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Design of a Cellular Manufacturing System for a Product Oriented Plant
B. Mahadevan1, S.Venkataramanaiah2, Janat Shah3
Abstract
Traditional approach to gearing up a cellular manufacturing system
(CMS) is to consider only two dimensions, viz, “machine –
component”. However, when the problem is addressed in a context of
“product – part – machine” dimension, it provides a different
perspective to the problem. The current study is an effort in this
direction. A formulation of the CMS problem to a product oriented
plant as well as a solution procedure has been proposed. A measure for
product ownership has been proposed. Practitioners often face the
difficult choice of what machines to dedicate to the cells and what to
keep centralised. The study has provided a quantitative basis for
resolving this conflict. The results show that while a high product
ownership can guarantee a high component ownership, the reverse
does not. The results underscore the need for including the “product”
dimension to the CMS design problem.
Third International Conference on
Operations & Quantitative Management (ICOQM – III) Sydney, December 17 – 20, 2000
1 Associate Professor, Production & Operations Management, Indian Institute of Management
Bangalore, Bannerghatta Road, Bangalore 560 076. INDIA. email:[email protected] 2 Lecturer, Industrial Engineering Division, College of Engineering, Guindy, Anna
University, Chennai – 600 025, INDIA, email:[email protected] 3 Associate Professor, Production & Operations Management, Indian Institute of Management
Bangalore, Bannerghatta Road, Bangalore 560 076. INDIA. email:[email protected]
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Design of a Cellular Manufacturing System for a Product Oriented Plant
1. Introduction
Cellular Manufacturing Systems (CMS) have become the accepted philosophy for
addressing the structural issues in a manufacturing system. Over the last three decades
considerable research effort has gone into studying various aspects of designing a
CMS. The importance of a good design is underscored by the fact that a properly
designed structure is the basis for other tactical decisions related to managing the
manufacturing system in the short run. Moreover, redesigning the system often is
prohibitively costly and often not feasible.
In a CMS design, an assessment of the similarity among the parts that are
manufactured or among the machines (or more generally the resources) required for
manufacturing forms the predominant basis for cell design. The similarity could be
based on the requirement of machines, the process sequence, the design attributes
such as shape or the extent the resources are required. A variety of similarity
measures are employed for this purpose. The readers are referred to Shafer and
Rogers (1993) for more details.
Based on a suitable logic, the parts and/or the machines are grouped in a disjoint
fashion to form part families and/or machine groups. Each part family is assigned to a
machine group. However, such a grouping has seldom been perfect in reality for a
variety of reasons. Hence the goodness of cell design is often measured on the basis of
an assessment of “within cell” and “across cell” processing the part families undergo
as a result of the design. In the past, researchers have employed several measures for
this purpose (see Sarker and Mondel, 1999 for a survey).
A typical manufacturing system manufactures a set of products. These products in
turn are assembled from several parts, which are partly manufactured in house. While
designing a cellular manufacturing system for the manufactured parts, a cell designer
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has two options to consider. The first one is to gear up cells based purely on the part –
machine information. The second option is to also consider the products to which the
parts belong as an important input to the problem at the design stage. It appears that
in an era of customer focused approach to business, the second approach is more
attractive. Nevertheless, bulk of the work in the area of CMS design in the last two
decades more attention has been given to the first option. Research, often, was
confined to the use of binary input machine – component incident matrices and
solving problems of size not more than 40 machines and 100 parts. However, there is
a noticeable trend in recent times towards use of other production information such as
processing times, capacity, set-up time and sequence of operations. Moreover, there
seems to be an increased emphasis towards solving real life problems of large size.
This paper makes an attempt to incorporate product related information into CMS
design and address the new concerns arising out of such a design. To our knowledge,
no other work in the past has considered this issue explicitly except Sheu and
Krajewski (1996).
Miltenburg and Montazemi (1993) discussed the problems in using existing solution
methodologies in solving a large real life problem. They reported that the
computational requirements were excessive and many cells were infeasible. They
proposed a method to systematically decompose the problem size. After identifying
the candidate parts for CMS, they recommended the use of one of the existing
methods to gear up the cells. The procedure suffers from a drawback that parts of a
product get eventually assigned to several production systems. In such a situation,
production planning and control becomes complex.
Harhalakis et al. (1996) proposed a heuristic to solve the cell design and layout
problem and applied it to a manufacturer of radar antennas. They considered machine
capacity and cell size constraints and emphasised the need for excluding a few
machines from cell design from a practical viewpoint. These arise due to very low
utilisation of machines in each cells and high cost of duplication.
Cantamessa and Turroni (1997) identified the need for incorporating safety,
technological, organisational and economic factors in a CMS design problem and
proposed an AHP supported clustering algorithm. Initially, the AHP model helps in
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identifying and exercising certain tradeoffs in the CMS design process. The procedure
identifies reminder cells and parts that require either outsourcing or changes in
process plan. Further, the cluster algorithm identifies part families and machine
groups.
Lee and Chen (1997) proposed a heuristic that seeks to balance two measures, inter-
cell movement and workload balance subject to capacity availability and cell size
restrictions. Using a three-stage procedure, they allocate machines and parts to each
cell. They reported testing the procedure for an industry size problem of 60 machine
types and 180 parts.
