JP2018180694A - Arithmetic unit, method and program for nonlinear programming problem - Google Patents

Abstract

PROBLEM TO BE SOLVED: To prevent a problem scale from increasing to obtain an optimum solution with high accuracy within a practically reasonable time when solving a nonlinear programming problem.SOLUTION: A mixture planning device 100 functions as an arithmetic unit for a nonlinear programming problem to solve the nonlinear programming problem which minimizes or maximizes an objective function under constraints including a linear constraint and a nonlinear constraint, and makes a mixture plan with a mixture ratio as a decision variable in a manufacturing process of mixing a plurality of raw materials having different properties to manufacture a product. A nonlinear function representing the nonlinear constraint is a function having, as an independent variable, an intermediate variable represented by linear combination of the decision variable in the nonlinear programming problem. A constraint setting unit 103 sets upper and lower limit values of the intermediate variable and approximates the nonlinear function within a range between the upper and lower limit values by a piecewise linear function and sets constraints, and the nonlinear programming problem is solved under the constraints set by the setting means.SELECTED DRAWING: Figure 1

Description

  The present invention relates to a computing device, method, and program of a non-linear programming problem for solving a non-linear programming problem which minimizes or maximizes an objective function under constraints including linear and non-linear constraints.

A mathematical programming problem is one that minimizes or maximizes the objective function under constraints. Among them, those including a nonlinear function in at least one of a constraint and an objective function are called a nonlinear programming problem.
For example, in the process of producing coke and sintered ore in the iron and steel industry, a plurality of raw materials held by each plant are blended to produce a product. In this case, it is necessary to formulate a compounding plan so as to meet the target product characteristic value and production volume. In addition, it is necessary to formulate a blending plan to satisfy the operation conditions in the manufacturing process.
As described above, the constraints for setting the characteristic value, the production amount, and the operation condition of the target product in many cases include a function in which it is difficult to mathematically obtain an optimal solution. For example, an estimation equation for predicting the quality of coke includes an equation representing structural characteristics such as coke strength, but in this estimation equation, a nonlinear function based on experiment or theory is used. As such, many of the mathematical programming problems in the manufacturing process become complex nonlinear programming problems, and it has been difficult to obtain an accurate solution with high accuracy in practical time.

JP, 2009-175804, A

Applied Mathematical Programming, Stephen P. Bradley, Addison-Wesley, 1977 Algorithm for representing multivariable functions by sum of one-variable functions I, Kiyotaka Yamamura, Yasuko Murayama, IEICE Technical Report, CAS, Circuits and Systems 93 (102), 67-74, 1993-06-19 An Algorithm for Representing Multivariate Functions by Sum of One-Variable Functions II, Kiyotaka Yamamura, Yasuko Murayama, IEICE Technical Report, CAS, Circuits and Systems 93 (102), 75-82, 1993-06-19 Tetsuya Fujie, "Introduction to Formulation by Integer Programming," Operations Research: Science of Management, vol. 57, no. 4, pp. 190-197, Apr. 2012.

Several proposals have been made as a method for solving nonlinear programming problems.
For example, in Patent Document 1, a simulator for creating a combination plan for mixing a plurality of types of blended raw materials, a mathematical model model representing supply / demand balance constraints of blended raw materials, and a mathematical model representing property constraints after mixing are constructed. A model construction unit, and a planning unit that performs optimization calculation based on a predetermined objective function using a mathematical model built by the model construction unit, and calculates an instruction to the simulator, and is a mathematical expression that represents the property constraint after mixing When the model includes a non-linear equation, a linear equation is introduced instead of the non-linear equation to formulate the equation model, and the solution by the planning unit using the equation model including the line equation is the non-linear equation A method is disclosed for confirming whether or not a mathematical expression model including.
Also, Non-Patent Document 1 shows a method for introducing auxiliary variables to a nonlinear function, converting the function into a separable form, and approximating it with a piecewise linear function to formulate it as a mixed integer programming problem. There is.

However, in the method of Patent Document 1, since the nonlinear programming problem is simplified and approximated to the linear programming problem, an approximation error may occur and the solution accuracy may be degraded.
Further, in the method of Non-Patent Document 1, the formulation of a nonconvex function requires definitions of 0 and 1 variables in accordance with the number of divisions of the non-linear function. This problem is NP-hard, and if the definition range of variables is too wide or the number of divisions is too large, the size of the problem may increase exponentially and it may not be possible to solve in practical time.

  The present invention has been made in view of the above points, and in solving a nonlinear programming problem which minimizes or maximizes an objective function under constraints including linear constraints and nonlinear constraints, The objective is to avoid an increase in the problem size and to obtain an accurate solution with high accuracy in practical time.

