Multicriteria Group Decision Making in a Linguistic Environment
Multicriteria Group Decision Making in a Linguistic Environment Chunqiao Tan1, Hepu Deng2, and Junyan Liu3 1
School of Business, Central South University, Changsha 410083, China [email protected] 2 School of Business Information Technology and Logistics, RMIT University GPO Box 2476 V, Melbourne 3001 Victoria, Australia [email protected] 3 Armored Force Engineering Institute, Beijing 100072, China [email protected]
Abstract. This paper presents a method for solving the multicriteria group decision making problem in an uncertain linguistic environment. To ensure an effective aggregation of the uncertain linguistic variables in multicriteria group decision making for adequately modeling the uncertainty and imprecision of the decision making process, an uncertain linguistic Choquet integral operator is proposed. A method based on the proposed linguistic Choquet integral operator is developed. An application of the developed method for evaluating the university faculty on promotion is given that shows the developed method is effective for solving the multicriteria group decision making problem in a linguistic environment. Keywords: Aggregation operators, Multicriteria group decision making, Choquet integral, Linguistic variables.
1 Introduction Decision making is the process of selecting the best alternative from all the available alternatives in a given situation. With the increasing complexity of the socio-economic environment, several decision makers are usually involved in considering all the relevant aspects of a decision making problem so that multicriteria group decision making problems are often present. Uncertainty and imprecision are always present in multicriteria group decision making. This is because crisp data are usually inadequate to model the real-life phenomena in real-life situations. Furthermore, decision makers (DMs) may not possess a sufficient level of knowledge of the problem so that they may not be unable to discriminate explicitly the degree to which one alternative is better than another. As a result, linguistic assessments represented in the form of linguistic variables [1] are often used for adequately modeling the uncertainty and imprecision in the human decision making process. However, in many situations, the DMs are willing or able to provide only uncertain linguistic information because of time pressure, lack of knowledge, or data, and their limited expertise related to the problem domain. To adequately aggregate such uncertain linguistic information, uncertain linguistic H. Deng et al. (Eds.): AICI 2011, Part I, LNAI 7002, pp. 508–516, 2011. © Springer-Verlag Berlin Heidelberg 2011
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aggregation operators are proposed [2, 3]. Such operators, however, are used in multicriteria group decision making under the assumption of the mutual preferential independence between DMs in the linguistic evaluation process. But many real-world problems often have an interdependent property between the preferences of DMs in the decision making process. The Choquet integral [4] as an extension of the additive aggregation operators is widely used to minic the human decision process. The popularity of the Choquet integral is that it coincides with the Lebesgue integral in which the measure is additive. The Choquet integral is able to perform aggregation when the mutual preferential independence between DMs is violated [4-6]. In this paper, we develop a new decision making method for multi-criteria group decision making under uncertain linguistic environment by means of uncertain linguistic Choquet integral operator. In what follows, the interval number and Choquet integral are reviewed first in Section 2. In Section 3 the uncertain linguistic Choquet integral operator are proposed, leading to the development of a method for solving the multicriteria group decision making problem with uncertain linguistic information in Section 4. In Section 5, an example is presented for illustrating the applicability of the method for solving the multicriteria group decision making problem in a linguistic environment.
2 Preliminaries: Interval Numbers and Choquet Integral Given r−, r+ ∈R+ and r− ≤ r+, the closed interval [r−, r+] defines an interval number r = [r−, r+] = {r| r− ≤ r ≤ r+ }. Suppose that I(R+) = { r : [r−, r+] ⊂ R+} is the set of interval numbers. Let a = [a−, a+] and b = [b−, b+] be any two interval numbers, the following basic operations are valid [7]. (a) a + b = [a− + b−, a+ + b+], (b) a × b = [a−b−, a+b+], (c) λa = [λa−, λa+] for λ > 0. To facilitate the comparison of two interval numbers, a possibility-based measure is developed in the following for comparing each pair of interval-value variables. Definition 1. Let a = [a−, a+] and b = [b−, b+] be any two interval numbers, the degree of possibility of a > b is defined as
P(a > b ) =
max{0, a + − b − } + max{0, a − − b + } max{0, a − b } + max{0, a − − b + } + max{0, b + − a − } + max{0, b − − a + } +
−
Based on the definition above, there are three propositions listed as follows:
a ≥ b if P(a > b ) ≥ 0.5 , a > b if P(a > b ) > 0.5 , a = b if P(a > b ) = 0.5 . Let X be the universal set and Σ b e a σ-algebra of subsets of X. (X, Σ) is called a measurable space[8]. If X is finite, P(X), the power set of X, is usually taken as Σ. Definition 2. A fuzzy measure on a set X is a set function μ : P(X) → [0, 1], satisfying the following conditions:
(1) μ (φ) = 0, μ (X) = 1 (boundary conditions) (2) If A, B ∈ P(X) and A ⊆ B then μ (A) ≤ μ (B) (monotonicity).
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In order to determine a fuzzy measure on X = {x1, x2,…, xn}, we generally need to find 2n−2 values. To reduce the complexity of a fuzzy measure, Sugeno [9] introduces λ-fuzzy measures g which satisfies the following additional property: g(A∪B) = g(A) + g(B) + λg(A) g(B)
(1)
where −1 < λ < ∞ for all A, B ∈ P(X) and A ∩ B = φ. If X = {x1,…, xn}, ni = 1 xi = X . The λ-fuzzy measure g satisfies following Eq.(2). 1 n ( [1 + λg ( xi )] − 1) if λ ≠ 0, n λ i =1 (2) g ( X ) = g ( xi ) = n i =1 if λ = 0, g ( xi ) i = 1 where xi ∩ xj = φ for all i, j= 1, 2, …, n and i≠ j. It can be noted that g(xi) for a subset with a single element xi is called a fuzzy density, and can be denoted as gi = g(xi). Especially for every subset A ∈ P(X), we have
∏
1 ( [1 + λg ( xi )] − 1) if λ ≠ 0, g ( A) = λ i∈ A g ( xi ) if λ = 0. i∈ A
∏
(3)
Based on Eq. (2), the value λ of can be uniquely determined from g(X) = 1, which is equivalent to solving the following Eq.(4).
