TOPSIS Method for Multiple Attribute Decision ... - Semantic Scholar
Journal of Convergence Information Technology Volume 5, Number 10. December 2010
TOPSIS Method for Multiple Attribute Decision Making with Incomplete Weight Information in Linguistic Setting Jianli Wei Department of Social Sciences, Hebei University of Engineering, Handan 056038, China E-mail: [email protected] doi:10.4156/jcit.vol5. issue10.23
Abstract The aim of this paper is to investigate the multiple attribute decision making problems with linguistic information, in which the information about attribute weights is incompletely known, and the attribute values take the form of linguistic variables. We develop a new method to solve linguistic MADM with incomplete weight. In order to get the weight vector of the attribute, we establish an optimization model based on the basic ideal of traditional TOPSIS, by which the attribute weights can be determined. Based on this model, we develop a TOPSIS method to rank alternatives and to select the most desirable one(s). Finally, an example is shown to highlight the procedure of the proposed algorithm at the end of this paper.
Keywords: Multiple Attribute Decision Making, Linguistic Variables, Attribute Weight, TOPSIS 1. Introduction Decision-making is the procedure to find the best alternative among a set of feasible alternatives. Sometimes, decision-making problems considering several criteria are called multiple attribute decision making (MADM) problems. An MADM problem with m alternatives and n attributes can be expressed in matrix format as follows:
G1
G2
a11 a 21 A= Am am1
a12 a22
A1 A2
am 2
Gn a1n a2 n , W = [ w1 amn
w2
wn ]
A1 , A2 , , Am are possible alternatives among which decision makers have to choose, G1 , G2 , , Gn are criteria with which alternative performance are measured, aij is the rating of
where
alternative
Ai with respect to criterion G j , w j is the weight of criterion G j .
In the process of MADM with linguistic information, sometimes, the attribute values take the form of linguistic variables, and the information about attribute weights is incompletely known because of time pressure, lack of knowledge or data, and the expert’s limited expertise about the problem domain. Therefore, it is necessary to pay attention to this issue. Xu [1] have investigated the multiple attribute decision making problems, in which the attribute values take the form of linguistic variables, and the information about attribute weights is incompletely known. To determine the attribute weights, some simple optimization models are established. Especially, for the situations where the information about the attribute weights is completely unknown, a simple and exact formula for obtaining the attribute weights is provided. Xu [2] have developed an interactive procedure for linguistic multiple attribute decision making with incomplete weight information. Wu and Chen[3] have developed the maximizing deviation method for group multiple attribute decision making with completely unknown weight information under linguistic environment. The aim of this paper is to develop a TOPSIS method to solve linguistic MADM with incomplete weight. The remainder of this paper is set out as follows. In the next section, we introduce some basic concepts and operational laws of linguistic variables and define some useful concepts. In Section 3 we
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TOPSIS Method for Multiple Attribute Decision Making with Incomplete Weight Information in Linguistic Setting Jianli Wei
introduce the MADM problem with linguistic information, in which the information about attribute weights is incompletely known, and the attribute values take the form of linguistic variables. Then, we establish an optimization model based on the basic ideal of traditional TOPSIS, by which the attribute weights can be determined. Based on this model, we develop a TOPSIS method to rank alternatives and to select the most desirable one(s). In Section 4, an illustrative example is pointed out. In Section 5 we conclude the paper and give some remarks.
Preliminaries Let= S {= si i 0,1, , T } be a linguistic term set with odd cardinality. Any label, si represents a possible value for a linguistic variable, and it should satisfy the following characteristics [4,5]: (1) The set is ordered: si
> s j , if i > j ; (2) There is the reciprocal operator: rec ( si ) = s j such that
(
)
i+ j = t ; (3) Max operator: max si , s j = si , if
si ≥ s j ; (4) Min operator: min ( si , s j ) = si , if
si ≤ s j . For example, S can be defined as = S {= s0 extremely poor ,= s1 very poor , = s2 poor , = s3 medium, = s4 good = , s5 very good = , s6 extremely good } To preserve all the given information, we extend the discrete term set S to a continuous term set
= S
{s
a
}
s0 ≤ sa ≤ sq , a ∈ [0, q ] , whose elements also meet all the characteristics above. If
sa ∈ S , then we call sa the original linguistic term, otherwise, we call sa the virtual linguistic term, q is a large positive integer. In general, the decision maker uses the original linguistic term to evaluate attributes and alternatives, and the virtual linguistic terms can only appear in calculation [6,7]. Definition= 1: Let R ( rij ) ( rij ∈ S ) be a linguistic decision matrix, then we call m× n
{ }
r + = ( r1+ , r2+ , , rn+ ) an positive ideal point of attribute values, where rj+ = max rij r − = ( r1− , r2− , , rn− ) an negative ideal point of attribute values,
i
and
where rj− = min {rij } , j = 1, 2, , n . i
Definition 2: Let
s1 = sα and s2 = sβ be two linguistic variables, then we call d ( s1 , s2 ) =
the distance between
α −β
(1)
T
s1 and s2 [1].
