 Research
 Open Access
Solving a mathematical model integrating unequalarea facilities layout and part scheduling in a cellular manufacturing system by a genetic algorithm
 Ahmad Ebrahimi^{1},
 Reza Kia^{1}Email author and
 Alireza Rashidi Komijan^{1}
 Received: 7 December 2015
 Accepted: 5 July 2016
 Published: 4 August 2016
Abstract
In this article, a novel integrated mixedinteger nonlinear programming model is presented for designing a cellular manufacturing system (CMS) considering machine layout and part scheduling problems simultaneously as interrelated decisions. The integrated CMS model is formulated to incorporate several design features including part due date, material handling time, operation sequence, processing time, an intracell layout of unequalarea facilities, and part scheduling. The objective function is to minimize makespan, tardiness penalties, and material handling costs of intercell and intracell movements. Two numerical examples are solved by the Lingo software to illustrate the results obtained by the incorporated features. In order to assess the effects and importance of integration of machine layout and part scheduling in designing a CMS, two approaches, sequentially and concurrent are investigated and the improvement resulted from a concurrent approach is revealed. Also, due to the NPhardness of the integrated model, an efficient genetic algorithm is designed. As a consequence, computational results of this study indicate that the best solutions found by GA are better than the solutions found by B&B in much less time for both sequential and concurrent approaches. Moreover, the comparisons between the objective function values (OFVs) obtained by sequential and concurrent approaches demonstrate that the OFV improvement is averagely around 17 % by GA and 14 % by B&B.
Keywords
 Cellular manufacturing system
 Machine layout
 Part scheduling
 Mixedinteger nonlinear programming
 Genetic algorithm
Background
Nowadays in modern competitive manufacturing environments, each company will be required to be capable of reacting quickly to sudden unpredictable changes in a market. Hence, flexibility and efficiency in production have been the main targets of many manufacturing systems such as flexible manufacturing systems (FMS) and justintime (JIT) production (Adeil et al. 1996; Selim et al. 1997). One of the major approaches to enhance both flexibility and efficiency is cellular manufacturing (CM), which is an important application of group technology (GT) that handle the formation of manufacturing cells in a way that each part family is processed using a machine cell (Wemmerlov and Hyer 1986). Three major considerable steps in a successful design of a cellular manufacturing system (CMS) are: (1) cell formation (CF) (i.e., to group parts with similar processing requirements into part families and machines to machine cells); (2) group layout (GL) (i.e., to assign machines to workstations within each cell, called intracell layout, and cells arrangements within shop floor, called intercell layout), and (3) group scheduling (GS) (i.e., scheduling of part families) (Jajodia et al. 1992; Wu et al. 2007b).
Studies integrating CF, machines layout, and parts scheduling decisions in CM design
References  Decisions  Integrating approach  Solution method 