∏
λ +1 =
n i =1
(1 + λgi ) .
(4)
As a generalization of the linear Lebesgue integral and the weighted means operator, the Choquet integral is defined as follows [6]. Definition 3. Let f be a positive real-valued function from X to R+, and μ be a fuzzy measure on X. The Choquet integral of f with respect to μ is defined as
Cμ ( f ) =
fdμ =
∞
μ (Fα )dα
(5)
0
where Fα = {x | f (x) ≥ α}(α ∈ [0, +∞)). If X = (x1, …, xn), without loss of generality we assume that subscript (⋅) indicates a permutation on X such that f(x(1)) ≤ f(x(2)) ≤ … ≤ f(x(n)), Then the discrete value of the Choquet integral is obtained Cμ ( f ) =
fdμ =
n
f (x
where A(i) = {x(i), x(i+1), …, x(n)}, A(n+1) = φ.
i =1
(i ) )[ μ ( A(i ) )
− μ ( A(i +1) )] ,
(6)
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Definition 4. An interval-valued function f : X →I(R+) is measurable if both f −(x)
= [ f ( x)]l , the left end point of interval f (x) , and f +(x) = [ f ( x)]r , the right end point of interval f (x) , are measurable function of x. Theorem 1. Let f : X →I(R+) be a measurable interval-valued function on X and μ be
a fuzzy measure on Σ. Then the Choquet integral of f with respect to μ is Cμ ( f ) =
fdμ = [ f
−
dμ ,
f
+
dμ ] ,
(7)
where f −(x) = [ f ( x)]l , the left end point of interval f (x) , and f +(x) = [ f ( x)]r , the right end point of interval f (x) ∀x∈X. According to Eq. (6) and Theorem 1, we can easily obtain the following conclusion. Proposition 1. Let f : X →I(R+) be a measurable interval-valued function on X and μ
be a fuzzy measure on P(X). If X = (x1, x2, …, xn), then the discrete Choquet integral of f with respect to μ can be expressed by n
Cμ ( f ) = fdμ = [
f − ( x(i ) )(μ ( A(i ) ) − μ ( A(i+1) )),
i=1
n
=
n
f
−
( x(i ) )(μ( A(i) ) − μ( A(i+1) ))]
i =1
f (x
(i) )(μ( A(i) ) − μ( A(i+1) ))
(8)
i=1
where subscript (⋅) indicates a permutation on X such that f ( x(1) ) ≤ ≤ f ( x( n ) ) ,
f ( x(i ) ) = [ f − ( x( i ) ), f + ( x(i ) )] , and A(i) = {x(i), x(i+1), …, x(n)}, A(n+1) = φ.
3 A Linguistic Choquet Integral Operator The linguistic approach is an approximate technique, which represents qualitative aspects as linguistic values by means of linguistic variables. Let us consider a finite and totally ordered discrete linguistic label set S = {sα | α = 0, …, t}, where sα represents a linguistic variable. To preserve all the given information, we extend the ~ discrete term set S to a continuous term set S = { sα | s0 < sα ≤ st, α ∈ [0, t]}. If sα ∈ S, then we call sα the original linguistic term, otherwise, we call sα the virtual term. In general, the decision maker uses the original linguistic terms in real decision making problems. In the real world, many decision making processes take place in an uncertain environment where the linguistic preference information provided by the experts does not take the form of precise linguistic variables, but value ranges can be obtained due to the experts’ vague knowledge about the preference degrees of one alternative over another. In the following, we give the definition of uncertain linguistic variable [3].
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~ Definition 5. Let s = [sα, sβ], where sα, sβ ∈ S , sα and sβ are the lower and the upper limits, respectively, we then call s the uncertain linguistic variable. Let S be the set of all uncertain linguistic variables. Consider any three uncertain linguistic variables s = [sα, sβ], s1 = [ sα1 , sβ1 ] , s2 = [ sα 2 , sβ 2 ] , and μ, μ1, μ2 ∈[0, 1], then their operational laws are defined as: (a) s1 ⊕ s2 = s2 ⊕ s1 = [ sα1 ⊕ sα 2 , s β1 ⊕ sβ 2 ] = [ sα1 +α 2 , sβ1 + β 2 ] ; (b) μs = [ μsα , μs β ] = [ s μα , s μβ ] , μ ( s1 ⊕ s2 ) = μs1 ⊕ μs2 , ( μ1 + μ 2 ) s = μ1s ⊕ μ 2 s . Definition 6. Let s1 = [ sα1 , sβ1 ] and s2 = [ sα 2 , sβ 2 ] be two uncertain linguistic
variables, then the degree of possibility of s1 > s2 is defined as max{0, β1 − α 2 } + max{0, α1 − β 2 } (9) max{0, β1 − α 2} + max{0, α1 − β 2 } + max{0, β 2 − α1} + max{0, α 2 − β1}
P( s1 > s2 ) =
we say s1 ≥ s2 if P( s1 > s 2 ) ≥ 0.5 ; s1 > s2 if P( s1 > s2 ) > 0.5 ; s1 = s2 if P( s1 > s2 ) = 0.5 . Based on the operational laws on uncertain linguistic variables, in the following we propose a new uncertain linguistic aggregation operator. Definition 7. Let ULCμ: S n → S , if n
ULC μ (s1 , s 2 , , sn ) = where α =
n
⊕
s(i ) [ μ ( A(i ) ) − μ ( A( i +1) )] = [ sα , s β ]
i = +1
(α i )[μ ( A(i ) ) − μ ( A( i +1) )] , β =
i =1
n
(β )[μ ( A
(i ) ) −
i
g ( A(i +1) )] , μ is a fuzzy
i =1
measure on ( s1 , s2 , , sn ) , subscript (⋅) indicates a permutation such that s(1) ≤ s( 2) ≤ ≤ s( n ) , si = [ sα i , sβ i ] ∈ S and A(i) ={(i), …, (n)}, A(n+1) = φ. Then ULC is called the uncertain linguistic Choquet integral operator. Proposition 2. Let g be a λ-fuzzy measure on ( s1 , s2 , , sn ) , then we have n
ULC g ( s1 , s2 ,, sn ) =
⊕
s(i ) [ g ( A(i ) ) − g ( A(i +1) )] = [ sα , s β ] .
