JP6834717B2 - Arithmetic logic units, methods and programs for nonlinear programming problems - Google Patents

Description

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

Mathematical programming problems are problems that minimize or maximize the objective function under constraints. Among them, a problem in which at least one of the constraint condition and the objective function includes a nonlinear function is called a nonlinear programming problem.
For example, in the coke and sinter manufacturing process in the steel industry, a plurality of raw materials owned by each factory are mixed to manufacture a product. In this case, a formulation plan must be formulated so as to satisfy the characteristic values and production volume of the target product. In addition, a formulation plan must be formulated to meet the operational requirements of the manufacturing process.
In this way, the characteristic values of the target product, the production amount, and the constraints for satisfying the operating conditions often include a function that is mathematically difficult to find the optimum solution. For example, the estimation formula for predicting the quality of coke includes a formula expressing structural characteristics such as coke strength, and a non-linear function based on experiments and theories is used for this estimation formula. As described above, many of the mathematical programming problems in the manufacturing process are complicated nonlinear programming problems, and it is difficult to obtain an optimal solution with high accuracy within a practical time.

JP-A-2009-175804

Applied Mathematical Programming, Stephen P. Bradley, Addison-Wesley, 1977 Algorithm for expressing multivariable functions as the sum of one-variable functions I, Kiyotaka Yamamura, Yasuko Murayama, Technical Report of the Society of Electronics, Information and Communication Engineers, CAS, Circuits and Systems 93 (102), 67-74, 1993-06-19 Algorithm for expressing multivariable functions as the sum of one-variable functions II, Kiyotaka Yamamura, Yasuko Murayama, Technical Report of the Society of Electronics, Information and Communication Engineers, 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 methods for solving nonlinear programming problems.
For example, in Patent Document 1, a simulator for creating a blending plan for mixing a plurality of types of blended raw materials, a mathematical model representing a supply-demand balance constraint of the blended raw materials, and a mathematical model representing a property constraint after mixing are constructed. It is equipped with a model construction unit and a planning unit that performs optimization calculations based on a predetermined objective function and calculates instructions to the simulator using a mathematical model constructed by the model construction unit, and is a mathematical formula that represents the property constraints after mixing. When the model contains a non-linear mathematical formula, a linear mathematical formula is introduced in place of the non-linear mathematical formula to formulate the mathematical model, and the solution result by the planning unit using the mathematical model including the linear mathematical formula is the non-linear mathematical formula. A method for confirming whether or not a mathematical model including is satisfied is disclosed.
Further, Non-Patent Document 1 shows a method of introducing an auxiliary variable into a nonlinear function, converting the function into a separable form, approximating it with a piecewise linear function, and formulating 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 accuracy of the solution may deteriorate.
Further, in the method of Non-Patent Document 1, in order to formulate a non-convex function, it is necessary to define 0 and 1 variables according to the number of divisions of the nonlinear function. This problem is NP-hard, and if the definition range of variables is too wide or the number of divisions increases too much, the scale of the problem increases exponentially, and there is a risk that it cannot be solved within practical time.

The present invention has been made in view of the above points, and when solving a nonlinear programming problem that minimizes or maximizes an objective function under a constraint condition including a linear constraint condition and a nonlinear constraint condition. The purpose is to avoid increasing the scale of the problem and to obtain a highly accurate optimum solution within the practical time.

