Introducing a novel method for materials

Materials and Design 30 (2009) 4396–4404

Contents lists available at ScienceDirect

Materials and Design
journal homepage: www.elsevier.com/locate/matdes

Introducing a novel method for materials selection in mechanical design using
Z-transformation in statistics for normalization of material properties
K. Fayazbakhsh, A. Abedian *, B. Dehghan Manshadi, R. Sarfaraz Khabbaz
Department of Aerospace Engineering, Sharif University of Technology, P.O. Box 11365-8639, Tehran, Iran

a r t i c l e

i n f o

Article history:
Received 10 January 2009
Accepted 4 April 2009
Available online 12 April 2009
Keywords:

H. Selection for material properties
H. Weighting and ranking factors
H. Performance indices

a b s t r a c t
Optimum materials selection is a very important task in design process of every product. There are various materials selection methods like Ashby’s method or digital logic methods such as DL and MDL. In the
present research work the Z-transformation method is proposed for scaling the material properties to
overcome the shortcoming of MDL method. The results show that despite the simple scaling function
used, the ranking procedure is as powerful as MDL method and even it is superior to MDL when it ranks
the less important materials existing among a list of candidate materials.
Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction
It is estimated that more than 40,000 useful metallic alloys and
probably close to that number of nonmetallic engineering materials such as plastics, ceramics and glasses, composite materials, and
semiconductors to exist. This large number of materials and enormous number of manufacturing processes available to the engineers, coupled with the complex relationships between different
selection parameters, often make the selection of materials for a given component a difficult task.
Mainly, based on quantitative and qualitative properties of
materials, different materials selection approaches have been proposed and practiced by different researchers. Ashby et al. [1] have
provided a comprehensive review of the strategies or methods for

materials selection. From this study three types of materials selection methodology could be identified. These are: (a) free searching
based on quantitative analysis with the most famous being the
graphical engineering selection method or the ranking method
[2,3], (b) checklist/questionnaire based on expertise capture like
the reports [4–8] which are knowledge-based and intelligent data
base systems or like [9] which is a structured set of questions that
eventually end up with an optimal design solution, and (c) inductive reasoning and analog procedure [10].
Since a materials selection process could be considered as a
decision making problem, the neural network, a kind of artificial
intelligence [11], or the fuzzy logic approach [12] have been used
for dealing with qualitative material properties. Also, in this direction some fuzzy multi-criteria decision making methods like [13]
* Corresponding author. Tel.: +98 21 66164947; fax: +98 21 66022731.
E-mail address: Abedian@sharif.edu (A. Abedian).
0261-3069/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.matdes.2009.04.004

or multiple attribute decision making methods like [14] or the
Technique of ranking Preferences by Similarity to the Ideal Solution method (TOPSIS) [15] have been proposed by other researchers. Furthermore, the environmental costs [16] or the dimensions
of technology, economy and environment [17] have been included
in the single decision making tools. Even, compatibility of a component with the linked components in an assembly has been considered [18] or the process of selection has been divided to two stages,

i.e. discrimination and optimization to minimize the number of
qualitative decisions [19]. However, most of these systems and
methods appear to be complex and knowledge intensive.
In this row, a variety of quantitative selection procedures have
also been developed to analyze the large amount of data involved
with the materials selection processes so that a systematic evaluation could be established. For example, Ashby [20,21] has introduced materials selection charts. He has also proposed a multiobjective optimization method for compromising between several
conflicting objectives [22]. Garton et al. [23] used Fatigue Property
Charts to select the optimal class or subclass of materials in minimum weight design for infinite fatigue life. Another approach to
materials selection problems is the Weighted Properties Method
(WPM). This numerical method ranks the candidate materials on
the basis of their performance indices calculated from simple
mathematics [24]. In cases where numerous material properties
are involved and the relative importance of each property is not
clear, determination of the weighting factors (a) can be largely
intuitive, so that the reliability of the selection method is highly
reduced. As a result, the Digital Logic (DL) approach was proposed
for determination of a [25]. The DL Method was then modified
by Manshadi et al. [26] due to existing flaws in the scaling procedure. This new method was proposed by considering nonlinear

