Expert Systems with Applications 38 (2011) 9032–9035

Contents lists available at ScienceDirect

Expert Systems with Applications

journal homepage: www.elsevier .com/locate /eswa

Short Communication

Approach for aggregating interval-valued intuitionistic fuzzy informationand its application to reservoir operation

Ke Xu, Jianzhong Zhou ⇑, Ran Gu, Hui QinCollege of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan, Hubei Province, China

a r t i c l e i n f o a b s t r a c t

Keywords:Multiple attribute decision makingInterval-valued intuitionistic fuzzyinformationAggregation operatorWeight

0957-4174/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.eswa.2011.01.011

⇑ Corresponding author. Tel.: +86 13607174132.E-mail addresses: xuke19841125@126.com (K

(J. Zhou).

In this article, we concentrate on the interval-valued intuitionistic fuzzy multiple attribute decision mak-ing problems with preference information on schemes and incomplete weight information. First someoperational laws, score function and accuracy function of interval-valued intuitionistic fuzzy numbersare defined, and based on these, a method for ranking interval-valued intuitionistic fuzzy numbers is pre-sented, then an optimization model based on the minimizing deviation method, by which the attributeweights can be determined, is established, so an approach for decision making with interval-valued intui-tionistic fuzzy information is developed, finally these are applied to practical reservoir operation and theoptimal scheme is obtained. This case study shows that the model and corresponding method are scien-tific and practical.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Atanassov (1986) introduced the concept of intuitionistic fuzzyset (IFS), which is a generalization of the concept of fuzzy set (Asai,1995). Since its appearance, the IFS theory has received more andmore attention; it has been used in a wide range of applications,such as, decision making, logic programming, medical diagnosis,pattern recognition, and so on. Then Bustince and Burillo (1996)showed that vague sets (Gau & Buehrer, 1993) are intuitionistic fuz-zy sets and Atanassov and Gargov (1989) further generalized the IFS,and introduced the notion of an interval-valued intuitionistic fuzzyset (IVIFS). After that, some researchers investigated the IVIFSs, suchas, the relations and operations of IVIFSs (Atanassov, 1994; Calvo,Mayor, & Mesiar, 2002; Xu, 2006), the correlation and correlationcoefficients of IVIFSs, the topology of IVIFSs, and so on.

In the process of MADM with intuitionistic fuzzy information,sometimes, the attribute values take the form of intuitionistic fuz-zy numbers, and the information about attribute weights is incom-pletely known or completely unknown. In such cases, it is suitableand convenient to express the decision makers’ preferences in aninterval-valued intuitionistic fuzzy number (IVIFN) (Xu, 2007).Therefore, it is necessary and interesting to pay attention to thegroup decision-making problems with interval-valued intuitionis-tic preference information. Therefore, it is necessary to pay atten-tion to this issue. In this article, the authors have developedsome operators (Herrera & Herrera-Viedma, 1997; Xu & DA,

ll rights reserved.

. Xu), jz.zhou@hust.edu.cn

2003) for aggregating interval-valued intuitionistic preferenceinformation and have develop an minimizing deviation model ofinterval-valued intuitionistic judgement matrix, and provide anapproach to solving the group decision-making problems wherethe decision makers provide their preference information overschemes (Herrera & Herrera-Viedma, 2000; Xu, 2002; Xu, 2003).

2. Preliminaries

2.1. Interval-valued intuitionistic fuzzy number

Definition 1 Atanassov and Gargov, 1989. Let Xw be an interval-valued intuitionistic fuzzy set (IVIFS) A in Xw is an object of thefollowing form:

A ¼ hx; ~lAðxÞ; ~tAðxÞijx 2 Xf g

where ~lAðxÞ : X ! ½0;1�, ~tAðxÞ ! ½0;1�, x 2 X, 0 6 supð~lAðxÞÞþsupð~tAðxÞÞ 6 1, x 2 X then the pair (~lAðxÞ, ~tAðxÞÞ is called an inter-val-valued intuitionistic fuzzy number (IVIFN).

