Projection Method for Material Selection Problem with ...

Advanced Materials Research Vol. 1046 (2014) pp 375-379 Online available since 2014/Oct/01 at www.scientific.net © (2014) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.1046.375

Submitted: 29.08.2014 Accepted: 30.08.2014

Projection Method for Material Selection Problem with Triangular Fuzzy Numbers Lianwu Yang1, a, Shixiao Xiao2, b*, Shaoliang Yuan1,c 1

School of Mathematics and Computer Science, Yichun University, Yichun 336000, China 2

Chengyi College, Jimei University, Xiamen 361021, China

a

b

c

[email protected], [email protected], [email protected] *Corresponding author

Keywords: material selection; multi-attribute decision making; projection method; triangular fuzzy number; coefficient of variation

Abstract. The aim of this paper is to develop a new method for the material selection problem, which is important for the product design. Material selection problem is actually a multi-attribute decision making (MADM) problem which contains many influence factors. The new material selection method is an extension of projection method with evaluation attribution values expressed with triangular fuzzy numbers. Coefficient of variation method is used to determine weights of evaluation attribute. A grinding wheel abrasive material selection problem is used to illustrate the effectiveness and practicability of the proposed method. Introduction A key step of product design process is material selection. Material selection problem is an important study field, which attracts many authors’ attention and study. To the evaluation attribute values expressed with crisp numbers, there are already many decision methods developed on the basis of multi-attribute decision making (MADM) methods. For example, TOPSIS method [1], VIKOR method [2], multi objective optimization on the basis of ratio (MOORA) method [3], quality function deployment (QFD) method [4], ELECTRE method [5], PROMETHEE method [6] and complex proportional assessment (COPRAS) method [7]. In some situations, some attribute measure values obtained are probabilly fuzzy numbers, such as interval numbers, triangular fuzzy numbers and linguistic variables because of the evaluation attributes given by subjectiveity. There are still few methods developed to deal with the material selection problems with attribute values expressed with fuzzy numbers. The Paper [8] studied the material selection using an interval 2-tuple linguistic VIKOR method. Paper [9-11] are respectively developed the VIKOR method,GRA method and projection method for materials selection problem with interval numbers. In this paper, we will develop a projection method for the material selection problem. As the key step of the projection method, the projection values are calculated for material selection problem with interval numbers between each alternative and positive ideal solution and negative ideal solution [12]. The remains of this paper is organized as follows. Section 2 first constructs the MADM model and then develops the projection method to the material selection problem with interval numbers. Section 3 provides an application of the proposed method to illustrate the feasibility and practicability of the proposed method. Finally, a conclusion is given in Section 4. Projection method for Material Selection with Interval Numbers In this section, we first recall some basic concepts of triangular fuzzy number and projection method, and then we will construct the MADM model for material selection problem with triangular fuzzy numbers. Definition 1 [12]. z
+ = ( z
1+ , z
2+ ,..., z
n+ )T is called the triangular fuzzy positive ideal point defined as follows (1) z +j = [ z +j L , z +j M , z +j U ] = [max zijL , max zijM , max zijU ] , j = 1, 2,..., n . i i i All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 120.203.125.251-16/10/14,13:45:43)

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Definition 2 [12]. Let α = (α1 , α 2 ,..., α n ) and β = ( β1 , β 2 ,..., β n ) be two vectors，the projection of vector α on β is defined as follows: n

n

∑α Pr j β (α ) =

j

βj

n

j =1 n

∑α

n

∑α

∑β

2 j

j =1

∑α 2 j

βj

(2)

n

j =1

2 j

j

j =1

=

∑β

j =1

2 j

j =1

Generally speaking, the greater value of Pr jβ (α ) means the closer of vector α and β .Set n

∑ (z Pr jz
+ ( z
i ) =

z +j L + zijM z +j M + zijU z +j U )