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K. Fayazbakhsh et al. / Materials and Design 30 (2009) 4396–4404

normalization combined with some modifications in the existing
digital logic approach.
Even though a good amount of research work had been carried
out on materials selection in the past, a simple and systematic scientific method or mathematical tool which can guide user organizations in taking a proper material selection decision is still a need.
Therefore, in this paper, a new normalization function for MDL
method is proposed. Besides, the capabilities of this new normalization function are discussed and compared with those of the MDL
through three example problems in mechanical design and lightweight naval structures.
2. Method description
In materials selection problems, for when several material
properties should be taken into account, the weighted property
method and later on the digital logic methods (DL & MDL) have
shown to be quite effective. Basically, in these methods, each material property is assigned a certain weight depending on its relative
importance to the others. Using DL approach, evaluations are arranged such that only two properties considered at a time. In comparing two properties or performance goals, the more important
goal is given a numerical value of (1) and the less important one
is given (0). However, for the modified digital logic (MDL) method,
a value of (1) is assigned to the less important property and the value of (3) to the more important one. In this method, two properties with equal importance receive equal numerical values of (2).
Then the weighting factor (a) for each property is found by summing up the positive decisions that every property receives divided

by the total positive decisions that all the material properties are
P
ai ¼ 1 for each material would be obtained.
given. In this way
Then the performance index c for each candidate material is
found by using the following equation:

c¼

n
X

ai Y i

ð1Þ

for definition of XC makes the process user knowledge dependent
and as was discussed in [26], for XC = Xmax/2, the nonlinear scaling
function would become indeterminate and as a result it is replaced
with a linear function.

In the following subsection a new normalization function is
proposed to replace the nonlinear normalization functions used
in MDL method. This simplifies the calculations, has most accordance with statistic normalization, and increases the reliability of
MDL method with eliminating the previously mentioned XC. It is
obvious that in MDL method, the normalization functions or scaling functions have no background in statistics and do not accord
with it in any way. For the new method, the same weighting factor
(a) as for MDL method is incorporated due to its ability in providing better comparison of material properties and their level of
importance compared to DL method.
3. The bases of the proposed normalization function
In this paper the Z-transformation in statistics science is proposed for standard scaling or normalization [27] of materials properties. The Z-value is a dimensionless quantity which is defined by
the following equation:

Z¼

ð2Þ

r

where X represents an individual raw score that is to be standardized, r is the standard deviation of the population, and l is the
mean of the population. Here, r and l are calculated by the following equations:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
N
u1 X
r¼t
ðX i  lÞ2
N i¼1

l¼

i¼1

where Yi is the scaled value of each property of a material with respect to the other candidate materials for the given product. This
scaling is done linearly or nonlinearly for DL and MDL methods,
respectively. Although MDL method has eliminated the weighting
and scaling flaws in WPM and DL methods, it may still need some
modifications to find higher power or efficiency in final ranking of
the candidate materials. These modifications may target the method’s complications, time consuming calculations, or user unfriendly
matters involved with the employed nonlinear scaling functions,

should either be claimed by some users. Furthermore, the dependency of the scaling or normalization functions to the definition
of the critical value for each material property (i.e. XC) and also its
ratio to the maximum value of the corresponding property may
justify the modification attempts. It should be noted that the need

Xl

ð3Þ

N
X1 þ X2 þ    þ XN 1 X
Xi
¼
N i¼1
N

ð4Þ

where N represents the number of candidate materials. The materials performance indices are calculated by Eq. (1), respectively [26].
It should be noted that in case, higher value of a property is of

interest, Eq. (2) will be considered. However, for the properties that
the lower value is of interest, Eq. (2) is used with a negative sign as
indicated in the following equation:

Y¼

Xl

ð5Þ

r

Here, for verification of the proposed function, three test cases are
studied and the results will be compared with the results obtained
by MDL method. The case studies are involved with materials selection for a cryogenic storage tank, the spar for wing structure of a human powered aircraft (HPA), and a high speed naval craft.

Table 1
Candidate materials properties for cryogenic storage tank (case I) [26].
Materials

1
Toughness indexa

2
Yield strength (MPa)

3
Young’s modulus (GPa)

4
Density (kg/m3)

5
Thermal expansionb

6
Thermal conductivityc

7
Specific heatd

Al 2024-T6
Al 5052-O
SS 301-FH
SS 310-3AH
Ti–6Al–4V
Inconel 718
70Cu–30Zn

75.5
95
770
187
179
239
273

420
91
1365
1120
875
1190
200

74.2
70
189
210
112
217
112

2800
2680
7900
7900
4430
8510
8530

21.4
22.1
16.9
14.4
9.4
11.5
19.9

1.55
1.38
0.167
0.126
0.067
1.30
1.21

669.4
669.4
334.7
334.7
376.6
292.9
251

a
b
c
d

Toughness index, T, is based on UTS, yield strength YS, and ductility e, at 196 °C. T = (UTS + YS)e/2.
Thermal expansion coefficient is given in 106/°C. The values are averaged between RT and 196 °C.
Thermal conductivity is given in 106 J/m2/m/°C/s.
Specific heat is given in J/kg/°C. The values are averaged between RT and 196 °C.

