CN117151307B - Layout optimization method based on hybrid linear programming - Google Patents
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Abstract
The invention provides a stock layout optimization method based on mixed linear programming, which adopts mixed linear programming to construct a mathematical model, and sorts rectangular components to be produced from large to small in length before cutting, sorts rectangular components with the same length from large to small in width, so as to complete data preprocessing, fully excavate the characteristics of data, facilitate understanding and realization, have low calculation cost and high speed, can be realized by personnel without advanced training, thereby remarkably saving expenditure and time, and is applied to the practical application of enterprises; that is, the invention optimizes the layout under the condition of comprehensively considering the length utilization rate and the width utilization rate, realizes the layout optimization of a large number of rectangular assemblies with various specifications, and improves the layout utilization rate and the layout efficiency in practical application.
Description
Technical Field
The invention belongs to the technical field of stock layout optimization, and particularly relates to a stock layout optimization method based on hybrid linear programming.
Background
The square stock layout optimization method can solve the problem of material loss in the cutting process, improves the total utilization rate of materials, and reduces the production cost. The square stock layout problem can be applied to various fields such as metal cutting and glass cutting in automobile manufacturing and aerospace, wood cutting in furniture manufacturing, cloth cutting in clothing design field, and the like.
The plate-type product is formed by combining and assembling a plurality of plate-type fittings generated by plane processing by taking a plate as a raw sheet. The products can be processed in a scattered way after being designed, can be assembled flexibly and have more styles.
The current general plate cutting mode is a three-stage layout mode, wherein the plates are cut and fed into square products in three stages; most of the existing plate cutting optimization methods focus on optimizing the original plate consumption, namely, the consumption of square raw materials is used as a target, and the utilization rate of the plate is improved. However, the plate optimization algorithm is easy to fall into local optimum under the condition of not carrying out data preprocessing, an initial clustering center needs to be determined before optimization, and the plate optimization algorithm is difficult to put into practical application of enterprises.
Disclosure of Invention
In order to solve the problems, the invention provides a layout optimization method based on hybrid linear programming, which takes the length utilization rate and the width utilization rate as optimization targets, realizes the layout optimization of a large number of rectangular assemblies with various specifications, and improves the layout utilization rate and the layout efficiency in practical application.
A layout optimization method based on hybrid linear programming comprises the following steps:
sequencing the lengths of rectangular assemblies which are required to be sequentially cut from plates with the same size from large to small, sequencing the rectangular assemblies with the same length from large to small, and sequentially numbering the rectangular assemblies according to the sequencing;
and sequentially discharging the rectangular components under the constraint condition that the length utilization rate and the width utilization rate of each plate are maximum, the sum of the lengths of the rectangular components of the discharged samples on each plate is not more than the length of the plate, and the sum of the widths of the rectangular components of the discharged samples on each plate is not more than the width of the plate, so as to sequentially obtain an optimal discharging scheme of the rectangular components on each plate, wherein when determining the optimal discharging scheme of the rectangular components on each plate which does not complete discharging, whether the current remaining non-discharged rectangular components are arranged on the plate which is currently discharging is sequentially judged according to the sequence from the small number to the large number.