Chandrasekharan and Rajagopalan (1993)  CF, GL  Sequential  Nonmetric multidimensional scaling 
Jajodia et al. (1992)  CF, GL  Sequential  Simulated annealing (SA) 
Salum (2000)  CF, GL  Sequential  Two phase method based on manufacturing lead time (MLT) 
Urban et al. (2000)  CF, GL  Sequential  Mathematical model based on quadratic assignment problem (QAP) and network flow problem (NFP) 
Alfa et al. (1992)  CF, GL  Concurrent  Mathematical modelSA 
Bazarganlari et al. (2000)  CF, GL  Concurrent  Threephase approach 
Wang et al. (2001)  CF, GL  Concurrent  Mathematical modelSA 
Arvindh and Irani (1994)  CF, GL  Concurrent  An integrated framework 
Akturk (1996)  CF, GL  Concurrent  Mathematical model 
Chiang and Lee (2004)  CF, GL  Concurrent  SA 
Mahdavi et al. (2008)  CF, GL  Concurrent  Heuristic based on flow matrix 
Ahi et al. (2009)  CF, GL  Concurrent  Multiple attribute decision making (MADM) 
Wu et al. (2006)  CF, GL  Concurrent  Genetic algorithm (GA) 
Wu et al. (2007a)  CF, GL  Concurrent  Genetic algorithm (GA) 
Wu et al. (2007b)  CF, GL, GS  Concurrent  Mathematical model–Hierarchical GA (HGA) 
Mahdavi and Mahadevan (2008)  CF, GL  Concurrent  Heuristic approach 
Solimanpur et al. (2004b)  CF, GL  Concurrent  Mathematical model based on QAPAnt colony optimization (ACO) 
Castillo and Westerlund (2005)  CF, GL  Concurrent  An eaccurate model 
Jolai et al. (2012)  CF, GL  Concurrent  Electromagnetism algorithm 
Xie and Sahinidis (2008)  CF, GL  Concurrent  Branch and bound 
Javadi et al. (2013)  CF, GL  Concurrent  Mathematical model 
Mohammadi and Forghani (2014)  CF, GL  Concurrent  Genetic algorithm (GA) 
Sridhar and Rajendran (1993)  CF, GS  Sequential  Hybrid SA 
Solimanpur et al. (2004a)  CF, GS  Sequential  Twostage heuristic algorithm 
Atmani et al. (1995)  CF, GS  Concurrent  Mathematical model 
Franca et al. (2005)  CF, GS  Concurrent  Evolutionary algorithm 
Reddy and Narendran (2003)  CF, GS  Concurrent  Heuristic 
Leung et al. (2007)  CF, GS  Concurrent  Heuristic 
Lin et al. (2009)  CF, GS  Concurrent  Tabu search (TS)–GA–SA 
Hendizadeh et al. (2008)  CF, GS  Concurrent  Tabu Search (TS) 
TavakkoliMoghaddam et al. (2008)  CF, GS  Concurrent  GA and Memetic algorithm 
TavakkoliMoghaddam et al. (2010)  CF, GS  Concurrent  Scatter search (SS) 
Chen and Cao (2004)  CF, GS  Concurrent  Mathematical model–TS 
CF, GL, GS  Sequential  Mathematical model–GA 
Due to the complexity and NPcomplete nature of CF, GL, and GS decisions, most researchers have addressed two or three decisions sequentially or independently. However, the benefits gained from CMS implementation are highly affected by how three stages of the CMS design have been performed in collaboration with each other. Hence, all of these decisions should be addressed concurrently with the intention of obtaining the best results (Alfa et al. 1992; Bazarganlari et al. 2000).
Ranjbar and Najafian Razavi (2012) proposed a new approach to concurrently make the layout and scheduling decisions in a job shop environment and developed a hybrid metaheuristic approach based on the scatter search algorithm. Ripon and Torresen (2013) presented a multiobjective evolutionary method based on a hybrid genetic algorithm by incorporating variable neighborhood search for solving job shop scheduling problem (JSSP) that considers transportation delays and facility layout planning (FLP) as an integrated problem. Halat and Bashirzadeh (2014) developed a concurrent approach for job shop cell scheduling to minimize the makespan in an integer linear programming model by considering exceptional elements, intercellular moves, intercellular transportation times, and sequencedependent family setup times. They also developed a heuristic approach based on the genetic algorithm.
Wu et al. (2007b) have extended the mathematical models proposed in Wu et al. (2006, 2007a) to develop a new one which integrated three mentioned decisions. Then, they developed a hierarchical genetic algorithm (HGA) to solve the integrated cell design problem. The deficiencies of that model are inaccuracy in determining the layout of cells and probable overlapping of the cells. Tang et al. (2010) developed a scatter search approach to solve a nonlinear mathematical programming model for the problem of parts scheduling in a CMS by considering exceptional parts by minimizing the total weighted tardiness.
Arkat et al. (2012a) have promoted a similar integration by proposing two mathematical models, the first one integrates cellular layout with cell formation to determine optimal cell configuration and the layout of machines and cells in order to minimize the total movement costs. Considering the results obtained by solving this model integrating cellular configuration and layout, the cell scheduling problem becomes a job shop scheduling problem with transportation times. The second model is based on the concurrent design and integrates the GS problem with CF and GL problems. Also, two genetic algorithms were developed to solve the realsized problems. Arkat et al. (2012b) presented a multiobjective model to make decisions about cell formation, cellular layout and operation sequence simultaneously. The first objective was to minimize total transportation cost of parts and the second objective was to minimize makespan. A multiobjective genetic algorithm was used to solve the model. Zeng et al. (2015) proposed a twostage GAbased heuristic algorithm to solve a nonlinear mathematical programming model to determine the sequences of the exceptional parts to be transferred via an automated guided vehicle (AGV) in order to minimize the process makespan.
As the main aim of this article, regarding the articles reviewed above, is proposing a new mathematical model with consisting of important manufacturing features such as operation sequence, processing time, transferring time, intracell layout and parts scheduling. The presented model is different from the existing models available in the literature because of incorporating some important design aspects simultaneously. In the first aspect, the layout of machines with unequalareas in the cells is not restricted to linear type. However, dimensions of cells are predetermined by a system designer in a shop floor with a continuous area. In the second aspect, process routings for part types can be flexible. In the third aspect, time and cost of movement are depended on three factors: movement distance, part type, and movement type (intercell or intracell). In the fourth aspect, despite the fact that three ingredients have been formulated in the objective function including makespan, penalty cost, and material handling costs, however, only one, two, or three of them can be used to form cells based on the desired objective. Finally, in the fifth aspect, the CF, machines layout, and parts scheduling decisions are made simultaneously by an integrated model.
Another aim of this article is developing an efficient genetic algorithm enhanced by a matrixbased chromosome structure consisting of two sections for layout and scheduling, a heuristic procedure generating initial feasible solutions, a procedure calculating the fitness functions of generated solutions, and efficient crossover and mutation operators in order to determine three interrelated stages in designing a CMS simultaneously.
The remainder of this article is organized as follows. In “Mathematical model” section, a mathematical model integrating CMS, machines layout and parts scheduling decisions is formulated. The development of the designed GA is discussed in “Designed genetic algorithm for the integrated layout and scheduling model of a CMS” section. “Experimental results” section illustrates the test problems that are utilized to investigate the features of the proposed model and the performance of the developed algorithm. Finally, a conclusion is given in “Conclusion” section.
Mathematical model
Model assumptions
 1.
Each part type has several operations which should be processed in a given sequence. Also, the processing capabilities and processing times of partoperations for each machine type are known and deterministic.
 2.
All machine types are assumed to be multipurpose ones which are capable of performing one or more operations without imposing a reinstalling cost. In like manner, each operation of a part type can be performed on different machine types with different processing times. This feature providing flexibility to process plan of parts has been known as alternative process routings. Nevertheless, a part operation should be processed by only one of those machines which are capable of processing that operation.
 3.
A machine cannot process more than one part at the same time.
 4.
Parts are moved individually by material handling devices between and within cells. Intercell movement happens whenever successive operations of a part type are carried out in different cells. Also, the intracell movement happens whenever successive operations of a part type are processed on different machines in the same cell.
 5.
The rectangular facilities of unequalareas can be located anywhere in the cells having a predetermined shape with a continual space without any overlaps. In fact, the intercell layout and distances between cells are given and intracell layout is determined by the model.
 6.
Each planar cell has a rectangular shape whose length and width is known in advance. Also, the number of cells to be formed is given.
 7.
The maximum and minimum limit of the cell size in terms of the number of machines is known.
 8.
The loading and unloading point is at the center of each machine.
 9.
Machines have a predetermined orientation (i.e., machines may be located either horizontally or vertically). The machine is horizontally located if the longer side of the machine is parallel to the xaxis. On the contrary, the machine is vertically located if the longer side is parallel to the yaxis.
 10.
The rectilinear distance between the centers of two facilities i and j with coordinates (x _{ i }, y _{ i }) and (x _{ j }, y _{ j }) is considered to be the distance norm: d _{ ij } = x _{ i } − x _{ j } + y _{ i } − y _{ j }.
 11.
Once an operation of a part starts to be processed on a machine, it cannot be interrupted before being completed.
 12.
Due date is determined for each part. As a result, tardiness penalty is incurred for each part type per time unit if it is not completed before its due date.
 13.
Cost and time of handling a part between two locations in the same cell or between different cells depend on three factors: the distance between locations of machines, type of part, and type of movement (intercell or intracell). Hence, for each part type, three coefficients per distance unit are considered: movement time, intercell movement cost and intracell movement cost.
 14.
Machine setup time is negligible.
 15.
The machines will never breakdown and be available throughout the scheduling period.
The notations used in the model are presented below:
Sets
 \(i = \left\{ {1,2, \ldots ,P} \right\}\) :

