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J Intell Manuf (2008) 19:7185
DOI 10.1007/s10845-007-0046-4
A methodology to incorporate product mix variationsin cellular manufacturing
Amit Bhandwale Thenkurussi Kesavadas
Published online: September 2007 Springer Science+Business Media, LLC 2007
Abstract The identification of part families and
machine groups that form the cells is a major step in thedevelopment of a cellular manufacturing system and,
consequently, a large number of concepts, theories and
algorithms have been proposed. One common assump-
tion for most of these cell formation algorithms is that
the product mix remains stable over a period of time.
In todays world, the market demand is being shaped
by consumers resulting in a highly volatile market. This
has given rise to a new class of products characterized
by low volume and high variety. To incorporate prod-
uct mix changes into an existing cellular manufactur-
ing system many important issues have to be tackled.
In this paper, a methodology to incorporate new partsand machines into an existing cellular manufacturing
system has been presented. The objective is to fit the
new parts and machines into an existing cellular manu-
facturing system thereby increasing machine utilization
and reducing investment in new equipment.
Keywords Group technology Cellular
manufacturing Product mix variations
Introduction
Group Technology (GT) is a philosophy for identifying
and exploiting similarities of product design and manu-
facturing processes throughout the manufacturing cycle.
A. Bhandwale T. Kesavadas (B)Department of Mechanical & Aerospace Engineering,University at Buffalo, 1006 Furnas Hall,Buffalo, NY 14260, USAe-mail: [email protected]
One of the objectives is to increase customization lead-
ing to a higher product variety with a lower product vol-ume. Cellular Manufacturing (CM) is the application of
the GT concept to manufacturing. This involves group-
ing similar parts together into part families which are to
be processed by dedicated clusters of machines/manu-
facturing processes called cells. The origins of GT/CM
can be traced back to as early as 1940 when it was pio-
neered on a large scale by the Russians, British and
Germans. Decades of research have brought this field
to its current state and have helped prove that adop-
tion of CM reduces setup times, in-process inventory,
tooling, and enhances product quality. A large number
of researchers have put forth numerous concepts, the-ories and algorithms to solve the Cell Formation (CF)
problem. But, there is one aspect of the CF problem
which very few researchers have delved into. What if
new part families are introduced into an existing CM
system? Is it then expedient to alter the existing lay-
out? If yes, how do we assign the new part families into
the existing layout? If no, should new cells be formed
for the new part families? What if the new part fami-
lies require new machines? The redesign of such a sys-
tem involves issues such as reformation of part families
and machine groups, relocation expenses for existing
machines and investment of new machines and materialhandling equipment. In this paper, some of these issues
have been addressed and a methodology to incorporate
product mix changes has been proposed. In this section
the problem domain is described and issues associated
with product mixvariations are discussed. In Sect. Prior
work," previous work done and applications developed
are discussed. In Sect. A methodology to incorporate
product mix variations," a formal methodology to incor-
porate product mix changes in a CM system is presented.
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In Sect. Performance evaluation," performance evalu-
ation in the case of product mix variations is discussed.
In Sect. Implementation," the proposed methodology
is implemented on large datasets with high product mix
changes and in Sect. Mathematical computations," the
computations involved are described.
Prior work
Problems encountered upon introduction of new parts
and machines
In manufacturing, we come across a variety of part
classes; two of which includethe high-volume, low-
variety parts and the medium/low-volume, mid-variety
parts. The high-volume, low-variety parts constitute the
commodity items for which there typically is a large
and steady demand. Alternatively, the medium/low-vol-
ume, mid-variety parts tend to be special order itemsfor which the demand is typically unsteady. These parts
often result from the need to meet the requirements of
a large and varied customer base. For these parts, flex-
ibility and production volume is of prime importance
(Singh, 1996). One of the implied assumptions in the
modeling and development of CM systems is that the
product mix remains stable over time. Over the years,
there has been a shift of power in the global econ-
omy, in shaping the market demand, from producers to
consumers. Consequently, manufacturers must continu-
ously respond to market changes. Also, over a period of
time, design changes take place and many existing partsare replaced by variants or new parts. This necessitates
allocation of resources such as machines, material han-
dling equipment, jigs, fixtures, and personnel to manu-
facture these parts. To incorporate product mix changes
in a CM system raises many issues.
Review of prior work
Very few researchers have addressed issues regarding
product mix and their subsequent handling. A syntac-
tic pattern recognition approach has been developed by
Wu, Venugopal and Barash (1986). Their method uses
operation sequences to determine distances between the
part families. Operation sequences on machines are rep-
resented by numerical strings. The Levenshtein distance
(Fu, 1998) between two such strings x and y is the small-
est number of transformations required to derivey from
x. Such strings are used to form dendograms which then
can be grouped at various threshold levels to give differ-
ent cell designs. This approach is similar to the Similarity
Coefficient Approach (McAuley, 1972). In case of the
introduction of a new part family, Levenshtein distances
are calculated between the new part family andthe exist-
ing cells. The new part family is assigned to that cell with
which it shares the minimum Levenshtein distance. This
method does not address introduction of multiple new
parts and new machines. A large number of machines
will result in larger operation sequences (strings) and a
large number of parts will increase the number of com-parisons that are required between two such operation
sequences.
