JP2020149607A - Material blend planning method, coke manufacturing method, solving method for mixed integer programming problem and material blend planning device - Google Patents

Abstract

To provide a material blend planning method, a coke manufacturing method, a solving method for a mixed integer programming problem and a material blend planning device, whereby a material blend programming problem formulated as a mixed integer programming problem including a non-linear variable can be solved highly speedily when a material blend ratio at which an evaluation function is optimum with a constraint condition satisfied is planned.SOLUTION: A material blend planning method comprises: a vectorizing step in which one or more elements included in a non-linear term of a non-linear constraint are converted into different variables to be vectorized; an easing constraint condition formation step in which a constraint condition relating to a vector variable is added; a first optimization step in which a predetermined number of values for a to-be-set vector variable are set and the values of the vector variable are retrieved; a second optimization step in which a plurality of linear plan problem groups are composed to search for a value of a vector variable that is improved in operation index evaluation reference; and a final solution determination step in which a final solution is obtained by a mixed integer programming problem.SELECTED DRAWING: Figure 4

Description

The present invention relates to a raw material compounding plan creating method, a coke manufacturing method, a method for solving a mixed integer programming problem, and a raw material compounding plan creating device.

In recent years, in the steelmaking process, optimization of production plans and distribution plans in each process and automation of planning have been promoted. In the raw material process, there is a yard placement problem that determines in which yard the raw materials loaded from the ship loaded with the raw materials are loaded, and a raw material transportation problem that determines when to use which belt conveyor series the raw materials on the yard or pile to be delivered. There are planning problems such as the problem of mixing raw materials to be charged into the blast furnace.

In addition, one of the raw material compounding problems is the ore compounding problem. This ore blending problem is a problem of considering the blending ratio of each brand so as to minimize the cost after satisfying the constraint conditions such as the property constraint required by the blast furnace. Regarding this ore compounding problem, in Patent Document 1, when creating a raw material purchasing plan and a usage plan that minimizes the cost of purchasing and manufacturing raw materials while observing the restrictions on the properties of the product and the capacity of the manufacturing equipment. , A method of performing a search using a constraint expression and an evaluation function as vector variables even when they are given as non-linear expressions is disclosed.

JP-A-2018-77840

However, in actual operation, there is often an upper limit on the number of brands that can be used for compounding due to equipment restrictions and the like. Patent Document 1 discloses a method of replacing a non-linear programming problem with a linear programming problem. However, if the number of issues is limited, the compounding problem becomes a mixed integer programming problem including nonlinear variables, so the same processing is performed. Even if is applied, the solution becomes much more difficult than the linear programming problem.

Further, the method disclosed in Patent Document 1 is generally established in a linear programming problem that can be solved at high speed, but in a mixed integer programming problem that requires a much longer calculation time than a linear programming problem, an optimum solution is obtained. There was the problem that it was difficult to obtain.

The present invention has been made in view of the above, and has been formulated as a mixed integer programming problem including non-linear variables in planning the blending ratio of raw materials for which the evaluation function is optimal after satisfying the constraints. It is an object of the present invention to provide a raw material blending plan creation method, a coke manufacturing method, a method for solving a mixed integer programming problem, and a raw material blending plan creation device capable of solving a raw material blending plan problem at high speed.

In order to solve the above-mentioned problems and achieve the object, the raw material blending plan preparation method according to the present invention includes linear constraints including production amount constraints and inventory constraints for each brand of a plurality of raw materials, and after blending the raw materials. Constraints including the non-linear constraint given by the non-linear formula including the property constraint and the integer constraint including the upper limit of the number of brands of the raw material used for blending are set, and the operation index related to the blending of the raw material is used as an evaluation standard. A method for creating a raw material blending plan for determining the blending amount of each brand of the raw materials based on the constraint conditions and the evaluation criteria, and one or more elements included in the non-linear term with respect to the non-linear constraint. Is replaced with a different variable to vectorize it into a vector variable, and the integer constraint is converted to a continuous variable constraint, the non-linear constraint is linearized using the vector variable, and a constraint condition for the vector variable is added. By doing so, a vector that includes the relaxation constraint creation step for creating the relaxation constraint, the linear constraint, and the relaxation constraint, and is set so that the optimization problem using the operation index as the evaluation criterion has a solution. A first optimization step of searching for the value of the vector variable by setting a predetermined number of variable values, and all or some of the vector variables searched in the first optimization step. On the other hand, the value of the vector variable that includes the linear constraint and the relaxation constraint condition and constitutes a plurality of linear planning problem groups using the operation index as an evaluation standard, and the evaluation standard of the operation index is improved. A second optimization step for searching for, and a final solution determination step for obtaining the final solution by a mixed integer programming problem, which is constructed based on one vector variable value determined in the second optimization step. Including.

Further, in the method for creating a raw material compounding plan according to the present invention, in the above invention, the first optimization step is an element including the linear constraint and the relaxation constraint condition and included in the non-linear term, and is a vector. The element replaced by the element of the variable constitutes a linear programming problem in which the difference in value from the element of the vector variable corresponding to the element included in the non-linear term is used as the penalty term and the sum of the penalty terms is used as the evaluation function. Steps to be performed, a plurality of values of the vector variables are set, a plurality of linear planning problem groups are constructed, and the evaluation function of the sum of the penalty terms gradually approaches 0 while finding the solution of the plurality of linear planning problem groups. As described above, it has a step of searching the vector variable by a heuristic solution method.