The gist of the present invention for solving the above problems is as follows.
[1] A non-linear programming problem computing device for solving a non-linear programming problem that minimizes or maximizes an objective function under a constraint condition including a linear constraint condition and a non-linear constraint condition.
The nonlinear function representing the nonlinear constraint condition is a function having an intermediate variable represented by a linear combination of the determinants of the nonlinear programming problem as an independent variable.
A setting means for setting the upper and lower limit values of the intermediate variable and approximating the nonlinear function with a piecewise linear function within the range of the upper and lower limit values to set a constraint condition.
It is provided with a solving means for solving the nonlinear programming problem under the constraint conditions set by the setting means.
The setting means is characterized in that the upper and lower limit values of the intermediate variable are set by solving a mathematical programming problem using the intermediate variable as an objective function by removing the nonlinear constraint condition from the constraint condition. A computing device for nonlinear programming problems.
[2] The setting means is
An auxiliary variable is introduced into the nonlinear function to convert it into a separable function expressed in the form of the sum of the one-variable functions.
It is characterized in that the upper and lower limits of the intermediate variable used for the converted separable function are set, and the constraint conditions are set by approximating the nonlinear functions constituting the converted separable function with a piecewise linear function. The computing device for the nonlinear programming problem according to [1].
[3] The setting means is characterized in that when the nonlinear function is approximated by a piecewise linear function, at least one of a convex combination formulation and a formulation using a special order set of type 2 is performed [3]. The arithmetic unit for the nonlinear programming problem according to 1] or [2].
[4] An approximation calculated value obtained by substituting the value of the coefficient of determination obtained by the solution means into the function after the piecewise linear approximation by the setting means.
The determination variable obtained by the solution means is compared with the theoretically calculated value obtained by substituting the nonlinear constraint condition for the nonlinear function.
It is characterized in that the number of divisions of the piecewise linear function is changed and the processing by the setting means and the solving means is repeated until the difference between the approximate calculated value and the theoretical calculated value satisfies a predetermined condition [1]. The arithmetic unit for the nonlinear programming problem according to any one of [3].
[5] Any of [1] to [4], wherein the nonlinear programming problem is a compounding planning problem in which a compounding ratio is a determinant in a manufacturing process in which a plurality of raw materials are compounded to manufacture a product. An arithmetic unit for the nonlinear programming problem described in one of them.
[6] A linear constraints and a method of calculating the nonlinear programming problem for solving the computing device a nonlinear programming problem to minimize or maximize the objective function under constraints including the nonlinear constraints,
The nonlinear function representing the nonlinear constraint condition is a function having an intermediate variable represented by a linear combination of the determinants of the nonlinear programming problem as an independent variable.
The setting means of the arithmetic unit sets the upper and lower limit values of the intermediate variable, approximates the nonlinear function with a piecewise linear function within the range of the upper and lower limit values, and sets a constraint condition.
The solving means of the arithmetic unit has a solving step of solving the nonlinear programming problem under the constraint condition set.
In the setting step, the setting means sets the upper and lower limits of the intermediate variable by solving a mathematical programming problem using the intermediate variable as an objective function by removing the nonlinear constraint condition from the constraint condition. A method of calculating a nonlinear programming problem, characterized in that it is set.
[7] A program for solving a nonlinear programming problem that minimizes or maximizes an objective function under a constraint including a linear constraint and a nonlinear constraint.
The nonlinear function representing the nonlinear constraint condition is a function having an intermediate variable represented by a linear combination of the determinants of the nonlinear programming problem as an independent variable.
A setting process in which the upper and lower limit values of the intermediate variable are set, the nonlinear function is approximated by a piecewise linear function within the range of the upper and lower limit values, and a constraint condition is set.
A computer is made to execute a solution process for solving the nonlinear programming problem under the constraint conditions set.
The setting process is characterized in that the upper and lower limits of the intermediate variable are set by solving a mathematical programming problem using the intermediate variable as an objective function by removing the nonlinear constraint condition from the constraint condition. Program to be.

According to the present invention, when solving a nonlinear programming problem that minimizes or maximizes an objective function under a constraint condition including a linear constraint condition and a nonlinear constraint condition, it is possible to avoid an increase in the problem scale and practical time. It is possible to obtain an optimal solution with high accuracy within.

It is a figure which shows the functional structure of the compounding planning apparatus which concerns on embodiment. It is a flowchart which shows the process of the compounding plan planning method by the compounding plan planning apparatus which concerns on embodiment. It is a flowchart which shows the detail of the formulation process of the constraint condition in the flowchart of FIG. It is a figure which shows the example of the directed graph which shows the master-slave relationship between variables. It is a figure which shows the result of giving a label to the node of the directed graph of FIG. 4A. It is a characteristic figure which shows the relationship between the number of divisions of a piecewise linear approximation and a relative error in an Example. It is a characteristic diagram which shows the relationship between the number of divisions of the piecewise linear approximation in an Example, and the calculation time. It is a characteristic diagram which shows the comparison result of the relationship between the compounding amount changeable ratio and the cost improvement effect in an Example.

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

When formulating a nonlinear function by piecewise linear approximation, it is necessary to divide the number of pieces of the piecewise linear function into small pieces in order to improve the approximation accuracy of the piecewise linear function. However, as the number of divisions increases, the calculation load increases, so the number of divisions must be given the minimum value that satisfies the required accuracy. Here, the number of divisions of the piecewise linear approximation function required to maintain the calculation accuracy is determined by the range of variables that construct the piecewise linear approximation function. If the range of variables that can be taken is wide, many divisions are required, and if it is small, the formulation can be completed with a small number of divisions. Therefore, the present inventors reduced the search range of the solution of the optimization problem after the formulation by obtaining the minimum / maximum value of the variable for constructing the nonlinear function in advance using the linear programming method. .. As a result, we have established a technology to realize high-speed and high-precision planning for the compounding planning problem of this process.

In the present embodiment, an example in which the present invention is applied to a manufacturing process in which a product is manufactured by blending a plurality of raw materials having different properties, such as a manufacturing process of coke and sinter in the steel industry, will be described.