The gist of the present invention for solving the above problems is as follows.
[1] A computing device of a non-linear programming problem for solving a non-linear programming problem which minimizes or maximizes an objective function under constraints including a linear constraint and a non-linear constraint,
The non-linear function representing the non-linear constraint condition is a function having an intermediate variable represented by a linear combination of decision variables of the non-linear programming problem as an independent variable,
Setting means for setting upper and lower limit values of the intermediate variable and approximating the non-linear function with a piecewise linear function within the range of the upper and lower limit values, and setting a constraint condition;
A solution solving means for solving the nonlinear programming problem under the constraint set by the setting means;
The setting means sets upper and lower limits by solving the mathematical programming problem using the intermediate variable as an objective function by removing the non-linear constraint condition from the constraint for the intermediate variable. A computing device for nonlinear programming problems.
[2] The setting means is
Introduce an auxiliary variable to the nonlinear function and convert it into a separable function represented in the form of a sum of univariate functions,
The upper and lower limit values of the intermediate variable used for the converted separable function are set, and a nonlinear function constituting the converted separable function is approximated by a piecewise linear function to set constraints. The arithmetic unit of the nonlinear programming problem according to [1].
[3] The setting means is characterized in that, when approximating a non-linear function by a piecewise linear function, at least one of a convex connection formulation and a formulation using a special ordered set of type 2 is performed [ The arithmetic unit of the nonlinear programming problem as described in 1] or [2].
[4] An approximate calculation value obtained by substituting the value of the decision variable obtained by the solution solving means into the function after the piecewise linear approximation by the setting means,
Comparing the theoretical variables obtained by substituting the decision variables obtained by the solution solving means into a non-linear function representing the non-linear constraint condition;
The number of divisions of the piecewise linear function is changed until the difference between the approximate calculation value and the theoretical calculation value satisfies a predetermined condition, and the processing by the setting means and the solution solving means is repeated [1]. The computing device of the nonlinear programming problem according to any one of [3].
[5] The above-mentioned nonlinear programming problem is a blending planning problem in which a blending ratio is a deciding variable in a manufacturing process of blending a plurality of raw materials to produce a product, any one of [1] to [4] Arithmetic unit of the nonlinear programming problem described in one or one.
[6] A method of computing a non-linear programming problem for solving a non-linear programming problem which minimizes or maximizes an objective function under constraints including a linear constraint and a non-linear constraint,
The non-linear function representing the non-linear constraint condition is a function having an intermediate variable represented by a linear combination of decision variables of the non-linear programming problem as an independent variable,
A setting step of setting upper and lower limit values of the intermediate variable and approximating the non-linear function with a piecewise linear function in the range of the upper and lower limit values to set a constraint condition;
The solving means has a solving step of solving the non-linear programming problem under the set constraints.
In the setting step, upper and lower limit values are set by solving the mathematical programming problem using the intermediate variable as an objective function by removing the non-linear constraint condition from the constraint condition for the intermediate variable. How to calculate nonlinear programming problems.
[7] A program for solving a nonlinear programming problem which minimizes or maximizes an objective function under constraints including linear constraints and nonlinear constraints,
The non-linear function representing the non-linear constraint condition is a function having an intermediate variable represented by a linear combination of decision variables of the non-linear programming problem as an independent variable,
Setting processing of setting upper and lower limit values of the intermediate variable and approximating the non-linear function with a piecewise linear function in the range of the upper and lower limit values, and setting constraints;
Causing a computer to execute a solution solving process for solving the nonlinear programming problem under the set constraint condition;
In the setting process, upper and lower limit values are set by solving the mathematical programming problem using the intermediate variable as an objective function by removing the non-linear constraint condition from the constraint for the intermediate variable. And the program to be.

  According to the present invention, when solving a non-linear programming problem which minimizes or maximizes the objective function under constraints including linear constraints and non-linear constraints, the problem size is prevented from increasing and the working time is avoided. It is possible to obtain an optimal solution with high accuracy.

It is a figure showing functional composition of a combination plan drafting device concerning an embodiment. It is a flowchart which shows the process of the planning method of the mixing | blending plan by the mixing | blending plan planning apparatus which concerns on embodiment. It is a flowchart which shows the detail of a formulation process of the constraint condition in the flowchart of FIG. It is a figure which shows the example of the directed graph which represents the master-detail relationship between variables. It is a figure which shows the result of labeling with respect to the node of the directed graph of FIG. 4A. It is a characteristic view showing the relation between the number of divisions of piecewise linear approximation and relative error in an example. It is a characteristic view showing the relation between the number of divisions of piecewise linear approximation and calculation time in an example. It is a characteristic view which shows the comparison result of the relationship between the compounding amount changeable ratio in an Example, and the cost improvement effect.

  Hereinafter, preferred embodiments of the present invention will be described with reference to the accompanying drawings.

  When formulating a non-linear function by piecewise linear approximation, it is necessary to finely divide the number of divisions of the piecewise linear function in order to increase the approximation accuracy of the piecewise linear function. However, since the computational load increases as the number of divisions increases, the number of divisions must have a minimum value that satisfies the required accuracy. Here, the number of divisions of the piecewise linear approximation function necessary to maintain the calculation accuracy is determined by the possible range of the variables constructing the piecewise linear approximation function. If the range of possible variables is wide, many divisions are required, and the smaller the number of divisions, the more formal the formulation. Therefore, the present inventors reduced the search range of the solution of the optimization problem after formulation by obtaining the minimum value / maximum value of the variable constructing the non-linear function in advance using linear programming. . Through this, we have established technology to realize high-speed, high-precision planning for the formulation planning problem of this process.

  In this embodiment, an example is described in which the present invention is applied to a manufacturing process for producing a product by blending raw materials having different properties, such as the manufacturing process of coke and sintered ore in the steel making industry.

