Regional area utilization rate optimization method

Abstract

The invention discloses a region area utilization rate optimization method, which belongs to the technical field of space planning, is used for planning and utilizing a fixed region area, and comprises the steps of constructing a target region grid map and configuring an auxiliary storage region, constructing a to-be-stored article set, setting an article rotation angle set, and constructing a feature identification grid group. Defining a feature identification grid variable, a feature identification grid center grid, a position grid variable and an occupation grid variable, constructing a position grid matrix and an occupation grid matrix, and constructing a target function; and constructing a constraint condition, establishing an optimization problem of the target function under the constraint condition, and solving to obtain an optimal layout of the to-be-stored articles. According to the method, the space utilization rate problem is converted into the nonlinear constraint optimization problem, and the area utilization rate maximization scheme is directly obtained by solving the nonlinear constraint optimization problem.

Description

Technical Field

[0001] This invention discloses a method for optimizing regional area utilization, belonging to the field of spatial planning technology. Background Technology

[0002] Against the backdrop of increasingly prominent resource constraints and continuously rising operating and warehousing costs, achieving optimal space utilization has become a core strategic requirement in modern industrial manufacturing, logistics warehousing, and container shipping. Two-dimensional area space optimization technology, as a key direction for addressing this requirement, aims to maximize storage capacity, operational efficiency, and resource value within limited physical space through systematic planning and intelligent algorithms. Its core evaluation indicator is the improvement of area utilization.

[0003] In the field of computational science, this technical problem can be formalized as a classic two-dimensional nesting (or packing) problem. Its core lies in how to place a series of items of given shapes without overlap within a fixed two-dimensional region, thereby maximizing the utilization of the area. This problem is widely found in various industrial processes, such as metal sheet cutting, textile blanking, container loading layout optimization, and integrated circuit physical design. As the problem size increases, the time required to find the optimal solution grows exponentially. To obtain a feasible solution within an acceptable timeframe, existing technical solutions generally assume that all items to be placed are regular rectangles. Under this assumption, the judgment of interference between items and the calculation of their positions are greatly simplified.

[0004] Although rectangular nesting algorithms have become relatively mature, they have limitations in practical industrial applications. In real-world scenarios, many items to be processed, such as irregularly shaped mechanical parts, non-standard shaped home appliances, and integrated circuit modules customized to meet functional and performance requirements, often have irregular outlines. Current methods commonly employ geometric approximation, replacing each irregular item with its smallest bounding rectangle. However, this approach ignores the true geometric contours and detailed features of the items, resulting in numerous unusable gaps and fragmented spaces in the nesting layout. This significantly reduces the overall area utilization rate compared to the theoretical maximum, leading to increased raw material loss, wasted warehousing and transportation space, and increased production costs. Summary of the Invention

[0005] The purpose of this invention is to provide a method for optimizing the utilization rate of area, in order to solve the problem in the prior art that ignores the real geometric outline and detailed features of the items, resulting in the overall area utilization rate being significantly lower than the theoretical potential maximum value, which in turn leads to an increase in raw material loss rate and waste of warehousing and transportation space, and increases production costs.

[0006] A method for optimizing regional area utilization includes: S1. Rasterize the target area into a target area raster map, construct a set of items to be stored, set a set of item rotation angles, with each item's rotation angle corresponding to a projection shape, construct a feature identifier raster group for each projection shape, using the upper left corner of the outer rectangle of the projection shape as the reference point, and using raster cells of the same size as in the target area raster map to accommodate the projection shapes of all items at any rotation angle; based on the target area raster map and the feature identifier raster group, configure an auxiliary storage area to store items that cannot be placed in the target area; S2. Define feature identifier raster variables, feature identifier raster center cells, position raster variables, and occupancy raster variables. Construct a position raster matrix using the position raster variables, construct an occupancy raster matrix using the occupancy raster variables, and construct an objective function based on the occupancy raster variables. S3. Construct constraints, including mutual exclusion constraints for rotational states, constraints for center cell positions, non-overlapping constraints, and binary constraints for raster variables. S4. Rewrite the objective function based on the relationship between the elements in the position grid matrix and the occupied grid matrix and the center grid position constraint, and establish the optimization problem of the rewritten objective function under the constraint condition. Solve the optimization problem under the constraint condition to obtain the maximum value of the objective function under the constraint condition and the corresponding independent variable value. Based on the corresponding independent variable value, obtain the optimal layout of the items to be stored.

[0007] S1 includes, S1.1, rasterizing the target area into... Target area raster map; S1 includes S1.2, constructing a set of items to be stored. , , Let the total number of items be denoted as . For item indexing, Set the set of item rotation angles ,correspond Let a projection shape be... For rotation angle index, , Each item corresponds to A projected shape, No rotation occurred. to For different rotation angles.