index of parts
 \(j,j^{{\prime }} = \left\{ {1,2, \ldots ,M} \right\}\) :

index of machines
 \(c,c^{{\prime }} = \left\{ {1,2, \ldots ,C} \right\}\) :

index of cells
 \(k = \left\{ {1,2, \ldots ,K_{p} } \right\}\) :

index of operations for part type p
 \(k^{\prime},k^{{\prime \prime }} = \left\{ {1,2, \ldots ,K_{m} } \right\}\) :

index of processing positions for machine type m
Model parameters
 \(Q\) :

factory costs per time unit
 \(Ply_{i}\) :

tardiness penalty for part type i per time unit
 \(dd_{i}\) :

due date for part type i
 \(CO_{i}\) :

intercell material handling cost for part type i per distance unit
 \(CI_{i}\) :

intracell material handling cost for part type i per distance unit
 \(TT_{i}\) :

material handling time for part type i per distance unit
 \(UBC\) :

upper cell size limit
 \(LBC\) :

lower cell size limit
 \(T_{kij}\) :

processing time of operation k of part type i on machine j
 \(a_{kij}\) :

1 if operation k of part type i can be processed on machine type j; 0 otherwise (i.e., \(a_{kij}\) is 1 if \(T_{kij} > 0\); 0 otherwise)
 \(L_{j}\) :

length of the horizontal side of machine type j
 \(H_{j}\) :

height of the vertical side of machine type j
 \(LX_{c}\) :

horizontal coordinate of left side of cell c
 \(RX_{c}\) :

horizontal coordinate of right side of cell c
 \(LY_{c}\) :

vertical coordinate of lower side of cell c
 \(UY_{c}\) :

vertical coordinate of upper side of cell c
 \(M\) :

a big positive number
Decision variables
 \(V_{jc}\) :

1 if machine j is assigned to cell c,0 otherwise
 \(Z_{ki}^{k'j}\) :

1 if kth operation of part type i is processed at k′th processing position on machine j, 0 otherwise
 \(CTM_{k'j}\) :

completion time of k′ th processing position of machine j
 \(CTP_{i}\) :

completion time of part type i
 \(C_{\rm{max} }\) :

makespan time
 \(\alpha_{j}\) :

horizontal coordinate of center of machine j
 \(\beta_{j}\) :

vertical coordinate of center of machine j
 \(x_{j}^{{\prime }}\) :

horizontal coordinate of left side of machine j
 \(x_{j}^{{\prime \prime }}\) :

horizontal coordinate of right side of machine j
 \(y_{j}^{{\prime }}\) :

vertical coordinate of lower side of machine j
 \(y_{j}^{{\prime \prime }}\) :

vertical coordinate of upper side of machine j
 \(R_{jj'}^{X}\) :

