A Mathematical Model for Integrating Cell Formation Problem with Machine Layout

Abstract

This paper deals with the cellular manufacturing system (CMS) that is based on group technology concepts. CMS is defined as identifying the similar parts that are processed on the same machines and then grouping them as a cell. The most proposed models for solving CMS are focused on cell formation problem while machine layout is considered in few papers. This paper addresses a mathematical model for the joint problem of the cell formation problem and the machine layout. The objective is to minimize the total cost of inter-cell and intra-cell (forward and backward) movements and the investment cost of machines. This model has also considered the minimum utilization level of each cell to achieve the higher performance of cell utilization. Two examples from the literature are solved by the LINGO Software to validate and verify the proposed model.

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A Mathematical Model for Integrating Cell Formation
Problem with Machine Layout
I. Mahdavi, M. M. Paydar, M. Solimanpur & M. Saidi-Mehrabad*
*
I. Mahdavi, Department of Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran
M. M. Paydar, Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran
M. Solimanpur, Faculty of Engineering, Urmia University, Urmia, West Azerbaijan Province, Iran
M. Saidi-Mehrabad, Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran
K
KE
EY
YW
WO
OR
RD
DS
S ABSTRACT
This paper deals with the cellular manufacturing system (CMS) that is
based on group technology concepts. CMS is defined as identifying the
similar parts that are processed on the same machines and then
grouping them as a cell. The most proposed models for solving CMS
are focused on cell formation problem while machine layout is
considered in few papers. This paper addresses a mathematical model
for the joint problem of the cell formation problem and the machine
layout. The objective is to minimize the total cost of inter-cell and
intra-cell (forward and backward) movements and the investment cost
of machines. This model has also considered the minimum utilization
level of each cell to achieve the higher performance of cell utilization.
Two examples from the literature are solved by the LINGO Software to
validate and verify the proposed model.
© 2010 IUST Publication, IJIEPR, Vol. 21, No. 2, All Rights Reserved.
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Cellular manufacturing (CM) is an application of
the group technology (GT) philosophy to designing
manufacturing systems. The main idea of GT is to
improve productivity of manufacturing system by
grouping parts and products with similar characteristics
into families and forming production cells with a group
of dissimilar machines and processes. Comprehensive
summaries and taxonomies of studies devoted to part-
machine grouping problems were presented in [1], [2],
[3] and [4]. One of the first problems encountered in
implementing CM is cell formation problem (CFP). 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 [5] and [6], tabu search [7]
and [8], neural network [9], mathematical
Corresponding author. M. Saidi-Mehrabad
Email: mehrabad@iust.ac.ir mmp_63_2004@yahoo.com
Paper first received April. 05. 2010 ,and in revised form August.
17. 2010.
programming [10] and [11] and simulated annealing
[12] and [13]. 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. CFP
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 back and forth between cells.
Such movements may not be accurately reflected
without properly using operation sequence data.
Different data structures provide different sets of
information and enable the cell designers to make
appropriate use of them while solving the CFP. A zero
one incidence matrix offers advantages of
computational simplicity for solving the CFP.
However, it is not possible to address issues pertaining
to machine utilization, inter-cell workload and layout
of machines within each identified cell. On the other
hand, using additional data pertaining to setup time,
process time and production volumes considering
Cell formation problem,
Machine layout,
Cell utilization,
Mathematical model.
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machine capacity enable cell designers to address these
issues using a much more complex solution
methodology. Use of sequence data for CFP provides
additional information to the cell designer. Sequence
data identifies the order in which jobs are processed in
a manufacturing system. Therefore, this information
could be used not only for identifying part families and
machine groups but also to arrive at the layout of
machines within each cell based on dominant flow
patterns within each cell. Despite this simple truth,
traditionally, the cell design problem and the layout
problem are treated in a discontinuous fashion. Thus,
only a few studies have attempted to resolve these
decisions concurrently [14], [15] and [16]. Besides
using sequence data for CFP, it is important to address
layout of machines as the required information for
identification of manufacturing cells.