Sheu and Krajewski (1996) studied the problem of grouping products based on certain
competitive priorities and similarities. A heuristic solution has been proposed to the
non-linear formulation of the problem. This approach will be useful in the case of an
organisation manufacturing a large variety of products. While it helps to address the
problem at an aggregate level, drilling down to the component level is important to
complete the design. Our study differs from this in this aspect. Furthermore, we
consider the problem of keeping the products separate as opposed to keeping them
together.
We note that none of the above studies considered the product to which the parts
belong at the time of cell design. While grouping parts that share similar
manufacturing requirements is very desirable and fundamental to the CMS design
problem, it is our opinion that grouping parts belonging to a product is far more
important. A part family consisting of parts from different products creates numerous
problems when it comes to planning and control of operations on the shop floor.
Firstly, such a group formation is an antithesis to the current customer oriented
thinking currently prevailing widely in business. Secondly, such a design often leads
to a clash of priorities and conflicts among the parts belonging to various products
within each group competing for the same set of constrained resources. For instance,
the part family may consist of components belonging to three or four products and
there will be a difficulty in handling the conflicting demands from final assembly
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shops belonging to these products. Thirdly, there is a loss of accountability and
ownership among the workers.
It is therefore necessary to address the CMS design problem with a product
perspective. Although the problem boils down to creating part families and machine
groups, the overall product perspective to the problem provides a new context and a
dimension to the problem. We define a product focused cell as one in which the cell
design begins with identifying the machines required for various products and ends
with identifying non-overlapping part families and machine groups. This will involve
a two-stage process. In the first stage, the machines to be dedicated to the products are
to be identified. In the second stage, the assigned machines and the parts of the
product are to be sub-divided into part families and machine cells. It is easy to note
that traditional approaches to cell design have adequately addressed the second stage
of this problem.
Solving the first stage calls for newer measures for assessing the goodness of cell
design. Moreover, use of pair-wise comparison to assess similarity between two parts
are not appropriate since the purpose is not to club two products together but to keep
them apart and design cells for them separately.
In a recent survey, Venkataramanaiah et al. (1999) pointed out several other areas that
need more attention in CMS design. Of particular interest is the fact that there have
not been too many efforts towards solving problems involving interval level data. A
majority of work in the area of CMS design has resulted in obtaining a block diagonal
structure using zero – one matrices through an appropriate solution methodology.
These methodologies range from simple matrix manipulation to complex
mathematical programming and graph theoretic procedures. Binary data offer several
computational advantages to solve the problem. However, the cell design obtained is
at best preliminary. Unless the capacity requirements are considered it is difficult to
assess the appropriateness of the design.
Moreover, real life problems are large. Case studies in the area of CMS design have
reported large problem sizes. Harhalakis et al. (1996) indicated a problem size of
3271 parts and 63 machine types (103 machines). Miltenburg and Montazemi (1993)
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reported a size of 5498 parts in their study. In a survey, Wemmerlov and Hyer (1989)
reported that the number of parts range from 300 to 300,000 parts and the median
value is 15,000 parts. Solving such large problems require some means of
decomposing the problem into stages. The two-stage product focused cell design
approach proposed above appears to serve such a purpose to handle large sizes.
We motivate the current research in view of the above discussions. We propose to
address the problem of gearing up product focused cells in a product oriented plant
(POP). We introduce the notion of POP and address the problem context in POP in
the next section. Further, we develop a formulation of the problem in section 3. We
also provide alternative formulations to reduce the problem size and complexity. In
section 4, we propose a heuristics for solving the problem. Based on a number of
randomly generated problems of varying complexity we provide an assessment of the
performance of the heuristic in section 5 before we draw certain conclusions in
section 6.
2. Product Oriented Plant Architecture
Manufacturing organisations differ from one another in a variety of ways. The
variations result on account of not only the nature of product and technology
employed but also on other dimensions such as volume and variety. However, the
CMS design problem is significantly affected by the volume and variety aspects of a
manufacturing organisation. Based on a study of several manufacturing firms,
Mahadevan (1999) identified three generic types of plant architecture existing in
practice and argued that the CMS design should consider the underlying differences
among the three.
The product oriented plant architecture (POP) is characterised by the existence of few
products and high volume of production. In such plants, fairly high level of resource
dedication to individual products is possible on account of high production volumes.
Typical examples of POP include automobile and auto-components manufacturers.
The other extreme in the volume – variety dimension is the Manufacturing operations
Oriented Plant (MOP) that produces a large variety of products but each in smaller
volumes. Air craft manufacturers, and manufacturers of high-end heat exchangers,
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turbines and earth moving equipment belong to this category. The third plant
architecture known as Turnover Oriented Plant (TOP) is also characterised by a large
variety. However, a few among them account for most of the revenue and/or
manufacturing activity. There are several mid-volume mid variety manufacturers who
will fall in this category. Gearing up a CMS will obviously have different
consideration among these three generic types of plants.
POP architecture possesses certain characteristics that merit closer attention while
designing cells. Since the driving force for organising the production system is the
products that are manufactured, the notion of product based cells is a significant
design requirement. Although a higher level of machine dedication could be achieved
at the shop floor due to high volume production of a few products perfect machine
balancing will still not be possible. This is due to the fact that it is not possible to add
machines in fractional capacities. Sharing of workload among different product cells
will become inevitable. In the absence of this, the investment and operating costs of
the system will tend to be higher.
