JP2020086706A - Arithmetic unit, method and program for nonlinear planning problem - Google Patents

Abstract

To acquire a highly accurate optimal solution within a practical time by appropriately setting an approximation range of piecewise linear approximation in solving a nonlinear planning problem.SOLUTION: A compounding planning device 100 forms a compounding plan with a compounding ratio as a determination variable in a manufacturing process for compounding raw materials having a plurality of different qualities to manufacture a product by a method for solving a non-linear planning problem. A constraint condition setting part 103 introduces an intermediate variable to a nonlinear function representing a nonlinear constraint condition to convert it into a separable function, and represents it in the form of the sum of a one-variable function. Also, the constraint condition setting part 103 generates a plurality of samples of the compounding ratio so as to satisfy a condition according to a nonlinear constraint by random sampling, and sets the upper and lower limit values of the intermediate variable by using the samples. Then, the constraint condition setting part 103 performs piecewise linear approximation of a function constituting the separable function within the range of the upper and lower limit values of the intermediate variable, and sets a constraint condition.SELECTED DRAWING: Figure 1

Description

The present invention relates to a nonlinear programming problem computing device, method, and program for solving a nonlinear programming problem that minimizes or maximizes an objective function under a constraint condition that includes a linear constraint condition and a nonlinear constraint condition.

The mathematical programming problem is a problem of minimizing or maximizing an objective function under a constraint condition. Among them, the one including a nonlinear function in at least one of the constraint condition and the objective function is called a nonlinear programming problem.
For example, in the manufacturing process of coke and sinter in the steel industry, a plurality of raw materials held by each factory are mixed to manufacture a product. In this case, a blending plan must be prepared so as to satisfy the target product characteristic value and production amount. In addition, a blending plan must be prepared so as to satisfy the operating conditions in the manufacturing process.
As described above, the constraint condition for satisfying the target characteristic value of the product, the production amount, and the operation condition often includes a function in which it is difficult to mathematically obtain an optimum solution. For example, an estimation formula for predicting the quality of coke includes a formula representing structural characteristics such as coke strength, and a non-linear function based on experiments and theory is used for this estimation formula. As described above, since many of the mathematical programming problems in the manufacturing process are complicated nonlinear programming problems, it is difficult to obtain a highly accurate optimal solution within the practical time.

JP, 2009-175804, A

Applied Mathematical Programming, Stephen P. Bradley, Addison-Wesley, 1977 Algorithm I for expressing multivariable functions by sum of one variable I, Kiyotaka Yamamura, Yasuko Murayama, IEICE technical report, CAS, Circuits and Systems 93(102), 67-74, 1993-06-19 Algorithm for expressing multi-variable functions by sum of one variable 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 a nonlinear programming problem.
For example, in Patent Document 1, a simulator for creating a compounding plan for mixing a plurality of types of compounding raw materials, a mathematical model expressing the supply and demand balance constraint of the compounding raw materials, and a mathematical model expressing the property constraint after mixing are constructed. A model construction section and a mathematical section model constructed by the model construction section are used to perform optimization calculation based on a predetermined objective function, and a planning section for calculating instructions to the simulator. When the model includes a non-linear mathematical expression, a linear mathematical expression is introduced instead of the non-linear mathematical expression to formulate the mathematical expression model, and the solution result by the planning unit using the mathematical expression model including the linear mathematical expression is the non-linear mathematical expression. A method for confirming whether or not a mathematical model including is satisfied is disclosed.

In Non-Patent Document 1, a method of introducing an auxiliary variable (also referred to as an intermediate variable) to a non-linear function, converting it into a separable function, and then approximating it with a piecewise linear function to formulate it as a mixed integer programming problem It is shown. Here, the separable function is a multivariable function that can be expressed in the form of a sum of univariate functions. It is known that the number of function evaluations in piecewise linear approximation is significantly reduced by converting a non-linear function into a separable function (Non-Patent Documents 2 and 3). The piecewise linear function is a function that is defined piecewise, and each section is a linear function. Approximating a non-linear function with a piecewise linear function is called piecewise linear approximation. Hereinafter, the entire range in which the nonlinear function is approximated by the piecewise linear function is referred to as an approximation range, and each section approximated by the piecewise linear function (each section approximated by one linear function) is referred to as an approximate section.