Further, in the method for creating a raw material compounding plan according to the present invention, in the above invention, the plurality of linear programming problem groups in which the second optimization step is composed of the second optimization step are described. It has a step of searching the vector variable by a heuristic solution method using the operation index as an evaluation standard while finding a solution of a linear programming problem group, and determining one vector variable value.

Further, in the method for creating a raw material compounding plan according to the present invention, in the above invention, the final solution determination step is a mixture in which the nonlinear constraint is linearized based on the vector variable value obtained in the second optimization step. Derivation of the solution of the integer programming problem and determining the final solution.

Further, in the method for creating a raw material compounding plan according to the present invention, in the above invention, the heuristic solution method includes a particle swarm optimization method or a genetic algorithm.

Further, in the method for creating a raw material compounding plan according to the present invention, in the above invention, the raw material is coal, and the non-linear constraint includes a coke strength estimation formula.

In order to solve the above-mentioned problems and achieve the object, the coke manufacturing method according to the present invention manufactures coke based on the raw material blending plan prepared by the raw material blending plan preparation method according to the above invention.

In order to solve the above-mentioned problems and achieve the object, the method for solving the mixed integer programming problem according to the present invention includes linear constraints given in linear form, non-linear constraints given by non-linear equations, and integer constraints. It is a method of solving a mixed integer planning problem that solves a mixed integer planning problem based on the constraint condition and the evaluation standard by setting a constraint condition to be included, giving a predetermined index as an evaluation standard, and for the non-linear constraint. , One or more elements included in the non-linear term are replaced with different variables and vectorized to make a vector variable, and the integer constraint is converted into a continuous variable constraint, and the non-linear constraint is used. Optimum including the relaxation constraint creation step of creating a relaxation constraint by linearizing and adding the constraint on the vector variable, the linear constraint and the relaxation constraint, and using the predetermined index as an evaluation criterion. In the first optimization step of searching for the value of the vector variable by setting a predetermined number of values of the vector variables to be set so that the conversion problem has a solution, and the first optimization step For all the searched vector variables or some of the vector variables, a plurality of linear planning problem groups including the linear constraint and the relaxation constraint condition and using the predetermined index as an evaluation criterion are formed, and the predetermined A mixed integer composed based on a second optimization step for searching the value of the vector variable for which the evaluation criterion of the index is improved and one vector variable value determined in the second optimization step. Includes a final solution determination step to obtain the final solution by the planning problem.

In order to solve the above-mentioned problems and achieve the object, the raw material blending planning apparatus according to the present invention has linear constraints including production amount constraints and inventory constraints for each brand of a plurality of raw materials, and after blending the raw materials. Constraints including the non-linear constraint given by the non-linear formula including the property constraint and the integer constraint including the upper limit of the number of brands of the raw materials used for blending are set, and the operation index related to the blending of the raw materials is used as an evaluation standard. A raw material compounding plan creating device that determines the blending amount of each brand of the raw material based on the constraint conditions and the evaluation criteria, and is one or more elements included in the non-linear term with respect to the non-linear constraint. Is replaced with a different variable to vectorize it into a vector variable, and the integer constraint is converted into a continuous variable constraint, the non-linear constraint is linearized using the vector variable, and a constraint condition related to the vector variable is added. By doing so, a relaxation constraint condition creating means for creating a relaxation constraint condition, and a vector that includes the linear constraint and the relaxation constraint condition and is set so that the optimization problem using the operation index as an evaluation criterion has a solution. A first optimization means for setting a predetermined number of variable values and searching for the value of the vector variable, and all vector variables or some vector variables searched for by the first optimization means. On the other hand, the value of the vector variable that includes the linear constraint and the relaxation constraint condition and constitutes a plurality of linear planning problem groups using the operation index as an evaluation standard, and the evaluation standard of the operation index is improved. A second optimization means for searching for, and a final solution determination means for obtaining the final solution by a mixed integer programming problem, which is constructed based on one vector variable value determined by the second optimization means. Be prepared.

According to the present invention, it is possible to solve a raw material compounding planning problem formulated as a mixed integer programming problem including nonlinear variables at high speed. That is, it is possible to solve the raw material blending planning problem including restrictions on the upper limit of the number of brands of raw materials used for blending and the properties of the raw materials after blending at high speed.

FIG. 1 is a schematic diagram showing a solution area and a feasible area of a linear programming problem. FIG. 2 is a schematic diagram showing a solution area and a feasible area of a mixed integer programming problem. FIG. 3 is a schematic diagram showing a schematic configuration of a raw material compounding plan creating apparatus according to an embodiment of the present invention. FIG. 4 is a flowchart showing the flow of the raw material compounding plan creating method according to the embodiment of the present invention. FIG. 5 is a flowchart showing the details of the relaxation condition solving step in the raw material compounding plan creating method according to the embodiment of the present invention. FIG. 6 is a diagram for explaining an outline of the relaxation constraint condition creation step. FIG. 7 is a diagram for explaining the outline of the particle swarm optimization method. FIG. 8 is a flowchart showing the details of the true planning problem solving step in the raw material compounding plan creating method according to the embodiment of the present invention. FIG. 9 is an example of the raw material blending plan creating method according to the embodiment of the present invention, and is a graph showing the transition of the evaluation value when the raw material blending plan is drafted.

