Differential Evolution for Multiobjective Optimization of Process
Design Problems
Antonio Ochoa-Robles, Catherine Azzaro-Pantel and Serge Domenech
Université de Toulouse, Laboratoire de Génie Chimique, INP-ENSIACET,
4 Allée Emile Monso, 31432 Toulouse cedex 4, BP 84234, France
Keywords: Differential Evolution, Stopping Criterion, Evaluations, Structural Problems.
Abstract: Optimization is a highly important area in chemical engineering, particularly for process design that is
generally formulated as a mixed and non-linear problem with several competing objectives. A way to tackle
the problem is to couple multiobjective optimization based on evolutionary algorithms with a process
simulator. This situation may yet lead to prohibitive computational time as the number of objectives
increases. In this paper, the potential of multiobjective differential evolution (MODE) is tested with three
different stopping criteria. The performance of MODE is compared with the results obtained with a variant
of NSGA II. The performance metric is based on the number of evaluations used to get the Pareto front. The
results show that the combination of an efficient algorithm and the stopping criterion helps to reduce the
optimization time but its choice may affect the results. As far as multiobjective is concerned, it must be
emphasized that the final solution is the result of compromise that the decision maker must be aware.
1 INTRODUCTION
Process design is a key activity in the chemical
engineering field for implementing new
technologies, creating new facilities, or retrofitting
existing processes. If the traditional design approach
incorporates economic objectives, process systems
design has come to include more performance
measures, such as environment, safety,
controllability, and flexibility. This kind of problems
can be generally modelled as mixed integer
nonlinear programming (MINLP) formulations,
involving continuous and integer variables. This
class of mathematical problems generally involves
non-convexities, which are related to the problem
formulation concerning both the objective
function(s) and/or the set of constraints. The
inherent combinatorial nature of the problem
contributes to its complexity. In that context,
evolutionary algorithms (EAs) have received a lot of
attention for solving nonlinear multimodal problems
(Angira and Babu, 2006). They are also particularly
attractive to capture the multiobjective nature of the
criteria. Among the methods that have reported in
the dedicated literature, multiobjective optimization
(MOO) (Rangaiah, 2009) and particularly
evolutionary algorithms constitute a promising
approach to tackle the problem.
The early design stage implies the evaluation of
the various alternatives that can be used to produce a
chemical product involving several reaction routes
with various types of equipment and their
corresponding operating conditions. The importance
of early design activities has been addressed in
several recent studies. The problem is generally
solved by use of a process simulator for flowsheet
generation such Aspen (“Aspen One® -
AspenTech,” 2013), Hysis (“Aspen HYSYS® -
AspenTech,” 2013), Prosim (“PROSIM,” 2013).
If the computational time required for simulation
is quite acceptable (from several seconds to several
minutes for large size problems), the situation may
be quite different when performing optimization
where the various objectives must be evaluated
many times by successive use of the process
simulator. It must be emphasized that multiobjective
optimization does not lead to a single ideal solution
but to a set of compromise solutions (Jones et al.,
2002) that are generally represented through a Pareto
front as far as the objective functions are considered.
Problems such as the optimization of the process
of hydrodealkylation (HDA) of toluene, to produce
benzene were investigated previously (Ouattara et
226
Ochoa-Robles A., Azzaro-Pantel C. and Domenech S..
Differential Evolution for Multiobjective Optimization of Process Design Problems.
DOI: 10.5220/0004833102260232
In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 226-232
ISBN: 978-989-758-017-8
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
al., 2012) involving various objective functions
based on economic and ecological criteria evaluated
by coupling two simulators for chemical process and
utility requirement. Every evaluation involves a call
to the support software tools that makes the
optimization task quite long. Another example is the
selection of the synthesis strategy for a chemical
plant described by Grossmann (Papoulias and
Grossmann, 1983). The superstructure considers the
chemical and utility plant as the heat recovery
network. The objectives are to determine the
configuration of the plant, the heat exchanger
network and utility system that allows maximizing
the annual profit. Other examples can be mentioned
such as the supply chain management problem
presented in (Kallrath, 2000) as a multi-site, multi-
product, multi-period production/distribution
network planning system with the objective of
finding the best production schedule satisfying a
given demand.
