CHEMICAL ENGINEERINGTRANSACTIONS VOL. 51, 2016 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Guest Editors:Tichun Wang, Hongyang Zhang, Lei Tian Copyright © 2016, AIDIC Servizi S.r.l., ISBN978-88-95608-43-3; ISSN 2283-9216 A Method for Group Decision-making with Uncertain Preference Ordinals Based on Probability Matrix Minglin Jianga, Xiaowei Lina, Chen Youa ,Haiping Ren*b aSchool of Business ,Minnan Normal University.Zhangzhou,363000,P.R.China bSchool of Software, Jiangxi University of Science and Technology, Nanchang, 330013, P.R. China chinarhp@163.com In this paper, a new method to solve the group decision-making problems is proposed, in which the preference information on alternatives provided by decision makers is in the form of uncertain preference ordinals. This paper firstly gives two new definitions on the probability that the alternative is ranked in each position. Then, in order to process uncertain preference ordinals, two new definitions are used respectively to construct a decision matrix in the form of probabilities. On this basis, a weight probability matrix and a collective probability matrix on alternatives with regard to rank positions are constructed. Finally, an optimization model is built based on the collective probability matrix, and the ranking of alternatives can be obtained by solving the model. 1. Introduction Multiple criteria decision making (MCDM) is a discipline aimed at supporting decision maker who is faced with numerous and conflicting alternative to make an optimal decision (Pedrycz, 2013; Mardani et al., 2015). While group decision-making (GDM) is decision-making in groups consisting of multiple members. Multiple criteria group decision-making (MCGDM) problem involves a set of feasible alternatives that are evaluated on the basis of multiple, conflicting and non-commensurate criteria by a group of individuals. Meanwhile, it is a problem with extensive theoretical and practical backgrounds in industrial engineering (Soltani et al., 2015; Wijenayake, et al. 2016; Ravindran, 2016).In the real world, due to the complexity and uncertainty of decision- making problems, the limitation of cognition, the estimation inaccuracies and lack of decision makers’ knowledge, the preference rankings or ranking ordinals of alternatives provided by decision makers maybe in the form of uncertain preference ordinals. However, few approaches to solve MCGDM problems with preference rankings or uncertain preference ordinals can be found in the existing literature. The existing approaches have made to solve the GDM problems with uncertain preference ordinals on alternatives. This paper investigates the MCDM problems, where the preference information on alternatives provided by decision makers is in the form of uncertain preference ordinals. In order to solve these problems Fan et al. (2010) gave several definitions on uncertain preference ordinal, constructed a decision matrix in the form of probabilities, and built an optimization model based on the collective probability matrix. Fan’s approach can solve the GDM problem effectively. However, it is regarded that each ranking position of an uncertain preference ordinal on alternative has the same probability in Fan’s approach but not in line with laws of human cognition. For example, supposing as uncertain preference ordinal of an alternative on interval [3, 7] , the decision maker often thinks the probability of the alternative ranked in 5th position is higher than in 3th or 7th position (Xu, 2005). In fact, the closer a preference ordinal to the lower bound and upper bound of an uncertain preference ordinal, the smaller the probability that the alternative is ranked in the corresponding position. At the same time, the closer that preference ordinal in the centre of an uncertain preference ordinal, the larger probability that the alternative is ranked in the corresponding position. Focusing on the above object, this paper will firstly improve Fan’s model by giving two new