Multi - Criteria group decision making with picture linguistic numbers

 PLOWA (α1, . . . , αn, γ1, . . . , γm)
 =PLOWAδ (PLOWAω (α1, . . . , αn) ,
 PLOWA 0 (α , α , α ) = α,¯ (31)
 ω 1 2 3 PLOWAω0 (γ1, . . . , γm)) ,
where α¯ is determined as follows.  0 0   
 ω1 ωn ω1 ωm 1 1
 where  = 2 ,..., 2 , 2 ,..., 2 and δ = 2 , 2 .
 0 0 0
 θ (α¯) = ω1 × θ (β1) + w2 × θ (β2) + w3 × θ (β3) Proposition 6 shows some special cases of the
 = 0.2 × 4 + 0.55 × 2 + 0.25 × 2 = 2.4, PLOWA operator.
 µ (α¯) Proposition 6. Let (α1, . . . , αn) be a totally
  0 0 comparable collection of PLNs, and ω = (ω1, . . . , ωn)
 = w1 × θ (β1) × µ (β1) + w2 × θ (β2) × µ (β2)
  be the weight vector, then
 w0 × θ (β ) × µ (β ) /θ (α)
 + 3 3 3 ¯ (1) If ω = (1, 0,..., 0), then PLOWAω (α1, . . . , αn) =
 0.2 × 4 × 0.2 + 0.55 × 2 × 0.2 + 0.25 × 2 × 0.2 max {αi};
 = i=1,...,n
 2.4 (2) If ω = (0,..., 0, 1), then PLOWAω (α1, . . . , αn) =
 =0.2. min {αi};
 i=1,...,n
As a similarity, η (α¯) = 0.325 and ν (α¯) = 0.408. We (3) If ω j = 1, and ωi = 0 for all i , j, then
finally get PLOWAω (α1, . . . , αn) = β j where β j is the j-th largest
 of the collection of PLNs (α1, . . . , αn).
 PLOWAω (α1, α2, α3, α4) = hs2.4, 0.2, 0.325, 0.408i .
 Definition 15. Picture Linguistic hybrid averaging
 The PLOWA can be shown to satisfy the (PLHA) operator for PLNs is a mapping PLHA : ∆n →
properties of idempotency, boundary, monotonicity, ∆ defined as
commutativity and associativity. Let (α1, . . . , αn) be
a totally comparable collection of PLNs, and ω = 0 0
 PLHAw,ω (α1, . . . , αn) = ω1β1 ⊕ · · · ⊕ ωnβn;
(ω1, . . . , ωn) be the weight vector of the PLOWA
operator, then where ω is the associated weight vector of the
 0
(1) Idempotency: If αi = α for all i = 1,..., n, then PLHA operator, and β j is the j-largest of the totally
 comparable collection of ILNs (nw1α1,..., nwnαn)
 PLOWAω (α1, . . . , αn) = α; with w = (w1,..., wn) is the weight vector of the
 collection of PLNs (α1, . . . , αn).
(2) Boundary:
 The Proposition 7 gives the explicit formula for
 min {αi} ≤ PLOWAω (α1, . . . , αn) ≤ max {αi} ;
 i=1,...,n i=1,...,n PLHA operator.
48 P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52
 Proposition 7. Let (α1, . . . , αn) be a collection of
PLNs, ω = (ω1, . . . , ωn) be the associated vector of the
PLHA operator, and w = (w1,..., wn) be the weight PLHAu, (α1, . . . , αn, γ1, . . . , γm)
vector of (α1, . . . , αn), then PLHAw,ω (α1, . . . , αn) is a 
 =PLHAv,δ PLHAw,ω (α1, . . . , αn) ,
PLNs and 

