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Journal of Applied Operational Research (2009) 1(1), 30–38 ISSN 1735-8523
© 2009 Tadbir Institute for Operational Research, Systems Design and Financial Services Ltd. All rights reserved.
Designing a mathematical model for cell
formation problem using operation sequence
Mohammad Mahdi Paydar 1,* and Navid Sahebjamnia 2
1
2
Iran University of Science & Technology, Tehran, Iran
Mazandaran University of Science & Technology, Babol, Iran
Abstract. Cell formation problem (CFP) is an important problem in the design of a cellular manufacturing system. Most of
the methods have been proposed to solve the CFP based on machine-part incidence alone. However, other factors such as
production sequence and product volumes, if incorporated, can enhance the quality of the solutions. So an attempt have
been made to take into account the natural constraints of real-life production systems, such as operation sequences, minimum
and maximum numbers of cells and cell sizes. This paper presents a new mathematical model to solve a CFP based on operation
sequence with varied number of the cell that minimized the sum total cost of inter-cell and intra-cell movements simultaneously.
To validate the proposed model, some numerical example from existed literature has been solved with the linearized formulation.
Keywords: cell formation problem; inter-cell and intra-cell movement; operation sequence
* Received September 2009. Accepted November 2009
Introduction
Group technology (GT) has attracted a lot of attention from manufacturers because of its many applications and
positive impacts in the batch-type manufacturing system. One application of GT to production is cellular manufacturing (CM). Among the problems of CM, cell formation problem (CFP) is considered to be the foremost
problem in designing a cellular manufacturing system. Implementing a CM system can help organizations
achieve benefits in several ways, such as simplified planning and control procedures, reduced throughput times,
reduced work-in-process inventory, reduced set-up times and reduced material handling (Wemmerlov and Hyer
1989). Comprehensive summaries and taxonomies of studies devoted to part-machine grouping problems were
presented in Wemmerlov and Hyer 1986, Kusiak 1987, Selim et al. 1998, Mansouri et al. 2000, Yin and Yasuda
2006 and Papaioannou and Wilson 2010.
The CFP deals with the identification of part families and machine groups on which to process these parts. To
enable this, a basic relationship must be identified between a part and a set of machines, for example a part process
routing, where the latter is defined as the machines or work centers visited by a part type according to the sequence
and type of operations required. Part families can be formed such that all parts within a family are processed on
the same machine group. Similarly, machines can be grouped into cells if they process the same set of parts. In
the last three decades of research on CFP, researchers have mainly used zero - one machine component incidence
matrix as the input data for the problem. Many approaches that have been applied to the CFP include genetic
algorithms (Goncalves and Resende 2004 and Mahdavi et al. 2009), tabu search (Lozano et al. 1999 and Wu
* Correspondence: Mohammad Mahdi Paydar, Department of
Industerial Engineering, Iran University of Science & Technology,
Narmak 16846-13114 Tehran, Iran. E-mail: [email protected]
MM Paydar and N Sahebjamnia
31
2004), Neural network (Soleymanpour et al. 2002), mathematical programming (Albadawi et al. 2005 and
Mahdavi et al. 2007) and simulated annealing (Sofianopoulou 1997 and Wu et al. 2008).
One important manufacturing factor in the design of a CFP is the operation sequences of parts. The operation
sequence is defined as an ordering of the machines on which the part is sequentially processed (Vakharia and
Wemmerlov 1990). Despite a large number of published papers on CFP, very few authors have considered operation
sequence in calculating inter-cell material movement and intra-cell material movement.
Choobineh (1988) presented a two-stage procedure for the design of a cellular manufacturing system based on
the operation sequences. The first stage used a similarity coefficient to form part families. In the second stage, an
integer-programming model was developed to obtain machine cells. Wu and Salvendy (1993) considered a network
analysis method by using an undirected graph (network) to model the cell formation problem when taking into
account the operation sequences factor. Sarker and Xu (1998) presented a brief review of the methods of cell
formation based on the operation sequences. A number of operation sequence-based similarity/dissimilarity
coefficients were discussed in their research. They classified the methods of cell formation based on the operation
sequences into four kinds: mathematical programming, network analysis, the materials flow analysis method and
heuristics. Boulif and Atif (2006) addressed a branch-and-bound-enhanced genetic algorithm for cell formation
problem using a graph partitioning formulation of this problem. They considered some of the natural data inputs
and constraints encountered in real life production systems, such as operation sequence, maximum number of
cells, maximum cell size, and machine cohabitation and non-cohabitation.