In the next section we motivate our research by a
review of the literature pertaining to the use of
sequence data for CFP and methodologies.
2. Literature Review
Vakharia and Wemmerlov [17] presented a
heuristic approach for the machine cell design, where
machines within each cell are arranged along a linear
flow line. Irani et al. [18] used maximal spanning
arborescence as a graphic structure to integrate the
machine grouping, the intra-cell layout, and the inter-
cell layout. Sequential, two-phase mathematical
programming models were proposed to decompose the
joint problem. Liao [19] proposed a sequential three-
stage procedure, to determine the best part routing,
machine cells, and inter-cell layout for a line-type
cellular manufacturing system. Approaches that are
more complicated have also been developed to address
the joint problem. Arvindh and Irani [20] developed an
iterative approach to design a cellular manufacturing
system where one iterative loop deals with machine
and part grouping, and another iterative loop varies the
number of cells to find the best design. Akturk and
Balkose [21] described a multi-criterion clustering
approach that considers manufacturing attributes,
operational sequences, and within cell layout. Nair
Jayakrishnan and Narendran [22] addressed the task of
identifying machine-cells and component families on
the basis of production-sequence data. They defined a
new similarity coefficient that captures the ordinal
character of the data matrix and introduced a
quantitative criterion, with a weighing factor for
assessing the quality of the solution based on a non-
hierarchical clustering algorithm. Suresh et al. [23]
utilized fuzzy neural network approach to cell
formation using sequence data. Lee and Chiang [24]
considered the joint clustering-layout problem where
machine cells are to be located along the bi-directional
linear flow layout. They seek to minimize the actual
inter-cell flow cost, instead of the typical measure that
minimizes the number of inter-cell movements. A
three-phase approach, using the cut tree network
model, is developed to solve this joint problem. Chiang
and Lee [25] developed a genetic-based algorithm with
the optimal partition approach for the cell formation in
bi-directional linear flow layout, where the objective is
to minimize the actual inter-cell flow cost, instead of
the typical measure that optimizes the number of inter-
cell movements. Boulif and Atif [26] 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.
Wu et al. [15] used a hierarchical genetic algorithm to
form manufacturing cells and determine the group
layout of a CMS simultaneously. They have also
developed a new group mutation operator to increase
the mutation probability. Mahdavi et al. [16] developed
a heuristic algorithm based on flow matrix for cell
formation and layout design. The objective was to
make use of the valuable information about the flow
patterns of various jobs in a manufacturing system and
obtain relevant performance measures for the cell
design and layout problem. In this paper, the cell
formation and layout design are considered
simultaneously. We propose a new mathematical
model that utilizes the sequence data as input to the
problem and identifies machine cells and the layout of
machines within each cell. The objective is to
minimize the total costs of inter-cell and intra-cell
(forward and backward) movements and the
investment cost of machines in CM using sequence
data. Due to different minimum utilization level for
each cell, the proposed model presents different
scenarios of part-machine grouping. The approach is
illustrated by examples that are solved by the LINGO
Software and computational results are reported and
analyzed.
3. Problem Formulation
We consider several factors such as routing
(sequence), machine capacity, demand, and layout type
in the problem formulation. Routing is often presented
in a machine/part/sequence matrix, where the
component, aisj of the matrix indicates the operational
sequence (operation s) of part type i to be processed by
machine type j. Since the machine layout type (e.g.,
single row, U shape, multiple rows, or other
configurations) has significant impact on the part
transfer cost, it needs to be considered in the CFP.
Figure 1 shows the routing of part types based on
sequence data and the types of material handeling cost
considering distance between machines. The location
shows the place and layout of machines in cells. For
example, the location of machines in line layout in cell
1 is:
Machine 2 Æ Machine 5 Æ Machine 4
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Fig. 1. An integrated CMS
In this section, we formulate a mathematical model
based on sequence data in CMS. The proposed model
deals with the minimization of the integrated inter-cell
and intra-cell (forward and backward) movements cost
and the investment cost of machines.