1 if machine j is completely located right side of machine j′, 0 otherwise
 \(R_{jj'}^{Y}\) :

1 if machine j is completely located upper side of machine j′, 0 otherwise
 \(d_{jj'}\) :

distance between machine j and j′
Mathematical model
The mixinteger nonlinear mathematical model is presented as follow:
The objective function consists of three terms. Term (1.1) contains makespan time and calculates factory costs during makespan time. Term (1.2) is total tardiness penalties for all parts which have been completed after their due date. Terms (1.3) calculates the intercell and intracell material handling costs.
Constraints sets (2) ensure that each machine is assigned to only one cell. The cell size limits specified by a designer are enforced through Constraints (3) and (4). Constraint set (5) ensures that each operation of each part is processed by only one machine at one of its processing positions. Constraint set (6) ensures an operation of a part is processed by a machine at one of its processing positions provided that machine is capable of processing the related operation.
Constraints (7) compute completion time of the first processing position of a machine in which an operation of a part is processed. There are two cases: (1) the first operation of part iis processed at the first processing position of machine j. It can be simply understood that completion time of the first processing position of machine j, in this case, is equal to the processing time of the first operation of part i; (2) any operation of part i except the first one is processed at the first processing position of machine j. In this case, the part needs to be moved from the machine j′ processing the previous operation (i.e., k − 1) to the current machine j processing the current operation (i.e., k). Considering the distance between machines j and j′, this movement needs a handling time equal to \(\left( {d_{{jj^{\prime}}} \times TT_{i} } \right)\). Since the previous operation k − 1 of part i has been finished at the time \(CTM_{{k^{{\prime \prime }} j^{{\prime }} }}\), the part i will be ready for processing the operation k on machine j at time \(\left( {d_{{jj^{{\prime }} }} \times TT_{i} } \right) + CTM_{{k^{{\prime \prime }} j^{{\prime }} }}\). As a result, in this case, the completion time of the first processing position of machine j is when processing the operation k of part i on machine j is finished and it is equal to \(T_{kij} + \left( {d_{{jj^{{\prime }} }} \times TT_{i} } \right) + CTM_{{k^{{\prime \prime }} j^{{\prime }} }}\).
Constraints (8) compute completion time of any processing position except the first one of machine j in which an operation of a part is processed. Similarly, there are two cases: (1) the first operation of part i is processed at the processing position k′ of machine j. It can be simply understood that completion time of the processing position k′ of machine j, in this case, is equal to processing time of the first operation of part i on machine j plus the completion time of the previous processing position k′ − 1 of machine j; (2) any operation of part i except the first one is processed at the processing position k of machine j. In this case, as it was similarly explained for Constraint (7), the ready time to processes the operation k of part i on machine j is \(\left( {d_{{jj^{{\prime }} }} \times TT_{i} } \right) + CTM_{{k^{{\prime \prime }} j^{{\prime }} }}\). In addition, machine j should be idle to process that operation at its processing position k′. It means that the completion of the previous processing position k′ − 1 of machine j should have been reached. To satisfy the limitations of part readiness and machine idleness, the time which is equal to the maximum of \(CTM_{{k^{\prime}  1 j}}\) and \(\left( {d_{{jj^{{\prime }} }} \times TT_{i} } \right) + CTM_{{k^{{\prime \prime }} j^{{\prime }} }}\) is considered as an actual starting time to process that operation of part i. As a result, the completion time of the processing position k′ of machine j in which operation k of part j is processed is equal to \(T_{kij} + {\text{Max}}\left\{ {CTM_{{k^{\prime}  1 j}} ,\sum\nolimits_{{k^{\prime \prime } = 1}}^{{K_{m} }} {} \sum\nolimits_{{j^{\prime} = 1}}^{M} {Z_{k  1i}^{{k^{\prime \prime } j^{\prime } }} \left( {CTM_{{k^{\prime \prime } j^{\prime } }} + \left( {d_{{jj^{\prime}}} \times TT_{i} } \right)} \right)} } \right\} .\) Constraints (7) and (8) also enforce a machine to not process more than one part at the same time.
Constraint set (9) computes completion time for each part. Constraint set (10) returns makespan time based on the computed completion times of all parts.
Constraints (11) and (12) represent the horizontal and vertical coordinates of the center of each machine, respectively. In the other hand, Constraint sets (13) and (14) return the coordinates of four sides of a machine based on its center coordinates and its length and height. Constraint set (15) computes the rectilinear distance between the centers of two machines j and j′.
Constraint sets (16)–(19) ensure that machines assigned to a cell are placed with regard to coordinates of cell sides (vertically and horizontally). Constraint sets (20)–(24) ensure that machines do not overlap in the horizontal and vertical direction simultaneously. Finally, Constraint sets (25)–(29) are the logical binary and nonnegativity necessities on the binary and positive continuous decision variables.
Linearization techniques
The proposed model is nonlinear in both objective function and constraints sets. Hence, some linearization techniques are proposed to convert the nonlinear model into a linearized counterpart as follows:
Linearization of function Max in Eqs. (8) and (10) in constraint sets is exactly similar to that of Eq. (1.2).
Linearization of the other nonlinear terms multiplying binary variables in Eq. (1.3) is similarly done as explained at the above.
Linearization of the other nonlinear terms multiplying binary variables by continuous variables in Eqs. (7)–(9) is similarly done.
Designed genetic algorithm for the integrated layout and scheduling model of a CMS
The GA is an evolutionary search and optimization technique considering the design process as an evolutionary one. It seeks to find the best solution by generating a population of candidate individuals as the current parents. Using a selection mechanism, crossover and mutation operators, solutions (i.e., offsprings) with more fitness values are expected to be generated from the initial population of parents during successive generations. These generations continue until the algorithm finds an acceptable good solution or meets a terminating condition. Genetic algorithms have been implemented in a wide variety of engineering optimization applications (Gen and Cheng 1997; Man et al. 1999), including cellular manufacturing systems (Shiyas and Pillai 2014; Deljoo et al. 2010; Defersha and Chen 2008; Wu et al. 2007a, b; Kia et al. 2014; Vin and Delchambre 2014).
In this section, a genetic algorithm for solving the integrated layout and scheduling model of a CMS is employed. Principle factors for designing the employed GA are described as follows.
Chromosome structure
In the first section, each layout gene shows an integer number of set \(S_{j} = \left\{ {1,2, \ldots ,TS_{j} } \right\}\) determining the coordinate location of each machine. The definition of set \(S_{j}\) is discussed at below.
For example, in the above sample having a cell with dimension of \(10 \times 10\) m, the total number of CH’s is equal to \(TCH_{1} = 100 \times 2^{2} = 400\).
CH’s and CHG’s are numbered from the leftdown corner to the rightup corner according to the cell numbers. After numbering a cell, the continuous numbering CH’s and CHG’s starts from the next cell and is continued until the last cell.
The scheduling genes in the second section of each chromosome string are orderbased permutation of part operation. In this section, the value in each gene contains information related to part number (i) and part operation (k) that is shown as a decimal number (i.k) without having any mathematical value. The first number represents part number and the second one shows part operation.
In the following, the calculation formulas for defined parameters in chromosome structure are discussed:
The initial population
The next step after determination of chromosome structure is generating an initial population of chromosomes. In this section, the hierarchical procedure for filling the layout genes and scheduling genes of a chromosome is explained separately.