FIG. 1 shows the functional configuration of the compounding plan planning device 100 according to the embodiment. The blending planning device 100 functions as a calculation device for a nonlinear planning problem to which the present invention is applied, and uses a blending ratio as a determinant in a manufacturing process in which raw materials (brands) having different properties are blended to manufacture a product. , Develop a formulation plan that meets the target product characteristics, production volume, and operating conditions, and minimizes manufacturing costs. That is, the optimum blending ratio is obtained. In the formulation planning problem dealt with in this embodiment, at least a part of the constraint conditions is a non-linear constraint expressed by a non-linear function.

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

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

Reference numeral 103 denotes a constraint condition setting unit, and based on the data taken in by the input unit 101, the characteristic value of the target product, the production amount, and the constraint condition for satisfying the operation condition are formulated and set in the mathematical model.
Here, the constraint condition setting unit 103 includes a function conversion unit 103a and an upper / lower limit value setting unit 103b, sets the upper / lower limit values of the variables used for the nonlinear constraint, and classifies the nonlinear functions constituting the nonlinear constraint. It functions as a setting means for setting constraints by linear approximation.
More specifically, the function conversion unit 103a determines whether or not the nonlinear function representing the nonlinear constraint is a separable function. The separable function is a function in which the function f is expressed in the following form for _{any x = (x 1} , x ₂ , ..., X _n ) ^{T in the n-dimensional Euclidean space.}

If the nonlinear function is not represented by a separable function, an auxiliary variable is introduced to convert the nonlinear function into a separable function. This transformation makes it possible to represent all nonlinear functions in the form of the sum of one-variable functions.
The upper / lower limit value setting unit 103b sets the upper / lower limit values of the variables used for the separable function converted by the function conversion unit 103a. Among the mathematical models set above, the mathematical model relaxed by removing the non-linear constraint from the constraint condition is formulated into a mathematical model with the above variables for which the upper and lower limits should be obtained as the objective function, and the objective function is set. By solving the mathematical programming problem to be maximized and minimized, the upper and lower limits are set.
Then, when the one-variable function constituting the separable function converted by the function conversion unit 103a is a non-linear function, the constraint condition setting unit 103 sets the constraint condition by dividing and linearly approximating the one-variable function. ..
Note that the formulation here does not mean that the constraint condition setting unit 103 constructs the mathematical model itself from the beginning, but the data captured in step S1 is incorporated into the framework of the mathematical model of the constraint conditions set in advance. It means to reflect.

Reference numeral 104 denotes an objective function setting unit, and based on the data taken in by the input unit 101, the objective function that minimizes the manufacturing cost is formulated and set in the mathematical model.
The formulation referred to here does not mean that the objective function setting unit 104 constructs the mathematical model itself from the beginning, but the data captured in step S1 is incorporated into the framework of the mathematical model of the objective function that has been set in advance. It means to reflect.

Reference numeral 105 denotes a solution unit, which solves the compounding 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 or not the solution result by the solution unit 105 is appropriate. As a result, if the solution result is valid, the solution result is determined. If the solution result is not appropriate, the number of divisions 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 repeat the solution of the compounding planning problem.
As a mechanism for determining the validity of the solution result, the calculated value obtained by substituting the compounding ratio obtained by the solution unit 105 into the function after the division linear approximation by the constraint condition setting unit 103 and the original nonlinear constraint Compare with the theoretically calculated value obtained by substitution. Then, until the difference between the calculated value and the theoretically calculated value satisfies a predetermined condition, the number of divisions is changed by the division number setting unit 102, and the compounding plan by the constraint condition setting unit 103, the objective function setting unit 104, and the solution unit 105. Repeat the problem setting and solution.

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

Reference numeral 108 denotes a database, which stores data such as property information, operating condition information, cost information, etc. of each raw material necessary for formulating a blending plan. Although an example of fetching data 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 fetched. Absent.

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

Next, a method of formulating a blending plan by the blending plan planning device 100 will be described.
FIG. 2 is a flowchart showing a process of a blending plan planning method by the blending plan planning device 100 according to the embodiment.
In step S1, the input unit 101 takes in data such as property information, operating condition information, and cost information of each raw material necessary for formulating a blending plan from, for example, the database 108. For example, as shown in Table 1, information on the upper and lower limit values is taken in with respect to the blending ratio of each brand. In addition, although not specifically shown, information on the upper and lower limits regarding the properties of each issue is taken in.