3.1. Assumptions
The problem is considered under the following
assumptions.
1- The number of cells is known.
2- The upper bound and lower bound of the cells size
are known.
3- Consecutive operations of each part type are
performed on different machines in a given sequence.
Moreover, sequence of operations is important in the
calculation of intercellular and intracellular material
handling cost since it gives a more accurate count of
the number of times that a part either has to move
between cells or between machines(forward and
backward movements) within the same cell.
4- The processing times for all operations of part types
on different machine types are known and
deterministic.
5- Parts are moved between and within cells. 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 s of part type i
is processed on machine type j in cell k. If the next
operation, s + 1, of part type i is processed on any
machine but in another cell, then there is an inter-cell
movement. 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 s of part type i is processed on machine
type j in cell k. If the next operation, s + 1, of part type
i is processed on any machine but within the same cell,
then there is an intra-cell movement. To the best of our
knowledge, all studies considering this movement
supposed that intra-cell movement occurs between two
different machine types. But, in reality, it can occur
between same machine types on different locations in
one cell. We have considered this concept in our
model. Moreover, in the manufacturing systems, the
backward movement incurs more expenses, so its cost
is assumed greater than forward movement cost in the
proposed model.
6- We assume the type of layout is linear and all
machine types should be assigned to locations which
have same dimensions. Hence, the distance between
two machines assigned to two different locations is
calculated by subtracting location numbers of those
machines from each other. For instance, in Figure 2,
cell k has 7 locations with linear layout type. This
Figure shows the effect of operation sequence and
intra-cell layout on forward and backward movements
within a cell. Operations 1 and 2 of part type 1 must be
processed on machine type 3 in location 1, hence there
is no intra-cell movement. But operation 3 is processed
with machine type 3 in location 5. Because of
operations 2 and 3 of part type 1 are processed by two
machines of type 3 located in different locations 1 and
5, then a forward intra-cell movement occurs. The
movement distance of these consecutive operations is
equal to the distance between locations 1 and 5.
Operations 4 and 5 of part type 2 are processed on
machine type 5 in location 7 and machine type 4 in
location 2, respectively. Then a backward intra-cell
movement happens.
Cell k
Location 1 2 3 4 5 6 7
Machine 3 4 1 2 3 1 5
Par type 1 1,2 4 3
Part type 2 1,5 2 3 4
Fig. 2. Operation sequence and intra-cell layout in
cell k
7- The value of cell utilization is set by the designer
considering his experiences. This setting is based on a
trade off among all cells to obtain the best
configuration of machines in cells. Meanwhile in
cellular manufacturing systems, cells are formed in
different sizes, therefore investigating cell utilization in
all cells is important due to similarity coefficient. So,
decision maker set minimum utilization for each cell to
satisfy the formed cells in the cellular manufacturing
system.
8- The demand for each part type is given.
9- The capacity of each machine type is known.
Machines
2 5 4 1 3
1 0 1 0 0 2
3 2 1 3 0 0
5 1 0 2 0 0
6 0 1 2 0 0
2 0 0 0 1 2
4 0 0 1 2 3
Parts
7 0 0 0 1 2
1 2 3 1 2
Locations
Forward intra-cell cost
Backward intra-cell cost
Inte
r
-cell cost
(3 1) Unit cost−×
(2-1) Unit cost×
Unit cost
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I. Mahdavi, M. M. Paydar, M. Solimanpur & M. Saidi-Mehrabad A Mathematical Model for…
10- There are several machines of each type with
identical duplicates to satisfy capacity requirements
and reduce/eliminate inter-cell movement. The number
of duplicates of each machine type is constant over the
planning horizon.
11- The investment cost of each machine type is
known.
3.2. Indexing Sets
i index for part type (i =1, 2,…, P)
j,
j
′ index for machine type (j, =1,2...,M)
j′
k index for cells (k =1, 2,…, C)
s index for operations (s =1, 2,…,OPi)
l,l index for location of machine type ( l , =1,
2,…, Lk).