Step 1 The operations of all parts are permuted on machines randomly based on decimal numbers (i.k) in scheduling genes.

Step 2 A machine is chosen randomly among the machines that do the same operation based on Constraint (5). This means the information of the part and the operations that are related to it should appear only once in each chromosome.

Step 1 The value for all CH’s is calculated according to the dimension of existing cells and multiplier \(\upsigma\).

Step 2 One machine j is selected randomly from the unselected machines.

Step 3 CHG’s (i.e., series \(S_{j}\)) are calculated through formulas (44) and (45) for the selected machine j.

Step 4 A CHG from set \(S_{j}\) is selected randomly and is placed in the layout gene.

Step 5 The cell number where machine j is located is obtained through formulas (46) and (47) based on the CHG number or through formulas (48) and (49) based on CH number of CHG selected for machine j.

Step 6 The MH number is obtained through formulas (50) and (51) based on the cell number and the CHG number that the machine j has occupied.

Step 7 CH numbers of selected CHG for the machine j is obtained through formula (52) based on the MH number and cell number.
Calculation procedure of fitness function
In this section, the procedure of calculating objective function value is described. Since there are three different ingredients in the objective function, including makespan, tardiness penalty costs and intercell/intracell movements’ costs, they are calculated in three phases to evaluate the objective function for each chromosome.
In the first phase, for evaluating makespan ingredient of the objective function, it is needed some modifications on the chromosome to make the calculation of makespan possible. This is because by considering the constraints (7) and (8), the calculation of the completion time of each processing position k′ of machine j \((CTM_{{k^{\prime}j}} )\) is possible provided that two conditions:

Step 1 In the scheduling section of each chromosome, the genes for which it would be possible to evaluate their completion time are evaluated by considering two mentioned conditions.

Step 2 For each chromosome string, the first gene that its completion time has not been calculated yet is considered. Then, one of them is chosen randomly.

Step 3 the chromosome strength at was selected in step 2 is considered. Amongt he other genes that their completion time in that chromosome has not gained yet, the gene that has the minimum decimal number is chosen and is replaced with the selected gene in step 2. If there is more than one gene that has the minimum decimal number, one of them is chosen randomly and replaced with the selected gene in step 2.