[0008] S1 includes S1.3, using the top-left corner of the bounding rectangle of the projected shape as the reference point, and constructing grid cells of the same size as those in the target area grid map for each projected shape. OK The column's characteristic identifier is the grid group. It is an odd number greater than or equal to 3, and ; The feature grid group's coverage area is the smallest square that can accommodate the projected shape of all items at any rotation angle. For row index, , For column indexes, ; S1 includes S1.4, which involves expanding the target area raster map to the right based on the target area raster map and the feature identifier raster group. Columns, then expand downwards Okay, eventually we get Extended raster map, , , For row index, , For column indexes, The front of the expanded raster map The area that does not overlap with the target area grid map is used as an auxiliary storage area. The auxiliary storage area is used to store items that cannot be placed in the target area grid map.

[0009] S2 includes, S2.1, defining feature identifier raster variables, including setting the first... The first item In a grid group with a rotation angle feature identifier, the grid The corresponding feature identifier raster variable is , 0 indicates that the grid cell is not occupied, and 1 indicates that the grid cell is occupied; when and satisfy At that time, grid Centered on the grid.

[0010] S2 includes, S2.2, and the definition of the first. The first item In a raster map of a target area with various rotation angles, the raster... The position raster variable is , 0 represents a grid cell that is not the center cell, and 1 represents a grid cell that is the center cell. Definition of the first The first item In a raster map of a target area with various rotation angles, the raster... The occupied raster variable is , 0 indicates that the grid cell is not occupied, and 1 indicates that the grid cell is occupied; S2 includes, S2.3, and the construction of the first... The first item Position grid matrix of rotation angle : ; Construct the first The first item Occupied grid matrix of rotation angle : .

[0011] S2 includes S2.4, defining the objective function. : ; when At its maximum, the target area has the highest area occupancy rate.

[0012] S3 includes, S3.1, constructing mutually exclusive constraints for rotational states, where the sum of the position grid variables corresponding to all rotation angles of an item is 1: ; S3 includes S3.2, constructing the central grid position constraints, including setting... For the first The item When all projected shapes with various rotation angles are placed on the target area raster map, the maximum position value of the corresponding feature identifier raster center cell in the row direction is... For the first The item When all the projected shapes of various rotation angles are placed on the target area grid map, the maximum position value of the corresponding feature identifier grid center cell in the column direction; In the target area raster map, the location raster variable corresponding to a raster that cannot be used as a center cell has a value of 0, and the sum of the location raster variables corresponding to all raster cells that can be occupied by a center cell does not exceed 1. ; ; .

[0013] S3 includes S3.3, constructing non-overlapping constraints, including setting the matrix. The sum of the occupied grid matrices corresponding to all rotation angles of all items: ; In the corresponding target area raster map, the raster value does not exceed 1: ; In the formula, for The corresponding target area grid map grid The value; S3 includes S3.4, constructing binary constraints for raster variables. The value can be 0 or 1: .

[0014] S4 includes, S4.1, and The elements in The element representation in the code, combined with the central grid position constraint, is rewritten. : ; In the formula, For the rewritten objective function, For the first The number of grid cells occupied in the feature identifier grid corresponding to each item; Establish The optimization problem under constraints includes mutual exclusion constraints for rotational states, central grid position constraints, non-overlapping constraints, and binary constraints for grid variables.

[0015] S4 includes S4.2, and the solution. The optimization problem under constraints yields the following... Maximum value under constraints The value of the maximum value corresponds to the value of the independent variable, which includes the position of the center cell of the feature identifier grid and the rotation angle of the item, thus obtaining the optimal layout of the items to be stored.

[0016] Compared with the prior art, the present invention has the following advantages: The present invention transforms the space utilization problem into a nonlinear constraint optimization problem, solves the nonlinear constraint optimization problem, and directly obtains the scheme for maximizing area utilization; at the same time, compared with the traditional two-dimensional layout problem, the present invention can more conveniently handle the placement problem of irregularly shaped items and maximize the area utilization. Attached Figure Description

[0017] Figure 1 This is a schematic diagram of the vertical projection shape of the item to be stored. Figure 2 This is a schematic diagram of the feature grid markings on the bottom surface of the object when the L-shaped projection is rotated 0°. Figure 3 This is a schematic diagram of the feature grid markings on the bottom surface of the object when the L-shaped projection is rotated 90°; Figure 4 This is a schematic diagram of the feature grid on the bottom surface of the object when the L-shaped projection is rotated 180°; Figure 5This is a schematic diagram of the feature grid on the bottom surface of the object when the L-shaped projection is rotated 270°; Figure 6 This is a schematic diagram of the L-shaped object projection feature grid. Figure 7 This is a schematic diagram illustrating the assignment of L-shaped item feature identifier grid variables; Figure 8 This is a schematic diagram of an extended raster map; Figure 9 This is a schematic diagram of the optimal layout for storing items. Detailed Implementation

