2.2 Optimization Methods
25
and water production, to total revenue optimization in revenue management or the
optimal resource allocation in health management.
This chapter summarizes the basics of optimization methods starting with an
introduction on the common terminology in the field of optimization techniques.
Since there is no uniform classification of optimization methods in the literature,
the classification followed in this book is presented in section 2.2.2.
2.2.1 Terminology in Optimization
Optimization is seen as a basic tool in the field of Operations Research, concerning the development of solutions and algorithmic techniques for solving
mathematical problems under consideration of constraints or boundary conditions
which are limiting the allowed range of independent variables [Ba2012b, p. 3;
PR2002, p. XV; SM2009, p. 6]. An optimization model is a formal representation
of a planning problem that, in its simplest form, contains at least one alternative
set and one objective function that evaluates it. It is developed in order to be able
to determine optimal or suboptimal solutions with suitable methods [Do+2015,
p. 4]. Once an optimization problem and the variables influencing the optimization problem, the input parameters, and, if necessary, constraints setting bounds
for input parameters are defined, the way of evaluating the performance of the
problem needs to be found. The performance measure is the objective function
and the range of its possible values is the solution space [Fu2015, p. 1; Li1992,
p. 9]. The general parametric optimization problem where the objective function
is to be minimized is
min
x∈
f (x), x ∈ X ⊆ R
n
(2.3)
with f
objective function
x
decision variables
feasible region or constraint set
X
design space of the optimization problem
R n geometrical space of decision variables
Mathematical optimization is generally determining a minimum or maximum
of the objective function. Both extremes are summarized using the term optimum. There is an equivalence relationship (principle of duality) between the
25
and water production, to total revenue optimization in revenue management or the
optimal resource allocation in health management.
This chapter summarizes the basics of optimization methods starting with an
introduction on the common terminology in the field of optimization techniques.
Since there is no uniform classification of optimization methods in the literature,
the classification followed in this book is presented in section 2.2.2.
2.2.1 Terminology in Optimization
Optimization is seen as a basic tool in the field of Operations Research, concerning the development of solutions and algorithmic techniques for solving
mathematical problems under consideration of constraints or boundary conditions
which are limiting the allowed range of independent variables [Ba2012b, p. 3;
PR2002, p. XV; SM2009, p. 6]. An optimization model is a formal representation
of a planning problem that, in its simplest form, contains at least one alternative
set and one objective function that evaluates it. It is developed in order to be able
to determine optimal or suboptimal solutions with suitable methods [Do+2015,
p. 4]. Once an optimization problem and the variables influencing the optimization problem, the input parameters, and, if necessary, constraints setting bounds
for input parameters are defined, the way of evaluating the performance of the
problem needs to be found. The performance measure is the objective function
and the range of its possible values is the solution space [Fu2015, p. 1; Li1992,
p. 9]. The general parametric optimization problem where the objective function
is to be minimized is
min
x∈
f (x), x ∈ X ⊆ R
n
(2.3)
with f
objective function
x
decision variables
feasible region or constraint set
X
design space of the optimization problem
R n geometrical space of decision variables
Mathematical optimization is generally determining a minimum or maximum
of the objective function. Both extremes are summarized using the term optimum. There is an equivalence relationship (principle of duality) between the