Goal programming models for obtaining
the priority vector of incomplete
fuzzy preference relation
Z.S. Xu
College of Economics and Management, Southeast University, Nanjing, Jiangsu 210096, China Received 1 August 2003; accepted 1 October 2003
Available online 30 December 2003
Abstract
In this paper, we first define the concepts of incomplete fuzzy preference relation, additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation, and then propose two goal programming models, based on additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation respectively, for obtaining the priority vector of incomplete fuzzy preference relation. These two goal programming models are also extended to obtain the collective priority vector of several incomplete fuzzy pref-erence relations. Finally, two illustrative numerical examples are given to verify the developed models.
Ó 2003Elsevier Inc. All rights reserved.
Keywords: Incomplete fuzzy preference relation; Additive consistent incomplete fuzzy preference relation; Multiplicative consistent incomplete fuzzy preference relation; Goal programming model; Priority
1. Introduction
The use of preference relations is usual in decision making [1–18]. In the process of decision-making, the decision-maker (DM) generally shall compare
E-mail address: [email protected] (Z.S. Xu).
0888-613X/$ - see front matterÓ 2003Elsevier Inc. All rights reserved. doi:10.1016/j.ijar.2003.10.011
a set of n decision alternatives with respect to a single criterion and construct a preference relation, then derive the priority vector of the preference relation by some techniques. In a fuzzy context, a DM expresses his/her opinions using fuzzy preference relation [19–30] R¼ ðrijÞnn, whose element rij represents the
preference degree of the alternative i to j and satisfies rij2 ½0; 1, rijþ rji¼ 1.
The fuzzy preference relation R¼ ðrijÞnn necessitates the completion of all nðn 1Þ
2 judgements in its entire top triangular portion. Sometimes, however, it is
difficult to obtain such a fuzzy preference relation, especially for the fuzzy preference relation with high order, because of time pressure, lack of knowl-edge, and the DM’s limited expertise related with problem domain, the DM may develop an incomplete fuzzy preference relation in which some of the elements cannot be provided. Up to now, there is no approach of deriving the priority vector of incomplete fuzzy preference relation. Therefore, it is neces-sary to pay attention to this issue. In this paper, we first define the concepts of incomplete fuzzy preference relation, additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation, and then, based on additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relations, respectively, we propose two goal programming models for obtaining the priority vector of incomplete fuzzy preference relation. These two goal pro-gramming models are also extended to obtain the collective priority vector of several incomplete fuzzy preference relations, and finally, we give two numerical examples to illustrate the developed models.
2. Some concepts
For simplicity, we let M¼ f1; 2; . . . ; mg and N ¼ f1; 2; . . . ; ng.
Definition 2.1. Let R¼ ðrijÞnn be a preference relation, then R is called a fuzzy
preference relation [18–23], if
rij2 ½0; 1; rijþ rji¼ 1; rii¼ 0:5 for all i; j2 N :
Definition 2.2. Let R¼ ðrijÞnnbe a fuzzy preference relation, then R is called an
additive consistent fuzzy preference relation, if the following additive transi-tivity (given by Tanino [19]) is satisfied:
rij¼ rik rjkþ 0:5 for all i; j; k2 N :
Definition 2.3. Let R¼ ðrijÞnn be a fuzzy preference relation, then R is called a
multiplicative consistent fuzzy preference relation, if the following multiplica-tive transitivity (given by Tanino [19]) is satisfied:
rikrkjrji ¼ rkirijrjk for all i; j; k2 N :
In the following, we shall extend the above concepts to the situations where the preference information given by the DM is incomplete.
Definition 2.4. Let R¼ ðrijÞnn be a preference relation, then R is called an
incomplete fuzzy preference relation, if some of its elements cannot be given by the DM, which we denote by the unknown number x, and the others can be provided by the DM, which satisfy rij2 ½0; 1, rijþ rji¼ 1, rii¼ 0:5.