  FIG. 1 shows a functional configuration of the blending plan planning apparatus 100 according to the embodiment. The compounding plan drafting device 100 functions as a computing device for a nonlinear programming problem to which the present invention is applied, and uses a blending ratio as a decision variable in a manufacturing process of compounding raw materials (brands) having a plurality of different properties to produce a product. Create a combination plan that meets the target product characteristic values, production volume, and operating conditions, and minimizes manufacturing costs. In other words, the optimal blending ratio is determined. In the formulation planning problem dealt with in the present embodiment, at least a part of the constraint conditions is a non-linear constraint represented by a non-linear function.

  Reference numeral 101 denotes an input unit, which, for example, takes in data such as property information (information on properties and conditions) of each raw material, operation condition information, cost information, etc., which are necessary for drafting a blending plan from the database 108.

  Reference numeral 102 denotes a division number setting unit, which sets the number of divisions when piecewise linear approximation of the non-linear function is performed. The division number setting unit 102 may set the division number according to, for example, an instruction of the operator, or may automatically set the division number within the range of the predetermined division number.

Reference numeral 103 denotes a constraint condition setting unit, which formulates and sets characteristic values and production amounts of a target product and constraint conditions for satisfying operation conditions based on data acquired by the input unit 101 in an equation model.
Here, the constraint condition setting unit 103 includes a function conversion unit 103a and an upper and lower limit value setting unit 103b, sets upper and lower limit values of variables used for nonlinear constraints, and divides nonlinear functions constituting nonlinear constraints. The linear approximation is performed to function as setting means for setting constraints.
More specifically, the function conversion unit 103a determines whether the non-linear function representing the non-linear constraint is a separable function. The separable function is a function in which the function f is expressed in the following form with respect to any x = (x ₁ , x ₂ ,..., X _n ) ^T in n-dimensional Euclidean space.

If the non-linear function is not expressed as a separable function, an auxiliary variable is introduced to convert the non-linear function into a separable function. This transformation makes it possible to express all non-linear functions in the form of the sum of univariate functions.
The upper and lower limit value setting unit 103b sets upper and lower limit values of variables used for the separable function converted by the function conversion unit 103a. Of the above set mathematical model, for the mathematical model relaxed by removing the non-linear constraints from the constraints, the upper limit and lower limit are to be formulated into a mathematical model using the above variables as the objective function, and the objective function is Set upper and lower limits by solving mathematical programming problems to be maximized and minimized.
Then, when the one-variable function that constitutes the separable function converted by the function conversion unit 103a is a non-linear function, the restriction condition setting unit 103 sets the restriction condition by piecewise linear approximation of the one-variable function. .
Here, the term “formulation” does not mean that the constraint setting unit 103 constructs the formula model itself from the beginning, but the data captured in step S1 is included in the frame of the formula model of the constraint set in advance. It means to reflect.

Reference numeral 104 denotes an objective function setting unit, which formulates and sets an objective function that minimizes the manufacturing cost on a mathematical expression model based on the data acquired by the input unit 101.
Here, the term “formalization” does not mean that the objective function setting unit 104 constructs the mathematical expression model itself from the beginning, but the data acquired in step S1 is included in the framework of the mathematical expression model of the objective function set in advance. It means to reflect.

  A solution solver 105 solves the mixture planning problem set by the constraint condition setting unit 103 and the objective function setting unit 104.

Reference numeral 106 denotes a determination unit, which determines whether the solution solution by the solution solution unit 105 is valid. As a result, if the solution result is valid, the solution result is determined. If the solution result is not valid, the division number setting unit 102 changes the number of divisions, and the constraint condition setting unit 103, the objective function setting unit 104, and the solution unit 105 repeatedly solve the mixing scheme problem.
As a mechanism for determining the validity of the solution solution result, the calculated value obtained by substituting the blending ratio obtained by the solution solution unit 105 into the function after piecewise linear approximation by the constraint condition setting unit 103 and the original non-linear constraints. Compare with theoretical calculation value obtained by substitution. Then, the division number setting unit 102 changes the division number until the difference between the calculated value and the theoretical calculation value satisfies a predetermined condition, and the combination plan by the constraint condition setting unit 103, the objective function setting unit 104, and the solution unit 105 Repeat the problem setting and solving.

  Reference numeral 107 denotes an output unit, which outputs the solution solution result determined by the determination unit 106 to be valid. The output referred to here means, for example, storing and registering the solution solution result in a storage area (not shown), displaying it on the display 110, and sending it out to an external device via a network.

  Reference numeral 108 denotes a database, which stores data such as property information, operation condition information, cost information, and the like of each raw material which is required to make a blending plan. Although an example in which data is fetched from the database 108 is shown here, other than that, for example, data may be fetched from an external device via a network, or data directly input by the user may be captured. Absent.

  Reference numeral 109 denotes an input device such as a pointing device or a keyboard, and reference numeral 110 denotes a display.

Next, a method of drafting a blending plan by the blending plan planning device 100 will be described.
FIG. 2 is a flowchart showing processing of a method of creating a combination plan by the combination plan creating device 100 according to the embodiment.
In step S1, the input unit 101 takes in, for example, data such as property information, operation condition information, cost information, and the like of each raw material which is required to create a mixing plan from the database 108. For example, as shown in Table 1, upper and lower limit information is taken with respect to the blending ratio of each grade. Also, although not specifically shown, upper and lower limit information is taken as to the nature of each brand.

  In step S2, the division number setting unit 102 sets the division number K when piecewise linear approximating the non-linear function. When there are a plurality of non-linear functions to be piecewise linear approximation or a plurality of variables used for the separable functions, the number of divisions may be set for each non-linear function or for each variable used for the separable functions.

