Abstract
Over the last four decades of research, numerous cell formation algorithms have been developed and tested, still this research remains of interest to this day. Appropriate manufacturing cells formation is the first step in designing a cellular manufacturing system. In cellular manufacturing, consideration to manufacturing flexibility and productionrelated data is vital for cell formation. The consideration to this realistic data makes cell formation problem very complex and tedious. It leads to the invention and implementation of highly advanced and complex cell formation methods. In this paper an effort has been made to develop a simple and easy to understand/implement manufacturing cell formation heuristic procedure with considerations to the number of production and manufacturing flexibilityrelated parameters. The heuristic minimizes intercellular movement cost/time. Further, the proposed heuristic is modified for the application of principal component analysis and Taguchi’s method. Numerical example is explained to illustrate the approach. A refinement in the results is observed with adoption of principal component analysis and Taguchi’s method.
Introduction
In the present era, cutthroat competition, fluctuating demands, customization of product, very high initial investment and ever increasing manpower cost, are severely affecting the profit margins of manufacturing industry. The concept of cellular manufacturing is placed at high level on the agenda of manufacturing industries, not only to overcome but to excel in this situation. Cellular manufacturing is a wellmixed blend of manufacturing flexibility and production efficiency. It has the ability to deal with frequent changes in product mix and fluctuations in production volume. Due to its superior performance, it is considered as a feasible approach to realise mass customization philosophy (Lian et al. 2013). Cell formation (CF), group layout (GL) and group scheduling (GS) are the three major steps in cellular manufacturing (Fardis et al. 2013; Kia et al. 2013). Amongst these, CF is the foremost (Doulabi et al. 2009; Kumar and Sharma 2014) and key step (Krushinsky and Goldengorin 2012) in any cellular manufacturing problem. Cell formation deals with the identification of the part families with similar process requirements and allocating them to the machine cells for processing (Boutsinas 2013; Fardis et al. 2013; Kumar and Sharma 2014; Sarker 1996). Ideally manufacturing cell is to be formed in such a fashion that each manufacturing cell should act as an independent manufacturing unit. The essence of CF approaches is to eliminate/minimize the intercellular movement cost of parts (Arkat and Farahani 2012; Kumar and Sharma 2014; Lian et al. 2013; Selim et al. 1998). It can simply be achieved by duplicating the machines but duplication of machine involves large capital investment which ultimately adds to the product cost. Therefore, a manufacturing CF approach should provide an optimisation amongst these, without much complexity in approach. The cell formation approaches developed so far can be categorised as (Boutsinas 2013; Lian et al. 2013; Papaioannou and Wilson 2010; Kumar and Sharma 2014; Yasuda et al. 2005; Yin and Yasuda 2006) (i) Similarity coefficientbased methods (ii) Mathematical programmingbased methods (iii) Artificial intelligencebased approaches (iv) Heuristics/metaheuristics/hybrid metaheuristics, (v) Any combination of these. Amongst CF techniques similarity coefficientbased methods are more flexible and easy to implement (Yin and Yasuda 2006). A large number of cell CF approaches have been developed so far, majority of them do not consider productionrelated data (Boutsinas 2013; Won and Lee 2001). Susanto et al. (2009) revealed that 80 % of manufacturing CF approaches are focussed on the arrangement of binary partmachine incidence matrix, whereas more realistic and effective approaches could be developed by considering the various manufacturing flexibility and productionrelated data (Kumar and Sharma 2014). A little work is observed on CF considering any such data in a simple manner and some of them are summarised in Table 1, whilst on the other hand a large number of literature is available on binary matrixbased cell formation techniques. A statistical review of literature can be found in Reisman et al. (1997). The focus of researchers is shifting towards the development of metaheuristic techniques of CF. Study of metaheuristic techniques of CF can be found in Nourie et al. (2013) and Saeedi et al. (2010). The literature reflects the need for efforts to incorporate production and manufacturing flexibilityrelated data (realistic data) in CF procedures in a simple manner. Thus, to abridge this gap, in present study effort has been made to develop a simple CF heuristic approach with considerations to manufacturing flexibility and productionrelated parameters, namely production volume, operation