 PLHAw0,ω0 (γ1, . . . , γm) ,
 PLHAw,ω (α1, . . . , αn) =
 n
 P  0   0   0 0 
 w1 wn w1 wm
 * ω jθ β j µ β j where u = ,..., , ,..., ,  =
 j=1 2 2 2 2
 s n  , ,  ω0 ω0   
 P ω θ β0 n ω1 ωn 1 m 1 1
 j j P  0  ,..., , ,..., and v = δ = , .
 j=1 2 2 2 2 2 2
 ω jθ β j
 j=1 (32) We can prove that the PLWAA and PLOWA
 n     n    
 P 0 0 P 0 operators are two special cases of the PLHA
 ω jθ β j η β j ω jθ β j ν β j +
 j=1 j=1 operator as in Proposition 8.
 n , n ,
 P  0  P  0 
 ω jθ β j ω jθ β j  
 j=1 j=1 1 1
 Proposition 8. If ω = n ,..., n , the
where β0 is the j-largest of the totally comparable PLHA operator is reduced to the PLWAA
 j  
collection of ILNs (nw α ,..., nw α ). 1 1
 1 1 n n operator; and if w = n ,..., n , the PLHA
 operator is reduced to the PLOWA operator.
 Similar to PLWAA and PLOWA operators, the
PLHA operator is idempotent, bounded, monotonous,
commutative and associative. Let (α1, . . . , αn) be a 5. GDM under picture linguistic
collection of PLNs, ω = (ω , . . . , ω ) be the associated
 1 n assessments
vector of the PLHA operator, and w = (w1,..., wn) be
the weight vector of (α1, . . . , αn), then
(1) Idempotency: If αi = α for all i = 1,..., n, then Let us consider a hypothetical situation,
 in which A = {A1,..., Am} is the set of
 PLHAw,ω (α1, . . . , αn) = α;
 alternatives, and C = {C1,..., Cn} is the
 c
(2) Boundary: set of criteria with the weight vectorn =o
 (c1,..., cn). We assume that D = d1,..., dp
 α− PLHA (α , . . . , α ) α+;
 . w,ω 1 n . is a set of decision makers (DMs), and w =
    
 ∗ ∗ w ,..., w is the weight vector of DMs.
(3) Monotonicity: Let α1, . . . , αn be a collection of 1 p
 ∗
PLNs such that αi . αi for all i = 1,..., n, then Each DM dk presents the characteristic of
  ∗ ∗
 the alternative Ai with respect to the criteria
 PLHAw,ω α1, . . . , αn . PLHAw,ω (α1, . . . , αn) ;  
 (k)  
 C j by the PLN αi j = sθ α(k) , µα(k) , ηα(k) , να(k)
(4) Commutativity: i j i j i j i j
 (i = 1,..., m, j = 1,..., n, k = 1,..., p). The
  (k)
  
 decision matrix Rk is given by Rk = α
PLHAw,ω (α1, . . . , αn) = PLHAw,ω ασ(1), . . . , ασ(n) , i j m×n
 (k = 1,..., p). The alternatives will be ranked
where σ is any permutation on the set by the following algorithm.
 0  
 (k)
{1,..., n} and w = wσ(1),..., wσ(n) .
 Step 1. Derive the overall values αi of the
(5) Associativity: Consider an added alternatives Ai, given by the DM dk:
collection of PLNs (γ , . . . , γ ) with the
 1 m   
associated weight vector w0 = w0 ,..., w0 (k) (k) (k)
 1 m αi = PLWAAc αi1 , . . . , αin , (33)
 0
such that nw1α1 ≥ · · · ≥ nwnαn ≥ mw1γ1 ≥
 0
· · · ≥ mwmγm. We have for i = 1,..., m, and k = 1,..., p.
 P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52 49
 Table 1. Decision matrix R1
 C1 C2 C3
 A1 hs4, 0.6, 0.1, 0.2i hs4, 0.4, 0.2, 0.2i hs5, 0.2, 0.3, 0.5i
 A2 hs5, 0.7, 0.2, 0.1i hs4, 0.4, 0.1, 0.4i hs4, 0.5, 0.2, 0.3i
 A3 hs5, 0.3, 0.1, 0.4i hs5, 0.4, 0.3, 0.3i hs6, 0.7, 0.1, 0.2i
 A4 hs4, 0.6, 0.1, 0.2i hs4, 0.6, 0.1, 0.2i hs5, 0.3, 0.1, 0.5i
 Step 2. Derive the collective overall values Ai (i = 1, 2, 3, 4) by the PLHA operator with
αi by aggregating the individual overall values associated weight vector ω = (0.2, 0.5, 0.3).
 (1) (p)
αi , . . . , αi :
 α1 = hs4.40, 0.3965, 0.2045, 0.3438i ,
  