In step S3, based on the data acquired in step S1, the constraint setting unit 103 formulates and sets the characteristic value and the production amount of the target product as a target and the constraint for satisfying the operation condition in a mathematical model.
In the present embodiment, the constraints are divided into a linear constraint represented by a linear function as in equation (1) and a nonlinear constraint represented by a non-linear function as in equation (2).

Expression (1) is an expression representing a constraint that is linear among the constraints. In the equation (1), x is an M-dimensional column vector representing a blend ratio which is a decision variable, and the elements of the vector correspond to the blend ratio of each raw material. In equation (1), A is a matrix of N rows and M columns, and all elements of A have real values. b is an N-dimensional column vector, and takes real values.
In equation (2), g _i (x) is a non-linear function, and i represents the type of constraint. For example, the non-linear function g _i (x) represents a characteristic value with strong non-linearity such as coke strength in the case of steel making industry. The nonlinear function g _i (x) in equation (2) is a multivariable function, a nonconvex function, or a non-differentiable function.

  In applying the present invention, the constraints are formulated by dividing linear constraints and nonlinear constraints. First, add linear constraints to the mathematical model. Linear constraints can be added to mathematical models without particular difficulty.

Next, the formulation of nonlinear constraints is performed. In step S3, according to the flowchart shown in FIG. 3, the non-linear constraints are converted into separable functions, upper and lower limit values of variables used for the separable functions are set, and one-variable functions constituting the separable functions are non-linear functions. In some cases, the one-variable function is approximated by a piecewise linear function, and the approximated piecewise linear function is set as a constraint.
In step S301, the constraint setting unit 103 inputs a non-linear function g _i (x). In this embodiment, it is assumed that the non-linear function g _i (x) is expressed in the form of equation (3). x is an M-dimensional column vector representing a blend ratio which is a decision variable, and elements of the vector correspond to the blend ratio of each raw material. w ₁ , w ₂ , and w ₃ are M-dimensional vectors, and elements of the vectors have real values.

The non-linear function g _i (x) of Equation (3) is a nonconvex function and a multivariable function. In general, when applying piecewise linear approximation to a nonlinear nonconvex function, it is necessary to perform formulation using a special order set of type 2 (Special Order Set type 2: SOS 2) in addition to the convex connection formulation. . For this reason, the optimization problem which formulated this nonlinear function by piecewise linear approximation becomes an NP-hard problem.
Further, the non-linear function g _i (x) of equation (3) is an M-order multivariable function. Therefore, to apply the piecewise linear approximation without adding the equation transformation, the M-order hyperplane must constitute an approximation function. For example, consider the case where a non-linear function g _i (x) of equation (3) is divided into K piecewise linear functions for each variable. At this time, the number of variables necessary for piecewise linear approximation of the non-linear function g _i (x) is approximately K ^M and an order of power. For example, if the K = 50, M = 10, the number of variables needed to piecewise linear approximation are numerous and 50 ^10. Therefore, there is a high possibility that the formulation as it is can not be solved within the practical time. Therefore, there is a need for formulation of formulation that reduces the size of the problem.

Therefore, in step S302, the function conversion unit 103a of the constraint setting unit 103 introduces an auxiliary variable to the non-linear function g _i (x), and is a multivariable function that can be expressed in the form of a separable function, that is, a sum of single variable functions. Convert to The conversion to the separable function may use an algorithm presented in Non-Patent Document 2 or Non-Patent Document 3, or any other algorithm. When the non-linear function g _i (x) is converted to a separable function, it is expressed as equation (4). The non-linear function g _i (x) is a function having the intermediate variables v _{1 to} v ₄ represented by the linear combination of the decision variables x as independent variables.

It is assumed that the division number K is set for the non-linear function of equation (4). Although there are three non-linear functions in Equation (4), since each function is a one-variable function, the number of variables required for piecewise linear approximation may be K only. Therefore, the number of variables constituting the piecewise linear function is 3 × K and 3K. That is, by converting the non-linear function to the separable function as in equation (4), the number of variables constituting the piecewise linear function, which was required by K ³ before the conversion, is suppressed to 3 K after the conversion It can be said that.

In step S303, the constraint setting unit 103 clarifys the relationship between the variables with respect to the function converted in step S302, and prepares a directed graph in which variables in the function are set as nodes as preparation for performing formulation according to the order. Build In a directed graph, for each function, a branch is made starting at a variable serving as an input of the function and ending at a variable obtained through the function. For example, if there is a relation that a function has z = f (x, y), two branches (x, z) and (y, z) are connected. FIG. 4A is an example in which x and v _k are nodes, and the dependency of v _k is connected by a directed graph, with respect to equation (4). In the example of equation (4), v ₄ is a variable composed of variables v ₂ and v ₃ , and variable v ₅ is a variable composed of v ₁ and v ₄ . By using a directed graph as shown in FIG. 4A, it is possible to set a master-slave relationship between variables.

In step S304, the constraint setting unit 103 labels the nodes (variables) of the valid graph constructed in step S303. Since the digraph constructed in step S303 is an acyclic valid graph (tree structure), topological sorting is possible. The topological sort is to order each node so that every node precedes the node before the output edge. Topological sorting is performed on the directed graph constructed in step S303, and labels are sequentially added from 1 to the nodes located behind. FIG. 4B shows the result of topological sorting of the directed graph constructed in step S303 and labeling of nodes.
As mentioned above, the graphing of the relationship between variables for formulating a non-linear function and labeling were completed by step S303 and S304.
In the present embodiment, in the subsequent steps S305 to S314, the relationship of variables is added to the mathematical expression model by repeating the process in the order of the labels given in step S303.