Two important cost considerations when forming cells are the cost of inter-cell material movement and cost of
intra-cell material movement. An intra-cell transfer occurs when a part is moved between two processes that are
performed within the same cell. An inter-cell transfer occurs when a part is moved from one cell to another so the
next process required can be performed. Inter-cell transfers occur if a cell does not contain a process required by
a part that is produced in that cell. Despite a large number of published papers on cell formation, very few authors
have considered operation sequence in calculating inter-cell and intra-cell material movement (Jayaswal and
Adil, 2004).
Logendran (1991) developed an algorithm to form the cells by evaluating the inter- and intra-cell moves with
the operation sequences. He also indicated the impact of the sequence of operations in the cell formation problem.
Sankaran and Kasilingam (1993) formulate an integer programming model which allows for both small and large
cells within a single layout. In this model the intra-cell transfer costs of parts produced in a cell are based on the
size of the cell which is defined in terms of the number of machines in the cell. Adil and Rajamani (2000) explicitly
consider the cost of intra-cell transfers based on the number of machines assigned to a cell. Also, a methodology
is presented for estimating the costs of intra-cell and inter-cell transfers. Their approach considers part volumes
but does not consider machine capacities and processing loads of parts.
In this paper, we propose a linear mathematical programming model for cell formation problem in CM. The
objective of the model is to determine the optimal cell configuration with the minimization of the total cost of intercell and intra-cell movements. Also this approach has the flexibility to allow the cell designer to either identify
the required number of cells in advance.
Mathematical formulation
In this section, we formulate the mathematical model based on operation sequence in CFP. This proposed model
deals with the minimization of the integrated inter-cell and intra-cell movement cost.
Indexing sets
i
j
k
o
index for parts (i =1, 2,…, P)
index for machines (j=1,2...,M)
index for cells (k =1, 2,…, C)
index for operations belong to part i (o =1, 2,…,Oi)
Journal of Applied Operational Research Vol. 1, No. 1
32
Parameters
γ int er : Material handling cost between cells.
γ int ra : Material handling cost within cells.
NC: Minimum number of cells to be formed.
NF: Minimum number of part type must be assigned in each cell.
NM: Maximum number of machine types allowed in each cell.
S: a set of machine pairs
{( j , j ) / j , j
a
b
a
b
D: a set of machine pairs
{( j , j ) / j , j
c
d
c
d
}
∈ {1, 2,..., M } , j a ≠ j b , and j a cannot be placed in the same cell with j b .
}
∈ {1, 2,..., M } , j c ≠ j d , and j c should be placed in the same cell with j d .
aisj : 1, if operation s of part i is to be processed on machine j; 0, otherwise.
Decision variables
X iojk : 1, if operation s of part i is done on machine j in cell k; 0, otherwise.
Yjk : 1, if machine j is assigned to cell k; 0, otherwise.
Zik : 1, if part i is assigned to cell k; 0, otherwise.
CFk : 1, if cell k is to be formed; 0, otherwise.
