Selection of manufacturing process using graph theoretic approach

ORIGINAL ARTICLE

Selection of manufacturing process using graph theoreticapproach

Mohit Singh • I. A. Khan • Sandeep Grover

Received: 24 October 2011 / Revised: 17 January 2012 / Published online: 5 February 2012

� The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and

Maintenance, Lulea University of Technology, Sweden 2012

Abstract To manufacture a product, nowadays there are

many methods available in the market to manufacture them

and to earn more profits and best production which is the

prime focus of any manufacturing industry, it is necessary to

select only that type of manufacturing process which leads to

more profits, less scraps, and reworks, faster production rate,

good quality of production, employee satisfaction, customer

satisfaction, etc. So the aim of this paper is to judge the best

manufacturing process among various manufacturing pro-

cesses for manufacturing any product using graph theoretic

approach. The graph theoretic approach reveals a single

numerical index and accordingly it is possible to choose the

best manufacturing process. To apply the graph theoretic

approach the authors selected four factors namely: Quality,

Cost, Technical Capability, and Production. Based on these

factors and their co-factors a fish bone diagram is repre-

sented. While applying graph theoretic approach a digraph of

the characteristics is drawn which represented the factors and

co-factors affecting the selection of manufacturing process

and further the interdependency of the factors as well as their

inheritances has been identified and its representation in the

matrix form has been used for the calculation of numerical

index of the manufacturing process through its variable

permanent quality function. The technique is applicable

when there are more than options are available for manu-

facturing a product. An example is also shown in the last of

the paper to understand the application of graph theoretic

approach for the selection of best manufacturing process

among three processes.

Keywords Manufacturing organization � Selection

process � Graph theoretic approach

1 Introduction

The main goal of any organization is to achieve more and

more profits with effective customer satisfaction so that the

organization can play effectively in the market, and to fulfill

this goal it is necessary to select a proper manufacturing

system carefully. A company with an excellent manufac-

turing process produces a quality product which passes

stringent inspections and gains customer recognition. The

benefits of a good, solid manufacturing process are: quality

products, decreased labour cost, high employee morale,

positive image, higher profits, etc. It is therefore important to

develop designing aids which would assist designers in the

selection of materials and manufacturing processes (Perzyk

and Mefta 1998). The manufacturing process should be

flexible and precise enough so that during the fabrication it is

possible to reach the required tolerances. The manufacturing

process should also bear low cost, low scrap, and rapid in

order to be competitive (Jahazi and Hossein-Nejad 2004).

To predict or compare the performance of a manufac-

turing process, it is necessary to analyze various factors and

their effect. Therefore, a mathematical model is required to

correlate the different factors, sub-factors to evaluate and

M. Singh (&) � I. A. Khan

Department of Mechanical Engineering, Jamia Millia Islamia,

New Delhi, India

e-mail: [email protected]

I. A. Khan


S. Grover

Department of Mechanical Engineering, YMCAUST, Faridabad,

India


123

Int J Syst Assur Eng Manag (Oct-Dec 2011) 2(4):301–311

DOI 10.1007/s13198-012-0083-z

compare the manufacturing processes. The present works

undertakes the application of graph theoretic approach in

the manufacturing process in manufacturing industry.

2 Graph theoretic approach

A graph theoretic model is a versatile tool that has been

used in various applications. It helps to analyze and

understand the system as a whole by identifying system

and subsystem up to the component level. The conven-

tional approach of representation vis-a-vis block diagram,

flow diagram, schematic representation, etc., is suitable for

the visualisation of relationships/interactions but not for

any mathematical analysis. Whereas the mathematical

model developed by graph theoretic approach, considers

both the contribution of factor itself, i.e., the inheritance of

factor and extent of dependence among factors, i.e., their

interactions. This methodology starts from where conven-

tional representations end. This logical and systematic

approach uses well-documented applications of graph

theory (Narsingh 2000; Robinson and Foulds 1980).

Digraph representation is useful for modelling and visual

analysis. Matrix representation is useful in analyzing the

digraph model mathematically and for computer process-

ing. Permanent multinomial function characterizes the

system uniquely and the permanent value of a multinomial

represents the system by a single number, which is useful

for comparison, ranking, and optimum selection.

