A Soft-Computing Approach for Multi Criteria Project Selection Problem with Grey Number

Multi-criteria decision support systems are used in various fields of human activities. In every alternative multi-criteria decision making problem can be represented by a set of properties or constraints. The properties can be qualitative & quantitative. For measurement of these properties, there are different unit, as well as there are different optimization techniques. Depending upon the desired goal, the normalization aims for obtaining reference scales of values of these properties. This paper deals with the multi-attribute Complex Proportional Assessment of alternative. In order to make the appropriate decision and to make a proper comparison among the available alternatives Analytic Hierarchy Process (AHP) under fuzziness and COPRAS method with grey numbers has been used. The uses of AHP is for analysis the structure of the project selection problem is used under fuzziness and to assign the weights of the properties and the COPRAS-G method is used to obtain the final ranking and select the best one among the projects. To illustrate the above mention methods survey data on the expansion of optical fiber for a telecommunication sector is reused. The decision maker can also used different weight combination in the decision making process according to the demand of the system.COPRAS-G method is used to evaluate the overall efficiency of a project with the criterion values expressed in terms of intervals. It is based on the real conditions of decision making and applications of the grey number theory


INTRODUCTION
MCDM problems are encountered under various situations where a number of alternative and criteria need to be chosen based on a set of attributes. Decision making analysis is based on MCDM theory. The three founder of decision making analysis are Howard, Keeney and Raiffa. [1,2]. The analysis of the purpose is to be achieved by using attributed of effectiveness, which have different dimensions, different weight as well as different directions of optimization. The discrete attribute values can be normalized by applying different normalization method. According to Hwang and Yoon [3] Multicriteria decision making (MCDM) is applied to preferable decisions among available classified alternatives by multiple attributes. So MCDM is one of the most widely used decision methodology in project selection problems. The MCDM is a method that follows the analysis of several criteria, simultaneously. In this method economic, environmental, social and technological factors are considered for the selection of the project and for making the choice sustainable [4][5][6]. Several framework have been proposed for solving MCDM problems, namely Analytical Hierarchy Process[AHP] [7,8,9],Analytical Network Process[ANP] [10],which deals with decisions in absence of knowledge of the independence of higher level elements from lower level elements and about the independence of the elements within a level. Other framework available are data envelopment analysis ( DEA),Technique for order performance by similarity to ideal solution (TOPSIS) [11][12],VIKOR, COPRAS [13], with grey number, [14][15][16],Simple Additive weighting ( SAW) etc [17], LINMAP [18].With these techniques alternative ratings are measured, weight of the criteria are expressed in precise numbers .The projects' life cycle assessment is to be determined and the impact of all actors is to be measured. There are some mandatory axioms that the criteria describing feasible alternatives are dimensions which are important to determine the performance.

Grey Systems in Decision-Making
Grey system theory was developed by Deng [19] .This theory is very useful in decision making process. Grey system is based on Grey numbers. A grey number is a number whose exact value is unknown, but a range within which the value lies is known. There are various types of Grey numbers: i) Grey numbers with only lower limits.
ii) Grey numbers with only upper limits.
iii) Interval Grey numbers iv) Continuous Grey numbers v) Discrete Grey numbers vi) Black and white Grey numbers The theory of grey systems mainly consists of the grey system analysis, grey system modeling, grey decision making and control.

Fuzzy AHP Method
The fuzzy AHP technique can be viewed as an advanced analytical method developed from the traditional AHP. According to the method of Chang's (1992) [20] extent analysis, each criterion is taken and extent analysis for each criterion, i g 's performed on, respectively. Therefore, m extent analysis values for each criterion can be obtained by using following notation:  Step 1:-Perform the fuzzy addition operation of  (3) below, at the end step of calculation, new (l, m, and u) set is obtained and used for the next:- Where l is the lower limit value, m is the most promising value and u is the upper limit value and to obtain equation (2):- .m) values given as equation (3):- and then compute the inverse of the vector in the equation (3) and equation (4) is then obtained as:- l m u ) is defined as equation (5):- y≥x and x and y are the values on the axis of membership function of each criterion. This equation can be written as Step 3:-The degree possibility for a convex fuzzy number to be greater than k convex fuzzy number (7) is For k = 1, 2, 3.................n; k ≠ i. Then the weight vector is given by equation (8):- Where i A (i = 1, 2, 3 ...n) are n elements.
Step 4:-Via normalization, the normalized weight vectors are given in equation (9):- Where W is non-fuzzy numbers.

COPRAS-G METHOD
COPRAS-G method is used to evaluate the overall efficiency of a project with the criterion values expressed in terms of intervals. It is based on the real conditions of decision making and applications of the grey number theory. The COPRAS-G method uses a stepwise ranking and evaluating procedure of the alternatives in terms of significance and utility degree.
The steps of COPRAS-G method are as follows: Step1: Selecting the set of the most important attributes, describing the alternatives.
Step2: Constructing the decision making matrix  X  X = where ji x  is determined by wji (the smallest value, the lower limit) and bji ( the biggest value, the upper limit) Step 3: Determining weight of the attributes qi.
Step 4: Normalizing the decision making matrix  X Step 6: Calculating the sums Pj of the attribute values, whose larger values are more preferable. Step 7: Calculating the sums Rj of attribute values , whose smaller values are more preferable: Step 9: Calculating the relative weight of each alternative Qj: Step 11: Determine the priority of the project.
Step 12: Calculating the utility degree of each alternative:

Project Risk
There may be some external circumstances or event that cannot occur for the project to be successful. The external events are called project risks. If such type event is likely to happen, then it would be a risk. The aim of project selection is to minimize the risk criteria.
After identifying these criteria, their weights are found by AHP method. Five homogeneous experts help us to specify the weight.

Case Study of Proposed Model
According to expert's decision, the following matrix is formed and then by using Triangular Fuzzy Number the Fuzzy evaluation matrix is formed  The weight W = (0.29, 0.34, 0.23, 0.14)   So P1>P3>P5>P2>P4.
According to this combined approach P1 is the best project.

CONCLUSION
In this article, authors proposed a new methodology to provide a simple approach to assess alternative projects and select the best set of project by using a integrated approach of AHP under fuzziness and COPRAS methods with grey numbers. In conclusion, the COPRAS-G method has a promising future in project management field.