2.TOPSIS Method for Linguistic MADM Problems 2.1. Traditional TOPSIS Method TOPSIS (technique for order preference by similarity to an ideal solution) method is presented in Chen and Hwang [8], with reference to Hwang and Yoo n [9]. TOPSIS is a multiple criteria method to identify solutions from a finite set of alternatives. The basic principle is that the chosen alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. The procedure of TOPSIS can be expressed in a series of steps: (1) Calculate the normalized decision matrix. The normalized value bij is calculated as
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Journal of Convergence Information Technology Volume 5, Number 10. December 2010 n
= 2, , m, j 1, 2, , n ∑ aij2 ， i 1,=
bij = aij
(2)
j =1
(2) Calculate the weighted normalized decision matrix. The weighted normalized value
cij is
calculated as , i 1,= 2, , m, j 1, 2, , n c= ij = w j bij where w j is the weight of the
j th criterion G j , and
∑
n j =1
(3)
wj = 1 .
(3) Determine the positive ideal and negative ideal solution
{( = {c ,, c } = {( min c
)( j ∈ I ) , ( max c
)} j ∈ J )}
C+ = {c1+ ,, cn+ } = max cij j ∈ I , min cij j ∈ J
(4a)
C−
(4b)
i
− 1
− n
i
ij
i
i
ij
where I is associated with benefit criteria, and J is associated with cost criteria. (4) Calculate the separation measures, using the n-dimensional Euclidean distance. The separation of each alternative from the ideal solution is given as
d i+ =
∑ (c n
j =1
− c +j ) , i = 1, , m . 2
ij
(5a)
Similarly, the separation from the negative ideal solution is given as d i− =
∑ (c n
j =1
− c −j ) , i = 1, , m . 2
ij
(5) Calculate the relative closeness to the ideal solution. The relative closeness of the alternative with respect to A
+
(5b)
Ai
is defined as Ri = d i−
(d
− i
+ d i+ ) ,
i = 1, , m .
(6)
Since d i− ≥ 0 and d i+ ≥ 0 , then, clearly, Ri ∈ [ 0,1] . (6) Rank the preference order. For ranking DMUs using this index, we can rank DMUs in decreasing order. The basic principle of the TOPSIS method is that the chosen alternative should have the “shortest distance” from the positive ideal solution and the “farthest distance” from the negative ideal solution. The TOPSIS method introduces two “reference” points, but it does not consider the relative importance of the distances from these points.
2.2. Extended Traditional TOPSIS Method to Linguistic MADM Problems The following assumptions or notations are used to represent the MADM problems with incomplete weight information in linguistic setting: (1) The alternatives are known. Let A = { A1 , A2 , , Am } be a discrete set of alternatives;
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TOPSIS Method for Multiple Attribute Decision Making with Incomplete Weight Information in Linguistic Setting Jianli Wei
(2) The attributes are known. Let G = {G1 , G2 , , Gn } be a set of attributes; (3) The information about attribute weights is incompletely = w ( w1 , w2 , , wn ) ∈ H be the weight vector of attributes, where w j ≥ 0 ,
∑
n j =1
wj = 1 ,
known. Let j = 1, 2, , n ,
H is a set of the known weight information, which can be constructed by the
following forms [10-13], for
i ≠ j : Form 1. A weak ranking: wi ≥ w j ；Form 2. A strict ranking:
wi − w j ≥ α i , α i > 0 ；Form 3. A ranking of differences: wi − w j ≥ wk − wl , for j ≠ k ≠ l ; Form 4. A ranking with multiples: wi ≥ β i w j , 0 ≤ β i ≤ 1 ； Form 5. An interval form: α i ≤ wi ≤ α i + ε i , 0 ≤ α i < α i + ε i ≤ 1 . In the following, we will extend the TOPSIS method to solve multiple attribute decision making problems with incompletely known weight information in linguistic setting. Step1. Determine the linguistic positive ideal and linguistic negative ideal solution by definition 3. Linguistic positive ideal is: r + = ( r1+ , r2+ , , rn+ ) , where rj+ = max {rij } , j = 1, 2, , n .