However, in the method of Patent Document 1, since a nonlinear programming problem is simplified and approximated to a linear programming problem, an approximation error may occur and a solution with sufficient accuracy may not be obtained.

Further, in the method of Non-Patent Document 1, it is necessary to set an approximation range for each of the separated non-linear functions, but if an appropriate approximation range is not set, divergence of the piecewise linear function and deterioration of approximation accuracy may occur. There is.

The present invention has been made in view of the above points, and when solving a nonlinear programming problem, an approximate range of piecewise linear approximation is appropriately set, and a highly accurate optimal solution can be obtained within a practical time. The purpose is to do so.

The gist of the present invention for solving the above problems is as follows.
[1] A calculation device for a nonlinear programming problem for 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,
Setting means for introducing an intermediate variable into the non-linear function representing the non-linear constraint condition, converting it into a separable function, and then performing piecewise linear approximation, and setting the constraint condition,
A solving means for solving the nonlinear programming problem under the constraint condition set by the setting means,
The setting means generates a plurality of independent variable samples excluding intermediate variables from variables constituting the nonlinear function by random sampling, and uses the samples to set upper and lower limit values of the intermediate variable. An arithmetic device for a nonlinear programming problem, characterized by performing a piecewise linear approximation of a function forming the separable function in the range of the upper and lower limits.
[2] The setting means generates a plurality of independent variable samples excluding intermediate variables among variables constituting the nonlinear function so as to satisfy a condition conforming to the nonlinear constraint condition [1 ] The arithmetic device of the nonlinear programming problem described in the above.
[3] The setting means expresses the non-linear function in the form of a sum of univariate functions by introducing the intermediate variable, and calculates the non-linear programming problem described in [1] or [2]. apparatus.
[4] The nonlinear programming problem is a compounding planning problem in which a compounding ratio is a decision variable in a manufacturing process in which a plurality of raw materials are compounded to manufacture a product,
The setting means randomly generates a blending ratio by random sampling, and generates a plurality of samples of the blending ratio. [1] to [3] The nonlinear programming problem calculation according to any one of [1] to [3]. apparatus.
[5] The setting means repeats the setting of the upper and lower limit values of the intermediate variable and piecewise linear approximation for each function forming the separable function of [1] to [4]. An arithmetic device for the nonlinear programming problem described in any one.
[6] A calculation method of a nonlinear programming problem for 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,
A setting step in which the setting means introduces an intermediate variable into the non-linear function representing the non-linear constraint condition and converts it into a separable function, and then performs piecewise linear approximation to set the constraint condition;
The solution solving means has a solution solving step of solving the nonlinear programming problem under the constraint condition set in the setting step,
In the setting step, random sampling is performed to generate a plurality of independent variable samples excluding intermediate variables among the variables forming the nonlinear function, and the samples are used to set upper and lower limit values of the intermediate variable. A method for computing a non-linear programming problem, characterized by performing a piecewise linear approximation of a function that constitutes the separable function in the range of the upper and lower limits.
[7] A program for 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,
Setting means for setting a constraint condition by introducing an intermediate variable into a nonlinear function representing the nonlinear constraint condition, converting the variable into a separable function, and then performing a piecewise linear approximation,
Causing a computer to function as a solution finding means for solving the nonlinear programming problem under the constraint condition set by the setting means,
The setting means generates a plurality of independent variable samples excluding intermediate variables from variables constituting the nonlinear function by random sampling, and uses the samples to set upper and lower limit values of the intermediate variable. A program, characterized by performing a piecewise linear approximation of a function constituting the separable function in the range of the upper and lower limits.

According to the present invention, when solving a nonlinear programming problem, an approximate range of piecewise linear approximation can be appropriately set, and an accurate and highly accurate solution can be obtained within practical time.

It is a figure which shows the function structure of the formulation planning apparatus which concerns on embodiment. It is a flow chart which shows processing of a formulation method of a formulation plan by a formulation planning device concerning an embodiment. 3 is a flowchart showing details of constraint condition setting processing in the flowchart of FIG. 2. 4 is a flowchart showing details of random sampling in the flowchart of FIG. 3. It is a figure which shows the image which determines the range of a variable. Shows the result of obtaining the range of the intermediate variable v _4, a diagram showing the relationship between the intermediate variable v ₄ and 2log (v _4). It is a diagram showing the relationship between the intermediate variable v ₄ and 2log (v _4).