′l′
3.3. Parameters
inter-cell
γ
: material handling cost between cells.
intra-cell
f
γ
: forward material handling cost within cells.
intra-cell
b
γ
: backward material handling cost within
cells.
Min_ut k: minimum utilization of cell k.
Lk: lower bound of the number of machine type in cell
k.
Uk: upper bound of the number of machine type in cell
k.
Nj : number of machines of type j available for
allotment to cells.
tisj: processing time of sth operation of part type i with
machine type j.
Di: demand quantity of part type i.
Tj: the capacity of machine type j.
Cj : investment cost of machine type j.
ll k
d′: Distance between location and in cell k. ll′
rij: 1 if part type i is to be processed on machine type j;
0 otherwise.
aisj: 1 if operation s of part type i is to be processed on
machine type j; 0 otherwise.
3.4. Decision Variables
1 if operation of part is assigned to cell ,
0 otherwise.
th
isk
s
ik
X⎧
⎪
=⎨
⎪
⎩
1 if machine type is assigned to cell in location ,
0 otherwise.
jkl jk
Y⎧
=⎨
⎩
l
1 if part type is assigned to cell ,
0 otherwise.
ik ik
Z⎧
=⎨
⎩
3.5. Mathematical Model
3.5.1 Objective Function
We propose the objective function as follows:
M
in Z =
()
1
inter-cell ,1,
11111111
1OP L L
PPCMM
ikk
iiski
iiksjjll
OP X X Y Y
γ
−
′′
+
′′
========
⎡ ⎤
skjkljkl
⎛⎞
⎢ ⎥
⎜⎟
×−−
⎢ ⎥
⎜⎟
⎜⎟
⎢ ⎥
⎝⎠
⎣ ⎦
∑ ∑∑∑∑∑∑∑
1
intra-cell ,1,
11 1 111 1
OP L L
PC MM
ikk
ll k isk i s k jkl j kl
fik s jj lll
dXX YY
γ
−
′′
+
′′
== = ====+
′
⎛ ⎞
⎜ ⎟
+ +
⎜ ⎟
⎜ ⎟
⎝ ⎠
∑∑ ∑ ∑∑∑ ∑
1
intra-cell ,1,
11 1 111 1
OP L L
PC MM
ikk
ll k isk i s k jkl j kl
bik s jjl ll
dXX YY
γ
−
′′′
+
′′ ′
== = = ===+
⎛⎞
⎜⎟
⎜⎟
⎜⎟
⎝⎠
∑∑ ∑ ∑∑∑ ∑ 111
L
CM k
j
jkl
kjl
CY
===
+∑∑∑
3.5.2. Constraints
11
L
Mk
j
kl k
jlYL
==
≥
∑∑ k∀ (1)
11
L
Mk
j
kl k
jlYU
==
≤
∑∑ k∀ (2)
11
L
Ckjkl j
klYN
==
≤
∑∑ j
∀
(3)
11
M
jkl
jY
=
≤
∑ ,kl∀ (4)
1
1
C
ik
k
Z
=
=
∑ i∀ (5)
111
1
L
MC k
isk jkl isj
jkl
XYa
===
=
∑∑∑ ,is∀ (6)
11
OP
pijkl isk isj i j
isYXtDT
==
≤
∑∑ ,,
j
kl∀ (7)
111 111
min _ut
LL
PM PM
kk
ik jkl ij k ik jkl
ijl ijl
Z
Yr ZY
=== ===
≥
∑∑∑ ∑∑∑
k∀
(8)
{
}
,, 0,1
isk ik jkl
XZY∈ ,,,,ijlsk∀ (9)
The objective function considers minimizing the total
cost of inter-cell and intra-cell (forward and backward)
movements and investment cost of machines. The first
term computes the total cost of inter-cell movements.