Definition 2.5. Let R¼ ðrijÞnnbe an incomplete fuzzy preference relation, then
Ris called an additive consistent incomplete fuzzy preference relation, if all the known elements of R satisfy the additive transitivity rij¼ rik rjkþ 0:5.
Definition 2.6. Let R¼ ðrijÞnnbe an incomplete fuzzy preference relation, then
Ris called a multiplicative consistent incomplete fuzzy preference relation, if all the known elements satisfy the multiplicative transitivity rikrkjrji¼ rkirijrjk.
For the convenience of computation, we construct an indication matrix D¼ ðdijÞnn of the incomplete fuzzy preference relation R¼ ðrijÞnn, where
dij ¼
0; rij¼ x;
1; rij6¼ x:
In the following section, we shall develop two goal programming models, based on additive consistent incomplete fuzzy preference relation and multi-plicative consistent incomplete fuzzy preference relations respectively, for obtaining the priority vector of incomplete fuzzy preference relation.
3. Goal programming models Let w¼ ðw1; w2; . . . ; wnÞ
T
be the priority vector of the incomplete fuzzy preference relation R¼ ðrijÞnn; where wiP0; i2 N ; Pni¼1wi¼ 1.
(1) If R¼ ðrijÞnn is an additive consistent incomplete fuzzy preference
relation, then such a preference relation is given by
dijrij ¼ dij½0:5ðwi wjþ 1Þ; i; j2 N : ð1Þ
However, in the general case, Eq. (1) does not hold. Here, we shall relax Eq. (1) by looking for the priority vector of the incomplete fuzzy preference rela-tion R¼ ðrijÞnn that approximates Eq. (1) by minimizing the error eij, where
Thus, we can construct the following multi-objective programming model: ðMOP1Þ min eij ¼ dijjrij 0:5ðwi wjþ 1Þj; i; j2 N s:t: wiP0; i2 N ; Xn i¼1 wi¼ 1:
Solution to the above minimization problem is found by solving the following goal programming model:
ðLOP1Þ min J ¼X n i¼1 Xn j¼1;i6¼j ðsijdijþþ tijdij Þ s:t: dij½rij 0:5ðwi wjþ 1Þ dijþþ d ij ¼ 0; i; j2 N ; i 6¼ j; wiP0; i2 N ; Xn i¼1 wi¼ 1; dijþP0; dij P0; i; j2 N ; i 6¼ j; where dþ
ij is the positive deviation from the target of the goal eij, defined as
dijþ ¼ dij½rij 0:5ðwi wjþ 1Þ _ 0:
d
ij is the negative deviation from the target of the goal eij, defined as
dij ¼ dij½0:5ðwi wjþ 1Þ rij _ 0:
sij is the weighting factor corresponding to the positive deviation dijþ, tij is the
weighting factor corresponding to the negative deviation d ij.
Consider that all the goal functions eij ði; j 2 N Þ are fair, then we can set
sij¼ tij¼ 1, i; j 2 N , and then the model (LOP1) can be rewritten as
ðLOP2Þ min J ¼X n i¼1 Xn j¼1;i6¼j ðdþ ij þ d ijÞ s:t: dij½rij 0:5ðwi wjþ 1Þ dijþþ d ij ¼ 0; i; j2 N ; i 6¼ j; wiP0; i2 N ; Xn i¼1 wi¼ 1; dijþP0; dij P0; i; j2 N ; i 6¼ j:
By solving the model (LOP2), we can obtain the priority vector w¼ ðw1; w2; . . . ; wnÞ
T
of the incomplete fuzzy preference relation R¼ ðrijÞnn.
