induced quasi-arithmetic uncertain linguistic

February 7, 2013 11:6 WSPC/118-IJUFKS

S0218488513500049

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 21, No. 1 (2013) 55–77 c World Scientific Publishing Company

DOI: 10.1142/S0218488513500049

Int. J. Unc. Fuzz. Knowl. Based Syst. 2013.21:55-77. Downloaded from www.worldscientific.com by 213.85.255.5 on 06/12/14. For personal use only.

INDUCED QUASI-ARITHMETIC UNCERTAIN LINGUISTIC AGGREGATION OPERATOR

WEI YANG Department of Mathematics, School of Science, Xi’an University of Architecture and Technology, 710055, Xi’an, Shaanxi, China [email protected] Received 15 July 2011 Revised 3 September 2012 Induced quasi-arithmetic aggregation operators are considered to aggregate uncertain linguistic information by using order inducing variables. We introduce the induced correlative uncertain linguistic aggregation operator with Choquet integral and we also present the induced uncertain linguistic aggregation operator by using the DempsterShafer theory of evidence. The special cases of the new proposed operators are investigated. Many existing linguistic aggregation operators are special cases of our new operators and more new uncertain linguistic aggregation operators can be derived from them. Decision making methods based on the new aggregation operators are proposed and architecture material supplier selection problems are presented to illustrate the feasibility and efficiency of the new methods. Keywords: Choquet integral; Dempster-Shafer theory; uncertain linguistic variable; aggregation operator; decision making.

1. Introduction The quasi-arithmetic ordered weighted averaging (QOWA) operator1 is a generalization of the ordered weighted averaging (OWA) operator2 by using the quasiarithmetic means. Several aggregation operators including the OWA operator, the ordered weighted geometric (OWG) operator,3,4 generalized OWA (GOWA) operator,5– 7 are all the special cases of the QOWA operator. The induced QOWA (I-QOWA) operator is an extension of the QOWA operator that uses a reordering process based on order inducing variables in order to assess complex reordering processes.8 Xia et al.9 have developed several quasi-arithmetic aggregation operators for hesitant fuzzy information. Yang and Chen10 present some quasi-arithmetic intuitionistic fuzzy aggregation operators including the quasi-arithmetic intuitionistic fuzzy ordered weighted averaging operator, the quasi-intuitionistic fuzzy Choquet ordered averaging (QIFCOA) operator and the quasi-intuitionistic fuzzy ordered weighted averaging (BS-QIFOWA) operator based on the Dempster-Shafer belief 55



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structure. Though QOWA operator has been generalized to accommodate different kinds of information, it has not been used for uncertain linguistic variables. Uncertainty and fuzziness exist extensively in the decision making process since complex decision problem, time pressure, decision maker’s limit knowledge and expertise, fuzzy nature existing in the human thinking, etc. Linguistic labels rather than exact numerical values are more suitable to deal with uncertain and fuzzy information. Several types of linguistic models have been developed. The ordinal linguistic computational model11 was introduced, but information may be lost in this model by use of the round operator. In order to avoid information distortion and lose in the linguistic information processing, the 2-tuple linguistic computational model12 was developed by Herrera and Mart´ınez, in which linguistic information is represented by a linguistic term and a number. Xu13 introduced another linguistic model by extending the discrete term set to a continuous linguistic term set. Xu14 further generalized the continuous linguistic model to uncertain linguistic model. Among all the linguistic models, the uncertain linguistic model is more appropriate to depict uncertain and fuzzy information. Hence it has received increasing research attentions and many multiple decision making methods have been proposed. Uncertain linguistic ordered weighted averaging (ULOWA) operator and uncertain linguistic hybrid aggregation (ULHA) operator are proposed in Ref.14 and uncertain linguistic ordered weighted geometric (ULOWG) operator is presented in Ref.15 by Xu. Generalized induced uncertain linguistic ordered weighted averaging (GIULOWA) operator and generalized induced uncertain linguistic ordered weighted geometric (GIULOWG) operator are considered in Ref.16. The linguistic correlated averaging operator and linguistic correlated geometric operator are introduced in Ref. 17, where the linguistic labels are correlative. Uncertain linguistic hybrid geometric mean (ULHGM) operator is proposed by Wei.18 Uncertain linguistic weighted continuous interval extended ordered weighted averaging (ULWCEOWA) operator is presented by Liu et al.19 Many uncertain linguistic aggregation operators have been proposed by now, but each aggregation operator only focuses on one point view of the decision problem and we have not yet found uncertain linguistic aggregation operator by making use of quasi-arithmetic means. Hence, in this paper, we propose several quasi-arithmetic uncertain linguistic aggregation operators. Correlation of the information is an important aspect we should consider, which exists extensively in the decision making process. Choquet integral is introduced by Choquet20 and is a useful tool to model the inter-dependence or correlation, which has been studied and applied in the decision making.5,6,21 – 29 The induced Choquet ordered averaging (I-COA) operator is first presented in Ref. 23 and then it is generalized to aggregate matrices and vectors in Ref. 24. The generalized Choquet aggregation operator is introduced in Ref. 5. The induced generalized intuitionistic fuzzy Choquet integral (I-GIFCOA) operator is developed by Xu and Xia.6 The quasi-arithmetic intuitionistic fuzzy Choquet ordered averaging (QIFCOA) operator is presented in Ref.10. Yang and Chen29 aggregate correlated 2-tuple linguistic


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arguments by using Choquet integral. All of the above methods have not yet considered the correlated uncertain linguistic arguments, hence we aggregate correlated uncertain linguistic information by using Choquet integral. Uncertainty is another important issue we should consider in the selection of alternatives. The Dempster-Shafer theory of evidence30 – 32 plays a crucial role in providing a unifying framework for representing the uncertainty including the situations of risk and ignorance,33 which has been used in the decision making by several authors.33 – 42 The induced aggregation operators in situations that decision making with the Dempster-Shafer theory of evidence are suggested by Merig´ o and Casanovas.37 Induced intuitionistic fuzzy aggregation operators based on the Dempster-Shafer belief structure are developed in Ref. 6. The Dempser-Shafer theory is applied to the QOWA operator to aggregate the intuitionistic fuzzy information in Ref. 10. Merig´o and Casanovas38 presented the belief structure-linguistic ordered weighted averaging (BS-LOWA) operator and the belief structure-linguistic hybrid averaging (BS-LHA) operator. In this paper, we extend BS-LOWA operator to accommodate uncertain linguistic arguments by making use of quasi-arithmetic means. In decision making process, a complete spectrum of the decision problem should be considered in order to make proper decision making. But all the existing operators with uncertain linguistic information only focus on one point view of the decision problem. In this paper, we develop several new uncertain linguistic aggregation operators based on QOWA operator. We first introduce the induced quasiarithmetic uncertain linguistic ordered weighted averaging (I-QULOWA) operator, where correlative uncertain linguistic information is aggregated with the I-QOWA operator. Then we present the I-QULOWA operator with Dempster-Shafer belief structure. The new proposed operators have the following characteristics: uncertain and vague information is depicted by uncertain linguistic variables; correlation of the information is considered by Choquet integral and uncertainty existing in the decision making process is considered by Dempster-Shafer belief structure; I-QOWA operator is used to aggregate decision making information and many aggregation operators are special cases of the proposed operators, thus it can provide a whole previews of the decision making problem. The special cases of the new aggregation operators are studied and many existing uncertain linguistic aggregation operators are special cases of the proposed operators, more new uncertain linguistic aggregation operators have been presented. The multiple attribute decision making methods based on the new operators are proposed and the numerical examples are given to illustrate the feasibility and efficiency of the new algorithms. In order to do so, the rest of the paper is organized as follows. In Sec. 2, some concepts on uncertain linguistic variable have been reviewed. In Secs. 3 and 4, we develop the induced quasi-arithmetic uncertain linguistic Choquet ordered averaging (I-QULCOA) operator and the induced quasi-arithmetic uncertain linguistic ordered weighted averaging operator based on Dempster-Shafer belief structure (BSI-QULOWA), respectively. The special cases of the new operators have been


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developed and the decision making methods based on new operators have been presented. Numerical examples have been given to illustrate feasibility and efficiency of the new algorithms. Section 5 gives some concluding remarks.


