EXTENSIONS OF LINMAP MODEL FOR MULTI CRITERIA DECISION MAKING WITH GREY NUMBERS

. The linear programming technique for multidimensional analysis of preference, known as LINMAP is one of the existing well-known ideal seeking methods for multi attribute decision making problems. This method originally is proposed under crisp and deterministic circumstances. However, uncertainty is an indubitable property of decision making problems. In this paper, a new version of LINMAP-G is proposed where the decision maker’s judgments are expressed as grey numbers. Like original LINMAP method, the grey ideal solution and attributes weight vector is determined and alternatives are ranked according to their weighted distance from determined ideal point. Application of the proposed method is illustrated in two numerical examples.


Introduction
Decision makers always seek a criterion to appraise their decisions. In this context, decisionmaking methods arise when decision maker simultaneously envisages various criteria for evaluating his or her decisions favorite (Kuo et al. 2008). Such a problem is the subject of multiple criteria decision making (MCDM) methods Peng et al. 2011;Kou et al. 2012). This class is further divided into multi objective decision making (MODM) and multi attribute decision making (MADM) (Climaco 1997). A formalized definition of MADM problems can be stated as follows.
Let we have a nonempty and finite set of decision alternatives, i.e. 1 2 , , , m A A A  , and there are a finite set of goals, attributes or criteria, i.e. 1 2 , , , n C C C  , according to which the desirability of an alternative is to be judged. The aim of MADM is to determine the optimal alternative with the highest degree of desirability with respect to all relevant goals (Zimmerman 1987). An optimal alternative in MADM problems can be defined as an alternative A* that has the highest value in all decision-making criteria. Usually, MADM problems do not have an optimal solution in practice and current methods seek an alternative with the highest degree of satisfaction for decision makers. In last decades, MADM techniques have a wide application in different areas that concern with selection.
Multi attribute decision making methods require decision makers judgments and evaluations about alternatives performance regard to multiple attributes. These judgments are a subject of uncertainty. Indeed, decision makers do not have complete information about alternatives or their conditions regard to a certain attribute. Therefore, it will be so difficult for them to express their evaluations based on exact numbers. Hence, uncertainty contexts are widely applied in MADM problems.
Fuzzy set theory (FST) was developed by Zadeh (Zadeh 1965) as a generalized form of the classical set theory that assigns a membership degree to each element of a given set in a universe. FST is one of the well-known paradigms in studying systems with uncertainty. Bellman and Zadeh (Bellman, Zadeh 1970) have introduced the concept of decision making under fuzzy environment. Afterward, MADM techniques have been extended under fuzzy environment (Aouam et al. 2003;Yazdani et al. 2011;Xu 2004;Li, Yang 2004;Wang, Chuu 2004;Hu et al. 2004;Antucheviciene et al. 2011;Kersuliene, Turskis 2011;Brauers et al. 2011;Fouladgar et al. 2011;Balezentis, Balezentis 2011).
The Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) was developed by Srinivasan and Schocker (Srinivasan, Shocker 1973) as one of the MADM tech-niques that determines the preference order among a set of alternatives by determination of weight vector w and positive ideal solution (PIS) vector. However, the LINMAP can only deal with MADM problems in crisp environments. Xia et al. (Xia et al. 2006) developed LINMAP method for MADM problems under fuzzy environment. Li and Sun (Li, Sun 2007) developed LINMAP method for MADM problems with linguistic variables and incomplete preference information. Li (2008) also developed this method under intuitionistic fuzzy environment. Considering the simplicity and clearness of grey numbers in expressing the uncertainty and lack of knowledge, in this paper, a LINMAP method is extended with grey data. The rest of the paper is organized as follows: Section 2 briefly introduces the grey numbers and their operations. MADM problems with grey data are expressed in section 3. The extended grey LINMAP model and proposed decision process is introduced in section 4. The proposed method is illustrated with an example in section 5. Finally, the paper is concluded in section 6.

Grey numbers
As stated in previous section, a grey number can be indicated as a range. In fact, a number x is called an grey number when its exact value is unknown and only it is known that , where x is the lower bound and x is the upper bound, such that x x  . Arithmetic operations on interval numbers are introduced by Moore (Moore 1966 , ( ) ( ) min , , , ,max , , , x y x y xy x y xy x y xy x y xy The center, x C , and width, x W of a grey number x are defined as follows (Ishibuchi, Tanaka 1990):

Grey MADM problem definition
Consider an MADM problem that consist evaluation of a set of m alternatives regard to a set of n attributes. In classic form, ratings of alternatives regard to attributes are stated with crisp data. However in many situations, and due to uncertainty or lack of knowledge, the crispness of ratings is an unfair assumption. Therefore, suppose that the ratings values are expressed in form of grey numbers. If 1 2 , , , m A A A  are m possible alternatives and 1 2 , , , n C C C  criteria over which alternatives performance are measured, and ,  is the rating of alternative i A , with respect to criterion j C , then a grey MADM problem can be concisely defined in a decision matrix as follows.
In addition, the weigh vector of criteria is defined as The problem here is to rank the alternative set's elements.

Grey LINMAP
In this section, the proposed method of LINMAP-G (Srinivasan, Shocker 1973;Hwang, Yoon 1981) with grey data is developed. It is noted that the LINMAP-G method seeks a positive ideal solution (PIS) and a weight vector w that minimizes the distance of a set of preference relations among alternatives that are expressed priori by decision makers from unknown PIS.