A method for creating a raw material compounding plan, a method for producing coke, a method for solving a mixed integer programming problem, and a device for creating a raw material compounding plan according to an embodiment of the present invention will be described with reference to the drawings. The present invention is not limited to the embodiments described below.

First, the outline of the optimization problem of the raw material compounding plan to which the present invention is applied will be described. In the steel industry, it is required to mix raw materials such as various kinds of purchased iron ore and coal having various properties, and to keep the quality and properties after mixing within a certain range. Further, in this case, since the cost is judged as an important index, it is required to minimize the purchase cost, the manufacturing cost, and the like. Furthermore, it is required to plan the blending over a plurality of days while changing the blending ratio so that the stock of raw materials is not exhausted. In the present invention, the raw material compounding plan is formulated as an optimization problem. The problem of optimizing the raw material compounding plan targeted by the present invention can be expressed by, for example, the following equations (1) and (2).

Here, in the above equations (1) and (2), F is a subset of the n-dimensional vector space R ⁿ , and f is an evaluation function defined by R ⁿ .

FIG. 1 is a schematic diagram showing a solution area and a feasible area of a linear programming problem (LP). In a linear programming problem, the merit function and constraints are linear. The polygon in the figure shows the feasible region of the linear programming problem. When solving a linear programming problem, it is only necessary to search for the vertices of the polygon (see the dashed circle in the figure), so high-speed calculation is possible.

FIG. 2 is a schematic diagram showing a solution area and a feasible area of a mixed Integer programming problem (MIP). The mixed integer programming problem is different from the linear programming problem in which the feasible region is a convex polygon (see FIG. 1), and the feasible region is a discontinuous figure as shown in the figure. Therefore, when solving a mixed integer programming problem, it is necessary to search for points on non-continuous figures, which takes time for calculation. It is also complicated by the addition of non-linear variables to the mixed integer programming problem.

The value of the evaluation function in the above equation (1) can be a predetermined numerical value according to the optimization problem. For example, it can be a minimum value, a maximum value, or a predetermined predetermined value. In the following, the optimization problem will be considered by formulating a mathematical model as shown in the following equations (3) to (9), taking the raw material compounding problem which is the object of the present invention as an example.

In the above formulas (3) to (9), x _{i and j} are the blending amount of the brand i in each planning period j, s is the inventory amount of the brand i in each planning period j, C is the cost of the brand i, and N is each. sum of the amount in the plan period _j, d _i, e i, components _{k i} with stock _{i, y i,} when the _j whether 0-1 variable issue i is blended with the above formula (3) to (9) are equations showing the following.

Formula (3): Formula showing the cost of planning after blending (4): Formula showing restrictions on the blending amount to be satisfied in each planning period j (5): Formula showing inventory constraints for each brand (6) ), (7): Formula (8) showing the property constraint after blending: Formula showing the upper limit constraint on the number of brands used for blending (9): Formula showing whether or not brand i is used

In addition, there may be a plurality of property restrictions as in the above equation (6). Further, C, C ₁ and C ₂ in the above equations (3) to (9) are constants. Also, all variables and constants are non-negative real numbers.

Further, the evaluation function represented by the above formula (3) uses the cost of planning after compounding here, but other embodiments may be used. For example, the change width of the blending ratio with respect to the usage amount x _{i, j} of each raw material brand in the planned period adjacent to each other may be used, and the change width may be minimized. This is because, for example, the change in operating conditions is smaller when the method of blending the raw materials is similar than when the method of blending the raw materials changes significantly on a daily basis, so that the burden on the operation can be reduced. In any case, the evaluation function represented by the above formula (3) may be an operation index related to the blending of raw materials, and in some cases, the purpose may be to minimize or maximize the operation index.

Here, since the above equation (7) includes a quadratic nonlinear variable, if it is treated as it is, it becomes a nonlinear programming problem. Moreover, since the above equation (8) contains a 0-1 variable on the left side, it is very difficult to find it. Therefore, in the present invention, the non-linear variable of the above equation (7) is replaced with a vector variable (steps S31 and S32 in FIG. 5 described later), and y of the above equation (8) is approximated to a continuous variable of "0 or more and 1 or less". (Step S33 in the figure), a mathematical model as shown in the following equations (10) to (19) is formulated (step S34 in the figure).

Here, the vector variables g _j ^k in the above equations (14) and (15) indicate constants of approximate values of Σ e _i x _{i, j} in the planning period j. Further, p _j in the above formula (15) is a penalty variable indicating the difference between the approximate value.

By the above equations (14) and (15), the above equation (7) can be expressed in linear form. Further, the above equations (18) and (19) are equations in which the above equation (9) is relaxed, and the binary variable y of 0-1 is set as a continuous variable of 0 to 1, and is relaxed by a sufficiently large constant M. A method called "big-M method" is used. This time, M = Σ _j N _j .

Here, using the penalty variable p _j in the above formula (15), when replacing it with a non-linear term of the equation (7) to a specific fixed value, because there is no guarantee that a solution is obtained. Therefore, after adding the penalty variable p _j (p _j ≧ 0) as in the equation (15), the minimization problem is solved so that the penalty variable p _j becomes 0, and the above equations (3) to (3) to ( solution of 9) searches the candidate parameters as obtained (vector variable g _{j ^k).}

Further, mathematical model shown in the equation (10) to (19), since greatly affected by the value of vector variable g _j ^k in the formula (14), to the K prepared a vector of the combination of the constants to be set .. That, K pieces of relaxation problem is formulated, updates the value of the vector variable g _j ^k of J dimension by comparing the evaluation function value of (step S36 of FIG. 5 described later), the formula (10) The search is performed until the evaluation function becomes the minimum, that is, 0.