Hereinafter, preferred embodiments of the present invention will be described with reference to the accompanying drawings.
When solving a nonlinear programming problem that minimizes or maximizes the objective function under a constraint condition including a linear constraint condition (hereinafter, simply referred to as linear constraint) and a nonlinear constraint condition (hereinafter, simply referred to as non-linear constraint) When the non-linear function representing the non-linear constraint is a multivariable function, it is possible to convert the non-linear function into a separable function by introducing an intermediate variable and expressing it in the form of the sum of the one-variable functions. This reduces the dimension of the non-linear function and reduces the variables required for piecewise linear approximation.

In order to perform the piecewise linear approximation of the one-variable function after separating in this way, the approximation range must be determined. When the approximation range of the univariate function is increased, more approximation intervals are required to maintain the approximation accuracy, and there is a possibility that the non-linear function to be approximated may diverge. Therefore, it is desirable to determine an approximation range that is as narrow as possible in advance. However, the determination of the approximate range is as difficult as the original nonlinear programming problem because the nonlinear constraint must be taken into consideration.

Therefore, the present inventor has established a method for determining the approximate range of a univariate function. As will be described in detail below, random sampling generates a plurality of independent variable samples excluding the intermediate variable from the variables forming the nonlinear function. At that time, the sample of the independent variable is generated so as to satisfy the condition conforming to the nonlinear constraint. Then, by using these samples, the upper and lower limits of the intermediate variable are set and the approximation range of the univariate function is determined. As a result, when solving a nonlinear programming problem, the approximation range of piecewise linear approximation can be set appropriately, and a highly accurate optimal solution can be obtained within practical time.

In the embodiments described below, a manufacturing process for manufacturing a product by mixing raw materials (hereinafter, referred to as a brand) having a plurality of different properties, such as a manufacturing process of coke and a sintered ore in the iron manufacturing industry An example to which the invention is applied will be described.
FIG. 1 shows a functional configuration of a formulation planning apparatus 100 according to the embodiment. The blending plan planning apparatus 100 functions as a computing device for a nonlinear programming problem to which the present invention is applied, and in a manufacturing process in which a plurality of brands having different properties are blended to manufacture a product, the blending ratio is used as a decision variable, and a target is set. Formulate a compounding plan that satisfies the characteristic values, production amount, and operating conditions of the following products and minimizes the manufacturing cost. That is, the optimum blending ratio is obtained. In the compounding planning problem handled in this embodiment, at least a part of the constraint conditions is a non-linear constraint represented by a non-linear function.

The input unit 101 fetches data such as property information (information about properties and state) of each brand, operating condition information, cost information, and the like, which is necessary for making a formulation plan from the database 108, for example.

The division number setting unit 102 sets the division number for piecewise linear approximation of the nonlinear function. The division number setting unit 102 may set the division number according to, for example, an operator's instruction, or may automatically set the division number within a range of a predetermined division number.

The constraint condition setting unit 103 sets a constraint condition for satisfying a target product characteristic value, a production amount, and an operation condition based on the data captured by the input unit 101.
Here, the constraint condition setting unit 103 includes a function conversion unit 103a and an upper and lower limit value setting unit 103b.
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 with respect to arbitrary x=(x ₁ , x ₂ ,..., X _n ) ^T in the n-dimensional Euclidean space. When the non-linear function is not a separable function, an intermediate variable is introduced into the non-linear function to convert it into a separable function, which is expressed as a sum of univariate functions.

