JP2019082786A - Solution method for nonlinear programming problem, generation method for composition plan, and computer program, recording medium, apparatus, and composition operation method therefor - Google Patents

Abstract

To provide a fast solution method for a nonlinear programming problem.SOLUTION: A solution method for a nonlinear programming problem has: a vectorization step for variable which determines a term or mathematical expression included in a term as a new variable, changes a description of the nonlinear programming problem, and defines the new variable as a linear or higher-order vector variable; an initial solution generation step for a vector variable which generates a plurality of initial solutions of the vector variable; a formulation step for a linear programming problem which formulate the nonlinear programming problem, as a plurality of linear programming problems using linear variables, from a term or mathematical expression included in the nonlinear term determined as the new variable and a plurality of initial solutions; a solution step for the linear programming problem which finds solutions of the linear programming problems respectively; a solution modification step for vector variable which evaluates an object function for the respective solutions of the plurality of vector variables and modifying the solutions of the vector variables; and a selection step for the solutions of the vector variables which divides the solutions of the vector variables into a plurality of groups and selecting a solution of a vector variable to be used for subsequent calculation in each group.SELECTED DRAWING: Figure 5

Description

The present invention relates to a method for solving a non-linear programming problem which solves a non-linear programming problem including one or more variables having non-linearity, a computer program therefor, a recording medium, and a solution solving apparatus.
The present invention also relates to a method of creating a compounding plan, a computer program therefor, a recording medium, a compounding operation method, and a device for planning the blending ratio of raw materials for which the objective function becomes optimal after satisfying the constraint conditions. .

  Mathematical optimization, also called optimization problem, is a problem of analyzing the state in which the value is maximum or minimum for an objective function defined on a specific set obtained under constraints. In many industries including steel, for example, it is extremely important to minimize the production cost of goods as an objective function.

  The optimization problem can be divided into a linear programming problem (hereinafter abbreviated as LP) and a nonlinear programming problem (hereinafter abbreviated as NLP). The former is a linear function for both the constraint and the objective function, and the latter is a constraint or purpose At least one of the functions includes a non-linear function (e.g., a function having a second or higher order term).

  The linear programming problem began with the proposal of the simplex method by Danzig in 1963. A fast solution suitable for the problem has been proposed, and a large number of general-purpose solvers can be applied.

  However, it is very difficult to solve nonlinear programming problems exactly, and it is mainstream to perform linear approximation processing or solve using an approximation solution. For example, in Patent Document 1, even when a constraint expression and an objective function are given by a non-linear expression, a search is performed using it as a vector variable.

Japanese Patent Application No. 2016-212398

  However, Patent Document 1 searches for vector variables by applying a multipoint search method such as particle swarm optimization, and many unnecessary calculations are performed, which causes a problem in calculation speed.

  The present invention has been made in view of such conventional problems, and selects a solution of vector variables used for calculation and solves the nonlinear programming problem that enables high-speed calculation, its computer program, and recording medium , It aims at providing a solution solving device.

  In recent years, in the iron making process, optimization and planning automation of production plans and distribution plans in each process have been promoted. In the raw material process, the yard arrangement problem of deciding which yard to load the raw material loaded from the ship loaded with the raw material into, and the raw material conveyance problem of deciding which belt conveyor line is used to pay away the raw material on the yard or pile There are planning problems such as the problem of mixing raw materials into the blast furnace.

  One of the raw material blending problems is the ore blending problem. The ore blending problem is a problem of considering the blending ratio of each brand which minimizes the cost after satisfying the constraint conditions such as the property constraint required by the blast furnace.

  On the other hand, Patent Document 1 restricts the constraints on the properties of the product and the capacity of the manufacturing facilities, and the planning of the purchasing and use of the raw materials which minimizes the cost for purchasing and manufacturing the raw materials. Even when the equation and the objective function are given as non-linear equations, the search is performed using them as vector variables.

  However, Patent Document 1 searches for vector variables by applying a multipoint search method such as particle swarm optimization, and many unnecessary calculations are performed, which causes a problem in calculation speed.

  Therefore, the present invention has an object of providing a method of creating a compounding plan which makes it possible to perform high-speed calculation by selecting solutions of vector variables used for calculation, and providing a computer program, a recording medium, a blending operation method, and a producing device. Do.

  The above problems can be solved by the present invention described below.

[1] A method for solving a nonlinear programming problem which solves a nonlinear programming problem including one or more nonlinear terms,
A vectorization step of a variable which defines a part of terms or an equation included in the non-linear term as a new variable, changes a description of the non-linear programming problem, and defines the new variable as a one or more dimensional vector variable;
An initial solution creating step of a vector variable that creates a plurality of initial solutions of the vector variable;
Formulation of a linear programming problem that formulates the nonlinear programming problem as a plurality of linear programming problems using linear variables from some terms or formulas included in the nonlinear term determined as the new variable and the plurality of initial solutions Step, and
A step of solving a linear programming problem for finding solutions of a plurality of linear programming problems formulated in the step of formulating the linear programming problem;
A solution modification step of the vector variable, evaluating an objective function for each of the plurality of solutions of the vector variable determined in the solution solving step of the linear programming problem, and modifying the solution of the vector variable;
The solution of the vector variable is divided into a plurality of groups, and the solution of the vector variable used for the subsequent calculation is selected in each group, and the step of discarding the solution of the vector variable for speeding up calculation is provided.
Repeating the formulation step of the linear programming problem, the solving step of the linear programming problem, and the solution modification step of the vector variable using the solution of each of the vector variables modified in the solution modification step of the vector variable Solve the non-linear programming problem,
In the middle of the repetition, after the solution modification step of the vector variable, the solution selection step of the vector variable is appropriately performed and then the formulation step of the linear programming problem is performed to speed up the solution of the non-linear programming problem. A method of solving a non-linear programming problem characterized by
[2] In the method for solving a nonlinear programming problem according to the above [1],
The method of solving a non-linear programming problem, wherein the solution modifying step of the vector variable searches for the vector variable that can be executed while aiming at minimizing or maximizing the objective function.
[3] A computer program characterized by executing the method of solving nonlinear programming problems according to the above [1] or [2].
[4] A recording medium on which the computer program according to [3] is recorded and readable.
[5] An apparatus for solving nonlinear programming problems, which solves nonlinear programming problems including one or more nonlinear terms,
Input data capturing means for capturing input data such as target variables, objective functions, constraints, planning period, end conditions, number of initial solutions of vector variables,
A means for solving the non-linear programming problem for obtaining an optimum solution of the non-linear programming problem based on the input data taken in by the means for taking in the input data;
Providing a plan output means for outputting the optimal solution as a plan result,
In the means for solving the nonlinear programming problem,
A vectorization process of a variable that defines a part of terms or an equation included in the non-linear term as a new variable, changes a description of the non-linear programming problem, and defines the new variable as a vector variable of one or more dimensions;
Initial solution creation processing of the vector variable for creating a plurality of initial solutions of the vector variable;
Formulation of a linear programming problem that formulates the nonlinear programming problem as a plurality of linear programming problems using linear variables from some terms or formulas included in the nonlinear term determined as the new variable and the plurality of initial solutions Processing,
A solution process of a linear programming problem to obtain solutions of a plurality of linear programming problems formulated by the formulation process of the linear programming problem;
Performing a solution modification process of the vector variable, evaluating an objective function for each solution of a plurality of the vector variables determined by the solution process of the linear programming problem, and modifying the solution of the vector variable;
The formulation process of the linear programming problem, the solution process of the linear programming problem, and the solution modification process of the vector variable are repeated using the solution of each of the vector variables modified by the solution modification process of the vector variable. Solve the optimal solution of the nonlinear programming problem,
In the middle of the repetition, after the solution modification process of the vector variable,
The solutions of vector variables are divided into a plurality of groups as appropriate, and the solutions of vector variables used in subsequent calculations are selected within each group, and the solution of vector variables is sorted out for speeding up calculation. An apparatus for solving a non-linear programming problem characterized by speeding up solution of an optimal solution of the non-linear programming problem by performing formulation processing of the linear programming problem.
[6] In the apparatus for solving nonlinear programming problems according to the above [5],
The solution solving apparatus of the nonlinear programming problem characterized in that solution modification processing of the vector variable searches for the vector variable that can be executed while aiming at minimizing or maximizing the objective function.