(2) If R¼ ðrijÞnn is a multiplicative consistent incomplete fuzzy preference
relation, then such a preference relation is given by dijrij ¼ dij
wi
wiþ wj
By Definition 2.4 and Eq. (2), we have
dijrijwj¼ dijrjiwi; i; j2 N : ð3Þ
However, in the general case, Eq. (3) does not hold. Here, we shall relax Eq. (3) by looking for the priority vector of the incomplete fuzzy preference rela-tion R¼ ðrijÞnn that approximates Eq. (3) by minimizing the error eij, where
eij¼ jdijrijwj dijrjiwij ¼ dijjrijwj rjiwij; i; j2 N :
Thus, we can construct the following multi-objective programming model: ðMOP2Þ min eij ¼ dijjrijwj rjiwij; i; j2 N
s:t: wiP0; i2 N ;
Xn i¼1
wi¼ 1:
Solution to the above minimization problem is found by solving the following goal programming model:
ðLOP3Þ min J ¼X n i¼1 Xn j¼1;i6¼j ðsijdijþþ tijdij Þ s:t: dijðrijwj rjiwiÞ dijþþ d ij ¼ 0; i; j2 N ; i 6¼ j; wiP0; i2 N ; Xn i¼1 wi¼ 1; dijþP0; dij P0; i; j2 N ; i 6¼ j; where dþ
ij is the positive deviation from the target of the goal eij, defined as
dijþ ¼ dijðrijwj rjiwiÞ _ 0:
dij is the negative deviation from the target of the goal eij, defined as
dij ¼ dijðrjiwi rijwjÞ _ 0:
sij is the weighting factor corresponding to positive deviation dijþ, tij is the
weighting factor corresponding to the negative deviation d ij.
Consider that all the goal functions eij (i; j2 N ) are fair, thus we can set
sij¼ tij¼ 1, i; j 2 N , and then the model (LOP3) can be rewritten as
ðLOP4Þ min J ¼X n i¼1 Xn j¼1;i6¼j ðdþ ij þ d ijÞ s:t: dijðrijwj rjiwiÞ dijþþ d ij ¼ 0; i; j2 N ; i 6¼ j; wiP0; i2 N ; Xn i¼1 wi¼ 1; dijþP0; dij P0; i; j2 N ; i 6¼ j:
By solving the model (LOP4), we can also obtain the priority vector w¼ ðw1; w2; . . . ; wnÞ
T
of the incomplete fuzzy preference relation R¼ ðrijÞnn.
In the following, we shall extend the above models to obtain the collective priority vector of two or more incomplete fuzzy preference relations.
Suppose that there are m incomplete fuzzy preference relations Rk ¼
ðrijkÞnn ðk 2 MÞ, and v ¼ ðv1; v2; . . . ; vnÞ T
is their collective priority vector, where viP0, i2 N ; Pni¼1vi¼ 1. We also construct m indication matrices
Dk ¼ ðdijkÞnn ðk 2 MÞ of the incomplete fuzzy preference relations Rk ¼
ðrijkÞnn ðk 2 MÞ, where dijk¼ 0; rijk¼ x; 1; rijk6¼ x: Then, v¼ ðv1; v2; . . . ; vnÞ T
can be obtained by solving one of the following two models (LOP5) and (LOP6), which are the extensions of the models (LOP2) and (LOP4), respectively:
ðLOP5Þ min J ¼X m k¼1 Xn i¼1 Xn j¼1;i6¼j ðdijkþ þ d ijkÞ
s:t: dijk½rijk 0:5ðvi vjþ 1Þ dijkþ þ d ijk¼ 0; i; j2 N ; i 6¼ j; k 2 M; viP0; i2 N ; Xn i¼1 vi¼ 1; dijkþ P0; dijk P0; i; j2 N ; i 6¼ j; k 2 M; where dþ
ijk is the positive deviation, defined as
dijkþ ¼ dijk½rijk 0:5ðvi vjþ 1Þ _ 0:
d
ijk is the negative deviation, defined as
dijk ¼ dijk½0:5ðvi vjþ 1Þ rijk _ 0:
ðLOP6Þ min J ¼X m k¼1 Xn i¼1 Xn j¼1;i6¼j ðdijkþ þ d ijkÞ
s:t: dijkðrijkvjþ rjikviÞ dijkþ þ d ijk¼ 0; i; j2 N ; i 6¼ j; k 2 M; viP0; i2 N ; Xn i¼1 vi¼ 1; dijkþ P0; dijk P0; i; j2 N ; i 6¼ j; k 2 M;
where dijkþ is the positive deviation, defined as dijkþ ¼ dijkðrijkvj rjikviÞ _ 0:
d
ijk is the negative deviation, defined as
dijk ¼ dijkðrjikvj rijkviÞ _ 0:
4. Numerical examples
In this section, two numerical examples are given, which are constructed in order to understand more fully the developed models.
