\ {\displaystyle \sum _{k=1}^C{Y}_{jkh}}=1,\hspace{1em}j=\mathrm{1,2},\dots ,n,\forall h,\\ 1\le {\displaystyle \sum _{i=1}^m{X}_{ikh}}\le NM,\hspace{1em}k=\mathrm{1,2},\dots ,C,\forall h,\\ N{Z}_{ih}\ge x_{ih}-x_{ih+1}+y_{ih}-y_{ih+1},\hspace{1em}\forall i,h<H,\\ x_{ih}-x_{i^{\prime }h}+y_{ih}-y_{i^{\prime }h}\ge 1,\\ \begin{array}{c}{x}_{ih}\ge p_{kh}^1-N1-X_{ikh}\\ x_{ih}\le p_{kh}^2+N1-X_{ikh}\\ y_{ih}\ge q_{kh}^1-N1-X_{ikh}\\ y_{ih}\le q_{kh}^2+N1-X_{ikh}\end{array},\forall i,k,h,\\ \begin{array}{c}{p}_{kh}^1\ge 0\\ q_{kh}^1\ge 0\\ p_{kh}^2\le E\\ q_{kh}^2\le F\end{array},\hspace{1em}\forall k,h,\\ \begin{array}{c}{p}_{kh}^1-p_{lh}^2+N{A}_{kl}+N{B}_{kl}\ge 0,\\ p_{kh}^2-p_{lh}^1-N{A}_{kl}-N1-B_{kl}\le 0,\\ q_{kh}^1-q_{lh}^2+N1-A_{kl}+N{B}_{kl}\ge 0,\\ q_{kh}^2-q_{lh}^1-N1-A_{kl}-N1-B_{kl}\le 0,\\ 0\le k<l\le C,\end{array}\\ \begin{array}{c}{x}_{ih}-x_{i^{\prime }h}+N{A}_{i{i}^{\prime }h}+N{B}_{i{i}^{\prime }h}\ge 1\\ x_{i^{\prime }h}-x_{ih}-N{A}_{i{i}^{\prime }h}-N1-B_{i{i}^{\prime }h}\ge 1\\ y_{ih}-y_{i^{\prime }h}+N1-A_{i{i}^{\prime }h}+N{B}_{i{i}^{\prime }h}\ge 1\\ y_{i^{\prime }h}-y_{ih}-N1-A_{i{i}^{\prime }h}-N1-B_{i{i}^{\prime }h}\ge 1\end{array},\hspace{1em}\forall 1\le i<i^{\prime }\le M,\\ x{x}_{ih},x{x}_{i^{\prime }h},y{y}_{ih},y{y}_{i^{\prime }h},q_{i{i}^{\prime }h}^j,s_{ih}\ge 0,\end{array}$$where Γ is the adjustable parameter controlling the size of the uncertainty set.

3. Proposed Solution Algorithm

The literature approved that the CMS models are classified as NP-hard problems [13–19]. The high complexity of CMS in large-scale instances motivates several researchers to propose novel metaheuristics [20]. This study in addition to the Genetic Algorithm (GA) applies two recent nature-inspired metaheuristics: Keshtel Algorithm (KA) and Red Deer Algorithm (RDA). To improve the benefits of these recent and old metaheuristics, a novel hybrid algorithm is also developed to better address the proposed problem and to provide a comparison among these algorithms based on the solution time and quality. Next, the encoding plan to run the initial population of the metaheuristics is explained. Then, the proposed optimizers are introduced.

3.1. Solution Representation

Since all stochastic optimizers such as GA, KA, and RDA use a continuous search space, an encoding plan to transform it into a discrete area to confirm that the algorithm can address the constraints of our model is highly needed [21–23]. A general view of the encoding scheme is depicted in Figure 1. Then, the assignment of machines and details of the cell manufacturing to generate a feasible solution are given in Figure 2.

[figure omitted; refer to PDF]

Finally, to compute the objective functions (the fitness functions), a matrix with H rows and N columns is generated regarding the matrixes in Figure 2. The structure of the final matrix for the alignment is given in Figure 3.

[figure omitted; refer to PDF]

Table 1

Evaluation metrics to the performance of the algorithms (i.e., DM, SNS, DEA, and POD).