(7)
i
Linguistic negative ideal is: r − = ( r1− , r2− , , rn− ) ,
where rj− = min {rij } , j = 1, 2, , n .
(8)
i
Step2. Calculate the separation measures, using the n-dimensional Hamming distance. The separation of each alternative from the linguistic ideal solution is given as
di+ =
d ( rj+ , rij ) w j , i 1, , m . = ∑ n
(9)
j =1
Similarly, the separation from the linguistic negative ideal solution is given as
= di−
= d (r , r ) w , i ∑ n
j =1
ij
− j
j
1, , m .
(10)
The basic principle of the TOPSIS method is that the chosen alternative should have the “shortest distance” from the positive ideal solution and the “farthest distance” from the negative ideal solution. Obviously, for the weight vector given, the smaller
di+ and the larger di− , the better alternative Ai . +
−
But the information about attribute weights is incompletely known. So, in order to get the d i and d i , we establish the following multiple objective optimization models: n + min d d ( rj+ , rij ) w j , i 1, , m = = ∑ i j =1 n = = d ( rij , rj− ) w j , i 1, , m ( M-1) max di− ∑ j =1 subject to : w ∈ H Since each alternative is noninferior, so there exists no preference relation on the all the alternatives. Then, we may aggregate the above multiple objective optimization models with equal weights into the following single objective optimization model: m n + − = min d ∑∑ d ( rj , rij ) − d ( rij , rj ) w j ( M-2 ) =i 1 =j 1 subject to : w ∈ H
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Journal of Convergence Information Technology Volume 5, Number 10. December 2010
By solving the model ( M-2 ) , we get the optimal solution w = ( w1 , w2 , , wn ) , which can be used as the weight vector of attributes. Step 3. Utilize the weight vector w = ( w1 , w2 , , wn ) and by Eq. (4), we obtain the d i+ and d i− of the alternatives Ai ( i = 1, 2, , m ) ; Step 4. Calculate the relative closeness to the linguistic positive ideal solution. The relative closeness of the alternative
Ai with respect to A+ is defined as ci = d i−
(d
− i
+ di+ ) , i = 1, , m .
(5)
Since d i− ≥ 0 and d i+ ≥ 0 , then, clearly, ci ∈ [ 0,1] . Step 5. Rank all the alternatives Ai ( i = 1, 2, , m ) by using the relative closeness to the linguistic ideal solution ci ( i = 1, 2, , m ) and select the best one(s);
3. Illustrative Example Let us suppose there is an investment company, which wants to invest a sum of money in the best option (adapted from [14]). There is a panel with five possible alternatives to invest the money: ① A 1 is a car company; ② A 2 is a food company; ③ A 3 is a computer company; ④ A 4 is a arms company; ⑤ A 5 is a TV company. The investment company must take a decision according to the following four attributes: ① G 1 is the risk analysis; ② G 2 is the growth analysis; ③ G 3 is the social-political impact analysis; ④ G 4 is the environmental impact analysis. The five possible alternatives
Ai ( i = 1, 2, ,5 ) are to be evaluated using the linguistic term set
= S {= s0 extremely poor ,= s1 very poor , = s2 poor , = s3 medium, = s4 good = , s5 very good = , s6 extremely good } by the decision maker under the above four attributes, as listed in the following matrix.
