1. Introduction - Firenze University Press

-0.309x15.77x2 2.085x3 240x4 12.53x5 0.452x7 0.09x9 0
(12)
51.57x1157.1x2 210.64x3 70.33x4 13.69x5 4.77x7 34x8 29.3x9 0
(13)
51.37x1-175.71x2-234.65x3-78.15x4-15.37x5-5.41x7-39.21x8 28.9x9 0
(14)
The total feeding of fuel must satisfy the specific heat consumption. This value would be 3600
kJ/kg clinker, but the additions of mineralizers reduce to 3181 kJ/kg clinker. Eq. (15) presents the
specific heat consumption; the coefficients of equation are Lower Heating Value of fuels. Eq. (16)
denotes the consumption of used tires. Eq. (17) and (18) are the mineralizers’ limits for CaF2 and
CaSO4. These limits are obtained in work [8] that established 1 % (mass) of each mineralizer.
27670x 36425x 32100x 3181
32100x7 795
x 0.01634
8
x 0.01634
9
5 6 7
3. Optimization techniques
In this work, the nonlinear problem defined with multi-objective functions and constraints was
solved using Genetic Algorithms (GAs) and Sequential Quadratic Programming (SQP).
3.1. Genetic Algorithms (GAs)
Genetic algorithms (GAs) work on the principle of “survival of the fittest”. They have been
extensively applied by many optimization problems. In GAs, the decision variables are encoded in a
string form. The encoded solutions are called chromosomes and the elements of the chromosomes
are called genes.
Depending on the nature of the problem the encoded solution may include binary digits or real
numbers. An initial population is created and the fitness (the objective function value) of the
population members is evaluated. Genetic operators (mutation and crossover) are applied to keep
the gene pool diverse that aids the inclusion of better fitted members for quick convergence [12].
3.2. Sequential Quadratic Programming (SQP)
The main idea in SQP is to obtain a search direction by solving a quadratic program, that is, a
problem with a quadratic objective function and linear constraints. This approach is a generalization
of Newton’s method for unconstrained minimization [13], and it is used to solve a nonlinear
program in the following form:
Minimize f ( x)
Subject to g( x) 0
Where g is a vector of m constraint functions gi. Applying Newton’s method to the corresponding
optimality conditions, the Lagrange for this problem is obtained as:
266
(15)
(16)
(17)
(18)
(19)