Definition 2 Xu, 2007. Let ~a1 ¼ ð½a1; b1�; ½c1; d1�Þ and ~a2 ¼ ð½a2; b2�;½c2; d2�Þ be two IVIFNs, then their operational laws can w be definedas follows:

(1) ~a1 þ ~a2 ¼ ð½a1 þ a2 � a1a2; b1 þ b2 � b1b2�; ½c1c2; d1d2�Þ;(2) ~a1 � ~a2 ¼ ð½a1a2; b1b2�; ½c1 þ c2 � c1c2; d1 þ d2 � d1d2�Þ;(3) k~a1 ¼ ð½1� ð1� a1Þk;1� ð1� b1Þk�; ½ck

1; dk1�Þ, k > 0;

(4) ð~a1Þk ¼ ð½ak1, bk

1�; ½1� ð1� c1Þk;1� ð1� d1Þk�Þ, k > 0;

K. Xu et al. / Expert Systems with Applications 38 (2011) 9032–9035 9033

Xu showed more operational laws of IVIFNs as follows:

(1) ~a1 þ ~a2 ¼ ~a2 þ ~a1

(2) ~a1 � ~a2 ¼ ~a2 � ~a1

(3) kð~a1 þ ~a2Þ ¼ k~a1 þ k~a2, k P 0(4) ð~a1 � ~a2Þk ¼ ~ak

1 � ~ak2, k P 0

(5) k~a1 þ k~a2 ¼ kð~a1 þ ~a2Þ, k P 0(6) ~ak

1 � ~ak2=ð~a1 � ~a2Þk; k P 0

2.2. Aggregating operators of interval-valued intuitionistic fuzzyinformation

Definition 3 Xu, 2007. Let ~aj ¼ ð½aj; bj�; ½cj; bj�Þ ðj ¼ 1;2; . . . ;nÞ be aset of IVIFNs, and let IIFWA : Qn ? Q, if IIFWAx : ð~a1; ~a2; . . . ; ~anÞ ¼Pn

j¼1xj~aj ¼

1�Yn

j¼1

ð1� ajÞxj ;1�Yn

j¼1

ð1� ajÞxj

" #;Yn

j¼1

cwj

j ;Yn

j¼1

dwj

j

" #" #ð1Þ

where x = (x1,x2, . . .,xn)T be the weight vector of ~aj ðj ¼ 1;2; . . . ;nÞ, then IIFWA is called the interval-valued intuitionisticfuzzy weighted averaging (IIFWA) operator.

2.3. Comparison between interval-valued intuitionistic fuzzy number

For the ranking among interval-valued intuitionistic fuzzy num-ber, the score function and the accuracy function are given asfollows:

Definition 4 Xu, 2007. Let ~a ¼ ð½a; b�; ½c; d�Þ be a IVIFN, the scorefunction Sð~aÞ can be represented as follows, that is,

Sð~aÞ ¼ a� c þ b� d2

; Sð~aÞ 2 ½�1;1� ð2Þ

Definition 5 Xu, 2007. Let ~a ¼ ð½a; b�; ½c; d�Þ be a IVIFN, the accu-racy function Hð~aÞ can be represented as follows, that is,

Hð~aÞ ¼ aþ c þ bþ d2

; Hð~aÞ 2 ½0;1� ð3Þ

Based on the analysis above, a rank method among interval-valuedintuitionistic fuzzy number is given as follows:

Definition 6 Xu, 2007. Let ~a1 ¼ ð½a1; b1�; ½c1; d1�Þ and ~a2 ¼ ð½a2; b2�;½c2; d2�Þ be any two IVIFNs, and,

(1) if Sð~a1Þ < Sð~a2Þ, then ~a1 < ~a2;(2) when Sð~a1Þ ¼ Sð~a2Þ, if Hð~a1Þ ¼ Hð~a2Þ, then ~a1 ¼ ~a2; if

Hð~a1Þ < Hð~a2Þ, then ~a1 < ~a2.

Definition 7. Let ~a1 ¼ ð½a1; b1�; ½c1; d1�Þ and ~a2 ¼ ð½a2; b2�; ½c2; d2�Þ beany two IVIFNs, so the normalized Hamming distance of them isdefined as follows:

dða1; a2Þ ¼14ð a1 � a2j j þ b1 � b2j j þ c1 � c2j j þ d1 � d2j jÞ ð4Þ

3. The multi-attribute decision-making method based oninterval-valued intuitionistic fuzzy information

On the basis of the earlier theoretic analysis, we define thefollowing assumptions to solve the interval-valued intuitionisticfuzzy MADM problems with incomplete weight information: let

A ¼ fA1;A2; . . . Ang be a discrete set of schemes; C ¼fC1;C2; . . . ;Cng be a set of attributes; the weight information ofattribute Cj in scheme Ai is expressed by interval-valued intuition-istic fuzzy number ([aij,bij], [cij,dij]), which construct the interval-valued intuitionistic fuzzy decision matrix R = (rij)m � n = ([aij,bij],[cij,dij])m�n; the preference value from decision-makers to attri-butes also expressed by interval-valued intuitionistic fuzzy num-ber, that is ri = ([ai,bi], [ci,gi]).