Further, the objective function is:
length utilization of sheet materialMaximum:
utilization of the width of the sheet materialMaximum:
wherein,number representing rectangular component>=1, 2, …, N is the total number of matrix elements, +.>Indicating number->Length of rectangular component>Indicating number->L represents the length of the sheet material, W represents the width of the sheet material,indicating number->The coefficients of the rectangular components of (2) are 0 or 1 variables, where, if numbered +.>Rectangular assembly of (c)Is arranged on the plate which is currently being subjected to the typesetting, then +.>=1, otherwise->=0。
Further, the constraint condition includes a length constraint and a width constraint, wherein the length constraint is:
the width constraint is:
wherein,number representing rectangular component>=1, 2, …, N is the total number of matrix elements, +.>Indicating number->Length of rectangular component>Indicating number->L represents the length of the sheet material, W represents the width of the sheet material,indicating number->Moment of (2)The coefficient of the shape component is 0 or 1 variable, wherein, if the number +.>The rectangular components of (2) are arranged on the sheet material which is currently being subjected to the typesetting>=1, otherwise->=0。
Further, the method for determining the optimal layout scheme of the rectangular components on each plate comprises the following steps:
s1: judging whether the length of the current plate is smaller than the length of the smallest numbered rectangular component in the current remaining non-typesetting rectangular components, if not, taking the left upper corner of the current plate as a typesetting starting point, taking the length of the current smallest numbered rectangular component as a blanking boundary, determining a first strip where the current smallest numbered rectangular component is located along the length direction of the current plate, and entering a step S3; if the number is smaller than the preset number, the step S2 is carried out;
s2: re-selecting the rectangular components with the next number according to the sequence from the small number to the large number, and re-executing the step S1 until the length of the rectangular component with the current selected number is not more than the length of the current plate or the rectangular component with the maximum number is selected;
s3: judging whether the width of the residual plates except the plates occupied by the rectangular assembly with the current selected number in the first strip is not smaller than the width of the rectangular assembly with the next number with the current selected number, if so, determining a first stack where the rectangular assembly with the next number is located along the width direction of the residual plates, and entering a step S5; if not, entering a step S4;
s4: re-selecting the rectangular assembly with the next number according to the sequence from the small number to the large number, and re-executing the step S3 until the width of the rectangular assembly with the currently selected number is not more than the width of the current residual plate or the rectangular assembly with the largest number is selected;
s5: re-executing the step S3 by taking the residual plates with the first stack removed as new residual plates until the width of the rectangular assembly with the number selected currently is not greater than the width of the current residual plate or the rectangular assembly with the maximum number is selected, and then entering the step S6;
s6: and (5) re-executing the steps S1-S5 by taking the residual plate with the first strip cut out as a new current plate until the length of the rectangular assembly with the number selected currently is not greater than the length of the current plate or the rectangular assembly with the maximum number is selected.
Further, the rectangular assembly is arranged on the plate material in a manner that the plate material is arranged and fed into the rectangular assembly in three stages, wherein the plate material is cut into strips in the first stage, the strips are cut into stacks in the second stage, and the stacks are cut into the rectangular assemblies in the third stage.
Further, after the optimal layout scheme of the rectangular assemblies on the current plate is obtained, the next plate which is not subjected to layout is selected again, and the steps S1-S6 are executed again until the layout of all the rectangular assemblies is completed.
The beneficial effects are that:
1. the invention provides a stock layout optimization method based on mixed linear programming, which adopts mixed linear programming to construct a mathematical model, and sorts rectangular components to be produced from large to small in length before cutting, sorts rectangular components with the same length from large to small in width, so as to complete data preprocessing, fully excavate the characteristics of data, facilitate understanding and realization, have low calculation cost and high speed, can be realized by personnel without advanced training, thereby remarkably saving expenditure and time, and is applied to the practical application of enterprises; that is, the invention optimizes the layout under the condition of comprehensively considering the length utilization rate and the width utilization rate, realizes the layout optimization of a large number of rectangular assemblies with various specifications, and improves the layout utilization rate and the layout efficiency in practical application.
2. The invention provides a layout optimization method based on hybrid linear programming, which can determine the value of the coefficient of each rectangular assembly relative to each plate according to the length comparison and the width comparison between the current rectangular assembly and the current plate, further determine the length utilization rate and the width utilization rate of each plate, the overall utilization rate of each plate and the average utilization rate of all plates, and provide theoretical verification and guidance for the whole layout optimization process.
3. The invention provides a layout optimization method based on hybrid linear programming, which adopts a three-stage layout mode, wherein a plate is cut into strips in a first stage, the strips are cut into stacks in a second stage, the stacks are cut into rectangular components in a third stage, and the width of a cutting gap is not considered, so that a large number of rectangular component products with various specifications can be cut into the plates with the same specification.
Drawings
FIG. 1 is a flow chart of a stock layout optimization method based on hybrid linear programming provided by the invention;
FIG. 2 is a schematic diagram of a blanking method based on a three-stage accurate layout mode;
FIG. 3 is a schematic illustration of an exemplary sheet 1 layout scheme provided by the present invention;
FIG. 4 is a schematic illustration of an exemplary sheet 15 layout scheme provided by the present invention;
FIG. 5 is a schematic illustration of an exemplary sheet 30 layout scheme provided by the present invention;
FIG. 6 is a schematic view of an exemplary sheet 80 layout scheme provided by the present invention;
Detailed Description
In order to enable those skilled in the art to better understand the present application, the following description will make clear and complete descriptions of the technical solutions in the embodiments of the present application with reference to the accompanying drawings in the embodiments of the present application.