Tam (1990) has proposed an operation sequence
based weighted similarity coefficient for drawing on sim-
ilar patterns of operation sequences, not on machine
requirements, for part family formations. This similar-
ity coefficient is also based on the Levenshtein distance
measure of two sequences. These distance measures are
then converted to a similarity coefficient which is used to
group parts by applying the k-Nearest-Neighbor cluster-
ing procedure (Wong & Lane, 1983), a density linkage
clustering technique based on nonparametric probabil-ity density estimates. The new part is assigned to the
part family with which it shares the least distance or
most similarity. A threshold value can be decided upon
to aid the planner in decision making. If the distance
between the new part and its closest group exceeds the
threshold value, a new group is created for the new part.
This approach is not concrete enough. Why to create a
new family for a part which is visiting a cell also visited
by some other part family? A better approach would
have been to create a new part family when the new
part has no operations on any of the existing machines.
Seifoddini (1990) developed a probabilistic modelto overcome assumptions of deterministic demand for
parts. A variety of productmixes with different probabil-
ities of occurrence is used to give different part machine
incidence matrices, which are then used as input to an
existing grouping algorithm.
Rajamani and Szwarc (1994) presented a mathemat-
ical programming model for maximizing the profit asso-
ciated with reduced intercellular movement. The model
takes into account production data such as machine con-
sumption rates (labor, energy, and maintenance) and
costs associated with material handling, relocation, and
sale of parts. A computerized procedure was devel-
oped and examples of machine relocation have been
provided. But, machine relocation is still impractical if
frequent product mix changes occur and the demand
associated with the new parts is not stable. The model
has a high input data requirement and needs to be solved
each time a product mix change occurs.
Harhalakis, Harhalakis, Ioannou, Minis, and Nagi
(1994)have presented a methodology that aims to obtain
robust shop decompositions with satisfactory perfor-
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J Intell Manuf (2008) 19:7185 73
mance over a certain range of demand variation. Their
method takes into account the independent demand,
production capacity and operation sequences and tries
to minimize the material handling cost. The intermedi-
ate steps are mathematical formulations within them-
selves which renders the method as robust but complex.
Also, the method seems more suited for solving a CF
problem under random product demand rather thanincorporating a random product mix into an existing
CM system.
Seifoddini and Djasemmi (1997) have performed sen-
sitivity analysis of the performance of a CM system with
respect to changes in product mix. They defined a flex-
ibility range, calculated with a simulation model, which
represents the capability of the system in dealing with
product mix changes. They showed that changes in prod-
uct mix lead to the deterioration of the performance of a
CM system in terms of mean flow time and work In Pro-
gress (WIP) inventories. Wei and Gaither (1990) argued
that CM is relatively inflexible to changes in productmix and volumes. They indicated that CM is restricted
to parts that areof moderate and stable volume and thus
not subject to great variation. But, with improved pro-
cess planning and part standardization, a cellular setup
should be able to accommodate introduction of new
products.
Akright and Kroll (1998) presented various perfor-
mance measures, which gauge the potential effects the
addition of new part families on the overall cost of the
layout, to decide whether or not to change an existing
layout to incorporate a new part family. The decision
is related to the profit target of the particular machinecell in question and, therefore, the addition of a new
part family is expected have a significant impact on
the Profit Margin (PM). According to this method, if
the new part family results in incremental revenue to
the machine cell, the layout should be changed. If, in
spite of the incremental revenue, the PM decreases, the
layout should not be changed and outsourcing should
be considered as an alternative. This method has been
explained with the help of only one new part and the
possibility of the introduction of new machines has not
been considered.
Kao and Moon (1998) proposed a different approach
for part family formation and multiple-application set
(machine cells, cutting tool sets, and canned cycle sets)
formation using feature-based memory association per-
formed by neural networks. New and modified parts can
be assigned to the correct part family without having to
repeat the whole part clustering algorithm again.
Wicks and Reasor (1999) formulated the CF problem
that addresses the dynamic nature of the production
environment by considering a multi-period forecast of
product mix and demand during the formation of part
families and machine cells. The goal of the multi-period
formulation is to obtain a cellular design that continues
to perform well with respect to the design objectives
as the part population changes with time. This method
addresses both product mix as well demand but again
during the CF stage rather than incorporating it into an
existing CM system.Ko and Egbelu (2003) proposed a concept known as
Virtual Cellular Manufacturing System (VCMS) which
is suitable for production environments subjected to fre-
quent product mix changes. In VCMS, the shop floor
configuration is changed in response to changes in the
product mix over time. In addition, virtual manufactur-
ing cells are simply logical cells in which the machines
belonging to the same cell need not occupy the same
contiguous area. But, the CF process has to be repeated
each time the product mix changes or when the changes
are sufficiently significant to warrant a new cell layout.
There is no cost measure associated with all the machinerelocation that takes place upon changes in the product
mix.
A methodology to incorporate product mix
variations
When new part families are introduced, two cases arise
1. The new part family has all its operations on exist-
ing machines, i.e., no new machines are introduced.Such a case can be observed when a company intro-
duces a variant of an existing design.