Let the attribute weights vector is x = {x1,x2, . . .,xn}T, we canaggregate the attribute value of interval-valued intuitionistic fuzzydecision matrix according to IIFWA operator, the overall value ofthe schemes Ai where is x = {x1,x2, . . .,xn}T the weight vector ofattributes is obtained (Hong & Choi, 2000):

ri ¼ ð½ai; bi�; ½ci; di�Þ ¼ IIFWAxðri1; ri2; . . . ; rinÞ

¼ 1�Yn

j¼1

ð1� aijÞxi ;1�Yn

j¼1

ð1� bijÞxi

" #;

"

Yn

j¼1

cxiij ;Yn

j¼1

dxiij

" ##i ¼ 1;2; . . . ;m ð5Þ

Obviously, the greater ri is, the better the schemes Ai will be, inthe situation where the information about attribute weights iscompletely known, the schemes can be ranked.

Now the information about attribute weights is incompletelyknown, so we should determine the attribute weights first, nowthe set of incomplete attribute weights information is expressedas H.

Generally, there is deviation between the preference from deci-sion-makers to schemes and the preference from decision-makersto attributes, to obtain reasonable attribute weights vector, thedeviation must be minimal, therefore, and the minimizing devia-tion model ðM � 1Þ is constructed as follows:

min ds ¼Pmi¼1

Pnj¼1

dðrij;riÞxj; i ¼ 1;2; . . . ;m

¼ 14

Pmi¼1

Pnj¼1

xj aij � ai

�� ��þ bij � bi

�� ��þ cij � ci

�� ��þ dij � gi

�� ��� �

s:t: x 2 H;Pnj¼1

xj ¼ 1; xj P 0; j ¼ 1;2; . . . ;n

8>>>>>>>><>>>>>>>>:

ð6Þ

In this, d(rij,ri) represents the deviation between the preference ri

from decision-makers to schemes Ai and the preference rij fromdecision-makers to attributes Gi, ds is the summation of the devia-tion above.

Solving this model, the optimal attribute weights is obtained:

x�j ¼Pm

i¼1dðrij;riÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnj¼1

Pmi¼1dðrij;riÞ

� �q 2 ð7Þ

by normalizing it as a unit : xi ¼Pm

i¼1d rij;ri� �

Pnj¼1

Pmi¼1dðrij;riÞ

; j¼ 1;2; . . . ;n

ð8Þ

according to formula (5), Eq. (9) can be express as follows:

xj ¼Pm

i¼1ð aij � ai

�� ��þ bij � bi

�� ��þ jcij � cij þ jdij � gijÞPnj¼1

Pmi¼1ðjaij � aij þ jbij � bij þ jcij � cij þ jdij � gijÞ

j

¼ 1;2; . . . ;n ð9Þ

so the decision-making steps are as follows (Hwang & Yoon, 1981;Iyer, 2003; Xu, 2004):

Step 1: Construct the interval-valued intuitionistic fuzzy decisionmatrix R = (rij)m�n = ([aij,bij], [cij,dij]m�n then the preference

Table 1The table of decision-making indicator system.

Evaluationindex

Economic benefit Social benefit Ecological benefit

Evaluationterm

Guaranteerate ofgeneration

Averageannualgeneratedenergy

Generatedenergy ofcascadestation

Guaranteerate ofwaterdiversion

Guarantee rate ofupper supply forsocial andeconomy

Guarantee rate oflower supply forsocial andeconomy

Guarantee rateof lower supplyforenvironment

Guarantee rateof lower supplyfor ecosystem

Guarantee rateof lower supplyfor sandflushing

Unit % 108 kW h 108 kW h % 108 kW h 108 kW h % % 108 kW hSymbol P1 P2 P3 P4 P5 P6 P7 P8 P9

9034 K. Xu et al. / Expert Systems with Applications 38 (2011) 9032–9035

values ri = ([ai,bi], [ci,gi]) from decision-makers toschemes Ai are formed.

Step 2: As the result of incomplete weight information; obtain theoptimal attribute weights vector x according to the opti-mization model ðM � 1Þ.

Step 3: Obtain the comprehensive evaluation values zi of schemeAi.

Step 4: Calculate the score function result and the accuracy func-tion result of scheme Aiand rank according to it.