The invention provides a layout optimization method based on hybrid linear programming, which comprises the steps of firstly preprocessing data of a produced rectangular assembly and realizing mining of data characteristics of the rectangular assembly. The method is characterized in that a mathematical model with the maximum length utilization rate and the maximum width utilization rate is established for the length and the width based on mixed linear programming respectively, the maximum length utilization rate and the maximum width utilization rate are achieved, and a model solving method is established.
Specifically, the layout optimization method based on the hybrid linear programming comprises the following steps:
sequencing the lengths of rectangular assemblies which are required to be sequentially cut from plates with the same size from large to small, sequencing the rectangular assemblies with the same length from large to small, and sequentially numbering the rectangular assemblies according to the sequencing; for example, for a stock sheet of sheet material having a length of 2440mm and a width of 1220mm, it is required to cut 752 rectangular modules, as shown in table 1, given the number of partial rectangular modules ordered by length and width;
table 1 specification of a part rectangular assembly
Sequentially taking the maximum length utilization rate and the maximum width utilization rate of each plate as objective functions, and taking constraint conditions that the sum of the lengths of rectangular assemblies of the discharge samples on each plate is not more than the length of the plate and the sum of the widths of the rectangular assemblies of the discharge samples on each plate is not more than the width of the plate as well as sequentially obtaining an optimal discharge scheme of the rectangular assemblies on each plate, wherein when determining the optimal discharge scheme of the rectangular assemblies on each plate which does not complete the discharge samples, sequentially judging whether the current remaining non-discharge rectangular assemblies are arranged on the plate which is currently undergoing the discharge according to the sequence from the small number to the large number.
Further, the objective function is:
length utilization of sheet materialMaximum:
utilization of the width of the sheet materialMaximum:
wherein,number representing rectangular component>=1, 2, …, N is the total number of matrix elements, +.>Indicating number->Length of rectangular component>Indicating number->L represents the length of the sheet material, W represents the width of the sheet material,indicating number->The coefficients of the rectangular components of (2) are 0 or 1 variables, where, if numbered +.>The rectangular components of (2) are arranged on the sheet material which is currently being subjected to the typesetting>=1, otherwise->=0。
The constraint condition comprises a length constraint and a width constraint, wherein the length constraint is as follows:
the width constraint is:
further, as shown in fig. 1, the method for determining the optimal layout scheme of the rectangular components on each plate is as follows:
s1: judging whether the length of the current plate is smaller than the length of the smallest numbered rectangular component in the current remaining non-typesetting rectangular components, if not, taking the left upper corner of the current plate as a typesetting starting point, taking the length of the current smallest numbered rectangular component as a blanking boundary, determining a first strip where the current smallest numbered rectangular component is located along the length direction of the current plate, and entering a step S3; if the number is smaller than the preset number, the step S2 is carried out;
it should be noted that, for the current plate and the current rectangular component, if the current rectangular component meets the judgment condition in step S1, the current rectangular componentiWill be arranged on the board in the current stock layout and the current rectangular componentiCorresponding coefficients=1, otherwise current rectangular componentiCorresponding coefficient->=0, the determination process is formulated as follows:
s2: re-selecting the rectangular components with the next number according to the sequence from the small number to the large number, and re-executing the step S1 until the length of the rectangular component with the current selected number is not more than the length of the current plate or the rectangular component with the maximum number is selected;
s3: judging whether the width of the residual plates except the plates occupied by the rectangular assembly with the current selected number in the first strip is not smaller than the width of the rectangular assembly with the next number with the current selected number, if so, determining a first stack where the rectangular assembly with the next number is located along the width direction of the residual plates, and entering a step S5; if not, entering a step S4;
it should be noted that, for the current plate and the current rectangular component, if the current rectangular component meets the judgment condition in step S3, the current rectangular component will be arranged on the plate currently being subjected to the layout, and the number of the current rectangular component is assumed to be n+1, and the coefficient corresponding to the current rectangular component n+1 is assumed to be n+1Otherwise the coefficient corresponding to the current rectangular component n+1 +.>=0, the determination process is formulated as follows:
wherein,=1, 2, …, n, n being the total number of matrix elements that have been determined to be sampled to the current sheet, and n being the number of the most recent matrix element that has been determined to be sampled to the current sheet;
s4: re-selecting the rectangular assembly with the next number according to the sequence from the small number to the large number, and re-executing the step S3 until the width of the rectangular assembly with the currently selected number is not more than the width of the current residual plate or the rectangular assembly with the largest number is selected;
s5: re-executing the step S3 by taking the residual plates with the first stack removed as new residual plates until the width of the rectangular assembly with the number selected currently is not greater than the width of the current residual plate or the rectangular assembly with the maximum number is selected, and then entering the step S6;
s6: and (5) re-executing the steps S1-S5 by taking the residual plate with the first strip cut out as a new current plate until the length of the rectangular assembly with the number selected currently is not greater than the length of the current plate or the rectangular assembly with the maximum number is selected.