2. Basic Concepts As a preparation for introducing our new method, some relevant concepts are illustrated in this section. Suppose that S = {si | i = 0, . . . , t} is a finite and totally ordered discrete term set, where si represents a possible value for a linguistic variable. For example, a set of nine terms S can be expressed as S = {s1 = extremely poor, s2 = very poor, s3 = poor, s4 = slightly poor, s5 = fair, s6 = good, s7 = very good, s8 = extremely good, s9 = extremely good}. The above set satisfies the following properties: (i) The set is ordered: si ≥ sj , if i ≥ j; (ii) Max operator: max(si , sj ) = si , if i ≥ j; (iii) Min operator: min(si , sj ) = si , if i ≤ j; (iv) A negation operator: Neg(si ) = sj , such that j = g + 1 − i. In order to preserve the all the given information, the discrete term set S is extended to a continuous term set S¯ = {sα | s0 ≤ sα ≤ st , α ∈ [0, t]}. Let s˜ = [sα , sβ ], where sα , sβ are the lower and the upper limits, respectively, then we call s˜ the uncertain linguistic variable. Let Se be the set of all the uncertain variables. Let s˜ = [sα , sβ ], s˜1 = [sα1 , sβ1 ], s˜2 = [sα2 , sβ2 ] be three uncertain linguistic variables, and λ, λ1 , λ2 ∈ [0, 1]. Then the operational laws are defined as: (i) s˜1 ⊕ s˜2 = [sα1 , sβ1 ] ⊕ [sα2 , sβ2 ] = [sα1 ⊕ sα2 , sβ1 ⊕ sβ2 ] = [sα1 +α2 , sβ1 +β2 ]; (ii) λ˜ s = λ[sα , sβ ] = [λsα , λsβ ] = [sλα , sλβ ]; (iii) s˜1 ⊗ s˜2 = [sα1 , sβ1 ] ⊗ [sα2 , sβ2 ] = [sα1 ⊗ sα2 , sβ1 ⊗ sβ2 ] = [sα1 α2 , sα2 β2 ]; (iv) s˜λ = [sα , sβ ]λ = [sαλ , sβ λ ]. Let s˜1 = [sα1 , sβ1 ] and s˜2 = [sα2 , sβ2 ] be two uncertain variables, then the degree of possibility of s˜1 ≥ s˜2 is defined as β1 − α2 p(˜ s1 ≥ s˜2 ) = min max ,0 ,1 . (1) β1 − α1 + β2 − α2 From the above definition, we can easily get the following results: (i) 0 ≤ p(˜ s1 ≥ s˜2 ) ≤ 1, 0 ≤ p(˜ s2 ≥ s˜1 ) ≤ 1; (ii) p(˜ s1 ≥ s˜2 ) + p(˜ s2 ≥ s˜1 ) = 1.


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3. New Uncertain Linguistic Choquet Aggregation Operator In this section, induced quasi-arithmetic uncertain linguistic ordered averaging (I-QULOA) operator is introduced to accommodate correlated uncertain linguistic information.


3.1. Induced Quasi-Uncertain Linguistic Choquet Ordered Averaging (I-QULCOA) operator Definition 3.1.43 A fuzzy measure m on set X is a set function m : P (X) → [0, 1] satisfying the following axioms: (i) m(φ) = 0, m(X) = 1; (ii) B ⊆ C implies m(B) ≤ m(C), for all B, C ⊆ X; (iii) m(B ∪ C) = m(B) + m(C) + ρm(B)m(C) for all B, C ⊆ X and B ∩ C = φ, where ρ ∈ (−1, ∞). In the above definition, if ρ = 0, then the third condition reduces to the axiom of additive measure: m(B ∪ C) = m(B) + m(C)

for all B, C ⊆ X

and B ∩ C = φ .

If the elements of the B in X are independent, and we have X m(B) = m(xi ), for all B ⊆ X .

(2)

(3)

xi ∈B

Definition 3.2. Let X = {x1 , x2 , . . . , xn } be a finite set, m be a fuzzy measure on X, {, , . . . , } be a collection of 2-tuples on X, where ui is the order-inducing variables and s˜i is the uncertain linguistic arguments to be aggregated. An induced quasi-arithmetic uncertain linguistic Choquet ordered averaging (I-QULCOA) operator of dimension n is a function I-QULCOA: (R × ˜ n → S: ˜ S) I-QULCOA(, , . . . , ) X n = g −1 m(Aσ(i) ) − m(Aσ(i−1) ) g(˜ sσ(i) )

(4)

i=1

where (σ(1), σ(2), . . . , σ(n)) is a permutation of (1, 2, . . . , n) such that uσ(1) ≥ uσ(2) ≥ · · · ≥ uσ(n) , s˜σ(i) is s˜j value of pair having the ith largest ui (i = 1, 2, . . . , n), g is a strictly continuous monotonic function and Aσ(i) = {xσ(1) , xσ(2) , . . . , xσ(i) }, i ≥ 1, Aσ(0) = φ. We discuss some special cases of the I-QULCOA operator if some special sets and some functions g(x) are considered: (i) If m(A) = 1, for any A ∈ P (X), then I-QULCOA(, , . . . , ) = s˜σ(1) .

(5)


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(ii) If m(A) = 0 for any A ∈ P (X) and A 6= X, then I-QULCOA(, , . . . , ) = s˜σ(n) .

(6)


(iii) For any A, B ∈ P (X) such that |A| = |B|, if m(A) = m(B) and m(Aσ(i) ) = 1 ≤ i ≤ n, then I-QULCOA(, , . . . , ) X X n n 1 −1 1 −1 =g g (˜ sσ(i) ) = g g (˜ si ) . n n i=1 i=1

i n,

(7)

(iv) If (3) holds, then m(xσ(i) ) = m(Aσ(i) ) − m(Aσ(i−1) ), i = 1, 2, . . . , n .