The update of the value of the vector variable g _j ^k, for example, Particle Swarm Optimization (PSO: Particle Swarm Optimization) and Genetic Algorithms (GA: Genetic Algorithm) can be used heuristic method such. In the following, an example using the particle swarm optimization method will be described.

The algorithm of the particle swarm optimization method is a meta-heuristics inspired by the collective behavior of living organisms, and when an evaluation function to be searched is given, multiple particles share information with each other to find the optimum solution. It moves around in the search space. The particle swarm optimization method is characterized by the simplicity of such concepts and algorithms, the flexibility of operation, and the possibility of improvement. In the particle swarm optimization algorithm, the components of each nonlinear vector are updated according to the following equations (20) and (21).

In the above equations (20) and (21), a _m ^k is the update vector of the vector variable k ∈ ^K at time m ∈ M, b _m ^k is the position vector (coordinates) of the vector variable k at time m, and r and C are. The parameters given by random numbers are shown respectively. P _k is the personal best indicating the best position vector taken by each particle (each vector variable) k by the time m, and G _m is the best taken by all particles (all vector variables) by the time m. It shows the global best, which shows the position vector. When the problem setting is a minimization problem, the personal best _PK and the global best _Gm are updated when the objective function value becomes small.

In the present invention, the position vector b _m ^k corresponds to the vector variable g _j ^k at time m in the mathematical models of the above equations (10) to (19). In the mathematical models of the above equations (10) to (19), the termination condition is satisfied when the evaluation value (value of the evaluation function) is 0 or equal to or less than a predetermined minimum predetermined value (value that can be regarded as almost 0). When the end condition is satisfied (Yes in step S37 of FIG. 5), the search ends. The search and select the L particles upon completion, to set anew vector variable g _j ^k, unravel the following relaxation problem. Here, as the L particles, any L particles after the evaluation function is minimized may be selected for the vector variable searched by the algorithm of the particle swarm optimization method described above. Further, L may be selected from the one having the smallest evaluation function. Alternatively, all the parameters of the K particles used in the search may be used.

When the evaluation function value of the above equation (10) becomes 0, the above equations (14) and (15) have the same meaning as the above equation (7). Therefore, using vector variable _g ^{j k} selected above, Figure 5 a mathematical model of the equation (10) to (19), which (later switched to the mathematical model of the following formula (22) - (30) Steps S38 and S39). As a result, the lower bound of the original mathematical model (the above equations (3) to (9)) is solved (step S40 in the figure).

For even mathematical model shown in the equation (22) - (30), greatly influenced by the value of vector variable _g ^{j k} in the formula (26). Therefore, the vectors that are the combination of the constants to be set are L vectors obtained by the mathematical model of the above equations (10) to (19) and the particle swarm optimization method. That, L-number of relaxation problem is formulated, updates the value of the vector variable g _j ^k of J dimension by comparing the evaluation function value (step S41 of FIG. 5 described later).

The update of the value of the vector variable g _j ^k, can be used, for example a heuristic method such as particle swarm optimization and genetic algorithm. In the following, an example using the particle swarm optimization method will be described. The algorithm of the particle swarm optimization method is as described above, and the components of each nonlinear vector are updated according to the above equations (20) and (21). The procedure for updating the vector variable is as described above, except that the target mathematical model is the above equations (22) to (30).

That is, for the L position vectors _b ^{m k,} it solves a mathematical model of the above formula (22) - (30), a personal best _{P k} using (f (x) of formula (19)) the evaluation function value and global best _{G m} with updating the position vector _b ^{m k,} predetermined and end condition is satisfied (Yes in step S42 of FIG. 5 described later), and outputs a global best _{G m} (step S43 in FIG.) .. The end condition can be determined, for example, by a method of determining the number of repetitions, the amount of change in the evaluation function, or the like.

Then, the global best _{G m} output, solving mixed integer programming problem represented by mathematical models of the following formulas (31) - (38) (step S4 in FIG. 4).

However, k ∈ L ∈ K. By the above procedure, a mixed integer programming problem including nonlinear variables can be solved at high speed.

[Raw material formulation planning device]
The configuration of the raw material compounding plan creating apparatus according to the embodiment of the present invention will be described with reference to FIG. The raw material blending plan creating device 1 includes an input unit 10, an output unit 20, and a calculation unit 30.

The input unit 10 is an input means for the calculation unit 30, and is realized by an input device such as a keyboard, a mouse pointer, or a numeric keypad.

The output unit 20 is realized by a display device such as an LCD display or a CRT display, and displays a calculation result or the like based on a display signal input from the calculation unit 30.

The arithmetic unit 30 is realized by, for example, a processor including a CPU (Central Processing Unit) and a memory (main storage unit) including a RAM (Random Access Memory) and a ROM (Read Only Memory).

The arithmetic unit 30 loads the program into the work area of the main storage unit, executes the program, and controls each component or the like through the execution of the program to realize a function that meets a predetermined purpose. The calculation unit 30 functions as a solution unit 31 through the execution of the program.