The value of the manufacturing process index is iden-

tified using the unified structural approach called graph-

theoretic-methodology. The graph theoretic approach is

divided into three terms:

1. Digraph representation

2. Matrix representation

3. Permanent function representation

2.1 Digraph representation

A digraph is used to represent the factors and their inter-

dependencies in terms of nodes and edges. The nodes

represent the measure characteristic, whereas edges repre-

sent interdependence between them. An example of four

characteristics digraph is shown in Fig. 1.

2.2 Matrix representation

The digraph provides a visual representation which is

fruitful up to a limited extent. If the number of character-

istics are increases then the digraph becomes complex, so

to resolve this complexity the matrix representation is

developed. If the digraph contains N nodes, then its matrix

representation is of size N 9 N, in which diagonal ele-

ments represent the dependence among the characteristics.

The matrix is also known as the variable permanent matrix

(VPM) corresponding to four characteristics digraph is

given as

VPM¼

H1 H12 H13 H14

H21 H2 H23 H24

H31 H32 H3 H34

H41 H42 H43 H4

0BB@

1CCA

2.3 Permanent function representation

To determine the numerical index, the permanent of the

matrix, called as variable permanent function is used here.

The permanent function is obtained in a similar manner as

its determinant but with all signs positive. This expression

is representative of the manufacturing process and contains

all possible quality terms of the manufacturing organiza-

tion. The VPF expression corresponds to the four charac-

teristics digraph/VPM is represented by Eq. 1 as follows:

VPF ¼H1H2H3H4

þ h12h21H3H4 þ h13h31H2H4 þ h14h41H2H3

þ h23h32H1H4 þ h24h42 H1H3 þ h34h43 H1H2

þ h12h23h31H4 þ h13h32h21H4 þ h12h24h41H3

þ h14h42h21H3 þ h13h34h41H2 þ h14h43h31H2

þ h23h34h42H1 þ h24h43h32H1

þ h12h21h34h43 þ h13h31h24h42 þ h14h41h23h32



ð1Þ

3 Identification of attributes

To select the best manufacturing process using graph the-

oretic approach, it is necessary to identify the attributes

affecting the manufacturing process significantly. As per

the literature review, the attributes and their co-factors

Fig. 1 Four-characteristic quality digraph

302 Int J Syst Assur Eng Manag (Oct-Dec 2011) 2(4):301–311

123

affecting the selection of manufacturing process are as

following:

1. Quality: low defect rate, commitment to quality,

improved process capability

2. Cost: processing cost, maintenance cost, product cost

3. Technical Capability: flexibility, efficiency, mechanism

4. Production: production rate, packaging, delivery

Factors affecting the selection of manufacturing process

are represented by a fishbone diagram as shown in Fig. 2.

The fishbone diagram identifies the main factors as Cost,

Quality, Technical Capability and Production. Each branch

or bone shows main factors whereas the sub branches

connected to main branches shows the co-factors of main

factors. The diagram represents that how these factors and

co-factors affect the manufacturing process.

3.1 Quality

The term Quality in reference to manufacturing process

meant for such processes and methodologies that ensure

that the manufactured products meet the required quality

standards consistently. The process should produce goods

right at the first time, without any rework. In an imperfect

manufacturing process, a certain proportion of products

become defective due to poor production quality and

material defects, and subsequently defective products are

scrapped if they are not re-workable or it is not cost-

effective to do so (Sarker et al. 2008). The term Quality is

very crucial for the manufacturing processes. With so much

competition and such few margins, no manufacturing

industry can afford to spend time and money on rework.

Every activity in the industry costs money and so does

rework, but customers do not pay for rework. Customers

pay for the value addition by the company and if they see

more valuable additions by some other company being

offered at same or lower costs, they move to that company.

Hence to assure good quality to customers, quality pro-

cesses plays a significant role.

3.2 Cost

The manufacturing process should be cost effective, as the

competition is increasing at a very faster rate. The current

trend forces companies to produce low-cost and high-

quality products in order to maintain their competitiveness

at the highest possible level (Shehab and Abdalla 2001).