i = +1
If λ ≠ 0, then
α=
n
(α i ) ⋅ g (i )
i =1
n
∏
[1 + λg ( j ) ] , β =
n
( β i ) ⋅ g (i )
i =1
j =i +1
n
∏[1 + λg
( j) ] .
j =i +1
If λ = 0, then
α=
n
i =1
(α i ) ⋅ g (i ) , β =
n
(β ) ⋅ g i
(i )
,
i =1
where (⋅) indicates a permutation such that s(1) ≤ s( 2) ≤ ≤ s( n ) , si = [ sα i , sβ i ] .
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4 A Multicriteria Group Decision Making Method A multi-criteria group decision making problems can be described as follows: Let E = {e1, e2, …, el} is the set of the experts involved in the decision process; A = (x1, x2, …, xn is the set of the considered alternatives; C = (c1, c2, …, cm ) is the set of the criteria used for evaluating the alternatives. In the following we utilize the uncertain linguistic Choquet integral operator to propose an approach for multicriteria group decision making with uncertain linguistic information, which involves the following steps: Step 1. Let S be a given uncertain linguistic scale, for every alternative xi (i = 1, 2, …, n) with respect to criteria cj (j = 1, 2, …, m), each expert ek (k = 1, 2, …, l) is invited to express their individual evaluation or preference value, which takes the form of uncertain linguistic variable aijk = [ sα k , sβ k ] ∈ S ( i = 1, 2, …, n; j = 1, 2, …, ij
ij
m, k = 1, 2, …, l). Then we can obtain a decision making matrix as follow:
a11k , a12k , , a1km a k , a k , , a k k 21 22 2m . R = k k k an1 , an 2 , , anm Step 2. Confirm the fuzzy density gi = g(ci) of each criteria. According to Eq.(4), parameter λ1 of criteria can be determined. Step 3. Confirm the fuzzy density gi = g(ei) of each expert. According to Eq.(4), parameter λ2 of expert can be determined. Step 4. For decision making matrix R k , according to Definition 6, we rank these arguments aijk (j = 1, …, m) in the ith line such that aik(1) ≤ ≤ aik( m ) . Utilize the ULC operator to derive the individual overall preference value aik of alternative xi, k aik = ULC g (aik1 , aik2 , , aim ) = sik = [ sα k , sβ k ] i
i
where
α ik =
m
m
(α
k i ( j ) ) ⋅ g (c( j ) )
j =1
∏[1 + λ g (c 1
( h ) )] ,
β ik =
h = j +1
m
m
(β
k i ( j ) ) ⋅ g (c( j ) )
j =1
∏[1 + λ g (c 1
( h ) )] .
h = j +1
k
Step 5. Similar to step 4, all ai (k = 1, …, l) is reordered such that ai(1) ≤ ≤ ai(l ) . Using the ULC operator aggregates all aik (k = 1, …, l) into the collective overall preference value ai of the alternative xi. ai = ULC g (ai1 , ai2 , , ail ) = [ sα i , sβ i ] , where
αi =
l
j =1
(α i( j ) ) ⋅ g (e( j ) )
l
∏
[1 + λ2 g (e( h ) )] , β i =
h = j +1
l
j =1
( β i( j ) ) ⋅ g (e( j ) )
l
∏[1 + λ g (e 2
h = j +1
( h ) )] .
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Step 6. According to Definition 6, we rank these collective overall preference values ai (i = 1, 2, …, n). According to the order of ai , we rank the alternative xi (i = 1, 2, …, n), then to select the best one(s). Step 7. End.