L ij

(3)

j =1 n

∑ [( z

+L 2 j

) + (z

+M j

2

) + (z

+U j

2

) ]

j =1

where z
i = ( z
i1 , z
i 2 ,..., z
in )T , i = 1, 2,..., m Obviously, the greater value of Pr jz
( z
i ) , the more closer of alternative xi to the triangular fuzzy positive ideal point z
+ . That is to say, the alternative xi is better. Consider a material selection problem with a set of m candidate materials (alternatives) x = {x1 , x2 , , xm } and a set of n influence factors (evaluation attributers) O = {o1 , o2 , , on } . Suppose the evaluation attribute value of ith alternative xi ( i = 1, 2, , m ) with respect to jth attribute o j ( j = 1, 2, , n ) given by decision maker is a triangular fuzzy number, which can be denoted by a
ij = (aijl , bijm , ciju ) . Suppose w = ( w1 , w2 ,..., wn ) be the attribute weight vector which satisfies w j ≥ 0, j = 1, 2,..., n and w1 + w2 + ... + wn = 1 . Hence, the material selection model can be dealed with MADM model with the decision matrix D
= (a
ij )m×n . In the following, we will develop the triangular fuzzy projecton method to the material selection problem with the following steps: Step1. Establish triangular fuzzy number decision matrix D
= (a
ij )m×n and then normalize +

D
= (a
ij )m× n to the normalized decision making matrix R
= (r
ij ) m× n = ((rijl , rijm , riju )) m× n by using the method

proposed by Xu [13].Using the normalized method proposed by, we transform the the decision matrix D
= (a
ij )m× n into the normalized fuzzy decision matrix R
= (r
ij ) m× n = ((rijl , rijm , riju )) m× n with the following formulas: rijl =

aijl cujmax

, rijm =

aijm c mjmax

, riju =

aiju c ljmax

, i ∈ M , j ∈ I1

rijl =

and

where M = {1, 2,..., m} , clj max = max {aijl } , c mj max = max{aijm }, i i

cljmin aiju

, rijm =

c uj max = max{aiju }, i

c mjmin aijm

c ujmin

, riju =

aijl

(4)

, i ∈ M , j ∈ I2

c lj min = min{aijl }, i

c mj min = min{aijm }, i

c uj min = min{aiju } , I1 and I 2 are the subset of benefit and cost attribute, respectively. i

Step3. Calculate the attribute weights as follows: (i) Transform R
= (r
ij ) m×n into a crisp number decision matrix G = ( gij )m×n , where gij = (rijl + rijm + riju ) / 3 is obtained by the centroid defuzzification method proposed in paper [14] . (ii) The attribute weights are obtained by coefficient of variation method as [15]:Let g j = sj / g j 1 m ( gij − g j ) 2 , then the attribute weights can be calculated as w j = n , j = 1, 2,..., n . ∑ m i =1 s / g ∑ j j

sj =

1 m ∑ gij , m i =1

(5)

j =1

Step 4. Using the attribute weight vector w in Eq. (5) and the normalized decision matrix R
= (r
ij ) m× n , the weighted normal decision matrix can be obtained as Z
= ( z
ij )m× n = ( w j r
ij ) m× n . Step5. Determine the triangular fuzzy positive ideal point z
+ according to Eq.(1) ; Step6. Calculated projection of each alternative xi on the triangular fuzzy positive ideal point Pr jz
( z
i ) according to Eq. (3), and then rank all the alternatives according to the projection Pr jz
( z
i ) (i = 1, 2,..., m) with the rule: the larger of the value of Pr jz
( z
i ) , the better of the candidate material xi . +