This cost is incurred when consecutive operations of
the same part type are carried out in different cells.
is the number of operations of part type i and -1
indicates the total number of movements of part type i.
i
OP
i
OP
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)
Y
′
−
−
Therefore, the term shows the total number
of movements in the cellular manufacturing system.
Moreover, the term
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
represents the forward intra-cell material handling cost,
where the forward movement distance between
machines j and j’ assigned to locations l and l’ in cell k
is designated by . This cost is sustained when
consecutive operations of the same part types are
processed in the same cell but on two machines of
different types or even same type in forward layout, for
example machines j and j’ assigned to locations l and
l’ in forward mood process the operations s and s+1 of
part i. The third term of objective function represents
the backward intra-cell material handling cost, where
the backward movement distance between machines j
and j’ assigned to locations l and l’ is given by .
This cost is sustained when consecutive operations of
the same part types are processed in the same cell but
on two machines of different types or even same type
in backward layout, for example machines j and j’
assigned to locations u and u’ in backward mood
process the operations s and s+1 of part type i. The
fourth term represents the cost of all machines assigned
to cells.
(
1
i
iOP −
∑
,1,isk i s k jkl j kl
iksjjll
XX YY
′′
+
′′
∑∑∑∑∑∑∑
ll k
d′
ll k
d′
Inequalities (1) and (2) ensure the lower and upper
bound considerations for the number of machines to be
allocated to each cell. Inequality (3) ensures that the
number of machines available for a given type is not
bypassed. Inequality (4) ensures that each machine can
be allocated to only one location of each cell at most.
Equation (5) guarantees that each part must be assigned
to only one cell. Constraint (6) guarantees that each
operation will be assigned to a cell which contains the
required machine type. Inequality (7) ensures that
capacity limitation of each machine is satisfied.
Constraint (8) specifies minimum utilization of cells to
achieve a feasible and better arrangement of machines
and operations of parts. Relation (9) specifies that the
decision variables are binary.
3.6. Linearization of the Proposed Model
Obviously, the objective function and constraints
(6) - (8) are nonlinear. However, these terms can be
linearized without much difficulty as they are products
of binary variables. We need to introduce auxiliary
variables to replace these nonlinear terms with
additional constraints.
The required new variables can be defined by the
following equations
isklj isk jkl
OXY=
,,,,,, , 1,iskll jj isklj is k jkl
BOX
′′ ′
+
=
iklj ik jkl
VZY=
By considering the above equation, following
constraints should be added to the mathematical model:
1.5 0
isklj isk jkl
OXY−−+≥
,,,,iskl j∀ (10)
1.5 0
isklj isk jkl
OXY−−≤ ,,,,iskl j∀ (11)
,,,,,, , 1, 2.5 0
iskll j j isklj is k jkl
BOXY
′′ ′′
+
−− −+≥
,,,,, , 1,.., 1ikll j j s OP
′′
∀=
(12)
,,,,,, , 1,
2.5 0
iskll j j isklj is k jkl
BOXY
′′ ′′
+
−− −≤
,,,,, , 1,.., 1ikll j j s OP
′′
∀=
(13)
1.5 0
iklj ik jkl
VZY−− +≥ ,,,ikl j∀ (14)
1.5 0
iklj ik jkl
VZY−− ≤ ,,,ikl j∀ (15)
{
}
,,,,,,
,, 0,
iklj isklj i s k l l j j
VO B ′′
∈1′
,,,,,,iskll j j
′
∀ (16)
Based on defining the new binary variables, the linear
mathematical model is as follows:
M
in Z =
()
1
int ,, ,,,,
11111111
1OP L L
PPCMM
ikk
er cell ii
iiksjjll
OP B
γ
−
−′′
′′
