Fuzzy Dr Hegazi

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A Fuzzy Approach for Material Selection from a Manufacturing and

Application Viewpoint

Hesham A. Hegazi

+

, Tarek M. El-Hossainy*

+Assistant Professor, Department of Mechanical Engineering, The American University in

Cairo (AUC), Cairo 11511, EGYPT

Email: [email protected]*Assistant Professor, Department of Mechanical Design & Production

Faculty of Engineering, Cairo University, Giza, 12316, EGYPT

Email: [email protected]

Abstract

Material selection process constitutes a high level of vagueness and imprecision.

Selected material must fulfil the machining requirements as well as application needs.

Due to the lack of complete information, uncertainty, and imprecision for material

selection in such applications, a technique to perform selection calculations on

imprecise representations of parameters is presented. This technique is based on fuzzy

logic using fuzzy sets. Different materials alternatives are expressed in terms of fuzzy

orders of magnitude. Calculations based on fuzzy weighted average are performed to

produce the ratings among selected material alternatives. This technique is applied for

the different alternatives based on the manufacturing requirements and the application

requirements. At the end, the third fuzzy decision making process that combines thehighest attributes important for both manufacturing and application requirements is

presented. According to a predefined goal, materials are classified depending on the

nearest to this goal.

1. Introduction

The product material selection is affected by two basic aspects, machining and its

associated parameters and the required product technical specifications needed in the

market. Much of the decision-making in the real world takes place in an environment

in which the goals, the constraints and the consequences of possible actions are notknown precisely. Fuzzy analysis should be introduced to product development so that

decision-making in difficult situations is eased and product cost and quality are

improved while time-to-market is shortened. The procedure of evaluating of multiple

attributes was investigated by many researchers in the last and present decades [1, 2].

Many researchers worked on improving the decision making process, especially in

design, material selection and manufacturing. Evaluation of preliminary designs is

often necessary when the design alternative is only in the roughest concept stage

[3]. The underlying power of fuzzy set theory is that it uses linguistic variables, rather

than quantitative variables, to represent imprecise concepts [4]. The imprecision in

fuzzy models is therefore generally quite high.
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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El Baradie [5] described the development stages of a fuzzy logic model for metal

cutting. His model is based on the assumption that the relationship between the

hardness of a given material and the recommended cutting speed is imprecise, and can

be described and evaluated by the theory of fuzzy sets. The model has been applied to

data extracted from Machining Data Handbook and a very good correlation was

obtained between the handbook data and that predicted using the fuzzy logic model.He concluded that the strategy and action of the skilled machine tool operator when

selecting the cutting speed and feed rate for a given material can be described by the

theory of fuzzy sets, as his strategy and action are based on intuition and experience.

He added that the relationship between a given material hardness and the

recommended cutting speed can be described and evaluated by the fuzzy sets. He

ended that the fuzzy logic model proposed suggests the possibility of establishing the

strategy of machining data selection for a specific machining process.

Chen [6] used calculations based on fuzzy weighted average to produce the ratings

among design alternatives. He demonstrated his method in the bearing selection case

study where imprecise linguistic description of the design problem in a mannersimilar to human language can be accommodated. The problem was the selection of

the best bearings for a specific problem. In his implementation, a qualitative linguistic

description of a desired bearing is used as weights in the fuzzy weighted average

algorithm. The evaluation of alternatives in fuzzy numbers was ranked according to

preferability.

Thruston et al. [4] developed a procedure for the evaluation of multiple attributes in

the preliminary design stage. They demonstrated and compared two techniques: fuzzy

set analysis and multi-attribute utility analysis. The problem of preliminary material

selection for an automobile bumper beam was analysed to illustrate the application of

both analytical procedures. They recommended the use of fuzzy analysis in the

earliest stage of preliminary design evaluations. Utility analysis may be used in later

stages of preliminary design, where numerical qualification of attribute levels is

possible.

Zhao et al. [7] mentioned that with the increasing of global market competition and

dynamic change of market environment, consumer needs are more personal and

diversified, enterprise production is more flexible. At present, the production cycle

period of traditional manufacturing industry is long, delivering goods is not in time,

product quality is not good and resource are not used in reason. Because of the

phenomena, products are not met the requirements of market and lack of marketcompetition ability. They concluded that using improved fuzzy reform optimization

method, enterprise can develop product rapidly to satisfy consumer requirements and

have high quality, low cost, reasonable price and good service that is because it can

assign right task to right person in right time for shortening development time of

product.