(8)

In this case, the I-QULCOA operator reduces to the following form: X n −1 I-QULCOA(, , . . . , ) = g m(xi )g (˜ si ) . (9) i=1

(v) If m(A) =

|A| X

wi , for all A ⊆ X ,

(10)

i=1

where |A| is the number of the elements in the set A, then wi = m(Aσ(i) ) − m(Aσ(i−1) ), i = 1, 2, . . . , n , (11) Pn where w = (w1 , w2 , . . . , wn ), wi ≥ 0, i = 1, 2, . . . , n, and i=1 wi = 1. In this case, the I-QULCOA operator reduces to the I-QULOWA operator: X n −1 I-QULOWA(, , . . . , ) = g wi g (˜ sσ(i) ) . (12) i=1

(vi) If m(A) = Q

X

xi ∈H

m(xi ) ,

for all A ⊆ X ,

(13)

where Q is a basic unit-interval monotonic (BUM) function Q : [0, 1] → [0, 1], which has the following properties: (i) Q(0) = 0; (ii) Q(1) = 1; and for x > y, Q(x) ≥ Q(y), then X X wi = m(Aσ(i) ) − m(Aσ(i−1) ) = Q m(xσ(j) ) − Q m(xσ(j) ) , (14) j≤i

j 0, γ 6= 1, then I-QULCOA(, , . . . , ) X n = logγ (m(Aσ(i) ) − m(Aσ(i−1) ))γ s˜σ(i) .

(25)

i=1

3.2. Decision making based on the I-QULCOA operator Assume that there is a decision making problem with a collection of alternatives {A1 , A2 , . . . , Am } and the states of nature {C1 , C2 , . . . , Cn }. The payoff value s˜ij is got if the alternative Ai is selected under the state Cj . The decision making method based on the I-QULCOA operator is given as follows. Algorithm 1 Step 1. Calculate the fuzzy measure of attributes in C = {C1 , C2 , . . . , Cn } and attribute sets. Step 2. Determine the inducing matrix U = (uij )m×n . The order-inducing variables are used to represent the attitudinal character, which is complex since it involves the opinion of different members of the board of directors.8 Step 3. Utilize the I-QULCOA operator to aggregate the payoff values into a collective payoff value of each alternative. s˜i = I-QULCOA(, , . . . , ) X n −1 =g (m(Aσ(j) ) − m(Aσ(j−1) ))g(˜ siσ(j) ) ,

(26)

j=1

where (σ(1), σ(2), . . . , σ(n)) is a permutation of (1, 2, . . . , n) such that uiσ(1) ≥ uiσ(2) ≥ · · · ≥ uiσ(n) , s˜iσ(j) is the value s˜ij in with jth largest uij and Aσ(i) = {Cσ(1) , Cσ(2) , . . . , Cσ(i) }, Cσ(j) is the state corresponding to s˜iσ(j) . Step 4. According to the comparison method of uncertain linguistic variables introduced in Sec. 2, we rank the collective payoff values and rank the alternatives accordingly.


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Example 1 In this section, we consider a multiple attribute decision making problem with correlated uncertain linguistic evaluation values to illustrate the proposed algorithm. There is an architecture company wants to select the most appropriate supplier for one important material such as cement. After pre-evaluation, five alternatives A1 , A2 , . . . , A5 are remained for further evaluation. Four attributes: C1 -the price of the product, C2 -the quality of the product, C3 -the delivery time and C4 -the risk are taken into consideration. The expected evaluation values are given as uncertain linguistic variables (˜ sij )5×4 and the results are shown in Table 1. If our new method is applied to solve this problem, the concrete procedure would be as follows. Table 1.

Decision matrix.

C1

C2

C3

C4

A1

[s4 , s5 ]

[s5 , s7 ]

[s1 , s2 ]

[s7 , s9 ]

A2 A3

[s6 , s8 ] [s4 , s6 ]

[s5 , s6 ] [s5 , s6 ]

[s6 , s7 ] [s3 , s5 ]

[s2 , s4 ] [s6 , s7 ]

A4

[s7 , s8 ]

[s4 , s5 ]

[s1 , s3 ]

[s5 , s6 ]

A5

[s2 , s3 ]

[s3 , s5 ]

[s8 , s9 ]

[s4 , s5 ]

Step 1. Calculate the fuzzy measure of the attributes as follows: m(φ) = 0, m({C1 }) = 0.2, m({C2 }) = 0.15, m({C3 }) = 0.3, m({C4 }) = 0.35, m({C1 , C2 }) = 0.4, m({C1 , C3 }) = 0.45, m({C1, C4 }) = 0.55, m({C2 , C3 }) = 0.48, m({C2, C4 }) = 0.62, m({C3 , C4 }) = 0.45, m({C1 , C2 , C3 }) = 0.6, m({C1 , C2 , C4 }) = 0.7, m({C1 , C3 , C4 }) = 0.85, m({C2, C3 , C4 }) = 0.8, m({C1 , C2 , C3 , C4 }) = 1.0. Step 2. Determine the inducing matrix U = (uij )5×4 as in Table 2. Table 2.

Inducing variables.

C1

C2

C3

C4

A1

16

21

18

23

A2 A3

17 19

24 21

22 23

15 18

A4 A5

25 18

21 24

14 22

16 20

Step 3. Aggregate the payoff values with the I-QULCOA operator. Table 3 shows the results when we use different aggregation operators. Step 4. Rank the alternatives according to the collectives values. For example, in order to rank [s7 , s9 ], [s6 , s8 ], [s6 , s7 ], [s7 , s8 ], [s8 , s9 ], we construct a complementary


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Table 3.

Aggregated results.

Max

Min

I-ULCOA

I-ULCOG

I-ULCOQA

A1 A2

[s7 , s9 ] [s6 , s8 ]

[s1 , s2 ] [s2 , s3 ]

[s4.78 , s6.40 ] [s4.25 , s5.85 ]

[s4.0264 , s5.7035 ] [s3.7621 , s5.6162 ]

[s5.2230 , s6.8775 ] [s4.6422 , s6.0704 ]

A3 A4

[s6 , s7 ] [s7 , s8 ]

[s3 , s4 ] [s1 , s3 ]

[s4.68 , s6.10 ] [s4.00 , s5.30 ]

[s4.4922 , s6.0419 ] [s3.1559 , s4.9773 ]

[s4.8497 , s6.1563 ] [s4.5607 , s5.5946 ]

A5

[s8 , s9 ]

[s2 , s4 ]

[s4.77 , s5.92 ]

[s4.1923 , s5.4808 ]

[s5.3282 , s6.3467 ]

I-GULCOA5

I-GULCOA10

I-QULCOAsin( π x)

I-QULCOAcos( π x)

I-QULCOA2x

2

2

A1

[s5.8636 , s7.6286 ] [s6.3203 , s8.1536 ]

[s4.4070 , s5.4934 ]

[s5.1411 , s6.7043 ]

[s5.8329 , s7.7871 ]

A2 A3

[s5.2480 , s6.5796 ] [s5.5687 , s7.0301 ] [s5.2181 , s6.3087 ] [s5.5162 , s6.4973 ]

[s3.9898 , s5.4975 ] [s4.5352 , s5.9940 ]

[s4.5790 , s5.9891 ] [s4.8131 , s6.1338 ]

[s5.1375 , s6.6323 ] [s5.1570 , s6.3219 ]

A4

[s5.3792 , s6.2562 ] [s5.9917 , s6.8720 ]

[s3.6756 , s4.9462 ]

[s4.4774 , s5.5034 ]

[s5.2854 , s6.3074 ]

A5

[s6.4524 , s7.3191 ] [s7.1612 , s8.0587 ]

[s4.2383 , s5.2189 ]

[s5.1663 , s6.1620 ]

[s6.5173 , s7.5361 ]

matrix as 

0.5  0.25  P = 0  0.3333 0.6667

0.75 0.5 0.3333 0.6667 1

1 0.6667 0.5 1 1

0.6667 0.3333 0 0.5 1

 0.3333  0  , 0   0 0.5

Summing all elements in each line of matrix P , we have p1 = 3.25, p2 = 1.75, p3 = 0.8333, p4 = 2.5, and p5 = 4.1667. Then we rank Ai (i = 1, 2, . . . , 5) in descending order in accordance with the values of pi to get A5 ≻ A1 ≻ A4 ≻ A2 ≻ A3 . Here, ≻ means prefer to. Similarly, we can get the ranking results of other methods. The results are shown in Table 4. Table 4.