Typically, a POP will consist of sub-plants and each sub-plant will have one or more
cells dedicated to manufacturing components belonging to the product for which the
sub-plant is geared. Due to the above mentioned reasons, there will be inter-cell
moves between cells both within a sub-plant and across other sub-plants. Moreover,
there will be certain common facilities for all the sub-plants. The existence of
common facilities in POP is attributed to several reasons. The utilisation of certain
machines may not justify dedicating machines to a particular plant. The other reasons
for non-dedication include safety, health and environmental hazards.
The job of a cell designer is to gear up appropriate cells in POP taking the above
factors into consideration. Before proposing a formulation of the problem and a
solution methodology, we introduce a few definitions and measures relevant to the
CMS design problem in POP.
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Product ownership
Venkataramanaiah et al. (1999) demonstrated the inadequacy of the traditional
measures of similarity in gearing up product focused cells in POP. The emphasis in
POP is to understand how well the manufacturing requirements of all the parts
belonging to a particular product has been met with. Product ownership is a measure
proposed to capture this information. Product ownership, in a broader context, can be
defined as the extent to which the required manufacturing resources are dedicated to
produce parts belonging to a product.
Such a definition indicates that a high degree of product ownership will result in
greater dedication of resources to a set of products, better accountability and morale
on the part of the employees and fewer conflicts arising out of prioritising the use of
common resources. Furthermore, it greatly simplifies production planning and control
and minimises material handling. Furthermore, the notion of ownership seems to
capture several organisational issues related to manufacturing a range of products.
In order that the measure of product ownership is more useful and directly related to
the cell design problem, we propose a narrow definition. In this paper, we define
product ownership at a cell level as the ratio of the total processing time of the
product in the cell to the total processing time. Specifically,
If Pjg is the processing time of the components of product j assigned to cell g
Product Ownership for product j in Cell g is given by
POWjg = ∑g
jg
jgp
p (1)
It can be seen that at the cell level, the product ownership will vary from 0 to 1.
Furthermore, the sum of product ownership over all the cells will be equal to one. It
may be noted that the notion of product ownership has not been explicitly considered
in the design of cells in a CMS environment in the past. Since more than one product
may use the resources in a cell, the product which has the maximum ownership in a
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cell will claim ownership for the cell. Table 1 illustrates this point for a hypothetical,
6 cells, three product situation.
Component ownership
The definition of the component ownership is very similar to that of the product. The
context changes from product to a component. Specifically,
If Pkg is the processing time of component k in cell g
Component Ownership for component k in cell g is given by
COWkg = ∑g
kg
kgp
p (2)
We note that our definition of component ownership is similar to the quality index
measure proposed by Seiffodini and Djassemi (1996). However, we prefer to use the
term component ownership in order to maintain parity with the product ownership
measure.
L1, L2, and L3 machine categories
In a POP, the available machines can be categorised into three. L1 machines are those
that are available for complete dedication to each product in the requisite numbers.
The high volume and few variety scenario existing in a POP will always result in a
few machine types belonging to this category. L1 machines contribute 100% to the
product ownership as all the processing requirements on these machines are met
entirely in the sub-plant itself.
On the other hand, every manufacturing system has machines that are not amenable
for dedication to any particular product group for reasons mentioned earlier. Such
machines are designated as L2 machines. Typical examples include painting,
processes such as nitriding, sand blasting, and electroplating, and other machines
performing pre-manufacturing activities such a bar cutting, profile cutting, and
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shearing. Such machines prevent the products from attaining 100% product
ownership. It will be necessary to identify such machines and to keep them as
common facilities in a remainder cell (Cantamessa and Turroni, 1997, Harhalakis et
al. 1996).
The third category of machines falls in between the two. They are neither available in
plenty to dedicate to each cell nor too few in number to keep them in a remainder
cell.. These machines are designated as L3.
However, there are other reasons for including machines as L2. Of significant
consideration is the utilisation of the machine if dedicated to cells. The issue of
dedication Vs centralisation has not been resolved in an objective manner in
organisations. This has partly been due to a lack of understanding of the impact of
these choices on the system design and performance. It appears that exact
configuration of L2 and L3 machines can significantly affect the component and
product ownership. Hence, it is may be worthwhile to know the extent to which these
affect the chosen performance measures for CMS design. The mechanism to identify
L1, L2, and L3 machines and using them in the CMS design problem is given in
section 4.
It may be noted that while L1 machines may guarantee 100% product ownership, they
may not guarantee 100% component ownership. For instance, let us assume that three
cells are formed for product A and the L1 machines assigned to these cells. If a
particular L1 machine “i” is assigned only to cell 2 and Cell 3 and a component in
cell1 requires processing in machine “i”, then it may have to visit either cell 2 or cell
3, thereby reducing the component ownership for the component.