Figure 3 postulates that cost can be regarded as a function

of technical factors, such as cycle time, downtime, reject

rate, equipment and tooling requirements, or the material

used (Field et al. 2007). These technical factors, including

operational inefficiencies, drive the quantity of factor

resources that are required to produce a given level of

output for a given type of technology. Understanding the

effect of these underlying technical cost drivers can pro-

vide insight for managers and engineers as to what process

improvements are most critical to lower production costs

(Fuchs et al. 2006). It has been proposed that reducing the

number of parts in a product will result in cost savings

(Ernst 1987). IBM increased productivity by 700% after

Manufacturing process

Cost

Technical Capability

Production

Maintenance Cost

Product Cost

Flexibility

Efficiency

Mechanism

Production

Packaging

Delivery

Quality

Commitment to quality

Improved process capability

Low defect rateProcessing Cost

Fig. 2 Fish bone diagram of

manufacturing process

Fig. 3 Process-based cost

modeling framework (Field

et al. 2007)

Int J Syst Assur Eng Manag (Oct-Dec 2011) 2(4):301–311 303

123

reducing part count by two-thirds; Ford reduced the part

count in its door trim by 79%, assembly cost by 94%, and

material cost by 27% (Johnson and Kirchain 2009). Also

similar models have been developed to examine the pro-

duction of many technologies including both structural

(Fuchs et al. 2008; Ruffo et al. 2006) and electronic

components (Fuchs et al. 2006; Singer 2004).

3.3 Technical Capability

The term Technical Capability is an important factor which

influencing the selection of manufacturing process signifi-

cantly. Understanding the structure of a process and

quantifying process performance no doubt are essential for

successful quality improvement initiatives (Chien-Wei

et al. 2009). Growth accounting studies typically find a

very large proportion of growth is due to the ‘Solow

residual’ which includes technological progress (Best

2001). So, an important part of more rapid growth is the

ability of firms and their workers to learn how to upgrade

their production capabilities and access new markets (Te-

wari 2002).

3.4 Production

All manufacturing process has some strengths and weak-

nesses. But the prime motive is to determine an actual

balance between the desired production and cost because it

is obvious that, faster production and greater precision

generally increased the cost of product which is always

undesirable. So to find the best manufacturing process, it is

necessary to exercise for an accurate and precise manu-

facturing process which can provide desired production

without increasing the cost of product. The manufacturing

process should be of such type which can provide higher

production rate at low cost, easier and smooth packaging of

product. While selecting the manufacturing process it is

also essential to consider, that the product must reach to the

customer end within the specified period safely.

4 Application of graph theoretic approach

The single numerical index is resolute for selection of

suitable manufacturing process using graph theoretic

approach. As already discussed, the graph theoretic

approach has three steps as following:

1. Manufacturing process based digraph

2. Matrix representation of manufacturing process

3. Permanent function representation of manufacturing

process

4.1 Manufacturing process based digraph

A digraph (directed graph) is a graphical representation of

factors identified and interdependence between them. The

digraph carries nodes and edges. The nodes (Bi’s) represent

the factors identified and edges (bij’s) represent the inter-

action among different processes. Bi represents the inher-

itance of factors and bij represent the degree of dependence

of jth factor on ith factor. In the digraph bij is represented

as a directed edge from node i to node j. To show the

manufacturing process digraph, the four factors identified

are taken into consideration, i.e., Quality (B1), Cost (B2),

Technical Capability (B3), Production (B4). Based on the

interactions between the factors a digraph has been plotted

as shown in Fig. 4.

In the digraph represented above, the edges directed

from B1 to B2 and B2 to B1 shows that the nodes B1 and B2

are interdependent. It is obvious that good quality of

product may lead to increase the investments to maintain a

persistent quality level. These investments may increases

the cost of product. When the product cost is higher, it is

easy to maintain a proper quality level because due to

increased cost the manufacturers have sufficient funds to

maintain the quality level. Generally speaking, a higher

level product should be sold at a higher price due to more

costly or more extensive features (Freiesleben 2010). The

edges directed from B1 to B4 and B4 to B1 shows that nodes

B1 and B4 are also interdependent. Quality and production

control are vital activities for the profitability of modern

manufacturing companies, that are continuously facing

product specification and demand changes (Wiendahl et al.

2007). Quality control makes it possible to meet high

product quality standards, also reducing scraps and

reworks. Production control reduces the work in progress

(WIP) while meeting the target production rate (Colledani

and Tolio 2011).