[7] A method of creating a blending plan for planning blending proportions of raw materials for which an objective function is optimal after satisfying constraint conditions.
The combination plan is represented by a nonlinear programming problem including one or more nonlinear terms,
A step of capturing input data for capturing input data required to create a formulation plan;
A part of terms or an equation included in the non-linear term taken in the input data taking-in step is defined as a new variable, the description of the non-linear programming problem is changed, and the new variable is taken as one or more dimensional vector variables A vectorization step of the variables to be defined,
An initial solution creating step of a vector variable that creates a plurality of initial solutions of the vector variable;
Formulation of a linear programming problem that formulates the nonlinear programming problem as a plurality of linear programming problems using linear variables from some terms or formulas included in the nonlinear term determined as the new variable and the plurality of initial solutions Step, and
A step of solving a linear programming problem for finding solutions of a plurality of linear programming problems formulated in the step of formulating the linear programming problem;
A solution modification step of the vector variable, evaluating an objective function for each of the plurality of solutions of the vector variable determined in the solution solving step of the linear programming problem, and modifying the solution of the vector variable;
The solution of the vector variable is divided into a plurality of groups, and the solution of the vector variable used for the subsequent calculation is selected in each group, and the step of discarding the solution of the vector variable for speeding up calculation is provided.
Repeating the formulation step of the linear programming problem, the solving step of the linear programming problem, and the solution modification step of the vector variable using the solution of each of the vector variables modified in the solution modification step of the vector variable Solve the non-linear programming problem,
In the middle of the repetition, after the solution modification step of the vector variable, the solution selection step of the vector variable is appropriately performed and then the formulation step of the linear programming problem is performed to speed up the solution of the non-linear programming problem. How to make a recipe that is characterized by
[8] In the method of creating a combination plan described in the above [7],
The solution modification step of the vector variable is searching for the vector variable which can be executed while aiming at minimizing or maximizing the objective function.
[9] A computer program characterized by executing the method of creating a combination plan described in the above [7] or [8].
[10] A recording medium on which the computer program according to [9] is recorded and readable.
[11] A compounding operation method characterized in that compounding operation is performed based on the compounding plan created by the method of preparing the compounding plan described in [7] or [8] above.
[12] A device for creating a blending plan that plans blending ratios of raw materials for which an objective function is optimal after satisfying constraint conditions,
The combination plan is represented by a nonlinear programming problem including one or more nonlinear terms,
Input data capture unit that captures input data required to create a formulation plan,
A solving unit for a non-linear programming problem for obtaining an optimal solution of the non-linear programming problem based on the input data taken in by the taking-in unit of the input data;
And an output unit of a plan that outputs the optimal solution as a plan result,
In the solution part of the nonlinear programming problem,
Vectorization processing of a variable which defines a part of a term or formula contained in a nonlinear term as a new variable, changes the description of the nonlinear programming problem, and defines the new variable as a vector variable of one or more dimensions;
Initial solution creation processing of the vector variable for creating a plurality of initial solutions of the vector variable;
Formulation of a linear programming problem that formulates the nonlinear programming problem as a plurality of linear programming problems using linear variables from some terms or formulas included in the nonlinear term determined as the new variable and the plurality of initial solutions Processing,
A solution process of a linear programming problem to obtain solutions of a plurality of linear programming problems formulated by the formulation process of the linear programming problem;
Performing a solution modification process of the vector variable, evaluating an objective function for each solution of a plurality of the vector variables determined by the solution process of the linear programming problem, and modifying the solution of the vector variable;
The formulation process of the linear programming problem, the solution process of the linear programming problem, and the solution modification process of the vector variable are repeated using the solution of each of the vector variables modified by the solution modification process of the vector variable. Solve the non-linear programming problem,
In the middle of the repetition, after the solution modification process of the vector variable,
The solutions of vector variables are divided into a plurality of groups as appropriate, and the solutions of vector variables used in subsequent calculations are selected within each group, and the solution of vector variables is sorted out for speeding up calculation. An apparatus for creating a combination plan characterized by speeding up solution of the non-linear programming problem by performing formulation processing of the linear programming problem.
[13] In the preparation apparatus of the combination plan described in the above [12],
The solution modification processing of the vector variable is searching for the vector variable that can be executed while aiming at the minimization or maximization of the objective function.