Mathematical model
Objective function
We propose the objective function as:
⎛ P C Oi −1 M M
⎞⎤
⎡P
in ter
⎜
Min Z = γ
× ⎢ ∑ ( Oi − 1) − ∑ ∑ ∑ ∑ ∑ X iojk X i, o +1, j ′, k ⎟ ⎥ +
⎜
⎟⎥
⎢⎣i =1
⎝ i =1 k =1 o =1 j =1 j ′=1
⎠⎦
⎛ P C Oi −1 M M
⎞
γ int ra × ⎜ ∑ ∑ ∑ ∑ ∑ X iojk X i, o +1, j , k ⎟
⎜ i =1 k =1 o =1 j =1 j ′=1
⎟
⎝
⎠
(1)
Constraints
C
∑ Zik CFk = 1
∀i
(2)
Y jk CFk = 1 ∀j
k =1
C M
(3)
k =1
C
∑
∑ ∑ aioj X iojk = 1
∀i, o, j
(4)
k =1 j =1
C
∑ CF
k =1
k
≥ NC
(5)
MM Paydar and N Sahebjamnia
M
∑Y
j =1
jk
≤ NM × CFk ∀k
M
∑ Zik ≥ NF × CFk
∀k
33
(6)
(7)
j =1
( j a , jb ) , ∀k
Y j k + Y j k = 0 ( j c , j d ) , ∀k
Y j a k + Y jb k ≤ 1
c
d
X iojk , Zik , Y jk ∈ {0,1} ∀i, j , o, k
(8)
(9)
(10)
Objective function (1) is considered for minimizing the total sum of inter-cell and intra-cell movement costs.
The first term computes the total inter-cell movement cost, where Oi –1 indicates the total number of movements
of part i. Inter-cell movement is incurred whenever consecutive operations of the same part type are carried out in
different cells. For instance, assume that the operation o of part type i is processed on machine type j in cell k. If
the next operation, o + 1, of part type i is processed on any machine but in another cell, then there is an inter-cell
movement. Oi is the number of operations of part type i and Oi –1 indicates the total number of movements of
part type i. Therefore, the term
∑ i ( OPi − 1)
shows the total number of movements in the CFP. Moreover, the term
∑ i ∑ k ∑ o ∑ j ∑ j′ X iojk X i,o +1, j′,k
computes the total number of intra-cell movements in the manufacturing system. So, the first term calculates the
total number of intercellular cost, i.e., the total number of inter-cell movements is equal to the total number of
movements minus the total number of intra-cell movements. The second term of the objective function computes
the total intra-cell cost respectively. The intra-cell movement is incurred whenever consecutive operations of the
same part type are processed in the same cell. For instance, say that the operation o of part type i is processed on
machine type j in cell k. If the next operation, o + 1, of part type i is processed on any machine but within the
same cell, then there is an intra-cell movement.
Equation (2) guarantees that each machine must be assigned to one cell only. Equation (3) guarantees that
each part must be assigned to one cell only. Equation (4) guarantees that each operation of each part type which
is done by one machine must be allocated to one cell. Constraint (5) is for forcing at least NC cells to be formed.
Inequality (6) is for preventing the assignment of more than NM machines to each cell. Constraint (7) guarantees
that lower limit on number of parts in each cell. Equation (8) ensures that machine pairs included in S are not
placed in the same cell. Equation (9) is to ensure that machine pairs included in D should be placed in the same
cell. And relation (10) specifies that the decision variables are binary.
Properties of the model
Despite a large number of published papers on CFP , some authors have considered operation sequence in calculating
inter-cell and intra-cell material movements cell formation methods, without using operation sequence data, may
calculate inter-cell movement based on the number of cells that a part will visit in the manufacturing process.
However, the number of cells visited by the part can be less than the actual number of inter-cell movements since
the part may travel between cells. Such movements may not be accurately reflected without properly using operation
sequence data.
We do not assume that the number of cells is predetermined. In practice, designers usually do not know the
number of cells that would yield the best CMS design. Hence, in our approach, the optimal number of cells is
determined by the model, but the minimum number of cells and maximum number of cells can be formed is given.
Journal of Applied Operational Research Vol. 1, No. 1
34
Such requirements exist since some machines must be separated from each other while other machines must be
placed together due to technical and safety considerations. For example, machines that produce vibrations, dust,
noise, or high temperatures may need to be separated from electronic assembly and final testing. In other situations,
certain machines should be placed in the same cells. For example, a heat treatment station and a forging station
may be placed adjacent to each other for safety reasons. Machines that share a common resource or those that
require a particular operator’s skill may also be placed in a same cell.