5 An Example In this section, a problem of evaluating university faculty for tenure and promotion adapted from Bryson and Mobolurin [10] is used to illustrate the developed approach. For evaluation of university faculty for tenure and promotion, there are three criteria used at some universities which are c1: teaching, c2: research, and c3: service. Five faculty candidates (alternatives) xi(i = 1, 2, 3, 4, 5) are to be evaluated using the term set S = {s0 = extremely poor, s1 = very poor, s2 = poor, s3 = slightly poor, s4 = fair, s5 = slightly good, s6 = good, s7 = very good, s8 = extremely good}, by four experts ek(k = 1, 2, 3, 4) under these three criteria. The decision making matrix of expert ek is constructed as follows, respectively: [ s7 , s8 ] [ s5 , s 6 ] R 1 = [ s4 , s5 ] [ s7 , s8 ] [ s7 , s8 ] [ s6 , s 7 ] [ s4 , s 6 ] 3 R = [ s7 , s8 ] [ s6 , s 7 ] [ s5 , s 6 ]
[ s5 , s 6 ] [ s6 , s 7 ] [ s7 , s8 ] [ s3 , s5 ] [ s5 , s 7 ] [ s7 , s 8 ] [ s5 , s 6 ] [ s6 , s 7 ] [ s4 , s 5 ] [ s5 , s 7 ]
[ s5 , s 7 ] [ s5 , s 6 ] [ s6 , s8 ] [ s4 , s 6 ] [ s6 , s8 ] , R 2 = [ s6 , s 7 ] [ s6 , s8 ] [ s5 , s 6 ] [ s6 , s 7 ] [ s7 , s8 ]
[ s7 , s 8 ] [ s5 , s 7 ] [ s5 , s 6 ] [ s5 , s 6 ] [ s6 , s 7 ]
[ s5 , s 6 ] [ s5 , s 6 ] [ s6 , s 7 ] [ s7 , s8 ] 4 [ s5 , s 6 ] , R = [ s6 , s8 ] [ s4 , s 6 ] [ s7 , s 8 ] [ s4 , s 7 ] [ s4 , s 6 ]
[ s4 , s 5 ] [ s6 , s8 ] [ s6 , s8 ] , [ s6 , s 7 ] [ s4 , s 7 ]
[ s5 , s 7 ] [ s6 , s 8 ] [ s6 , s 7 ] [ s6 , s 7 ] [ s7 , s8 ] . [ s6 , s 7 ] [ s5 , s 6 ] [ s5 , s 7 ] [ s4 , s 5 ] [ s6 , s 8 ]
Step 1. Suppose that g(c1) = 0.30, g(c2) = 0.40, g(c3) = 0.20. Then λ parameter of criteria can be determined: λ1 = 0.37. Suppose that g(e1) = 0.3, g(e2) = 0.3, g(e3) = 0.3, g(e4) = 0.3, then the λ parameter of expert can be determined: λ2 = − 0.40. Step 2. For the first line of decision making matrix R 1 , we use Eq.(9) to rank these arguments a11j (j = 1, 2, 3) such that [s5, s6] ≤ [s5, s7] ≤ [s7, s8]. Utilize the ULC operator to derive the individual overall preference value aik of alternative xi: k aik = ULC g (aik1 , aik2 , , aim ) = [ sα k , s β k ] , i
3
3
1 1 1 , a12 , a13 )= where a11 = ULC g (a11
a
1 1( j )
j =1
i
⋅ g (c j )
∏ (1 + λ g
1 ( h ) ) =[s5.6, s6.82].
h = j +1
Similarly, we have a21 = [s5.63, s6.83], a31 = [s5.66, s6.89], a41 = [s4.64, s6.12], a51 = [s5.82, s7.3];
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a12 = [s5.5, s6.49], a22 = [s4.83, s6.83], a32 = [s5.52, s6.72], a42 = [s5.52, s6.82], a52 = [s5.74, s7.23]; a13 = [s6.09, s7.08], a23 = [s4.83, s6.2], a33 = [s5.99, s6.98], a43 = [s5.24, s6.24], a53 = [s4.77, s6.63]; a14 = [s5.4, s7.03], a24 = [s6.24, s7.67], a34 = [s6.2, s7.52], a44 = [s5.03, s6.4], a54 = [s4.26, s5.99]; Step 3. We use Eq.(9) to rank these arguments a1k ( k = 1, 2, 3, 4) such that a12 ≤ a11 ≤ a14 ≤ a13 . Using the ULC operator operator aggregates all aik into the collective overall preference value ai of the alternative xi: ai = ULC g (ai1 , ai2 , ai3 , ai4 ) = [ sα i , sβ i ] , 4
4
where a1 = ULC g (a11 , a12 , a13 , a14 ) =
k =1
a1( k ) ⋅ g (ek )
∏ (1 + λ g (e )) =[s 2
h
5.61, s6.82].
h = k +1
Similarly, we have a2 = [s5.41, s6.89], a3 = [s5.82, s7], a4 = [s5.1, s6.37], a5 = [s5.19, s6.79]. Step 4. According to the overall values ai of the alternative xi (i = 1, 2, 3, 4, 5), according to Eq.(9), we can obtain that a3 > a2 > a1 > a5 > a4 .Thus the order of five alternatives is x3, x2, x1, x5, x4. Hence the best is x3.
6 Conclusion Interactive phenomena between the preferences of DMs are often present in multicriteria group decision making in a linguistic environment. Conventional linguistic additive linear operators are inadequate for effectively aggregating such preference information. To address this issue, this paper has proposed an uncertain linguistic Choquet integral operator. Based on the proposed operator, a method is developed for solving the multicriteria group decision making problem in uncertain linguistic information. The prominent characteristic of the developed method is not only that all the aggregated preference information is expressed by uncertain linguistic variables, but also the interactive phenomena among criteria and preference of DMs are considered in the aggregation process, which can avoid losing and distorting the given preference information, and makes the final results accord with the real decision problems. Acknowledgments. This work was supported by the Funds of the National Natural Science Foundation of China (No. 70801064).