definitions on the probability that the alternative is ranked in each position, which is raised by Wang and Xu (2008). Then, to process uncertain preference ordinals, a matrix in the form of probabilities is constructed. Based on the probability matrix, a weight probability matrix and a collective probability matrix on alternatives with regard to DOI: 10.3303/CET1651105 Please cite this article as: Jiang M.L., Lin X.W., You C., Ren H.P., 2016, A method for group decision-making with uncertain preference ordinals based on probability matrix, Chemical Engineering Transactions, 51, 625-630 DOI:10.3303/CET1651105 625 rank positions are constructed. Furthermore, an optimization model is built based on the collective probability matrix, and the ranking of alternatives can be obtained by solving the model. 2. Preliminaries In this section, we will introduce some basic concepts related to multiple criteria group decision making problems, which the preference information on alternatives provided by decision makers is in the form of uncertain preference ordinals. Definition 1. (Fan et al, 2010) Let Z be the set of positive integer. An uncertain preference ordinal r is expressed in  , 1, ,L L Ur r r r  , where , 1, ,L L Ur r r Z  , L Ur r , Lr and Ur are the lower bound and upper bound of r .For simplicity, we express r as ,    L U r r r . Remark 1. Let m denotes the number of all alternatives in a GDM analysis as well as the total number of ranking positions. Let  1, 2, ,M m be the set of all the ranking positions, where 1, 2, , m denote that the ranking position is the 1 , 2 , ,st nd mth , respectively. If k , k M represents that the ranking position of an alternative is the kth ,then the smaller k is, the better the corresponding alternative will be. Thus, for uncertain preference ordinal ,L Ur r r    , ,  L U r r M . Remark 2. Consider an uncertain preference ordinal ,L Ur r r    .Let 1   r U L u r r , then ru denotes the number of possible ranking positions in r and it is also viewed as the uncertainty degree of r . Thus, the greater ru is, the greater the uncertainty degree of r will be. In multiple criteria group decision-making (MCGDM) process, let  1 2, , , mS s s s  2m  be a finite set of alternatives  1 2, , , nC c c c  2n  be a set of criteria,  1 2, , , n    is the weight vector of criteria ( 1, , )jc j n ,where 0j  , 1 1 n j j    , Let  1 2, , lE e e e  2l  be the set of decision makers,  1 2, , , l    be the weighting vector of decision makers, with 0 t   , 1 1 l t t    .Suppose the decision makers  1, ,te t l provide their preferences for alternatives in the form of uncertain preference ordinals, then, the following definitions are obtained. Definition 2. (You et al., 2013) Let Z  be the set of positive integer. An uncertain preference ordinal t ij r is expressed in  , 1, ,t L L Uij ijt ijt ijtr r r r  , where , 1, , L L U ijt ijt ijt r r r Z    , L U ijt ijt r r , t ij r indicates the preference information that the alternative i s satisfies the criteria j c given by the decision maker t e , Lijtr and U ijt r are the lower bound and upper bound of t ij r , 1, 2, ,i m , 1, 2, ,j n , 1, 2, ,t l . Especially if L U ijt ijt r r , then t ij r reduces to a ranking ordinal. For simplicity, we express t ij r as ,t L U ij ijt ijt r r r    . Remark 3. Let m denotes the number of all alternatives in a MCGDM analysis as well as the total number of ranking positions. Let  1, 2, ,M m be the set of all the ranking positions, where 1, 2, , m denote that the ranking position is the 1 , 2 , ,st nd mth , respectively, if k , k M represents that the ranking position of an alternative is the kth ,then the smaller k is, the better the corresponding alternative will be. Thus, for uncertain preference ordinal ,t L U ij ijt ijt r r r    , , L U ijt ijt r r M , the smaller Lijtr or U ijt r is, the better the ranking position of the alternative will be. If ,L Ur r r    , L r , Ur M is an uncertain preference ordinal on an alternative provided by the decision maker. Fan