Development of a cell formation heuristic by considering realistic data using principal component analysis and Taguchi’s method
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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.
Keywords
Cellular manufacturing Cell formation Manufacturing flexibility Production data Principal component analysis Taguchi’s methodIntroduction
Summary of work observed on cell formation using production or manufacturing flexibilityrelated data
Author and year  Parameter considered  Approach/remarks 

Kumar and Sharma (2014)  Operation sequence, production volume, intercell movement cost, part processing cost, alternate process plans  Proposed similarity coefficientbased heuristic 
Lian et al. (2013)  Multiple identical machines, processing time, setup time, machine capacity, production volume, cell size, alternative routes  Proposed genetic algorithm (GA)based procedure 
Gupta et al. (2012)  Operation sequence  Similarity coefficient, principal component analysis (PCA), Kmeans algorithm 
Kumar and Jain (2010)  Operation sequence, operation time, production volume, machine capacity  Proposed a PCAbased concurrent algorithm “APMOSTVC” 
Ahi et al. (2009)  Operational time, operation sequence  TOPSIS and SAW 
Pandian and Mahapatra (2009)  Operation sequence, operation time  Adaptive resonance theory, neural network 
Paydar and Sahebjamnia (2009)  Operation sequence  Proposed a linear mathematical programming model 
Susanto et al. (2009)  Sequence of operations, partvolume, alternative routes  Cmeans clustering algorithm, Hungarian (assignment) algorithm, linear programming model 
Garbie et al. (2008)  Alternative routings, processing time, machine capacity (reliability), machine capability (flexibility), production volume, part demand, number of operations done on each machine  Proposed similarity coefficientbased heuristic 
Muruganandam et al. (2008)  Demand of parts in different period, routing sequences, processing time, machine capacities  Proposed a GA based heuristic “PRABHA” 
Kumar and Jain (2008)  Operation sequence, operation time, production volume, intercellular travel loss  Proposed an algorithm “APOSTVUIT” based on average void values and PCA 
Masmoudi et al. (2008)  Alternative routes  Combined axiomatic design principles with experimental design technique, and PCA 
Kim et al. (2004)  Machine sequence of part routes, alternative routes, machine work load imbalance  Proposed a two phased heuristic algorithmbased on dissimilarity measure 
Mahesh and Srinivasan (2002)  Processing time, alternative routes  Branch and bound technique, a heuristic based multistage programming approach 
Mukattash et al. (2002)  Multiple parallel machines, processing time, alternative routes  Proposed three heuristics 
Won and Lee (2001)  Operation sequence and production volume, with extension for intercell material handling cost and processing times  Mathematical model that seeks to minimize the actual intercell flows 
Sofianopoulou (1999)  Operation sequence  Proposed simulated annealingbased algorithm 
Wicks and Reasor (1999)  Operation sequence, production volume  Proposed genetic algorithmbased procedure 
Nair and Narendran 1998  Operation sequence  Proposed measures of similarity and performance are incorporated in a nonhierarchical clustering algorithm: “CASE” 
Beaulieu et al. (1997)  Machines and material handling costs, machine utilization, alternative routeings, Intercell movement  Presented algorithm has two main resolution phases: formation of independent cells then introduction of intercell movements 
Beaulieu et al. (1993)  Work load balance, machine flexibility, routing flexibility  Developed heuristic 
Ahmed et al. (1991)  Production volume, material handling cost  Proposed heuristic for minimization of total material handling cost 
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
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.
Taguchi’s method

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.
 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.
B is the standardization matrix of the initial matrix A, B ^{T} is the transpose matrix of matrix B
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
Initial data for cell formation illustrative problem
Operation  Machine  Reliability  Operation time  Part  Alternate process plans (routings)  Interchangeability of operation  Production volume  Intercell movement time 

O1  M1  0.9  10  P1  O1 → O3 → O2 → O5  O1 ↔ O3  100  1 
M2  0.8  10  O2 → O4 → O1  O4 ↔ O2  
O2  M3  0.9  15  P2  O1 → O2 → O5  Nil  50  4 
M4  1.0  15  O3 → O1 → O2 → O4  O3 ↔ O1  
O3  M5  0.7  25  P3  O4 → O2 → O1 → O5  O2 ↔ O1  70  3 
O4  M6  0.9  20  P4  O4 → O3 → O2 → O3 → O1  O4 ↔ O2  65  2 
O5  M7  1.0  15  P5  O1 → O5 → O4  Nil  75  2 
M8  0.85  15  O3 → O4 → O5 → O4  Nil 
POI matrix for illustration
Part  P1  P2  P3  P4  P5  

