Multi-attribute decision making with generalized fuzzy

Journal of the Operational Research Society (2015) 1–11

© 2015 Operational Research Society Ltd. All rights reserved. 0160-5682/15 www.palgrave-journals.com/jors/

Multi-attribute decision making with generalized fuzzy numbers Guangxu Li1, Gang Kou2*, Changsheng Lin3, Liang Xu2 and Yi Liao2 1

University of Electronic Science and Technology of China, Chengdu, China; 2Southwestern University of Finance and Economics, Chengdu, China; and 3Yangtze Normal University, Chongqing, China This paper proposes a multi-attribute decision-making method with generalized fuzzy numbers (GFNs). In the proposed method, the distance between GFNs is calculated using the Hausdorff distance. Based on the maximizing deviation degree, the attribute weights are determined by a linear programming model. Furthermore, a ranking formula with a modified possibility degree is adopted to rank alternatives. A numerical example is introduced to validate the proposed model, and the results indicate that the proposed model offers a practical and effective way to meet the different assessment requirements of decision makers. Journal of the Operational Research Society advance online publication, 4 March 2015; doi:10.1057/jors.2015.1 Keywords: multi-attribute decision making; generalized fuzzy numbers; hausdorff distance; modified possibility degree

1. Introduction Multi-attribute decision making (MADM) can be used to rank alternatives under multiple criteria in various application areas (Yakowitz et al, 1993; Zavadskas et al, 2008; Sun, 2010; Kou and Lin, 2014). Many methods have been developed over the years. For example, Saaty (1980) proposed the Analytic Hierarchy Process (AHP), which is one of the most widely used MADM methods. Hwang and Yoon (1981) proposed the technique for order preference by similarity to ideal solution (TOPSIS) method to rank alternatives over multiple attributes. Gabus and Fontela (1972) proposed the method of Decision Making Trial and Evaluation Laboratory (DEMATEL) to collect group knowledge, analyse the inter-relationships among system factors, and visualize the structure by cause-effect relationship diagram. Peng et al (2008) proposed an MADM model with multi-criteria convex quadratic programming for credit data analysis. Barker and Zabinsky (2011) proposed an MADM model for reverse logistics using AHP. Chen (2012) proposed the integrated MADM methods, and gave the applications in the Inno-Qual performance system. A sensitivity analysis approach for MADM methods was performed in Triantaphyllou and Sanchez (1997). Kou et al (2012) proposed an evaluation of classification algorithms using multiple criteria decision making and rank correlation. Real-world decision problems often require subjective data provided by decision makers (Kou et al, 2014a, b). Since each *Correspondence: Gang Kou, School of Business Administration, Southwestern University of Finance and Economics, No. 555, Guanghuacun Street, Chengdu 610054, China. E-mail: [email protected]

decision maker has different preference structures, limited information and complex decision-making background, good decision-making models and decision makers must tolerate vagueness or ambiguity and be able to play roles in such situations (Yu, 2002). In order to precisely access the proper preference information, the decision makers should carry out either quantitative or qualitative analysis to determine the importance or the performance of criteria. The language variables or fuzzy variables are applied to reflect the attribute information, rather than crisp values. Fuzzy set theory, initially proposed by Zadeh (1965, 1978), has been extensively applied to objectively reflect the ambiguities in human judgment. One application of the fuzzy theory is in fuzzy multi-attribute decision making (FMADM). The fuzzy set theory is an efficient way to resolve the uncertainties of the available information in FMADM (Chen et al, 1992). Since the FMADM method is easily understandable, it has attracted widespread attention. A decision-making process in a fuzzy environment was considered in Bellman and Zadeh (1970). The integrated applications of fuzzy set theory, ANP and the mathematical programming method were described in Huang (2012). Chang et al (2011) applied fuzzy DEMATEL method to develop supplier selection criteria. Hatami-Marbini and Tavana (2011) proposed an alternative fuzzy outranking method by extending the ElectreI method to take into account the uncertain, imprecise and linguistic assessments provided by a group of decision makers. Sun and Lin (2009) used the fuzzy TOPSIS method to evaluate the competitive advantages of shopping websites. Wan and Li (2013) extended the Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) to solve heterogeneous MADM problems