2. The new part family has some operations on
machines not in the existing layout, i.e., new mach-
ines need to be introduced. Such a case can be
observed when a company introduces a new tech-
nology in the design.
According to Shafer and Rogers (1991), there are four
fundamental design objectives associated with CM: (1)
Setup time reduction, (2) Production of mutually sep-
arable clusters, (3) Minimize investment in new equip-ment, and (4) Maintain acceptable machine utilization
levels. In the method presented here, we try to conform
to some of the above objectives.
Objectives and features
To form mutually separable clusters (cells), if they
exist or keep any existing ones intact thereby adher-
ing to the second design objective.
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Table 1 Nomenclature m Number of rows (machines)
n Number of columns (parts)A [aij] Machine-part incidence matrix, where i = 1, 2, , m, j= 1, 2, , nB [bij] = [A
TX A] i,j= 1, 2, . . . , n matrix multiplication of AT and A
B [bij] = [AT* A] i,j= 1, 2, . . . , n matrix dot product of AT and A
Existing part families are not to be altered. Existing
machine groups are also not to be altered. This will
help in achieving the third and fourth design objec-
tives viz. minimizing investment in new equipment
and maintaining acceptable machine utilization lev-
els. If the introduction of new parts calls for the intro-
duction of new machines, the machines are to be
assigned to the existing cells before the parts. This is
because assigning new machines to existing cells, as
opposed to the creation of a new cell, will help reduce
costs associated with investment in new equipmentand labor.
Case 1 involves only new part assignment and can
be considered as a single problemPart Assignment
Problem (PAP). Case 2 handles new machine assign-
ment as well as new part assignment and can be
divided into two problemsMachine Assignment
Problem (MAP) and PAP.
The methodology presented here incorporates the
concept of the matrix dot product (Venugopal &
Narendran, 1993). Referring to Table 1, consider
matrix A having m rows and n columns. The trans-
pose of A, denoted by AT will have n rows and mcolumns. Dot Product is defined as the matrix multi-
plication of the transpose (AT) and the input matrix
(A) or vice-versa, where all elements greater than 0
are interpreted as 1s. The resulting matrix (B) will be
a 01 binary matrix of order nby n(if AT*A) ormby
m (if A * AT). Table 2 illustrates the dot product
concept.
The product B and the dot product B contain valu-
able information. The diagonal elements [bii] of B
indicate the maximum number of operations on each
part. Every other element [bij] represents the num-
ber of common operations for parts i and j. Eachelement of B can be interpreted as the similarity
between row i of A and column jof AT. It indicates
whether part i has a relationship with part jbased on
their visiting the same machine. For a part to qual-
ify as a non-bottleneck part there should be at least
one other part with which it shares the same num-
ber of operations (Nair & Narendran (1997)). B, in
Table 2, can be considered as a modified form of a
conventional similarity coefficient matrix.
Table 2 Concept of matrix dot product
A AT
Part Machine
Machine 1 2 3 4 Part 1 2 3 4 5
1 1 0 1 0 1 1 0 1 1 0
2 0 1 0 1 2 0 1 1 0 1
3 1 1 1 0 3 1 0 1 0 0
4 1 0 0 0 4 0 1 0 0 1
5 0 1 0 1
B = AT
X A B = AT *
A
Part Part
Part 1 2 3 4 Part 1 2 3 4
1 3 1 2 0 1 1 1 1 0
2 1 3 1 2 2 1 1 1 1
3 2 1 2 0 3 1 1 1 0
4 0 2 0 2 4 0 1 0 1
Table 3 PAPinitial groups
Cell 1 Cell 2 Cell 3
Machines 1, 2, 3 4, 5 6, 7, 8Parts 1, 2, 3, 4 5, 6, 7 8, 9, 10, 11
The part assignment problem (PAP)
Consider an existing CM system consisting of eight
machines and 11 parts forming three cells (Table 3).
Parts 12, 13, 14, and 15 are to be incorporated into the
system.
Step 1The input matrix is a typical machine-part
matrix representing the current cellular layout. It is
in a block-diagonalized form, i.e., with the 1s (oper-
ations) clustered along the matrix diagonal. The new
parts (12, 13, 14, and 15) are appended as shown in
Table 4. Rows represent machines and columns
represent parts.
Step 2Calculate the transpose (interchanging row and
column elements) of the input matrix as shown in Table
5. Now, the rows represent parts and columns represent
machines.
Step 3The dot productof the transpose and the input
matrix is calculated. This will yield a symmetric matrix
with 15 rows and 15 columns (Table 6) which represents
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Table 4 PAPinitial cell layout
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 1 1 1 1 1
2 1 1 1 1 1
3 1 1 1
4 1 1 1 15 1 1 1
6 1 1 1 1 1
7 1 1 1 1 1
8 1 1 1 1
parts
sen
ihcam
Table 5 PAPtranspose
1 2 3 4 5 6 7 8
1 1 1 1
2 1 1
3 1 1
4 1 1 1
5 1 1
6 1
7 1 1
8 1 1 1
9 1 1
10 1 1
11 1 1 1
12 1 1
13 1 1
14 1 1
15 1 1 1
machines
parts
the relationship between parts based on the machines
they visit.