3. Formulation of the CMS problem for POP
Notations
Indices
i index of machine types, i = 1,2,…,m
j index of product j = 1,2,…,p
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k index of components k = 1,2,…,c
g index of cells g = 1,2,…,G
Parameters
Si Set of components requiring machine i for processing
Sk Set of machines required for processing component k
Sj Set of components belonging to product j
tik Total time required for processing component k on machine type i
tij Total time required for product j on machine type i
= ∑ tik
k ∈ Sj
Ai number of units of machine type i available
Ci Available capacity per machine type i per unit time
L Upper limit on cell size
U Lower limit on cell size
Decision variables
Yjg = 1, if product j claims ownership to cell g
0, otherwise
Xikg = 1, if operation on machine type i of component k is assigned to cell g,
0, otherwise
Nig number of units of machine type i assigned to cell g
Powjg ownership of product j in cell g
Powg maximum ownership in cell g
Model P1
Maximise Z1 = ∑∑ Powg Yjg (3)
j g
Subject to:
∑ Xikg = 1, ∀ i, k (4) g
∑ Xikgtik ≤ NigCi ∀ i, g (5) k∈ Si
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∑ Nig ≤ Ai ∀ i (6) g
L ≤ ∑ Nig ≤ U ∀ g (7) i
Powjg = { ∑ ∑ Xikgtik }/ [ ∑ ∑ tik ] ∀ j,g (8) k∈ Sj i∈ Sk k∈ Sj i∈ Sk
Powg - Powjg ≤ M (1- Yjg) ∀ j,g (9)
∑Yjg = 1, ∀ g (10) j
Xikg ,Yjg ={0 or 1}, Nig ≥ 0 and integer, POWjg, POWg ≥ 0 (11)
Constraint 4 ensures that the operation of component k on machine type i is not split
across the cells. Constraint 5 ensures adequate capacity availability and 6 limits the
total number of machines of type i assigned to availability. 7 represents the cell size
constraints. Constraint 8 computes the product ownership in a cell and constraints 9
and 10 together ensure that the product that has the maximum ownership in a cell in
fact claims ownership of the cell. Finally, the objective function maximises the
product ownership.
The formulation has a non-linear objective function. It may be noted that the non-
linearity arises primarily from the constraint on cell size. Due to a restriction on the
number of machines allowed in a cell, it is likely that products may require more than
one cell to complete all its processing requirements. Hence it becomes necessary to
first identify the cells in which a product undergoes processing and add up all its
ownership in those cells to obtain the product ownership. The POWg and the Yjg
decision variables together identify these and result in non-linearity.
CMS design problems involving machine capacity constraints are shown to be NP
hard (Boctor, 1996). Moreover, typically the problems are also large. Hence the use of
a combination of alternative formulations, methods to decompose the problem in a
step-wise manner and solving the resultant problems for an approximate but close to
optimal solutions and greedy heuristics is inevitable. Chen and Heragu (1999)
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provided a step-wise decomposition approach to solve large size CMS design
problems. However, the use of Bender’s decomposition approach for solving the sub-
problems puts a limitation to the problem size. It appears that solving problems
involving more than 50 machine types optimally may be difficult.
Boctor (1996) proposed a decomposition procedure and alternative formulations to
solve the CMS design problem for obtaining good approximation solutions. Since the
quality of the proposed procedure depends on the part assignment method used, they
proposed a simulated annealing approach for part assignment. Typically, the number
of parts is several fold more than the number of machines in a CMS. The use of SA in
the solution procedure may pose computational severity especially while solving
problems involving large number of parts.
In this context, we provide an alternative equivalent formulation by relaxing the cell
size constraint and thereby linearising the objective function. In effect, this involves a
step-wise decomposition of the problem and solving them in two stages. In stage 1,
each product will have only one cell without any restriction on the cell size. In stage
2, using the solution obtained for each product in stage one, we solve a sub-problem
by re-introducing the cell size constraint.
The advantage of such an approach lies in the fact that we will be able handle
relatively large size problems. Such an approach is better than just using heuristic
algorithms for the original problem. This is due to the fact that most heuristic
approaches are good in forming large loose cells (Chen and Heragu, 1999). However,
when used to form large number of small cells these approaches are often inefficient
and produce inferior quality solutions (Chandrasekharan and Rajagopalan, 1989). The
relaxed problem P2 and the sub-problem P3 are as follows:
Model P2
Maximise Z2 = ∑POWj (12) j
13
Subject to:
(4), (5), (6), (8), (10), (11),
Powj - Powjg ≤ M (1- Yjg) ∀ j,g (13)
∑Yjg = 1, ∀ j (14) g
POWj ≥ 0 (15)
Constraint 13 fixes the ownership for product j based on maximum ownership and 14
ensures that each product is assigned to only one cell.
Model P3
Maximise Z3 = ∑Cowk (16)
Subject to:
(4), (5), (6), (7), (11),
Cowkg = { ∑ Xikgtik }/ [∑ tik ] ∀ k,g (17)
i∈ Sk i∈ Sk
COWk ≥ COWkg ∀ g (18)
COWkg ≥ 0 (19)
It may be noted that P2 differs from P1 in two ways. First the objective function
becomes linear and secondly the number of variables will be reduced. It may also be
noted that while P3 is structurally very similar to P1, the problem is much smaller and
different from P1. Specifically, the RHS for equation 6 for each sub-problem in P3
will be the outcome of the solution for respective product obtained through P2.
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Moreover the relevant sets of components (index k), and machines (index i) will also
vary from product to product and much smaller in number.
Although P3, the sub-problem will be considerably small in size both in terms of
variables and constraints, it is still large to solve if we consider real life situations.