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that involve intuitionistic fuzzy (IF) sets (IFSs), trapezoidal fuzzy numbers (TrFNs), intervals and real numbers. Chen et al (2005) proposed an FMADM approach for evaluating expatriate assignments. Yang et al (2008) proposed the integrated FMADM techniques with independent and interdependent relationships to select a vendor. Zheng et al (2012) proposed a trapezoidal fuzzy AHP method for work safety evaluation and early warning rating of hot and humid environments. Fenton and Wang (2006) researched the risk and confidence analysis for fuzzy multi-criteria decision making. Improved methods have also been discussed by various scholars (Pankaj et al, 2007; Chu and Lin, 2009; He et al, 2009; Wei et al, 2012; Wei et al, 2013). The existing methods consider only one type of fuzzy number. This paper uses a generalized fuzzy number (GFN), which represents a different fuzzy number when the parameter changes. Some graphs of GFNs are also given with different parameters to intuitively illustrate the features of GFNs. The Hausdorff distance, a measure proposed by Nadler (1978), is to a min-max distance. A major advantage of the Hausdorff distance measure is that the distance can be calculated without the explicit pairing of points in their respective data sets (Lin et al, 2003). Since this distance has better robustness on certain topics, some problems could be simplified by using the Hausdorff distance. This distance has been studied by many scholars. Huttenlocher et al (1993) provided algorithms for computing the Hausdorff distance between all possible relative positions of a binary image and a model. Chaudhuri and Rosenfeld, 1999 proposed a modified Hausdorff distance between fuzzy sets. The similarity measures of intuitionistic fuzzy sets based on the Hausdorff distance was researched in Hung and Yang (2004). Lin et al (2003) proposed a new spatially eigen-weighted Hausdorff distance measure for human face recognition. Kwon et al (2001) applied the modified Hausdorff distance in image matching, and obtained more accurate target matching results. Wan et al (2013) proposed the distance of triangular intuitionistic fuzzy numbers based on the Hausdorff distance. Xu and Xia (2011) proposed the Hausdorff distance measure for hesitant fuzzy sets. However, the Hausdorff distance is rarely used by MADM methods. This paper discusses the Hausdorff distance between GFNs and applies the Hausdorff distance in the linear programming model to determine the attribute weights based on the maximizing deviation. Many MADM methods have been developed on how to rank the alternatives based on the priority vector. For example, Peng et al (2011) proposed a fusion approach of MADM methods to rank multiclass classification algorithms. Soylu (2010) proposed a method to integrate PrometheeII with the Tchebycheff function for multi-criteria decision making. Ergu et al (2014) advanced a rapid MADM method to assess the key factors of risks and analyse the impacts and preferences of decision alternatives. Chen et al (2006) suggested a fuzzy approach for supplier evaluation and selection in supply chain management. Nakahara et al (1992) investigated a linear programming problem with interval coefficients and proposed a possibility

degree formula between interval fuzzy numbers. Xu (2002) proposed a ranking method based on possibility degree of triangular fuzzy numbers. However, the possibility degree formula developed in Nakahara et al (1992) and Xu (2002) can only be applied to interval fuzzy numbers or triangular fuzzy numbers. Carlsson and Fuller (2001) proposed the possibilistic mean value and variance of fuzzy numbers. Based on the possibilistic mean value of fuzzy numbers, this paper proposed a modified possibility degree formula, which can not only be applied to interval fuzzy numbers or triangular fuzzy numbers, but also applied to GFNs, and a ranking method based on the Hausdorff distance, GFNs and the modified possibility degree. A numerical example is used to demonstrate the applicability of the proposed method. This paper is organized as follows. Section 2 briefly describes the fuzzy set theories. A calculation method of attribute weights is proposed in Section 3. Section 4 presents the proposed method for the FMADM problem. Section 5 uses a numerical example to validate the proposed method. Section 6 concludes.

2. Definitions and theorems In this section, some basic definitions and theorems of fuzzy numbers are reviewed. The basic notations and definitions below will be used throughout the paper unless otherwise stated. ~ can be defined as Definition 1 A trapezoidal fuzzy number A ~ = ða; b; c; dÞ; 0 ⩽ a ⩽ b ⩽ c ⩽ d; if the membership A function μA~ : R ! ½0; 1 is defined as follows: 8x-a > ; a⩽x⩽b > > b-a > > > > < 1; b⩽x⩽c : μA~ = d-x > > > ; c⩽x ⩽ d > > d-c > > : 0; others Based on the above definition, the definition of GFN can be given as follows: ~ is given by A ~ = ða; b; c; dÞ ; n > 0, Definition 2 A GFN A n 0 ⩽ a ⩽ b ⩽ c ⩽ d, if the membership function μA~ : R ! ½0; 1 is defined as follows: 8 x - a n > > ; a⩽x⩽b > > b-a > > > > < 1; b ⩽ qx ⩽ c n : μA~ =  d-x > > > ; c⩽x⩽d > > d-c > > > : 0; others In order to intuitively illustrate the GFN, we assume a = 2, b = 6, c = 8, d = 12, and give the graphs of the GFN with different values of n as follows (Figures 1):

Guangxu Li et al—Multi-attribute decision making with generalized fuzzy numbers

Figure 1 GFN with different values of n.