========
⎡ ⎤
sklljj
⎛⎞
⎢ ⎥
⎜⎟
×−−
⎢ ⎥
⎜⎟
⎜⎟
⎢ ⎥
+
⎝⎠
⎣ ⎦
∑ ∑∑ ∑ ∑∑∑∑
1
int ,, ,,,,
11 1 111 1
OP L L
PC MM
ikk
ra cell ll k i s k l l j j
fik s jj lll
dB
γ
−
−′′′
′′
== = ====+
⎛⎞
⎜⎟
+
⎜⎟
⎜⎟
⎝⎠
∑∑ ∑ ∑∑∑ ∑
1
int ,,,,,,
11 1 111 1
OP L L
PC MM
ikk
ra cell ll k i s k l l j j
bik s jjlll
dB
γ
−
−′′′
′′ ′
== = ====+
⎛⎞
⎜⎟
+
⎜⎟
⎜⎟
⎝⎠
∑∑∑∑∑∑∑ 111
L
CM k
jjkl
kjl
CY
===
∑∑∑
So that, constraints (1)-(5), (9)-(16) and the new
version of constraints (6) - (8) are:
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1
L
MC k
isklj isj
jkl
Oa
===
=
∑∑∑ ,is∀ (17)
11
OP
piisklj isj i j
isOtDT
==
≤
∑∑ ,,
j
kl∀ (18)
111 111
min_ut
kk
LL
PM PM
iklj ij k iklj
ijl ijl
Vr V
=== ===
≥
∑∑∑ ∑∑∑ k∀ (19)
Now, the objective function becomes a 0-1 integer
linear programming model. All constraints in the
model are also linear. The number of variables and
number of constraints in the linearized model are
presented in Tables 1 and 2, respectively, based on the
variable indices.
Tab. 1. Number of variables in the linearized model Variable name Variable count
isk
X
P
OP C××
j
kl
Y
M
CL××
ik
Z
P
C×
iklj
V
P
CLM×××
isklj
O
P
OP C L M××××
,, ,, ,,iskll j j
B′′
22
(1)POP CLM×−×××
Tab. 2. Number of constraints in the linearized model
Equation
number Constraint count Equation
number Constraint count
(1) C (12) 22
(1)POP CLM×−×××
(2) C (13) 22
(1)POP CLM×−×××
(3)
M
(14)
P
CLM×××
(4) CL× (15)
P
CLM×××
(5) P (16)
[]
1(PCLM OPLM OP××× + +× × −1)
(9)
P
OP C P M M C L××+×+×× (17)
P
OP×
(10)
P
OP C L M×××× (18) CLM××
(11)
P
OP C L M×××× (19) C
4. Numerical Illustration
To verify the behavior of the proposed model, two
numerical examples are presented to illustrate
applicability of the proposed model when various
values of cell utilization level are defined by the
decision maker. These examples are solved by a branch
and-bound (B&B) method with the LINGO 8.0
Software.
Example 1.
Table 3 shows the sequence data pertaining to the
problem consisting of 5 machines and 7 parts in which
each part type is assumed to have two or three
operations that must be processed respectively as
numbered in the order with the processing time shown
in the parentheses. For instance, the first operation of
part 1 should be processed on machine 4 with
processing time 0.51 hours. In Table 3, the last three
columns include the machine information (i.e., number
of each machine type available, capacity and
investment cost for their single copy). The last row of
this table presents the demand of each part. Table 4
shows the input parameters for solving the above
problem with two different cell utilization levels. Also,
the distance between the locations of a cell is shown in
Table 5. run without the fourth term of objective
function (without cost of machines). Tables 7 and 8
show the solution of the model with the cost of
machine for the second and the third run.
Tab.3. 5 x 7 machines- part matrix in [27]
i
j 1 2 3 4 5 6 7 Nj Cj Tj
1 0 1 (0.33) 0 2 (0.72) 0 0 1 (0,57) 2 600 200
2 0 0 2 (0.52) 0 1 (0.62) 0 0 2 900 350
3 2 (0.31) 2 (0.44) 0 3 (0.37) 0 0 2 (0.22) 2 750 200
4 1 (0.51) 0 1 (0.63) 0 0 1 (0.52) 0 2 700 200
5 0 0 3 (0.4) 1 (0.61) 2 (0.35) 2 (0.25) 0 2 600 350
Di
80 110 140 95 120 80 135
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Tab. 4. Parameter setting for example 1
Parameter Cell
I Cell
II
Lk 2 2
Uk 4 4
Forward intra cell movement unit cost 3
Backward intra cell movement unit cost 11
Inter cell movement cost 35
Min_Utk (first run) (without machine cost) 0.4 0.4
Min_Utk (second run) (with machine cost) 0.4 0.4
Min_Utk (third run) (with machine cost) 0.4 1
Tab. 5. The location distance in a cell.