Wood et al. [8] developed a technique to perform design calculations on imprecise

representations of parameters, using fuzzy calculus. The fuzzy weighted average

technique is used to perform these calculations. They demonstrated the technique

using a simple mechanical design example. The problem was to design a mechanical

structure, attached to a wall at one end, while supporting an overhanging vertical pointload. Additional useful information that this method can provide, through the use of


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the -level measure, was the coupling between imprecise representations of designparameters (inputs) and the performance parameter results.

Koning [9] concentrated on two types of material-related reasoning that occur in

engineering design: selection and substitution. He represented a material selection

substitution system that is used as a part of a larger case-based design environment.

The material selection system helps the designer to adapt previous designs bysuggesting material substitutions that better suit the target application. The fuzzy sets

based representation in his system supported the following types of queries to the

material knowledge base: 1) given a material class, what is the range of possible

material property, 2) given an order of magnitude of a material property, what are the

corresponding material classes.

The selected material should meet in globally both the production requirements and

the market needs. This could improve the production performance and the product life

cycle. The fuzzy analysis should take place to select one or two best materials which

could identify as a possible solution for a decision making process and eventually the

product development can be significantly improved.

The purpose of the paper is the implementation of the fuzzy theory in the selection

from material alternatives according to manufacturing and application requirements.

2. Fuzzy Analysis and Computations

The nature of uncertainty in a problem is a very important point that engineers should

ponder prior to their selection of an appropriate method to express the uncertainty.

Fuzzy sets provide a mathematical way to represent vagueness in humanistic systems

[5].

2.1 Fundamentals

For a classic set Uwhose generic elements are denoted u. membership in a classic

subset Fof U is often viewed as a characteristic function F such that[5, 10, 11]:

F uiff u F

iff u F ( ) =

1

0(1)

(Note that, iff is short for if and only if.)

The characteristic function is generalized to a membership function that assigns to

every u U a value from the unit interval [0, 1] instead from the two-element set{0, 1}.

The membership functionFof a fuzzy set Fis a function

F: U [0, 1] (2)


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So, every element u from Uhas a membership degree F[0, 1].Fis completelydetermined by the set of tuples:

F = {(u, F(u)) : u U} (3)

For example, considering the fuzzy set Fas number close to 9.

IfU= {1, 5, 8, 13, 25}, the set Fcould be defined as a tabulation of its membership

function at each uU:

F= {(1, 0.2), (5, 0.5), (8, 0.9), (9, 1.0), (13, 0.5), (25, 0)} (4)

where

F = F(u1) / u1 + ........+ F(un) / un = (5)( )Fi

n

iu u=

1

/ i

Any countable or discrete universe Uallows a notation

F= Fu U

u( ) / u

u

(6)

but when U is uncountable or continuous, we will write

F= (7)FU

u( ) /

2.2 Fuzzy Analysis

The extension principle defines a fuzzy set Cand its membership function C y( )

,.....,1induced by a real function y = f(x1, ....., xr) and the fuzzy sets

with membership function

B ii, = r( )xi .

[{ } C x x r y x xr( ) sup min ( ), ( ),......., ( ),.......= 1 1 2 ] x (8)

where sup refers to the supremum achieved by choice of x1 , ....., xr. Thus the

definition of C y( ) requires the solution of a maximization problem for each value

ofy defined by f(x1 , ....., xr). Membership functions for fuzzy numbers can be

approximated using a number of -cuts which are a set of n intervals [ai, bi], i=1,....n over which (x) i for ai < x < bi, where i = (i - 1)/(n - 1). The fuzzyweighted average (FWA) algorithm developed by Dong, et al. [12] and illustrated its

application when the functiony = f(x1, ....., xr) is


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y

w x

w

i ii

n

ii

n= =

=

1

1

(9)

In describing a design alternative, one approach is to use a rating which describes the

desired levels of the attributes and further to attach weights to the ratings according to

the importance of those attributes. Real variables can be used to express the weights

and ratings by means of a figure of merit. If the weight for thejth attribute is wi, and

the rating for the jth attribute of the ith alternative is rij, it would be natural to

compute the weighted averages, ri, by the following equation [4, 6]:

r w r ii j ijj

k

= ==

121

, ,.....I (10)

for each alternative, and to rank the alternatives accordingly.