G1 A1 s4 A2 s3 R = A3 s6 A4 s1 A5 s2
G2
G3
s5
s3
s2 s3 s4 s0
s4 s4 s3 s3
G4 s3 s6 s1 s4 s6
The information about the attribute weights is partly known as follows:
H=
{0.15 ≤ w1 ≤ 0.25, 0.2 ≤ w2 ≤ 0.35, 0.35 ≤ w3 ≤ 0.55,
}
w4 ≥ 0.3 w= 1, 2, 3,= 4, ∑ j =1 w j 1 2 , w j ≥ 0, j 4
Then, we utilize the linguistic approach developed to get the most desirable alternative(s). Step 1. Utilize the model (M-2) to establish the following single-objective programming model:
min d ( w ) = 0.5w1 − 0.5w2 + 0.1667 w3 − 0.8333w4 Subject to : w ∈ H
Step 2. Solve this model, we get the weight vector of attributes: w = ( 0.15, 0.35, 0.3950, 0.1050 )
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TOPSIS Method for Multiple Attribute Decision Making with Incomplete Weight Information in Linguistic Setting Jianli Wei
Step 3. Utilize the weight vector + i
d and d
− i
w and by Eq. (4), we calculate the separation measures
of the alternatives Ai ( i = 1, 2, 3, 4, 5 ) :
= d1+ 0.1683, = d 2+ 0.2500, = d3+ 0.2042, = d 4+ 0.2842, = d5+ 0.4575 = d1− 0.4017, = d 2− 0.3200, = d3− 0.3658, = d 4− 0.2858, = d5− 0.1125 Step 4. Calculate the relative closeness to the linguistic positive ideal solution. = c1 0.7047, = c2 0.5614, = c3 0.6418, = c4 0.5015, = c5 0.1974 Step 5. Rank all the alternatives Ai ( i = 1, 2, , 5 ) in accordance with the relative closeness to the linguistic positive ideal solution ci ( i = 1, 2, 3, 4, 5 ) : A 1 desirable alternative is
A 3
A 2
A 4
A5 , and thus the most
A1 .
4. Conclusion With respect to the MADM problems, in which the information about attribute weights is incompletely known, and the attribute values take the form of linguistic variables. In order to get the attribute weight, we establish the multiple objective optimization models based on the basic ideal of the traditional TOPSIS. Then, by linear equal weighted method, the multiple objective optimization models can be transformed into a single-objective programming model. By solving the single-objective programming model, we can get the weight information and get the relative closeness to the linguistic positive ideal solution of all the alternatives. At last, a practical example is provided to illustrate the proposed method. Theoretical analysis and the numerical results have shown that the developed approach is straightforward and has no loss of information.
5. References [1] Z. S. Xu. “A method for multiple attribute decision making with incomplete weight information in linguistic setting”, Knowledge-Based Systems, vol 21, no. 8, pp.837–841, 2008. [2] Z. S. Xu, “An interactive procedure for linguistic multiple attribute decision making with incomplete weight information”, Fuzzy Optimization and Decision Making, vol 6, no. 1, pp.17-27, 2007. [3] Z. B. Wu, Y. H. Chen, “The maximizing deviation method for group multiple attribute decision making under linguistic environment”, Fuzzy Sets and Systems, vol 158, no. 14, pp. 1608-1617, 2007. [4] F. Herrera, L. Nartinez, “A 2-tuple fuzzy linguistic representation model for comput ing with words”, IEEE Transactions on Fuzzy Systems, vol 8, no.6, pp. 746-752, 2000. [5] F. Herrera, L. Nartinez, “A model based on linguistic 2-tuples for dealing with multigranularity hierarchical linguistic contexts in multiexpert decision-making”, IEEE Transactions on Systems, Man and Cybernetics-Part B: Cybernetics, vol 31, no. 2, pp. 227-234, 2001. [6] Z. S. Xu, ”Uncertain Multiple Attribute Decision Making: Methods and Applications”, Beijing: Tsinghua University Press, 2004. [7] Z. S. Xu, “A note on linguistic hybrid arithmetic averaging operator in multiple attribute group decision making with linguistic information”, Group Decision and Negotiation, Vol 15, No. 6, pp. 593-604, 2006. [8] P. Korhonen, J. Wallenius, S. Zionts, “Solving the discrete multiple criteria problem using convex cones”, Management Science, vol 30, pp.1336-1345, 1981. [9] C.L. Hwang, K. Yoon, “Multiple Attribute Decision Making Methods and Applications”, Springer, Berlin Heidelberg, 1981. [10] P. S. Park, S. H. Kim, W. C. Yoon, “Establishing strict dominance between alternatives with special type of incomplete information”, European Journal of Operational Research, vol 96, pp. 398-406, 1996. [11] P. S. Park, S. H. Kim, “Tools for interactive multi-attribute decision making with incompletely
