2  Fuzzy  sets  theory  and  fuzzy  TOPSIS method To deal with vagueness of human thought, ZADEH [17]  first  introduced  the  fuzzy  set  theory,  which  was oriented  to  the  rationality  of  uncertainty  due  to imprecision  or  vagueness。  A  major  contribution  of  the fuzzy  set  theory  is  its  capability  of  representing  vague data。 The theory also allows mathematical operators and programming to apply the fuzzy domain。 A fuzzy set is a class  of  objects  with  a  continuum  of  grades  of membership。  This  set  is  characterized  by  a  membership (characteristic)  function,  which  assigns  to  each  object  a grade  of  membership  ranging  between  zero  and  one。  A tilde  “~”  will  be  placed  above  a  symbol  if  the  symbol represents  a  fuzzy  set。  Triangular  fuzzy  numbers  are expressed  as indicate  the  smallest  possible  value,  the  most  promising value, and the largest possible value that describe a fuzzy event, respectively。 A triangular fuzzy number (TFN), n~, is shown in Fig。 1 [18]。 In  this  work,  the  importance  weights  of  various criteria  and  ratings  of  qualitative  criteria  are  considered as linguistic variables。 Linguistic  assessments  are  appropriate  for  the subjective  judgment  of  decision-makers,  so  that  we  use triangular fuzzy numbers to capture the vagueness of the linguistic  assessments  for  importance  weight  of  each criterion and rating。 Fuzzy  linguistic  variables  are  very  low  (VL),  low (L),  middle  low  (ML),  middle  (M),  middle  high  (MH), high (H) and very high (VH), where VL=(0,0,1), L=(0, 1, 3), ML=(1, 3, 5), M=(3, 5, 7), MH=(5, 7, 9), H=(7, 9, 10) and VH=(9, 10, 10) [17]。 The  FTOPSIS procedure  involves  carrying  out following steps [16, 13, 19]。 Step  1:  Form  a  committee  of  decision-makers  and then identify the evaluation criteria。 The decision makers use  the  linguistic  weighting  variables  to  assess  the importance of the criteria。 Step  2:  Choose  the  appropriate  linguistic  variables for  the  importance  weight  of  the  criteria  and  the linguistic ratings for alternatives with respect to criteria。 Step  3:  Aggregate  the  weight  of  criteria  to  get  the aggregated  fuzzy  weight  (j~)  of  criterion  Cj。  And  pool the decision makers’ opinions to get the aggregated fuzzy rating (x) of alternative A under criterion CStep 4: Establish decision matrix。 In a decision committee that has K decision makers, fuzzy  rating  of  each  decision  maker  D can  be represented  as  triangular  fuzzy  number The aggregated fuzzy rating can be defined asStep  5:  Establish  a  normalized  matrix R To  avoid  the  complicated  normalization  formula used in classical TOPSIS, the linear scale transformation can be used to transform the various criteria scales into a comparable  scale。  Therefore,  it  is  possible  to  obtain  the normalized  fuzzy  decision  matrix  denoted  by Where  B  and  C  are  the  set  of  benefit  criteria  and cost criteria, respectively。 Step 6: Criteria weighted matrix。 It  cannot  be  assumed  that  each  evaluation  criterion is  of  equal  importance  because  the  evaluation  criteria have various meanings。Step  7:  Construct  the  normalized  weighted  fuzzy decision matrix。 Considering  the  different  importance  of  each criterion, the weighted normalized fuzzy-decision matrix is formed as follows。 Step  8:  Calculate  the  separation  measure  from  the ideal  solutions  (FPIS)  and  the  negative  ideal  solutions (FNIS)。 According  to  the  weighted  normalized  fuzzy- decision  matrix,  normalized  positive  triangular  fuzzy numbers  can  also  approximate  the  elements Then,  the  fuzzy  positive  ideal  solution  (FPIS,  A Step  9:  Calculate  the  distance  of  each  alternative from FPIS and FNIS。 Step 10: Calculate the closeness coefficient of each alternative。 A  closeness  coefficient is  defined  to determine  the  order  of  all  possible  alternatives。  The closeness coefficient represents the distances to the fuzzy positive  ideal  solution and  fuzzy  negative  ideal solution simultaneously。 The closeness coefficient of