Step 4In the matrix obtained after Step 3, if all col-
umns of row i have a 1 (and therefore all rows of col-
umn jbecause of symmetry), it indicates that part i has
a relationship with every other part in the dataset (with
respectto their visiting the same machine). Part i hinders
the formation of mutually separable clusters and hence
warrants removal of the ith row and column from fur-
ther consideration. Part i is referred to as an excep-
tional part. Ignoring bottleneck machines/exceptional
parts during block diagonalization is a common practice
and has been adopted by King and Nakornchai (1982).
In this method, it is accomplished by making all its ele-
ments 0 so that after the succeeding operations sort (see
Step 5), it gets pushed to the last row and column of
the matrix. This serves a dual purpose in that the part is
no more involved in following steps but its presence in
the matrix is a reminder that it needs to be dealt with
after initial part families and machine groups have been
Table 6 PAPdot product
parts
parts
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 1 1 1 1 1 1
2 1 1 1 1 1
3 1 1 1 1 1 1
4 1 1 1 1 1 1
5 1 1 1 16 1 1 1 1
7 1 1 1 1
8 1 1 1 1 1 1
9 1 1 1 1 1 1
10 1 1 1 1 1 1
11 1 1 1 1 1 1
12 1 1 1 1 1 1
13 1 1 1 1 1 1
14 1 1 1 1
15 1 1 1 1 1 1 1 1 1 1
Table 7 PAPoperations sortparts
strap
15 1 3 4 8 9 1 0 11 12 13 2 5 6 7 1 4
15 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1
3 1 1 1 1 1 1
4 1 1 1 1 1 1
8 1 1 1 1 1 1
9 1 1 1 1 1 1
10 1 1 1 1 1 1
11 1 1 1 1 1 1
12 1 1 1 1 1 1
13 1 1 1 1 1 1
2 1 1 1 1 15 1 1 1 1
6 1 1 1 1
7 1 1 1 1
14 1 1 1 1
formed. From Table 6, we see that there are no such
parts and hence, we go to Step 5.
Step 5The rows of the matrix obtained from Step 4
are arranged in decreasing order of the number of oper-
ations, i.e., 1s, from top to bottom. Next, the columns
are arranged in decreasing order of the number of oper-
ations from left to right. These two operations result in
the formation of an ordered matrix with the 1s collect-
ing across and around the diagonal. This is called an
Operations Sort and is carried out so as to group parts
with the same number of operations (Table 7).
Step 6At this stage, the blocks of 1s in the matrix
may not result in mutually separable clusters. This is
because the parts were sorted based only on the number
of operations without considering the similarity between
them. The mutually in-separable clusters is a result of
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Table 8 PAPexceptional part identification and removal
parts
parts
1 3 4 8 9 10 11 12 13 2 5 6 7 14 15
1 1 1 1 1 1
3 1 1 1 1 1
4 1 1 1 1 1
8 1 1 1 1 1
9 1 1 1 1 110 1 1 1 1 1
11 1 1 1 1 1
12 1 1 1 1 1
13 1 1 1 1 1
2 1 1 1 1 1
5 1 1 1 1
6 1 1 1 1
7 1 1 1 1
14 1 1 1 1
15 0
exceptional elements which were not detected in Step4 simply because they have a relationship with mostof
the other parts, but not all, i.e., they have operations
on most of the machines and hence require processing
in multiple cells. There is an easy way to identify these
exceptional parts. The user has to inspect the matrix at
the end of Step 5. If there is an exceptional part, it will
collect at the topmost row and leftmost column in the
matrix. This is so because that part has more operations
(1s) than any other part and hence, after an operations
sort, will end up at the top of the matrix. This part is
eliminated from further consideration by making all its
elements 0 (Table 8). Then, an operations sort is carriedout again. These two steps, exceptional part identifica-
tion/removal and operations sort are repeated for every
exceptional part.
Step 7The dataset at the end of the recursive Step 6
is inspected. If there are a number of parts with similar
number of operations, the algorithm will give a partially
ordered matrix as shown in Table 9. Parts 2, 4 and 12
should have collected below parts 1, 3 and 11. But, as
parts 5, 6, 7, 8, 10, 2, 4 and 12 share the same number
of operations, the algorithm cannot distinguish between
them. Hence, a Precedence Sort is performed to group
similar parts together. The matrix is scanned from left to
right, two rows at a time and every element is compared.
If both rows have a 1 or 0, the scan proceeds to the next
element. If the first row in the pair has a 0 and the sec-
ond one has a 1, the second row has precedence over the
first and hence the positions of both rows and their ele-
ments are swapped. Here, there is no need to compare
the remaining elements and hence the procedure is like
a very quick sort. As the matrix is symmetric, a Prece-
dence Sort on the rows is enough because the columns
Table 9 PAPprecedence sort
parts
strap
1 3 4 12 2 10 11 8 13 9 5 6 7 14 15
1 1 1 1 1 1
3 1 1 1 1 1
4 1 1 1 1 1
12 1 1 1 1 1
2 1 1 1 1 110 1 1 1 1 1
11 1 1 1 1 1
8 1 1 1 1 1
13 1 1 1 1 1
9 1 1 1 1 1
5 1 1 1 1
6 1 1 1 1
7 1 1 1 1
14 1 1 1 1
15 0
Table 10 PAPprocess based layoutparts
senihcam
1 3 4 1 2 2 5 6 7 1 4 10 11 8 13 9 15
1 1 1 1 1 1
2 1 1 1 1 1
3 1 1 1
4 1 1 1 1
5 1 1 1
6 1 1 1 1 1
7 1 1 1 1 1
8 1 1 1 1
get arranged automatically. Table 10 represents Table 9
after a precedence sort.