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

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

Equation (1) is an equation representing a constraint condition that is linear among the constraint conditions. X in the formula (1) is an M-dimensional column vector representing the mixing ratio, which is a coefficient of determination, and the elements of the vector correspond to the mixing ratio of each raw material. It is assumed that A in the equation (1) is a matrix of N rows and M columns, and all the elements of A take real values. b is an N-dimensional column vector and takes a real value.
_{G i} (x) in Eq. (2) 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 steel industry. _{The nonlinear function g i} (x) in Eq. (2) may be a multivariable function, a non-convex function, or a non-differentiable function.

In applying the present invention, the constraints are divided into linear constraints and non-linear constraints and formulated. First, add a linear constraint to the mathematical model. Linear constraints can be added to the mathematical model without any difficulty.

Next, the non-linear constraint is formulated. In step S3, the non-linear constraint is converted into a separable function according to the flowchart shown in FIG. 3, the upper and lower limits of the variables used for the separable function are set, and the one-variable function constituting the separable function is a non-linear function. 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 condition.
In step S301, the constraint condition setting unit 103 _{inputs the nonlinear function g i} (x). In the present embodiment, the nonlinear function g _i (x) is, consider the case represented in the form of equation (3). x is an M-dimensional column vector representing the mixing ratio, which is a coefficient of determination, and the elements of the vector correspond to the mixing ratio of each raw material. It _{is assumed that w 1} , w ₂ , and w ₃ are M-dimensional vectors, and the elements of the vector take real values.

_{The nonlinear function g i} (x) in Eq. (3) is a non-convex function and a multivariable function. Generally, when applying a piecewise linear approximation to a nonlinear non-convex function, it is necessary to perform a formulation using a special ordered set of type 2 (Special Order Set type2: SOS2) in addition to the convex combination formulation. .. Therefore, the optimization problem in which this nonlinear function is formulated by piecewise linear approximation becomes an NP-hard problem.
_{The nonlinear function g i} (x) in Eq. (3) is a multivariable function of order M. Therefore, if we try to apply the piecewise linear approximation without adding the equation transformation, we have to construct the approximation function on the M-th order hyperplane. _{For example, consider the case where the nonlinear function g i} (x) of the equation (3) is divided into K pieces for each variable to form a piecewise linear function. At this time, the number of variables required for piecewise linear approximation of the _{nonlinear function g i} ^{(x) is approximately K M} , which is on the order of powers. For example, when K = 50 and M = 10, the number of variables required for piecewise linear approximation is as large as ^{50 10.} Therefore, it is highly possible that the formulation as it is cannot be solved within the practical time. Therefore, it is necessary to devise a formulation to reduce the scale of the problem.

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

It is assumed that the division number K is set for the nonlinear function of the equation (4). There are three non-linear functions in equation (4), but since each function is a one-variable function, the number of variables required for piecewise linear approximation is only K. Therefore, the number of variables constituting the piecewise linear function is 3 × K, which is 3K. That is, by converting the nonlinear function into a separable function as in Eq. (4), the number of variables constituting the piecewise linear function, which required ^{3 K before the conversion, is suppressed to 3K after the conversion.} It can be said that

In step S303, the constraint condition setting unit 103 clarifies the relationship between the variables for the function converted in step S302, and prepares for formulating in order, a directed graph in which the variables in the function are nodes. To build. In the directed graph, each function is branched with the variable that is the input of the function as the start point and the variable obtained through the function as the end point. For example, if a function has a relationship of z = f (x, y), two branches (x, z) and (y, z) are stretched. _{FIG. 4A is an example in which x and v k} are nodes and _{the dependency relations of v k} are connected by a directed graph with respect to the 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, a master-slave relationship between variables can be set.

In step S304, the constraint condition setting unit 103 performs labeling to the node of a directed graph constructed in step S303 (variable). Directed graph constructed in step S303, since a non-cyclic a directed graph (tree structure), it is possible to topological sort. Topological sorting is the ordering of each node so that every node comes before the node beyond its output side. Topological sort is performed on the directed graph constructed in step S303, and labels are assigned from 1 in order from the node behind. FIG. 4B shows the result of topological sorting of the directed graph constructed in step S303 and labeling of the nodes.
As described above, in steps S303 and S304, the graphing and labeling of the relationship between the variables for formulating the nonlinear function is completed.
In the present embodiment, in the subsequent steps S305 to S314, the relation of variables is added to the mathematical model by repeating the processing in the order of the labels given in step S303.

In step S305, the constraint condition setting unit 103 starts iterative calculation for all the variables in ascending order of the node label k for the node set having the tree structure. Hereinafter, the variable corresponding to the node k is _uk (k = 1, ..., M + 6). In addition, hereinafter, in a node set having a tree structure, a node adjacent to a certain node and existing below it is referred to as a child node. Also, when viewed from a certain node, any of its child nodes and all child nodes beyond it is called a descendant node.