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Journal of Convergence Information Technology Volume 5, Number 10. December 2010
identified information”, European Journal of Operational Research,vol 98, pp. 111-123, 1997. [12] S. H. Kim, S. H. Choi, J. K. Kim, “An interactive procedure for multiple attribute group decision making with incomplete information: rangebased approach”, European Journal of Operational Research, vol 118, pp. 139-152, 1999. [13] S. H. Kim, B. S. Ahn, “Interactive group decision making procedure under incomplete information”, European Journal of Operational Research, vol 116, pp. 498-507, 1999. [14] F. Herrera, E. Herrera-Viedma, “Linguistic decision analysis: steps for solving decision problems under linguistic information”, Fuzzy Sets and systems, vol. 115, no. 10, pp. 67-82, 2000. [15] J.M. Wang, “Robust optimization analysis for multiple attribute decision making problems with imprecise information”, Annals of Operations Research, 2010. In press. [16] P.D. Liu, Y. Su, “The extended TOPSIS based on trapezoid fuzzy linguistic variables”, Journal of Convergence Information Technology, vol. 5, no. 4, pp.38-53, 2010. [17] Nabendu Chaki, Bidyut Biman Sarkar , "Virtual Data Warehouse Modeling Using Petri Nets for Distributed Decision Making ", JCIT: Journal of Convergence Information Technology, vol. 5, no. 5, pp. 8 -21, 2010. [18] G. W. Wei, Uncertain linguistic hybrid geometric mean operator and its Application to group decision making under uncertain linguistic environment, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems,vol.17, no.2, pp.251-267, 2009. [19] G.W. Wei, “A method for multiple attribute group decision making based on the ET-WG and ET-OWG operators with 2-tuple linguistic information”, Expert Systems with Applications, vol. 37, no. 12, pp. 7895-7900, 2010. [20] G. W. Wei, R. Lin, X.F. Zhao, H.J. Wang, “Models for multiple attribute group decision making with 2-tuple linguistic assessment information”, International Journal of Computational Intelligence Systems, vol.3, no.3, pp.315-324, 2010.
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TOPSIS Method for Multiple Attribute Decision Making with Incomplete Weight Information in Linguistic Setting Jianli Wei Department of Social Sciences, Hebei University of Engineering, Handan 056038, China E-mail: [email protected] doi:10.4156/jcit.vol5. issue10.23
Abstract The aim of this paper is to investigate the multiple attribute decision making problems with linguistic information, in which the information about attribute weights is incompletely known, and the attribute values take the form of linguistic variables. We develop a new method to solve linguistic MADM with incomplete weight. In order to get the weight vector of the attribute, we establish an optimization model based on the basic ideal of traditional TOPSIS, by which the attribute weights can be determined. Based on this model, we develop a TOPSIS method to rank alternatives and to select the most desirable one(s). Finally, an example is shown to highlight the procedure of the proposed algorithm at the end of this paper.
Keywords: Multiple Attribute Decision Making, Linguistic Variables, Attribute Weight, TOPSIS 1. Introduction Decision-making is the procedure to find the best alternative among a set of feasible alternatives. Sometimes, decision-making problems considering several criteria are called multiple attribute decision making (MADM) problems. An MADM problem with m alternatives and n attributes can be expressed in matrix format as follows:
G1
G2
a11 a 21 A= Am am1
a12 a22
A1 A2
am 2
Gn a1n a2 n , W = [ w1 amn
w2
wn ]
A1 , A2 , , Am are possible alternatives among which decision makers have to choose, G1 , G2 , , Gn are criteria with which alternative performance are measured, aij is the rating of
where
alternative
Ai with respect to criterion G j , w j is the weight of criterion G j .