References 1. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Part 1, Information Sciences 8, 199–249 (1975); Part 2, Information Sciences 8, 301–357 (1975); Part 3, Information Sciences 9, 43–80 (1975) 2. Xu, Z.S.: Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment. Information Sciences 168, 171–184 (2004)
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3. Xu, Z.S.: An approach based on the uncertain LOWG and induced uncertain LOWG operators to group decision making with uncertain multiplicative linguistic preference relations. Decision Support Systems 41, 488–499 (2006) 4. Grabisch, M., Murofushi, T., Sugeno, M.: Fuzzy Measure and Integrals. Physica-Verlag, NewYork (2000) 5. Marichal, J.L.: An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. IEEE Transactions on Fuzzy Systems 8, 800–807 (2000) 6. Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge (2009) 7. Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, London (1983) 8. Halmos, P.R.: Measure Theory. Van Nostrand, NewYork (1967) 9. Sugeno, M.: Theory of fuzzy integral and its application, Doctorial Dissertation, Tokyo Institute of Technology (1974) 10. Bryson, N., Mobolurin, A.: An action learning evaluation procedure for multiple criteria decision making problems. European Journal of Operational Research 96, 379–386 (1995)
School of Business, Central South University, Changsha 410083, China [email protected] 2 School of Business Information Technology and Logistics, RMIT University GPO Box 2476 V, Melbourne 3001 Victoria, Australia [email protected] 3 Armored Force Engineering Institute, Beijing 100072, China [email protected]
Abstract. This paper presents a method for solving the multicriteria group decision making problem in an uncertain linguistic environment. To ensure an effective aggregation of the uncertain linguistic variables in multicriteria group decision making for adequately modeling the uncertainty and imprecision of the decision making process, an uncertain linguistic Choquet integral operator is proposed. A method based on the proposed linguistic Choquet integral operator is developed. An application of the developed method for evaluating the university faculty on promotion is given that shows the developed method is effective for solving the multicriteria group decision making problem in a linguistic environment. Keywords: Aggregation operators, Multicriteria group decision making, Choquet integral, Linguistic variables.
1 Introduction Decision making is the process of selecting the best alternative from all the available alternatives in a given situation. With the increasing complexity of the socio-economic environment, several decision makers are usually involved in considering all the relevant aspects of a decision making problem so that multicriteria group decision making problems are often present. Uncertainty and imprecision are always present in multicriteria group decision making. This is because crisp data are usually inadequate to model the real-life phenomena in real-life situations. Furthermore, decision makers (DMs) may not possess a sufficient level of knowledge of the problem so that they may not be unable to discriminate explicitly the degree to which one alternative is better than another. As a result, linguistic assessments represented in the form of linguistic variables [1] are often used for adequately modeling the uncertainty and imprecision in the human decision making process. However, in many situations, the DMs are willing or able to provide only uncertain linguistic information because of time pressure, lack of knowledge, or data, and their limited expertise related to the problem domain. To adequately aggregate such uncertain linguistic information, uncertain linguistic H. Deng et al. (Eds.): AICI 2011, Part I, LNAI 7002, pp. 508–516, 2011. © Springer-Verlag Berlin Heidelberg 2011
Multicriteria Group Decision Making in a Linguistic Environment
509
aggregation operators are proposed [2, 3]. Such operators, however, are used in multicriteria group decision making under the assumption of the mutual preferential independence between DMs in the linguistic evaluation process. But many real-world problems often have an interdependent property between the preferences of DMs in the decision making process. The Choquet integral [4] as an extension of the additive aggregation operators is widely used to minic the human decision process. The popularity of the Choquet integral is that it coincides with the Lebesgue integral in which the measure is additive. The Choquet integral is able to perform aggregation when the mutual preferential independence between DMs is violated [4-6]. In this paper, we develop a new decision making method for multi-criteria group decision making under uncertain linguistic environment by means of uncertain linguistic Choquet integral operator. In what follows, the interval number and Choquet integral are reviewed first in Section 2. In Section 3 the uncertain linguistic Choquet integral operator are proposed, leading to the development of a method for solving the multicriteria group decision making problem with uncertain linguistic information in Section 4. In Section 5, an example is presented for illustrating the applicability of the method for solving the multicriteria group decision making problem in a linguistic environment.
2 Preliminaries: Interval Numbers and Choquet Integral Given r−, r+ ∈R+ and r− ≤ r+, the closed interval [r−, r+] defines an interval number r = [r−, r+] = {r| r− ≤ r ≤ r+ }. Suppose that I(R+) = { r : [r−, r+] ⊂ R+} is the set of interval numbers. Let a = [a−, a+] and b = [b−, b+] be any two interval numbers, the following basic operations are valid [7]. (a) a + b = [a− + b−, a+ + b+], (b) a × b = [a−b−, a+b+], (c) λa = [λa−, λa+] for λ > 0. To facilitate the comparison of two interval numbers, a possibility-based measure is developed in the following for comparing each pair of interval-value variables. Definition 1. Let a = [a−, a+] and b = [b−, b+] be any two interval numbers, the degree of possibility of a > b is defined as
P(a > b ) =
max{0, a + − b − } + max{0, a − − b + } max{0, a − b } + max{0, a − − b + } + max{0, b + − a − } + max{0, b − − a + } +
−
Based on the definition above, there are three propositions listed as follows:
a ≥ b if P(a > b ) ≥ 0.5 , a > b if P(a > b ) > 0.5 , a = b if P(a > b ) = 0.5 . Let X be the universal set and Σ b e a σ-algebra of subsets of X. (X, Σ) is called a measurable space[8]. If X is finite, P(X), the power set of X, is usually taken as Σ. Definition 2. A fuzzy measure on a set X is a set function μ : P(X) → [0, 1], satisfying the following conditions:
(1) μ (φ) = 0, μ (X) = 1 (boundary conditions) (2) If A, B ∈ P(X) and A ⊆ B then μ (A) ≤ μ (B) (monotonicity).