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~ Based on the graphs of the GFN, we can discuss the GFN A ~ is a trapezoidal from three perspectives. First, if n = 1, then A ~ is a positive fuzzy number, and if n = 1 and b = c, then A triangular fuzzy number. Therefore, a triangular fuzzy number and a trapezoidal fuzzy number are special forms of a GFN. Second, if n > 1, the left and right branches contract for the membership function of the GFN. Third, if 0 < n < 1, the left and right branches expand for the membership function of the GFN. Moreover, we can obtain some characteristics of the GFN. For example, when n increases, the fuzzy degrees of the GFN decrease, which results in larger differences in the assessments. Therefore, the GFN could be suitable for all kinds of fuzzy environments. When the GFN is applied in the decision-making process, the subjectivity of selecting the model parameters is reduced, and the robustness of the model is improved. ~ = ða1 ; b1 ; c1 ; d1 Þ ; B ~= Now, given any two GFNs A n ða2 ; b2 ; c2 ; d2 Þn and a positive real number λ, some main ~ and B ~ can be expressed as operations of the fuzzy numbers A follows: 1. 2. 3. 4. 5.

~ B ~  ða1 + a2 ; b1 + b2 ; c1 + c2 ; d1 + d2 Þ ; A n ~ B ~  ða1 a2 ; b1 b2 ; c1 c2 ; d1 d2 Þn ; A ~  ðλa1 ; λb1 ; λc1 ; λd1 Þ ; λA n ~ =B ~  ða1 = d2 ; b1 = c2 ; c1 = b2 ; d1 = a2 Þn : A Manhattan distance between GFNs Z1    +    A ~ -B ~ B ~ -B ~ λ-  + A ~ λ+  ~ = dM A; λ λ

( )   ~ ~ dH A; B = sup max sup inf d ðx; yÞ; sup inf d ðx; yÞ : ~λ ~ λ y2B x2A

λ2½0; 1

~λ ~ λ x2B y2A

(2)

~ and B, ~ λ and B ~ A ~ λ are non-empty For any two fuzzy sets A bounded closed intervals contained in X. They can be ~ λ = ½A ~-; A ~ +  and B ~ λ = ½B ~ λ- ; B ~ λ+ ; respecdenoted by A λ λ ~ - and B ~ λ- are, respectively, the lower tively, where A λ ~ + and B ~ λ+ are, bounds of the closed interval, and A λ respectively, the upper bounds of the closed interval. In this case, we have (

)

max sup inf dðx; yÞ; sup inf d ðx; yÞ ~λ ~ λ y2B x2A

~λ ~ λ x2B y2A

  + 

 ~ -B ~ -B ~ λ- ; A ~ λ+  ; = max A λ λ

ð3Þ

By (2) and (3), the distance definition between fuzzy sets is given as follows: Definition 4 The Hausdorff distance between any two fuzzy ~ and B ~ is defined by sets A   + 

   ~ -B ~ -B ~ B ~ λ- ; A ~ λ+  : ~ = sup max A dH A; λ λ

(4)

λ2½0; 1

0

   n  1 ðb 1 - b 2 Þ + ða1 - a2 Þ dλ =  n+1 n+1    n  1 +  ðc 1 - c 2 Þ + ðd1 - d2 Þ n+1 n+1 In the main operation (5), the Manhattan distance between GFNs is complicated and the Euclid distance between GFNs even more so. In order to shorten the process of calculation, the Hausdorff distance between GFNs is proposed in the paper. In Nadler (1978), the Hausdorff distance is a measure of the extent to which two non-empty compact (closed and bounded) sets A and B in a metric space S resemble each other with respect to their positions. As a min-max distance, the Hausdorff distance can elaborate further on the correspondence of two sets and can be applied as the distance of two fuzzy sets. It can also measure the maximum degree of mismatch between two sets and be more robust; some problems can be simplified using the Hausdorff distance. Definition 3 For any two sets A and B, the Hausdorff distance H(A, B) is defined by H ðA; BÞ = max ðhðA; BÞ; hðB; AÞÞ;

~ and B, ~ the Based on Definition 3, for any two fuzzy sets A ~ BÞ ~ between the two fuzzy sets Hausdorff distance dH ðA; can be defined as follows:

(1)

where h(A, B) = maxa ∈ A minb ∈ B ||a − b||, ||·|| is a norm in some spaces, such as L2 and H2.