Table 2 presents the statistics on the number of variables and constraints for the three
models both for a hypothetically small and a moderately sized real life problems. The
computations are based on the minimum number of cells that could be geared up
keeping in mind cell size constraints. Table 2 shows that while solving hypothetically
small problems are possible due to reduced problem size, in the case of real life
problems it is impossible. We propose a heuristic procedure to solve the problem.
4. A heuristic procedure for the CMS problem in POP
The heuristic procedure for the CMS problem in POP has three stages. In stage 1, an
attempt is made to bring down the problem size by identifying L1, L2 and L3
machines. In the second stage, a heuristic procedure is developed for solving P2.
Using the solution obtained in the second stage, a heuristic procedure for solving P3, a
set of sub-problems is developed in the third stage. For reasons discussed in the
previous section, L2 machines are not considered any more for the CMS problem and
L1 machines are considered only in the third stage. The three stages are detailed
below.
Stage 1. Segregating the machine types into L1, L2, and L3
The procedure for identifying the L1 machine types essentially involves computing
the total number of machines required if each product is dedicated with the requisite
number of each machine type and comparing it with the available. The motivation for
this exercise stems from the fact that such machines ensure 100% ownership for the
products and hence can be deferred to stage 3 for further consideration. This reduces
the problem size to that extent.
The procedure identifies the L2 machines in two ways. First all singleton machines,
which are potential candidates are examined for their utilisation if dedicated to any
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one of the product. If the maximum utilisation is lower than a lowest cut-off value α,
then the machine is considered unsuitable for inclusion in any cell. In the case of more
than one machine, if all the products stake more or less an equal claim, then there is
no basis for including them in any of the product cells. This is assessed by computing
the difference between the maximum and the minimum utilisations of a machine and
comparing it with the average utilisation using a parameter β. The relationship
between the machine dedication parameters and the machine utilisation can be
expressed as follows:
If Uij denote the utilisation of machine type ‘i’ by product ‘j’ and Uimax, Ui
min, and
Uiavg denote the maximum, minimum and average utilisation for each machine type
‘i’, then the following conditions are to be satisfied to dedicate machine type ‘i’ to
the cells:
Uimax ≥ α (20)
(Uimax – Ui
min)/ Uiavg ≥ β (21)
The magnitude of the parameters α and β (these could be varied between 0 and 1)
seeks to resolve the frequently encountered problem of machine dedication Vs
centralisation. It may be noted that a lower value of these parameters will favour
machine dedication and a higher value will promote centralisation. These two
parameters are organisational realities that play a crucial role in not only influencing
the cell design but also the day to day monitoring and control of the cells. Proper
consideration of these will thus avoid potential conflicts arising out of clash of
priorities and misplaced ownership of the machines by the product cells. This is
consistent with the view often held in CMS design that the conversion of a traditional
manufacturing system into CMS is more an organisational problem than a
computational one (Cantamessa and Turroni, 1997).
All the other machines are identified as L3. It may be easily noted that it finally calls
for solving P2 with L3 machines and P3 with L1 and L3 machines. There will be a
loss of product and component ownership due to L2 machines.
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The procedure for identifying L1, L2, and L3 is outlined in the following steps:
Step 1. Compute for each product j and machine type i:
(a) Number of machines of type i required by product j
+
=
i
ijij C
tR where +. is the smallest integer greater than . .
(b) Maximum number of machines of type i required ∑=j
iji RR
(c) Ri – Ai
(d) Utilisation of machine i by product j Uij = tij/Ci
(e) Maximum utilisation for machine type i, )(maxmaxij
ji UU =
(f) Minimum utilisation for machine type i, )(minminij
ji UU =
(g) Average utilisation for machine type i, j
UU j
ijavgi
∑=
(h) Set the machine dedication parameters α and β.
Step 2. If the machine list is empty go to step 5
Else let the machine in the top of the list be i.
Step 3. (a) If Ri – Ai ≤ 0, include machine type i in list L1.
(b) If Ri – Ai < P – 1,
include machine type i in list L3 if (20) and (21) are satisfied
else include machine type i in list L2.
(c) If Ri – Ai = P – 1,
include machine type i in list L3 if (21) is satisfied
else include machine type i in list L2.
Step 4. Delete machine i from the machine list and go to step 2.
Step 5. Stop.
Stage 2. Procedure for solving P2 for the L3 machines
The heuristic has three steps. Initially, the machines are assigned to product cells on
the basis of machine criticality to each product. It can be easily seen assigning a
machine to a product with a higher value of criticality, greatly increases the chances
17
of improving the product ownership. Once the machines are assigned, the operations
of the components are assigned to the cells and the product ownership calculated in
the second step. The third step investigates the improvement opportunities through
machine reassignment and exchange thereby fine tuning the cell design in order to
maximise the product ownership.
Machine Allocation Rule
Step 1. For every machine i and product j, compute machine criticality CRij
where
=
i
ijij A
tCR .
Step 2. If list L3 is empty go to Step 5
Step 3. Select product j’ such that j’ = )(max ijj
CR
If Rij’ ≥ Ai, assign Ai machines to product cell j, Ai = 0.
If Rij’ < Ai, assign Rij’ machines to product cell j, Ai = Ai – Rij’.
Step 4. CRij’ = 0. If Ai = 0, delete machine i from the list, go to step 2
Else go to step 3.
Step 5. Stop.