4398

K. Fayazbakhsh et al. / Materials and Design 30 (2009) 4396–4404

adequate toughness at the operating temperature, and also the
material should be sufficiently strong and stiff.
Table 1 presents the properties of the candidate materials for
the cryogenic tank. Also, Table 2 shows the calculations for a. It
should be noted that here, the same weighting factors as MDL
method are used for the Z-transformation method. Finally, the performance index (c) for each material is calculated using Eq. (1). It
should be stated that here Yi is calculated using Eqs. (2) or (5) for
the Z-transformation method. It is reminded again that linear scaling and nonlinear scaling functions are used for calculating Yi by DL

4. Verification of the method
4.1. Test case (I): cryogenic storage tank
This example problem is one of the problems that is widely tested
by other materials selection methods. This is reanalyzed here using
the Z-transformation method. Since this tank is designed for transportation of liquid nitrogen, the candidate materials should have
good weldability and processability, lower density and specific heat,
smaller thermal expansion coefficient and thermal conductivity,

Table 2
Application of modified digital logic method to cryogenic storage tank (case I).
Goals

Number of possible decisions

Toughness
Yield strength
Young’s modulus
Density
Thermal expansion coefficient
Thermal conductivity
Specific heat

1

2

3

4

5

6

3
1

3

3

3

3

3

1
1

7

8

9

10

11

3
1

1

2

3

3

12

1
3

3
1

2
1

13

1

3

15

16

3
1
1

1

17

18

19

20

21

3

3
1

1

14

3

3
3
1

1
1

1

3
1

2
2

Positive
decisions

Weighting
factors (a)

18
13
10
16
13
7
7

0.214
0.155
0.199
0.19
0.155
0.083
0.083

Table 3
Performance index and ranking of candidate materials for cryogenic storage tank using MDL and Z-transformation method (case II).
Materials

The method of Manshadi et al. [26]

Al-2024-T6
Al-5052-O
SS 301-FH
SS 310-3AH
Ti–6Al–4V
Inconel 718
70Cu–30Zn

Z-transformation method

Performance index (c)

Rank

Performance index (c)

Rank

1.2
8.8
47.7
31.9
43.5
33.5
3.1

5
7
1
4
2
3
6

0.56
0.65
0.78
0.26
0.39
0.27
0.49

6
7
1
4
2
3
5

Table 4
Scaled property values performed by MDL method for cryogenic storage tank (case I).
Materials

1
Y1
Toughness

2
Y2
Yield strength

3
Y3
Young’s modulus

4
Y4
Density

5
Y5
CTE

6
Y6
Thermal conductivity

7
Y7
Specific heat

Al 2024-T6
Al 5052-O
SS 301-FH
SS 310-3AH
Ti–6Al–4V
Inconel 718
70Cu–30Zn

33.3
22.9
100
12.1
9.6
26.2
34.1

3.3
68.4
100
81.7
59.9
87.3
39.3

15.2
19.4
80.7
95.3
20
100
20

90.1
100
38.4
38.4
14
43
43.1

25.5
28.1
2.7
17.2
100
54.1
19.2

53.9
50.1
47.9
63.9
100
47.9
45.6

17.6
17.6
55.5
55.5
39.9
74.9
100

Sum

125

224

281

41.2

95.8

14.4

290

Table 5
Scaled property values performed by Z-transformation method for cryogenic storage tank (case I).
Materials

1
Y1
Toughness

2
Y2
Yield strength

3
Y3
Young’s modulus

4
Y4
Density

5
Y5
CTE

6
Y6
Thermal conductivity

7
Y7
Specific heat

Al 2024-T6
Al 5052-O
SS 301-FH
SS 310-3AH
Ti–6Al–4V
Inconel 718
70Cu–30Zn

0.84
0.75
2.34
0.33
0.37
0.10
0.06

0.70
1.39
1.30
0.78
0.26
0.93
1.16

1.13
1.21
0.83
1.18
0.49
1.30
0.49

1.33
1.37
0.72
0.72
0.67
0.96
0.97

1.07
1.22
0.08
0.46
1.55
1.09
0.74

1.16
0.89
1.06
1.13
1.23
0.75
0.62

1.54
1.54
0.51
0.51
0.26
0.77
1.03

Sum

0

0

0

0

0

0

0

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K. Fayazbakhsh et al. / Materials and Design 30 (2009) 4396–4404