Among multiobjective evolutionary approaches,
Genetic Algorithms (GAs) constitute a quite popular
method used in engineering field, particularly in the
chemical engineering community (Abbass et al.,
2001). One of the most efficient genetic algorithms
is NSGA II Non-dominated Sorting Genetic
Algorithm (Deb et al., 2002) an upgrade version of
NSGA which estimates the density of solutions
surrounding a particular one, in order to perform a
scanning of the solution space.
The design optimization time obviously depends
on the number of the successive evaluations of the
possible solutions by use of the process simulator.
In that context, the formulation of an effective
criterion is necessary in the case of the
multiobjective optimization problem as judging the
advance of the optimization. If the selection of an
appropriate criterion has been identified as one of
the fundamental topics, it must be highlighted that
this issue has not been solved properly.
The objective of this work is twofold: first, the
potential of Differential Evolution (DE) is
investigated since DE has been successful in the
solution of a variety of continuous single-objective
optimization problems in which it has shown great
robustness and a very fast convergence. Recently,
there have been successful proposals to extend DE
to MOO (Robič and Filipič, 2005). A multi-
objective differential evolution algorithm was thus
implemented. The second objective is to evaluate
different stopping criteria for reducing the number of
evaluations. For this purpose, some benchmark
problems and a chemical engineering problem are
tested.
This paper is divided into 5 sections. Section 2 is
devoted to main concepts of differential evolution.
Section 3 describes the solution strategy. Section 4
discusses the results obtained with test problems.
Section 5 concerns the application to a small-size
structural problem for process design. Finally,
conclusions and perspectives are proposed.
2 DIFFERENTIAL EVOLUTION
(DE)
Differential Evolution (DE) is an evolutionary
algorithm proposed by Price (Price, 1996) using
vectors to perturb the best solution found so far
together with mutation and crossover. It needs three
parameters, i.e., population size , scaling constant
and crossover constant .
The details on DE algorithm, various strategies
of DE and wide range of applications in various
engineering areas are well documented in literature
(Angira and Babu, 2006)(Onwubolu and Babu,
2004). Only the principles are presented here for the
sake of brevity.
The procedure is rather simple. The first step is
to initialize the population for every variable in
the dimension and evaluate the fitness of each
individual within the boundary constraints (upper
and lower bounds and L, respectively), such as:
0,1
,
1,…,;
1,…,
(1)
Each iteration consists in 4 steps. First, three
individuals of the population are randomly selected
(
,
,
); they must be mutually different and also
different of the current vector . Secondly, a trial
vector is created according to equation (2), where
0,1
is a randomly generated number and
is a randomly selected variable.
,
,
,
0,1
⋁
(2)
Step 3 checks the boundary constraints; if a value is
out of the boundary zone, it is calculated again
according to:
∉
,
0,1
.
(3)
Finally, if the trial vector is inferior or equal to the
current one, the trial individual replaces the current
individual.
Some guidelines for the use of DE are proposed
in (Storn, 1996):
At initialization step, the population should be
DifferentialEvolutionforMultiobjectiveOptimizationofProcessDesignProblems
227
spread as much as possible over the objective
function surface.
Most often, the crossover probability CR
(∈
0,1
must be considerably lower than 1. If
no convergence can be achieved, a value of CR
within
0.8,1
is yet recommended
For many applications, a size of the population
corresponding to 10 times the size of the problem
(10) is a good choice. F is usually
chosen within the interval
0.5,1
.
Some differences between DE and GA (Abbass et
al., 2001) can be highlighted:
In GAs, crossover is carried out between two
parents and the child is a recombination of both
of them, while in DE, three parents are selected
and the child is only the perturbation of one of
them.
The new child only replaces a randomly selected
vector of the population when it is better. In GA,
the children replace the parents with some
probability regardless of their fitness.
2.1 Stopping Criterion
As mentioned in the motivation of this work, the
stopping criterion is of major importance to
guarantee that the solution (or the set of solutions)
obtained so far is of acceptable quality regarding the
numerical effort. Various scenarios can be used
among others (Martí et al., 2007):
the solution yielded so far is satisfactory;
the method is able to produce a solution : it is yet
not satisfactory but a better one will not be
produced;
the method is unable to “converge” to any
solution;
there is no progress in the search of a new
solution.