Ranking of the alternatives.

Rankings

Rankings

Max Min

A5 ≻ A1 ≻ A4 ≻ A2 ≻ A3 A3 ≻ A5 ≻ A2 ≻ A4 ≻ A1

I-GULCOA5 I-GULCOA10

A5 ≻ A1 ≻ A2 ≻ A4 ≻ A3 A5 ≻ A1 ≻ A4 ≻ A2 ≻ A3

I-ULCOA

A1 ≻ A5 ≻ A3 ≻ A2 ≻ A4

I-QULCOAsin( π x)

A3 ≻ A1 ≻ A2 ≻ A5 ≻ A4

I-ULCOG

A3 ≻ A1 ≻ A5 ≻ A2 ≻ A4

I-QULCOAcos( π x)

A1 ≻ A5 ≻ A3 ≻ A2 ≻ A4

I-ULCOQA

A1 ≻ A5 ≻ A3 ≻ A2 ≻ A4

I-QULCOA2x

A5 ≻ A1 ≻ A2 ≻ A4 ≻ A3

2

2

From the results we can see that different aggregation operators can result different best alternatives since each aggregation operator focuses on one point of view. In the real decision making process, the decision maker can choose the corresponding aggregation operator according to his own interests and tastes.


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4. Induced Quasi-Arithmetic Aggregation Operator Based on the Dempster-Shafer Belief Structure


The D-S theory of evidence was introduced by Dempster30,31 and Shafer.32 It is an useful tool to deal with the problem that has some uncertainty in our knowledge. It can provide an unifying framework for representing various types of uncertainties including the situations of risk and ignorance.33 4.1. The induced QULOWA operator based on the Dempster-Shafer belief structure The concept of the Dempster-Shafer belief structure is given as follows: Definition 4.1. A Dempster-Shafer belief structure defined on a space X = {x1 , x2 , . . . , xn } consists of a collection of r non-null subsets of X, Bj (j = 1, 2, . . . , r) called focal elements, and a mapping p called the basic probability assignment, defined as p : 2X → [0, 1] such that: (i) p(Bj ) ∈ [0, 1]; Pr (ii) j=1 p(Bj ) = 1;

(iii) p(A) = 0, ∀A 6= Bj . As described above, some traditional cases of the uncertainty can be represented by the Dempster-Shafer belief structure. If the belief structure consists of n focal elements such that Bj = {xj }, then we are in the situation that making decision under risk environment as p(Bj ) = pj = prob{xj }. If the belief structure consists of only one focal element which comprise all the states of nature (B1 = {B1 , B2 , . . . , Bn }), we have to make decision under ignorance environment. Definition 4.2. Let X = {x1 , x2 , . . . , xn } be a fixed set, {, , . . . , } be a collection of 2-tuples on X, where ui (i = 1, 2, . . . , n) are the order-inducing variables and s˜i (i = 1, 2, . . . , n) are uncertain linguistic arguments, and M = (Mk |Mk = {|xi ∈ Bk , i = 1, 2, . . . , n}, k = 1, 2, . . . , r) = ({, , . . . , } | k = 1, 2, . . . , r) be a collection of uncertain linguistic arguments with r focal elements Bk (k = 1, 2, . . . , r). A BSI˜ r → S˜ QULOWA operator of dimension r is a function BSI-QULOWA: (R × S) defined as BSI-QULOWA(M ) = g2−1

X r

k=1

X qk wkj g1 (˜ skσ(j) ) , p(Bk )g2 g1−1

(27)

j=1

where qk is the numbers of elements in Bk , (σ(1), σ(2), . . . , σ(qk )) is a permutation of (1, 2, . . . , qk ) such that ukσ(1) > ukσ(2) > · · · > ukσ(qk ) , Wk = (wk1 , wk2 , . . . , wkqk ) is Pqk the weighting vector for the kth element Bk with j=1 wkj = 1, wkj ≥ 0 and p(Bk ) is the basic probability assignment. g1 and g2 are strictly monotonic functions.


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We discuss some special cases of the BSI-QULOWA operator: (i) If Bk = {xk }, k = 1, 2, . . . , n, then M = ({}, {}, . . . , {}). The BSI-QULOWA operator reduces to the quasi-arithmetic uncertain linguistic weighted averaging (QULWA) operator. X n QULWA(M ) = g2−1 p(Bk )g2 (˜ sk ) . (28) Int. J. Unc. Fuzz. Knowl. Based Syst. 2013.21:55-77. Downloaded from www.worldscientific.com by 213.85.255.5 on 06/12/14. For personal use only.

k=1

(ii) If only one focal element is consisted in the belief structure, that is B1 = X = {x1 , x2 , . . . , xn }, then M = {, , . . . , } and the BSI-QULOWA operator reduces to the induced quasi-arithmetic uncertain linguistic ordered weighted averaging (I-QULOWA) operator. X n −1 I-QULOWA(M ) = g2 wj g2 (˜ sσ(j) ) , (29) j=1

where s˜σ(j) is the value of s˜i in with jth largest ui . If g2 (x) = x, then I-QULOWA operator reduces to the induced uncertain linguistic ordered weighted averaging (I-ULOWA) operator: I-ULOWA(M ) =

n X

wj s˜σ(j) .

(30)

j=1

(iii) If g1 (x) = g2 (x) for all x, the BSI-QULOWA operator reduces to X qk r X −1 BSI-QULOWA(M ) = g p(Bk ) wkj g (˜ skσ(j) ) .

(31)

j=1

k=1

Especially, if g1 (x) = g2 (x) = x, the BSI-QULOWA operator reduces to the BSI-ULOWA operator qk r X X BSI-ULOWA(M ) = p(Bk ) (wkj s˜kσ(j) ) . (32) j=1

k=1

(iv) If g1 (x) = xλ1 , g2 (x) = xλ2 , λ1 > 0, λ2 > 0, the BSI-QULOWA operator reduces to X X λ2 /λ1 1/λ2 qk r 1 BSI-GULOWA(M ) = p(Bk ) wkj s˜λkσ(j) . (33) k=1

j=1

If g1 (x) = g2 (x) = xλ , the BSI-QULOWA operator reduces to the following BSI-QULOWA(M ) =

r X k=1

p(Bk )

qk X j=1

wkj s˜λkσ(j)

1/λ

.

(34)


S0218488513500049


67

(v) If g1 (x) → x0 , g2 (x) = xλ2 , λ2 > 0, we get the BSI-QULOWA(M ) =

X r k=1

Y λ2 1/λ2 qk wkj p(Bk ) s˜kσ(j) .

(35)

j=1


If only one focal element consists in the belief structure, that is B1 = X = {x1 , . . . , xn }, p(X) = 1, then the BSI-QULOWA operator reduces to the induced uncertain linguistic ordered weighted geometric (I-ULOWG) operator. I-ULOWG(M ) =

n Y

w

j s˜σ(j) .

(36)

j=1

If g1 (x) → x0 , g2 (x) = x, we get qk r X Y wkj BSI-QULOWA(M ) = p(Bk ) (˜ skσ(j) ) .

