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Integrated Asset Management and Optimization Workflows 205

conflicting, cost functions. An example of conflicting objective functions is an
NPV of an asset, of which exploitation is considered as an “opportunity”, and
some measure of the risk, associated with that exploitation (Echeverria Ciaurri
et al., 2012). Such optimization problems involving multiple conflicting
objectives are often addressed by “aggregating the objectives into a scalar func-
tion and solving the resulting single-objective optimization problem”
(Schulze-Riegertetal.,2007). In multiobjective optimization, the selection
of weights corresponding to specific components of single objective function
(see Eq. 6.1) is omitted by splitting the objective function into several com-
ponents, which are optimized simultaneously. The objective function now
takes the form F(x)¼{f 1 (x),f 2 (x),…,f k (x),…,f M (x)} and the optimization
problem is defined as (Schulze-Riegert et al., 2007; Hutahaean et al., 2015)

minimizeF xðÞ
l
subjectto h x k h u k (6.5)
k
x ¼ x 1 , x 2 , …, x k , …, x N g
f
M
N
where F(x):ℜ !ℜ ,x¼{x 1 ,x 2 ,…,x k ,…,x N } is the vector of the N var-
l u
iables in the parameterization, M is the number of objectives, and h k and h k ,
respectively, correspond to the lower and upper boundaries of each variable.
In contrast to single-objective optimization, the task now becomes find-
ing a set of optimal solutions, also referred to as the Pareto optimal set, usu-
ally represented as a Pareto front (Fig. 6.3). Because different objectives in
multiobjective optimization are not comparable, the concept of Dominance
and Pareto optimality applies (Hutahaean et al., 2015). Fig. 6.3 shows this
concept, while Fig. 6.4 gives an example of the objective space behavior
for several iterations of a history-matching workflow, which combines
the joint misfit minimization of field watercut and field static pressure, using
the multiobjective genetic algorithm (MOGA) (Kam et al., 2016; Ferraro
and Verga, 2009) with the population of 40 model realizations and by
parameterizing reservoir permeability and water saturation.
Despite the success the multiobjective algorithms have demonstrated in
optimization problems, such as history matching of reservoir simulation
models, their performance is known to reduce substantially when the num-
ber of objectives in multiobjective function exceeds three. Hutahaean et al.
(2017) refer to such problems as many-objective problems (MaOP), where
Pareto-based algorithms become significantly less effective in discriminating
between solutions. This compromises the concepts of Dominance and
Pareto optimality as well as the convergence of the search procedure.
Hutahaean et al. (2017) identify various possible conflicts between the