~ = ða; b; c; dÞ ; 0 ⩽ a ⩽ b ⩽ c ⩽ d; n > 0 its For a GFN A n ~ λ = ½ðb - aÞλ1 = n + a; ðc - dÞ λ-cut could be denoted by A λ1 = n + d; and based on the above analysis, the proposition about the Hausdorff distance between two GFNs can be given as follows: ~ = ða1 ; b1 ; c1 ; d1 Þ and B ~ = ða2 ; b2 ; c2 ; Proposition 1 Let A n d2 Þn , and let n > 0 be two GFNs, respectively; then, the Hausdorff distance between two GFNs is given as follows:   ~ B ~ = maxfja1 - a2 j; jb1 - b2 j; jc1 - c2 j; jd1 - d2 jg: (5) DH A;

~ = ða1 ; b1 ; c1 ; d1 Þ and B ~ = ða2 ; b2 ; c2 ; d2 Þn be Proof: Let A n ~ λ = ½ðb1 - a1 Þλ1 = n + a1 ; two GFNs, and let the λ-cut be A ~ λ = ½ðb2 - a2 Þλ1 = n + a2 ; ðc2 - d2 Þλ1 = n ðc1 - d1 Þλ1 = n + d1 ; B + d2 : Then, we obtain     1 1 A ~ -B ~ λ-  = ðb1 - a1 Þλn + a1 - ðb2 - a2 Þλn - a2  λ   1   = ðb1 - a1 - b2 + a2 Þλn + ða1 - a2 Þ   +   1 1 A ~ -B ~ λ+  = ðc1 - d1 Þλn + d1 - ðc2 - d2 Þλn - d2  λ   1   = ðc1 - d1 - c2 + d2 Þλn + ðd1 - d2 Þ

Guangxu Li et al—Multi-attribute decision making with generalized fuzzy numbers

If the values of all alternatives under attribute ~x′ij have larger differences, this shows that the attribute plays an important role in ranking the alternatives. From the above analysis, the greater deviation degree of the attribute should be given greater sort weight. Therefore, the choice of the attribute weight vector should make all attributes maximize the overall deviation degree of all alternatives. Owing to the uncertainty and complexity of the objectives and the fuzziness of the human mind, weights of the attributes are given by the partial information. For example, the value of weight is given by an interval. For the weight ωj ∈ [aj, bj], 0 ⩽ aj ⩽ bj ⩽ 1, a linear programming model is constructed as follows:

If we put the above formula into formula (4), we have  n 1   dH = sup max ðb1 - a1 - b2 + a2 Þλn + ða1 - a2 Þ; λ2½0; 1

o  1   ðc1 - d1 - c2 + d2 Þλn + ðd1 - d2 Þ n   1 1   = sup max ðb1 - b2 Þλn + ða1 - a2 Þ 1 - λn ; λ2½0; 1

  o 1 1   ðc1 - c2 Þλn + ðd1 - d2 Þ 1 - λn  = maxfja1 - a2 j; jb1 - b2 j; jc1 - c2 j; jd1 - d2 jg

Therefore, formula (5) holds. □

P : max DðωÞ = max

In this section, according to the maximizing deviation degree, the attribute weights can be determined by the linear programming model. Let us use x = {x1, x2, …, xm} to represent the alternatives and c = {c1, c2, …, cn} to represent the evaluation attributes. The attributes are additively independent. ~xij is the assessed value of attribute cj of alternative xi, and is expressed in a GFN in this paper. The different values of ~xij can be represented by a matrix ~ = ð~xij Þ V m ´ n , which is called the decision-making matrix. The vector of attribute weights is ω = {ω1, ω2, …,ωn}. To eliminate the difference of the attribute indexes on the dimension, each attribute index is normalized 8 ~xij > > > < xj+ ^ 1 8i 2 M; j 2 I1 ; ~x′ij = (6) > xj> > : ^ 1 8i 2 M; j 2 I2 : ~xij where I1 is associated with a set of benefit criteria, and I2 is associated with a set of cost criteria and M = {1, 2, …, m}. ~ = ½~x′ij m ´ n . Because Let the normalized decision matrix be V′ ′ ~xij is a fuzzy number, it is difficult to contrast these. For this reason, the Hausdorff distance is used to represent their deviation degree. Let Dð~x′ij ; ~x′kj Þ be the deviation degree ~ where between ~x′ij and ~x′kj in the normalized decision matrix V′, Dð~x′ij ; ~x′kj Þ is given by Definition 4. Apparently, the greater the deviation between attributes, the higher the value of the deviation degree. For attribute ~x′j ; the deviation degree between alternative xi and any other alternative could be calculated as follows: m   X (7) Dij = D ~x′ij ; ~x′kj ; i 2 M; j 2 N; k=1

then the total deviation degree could be calculated as follows: m X i=1

Dij =

n X

Dj ωj

j=1

3. Attribute weights

Dj =

5

m X m   X D ~x′ij ; ~x′kj ; i 2 M: i=1 k=1

(8)