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In order to determine a fuzzy measure on X = {x1, x2,…, xn}, we generally need to find 2n−2 values. To reduce the complexity of a fuzzy measure, Sugeno [9] introduces λ-fuzzy measures g which satisfies the following additional property: g(A∪B) = g(A) + g(B) + λg(A) g(B)
(1)
where −1 < λ < ∞ for all A, B ∈ P(X) and A ∩ B = φ. If X = {x1,…, xn}, ni = 1 xi = X . The λ-fuzzy measure g satisfies following Eq.(2). 1 n ( [1 + λg ( xi )] − 1) if λ ≠ 0, n λ i =1 (2) g ( X ) = g ( xi ) = n i =1 if λ = 0, g ( xi ) i = 1 where xi ∩ xj = φ for all i, j= 1, 2, …, n and i≠ j. It can be noted that g(xi) for a subset with a single element xi is called a fuzzy density, and can be denoted as gi = g(xi). Especially for every subset A ∈ P(X), we have
∏
1 ( [1 + λg ( xi )] − 1) if λ ≠ 0, g ( A) = λ i∈ A g ( xi ) if λ = 0. i∈ A
∏
(3)
Based on Eq. (2), the value λ of can be uniquely determined from g(X) = 1, which is equivalent to solving the following Eq.(4).
∏
λ +1 =
n i =1
(1 + λgi ) .
(4)
As a generalization of the linear Lebesgue integral and the weighted means operator, the Choquet integral is defined as follows [6]. Definition 3. Let f be a positive real-valued function from X to R+, and μ be a fuzzy measure on X. The Choquet integral of f with respect to μ is defined as
Cμ ( f ) =
fdμ =
∞
μ (Fα )dα
(5)
0
where Fα = {x | f (x) ≥ α}(α ∈ [0, +∞)). If X = (x1, …, xn), without loss of generality we assume that subscript (⋅) indicates a permutation on X such that f(x(1)) ≤ f(x(2)) ≤ … ≤ f(x(n)), Then the discrete value of the Choquet integral is obtained Cμ ( f ) =
fdμ =
n
f (x
where A(i) = {x(i), x(i+1), …, x(n)}, A(n+1) = φ.
i =1
(i ) )[ μ ( A(i ) )
− μ ( A(i +1) )] ,
(6)
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Definition 4. An interval-valued function f : X →I(R+) is measurable if both f −(x)
= [ f ( x)]l , the left end point of interval f (x) , and f +(x) = [ f ( x)]r , the right end point of interval f (x) , are measurable function of x. Theorem 1. Let f : X →I(R+) be a measurable interval-valued function on X and μ be
a fuzzy measure on Σ. Then the Choquet integral of f with respect to μ is Cμ ( f ) =
fdμ = [ f
−
dμ ,
f
+
dμ ] ,
(7)
where f −(x) = [ f ( x)]l , the left end point of interval f (x) , and f +(x) = [ f ( x)]r , the right end point of interval f (x) ∀x∈X. According to Eq. (6) and Theorem 1, we can easily obtain the following conclusion. Proposition 1. Let f : X →I(R+) be a measurable interval-valued function on X and μ
be a fuzzy measure on P(X). If X = (x1, x2, …, xn), then the discrete Choquet integral of f with respect to μ can be expressed by n
Cμ ( f ) = fdμ = [
f − ( x(i ) )(μ ( A(i ) ) − μ ( A(i+1) )),
i=1
n
=
n
f
−
( x(i ) )(μ( A(i) ) − μ( A(i+1) ))]
i =1
f (x
(i) )(μ( A(i) ) − μ( A(i+1) ))
(8)
i=1
where subscript (⋅) indicates a permutation on X such that f ( x(1) ) ≤ ≤ f ( x( n ) ) ,
f ( x(i ) ) = [ f − ( x( i ) ), f + ( x(i ) )] , and A(i) = {x(i), x(i+1), …, x(n)}, A(n+1) = φ.
3 A Linguistic Choquet Integral Operator The linguistic approach is an approximate technique, which represents qualitative aspects as linguistic values by means of linguistic variables. Let us consider a finite and totally ordered discrete linguistic label set S = {sα | α = 0, …, t}, where sα represents a linguistic variable. To preserve all the given information, we extend the ~ discrete term set S to a continuous term set S = { sα | s0 < sα ≤ st, α ∈ [0, t]}. If sα ∈ S, then we call sα the original linguistic term, otherwise, we call sα the virtual term. In general, the decision maker uses the original linguistic terms in real decision making problems. In the real world, many decision making processes take place in an uncertain environment where the linguistic preference information provided by the experts does not take the form of precise linguistic variables, but value ranges can be obtained due to the experts’ vague knowledge about the preference degrees of one alternative over another. In the following, we give the definition of uncertain linguistic variable [3].