When a non-linear function is converted into a separable function, one non-linear function is represented by a plurality of univariate functions. Here, each univariate function may include a non-linear function. Therefore, for formulation, piecewise linear approximation must be performed for each univariate function. However, the range that the variable can take is not defined for the intermediate variable used when separating the nonlinear function. Therefore, if an intermediate variable exists in the independent variable of the piecewise linear approximation, the approximation range of the piecewise linear function must be determined in advance to determine the range that the intermediate variable can take. Here, if the approximation range of the univariate function is increased, more approximation intervals are required to maintain the approximation accuracy, and depending on the shape of the nonlinear function, the area in which the nonlinear function diverges (becomes ±∞ values) It can be an approximation target. In order to avoid these problems, it is desirable to predetermine the approximation range as narrow as possible while maintaining the approximation accuracy.
Therefore, the upper and lower limit value setting unit 103b uses random sampling to determine the independent variables (for example, the compounding ratio that is a decision variable) excluding the intermediate variables from the variables that form the nonlinear function so as to satisfy the conditions conforming to the nonlinear constraints. The upper and lower limit values of the intermediate variable are set by generating a plurality of samples and using these samples.
Then, the constraint condition setting unit 103 sets in the upper and lower limit value setting unit 103b when the function (the one-variable function in this embodiment) forming the separable function converted by the function converting unit 103a is a non-linear function. Within the range of upper and lower limits, the univariate function is piecewise linearly approximated to set the constraint condition.

The objective function setting unit 104 sets an objective function that minimizes the manufacturing cost based on the data captured by the input unit 101.

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

The determination unit 106 determines whether or not the solution result obtained by the solution solution unit 105 is valid. As a result, if the solution result is valid, the solution result is confirmed. When the solution result is not valid, 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 finding unit 105 repeat the solution of the mixture planning problem.
As a mechanism for determining the appropriateness of the solution result, the calculated value obtained by substituting the mixture ratio obtained by the solution solving unit 105 into the function after the piecewise linear approximation by the constraint condition setting unit 103 and the original nonlinear constraint The theoretical calculation value obtained by substitution is compared. Then, until the difference between the calculated value and the theoretical calculated value satisfies the predetermined condition, the number of divisions is changed by the number-of-divisions setting unit 102, and the combination plan by the constraint condition setting unit 103, the objective function setting unit 104, and the solution finding unit 105. Repeat the problem setting and solution.

The output unit 107 transmits the solution result obtained by the determination unit 106 to the output device 110.

The database 108 stores data such as property information of each brand, operating condition information, cost information, etc., which is necessary for making a formulation plan. It should be noted that the example in which the data is fetched from the database 108 has been shown here, but other than that, for example, the data may be fetched from an external device via a network, or the data directly input by the user may be fetched. Absent.

The input device 109 is a pointing device, a keyboard, or the like.
The output device 110 may be a display device such as a display that outputs the solution result transmitted from the output unit 107, an audio output device such as a speaker that outputs audio information, or the like. Further, the output device 110 may be a storage device or the like that stores and registers the solution result transmitted from the output unit 107 in a storage area (not shown).

Next, a method of formulating a composition plan by the composition planning apparatus 100 will be described.
FIG. 2 is a flow chart showing the processing of the method for formulating a blending plan by the blending plan drafting apparatus 100 according to the embodiment.
In step S1, the input unit 101 fetches data such as property information of each brand, operating condition information, cost information, etc., which is necessary for formulating a blending plan from the database 108, for example. For example, as shown in Table 1, information on upper and lower limit values regarding the blending ratio of each brand is fetched.

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

In step S3, the constraint condition setting unit 103 sets constraint conditions for satisfying the target product characteristic value, the production amount, and the operation condition based on the data captured in step S1.
In the present embodiment, the constraint condition is divided into a linear constraint represented by a linear function as in Expression (1) and a nonlinear constraint represented by a nonlinear function as in Expression (2).

Expression (1) is an expression that expresses a linear constraint condition among the constraint conditions. X in the equation (1) is an M-dimensional column vector that represents the mixing ratio that is a decision variable, and the elements of the vector correspond to the mixing ratio of each brand. It is assumed that A in Expression (1) is a matrix with N rows and M columns, and all elements of A take real values. b is an N-dimensional column vector and takes a real value.
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 having strong non-linearity such as coke strength in the steel industry. The nonlinear function g _i (x) in Expression (2) is a multivariable function, a non-convex function, or a non-differentiable function. d _i is a constant and represents the lower limit of g _i (x).

When applying the present invention, the constraint condition is divided into a linear constraint and a non-linear constraint for formulation. First, add a linear constraint to the mathematical model. Linear constraints can be added to mathematical models without particular difficulty.

Next, the formulation of the nonlinear constraint is performed. In step S3, an intermediate variable is introduced into a non-linear function representing a non-linear constraint and converted into a separable function according to the flowchart shown in FIG. Then, when the upper and lower limit values of the intermediate variable are set and the function forming the separable function (univariate function in this embodiment) is a non-linear function, the univariate function is piecewise linearly approximated and approximated. Set a piecewise linear function as a constraint.