Z-transformation provides the same ranking of the candidate
materials as MDL method except for the two of the candidate
materials, i.e. choices 5 and 6 of MDL method are switched by

and MDL methods, respectively [26]. Table 3 presents the calculated c and the corresponding ranking of the candidate materials
using both of the applied methods. As it is seen, interestingly, the
Table 6
Candidate materials properties for HPA (case II) [26].
Materials

1
Price

2
Tensile strength (MPa)

3
Young’s modulus (GPa)

4
Density (kg/m3)

5
Compressive strength (MPa)

6
Creep resistance

Al-7075-T6
Al-2024-T4
Ti–6Al–4V
Ti–2Fe–3Al–10V
E-glass 73%–Epoxy
E-glass 56%–Epoxy
E-glass 65%–Polyester
S-glass 70%–Epoxy continuous fiber
S-glass 70%–Epoxy Fabric
Carbon 63%–Epoxy
Aramid 62%–Epoxy
Balsa

3.5
3.5
21
22
2.6
2.5
2.5
9
8
45
20
6

581
425
1008
1295
1642
1028
340
2100
680
1725
1311
28.5

70.0
72.5
112.0
120.0
55.9
42.8
19.6
62.3
22.0
158.7
82.7
7.0

2600
2600
4400
4500
2170
1970
1800
2110
2110
1610
1380
220

581
425
1008
1295
410
290
90
550
180
900
300
17.5

Good
Good
Excellent
Excellent
Average
Weak
Weak
Average
Average
Average
Average
Average

Table 7
Application of modified digital logic method to HPA (case II).
Goals

Price
Tensile strength
Young’s modulus
Density
Compressive strength
Creep resistance

Number of possible decisions
1

2

3

4

5

1
3

1

1

2

2

3

6

7

8

9

1
3

1

3

3

3

1
3

3
2

10

1
2

11

3

12

14

3
1

3

1

1

Positive decisions

Weighting factors (a)

7
11
13
15
7
7

0.116
0.183
0.216
0.250
0.116
0.116

15

3

1
1

13

2
2

Table 8
Performance index and ranking of candidate materials for HPA using MDL and Z-transformation method (case II).
Materials

The method of Manshadi et al. [26]

Al-7075-T6
Al-2024-T4
Ti–6Al–4V
Ti–2Fe–3Al–10V
E-glass 73%–Epoxy
E-glass 56%–Epoxy
E-glass 65%–Polyester
S-glass 70%–Epoxy continuous fiber
S-glass 70%–Epoxy Fabric
Carbon 63%–Epoxy
Aramid 62%–Epoxy
Balsa

Z-transformation method

Performance index (c)

Rank

Performance index (c)

Rank

20.9
17.7
31.5
38.5
31.8
11.6
12
35.2
4.9
46.1
25.3
7.3

7
8
5
2
4
9
12
3
10
1
6
11

0.007
0.090
0.030
0.216
0.173
0.196
0.548
0.341
0.401
0.591
0.182
0.292

7
8
6
3
5
9
12
2
11
1
4
10

Table 9
Scaled property values performed by MDL method for HPA (case II).
Materials

1
Y1
Price

2
Y2
Tensile strength

3
Y3
Young’s modulus

4
Y4
Density

5
Y5
Compressive strength

6
Y6
Creep resistance

Al-7075-T6
Al-2024-T4
Ti–6Al–4V
Ti–2Fe–3Al–10V
E-glass 73%–Epoxy
E-glass 56%–Epoxy
E-glass 65%–Polyester
S-glass 70%–Epoxy continuous fiber
S-glass 70%–Epoxy Fabric
Carbon 63%–Epoxy
Aramid 62%–Epoxy
Balsa

83.1
83.1
2.1
4.2
98
100
100
36.8
42.5
33.2
0
56.5

40.7
56.3
0.8
27.8
59.5
2.7
64.9
100
31
67
29.3
97

12.7
15.8
58.8
66.5
5.5
24
61.6
3
57.4
100
27.8
85.4

5.2
5.2
13.7
14.5
11.9
15.5
18.9
12.9
12.9
23.1
28.9
100

76.8
67.7
92.7
100
66.7
56.7
23
75.2
42.9
89.5
57.7
22.8

50
50
100
100
0
50
50
0
0
0
0
0

Sum

560.5

3.05

50.7

206.4

726

200

4400

K. Fayazbakhsh et al. / Materials and Design 30 (2009) 4396–4404

the members of each column of Table 5 is the same and equal to
zero. This is expected because the Z-transformation method has
its roots in statistics and the standard scaling performed by the
method must have such a characteristic. However, as Table 4
shows, the summations of members of each column, which are
the scaled properties of the candidate materials performed by nonlinear functions of MDL method, are neither zero nor equal to one
another. This fact under some conditions may adversely affect the
calculation of performance index (c) which uses the scaled property values and the weighting factors (a), see Eq. (1). When sums
of the scaled property values of the candidate materials, as explained above, are not equal, the material property corresponding
to the column with higher sum value will have more influence on