In the process of MADM with linguistic information, sometimes, the attribute values take the form of linguistic variables, and the information about attribute weights is incompletely known because of time pressure, lack of knowledge or data, and the expert’s limited expertise about the problem domain. Therefore, it is necessary to pay attention to this issue. Xu [1] have investigated the multiple attribute decision making problems, in which the attribute values take the form of linguistic variables, and the information about attribute weights is incompletely known. To determine the attribute weights, some simple optimization models are established. Especially, for the situations where the information about the attribute weights is completely unknown, a simple and exact formula for obtaining the attribute weights is provided. Xu [2] have developed an interactive procedure for linguistic multiple attribute decision making with incomplete weight information. Wu and Chen[3] have developed the maximizing deviation method for group multiple attribute decision making with completely unknown weight information under linguistic environment. The aim of this paper is to develop a TOPSIS method to solve linguistic MADM with incomplete weight. The remainder of this paper is set out as follows. In the next section, we introduce some basic concepts and operational laws of linguistic variables and define some useful concepts. In Section 3 we
- 181 -
TOPSIS Method for Multiple Attribute Decision Making with Incomplete Weight Information in Linguistic Setting Jianli Wei
introduce the MADM problem with linguistic information, in which the information about attribute weights is incompletely known, and the attribute values take the form of linguistic variables. Then, we establish an optimization model based on the basic ideal of traditional TOPSIS, by which the attribute weights can be determined. Based on this model, we develop a TOPSIS method to rank alternatives and to select the most desirable one(s). In Section 4, an illustrative example is pointed out. In Section 5 we conclude the paper and give some remarks.
Preliminaries Let= S {= si i 0,1, , T } be a linguistic term set with odd cardinality. Any label, si represents a possible value for a linguistic variable, and it should satisfy the following characteristics [4,5]: (1) The set is ordered: si
> s j , if i > j ; (2) There is the reciprocal operator: rec ( si ) = s j such that
(
)
i+ j = t ; (3) Max operator: max si , s j = si , if
si ≥ s j ; (4) Min operator: min ( si , s j ) = si , if
si ≤ s j . For example, S can be defined as = S {= s0 extremely poor ,= s1 very poor , = s2 poor , = s3 medium, = s4 good = , s5 very good = , s6 extremely good } To preserve all the given information, we extend the discrete term set S to a continuous term set
= S
{s
a
}
s0 ≤ sa ≤ sq , a ∈ [0, q ] , whose elements also meet all the characteristics above. If
sa ∈ S , then we call sa the original linguistic term, otherwise, we call sa the virtual linguistic term, q is a large positive integer. In general, the decision maker uses the original linguistic term to evaluate attributes and alternatives, and the virtual linguistic terms can only appear in calculation [6,7]. Definition= 1: Let R ( rij ) ( rij ∈ S ) be a linguistic decision matrix, then we call m× n
{ }
r + = ( r1+ , r2+ , , rn+ ) an positive ideal point of attribute values, where rj+ = max rij r − = ( r1− , r2− , , rn− ) an negative ideal point of attribute values,
i
and
where rj− = min {rij } , j = 1, 2, , n . i
Definition 2: Let
s1 = sα and s2 = sβ be two linguistic variables, then we call d ( s1 , s2 ) =
the distance between
α −β
(1)
T
s1 and s2 [1].
2.TOPSIS Method for Linguistic MADM Problems 2.1. Traditional TOPSIS Method TOPSIS (technique for order preference by similarity to an ideal solution) method is presented in Chen and Hwang [8], with reference to Hwang and Yoo n [9]. TOPSIS is a multiple criteria method to identify solutions from a finite set of alternatives. The basic principle is that the chosen alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. The procedure of TOPSIS can be expressed in a series of steps: (1) Calculate the normalized decision matrix. The normalized value bij is calculated as
- 182 -
Journal of Convergence Information Technology Volume 5, Number 10. December 2010 n
= 2, , m, j 1, 2, , n ∑ aij2 ， i 1,=
bij = aij
(2)
j =1
(2) Calculate the weighted normalized decision matrix. The weighted normalized value
cij is
calculated as , i 1,= 2, , m, j 1, 2, , n c= ij = w j bij where w j is the weight of the
j th criterion G j , and
∑
n j =1
(3)
wj = 1 .