The edge directed from B3 to B1 shows that node B1 is

affected by node B3. It is a proven fact that quality level of

any product can be raised by an effective and updated

Fig. 4 Manufacturing process based digraph


123

Technical Capability. In most circumstances, high perfor-

mance firms would have stronger capabilities as compared

to low performance firms (Yam et al. 2010). Technological

innovation capability is a comprehensive set of character-

istics of an organization that facilities and supports its

technological innovation strategies (Burgelman et al.

2004). They are a kind of special assets or resources that

include technology, product, process, knowledge, experi-

ence and organization (Guan and Ma 2003). The edge

directed from B3 to B4 shows that node B4 is affected by

node B3. The production of any organization is greatly

influenced by technical capabilities of the organization. As

the technical capabilities smoothness the production sys-

tem. If the technical capabilities of any organization are not

competent enough then, it is difficult for such organizations

to run the production system effectively. Due to rapid

technological change, high-tech industries typically require

a certain level of timely and dynamic strategy (Yasuda

2005).

The edges directed from B2 to B4 and B4 to B2 shows

that node B2 and node B4 are interdependent. To get good

quality production an organization needs good quality of

raw material, machinery, technology, skilled labour etc.

and all this leads to increase the cost of product. Also if the

production of product is not satisfying then rework or

rejection may lead to increase the cost. Higher product

variety leads to higher forecast errors, excessive inventory

for some products and shortages for others, higher over-

head and administrative costs, and higher manufacturing

costs due to more specialized processes, materials,

changeovers, and quality assurance methods (Lee and

Billington 1994). The edge directed from B2 to B3 shows

that node B3 is affected by node B2. As already discussed

between node B1 and B3, to enhance the quality of product,

it is essential to have effective and updated technical

capabilities. To get similar technical capabilities the cost of

product will increase due to the expense incurred in

upgrading and maintaining the technical capabilities.

The graphical representation of factors given through

the digraph is suitable for visual analysis. But it becomes

complex when a number of nodes in a given case (i.e., the

factors) increase. Moreover, it is not suitable for computer

processing. For this, the digraph is represented in matrix

form.

4.2 Matrix representation of manufacturing process

As already discussed in the previous section that the

graphical representation of factors is complex. So to reduce

the complexity of the digraph it is proposed to present a

manufacturing process digraph in the form of a matrix

which can be further resolved with the help of software.

The matrix of Manufacturing Process is an N 9 N

matrix, which considers all the factors (Bi’s) affecting the

manufacturing process and their relative interdependencies

(bij’s). The matrix representation of manufacturing process

is a one-to-one representation of the digraph. The matrix of

manufacturing process considering all the factors can be

represented as:

1 2 3 4 Factors

P1 ¼

B1 b12 b13 b14

b21 B2 b23 b24

b31 b32 B3 b34

b41 b42 b43 B4

0BBB@

1CCCA

1

2

3

4

ð2Þ

In the above mentioned matrix the diagonal elements

represents the inheritance of factors whereas the off-

diagonal elements represents the interaction between the

factors. The row of the matrix shows the inheritance of a

factor and its influence on other factors. And the column of

the matrix shows inheritance of factor and its dependency

on other factors. According to the digraph the

manufacturing process matrix can be represented as:

P1 ¼

B1 b12 0 b14

b21 B2 b23 b24

b31 0 B3 b34

b41 b42 0 B4

0BB@

1CCA ð3Þ

In the above matrix the some off-diagonal elements are

zero as there is no edge directed between them in the

manufacturing process digraph.

The value of off-diagonal elements can be determined

from Table 1 and to determine the value of diagonal ele-

ment, it is proposed to represent digraph for each factor

with their co-factors. The value of diagonal elements can

be determined through Table 2 also, but to get more pre-

cise value it is suggested to plot the digraph for each factor.

The value of diagonal elements can also be determined

by Table 2.

Now,

P1 ¼

B1 5 0 4

5 B2 4 3

4 0 B3 2

3 2 0 B4

0BB@

1CCA ð4Þ


To find out a single numerical index, the permanent of the

matrix, called as variable permanent quality function of

manufacturing process is used here. The permanent func-

tion is obtained in a similar manner as its determinant but

with keeping all signs positive. This expression is repre-

sents the rating of manufacturing process and contains all

possible quality terms of the manufacturing process. The


123

VPF expression corresponds to the four-characteristic

digraph/VPM is given as.