Linearizing the objective function
The objective function and the constraints (2) and (3) in the model are nonlinear equations. To linearize the term
X iojk X i,o +1, j ′, k in the objective function, we need to introduce one auxiliary variable to replace this nonlinear
term with additional constraints. The required new variable can be defined by the following equations:
Piojj ′k = X iojk X i, o +1, j ′, k
By considering the above equation, following constraints should be added to the mathematical model:
Piojj ′k ≥ X iojk + X i, o +1, j ′, k − 1
∀i, j , j ′, k , o = 1,..., Oi − 1
(11)
Piojj ′k ∈ {0,1}
(12)
∀i, j , j ′, k , o = 1,..., Oi − 1
In the next step, to transform the constraints into the linear form, two new variables are introduced as follows:
Wik = Zik CFk
V jk = Y jk CFk
where the following constraints must be added to the original model.
Wik − Zik − CFk + 1.5 ≥ 0
1.5Wik − Zik − CFk ≤ 0
V jk − Y jk − CFk + 1.5 ≥ 0
1.5V jk − Y jk − CFk ≤ 0
∀ i, k
(13)
∀ i, k
(14)
∀ j, k
(15)
∀ j, k
(16)
Now, the new version of constraints (2) and (3) are as follows:
⎛ P C Oi −1 M M
⎞⎤
⎛ P C Oi −1 M M
⎞
⎡P
Min Z = γ in ter × ⎢ ∑ ( Oi − 1) − ⎜ ∑ ∑ ∑ ∑ ∑ Piojj ′k ⎟ ⎥ + γ int ra × ⎜ ∑ ∑ ∑ ∑ ∑ Piojj ′k ⎟
⎜ i =1 k =1 o =1 j =1 j ′=1
⎟⎥
⎜ i =1 k =1 o =1 j =1 j ′=1
⎟
⎢⎣i =1
⎝
⎠⎦
⎝
⎠
Subject to:
(4)-(16).
Numerical illustration
In this section, our computational results are presented to verify the proposed model. Two data set collected from
the literature are solved by running LINGO package on a PC Pentium IV 2.1 GHz with 512 Mb of RAM.
The first example exists in Tsai and Lee 2006 which it consists of 11 part types and 7 machines. Table 1 shows
the part-machine incidence matrix to which the operation sequence information has been added. Also, Table 2
shows the parameters for solving the Example 1. The cell formation of this example is shown in Table 3. The total
number of intra-cell movements in cells is 10 and the total number of inter-cell movements is 4. Therefore the
objective function value for this problem is 44.
MM Paydar and N Sahebjamnia
35
Table 1. Part-machine matrix, Example 1.
Part
1
2
3
4
5
6
7
8
9
10
11
1
2
2
1
1
Machine
3 4 5
7
2
1
2
2
2
6
3
2
3
1
1
1
2
1
2
2
1
1
2
1
2
1
3
Table 2. Parameter setting.
Parameters
γ
value
6
int er
γ int ra
NC
C
NF
NM
Pair of machines that should not be located in the same cell
Pair of machines that should be located in the same cell
2
2
3
1
4
(3,4)
(2,3)
Table 3. The cell formation to example 1.
Part
4
5
8
10
1
2
3
6
7
9
11
4
3
2
2
2
6
1
0
0
0
3
3
Machine
7 1 2
0
1
1
1
2 1
0 1
1 0
0 2
2 0
0 2
1 0
3
2
5
0
2
0
1
0
1
2
0
0
2
0
1
0
0
If we change the input parameters, the optimal solution can be changed. For example, maximum number of
machine types can be assigned in each cell (NM) changed to 6, the optimal cell configuration as shown in Table4.
In this case, the total number of intra-cell movements in cells is 13 and the total number of inter-cell movements is 1.
Therefore the objective function value for this problem is 32. As a result, we find the objective function is improved.
Journal of Applied Operational Research Vol. 1, No. 1
36
Table 4. The cell formation to part-machine 11 × 7.
1
2
0
1
0
0
2
0
1
1
2
3
4
6
7
9
11
5
8
10
Part
Machine
3 5 6
0 0 0
2 0 0
0 2 3
2 0 1
1 0 0
0 1 0
1 0 0
2 0 3
2
1
1
0
0
2
0
2
0
4
7
3
2
2
2
1
1
1
In the second example, from Jayakrishnan Nair and Narendran 1998, 20 parts are processed on 8 machines as
shown in Table 5. And the input parameters for Example 2 are given in Table 6. Table 6 includes obtained results
from our proposed model is solved optimally by Lingo package solver.