Location
1 2 3 4
1 - 1 2 3
2 1 - 1 2
3 2 1 - 1
Location
4 3 2 1 -
Tab. 6. The cell formation of the first run without
machine cost
Machine
4 5 1 3 3 4 2 5
1 1 0 0 2
P 2 0 0 1 2
a 4 0 1 2 3
r 6 1 2 0 0
t 7 0 0 1 2
3
0 1 2 3
5
0 0 1 2
Tab.7. The cell formation with min_ut1 = 0.4,
min_ut2 = 0.4 (second run)
Machine
4 2 5 1 3
3 1 2 3
P 5 0 1 2
a 6 1 0 2
r 1 1 0 2
t 2 1 2
4 1 2 3
7 1 2
Tab. 8. The cell formation with min_ut1 = 0.4,
min_ut2= 1 (third run)
Machine
4 2 5 1 3
1 1 0 0 2
P 3 1 2 3
a 5 0 1 2
r 6 1 0 2
t 2 1 2
4 1 2 3
7 1 2
Tab. 9. 7 × 14 machine- part matrix
j
i 1 2 3 4 5 6 7 Di
1 0 3(.25) 0 0 0 2(.67) 1(.52) 200
2 2(.43) 0 0 0 1(.56) 0 0 130
3 0 0 0 1(.78) 0 0 0 90
4 0 2(.66) 0 0 0 0 1(.35) 85
5 1(.45) 3(.26) 0 0 0 0 2(.77) 110
6 0 3(.24) 0 0 0 2(.18) 1(.55) 125
7 0 1(.85) 0 2(.42) 0 0 0 145
8 1(.33) 0 0 0 2(.66) 3(.15) 0 110
9 1(.86) 0 0 2(.4) 0 0 3(.24) 95
10 0 0 1(.72) 0 0 0 2(.6) 60
11 0 0 2(.64) 1(.21) 0 0 0 80
12 0 0 2(.5) 1(.15) 0 0 0 190
13 0 0 1(.9) 0 0 0 0 65
14 1(.32) 0 0 0 2(.3) 0 0 115
Nj
2 2 2 2 2 2 2
Cj
400 550 320 600 240 520 200
T
j
250 300 250 360 300 340 400
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I. Mahdavi, M. M. Paydar, M. Solimanpur & M. Saidi-Mehrabad A Mathematical Model for…
Example2.
In solving this example, we consider 7 different types
of machines, 14 part types from the literature (Wu et al.
2007). The input data of this example are given in
Tables 9 and 10.
The cells generated and the part assignment to various
cells for minimum cell utilization (0.5, 0.5, and 0.5) is
given in Table 11. For second run in example 2, the
material handling cost between cells has been
increased for the proposed model to 110 and the results
have been shown in Table 12.
Tab. 10. Parameter setting for example 2.