VPF ¼B1B2B3B4

þ b12b21B3B4 þ b13b31B2B4 þ b14b41B2B3

þ b23b32B1B4 þ b24b42B1B3 þ b34b43B1B2

þ b12b23b31B4 þ b13b32b21B4 þ b12b24b41B3

þ b42b21B3 þ b13b34b41B2 þ b14b43b31B2

þ b23b34b42B1 þ b24b43b32B1

þ b12b21b34b43 þ b13b31b24b42 þ b14b41b23b32



ð5Þ

4.4 Determination of value of diagonal elements

4.4.1 Quality based digraph

The factor B1, i.e., Quality is having three co-factors: low

defect rate (Q1), commitment to quality (Q2), improved

process capability (Q3) and to determine the interdepen-

dency between them, a quality based digraph is represented

as shown in Fig. 5.

To simplify the complexity of digraph shown in Fig. 5,

it is required to convert the digraph into a matrix as shown

in matrix 4.

B1 ¼Q1 q12 q13

q21 Q2 q23

q31 q32 Q3

0@

1A ð6Þ

B1 ¼4 4 0

5 3 4

3 3 4

0@

1A ð7Þ

The value of off-diagonal elements can be determined

through Table 1 and the value of diagonal elements can be

determined through Table 2.

To determine the value of factor B1, it is proposed to

find out the permanent function of the matrix using Eq. 1.

B1 ¼ 224

4.5 Cost based digraph

The factor B2, i.e., Cost is having three co-factors: pro-

cessing cost (C1), maintenance cost (C2), product cost (C3)

and to determine the interdependency between them, a cost

based digraph is represented as shown in Fig. 6.

Matrix representation of Cost based digraph is as

follows:

B2 ¼C1 c12 c13

c21 C2 c23

c31 c32 C3

0@

1A ð8Þ

B2 ¼4 0 4

0 4 3

0 0 3

0@

1A ð9Þ

So the permanent function of the matrix using Eq. 1 is

48, and,

B2 ¼ 48

4.6 Technical Capability based digraph

The factor B3, i.e., Technical Capability is having three co-

factors: Flexibility (T1), Efficiency (T2), Mechanism (T3)

Table 1 Value of interdependency of factors (Bij)

S. no. Qualitative measure

of interdependency

Assigned value

of factor

1 Very strong 5

2 Strong 4

3 Medium 3

4 Weak 2

5 Very weak 1

Table 2 Value of inheritance of factors (Bi)

S. no. Qualitative measure

of factor

Assigned value

of factor

1 Extremely low 1

2 Low 2

3 Below average 3

4 Average 4

5 Above average 5

6 High 6

7 Extremely high 7

Q1

Q2 Q3

Fig. 5 Quality based digraph

(B1)

C1

C2 C3

Fig. 6 Cost based digraph (B2)


123

and to determine the interdependency between them, a

Technical Capability based digraph is represented as

shown in Fig. 7.

Matrix representation of Technical Capability based

digraph is as follows:

B3 ¼T1 t12 t13

t21 T2 t23

t31 t32 T3

0@

1A ð10Þ

B3 ¼4 4 3

0 5 4

0 3 5

0@

1A ð11Þ


148, and,

B3 ¼ 148

4.7 Production based digraph

The factor B4, i.e., Production is having three co-factors:

Production rate (P1), Packaging (P2), Delivery (P3) and to

determine the interdependency between them, a production

based digraph is represented as shown in Fig. 8.

Matrix representation of Technical Capability based

digraph is as follows:

B4 ¼P1 p12 p13

p21 P2 p23

p31 p32 P3

0@

1A ð12Þ

B4 ¼5 4 5

0 3 3

2 2 5

0@

1A ð13Þ


159, and,

B4 ¼ 159


The single numerical index of the manufacturing process

matrix can be determined using Eq. 3 by filling the value of

factor B1, B2, B3 and B4 in matrix (3).