Further, after the optimal layout scheme of the rectangular assemblies on the current plate is obtained, the next plate which is not subjected to layout is selected again, and the steps S1-S6 are executed again until the layout of all the rectangular assemblies is completed.
It should be noted that the three-stage layout mode is adopted in the layout mode of the invention, and the plate layout is fed into a rectangular component in three stages; as shown in fig. 2, the sheet material is cut into strips in a first stage, the strips are cut into stacks in a second stage, and the stacks are cut into rectangular component products in a third stage; the three-stage layout mode is divided into a three-stage inaccurate layout mode, a three-stage homogeneous layout mode and a three-stage homogeneous layout mode, wherein the two latter layout modes can cut square components with accurate specifications, so that the two latter layout modes belong to the three-stage accurate layout modes.
That is, the invention adopts a three-stage pattern layout mode, does not consider the width of a cutting gap, and is particularly suitable for application scenes in which a large number of rectangular component products with various specifications are required to be cut from plates with the same specification.
After finishing the layout of all the rectangular components on each plate, for the process of obtaining the optimal layout scheme of the first plate, the process of obtaining the optimal layout scheme of the subsequent plate, and the process of obtaining the optimal layout scheme of the subsequent plate, wherein the optimal layout scheme of the subsequent plate only traverses the remaining rectangular components without layout, the process of determining the coefficients corresponding to the rectangular components for the remaining plates except the first plateWhen the coefficient corresponding to the rectangular component of the plate which has been discharged before is directly assigned to 0, the coefficient is not required to be determined through condition judgment.
Based on the method, after the optimal layout scheme of the rectangular assemblies corresponding to each plate is obtained, the coefficients of the rectangular assemblies relative to each plate are determined accordinglyIs taken from (a)The value can further determine the length utilization rate and the width utilization rate of each plate, average the length utilization rate and the width utilization rate to obtain the overall utilization rate of each plate, and finally average the overall utilization rate of all plates to obtain the average utilization rate of all plates.
For example, for a raw sheet of plate with the length of 2440mm and the width of 1220mm, a 752 rectangular assembly is required to be cut, and the sheet is divided by adopting the method, so that a 775 square assembly with required specification can be cut out by 89 sheets; as shown in fig. 3-6, which show the cutting schemes of the 1 st, 15 th, 30 th and 80 th sheets, table 2 shows the utilization rates of the four sheets and the utilization rates of all sheets, it is obvious that the utilization rate of each sheet reaches a higher level.
Table 2 single sheet utilization and all sheet utilization
In summary, the invention provides a layout optimization method based on hybrid linear programming, which can be summarized as the following steps:
1. data preprocessing is carried out, and the characteristics of the data are fully mined.
2. And respectively establishing a mathematical model aiming at the maximum length utilization rate and the maximum width utilization rate for the length and the width based on the mixed linear programming.
3. And designing constraint conditions of the model, and constructing a model solving method.
Therefore, the invention realizes the layout optimization of a large number of rectangular assemblies with various specifications under the condition of comprehensively considering the length utilization rate and the width utilization rate, and improves the layout utilization rate and the efficiency in practical application.
Of course, the present invention is capable of other various embodiments and its several details are capable of modification and variation in light of the present invention by one skilled in the art without departing from the spirit and scope of the invention as defined in the appended claims.
