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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:mahadev@iimb.ernet.in 2

Lecturer, Industrial Engineering Division, College of Engineering, Guindy, Anna University, Chennai – 600 025, INDIA, email:svr@annauniv.edu

3

Associate Professor, Production & Operations Management, Indian Institute of Management Bangalore, Bannerghatta Road, Bangalore 560 076. INDIA. email:janat@iimb.ernet.in

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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 inter-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 jg p 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 kg p 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 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 sub-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)

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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 Ui max

, 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 – Uimin)/ 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 ij ij C t

R where



. + is the smallest integer greater than



. .

(b) Maximum number of machines of type i required =∑

j ij

i R

R

(c) Ri – Ai

(d) Utilisation of machine i by product j Uij = tij/Ci

(e) Maximum utilisation for machine type i, max max( ij) j

i U

U =

(f) Minimum utilisation for machine type i, min min( ij) j

i U

U =

(g) Average utilisation for machine type i,

j U U j ij avg i ∑ = (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

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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 ij ij A t CR .

Step 2. If list L3 is empty go to Step 5

Step 3. Select product j’ such that j’ = max( ij) j

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( jg)

g

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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

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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.

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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.

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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

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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.

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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

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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

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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

α αα α 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.

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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.

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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.

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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)

Owne rs hip (%) Product 1 Product 2 Product 3 Product 4 Product 5 Average 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)

Owne rs hip (%) Product 1 Product 2 Product 3 Product 4 Product 5 Average

Figure

Table 2. Problem complexity for the three proposed models
Table 4. Solution to the P2 problem
Table 7. Product ownership for various values of machine dedication parameter (α αα α)
Table 8. Product ownership for various values of machine dedication parameter ( ββββ )
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References

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