[0018] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention are described clearly and completely below. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0019] A method for optimizing regional area utilization includes: S1. Rasterize the target area into a target area raster map, construct a set of items to be stored, set a set of item rotation angles, with each item's rotation angle corresponding to a projection shape, construct a feature identifier raster group for each projection shape, using the upper left corner of the outer rectangle of the projection shape as the reference point, and using raster cells of the same size as in the target area raster map to accommodate the projection shapes of all items at any rotation angle; based on the target area raster map and the feature identifier raster group, configure an auxiliary storage area to store items that cannot be placed in the target area; S2. Define feature identifier raster variables, feature identifier raster center cells, position raster variables, and occupancy raster variables. Construct a position raster matrix using the position raster variables, construct an occupancy raster matrix using the occupancy raster variables, and construct an objective function based on the occupancy raster variables. S3. Construct constraints, including mutual exclusion constraints for rotational states, constraints for center cell positions, non-overlapping constraints, and binary constraints for raster variables. S4. Rewrite the objective function based on the relationship between the elements in the position grid matrix and the occupied grid matrix and the center grid position constraint, and establish the optimization problem of the rewritten objective function under the constraint condition. Solve the optimization problem under the constraint condition to obtain the maximum value of the objective function under the constraint condition and the corresponding independent variable value. Based on the corresponding independent variable value, obtain the optimal layout of the items to be stored.

[0020] S1 includes, S1.1, rasterizing the target area into... Target area raster map; S1 includes S1.2, constructing a set of items to be stored. , , Let the total number of items be denoted as . For item indexing, Set the set of item rotation angles ,correspond Let a projection shape be... For rotation angle index, , Each item corresponds to A projected shape, No rotation occurred. to For different rotation angles.

[0021] S1 includes S1.3, using the top-left corner of the bounding rectangle of the projected shape as the reference point, and constructing grid cells of the same size as those in the target area grid map for each projected shape. OK The column's characteristic identifier is the grid group. It is an odd number greater than or equal to 3, and ; The feature grid group's coverage area is the smallest square that can accommodate the projected shape of all items at any rotation angle. For row index, , For column indexes, ; S1 includes S1.4, which involves expanding the target area raster map to the right based on the target area raster map and the feature identifier raster group. Columns, then expand downwards Okay, eventually we get Extended raster map, , , For row index, , For column indexes, The front of the expanded raster map The area that does not overlap with the target area grid map is used as an auxiliary storage area. The auxiliary storage area is used to store items that cannot be placed in the target area grid map.

[0022] S2 includes, S2.1, defining feature identifier raster variables, including setting the first... The first item In a grid group with a rotation angle feature identifier, the grid The corresponding feature identifier raster variable is , 0 indicates that the grid cell is not occupied, and 1 indicates that the grid cell is occupied; when and satisfy At that time, grid Centered on the grid.

[0023] S2 includes, S2.2, and the definition of the first. The first item In a raster map of a target area with various rotation angles, the raster... The position raster variable is , 0 represents a grid cell that is not the center cell, and 1 represents a grid cell that is the center cell. Definition of the first The first item In a raster map of a target area with various rotation angles, the raster... The occupied raster variable is , 0 indicates that the grid cell is not occupied, and 1 indicates that the grid cell is occupied; S2 includes, S2.3, and the construction of the first... The first item Position grid matrix of rotation angle : ; Construct the first The first item Occupied grid matrix of rotation angle : .

[0024] S2 includes S2.4, defining the objective function. : ; when At its maximum, the target area has the highest area occupancy rate.

[0025] S3 includes, S3.1, constructing mutually exclusive constraints for rotational states, where the sum of the position grid variables corresponding to all rotation angles of an item is 1: ; S3 includes S3.2, constructing the central grid position constraints, including setting... For the first The item When all projected shapes with various rotation angles are placed on the target area raster map, the maximum position value of the corresponding feature identifier raster center cell in the row direction is... For the first The item When all the projected shapes of various rotation angles are placed on the target area grid map, the maximum position value of the corresponding feature identifier grid center cell in the column direction; In the target area raster map, the location raster variable corresponding to a raster that cannot be used as a center cell has a value of 0, and the sum of the location raster variables corresponding to all raster cells that can be occupied by a center cell does not exceed 1. ; ; .