In step S306, the constraint condition setting unit 103 determines whether or not a child node exists in the node k. If there is a child node to node k, the process proceeds to step S307, if the child node does not exist, at step S313, it sets the upper and lower limit values of the variable u _k. In the example of the equation (4), since the variables (x ₁ , x ₂ , ..., X _M ) correspond to the nodes having no child nodes, the process proceeds to step S313. On the other hand, the variables (v ₁ , v ₂ , ..., V _n ) are the nodes in which the 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, the upper and lower limit values are set for the node k having no child node. The upper and lower limit values set the upper and lower limit values of the node k given in advance. In a blending plan like this embodiment, the upper and lower limit values are often predetermined for the _{variables (x 1} , x ₂ , ..., X _{M) representing the blending ratio, and if it is given as a set value,} Good. In the present embodiment, the upper and lower limit values set in Table 1 are set in variables (x ₁ , x ₂ , ..., X _M ). By determining the upper and lower limits of all nodes that do not have child nodes in this way, all the child nodes become bounded variables in the formulation of the nodes that appear in the subsequent calculations. This makes it possible to add functions that need to be represented by bounded variables, such as piecewise linear functions, to the 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 limit values of the node k. For example, in the blending planning problem in the present embodiment, the blending ratio does not become 0% or less or 100% or more, so the lower limit value of the node k may be set to 0 and the upper limit value may be set to 100.

Next, consider the case where the child node exists in the node k in step S306 and the process proceeds to step S307. In step S307, the constraint condition setting unit 103 determines whether the relationship between the node k and all the descendant nodes of the node includes all linear relational expressions or even one non-linear relational expression. If only the linear relational expression is included, the process proceeds to step S308, and if the non-linear relational expression is included, the process proceeds to step S310. _{In the example of equation (4), all the progeny nodes of the variables (v 1} , v ₂ , v ₃ , v ₄ ) corresponding to the nodes M + 1 to M + 4 in FIG. 4B are connected by a linear relational expression. , Step S308. _{Since the variable (v 5} ) corresponding to the nodes M + 5 and M + 6 in FIG. 4B _{and the progeny node of gi} (x) include the non-linear relational expression, the process proceeds to step S309.

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

In step 309, the upper / lower limit value setting unit 103b of the constraint condition setting unit 103 sets the upper / lower limit value of the variable corresponding to the node k. Specifically, from the constraints of the nonlinear programming problem and linear constraints removal of the non-linear constraints, and equality constraints constraints representing the relationship of all descendant nodes of the variables u _k and k, the purpose of the variable u _k Construct a mathematical programming problem as a function, and maximize and minimize the problem. By thus solving the mathematical programming problem, defining a range of variables u _k, that is, to determine the upper and lower limit values.
Equation (6), equation (7), the equation (8), the minimum (minimum value) of u _k, shows a linear programming problem to determine the maximum (upper limit).

This linear programming problem is composed of the constraints of Eqs. (1) and (2) with the nonlinear constraints removed. By solving the mathematical programming problem for the purpose function variables u _k using such linear constraints can be determined to define the scope of the variables u _k.
In this case, the definition range of the variables u _k obtained in step S309, the since the extent obtained by solving a kind of relaxation problem by removing the nonlinear constraint from the original problem, a range which can be taken of the variable u _k in the original problem Do not always match. However, the range that can be taken of the variable u _k, by obtaining in advance by the present method, the formulation of the piecewise linear approximation function to be described later, reducing the number 0, 1 variables needed to maintain the accuracy of the piecewise linear approximation Therefore, it is expected that the calculation accuracy will be improved and the calculation time will be shortened.

Then in step S313, the lower limit setting section 103b on the constraint condition setting unit 103 sets an optimal solution obtained in step S309 as the upper and lower limits of the variable u _k.
The above is the formulation method when the relationship between the node k and all the descendant nodes of the node is expressed by a linear relational expression in step S307.

Next, in step S307, consider the case where the relationship between the node k and all the descendant nodes is not expressed only by the linear relational expression. For example, in Equation (4), since the variable (v ₅₎ and descendant nodes of g _i (x) is not represented only linear relations, the process proceeds to step S310. If node k and all descendant nodes of that node are not represented only by linear relational expressions, nonlinear constraints are required for the problem of finding the upper and lower limits of variables, so as in steps S308 to S309, It is not possible to set the upper and lower limits of a variable using linear programming. Therefore, in the subsequent steps S310 to S312, attention is paid only to the linearity / non-linearity of the node k, the upper and lower limits of the variables are set, and the formula is formulated into the mathematical model. By sequentially formulating the relationship between the parent node and the child node in this way, we aim to formulate the entire nonlinear function into a mathematical model.