In addition, the method of solving a non-linear programming problem described in the above [1] preferably has the following aspect.
A method for solving a non-linear programming problem which solves a non-linear programming problem including at least one non-linear term in a constraint and / or an objective function,
While defining some terms or formulas included in the non-linear term as one or more new variables, the description formula of the non-linear programming problem is changed by the new variables, and the one or more new variables are first-order A vectorization step of variables defined as vector variables of original and higher
An initial solution creating step of a vector variable that creates a plurality of initial solutions of the vector variable;
A replacement of a part of a term or equation included in the non-linear term with a new variable is added as a constraint equation, and each of a plurality of solutions of the vector variable is substituted into a description equation of the modified non-linear programming problem, A linear programming problem formulation step which is formulated as a plurality of linear programming problems using linear variables;
A step of solving a linear programming problem for finding solutions of a plurality of linear programming problems formulated in the step of formulating the linear programming problem;
A solution modification step of the vector variable, evaluating an objective function for each of the plurality of solutions of the vector variable determined in the solution solving step of the linear programming problem, and modifying the solution of the vector variable;
Selecting solutions of vector variables for speeding up calculation, dividing solutions of vector variables into a plurality of groups, and selecting solutions of vector variables used for subsequent calculation in each group;
Repeating the formulation step of the linear programming problem, the solving step of the linear programming problem, and the solution modification step of the vector variable using the solution of each of the vector variables modified in the solution modification step of the vector variable Solve the non-linear programming problem,
In the middle of the repetition, after the solution modification step of the vector variable, the solution selection step of the vector variable is appropriately performed and then the formulation step of the linear programming problem is performed to speed up the solution of the non-linear programming problem. A method of solving a non-linear programming problem characterized by

  According to the present invention, it is possible to provide a method of solving a non-linear programming problem that enables high-speed calculation by efficiently searching vector variables, a computer program, a recording medium, and a solution solution apparatus thereof.

  Further, according to the present invention, there is provided a method of creating a blending plan of a blending plan which enables high-speed calculation by efficiently searching for vector variables, a computer program, a recording medium, a blending operation method, and a creating apparatus. Can.

It is a schematic diagram which shows the solution space and feasible area | region of a linear programming problem. It is a schematic diagram which shows the solution space and feasible area | region of a nonlinear programming problem. It is a schematic diagram which shows the search process in a nonlinear programming problem. It is a figure which shows the structural example of the solution-solving apparatus of the nonlinear programming problem which concerns on this invention. It is a figure which shows the process example of the solution method of the nonlinear programming problem which concerns on this invention. It is the schematic which shows the processing flow until iron ore is charged to a blast furnace. It is a figure which shows the structural example of a mixing | blending plan production apparatus and a high-order system. It is a figure showing the example of processing of the preparation method of the combination plan concerning the present invention. It is a figure showing an algorithm of K-means method. It is a figure which shows the convergence process of the solution in an Example.

  Hereinafter, embodiments of the present invention will be described with reference to the drawings and formulas. First, the nonlinear programming problem targeted by the present invention can be expressed as follows.

  FIG. 1 is a schematic view showing a solution space and feasible region of a linear programming problem. The objective function and the constraint are linear, and the polygon in FIG. 1 indicates the feasible region of the linear programming problem (LP). When applying LP, high-speed calculation is possible because it is only necessary to search for polygon vertices. Assuming that the objective function is a downward-sloping straight line, the optimal solution of LP in this example is a vertex surrounded by a black circle.

  FIG. 2 is a schematic diagram showing the solution space and feasible region of the nonlinear programming problem. And FIG. 3 is a schematic diagram which shows the search process in a nonlinear programming problem.

  The nonlinear programming problem (NLP) is much more complicated than LP, and unlike the case of LP in FIG. 1 where the feasible region is a convex polygon, it is complicated curves and surrounded by curves as shown in FIG. It has become. And, when solving NLP, it is necessary to search for a point on the boundary of this area.

  In the example of FIG. 2, the global optimum solution is a point surrounded by a black circle. However, in the search process, the solution often converges to a point of a concave portion called a local optimum solution surrounded by a dashed circle in FIG. Once the solution converges to the local optimal solution, it is difficult to remove the local optimal solution and update the solution, and it is difficult to derive a global optimal solution. In addition, since the problem itself is complicated, the calculation time becomes long.

  The value of the objective function can be a predetermined value depending on the optimization problem. For example, the minimum value or the maximum value, or a predetermined value set in advance can be used. Here, as an example, consider the following optimization problem.

Since the above equation (5) includes the second-order term of x _{i on} the left side, handling it as it is a complicated problem. Therefore, in the present invention, one item on the left side of the equation (5) is made constant as in the following equation (6).

  By making one item on the left side of equation (5) constant as equation (6), the constraint condition of equation (5) can be described by a linear expression, and equation (6) can be regarded as an equality constraint. For example, the non-linear programming problem is a simple linear programming problem and can be easily solved. However, although the number of non-linear terms to be constant is one in the above example, a plurality of non-linear terms can be selected in consideration of the nature of the objective function and the constraints.

  In this case, since the selected variable is treated as a vector, this operation is hereinafter referred to as vectorization. Also, some terms or formulas included in the vectorized non-linear terms are referred to as vector variables or linearization variables.

  However, the search range of the original optimization problem is narrowed by changing the constraint condition in accordance with the constantization, and it is conceivable that the solution converges to the local solution. Therefore, in the present invention, an evaluation function is obtained by preparing a large number of constant terms on the right side of the equation (6) (a predetermined number j determined appropriately according to the problem), formulating a large number of linear programming problems, and solving the problems. By comparing the values (the values of the objective function), it is possible to search for a method of giving a constant term that makes the evaluation function value smaller.

  Although the above problem example is a very simple example in which there is only one equation including a quadratic term, the present invention can be more complicated by making a plurality of variables constant as described above. Nonlinear programming problems can be replaced by simple linear programming problems. For example, in the case where there are a plurality of mathematical expressions including non-linear terms of second order or higher in the constraints, variables to be constantized can be selected for each of the mathematical expressions and linearized. In addition, even when there are a plurality of quadratic or higher order terms in one equation, it is possible to replace the term with a simple linear programming problem by appropriately selecting the term and linearizing the term.

  FIG. 4 is a view showing an example of the configuration of an apparatus for solving nonlinear programming problems according to the present invention. In the figure, 01 represents a means for taking in input data, 02 represents a means for solving nonlinear programming problems, and 03 represents a means for outputting plans.