(37)

j=1

k=1

(vi) If g1 (x) = xλ1 , λ1 > 0, g2 (x) → x0 , the BSI-QULOWA operator reduces to qk r X Y

BSI-QULOWA(M ) =

1 wkj s˜λkσ(j)

j=1

k=1

1/λ1 p(Bk )

.

(38)

If g1 (x) = x, g2 (x) → x0 , the BSI-QULOWA operator reduces to BSI-QULOWA(M ) =

qk r X Y

wkj s˜kσ(j)

j=1

k=1

p(Bk )

.

(39)

(vii) If g1 (x) → x0 , g2 (x) → x0 , we get BSI-QULOWA(M ) =

qk r Y Y

k=1

j=1

w

kj s˜kσ(j)

p(Bk )

.

(40)

(viii) If g1 (x) = xλ , g2 (x) = sin( π2 x), then 2 BSI-QULOWA(M ) = arcsin π

X r

k=1

X qk 1/λ π λ p(Bk ) sin wkj s˜kσ(j) , (41) 2 j=1

If λ → 0, then we get 2 BSI-QULOWA(M ) = arcsin π

Y r k=1

X 1/λ p(Bk ) qk π λ sin wkj s˜kσ(j) , 2 j=1 (42)

here, g2 (x) = sin( π2 x) can be replaced by g2 (x) = cos( π2 x) or g2 (x) = tan( π2 x).


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W. Yang

(ix) If g1 (x) = xλ , g2 (x) = γ x , γ > 0, γ = 6 1, then X r Pqk λ 1/λ BSI-QULOWA(M ) = logγ p(Bk )γ ( j=1 wkj s˜kσ(j) ) .

(43)

k=1

If λ → 0, then we get BSI-QULOWA(M ) = logγ

Y r

(γ (

Pqk

j=1

1/λ wkj s ˜λ p(Bk ) kσ(j) )

)


k=1

.

(44)

(x) If g1 (x) = sin( π2 x), g2 (x) = xλ , then X X λ 1/λ qk r 2 π BSI-GULOWA(M ) = p(Bk ) arcsin wkj sin s˜kσ(j) . π 2 j=1 k=1

(45) If λ → 0, then we get BSI-GULOWA(M ) =

r Y 2

π

k=1

arcsin

X qk

wkj sin

j=1

π s˜kσ(j) 2

p(Bk )

. (46)

here, g1 (x) = sin( π2 x) can be replaced by g1 (x) = cos( π2 x) or g1 (x) = tan( π2 x). (xi) If g1 (x) = sin( π2 x), g2 (x) = γ x , γ > 0, γ 6= 1, then X r Pqk 2 (π arcsin( j=1 wkj sin( π s ˜kσ(j) ))) 2 BSI-GULOWA(M ) = logγ p(Bk )γ , (47) k=1

sin( π2 x)

here, g1 (x) = can be replaced by g1 (x) = cos( π2 x) or g1 (x) = tan( π2 x). (xii) If g1 (x) = γ x , γ > 0, γ 6= 1, g2 (x) = xλ , then λ 1/λ X X qk r BSI-GULOWA(M ) = p(Bk ) logγ wkj γ s˜kσ(j) . (48) j=1

k=1

If λ → 0, then we get BSI-GULOWA(M ) =

r Y

logγ

k=1

X qk j=1

wkj γ

s ˜kσ(j)

p(Bk )

.

(49)

(xiii) If g1 (x) = γ x , γ > 0, γ 6= 1, g2 (x) = sin( π2 x), then X X qk r 2 π s ˜kσ(j) BSI-GULOWA(M ) = arcsin p(Bk ) sin logγ wkj γ . π 2 j=1 k=1

(50) (xiv) If g1 (x) = γ1x , g2 (x) = γ2x , γ1 , γ2 > 0, γ1 6= 1, γ2 6= 1, then X s ˜kσ(j) r Pqk logγ1 j=1 wkj γ1 BSI-QULOWA(M ) = logγ2 p(Bk )γ2 . k=1

(51)


S0218488513500049


69

4.2. Decision making based on the BSI-QULOWA operator


Consider a decision making problem. In this problem, {A1 , A2 , . . . , Am } is the set of the alternatives, C = {C1 , C2 , . . . , Cn } is the set of the states of nature, and s˜ij is the payoff value if the alternative Ai is selected under the state Cj . p is the Pr belief structure and {B1 , B2 , . . . , Br } are the focal elements with k=1 p(Bk ) = 1. In order to select the best alternative, we design the following new decision making method: Algorithm 2 Step 1. Calcualte the induced decision matrix U = (uij )m×n to represent the attitudinal character of the decision maker (using the method as that in Step 1 of Algorithm 1). Step 2. Determine the belief structure p and the focal elements Mik = (1) (1) (2) (2) { | Cj ∈ Bk , j = 1, 2, . . . , t} = {, , . . . , (qk ) (qk ) }. (1)

(2)

(q )

Step 3. Determine the weighting vector Wqk = (wqk , wqk , . . . , wqkk ) by utilizing one of the existing methods to aggregate arguments in Mik . Step 4. Calculate the aggregate payoff Vik by the I-QULOWA operator for the focal element Bk as follows X qk (σ(j)) −1 (j) wqk g1 (˜ sik ) , (52) Vik = g1 j=1

(σ(j)) s˜ik

(l) s˜ik

(l)

(l)

(l)

where is the value of the pair with the jth largest uik . We can use different g1 (x) to aggregate the payoff values, such as the I-ULOWA operator, the I-ULOWG operator, the I-GULOWA2 operator, the I-GULOWA5 operator, the I-GULOWA10 operator, the I-GULOWA20 operator, the I-QULOWAsin( π2 x) operator, the I-QULOWAcos( π2 x) operator, the I-QULOWAtan( π2 x) operator, the I-QULOWA2x operator, the I-QULOWA5x operator, the I-QULOWA10x operator, etc. Step 5. Utilize the BSI-QULOWA operator to aggregate the payoff values to get a collective payoff value Hi as follows X r −1 Hi = g 2 p(Bk )g2 (Vik ) , (53) j=1

we can use different g2 (x) to aggregate the payoff values into the collective payoff value. We can use the ULWA operator, the ULWG operator, the GULWA2 operator, the GULWA5 operator, the GULWA10 operator, the GULWA20 operator, the QULWAsin( π2 x) operator, the QULWAcos( π2 x) operator, the QULWAtan( π2 x) operator, the QULWA2x operator, the QULWA5x operator, the QULWA10x operator, etc. Step 6. According to the collective payoff values, we rank the alternatives and select the alternative with the largest collective payoff value.


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W. Yang


Example 2 In Example 1, if the following attributes are used to evaluate alternatives, we then rank alternatives by Algorithm 2: C1 −the product price, C2 −the freight cost, C3 −the technological and R&D support, C4 −the service performance of supplier, C5 −the performance history, C6 −the production facility and capacity, C7 −the geographical location, C8 −the risk factor. The possible values evaluating the alternatives with respect to the attributes are given as uncertain linguistic variables and given in Table 5. Table 5.

Uncertain linguistic decision matrix.

