Grey Relationsical Analysis: The Process Of Grey Relational Analysis

d_ij=x_ij/(∑_(i=1)^m▒x_ij ) q_j (6)
Where;
x_ij is the value that corresponds measure of performance of the i -th alternative and j -th attribute and q_j represents the weight of ach attribute. d_ij represents dimensionless weighted value. The weights of attributes can be calculated using Equation (7). q_j= ∑_(i=1)^m▒d_ij (7)
The alternatives are distinguished by beneficial (maximizing) attributes and cost (minimizing) attributes.
〖s+〗_i= ∑_(j=1)^n▒d_ij (8)
〖s-〗_i= ∑_(j=1)^n▒〖d_ij (9)〗
Where;
〖s+〗_i refers to sum of elements in the weighted normalized matrix that corresponds to beneficial attributes. On the other hand, 〖s-〗_i refers to sum of elements in the weighted normalized matrix that corresponds to cost attributes.

N_j=Q_i/Q_max *100% (11)

Grey Relational Analysis
The process of grey relational analysis is divided into four steps are (Kuo et al 2008):
Grey Relational generating is normalization process for performance attributes. Equation (12) is used to normalize beneficial attributes (the higher value the better option). Equation (13) is used to normalize non-beneficial attributes (the lower value the better option). Equation (14) is used to normalize attributes where the closer to the desired value (x_j*) the better option. y_ij=(x_ij-min⁡{x_ij,i=1,2,………m})/(max⁡{x_ij,i=1,2,………m}-min⁡{x_ij,i=1,2,………m}) (12) y_ij=(max⁡{x_ij,i=1,2,………m}-x_ij)/(max⁡{x_ij,i=1,2,………m}-min⁡{x_ij,i=1,2,………m}) (13) y_ij=(〖|x〗_ij-x_j*|)/(max⁡{x_ij,i=1,2,………m}-min⁡{x_ij,i=1,2,………m}) (14)

Reference sequence generation is the second step where performance values are defined within the range [0,1]. For cost category is the lowest value while benefit category is the highest value. Grey Relational coefficient generation is the third step. The aim of this step is to determine whose compatibility sequence is closest to the reference sequence. Grey relational coefficient is calculated using equation

The grey relational grade is calculated using equation (15). The best alternative is the alternative with the highest relational grade. r(y_0,y_i)= ∑_(j=1)^n▒〖w_j*〗 γ(y_0j,y_ij) (19)

TOPSIS Technique
TOPSIS decision making technique is divided into five main steps (Triantaphyllou 1998):
The decision matrix is normalized where the purpose of this step is to convert performance attributes into non-dimensional ones. r_ij=x_ij/(∑_(i=1)^m▒〖x²〗_ij ) (20)
The weighted normalized matrix is obtained using equation (21). v_ij=r_ij* w_j (21)
The ideal and negative ideal solutions are determined. A* indicates the most preferable alternative or ideal solution. On the contrary, A- indicates the least preferable alternative or negative ideal solution. For benefit criteria, decision maker wants to obtain the maximum value among all alternatives. On the other hand, the decision maker wants to obtain minimum value among all alternatives for cost criteria.
A*={(max v_ij│j Є J ),(min v_ij│j Є J^' ),i=1,2,3,……….M}={〖v*〗_1,〖v*〗_2………..〖v*〗_N} (22)
A-={(min v_ij│j Є J ),(max v_ij│j Є J^' ),i=1,2,3,……….M}={〖v-〗_1,〖v-〗_2………..〖v-〗_N}