[How to create a raw material formulation plan]
A method for creating a raw material compounding plan according to an embodiment of the present invention will be described with reference to FIGS. 4 to 8. In the raw material blending plan creation method according to the present embodiment, in the raw material blending plan formulated as a mixed integer programming problem including nonlinear variables, the blending ratio of the raw materials for which the evaluation function is optimal is determined after satisfying the constraints. Is the way to. In the raw material blending plan creation method according to the present embodiment, specifically, constraint conditions including linear constraint, non-linear constraint and integer constraint are set, and an operation index related to raw material blending is given as an evaluation standard, and the constraint condition and evaluation are performed. Based on the standard, the blending amount of each brand of raw material is determined.

Here, examples of the above-mentioned raw material include coal. For example, coke can be produced by creating a raw material blending plan by the method for creating a raw material blending plan according to the present embodiment, blending a plurality of brands of coal according to the raw material blending plan, and then carbonizing in a coke oven.

Further, as the linear constraint described above, for example, a production volume constraint relating to a planned production volume of coke, an inventory constraint for each brand of a plurality of raw materials, and the like can be mentioned. Further, the non-linear constraint described above is given by a non-linear formula, and indicates a property constraint after blending raw materials of a plurality of brands. Further, the above-mentioned integer constraint indicates an upper limit of the number of brands of raw materials used for blending.

As will be described later, the raw material compounding plan creation method according to the present embodiment includes a vectorization step S31, a relaxation constraint condition creation step S33, a linear constraint and a relaxation constraint condition, and is optimized using an operation index as an evaluation standard. The first optimization step (first mitigation problem creation step S34 to vector variable) in which the values of the vector variables to be set are set in a predetermined number in advance so that the problem has a solution, and the values of the vector variables are searched. Update step S36) and the first optimization step are repeated until a predetermined end condition is satisfied (first mitigation problem repetition step S37), and a vector variable group when the end condition is satisfied is determined. The vector variable setting step S38 is performed.

Subsequently, for all vector variables or some vector variables searched in the first optimization step, a plurality of linear programming problem groups including linear constraints and relaxation constraints and using the operation index as an evaluation criterion are set. The second optimization step (vector variable setting step S38 to vector variable update step S41) for searching for the value of the vector variable that is configured and the evaluation standard of the operation index is improved is performed, and a predetermined termination condition is satisfied. The second optimization step is repeated until (second mitigation problem repetition step S42), and the mixture is composed based on one vector variable value determined in the second optimization step when the end condition is satisfied. The final solution determination step (true planning problem solving step S4) for obtaining the final solution by the integer programming problem is performed.

Hereinafter, the details of the raw material blending plan preparation method according to the present embodiment will be described. As shown in FIG. 4, the raw material compounding plan creation method according to the present embodiment includes input data acquisition step S1, constraint condition acquisition step S2, relaxation problem solving step S3, true planning problem solving step S4, and results. Output step S5 is performed in this order. In addition, each step shown in the figure is mainly carried out by the solution unit 31 of the calculation unit 30.

(Input data acquisition step S1)
In the input data acquisition step S1, the input data necessary for drafting the raw material blending plan is acquired via the input unit 10. Examples of this input data include information on the properties of stocks used in inventory, arrival, operation planning, and compounding planning.

(Restriction condition import step S2)
In the constraint condition incorporation step S2, a plurality of constraint conditions necessary for formulating a raw material blending plan, such as properties required after blending the raw materials, upper and lower limits of inventory, and operating conditions, are captured via the input unit 10.

(Mitigation problem solving step S3)
In the relaxation problem solving step S3, a problem in which the mixed integer programming problem including non-linear variables is relaxed is formulated based on the input data captured in the input data acquisition step S1 and the constraint conditions captured in the constraint condition acquisition step S2. , To solve. The details of the mitigation problem solving step S3 will be described later.

(True plan problem solving step S4)
In the true programming problem solving step S4, a mixed integer programming problem including the original nonlinear variable is formulated and solved based on the result obtained by solving the relaxation problem in the relaxation problem solving step S3. That is, in the true programming problem solving step S4, the solution of the mixed integer programming problem in which the nonlinear constraint is linearized is derived based on the vector variable value obtained in the relaxation problem solving step S3, and the final solution is determined. The details of the true planning problem solving step S4 will be described later.

(Result output step S5)
In the result output step S5, the result obtained in the true planning problem solving step S4 is output via the output unit 20.

<Details of mitigation problem solving step S3>
Specifically, as shown in FIG. 5, the relaxation problem solving step S3 includes a vectorization step S31, a vector variable combination creation step S32, a relaxation constraint condition creation step S33, and a first relaxation problem creation step S34. , The first mitigation problem optimization step S35, the vector variable update step S36, and the first mitigation problem repetition step S37 are performed to determine the vector variables corresponding to the true optimization problem, and then the second The vector variable setting step S38 of the mitigation problem is carried out. Then, subsequently, the second mitigation problem creation step S39, the second mitigation problem optimization step S40, the vector variable update step S41, the second mitigation problem repetition step S42, and the mitigation solution output step S43 are performed. Do it in order.

(Vectorization step S31)
In the vectorization step S31, the nonlinear term included in one or more nonlinear constraints or some variables (elements) included in the nonlinear term are replaced with different variables, and the replaced variables are vectorized. In vectorization step S31, specifically, a nonlinear variable (the above formula (7) see) captured by the input data acquisition steps, "vector variable g _j ^k" one or more dimensions of a vector variable (the formula (14) (See) and vectorize.

(Step S32 for creating a combination of vector variables)
In combination creating step S32 in vector variable, created in the vectorization step S31 the vectorized nonlinear terms in the corresponding variable (see "vector variable g _j ^k" in the formula (14)), only the number of combinations of predetermined To do.