  In the input data fetching means 01, target variables, objective functions, constraints, planning period, end conditions, the number of initial solutions of vector variables, etc. are fetched as input data.

Then, in the means 02 for solving the non-linear programming problem, a large number of mathematical programming problems combining the input data taken in from the above-mentioned input data taking means 01 and the particle swarm optimization method described later and the linear programming are formulated, Repeat the solution of to obtain the optimal solution.
The plan output means 03 outputs, as a plan, the best solution that minimizes the objective function solved by the non-linear programming solution means 02.

  The above-described apparatus may be configured by a computer, may be configured by one computer, or may be configured by different computers to transmit and receive information. The processing of the solution method to be described later is executed by a computer program installed in a computer. Furthermore, the computer program may be recorded on a recording medium such as a CD or a magnetic tape so as to be readable.

  FIG. 5 is a diagram showing an example of processing of the method of solving nonlinear programming problems according to the present invention. Here, the steps of the input data acquisition and the planning output, which are the first and last processing of the whole, are omitted, and the processing of the solution solving means 02 of the nonlinear programming problem is shown. The non-linear programming problem solving means 02 includes: vectorization means for variables, initial solution preparation means for vector variables, formulation means for linear programming problems, solution means for linear programming problems, solution modification means for vector variables, A means for selecting and selecting solutions of vector variables is provided.

  In the vectorization step of the variable Step 101, the vectorization means of the variable appropriately selects some terms or formulas included in the non-linear term, determines them as new variables, changes the description of the non-linear problem, and changes the new variables. Performs vectorization processing of variables defined as vector variables of one or more dimensions. That is, some terms or formulas included in the non-linear term to be vectorized are determined. When there are a plurality of terms or formulas to be vectorized, they are multi-dimensional vectorized according to the number of the terms or formulas. When there are a plurality of expressions including non-linear terms, the non-linear terms are appropriately selected and vectorized respectively.

  In the initial solution creating step of the vector variable Step 102, the initial solution creating means of the vector variable performs an initial solution creating process of the vector variable for creating the combination of the vectorized variables defined in the vectorization step Step 101 of the variable. Here, vector variables and their combinations are called initial solutions. The number of combinations is the number set by the input data acquisition means 01. That is, there are as many initial solutions as the number of combinations.

  In the formulation step Step 103 of the linear programming problem, the formulation means of the linear programming problem performs a formulation process of the linear programming problem in which the vector variable defined in the vectorization step Step 101 is a constraint of the linear programming problem. Then, the optimization problem with the objective function and the constraint as the constraint equation is formulated as a linear programming problem for the number of objects set in advance by the input data acquisition means 01.

  In the step of solving linear programming problems Step 104, the means of solving linear programming problems is linear for solving each linear programming problem and obtaining an objective function for the plurality of linear programming problems created in the step of formulation linear programming problems Step 103 Solve the planning problem. Then, the combination of linear variables that minimizes the objective function is determined.

  In the vector variable solution modification step Step 105, the vector variable solution modification means performs a vector variable solution modification process for updating the vector variable based on the solution obtained in the linear programming problem solving step Step104. In the process at this time, a metaheuristic method is used which searches for the vector variable that can be executed while aiming for the minimization or maximization of the objective function. Preferably, techniques such as particle swarm optimization (PSO), genetic algorithm (GA), and tabu search are used. These are methods for efficiently searching for solutions while appropriately selecting good candidates from a large number of candidates. Although the example which applied the particle swarm optimization method is described below, it is not this limitation.

  In the step of solving the linear programming problem Step 104 and the step of changing the solution of the vector variable Step 105, the value of the objective function which is the smallest (if the metaheuristic method aiming to minimize the objective function is applied) A combination of variables is sought. Although the value of the objective function obtained here is the smallest among the given combinations of vector variables, it is not necessarily the smallest value as the initial non-linear programming problem. Therefore, starting from the combination of vector variables that minimize the value of the objective function, the combination of vector variables is further created and calculation is performed.

For example, a new combination of linear variables is created between the combination of linearization variables in which the value of the objective function is the smallest and the combination of linearization variables in which the value of the objective function is the second smallest, and these linear variables Solve the linear programming problem again by the combination of
Here, the number of combinations of linearization variables is included in the number set by the input data acquisition means 01.

  The vector variable solution selection step Step 106, the vector variable solution selection means divides the vector variable solution into a plurality of groups, and discards a vector variable solution to be used in subsequent calculations in each group. Perform selection processing of solution of variable. The selection process of solutions of vector variables in Step 106 can be performed by an appropriate method. For example, solutions of vector variables are divided into a plurality of groups, and solutions are sorted according to the evaluation function value in each group. can do. For example, the evaluation function values are good (for example, the evaluation function value is small if the objective function is to obtain the minimum value, the evaluation function value is large if the objective function is to obtain the maximum value) It is possible to select a predetermined number of solutions and not to select other solutions. Moreover, it is preferable to use clustering for grouping in Step 106. Although an example in which the K-means method is applied is described here, any clustering method such as the nearest distance method (Nearest Neighbor (NN method)), the K-NN method, the Ward method, and the longest distance method can be applied. .

  In the repeat determination step Step 107, the repeat determination means determines whether or not the calculation of the objective function satisfies the end condition set in advance by the input data take-in means 01. If the end condition is satisfied, the process is returned to the plan output means 03. If the end condition is not satisfied, the process returns to the linear programming problem formulation step Step 103, and the subsequent processing steps are repeated. As the end condition, for example, the calculation time, the number of loops, the upper limit of the number of updates, and the like can be mentioned.

  Note that the solution selection step Step 106 of the vector variable may be performed at an appropriate timing. For example, after the loop of the formulation step Step103 of the linear programming problem, the solution step Step104 of the linear programming problem, and the solution modification step Step105 of the vector variable is performed any number of times, the solution step of the vector variable is sorted out Step106. It is also preferable that the solution of the vector variable is performed at the timing when the evaluation function value converges to a certain degree (for example, the timing when the personal best described later stays at the same position to a certain degree). The selection step Step 106 may be performed. Since the solution of the vector variable is carried out at Step 106, the solution of the vector variable is selected, the number of solutions of the vector variable used for the subsequent calculation is reduced, and the calculation is speeded up.