= max

n X m X m   X D ~x′ij ; ~x′kj ωj ; j=1 i=1 k=1

s:t:

n X

 ωj = 1; ωj 2 aj ; bj ; ωj ⩾ 0; j 2 N:

ð9Þ

j=1

The attribute weights ω can be calculated by solving Equation (9). Let the attribute weights be ω = (ω1, ω2, …, ωn)T; the weighted normalized decision matrix vectors can be calculated as follows: n X 0 ~xij ωj ; i 2 M: (10) Z~i = j=1

4. Ranking of the alternatives In this section, based on the modified possibility degree, a ranking method of the complementary judgment matrix is proposed to rank the alternatives. In Carlsson and Fuller (2001), the definition of interval possibilistic mean value of fuzzy number is given as follows: ~-; A ~ +  be λ-cut of fuzzy number A; ~ ~ λ = ½A Definition 5 Let A λ λ ~ the interval possibilistic mean value of A is given by 2 1 3 Z Z1   ~ = 42 λA ~ - dλ; 2 λA ~ + dλ5: E A λ λ 0

0

In Nakahara et al (1992), a possibility degree formula between interval fuzzy numbers is given as follows: ~ and B ~ be two interval fuzzy numbers; the Definition 6 Let A ~ ⩾B ~ is given by possibility degree of A ( "   ~ ~ p A ⩽ B = min max 

# ) ~~ λ+ - A B λ   + ; 0 ; 1 : ~+ -A ~- + B ~ λ~λ - B A λ λ

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Based on definitions 5 and 6, the modified possibility degree between two fuzzy sets can be given as follows: ~ and B, ~ the possibility Definition 7 For any two fuzzy sets A ~ ~ degree of A ⩾ B is given by   ~⩾B ~ = maxf0; f1 ðλÞ - maxff2 ðλÞ; 0gg ; p A f1 ðλÞ

(11)

where Z1 f1 ðλÞ =

 +  ~ -A ~- +B ~ λ+ - B ~ λ- dλ; λ A λ λ

0

Z1 f2 ðλÞ =

 +  ~ - dλ; ~λ - A λ B λ

0

~ λ and B ~ λ are non-empty bounded closed intervals and A ~ - and B ~ λ- are the lower bounds of the contained in X, A λ ~ + and B ~ λ+ are the upper closed interval , respectively, and A λ bounds of the closed interval, respectively. Moreover, the conclusion of the improved possibility degree is given as follows:   ~⩾B ~ ⩽ 1; 0⩽p A (12)     ~ ⩾B ~ = 1: ~ +p B ~⩾A p A

(13)

Proof Expression (12) obviously holds, and therefore we only need to prove conclusion (13). By formula (11), we have   ~ = maxf0; f1 ðλÞ - maxff3 ðλÞ; 0gg ; ~ ⩾A p B f1 ðλÞ where Z1 f1 ðλÞ =

 +  ~ -A ~- +B ~ λ+ - B ~ λ- dλ; λ A λ λ

0

Z1 f3 ðλÞ =

 +  ~ -B ~ λ- dλ: λ A λ

0

Then we obtain     ~ ⩾B ~ = maxf0; f1 ðλÞ - maxff2 ðλÞ; 0gg ~ +p B ~ ⩾A p A f 1 ðλ Þ + =

maxf0; f1 ðλÞ - maxff3 ðλÞ; 0gg f 1 ðλ Þ

f2 ðλÞ + f3 ðλÞ f1 ðλÞ = =1 f1 ðλÞ f1 ðλÞ

Consequently, the conclusion in equation (13) holds. h□

In Orlovsky (1978), the definition of a complementary judgement matrix is given as follows: Definition 8 Let A = (xij)n × n be a matrix; if xij + xji = 1, the matrix is called a complementary judgement matrix. For a given fuzzy performance matrix, P = (pij)m × m is defined as a possible matrix. By formula (13), we find that P is a complementary judgement matrix. Using the ranking formula in Xu (2001), we have ! m X 1 m (14) vi = pij + - 1 ; i 2 M; mðm - 1Þ j = 1 2 That is, the ranking vector v = (v1, v2, … vm) about the possible matrix P can be calculated. Then, the alternatives are ranked in accordance with the size of the various components of the vector. Based on previous discussions, a five-step hybrid procedure for MADM is proposed as follows: Step 1: Set up decision matrix based on GFNs and normalize decision-making matrix. Step 2: Construct the linear programming model based on maximizing the deviation degree and calculate the attribute weights. Step 3: Calculate the weighted normalized decision matrix vectors. Step 4: Calculate the pair-wise comparisons matrix based on the possibility degree. Step 5: Rank the alternatives based on ranking formula (14).

