3.3 Lp-Metric
There are various assessment methods for multi-objective problems. Choosing a
suitable method depends on the time limit, information, and preferences of a
decision-maker. One of the easiest methods is the Lp-metric. In this method, the
purpose is to minimize the deviation of ideal solutions from the existing objective
functions (see Eq. 5.10):
Min L À P ¼
X H
k¼1
γ k
f k y
Ã
k
À Á À f k y k
ð Þ
f k y Ã
k
À Á
"
# p
(
) 1= p
ð5:10Þ
where y
Ã
k shows the ideal solution in optimizing the k
th objective and γ k shows the
importance (weight) of the k
th objective (γ k > 0). Moreover, 1 p 1 shows the
amount of emphasis on the existing deviations so that if p goes up, the emphasis is
greater on the deviations (Cao and Zhang 2009). After rewriting the coverage model,
it is as the following:
Min L À P ¼ γ 1
Z
Ã
1 À Z 1
Z
Ã
1
! p
þ 1 À γ 1
ð
Þ
Z
Ã
2 À Z 2
Z
Ã
2
! p
&
' 1= p
ð5:11Þ
Subject to:
X m
i¼1
a di y i ! 1
for all d
ð5:12Þ
y i 2 0, 1
f g
ð5:13Þ
An advantage of using this model is the normalization it does for the objective
functions. That is, although the two objectives functions used in this study are of two
different scales, the normalization done in the Lp model enables us to form a singleobjective model.
4 Case Study
In this section, we employ the proposed methodology to determine the appropriate
location of bioethanol distribution centers in Iran. To this end we first screen the
criteria presented in Table 5.1. By screening the criteria, the discrimination power of
decision criteria increases (Kheybari et al. 2019a). For this purpose, the opinion of
six experts collected by a five-point Likert scale questionnaire is employed. After
aggregating the questionnaires, due to existing approximate balance among the
sub-criteria in each dimension of sustainability, the number of 3 (out of 5) is selected
82
S. Kheybari and A. Pooya
There are various assessment methods for multi-objective problems. Choosing a
suitable method depends on the time limit, information, and preferences of a
decision-maker. One of the easiest methods is the Lp-metric. In this method, the
purpose is to minimize the deviation of ideal solutions from the existing objective
functions (see Eq. 5.10):
Min L À P ¼
X H
k¼1
γ k
f k y
Ã
k
À Á À f k y k
ð Þ
f k y Ã
k
À Á
"
# p
(
) 1= p
ð5:10Þ
where y
Ã
k shows the ideal solution in optimizing the k
th objective and γ k shows the
importance (weight) of the k
th objective (γ k > 0). Moreover, 1 p 1 shows the
amount of emphasis on the existing deviations so that if p goes up, the emphasis is
greater on the deviations (Cao and Zhang 2009). After rewriting the coverage model,
it is as the following:
Min L À P ¼ γ 1
Z
Ã
1 À Z 1
Z
Ã
1
! p
þ 1 À γ 1
ð
Þ
Z
Ã
2 À Z 2
Z
Ã
2
! p
&
' 1= p
ð5:11Þ
Subject to:
X m
i¼1
a di y i ! 1
for all d
ð5:12Þ
y i 2 0, 1
f g
ð5:13Þ
An advantage of using this model is the normalization it does for the objective
functions. That is, although the two objectives functions used in this study are of two
different scales, the normalization done in the Lp model enables us to form a singleobjective model.
4 Case Study
In this section, we employ the proposed methodology to determine the appropriate
location of bioethanol distribution centers in Iran. To this end we first screen the
criteria presented in Table 5.1. By screening the criteria, the discrimination power of
decision criteria increases (Kheybari et al. 2019a). For this purpose, the opinion of
six experts collected by a five-point Likert scale questionnaire is employed. After
aggregating the questionnaires, due to existing approximate balance among the
sub-criteria in each dimension of sustainability, the number of 3 (out of 5) is selected
82
S. Kheybari and A. Pooya