In step S301, the constraint condition setting unit 103 inputs the nonlinear function g _i (x). In the present embodiment, the case where the non-linear function g _i (x) is expressed in the form of Expression (3) will be considered as an example. x is an M-dimensional column vector representing the mixture ratio which is a decision variable, and the element of the vector corresponds to the mixture ratio of each brand. w ₁ , w ₂ , and w ₃ are M-dimensional vectors, and the elements of the vectors have real values. In the present embodiment, it is assumed that the dimension of each w _i is 5 and the values are given as shown in Table 2. w _i represents, for example, a parameter representing the property or component of each brand.

In step S302, the constraint condition setting unit 103 converts the nonlinear function g _i (x) into a separable function by introducing an intermediate variable, and expresses it in the form of the sum of the one-variable functions. 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. When the non-linear function g _i (x) is converted into a separable function, it is expressed as in equation (4). The non-linear function g _i (x) is a function having the intermediate variables v _{1 to} v ₅ represented by the linear combination of the decision variable x as independent variables.

Here, in order to perform a piecewise linear approximation of a function (a one-variable function in this embodiment) that constitutes a separable function, the range (upper and lower limit values) of the intermediate variables v ₁ , v ₄ , and v ₅ which are independent variables of the nonlinear function. ) Must be set. In the example of Equation (4), the intermediate variable v _2, v ₃ is the independent variable of the intermediate variable v _4, a linear relationship with the intermediate variable v _4. Therefore, approximation error due to piecewise linear approximation and divergence of the function do not occur. In other words, it is not necessary to determine the approximate range, so it is excluded from the consideration of this embodiment.

In step S303, the constraint condition setting unit 103 randomly generates a mixture ratio by random sampling, and generates a plurality of mixture ratio samples. At that time, the sample of the mixing ratio is generated so as to satisfy the condition conforming to the nonlinear constraint. Then, the range of each intermediate variable is set by using these samples.
Details of the random sampling in step S303 will be described with reference to FIG.
Here, d ^rs _i is the threshold value of the nonlinear function g _i (x) when performing random sampling. When the condition of the threshold value d ^rs _i is strict, there is a possibility that the feasible region that should be solved by optimization originally may be lost by sampling. Therefore, normally, the value of the threshold value d ^rs _i gives a value relaxed from the threshold value d _i of the original nonlinear constraint. For example, since the threshold value d _i is 500, the threshold value d ^rs _i may be given a value of 500 or less that relaxes the nonlinear constraint. In this embodiment, both the threshold values d _i and d ^rs _i are set to 500.
In order to accurately determine the range of the intermediate variable v _k, the range of the intermediate variable v _k must be kept from the constraint condition expressions (1) and (2) of the main problem. However, the calculation is difficult because it is necessary to consider the non-linear function of equation (2). If the range of variables greatly deviates from the actual operating range, there is a possibility that the function of the non-linear constraint relating to the quality of the compounded brand itself is undefined and piecewise linear approximation itself cannot be performed.
Therefore, the inventor considered that when setting the range of the intermediate variable v _k , the range should be limited by using a constraint condition that approximates the nonlinear function. FIG. 5 shows an image for determining the range of variables. It was considered that the range of variables could be determined by generating random numbers of mixture ratios for nonlinear constraints, extracting samples satisfying the conditions according to nonlinear constraints, and approximating the sample results with a convex function.

In step S401, random sampling is repeated until a fixed number of samples are obtained.
In step S402, a mixture ratio is randomly generated within the upper and lower limit values of the mixture ratio of each brand given as in the example of Table 1.
In step S403, it is determined whether or not the total value of the mixture proportions of all the brands generated in step S402 is 100% of the planned value. If it is 100%, the process proceeds to step S406, and if not, step S404. Move to.
In step S404, a brand is randomly selected.
In step S405, the blending amount of the brand selected in step S404 is adjusted so that the total value of the blending ratios of all the brands approaches 100% of the planned value. Specifically, the compounding amount of the selected brand is set to x, the adjusted compounding amount is set to x′, and the upper and lower limits of the mixing ratio of the brand are set to x _min and x _max , respectively. Next, when the total value of the blending ratios of all brands exceeds 100%, a random value between x _min and x is generated, and the generated value is adopted as x′. Further, when the total value of the blending ratios of all the brands is less than 100%, a random value between x and x _max is generated, and the generated value is adopted as x′. Then, the process proceeds to step S403, and it is determined again whether or not the total value of the mixing ratios is 100%. By repeating the above procedure, it is possible to obtain a sample in which the total value of the blending ratios of all brands is 100% of the planned value.