C1

C2

C3

C4

C5

C6

C7

C8

A1

[s4 , s5 ]

[s5 , s7 ]

[s3 , s4 ]

[s6 , s8 ]

[s2 , s3 ]

[s7 , s8 ]

[s3 , s4 ]

[s2 , s4 ]

A2

[s6 , s7 ]

[s7 , s8 ]

[s8 , s9 ]

[s1 , s2 ]

[s4 , s5 ]

[s3 , s5 ]

[s5 , s6 ]

[s3 , s4 ]

A3 A4

[s2 , s4 ] [s5 , s7 ]

[s4 , s5 ] [s3 , s5 ]

[s5 , s6 ] [s4 , s6 ]

[s7 , s9 ] [s4 , s5 ]

[s5 , s6 ] [s6 , s8 ]

[s4 , s6 ] [s6 , s7 ]

[s4 , s5 ] [s3 , s4 ]

[s3 , s5 ] [s7 , s8 ]

A5

[s7 , s8 ]

[s6 , s7 ]

[s2 , s3 ]

[s4 , s6 ]

[s5 , s7 ]

[s4 , s5 ]

[s1 , s2 ]

[s8 , s9 ]

The experts analyze the problem and obtain the following probabilistic information represented by the following belief function p: p(B1 ) = p({C1 , C3 , C4 , C7 }) = 0.4, p(B2 ) = p({C1 , C5 , C6 }) = 0.3, p(B3 ) = p({C2 , C5 , C8 }) = 0.3. Step 1. Calculate the inducing matrix U = (uij )5×8 as in Table 6. Table 6.

Inducing variables.

C1

C2

C3

C4

C5

C6

C7

C8

A1

18

24

22

17

29

13

15

21

A2 A3

19 15

27 17

16 23

22 25

13 14

29 19

24 21

20 26

A4 A5

24 14

18 23

25 17

19 28

15 19

21 16

27 12

17 20

(1)

(1)

Step 2. Determine Mik = { | Cj ∈ Bk , j = 1, 2, . . . , t} = {, (2) (2) (q ) (q ) , . . . , } as follows: A1 : M11 = {, , , }, M12 = {, , }, M13 = {, , }. A2 : M21 = {, , , }, M22 = {, , }, M23 = {, , }.


S0218488513500049



71

A3 : M31 = {, , , }, M32 = {, , }, M33 = {, , }. A4 : M41 = {, , , }, M42 = {, , }, M43 = {, , }. A5 : M51 = {, , , }, M52 = {, , }, M53 = {, , }. Step 3. Utilize the existing method to determine the weight vector as: W3 = (0.3, 0.4, 0.3), W4 = (0.2, 0.3, 0.3, 0.2) . Step 4. Calculate the aggregated payoff Vik for Ai by using the I-ULOWA operator, the I-ULOWG operator, the I-GULOWA2 operator, the I-GULOWA5 operator, the I-GULOWA10 operator, the I-GULOWA20 operator, the I-QULOWAsin( π2 x) operator, the I-QULOWAcos( π2 x) operator, the I-QULOWA2x operator, the I-QULOWA5x operator, respectively. The results are shown in Tables 7 and 8. Here, we can use other operators, such as the I-GULOWA3 operator, the I-GULOWA30 operator, the I-QULOWAtan( π2 x) operator, the I-QULOWA3x operator, the I-QULOWA10x operator, the I-QULOWA20x operator, etc. Table 7. I-ULOWA

I-ULOWG

Aggregated payoff for all Vik . I-GULOWA2

I-GULOWA5

I-GULOWA10

V11

[s4.2 , s5.5 ]

[s4.0264 , s5.2655 ]

[s4.3818 , s5.7533 ]

[s5.3292 , s7.0999 ]

[s5.3292 , s7.0999 ]

V12 V13

[s4.3 , s5.3 ] [s3.2 , s4.9 ]

[s3.8429 , s4.9391 ] [s2.8854 , s4.5898 ]

[s4.7223 , s5.6480 ] [s3.5214 , s5.2058 ]

[s6.2090 , s7.1011 ] [s4.5623 , s6.3890 ]

[s6.2090 , s7.1011 ] [s4.5623 , s6.3890 ]

V21

[s4.7 , s5.7 ]

[s3.5798 , s4.9014 ]

[s5.3759 , s6.2690 ]

[s6.8719 , s7.7623 ]

[s6.8719 , s7.7623 ]

V22 V23

[s4.5 , s5.8 ] [s4.5 , s5.5 ]

[s4.3153 , s5.7203 ] [s4.2169 , s5.2655 ]

[s4.6797 , s5.8822 ] [s4.8062 , s5.7533 ]

[s5.4821 , s6.4195 ] [s6.2084 , s7.0999 ]

[s5.4821 , s6.4195 ] [s6.2084 , s7.0999 ]

V31 V32

[s4.5 , s5.9 ] [s3.5 , s5.2 ]

[s4.1642 , s5.6806 ] [s3.2413 , s5.1017 ]

[s4.7854 , s6.1400 ] [s3.7283 , s5.2915 ]

[s5.9928 , s7.6851 ] [s4.4783 , s5.7078 ]

[s5.9928 , s7.6851 ] [s4.4783 , s5.7078 ]

V33

[s4.0 , s5.3 ]

[s3.9233 , s5.2811 ]

[s4.0743 , s5.3198 ]

[s4.4949 , s5.4923 ]

[s4.4949 , s5.4923 ]

V41 V42

[s4.1 , s5.7 ] [s5.7 , s7.3 ]

[s4.0378 , s5.5870 ] [s5.6806 , s7.2861 ]

[s4.1593 , s5.8052 ] [s5.7184 , s7.3144 ]

[s4.5080 , s6.3407 ] [s5.8286 , s7.4403 ]

[s4.5080 , s6.3407 ] [s5.8286 , s7.4403 ]

V43

[s5.5 , s7.1 ]

[s5.1835 , s6.9479 ]

[s5.7533 , s7.2319 ]

[s6.4830 , s7.7227 ]

[s6.4830 , s7.7227 ]

V51 V52

[s3.7 , s4.9 ] [s5.2 , s6.5 ]

[s2.9124 , s4.2648 ] [s5.0588 , s6.3686 ]

[s4.3932 , s5.4681 ] [s5.3479 , s6.6257 ]

[s6.2075 , s7.1188 ] [s6.2301 , s7.2671 ]

[s6.2075 , s7.1188 ] [s6.2301 , s7.7671 ]

V53

[s6.5 , s7.8 ]

[s6.3734 , s7.7403 ]

[s6.6257 , s7.8613 ]

[s7.3346 , s8.3067 ]

[s7.3346 , s8.3067 ]

Step 5. Aggregate payoff Vik for all Ai by the ULWA operator to get Table 9, the ULWG operator to get Table 10 and the GULWA2 operator to get Table 11. Here, we can use other aggregation operators, such as the GULWA5 operator, the GULWA10


72

W. Yang Table 8. I-GULOWA20


S0218488513500049

Aggregated payoff for all Vik .