Example 4.1. For a decision-making problem, there are six decision alternatives Ai ði ¼ 1; 2; . . . ; 6Þ. The DM provides his/her preferences over these six decision
alternatives, and gives an incomplete fuzzy preference relation as follows:
R¼ 0:5 0:4 x 0:3 0:8 0:3 0:6 0:5 0:6 0:5 x 0:4 x 0:4 0:5 0:3 0:6 x 0:7 0:5 0:7 0:5 0:4 0:8 0:2 x 0:4 0:6 0:5 0:7 0:7 0:6 x 0:2 0:3 0:5 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 :
(1) By the model (LOP2), we get the priority vector w of the incomplete fuzzy preference relation R as follows:
w¼ ð0:144; 0:192; 0:133; 0:267; 0:142; 0:122ÞT: Thus the ranking of these six alternatives is
A4 A2 A1 A5 A3 A6
and the best alternative is A4.
(2) By the model (LOP4), we get the priority vector w of the incomplete fuzzy preference relation R as follows:
w¼ ð0:156; 0:179; 0:155; 0:200; 0:154; 0:156ÞT: Thus the ranking of these six alternatives is
A4 A2 A1¼ A6 A3 A5
and the best alternative is A4.
Example 4.2. For a decision-making problem, suppose that there are four decision alternatives Aiði ¼ 1; 2; 3; 4Þ and three DMs ei ði ¼ 1; 2; 3Þ. The DMs
provide their preferences over these four decision alternatives, and give three incomplete fuzzy preference relations as follows, respectively:
R1¼ 0:5 0:6 x 0:7 0:4 0:5 0:2 0:8 x 0:8 0:5 0:4 0:3 0:2 0:6 0:5 2 6 6 4 3 7 7 5; R2¼ 0:5 0:8 0:4 x 0:2 0:5 0:3 0:6 0:6 0:7 0:5 0:3 x 0:4 0:7 0:5 2 6 6 4 3 7 7 5; R3¼ 0:5 0:3 0:4 0:6 0:7 0:5 x 0:5 0:6 x 0:5 0:7 0:4 0:5 0:3 0:5 2 6 6 4 3 7 7 5:
(1) By the model (LOP5), we get the collective priority vector v of the incomplete fuzzy preference relations Rk ðk ¼ 1; 2; 3Þ as follows:
v¼ ð0:276; 0:225; 0:300; 0:199ÞT:
Thus the ranking of these four alternatives is A3 A1 A2 A4
and the best alternative is A3.
(2) By the model (LOP6), we get the collective priority vector v of the incomplete fuzzy preference relation Rk ðk ¼ 1; 2; 3Þ as follows:
v¼ ð0:265; 0:236; 0:276; 0:223ÞT:
Thus the ranking of these four alternatives is A3 A1 A2 A4
and the best alternative is A3.
5. Concluding remarks
In this paper, we have defined some concepts such as incomplete fuzzy preference relation, additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation. We have proposed two goal programming models, based on additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation, respectively, for obtaining the priority vector of incomplete fuzzy preference relation, and then extended these two goal pro-gramming models to obtain the collective priority vector of several incomplete
fuzzy preference relations. In the future, we shall focus on the application of incomplete fuzzy preference relations in fuzzy decision making.
Acknowledgements
This research has been partially supported by the National Natural Science Foundation of China (NSFC) under Project 79970093. The author is very grateful to four anonymous referees for their valuable comments and suggestions.
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