Z-transformation method. The reason for this ranking alternation
will be explained latter. However, this finding (i.e. similarity in
ranking of the materials by the two methods) is very exciting because the scaling done here is very easy and straight forward compared to MDL method. It should be highlighted that this small
difference in ranking by the two methods is in fact another important advantage of the Z-transformation over MDL method which
for clear explanation the second test case (i.e. the human powered
aircraft (HPA)) should be discussed first. However, at this stage, for
better understanding of the subject matter, only a brief discussion
on the scaled property values provided by the two methods is presented. Here, Tables 4 and 5 present Yi for the candidate materials
by MDL and Z-transformation, respectively. As it is seen, the sum of

Table 10
Scaled property values performed by Z-transformation method for HPA (case II).
Materials

1
Y1
Price

2
Y2
Tensile strength

3
Y3
Young’s modulus

4
Y4
Density

5
Y5
Compressive strength

6
Y6
Creep resistance

Al-7075-T6
Al-2024-T4
Ti–6Al–4V
Ti–2Fe–3Al–10V
E-glass 73%–Epoxy
E-glass 56%–Epoxy
E-glass 65%–Polyester
S-glass 70%–Epoxy continuous fiber
S-glass 70%–Epoxy Fabric
Carbon 63%–Epoxy
Aramid 62%–Epoxy
Balsa

0.7
0.7
0.72
0.8
0.78
0.78
0.78
0.26
0.34
2.68
0.64
0.50

0.72
0.98
0.01
0.47
1.05
0.02
1.12
1.81
0.56
1.19
0.50
1.64

0.03
0.09
1.01
1.19
0.30
0.61
1.15
0.15
1.09
2.10
0.32
1.44

0.27
0.27
1.86
1.94
0.10
0.28
0.43
0.16
0.16
0.60
0.80
1.82

0.21
0.21
1.36
2.13
0.25
0.58
1.11
0.12
0.87
1.07
0.55
1.31

0.71
0.71
1.77
1.77
0.35
1.41
1.41
0.35
0.35
0.35
0.35
0.35

Sum

0

0

0

0

0

0

Table 11
Candidate materials properties for high speed craft (case III) [28].
Materials

Grade A Steel
Single Skin Aluminum
(A5086-H34)
Aluminum Sandwich
(honeycomb core)
LASCOR Steel
Composite (CFRP) Carbon
w/Vinyl Ester Resin
DUCTAL (UHP2C)

1
Yield strength
(MPa)

2
Young’s modulus
(GPa)

3
Fire
resistance

4
Repairability

5
Resistance to
corrosion

6
Fabrication
cost

7
Risk

8
Density
(kg/m3)

9
Overall
potential for
weight saving

234.4
137.9

204.1
67

High
Low

Very high
High

Low
High

Avg.
Low

Low
Avg.

7800
2700

None
High

268.9

67

Avg.

Avg.

High

Avg.

Avg.

1800

Very high

204.1
227.5

High
Low

Avg.
Avg.

High
Very high

Very high
Very high

High
Avg.

5200
1800

High
Very high

53.9

Very high

Very high

Very high

Very low

Very
high

2500

High

379.2
1496.2
220.6

Table 12
Quantitative values for material properties using Rao’s fuzzy conversion scale for high speed craft (case III).
Materials

Grade A Steel
Single Skin Aluminum
(A5086-H34)
Aluminum Sandwich
(honeycomb core)
LASCOR Steel
Composite (CFRP) Carbon
w/Vinyl Ester Resin
DUCTAL (UHP2C)

1
Yield strength
(MPa)

2
Young’s modulus
(GPa)

3
Fire
resistance

4
Repairability

5
Resistance to
corrosion

6
Fabrication
cost

7
Risk

8
Density
(kg/m3)

9
Overall
potential for
weight saving

234.4
137.9

204.1
67

0.665
0.335

0.745
0.665

0.335
0.665

0.5
0.335

0.335
0.5

7800
2700

0
0.665

268.9

67

0.5

0.5

0.665

0.5

0.5

1800

0.745

204.1
227.5

0.665
0.335

0.5
0.5

0.665
0.745

0.745
0.745

0.665
0.5

5200
1800

0.665
0.745

53.9

0.745

0.745

0.745

0.335

0.745

2500

0.665

379.2
1496.2
220.6

4401

K. Fayazbakhsh et al. / Materials and Design 30 (2009) 4396–4404

the performance index (c) than the expected amount warranted by
its weighting factor. The effect of this phenomenon will be explained in details in the next subsection.