3 SOLUTION STRATEGY
3.1 Principles
The algorithm used is the one suggested by Price
(Price, 1996) that was adapted for considering mixed
variables and a multiobjective formulation. By lack
of place, a major attention is only paid to the
stopping criterion in what follows.
The binary variables are taken into account using
the strategy of Angira (Angira and Babu, 2006), that
means that the variable is handled as a continuous
one, within the interval (Feoktistov, 2006) with a
rounded value (Feoktistov, 2006). Every variable
that is modified during the optimization process
must be within its definition domain; otherwise, it is
initialized again.
The violation of constraints is calculated and
used in the adaptation comparison procedure
according to Deb’s criterion, that means that a vector
A dominates B if one of the next conditions is
achieved (Deb et al., 2002):
A is feasible and B not;
A and B are not feasible but the violation of the
constraints is lower in A than in B;
A and B are feasible but A dominates B.
3.2 MGMB
The MGMB criterion (from the initials of the
authors (Martí et al., 2007)) is based on the
comparison of the set of non-dominated solutions of
two iterations. A progress indicator (
) is to be
calculated indicating or not an evolution of the
population. For example, a value of
equal to 1
means that the last population is better than the
previous one. A value of
equal to 0, means that
there is no progress and a value of
equal to -1 is
the sign of deterioration of the population. A
correction step
is considered to take into account
the influence of the changes.
3.3 Consolidation Ratio (CoR)
This is a convergence metric that can be used as a
stopping criterion. The consolidation ratio is the
fraction of the population at the generation ∆
(∆represents a kind of observation step) that has
evolved up to the current generation .
This is calculated as the ratio of the number of
non-dominated individuals () in the generation
∆ present in the generation and the non-
dominated of the last generation (), expressed as:
∆
(4)
In the early stages of the algorithm, a large fraction
of non-dominated will not remain in the last
population that will result in a low value for CoR,
while the quantity of non-dominated individuals
after several generations that will remain in the
population will be higher, leading to a CoR ratio
close to 1 (no changes of non-dominated vectors).
ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems
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3.4 Improvement Ratio (IR)
This ratio represents the proportion of the population
∆ dominated by the population of size
, as
expressed by:
∆
(5)
The initial value of this ratio is equal to unity, which
means that every solution is dominated by the latest
population, while when convergence is achieved this
value is equal to zero.
3.5 Final Selection with TOPSIS
The final selection process is made using a
multicriteria decision-making process that takes into
account the optimal alternatives found in the Pareto
front. These alternatives are found to be non-
dominated solutions near optimal value, and
although the decision maker may use judgment to
make the final selection from the alternatives, a
formal method based on TOPSIS (Technique for
Order of Preference by Similarity to Ideal Solution)
was adopted (Lai et al., 1994); (Ren et al., 2010).
This method is based on the idea of choosing the
best alternative solution from a set by analyzing the
shortest geometric distance from the positive ideal
solution and the longest distance from the negative
ideal solution. It also requires weights to be assigned
per criterion and normalizes the information, so that
the various alternatives are ranked. Although other
ranking and classification methods exist, TOPSIS
has proven its efficiency in the final alternative
selection process obtained through GA (Gomez et
al., 2010) and is used here after MODE process. The
same weight was allocated to each criterion in the
experimental study. The approach for each stopping
criterion will be tested based on the solution that
obtained the top rank by TOPSIS.
4 VALIDATION
The test problems selected to evaluate the
performance of the algorithm are the classical SRN
and TNK problems used in previous works (Deb et
al., 2002). The formulation of the two problems is
presented in Table 2 as well as the Pareto fronts of
SRN and TNK. First, to validate the algorithm, the
classical DE algorithm was used without any
specific stopping criterion (the algorithm stops when
the maximum number of generation is reached) and
the results were compared with the solutions
obtained by previous researchers with other
algorithms. The parameters used were CR=0.6,
F=0.8, NP=200 and 100 generations They can be
visualized in Fig. 1 and 2 and are in agreement with
the results obtained with deterministic methods.
Then, the three stopping criteria are considered
for DE i.e., the so-called MGMB (Martí et al.,
2007), consolidation ratio (Goel and Stander, 2010)
and improvement ratio (Goel and Stander, 2010) in
combination with the maximum number of
generations.
