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The uncertain multi-attribute decision making (UMADM) problem that the information about attribute weights is known partly and the decision maker (DM) has preference information on alternatives in the form of interval numbers complementary judgment matrix is investigated. Firstly, based on interval numbers complementary matrix the objective decision information is uniformed by using a translation function and then a goal programming model is established. The attribute weights are obtained by solving the model and thus the overall values of every alternative are gained. Secondly, by using an existing priority formula of possibility the alternatives are ranked. An approach based on the goal programming model and possibility is proposed. It is characterized by simple operation and easy to implement on computer. Finally, a practical example is illustrated to show the feasibility and availability of the developed method. Introduction Because of the complexity of objective thing and the uncertainty of DM’s understandings, moreover, the decision maker take part in the process of decision-making, that is the subjective preference information of DM is often used to rank alternatives, this makes the UMADM problems with uncertain preference information on alternatives become a hot topic of research in Refs [1-7]. At present, these problems have been investigated and some methods of solving them have been presented by some scholar. The UMADM problems that the information about attribute weights is unknown completely and the DM has interval numbers utility preference on alternatives are studied in Refs [2-3], the linear programming and least deviation method are proposed respectively. The UMADM problems that the attribute weights are interval numbers and DM’s preference relation on alternatives is in the interval numbers reciprocal or complementary judgment matrix judgment matrix are investigated in Refs [4-5], the attribute weights are obtained by establishing a linear programming model, an approach based on interval positive ideal point of alternatives and projection to rank alternatives is presented in Ref [4]. The method based on based on the additive consistency of interval numbers complementary judgment matrix and C-OWA operator is presented in Ref [5]. With respect to UMADM problems that decision-making information are interval numbers and preference information on alternatives, the method of grey relational analysis is proposed in Ref [6]. The UMADM problems is investigated,in which the DM has preference information on alternatives in the form of interval reciprocal judgment matrix and the information about attribute weights take the form of interval complementary judgment matrix, a maximizing model based on the consistency of preference information is established to asses the weights of attributes in Ref [7]. But the interval attribute vales are made exact numbers, so there exist the loss of decision making information. Based on the above references a new approach will be proposed to solve the UMADM problems that the information about attribute weights is known partly and DM’s preference information on alternatives take the form of interval numbers complementary judgment matrix in this paper. Firstly, we makes the objective decision information uniform based on interval numbers complementary matrix by using a translation function and then a goal programming model is established. The attribute weights are obtained by solving the model and thus the overall values of every alternative are gained. Secondly, by using an existing priority formula of possibility the 4th National Conference on Electrical, Electronics and Computer Engineering (NCEECE 2015) © 2016. The authors Published by Atlantis Press 792 alternatives are ranked. An approach based on the goal programming model and possibility is proposed. It is characterized by simple operation and easy to implement on computer. Finally, a practical example is illustrated to show the feasibility and availability of the developed method. Preliminary knowledge Definition 1 Let [ , ] { 0 } L R L R a a a x a x a = = ≤ ≤ ≤  , then a is called an interval number. Specially, a is degenerated to be a real number if L R a a = in Ref [2]. In order to compare interval numbers, the following concept is introduced. Definition 2 Let [ , ] L R a a a =  , [ , ] L R b b b =  , Then ( ) p a b ≥   is called possibility of a b ≥   in Ref [2],where ( ) min{max[ ,0],1} ( ) ( ) R L R L R L a b p a b a a b b − ≥ = − + −   (1) The basic model of the UMADM problem which will be studied in this paper can be represented as follows:  Let 1 2 { , , , } n X x x x =  be a set of ( 2) n ≥ feasible alternatives, and denote {1,2, , } N n =  ;  Let 1 2 { , , , } m S s s s =  be a set of ( 2) m ≥ attributes , and denote {1,2, , } M m =  ;  Let 1 2 ( , , , ) T m ω ω ω ω =  be the vector of weights, such that 0, R L k k k ω ω ω ≥ ≥ ≥ 1 1 1 1, 1, 1 m m m L R k k k k k k ω ω ω = = = ≤ ≥ = ∑ ∑ ∑ , where L k ω and R k ω are the left and right bounds of k ω , respectively. And k ω is the weight of attribute k s ;  Let   ( ) ij m n A a × = be an objective decision matrix, where [ , ] L R ij ij ij a a a =  is the attribute value for the alternative j x with respect to attribute i s , , i M j N ∈ ∈ ;  Let ( ) lj n n B b × =   be the DM’s uncertain complementary preference matrix on alternatives, where lj b denotes a ratio of preference degree for the alternative l x over j x , and such that [ , ] L R lj lj lj b b b =  , 1 L R R L lj jl lj jl b b b b + = + = , , l j N ∈ . In the general case, there is benefit and cost attribute values in the UMADM, and the different attribute values may be the different dimension. For the convenience of decision-making, we need to deal with all attributes in dimensionless units and normalize each attribute value. This can be achieved by normalizing  ij a in the matrix   ( ) ij m n A a × = into a corresponding element ij r in the matrix  ( ) ij m n R r × =  with the following formulas in Ref [2],where [ , ] L R ij ij ij r r r =  , , i M j N ∈ ∈ .

Approach for uncertain multi-attribute decision-making with interval numbers complementary preference relation on alternatives under partial weight information