(3) Determine the positive ideal and negative ideal solution
{( = {c ,, c } = {( min c
)( j ∈ I ) , ( max c
)} j ∈ J )}
C+ = {c1+ ,, cn+ } = max cij j ∈ I , min cij j ∈ J
(4a)
C−
(4b)
i
− 1
− n
i
ij
i
i
ij
where I is associated with benefit criteria, and J is associated with cost criteria. (4) Calculate the separation measures, using the n-dimensional Euclidean distance. The separation of each alternative from the ideal solution is given as
d i+ =
∑ (c n
j =1
− c +j ) , i = 1, , m . 2
ij
(5a)
Similarly, the separation from the negative ideal solution is given as d i− =
∑ (c n
j =1
− c −j ) , i = 1, , m . 2
ij
(5) Calculate the relative closeness to the ideal solution. The relative closeness of the alternative with respect to A
+
(5b)
Ai
is defined as Ri = d i−
(d
− i
+ d i+ ) ,
i = 1, , m .
(6)
Since d i− ≥ 0 and d i+ ≥ 0 , then, clearly, Ri ∈ [ 0,1] . (6) Rank the preference order. For ranking DMUs using this index, we can rank DMUs in decreasing order. The basic principle of the TOPSIS method is that the chosen alternative should have the “shortest distance” from the positive ideal solution and the “farthest distance” from the negative ideal solution. The TOPSIS method introduces two “reference” points, but it does not consider the relative importance of the distances from these points.
2.2. Extended Traditional TOPSIS Method to Linguistic MADM Problems The following assumptions or notations are used to represent the MADM problems with incomplete weight information in linguistic setting: (1) The alternatives are known. Let A = { A1 , A2 , , Am } be a discrete set of alternatives;
- 183 -
TOPSIS Method for Multiple Attribute Decision Making with Incomplete Weight Information in Linguistic Setting Jianli Wei
(2) The attributes are known. Let G = {G1 , G2 , , Gn } be a set of attributes; (3) The information about attribute weights is incompletely = w ( w1 , w2 , , wn ) ∈ H be the weight vector of attributes, where w j ≥ 0 ,
∑
n j =1
wj = 1 ,
known. Let j = 1, 2, , n ,
H is a set of the known weight information, which can be constructed by the
following forms [10-13], for
i ≠ j : Form 1. A weak ranking: wi ≥ w j ；Form 2. A strict ranking:
wi − w j ≥ α i , α i > 0 ；Form 3. A ranking of differences: wi − w j ≥ wk − wl , for j ≠ k ≠ l ; Form 4. A ranking with multiples: wi ≥ β i w j , 0 ≤ β i ≤ 1 ； Form 5. An interval form: α i ≤ wi ≤ α i + ε i , 0 ≤ α i < α i + ε i ≤ 1 . In the following, we will extend the TOPSIS method to solve multiple attribute decision making problems with incompletely known weight information in linguistic setting. Step1. Determine the linguistic positive ideal and linguistic negative ideal solution by definition 3. Linguistic positive ideal is: r + = ( r1+ , r2+ , , rn+ ) , where rj+ = max {rij } , j = 1, 2, , n .
(7)
i
Linguistic negative ideal is: r − = ( r1− , r2− , , rn− ) ,
where rj− = min {rij } , j = 1, 2, , n .
(8)
i
Step2. Calculate the separation measures, using the n-dimensional Hamming distance. The separation of each alternative from the linguistic ideal solution is given as
di+ =
d ( rj+ , rij ) w j , i 1, , m . = ∑ n
(9)
j =1
Similarly, the separation from the linguistic negative ideal solution is given as
= di−
= d (r , r ) w , i ∑ n
j =1
ij
− j
j
1, , m .
(10)
The basic principle of the TOPSIS method is that the chosen alternative should have the “shortest distance” from the positive ideal solution and the “farthest distance” from the negative ideal solution. Obviously, for the weight vector given, the smaller
di+ and the larger di− , the better alternative Ai . +
−
But the information about attribute weights is incompletely known. So, in order to get the d i and d i , we establish the following multiple objective optimization models: n + min d d ( rj+ , rij ) w j , i 1, , m = = ∑ i j =1 n = = d ( rij , rj− ) w j , i 1, , m ( M-1) max di− ∑ j =1 subject to : w ∈ H Since each alternative is noninferior, so there exists no preference relation on the all the alternatives. Then, we may aggregate the above multiple objective optimization models with equal weights into the following single objective optimization model: m n + − = min d ∑∑ d ( rj , rij ) − d ( rij , rj ) w j ( M-2 ) =i 1 =j 1 subject to : w ∈ H
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By solving the model ( M-2 ) , we get the optimal solution w = ( w1 , w2 , , wn ) , which can be used as the weight vector of attributes. Step 3. Utilize the weight vector w = ( w1 , w2 , , wn ) and by Eq. (4), we obtain the d i+ and d i− of the alternatives Ai ( i = 1, 2, , m ) ; Step 4. Calculate the relative closeness to the linguistic positive ideal solution. The relative closeness of the alternative
Ai with respect to A+ is defined as ci = d i−
(d
− i
+ di+ ) , i = 1, , m .