Step 4 The first, second and third steps are repeated until the completion times of all genes \((CTM_{{k^{\prime}j}} )\) are obtained. Following these steps and considering the constraints (9) and (10), the \((CTP_{i} )\) values and the makespan value \((C_{\hbox{max} } )\) are returned.
In the second phase, by considering the amounts of the layout genes of each chromosome (CHG’s) and formulas (43)–(54), the following values are obtained: the cell number, MH number, coordinates of each machine and the distances between machines.
In the third phase, the calculation of fitness function of each chromosome is gained by considering the objective function and the information obtained from the first and second phases.
Selection
 1.
At first, all individuals of a population are ranked based on their fitness function value (i.e., objective function value) increasingly. An individual with rank n is given value \(1 /\sqrt n\).
 2.
Next, total measure amounts of the whole population are equal with the number of needed parents for the production of the next generation.
 3.
The value \(1 /\sqrt n\) is placed in the interval (0, 1] and correctness coefficient \((\alpha )\) is obtained from the below formula.
By multiplying the value \(1 /\sqrt n\) of each individual bycoefficient \((\alpha )\), a scaling number is given toeach individual which is used by roulette wheel rule for selecting parents
Reproduction operators
Offsprings in each new generation are created using recombination operators (i.e., crossover and mutation) as described below:
Crossover
Crossover operator designed in this algorithm makes two offsprings from two selected parents by considering each chromosome string separately. To implement this operator, three crossover points are selected on each string. Clearly, the first crossover point is between the first and the second gene, and two other crossover points are selected after the first crossover point, randomly. This crossover acts in two phases as described below.

Step 1 Along the length of the two parents, two crossover points are chosen in similar points, randomly. Then, the length of first offspring is calculated by the below formula:where \(\omega\) is the number of genes that are common along the two chosen points of the first parent and the whole chromosome string of the second parent.$$\begin{aligned} & {\text{length}}\;{\text{of}}\;{\text{chromosome}}\;{\text{string}}\;{\text{of}}\;{\text{the}}\;{\text{first}}\;{\text{offspring}} \\ & \quad = {\text{``distance}}\;{\text{between}}\;{\text{two}}\;{\text{random}}\;{\text{points''}}\, + \,{\text{``length}}\;{\text{of}}\;{\text{chromosome}}\;{\text{of}}\;{\text{the}}\;{\text{second}}\;{\text{parent''}}\,  \,\omega \\ \end{aligned}$$

Step 2 The section between two random points of the first parentis copied into the first offspring.

Step 3 Starting from the second crossover point of the second parent, the other unused numbers in an order that they appear are copied in the second parent. If it reaches to the end of the string, it starts from the beginning and copies the remaining genes in order.

Step 4 In this step, the correction of the offsprings is made. The random choice of a machine among the machines that process the same operations based on constraint (5) means each part information and its related operations (i,k) should appear only once in each chromosome.

Step 5 The second offspring is produced in the same way by reversing the parents’ role.
Mutation
Mutation operator in the layout and scheduling genes of each offspring operates separately by mutation probability \(P_{m}\) and \(P_{{m^{{\prime }} }}\) respectively, as described in the two following phases.
In the first phase, a random resetting mutation operator is done on the layout gene. A random number in the interval \(\left( {0 1} \right]\) is generated and compared with \(P_{m}\). If the random number is smaller than \(P_{m}\), one of thechromosome strings will be selected by chance and an amount of allowable set \(S_{j}\) will be selected in that situation and other amounts of different offspring genes will be changed in an order according to the steps that were explained in “The initial population” section, and new amounts will be replaced.
In the second phase, swap mutation operator is done on the scheduling genes. At first, a random number is produced for each chromosome string. Then, if the random number is smaller than \(P_{{m^{{\prime }} }}\), two positions of genes will be randomly selected and their amounts will be replaced by each other.
Termination criterion
Termination criterion in the proposed algorithm is the number of iterations. Therefore, the production of new generations will be continued until the number of iterations is reached. That value depends on the problem size.
Experimental results
Two illustrative numerical examples
In this section, two experiments are performed to validate the proposed model and compare the effects of the sequential and concurrent approach in the CM design.
To verify the proposed model and reveal that the concurrent integration of parts scheduling with CF and machines layout is more effective than sequential approach, two smallsized examples are solved by a Branch and Bound (B&B) method under Lingo 8.0 software on an Intel(R) core(TM) i5 CPU 2.6 GHz personal computer with 4.00 GB RAM.
In the sequential approach, the proposed model is solved to find the optimal values of decision variables for CF and CL. Then, using the obtained solution the optimal solution of the scheduling problem is determined. To implement this approach, the main model is solved by excluding the components (1.1) and (1.2) associating with makespan and tardiness from the objective function in the first step. Then, in the second step, the main model is solved by regarding the values obtained for decision variables of CF and CL as input parameters and excluding the component (1.3) associating with material handling costs from the objective function.
In the concurrent approach, the proposed model is optimally solved including all three components in the objective function to find the optimal solution of integrating parts scheduling with CF and machines layout simultaneously. Finally, the objective function values (OFV) obtained for the optimal solutions of both approaches are compared.
In these two numerical examples, the data is randomly generated which includes process routing for each part type and processing time for each operation. Machines are selected randomly for each process routing and the processing times of operations are random integer numbers between 2 and 30. The machines should be assigned to two cells. The due date for all part types is 40 min. The material handling time for all part types per distance unit is 3 min. The intercell and intracell material handling costs for all part types per distance unit are 5$ and 2$, respectively. The tardiness penalty for all part types per time unit is 3$. The factory costs per time unit are 25$.
Processing times of part operations for the first example
Machines  P1  P2  P3  P4  