In step S305, the constraint setting unit 103 repeatedly starts calculations for all variables in the order of small node labels k with respect to a node set having a tree structure. Hereinafter, variables corresponding to the node k are u _k (k = 1,..., M + 6). Also, hereinafter, in a node set having a tree structure, a node adjacent to and below a certain node is called a child node. Also, from the viewpoint of a certain node, any of its child node and all of its descendant nodes is called a descendant node.

In step S306, the constraint setting unit 103 determines whether a child node exists in the node k. If there is a child node at node k, the process proceeds to step S307. If there is no child node, upper and lower limit values of variable u _k are set at step S313. In the example of the equation (4), since the variables (x ₁ , x ₂ ,..., X _M ) correspond to nodes having no child node, the process proceeds to step S313. On the other hand, variables (v ₁ , v ₂ ,..., V _n ) are nodes in which child nodes exist, and the process proceeds to step S307.

First, consider the case where the child node does not exist in the node k in step S306 and the process proceeds to step S313. In this case, in step S313, upper and lower limit values are set for node k having no child node. The upper and lower limit values set upper and lower limit values of the node k given in advance. In the blending plan as in this embodiment, upper and lower limit values are often determined in advance in variables (x ₁ , x ₂ ,..., X _M ) representing blending proportions. Good. In the present embodiment, the upper and lower limit values set in Table 1 are set as variables (x ₁ , x ₂ ,..., X _M ). Thus, by determining the upper and lower limits of all nodes having no child node, all the child nodes become bounded variables in the formulation of the node appearing in the subsequent calculation. This makes it possible to add a function that needs to be represented by a bounded variable, such as a piecewise linear function, to a mathematical model. If the node k is not a bounded variable, the maximum range that the node k can take may be set as the upper and lower limits of the node k. For example, in the blending planning problem in the present embodiment, the blending ratio is never 0% or 100% or more, so the lower limit value of the node k may be set to 0 and the upper limit value to 100.

Next, consider the case where a child node exists at node k in step S306 and the process proceeds to step S307. In step S307, the constraint setting unit 103 determines whether all the relationships between the node k and all the descendant nodes of the node k include a linear relationship or a non-linear relationship. In the case of only the linear relational expression, the process proceeds to step S308, and in the case of including the non-linear relational expression, the process proceeds to step S310. In the example of equation (4), all descendant nodes of variables (v ₁ , v ₂ , v ₃ , v ₄ ) corresponding to nodes M + 1 to M + 4 in FIG. 4B are connected by a linear relational expression. , And proceeds to step S308. Since the variables (v ₅ ) and the descendant nodes of g _i (x) corresponding to the nodes M + 5 and M + 6 in FIG. 4B include non-linear relations, the process proceeds to step S309.

In step S308, the constraint setting unit 103 adds a linear relational expression between the node k and the child node to the mathematical expression model. Specifically, equation (5) is added to the mathematical expression model as a constraint. Expression (5) represents a linear relationship between the node k and the child nodes of the node k, S _k is a set of all the child nodes of the node k, and w _jk is a coefficient representing the relationship between the node and the child nodes. In the example of variable v _{1 in} equation (4), the child nodes of v ₁ are variables (x ₁ , x ₂ ,..., X _M ), and the relationship between v ₁ and child nodes is v ₁ = w _{Since 1} ^T x is connected, v ₁ = w ₁ ^T x is set as a constraint.

In step 309, the upper and lower limit value setting unit 103b of the constraint condition setting unit 103 sets the upper and lower limit values of the variable corresponding to the node k. Specifically, linear constraints that remove nonlinear constraints from constraints of nonlinear programming problems and equality constraints that represent the relationship between all descendant nodes of variables u _k and k are the constraints, and the purpose of variable u _k is Construct a mathematical programming problem as a function and maximize and minimize the problem. By solving the mathematical programming problem in this way, the definition range of the variable u _k , that is, the upper and lower limit values can be obtained.
Equations (6), (7) and (8) show a linear programming problem for finding the minimum (lower limit) and the maximum (upper limit) of u _k .

This linear programming problem is constructed by removing non-linear constraints from the constraints of Equations (1) and (2). Thus, the definition range of the variable u _k can be obtained by solving the mathematical programming problem using the variable u _k as the objective function by using the linear constraint.
In this case, the definition range of the variable u _k obtained in step S309 is a range obtained by solving a kind of relaxation problem obtained by removing the non-linear constraints from the original problem, so the possible range of the variable u _k in the original problem and Do not necessarily match. However, by predetermining the possible range of the variable u _{k according} to the present method, the number of 0, 1 variables necessary for maintaining the accuracy of the piecewise linear approximation is reduced in the formulation of the piecewise linear approximation function described later Can be expected to improve the calculation accuracy and reduce the calculation time.

Subsequently, in step S313, the upper and lower limit value setting unit 103b of the constraint condition setting unit 103 sets the optimum solution obtained in step S309 as the upper and lower limit values of the variable u _k .
The above is the formulation method in the case where the relationship between the node k and all the descendent nodes of the node is represented by a linear relationship at step S307.