5. Numerical example A numerical example is considered in this section. Assume that a car company needs to purchase some auto parts, and there are four auto parts suppliers, A1, A2, A3, and A4. Five attributes, C1 (price of the product), C2 (level of environmental protection), C3 (quality of the product), C4 (level of suppliers service), C5 (time of reaction), are taken into consideration. During the decision-making process, because the social environment is rather complex and the views of decision makers are usually uncertain, vague and ambiguous, people are usually reluctant or unable to assign accurate values in the evaluation process. They prefer to provide their evaluations in linguistic terms. The results of the assessment of attributes after analysing and assessing these suppliers are shown in Table 1. The partial weight information of the attributes is given as follows: 0:2 ⩽ ω1 ⩽ 0:3; 0:25 ⩽ ω2 ⩽ 0:35; 0:15 ⩽ ω3 ⩽ 0:25; 0:1 ⩽ ω4 ⩽ 0:2; 0:15 ⩽ ω5 ⩽ 0:25: Because each attribute is described by linguistic variables, we should convert the linguistic variables into the corresponding

Guangxu Li et al—Multi-attribute decision making with generalized fuzzy numbers

A1

A2

A3

A4

By solving the linear programming model, the attribute weights could be given by ω = (ω1, ω2, ω3, ω4, ω5)T = (0.25, 0.25, 0.15, 0.2, 0.15)T.

High High Medium Medium Low

Extremely high Medium High Low High

Medium Low Medium Very high High

Low Medium Medium High Medium

Step 4: Calculate the weighted normalized decision matrix vectors by formula (10); we obtain Z~1 = ð0:42; 0:62; 0:77; 0:97Þ ;

Table 1 Decision matrix based on linguistic variables

C1 C2 C3 C4 C5

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n

Z~2 = ð0:41; 0:59; 0:72; 0:85Þn ; Z~3 = ð0:47; 0:69; 0:84; 1:00Þn ;

~ Table 2 Decision matrix V

C1 C2 C3 C4 C5

A1

A2

A3

A4

Z~4 = ð0:48; 0:73; 0:88; 1:00Þn :

(5, 6, 7, 8)n (5, 6, 7, 8)n (4, 5, 5, 6)n (4, 5, 5, 6)n (2, 3, 4, 5)n

(8, 9, 10, 10)n (4, 5, 5, 6)n (5, 6, 7, 8)n (2, 3, 4, 5)n (5, 6, 7, 8)n

(4, 5, 5, 6)n (2, 3, 4, 5)n (4, 5, 5, 6)n (7, 8, 8, 9)n (5, 6, 7, 8)n

(2, 3, 4, 5)n (4, 5, 5, 6)n (4, 5, 5, 6)n (5, 6, 7, 8)n (4, 5, 5, 6)n

~ = ða; b; c; dÞ ;0 ⩽ a ⩽ b ⩽ c ⩽ d, n > 0, its λ-cut For a GFN A n ~ λ = ½ðb - aÞλ1 = n + a; ðc - dÞλ1 = n + d; and could be denoted by A thus we have

GFNs. Let there be seven kinds of linguistic variables: l1 (extremely high), l2 (very high), l3 (high), l4 (medium), l5 (low), l6 (very low), l7 (extremely low). Their corresponding GFNs are given by (8, 9, 10, 10)n, (7, 8, 8, 9)n, (5, 6, 7, 8)n, (4, 5, 5, 6)n, (2, 3, 4, 5)n, (1, 2, 2, 3)n, (0, 0, 1, 2)n. When the improved algorithm is applied to solve this supplier selection problem, the specific calculating steps are shown as follows. Step 1: Translate the fuzzy linguistic terms into GFNs and set ~ as listed in Table 2. up decision matrix V C1 (the price of the product) is the cost criterion and the other attributes are benefit criteria. To eliminate the difference of the attribute indicators on the dimension, each attribute index is normalized. ~ by some main Step 2: Normalize decision-making matrix V operations of the fuzzy numbers and (6), and then ~ as obtain the standardized decision-making matrix V′ listed in Table 3. Step 3: Construct the linear programming model based on maximizing the deviation degree and then calculate the attribute weights by Equation (9); we have P : maxð4:00ω1 + 2:66ω2 + 1:02ω3 + 4:24ω4 + 2:96ω5 Þ 8 0:2 ⩽ ω1 ⩽ 0:3 > > > > > > > 0:25 ⩽ ω2 ⩽ 0:35 > > > > > > < 0:15 ⩽ ω3 ⩽ 0:25 s:t:

> > 0:1 ⩽ ω4 ⩽ 0:2 > > > > > > > 0:15 ⩽ ω5 ⩽ 0:25 > > > > : ω1 + ω2 + ω3 + ω4 + ω5 = 1

h i 1 1 Z~1λ = 0:42 + 0:2λn ; 0:97 - 0:2λn ; h i 1 1 Z~2λ = 0:41 + 0:18λn ; 0:85 - 0:13λn ; h i 1 1 Z~3λ = 0:47 + 0:22λn ; 1 - 0:16λn ; h i 1 1 Z~4λ = 0:48 + 0:25λn ; 1 - 0:12λn :

Step 5: Calculate the pair-wise comparisons matrix based on the possibility degree. Then, calculate the possibility degree between Z~i and Z~j by formula (11); we have   0:18n + 0:28   0:08n + 0:25 ; p Z~1 ⩾ Z~3 = ; p Z~1 ⩾ Z~2 = 0:28n + 0:495 0:3n + 0:54   0:04n + 0:245   0:03n + 0:19 p Z~1 ⩾ Z~4 = ; p Z~2 ⩾ Z~3 = ; 0:3n + 0:535 0:28n + 0:485   maxf0:185 - 0:01n; 0g ; p Z~2 ⩾ Z~4 = 0:28n + 0:48   0:11n + 0:26 p Z~3 ⩾ Z~4 = : 0:3n + 0:525

For pðZ~2 ⩾ Z~4 Þ, if 0 < n < 18.5, then pðZ~2 ⩾ Z~4 Þ = ðð0:185 - 0:01nÞ = ð0:28n + 0:48ÞÞ; and if n ⩾ 18.5, then pðZ~2 ⩾ Z~4 Þ = 0. Because of the particularity of a GFN, we should discuss the pair-wise comparisons matrix. For a GFN, the membership function is different when n is given different values. For these reasons, based on the formula of the possibility degree, the results of different pair-wise comparisons matrices could be given as follows.

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~0 Table 3 Normalized decision matrix V

C1 C2 C3 C4 C5

A1

A2

A3

A4

(0.25, 0.43, 0.67, 1.00)n (0.63, 0.86, 1.00, 1.00)n (0.50, 0.71, 0.83, 1.00)n (0.44, 0.63, 0.63, 0.86)n (0.25, 0.43, 0.67, 1.00)n

(0.20, 0.30, 0.44, 0.63)n (0.50, 0.71, 0.83, 1.00)n (0.63, 0.86, 1.00, 1.00)n (0.22, 0.38, 0.50, 0.71)n (0.63, 0.86, 1.00, 1.00)n

(0.33, 0.60, 0.80, 1.00)n (0.25, 0.43, 0.67, 1.00)n (0.50, 0.71, 0.83, 1.00)n (0.78, 1.00, 1.00, 1.00)n (0.63, 0.86, 1.00, 1.00)n

(0.40, 0.75, 1.00, 1.00)n (0.50, 0.71, 0.83, 1.00)n (0.50, 0.71, 0.83, 1.00)n (0.56, 0.75, 0.88, 1.00)n (0.50, 0.71, 0.83, 1.00)n

Figure 2 Ranking results under different values of n.

First, let 0 < n < 18.5, and calculate the pair-wise comparisons matrix by formula (11); we have 2 0:5

6 6 6 0:1n + 0:215 6 6 6 0:28n + 0:495 A=6 6 0:22n + 0:29 6 6 0:3n + 0:54 6 4 0:26n + 0:29 0:3n + 0:535

0:18n + 0:28 0:08n + 0:25 0:28n + 0:495 0:3n + 0:54 0:03n + 0:19 0:5 0:28n + 0:485 0:25n + 0:295 0:5 0:28n + 0:485 0:295 + 0:29n 0:19n + 0:265 0:28n + 0:48 0:3n + 0:525

0:04n + 0:245 3 0:3n + 0:535 7 7 0:185 - 0:01n 7 7 7 0:28n + 0:48 7 7: 0:11n + 0:26 7 7 0:3n + 0:525 7 7 5 0:5

Second, let n ⩾ 18.5, and calculate the pair-wise comparisons matrix by formula (11); we obtain 2 0:5 6 6 6 0:1n + 0:215 6 6 6 0:28n + 0:495 A=6 6 0:22n + 0:29 6 6 0:3n + 0:54 6 4 0:26n + 0:29 0:3n + 0:535

0:18n + 0:28 0:08n + 0:25 0:28n + 0:495 0:3n + 0:54 0:03n + 0:19 0:5 0:28n + 0:485 0:25n + 0:295 0:5 0:28n + 0:485 0:19n + 0:265 1 0:3n + 0:525

0:04n + 0:245 3 0:3n + 0:535 7 7 7 7 0 7 7 7: 0:11n + 0:26 7 7 0:3n + 0:525 7 7 5 0:5