Next, an application example of the present invention (a method of creating a blending plan of raw materials) will be described.
In the iron and steel industry, raw materials such as iron ore and coal having various properties purchased and mixed are mixed, but from the ground of blast furnace operation, it is required to keep the quality and properties after mixing raw materials within a certain range ing. At this time, since cost is judged as an important index, it is required to minimize purchase cost, manufacturing cost and the like. Furthermore, it is required to plan the blending over several days while changing the blending ratio so that the stock of raw materials is not exhausted. Although the formulation plan is created with all the raw materials as variables, the present invention will be described below by taking the creation of an optimal formulation plan of iron ore as an example.

  FIG. 6 is a schematic view showing a process flow until iron ore is charged into a blast furnace. In the figure, 201 represents a berth, 202 represents an ore yard, 203 represents an ore pile, 204 represents a sintering machine, and 205 represents a blast furnace.

  Iron ore which is a raw material of a blast furnace loaded on a ship entering the berth 201 is unloaded by a cargo handling device called an unloader and stacked in the ore yard 202. The iron ore that contains significantly different components depending on the produced mine is then blended to be the desired component in consideration of the quality such as hardness of sintered ore, and the property index of all iron ore input to the blast furnace. , To the ore pile 203.

  Among the iron ore to be discharged, powder ore that adversely affects the air permeability of the blast furnace is discharged to the sintering machine 204, and is sintered to be sintered ore, and is discharged to the blast furnace 205. In addition, lump ore and treated ore are discharged to the blast furnace 205 as it is. The present invention is applied to preparation of a compounding plan when delivering from the ore yard 202 to the ore pile 203.

  FIG. 7 is a diagram showing an example of the configuration of a blending plan creating device and a host system. In the figure, reference numeral 301 denotes an input data acquisition unit, 302 denotes a solution solver for nonlinear programming problems, 303 denotes a plan output unit, 304 denotes a data output unit of each process, 305 denotes a process computer, and 306 denotes a business computer.

  The compounding plan creation apparatus according to the present invention mainly includes an input data fetching unit 301, a non-linear programming problem solving unit 302, and a planning output unit 303.

  In the input data import section 301, planned arrival date and expected arrival quantity of iron ore, component, cost, inventory amount on ore yard, ingredient of raw material to be mixed as auxiliary material to be charged, cost, planned delivery of blast furnace Various input data such as amount, mass ore dispensable amount, planned production amount of sintered ore, planned period, end condition, number of initial solution of vector variable (such as calcination yield) are taken.

  Various input data are sent from the data output unit 304 of each process such as the berth 201, the ore yard 202, the ore pile 203, the sintering machine 204, and the blast furnace 205 shown in FIG. Then, it is sent to the input data acquisition unit 301.

  The non-linear programming problem solving unit 302 formulates a large number of mathematical programming problems combining the particle swarm optimization method and the linear programming method with the input data taken in from the input data taking-in unit 301, and solves the linear programming problem (for example, , Minimize costs, etc.) to obtain an optimal solution.

  The plan output unit 303 sends the optimum solution obtained by the non-linear programming problem solving unit 302 to the process computer 305 as a planning result.

  The above-described apparatus may be configured by a computer, may be configured by one computer, or may be configured by different computers to transmit and receive information. The processing of the solution method to be described later is executed by a computer program installed in a computer. Furthermore, the computer program may be recorded on a recording medium such as a CD or a magnetic tape so as to be readable.

FIG. 8 is a view showing a processing example of a method of creating a combination plan according to the present invention. In this figure, the processing steps for the acquisition of input data and the output of a plan, which are the first and last processing of the entire processing, are omitted, and the processing of the non-linear programming problem by the solution section 302 is shown. The non-linear programming problem solution part 302 comprises: vectorization means of variables, initial solution preparation means of vector variables, formulation means of linear programming problems, solution means of linear programming problems, solution modification means of vector variables, A means for selecting and selecting solutions of vector variables is provided.
It should be noted that in the input data import, various input data such as the planned arrival date and arrival amount of iron ore mentioned above are fetched from the upper system, and the plan output processing unit 303 outputs the best solution of the combination to the upper system .

  First, in the variable vectorization step Step 401, the vectorization unit of variables appropriately selects some terms or formulas included in the non-linear terms and determines them as new variables, changes the description of the non-linear programming problem, and Perform vectorization processing of variables that define various variables as vector variables of one or more dimensions. That is, some terms or formulas included in the non-linear term to be vectorized are determined. When there are a plurality of terms or formulas to be vectorized, they are multi-dimensional vectorized according to the number of the terms or formulas. When there are a plurality of expressions including non-linear terms, the non-linear terms are appropriately selected and vectorized respectively.

  In the initial solution creating step of the vector variable Step 402, the initial solution creating means of the vector variable creates a plurality of initial solutions of the vector variable defined in the vectorization step Step 401 of the variable by the number set in advance by the input data loading unit 301. Perform initial solution creation processing of vector variables. The number can be freely increased or decreased. Here, the number of initial solutions is appropriately determined in consideration of the calculation time in the initial solution modification step and the iterative determination step Step 407 described below, the accuracy of the solution to be obtained, and the like. Specifically, although depending on the calculation conditions, it is appropriate to set the upper limit to about 1000 times the vector variable.

  In the step of formulating the linear programming problem Step 403, the formulating means of the linear programming problem is a constraint for avoiding stock shortage of each raw material, a constraint on composition and properties of main raw materials after blending, and slag for maintaining slag fluidity. Component constraint, vectorization step of variables A formulation process of a linear programming problem is performed in which the variable defined in Step 401 is created as a constraint of the linear programming problem. Then, an optimization problem in which a function representing the cost of the blending plan is an objective function and the constraint is a constraint equation is formulated as a linear programming problem for the number of vector variables set in advance.

  In the step of solving linear programming problems Step 404, the means of solving linear programming problems is linear for solving each linear programming problem and obtaining an objective function for a plurality of linear programming problems created in the step of formulation linear programming problems Step 403. Solve the planning problem. Then, the combination of linear variables that minimize the objective function is determined.

