Fuzzy Logic What is that? Prof. Dr. T. Nouri [email protected].
Fuzzy Dr Hegazi
Transcript of 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.
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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 approximate triangular plot of the membership functions is given in Figure 3.
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 approximate triangular plot of the membership functions is given in Figure 4.
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
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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.
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Figure 3 Membership function of alternative rating and goal with respect to
application requirements.
Figure 4 Membership function of alternative rating and goal with respect to
manufacturing and application requirements.
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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
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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.