In step S310, the constraint condition setting unit 103 determines whether or not 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 relational expression, the process proceeds to step S311. If the child node of the node k cannot be represented by the linear relational expression alone and includes at least one non-linear relational expression, the piecewise linear approximation is used. In order to carry out the formulation, the process proceeds to step S312. In the example of Equation (4), variable (v ₅₎ are nonlinear function of the variables _{v 1, v 4, g i} (x) , so is represented by a nonlinear function of the variables v _5, the process proceeds to step S312.

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

In step S312, the constraint condition setting unit 103 formulates the nonlinear relational expression between the node k and the child node of the node k into a mathematical model. In this embodiment, the nonlinear function is approximated by a piecewise linear function, and the piecewise linear function is added to the mathematical model as a constraint condition by the convex combination formulation method. The convex combination formulation is a piecewise linear approximation to the nonlinear function y = f (x), and when the section of the piecewise linear function is set as (x _i , y _i ) i = 1, ..., K. , Formulating the relationship as in equation (9).

Equation (9) corresponds to finding a vector _{whose weight t k} is 1 for a vector pointing to a section of a piecewise linear function from the origin. If f (x) is a convex function and the problem minimizes f (x), then since the solution is on a line segment, _{only two consecutive t ks} always take positive values. Otherwise, two non-adjacent t _ks may be positive and such a solution needs to be removed. To remove such a solution, we can add a constraint to the set of _{t k that "only two adjacent t ks at the} highest have positive values". High I or only two adjacent t _k is variable group such as a positive value _{(t 1, t 2, ···} , t K) is called the SOS2 variable group. The SOS2 variable group may be formulated by using the integer programming method, or a dedicated function for defining the SOS2 set provided in the 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. In this way, in addition to the convex combination formulation, the SOS2 variable group is defined, and the non-convex nonlinear function is formulated as a mixed integer programming problem.
Then in step S313, the lower limit setting section 103b on the constraint condition setting unit 103 sets the upper limit value of the variable u _k 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 when all the progeny nodes of the node k include at least one nonlinear 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.
Then, in step S314, if g _i (x) ≧ 0 is added to the mathematical model, the constraint condition including the non-linear constraint can be formulated in the mathematical model.
The above is the details of step S3. Conventionally, in order to formulate a nonlinear function used for formulating a blending plan using a piecewise linear approximation, it has been necessary to prepare and formulate a piecewise linear function only for the dimension of the variable representing the blending ratio. On the other hand, according to the procedure of the present embodiment, a calculation graph is constructed with the decision variable used in the nonlinear constraint equation as a node, and the relation between the nodes (linear / nonlinear) is formulated. This has made it possible to significantly reduce the number of variables required to perform a compartmentalized linear approximation of a nonlinear function. Furthermore, the range of variables for which piecewise linear functions are constructed was solved in advance using the linear programming method to solve the variable minimization / maximization problem. This makes it possible to narrow the definition range of the variables that construct the piecewise linear approximation function, and reduces the number of 0 and 1 variables required to maintain the accuracy of the piecewise linear approximation function. Therefore, we were able to build a model that can be solved within practical time.

Returning to FIG. 2, in step S4, the objective function setting unit 104 formulates and sets an objective function that minimizes the cost related to manufacturing in a mathematical model based on the data captured in step S1. The objective function may be a linear function or a non-linear function. When the objective function is represented by a nonlinear function, the nonlinear function is converted into a separable function as in the case of the constraint condition, the upper and lower limits of the variables are set, and the nonlinear function is divided and linearly approximated to obtain the objective function. You can set it.

In step S5, the solution unit 105 solves the compounding planning problem, which is the nonlinear planning problem set in steps S3 and S4.

In step S6, the determination unit 106 determines whether or not the solution result in step S5 is appropriate. Since piecewise linear approximation is an approximation method, it is necessary to confirm that an approximation error always occurs and the approximation error is 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 the piecewise linear approximation in step S3 and the original nonlinear constraint. Compare with the obtained theoretically calculated values. Then, until the difference between the calculated value and the theoretically calculated value satisfies a predetermined condition, the number of divisions K is changed in step S2, and the setting and solving of the nonlinear programming problem in steps S3 to S5 are repeated.