Component operations allocation
Once the machine allocation is done using the above rule, the component operations
are allocated to the respective machines to the extent of availability. If sufficient
capacity is not available in the parent cell, the operations are allocated to machines
available in other cells. Since the logic is straightforward, we do not give a step by
step algorithmic logic for this procedure.
Compute the product ownership for each product in each cell. For each product j
designate cell g as its cell such that POWj is )(max jgg
POW .
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Machine reassignment/exchange
Step 1. Let i be the machine considered for reassignment from product cell j1
to product cell j2. Let the product ownership before and after the
reassignment be POWj1 and POWj1’ in product cell j1 and POWj2 and
POWj2’ in product cell j2 respectively. Reassign machines i only if
(POWj2’ – POWj2) > (POWj1 – POWj1’).
Step 2. Repeat step 1 for all machines in a cell with and all other cells. If no
more improvement is possible go to step 3.
Step 3. Let machine i1 be considered for exchange from product cell j1 with
machine i2 in product cell j2. Let the product ownership before and
after the exchange be POWj1 and POWj1’ in product cell j1 and POWj2
and POWj2’ in product cell j2 respectively. Exchange machines i1 and
i2 only if (POWj2’ – POWj2) + (POWj1’ – POWj1) > 0.
Step 4. Repeat step 3 for all possible pair-wise cells and respective pair-wise
machines in the cells. If no more improvement is possible go to step 5.
Step 5. Compute the product ownership for each product in each cell.
Designate the cell to product j where the product has maximum
ownership. Stop.
Stage 3. Procedure for solving P3 for the L1 and L3 machines
The heuristic for solving P3 employs the cellular similarity (CS) coefficient proposed
by Luong (1993). The heuristic employs a modified version of their algorithm in order
to accommodate the interval level data. The first part generates efficient seed cells.
Initially, each part is assumed to form a cell by itself. Based on CS, the cells are
progressively merged without violating the cell size constraint. Infinite capacity
availability of machines is assumed at this stage. When no more mergers are possible,
either due to cell size constraints or non-similarity between cells, the second part of
the heuristic is triggered. During the second part, the machines are allocated to the
cells formed at the end of the first part only to the extent of availability.
Consequently, the allocation of parts may not have been appropriate. Hence, by
computing the component ownership, the parts are re-assigned to appropriate cells.
An iterative procedure alternates between possible machine reallocation and
19
component reallocation until no more improvement in component ownership is
possible. The heuristic is enumerated in the following steps:
Formation of seed cells (Un-capacitated machine allocation)
Step 1. Set the number of cells to the number of parts. For each cell, compute
{parts} and {machines}. Compute [CS], the matrix of similarity for all
pairs of cells. CSij = 0 if i=j.
Step 2. Merge cells i and j if CSij = 1. Delete cell j. Update {parts}i and
{machines}i and [CS] Repeat the procedure until all such cells are
merged.
Step 3. From the available cells, select two cells with the highest CS. Let the
selected cells be i and j.
Step 4. If no more cells exist with CSij > 0 go to step 6.
Step 5. If the number of machines required after the merge ≤ U, merge cells i
and j. Delete cell j. Update {parts}i and {machines}j and [CS]. Else
CSij = 0.
Go to step 3.
Step 6. Store the current solution and proceed to the next step.
Formation of final cells (Capacitated machine allocation)
Step 7. For the solution obtained compute the number of machines required for
each machine type. If for every machine type, the number required is
less than or equal to available go to step 9. Else go to step 8.
Step 8. Let the machine that has excess requirement over availability be
designated as ‘i’.
(a) Assign the required number of machine type i to the cell that has the
maximum requirement of the machine. Update the balance machine
available for allocation.
(b) Repeat step (a) until all machines of type i are assigned.
(c) Remove machine type i from all other cells where it is required.
(d) Go to step 7.
20
Step 9. Compute component ownership for all the parts. Assign the
components to the cells based on the maximum ownership.
Step 10. If there is a change in component assignment to the cells go to step 7.
Else go to step 11.
Step 11. Stop. The current solution is the best solution for the problem.
5. Computational results
The base case for the problem considered for this study consists of five product
groups, 150 components and 30 machine types. It is assumed that each product has
thirty components. Based on the desired density of the input matrix, and the
maximum number of machines a component can visit, the routing for the component
and the processing time are generated using a random shop simulator that has been
developed specifically for this problem. The processing times are assumed to vary
uniformly between 15 and 35 time units. The number of machines available in each
machine type is computed based on the desired machine utilisation factor. For a
desired density of 20% and a machine utilisation value of 80%, the base case problem
has 97 machines with an average utilisation of 75.63% and an actual matrix density of
16.36%.
Table 3 shows the L1, L2 and L3 machines for the problem as computed using the
first stage of the heuristic. Table 4 has the details on the solution to the P2 problem
using the proposed heuristic and table 5 has the solution for problem P3. The base
case has been solved for α = 0.50 and a β = 0.25. The cell size is restricted between
four and 10.
In addition to the 11 cells exclusively dedicated to the products, there will also be
remainder cells comprising of the L2 machines identified earlier. It may be noted that
a higher percentage of L3 machines has resulted in both lower product ownership and
component ownership. It may be therefore be interesting to know the role played by
the two machine dedication parameters in determining the product and component
ownership.