The DE procedure is compared with a variant of
NSGA II developed for mixed problems and
implemented in the Multigen environment (Gomez
et al., 2010). The stopping criterion proposed in
Multigen (in addition to the maximum number of
generations) consists in comparing the Pareto fronts
associated with non-dominated solutions for
populations and , where the period ∈
10,20,30,40,50 for example. If the union of the
two fronts provides a single non dominated front, the
procedure stops; else the iterations continue.
Table 1: Problem formulation for the test functions.
SRN TKN
2
1
2
9
1
225
3
10
∈
20,20
1
0.1cos
16tan
⁄
0
0.5
0.5
0.5
∈
0,
Figure 1: Pareto front of
the test problem SRN.
Figure 2: Pareto front of
the test problem TNK.
Three scenarios for DE and one for NSGA-II are
tested as shown in Table 2. The period represents the
time of observation and application of the stopping
criterion. Every problem is analyzed relative to the
number of evaluations performed for each stopping
criterion and procedure. The solutions obtained after
TOPSIS application are also analyzed which can be
viewed as another validation. The solutions concern
DifferentialEvolutionforMultiobjectiveOptimizationofProcessDesignProblems
229
both the values of the objective functions and the
associated variables.
Table 2: Scenarios for DE and NSGA-II.
DE
Test 1
DE
Test 2
DE
Test 3
NSGA II
Individuals 100 100 100 100
Maximum number
of generations
200 200 200 200
CR 1 0.6 0.4 0.9
F 0.8 0.8 0.8
0.5
(mutation)
Period 10 10 5 -
By lack of place, the Pareto front obtained for
each problem is not presented here. All the fronts
exhibit similar behaviors as previously seen in Fig. 1
and 2. It must be said that the curves are overlaid in
the domain with common intersection each other.
For SRN problem(see Table 1), the solutions
obtained by DE-TOPSIS exhibit a similar behavior
both for criteria and variables. The order of
magnitude of criteria and variables is quite different
with NSGA-II. It must be emphasized that for
NSGA II, the algorithm ends because the maximum
number of generations is reached. Regarding the
objective functions, the stopping criteria IR and CoR
require yet a higher number of evaluations (Figure
3) than MGMB.
For TNK problem (see Table 1), the order of
magnitude of the objective functions is quite similar,
in fact all the selected solutions are non-dominated
between them. Considering the number of
evaluations, it is interesting to see that the MGMB
requires around 9 times lower evaluations than the
other criteria, which can be of practical importance
in solving real problems (Figure 4). Considering the
homogeneity of the selected solutions obtained after
DE-TOPSIS, IR shows almost no difference, while
CoR and MGMB have a larger deviation. This can
be attributed to scattered points in the Pareto front
obtained for each test, thus giving different ranking
after application of TOPSIS method.
5 APPLICATION TO A
STRUCTURAL DESIGN
PROBLEM
This problem is a bicriteria one proposed by
Papalexandri and Dimkou (Papalexandri and
Dimkou, 1998). It consists of 3 continuous variables,
3 binary variables and two objective functions. The
formulation can be expressed as follows:
,
3
2
(6)
,
2
3
2
2
(7)
,
3
2
0
(8)
,
4
2
40
7
0
(9)
,
2
3
7
0
(10)
,
1012
0
(11)
,
102
0
(12)
Table 3: Selected solutions for SRN and TNK problem using DE-TOPSIS.
SRN TNK
NSGA-II
0,0282 18,4389 310,003 -303,861 0,043 1,039 0,043 1,039
IR Test 1
-2,418 -11,666 181,939 -182,183 1,029 0,063 1,029 0,063
IR Test 2
-2,539 -12,499 204,821 -205,07 1,006 0,088 1,006 0,088
IR Test 3
-2,924 14,1161 198,274 -198,344 1,022 0,072 1,022 0,072
CoR Test 1
-2,288 -8,6635 113,773 -113,978 1,006 0,088 1,006 0,088
CoR Test 2
-2,165 -12,371 198,117 -198,254 0,957 0,138 0,957 0,138
CoR Test 3
-2,351 -12,292 197,614 -197,842 0,089 1,004 0,089 1,004
MGMB Test 1
-2,066 13,1749 166,762 -166,824 0,108 1,011 0,108 1,011
MGMB Test 2
-2,287 14,1367 192,948 -193,152 0,068 1,026 0,068 1,026
MGMB Test 3
-3,176 -12,307 205,862 -205,656 0,066 1,042 0,066 1,042
ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems
230
Figure 3: Number of
evaluations for SRN
problem.