Step 8Output the part families. From Table 9, these
are {1, 2, 3, 4, 12}, {8, 9, 10, 11, 13}, {5, 6, 7, 14}, and the
exceptional set is {15}.
Step 9At the end of Step 8, the part families and
machine groups are known. But, which part family visits
which machine group is still to be determined i.e., the
part families have to be assigned to the machine groups
to form the cells. This is taken care of by the algorithm
which re-arranges the columns of the machine-part inci-
dence matrix according to the order in Table 9 (i.e., 1 3
11 2 4 12 8 10 5 6 7 9). This way, the solution of the part
assignment problem can be represented as a machine-
part matrix. After assigning part families to the machine
groups, part 9 appears as the last column. This part has
to be dealt with and is usually assigned to the cell in
which it has the maximum number of its operations.
Step 10Exceptional elements will be treated accord-
ing to nature of the layout desired. For a process based
layout, part 15 is assigned to part family {8, 9, 10, 11, 13}
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Table 11 PAPprocess based groups
Cell 1 Cell 2 Cell 3
Machines 1, 2, 3 4, 5 6, 7, 8Parts 1, 2, 3, 4, 12 5, 6, 7, 14 8, 9, 10, 11, 13
Table 12 PAPproduct based layout
parts
machines
1 3 4 12 2 5 6 7 14 10 11 8 13 9 15
1 1 1 1 1 1
2 1 1 1 1
3 1 1 1
4 1 1 1 1
5 1 1 1
6 1 1 1 1 1
7 1 1 1 1 1
8 1 1 1 1
2* 1
Table 13 PAPproduct based groups
Cell 1 Cell 2 Cell 3
Machines 1, 2, 3 4, 5 6, 7, 8, 2Parts 1, 2, 3, 4, 12 5, 6, 7, 14 8, 9, 10, 11, 13
and routed to the other cell for operation on machine 2
(Table 11).
For a product based layout, another machine of type 1 is
assigned to Cell 2 so that part 11 is completely processedin a single cell (Table 12).
Thedecision regarding exceptionalelements needscare-
ful deliberation. Duplicating machine 2 just for the sake
of one operation on part 15 might not be economically
feasible (Table 13). On the other hand, if part 15 is a
high revenue part, the cost associated with duplicating
machine 2 might be recovered over a period of time.
Otherwise, the best option would be to assign the part
to thecell where most of its operations will be performed
and route it to the other cell(s) for the remaining oper-
ations.
The machine assignment problem (MAP)
Consider an existing cellular layout consisting of 8
machines and 11 parts forming three cells. Parts 12, 13,
14 and 15 are to be incorporated into the above layout.
But, these parts have operations on machines not pres-
ent in the existing layout (Table 14). The new machines
9 and 10 have to be assigned to the cells before incorpo-
rating the new parts.
Table 14 MAPinitial groups
Cell 1 Cell 2 Cell 3
Machines 1, 2, 3 4, 5 6, 7, 8Parts 1, 2, 3, 4 5, 6, 7 8, 9, 10, 11
Table 15 MAPinitial cell layout
parts
senihcam
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 1 1 1 1 1
2 1 1 1 1 1
3 1 1 1
4 1 1 1 1
5 1 1 1
6 1 1 1 1 1
7 1 1 1 1
8 1 1 1 1
9 1
10 1 1
Step 1The input matrix is a typical machine-part
matrix representing the current cellular layout. The new
parts (12, 13, 14, and 15) are appended as shown in
Table 15. Rows represent machines and columns repre-
sent parts.
Step 2The machines are assigned using the Aver-
age Linkage Clustering (ALC) approach (Seifoddini,
1989a). This method defines the similarity coefficient
between a single machine and a machine group as the
average of the similarity coefficients of the singlemachine with all members of the machine group.