Next, in step S307, consider the case where the relationship between node k and all descendent nodes is not represented by a linear relationship alone. For example, in the equation (4), since the descendant nodes of the variable (v ₅ ) and g _i (x) can not be represented only by the linear relational expression, the process proceeds to step S310. In the case where the node k and all the descendant nodes of the node k are not represented by the linear relational expression alone, nonlinear constraints are also required for the problem of finding the upper and lower limits of the variable, as in steps S308 to S309. It is not possible to set upper and lower limits of variables using linear programming. Therefore, in the subsequent steps S310 to S312, focusing on only the linear / nonlinearity of the node k, upper and lower limits of variables are set, and formulation into a mathematical expression model is performed. Thus, by formulating the relationship between parent nodes and child nodes one by one, it is aimed to formulate the whole non-linear function into a mathematical model.

In step S310, the constraint setting unit 103 determines whether the node k and its child nodes are represented by a linear relational expression. If the child node of the node k is represented by a linear relation, the process proceeds to step S311, and if the child node of the node k can not be expressed only by the linear relation but includes at least one nonlinear relation, the piecewise linear approximation is used. In order to formulate, it progresses to step S312. In the example of equation (4), variable (v ₅ ) is a non-linear function with variables v ₁ and v _4, and g _i (x) is represented by a non-linear function with variable v ₅ .

In step S311, the constraint setting unit 103 adds a linear relational expression of the node k and a child node of the node k as a constraint to the mathematical expression model.
Subsequently, in step S313, the upper and lower limit value setting unit 103b of the constraint condition setting unit 103 sets the upper and lower limit values of the variable u _k which is a linear function. The upper and lower limit values of the variable u _k are set analytically by determining the maximum value and the minimum value from the upper and lower limit values of the variable on the right side of the linear function.

In step S312, the constraint setting unit 103 formulates the nonlinear relationship between the node k and the child nodes of the node k into a mathematical expression model. In the present embodiment, the non-linear function is approximated by a piecewise linear function, and the piecewise linear function is added to the mathematical expression model as a constraint by the convex coupling formulation method. The convex coupling formulation is a piecewise linear approximation to a non-linear function y = f (x), and the nodes of the piecewise linear function are set as (x _i , y _i ) i = 1,. , Formula (9) to formulate the relationship.

Expression (9) corresponds to finding a vector whose weight t _k is 1 for a vector pointing from the origin to a node of the piecewise linear function. If f (x) is a convex function and a problem that minimizes f (x), then the solution is on a certain line segment, so only two consecutive t _k have positive values. Otherwise, two non-adjacent t _k may be positive, and such a solution needs to be removed. To remove such solutions, the set of t _k, "only t _k to high but if two adjacent takes a positive value" may be added constraints. A group of variables (t ₁ , t ₂ ,..., T _K ) in which only two or more adjacent t _k have positive values are called SOS2 group of variables. For the formulation of the SOS two variable group, it may be formulated using integer programming, or a dedicated function for defining the SOS set prepared in a standard solver may be used. An example of formulating the SOS2 variable group using integer variables is shown in equation (10). The details are described in Non-Patent Document 4. Thus, in addition to the convex connection formulation, the SOS2 variable group is defined to formulate a nonconvex nonlinear function as a mixed integer programming problem.
Subsequently, in step S313, the upper and lower limit value setting unit 103b of the constraint condition setting unit 103 sets the upper and lower limit values of the variable u _k which is a non-linear function. Upper and lower limits of the variable u _k sets the upper and lower limits of the piecewise linear function that approximates the variable u _k.
The above is the formulation method in the case where all the descendent nodes of the node k include at least one non-linear relational expression.

By repeating the above steps S306～S313, for all variables u _k, and a mathematical model of the relationship between variables, it is possible to set upper and lower limit values further for all variables u _k.
And if g _i (x) 制約 0 is added to the mathematical expression model in step S 314, the constraints including non-linear constraints can be formulated into the mathematical expression model.
The above is the details of step S3. Heretofore, in order to formulate a non-linear function used for formulation of a compounding plan by using a piecewise linear approximation, it has been required to formulate a piecewise linear function by the dimension of the variable representing the compounding ratio. On the other hand, according to the procedure of the present embodiment, construct a calculation graph with nodes as decision variables used in non-linear constraint expressions as nodes, and carry out formulation for the relationship between nodes (linear and non-linear) It has become possible to greatly reduce the number of variables needed to piecewise linearize a non-linear function. Furthermore, the range of variables for constructing the piecewise linear function was solved in advance for the minimization / maximization problem of variables using linear programming. This makes it possible to narrow the definition range of the variables constructing the piecewise linear approximation function, and reduces the number of 0, 1 variables necessary to maintain the precision of the piecewise linear approximation function. Therefore, it was possible to construct a model that can be solved in practical time.

  Referring back to FIG. 2, in step S4, the objective function setting unit 104 formulates and sets an objective function that minimizes the cost for manufacturing on a mathematical expression model based on the data acquired in step S1. The objective function may be a linear function or a non-linear function. If the objective function is represented by a nonlinear function, convert the nonlinear function into a separable function as in the case of the constraints, set upper and lower limits of variables, and piecewise linearize the nonlinear function to obtain the objective function. It should be set.

  In step S5, the solution unit 105 solves the blending planning problem which is the non-linear programming problem set in step S3 and step S4.