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~ Definition 5. Let s = [sα, sβ], where sα, sβ ∈ S , sα and sβ are the lower and the upper limits, respectively, we then call s the uncertain linguistic variable. Let S be the set of all uncertain linguistic variables. Consider any three uncertain linguistic variables s = [sα, sβ], s1 = [ sα1 , sβ1 ] , s2 = [ sα 2 , sβ 2 ] , and μ, μ1, μ2 ∈[0, 1], then their operational laws are defined as: (a) s1 ⊕ s2 = s2 ⊕ s1 = [ sα1 ⊕ sα 2 , s β1 ⊕ sβ 2 ] = [ sα1 +α 2 , sβ1 + β 2 ] ; (b) μs = [ μsα , μs β ] = [ s μα , s μβ ] , μ ( s1 ⊕ s2 ) = μs1 ⊕ μs2 , ( μ1 + μ 2 ) s = μ1s ⊕ μ 2 s . Definition 6. Let s1 = [ sα1 , sβ1 ] and s2 = [ sα 2 , sβ 2 ] be two uncertain linguistic
variables, then the degree of possibility of s1 > s2 is defined as max{0, β1 − α 2 } + max{0, α1 − β 2 } (9) max{0, β1 − α 2} + max{0, α1 − β 2 } + max{0, β 2 − α1} + max{0, α 2 − β1}
P( s1 > s2 ) =
we say s1 ≥ s2 if P( s1 > s 2 ) ≥ 0.5 ; s1 > s2 if P( s1 > s2 ) > 0.5 ; s1 = s2 if P( s1 > s2 ) = 0.5 . Based on the operational laws on uncertain linguistic variables, in the following we propose a new uncertain linguistic aggregation operator. Definition 7. Let ULCμ: S n → S , if n
ULC μ (s1 , s 2 , , sn ) = where α =
n
⊕
s(i ) [ μ ( A(i ) ) − μ ( A( i +1) )] = [ sα , s β ]
i = +1
(α i )[μ ( A(i ) ) − μ ( A( i +1) )] , β =
i =1
n
(β )[μ ( A
(i ) ) −
i
g ( A(i +1) )] , μ is a fuzzy
i =1
measure on ( s1 , s2 , , sn ) , subscript (⋅) indicates a permutation such that s(1) ≤ s( 2) ≤ ≤ s( n ) , si = [ sα i , sβ i ] ∈ S and A(i) ={(i), …, (n)}, A(n+1) = φ. Then ULC is called the uncertain linguistic Choquet integral operator. Proposition 2. Let g be a λ-fuzzy measure on ( s1 , s2 , , sn ) , then we have n
ULC g ( s1 , s2 ,, sn ) =
⊕
s(i ) [ g ( A(i ) ) − g ( A(i +1) )] = [ sα , s β ] .
i = +1
If λ ≠ 0, then
α=
n
(α i ) ⋅ g (i )
i =1
n
∏
[1 + λg ( j ) ] , β =
n
( β i ) ⋅ g (i )
i =1
j =i +1
n
∏[1 + λg
( j) ] .
j =i +1
If λ = 0, then
α=
n
i =1
(α i ) ⋅ g (i ) , β =
n
(β ) ⋅ g i
(i )
,
i =1
where (⋅) indicates a permutation such that s(1) ≤ s( 2) ≤ ≤ s( n ) , si = [ sα i , sβ i ] .
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4 A Multicriteria Group Decision Making Method A multi-criteria group decision making problems can be described as follows: Let E = {e1, e2, …, el} is the set of the experts involved in the decision process; A = (x1, x2, …, xn is the set of the considered alternatives; C = (c1, c2, …, cm ) is the set of the criteria used for evaluating the alternatives. In the following we utilize the uncertain linguistic Choquet integral operator to propose an approach for multicriteria group decision making with uncertain linguistic information, which involves the following steps: Step 1. Let S be a given uncertain linguistic scale, for every alternative xi (i = 1, 2, …, n) with respect to criteria cj (j = 1, 2, …, m), each expert ek (k = 1, 2, …, l) is invited to express their individual evaluation or preference value, which takes the form of uncertain linguistic variable aijk = [ sα k , sβ k ] ∈ S ( i = 1, 2, …, n; j = 1, 2, …, ij
ij
m, k = 1, 2, …, l). Then we can obtain a decision making matrix as follow:
a11k , a12k , , a1km a k , a k , , a k k 21 22 2m . R = k k k an1 , an 2 , , anm Step 2. Confirm the fuzzy density gi = g(ci) of each criteria. According to Eq.(4), parameter λ1 of criteria can be determined. Step 3. Confirm the fuzzy density gi = g(ei) of each expert. According to Eq.(4), parameter λ2 of expert can be determined. Step 4. For decision making matrix R k , according to Definition 6, we rank these arguments aijk (j = 1, …, m) in the ith line such that aik(1) ≤ ≤ aik( m ) . Utilize the ULC operator to derive the individual overall preference value aik of alternative xi, k aik = ULC g (aik1 , aik2 , , aim ) = sik = [ sα k , sβ k ] i
i
where
α ik =
m
m
(α
k i ( j ) ) ⋅ g (c( j ) )
j =1
∏[1 + λ g (c 1
( h ) )] ,
β ik =
h = j +1
m
m
(β
k i ( j ) ) ⋅ g (c( j ) )
j =1
∏[1 + λ g (c 1
( h ) )] .
h = j +1
k
Step 5. Similar to step 4, all ai (k = 1, …, l) is reordered such that ai(1) ≤ ≤ ai(l ) . Using the ULC operator aggregates all aik (k = 1, …, l) into the collective overall preference value ai of the alternative xi. ai = ULC g (ai1 , ai2 , , ail ) = [ sα i , sβ i ] , where
αi =
l
j =1
(α i( j ) ) ⋅ g (e( j ) )
l
∏
[1 + λ2 g (e( h ) )] , β i =
h = j +1
l
j =1
( β i( j ) ) ⋅ g (e( j ) )
l
∏[1 + λ g (e 2
h = j +1
( h ) )] .
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Step 6. According to Definition 6, we rank these collective overall preference values ai (i = 1, 2, …, n). According to the order of ai , we rank the alternative xi (i = 1, 2, …, n), then to select the best one(s). Step 7. End.