4. The application of the decision-making method for reservoiroperation

In above sections, the theory of interval-valued intuitionisticfuzzy multiple attribute decision making has been described. Com-prehensive evaluation of reservoir operation schemes is a complexmulti-hierarchy and multi-index evaluation problem with incom-plete weight information. To solve the problem of decision-makingfor reservoir operation, the model will be constructed in thissection.

4.1. The schemes of reservoir operation and the decision-makingindicators

It is water resources system that is the system led by Jiudianxiareservoir in complex condition and multipurpose use with Tao Riverbasin and cascaded power stations in Tao River. The Jiudianxiareservoir has been designed for many purposes, such as powergeneration, irrigation, total water supply for industry, agriculture,residents and environment etc. Because lower Tao River is in losshilly-gully region and serious water–soil erosion areas andconsidering sediment transporting balance, the water supply forecosystem and environment accounts for high proportion of totalsupply, including lower supply for ecosystem, supply for channelenvironment, water for sand flushing etc.

Based on different planning of water use in Tao River basin anddifferent water transfer schemes of water supply project from TaoRiver, five reservoir operation schemes are made for different sat-isfaction degrees of water requirement:

Scheme 1: With the objectives of the maximum plant output,satisfaction of the water use in this basin, upper and lower sup-ply for society and economy.Scheme 2: With the objectives of the maximum plant output,satisfaction of the water use in this basin, upper and lower sup-ply for society and economy, lower supply for ecosystem.Scheme 3: With the objectives of the maximum plant output,satisfaction of the water use in this basin, upper and lower sup-ply for society and economy, total supply for ecosystem andenvironment at level and flood period, 90% of which is supplyfor sand flushing at low water period.Scheme 4: With the objectives of the maximum plant output,satisfaction of the water use in this basin, upper and lower sup-ply for society and economy, total supply for ecosystem and

environment at level and flood period, 50% supply for sandflushing at low water period.Scheme 5: With the objectives of the maximum plant output,satisfaction of the water use in this basin, upper and lower sup-ply for society and economy, total supply for ecosystem andenvironment at level and flood period.

Table 1 shows the decision-making indicators for the schemesof reservoir operation.

4.2. The decision steps for reservoir operation based on aggregatinginterval-valued intuitionistic fuzzy information

There are five reservoir operation schemes (A1, A2, A3, A4, A5)for evaluated and three evaluation index C1–C3 (economic benefit,social benefit, ecological benefit), the attribute weights cannot bedetermined completely, the known part is as follows:

H¼ 0:56x1 60:75;0:676x2 60:9;0:656x3 60:8;xj P 0; j¼1;2;3;X3

j¼1

xj ¼1

( )

By using statistical methods, the satisfy degree and dissatisfydegree from schemes Ai to attributes Cj are obtained and the intui-tionistic fuzzy appraisal matrix is constructed as follows:

ð½0:5;0:7�; ½0:1;0:2�Þð½0:7;0:8�; ½0:1;0:2�Þð½0:3;0:4�; ½0:3;0:5�Þ

ð½0:5;0:7�; ½0:2;0:3�Þð½0:7;0:8�; ½0:1;0:2�Þð½0:4;0:6�; ½0:2;0:4�Þ

ð½0:2;0:4�; ½0:4;0:6�Þð½0:6;0:7�; ½0:1;0:3�Þð½0:1;0:4�; ½0:5;0:6�Þ

ð½0:3;0:6�; ½0:3;0:4�Þð½0:5;0:6�; ½0:2;0:3�Þð½0:3;0:6�; ½0:3;0:4�Þ

ð½0:5;0:7�; ½0:2;0:3�Þð½0:7;0:8�; ½0:1;0:2�Þð½0:6;0:8�; ½0:1;0:2�Þ

2666666664

3777777775

Similarly, the satisfy degree and dissatisfy degree from deci-sion-makers to schemes Ai are obtained, that is:

r1 ¼ ð½0:2;0:3�; ½0:4;0:5�Þ; r2 ¼ ð½0:3;0:4�; ½0:3;0:5�Þ; r3

¼ ð½0:5;0:6�; ½0:1;0:3�Þ; r4 ¼ ð½0:4;0:5�; ½0:2;0:4�Þ; r4

¼ ð½0:7;0:8�; ½0:1;0:2�Þ

according to model ðM � 1Þ, the optimization model is establishedas follows:

min DðxÞ ¼ 0:975x1 þ 1:25x2 þ 0:825x3

s:t:x 2 H

�

solving this model, the optimal attribute weights vector is obtained:x⁄ = (0.50,0.67,0.65)T according to Eq. (5), the comprehensive eval-uation values zi of scheme Ai (1,2, . . .,m) are obtained:

z1 ¼ ð½0:7497;0:9160�; ½0:0309;0:0968�Þ

z2 ¼ ð½0:7736;0:8973�; ½0:0336;0:1027�Þ

z3 ¼ ð½0:5480;0:7519�; ½0:0862;0:2481�Þ

z4 ¼ ð½0:5830;0:8113�; ½0:0852;0:1556�Þ

z4 ¼ ð½0:8260;0:9345�; ½0:0214;0:0655�Þ

K. Xu et al. / Expert Systems with Applications 38 (2011) 9032–9035 9035

calculate the score function result of the comprehensive evaluationvalue of scheme Ai:

Sðz1Þ ¼ 0:7690; Sðz2Þ ¼ 0:7673; Sðz3Þ ¼ 0:4849Sðz4Þ ¼ 0:5767; Sðz5Þ ¼ 0:8368

Rank the scheme by the S(zi), we have: A5 > A1 > A2 > A4 > A3, so theranking about comprehensive evaluation of the schemes of reser-voir operation is obtained, form high to low are: Scheme 5,Scheme 1, Scheme 2, Scheme 4, Scheme 3, and the optimal schemeis Scheme 5.

5. Conclusions

In this paper, we developed the interval-valued intuitionisticfuzzy weighted averaging (IIFWA) operator and built a multiattrib-ute decision-making model with uncertain weight. We presenteddetails for the connotation and the algorithm steps of IIFWA oper-ator, and then the model is applied to the reservoir operation andproved to be feasible, the present analysis can be extended to othertypes of data form and combination with other operators etc.,which may be areas of future research.

Acknowledgment

This work is granted by the Special Research Foundation for thePublic Welfare Industry of the Ministry of Science and Technologyand the Ministry of Water Resources (No. 200701008), NationalS&T Supported Plan of China (No. 2008BAB29B08) and the NationalNatural Science Foundation of China (50539140).

References

Asai, K. (1995). Fuzzy system for management, Ohmsha.Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1),

87–96.Atanassov, K. (1994). Operators over interval-valued intuitionistic fuzzy sets. Fuzzy

Sets and Systems, 64(2), 159–174.Atanassov, K., & Gargov, G. (1989). Interval-valued intuitionistic fuzzy sets. Fuzzy

Sets and Systems, 31(3), 343–349.Bustince, H., & Burillo, P. (1996). Vague sets are intuitionistic fuzzy sets. Fuzzy Sets

and Systems, 79(3), 403–405.Calvo, T., Mayor, G., & Mesiar, R. (2002). Aggregation operators: New trends and

applications. Heidelberg, Germany: Physica-Verlag.Gau, W. L., & Buehrer, D. J. (1993). Vague sets. IEEE Transactions on Systems, Man, and

Cybernetics, 23(2), 610–614.Herrera, F., & Herrera-Viedma, E. (1997). Aggregation operators for linguistic

weighted information. IEEE Transactions on Systems, Man and Cybernetics, 27(5),646–656.

Herrera, F., & Herrera-Viedma, E. (2000). Decision analysis: Steps for solvingdecision problems under linguist information. Fuzzy Sets and Systems, 115(2),67–82.

Hong, D. H., & Choi, C. H. (2000). Multi-criteria fuzzy decision-making problemsbased on vague set theory. Fuzzy Sets and Systems, 114(1), 103–113.

Hwang, C. L., & Yoon, K. S. (1981). Multiple attribute decision making: Methods andapplications. Berlin: Springer.

Iyer, N. S. (2003). A family of dominance rules for multi attributes decision-makingunder uncertainty. IEEE Transactions on Systems, Man, and Cybernetics Part A, 33,441–450.

Xu, Z. S. (2002). A new method solving multi attribute decision making withuncertainty. Journal of Systems Science, 17(3), 176–181.

Xu, Z. S. (2003). A direct approach to group decision making with uncertain additivelinguistic preference relations. Technical Report.

Xu, Z. S. (2004). The uncertain multi-attribute decision-making method and itsapplication. Beijing: Tsinghua University Press.

Xu, Z. S. (2006). On correlation measures of intuitionistic fuzzy sets. Lecture Notes inComputer Science. Berlin: Springer-Verlag. pp. 16–24.

Xu, Z. S. (2007). Intuitionistic fuzzy aggregation operators. IEEE Transactions onFuzzy Systems, 15(6), 1179–1187.

Xu, Z. S., & DA, Q. L. (2003). An overview of operators for aggregating information.International Journal of Intelligent Systems, 18(9), 953–969.