VPF ¼ 224� 48� 148� 159

þ 5� 5� 148� 159þ 0� 4� 48� 159

þ 4� 3� 48� 148þ4� 0� 224� 159þ 3� 2

� 224� 148þ 2� 0� 224� 48þ 5� 4

� 4� 159þ 0� 0� 5� 159þ 5� 3� 3� 148

þ 4� 2� 5� 148þ 0� 2� 3� 48þ 4� 0� 4

� 48þ 4� 2� 2� 224þ 3� 0� 0� 224

þ 5� 5� 2� 0þ 0� 4� 3� 2þ 4� 3� 4� 0

þ 5� 4� 2� 3þ 4� 0� 0� 5þ 0� 2� 2� 5

þ 5� 3� 0� 4þ 4� 2� 4� 4þ 0� 0� 3� 3

VPF ¼ 253917656

5 An illustrative example

To exhibit the proposed methodology, an example is taken

considering mass production of a mild steel cylindrical bar

of [ 20 mm and length 50 mm. For manufacturing the

cylindrical bar, three different manufacturing methods are

proposed as following:

I. CNC lathe

II. Casting

III. Extrusion

The following assumptions have been made to manu-

facturing process:

1. All the necessary machinery and equipments required for

each manufacturing process mentioned above is available

within the Industry, no further investment is required.

2. The cylindrical bar can be produced from any of the

manufacturing process, as there is no any restriction

from the customer end. So the required bar can be

produced from any of the process.

3. All Technical capabilities and expertise required for each

manufacturing process are available within the Industry.

6 Identification of attributes for selection

of manufacturing process

To select the best manufacturing process among three

manufacturing methods, it is necessary to identify the

T1

T2 T3

Fig. 7 Technical Capability

based digraph

P1

P2 P3

Fig. 8 Production based

digraph


123

Table 3 Application of graph theoretic approach


123

Table 3 continued


123

attributes affecting the manufacturing process significantly.

The attributes and their co-factors affecting the selection of

manufacturing process are as following:

1. Quality—low defect rate, less scrap and wastage,

commitment to quality

2. Cost—processing cost, maintenance cost, product cost

3. Production—time taken per piece, packaging, system

complexity

7 Application of graph theoretic approach

To identify the single numerical index for the selection of

best manufacturing, the graph theoretic approach is applied.

The graph theoretic approach has following three steps:

• Digraph representation

• Matrix representation

• Permanent function representation

The digraph representation and matrix representation for

all the three processes taken here will remain same but

values will differ accordingly from process to process.

Table 3 represents the application of graph theoretic

approach for the selection of best process among three

processes, i.e., CNC Lathe, Casting and Extrusion.

The Permanent function representation of the different

processes which reveals a single numerical index for the

selection of best process are as following:

Process I CNC Latheð Þ ¼ 674168

Process II Castingð Þ ¼ 4052

Table 3 continued


123

Process III Extrusionð Þ ¼ 107910

The above calculations shows that production of a mild steel

cylindrical bar through Process I (CNC Lathe) is much better

and effective process than other processes as the permanent

function is 674168 which is higher than the permanent function

of Process II (Casting) and Process III (Extrusion)

8 Conclusions

The paper shows methodology for selecting best feasible

manufacturing process in general using graph theoretic

approach which reveals a single numerical index. The

index is helpful in assessing the best manufacturing pro-

cess. The factors/attributes chosen for the selection and

assessment of best manufacturing process are Quality,

Cost, Technical Capability, Production. A fishbone dia-

gram is drawn to represent the factors and co-factors which

affects the selection of manufacturing process. The graph

theoretic approach is divided into three stages namely:

digraph representation, matrix representation and perma-

nent function representation. Digraph representation is a

graphical representation of identified factors and interde-

pendence between them. As the digraph is a visual repre-

sentation and it seems complex with higher number of

factors, So to reduce the complexity, the digraph is con-

verted into a matrix. To find out a single numerical index,

the permanent of the matrix, called as variable permanent

quality function of manufacturing process is determined.

To determine the value of factors, i.e., B1 (Quality), B2

(Cost), B3 (Technical Capability) and B4 (Production),

digraphs are drawn for each factor so that precise values of

the factors can be determined.

An example is also explained for understanding the

application of graph theoretic approach. In this example,

three processes (i.e., CNC Lathe, Casting, Extrusion) are

mentioned for mass production of a mild steel cylindrical bar

of [ 20 mm and length 50 mm. The attributes chosen for the

selection of best process are Quality, Cost and Production.

The result shows that permanent function representation of

CNC lathe process is 674168 which is higher than the per-

manent function of casting and extrusion process. So, the

results proves that the CNC lathe process is much better and

effective process than casting and extrusion process.

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Selection of manufacturing process using graph theoretic approach

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