S[B,Cell(AD)] = {S[B,A] + S[B,D]}/2 (4.1)
where S[B,Cell(AD)] is the similarity coefficient between
machines B and Cell AD; S[B,A] is the similarity coeffi-
cient between machines B and A; S[B,D] is the similarity
coefficient between machines B and D
Sij =
N
k=1
Xijk
N
k=1
Yik + Zjk Xijk
(4.2)
where Xijk = operation on part k performed on both
machine i and j; Yik = operation on part k performed
on machine i; Zjk = operation on part k performed on
machine j
Cell 1
S(9, 1) = 1/(5 + 1 1) = 0.2
S(9, 2) = 1/(5 + 1 1) = 0.2
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Table 16 MAPnew machine assignment
parts
machin
es
1 2 3 4 5 6 7 8 9 1 0 11 12 13 14 15
1 1 1 1 1 1
2 1 1 1 1 1
3 1 1 1
9 1
4 1 1 1 15 1 1 1
6 1 1 1 1 1
7 1 1 1 1
8 1 1 1 1
10 1 1
S(9, 3) = 0/(3 + 1 0) = 0
S(9, Cell1) = {S(9,1) + S(9,2) + S(9,3)}/3
= (0.2 + 0.2 + 0)/3 = 0.133
Cell 2 S(9, 4) = 0/(4 + 1 0) = 0
S(9, 5) = 0/(4 + 1 0) = 0
S(9,Cell2) = {S(9,4) + S(9,5)}/2 = 0
Cell 3
S(9, 6) = 0/(5 + 1 0) = 0
S(9, 7) = 0/(4 + 1 0) = 0
S(9, 8) = 0/(4 + 1 0) = 0
S(9,Cell3) = {S(9, 6) + S(9, 7) + S(9, 8)}/3 = 0
As, S(9, Cell 1) > S(9, Cell 2) and S(9, Cell 3), machine 9 isassigned to Cell 1. Similarly, S(10,Cell 3) > S(10,Cell 1)and
S(10,Cell 3). Hence, machine 10 is assigned to Cell 3. The
input matrix is modified by appending machines 9 and
10 below machines 3 and 8 respectively (Table 16).
Step 3Calculate the Transpose of the input matrix
(AT) as shown in Table 17.
Step 4Take the Dot Product of the transpose and the
input matrix (AT* A). The resulting matrix (B) will be of
order 11 11 matrix and represents part families (Table
18).
Step5Identify rows (and by symmetry, columns) which
have 1s in all columns (rows). These are the exceptional
parts as they have a relationship with every other part
in he dataset. Remove these parts from the data set.
Step 6In the modified matrix, sort rows in decreasing
order of the number of operations from top to bottom.
Repeat the process for the columns (Table 19).
Step 7Inspect the data set at the end of Step 6. If a
block diagonal structure is obtained, go to Step 9, else
check for exceptional elements. Part 15 is an exceptional
part as it has a relationship with most of the parts in the
Table 17 MAPtranspose
parts
parts
1 2 3 9 4 5 6 7 8 10
1 1 1 1
2 1 1
3 1 1
4 1 1 1
5 1 1
6 1
7 1 1
8 1 1 1
9 1 1
10 1 1
11 1 1 1
12 1 1 1
13 1 1 1
14 1 1
15 1 1 1
Table 18 MAPdot product
parts
parts
1 2 3 4 5 6 7 8 9 1 0 11 12 13 14 15
1 1 1 1 1 1 1
2 1 1 1 1 1
3 1 1 1 1 1 1
4 1 1 1 1 1 1
5 1 1 1 1
6 1 1 1 1
7 1 1 1 1
8 1 1 1 1 1 1
9 1 1 1 1 1 1
10 1 1 1 1 1
11 1 1 1 1 1 1
121 1 1 1 1 1
13 1 1 1 1 1 1
14 1 1 1 1
15 1 1 1 1 1 1 1 1 1
Table 19 MAPoperations sort
parts
parts
15 1 3 4 8 9 1 1 12 13 2 10 5 6 7 1 4
15 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1
3 1 1 1 1 1 1
4 1 1 1 1 1 1
8 1 1 1 1 1 1
9 1 1 1 1 1 1
11 1 1 1 1 1 1
12 1 1 1 1 1 1
13 1 1 1 1 1 1
2 1 1 1 1 1
10 1 1 1 1 1
5 1 1 1 1
6 1 1 1 1
7 1 1 1 1
14 1 1 1 1
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Table 20 MAPexceptional element identification and removal
parts
parts
1 3 4 8 9 1 1 12 13 2 10 5 6 7 14 15
1 1 1 1 1 1
3 1 1 1 1 1
4 1 1 1 1 1
8 1 1 1 1 19 1 1 1 1 1
11 1 1 1 1 1
12 1 1 1 1 1
13 1 1 1 1 1
2 1 1 1 1 1
10 1 1 1 1 1
5 1 1 1 1
6 1 1 1 1
7 1 1 1 1
14 1 1 1 1
15 0
Table 21 MAPprecedence sort
parts
parts
1 3 4 12 2 11 8 13 9 10 5 6 7 14 15
1 1 1 1 1 1
3 1 1 1 1 1
4 1 1 1 1 1
12 1 1 1 1 1
2 1 1 1 1 1
11 1 1 1 1 1
8 1 1 1 1 1
13 1 1 1 1 1
9 1 1 1 1 1
10 1 1 1 1 1
5 1 1 1 1
6 1 1 1 1
7 1 1 1 1
14 1 1 1 1
15 0
dataset andis hence removed (Table 20). This is followed
by another Operations Sort.
Step 8Inspect the dataset at the end of Step 7. If a
block diagonal structure is not obtained, sort the rows
(columns) by precedence of 1s from left to right (top tobottom). This is called a Precedence Sort(Table 21).