21
In order to evaluate the performance of the proposed method with the traditional
method of cell formation, the number of products was set to 1. In this scenario, we
can directly solve P3 with 150 components and 30 machine types. The base case
problem was solved using the heuristic proposed for P3. Table 6 shows the results
obtained. The solution has 12 cells and components of each product were distributed
throughout the cells depending on the machining similarity they shared with the other
components. This resulted in an improvement in the average component ownership at
the expense of product ownership. Cells 2 and 7 have several one off machines that
could potentially form a reminder cell. This has resulted in poor component
ownership in these cells. The results underscore the fact that high component
ownership need not ensure high product ownership. The results therefore clearly
indicate the need for gearing up product focused cells.
The machine dedication parameters and the L1, L2 and L3 machines are not only the
crucial parameters of the proposed formulation and the solution procedure but also
organisational realities. In order to understand the impact of these on the product
ownership, we conducted a series of experiments by varying these parameters from
the base case. Specifically the following additional experiments were carried
(a) Maintaining the system configuration at the base case level and studying the
performance of the system for various values of α
(b) Studying the performance of the system for various values of β
(c) Generating alternative matrices with varying percentages of L1, L2 and L3. The
focus was more on L1 and L3. Hence the percentage of L2 machines was kept as
low as possible in all these cases
The results of these studies are presented in tables 7, 8, and 9. Figures 1 and 2 are the
graphical representation of tables 7 and 8. The results show that of the two machine
dedication parameters α and β, the latter plays an important role in the performance of
the systems. This evident from the fact that the base case solution was vastly
improved by changing the β value from 0.25 to 0.20. The percentage of L3 machines
significantly falls down with increasing values of β. The results are not surprising
though. The parameter β is used to resolve the case of multiple machines. Hence
centralising them could significantly reduce the product ownership and the component
22
ownership. The results indicate that whenever the requirements of multiple machines
are nearly similar across the products, allowing the solution procedure to perform the
allocation to the products may yield better product ownership.
A higher percentage of L1 machines obviously results in a higher product ownership.
However the results further indicated that a higher product ownership invariably
results in higher component ownership also. In contrast to this, the comparison of the
base case results with the traditional CMS problem showed that the vice versa does
not hold good. This has underscored the need for a different approach to solve POP
problems.
6. Conclusions
This study has addressed the problem of gearing up a product oriented plant
architecture. Traditional approach to CMS design considers only the “machine –
component” dimensions of the problem. However, when a third dimension namely
“product” is added to the problem, it not only provides a different context to the
problem but also becomes more complicated to solve. We have proposed alternative
formulations of the problem and a heuristic solution procedure.
Our experiments with the proposed formulation shows that while a high product
ownership can result in high component ownership, the reverse is not true. We have
provided certain quantification of the often encountered problem of what to dedicate
to the cells and what to centralise while designing the cells. These are organisational
realities that every manager would face while solving the POP problem.
Acknowledgements
This research was partly supported by the research grant from The Department of
Science & Technology, Ministry of HRD, Government of India and The Centre for
Asia and Emerging Economies, The Amos Tuck School of Business Administration,
Hanover, NH 03755, USA.
23
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25
Ownership in various cells Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Cell 6 Remarks
Product A 0.49 0.45 0.02 0.03 0.01 Claims ownership in 1 and 2. Product ownership = 0.94
Product B 0.10 0.35 0.43 0.12 Claims ownership in 3 and 5. Product ownership = 0.78
Product C 0.02 0.47 0.51 Claims ownership in 4 and 6. Product ownership = 0.98
Table 1. An illustrative example of product ownership in POP
Hypothetically small sized
problem
Moderately sized real life
problem
Problem
description
No. of products: 3