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

An example of solving a non-linear programming problem, which is a non-linear programming problem, will be described for a coal compounding plan by using the method described above.
In the coal compounding plan, the compounding ratio is used as a determinant, and the non-linear programming problem of formulating a compounding plan that satisfies the characteristic values, production volume, and operating conditions of the target product and minimizes the manufacturing cost is as follows. It was formulated as the formulas (11) to (15) of.
i: Subscript related to brand name j: Subscript related to constraints x _i : Coal mixing ratio (coefficient of determination)
p _i: coke production cost (coal purchasing cost + transport costs + operating costs)
a _ij : Ratio of coking coal components (measured value)
g _i (x ₁ , x ₂ , ..., x _N ): Coke quality estimation formula (non-linear)
LB _j , UB _j : Upper and lower limits of quality U _i ^L , U _i ^U : Upper and lower limits of blending ratio

Although the formulation plan for a single period is shown here, the formulation of the formulas (11) to (15) may be extended to a multi-period formulation plan. In order to extend to a multi-period formulation plan, constraints on inventory volume in the previous and next periods, multi-period usage plans, and raw material acceptance plans are required, but all of these constraints are applied to the mixed integer programming method. Therefore, if a compounding plan for a single period can be formulated, a compounding plan for multiple periods can also be formulated.

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

The constraint condition of the equation (12) represents a linear constraint on quality, and includes, for example, a constraint on volatile matter, sulfur content, and the like. a _ij represents the ratio of the components of coal, and the value obtained by weighted averaging the value with the blending ratio is given as a constraint condition.
The constraints of equation (13) represent non-linear constraints on quality, including, for example, coke cold strength (coke DI) and expansion pressure constraints. The coke DI is an index showing the strength of coke obtained after coking coal in a coke oven. Coke is required to have a certain strength or higher in order to ensure air permeability in a blast furnace, and coke DI is often used as an index of the strength. The expansion pressure is an index showing the pressure applied to the coke oven by the gas generated when coal is baked and hardened in the coke oven. If the expansion pressure is high, it will damage the core of the coke oven, so a value as low as possible is desirable.
The constraint condition of the formula (14) represents a constraint on the total value of the blending ratio.
The constraint condition of the formula (15) represents the upper and lower limit constraints regarding the blending ratio of coal.

In the mathematical model as described above, equation (13) corresponds to the nonlinear constraint of equation (2). Further, the equations (12), (14) and (15) correspond to the linear constraints of the equation (1). In this embodiment, the coke DI and expansion pressure contained in the formula (13) were formulated according to the flowchart shown in FIG.

Table 2 shows the values of the blending ratios obtained by optimizing according to the embodiment under the conditions of the blending ratio (Table 1) and the properties. As shown in Table 2, the optimization results comply with the compounding ratio restrictions in Table 1. In addition, the optimization results complied with all the constraints, and the correct optimization was performed.

FIG. 5 shows the result of performing the optimization calculation for each number of divisions by changing the number of divisions of the piecewise linear approximation. The horizontal axis represents the number of divisions, and the number of divisions was optimized by increasing the number of divisions from 10 to 80 in increments of 10. The vertical axis is the relative error [%] between the calculated value and the theoretically calculated value, and the value obtained by dividing the difference between the calculated value and the theoretically calculated value by the theoretically calculated value is expressed as a percentage. From FIG. 5, both the coke DI and the relative error of the expansion pressure decrease as the number of divisions increases. Further, 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 operational 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 for 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, the calculation time increases as the number of divisions increases, but under a calculation environment using a general-purpose computer, the calculation time is within 60 seconds for all optimizations from the number of divisions 10 to 80, which is practical. I was able to solve it in a calculation time that did not cause any problems.

Next, FIG. 7 shows the result of investigating the cost improvement effect when the constraint condition of the blending ratio is changed. The number of divisions of the piecewise linear approximation was fixed at 50. In this embodiment, the amount of compounding at a certain point in time is set as the standard compounding, and the change in cost when the ratio of allowing the change of the compounding from the standard compounding is gradually increased is illustrated. Here, the ratio that allows the change of the compounding amount from the standard compounding amount is called the compounding amount changeable ratio. The horizontal axis represents the blending amount changeable ratio, and the blending amount changeable ratio was changed from 0% to 80%. The vertical axis represents the cost improvement effect [%], and the cost improvement effect with respect to the standard is expressed as a percentage based on the case where the compounding 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 of 3 times or more as compared with the comparative method (algorithm of Patent Document 1), indicating that the present invention is sufficiently useful. ..

Although the present invention has been described above together with the embodiments, the above-described embodiments merely show examples of embodiment of the present invention, and the technical scope of the present invention is interpreted in a limited manner by these. It must not be. That is, the present invention can be implemented in various forms without departing from the technical idea or its main features.
In the present embodiment, the production of coke and sinter in the steel industry is mentioned as a manufacturing process for manufacturing a product by blending a plurality of raw materials having different properties. For example, blending in oil refining, cement manufacturing, etc. The present invention may be applied to planning.
Furthermore, the present invention is not limited to the formulation planning problem, but is widely applicable to the nonlinear planning problem. For example, when designing a building, there is a problem of optimizing the thickness of columns while ensuring the required strength. At this time, if 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 to such cases, it is possible to avoid an increase in the scale of the problem and to obtain an optimal solution with high accuracy within a practical time.