In step S406, it is determined whether or not the sample obtained in the current loop satisfies g _i (x)≧d ^rs _i which is a condition conforming to the nonlinear constraint. If g _i (x)≧d ^rs _i is not satisfied, the process returns to step S402 and the process is repeated. If g _i (x)≧d ^rs _i is satisfied, the process proceeds to step S407, the sample obtained in this loop is adopted as a valid sample, and then the process returns to step S402 to proceed to the generation of the next sample. .. When a sufficient number of valid samples are prepared, the random sampling in step S303 ends.

Table 3 shows an example in which 2000 samples with a mixing ratio satisfying g _i (x)≧d ^rs _i =500 are generated by the random sampling in step S303.

Returning to FIG. 3, in step S304, the constraint condition setting unit 103 starts iteration of formulation by piecewise linear approximation for each function (univariate function in this embodiment) forming a separable function. To do. The relationship between the functions forming the separable function can be represented by an acyclic digraph, and each function can be labeled so that any function is arranged so that it comes before any function ahead of its output side. In step S304, iterative calculation is performed according to this arrangement order. For example, in the example of Expression (4), the formulation is performed in the order of the intermediate variables v _{1 to} v ₅ .

In step S305, the constraint condition setting unit 103 determines the approximate range of the function (univariate function in this embodiment) that constitutes the separable function. For the approximation range, the sample generated in step S303 is substituted into the function, and the value of the intermediate variable for the number of samples is calculated. Then, the maximum value is set as the upper limit value of the intermediate variable, and the minimum value is set as the lower limit value. 6 substitutes the values of the samples to the intermediate variable v ₄ of the formula (4) is a diagram showing the result of obtaining the range of the intermediate variable v _4, the intermediate variable v _4, the intermediate variable v ₅ of formula is a diagram showing a relationship between a part 2log (v _4). The range of the intermediate variable v ₄ can be calculated as 7.74≦v ₄ ≦14.29.

In step S306, the constraint condition setting unit 103 approximates a function (a one-variable function in this embodiment) forming a separable function with a piecewise linear function and adds it to the mathematical model. As a method of adding the piecewise linear function to the mathematical model, the method described in Non-Patent Document 4 may be used, or any other method may be used. In the present embodiment, the non-convex non-linear function is subjected to piecewise linear approximation, and thus it is added to the mathematical model using the convex coupling formulation method. The convex coupling formulation means piecewise linear approximation to a non-linear function y=f(x), and a section of the piecewise linear function is set as (x _i , y _i ) i=1,..., K. , To formulate the relationship as shown in Expression (5).

When the function forming the separable function is a linear function, the linear function may be added as a mathematical model without needing the piecewise linear approximation.
By repeating the above steps S304 to S306, it is possible to formulate the relationship between variables for all intermediate variables, and set upper and lower limit values for all intermediate variables.
Then, in step S307, by adding g _i (x)≧d _i to the mathematical model, it is possible to set a constraint condition including a nonlinear constraint.
The above is the details of the constraint condition setting process in step S3.

Returning to FIG. 2, in step S4, the objective function setting unit 104 sets an objective function that minimizes the manufacturing cost 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 non-linear function, the objective function may be set by converting the non-linear function into a separable function and then performing piecewise linear approximation as in the case of the constraint condition.

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

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

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

By using the method described above (hereinafter referred to as this method), when solving a nonlinear programming problem that minimizes or maximizes the objective function under constraint conditions including linear constraints and nonlinear constraints, By appropriately setting the approximation range of approximation, it is possible to obtain a highly accurate optimum solution within the practical time.