In step S6, the determination unit 106 determines whether the solution result in step S5 is valid. Since piecewise linear approximation is an approximation method to the last, approximation errors always occur, and it is necessary to confirm whether the approximation errors are sufficiently small.
As a mechanism for determining the validity of the solution result, the compounding ratio obtained in step S5 is substituted into the calculated value obtained by substituting the function after piecewise linear approximation in step S3 and the original nonlinear constraint. The obtained theoretical calculation values are compared. Then, the division number K is changed in step S2 until the difference between the calculated value and the theoretical calculated value satisfies a predetermined condition, and the setting and the solution of the nonlinear programming problem in steps S3 to S5 are repeated.

  In step S7, the output unit 107 outputs the solution solution result determined to be valid in step S6.

An embodiment for solving a blending planning problem, which is a non-linear programming problem, will be described by using the method described above and targeting the blending planning of coal.
Coal blending planning uses the blending ratio as the deciding variable, and meets the target product characteristic value, production volume, and operating conditions, and as a non-linear planning problem to formulate a blending plan that minimizes manufacturing costs, Formula (11) to formula (15)
i: Index related to brand name j: Index related to constraint x _i : Coal blend ratio (decision variable)
p _i : Coke production cost (coal purchasing cost + transportation cost + operation cost)
a _ij : Ratio of components of raw coal (measured value)
g _i (x ₁ , x ₂ , ..., x _N ): Coke quality estimation equation (nonlinear)
LB _j , UB _j : Upper and lower limit value of quality U _i ^L , U _i ^U : Upper and lower limit value of blending ratio

  Here, although the combination plan in a single period is represented here, the formulation of the equations (11) to (15) may be extended to a multiple period combination plan. In order to extend to multi-period blending planning, it is necessary to have stock conditions in the previous and subsequent periods, multi-period usage planning, and constraints on raw material acceptance planning, all of which are mixed integer programming It is possible to formulate a multi-period compounding plan if it is possible to formulate a single-period compounding plan.

  The objective function of equation (11) represents the cost of coke production. Although the objective function is given as a linear function in this embodiment, it may be a non-linear function.

The constraints in equation (12) represent linear constraints on quality, and include, for example, constraints on volatile content, sulfur content, and the like. a _ij represents the ratio of components of coal, and the value obtained by weighted average of the values by the mixture ratio is given as a constraint.
The constraints in equation (13) represent nonlinear constraints on quality, including, for example, constraints on coke cold strength (coke DI) and expansion pressure. The coke DI is an index representing the strength of coke obtained after the coal is sintered in a coke oven. Coke is required to have a certain strength or more to secure air permeability in blast furnaces, and coke DI is often used as an index of the strength. Moreover, expansion pressure is a parameter | index showing the pressure applied in a coke oven by the gas which generate | occur | produces when it hardens coal in a coke oven. If the expansion pressure is high, it is desirable that the value be as low as possible in order to damage the furnace body of the coke oven.
The constraint of equation (14) represents a constraint on the total value of the blend ratios.
The constraint of Formula (15) represents the upper and lower limits regarding the blending ratio of coal.

  In the mathematical expression model as described above, the equation (13) corresponds to the nonlinear constraint of the equation (2). Further, the equations (12), (14), and (15) correspond to the linear constraints of the equation (1). In this example, the formulation was performed according to the flowchart shown in FIG. 3 for the coke DI and the expansion pressure included in the equation (13).

  In Table 2, optimization is performed according to the embodiment under the blending ratio (Table 1) and the constraint conditions of the properties, and the values of the blending ratio obtained are shown. As shown in Table 2, the optimization results comply with the blending ratio constraints of Table 1. Also, the optimization results comply with all constraints, and the correct optimization was performed.

  FIG. 5 shows the result of performing optimization calculation for each division number while changing the division number of the piecewise linear approximation. The horizontal axis represents the number of divisions, and optimization was performed by increasing the number of divisions from 10 to 80 in 10 steps. The vertical axis represents the relative error [%] between the calculated value and the theoretical calculated value, and represents the value obtained by dividing the difference between the calculated value and the theoretical calculated value by the theoretical calculated value as a percentage. From FIG. 5, both the relative error of coke DI and expansion pressure decrease as the number of divisions increases. Furthermore, the relative error was 0.1% or less under all conditions, and there was almost no error when the number of divisions exceeded 30. The magnitude of this error is sufficiently small compared to the operation variation of coke DI and expansion pressure, and does not pose a problem in practice.

  Next, FIG. 6 changes the number of divisions of the piecewise linear approximation, and shows the calculation time required for the optimization calculation at each number of divisions. The horizontal axis represents the number of divisions, and the vertical axis represents the calculation time [sec]. As shown in FIG. 6, although the calculation time increases as the number of divisions increases, in a calculation environment using a general-purpose computer, the calculation time is within 60 seconds in all optimizations from 10 to 80 divisions, It was possible to solve it in calculation time which does not become upper problem.

  Next, FIG. 7 shows the result of investigation of the cost improvement effect when the restriction condition of the blending ratio is changed. The division number of the piecewise linear approximation was fixed at 50. In this example, the blending amount at a certain point was set as the standard blending, and the change in cost when the ratio of changing the blending from the standard blending was gradually increased was illustrated. Here, the ratio that allows the change of the mixing ratio from the reference mixing amount is called the mixing amount changeable ratio. The horizontal axis represents the mixing ratio changeable ratio, and the mixing ratio changeable ratio was changed from 0% to 80%. The vertical axis represents the cost improvement effect [%], and represents the cost improvement effect relative to the standard as a percentage based on the case where the mixture ratio changeable ratio is 0%. As shown in FIG. 7, the invention method to which the present invention is applied is expected to have a cost improvement effect three or more times higher than that of the comparison method (the algorithm of Patent Document 1), showing that the present invention is sufficiently useful. .