sequence, intercell movement cost/time, alternate process plans (routing flexibility), identical machines and operation sequence for a part (operation flexibility). Further considerations to machine capacity and machine reliability are also given. In proposed heuristic, considerations to intercell movement time/cost has given precedence over part processing cost/time as part processing cost does not much affect the intercellular movement. The proposed heuristic approach cannot be limited to any particular similarity coefficientbased or other clustering approach. It can work well with any clustering approach with some modifications. Its ability to adopt modern statistical tools like principal component analysis (PCA), and Taguchi’s method, with little modifications is demonstrated.
The outline of rest of the paper is as follows: "Methodology" explains the methodology and proposed heuristic for solving the CF problem. "Implementation and illustration" illustrates implementation of proposed heuristic through a numerical problem. In discussion and analysis part i.e. "Discussion and analysis", results of clustering algorithm used, are compared with some wellknown CF algorithms. Further in this section, proposed algorithm is modified for the implementation of PCA and Taguchi’s method, whilst conclusions are drawn in "Conclusions".
Methodology
The proposed heuristic is a development in the work of Kumar and Sharma (2014). The simple logic used for consideration to operation sequence is that a machine could add maximum one intercell move per part if it is either at starting or at ending position of the operation sequence of a particular part, otherwise it could add maximum two intercell moves (Won and Lee 2001). Taking inspiration of Leem and Chen (1996) the concept of partoperation incidence (POI) matrix is used in proposed CF procedure. Albadawi et al. (2005), Hachicha et al. (2006, 2008a) highlighted the application of PCA in solving CF problems. Hachicha et al. (2008b) used Taguchi’s method along with PCA in route selection of CF problems. Applications of PCA and Taguchi’s method are introduced in modification of proposed heuristic.
The methodology adopted is discussed under four subtitles namely proposed CF heuristic procedure, commonality score/similarity coefficientbased clustering algorithm, PCA and Taguchi’s method.
Proposed CF heuristic procedure
The proposed CF heuristic procedure is explained with the help of a selfexplanatory flow chart presented in Fig. 1. The essence of proposed heuristic procedure is to minimize the intercellular movement time or cost
Commonality scorebased clustering algorithm
Though any similarity scorebased clustering algorithm can be used with the proposed procedure, the commonality scorebased clustering algorithm used is discussed in illustration in this section. The commonality score is used to deduce similarity matrix amongst all possible machine pairs.
Step 1: Compute the similarity coefficient amongst all possible machine pairs from the data matrix by using a variant of Jaccard’s similarity coefficient, proposed by Kumar and Sharma (2014), is reproduced in Eq. 1.
where a is the sum of elements common to both machines in concerned machine pair (in this case, maximum possible intercell movement cost of parts visiting both machines) \( a = \mathop \sum \nolimits_{k = 1}^{k = n} a_{{({\text{ij}})k}} \), a _{(ij)k } is the elements common to both machines M _{i} and M _{j}, for k = 1 to k = n parts. b is the sum of values of elements concerned to only first machine in pair (in this case, maximum possible intercell movement cost of parts visiting only first machine) \( b = \mathop \sum \nolimits_{k = 1}^{k = n} b_{{({\text{i}})k}} \), b _{(i)k } is the elements concerned to machine M _{i} but not machine M _{j}, for k = 1 to k = n parts. c is the sum of values of elements concerned to only second machine in pair (in other terms, maximum possible intercell movement cost of parts visiting only second machine) \( c = \mathop \sum \nolimits_{k = 1}^{k = n} c_{{({\text{j}})k}} \), c _{(j)k } is the elements concerned to machine M _{j} but not machine M _{i}, for k = 1 to k = n parts.
Step 2: Group machine pair having highest value of commonality score, and transform this machine pair into a machine unit M _{r} having elements M_{(i,j)r} (Kumar and Sharma 2014).
where M _{(i, j)r is the} corresponding elements of machine unit M _{r}, obtained after transformation of machine M _{i} and machine M _{j} into a single machine unit, a _{i(r)} and a _{j(r) is the} corresponding elements of machine M _{i} and M _{j}, respectively.
Taguchi’s method
This is a powerful statistical method for improving the performance of the design, process and product by optimizing process parameters. It looks for a mean performance characteristic value close to target value rather than a value within a certain range (Eşme 2009). It is made up of following three design procedures:

System design: to find the suitable working levels of the design factors.

Parameter design: to determine the factor levels for the optimum performance of the product or process.

Tolerance design: to refine the results of parameter design by narrowing the tolerance levels of factors that have significant effects on the product or process under study.
Special design of orthogonal arrays (OAs) are utilized in parameter design for minimising the time, cost and number of experiments. Analyses of variance (ANOVA) and the signaltonoise (S/N) ratio are used to analyse the experimental data and find the optimal parameter combination. A good amount of literature (Eşme 2009; Ghosh and Dan 2011; Hachicha et al. 2008b; Hadighi et al. 2013; Kamaruddin et al. 2004; Seenivasan et al. 2014; Unal and Dean 1991, and others) is available on Taguchi’s method and its implementation. The procedure includes the following three steps:

1.
Planning experiment

Determination of the control factors, noise factors and quality or performance measure responses of the product or process.

Determination of the levels of each factor.

Selection of a most suitable OA table. It depends on the number of factors and interactions, and the number of levels for the factors.


2.
Implementing experiment.

3.
Analysing and examining result.

Determination of the parameters signification (ANOVA).

Conduct a main effect plot analysis to determine the optimal level of the control factors.

Execute a factor contribution rate analysis.

Confirm experiment and plan future application.

The intended use of Taguchi’s method is in the selection of best part routings for each part type. The essence of proposed CF procedure is to minimize the intercell movement time/cost, which cannot be used as a response measure for Taguchi’s method due to large dimensions of variables. To overcome this situation, PCA, a dimensionreduction technique is employed.
Principal component analysis
Principal component analysis is the most widely used dimensionreduction statistical technique. It investigates the largely widespread data in many areas of science and industry. It provides a condensed description (Hachicha et al. 2008a; Kumar and Jain 2010), in order to model the total variance of the original data set, through new uncorrelated variables known as principal components. These components recover as much variability in the data as possible and account for near total variance of the data. Principal component analysis is recommended for large sample sizes (Gupta et al. 2012; Hachicha et al. 2008a; Mehrjoo and Bashiri 2013).The usual progression of PCA starts with the eigenvalues and eigenvector of semidefinite matrix. A brief description on implementation of PCA is as follows:
Let, the initial matrix (A) be a semidefinite matrix, in which rows and columns stand for part (P) and machines (M) respectively, having the information like partmachine incidence, operation sequence, production volume and intercell movement time/cost. Since CF problem is a dimensionreduction problem in which a number of interrelated machines and parts are to be grouped into a smaller set of independent cells, the application of principal components analysis can give a very good solution as mentioned by (Albadawi et al. 2005; Gupta et al. 2012; Hachicha et al. 2006, 2008b) quickly.
The eigenvector equation where the terms λ _{1} ≥ λ _{2} ≥…≥ λ _{n} are the nonnegative real roots of the determinant polynomial equation of degree m, given as:
B is the standardization matrix of the initial matrix A, B ^{T} is the transpose matrix of matrix B
To obtain a model with the first and the second principal components the principal components analysis was performed on the mean centred data, this model explained the recovered cumulated percentage (CP) of the variance in the data as follows:
For more details of PCA method, and its application in cell formation relevant literature such as (Albadawi et al. 2005; Chattopadhyay et al. 2012; Gupta et al. 2012; Hachicha et al. 2006, 2008a, b; Llin et al. 2010; Kumar and Jain 2010; Mehrjoo and Bashiri 2013; Min et al. 2014) and others can be referred.
In this work, PCA finds its application in two ways, first in reduction of variability for implementation of Taguchi’s method in selection of alternate route, second in the making of operation and part families through graphical analysis.
Implementation and illustration
The proposed heuristic CF procedure is implemented on an arbitrarily designed CF problem illustrated in "Illustrative problem".
Illustrative problem
For illustration purpose, a cell formation problem of five operation and five different parts with random data has been developed and given in Table 2. Operation ‘O1’ can be performed either on machine ‘M1’ or ‘M2’. Reliability of machine ‘M1’ and ‘M2’ is 0.9 and 0.8, respectively. The operation time (processing time of any part for operation ‘O1’) is 10 units. Part ‘P1’ can be made by two alternate process plans either through ‘O1 → O3 → O2 → O5' or ‘O2 → O4 → O1'. Production volume required for part ‘P1’ is 100 units. Per move intercell movement time for part ‘P1’ is 1 unit. In the operation sequence (alternate process plans) of part ‘P1’ the order of operation ‘O1’ and operation ‘O3’ can be interchanged. Rest of the data can be explained in similar manner.