3. Selection Method Based on Manufacturing Requirements

The selection process considers six attributes: power consumed, tool life, surface

roughness, production rate, production cost, and machining accuracy. On the other

hand, the designer might easily be able to describe the attributes of an alternative as

Very High, High, Low,... in relation to other alternatives. These attributes will be

evaluated for six candidates materials, and a fuzzy rating will be calculated from Eq.

10. Table A1 shows the properties of the selected materials. Table A2 in Appendix

(A)

shows the main characteristics of these materials, while Table A3 summarizes the

general applications of these materials. All attributes may assume a fuzzy value as

defined in Table 1, which gives the summary of the fuzzy number assignments.

Seven levels will be used: very low (VL), low (L), low to middle (ML), middle (M),

middle to high (MH), high (H), and very high (VH). Then the universe of discourse U

will be expressed as the finite set of fuzzy numbers U = { U1, U2 , ....., U7} where~

,

~

,......

~

.U VL U L U VH 1 2 7= = = Membership functions characterize thefuzziness in a fuzzy set-whether the elements in the set are discrete or continuous- in a

graphical form for eventual use in the mathematical formalism of fuzzy set theory.

But the shapes used to describe the fuzziness have few restrictions indeed. It might be

claimed that the rules used to describe fuzziness graphically are also fuzzy[5]. It is

usual to have functions with straight lines, instead of functions with curves. The

membership functions will be defined as a triangular in shape, generally following the

approach given by [4, 6] as shown in Figure 1, for a variable (x) ranging from 0 to 1.

For k= 1:


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U xx or x

x x1

0 0

1 6 0 1 6( )

/

/=

1 6

for k= 2, 3, 4, 5, 6

Uk x

x k or x k

x k k x k

k x k x k

( )

( ) / /

( ) ( ) / ( ) /

( ) / /

=

0 2 6

6 2 2 6 1 6

6 1 6 6

6

x 1

and for k= 7:

(11)U xx or

x x7

0 5 6

6 5 5 6 1( )

/

/=

Table 1: Fuzzy description of attributes based on manufacturing requirements.

Power

Consumed

Tool

life

Surface

Roughness

Production

Rate

Production

Cost

Machining

Accuracy

Carbon Steel

AISI 1050,

0.54% C, Q & T

H ML H M H H

Alloy Steel

AISI 4140,

0.4% C, Q & T

VH L MH M VH H

Gray Cast IronASTM Class 60

MH ML VH L MH L

Aluminium Bronze

Heat Treated

L M L VH ML M

Tin Bronze

Chill Cast

VL H VL VH L M

Aluminium

2024 T4

L M VL VH ML MH

VL: Very Low, L: Low, ML: Low to Middle, M: Middle, MH: Middle to High,

H: High, and VH: Very High.

Table 2:Fuzzy descriptions of the goal based on manufacturing requirements.

Power

Consumed

Tool

life

Surface

Roughness

Production

Rate

Production

Cost

Machining

Accuracy

Importance M ML H L H VH

Goal VL VH VL VH VL VH

When starting a new design, the designer needs to specify the requirements for the

power consumed, tool life, surface roughness, production rate, production cost, andmachining accuracy. The levels of the attributes of each alternative are described

AlternativesCriteria

Criteria


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using the set of fuzzy numbers given above; in addition, the importance of each

alternative is weighted using the same universe of discourse as shown in Table 2. The

idea of selecting the best alternative is based on finding the alternative which is closed

to a fuzzy goal. The description of the fuzzy goal is shown in Table 2.

The membership functions of the fuzzy rating ri can now be computed for eachalternative material, from the extended fuzzy number multiplication and summation.

Since all of the membership functions of the fuzzy numbers are triangular, the exact

computation is straight forward, although the algebra can become tedious [4]. The

exact value of membership functions are given in equation (B5, B7), and it is only for

the first alternative of Carbon Steel AISI 1050 Q & T, the computations for the other

alternatives can be done in the same manner as described in Appendix (B).