Normalization of Grey decision matrix
An intrinsic aspect of MADM problems is that different attributes have different dimensions that make their comparison impossible. Therefore, an initial step before the decision making process, is to normalize the grey decision matrix, defined in previous section. Different procedures are introduced to normalize grey decision matrix. In this paper the method proposed in Jahanshahloo et al. (Jahanshahloo et al. 2006) is applied by modifications. If attribute j is as profit (maximization) type, its normalized values are calculated as: Otherwise, if attribute j is as cost (minimization) type, its normalized values are calculated as:

Grey LINMAP modeling process
The main idea of LINMAP-G method is to determine an unknown PIS vector, like . Then, the best alternative is chosen as the nearest one to this PIS vector. A note here is to define the distance between two grey vectors. If are two grey vectors, and is a weight vector, the weighted grey numbers Euclidean distance is defined as follows: The inconsistency between alternatives k A and l A ranking based on k s and l s with preference relations that are determined by decision maker is measured by an inconsistency In fact, the alternatives k A and l A ranking is consistent with decision maker's preferences if k l s s Now, the total inconsistency index is defined as follows: Similarly, the consistency index is defined as follows: Eq. (13) can be written as follows: Therefore, total consistency index is defined as follows: Note that whether l k s s ≤ or l k s s  , the following relation is hold: The grey LINMAP-G model to determine the PIS ( ) , , , n W w w w =  can be constructed as follows: . . 1,2, , where, h is a constant that is determined by decision maker. Also, 0 ε  is a sufficiently small real value that guarantees that obtained weights are greater than zero. The objective of model (17) is to maximize the consistency index C, while it will be greater than I at least as pre-determined value of h.
Using Eq. (15) and (16), the model (17) The variable kl λ is introduced as follows: For each pair ( ) , k l ∈Ω the following relations are hold: Substituting Eq. (19) -(21), the model (18) is transformed to the following model: The final LINMAP-G model is achieved by acquisition of a corresponding relation for k l s s − . Using Eq. (9) and the definition of variable i s , this relation is obtained as follows: The model (22) is now transformed into the following model, which is called grey LINMAP-G model. where, and j j j v w x * = . (26) Note that constraints 1,2, , , , , n W w w w =  are determined. The optimal weights of attributes can be determined after normalization of weight vector W. Finally, the ranking of alternatives are specified by calculation of i s variables for all alternatives and ascending sort of these values. Figure 1 shows an algorithm about the decision making process with LINMAP-G method. It is possible that decision maker has some viewpoints regard to weight vector, such that he/ she do not want none of the attribute's weights be greater than other's weights. This set of constraints can be added to model as , , 1,2, , ,

Numerical example
In this section, two numerical examples are solved by proposed LINMAP-G decision making process.

Ranking of constructing projects
The first study is done on a relatively small instance. This example is about a company that wants to rank its target market sectors. The company's market is divided into five different sectors A, B, C, D, and E that are evaluated based on four attributes: three attributes include (1) market size, (2) market growth and (3) consistency with company's mission as profit attributes and a (4) structural risk attribute as cost attributes. The grey decision matrix is constructed as follows (see Table 1). . The first step is to normalize the decision matrix. Attributes 1-3 are normalized based on Eq. (7) and the attribute 4 by Eq. (8). The normalized decision matrix is shown in Table 2.  In the next step, the grey LINMAP-G method is developed according to Eq. (24) as follows. Assume that decision maker wants that none of the attributes weights be more than three times greater than the others. Note that 1 h = and 0.001 ε = .
The optimal solution of the above model is as follows: Therefore, the ranking order of alternatives are D  E  A = C  B.

Conclusions
Constructing project selection is an important issue, according to high risks and costs of mistakes. Therefore decision making based on multiple attributes eases it to prevent these likely problems. In construction evaluation problems, the decision maker selects some alternatives among different ones and it is necessary to consider different qualitative and quantitative criteria. Evaluation of a set of alternatives regard to a set of quantitative or qualitative attributes is the main concentration of multi attribute decision making problems. In crisp MADM algorithms, subjective judgments and qualitative measures are translated into crisp numbers. This transformation means that decision maker ignores the uncertainty and ambiguity of his/her thinking and believes. Therefore some frameworks are presented to handle these uncertainties. According to uncertainty in the real world, we tried to calculate these parameters by uncertain data and in this paper, a new version of LINMAP method, originally presented in (Srinivasan, Shocker 1973), where decision maker's judges are expressed as grey number is proposed. The proposed method ranks alternatives by solving a linear programming that determines the attributes weight vector and an ideal solution. Then, the alternatives are ranked regard to their distances from PIS by specified weights. Application of the developed method is shown in two constructing examples that one of them was about ranking a set of various constructing projects for a developer company and another example was about ranking a set of constructing contractors. This suggests that the proposed method can be applied in different multi attribute decision making problems which contain uncertainty and ill-defined data and decision maker has not determined attributes weights priori.  Structures (1973) andDr Sc. (1987) in Building Technology and Management. He is a member of the Lithuanian and several foreign Academies of Sciences. He is Doctore Honoris Causa at Poznan, Saint-Petersburg, and Kiev universities as well as a member of international organizations; he has been a member of steering and programme committees at many international conferences. E. K. Zavadskas is a member of editorial boards of several research journals. He is the author and co-author of more than 400 papers and a number of monographs in Lithuanian, English, German and Russian. Research interests are: building technology and management, decision-making theory, automation in design and decision support systems.