1. The
purposes and tasks.
Charge components calculation is the
definition of an optimum combination various on a chemical compound and
cost of materials (including pig metal, addition alloy, return of own
manufacture, iron bar, metal shaving, pure metals) and demands
observance of following basic conditions:
- all the elements defined by alloy
type should be included in the list of controllable chemical elements;
- the maintenance of alloying
chemical elements should be limited by minimal and maximal edges
admissible for the given alloy (are defined corresponding to the State
Standard);
- the admixture should be limited to
as much as possible admissible maintenance of the given elements in an
alloy;
- the cost of furnace charge should
be minimal.
2. The
calculation of charge.
With the aim to decrease the
cost of charge, the basic set of components contains the cheaper
materials as well: iron bar, metal shaving, secondary alloys, return of
own manufacture, etc. Thus the maintenance of the given components in a
furnace charge should be limited for the following reasons:
a) the percent of return in
almost all of the cases is the same and depends on a kind of alloy, the
way of moulding, type of melting unit, etc.;
b) iron bar, metal shaving
and other cheap components of charge, as a rule, are characterized by
the raised degree of harmful impurity, that negatively influences on
smelted alloy quality;
c) the ability of using one
or other cheap component is often defined by its presence at the
moment.
Thus, the problem of charge
structure optimization can be formally presented as a problem of search
for an extremum (minimum) of scalar linear fitness function F (cost) of
a vector of operated variables (the maintenance of furnace charge
components) X = (X1..., Xn):
F (X) = C1*X1
+...+ Cj*Xj
+... + Cn*Xn -> Ext (Min);
At linear functional
restrictions:
Ai1*X1
+... + Aij*Xj
+... +
Ain*Xn {Ri} Bi,
i = 1,..,m.
And direct restrictions of
operated parameters:
Xj
0, j = 1,...,n,
where Ri - attitudes <;>; = or their
combinations;
Aij - the factors defined by the maintenance
of chemical elements in
charge; Bi - the factors defined by the
maintenance of chemical
elements in an alloy.
The described task could be
solved if the initial data contains the information about components of
furnace charge: quantity, costs, chemical compound, losses of elements
in concrete melting unit, restrictions on a chemical compound of an
alloy and componental structure of charge. Data about chemical compound
of foundry alloys is presented in the State Standard for alloys,
charging materials - in certificates of
factories-suppliers. The size of each element losses depends
on an initial condition of charge and concrete conditions of melt, but
in according to the difficulty of such account in calculations average
value of loss, irrespective of charge structure, but according to the
condition of charge (compact or disperse charge) is usually
applied.
Search for a minimum of
linear fitness function F(X) can be carried out with numerical methods
of linear programming for decision of extreme problems. So, the purpose
of the current work was the efficiency analysis of numerical linear
programming methods for the decision of a problem of charge structure
optimization and development on this basis of the software realizing
algorithm of the decision of a multiple problem of minimization of
charge cost under set conditions of alloy reception of a certain
chemical compound from unlimited set of initial components of furnace
charge.
The roots of a task can be
found by a direct search of all possible combinations of a vector
operated variables X with some set accuracy X. However realization of
such algorithm is defined by necessity of usage of the enclosed cyclic
structures essentially increasing duration of calculation.
The most effective
development of universal system of an optimum structure choice of
charge, which is not limited by quantity of components included in
calculation of furnace charge, is usage of a simplex-method of linear
programming. The given method realizes rational search for basic
admissible roots as a final iterative process. The improving value of
the fitness function on each step of calculations is
necessary.
The usage of a simplex-method
for charge structure optimization problem assumes preliminary reduction
of its formal statement to a standard initial form with n non-negative
variables (X1..., Xn) where minimization of
a linear fitness function F(X) is required with m linear restrictions.
The formal model of the
general linear programming problem of minimization the charge cost can
be expressed in the matrix form as follows:
F(X) = CX -> min AX
= BX 0,
where C - a vector of fitness function coefficients, size n; A
- a matrix of coefficients functional restrictions, size (m x n); B -
vector-column of free members of restrictions, size m; X - expanded
vector of operated variables. In described initial model a vector C and
matrix A are expanded concerning initial statement in factors 0 or 1 to
coordinate their sizes with expanded vector X. Unlike a method of
direct search of all possible combinations of vector X, the
simplex-method realizes the directed search of admissible basic
decisions on corresponding points of admissible decisions in the form
of iterative process where every step the value of criterion function
strictly decrease. Transition between extreme points of admissible
decisions is carried out according to simple linearly-algebraic
transformations of restrictions system. As number of extreme points of
admissible decisions certainly, and criterion function is linear, that,
touching extreme points in a direction of decrease of criterion
function, the simplex-method for final number of steps converges to a
global minimum.
The developed program will
allow to choose from the list various (available in a database or set
by the user independently) charge materials (including pig metal,
addition alloy any structure, return of own manufacture, iron bar,
metal shaving, pure metals) such set which would answer conditions of
the minimal cost and there corresponded to the set chemical compound,
certain State Standard for an alloy. Thus the admissible maintenance of
harmful impurity in an alloy also is considered. Final results of
calculation data on the calculated structure of charge materials in
view of an intoxication, a lump of furnace charge represent, and also
final structure of an alloy in view of restriction on impurity and
optimum cost of furnace charge.
The algorithm used for
calculations a simplex-method will allow to receive the optimum
decision at the limited quantity of admissible basic decisions
practically instantly, even at inclusion in calculation of all
components available in a database of charge.
In
beginning
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