Instances$\mathbf{M}\times \mathbf{P}$	DM
	GA	KA	RDA	H-RDKGA
$3\ast 5$	15861	14389	16452	14015
$4\ast 6$	18753	17275	19743	16527
$5\ast 8$	20213	19833	21872	18817
$6\ast 9$	22916	21806	33112	20763
$7\ast 11$	25817	24319	39671	23917
$8\ast 13$	27918	26518	43749	24008
$10\ast 12$	32650	31997	55761	29879
$11\ast 13$	38650	36521	57144	34699
$12\ast 15$	47840	47003	60195	45810
Instances$\mathbf{M}\times \mathbf{P}$	SNS
	GA	KA	RDA	H-RDKGA
$3\ast 5$	2498	2267	1748	2699
$4\ast 6$	6122	7210	5426	7495
$5\ast 8$	7445	7296	6948	8155
$6\ast 9$	3485	3105	2915	4039
$7\ast 11$	2143	1834	7501	2867
$8\ast 13$	1077	1282	675	2049
$10\ast 12$	5482	4912	4466	4288
$11\ast 13$	6388	5187	5514	6382
$12\ast 15$	6237	5853	6432	7528
Instances$\mathbf{M}\times \mathbf{P}$	DEA
	GA	KA	RDA	H-RDKGA
$3\ast 5$	0.18	0.16	0.12	0.15
$4\ast 6$	0.20	0.12	0.18	0.12
$5\ast 8$	0.24	0.22	0.26	0.18
$6\ast 9$	0.28	0.14	0.22	0.14
$7\ast 11$	0.16	0.26	0.18	0.16
$8\ast 13$	0.24	0.12	0.12	0.19
$10\ast 12$	0.18	0.14	0.20	0.22
$11\ast 13$	0.26	0.18	0.14	0.18
$12\ast 15$	0.14	0.22	0.20	0.35
Instances$\mathbf{M}\times \mathbf{P}$	POD
	GA	KA	RDA	H-RDKGA
$3\ast 5$	0.16	0.22	0.14	0.22
$4\ast 6$	0.18	0.18	0.19	0.21
$5\ast 8$	0.22	0.20	0.10	0.18
$6\ast 9$	0.15	0.14	0.11	0.16
$7\ast 11$	0.17	0.18	0.16	0.12
$8\ast 13$	0.19	0.14	0.12	0.18
$10\ast 12$	0.22	0.16	0.14	0.12
$11\ast 13$	0.22	0.18	0.14	0.16
$12\ast 15$	0.20	0.16	0.08	0.22

Then, the acquired results for every problem are converted to the Relative Percentage Deviation (RPD) computed by$$\begin{array}{}\text{(16)}& \text{RPD}=\frac{{\text{Alg}}_{\text{sol}}-{\text{Best}}_{\text{sol}}}{{\text{Best}}_{\text{sol}}},\end{array}$$where ${\text{Alg}}_{\text{sol}}$ is the output of the algorithm and ${\text{Best}}_{\text{sol}}$ is the best value ever found in the problem size. It should be noted that the lower value for the RPD is preferred.

4. Tuning of Algorithms and Comparison Studies

In this section, first, using the design of the experimental approach, the parameters of the aforementioned algorithms are tuned. Then, a comprehensive study is done to evaluate the performances of the algorithms.

4.1. Tuning of Metaheuristics

As all metaheuristics have some controlling parameters, tuning is needed satisfactorily. Here, based on the concept of the Design of Experiment (DOE), all algorithms have been calibrated. This method can analyze the impact of different candidate values on the parameters of the algorithms and evaluate the behavior of the algorithms. Without a good calibration of the parameters, the behavior of the metaheuristics is not reliable.

To do the tuning, the parameters of the given metaheuristics are considered. With regard to the DOE method, a full factorial method to analyze all possible experiments with regard to the levels is done. The levels and tuned values for each parameter are given in Table 2. It should be noted that all candidate level values are taken from similar studies in the literature [18, 21, 26].

Table 2

Tuning of metaheuristics.

Metaheuristic	Parameters	Levels			Tuned value
		−1	0	+1
GA	Population size	100	150	200	200
	Maximum number of iterations	300	500	700	500
	Rate of mutation	0.05	0.15	0.25	0.15
	Rate of crossover	0.6	0.7	0.8	0.8
KA	Population size	100	150	200	100
	Maximum number of iterations	300	500	700	300
	Percentage of N1	0.1	0.2	0.3	0.1
	Percentage of N2	0.4	0.5	0.6	0.6
	Maximum number of swirlings	5	10	15	10
RDA	Population size	100	150	200	150
	Maximum number of iterations	300	500	700	700
	Number of males	15	25	30	25
	Alpha	0.5	0.6	0.7	0.6
	Beta	0.7	0.8	0.9	0.7
	Gamma	0.8	0.9	1	0.8
H-RDKGA	Population size	100	150	200	150
	Maximum number of iterations	300	500	700	500
	Number of males	15	25	30	30
	Maximum number of swirlings	5	10	15	15

4.2. Comparison among Employed Metaheuristics

To do the comparison among the employed metaheuristics, nine test studies are benchmarked from the literature [15]. These tests are selected from small-scale to large-scale instances. As the model of our work is novel and differs from previous works, no comparison between our results and previous studies is done. Accordingly, we compare our metaheuristics with each other, as well as the results of the exact solver.