1  2  1  2  1  2  1  2  
M1  18  15  3  11  
M2  24  5  10  6  
M3  5  19  8  20 
Processing times of part operations for the second example
Machine  P1  P2  P3  P4  

1  2  3  1  2  3  1  2  3  1  2  3  
M1  7  5  25  17  
M2  29  14  30  3  
M3  25  28  12  8  
M4  3  19  8  11 
Machines dimensions and coordinates of cells partitions for the first example
Parameter  Machines’ information  Cells’ information  

\(L_{j}\)  \(W_{j}\)  \(LX_{c}\)  \(UX_{c}\)  \(LY_{c}\)  \(\varvec{UY}_{\varvec{c}}\)  
M1  4  2  C1  3  8  3  13 
M2  2  3  C2  11  17  5  11 
M3  3  5 
Machines dimensions and coordinates of cells partitions for the second example
Parameter  Machines’ information  Cells’ information  

\(L_{j}\)  \(W_{j}\)  \(LX_{c}\)  \(UX_{c}\)  \(LY_{c}\)  \(\varvec{UY}_{\varvec{c}}\)  
M1  4  2  C1  2  7  2  10 
M2  2  3  C2  10  15  6  14 
M3  5  3  
M4  2  2 
Now, the solution obtained for the first example is discussed. This example consists of four part types and three machine types. Each part type requires two operations that each one can be processed by only one machine type selected from two alternative machine types with different processing times. For instance, there are two process routings for part type 1, the first operation of part type 1 can be processed either on machine type 1 or machine type 2 while the second one can be done only on machine type 3. Also, there are four, one and two process routings for part types 2, 3 and 4, respectively.
Objective function value and its cost components for the first example
Cost  Value  

Sequential approach  Concurrent approach  Improvement by concurrent approach  
Factory costs × C _{max}  25 × 48.5 = 1212.5  25 × 41.5 = 1037.5  175 (16.7 %) 
Tardiness penalty  25.5  6  19.5 (325 %) 
Material handling  54.5  91.5  −37 (67.9 %) 
Total (OFV)  1292.5  1135  157.5 (13.9 %) 
Since in the sequential approach, the objective function is optimized without components related to makespan and tardiness, the model is able to find the optimal value for material handling cost equal to 54.5. However, when the components related to makespan and tardiness are included in the objective function in the concurrent approach, the previous optimal material handling cost increase due to the effect of scheduling parts on machines layout. Switching from sequential approach to concurrent approach reduces the value of makespan from 48.5 to 41.4 and the value of tardiness penalty from 25.5 to 6, which are remarkable improvements for these components. On the other hand, incorporating parts scheduling in the concurrent approach increases the value of material handling cost from 54.5 to 91.5. Totally, OFV is improved about 14 % by switching from sequential approach to concurrent approach. This achievement was expectative since simultaneous decisions making about interrelated decisions machines layout and parts scheduling brings the capability for the model to optimize all components of the objective function as an optimal strategy in designing a CMS.
Next, the second example consists of four part types and three machine types. Each part type requires three operations that each one can be performed by at most two machine types. There are two process routings for each part type.
Objective function value and its cost components for the second example
Cost  Value  

Sequential approach  Concurrent approach  Improvement by concurrent approach  
Factory costs × C _{max}  25 × 98 = 2450  25 × 90 = 2250  200 (8.9 %) 
Tardiness penalty  505.5  472.5  33 (7 %) 
Material handling  176.5  188.5  −12 (6.8 %) 
Total (OFV)  3132  2911  221 (7.6 %) 
A similar improvement in the obtained OFV is observed for the second example in Table 7 as it was expected due to the advantage of concurrent approach.
As can be seen in Figs. 12 and 13, when makespan and tardiness components are included in the objective function in the concurrent approach, cell formation, machines layout, and parts scheduling change completely in comparison with sequential approach. It can be concluded that even if factory cost is very high forcing to complete all parts as soon as possible, one should group machines, locate them and schedule operations simultaneously.
To conclude, by comparing the OFVs obtained for examples 1 and 2 in the sequential and concurrent approaches, it is recognized that if cells are configured, machines are located and operations are scheduled sequentially, the optimum strategy with the minimum costs cannot be reached.
Evaluation of the proposed GA
Comparison of GA solutions and exact solutions
Problem no.  No. of machines  No. of parts  No. of operations per part  No. of processing positions per machine  No. of cells  Sequential approach  Solution GAP  

B&B  GA  
OFV  Time (s)  Best OFV  Average OFV  Time (s)  
1  3  4  2  4  2  1292.5*  16  1292.5  1292.5  1.6  – 
2  4  4  3  3  2  3474.5*  276  3474.5  3474.5  3.9  – 
3  4  4  3  4  2  3132*  20,350  3132  3132  14.6  – 
4  4  5  3  4  2  3688.5*  38,405  3688.5  3700.5  15.4  – 
5  5  6  3  4  2  5010.5  43,200  4798  4851.7  18.5  −3.3 
6  6  8  3  4  2  6490.7  43,200  6206.5  6456.5  23.3  −4.6 
7  8  10  4  5  3  13,728.6  43,200  12,802  13,588.4  33.8  −7.2 
8  10  12  5  6  3  22,347.4  43,200  20,997  21,190.8  42.4  −6.4 
9  15  25  3  5  4  NA  43,200  30,677  30,152.7  61.1  NA 
10  20  40  6  6  NA  43,200  86,110  77,090.3  102.5  NA 
Problem no.  No. of machines  No. of parts  No. of operations per part  No. of processing positions per machine  No. of cells  Concurrent approach  Solution GAP  OFV Improvement B&B (%)  OFV Improvement GA (%)  