et al (2010) regarded that the alternative could be ranked in position , 1, ,L L Ur r r with the same possibility, and gave the definition of the probability vector. Definition 3. (Fan et al, 2010) Let ,L Ur r r    ( L r , Ur M ) be an uncertain preference ordinal on an alternative provided by the decision maker. Then, the probability vector on r is represented by  1 2, , , mr r r rp p p p and the elements of rp are given by 626 0, 1,2, , 1; 1 , =r , 1, , ; 0, = 1, 2, , . L k U L L U r r U U k r p r k r r u k r r m            (1) Where k r p ( 1, 2, ,k m ) denotes the probability that the alternative is ranked in the kth position, satisfies 1 1 m k r k p   and 0 1 k r p  ， 1, 2, ,k m . Definition 4. Let ,L Ur r r    ( L r , Ur M ) be an uncertain preference ordinal on an alternative provided by the decision maker, then, the probability that the alternative is ranked in each position in ,L Ur r   is represented by: 1 12 h L n h n C v     , , 1, ,h L L U  , 1n U L   (2) where 0 h v  , 1 U h h L v   ,and 0 0 1 0 1C C  .That is, the alternative could be ranked in position , 1, , L L U r r r with possibility 112 h L n n C    . Definition 5. Let ,L Ur r r    ( L r , Ur M ) be an uncertain preference ordinal on an alternative provided by the decision maker, then, the probability that the alternative is ranked in each position in ,L Ur r   is represented by:     2 2 2 2 2 2 , , 1, , n n n n h u h h u U j L e v h L L U e             (3) where     211 1 1 , , 1 2 2 U n n h L n n n u h u h h L n n             . Remark 4. It is easy to improve that h v in Definition 4 has the following well-known properties:1) h v is symmetrical, i.e., 1 ( 1, 2, , ).h n hv v h n   2) the probability vector is 1 2( , , , ) T n v v v v ,when 2 1n k  ,which satisfy 1 2 1 1 3 1 2 2 2 n n n n n v v v v v v v             (4) when 2n k ,which satisfy 1 2 1 1 1 2 2 2 = n n n n n v v v v v v v           (5) While h v in Definition 5 has a same property as h v . Consider an uncertain preference ordinal , , ,t L U L U ij ijt ijt ijt ijt r r r r r M    and motivated by Definition 4 and Definition 5, we have the following definitions. Definition 6. Let , , ,t L U L U ij ijt ijt ijt ijt r r r r r M    .Then, the probability vector on t ij r is represented by  1 2, , ,t t t t ij ij ij ij m r r r r p p p p and the elements of t ijr p are given by ~ ~ 1, 2, , 1; , 1, , , , 0, , 2 0 1, , 1; 1, 2, , ., k t ij L ijt L L U ijt ijt ijt U U ij h L t ijt U L r U L k r r r r h L L U L r r p m c k k                      (6) 627 Where denotes t ij k r p denotes the probability that the alternative is ranked in the k th position, such that 1 1t ij m k r k p   and 0 1, 1, 2, , .t ij k r p k m   Definition 7. Let , , , , , 1, , 1.t L U L U ij ijt ijt ijt ijt r r r r r M h L L U L        Then, the probability vector on t ij r is represented by  1 2, , ,t t t t ij ij ij ij m r r r r p p p p and the elements of t ijr p are given by     2 2 2 2 2 2 0, 1, 2, 1; , , 1, , , 1; 0 , 1, 2, , . n n t ij n n L ijt h u k L L U ijt ijt ijtr h u U j L U U ijt ijt k r e p k r r r h h L e k r r m                            (7) where t ij k r p denotes the probability that the alternative is ranked in the k th position, such that 1 1t ij m k r k p   and 0 1, 1, 2, , .t ij k r p k m   Remark 5. If L U ijt ijt r r for uncertain preference ordinal , , ,t L U L Uij ijt ijt ijt ijtr r r r r M    , i.e. , t ij r reduced to a ranking ordinal, then the elements of probability vector  1 2, , ,t t t t ij ij ij ij m r r r r p p p p on tijr are given by ~ ~ 1, 2, , 1; ; 1 0, 1, 0, , 2, , . k t ij L ijt L U ijt ijt U U ijt ijt r k r r rp k r mk r            (8) Definition 8. Let 1 2, , , t t t i i in r r r be n uncertain preference ordinals and   2 2 2 2 1 2, , , ,t t t t i i i i m r r r r p p p p , t inr p   1 2, , ,t t t in in in m r r r p p p be the corresponding probability vectors. Let  1 2, , , nw w w w be a weight vector, where i w denotes the weight of i r such that 1 1 n j j w   and 0 1, 1, 2, , .jw j n   Then, the