used as a criterion to select a subset of PCs for further analysis
(Jolliffe, 2002).
In the context of environmental engineering, PCA can be used
to analyze relationships between environmental criteria.
Gutie
´
rrez et al. (2010) explored the combined use of LCA and
PCA to uncover and visualize the structure of large multidimen-
sional data sets in the context of waste water treatment plants.
After applying PCA, they observed strong correlations between
some LCA metrics. They applied the same approach to mussel
cultivation and waste electrical and electronic equipment
(Gutie
´
rrez et al., 2010).
PCA can be effectively employed for selecting a few variables
among a wider set of correlated variables. In the context of MOO,
this property is certainly useful as it allows identifying redundant
objectives that can be removed from the model, thereby alleviat-
ing its complexity.
To the best of our knowledge, Deb and Saxena (2005) were the
first to use PCA in MOO. They proposed some heuristic rules based
on PCA that allow identifying redundant objectives from a set of
feasible points of the MOO. These rules are based on the analysis
of the components of the eigenvectors of the correlation matrix.
The authors coupled this technique with evolutionary algorithms
(i.e., elitist non-dominated sorting genetic algorithm, NSGA-II) in
order to improve their performance.
In contrast to the work of Deb and Saxena (2005), we propose
herein to integrate PCA with a rigorous MOO solution method
(i.e., epsilon constraint coupled with branch and cut) that guar-
antees the global optimality of the solutions found. As will be
discussed later in the paper, this approach has the advantage
of providing solutions that are globally optimal within a desired
tolerance in shorter CPU times, as opposed to the approach by
Deb and Saxena (2005) that is unable to guarantee the optimality
of the solutions found. An additional novelty of this work is that,
to the best of our knowledge, it is the first contribution that
makes use of PCA for reducing the number of objectives in an
environmental engineering MOO problem.
2.3. Combined used of PCA and MOO
We next present a PCA based epsilon constraint procedure to
solve MOO problems arising in environmental engineering. The
method is applied in an iterative manner as follows. First, some
Pareto solutions of the problem with all the original objectives are
generated using the epsilon constraint method. PCA is then
applied to these high-dimensional points in order to identify
redundant environmental metrics that can be omitted. After
removing redundant environmental objectives, the epsilon con-
strain method is applied again to the reduced space model,
generating new solutions that will be analyzed using PCA. The
algorithm proceeds in this manner until no further reductions in
the dimension of the MOO problem are possible. The detailed
steps of the algorithm are as follows:
1. Initialization.
(a) Set a threshold cut TC above which no more PCs will be
considered, and a number of iterations it for the epsilon
constraint method.
(b) Optimize each individual objective f
i
A F separately. Store
the best and worst values (
f
i
and f
i
, respectively) of each
objective function obtained in these optimizations.
2. Apply the epsilon constraint method to model MO.
(a) Choose an objective f
i
as main objective and transfer the
remaining ones (i.e., f
i
0
a f
i
) to auxiliary constraints, giving
rise to problem MOEC:
MOECðx, yÞ¼min f
i
ðx, yÞ
s:t: f
i
0
ðx, yÞ r
e
n
i
0
8f
i
0
A F\f
i
n ¼ 1, ..., N
hðx, yÞ¼0
gðx, yÞ r 0
xA R, yA f0; 1gð4Þ
(b) Calculate the epsilon parameters (
e
n
i
0
) for each objective
f
i
0
a f
i
by splitting the interval [f
i
0
, f
i
0
] into N1 sub-
intervals.
(c) Solve MOEC for each set of epsilon parameters (a total of
N
9F91
problems must be solved).
(d) Generate matrix M (with dimension N
9F91
9F9) by stor-
ing in each row the values of the 9F9 objectives associated
with each solution n of problem MOEC.
3. Compute the PCs of matrix M.
(a) Filter matrix M by eliminating repeated, dominated or
infeasible solutions.
(b) Standardize the filtered matrix M so as to make its
centroid equal to 0. For this, we subtract the mean of each
column
m
i
from each data point in the matrix.
(c) If the magnitudes of the 9F9 objectives are different, divide
each data point in the matrix by the standard deviation of
the corresponding column
s
i
and compute the correlation
matrix R of the standardized matrix M. Otherwise, com-
pute the covariance matrix
S
of M.
(d) Determine the eigenvectors and eigenvalues of R (or
S
).
Order them in decreasing order of their eigenvalues and
assign PCs recursively. That is, let the first eigenvector be
the first PC; the second eigenvector be the second PC and
so on so forth. Let the variance explained by each PC be
equal to the associated eigenvalue (VAR
j
¼
l
j
).
4. Apply the objective reduction heuristic proposed by Deb and
Saxena (2005).
(a) Define an alternative set of objectives F
0
¼ |.
(b) Calculate the portion of the total variance explained by the
first k PCs (CVAR
k
) for k ¼ 1, ..., 9F9. Retain for further
analysis the minimal subset of PCs with a total cumulative
variance above TC. Fathom the rest of PCs.
(c) Add to F
0
the objectives with the most positive and most
negative contribution to the eigenvector of the first PC.
(d) For the remaining PCs, proceed as follows. If
l
j
r 0:1, add
to F
0
the objective with the highest contribution in
absolute value. If
l
j
4 0:1:
i. If all the components of the PC are positive, add to F
0
the objective corresponding to the highest component
of the eigenvector.
ii. If all the components of the PC are negative, add all the
objectives to F
0
.
iii. If none of the previous two cases apply, proceed as
follows. Let mp
j
be the most positive component of the
PC under consideration and mn
j
its most negative one.
A. If mp
j
o 0:99mn
j
9, add the objective corresponding
to mn
j
to F
0
.
B. If 0:99mn
j
9r mp
j
o 9mn
j
9, add to F
0
the objectives
associated with both mn
j
and mp
j
.
C. If 0:89mp
j
9r mn
j
o 9mp
j
9, add to F
0
the objectives
associated with both mn
j
and mp
j
.
D. If none of the previous three cases apply, add to F
0
the objective associated with mp
j
.
5. If F
0
a F, make F ¼ F
0
and go to step 2. Otherwise stop.
2.3.1. Remarks
In step 1.a, it is recommended to use a large value of TC
(refer to Deb and Saxena (2005) for further discussion on this
issue).
C. Pozo et al. / Chemical Engineering Science 69 (2012) 146–158148
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