A triangular approximation of the equation (B5), and (B7) would be extremely close

and more than adequate for the comparison of alternatives [6]. After evaluation, the

membership functions for the alternative ratings are respectively represented by the

triplets:

U(ra):

36

149,

36

102,

36

61; U(rb):

36

140,

36

103,

36

62; U(rc):

36

111,

36

73,

36

39

U(rd):

36

86,

36

48,

36

16; U(re):

36

76,

36

39,

36

14; U(rf):

36

86,

36

49,

36

21;

U(rGoal):

36

82,

36

54,

36

30(12)

The approximate triangular plot of the membership functions is given in Figure 2.

The membership functions for the fuzzy rating readily indicates that the Aluminium

2024 T4 alternative is the closed to the goal, so it is the preferable choice which

ranked first by the fuzzy scheme. The Aluminium Bronze Heat Treated ranked

second, followed by Tin Bronze Chill Cast, followed by Gray Cast Iron ASTM Class

60, followed by Carbon Steel AISI 1050 Q & T, then the Alloy Steel AISI 4140 Q &

T.

4. Selection Method Based on Application Requirements

In case of considering the application requirements, the selection process considers

six attributes: material cost, wear resistance, heat resistance, specific gravity, fatigue

resistance, and corrosion resistance. On the other hand, the designer might easily be

able to describe the attributes of an alternative as Very High, High, Low,... in

relation to other alternatives. These attributes will be evaluated for six candidates

materials, and a fuzzy rating will be calculated from Eq. 10. All attributes may assume

a fuzzy value as defined in Table 3, which gives the summary of the fuzzy number

assignments.

The levels of the attributes of each alternative are described using the set of fuzzy

numbers given above; in addition, the importance of each alternative is weighted


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using the same universe of discourse as shown in Table 4. The idea of selecting the

best alternative is based on finding the alternative which is closed to a fuzzy goal. The

description of the fuzzy goal is shown in Table 4.

Table 3: Fuzzy description of attributes based on application requirements.

Material

Cost

Wear

Resistance

Heat

Resistance

Specific

Gravity

Fatigue

Resistance

Corrosion

Resistance

Carbon Steel

AISI 1050,

0.54% C, Q & T

M MH H VH MH M

Alloy Steel

AISI 4140,

0.4% C, Q & T

H H VH VH H MH

Gray Cast Iron

ASTM Class 60

L H MH H L M

Aluminium Bronze

Heat Treated

VH M M VH MH VH

Tin Bronze

Chill Cast

VH L M VH MH VH

Aluminium

2024 T4

H L VL VL L H

VL: Very Low, L: Low, ML: Low to Middle, M: Middle, MH: Middle to High,

H: High, and VH: Very High.

Table 4:Fuzzy descriptions of the goal based on application requirements.

Material

Cost

Wear

Resistance

Heat

Resistance

Specific

Gravity

Fatigue

Resistance

Corrosion

Resistance

Importance H VH VH M M ML

Goal VL VH VH VL VH VH

A triangular approximation of the equation (A5), and (A7) would be extremely close

and more than adequate for the comparison of alternatives[6]. After evaluation, the

membership functions for the alternative ratings are respectively represented by the

triplets:

U(ra):

36

146,

36

105,

36

61; U(rb):

36

171,

36

132,

36

82; U(rc):

36

122,

36

83,

36

45

U(rd):

36

146,

36

108,

36

61; U(re):

36

134,

36

96,

36

51; U(rf):

36

84,

36

44,

36

20;

U(rGoal):

36

124,

36

102,

36

65(13)

Alternatives

Criteria

Criteria


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The membership functions for the fuzzy rating readily indicates that the Carbon Steel

AISI 1050 Q & T alternative is the closed to the goal, so it is the preferable choice

which ranked first by the fuzzy scheme. Both Aluminium Bronze Heat Treated and

Tin Bronze Chill Cast ranked second, followed by Gray Cast Iron ASTM Class 60,

followed by Alloy Steel AISI 4140 Q & T, then Aluminium 2024 T4.