Step 1: It is dedicated to the deduction of POI matrix from the problem data. If a particular operation is required for procession of a particular part by a particular process plan, put ‘1’, otherwise ‘0’ in corresponding cell of POI matrix. Thus obtained POI matrix with alternative process plans is presented in Table 3.
Step 2: For consideration of production volume, multiply the elements of POI matrix (Table 3) with the corresponding production volume. Thus, modified POI matrix is given in Table 4.
Step 3: Operation sequence is considered by introducing maximum intercell moves that could be generated by a machine according to its position in operation sequence. For these elements of matrix in Table 4, are multiplied by the maximum intercell moves that can be generated by corresponding operation as per its position in operation sequence of particular part. Thus, modified matrix accounts for production volume and operation sequence. This matrix is called maximum possible intercell moves matrix and presented in Table 5.
Step 4: For consideration of intercell movement time, multiply each element of the matrix presented in Table 5 with corresponding per part per move intercell movement time. Thus the obtained maximum possible intercell movement time matrix is presented in Table 6. As explained in Fig. 1, the intercell movement cost may be considered on the same line as intercell movement time.
Step 5: For formation of operation cells, any clustering algorithm is to be applied in the matrix given in Table 6. In this case, the similaritybased algorithm discussed in “Commonality scorebased clustering algorithm” is applied. The matrix after application of clustering algorithm is given in Table 7. In this case operation ‘O1’ and ‘O2’ are clubbed in an operation cell whilst operation ‘O3’, ‘O4’ and ‘O5’ are clubbed into another operation cell. Minimum value of maximum possible intercellular movement time is calculated for each process plan. The operation cell in which the part is producing this minimum value of maximum possible intercell movement is also noted in the last column of Table 7.
Step 6: The use of interchangeability of operation i.e. operation flexibility for a particular part is made here. After its consideration, minimum value of maximum possible intercellular movement time is calculated for each part. Further, for this minimum value the operation cell to which the corresponding part should be assigned is also noted and presented in Table 8.
Step 7: At this stage selection of process plan and operation cell of a particular part is made by the comparison of Tables 7 and 8. The conditions for selection of process plan and operation cell are only one process plan for a particular part, and minimum intercellular movement time. Process plans 2, 3 and 8 are selected for parts ‘P1’, ‘P2’ and ‘P5’, respectively. Part ‘P3’ and ‘P4’ each has only one process plan. The selected process plan and corresponding operation cells are highlighted in Table 9.
At this stage, total intercell moves and intercell movement cost for required production volume are 1,210 and 585 units, respectively.
Step 8: Assignment of machines to operation cells on the basis of their production capacity and reliability is to be made at this stage. If processing time of a part is 10 min on a reliable machine, it would be 10/0.8 = 12.5 min for 80 % reliable machine. For this illustrative problem, 8 h production capacity after reliability considerations for each machine ‘M1’, ‘M2’, ‘M3’, ‘M4’, ‘M5’, ‘M6’, ‘M7’, ‘M8’ is found to be 129.6, 115.2, 86.4, 96 and 81.6 parts, respectively. Further, it is observed that operation ‘O5’ can be performed on two machines ‘M7’ and ‘M8’ with 8 h production capacity of 96 and 81.6 parts, respectively. It is observed that the processing requirement of operation ‘O5’ for parts assigned to its operation cell (operation cell 2) is 75 whilst for parts assigned to another operation cell is 120. It is advisable to assign machine ‘M8’ and ‘M7’ to ‘operation cell 2’ and ‘operation cell 1’ respectively, based on their production capacity and processing requirement. Final manufacturing cell is shown in Table 10.
Thus, total intercell moves and intercell movement cost for required production volume are 300 and 570 units, respectively.
Discussion and analysis
The discussion and analysis is performed in two subsections. In first subsection results from clustering algorithm used, are compared with the results of some wellestablished binary matrixbased CF methods. In second subsection, the modified proposed CF procedure for adoption PCA and Taguchi’s method is presented.
Comparison of results of clustering algorithm used