Table 5. Component machine-matrix, example 2.
Part
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1
0
1
2
0
0
0
0
1
1
0
3
0
1
1
0
1
3
0
1
0
2
0
0
1
1
0
1
4
0
0
0
0
0
0
2
0
0
0
2
0
2
3
0
2
0
0
0
0
0
2
3
0
2
0
2
3
0
2
1
0
2
0
Machines
4 5 6
0 2 1
0 0 0
5 0 0
2 0 0
0 2 1
2 5 0
2 0 0
0 0 0
0 0 2
2 3 1
0 0 0
0 1 3
0 0 0
0 0 0
0 1 2
0 0 0
0 2 0
1 0 0
0 0 0
1 0 3
7
0
0
3
3
0
3
3
0
0
0
1
2
0
0
0
0
0
4
0
4
8
0
0
4
4
0
4
1
0
0
0
0
0
0
0
0
0
0
3
0
5
MM Paydar and N Sahebjamnia
37
Table 6. Parameter setting.
Parameters
γ
value
6
int er
γ int ra
NC
C
NF
NM
Pair of machines that should not be located in the same cell
Pair of machines that should be located in the same cell
2
2
5
2
4
(1,5)
(2,4), (2,7)
Table 7. The cell configuration to example 2.
Parts
2
8
9
11
13
14
16
17
19
3
4
6
7
18
20
1
5
10
12
15
1
1
1
1
3
1
1
1
3
1
2
0
0
0
0
0
0
0
0
0
0
3
2
2
3
2
2
3
2
1
2
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
2
0
0
0
1
1
1
4
2
2
0
0
0
0
0
Machines
4 7 8
0 0 0
0 0 0
0 0 0
0 1 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
5 3 4
2 3 4
4 3 4
2 3 1
1 4 3
1 3 4
0 0 0
0 0 0
2 0 0
0 2 0
0 0 0
5
0
0
0
0
0
0
0
2
0
0
0
1
0
0
0
2
2
3
1
1
6
0
0
2
0
0
0
0
0
0
0
0
0
0
0
5
1
1
1
3
2
Conclusion
Cell formation is one of the main problems to be solved in the design of a cellular manufacturing system. In this
paper, we have proposed a mathematical model for cell formation which the objective of the model is to minimize the
sum total cost of inter-cell and intra-cell movements simultaneously. We considered some of the natural data
inputs and constraints encountered in real life production systems, such as operation sequence, maximum number
of cells, maximum cell size. This approach has the flexibility to allow the cell designer to either identify the
required number of cells. To verify the behavior of the proposed model, two examples are presented to illustrate
the applicability of the proposed model. These examples are solved by a branch and bound (B&B) method with
the LINGO 8.0 software package. Further researches on the proposed model may be attempted in future studies
by incorporating the following issues:
• Application of meta-heuristic approaches e.g. simulated annealing, genetic algorithm, etc to solve the proposed
model for real-sized problems.
• Consideration of layout of machines to precisely calculate the material handling cost.
38
Journal of Applied Operational Research Vol. 1, No. 1
References
Adil, G. K., Rajamani, D., 2000. The trade-off between intracell and intercell moves in group technology cell
formation. Journal of Manufacturing Systems, 19(5),
305–317.
Albadawi Z., Bashir H. A., Chen M., 2005, A mathematical
approach for the formation of manufacturing cells.
Computer and Industrial Engineering, 48, 3-21.
Boilif, M, Atif, K., 2006, A new branch-&-boundenhanced genetic algorithm for the manufacturing cell
formation problem. Computers & Operations Research,
33, 2219–2245.
Choobineh, F., 1988, A framework for the design of cellular manufacturing systems. International Journal of
Production Research, 26, 1161-1172.
Goncalves, J., Resende, M. 2004. An evolutionary algorithm for manufacturing cell formation. Computers and
Industrial Engineering, 47, 247-273.