  In the vector variable solution modification step Step 405, the vector variable solution modification means performs a vector variable solution modification process for updating the vector variable based on the solution obtained in the linear programming problem solution step Step 404. In the process at this time, a metaheuristic method is used which searches for the vector variable that can be executed while aiming for the minimization or maximization of the objective function. Preferably, methods such as particle swarm optimization (PSO), genetic algorithm (GA), and tabu search are used. These are methods for efficiently searching for solutions while appropriately selecting good candidates from a large number of candidates. Although the example which applied the particle swarm optimization method is described below, it is not this limitation.

  In the step of solving the linear programming problem Step 404 and the step of changing the solution of the vector variable Step 405, the value of the objective function which is the smallest (if the metaheuristic method aiming to minimize the objective function is applied) A combination of variables is sought. Although the value of the objective function obtained here is the smallest among the given combinations of vector variables, it is not necessarily the smallest value as the initial non-linear programming problem. Therefore, starting from the combination of vector variables that minimize the value of the objective function, the combination of vector variables is further created and calculation is performed.

For example, a new combination of linear variables is created between the combination of linearization variables in which the value of the objective function is the smallest and the combination of linearization variables in which the value of the objective function is the second smallest, and these linear variables Solve the linear programming problem again by the combination of
Here, the number of combinations of linearization variables is included in the number set by the input data acquisition unit 301.

  The vector variable solution selection step Step 406, the vector variable solution selection means divides the vector variable solution into a plurality of groups, and discards a vector variable solution to be used in subsequent calculations within each group. Perform selection processing of solution of variable. The selection process of solutions of vector variables in this Step 406 can be performed by an appropriate method. For example, solutions of vector variables are divided into a plurality of groups, and solutions are sorted according to the evaluation function value in each group. can do. For example, the evaluation function values are good (for example, the evaluation function value is small if the objective function is to obtain the minimum value, the evaluation function value is large if the objective function is to obtain the maximum value) It is possible to select a predetermined number of solutions and not to select other solutions. In addition, it is preferable to use clustering for grouping in Step 406. Although an example in which the K-means method is applied is described here, any clustering method such as the nearest distance method (Nearest Neighbor (NN method)), the K-NN method, the Ward method, and the longest distance method can be applied. .

  In the iterative determination step Step 407, the iterative determination means determines whether the calculation of the objective function satisfies the termination condition set in advance by the input data fetching unit 301. If the end condition is satisfied, the process returns to the plan output unit 303. If the end condition is not satisfied, the process returns to the linear programming problem formulation step Step 403, and the subsequent processing steps are repeated. As the end condition, for example, the calculation time, the number of loops, the upper limit of the number of updates, and the like can be mentioned.

  Note that the solution selection and selection step Step 406 of the vector variable may be performed at an appropriate timing. For example, after the loop of the formulation step Step 403 of the linear programming problem, the solution solving step Step 404 of the linear programming problem, and the solution modification step Step 405 of the vector variable is performed any number of times, the solution selection of the vector variable step Step 406 is implemented. It is also preferable that the solution of the vector variable is performed at the timing when the evaluation function value converges to a certain extent (for example, the timing when the personal best described later remains at the same position to a certain extent). The selection step Step 406 may be performed. By performing the step of discarding the solution of the vector variable Step 406, the solution of the vector variable is sorted out, the number of solutions of the vector variable used for the subsequent calculation is reduced, and the calculation is speeded up.

  Here, the particle swarm optimization method will be further described.

  The particle swarm optimization algorithm is a metaheuristics inspired by the collective behavior of organisms, and given an objective function to be searched for, an optimal solution is sought while multiple particles share information with each other. Move around in the search space. It features the simplicity of the concept and algorithm, the flexibility of operation, and the possibility of improvement.

  The particle swarm optimization algorithm updates the components of each vector variable according to the following equation.

The personal best represented by p _k (Pbest) is a position vector giving the best value of the objective function to the current generation of the k-th vector variable, further global best represented by g (gbest) Is the personal best of all vector variables, and is defined as (9) and (10) shown below. Here, F (·) indicates an objective function, and in this case, for example, a case where a minimum value is obtained as the best value of the objective function is shown. That is, it is shown that the value of the objective function taken in all the current generations of the particle k is the personal best or higher in the equation (9), and the objective function value of the global best g in the equation (10) Indicates that it is less than or equal to the personal best p _k objective function value of any particle k.

Here, the position vector z corresponds to the variable vectorized in Step 401 (Step 101), and the velocity vector v corresponds to the direction of creating a new combination of linearization variables performed in the step of modifying the vector variable Step 405 (Step 105). Do. Then, p _k is the linear combination of variables to obtain the minimum value of the objective function in combination any vector variable k is taken to generation l, g is the smallest objective function among all p _k It is a combination from which values can be obtained. The calculation of the objective function is performed as shown in the following (11) and (12), and p _k and g are updated each time the value of the objective function is obtained. Equation (11) is a personal best update equation, and equation (12) is a global best update equation.

F (z) is an objective function of the LP solution when β _j is z in the equations (3) to (6) of the above example. The position vector z is changed to sequentially calculate so that F is minimized. Then, let z be the minimum of F be p, and F (p). Then, further calculation is performed, and p is switched whenever F becomes minimum.

  F (g) is the smallest F among all F (p). When a minimum p is obtained, g is replaced by p to obtain F (g).

  F (g) is the smallest F among all F (p) (there are multiple personal vests for multiple particles). When a minimum p is obtained, g is replaced by p to obtain F (g).

  F (z) is determined by the combination of z, however, it is unclear in which direction there is a combination of z that minimizes F (z). Therefore, F (g) is updated every time the minimum value of F (p) is obtained, assuming that the minimum F (p) at a certain point in time is F (g). That is, if F (g) becomes smaller, there is a high possibility that there is an optimal solution in that direction, and by updating F (g), the possibility that the optimal solution can be reached becomes high.

  That is, in the present invention, vector variables are searched using an algorithm of particle swarm optimization method so that the objective function value becomes smaller. When a particle becomes an "excellent particle" corresponding to Gbest, a particle group is dragged by the "excellent particle" and approaches an optimal solution.

  Although the example in which the objective function is minimized has been described above, the same procedure can be used to maximize the objective function. In this case, the combination of variables that maximizes the objective function may be determined. In addition to minimizing or maximizing the value of the objective function, it is also possible to set the objective function to a predetermined value. Also in this case, a combination of vector variables having an objective function as a predetermined value may be obtained similarly.

