Page 281 - Materials Chemistry, Second Edition

13.2 Decision-making under multi-type data condition 279
C 1 C 2 ⋯ C N
A 1 z z ⋯ z
11
12
1N
Z ¼ A 2 z z ⋯ z (13.30)
22
2N
21
⋮ ⋮ ⋮ ⋱ ⋮
A M z z ⋯ z
M1
M2
MN
h i h i
U U
L L
z ¼ z L z U ¼ ω y ¼ ω y ω y (13.31)

ij ij ij j ij j ij j ij
Step 5: Determining the sustainability ranking of the alternatives. The ideal solutions with
respect to the criteria can be determined by Eq. (13.32).

M
M
max z U max z U ⋯ max z U (13.32)
M
i1
iN
i2
i¼1 i¼1 i¼1
n o
M
where max z U j ¼ 1,2,⋯,N represents the best ideal solution with respect to the jth
ij
criterion. i¼1
This study develops a goal programming model for selecting the best alternative or the
most sustainable alternative among multiple choices based on the work of Ren et al.
(2015b). The principle of this model is that the best alternative should be the choice that is
the closest to the ideal solution. In other words, the best alternative should have the shortest
distance to the ideal solution.
The objective function is to minimize the total distance to the ideal solution, as presented in
Eq. (13.33).
L
N
X g + g U !
j
j
Min (13.33)
2
j¼1
with the following constraints, including goal constraints, 0-1 constraint, and the selection
constraint (Ren et al., 2015a, b).
Goal constraints:
M
X h L U i h L U i M n U o M n U o
z z p i + g g ¼ max z max z j ¼ 1,2,⋯,T (13.34)
ij ij j j ij ij
i¼1 i¼1
i¼1
The constraints presented in Eq. (13.34) can be further rewritten into:
M
X L L M n U o
z p i + g ¼ max z j ¼ 1,2,⋯,T (13.35)
ij j ij
i¼1 i¼1
M
X U U M n U o
z p i + g ¼ max z j ¼ 1,2,⋯,T (13.36)
ij j ij
i¼1 i¼1
0-1 constraint:

1 if the ith alternative has been selected asthe best alternative
(13.37)
0 otherwise
p i ¼