Results from clustering algorithm used, are compared with the results of some wellestablished binary matrixbased CF methods. These methods were compared and found better than several other methods in the studies made by their respective authors. The comparison of results is summarized in Table 11. The comparison establishes that clustering algorithm used in proposed procedure, is comparable to other contemporary algorithms. The abbreviations used in Table 11 are explained as EE → number of exceptional elements: The number of machinepart cells that remain outside the diagonal blocks. These offdiagonal nonzero entries are called exceptional elements (Albadawi et al. 2005; Murugan and Selladurai 2011). GE → grouping efficiency: It incorporates both machine utilization and intercell movement. It is defined as the sum of two weighted functions, one for machine utilization and other for intercell movement with a weight factor; recommendation is in favour of equal weight for both functions (Chandrasekharan and Rajagopalan 1986; Nair and Narendran 1998; Sarker 1996; Wang 2003). GEF → grouping efficacy, GEF = \( \frac{{N_{1}  N_{1}^{out} }}{{N1  N_{0}^{in} }}, \) where, N _{1} is the Total number of 1 s in the clustered machinepart incidence matrix, N ^{out}_{1} → Total number of 1 s outside the diagonal block in clustered matrix, N ^{in}_{0} is the Total number of 0 s inside the diagonal block of clustered matrix (Albadawi et al. 2005; Nair and Narendran 1998; Sarker 1996; Wang 2003). GI → Grouping Capability Index: GI = 1 − \( \frac{e}{o}, \) where, e → number of exceptional elements, o → total number of 1 s (i.e. number of operations) in the matrix (Yin and Yasuda 2006). GM → grouping Measure: It is the difference between the measure of utilization of machines and measure of intercell movement i.e. {(ratio of the number of 1 s to the number of total elements in the diagonal block) − (ratio of the number exceptional elements to the total number of 1 s in the matrix)} (Miltenburg and Zhang 1991; Yin and Yasuda 2006).
Modified proposed heuristic
Few modifications are incorporated in proposed heuristic for the application of PCA and Taguchi’s method. The modified CF heuristic is presented in a selfexplanatory flow chart shown in Fig. 2.
Route selection through PCA, and Taguchi’s method
In the modified proposed methodology, route selection is performed on the basis of data available in maximum possible intercell movement time matrix (Table 6). In this section route/process plan selection is made on the application of PCA ad Taguchi’s method and rest of the work is done as per the method explained in "Proposed CF heuristic procedure" and illustration in "Implementation and illustration". Part ‘P1’, ‘P2’ and ‘P5’ each has two possible process plans, whilst parts ‘P3’ and ‘P4’ have no choice in process plans. For application of Taguchi’s method, only parts ‘P1’, ‘P2’ and ‘P5’ are considered with two level each namely ‘1’ and ‘2’ for ‘P1’, ‘3’ and ‘4’ for ‘P2’, ‘7’ and ‘8’ for ‘P8’. For application and analysis of Taguchi’s method a software package “MINITAB 16” is used. Value of cumulative percentage of variance is calculated through PCA, with the help of “MATLAB 13a. The OA and CP measure is presented in Table 12. Cumulated percentage measure is calculated separately for each serial number by making a matrix having routes according to Table 12, and data of Table 6. The optimum route (level) for each part (factor) is the level having the highest value of CP measure in the experimental region. Based on the main effect plot, shown in Fig. 3, the optimal route of each part is 1, 4, 8 for parts P1, P2, P5. respectively. The route selection through Taguchi’s method is also verified by interactions plot between CP measure and process plan drawn through ANOVA analysis using “MINITAB 16” and presented in Fig. 4.
After route selection though PCA and Taguchi’s method, the maximum possible intercell movement time matrix of Table 6 is reduced to maximum possible intercell movement time matrix after route selection presented in Table 13. The clustered matrix before machine assignment and operation flexibility, and final clustered matrix is shown in Tables 14 and 15, respectively.
Thus, total intercell moves and intercell movement cost for required production volume are 285 and 640 units, respectively.
In comparing the two solutions of same illustrative problem presented in Tables 10 and 15 (route selection through PCA and Taguchi’s method), it is observed that the implementation of PCA and Taguchi’s method only in route selection decreases the total intercell moves whilst a slight increase in total intercell movement cost is also there.
Clustering of operations with PCAbased graphical analysis
In this section the objective is clustering of operations into groups. After route selection in "Route selection through PCA, and Taguchi’s method", clustering of operations are performed on maximum possible intercell movement time matrix after route selection (Table 13) by the application of PCA in the lines of Hachicha et al. (2008a) with the help commercially available package XLSTAT 2014. Any other statistical package like SPSS, SPLUS, SAS, SPAD, etc. may also be used. Two principal components are sufficient to analyse correlation between elements (operations and parts).The use of graphical analysis is based on a twodimensional scatter plot (Fig. 5) where each machine is represented by a line from the origin and the axis are two eigenvalues contributing maximum amount of variance. Following four principal situations for the classification of machines are possible:

Two neighbouring operations with a small angle distance measure → Operations belong to the same cell. (‘O4’ and ‘O5’, ‘O1’ and ‘O2’ in Fig. 5).

Two operations with angle distance measurement between them is almost 180°. → Operations may not belong to the same cell.

Two operations for which the angle distance measurement between them is almost 90°. → Operations are independent and do not belong to the same cell (‘O2’ and O3 in Fig. 5).

If none of the above three cases are verified, the operation is not affected to any cell. → An exceptional operation. Since the objective is to group operations with minimum angle distance, Operation Oi, which has the smallest angle distance with Ok, is assigned to the operation group containing O_{i} and O_{ k }.
For illustrative problem two operation cells are identified having facility for operation ‘O1’, ‘O2’ and ‘O3’, ‘O4’, ‘O5’. The final clustered matrix would be same as Table 15. Further on the similar lines of Hachicha et al. (2008a) part may also be assigned through PCA.
Conclusions
In the availability of large number of highly complex and sophisticated manufacturing cell formation heuristics, the paper successfully proposed a simple and easy way to implement heuristic procedure having the ability to handle a large number of production and flexibilityrelated parameters namely production volume, operation sequence, intercell movement cost/time, alternate process plans (routing flexibility), identical machines and operation sequence for a part (operation flexibility) with considerations to machine capacity and machine reliability. The proposed procedure is producing good results compared to highly sophisticated methods. Further, the modified proposed CF procedure shows its ability to go with modern statistical tools like PCA and Taguchi’s method. Following salient features may be observed in proposed CF procedure:

It is computationally very simple and conceptually easy to understand.

It has the ability to consider a number of production and manufacturing flexibilityrelated data.

The relationship between the machines are found on the basis of commonality score.

The proposed CF procedure can also be implemented with any other compatible clustering algorithm.

It can be used for both cases, binary and nonbinary.

It minimizes the intercellular movement cost/time

It is adaptable for more sophisticated techniques like PCA, Taguchi’s method and others

Use of modern statistical and computational tools extend the applicability of proposed heuristic from mid to large size flexible manufacturing system.
Scope for further work, one hand lies in considerations to other/more production and manufacturing flexibility parameters like decisions on number of manufacturing cells and size, work imbalance, machine flexibility, etc. for the development of more realistic, efficient and effective simple CF procedure; on the other hand, in the development of simple procedures for simultaneous assignments of machine groups and part families.
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Acknowledgments
The authors are indebted to the unknown reviewers for their critical review and the pointing suggestions that enabled us to bring out the present form of the work. We are, indeed, thankful to them.
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Kumar, S., Sharma, R.K. Development of a cell formation heuristic by considering realistic data using principal component analysis and Taguchi’s method. J Ind Eng Int 11, 87–100 (2015). https://doi.org/10.1007/s4009201400933
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Keywords
 Cellular manufacturing
 Cell formation
 Manufacturing flexibility
 Production data
 Principal component analysis
 Taguchi’s method