Jayakrishnan Nair, G. J., Narendran, T. T., 1998, CASE: A
clustering algorithm for cell formation with sequence
data. International Journal of Production Research, 36,
157 – 179.
Jayaswal, S., Adil, G.K., 2004. Efficient algorithm for cell
formation with sequence data, machine replications
and alternative process routings. International Journal
of Production Research, 19, 281–294.
Kusiak, A., The generalised group technology concept. International Journal of Production Research, 1987, 25,
561-569.
Logendran, R., 1991, Impact of sequence of operations and
layout of cells in cellular manufacturing. International
Journal of Production Research, 29, 375-390.
Lozano, S., Adenso-Diaz, B., Eguia, I., Onieva, L., 1999 A
one-step tabu search algorithm for manufacturing cell
design. Journal of the Operational Research Society,
50, 509-16.
Mahdavi, I., Javadi, B. F., Alipour, K., Slomp, J. 2007. Designing a new mathematical model for cellular manufacturing system based on cell utilization. Applied
Mathematics and Computation, 190, 662- 670.
Mahdavi, I., Paydar, M.M., Solimanpur, M., Heidarzade,
M., 2009, Genetic algorithm approach for solving a
cell formation problem in cellular manufacturing, Expert Systems with Applications, 36, 6598–6604.
Mansouri, S.A., Moattar Husseini, S.M. and Newman,
S.T., 2000, A review of the modern approach to multicriteria cell design. International Journal of Production
Research, 38, 1201-1218.
Morris ,J.S., Tersine ,R.J., 1990, A simulation analysis of
factors influencing the attractiveness of group technology cellular layouts, Management Science 36 (12),
1567-1578.
Papaioannou, G., Wilson, J.M., 2010, The evolution of cell
formation problem methodologies based on recent
studies (1997–2008): Review and directions for future
research, European Journal of Operational Research,
206 (3), 509-521.
Sankaran, S., Kasilingam, R. G., 1993. On cell size and
machine requirements planning in group technology
systems. European Journal of Operations Research, 69,
373–383.
Sarker, B.R. and Xu,Y., (1998), Operation sequences-based
cell formation methods: a critical survey. Production
Planning and Control, 9, 771-783.
Selim, H. M., Askin, R. G., Vakharia, A. J. 1998. Cell formation in group technology: Review, evaluation and
directions for future research. Computers and Industrial
Engineering, 34, 3-20.
Sofianopoulou S., 1997, Application of simulated annealing to a linear model for the formation of machine cells
in group technology. International Journal of Production Research, 35, 501-11.
Soleymanpour, M., Vrat, P., Shanker, R., 2002, A transiently chaotic neural network approach to the design
of cellular manufacturing. International Journal of Production Research, 40(10), 2225-2244.
Tsai,C.C., Lee, C.Y., (2006), Optimization of manufacturing cell formation with a multi-functional mathematical programming model, International Journal advanced Manufacturing Technology, 30,309–318.
Vakharia, A. J. Wemmerlov, U., (1990), Designing a cellular manufacturing system: a materials flow approach
based on operation sequences. IEE Transactions, 22,
84-97.
Wemmerlov, U., Hyer, N. L., 1986, Procedures for the
part-family/ machine group identification problem in
cellular manufacturing. Journal of Operations Management, 6(2), 125–147.
WemmerloK, U. Hyer, V. N., (1989), Cellular manufacturing in the U.S. industry: A survey of users, International Journal of Computer Integrated Manufacturing
27 (9), 1511-1530.
Wu, N. and Salvendy, G., (1993), A modified network approach for the design of cellular manufacturing systems. International Journal of Production Research, 31,
1409-1421.
Wu, T., Low, C., Wu, W., 2004. A tabu search approach to
the cell formation problem. International Journal of
Advanced Manufacturing Technology, 23, 916–24.
Wu, T. H., Chang, C. C., Chung, S. H. 2008. A simulated
annealing algorithm for manufacturing cell formation
problems. Expert Systems with Applications, 34, 16091617.
Yin, Y., Yasuda, K., 2006, Similarity coefficient methods
applied to the cell formation problem: A taxonomy and
review. International Journal of Production Economics,
101, 329-352.