As a comparative example, a method of calculating the upper and lower limit values of the intermediate variable from the ascending order of the labels of the directed graph of the functions forming the separable function can be considered without using this method. For example, in the case of Expression (4), consider a situation in which the upper and lower limit values are sequentially determined with the intermediate variables v _{1 to} v ₅ . When the value shown in Table 1 is given to the constraint condition of the mixing ratio of each brand and the value shown in Table 2 is given to each vector w _i , the range of v ₁ , v ₂ and v ₃ is shown in Table 4. Like

Here, the range of v ₄ having v ₂ and v ₃ as independent variables is calculated as −16.45≦v ₄ ≦19.16 based on the range of v ₂ and v ₃ shown in Table 4 ( The range shown as a simple calculation in FIG. 7). However, since v ₅ is a log function of v ₄ , as shown in FIG. 7, the function is undefined in the range of v ₄ ≦0. Even if the definition area of v ₄ is limited to the positive range, if v ₄ =0 is included in the approximation range, the definition area of v _{5 in} the subsequent stage needs to be defined from minus infinity. Therefore, the approximation accuracy is significantly deteriorated.
In contrast, when using this technique, the range of v ₄ as described above is limited to a positive range and 7.74 ≦ v ₄ ≦ 14.29, functions not diverge. Therefore, it was confirmed that by using this method, the approximation range can be appropriately determined and a piecewise linear function with high approximation accuracy can be constructed.

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

100: Mixing plan formulation device 101: Input unit 102: Division number setting unit 103: Constraint condition setting unit 104: Objective function setting unit 105: Solving unit 106: Judgment unit 107: Output unit

Claims (7)

An arithmetic device for a nonlinear programming problem for 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,
Setting means for introducing an intermediate variable into the non-linear function representing the non-linear constraint condition, converting it into a separable function, and then performing piecewise linear approximation, and setting the constraint condition,
A solving means for solving the nonlinear programming problem under the constraint condition set by the setting means,
The setting means generates a plurality of independent variable samples excluding intermediate variables from variables constituting the nonlinear function by random sampling, and uses the samples to set upper and lower limit values of the intermediate variable. An arithmetic device for a nonlinear programming problem, characterized by performing a piecewise linear approximation of a function forming the separable function in the range of the upper and lower limits. The said setting means produces|generates the some sample of the independent variable which excluded the intermediate variable among the variables which comprise the said non-linear function so that the conditions according to the said non-linear constraint condition may be satisfied. Computing device for the nonlinear programming problem. The arithmetic unit for a nonlinear programming problem according to claim 1 or 2, wherein the setting means expresses the nonlinear function in the form of a sum of univariate functions by introducing the intermediate variable. The non-linear programming problem is a compounding planning problem in which a compounding ratio is a decision variable in a manufacturing process in which a plurality of raw materials are compounded to manufacture a product,
4. The non-linear programming problem computing device according to claim 1, wherein the setting unit randomly generates a blending ratio by random sampling to generate a plurality of blending ratio samples. .. 5. The setting unit repeats the setting of the upper and lower limit values of the intermediate variable and piecewise linear approximation for each function forming the separable function, according to any one of claims 1 to 4. A computing device for the described nonlinear programming problem. A calculation method of a nonlinear programming problem for 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,
A setting step in which the setting means introduces an intermediate variable into the non-linear function representing the non-linear constraint condition and converts it into a separable function, and then performs piecewise linear approximation to set the constraint condition;
The solution solving means has a solution solving step of solving the nonlinear programming problem under the constraint condition set in the setting step,
In the setting step, random sampling is performed to generate a plurality of independent variable samples excluding intermediate variables among the variables forming the nonlinear function, and the samples are used to set upper and lower limit values of the intermediate variable. A method for computing a non-linear programming problem, characterized by performing a piecewise linear approximation of a function that constitutes the separable function in the range of the upper and lower limits. A program for 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,
Setting means for introducing an intermediate variable into the non-linear function representing the non-linear constraint condition, converting it into a separable function, and then performing piecewise linear approximation, and setting the constraint condition,
Causing a computer to function as a solution finding means for solving the nonlinear programming problem under the constraint condition set by the setting means,
The setting means generates a plurality of independent variable samples excluding intermediate variables from variables constituting the nonlinear function by random sampling, and uses the samples to set upper and lower limit values of the intermediate variable. A program, characterized by performing a piecewise linear approximation of a function forming the separable function in the range of the upper and lower limits.

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