I-QULOWAsin( π x)

I-QULOWAcos( π x)

2

2

I-QULOWA2x

I-QULOWA5x

V11

[s5.6496 , s7.5327 ]

[s4.0741 , s5.1478 ]

[s4.3429 , s5.6540 ]

[s4.7655 , s6.5361 ]

[s5.2826 , s7.2582 ]

V12 V13

[s6.5910 , s7.5327 ] [s4.7761 , s6.6865 ]

[s3.9913 , s4.8752 ] [s3.0772 , s4.5948 ]

[s4.6296 , s5.5318 ] [s3.8422 , s5.1213 ]

[s5.5236 , s6.5236 ] [s3.9250 , s5.8679 ]

[s6.2587 , s7.2587 ] [s4.4381 , s6.4351 ]

V21

[s7.3832 , s8.3083 ]

[s4.1458 , s4.9263 ]

[s5.2520 , s6.1019 ]

[s6.2743 , s7.2743 ]

[s7.0409 , s8.0409 ]

V22 V23

[s5.7314 , s6.6871 ] [s6.5910 , s7.5327 ]

[s4.3570 , s5.6668 ] [s4.2434 , s5.1478 ]

[s4.6400 , s5.8501 ] [s4.7248 , s5.6540 ]

[s5.0356 , s6.1375 ] [s5.5361 , s6.5361 ]

[s5.4527 , s6.4669 ] [s6.2582 , s7.2582 ]

V31

[s6.4594 , s8.3043 ]

[s4.2765 , s5.4903 ]

[s4.7235 , s6.0261 ]

[s5.3505 , s7.0704 ]

[s6.0434 , s8.0091 ]

V32 V33

[s4.7106 , s5.8488 ] [s4.7115 , s5.6662 ]

[s3.4018 , s5.0962 ] [s3.9563 , s5.2753 ]

[s3.7010 , s5.2669 ] [s4.0618 , s5.3134 ]

[s4.0000 , s5.4854 ] [s4.2016 , s5.3785 ]

[s4.3707 , s5.6990 ] [s4.4181 , s5.4899 ]

V41 V42

[s4.7124 , s6.6061 ] [s5.8972 , s7.5892 ]

[s4.0633 , s5.5435 ] [s5.6723 , s7.2407 ]

[s4.1491 , s5.7699 ] [s5.7123 , s7.3052 ]

[s4.2630 , s6.0704 ] [s5.7655 , s7.3785 ]

[s4.4430 , s6.3816 ] [s5.8295 , s7.4899 ]

V43

[s6.6978 , s7.8586 ]

[s5.1784 , s6.6806 ]

[s5.6865 , s7.1676 ]

[s6.1859 , s7.5607 ]

[s6.5182 , s7.7805 ]

V51 V52

[s6.5910 , s7.5334 ] [s6.5914 , s7.5579 ]

[s3.3216 , s4.3397 ] [s5.0249 , s6.2100 ]

[s4.2719 , s5.3142 ] [s5.2997 , s6.5681 ]

[s5.4330 , s6.5361 ] [s5.7655 , s7.0000 ]

[s6.2555 , s7.2685 ] [s6.2826 , s7.3707 ]

V53

[s7.6427 , s8.6012 ]

[s6.2073 , s7.4540 ]

[s6.5651 , s7.8158 ]

[s7.0356 , s8.1375 ]

[s7.4527 , s8.4669 ]

Table 9.

Aggregated payoffs for all Ai by the ULWA operator.

I-ULOWA

I-ULOWG

I-GULOWA2

I-GULOWA5

I-GULOWA10

A1

[s3.9300 , s5.2600 ]

[s3.6290 , s4.9649 ]

[s4.2258 , s5.5574 ]

[s4.8775 , s6.2830 ]

[s5.3631 , s6.8870 ]

A2

[s4.5800 , s5.7500 ]

[s3.9916 , s5.3177 ]

[s4.9961 , s6.0798 ]

[s5.7085 , s6.6776 ]

[s6.2559 , s7.1607 ]

A3 A4

[s4.0500 , s5.5100 ] [s5.000 , s6.6000 ]

[s3.8151 , s5.3871 ] [s4.8744 , s6.5050 ]

[s4.2549 , s5.6394 ] [s5.1052 , s6.6859 ]

[s4.6934 , s6.0150 ] [s5.3154 , s6.8866 ]

[s5.0891 , s6.4341 ] [s5.4967 , s7.0852 ]

A5

[s4.8700 , s6.1300 ]

[s4.4952 , s5.8317 ]

[s5.2085 , s6.3998 ]

[s5.8425 , s6.9161 ]

[s6.5524 , s7.5196 ]

I-GULOWA20

I-QULOWAsin( π x)

I-QULOWAcos( π x)

I-QULOWA2x

I-QULOWA5x

A1 A2

[s5.6700 , s7.2788 ] [s6.6500 , s7.5903 ]

[s3.7502 , s4.9001 ] [s4.2385 , s5.2149 ]

[s4.1707 , s5.4576 ] [s4.9095 , s5.8920 ]

[s4.7411 , s6.3319 ] [s5.6812 , s6.7118 ]

[s5.3221 , s7.0114 ] [s6.3296 , s7.3339 ]

A3

[s5.4104 , s6.7762 ]

[s3.9180 , s5.3076 ]

[s4.2183 , s5.5845 ]

[s4.6007 , s6.0873 ]

[s5.0540 , s6.5603 ]

A4 A5

[s5.6635 , s7.2768 ] [s6.6233 , s7.5846 ]

[s4.8805 , s6.3938 ] [s4.6983 , s5.8351 ]

[s5.0793 , s6.6498 ] [s5.2682 , s6.4408 ]

[s5.2906 , s6.9099 ] [s6.0135 , s7.1557 ]

[s5.4815 , s7.1337 ] [s6.6228 , s7.6587 ]

2

2

operator, the GULWA20 operator, the QULWAsin( π2 x) operator, the QULWAcos( π2 x) operator, the QULWAtan( π2 x) operator, the QULWA2x operator, the QULWA5x operator, the QULWA10x operator, the QULWA20x operator, etc. Step 6. The alternatives are ranked according to the collective payoff values and the results are shown in Table 12. From the results we can see that different aggregation operators can result different best alternatives. Since each aggregation operator focuses on one point of


S0218488513500049



Table 10.

73

Aggregated payoffs for all Ai by the ULWG operator.

I-ULOWA

I-ULOWG

I-GULOWA2

I-GULOWA5

I-GULOWA10

A1 A2

[s3.8984 , s5.2540 ] [s4.5740 , s5.7475 ]

[s3.5927 , s4.9569 ] [s3.9769 , s5.3106 ]

[s4.1968 , s5.5524 ] [s4.9864 , s6.0713 ]

[s5.6262 , s7.2682 ] [s6.2289 , s7.1543 ]

[s5.3251 , s6.8790 ] [s6.2289 , s7.1338 ]

A3 A4

[s4.0283 , s5.5008 ] [s4.9429 , s6.5573 ]

[s3.7942 , s5.3813 ] [s4.8213 , s6.4592 ]

[s4.2309 , s5.6248 ] [s5.0439 , s6.6459 ]

[s5.0374 , s6.3551 ] [s5.4299 , s7.0577 ]

[s5.0374 , s6.3551 ] [s5.4299 , s7.0577 ]

A5

[s4.7608 , s6.0356 ]

[s4.2783 , s5.6710 ]

[s5.1566 , s6.3469 ]

[s6.2999 , s7.2894 ]

[s6.5332 , s7.5024 ]

I-GULOWA20

I-QULOWAsin( π x)

I-QULOWAcos( π x)

I-QULOWA2x

I-QULOWA5x

A1

[s5.6262 , s7.2682 ]

[s3.7222 , s4.8947 ]

[s4.1432 , s5.4528 ]

[s4.7000 , s6.3244 ]

[s5.2753 , s7.0009 ]

A2 A3

[s6.6140 , s7.5599 ] [s5.3450 , s6.6654 ]

[s4.2376 , s5.2059 ] [s3.9006 , s5.3050 ]

[s4.9016 , s5.8890 ] [s4.1958 , s5.5730 ]

[s5.6572 , s6.6943 ] [s4.5604 , s6.0359 ]

[s6.2946 , s7.3043 ] [s4.9917 , s6.4572 ]

A4

[s5.6011 , s7.2550 ]

[s4.8299 , s6.3517 ]

[s5.0197 , s6.6095 ]

[s5.2187 , s6.8746 ]

[s5.4075 , s7.1059 ]

A5

[s6.6231 , s7.5844 ]

[s4.5368 , s5.6837 ]

[s5.1844 , s6.3577 ]

[s5.9766 , s7.1253 ]

[s6.6015 , s7.6410 ]

2

Table 11.