Figure 4: Number of
evaluations for TNK
problem.
,
20
0
(13)
,
40
0
(14)
,
17
0
(15)
,
25
0
(16)
∈
100,100
(17)
∈
0,1
(18)
For the three stopping criteria, the previous
conditions were applied (see Table 2). A similar
analysis as the one previously adopted for SRN and
TNK is carried out.
For the Improvement Ratio (Fig. 13) all the runs
exhibit similar results and the slight discrepancy that
is observed can be attributed to the different choice
in parameter settings.
For the MGMB criterion (Fig. 14) the behavior is
quite similar for all tests, the best performance being
obtained by test 3.
For the Consolidation Ratio (Fig. 15) the
performance of the three tests is quite similar,
meaning that this criterion leads to quasi-identical
results whatever the intrinsic parameters of the
procedure.
Regarding the number of evaluations of the
objective functions (Fig. 16), the stopping criterion
MGMB leads to the best performance once more. A
slight difference is observed between IR and CoR,
with a lower number of evaluations than NSGA-II.
Figure 5: Pareto front
with Improvement Ratio
as stopping criterion.
Figure 6: Pareto front
with MGMB as stopping
criterion.
Figure 7: Front de Pareto
using Consolidation Ratio
as stopping criterion.
Figure 8: Number of
evaluations for every
stopping criterion.
Table 4) shows that significant differences in
solutions are observed. For IR, CoR and MGMB
Test 1 and 2, the results for function f_1 are better
that those proposed by NSGA-II. Such a situation
never occurs for f_2 since they are out-performed by
the NSGA-II. Test 2 provides consistent results for
all criteria, so the configuration CR=0.6 and F=0.8,
with a period of 10 generations seems the more
appropriate for this problem. As far as the number of
evaluations is concerned, MGMB is more
performing. Yet, if the evaluation functions lead to
similar performances, the corresponding set of
TOPSIS analysis variables is not the same. The final
choice of the decision maker may also consider the
difficulty of implementation of a solution over
another one as an effective lever.
Table 4: Selected solutions for SRN and TNK problem using DE-TOPSIS for the small-design problem.
NSGA-II
0,0919 39,9759 -1,7991 0 0 0 42,9538 -41,7666
IR Test 1
0,1980 36,4077 -0,6076 0 0 1 34,2613 -35,9761
IR Test 2
0,1969 37,6330 -0,0376 0 0 1 35,1213 -36,6318
IR Test 3
0,4251 35,4294 -0,9154 0 0 0 35,3535 -36,1641
CoR Test 1
-0,0011 40,1225 -1,2869 0 0 1 39,7819 -40,4093
CoR Test 2
0,1405 36,7248 -0,3565 0 0 1 34,4699 -36,0615
CoR Test 3
0,2955 36,8305 -0,4736 0 0 1 34,3429 -36,2167
MGMB Test 1
0,2727 39,4125 -1,3556 0 0 1 38,5809 -39,6938
MGMB Test 2
0,9297 35,2025 -2,7510 0 0 0 35,3495 -36,0356
MGMB Test 3
-0,4732 10,8265 -0,8724 0 0 1 49,3434 -41,3825
DifferentialEvolutionforMultiobjectiveOptimizationofProcessDesignProblems
231
6 CONCLUSIONS AND
PERSPECTIVES
A differential evolution algorithm with several
stopping criteria was developed. Its performance
was compared with the results obtained by a variant
of NSGA II implemented in previous works. Results
show that, every proposed stopping criterion
obtained similar results as done by NSGA-II. But,
the use of the MGMB criterion implies a lower
number of evaluations as compared with IR and
CoR. Nevertheless, no stopping criterion is the
panacea. Its choice must be a compromise between
the required gain and the computational effort. This
study will now be applied to a large size chemical
engineering design problem which involves the
evaluation of every proposed solution with a
simulator. Even if MGMB appears to be a good
candidate, its robustness must be now investigated
as far as multiple variable-mapping is concerned.
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