No. of components per product: 20
No. of machine types: 10
Total number of machines: 20
Cell size restriction (2 – 4)
L1 machines = 30%
L2 machines = 10%
No. of products: 5
No. of components per product: 60
No. of machine types: 40
Total number of machines: 80
Cell size restriction (2 – 8)
L1 machines = 30%
L2 machines = 10%
P1 P2 P3 P1 P2 P3
No. of 0 – 1
variables
3015 1089 400 120,050 36,025 2,520
No. of integer
variables
50 18 10 400 120 40
No. of
constraints
700 417 199 12,716 7,429 1,466
Table 2. Problem complexity for the three proposed models
26
Machine Category
Machine Types % of the total machine types
L1 4, 7, 27, 28, 29 16.67 L2 2, 17, 19, 20, 21, 22, 23, 24, 25, 26, 30 36.67 L3 1, 3, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18 46.67
Table 3. L1, L2, and L3 machines for the base case problem
Product Machines assigned* Total Ownership
Product 1 1, 3, 5(2), 6, 8, 9, 10, 11, 12, 12, 14, 16 13 77.95% Product 2 1(2), 3(2), 5(2), 6, 8, 9, 10, 11, 12, 13, 15, 16, 18 16 77.66% Product 3 1, 3, 5, 6, 8, 9, 10, 11, 12, 15 10 74.89% Product 4 1(2), 3, 5, 6, 8, 9(2), 10, 11, 12, 13, 15, 16 14 81.79% Product 5 1, 3, 5, 6, 8(2), 9, 10, 11, 14, 18 11 71.24%
Table 4. Solution to the P2 problem
* The number in parenthesis indicates the quantity (if more than one) allotted to the
product
Product Cell Cell configuration Component
ownership No. of machines
assigned No. of components
assigned Average(%)
Product 1 Cell 1 Cell 2
7 8
12 18
61.19
Product 2 Cell 1 Cell 2 Cell 3
8 5 7
16 5 9
58.57
Product 3 Cell 1 Cell 2
4 10
7 23
66.19
Product 4 Cell 1 Cell 2
8 8
19 11
71.57
Product 5 Cell 1 Cell 2
10 4
23 7
55.76
Table 5. Solution to the P3 problem
27
Cell
Configuration Ownership
No. of m/c s
No. of compo-nents
Product 1
Product 2
Product 3
Product 4
Product 5
Compo-nent
Cell 1 8 16 12.39 3.94 14.95 11.53 10.00 77.60 Cell 2 8 10 2.97 2.66 10.00 16.55 56.29 Cell 3 10 10 13.20 6.52 10.00 6.91 6.75 58.99 Cell 4 10 15 10.10 10.12 15.54 9.60 5.00 82.83 Cell 5 9 18 19.00 11.26 8.32 17.00 3.55 89.82 Cell 6 6 7 2.03 6.81 3.53 4.17 58.44 Cell 7 8 8 10.00 9.71 10.00 15.66 5.01 42.50 Cell 8 8 14 5.66 14.36 7.29 2.98 7.95 78.09 Cell 9 6 10 9.97 2.93 6.50 2.99 3.56 78.75 Cell 10 6 11 5.74 3.37 6.97 3.00 12.33 80.49 Cell 11 8 15 3.40 15.08 9.06 5.01 8.01 59.60 Cell 12 10 16 8.51 12.92 8.69 11.79 17.12 96.54
Table 6. Solution for the traditional CMS problem
Machines Product Ownership
αααα L2 (%) L3 (%) P1 P2 P3 P4 P5 Avg.
0.10 10.00 73.33 80.69 88.53 77.70 85.09 71.94 80.79
0.20 16.67 66.67 80.69 88.53 77.70 85.09 71.24 80.37
0.30 26.67 56.67 80.69 88.53 77.70 81.79 71.24 79.33
0.40 30.00 53.33 80.69 83.22 77.70 81.79 71.24 78.93
0.50 36.67 46.67 77.95 77.66 74.89 81.79 71.24 76.71
1.00 56.67 26.67 60.00 60.41 64.76 65.15 67.11 63.49
Cell size restriction: 4 – 10 machines. β = 0.25.
Table 7. Product ownership for various values of machine dedication parameter (αααα).
28
Machines Product Ownership
ββββ L2 (%) L3 (%) P1 P2 P3 P4 P5 Avg.
0.10 33.33 50.00 84.54 84.10 80.93 87.96 77.41 82.99
0.20 33.33 50.00 84.54 84.10 80.93 87.96 77.41 82.99
0.30 36.67 46.67 77.95 77.66 74.89 81.79 71.24 76.71
0.40 40.00 33.33 71.64 72.29 68.21 73.58 65.05 70.15
0.50 50.00 33.33 56.72 57.59 61.66 58.84 60.28 59.02
1.00 73.33 10.00 22.86 18.19 21.33 19.57 29.58 22.31
Cell size restriction: 4 – 10 machines. α = 0.50.
Table 8. Product ownership for various values of machine dedication parameter (ββββ).
29
Sl. No. Machines Ownership*
L1 (%) L3 (%) P1 P2 P3 P4 P5 Avg.
1 70.00 30.00 Product
Component
98.61
94.65
96.64
84.61
93.89
84.73
97.05
78.52
100.00
75.59
96.85
83.62
2 56.67 40.00 Product
Component
95.24
82.42
94.23
83.49
94.85
88.17
95.16
70.34
98.13
81.19
95.52
81.12
3 46.67 50.00 Product
Component
95.45
91.45
90.34
87.65
95.24
87.99
93.34
72.16
93.83
76.20
93.64
83.09
4 36.67 53.33 Product
Component
69.46
63.81
75.54
69.92
83.48
74.71
94.83
80.47
89.53
78.82
82.65
73.55
5 16.67 70.00 Product
Component
83.04
64.60
81.37
59.74
88.81
67.42
85.72
68.12
90.93
61.10
85.97
64.20
6 3.33 86.67 Product
Component
86.35
64.15
76.98
56.95
85.45
60.46
77.52
58.98
83.71
61.16
82.00
60.38
Cell size restriction: 4 – 10 machines. α = 0.50, β = 0.25.
* The component ownership measure for each product is the average of all the ownership of the components belonging to the product.
Table 9. Effect of L1 and L3 machines on product and component ownership
30
Sensitivity of machine dedication parameter (αααα)
Sensitivity of machine dedication parameter (ββββ)
0
10
20
30
40
50
60
70
80
90
100
0.10 0.20 0.30 0.40 0.50 1.00
Machine dedication parameter (alpha)
Ow
ners
hip
(%)
Product 1Product 2Product 3Product 4Product 5Average
0
10
20
30
40
50
60
70
80
90
100
0.10 0.20 0.30 0.40 0.50 1.00
Machine dedication parameter (beta)
Ow
ners
hip
(%)
Product 1Product 2Product 3Product 4Product 5Average