 (1) (p) h i
 αi = PLHAw,ω αi , . . . , αi , (34) α2 = s4.57, 0.3481, 0.1428, 0.4040 ,
 α = hs , 0.3628, 0.1666, 0.4050i ,
   3 5.32
where ω = ω1, . . . , ωp is the weight vector α4 = hs5.16, 0.4098, 0.1510, 0.3948i .
of the PLHA operator (i = 1,..., m).
 Step 3. Calculate the scores h (αi), first Step 3. By eq. (12),
accuracies H1 (αi) and second accuracies
 h (α1) = 0.2318, h (α2) = −0.2556
H2 (αi) (i = 1,..., m), rank the alternatives
 −
by using Definition 9 (the alternative Ai1 is h (α3) = 0.2246, h (α4) = 0.078.
called to be better than the alternative Ai ,
 2 By Definition 9,
denoted by Ai1 > Ai2 , iff αi1 > αi2 , for all
i1, i2 = 1,..., m).
 h (α1) > h (α4) > h (α3) > h (α2)
6. An illutrative example then A1 > A4 > A3 > A2.
 This situation concerns four alternative 7. Conclusion
enterprises, which will be chosen by
three DMs whose weight vector is w = In this paper, motivated by picture fuzzy
(0.3, 0.4, 0.3). The enterprises will be sets and linguistic approaches, the notion
considered under three criteria C1, C2 and C3. of picture linguistic numbers are first
Assume that the weight vector of the criteria defined. We propose the score, first accuracy
is c = (0.37, 0.35, 0.28). Three decision and second accuracy of picture linguistic
matrices are listed in Tabs. 1, 2 and 3. numbers, and propose a simple approach
 Step 1. Using explicit form of the PLWAA for the comparison between two picture
operation given in Eq. 23, we obtain overall linguistic numbers. Simultaneously, the
 (k)
values αi of the alternatives Ai given by the operation laws for picture linguistic numbers
DMs dk (i = 1, 2, 3, 4 and k = 1, 2, 3) as are given and the accompanied properties are
in Tab. 4. studied. Further, some aggregation operators
 Step 2. Aggregate all the individual overall are developed: picture linguistic arithmetic
 (1) (2) (3)
values αi , αi and αi of the alternatives averaging, picture linguistic weighted
50 P.H. Phong, B.C. Cuong / VNU Journal of Science: Comp. Science & Com. Eng., Vol. 32, No. 3 (2016) 39–52
 Table 2. Decision matrix R2
 C1 C2 C3
 A1 hs4, 0.7, 0.1, 0.2i hs6, 0.2, 0.2, 0.5i hs4, 0.7, 0.2, 0.1i
 A2 hs3, 0.2, 0.2, 0.6i hs5, 0.5, 0.1, 0.2i hs5, 0.3, 0.1, 0.4i
 A3 hs4, 0.2, 0.1, 0.5i hs7, 0.2, 0.2, 0.6i hs5, 0.1, 0.2, 0.6i
 A4 hs5, 0.7, 0.2, 0.1i hs5, 0.2, 0.1, 0.7i hs4, 0.6, 0.1, 0.2i
 Table 3. Decision matrix R3
 C1 C2 C3
 A1 hs4, 0.6, 0.3, 0.1i hs6, 0.2, 0.3, 0.5i hs5, 0.2, 0.1, 0.7i
 A2 hs3, 0.2, 0.2, 0.5i hs5, 0.2, 0.1, 0.6i hs6, 0.2, 0.2, 0.6i
 A3 hs5, 0.3, 0.2, 0.5i hs7, 0.8, 0.1, 0.1i hs5, 0.2, 0.2, 0.5i
 A4 hs3, 0.7, 0.1, 0.2i hs5, 0.2, 0.2, 0.5i hs6, 0.3, 0.1, 0.6i
 (k)
 Table 4. Overall values αi of the alternatives Ai given by the DMs dk (i = 1, 2, 3, 4; k = 1, 2, 3)
 d1 d2 d3
 A1 hs4.28, 0.4037, 0.1981, 0.2981i hs4.70, 0.4766, 0.1685, 0.3102i hs4.98, 0.3189, 0.2438, 0.4373i
 A2 hs4.37, 0.5526, 0.1680, 0.2474i hs4.26, 0.3561, 0.1261, 0.3700i hs4.54, 0.2000, 0.1615, 0.5756i
 A3 hs5.28, 0.4604, 0.1663, 0.3032i hs5.33, 0.1737, 0.1722, 0.5722i hs5.70, 0.4904, 0.1570, 0.3281i
 A4 hs4.28, 0.5019, 0.1000, 0.2981i hs5.28, 0.4070, 0.1682, 0.3917i hs4.54, 0.3593, 0.1385, 0.4637i
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