(5)
Since d i− ≥ 0 and d i+ ≥ 0 , then, clearly, ci ∈ [ 0,1] . Step 5. Rank all the alternatives Ai ( i = 1, 2, , m ) by using the relative closeness to the linguistic ideal solution ci ( i = 1, 2, , m ) and select the best one(s);
3. Illustrative Example Let us suppose there is an investment company, which wants to invest a sum of money in the best option (adapted from [14]). There is a panel with five possible alternatives to invest the money: ① A 1 is a car company; ② A 2 is a food company; ③ A 3 is a computer company; ④ A 4 is a arms company; ⑤ A 5 is a TV company. The investment company must take a decision according to the following four attributes: ① G 1 is the risk analysis; ② G 2 is the growth analysis; ③ G 3 is the social-political impact analysis; ④ G 4 is the environmental impact analysis. The five possible alternatives
Ai ( i = 1, 2, ,5 ) are to be evaluated using the linguistic term set
= S {= s0 extremely poor ,= s1 very poor , = s2 poor , = s3 medium, = s4 good = , s5 very good = , s6 extremely good } by the decision maker under the above four attributes, as listed in the following matrix.
G1 A1 s4 A2 s3 R = A3 s6 A4 s1 A5 s2
G2
G3
s5
s3
s2 s3 s4 s0
s4 s4 s3 s3
G4 s3 s6 s1 s4 s6
The information about the attribute weights is partly known as follows:
H=
{0.15 ≤ w1 ≤ 0.25, 0.2 ≤ w2 ≤ 0.35, 0.35 ≤ w3 ≤ 0.55,
}
w4 ≥ 0.3 w= 1, 2, 3,= 4, ∑ j =1 w j 1 2 , w j ≥ 0, j 4
Then, we utilize the linguistic approach developed to get the most desirable alternative(s). Step 1. Utilize the model (M-2) to establish the following single-objective programming model:
min d ( w ) = 0.5w1 − 0.5w2 + 0.1667 w3 − 0.8333w4 Subject to : w ∈ H
Step 2. Solve this model, we get the weight vector of attributes: w = ( 0.15, 0.35, 0.3950, 0.1050 )
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TOPSIS Method for Multiple Attribute Decision Making with Incomplete Weight Information in Linguistic Setting Jianli Wei
Step 3. Utilize the weight vector + i
d and d
− i
w and by Eq. (4), we calculate the separation measures
of the alternatives Ai ( i = 1, 2, 3, 4, 5 ) :
= d1+ 0.1683, = d 2+ 0.2500, = d3+ 0.2042, = d 4+ 0.2842, = d5+ 0.4575 = d1− 0.4017, = d 2− 0.3200, = d3− 0.3658, = d 4− 0.2858, = d5− 0.1125 Step 4. Calculate the relative closeness to the linguistic positive ideal solution. = c1 0.7047, = c2 0.5614, = c3 0.6418, = c4 0.5015, = c5 0.1974 Step 5. Rank all the alternatives Ai ( i = 1, 2, , 5 ) in accordance with the relative closeness to the linguistic positive ideal solution ci ( i = 1, 2, 3, 4, 5 ) : A 1 desirable alternative is
A 3
A 2
A 4
A5 , and thus the most
A1 .
4. Conclusion With respect to the MADM problems, in which the information about attribute weights is incompletely known, and the attribute values take the form of linguistic variables. In order to get the attribute weight, we establish the multiple objective optimization models based on the basic ideal of the traditional TOPSIS. Then, by linear equal weighted method, the multiple objective optimization models can be transformed into a single-objective programming model. By solving the single-objective programming model, we can get the weight information and get the relative closeness to the linguistic positive ideal solution of all the alternatives. At last, a practical example is provided to illustrate the proposed method. Theoretical analysis and the numerical results have shown that the developed approach is straightforward and has no loss of information.
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