[0026] S3 includes S3.3, constructing non-overlapping constraints, including setting the matrix. The sum of the occupied grid matrices corresponding to all rotation angles of all items: ; In the corresponding target area raster map, the raster value does not exceed 1: ; In the formula, for The corresponding target area grid map grid The value; S3 includes S3.4, constructing binary constraints for raster variables. The value can be 0 or 1: .

[0027] S4 includes, S4.1, and The elements in The element representation in the code, combined with the central grid position constraint, is rewritten. : ; In the formula, For the rewritten objective function, For the first The number of grid cells occupied in the feature identifier grid corresponding to each item; Establish The optimization problem under constraints includes mutual exclusion constraints for rotational states, central grid position constraints, non-overlapping constraints, and binary constraints for grid variables.

[0028] S4 includes S4.2, and the solution. The optimization problem under constraints yields the following... Maximum value under constraints The value of the maximum value corresponds to the value of the independent variable, which includes the position of the center cell of the feature identifier grid and the rotation angle of the item, thus obtaining the optimal layout of the items to be stored.

[0029] The following description, in conjunction with the accompanying drawings and embodiments, provides further details. Figure 1 As shown, assuming the vertical projection shapes of the items in the set encompass both regular and irregular shapes, this embodiment of the invention uses L-shaped, T-shaped, Z-shaped, rectangular, and square items as examples, setting up 7 items, among which... Two shaped items, Two shaped items, One rectangular item, one long rectangular item, and one square item are included. Since the items can be rotated according to the actual situation, this invention considers clockwise rotation at fixed angles of 0 degrees, 90 degrees, 180 degrees, and 270 degrees. Each item corresponds to four projected shapes. A 0-degree rotation represents no rotation and is defined as the first rotation angle. 90-degree, 180-degree, and 270-degree rotations are the second, third, and fourth rotation angles, respectively. Taking an item with an L-shaped vertical projection as an example, its four projected shapes are as follows: Figure 2 , Figure 3 , Figure 4 and Figure 5 As shown. The collection of items... Each item corresponds to A projected shape.

[0030] Using the top left corner of the bounding rectangle of the projected shape as the reference starting point, for Each projection shape is constructed independently. Feature identifier grid group, the grid group's coverage area is the smallest square that can accommodate the projected shape of all items at any rotation angle, such as Figure 6 As shown, the shaded area represents the shape of the object's projection. , .

[0031] Define a feature identifier raster variable to characterize the occupancy status of the raster cell, such as... Figure 7 As shown, Feature identifier grid in grid group This is called the central lattice.

[0032] Target area rasterized To facilitate the management of items that cannot be placed in the target area, this method configures an auxiliary storage area for centrally storing such items. Specifically, the area grid map is first expanded three columns to the right, then two rows downwards, resulting in the final... raster map, For row index, , For column indexes, .like Figure 8 As shown, the white area is the target area, and the gray area on the right is the auxiliary storage area.

[0033] Define region grid variables. Each rotation angle of each item corresponds to two sets of region grid variables: position grid variable and occupancy grid variable. The position grid variable is used to reflect the position of the center grid, and the occupancy grid variable is used to reflect the item's occupancy of the region grid.

[0034] Construct the location raster matrix and the occupancy raster matrix: ; ; Define the objective function To maximize the area occupancy of the target region, it is necessary to maximize the area grid occupancy, which is the sum of all occupied grid variables in the target region. Reaching the maximum value: .

[0035] To maximize the utilization of the area, the following four constraints must be met. The first constraint is mutual exclusion of rotation states: each item can only choose one of the four rotation angles; that is, the four rotation angles cannot occur simultaneously. Therefore, the sum of the position grid variables corresponding to the four rotation angles of an item must be 1. The specific constraint is as follows: .

[0036] The center grid position constraint, to ensure that the item falls completely into the target area or auxiliary storage area, can be achieved by limiting the position of the center grid within the area grid. The values ​​of p and q depend on the values ​​of the variables in the feature identifier grid. Specifically, if the variables in the third row of the feature identifier grid contain non-zero elements, then... Otherwise, if the second row of the feature identifier raster contains a non-zero element, then ;otherwise, If the third column of the feature identifier grid contains a non-zero element, then Otherwise, if the second column of the feature identifier grid contains a non-zero element, then ;otherwise, The sum of the position raster variables corresponding to the raster cells that can be occupied by the center cell in a region raster does not exceed 1, while the position raster variables corresponding to the region raster cells that cannot be occupied by the center cell are directly set to 0. The specific constraints are as follows: ; ; .