The arithmetic unit for the nonlinear programming problem to which the present invention is applied is realized by, for example, a computer device equipped with a CPU, ROM, RAM, and the like.
The present invention also provides software (programs) that realize the functions of the present invention to a system or device via a network or various storage media, and the computer of the system or device reads and executes the program. It is feasible.

100: Blending plan planning device 101: Input unit 102: Number of divisions setting unit 103: Constraint condition setting unit 104: Objective function setting unit 105: Solving unit 106: Judgment unit 107: Output unit

Claims (7)

A non-linear programming problem arithmetic device for solving a nonlinear programming problem that minimizes or maximizes the objective function under constraints including linear and non-linear constraints.
The nonlinear function representing the nonlinear constraint condition is a function having an intermediate variable represented by a linear combination of the determinants of the nonlinear programming problem as an independent variable.
A setting means for setting the upper and lower limit values of the intermediate variable and approximating the nonlinear function with a piecewise linear function within the range of the upper and lower limit values to set a constraint condition.
It is provided with a solving means for solving the nonlinear programming problem under the constraint conditions set by the setting means.
The setting means is characterized in that the upper and lower limit values of the intermediate variable are set by solving a mathematical programming problem using the intermediate variable as an objective function by removing the nonlinear constraint condition from the constraint condition. A computing device for nonlinear programming problems. The setting means is
An auxiliary variable is introduced into the nonlinear function to convert it into a separable function expressed in the form of the sum of the one-variable functions.
It is characterized in that the upper and lower limits of the intermediate variable used for the converted separable function are set, and the constraint conditions are set by approximating the nonlinear functions constituting the converted separable function with a piecewise linear function. The computing device for the nonlinear programming problem according to claim 1. The setting means is characterized in that, when the nonlinear function is approximated by a piecewise linear function, at least one of a convex combination formulation and a formulation using a special order set of type 2 is performed. 2. The computing device for the nonlinear programming problem according to 2. The approximate calculation value obtained by substituting the value of the coefficient of determination obtained by the solution means into the function after the piecewise linear approximation by the setting means, and
The determination variable obtained by the solution means is compared with the theoretically calculated value obtained by substituting the nonlinear constraint condition for the nonlinear function.
Claim 1 is characterized in that the number of divisions of the piecewise linear function is changed and the processing by the setting means and the solving means is repeated until the difference between the approximate calculated value and the theoretical calculated value satisfies a predetermined condition. The arithmetic unit for the nonlinear programming problem according to any one of items 3 to 3. The non-linear programming problem according to any one of claims 1 to 4, wherein the nonlinear programming problem is a compounding planning problem in which a compounding ratio is used as a determinant in a manufacturing process for producing a product by blending a plurality of raw materials. Arithmetic logic unit for nonlinear programming problems. A calculation method of the nonlinear programming problem for solving the nonlinear programming problem to minimize or maximize the objective function under constraints including linear constraints and non-linear constraint by the arithmetic unit,
The nonlinear function representing the nonlinear constraint condition is a function having an intermediate variable represented by a linear combination of the determinants of the nonlinear programming problem as an independent variable.
The setting means of the arithmetic unit sets the upper and lower limit values of the intermediate variable, approximates the nonlinear function with a piecewise linear function within the range of the upper and lower limit values, and sets a constraint condition.
The solving means of the arithmetic unit has a solving step of solving the nonlinear programming problem under the constraint condition set.
In the setting step, the setting means sets the upper and lower limits of the intermediate variable by solving a mathematical programming problem using the intermediate variable as an objective function by removing the nonlinear constraint condition from the constraint condition. A method of calculating a nonlinear programming problem, characterized in that it is set. A program for solving nonlinear programming problems that minimize or maximize the objective function under constraints, including linear and nonlinear constraints.
The nonlinear function representing the nonlinear constraint condition is a function having an intermediate variable represented by a linear combination of the determinants of the nonlinear programming problem as an independent variable.
A setting process in which the upper and lower limit values of the intermediate variable are set, the nonlinear function is approximated by a piecewise linear function within the range of the upper and lower limit values, and a constraint condition is set.
A computer is made to execute a solution process for solving the nonlinear programming problem under the constraint conditions set.
The setting process is characterized in that the upper and lower limits of the intermediate variable are set by solving a mathematical programming problem using the intermediate variable as an objective function by removing the nonlinear constraint condition from the constraint condition. Program to be.

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