Process plan  1  2  3  4  5  6  7  8 
O1  1  1  1  1  1  1  1  0 
O2  1  1  1  1  1  1  0  0 
O3  1  0  0  1  0  1  0  1 
O4  0  1  0  1  1  1  1  1 
O5  1  0  1  0  1  0  1  1 
POI matrix after consideration to production volume
Part  P1  P2  P3  P4  P5  

Process plan  1  2  3  4  5  6  7  8 
O1  100  100  50  50  70  65  75  0 
O2  100  100  50  50  70  65  0  0 
O3  100  0  0  50  0  65  0  75 
O4  0  100  0  50  70  65  75  75 
O5  100  0  50  0  70  0  75  75 
Maximum possible intercell moves matrix
Part  P1  P2  P3  P4  P5  

Process Plan  1  2  3  4  5  6  7  8 
O1  100  100  50  100  140  65  75  0 
O2  200  100  100  100  140  130  0  0 
O3  200  0  0  50  0  260  0  75 
O4  0  200  0  50  70  65  75  210 
O5  100  0  50  0  70  0  150  150 
Maximum possible intercell movement time matrix
Part  P1  P2  P3  P4  P5  

Process plan  1  2  3  4  5  6  7  8 
O1  100  100  200  400  420  130  150  0 
O2  200  100  400  400  420  260  0  0 
O3  200  0  0  200  0  520  0  150 
O4  0  200  0  200  210  130  150  420 
O5  100  0  200  0  210  0  300  300 
Matrix after clustering of operation cells
Part  P1  P2  P3  P4  P5  

Process plan  1  2  3  4  5  6  7  8 
Operation cell 1  
O1  100  100  200  400  420  130  150  0 
O2  200  100  400  400  420  260  0  0 
Operation cell 2  
O3  200  0  0  200  0  520  0  150 
O4  0  200  0  200  210  130  150  420 
O5  100  0  200  0  210  0  300  300 
Max. Intercellular movement time (min value)  300  200  200  400  420  390  150  0 
Part in operation cell  Any  Any  1  1  1  2  2  2 
Matrix after clustering of operation cells and operation flexibility
Part  P1  P2  P3  P4  P5  

Process plan  1  2  3  4  5  6  7  8 
Operation cell 1  
O1  200  100  200  200  420  130  0  0 
O2  200  200  400  400  420  130  150  0 
Operation cell 2  
O3  100  0  0  400  0  520  0  150 
O4  0  100  0  200  210  260  150  420 
O5  100  0  200  0  210  0  300  300 
Max. intercellular movement time (min. value)  200  100  200  600  420  260  150  0 
Part in operation cell  1  1  1  any  1  2  2  2 
Operation cell after selection of process plan and part assignment
At this stage, total intercell moves and intercell movement cost for required production volume are 1,210 and 585 units, respectively.
Final clustered matrix containing manufacturing cell with assigned machines and parts
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
Performance comparison of clus tering algorithm used against some established binary matrixbased CF methods
Source of problem  Size of problem (part × machine)  Performance measure  

Procedure used  Source author’s method  
EE  GE  GEF  GI  GM  EE  GE  GEF  GI  GM  
Elbenani and Ferland (2012)  8 × 6  6  88.89  67.44  74.54  68.35  6  88.89  67.44  74.54  68.35 
Gupta et al. (2012)  11 × 7  1  80.3  62.5  70.3  54.29  1  80.3  62.5  70.3  54.29 
Ghosh and Dan (2011)  7 × 5  3  54.3  69.6  85  75.6  3  54.3  69.6  85  75.6 
Doulabi S H et al. (2009)  8 × 6  2  87.06  76.92  77.78  74.24  2  87.06  76.92  77.78  74.24 
Hachicha et al. (2006)  11 × 7  2  86.1  70.37  72.41  66.47  2  86.1  70.37  72.41  66.47 
Albadawi et al. (2005)  20 × 8  9  95.8  85.2  85.2  1  9  95.8  85.2  85.2  1 
Modified proposed heuristic
Route selection through PCA, and Taguchi’s method
The L_{8} orthogonal array and CP measure
S. no.  P1  P2  P5  CP 

1  1  3  7  91.36 
2  1  3  8  92.4 
3  1  4  7  92.58 
4  1  4  8  97.13 
5  2  3  7  71.34 
6  2  3  8  88.63 
7  2  4  7  91.36 
8  2  4  8  99.58 
Maximum possible intercell movement time matrix after route selection
Part  P1  P2  P3  P4  P5 

Process plan  1  4  5  6  8 
O1  100  400  420  130  0 
O2  200  400  420  260  0 
O3  200  200  0  520  150 
O4  0  200  210  130  420 
O5  100  0  210  0  300 
Clustered Matrix before machine assignment and operation flexibility
Final clustered matrix after route selection through PCA and Taguchi’s method
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

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

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.
Notes
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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