both of the mentioned reasons were then taken care of by the modifications considered in MDL method.
Here, this example is resolved by using MDL and the Z-transformation methods to shed more light on the advantages of the newly
proposed method. Tables 6 and 7 show the details of the candidate
materials for the mentioned application and calculations of the
weighting factors for MDL method, respectively. Also, calculation
of performance indices and ranking of the candidates materials
done by MDL and Z-transformation methods are shown in Table
8. A quick look at this table shows that except choices 1, 7, 8, 9,
and 12, the rest of the rankings done by MDL and Z-transformation
methods are different. For example, materials ranked 2 and 3 by
MDL are switched by Z-transformation method. Also, E-glass
73%–Epoxy is ranked 4th by MDL while the 4th ranked material
by Z-transformation method is Aramid 62%–Epoxy. These could

4.2. Test case (II): spar of a human powered aircraft (HPA)
This test example was used for explaining the advantage of MDL
over DL method in [26] where it was shown that balsa was
wrongly preferred over the titanium alloys for the spar application
by DL method. This was then shown to happen due to linear scaling
of the properties done by DL method. In fact, the method does not
adequately emphasize on the large or small differences that may
exist between the properties of the candidate materials. Also, the
way of calculation of the weighting factors (a) by DL method was
found to be responsible for this problem. It should be noted that

Table 13-1
Application of modified digital logic method to high speed craft (case III) (1–15).
Goals
Yield strength
Young’s modulus
Fire resistance
Repairability
Resistance to corrosion
Fabrication cost
Risk
Density
Overall potential

1
3
1

2
3

3
3

4
3

5
2

6
3

7
1

8
1

1

9

10

11

12

13

14

15

3
1

3

3

2

3

1

1

1

1
1

1
2

2
1

1
3

3
3

3

Table 13-2
Application of modified digital logic method to high speed craft (case III) (16–36).
Goals
Fire resistance
Repairability
Resistance to corrosion
Fabrication cost
Risk
Density
Overall potential

16
1
3

17
3

18
1

19
3

20
1

21
1

1

22

23

24

25

26

1
3

1

2

1

1

3

3
1

27

28

29

30

1
3

1

1

1

2
3

3
3

3

31

32

33

3
1

1

1

3
3

3
3

34

35

1
3

1

3

36

1
3

3

Table 13-3
Application of modified digital logic method to high speed craft (case III).
Goals

Positive decisions

Weighting factors (a)

Critical value (XC)

Yield strength
Young’s modulus
Fire resistance
Repairability
Resistance to corrosion
Fabrication cost
Risk
Density
Overall potential

19
17
12
11
10
18
11
22
24

0.132
0.118
0.0833
0.0763
0.0694
0.125
0.0763
0.153
0.167

34
10,000
0.335
0.665
0.665
0.335
0.335
2.7
0.665

Table 14
Performance index and ranking of candidate materials for high speed craft using MDL and Z-transformation method (case III).
Materials

Grade A Steel
Single Skin Aluminum (A5086-H34)
Aluminum Sandwich (honeycomb core)
LASCOR Steel
Composite (CFRP) Carbon w/Vinyl Ester Resin
DUCTAL (UHP2C)

The method of Manshadi et al. [26]

Z-transformation method

Performance index (c)

Rank

Performance index (c)