(Relaxation constraint condition creation step S33)
In the relaxation constraint condition creation step S33, the integer constraint is converted into a continuous variable constraint. In the relaxation constraint condition creation step S33, the original constraint condition (the above equation) is about the constraint condition to be satisfied by the constraint condition fetched in the constraint condition import step S2 and the plurality of vector variables created in the vector variable combination creation step S32. A set of constraint conditions defined in a relaxed form (see (3) to (9)) is created for the number of combinations of the vector variables. In the relaxation constraint condition creation step S33, the relaxation constraint condition is created by converting the integer constraint into a continuous variable constraint, linearizing the non-linear constraint using the vector variable, and adding the constraint condition related to the vector variable.

In the relaxation constraint condition creation step S33, specifically, the 0-1 variables (see the above equation (8)) included in the original constraint conditions (see the above equations (3) to (9)) are set to 0 to 1. By approximating a continuous variable, it is changed to a linear planning problem (linear constraint problem). Then, as shown in FIG. 6, the group of approximate linear programming problems that are the lower bound of the original problem is solved, and the neighborhood of the optimum solution of the lower bound is searched. As a result, the original problem can be solved efficiently. By changing to a linear programming problem, the solution region becomes a polygon as in FIG. 1 described above. Therefore, it is only necessary to search for the vertices of the polygon, and high-speed calculation becomes possible.

(First mitigation problem creation step S34)
In the first relaxation problem creation step S34, the evaluation function value improves as the relaxation constraint condition created in the relaxation constraint condition creation step S33 and the original constraint condition (see the above equations (3) to (9)) are satisfied. A set of mitigation problems (see equations (10) to (19) above) composed of evaluation functions defined by such linear variables is created for the number of combinations of vector variables.

In the first relaxation problem creation step S34, specifically, the elements including the linear constraint and the relaxation constraint condition and included in the nonlinear term, which are replaced with the elements of the vector variable, become the nonlinear term. A linear programming problem is constructed in which the difference between the values of the vector variables corresponding to the included elements and the elements of the vector variable is used as the penalty term, and the sum of the penalty terms is used as the evaluation function.

(First mitigation problem optimization step S35)
In the first mitigation problem optimization step S35, a set of mitigation problems created in the first mitigation problem creation step S34 is solved.

(Vector variable update step S36)
In the vector variable update step S36, the components of the vector variable are updated with respect to the relaxation solution set obtained in the first relaxation problem optimization step S35. Specifically, in the vector variable update step S36, according to a predetermined algorithm based on the relaxation solution and the evaluation value (here, equation (10)) obtained in the first relaxation problem optimization step S35. , Update the coordinates (position vector) of the vector variable. For example, a particle swarm optimization method is used as an algorithm for updating the coordinates of this vector variable.

In the particle swarm optimization method, as shown in FIG. 7, the coordinates are stochastically updated in the current best (personal best, global best) coordinate direction while sharing the coordinate and evaluation value information for each search unit. To do. This searches for better coordinates of the search unit. The search unit in the figure corresponds to the "vector variable" in the present embodiment.

As described above, in the first mitigation problem optimization step S35 and the vector variable update step S36, a plurality of vector variable values are set, a plurality of linear programming problem groups are constructed, and solutions of the plurality of linear programming problem groups are obtained. At the same time, the vector variable is searched by the heuristic solution so that the evaluation function of the sum of the penalty terms approaches 0.

(First mitigation problem repetition step S37)
In the first mitigation problem repetition step S37, the first mitigation problem optimization step S35 and the vector variable update step S36 are repeated for the combination of the vector variables updated in the vector variable update step S36. At step S37 repeat first relaxation problem, specifically, the first relaxation problem optimization step is not satisfied a predetermined end condition (less than the predetermined value sum is zero or a minimum of penalty variable p _j) Returning to S35, if the end condition is satisfied, the iterative process is ended, and the process proceeds to the vector variable setting step S38 of the second mitigation problem.

(Vector variable setting step S38)
In vector variable setting step S38, the acquired by the number of predetermined (the "vector variable g _j ^k" reference in the formula (26)) resulting vector variable as the optimal solution of the first relaxation problem, set.

(Second mitigation problem creation step S39)
In the second mitigation problem creation step S39, as described above, the vector variable at the time when the solution of the mitigation problem created in the first mitigation problem creation step S34 (hereinafter referred to as “mitigation solution”) becomes the optimum solution. to use the g _j ^k. Using this vector variable, the relaxation is replaced with the original constraint condition (see the above equations (3) to (9)) before relaxation in the first relaxation problem creation step S34 and the predetermined true evaluation function. A set of problems (see the above equations (22) to (30)) is created for the number of combinations of the vector variables.

(Second mitigation problem optimization step S40)
In the second mitigation problem optimization step S40, a set of mitigation problems created in the second mitigation problem creation step S39 is solved.

(Vector variable update step S41)
In the vector variable update step S41, the components of the vector variable are updated with respect to the relaxation solution set obtained in the second relaxation problem optimization step S40. Specifically, in the vector variable update step S41, the coordinates (position vector) of the vector variable are obtained according to a predetermined algorithm based on the relaxation solution and the evaluation value obtained in the second relaxation problem optimization step S40. ) Is updated. For example, a particle swarm optimization method is used as an algorithm for updating the coordinates of this vector variable. This particle swarm optimization method is as described above.