Step 9Part families and exceptional parts obtained
Part families {1, 2, 3, 4, 12} {8, 9, 10, 11, 13} {5, 6, 7, 14}
Exceptional part {15}
Step 10Arrange the columns (part families) of the
modified input matrix according to the part families
obtained from Step 9 keeping the rows (machines)
unchanged (Table 22).
Table 22 MAPprocess based layout
parts
senihcam
1 3 4 12 2 5 6 7 14 11 8 13 9 10 15
1 1 1 1 1 1
2 1 1 1 1 1
3 1 1 1
9 1
4 1 1 1 1
5 1 1 1
6 1 1 1 1 1
7 1 1 1 1
8 1 1 1 1
10 1 1
Table 23 MAPprocess based groups
Cell 1 Cell 2 Cell 3
Machines 1, 2, 3, 9 4, 5 6, 7, 8, 10Parts 1, 2, 3, 4, 12 5, 6, 7, 14 8, 9, 10, 11, 13, 15
Table 24 MAPproduct based layout
parts
senihcam
1 3 4 12 2 5 6 7 1 4 11 8 13 9 10 15
1 1 1 1 1 1
2 1 1 1 1
3 1 1 1
9 1
4 1 1 1 1
5 1 1 1
6 1 1 1 1 1
7 1 1 1 1
8 1 1 1 1
10 1 1
2* 1
Table 25 MAPproduct based groups
Cell 1 Cell 2 Cell 3
Machines 1, 2, 3, 9 4, 5 6, 7, 8, 10, 2Parts 1, 2, 3, 4, 12 5, 6, 7, 14 8, 9, 10, 11, 13, 15
Step 11Exceptional elements are treated according to
the nature of layout desired. If a process based layout
is needed, the corresponding part is assigned to the cell
where it has the maximum number of operations. Hence,
part 15 is assigned to Cell 3 (Table 23).Ifaproduct based
layout is desired, machine 2 is duplicated and assigned
to Cell 3 (Tables 24 & 25).
This method will also work if machine groups and part
families need to be formed afresh due to the necessity for
machine relocation. The machine-part incidence matrix
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Table 26 Introduction of new parts into an existing cellular system
is modified using the new machine locations and solved
using the PAP.
Performance evaluation
For the PAP, visual identification is one way of checking
the results obtained from the methodology. As long as
the existing setup is not disturbed and there is no signifi-
cant increase in the number of exceptional elements, we
can say that the results obtained are correct. There areno standard examples or any in published literature to
test on as the papers which have addressed product mix
variations have not tried it out on a large dataset. For the
MAP, introduction of new parts calls for introduction of
new machines. These are visited only by the new parts
due to which the number of operations maybe signifi-
cantly less than that on the existing machines. When
these new machines are assigned to the existing cells,
the number of voids in the cells will mostly increase
(exception would be introduction of a large number of
parts and few machines so that number of operations
of the new and old machines are nearly equal). Hence,
any grouping measure will give a low value for the new
groupings implying that the new layout is worsethan the
original. Our objective is to incorporate new machines
and parts into an existing cellular layout thereby min-
imizing investment in new equipment, avoiding relo-
cation of existing machines and avoiding intercellular
movement of parts by forming mutually separable clus-
ters. Hence, a grouping measure will not reflect quality
of the solution accurately. A cost based measure can be
adopted to show that assigning machines to existing cells
helps save investment in new equipment as well as mate-
rial handling costs rather than placing new machines in
a separate cell and routing the parts between cells.
Implementation
This section tests the solutions on large datasets to dem-
onstrate its effectiveness and capability in handling largeproduct mix variations. Here, the methodology has been
implemented on systems with high product mix changes.
The examples in Sects. Example 1" and Example 2"
have been taken from cell formation problems that
appeared in the literature with new parts and machines
introduced to suit the case at hand.
PAP
Example 1
Consider an existing CM system consisting of 20
machines and 35 parts forming four cells. Seven new
parts are introduced signifying a 20% increase in prod-
uct mix (Table 26). This problem has been adopted from
Chandrasekharan and Rajagopalan (1986).
From the solution obtained (Table 27), it can be seen
that the new parts have been incorporated into the exist-
ing system without altering the existing part families
and machine groups. Comparing the solution (Table 28)
with the input (Table 29), it can be seen that the parts
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Table 27 New cellular manufacturing system
Table 28 New groups
Machines Parts
Cell 1 1, 2, 3, 4, 5 1, 2, 3, 4, 5, 6, 7, 8, 9, 35, 39Cell 2 6, 7, 8, 9, 10 10, 11, 12, 13, 14, 15, 16, 17, 18, 36Cell 3 11, 12, 13, 14, 15 19, 20, 21, 22, 23, 24, 25, 26, 27, 38, 40Cell 4 16, 17, 18, 19, 20 28, 29, 30, 31, 32, 33, 34, 37, 41
Table 29 Initial groups
Machines Parts
Cell 1 1, 2, 3, 4, 5 1, 2, 3, 4, 5, 6, 7, 8, 9Cell 2 6, 7, 8, 9, 10 10, 11, 12, 13, 14, 15, 16, 17, 18Cell 3 11, 12, 13, 14, 15 19, 20, 21, 22, 23, 24, 25, 26, 27Cell 4 16, 17, 18, 19, 20 28, 29, 30, 31, 32, 33, 34
are assigned to the part families without disturbing the
existing setup.