Rank

3.22
0
8.07
4.5
28.66
33.15

5
4
6
3
2
1

0.485
0.017
0.164
0.107
0.188
0.245

6
3
5
4
2
1

4402

K. Fayazbakhsh et al. / Materials and Design 30 (2009) 4396–4404

be fairly explained by considering Tables 9 and 10 which illustrate
the scaled property values obtained by nonlinear scaling functions
of MDL and Z-transformation methods, respectively. As it is seen in
these Tables, like the previous case study, sum of the column members or scaled values of each one of the properties are the same and
equal to zero for the Z-transformation, while they are different for
each property for MDL method. But, how this phenomenon affects
the ranking is explained here. As Table 7 shows, for HPA example,
the density with a = 0.25 appears to be the most important factor
for c calculation for this test case. However, based on Table 9,
sum of the scaled density of the candidate materials (206.4) is lower compared to sums of the two other scaled properties (i.e. price
and compressive strength). This can strongly affect the ranking
by MDL. As it is seen, MDL ranks Ti–2Fe–3Al–10V second and Sglass 70%–Epoxy continuous fiber third despite the larger density
of the earlier compared to the latter (i.e. more than twice). In other
words, when a is not the only factor influencing c calculations the
price property with a = 0.116 and sum scaled values of 560 (see Table 9) may have more influence on the calculations than the density. However, Z-transformation replaces the above ranking
because with this method only a values affect the c calculations
due to standard scaling of the properties of the candidate materials
which provides equal values for sum of the scaled properties (all
are zero) as given in Table 10.
With the same reasoning, the Z-transformation method ranks
Aramid 62%–Epoxy better than E-glass 73%–Epoxy while it is opposite in MDL method. This is done despite much higher price and
smaller tensile strength of Aramid–Epoxy composite compared to
E-glass–Epoxy. The much larger density of E-glass 73%–Epoxy,
and also its corresponding a = 0.25 do not help MDL ranking

because of the method’s inherent problem with property scaling
explained earlier.
It should also be mentioned that negative sum values of the
scaled properties make the situation worst. This could be better
understood if one compares the sums of the scaled property values
for the two HPA and cryogenic tank examples. As it is seen in Table
4, there is no negative sum value for the cryogenic tank and also
sum values of all the scaled properties are not much different.
However, one negative sum value and large difference between
sums of the scaled values for HPA are seen in Table 9. That is
why a little or no serious effect seen on the ranking results for
the cryogenic tank by both methods. It should be noted that as explained in the previous subsection, only materials ranked 5 and 6
by MDL method for the cryogenic tank are replaced by the Z-transformation method. The MDL method ranks 70Cu–30Zn 6 despite
its much better toughness and Young’s modulus compared to Al
2024-T6 which is ranked 5. As the calculations show, the respective a values for the mentioned properties are 0.214 and 0.199
which are the highest a values for the cryogenic tank. Again, this
change in materials ranking by MDL method occurs due to the
way of scaling done by this method which makes sum values of
the scaled properties to be different.
4.3. Test case (III): high speed naval craft
‘‘Typically, the structural weight of a ship is about one-third of
its displacement. Thus, making the potential for substantial weight
saving when considering light-weight materials over traditional
steel construction” [28]. High Performance light-weight materials
can provide as much as 40% reduction in a ship structural weight

Table 15
Scaled property values performed by modified digital logic method for high speed craft (case III).
Materials

1
Y1
Yield
strength

2
Y2
Young’s
modulus

3
Y3
Fire
resistance

4
Y4
Repairability

5
Y5
Resistance
to corrosion

6
Y6
Fabrication
cost

7
Y7
Risk

8
Y8
Density

9
Y9
Overall potential
for weight saving

Grade A Steel
Single Skin Aluminum
(A5086-H34)
Aluminum Sandwich
(honeycomb core)
LASCOR Steel
Composite (CFRP) Carbon
w/Vinyl Ester Resin
DUCTAL (UHP2C)

21.3
0

82
0

100
0

72.4
0

46.9
0

100
0

72.6
0

72.6
0

100
0

27

42.7

48.9

0

46.9

0

100

100

100

41.3
100

82
0

48.9
48.9

0
100

68.2
68.2

32.1
0

56.6
100

56.6
100

0
100

18.9

100

100

100

100

41.2

12

12

0

Sum

209.5

265.3

309.7

57.4

132.7

124.1

33.7

90.8

109

Table 16
Scaled property values performed by Z-transformation method for high speed craft (case III).
Materials

1
Y1
Yield
strength

2
Y2
Young’s
modulus

3
Y3
Fire
resistance

Grade A Steel
Single Skin Aluminum
(A5086-H34)
Aluminum Sandwich
(honeycomb core)
LASCOR Steel
Composite (CFRP) Carbon
w/Vinyl Ester Resin
DUCTAL (UHP2C)