+

+

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Verification of the Proposed Method The example discussed in [16] is used to illustrate the feasibility and practicability of the proposed method. A manufacture wants to select the most suitable gear material from a set of candidate materials: x1 (Titanium carbide ), x2 (Tungsten carbide), x3 (Cubic boron nitride), x4 (Aluminum oxide), x5 (Synthetic polycrystal diamond), x6 (Silicon carbide), x7 (Boron carbide) and x8 (Yttria stabilized zirconia). Seven evaluation attributes are considered in this example, and they are Knoop hardness ( KHN ) ( o1 ),Modulus of elasticity ( Gpa ) ( o2 ),Compressive strength ( Mpa ) ( o3 ),Shear strength ( Mpa ) ( o4 ),Thermal conductivity ( W/mk ) ( o5 ),Fracture toughness ( MPa-m1/2 ) ( o6 ), Grinding wheel abrasive material cost ( USD/kg ) ( o7 ).Here, all attributes are benefit attribute except o7 . The attribute measure values are expressed with triangular fuzzy numbers and shown in Table 1. Table1. Fuzzy decision matrix for grinding wheel abrasive material selection problem o1

o2

o3

o4

o5

o6

o7

x1

( 2700, 3200, 3700 )

( 391, 451, 511 )

( 2925, 3475, 4025 )

( 581, 756, 931 )

( 12, 17, 22 )

(2.65,4.15,5.65)

( 12, 18, 24 )

x2

( 2000, 2400, 2800 )

( 590, 690, 790 )

( 4275, 4975, 5675 )

( 1099, 1324, 1549 )

( 68, 98, 128 )

( 2.2, 3, 3.8 )

( 45, 60, 75 )

x3

( 4400,5000, 5600 )

( 725, 850, 975 )

( 6000, 6900, 7800 )

( 1282, 1532, 1782 )

( 9, 13, 17 )

( 3.55, 4.5, 5.45 )

( 714, 864, 1014 )

x4

( 2600,3000, 3400 )

( 350, 400, 450 )

( 3200, 3800, 4400 )

( 729, 879, 1029 )

( 21, 30, 39 )

( 3.15, 4, 4.85 )

( 107, 152, 197 )

x5

( 7300, 8000, 8700 )

( 818, 953, 1088 )

( 5900, 6700, 7500 )

( 4188, 4688, 5188 )

( 950, 1200, 1450 )

( 6.45, 8.6, 10.8 )

( 1050, 1300, 1550 )

x6

( 2150, 2550, 2950 )

( 370, 440, 510 )

( 3950, 4600, 5250 )

( 400, 480, 560 )

( 150, 200, 250 )

( 2.6, 3.1, 3.6 )

( 6.5, 10, 13.5 )

x7

( 2400, 2800, 3200 )

( 385, 460, 535 )

( 1421, 1721, 2021 )

( 425, 600, 775 )

( 55, 90, 125 )

( 1.95, 2.5, 3.05 )

( 36, 50, 64 )

x8

( 900, 1200, 1500 )

( 115, 160, 205 )

( 1350, 1750, 2150 )

( 495, 620, 745 )

( 1.4, 2.2, 3 )

( 5.75, 8.2, 10.7 )

( 33, 45, 57 )

In the following, we will give main calculation results of the proposed. The attribute weights are clacluated by using coefficient of variance method, and obtained as w1 = 0.0840 , w2 = 0.0669 , w3 = 0.0655 , w4 = 0.1446 , w5 = 0.2782 , w6 = 0.1393 and w7 = 0.2217 . The projection Pr jz
( z
i ) of each alternative are calculated as Pr jz
( z
1 ) = 1.3995 , Pr jz
( z
2 ) = 1.0857 , Pr jz
( z
3 ) = 1.5099 , Pr jz
( z
4 ) = 0.8931 , Pr jz
( z
5 ) = 3.3258 , Pr jz
( z
6 ) = 1.0429 , Pr jz
( z
7 ) = 0.7911 and Pr jz
( z
8 ) = 0.5124 .Then the ranking order of these candidate materials accoring the decreasing order of the projection Pr jz
( z
i ) is obtained as 5 - 3 -1 - 2 - 6- 4 -7 - 8, and the desirable candidate material is x5 ( Synthetic polycrystal diamond), and this final selection result is also in agreement with the one obtained in ([16, 17]). +