As the metaheuristics are naturally random, we run each algorithm 10 times and the best, the worst, the average, and the standard deviation of solutions among runs are reported. An average of the computational time of the algorithms is noted. To check the validation of the results, an exact solver implemented by GAMS software is used. Table 3 provides all the results. It should be noted that the exact solver is not able to find a solution for the large-scale instances after one hour. But all the metaheuristics can solve the problem in a few minutes.

Table 3

Comparison of algorithms.

Algorithm		Test problems
		P1	P2	P3	P4	P5	P6	P7	P8	P9
EX	B	24283	28641	84180	119046	218907	476036	—	—	—
	CPU	18	64	201	836	1872	3315	3600	3600	3600
GA	B	24283	28641	84180	119046	221470	476124	694902	952906	120094
	W	27925	32937	98482	139038	257709	572564	812835	1118365	138304
	A	24283	28641	85637	120903	224095	497882	706813	972491	120265
	SD	4205	4960	15740	22090	40414	101164	129847	180812	20929
	CPU	22	17	22	32	42	65	79	96	102
KA	B	24283	28641	84180	119046	219968	480789	701712	962243	121270
	W	24525	28927	85021	123807	228766	504828	736797	1010355	127333
	A	24428	28812	84684	121902	225246	495212	722763	991110	124907
	SD	194	230	677	3834	7085	19358	28254	38745	4882
	CPU	18	15	20	28	38	58	72	88	92
RDA	B	24283	28641	84180	119046	219850	476362	687953	943376	118893
	W	24524	28926	85021	122617	226445	490652	708591	971677	122459
	A	24451	28840	84768	121545	224466	486365	702399	963186	121389
	SD	160	190	560	2382	4399	9532	13767	18879	2378
	CPU	24	18	26	33	43	66	78	95	106
H-RDKGA	B	24283	28641	84180	119046	218907	476101	681004	933847	117692
	W	24283	29213	85863	121426	223285	485623	694624	952523	120045
	A	24283	28927	85021	120236	221096	480862	687814	943185	118868
	SD	0	286	841	1190	2189	4761	6810	9338	1176
	CPU	22	16	20	26	40	62	72	90	94

B = best, W = worst, A = average, SD = standard deviation, EX = exact solver, and CPU = run time (seconds).

Generally, the behavior of the algorithms in the criterion of the solution time is very close. Both hybrid algorithms and KA have a neck-and-neck competition. However, the KA is slightly better than all the metaheuristics. Without a doubt, the proposed H-RDKGA outperforms other algorithms and its solution is very close to the global solution based on the results.

Finally, the results indicate that, based on the average of the standard deviation of the results and the gaps of the algorithm in the interval plot, the proposed hybrid algorithm, that is, H-RDKAGA, is highly better than other algorithms and outperforms the best. After this algorithm, a slight difference exists between the KA and the RDA, but the RDA is better. The last algorithm is the GA, as it has the weakest performance in this comparison.

5. Conclusion

Inter/intracell layouts and dynamic cell formation in a steady space were investigated concurrently in this paper by a novel mixed-integer nonlinear programming model. This model was performed somehow to minimize the cost of the number of exceptional elements (EEs), parts total transportation expense, and cell redesigning. As the model was an NP-Hard problem, first, three metaheuristic algorithms are proposed for optimization. In the next step, to further improve the solutions, a new hybrid metaheuristic algorithm combining the results of the three metaheuristics is proposed. To combine and improve the solutions of the algorithms, machine learning approaches are employed. More precisely, combining the merits of the aforementioned algorithms, the new metaheuristic algorithm is proposed.

Several recommendations can be proposed for better orientations in this study. Merging the proposed model for instance with a scheduling problem would be interesting. Also, to overcome the uncertainty, a two-stage or multistage stochastic programming method could be used. Applying more in-depth analyses by other large-scale optimization problems could be another approach from the aspect of the new suggested hybrid algorithm. Lastly, to evaluate the outcomes of the offered algorithms, new metaheuristics can be proposed.

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