  In the iron and steel industry, it is required to mix raw materials such as various types of iron ore and coal having various properties purchased and to keep the quality and properties after mixing within a certain range. Moreover, since cost is judged as an important parameter | index in this case, the minimization of a purchase cost, a manufacturing cost, etc. is calculated | required. Furthermore, it is required to plan the blending over several days while changing the blending ratio so that the stock of raw materials is not exhausted. Here, in particular, an application example of the present invention for producing an optimum blending plan of iron ore is shown.

  Examples will be described below. First, a detailed example of the constraint equation will be described. In the present embodiment, the preset combination plan creation period is further divided into a plurality of plan units and handled. For example, if the planning period is set to six months, one month is considered as the planning unit. In the following description, this setting is used.

  The constraint condition (constraint equation) will be specifically described. The stock constraint is given by the formula shown below.

  When constantizing some terms or formulas included in the non-linear terms, the above formula becomes a constraint. The firing yield will be described as an example. The firing yield means the yield when the powder ore, which is a powdery iron ore, is sintered to form a sintered ore.

  When sintered ore is charged into a blast furnace, if the strength of the sintered ore is low, it will be pulverized and air permeability of the blast furnace will be deteriorated. For this reason, raising the strength of sintered ore is very important in operation.

  Therefore, when making sinter ore, add the limestone by-products to the powder ore and sinter it to increase the strength of the sinter ore. The formula of derivation of the calcination yield is shown below.

That is, the sintered ore production amount is obtained by multiplying the raw material input amount obtained by adding the auxiliary raw material blending amount to the powder ore amount x _{i, m} and multiplying the firing yield.

  Since the amount x of powder ore to be blended is a variable, the calcination yield has nonlinearity from the above two equations. Here, the above two equations can be treated as a linear constraint equation of the powder ore blending amount sum, if the values of the auxiliary raw material blending amount and the firing yield are made constant in the vectorization step Step 401 of the variable. By performing the same operation on some terms or mathematical expressions included in other non-linear terms, linearization of constraints can be performed.

  Next, a detailed example of the objective function is shown below.

  The first item represents the ore cost, and the second item represents the change in the amount of payout for each planned unit period. In the iron and steel manufacturing industry, if the daily amount of withdrawal fluctuates greatly, it becomes the source of operation problems, so it is desirable that the difference in the daily amount of compounding be small. Therefore, in this example, in addition to the cost of the ore, a change in the amount of payout for each planned unit period is added as an operation constraint to form an objective function.

  The plan period is set to six months with one month as the plan unit, and a solution that minimizes the sum of the above two terms is sought.

Here, clustering using the K-means method will be described.
FIG. 9 is a diagram showing an algorithm of the K-means method. Although the case where it divides into 3 groups as an example is described here, it is not the limitation. Each vector variable is randomly divided into 3 groups (3 clusters). Next, the vector coordinates of the center of gravity for each cluster are calculated. The calculation method is a simple average of each direction of the coordinates of cluster elements. Subsequently, the cluster is changed to a group close to the position of the center of gravity. After cluster transfer of each vector variable is completed, the position of the center of gravity is updated for a new cluster, and then changing of the cluster and updating of the center of gravity are repeated, and the process ends if there is no movement of the vector variable between clusters. Note that the distance between two points is calculated by Euclidean distance.

  In the present embodiment, in the above-mentioned vector variable solution selection step Step 406, the vector variable solution selection means divides the vector variable into a plurality of clusters by clustering, and calculates only the variables having good evaluation values for each cluster. Process to continue. Although FIG. 9 shows an example in which the upper half variable having a good evaluation value is held in each cluster and the remaining calculation is stopped, the method of selecting particles for stopping the calculation is not limited to this.

  FIG. 10 is a diagram showing the convergence process of the solution in this embodiment. Calculated using a commercially available solver, the vertical axis indicates the total calculation time (minutes) of the computer or the non-dimensionalized cost merit (evaluation value), and the horizontal axis indicates the number of repetitions of the loop (Step 403 to Step 406). The calculation is performed with a planned period of 6 months, a planned unit period of 1 month, and 2,000 particles. As shown in FIG. 10, it can be seen that the calculation speed is improved compared to the conventional method (Patent Document 1), and the evaluation value is also comparable.

01 Input data acquisition means 02 Non-linear programming problem solving means 03 Planning output means 201 Berth 202 ore yard 203 Ore pile 204 Sintering machine 205 Blast furnace 301 Input data acquisition part 302 Non-linear programming problem solution part 303 Planning output part 304 Data output unit of each process 305 Process computer 306 Business computer

Claims (13)