5. Selection Method Based on Manufacturing and Application Requirements

In case of considering both manufacturing and application requirements, the selection

process considers the highest importance. In case of the manufacturing requirements,

surface roughness, and production cost are ranked high, while machining accuracy is

raked very high. In case of application requirements, material cost is ranked high,

while wear resistance and heat resistance are ranked very high. The new combined

decision matrix Table 5 shows the selected six attributes based on the above selection

criteria. Table 6 shows the importance of each attribute and a goal considered as areference for this combined selection.

Table 5: Fuzzy description of attributes based on manufacturing and

application requirements.

Surface

Roughness

Production

Cost

Machinin

g

Accuracy

Material

Cost

Wear

Resistance

Heat

ResistanceAlternatives

Criteria

Carbon Steel

AISI 1050,0.54% C, Q & T

H H H M MH H

Alloy Steel

AISI 4140,

0.4% C, Q & T

MH VH H H H VH

Gray Cast Iron

ASTM Class 60

VH MH L L H MH

Aluminium Bronze

Heat Treated

L ML M VH M M

Tin Bronze

Chill Cast

VL L M VH L M

Aluminium

2024 T4

VL ML MH H L VL


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Table 6:Fuzzy descriptions of the goal based on manufacturing and application

requirements.

Surface

Roughness

Production

Cost

Machinin

g

Accuracy

Material

Cost

Wear

Resistance

Heat

Resistance

Criteria

Importance H MH VH ML H M

Goal VL VL VH VL VH VH

After evaluation, the membership functions for the alternative ratings are respectively

represented by the triplets:

U(ra):

36

168,

36

116,

36

70; U(rb):

36

174,

36

127,

36

77; U(rc):

36

135,

36

91,

36

51

U(rd): 36109,

3667,

3630 ; U(re): 36

86,3648,

3619 ; U(rf): 36

85,3647,

3622 ;

U(rGoal):

36

110,

36

84,

36

55(14)


The membership functions for the fuzzy rating readily indicates that the Gray Cast

Iron ASTM Class 60 alternative is the closed to the goal, so it is the preferable choice

which ranked first by the fuzzy scheme. The Aluminium Bronze Heat Treated ranked

second, followed by Carbon Steel AISI 1050 Q & T, followed by Tin Bronze Chill

Cast, followed by Aluminium 2024 T4, then Alloy Steel AISI 4140 Q & T.

7. Conclusion

This paper has detailed the implementation of fuzzy rating in the process of material

selection when considering manufacturing and application. In the implementation, a

qualitative linguistic description of a desired material type is used as weights in the

fuzzy weighted average algorithm. It gives a simple and strong way for the selection

of an alternative when the attributes are imprecise. The application of the fuzzy theory

was practically applied to the selection of the proper material among six attributes

depending on manufacturing and application requirements. Based on eachrequirement and the defined goal, attributes are ranked with respect to the goal. A

combined attributes is defined based on the highest and the very highest importance in

both manufacturing and application requirements. The output ranked materials in this

case study can be the optimum decision of selecting materials for general engineering

products such as gears, cams, shafts, pulleys, etc... Similar analysis can be

implemented for different applications, manufacturing processes and different

alternative materials.


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References

1. Osman T. A., Hegazi H. A., A Fuzzy Approach for the Selection of Power

Trasmission Systems in the Preliminary Design Stage, Journal of Engineering and

Applied Science, Vol. 46, No. 4, pp. 679-692, Cairo University, 1999.

2. Kimura, I., Grote, K.-H., Design Decision-Making in the Early Stages ofCollaborative Engineering, Proceeding of the Design Automation Conf.,

Montreal, Canada, Paper No. DETC2002/DAC-34034, 2002.

3. Thurston D. L., Carnahan J. V., Fuzzy Ratings and Utility Analysis in Preliminary

design Evaluation of Multiple Attributes. Trans. ASME, J. Mech. Des. 114, pp.

648-658, 1992.

4. Ross T. J., Fuzzy Logic With Engineering Applications. McGraw-Hill Inc, 1995.

5. El Baradie M. A., A Fuzzy Logic Model For Machining Data Selection, Int. J.

Mech. Tools Manufact. Vol. 37, No. 9, pp. 1353-1372, 1997.