Parameter Cell I Cell II Cell III
Lk
2 2 2
Uk
4 4 4
Forward intra cell movement unit
cost 7
Backward intra cell movement unit
cost 20
Inter cell movement cost 70
Min_Utk (first run ) 0.5 0.5 0.5
Min_Utk (second run) 0.5 0.5 0.5
Tab. 11. The cell formation with min_ut1 = 0.5, min_ut2 = 0.5, min_ut3 = 0.5
Tab. 12. The cell formation for second run with the increased inter cell movement cost
Part
3 9 10 11 12 13 2 8 14 1 4 5 6 7
M 4 1 2 0 1 1 0 2
a 3 0 0 1 2 2 1
c 7 0 3 2 0 0 0
h 1 1 2 1 1 1
i 5 1 2 2
n 7 1 1 2 1 0
e 6 3 2 0 0 2 0
2 3 2 3 3 1
5. Discussion
Table 13 includes obtained results from the Lingo
solver in terms of the intra-cell forward and backward,
inter-cell, machine investment cost and objective
function value (OFV) for first and second runs in
Example 1. The results show that if we exclude the
machine investment cost in the objective function, then
we face with more real cost in the system. This is due
to using more machines for reducing inter-cell
movement costs. Moreover, the number of voids has
been increased, while the number of exceptional
elements is zero. Therefore, with same lower bound of
cell utilization, we achieved better results for the cell
formation on total real cost and voids, when we
consider the machine investment cost in the objective
function. Total real cost includes material handling cost
and machine investment cost, whenever the objective
function is either with or without machine investment
cost. In the first run the value of OFV is 33, but we
added the machine investment costs resulting the total
real cost of 33+5600=5633. In the second run the value
of OFV equals to the total real cost of 3644.
Part
1 4 5 6 7 10 3 11 12 13 2 8 9 14
M 7 1 1 2 1 0 2 3
a 6 2 0 0 2 0 0 3
c 2 3 2 3 3 1 0
h 4 2 1 1 1 0 2
i 3 1 0 2 2 1
n 1 1 2 1 1 1
e 5 1 2 0 2
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The utilization concept is considered as number of non-
zero components of each cell divided by whole
components of that cell, and it is useful for decision
maker.
By changing utilization of cell II from 0.4 to 1, the
solution of Table 8 is formed, and new part family has
been obtained. The results of two runs for Example 2
in Tables 12 and 13 illustrate the superiority of the
proposed model, when the inter-cell material handling
cost has been increased. Table 14 shows that the
number of machines as a decision variable can be
added to reduce the number of exceptional elements
though the number of voids increases from 9 to 12.
Tab. 13. Computational results of 5×7 machine-part
problem
With machine
cost Without machine
investment cost
Forward intra cell cost 24 33
Backward intra cell cost 0 0
Inter-cell cost 70 0
Machine investment cost 3550 5600
OFV 3644 33
Total real cost 3644 5633
Voids 3 12
Exceptional elements 2 0
Machine added 0 3
Tab.14. Computational results of 7×14 machine-part
problem
First run Second run
Inter-cell cost 420 440
OFV 3754 3770
Voids 9 12
Exceptional elements 6 4
Machine added 0 1
6. Conclusions
In this paper, we proposed a new mathematical
model which addresses the joint problems of the cell
formation and machine layout in cellular
manufacturing based on sequence data under cell
utilization levels. Then, we used a transformation
approach to convert the non-linear model to a linear
programming. The previous methods to cellular
manufacturing do not consider both the cell formation
problem and layout design in a same linear
mathematical model. We have considered inter-cell
movement cost and forward and backward intra-cell
movement cost parameters and also the minimum of
the cell utilization, which is based on the designer view
point. As it is known, in the manufacturing systems
with linear layout of the machines, the parts backward
movements incur more expenses than what we
considered in the proposed model.
The advantages of this study with respect to the recent
studies were as follows:
Consideration of machine layout in cellular
manufacturing.
Calculation of forward and backward intra-cell
material handling costs by considering the
operation sequence and the distance between the
locations assigned to machines.
Calculation of the cost of intra-cell material
handling between same machine types in different
locations accurately.
Two problems have been adopted from the literature
and solved by the Lingo 8.0 at different cell utilization
levels. We compared the results of cell formation
between two cases, i.e., with and without machine
investment cost as a term of objective function.
Moreover, the role of machine replication in the system
is shown in example 2. Application of meta-heuristics
like tabu search, simulated annealing and genetic
algorithm can be investigated in future researches to
solve large-sized problems.
Acknowledgement
Iran National Science Foundation (INSF) for the
financial support of this work.
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