Step 6: Calculate the ranking vector R under different n values by the ranking formula (14); we obtain the results shown in Table 4. In order to better illustrate the results, the ranking of the alternatives under different n values could be shown in Figure 2. According to the results, the best alternative is A4, and the final assessment values of the four alternatives are more apparent with the increase in n values. To illustrate the effectiveness of the Hausdorff distance between GFNs, a comparative analysis between the Hausdorff distance and the subtraction of two fuzzy numbers is shown in Table 5. It can be seen from Table 5 that the ranking results of the proposed method with the Hausdorff distance are consistent with the subtraction of two fuzzy numbers. Moreover, since the Hausdorff distance can measure the maximum degree of mismatch between two sets, it can simplify the problems. Table 6 summarizes the results of the comparison between the proposed model and the FMADM, which was developed in Hadi-Vencheh and Mokhtarian (2011). It can be seen from Table 6 that the two methods generate the same ranking results. It can also be seen that the final assessment values of the four alternatives generated by the proposed model are more apparent with the increase in n values

Guangxu Li et al—Multi-attribute decision making with generalized fuzzy numbers

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Table 4 Ranking vectors and ranking results n n = 0.01 n = 0.1 n = 0.5 n=1 n=2 n = 18 n = 19 n = 100

Ranking vector R

Ranking results

(0.2487, 0.2256, 0.2619, 0.2639) (0.2469, 0.2224, 0.2633, 0.2674) (0.2408, 0.2113, 0.2682, 0.2797) (0.2356, 0.2020, 0.2723, 0.2900) (0.2294, 0.1909, 0.2773, 0.3024) (0.2153, 0.1664, 0.2885, 0.3299) (0.2151, 0.1662, 0.2886, 0.3301) (0.2126, 0.1642, 0.2906, 0.3327)

A4 > A3 > A1 > A2 A4 > A3 > A1 > A2 A4 > A3 > A1 > A2 A4 > A3 > A1 > A2 A4 > A3 > A1 > A2 A4 > A3 > A1 > A2 A4 > A3 > A1 > A2 A4 > A3 > A1 > A2

Table 5 Comparison proposed model with another distance

Total deviation degree Attribute weights Ranking results

Hausdorff distance

The subtraction of two fuzzy numbers

(4.00, 2.66, 1.02, 4.24, 2.96) (0.25, 0.25, 0.15, 0.2, 0.15)T A4 > A3 > A1 > A2

(10.24, 6.84, 2.7, 13.08, 7.74) (0.25, 0.25, 0.15, 0.2, 0.15)T A4 > A3 > A1 > A2

Table 6 Comparison proposed model with another FMADM

Ranking vector (n = 1) Ranking results

The proposed method

FMADM in (Hadi-Vencheh and Mokhtarian, 2011)

(0.2356, 0.2020, 0.2723, 0.2900) A4 > A3 > A1 > A2

(0.2447, 0.238, 0.255, 0.2623) A4 > A3 > A1 > A2

than the FMADM. In the proposed method, by comparing the sorted results under different n values, it could be shown that the differences in the final assessment values of the four alternatives are more apparent with the increase in n values. In other words, the differences among the alternatives are more obvious when using the proposed model. These results are suitable for the characteristics of the GFN, and these also reflect the original intention that we introduce the GFNs. In the decision-making process, decision makers could be able to adjust parameters according to the actual situation. Moreover, the proposed model uses interval values for the attribute weights. It can not only meet the different assessment requirements of decision makers from weight adjustment, but also improve the robustness and adaptability of decisionmaking methods. These results will also be helpful for further study of dynamic or interactive MADM.

Furthermore, linguistic terms and GFNs are also used in the evaluation process. Because the membership function of the GFN has different forms, the subjectivity of selecting the model parameters is reduced, and the robustness of the model is improved. This study chooses the Hausdorff distance for the proposed model because within a certain error range it can elaborate further on the correspondence of two fuzzy sets and reduce the process of calculation. In future research, we will focus on how to solve the dynamic FMADM methods with GFNs and how to aggregate expert opinions in dynamic group decision making with other fuzzy numbers. Moreover, simulation is a better way to show the robustness of MADM, and this is one of our future research directions. Acknowledgements —This work was supported in part by grants from the National Natural Science Foundation of China (#71222108, # 71471149, # 71433001), and the Research Fund for the Doctoral Program of Higher Education (#20120185110031).

6. Conclusions References This paper proposes a new fuzzy decision-making method for decision-making problems and finds reasonable decision-making alternatives. In the proposed model, the attribute weights are determined by constructing a linear programming model, and an improved priority is proposed to calculate the possibility degree for the FMADM problem. The experimental result indicates that the proposed model is practical and effective.

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Received 6 January 2014; accepted 6 January 2015 after one revision