A method for solving a nonlinear programming problem which solves a nonlinear programming problem including one or more nonlinear terms,
A vectorization step of a variable which defines a part of terms or an equation included in the non-linear term as a new variable, changes a description of the non-linear programming problem, and defines the new variable as a one or more dimensional vector variable;
An initial solution creating step of a vector variable that creates a plurality of initial solutions of the vector variable;
Formulation of a linear programming problem that formulates the nonlinear programming problem as a plurality of linear programming problems using linear variables from some terms or formulas included in the nonlinear term determined as the new variable and the plurality of initial solutions Step, and
A step of solving a linear programming problem for finding solutions of a plurality of linear programming problems formulated in the step of formulating the linear programming problem;
A solution modification step of the vector variable, evaluating an objective function for each of the plurality of solutions of the vector variable determined in the solution solving step of the linear programming problem, and modifying the solution of the vector variable;
The solution of the vector variable is divided into a plurality of groups, and the solution of the vector variable used for the subsequent calculation is selected in each group, and the step of discarding the solution of the vector variable for speeding up calculation is provided.
Repeating the formulation step of the linear programming problem, the solving step of the linear programming problem, and the solution modification step of the vector variable using the solution of each of the vector variables modified in the solution modification step of the vector variable Solve the non-linear programming problem,
In the middle of the repetition, after the solution modification step of the vector variable, the solution selection step of the vector variable is appropriately performed and then the formulation step of the linear programming problem is performed to speed up the solution of the non-linear programming problem. A method of solving a non-linear programming problem characterized by In the method of solving nonlinear programming problems according to claim 1,
The method of solving a non-linear programming problem, wherein the solution modifying step of the vector variable searches for the vector variable that can be executed while aiming at minimizing or maximizing the objective function. A computer program for executing the method of solving nonlinear programming problems according to claim 1 or 2. A recording medium on which the computer program according to claim 3 is recorded and readable. An apparatus for solving nonlinear programming problems, which solves nonlinear programming problems including one or more nonlinear terms, comprising
Input data capturing means for capturing input data such as target variables, objective functions, constraints, planning period, end conditions, number of initial solutions of vector variables,
A means for solving the non-linear programming problem for obtaining an optimum solution of the non-linear programming problem based on the input data taken in by the means for taking in the input data;
Providing a plan output means for outputting the optimal solution as a plan result,
In the means for solving the nonlinear programming problem,
A vectorization process of a variable that defines a part of terms or an equation included in the non-linear term as a new variable, changes a description of the non-linear programming problem, and defines the new variable as a vector variable of one or more dimensions;
Initial solution creation processing of the vector variable for creating a plurality of initial solutions of the vector variable;
Formulation of a linear programming problem that formulates the nonlinear programming problem as a plurality of linear programming problems using linear variables from some terms or formulas included in the nonlinear term determined as the new variable and the plurality of initial solutions Processing,
A solution process of a linear programming problem to obtain solutions of a plurality of linear programming problems formulated by the formulation process of the linear programming problem;
Performing a solution modification process of the vector variable, evaluating an objective function for each solution of a plurality of the vector variables determined by the solution process of the linear programming problem, and modifying the solution of the vector variable;
The formulation process of the linear programming problem, the solution process of the linear programming problem, and the solution modification process of the vector variable are repeated using the solution of each of the vector variables modified by the solution modification process of the vector variable. Solve the optimal solution of the nonlinear programming problem,
In the middle of the repetition, after the solution modification process of the vector variable,
The solutions of vector variables are divided into a plurality of groups as appropriate, and the solutions of vector variables used in subsequent calculations are selected within each group, and the solution of vector variables is sorted out for speeding up calculation. An apparatus for solving a non-linear programming problem characterized by speeding up solution of an optimal solution of the non-linear programming problem by performing formulation processing of the linear programming problem. In the apparatus for solving nonlinear programming problems according to claim 5,
The solution solving apparatus of the nonlinear programming problem characterized in that solution modification processing of the vector variable searches for the vector variable that can be executed while aiming at minimizing or maximizing the objective function. This is a method of creating a blending plan that plans blending ratios of raw materials for which the objective function is optimal after satisfying the constraint conditions.
The combination plan is represented by a nonlinear programming problem including one or more nonlinear terms,
A step of capturing input data for capturing input data required to create a formulation plan;
A part of terms or an equation included in the non-linear term taken in the input data taking-in step is defined as a new variable, the description of the non-linear programming problem is changed, and the new variable is taken as one or more dimensional vector variables A vectorization step of the variables to be defined,
An initial solution creating step of a vector variable that creates a plurality of initial solutions of the vector variable;
Formulation of a linear programming problem that formulates the nonlinear programming problem as a plurality of linear programming problems using linear variables from some terms or formulas included in the nonlinear term determined as the new variable and the plurality of initial solutions Step, and
A step of solving a linear programming problem for finding solutions of a plurality of linear programming problems formulated in the step of formulating the linear programming problem;
A solution modification step of the vector variable, evaluating an objective function for each of the plurality of solutions of the vector variable determined in the solution solving step of the linear programming problem, and modifying the solution of the vector variable;
The solution of the vector variable is divided into a plurality of groups, and the solution of the vector variable used for the subsequent calculation is selected in each group, and the step of discarding the solution of the vector variable for speeding up calculation is provided.
Repeating the formulation step of the linear programming problem, the solving step of the linear programming problem, and the solution modification step of the vector variable using the solution of each of the vector variables modified in the solution modification step of the vector variable Solve the non-linear programming problem,
In the middle of the repetition, after the solution modification step of the vector variable, the solution selection step of the vector variable is appropriately performed and then the formulation step of the linear programming problem is performed to speed up the solution of the non-linear programming problem. How to make a recipe that is characterized by In the method of creating a combination plan according to claim 7,
The solution modification step of the vector variable is searching for the vector variable which can be executed while aiming at minimizing or maximizing the objective function. A computer program characterized by executing the method of creating a combination plan according to claim 7 or 8. A recording medium, wherein the computer program according to claim 9 is recorded and readable. A blending operation method comprising performing blending operation based on the blending plan created by the blending plan creating method according to claim 7 or 8. A blending plan creation device that plans blending ratios of raw materials for which an objective function is optimal after satisfying constraint conditions.
The combination plan is represented by a nonlinear programming problem including one or more nonlinear terms,
Input data capture unit that captures input data required to create a formulation plan,
A solving unit for a non-linear programming problem for obtaining an optimal solution of the non-linear programming problem based on the input data taken in by the taking-in unit of the input data;
And an output unit of a plan that outputs the optimal solution as a plan result,
In the solution part of the nonlinear programming problem,
Vectorization processing of a variable which defines a part of a term or formula contained in a nonlinear term as a new variable, changes the description of the nonlinear programming problem, and defines the new variable as a vector variable of one or more dimensions;
Initial solution creation processing of the vector variable for creating a plurality of initial solutions of the vector variable;
Formulation of a linear programming problem that formulates the nonlinear programming problem as a plurality of linear programming problems using linear variables from some terms or formulas included in the nonlinear term determined as the new variable and the plurality of initial solutions Processing,
A solution process of a linear programming problem to obtain solutions of a plurality of linear programming problems formulated by the formulation process of the linear programming problem;
Performing a solution modification process of the vector variable, evaluating an objective function for each solution of a plurality of the vector variables determined by the solution process of the linear programming problem, and modifying the solution of the vector variable;
The formulation process of the linear programming problem, the solution process of the linear programming problem, and the solution modification process of the vector variable are repeated using the solution of each of the vector variables modified by the solution modification process of the vector variable. Solve the non-linear programming problem,
In the middle of the repetition, after the solution modification process of the vector variable,
The solutions of vector variables are divided into a plurality of groups as appropriate, and the solutions of vector variables used in subsequent calculations are selected within each group, and the solution of vector variables is sorted out for speeding up calculation. An apparatus for creating a combination plan characterized by speeding up solution of the non-linear programming problem by performing formulation processing of the linear programming problem. In the preparation apparatus of the combination plan according to claim 12,
The solution modification processing of the vector variable is searching for the vector variable that can be executed while aiming at the minimization or maximization of the objective function.