0.471
0.676

0.887
0.925

0.763
1.265

1.209
0.497

2.161
0.203

0.072
0.962

0.398

0.925

0.251

0.971

0.203

0.164
2.21

0.887
1.202

0.763
1.265

0.971
0.971

0.501

1.127

1.254
0

Sum

0

0

4
Y4
Repairability

5
Y5
Resistance to
corrosion

6
Y6
Fabrication
cost

7
Y7
Risk

8
Y8
Density

9
Y9
Overall potential
for
weight saving

1.560
0.309

1.907
0.535

2.215
0.321

0.072

0.309

0.807

0.626

0.203
0.776

1.250
1.250

0.941
0.309

0.731
0.807

0.321
0.626

1.209

0.776

1.393

1.547

0.490

0.321

0

0

0

0

0

0

K. Fayazbakhsh et al. / Materials and Design 30 (2009) 4396–4404

when compared with traditional plate and beam steel construction. In this case, the system is a light-weight high-speed vessel.
In particular, the materials selection is sought for the structural
components of the ship such as hull plating, superstructure panels,
decks, and beams. Therefore the participant material properties
are: yield strength, Young’s modulus, fire resistance, repairability,
resistance to corrosion, fabrication cost, risk, mass density, and
overall potential for weight savings. It should be noted that this
example problem has been already solved by Torrez [28] using
MDL method. Here, the Z-transformation is employed to show
the power of the method. The properties of the candidate materials
and the corresponding quantitative values for the qualitative properties are presented in Tables 11 and 12. As stated in [28], Rao’s
fuzzy score conversion scale has been used to convert the qualitative values to quantitative values. Here, there are nine properties
(n = 9), therefore the number of possible decisions would be
N = n(n  1)/2 = 36. Tables 13-1–13-3 present the weighting factor
calculation for MDL method along with XC for each property. Finally, Table 14 illustrates ranking of the candidate materials for both
MDL and Z-transformation methods.
Interestingly, the first and second choices of materials are the
same for both methods. However, due to the reasons discussed before, choices 3 and 4 and also 5 and 6 made by MDL are switched
by the Z-transformation method. For example, MDL picks up LASCOR Steel as the third choice instead of Single Skin Aluminum
which is ranked third by Z-transformation. This is done despite
higher density of LASCOR (almost twice) compared to that of the
Single Skin Aluminum. It should be noted that three of the properties for this material, i.e. fire resistance, Young’s modulus, and yield
strength, are superior to the Single Skin Aluminum candidate.
However, Z-transformation method replaces the ranking of these
two materials. This seems to be done because of the absolute superiority of the material density. It could be easily understood if one
compares the scaled property values done by MDL and Z-transformation methods, which are shown in Tables 15 and 16, respectively. As it is seen, sum of the materials’ scaled density by MDL
is 90.8, while for fire resistance, Young’s modulus, and yield
strength this value is 309.7, 265.3, and 209.5, respectively. These
values are at least 2–3 times of the same quantity for density. This
huge difference does not allow the value of a for density (a = 0.153)
to play a reasonable role in calculating the performance index for
the Single Skin Aluminum. Also, the huge negative sum value of
the scaled fabrication cost (with a = 0.125) adversely affects the
MDL ranking. As a result, the differences in ranking of candidate
materials performed by both methods are more pronounced than
the last two test cases.

5. Concluding remarks
In comparison to MDL method, the Z-transformation method:
1. Technically uses a very simple function for scaling the material
properties.
2. Uses a similar function for scaling the properties that their minimum are important as those which their maximum are important with only multiplying the function with a negative sign.
3. There is no need for defining XC as in MDL method. This greatly
reduces the dependency of the method to the user knowledge.
4. With eliminating XC, there is no indeterminate point for the
scaling function. It is reminded that at XC = Xmax/2 the nonlinear
scaling functions for MDL method were found indeterminate
and the user was obliged to perform linear scaling at this point.
5. Interestingly, both methods act closely in ranking the candidate
materials. For example, in all three test examples the materials
ranked first are the same by both methods. In fact, both meth-

4403

ods are capable of readily identifying the materials with some
distinguished properties from a bunch of candidate materials.
This similarity in ranking results for some test cases even happens for most of the candidate materials. However, the advantages of Z-transformation method is more illustrated when
ranking materials for which it is not very easy to identify any
superiority in properties of the candidate materials. This advantage of the Z-transformation method is more appreciated by the
designers because most of the time the trades off between some
properties, which are the result of designers engineering knowledge, pursue them to make use of the lower ranked materials. In
such situations, Z-transformation method is more reliable than
the MDL method. This is done due to more proper scaling of the
material properties by the Z-transformation method. This is
because the method has its scaling roots in statistics. As a result,
a factor would be the only parameter that influences the calculations of performance index (c). It should be noted that based
on the reviewed examples in this study, sometimes the border
of this advantage of Z-transformation method is distributed to
even second ranked choice materials done by other methods.
So, it makes worthy to consider this new method. However, if
only the first ranked material from among a list of candidate
materials is of interest, the choice of the ranking method is
not of much importance.

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