6. Chen Y. H., Fuzzy Ratings in Mechanical Engineering Design--Application to

Bearing Selection. Proc. IMechE, Part B, J. of Engineering Manufacture 210, pp.

49-53, 1996.

7. Zhao, Y., Cha, J., Zhang, J., Fuzzy Reform and Optimization of Design Task in

Concurrent Engineering. Proceeding of the Design Automation Conf., Pittsburgh,

Pennsylvania, U.S.A., Paper No. DETC2001/DAC-21158, 2001.

8. Wood K. L., Antonsson E. K., Computations With Imprecise Parameters in

Engineering Design: Background and Theory. Trans. ASME, J. Mechanisms,

Transmissions, and Automn in Des. 111, pp. 616-625, 1989.

9. Koning J., A Fuzzy Approach to Material Selection in Mechanical Design.

Concurrent Engg, Research and Applications 3(4), pp. 271-279, 1995.

10. Evbuomwan N F O, Sivaloganathan S, Jebb A, A Survey of Design Philosophies,

Models, Methods, and Systems. Proc. IMechE, Part B, J. of EngineeringManufacture 210, pp. 301-320, 1996.

11. Driankov D., Hellendoorn H, Reinfrank M, An Introduction to Fuzzy Control.

Springer-Verlag, 1996.

12. Dong W. M., Wong F S, Fuzzy Weighted Averages and Implementation of the

Extension Principle. Fuzzy Sets and System 21, pp. 183-199, 1987.

13. Mott, R.L., Machine Elements in Mechanical Design. Charles E. Merrill, 2004.

14. Bralla, J.G., Design for Manufacturability Handbook. McGraw-Hill, USA,

1999.

15. Machining Data Handbook. Machinability Data Centre Metcut Research

Associates Inc., Cincinnati, Ohio, 1972.

16. Show, M.C., Metal Cutting Principles. Oxford University Press, Inc., Oxford,New York, USA, 2005.

17. Meyers, A.R., Slattery, T.J., Basic Machining Reference Handbook. Industrial

Press Inc., New York, USA, 1988.

18. Deutschman, A.D., Michels, W.J., Wilson, C.E., Machine Design: Theory and

Practice. Macmillan Publishing Co., Inc., New York, USA, 1975.


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Figure 1 Membership function for fuzzy numbers.

Figure 2 Membership function of alternative rating and goal with respect to

manufacturing requirements.


13/17


application requirements.


manufacturing and application requirements.


14/17

Appendix A: Properties of the selected materials:

Table A1 shows the material alternatives properties, while Table A2 and Table A3

present some characteristics and some applications for the material alternatives

respectively.

Table A1: Material properties [13, 14].

Hardness

(BHN)

Tensile

strength,

(MPa)

Yield

strength,

(MPa)

Specific

Gravity

Melting point

(C)

Carbon Steel

AISI 1050,

0.54% C, Q & T

212-248 655 380 7.7 1480-1520

Alloy Steel

AISI 4140,

0.4% C, Q & T

223-262 725 550 7.7 1430-1510

Gray Cast IronASTM Class 60

223 414 --- 7.2 1350-1400

Aluminium Bronze

Heat Treated121 550 276 7.7 855-1060

Tin Bronze

Chill Cast80 310 165 7.9 800-950

Aluminium

2024 T4120 469 324 2.8 485-660

Criteria

Alternatives

Table A2: Some characteristics for different material alternatives [15 - 18].

CharacteristicsCarbon Steel

AISI 1050,

0.54% C, Q & T

-AISI 1050 Medium carbon steel having 0.54% C

-Oil-quenched from 815 C, tempered at 593 C

-AISI 1050 posses 50% machinability

Alloy Steel

AISI 4140,

0.4% C, Q & T

-AISI 4140 material having 0.4% C, known by Chromium-molybdenum

steel: 0.95% Cr, 0.2% Mo-Oil-quenched from 843 C, tempered at 649 C

-AISI 4140 posses 60% machinability, high hardness at high temperature

(greater hot-hardness)