B&B  GA  
OFV  Time (s)  Best OFV  Average OFV  Time (s)  
1  3  4  2  4  2  1135*  185  1135  1112.5  2.9  –  13.9  13.9 
2  4  4  3  3  2  3089*  1203  3089  3089  7.6  –  12.5  12.5 
3  4  4  3  4  2  2911*  29,900  2911  2911  18.2  –  7.6  7.6 
4  4  5  3  4  2  3557.3  43,200  3520  3691.7  19.2  −1.1  3.7  4.8 
5  5  6  3  4  2  4556.7  43,200  4323.5  4482.7  23.3  −5.4  10  11 
6  6  8  3  4  2  5764.2  43,200  5486  5543.9  31.4  −5.1  12.6  13.1 
7  8  10  4  5  3  10,582.5  43,200  9957.5  10,233.8  56.7  −6.3  29.7  28.6 
8  10  12  5  6  3  18,632.4  43,200  17,453  18,255.6  68.47  −6.8  19.9  20.3 
9  15  25  3  5  4  NA  43,200  23,543  24,040.3  108.3  NA  NA  30.3 
10  20  40  6  6  NA  43,200  67,872  68,253.6  221.1  NA  NA  26.9 
As it can be seen from Table 8, GA has found optimal solutions for problems 1–4 for sequential approach and optimal solutions for problems 1–3 for the concurrent approach in which optimal solutions are obtained by B&B method. No feasible solution is found by B&B for problems 9 and 10 in both approaches due to the complexity of the model. Furthermore, for the rest of problems in both approaches, the best solutions found by GA are better than the solutions found by B&B in much less time. On average, the comparisons between the OFVs obtained by sequential and concurrent approaches indicate that the OFV improvement is around 17 % by GA and 14 % by B&B.
These promising results obtained by the proposed GA prove the efficiency of the designed algorithm enhanced by the matrixbased chromosome structure. Furthermore, the developed GA is elaborately designed to create feasible solutions by using some defined formulas, efficient crossover and mutation operators and established procedures calculating fitness function and generating initial population.
Conclusion
In this article, a novel integrated mathematical model has been formulated for designing a cellular manufacturing system considering three problems simultaneously: cell formation, intracell layout, and parts scheduling. The results show that considering these three significant decisions in a simultaneous manner contributes to a successful CM implementation in the manufacturing environment. All these problems have been optimized due to three components in the objective function including makespan time, tardiness penalty, and intercell and intracell material handling cost. By investigating two integration approaches, namely sequential and concurrent, it was revealed that to reach an optimal solution, all stages of CMS (CF, machine layout, and part scheduling) must be designed simultaneously. It In sequential approach, since cells are configured at first in order to decrease intercell and intracell movements costs, and finally operations are scheduled in order to optimize time factors in the objective function consisting of tardiness and makespan, the global optimal solution is not reached, although it is attainable in a concurrent approach. The advantages of the proposed model were as follows: designing layout of unequalarea machines in cells with continuous space, introducing alternative process routings for parts with different operation sequence, exerting the effects of distances between locations or cells on part scheduling, becoming both cost and time of part movements dependent on (1) traveled distance, (2) part type and (3) movement type (intercell and intracell), computing the completion time of each part by considering: (1) movement times, (2) processing times and (3) waiting times, and finally, determining three interrelated designing stages (i.e., CF, machines layout, and parts scheduling) in designing a CMS simultaneously. For transforming mixedinteger nonlinear programming formulation into a mixedinteger linear counterpart, some linearization techniques have been proposed. In order to verify the performance of the proposed model, two numerical examples have been solved. It was revealed that concurrent approach surpasses sequential approach in improving the quality of the obtained solutions in the CM system design.
Because of the complexity of the proposed model, Lingo software cannot obtain the optimum solution for medium or largesized problems. Hence, a genetic algorithm has been developed that its excellent advantages were as follows: determining the exact location of machines in continuousarea cells; computing exact completion time of each operation of each part type, proposing heuristic crossover and mutation operator. The obtained results show that the best solutions found by GA are better than those found by Lingo in much less time especially as the size of problem increases. Incorporating other features, such as uncertainty in part demands, machine time capacity and cost coefficients, integrating with reliability and labor issues and considering dynamic issues will be left to future research.
Declarations
Authors’ contributions
AE carried out the literature review studies, designed the mathematical model and the solution approach, and drafted the manuscript. RK participated in the design of the mathematical model and performed the numerical experiments. ARK conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
Acknowledgements
This work was financially supported by Islamic Azad University, Firoozkooh Branch. The authors would like to sincerely thank the anonymous referees for their thorough reviews of the early versions of this paper and their valuable comments.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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