5 An Example In this section, a problem of evaluating university faculty for tenure and promotion adapted from Bryson and Mobolurin [10] is used to illustrate the developed approach. For evaluation of university faculty for tenure and promotion, there are three criteria used at some universities which are c1: teaching, c2: research, and c3: service. Five faculty candidates (alternatives) xi(i = 1, 2, 3, 4, 5) are to be evaluated using the term set S = {s0 = extremely poor, s1 = very poor, s2 = poor, s3 = slightly poor, s4 = fair, s5 = slightly good, s6 = good, s7 = very good, s8 = extremely good}, by four experts ek(k = 1, 2, 3, 4) under these three criteria. The decision making matrix of expert ek is constructed as follows, respectively: [ s7 , s8 ] [ s5 , s 6 ] R 1 = [ s4 , s5 ] [ s7 , s8 ] [ s7 , s8 ] [ s6 , s 7 ] [ s4 , s 6 ] 3 R = [ s7 , s8 ] [ s6 , s 7 ] [ s5 , s 6 ]
[ s5 , s 6 ] [ s6 , s 7 ] [ s7 , s8 ] [ s3 , s5 ] [ s5 , s 7 ] [ s7 , s 8 ] [ s5 , s 6 ] [ s6 , s 7 ] [ s4 , s 5 ] [ s5 , s 7 ]
[ s5 , s 7 ] [ s5 , s 6 ] [ s6 , s8 ] [ s4 , s 6 ] [ s6 , s8 ] , R 2 = [ s6 , s 7 ] [ s6 , s8 ] [ s5 , s 6 ] [ s6 , s 7 ] [ s7 , s8 ]
[ s7 , s 8 ] [ s5 , s 7 ] [ s5 , s 6 ] [ s5 , s 6 ] [ s6 , s 7 ]
[ s5 , s 6 ] [ s5 , s 6 ] [ s6 , s 7 ] [ s7 , s8 ] 4 [ s5 , s 6 ] , R = [ s6 , s8 ] [ s4 , s 6 ] [ s7 , s 8 ] [ s4 , s 7 ] [ s4 , s 6 ]
[ s4 , s 5 ] [ s6 , s8 ] [ s6 , s8 ] , [ s6 , s 7 ] [ s4 , s 7 ]
[ s5 , s 7 ] [ s6 , s 8 ] [ s6 , s 7 ] [ s6 , s 7 ] [ s7 , s8 ] . [ s6 , s 7 ] [ s5 , s 6 ] [ s5 , s 7 ] [ s4 , s 5 ] [ s6 , s 8 ]
Step 1. Suppose that g(c1) = 0.30, g(c2) = 0.40, g(c3) = 0.20. Then λ parameter of criteria can be determined: λ1 = 0.37. Suppose that g(e1) = 0.3, g(e2) = 0.3, g(e3) = 0.3, g(e4) = 0.3, then the λ parameter of expert can be determined: λ2 = − 0.40. Step 2. For the first line of decision making matrix R 1 , we use Eq.(9) to rank these arguments a11j (j = 1, 2, 3) such that [s5, s6] ≤ [s5, s7] ≤ [s7, s8]. Utilize the ULC operator to derive the individual overall preference value aik of alternative xi: k aik = ULC g (aik1 , aik2 , , aim ) = [ sα k , s β k ] , i
3
3
1 1 1 , a12 , a13 )= where a11 = ULC g (a11
a
1 1( j )
j =1
i
⋅ g (c j )
∏ (1 + λ g
1 ( h ) ) =[s5.6, s6.82].
h = j +1
Similarly, we have a21 = [s5.63, s6.83], a31 = [s5.66, s6.89], a41 = [s4.64, s6.12], a51 = [s5.82, s7.3];
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a12 = [s5.5, s6.49], a22 = [s4.83, s6.83], a32 = [s5.52, s6.72], a42 = [s5.52, s6.82], a52 = [s5.74, s7.23]; a13 = [s6.09, s7.08], a23 = [s4.83, s6.2], a33 = [s5.99, s6.98], a43 = [s5.24, s6.24], a53 = [s4.77, s6.63]; a14 = [s5.4, s7.03], a24 = [s6.24, s7.67], a34 = [s6.2, s7.52], a44 = [s5.03, s6.4], a54 = [s4.26, s5.99]; Step 3. We use Eq.(9) to rank these arguments a1k ( k = 1, 2, 3, 4) such that a12 ≤ a11 ≤ a14 ≤ a13 . Using the ULC operator operator aggregates all aik into the collective overall preference value ai of the alternative xi: ai = ULC g (ai1 , ai2 , ai3 , ai4 ) = [ sα i , sβ i ] , 4
4
where a1 = ULC g (a11 , a12 , a13 , a14 ) =
k =1
a1( k ) ⋅ g (ek )
∏ (1 + λ g (e )) =[s 2
h
5.61, s6.82].
h = k +1
Similarly, we have a2 = [s5.41, s6.89], a3 = [s5.82, s7], a4 = [s5.1, s6.37], a5 = [s5.19, s6.79]. Step 4. According to the overall values ai of the alternative xi (i = 1, 2, 3, 4, 5), according to Eq.(9), we can obtain that a3 > a2 > a1 > a5 > a4 .Thus the order of five alternatives is x3, x2, x1, x5, x4. Hence the best is x3.
6 Conclusion Interactive phenomena between the preferences of DMs are often present in multicriteria group decision making in a linguistic environment. Conventional linguistic additive linear operators are inadequate for effectively aggregating such preference information. To address this issue, this paper has proposed an uncertain linguistic Choquet integral operator. Based on the proposed operator, a method is developed for solving the multicriteria group decision making problem in uncertain linguistic information. The prominent characteristic of the developed method is not only that all the aggregated preference information is expressed by uncertain linguistic variables, but also the interactive phenomena among criteria and preference of DMs are considered in the aggregation process, which can avoid losing and distorting the given preference information, and makes the final results accord with the real decision problems. Acknowledgments. This work was supported by the Funds of the National Natural Science Foundation of China (No. 70801064).
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