As described above, in the second mitigation problem optimization step S40 and the vector variable update step S41, while obtaining the solutions of the plurality of linear programming problem groups for the plurality of linear programming problem groups, the operation index is used as the evaluation standard. A vector variable is searched by a heuristic solution to determine one vector variable value.

(Second mitigation problem repetition step S42)
In the second mitigation problem repetition step S42, the second mitigation problem optimization step S40 and the vector variable update step S41 are repeated for the combination of the vector variables updated in the vector variable update step S41. In the second mitigation problem repetition step S42, specifically, if the predetermined end condition (number of repetitions) is not satisfied, the process returns to the second mitigation problem optimization step S40, and if the end condition is satisfied, the process is repeated. Is completed, and the process proceeds to the relaxation solution output step S43.

(Relaxation output step S43)
In the relaxation solution output step S43, the relaxation solution set is output in the form of associating the position vector of the vector variable with the relaxation solution and the evaluation value, and the process proceeds to the true planning problem solving step S4.

<Details of True Planning Problem Solving Step S4>
In the true planning problem solving step S4, specifically, as shown in FIG. 8, the vector variable setting step S51, the true planning problem creation step S52, and the true planning problem optimization step S53 are performed in this order. Do.

(Vector variable setting step S51)
In vector variable setting step S51, it acquires a vector variable of global best G _m mitigation solution output in relaxed solution output step S43, the process proceeds to create step S52 real programming problem.

(True plan problem creation step S52)
In the true planning problem creation step S52, the vector variables acquired in the vector variable setting step S51 are composed of the original constraint conditions (the above equations (3) to (9)) and a predetermined evaluation function. Create a true planning problem (see equations (31)-(38) above). In the true planning problem creation step S52, specifically, the constraint including the input data and the constraint condition acquired in the input data acquisition step S1 and the constraint condition acquisition step S2 and the non-linear variable created in the vector variable setting step S51. Establish constraints by replacing the conditions with linear form. Then, a true planning problem defined by a predetermined evaluation function is formulated.

(True planning problem optimization step S53)
In the true planning problem optimization step S53, the true planning problem (mathematical model) created in the true planning problem creation step S52 is solved, and the process proceeds to the result output step S5.

According to the raw material blending plan creation method according to the present embodiment as described above, the raw material blending planning problem formulated as a mixed integer programming problem including nonlinear variables can be solved at high speed. That is, it is possible to solve the raw material blending planning problem including restrictions on the upper limit of the number of brands of raw materials used for blending and the properties of the raw materials after blending at high speed.

An example of the method for creating a raw material compounding plan according to the present embodiment will be described with reference to FIG. The figure is a graph showing the transition of the evaluation value when the raw material compounding plan is made by the method for creating the raw material compounding plan according to the present embodiment. In this example, a coal blending plan for blending 50 or more types of coking coal was formulated for 3 months.

In this example, the coke intensity estimation formula defined as a function of quadratic or higher is used as the nonlinear constraint. Further, in this embodiment, the termination condition in the repeating step S42 the second relaxation problem was 10 loops, the vector variable setting step S51, it acquires a vector variable of global best G _m, Optimization step S53 real programming problem , The calculation time was set to 60 seconds with a commercially available personal computer. Moreover, since the method according to the present invention is affected by the initial value of the vector variable, numerical experiments were carried out 10 times.

As shown in FIG. 9, it can be seen that the calculation converges at high speed by using the method according to the present invention. That is, the calculation of about 15 minutes (900 seconds) is sufficient. In the method according to the present invention, the evaluation function is switched by nature, so that the plot during the solution of the first mitigation problem is deleted in the figure.

Tables 1 and 2 show a part of the raw material compounding plan output as a solution in this embodiment. As shown in the table, it can be seen that a feasible solution that satisfies all the set constraints is obtained.

The method for creating a raw material compounding plan, the method for producing coke, the method for solving a mixed integer planning problem, and the device for creating a raw material compounding plan according to the present invention have been specifically described above with reference to embodiments and examples for carrying out the invention. The purpose of the invention is not limited to these descriptions, and must be broadly interpreted based on the description of the scope of claims. Needless to say, various changes, modifications, etc. based on these descriptions are also included in the gist of the present invention.

For example, in the present invention, an example in which the variable is replaced so as to perform linearization for the squared term of the variable as a non-linear term is shown, but the present invention is not limited to this. By replacing the non-linear term with another variable, the non-linear constraint should be apparently linear.

Further, the raw material blending plan creating device may be configured by one computer, or may be configured by a plurality of computers so that information can be exchanged between the computers. Further, the above-mentioned method for creating a raw material compounding plan can be executed by a computer program mounted on a computer. This computer program may be recorded on a recording medium such as a CD or a magnetic tape and made readable.