Claims (3)
1. The stock layout optimization method based on the hybrid linear programming is characterized by comprising the following steps of:
sequencing the lengths of rectangular assemblies which are required to be sequentially cut from plates with the same size from large to small, sequencing the rectangular assemblies with the same length from large to small, and sequentially numbering the rectangular assemblies according to the sequencing;
sequentially taking the maximum length utilization rate and the maximum width utilization rate of each plate as objective functions, and taking constraint conditions that the sum of the lengths of rectangular assemblies of the discharge samples on each plate is not more than the length of the plate and the sum of the widths of the rectangular assemblies of the discharge samples on each plate is not more than the width of the plate as discharging, sequentially obtaining an optimal discharge scheme of the rectangular assemblies on each plate, wherein when determining the optimal discharge scheme of the rectangular assemblies on each plate which does not complete the discharge, sequentially judging whether the current remaining non-discharge rectangular assemblies are arranged on the plate which is currently undergoing the discharge according to the sequence from the small number to the large number;
the objective function is:
length utilization of sheet materialMaximum:
utilization of the width of the sheet materialMaximum:
wherein,number representing rectangular component>=1, 2, …, N is the total number of rectangular components, +.>Indicating number->Length of rectangular component>Indicating number->L represents the length of the sheet material, W represents the width of the sheet material, +.>Indicating number->The coefficients of the rectangular components of (2) are 0 or 1 variables, where, if numbered +.>The rectangular components of (2) are arranged on the sheet material which is currently being subjected to the typesetting>=1, otherwise->=0；
The method for determining the optimal layout scheme of the rectangular components on each plate comprises the following steps:
s1: judging whether the length of the current plate is smaller than the length of the smallest numbered rectangular component in the current remaining non-typesetting rectangular components, if not, taking the left upper corner of the current plate as a typesetting starting point, taking the length of the current smallest numbered rectangular component as a blanking boundary, determining a first strip where the current smallest numbered rectangular component is located along the length direction of the current plate, and entering a step S3; if the number is smaller than the preset number, the step S2 is carried out;
s2: re-selecting the rectangular components with the next number according to the sequence from the small number to the large number, and re-executing the step S1 until the length of the rectangular component with the current selected number is not more than the length of the current plate or the rectangular component with the maximum number is selected;
s3: judging whether the width of the residual plates except the plates occupied by the rectangular assembly with the current selected number in the first strip is not smaller than the width of the rectangular assembly with the next number with the current selected number, if so, determining a first stack where the rectangular assembly with the next number is located along the width direction of the residual plates, and entering a step S5; if not, entering a step S4;
s4: re-selecting the rectangular assembly with the next number according to the sequence from the small number to the large number, and re-executing the step S3 until the width of the rectangular assembly with the currently selected number is not more than the width of the current residual plate or the rectangular assembly with the largest number is selected;
s5: re-executing the step S3 by taking the residual plates with the first stack removed as new residual plates until the width of the rectangular assembly with the number selected currently is not greater than the width of the current residual plate or the rectangular assembly with the maximum number is selected, and then entering the step S6;
s6: and (5) re-executing the steps S1-S5 by taking the residual plate with the first strip cut out as a new current plate until the length of the rectangular assembly with the number selected currently is not greater than the length of the current plate or the rectangular assembly with the maximum number is selected.
2. The stock layout optimization method based on hybrid linear programming as claimed in claim 1, wherein the constraint condition includes a length constraint and a width constraint, wherein the length constraint is:
the width constraint is:
wherein,number representing rectangular component>=1, 2, …, N is the total number of matrix elements, +.>Indicating number->Length of rectangular component>Indicating number->L represents the length of the sheet material, W represents the width of the sheet material, +.>Indicating number->The coefficients of the rectangular components of (2) are 0 or 1 variables, where, if numbered +.>The rectangular components of (2) are arranged on the sheet material which is currently being subjected to the typesetting>=1, otherwise->=0。
3. The hybrid linear programming based layout optimization method of claim 1, wherein the layout of the rectangular components on the sheet material is to blanking the sheet material layout into rectangular components in three stages, wherein the sheet material is cut into strips in a first stage, the strips are cut into stacks in a second stage, and the stacks are cut into rectangular components in a third stage.
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| 基于两阶段分块式长板矩形优化排样;姜永亮;张亚敏;;锻压技术(第01期);全文 * |
| 矩形件优化排样算法研究;郭文文 等;《现代制造工程》;第2-3部分 * |
| 面向零件的人造板材矩形件锯切排样数学建模及遗传算法求解;张国梁;蔡小娜;侯晓鹏;赵旦;周玉成;葛浙东;;林业科学(第05期);全文 * |
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