2

Aggregated payoffs for all Ai by the GULWA2 operator.

I-ULOWA

I-ULOWG

I-GULOWA2

I-GULOWA5

I-GULOWA10

A1

[s3.9592 , s5.2659 ]

[s3.6623 , s4.9728 ]

[s4.2532 , s5.5624 ]

[s5.4010 , s6.8947 ]

[s5.4010 , s6.8947 ]

A2

[s4.5810 , s5.7525 ]

[s4.0059 , s5.3249 ]

[s5.0060 , s6.0885 ]

[s6.2824 , s7.2007 ]

[s6.2824 , s7.1824 ]

A3 A4

[s4.0712 , s5.5193 ] [s5.0543 , s6.6412 ]

[s3.8348 , s5.3929 ] [s4.9257 , s6.5494 ]

[s4.2790 , s5.6542 ] [s5.1633 , s6.7246 ]

[s5.1423 , s6.5152 ] [s5.5614 , s7.1120 ]

[s5.1423 , s6.5152 ] [s5.5614 , s7.1120 ]

A5

[s4.9750 , s6.2215 ]

[s4.6927 , s5.9833 ]

[s5.2602 , s6.4521 ]

[s6.3025 , s7.2937 ]

[s6.5724 , s7.5375 ]

I-GULOWA20

I-QULOWAsin( π x)

I-QULOWAcos( π x)

I-QULOWA2x

I-QULOWA5x

A1 A2

[s5.7134 , s7.2891 ] [s6.6852 , s7.6201 ]

[s3.7762 , s4.9055 ] [s4.2394 , s5.2241 ]

[s4.1967 , s5.4622 ] [s4.9175 , s5.8950 ]

[s4.7813 , s6.3391 ] [s5.7051 , s6.7292 ]

[s5.3687 , s7.0216 ] [s6.3638 , s7.3629 ]

A3

[s5.4777 , s6.8905 ]

[s3.9348 , s5.3101 ]

[s4.2407 , s5.5902 ]

[s4.6419 , s6.1402 ]

[s5.1182 , s6.6666 ]

A4 A5

[s5.7249 , s7.2981 ] [s6.6235 , s7.5849 ]

[s4.9296 , s6.4350 ] [s4.8526 , s5.9809 ]

[s5.1357 , s6.6887 ] [s5.3531 , s6.5241 ]

[s5.3592 , s6.9442 ] [s6.0522 , s7.1870 ]

[s5.5531 , s7.1610 ] [s6.6450 , s7.6770 ]

2

2

view, the new proposed aggregation operators can provide decision maker different aspects of the decision problem. In real decision making process, decision maker can choose the corresponding operator according to the preference of the decision maker, the actual needs and the interests. 5. Conclusion Uncertain linguistic variables are used to evaluate vague and uncertain information. Correlation and decision making under uncertainty are important aspects we should consider in the decision making process. In this paper, some new aggregation


74

W. Yang Table 12.


S0218488513500049

Ranking of the alternatives.

ULWA

ULWG

GULWA2

I-ULOWA

A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1

A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1

A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1

I-ULOWG I-GULOWA2

A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1 A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1

A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1 A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1

A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1 A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1

I-GULOWA5

A5 ≻ A2 ≻ A4 ≻ A1 ≻ A3

A5 ≻ A2 ≻ A4 ≻ A1 ≻ A3

A5 ≻ A2 ≻ A4 ≻ A1 ≻ A3

I-GULOWA10 I-GULOWA20

A5 ≻ A2 ≻ A4 ≻ A1 ≻ A3 A2 ≻ A5 ≻ A1 ≻ A4 ≻ A3

A5 ≻ A2 ≻ A4 ≻ A1 ≻ A3 A5 ≻ A2 ≻ A1 ≻ A4 ≻ A3

A5 ≻ A2 ≻ A4 ≻ A1 ≻ A3 A2 ≻ A5 ≻ A4 ≻ A1 ≻ A3

I-GULOWAsin( π x)

A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1

A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1

A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1

I-GULOWAcos( π x)

A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1

A4 ≻ A5 ≻ A2 ≻ A3 ≻ A1

A5 ≻ A4 ≻ A2 ≻ A3 ≻ A1

I-GULOWA2x

A5 ≻ A2 ≻ A4 ≻ A1 ≻ A3

A5 ≻ A2 ≻ A4 ≻ A1 ≻ A3

A5 ≻ A2 ≻ A4 ≻ A1 ≻ A3

I-GULOWA5x

A5 ≻ A2 ≻ A4 ≻ A1 ≻ A3

A5 ≻ A2 ≻ A4 ≻ A1 ≻ A3

A5 ≻ A2 ≻ A4 ≻ A1 ≻ A3

2

2

operators are introduced with the aid of quasi-arithmetic means and induced variables. At the same time, some special characteristics are also considered including correlation and decision making under uncertainty. First we consider the correlated uncertain linguistic information with I-QOWA operator. Then we consider the decision making under uncertainty with I-QOWA operator by making use of the Dempster-Shafer theory of evidence. Comparing with the existing aggregation operators, our new aggregation operators have the following characteristics: the alternative evaluation values are given as uncertain linguistic arguments which are more appropriate to depict uncertain and fuzzy information; the order of the uncertain linguistic arguments to be aggregated are determined by the induced variables, in which the preference of decision makers can be reflected; the correlation of the aggregated arguments is considered by making use of Choquet integral; the uncertainty existing in the decision making process is modeled by Dempster-Shafer belief structure; many existing linguistic aggregation operators are special cases of our new proposed operators and more new aggregation operators can be derived from the new operators. We study some of their special cases and the decision making methods based on the proposed methods are presented. Finally, the illustrative examples have been given to show the feasibility and efficiency of the developed method. The numerical results show that different aggregation operators can result in different optimal alternatives. Since each aggregation operator focuses on one point of view of the decision problem, the new aggregation operators can provide decision maker a more complete view of the decision problem. Hence, the decision maker can select the desirable alternative according to the actual needs and his own interest in the real decision making process. In further research, we will generalize the proposed methods to include multiple decision makers involved to get multiple attribute group decision making methods and we will apply the new aggregation operators in other more complicated real decision making problems.


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75

Acknowledgments


We would like to thank the anonymous referees for valuable comments that have improved the quality of the paper. This work is partly supported by Department of Education Fund of Shaanxi Province, China (No. 12JK1000), Young fund of Xi’an University of Architecture and Technology (No. QN1245), Foundation fund of Xi’an University of Architecture and Technology (No. JC1009) and the Work Foundation of Xi’an University of Architecture and Technology (No. RC1236).

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