1 Raw material compounding plan creation device 10 Input unit 20 Output unit 30 Calculation unit 31 Solution unit

Claims (9)

The linear constraint including the production amount constraint and the inventory constraint for each brand of a plurality of raw materials, the non-linear constraint given by the non-linear formula including the property constraint after blending of the raw materials, and the upper limit of the number of brands of the raw materials used for blending. A raw material that includes an integer constraint and a constraint condition that includes, gives an operation index related to the blending of the raw material as an evaluation standard, and determines the blending amount of each brand of the raw material based on the constraint condition and the evaluation standard. It is a method of creating a formulation plan,
For the nonlinear constraint, a vectorization step in which one or more elements included in the nonlinear term are replaced with different variables to vectorize them into vector variables, and
A relaxation constraint condition creation step for creating a relaxation constraint condition by converting the integer constraint into a continuous variable constraint, linearizing the nonlinear constraint using the vector variable, and adding a constraint condition for the vector variable.
The value of the vector variable to be set is set by a predetermined number, and the vector variable is set so that the optimization problem including the linear constraint and the relaxation constraint condition and using the operation index as an evaluation criterion has a solution. The first optimization step to search for the value of
For all vector variables or some vector variables searched in the first optimization step, a plurality of linear programming problem groups including the linear constraint and the relaxation constraint condition and using the operation index as an evaluation criterion. The second optimization step of searching for the value of the vector variable, which constitutes the above and improves the evaluation criteria of the operation index,
A final solution determination step for obtaining a final solution by a mixed integer programming problem, which is constructed based on one vector variable value determined in the second optimization step.
How to make a raw material formulation plan including. The first optimization step is
An element that includes the linear constraint and the relaxation constraint and is included in the nonlinear term and is replaced by the element of the vector variable is an element of the vector variable corresponding to the element included in the nonlinear term. A step of constructing a linear programming problem in which the difference between the values of is a penalty term and the sum of the penalty terms is an evaluation function, and
The vector is set so that a plurality of values of the vector variables are set, a plurality of linear programming problem groups are constructed, a solution of the plurality of linear programming problem groups is obtained, and the evaluation function of the sum of the penalty terms approaches 0. Steps to search for variables by heuristic solution and
The method for creating a raw material compounding plan according to claim 1. In the second optimization step, the solution of the plurality of linear programming problem groups is obtained for the plurality of linear programming problem groups composed of the second optimization step, and the operation index is used as an evaluation standard. The method for creating a raw material compounding plan according to claim 1 or 2, further comprising a step of searching the vector variable by a heuristic solution method and determining one vector variable value. The final solution determination step derives the solution of the mixed integer programming problem in which the nonlinear constraint is linearized based on the vector variable value obtained in the second optimization step, and determines the final solution. The method for creating a raw material compounding plan according to any one of claims 3. The method for creating a raw material compounding plan according to claim 2 or 3, wherein the heuristic solution method includes a particle swarm optimization method or a genetic algorithm. The raw material is coal
The nonlinear constraint includes a coke intensity estimation equation.
The method for creating a raw material compounding plan according to any one of claims 1 to 5. A coke manufacturing method for producing coke based on a raw material blending plan created by the raw material blending plan creating method according to any one of claims 1 to 6. A constraint condition including a linear constraint given in a linear form, a non-linear constraint given by a non-linear expression, and an integer constraint is set, a predetermined index is given as an evaluation criterion, and based on the constraint condition and the evaluation criterion, Solving mixed integer programming problems This is a method for solving mixed integer programming problems.
For the nonlinear constraint, a vectorization step in which one or more elements included in the nonlinear term are replaced with different variables to vectorize them into vector variables, and
A relaxation constraint condition creation step for creating a relaxation constraint condition by converting the integer constraint into a continuous variable constraint, linearizing the nonlinear constraint using the vector variable, and adding a constraint condition for the vector variable.
The value of the vector variable to be set is set by a predetermined number in advance so that the optimization problem including the linear constraint and the relaxation constraint condition and using the predetermined index as the evaluation criterion has a solution, and the vector The first optimization step to search for the value of a variable,
For all vector variables or some vector variables searched in the first optimization step, a plurality of linear programming problems including the linear constraint and the relaxation constraint condition and using the predetermined index as an evaluation criterion. A second optimization step of forming a group and searching for the value of the vector variable whose evaluation criteria of the predetermined index are improved, and
A final solution determination step for obtaining a final solution by a mixed integer programming problem, which is constructed based on one vector variable value determined in the second optimization step.
How to solve mixed integer programming problems including. The linear constraint including the production amount constraint and the inventory constraint for each brand of a plurality of raw materials, the non-linear constraint given by the non-linear equation including the property constraint after blending of the raw materials, and the upper limit of the number of brands of the raw materials used for blending. A raw material that includes an integer constraint and a constraint condition that includes, gives an operation index related to the blending of the raw material as an evaluation standard, and determines the blending amount of each brand of the raw material based on the constraint condition and the evaluation standard. It is a compounding plan making device,
With respect to the non-linear constraint, a vectorizing means that replaces one or more elements included in the non-linear term with different variables and vectorizes them into vector variables.
A relaxation constraint condition creation means for creating a relaxation constraint condition by converting the integer constraint into a continuous variable constraint, linearizing the non-linear constraint using the vector variable, and adding a constraint condition related to the vector variable.
The value of the vector variable to be set is set by a predetermined number, and the vector variable is set so that the optimization problem including the linear constraint and the relaxation constraint condition and using the operation index as an evaluation criterion has a solution. The first optimization method to search for the value of
For all vector variables or some vector variables searched by the first optimization means, a plurality of linear programming problem groups including the linear constraint and the relaxation constraint condition and using the operation index as an evaluation criterion. A second optimization means for searching for the value of the vector variable, which constitutes the above and improves the evaluation criteria of the operation index.
A final solution determining means for obtaining a final solution by a mixed integer programming problem, which is constructed based on one vector variable value determined by the second optimization means.
A raw material compounding plan making device equipped with.