+

+

+

+

+

+

+

+

+

Conclusion Material selection is a key step in the product design process of manfacturing, which is an important commercial activity carried out by companies that sell products to customers. Though many MADM methods have already developed or proposed for material selection problem, but due to the complex and uncertain information occured in the process, many evaluation attributes can not or hard to be described by crisp numbers. Few methods developed for the material selection problem with fuzzy numbers. Thus in this paper, we developed the projection method to the material selection problem in whcih attribute values are expressed with triangular fuzzy numbers. The weughts of

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attribute are very important in the decision process, and thus we use coefficient of variance method to determine the attribute weights, which is an objective method, and thus it can overcome the subjective randomness of subjective weights method. The proposed method is easy and can be applied other MADM problems. Acknowledgement This work was supported in part by National Natural Science Foundation of China (No. 11361067), Business intelligence and big data analysis laboratory of Chengyi College, Jimei University. References [1] A. Shanian and O. Savadogo. TOPSIS multiple-criteria decision support analysis for material selection of metallic bipolar plates for polymer electrolyte fuel cell. J. Power Sources. Vol. 159 (2006), p.1095-1104. [2] H. C. Liu, L. X. Mao, Z. Y. Zhang and P. Li. Induced aggregation operators in the VIKOR method and its application in material selection. Applied Mathematical Modelling, Vol. 37 (2013), p.6325-6338. [3] P. Karande and S. Chakraborty. Application of multi-objective optimization on the basis of ratio analysis (MOORA) method for materials selection. Materials & Design, Vol. 37 (2012), p.317-324 [4]A. Kasaei, A. Abedian and A.S. Milani .An application of Quality Function Deployment method in engineering materials selection.Materials & Design, Vol.55 (2014), p.912-920. [5] A. Shanian and O. Savadogo. A material selection model based on the concept of multiple attribute decision making. Materials & Design Vol.27 (2006), p.329-337. [6] A. H. Peng and X. M. Xiao. Material selection using PROMETHEE combined with analytic network process under hybrid environment. Materials & Design, Vol.47 (2013), p. 643-652. [7] P. Chatterjee, V. M. Athawale and S. Chakraborty. Materials selection using complex proportional assessment and evaluation of mixed data methods. Materials & Design vol. 32 (2011), p.851-860. [8] H. C. Liu, L. Liu and J. Wu. Material selection using an interval 2-tuple linguistic VIKOR method considering subjective and objective weights. Materials & Design, Vol. 52 (2013), p.158-167. [9] A. Jahan and K. L. Edwards. VIKOR method for material selection problems with interval numbers and target-based criteria. Materials & Design, Vol. 47 (2013), p. 759-765. [10]L. W. Yang. Grey relation analysis method for material selection problem with interval numbers. Advanced Materials Research, Vol. 1014 (2014), p.492-496. [11]L. W. Yang. Projection method for material selection problem with interval numbers. Advanced Materials Research, Vol. 1022 (2014), p.14-17. [12] Y. M. Wang. A new method for multi-indices decision and evaluation-a project method. Journal of Systems Engineering and Electronics, Vol. 21 (1999), p.1-4. [13] Z.S. Xu. Uncertain Multiple Attribute Decision Making Methods and Applications. (Tsinghua University Press, Beijing 2004). [14] R. R. Yager. A procedure for ordering fuzzy subsets of the unit interval. Information Sciences, Vol.24 (1981), p.143-161.

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[15] B. H. Men and C. Liang. Attribute recognition model-based variation coefficient weight for evaluating water quality, Journal of Harbin Institute of Technology, Vol.37 (2005), p.1373-1375. [16] S. R. Maity and S. Chakraborty. Grinding wheel abrasive material selection using fuzzy TOPSIS method, Materials and Manufacturing Processes, Vol. 28 (2013), p.408-417 [17] J. H. Leng. Grinding wheel abrasive in material selection using fuzzy grey relation analysis method. Advanced Materials Research, Vol. 1022 (2014), p.18-21.

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Projection Method for Material Selection Problem with Triangular Fuzzy Numbers 10.4028/www.scientific.net/AMR.1046.375