As mentioned above, although the present invention was explained with an embodiment, the above-mentioned embodiment shows only an example of the embodiment in the case of carrying out the present invention, and the technical scope of the present invention is interpreted restrictively by these. It is a must-have. That is, the present invention can be implemented in various forms without departing from the technical concept or the main features thereof.
In the present embodiment, the production process of coke or sintered ore in the iron making industry is mentioned as a production process of producing a product by blending raw materials having a plurality of different properties. For example, blending in petroleum refining, cement production, etc. The present invention may be applied to planning.
Furthermore, the invention is not limited to formulation problems but is broadly applicable to non-linear programming problems. For example, when designing a construction, there is a problem of optimizing the thickness of columns after securing necessary strength. At this time, assuming that the thickness of many columns is a variable, the calculation of the strength is a non-linear function of the variable. By applying the present invention also to such a case, it is possible to avoid an increase in the scale of the problem and to obtain an accurate solution with high accuracy in practical time.

The arithmetic device of the nonlinear programming problem to which the present invention is applied is realized by, for example, a computer device provided with a CPU, a ROM, a RAM, and the like.
Furthermore, the present invention can also be implemented by supplying software (program) for realizing the functions of the present invention to a system or apparatus via a network or various storage media, and a computer of the system or apparatus reading and executing the program. It is feasible.

100: formulation planning device 101: input unit 102: division number setting unit 103: constraint condition setting unit 104: objective function setting unit 105: solution solving unit 106: determination unit 107: output unit

Claims (7)

A computing device for a nonlinear programming problem for solving a nonlinear programming problem which minimizes or maximizes an objective function under constraints including linear constraints and nonlinear constraints.
The non-linear function representing the non-linear constraint condition is a function having an intermediate variable represented by a linear combination of decision variables of the non-linear programming problem as an independent variable,
Setting means for setting upper and lower limit values of the intermediate variable and approximating the non-linear function with a piecewise linear function within the range of the upper and lower limit values, and setting a constraint condition;
A solution solving means for solving the nonlinear programming problem under the constraint set by the setting means;
The setting means sets upper and lower limits by solving the mathematical programming problem using the intermediate variable as an objective function by removing the non-linear constraint condition from the constraint for the intermediate variable. A computing device for nonlinear programming problems. The setting means is
Introduce an auxiliary variable to the nonlinear function and convert it into a separable function represented in the form of a sum of univariate functions,
The upper and lower limit values of the intermediate variable used for the converted separable function are set, and a nonlinear function constituting the converted separable function is approximated by a piecewise linear function to set constraints. The arithmetic unit of the nonlinear programming problem according to claim 1.   The setting means performs at least one of a convex combination formulation and a formulation using a special ordered set of type 2 when approximating a non-linear function by a piecewise linear function. The arithmetic unit of the nonlinear programming problem according to 2. An approximate calculation value obtained by substituting the value of the decision variable obtained by the solution solving means into the function after the piecewise linear approximation by the setting means;
Comparing the theoretical variables obtained by substituting the decision variables obtained by the solution solving means into a non-linear function representing the non-linear constraint condition;
The number of divisions of the piecewise linear function is changed until the difference between the approximate calculation value and the theoretical calculation value satisfies a predetermined condition, and the processing by the setting means and the solution solving means is repeated. A computing device for the nonlinear programming problem according to any one of the items 1 to 3.   The said non-linear programming problem is a compounding planning problem which makes a compounding ratio a decision variable in the manufacturing process which mix | blends a several raw material and manufactures a product, It is characterized by the above-mentioned. A computing device for nonlinear programming problems. A method of computing a nonlinear programming problem for solving a nonlinear programming problem which minimizes or maximizes an objective function under constraints including linear constraints and nonlinear constraints.
The non-linear function representing the non-linear constraint condition is a function having an intermediate variable represented by a linear combination of decision variables of the non-linear programming problem as an independent variable,
A setting step of setting upper and lower limit values of the intermediate variable and approximating the non-linear function with a piecewise linear function in the range of the upper and lower limit values to set a constraint condition;
The solving means has a solving step of solving the non-linear programming problem under the set constraints.
In the setting step, upper and lower limit values are set by solving the mathematical programming problem using the intermediate variable as an objective function by removing the non-linear constraint condition from the constraint condition for the intermediate variable. How to calculate nonlinear programming problems. A program for solving a non-linear programming problem which minimizes or maximizes an objective function under constraints including linear constraints and non-linear constraints.
The non-linear function representing the non-linear constraint condition is a function having an intermediate variable represented by a linear combination of decision variables of the non-linear programming problem as an independent variable,
Setting processing of setting upper and lower limit values of the intermediate variable and approximating the non-linear function with a piecewise linear function in the range of the upper and lower limit values, and setting constraints;
Causing a computer to execute a solution solving process for solving the nonlinear programming problem under the set constraint condition;
In the setting process, upper and lower limit values are set by solving the mathematical programming problem using the intermediate variable as an objective function by removing the non-linear constraint condition from the constraint for the intermediate variable. And the program to be.

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