Example 2
Consider a randomly generated problem involving 22
parts and 14 machines forming four cells. There are two
machines of type 8, one in Cell 2 and the other in Cell
3. Eight new parts (36% increase in product mix) have
been introduced which have to be assigned to the exist-
ing cells (Table 30).
From the solution obtained (Tables 31 & 32), it can be
seen that the new parts have been incorporated into the
existing system without altering the existing part fami-
lies and machine groups.
MAP
Example 1
Consider the same CM system as before consisting of
20 machines and 35 parts forming four cells. Seven new
parts are introduced signifying a 20% increase in prod-
uct mix (Table 33). But, these parts have some opera-
tions which cannot be carried out on the existing
machines and hence new machines are also introduced.This problem has been adopted from Chandrasekharan
and Rajagopalan (1986).
From the solution obtained (Tables 34 & 35), it can be
seen that the new parts and machines have been incor-
porated into the existing system without altering the
existing part families and machine groups.
Example 2
Consider an existing system consists of 12 machines and
23 parts forming four cells. Eleven new parts are intro-
duced which have some operations on machines notpresent in the existing layout (Table 36). So, four new
machines are introduced and three of the new parts have
operations only on the new machines. The new groups
and layout are as shown in Tables 37 and 38, respectively.
Mathematical computations
For PAP, the machine-part matrix processing comprises
the main computational part. The method consists of
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Table 30 Introduction of new parts into an existing cellular system
Table 31 New cellular manufacturing system
straightforward multiplication of Boolean matrices. If
A = [aij] is an m n matrix and B = [bjk] is an n pmatrix, then their matrix product C = AB is the m p
matrix C = [cik]. Consider the scenario of square matri-
ces giving m = n = p. To multiply n n matrices a simple
matrix multiplication algorithm performs n3 multiplica-
tions and n2(n 1) additions giving a running time of(n3) (Cormen, Leiserson, and Rivest, 1990).
The sorting operations are ofexchange type and work
by exchanging pairs of items until the sequence is sorted.
The Bubble Sort is one of the simplest exchange type
sorting methods. It works by comparing each element
in the sequence with the element next to it, and swap-
ping them if the second one is greater than the first.
This process is repeated until it makes a pass all the way
through the sequence without swapping any item i.e.,
Table 32 New groups
Machines Parts
Cell 1 1, 2, 3 1, 2, 3, 4, 5, 6, 7, 8, 26, 27, 32Cell 2 4, 5, 6, 8 9, 10, 11, 12, 13, 24, 30, 31Cell 3 7, 8, 9, 10, 11 14, 15, 16, 17, 18, 19, 23, 28, 29Cell 4 12, 13 20, 21, 22, 33, 34
all elements are in the correct order. This results in the
largervalues bubbling totheendofthelist,whilesmaller
values sink towards the beginning of the list. The worst
case scenario occurs when elements have to be swapped
at every iteration giving a running time of (n2). The
bubble sort is very easy to understand and implement
and works very well on partially sorted sequences. The
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Table 33 Introduction of new parts and machines
Table 34 New cellular manufacturing system
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Table 35 New groupsMachines Parts
Cell 1 1, 2, 3, 4, 5, 22 1, 2, 3, 4, 5, 6, 7, 8, 9, 35, 39Cell 2 6, 7, 8, 9, 10, 24 10, 11, 12, 13, 14, 15, 16, 17, 18,36Cell 3 11, 12, 13, 14, 15, 23 19, 20, 21, 22, 23, 24, 25, 26, 27, 38, 40Cell 4 16, 17, 18, 19, 20, 21 28, 29, 30, 31, 32, 33, 34, 37, 41
Table 36 Introduction of new parts and machines
Table 37 New groupsMachines Parts
Cell 1 1, 2, 3, 13 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 34Cell 2 4, 5, 6 10, 11, 12, 13, 14, 28, 29Cell 3 7, 8, 9, 14, 15, 16 15, 16, 17, 18, 19, 20, 25, 26, 31, 32, 33
Cell 4 10, 11, 12 21, 22, 23, 30
Table 38 New cellular manufacturing system
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Quicksort is another exchange type sorting algorithm
which works in a divide-and-conquer style solving a
given problem by splitting it into two or more smaller
sub-problems, recursively solving each of the sub-prob-
lems, and then combining the solutions to the smaller
problems to obtain a solution to the original one. This
algorithm is faster than a bubble sort but is fairly tricky
to implement and debug and is highly recursive. Also,it is not very efficient on partially sorted sequences. A
detailed comparison of Bubble Sort and Quicksort can
be found in [1] and [2].
The Operations Sort and Precedence Sort can be con-
sidered as bubble sorts. As the dataset is partially sorted
to begin with, a worst case situation never arises. Hence,
the running time will always be less than (n2).
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1 http://www.math-info.univ-paris5.fr/ ycart/mst/mst031/Group10/first.html
2 http://people.cs.uct.ac.za/bmerry/manual/algorithms/sorting.html