overall probability vector on 1 2 , , ,t t t i i inr r r p p p is represented by   0 0 0 0 1 2, , ,t t t t i i i i m r r r r p p p p and the elements of 0 t ir p are given by 0 1 , 1, 2, ,   t t i ij n k k jr r j p w p k m (9) 3. The proposed approach In this section, this paper will present a handing method for MCGDM problems with uncertain preference ordinals. Firstly, it gives a brief description of the MCGDM problems with uncertain preference ordinals. Then, a probability matrix, the voting information matrix, the collective voting information matrix and an optimization model are constructed. Finally, an algorithm for determining the ranking position of each alternative is given. Let   1 2, , , 2mS s s s m  be a finite set of alternatives,  1 2, , , ( 2)nC c c c n  be a set of criteria, whose weight vector is  1 2, , , nw w w w , where 0( 1, , )jw j n  , 1 1 n j j w   and  1 2, , , ( 2)lE e e e l  be the set of decision makers whose weight vector is  1 2, , , l    ,where 0, 1, ,t t l   ,and 1 1 l t t    . Suppose the decision makers  1, ,te t l provide their preferences for alternatives in the form of uncertain preference ordinals, i.e., , , ,t L U L U ij ijt ijt ijt ijt r r r r r M    , where t ij r denotes the preference information that the alternative i s satisfies the criteria j c given by the decision maker t e , 1, 2, , , 1, 2, , , 1, 2, ,i m j n t l   .The problem concerned in this paper is to rank alternatives or to select the most desirable alternative(s) among a 628 finite set S based on uncertain preference ordinals t ij r .The method is described as follow: Let  1 2, , ,t t t tnij ij ij ijp p p p be probability vector on uncertain preference ordinals t ij r .It can be determined according to Definition 2.6 or Definition 2.7, where tk ij p denotes the probability that criteria j c is ranked in the kth position, such that 1 1 m tk ij k p   and 0 1, 1, 2, , ; tk ij p i m   1, 2, , .j n For the convenience of analysis, the decision matrix in the form of probabilities based on t ij p is constructed as follows:   1 2 1 11 11 1 2 21 22 2 1 2 n t t t n t t t t n ij m n t t t m m m mn c c c s p p p s p p p p p s p p p                (10) Using Eq.(10), the elements of the ith row of the probability matrix tP and the weight vector w are aggregated to form the weight probability vector on alternative i s ,which takes the weight of the criteria  1, 2, ,jc j n in to consideration, and it is given by 1 , 1, 2, , ; 1, 2 , , ,tk l tk i j jk j i kq w jp m n     (11) Through Eq.(11), it can be easily seen that m 1 1tk i k q   .Based on the obtained vectors  1, 2, , tk i q i m ,the weight probability matrix t ( )tk mi m qQ   can be constructed, i.e.,   1 2 1 1 1 1 1 2 2 2 2 2 1 2 1 2 t t tm t t tm t tk i m m t t tm m m m m m s q q q s q q q Q q s q q q                (12) The elements of the weight probability matrix tQ and vector  are aggregated to form the collective probability vector on alternative i s , i.e.,  1 2, , , mi i i i    , where k i  denotes the collective results by all the decision makers that alternative i s is ranked in the k th position and it is given by k 1 , 1, 2, ,, l tk i t i t i k mq     . Based on vectors k i  ,this paper constructs the following collective probability matrix  ,i.e., 1 2 1 1 1 1 1 2 2 2 2 2 1 2 m 1 2 m s s = s m m m m m m                       (13) Based on collective probability matrix  , this paper attempts to develop a method to determine the ranking position of each alternative that one alternative is only ranked in one ranking position. The description of this method is given below. Let  , 1, 2, ,ikb i k m be 0-1variable, where 1 k i b  represents that alternative i s is ranked in the k th position and 0k i b  , otherwise. The total probability that m alternatives are ranked in m positions can be expressed as 1 1 k k m m i i i k b    , where 1 1, 2, , 1( )k m i i k mb   and 1 1, 2, , 1( )k m i k i mb   .To rank alternatives or select the best alternative(s), we can construct the following optimization model max z= 1 1 m m k k i i i k b     (14) Subject to 1 1, 1, 2, , m k i i b k m    , 1 1, 1, 2, , m k i i b i m    , 0 1, , 1, 2, , k ib or i k m  . 