-Chromium increases depth-hardenability, provide abrasion-resistance,

and corrosion-resistance

-Molybdenum have high-temperature tensile and creep strengths

Gray Cast Iron

ASTM Class 60

-Gray cast iron have 2.8-3.6% C

-Gray cast iron: Cheapness, low melting temperature (1150-1250 C),

easily machined, natural lubricant, vibration damping quality, sliding

quality, good machinability, wear resistance, soft (BHN=180-240)

Aluminium Bronze

Heat Treated

-Aluminium Bronze have 90-95% bronze-5-10%aluminium

-Aluminium: noncorrosive-1/3 weight of steel

-Bronze: is basically an alloy of copper and tin. It possesses superior

mechanical properties and corrosion resistance

Tin Bronze

Chill Cast

-Tin: excellent resistance to corrosion

-Bronze: is basically an alloy of copper and tin. It possesses superiormechanical properties and corrosion resistance

Aluminium

2024 T4

-Aluminum alloy 2024-T4 very good machinability, excellent surface

finish, for light duty applications

Alternatives

Criteria


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Table A3: Some applications for different material alternatives [18].

Applications

Carbon Steel

AISI 1050,

0.54% C, Q & T

Shafts, gears, forging

Alloy Steel

AISI 4140,

0.4% C, Q & T

Gears, shafts, forgings

Gray Cast Iron

ASTM Class 60

Automobile cylinders and pistons, machine castings, water main pipes,

gears

Aluminium Bronze

Heat Treated

Gears, machine parts, bearings, washers, chemical plant equipment,

marine propellers, pump casings, chains and hooks

Tin Bronze

Chill CastAutomotive parts, aircraft, shafts, gears, bearings, piston rings, bushings

Aluminium

2024 T4Aircraft structures, wheels, machine parts, screw machine products

Alternatives

Criteria


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Appendix B: Arithmetic calculations on Fuzzy

Fuzzy numbers can be represented by using triangular membership functions. It can

be represented by a triplet which includes the lower limit of the support, the mode,

and the upper limit of the support: (l, m, n). The addition of two triangular fuzzy

numbers is done as follows:

(l1, m1, n1) (l2, m2, n2) = (l1+l2, m1+m2, n1+ n2) (B1)

The multiplication of two triangular fuzzy numbers will not generally produce a

triangular fuzzy number, but rather one which is approximately triangular as follows:

(l1, m1, n1) (l2, m2, n2) (l1l2, m1m2, n1 n2) (B2)

Consider a particular level for the desired membership function (for example the

Carbon Steel Q & T alternative), Ra(r) = , express each of the membershipfunctions in terms of , and distinguish between the increasing and the decreasingportions of the membership function of the fuzzy numbers.

~:

( ) /

~:

/

( ) /

~:

( ) /

( ) /

~:

( ) /

( ) /

~:

( ) /

( ) /

~:

( ) /

( ) /

~:

( ) /

U x U x U x

U x U x U x

U x

1 1 2 2 3 3

4 4 5 5 6 6

7 7

0

1 6

6

2 6

1 6

3 6

2 6

4 6

3 6

5 6

4 6

6 6

5 6

1

+

+

+

+

+

(B3)

In case of Carbon Steel Q & T in the manufacturing decision matrix for example, for

the increasing portion of the membership function, the weighted average gives:

ra = [(+4)(+2) + (+1)(+1) + (+4)(+4) + (+2) + (+4)(+4) + (+4)(+5)] / 36 (B4)

and since the non-linear program has a solution when Ra(r) = , Eq. (B4) is solvedfor , and taking the positive root yields:

Ra(r) = -35 / 12 + (864 r /144 - 239/144)1/2

61 / 36 ra 102 / 36 (B5)

For the decreasing portion of the membership function

ra = [(6-)(4-) + (3-)(3-) + (6-)(6-) + (4-)(2-) + (6- )(6-) + (6-)6] / 36(B6)

and


17/17

Ra(r) = 52 / 10 - (720 r /100 - 276 / 100)1/2

102 / 36 ra 149 / 36 (B7)

Then the exact membership function of Carbon Steel Q & T is shown as:

+

=

36

1490

36

149

36

102

100

276

100

720

10

52

36

102

36

61

144

239

144

864

12

35

36

610

)(

a

a

a

a

Ra

r

rr

rr

r

r (B8)

Outside this interval the membership function is zero.

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Fuzzy Dr Hegazi