629 Existing mathematical optimization software can be used to solve model (14). In summary, we give an algorithm to determine the ranking position of alternatives and its steps are presented as follows: step1: Calculate probability vectors t ij p by Eq.(7) or Eq.(8) based on , 1, 2, , ; 1, 2, ,t ij r i n t l  . step2: Construct probability matrix  t tij m n P p   based on t ij p , 1, 2, , ; 1, 2, , .i m j n  step3: Construct the weight probability matrix t tk i m m Q q      by Eq.(11). step4: Construct collective probability matrix k i m m        based on t tk i m m Q q      . step5: Build the optimization model (14) based on matrix  and solves it by Hungarian method. step6: Determine the ranking position of each alternative based on the obtained optimal solution(s) of model (14) and record the probability of ranking position of alternative based on matrix  . 4. Conclusion In multiple criteria group decision-making situations that the decisions makers can not give the exact value, the decision makers may be suitably expressed with preference ordinals. Focusing on this problem, this paper proposes a new method to solve the MCDM problems that the preference information is in the form of uncertain preference ordinal. It improves the method proposed by Fan, and is more in line with the law of general human cognition. First, it develops two normal distribution-based methods to determine the probability that the alternative is ranked in each position. Then, in order to process uncertain preference ordinals, a matrix in the form of probabilities is constructed. Furthermore, a weight probability matrix and a collective probability matrix on alternatives with regard to ranking positions are constructed. Finally, an optimization model is built based on the collective probability matrix, and the ranking of alternatives can be obtained by solving the model. The methods proposed in this paper may also be used in MCDM problems with other preferences information formats, e.g., interval and set-values, uncertain linguistic variables, etc. Acknowledgment This paper was supported by Soft Science Project of Fujian Natural Science Foundation (no. 2016R0074). References Fan Z. P., Liu Y., 2010, An approach to solve group decision-making problems with ordinal interval numbers, IEEE Transaction on Systems, Man and Cybernetics, Part B: Cybernetics, 40(5), 1413-1423. DOI:10.1109/TSMCB.2009.2039477 Fan Z. P., Yue Q., Feng B., Liu Y., 2010, An approach to group decision-making with uncertain preference ordinals, Computers and Industrial Engineering, 58(1), 51-57, DOI:10.1007/s11518-011-5185-7 Mardani, A., Jusoh, A., Zavadskas, E. K. 2015, Fuzzy multiple criteria decision-making techniques and applications-Two decades review from 1994 to 2014, Expert Systems with Applications, 42(8), 4126-4148, DOI:10.1016/j.eswa.2015.01.003 Pedrycz W. 2013, Analysis and Design of Intelligent Systems, Granular Computing, CRC Press, Boca Raton, Francis. Ravindran, A. R. 2016, Multiple Criteria Decision Making in Supply Chain Management, CRC Press, Boca Raton, Francis. Soltani, A., Hewage, K., Reza, B., Sadiq, R. 2015, Multiple stakeholders in multi-criteria decision-making in the context of Municipal Solid Waste Management: A review. Waste Management, 35, 318-328, DOI:10.1016/j.wasman.2014.09.010 Wang Y., Xu Z. S., 2008, A new method of giving OWA weights, Mathematics in Practice and Theory, 38(3), 51-61. Wijenayake, W. K., Amarasinghe, U. S., De Silva, S. S. 2016, Application of a multiple-criteria decision making approach for selecting non-perennial reservoirs for culture-based fishery development: Case study from Sri Lanka, Aquaculture, 459, 26-35, DOI:10.1016/j.aquaculture.2016.03.019 Xu Z. S. 2005, An Overview of methods for determining OWA weights, International Journal of Intelligent Systems, 20: 843-865, DOI: 10.1002/int.20097 630