[0037] The non-overlapping constraint ensures that items are placed without overlapping; each grid cell within the target area can be occupied at most once. That is, the sum of the grid matrices corresponding to all rotation angles of all items, the matrix All elements within the target region must not exceed 1. Specific constraints include: ; The occupied grid matrix The values ​​of the elements are related to the position raster variable and the feature identifier raster variable: ; ; ; ; ; ; ; ; .

[0038] Position raster variable The value can be either 0 or 1, that is: .

[0039] Since the sum of the raster variables occupied by each item is consistent with the sum of the raster variables of the item's feature identifier, the objective function can be further rewritten under the constraint of the center grid position as follows: ; Establish Optimization problems under constraints.

[0040] Solve using the interior point method The optimization problem under constraints can be obtained as follows: The maximum value under constraints and the corresponding value of the independent variable when the objective function reaches its maximum value. This independent variable can provide the position of the center cell of the item feature grid, and can also obtain the item's rotation angle information, thus obtaining the optimal layout of the items to be stored, such as... Figure 9 As shown.

[0041] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for optimizing regional area utilization, characterized in that, include: S1. Rasterize the target area into a target area raster map, construct a set of items to be stored, set a set of item rotation angles, with each item's rotation angle corresponding to a projection shape, construct a feature identifier raster group for each projection shape, using the upper left corner of the outer rectangle of the projection shape as the reference point, and using raster cells of the same size as in the target area raster map to accommodate the projection shapes of all items at any rotation angle; based on the target area raster map and the feature identifier raster group, configure an auxiliary storage area to store items that cannot be placed in the target area; S2. Define feature identifier raster variables, feature identifier raster center cells, position raster variables, and occupancy raster variables. Construct a position raster matrix using the position raster variables, construct an occupancy raster matrix using the occupancy raster variables, and construct an objective function based on the occupancy raster variables. S3. Construct constraints, including mutual exclusion constraints for rotational states, constraints for center cell positions, non-overlapping constraints, and binary constraints for raster variables. S4. Rewrite the objective function based on the relationship between the elements in the position grid matrix and the occupied grid matrix and the center grid position constraint, and establish the optimization problem of the rewritten objective function under the constraint condition. Solve the optimization problem under the constraint condition to obtain the maximum value of the objective function under the constraint condition and the corresponding independent variable value. Based on the corresponding independent variable value, obtain the optimal layout of the items to be stored.

2. The method for optimizing regional area utilization according to claim 1, characterized in that, S1 includes, S1.1, rasterizing the target area into... Target area raster map; S1 includes S1.2, constructing a set of items to be stored. , , Let the total number of items be denoted as . For item indexing, Set the set of item rotation angles ,correspond Let a projection shape be... For rotation angle index, , Each item corresponds to A projected shape, No rotation occurred. to For different rotation angles.

3. The method for optimizing regional area utilization according to claim 2, characterized in that, S1 includes S1.3, using the top-left corner of the bounding rectangle of the projected shape as the reference point, and constructing grid cells of the same size as those in the target area grid map for each projected shape. OK The column's characteristic identifier grid group, It is an odd number greater than or equal to 3, and ; The feature grid group's coverage area is the smallest square that can accommodate the projected shape of all items at any rotation angle. For row index, , For column indexes, ; S1 includes S1.4, which involves expanding the target area raster map to the right based on the target area raster map and the feature identifier raster group. Columns, then expand downwards Okay, eventually we get Extended raster map, , , For row index, , For column indexes, The front of the expanded raster map The area that does not overlap with the target area grid map is used as an auxiliary storage area. The auxiliary storage area is used to store items that cannot be placed in the target area grid map.

4. The method for optimizing regional area utilization according to claim 3, characterized in that, S2 includes, S2.1, defining feature identifier raster variables, including setting the first... The first item In a grid group with a rotation angle feature identifier, the grid The corresponding feature identifier raster variable is , 0 indicates that the grid cell is not occupied, and 1 indicates that the grid cell is occupied; when and satisfy At that time, grid Centered on the grid.

5. The method for optimizing regional area utilization according to claim 4, characterized in that, S2 includes, S2.2, and the definition of the first. The first item In a raster map of a target area with various rotation angles, the raster... The position raster variable is , 0 represents a grid cell that is not the center cell, and 1 represents a grid cell that is the center cell. Definition of the first The first item In a raster map of a target area with various rotation angles, the raster... The occupied raster variable is , 0 indicates that the grid cell is not occupied, and 1 indicates that the grid cell is occupied; S2 includes, S2.3, and the construction of the first... The first item Position grid matrix of rotation angle : ； Construct the first The first item Occupied grid matrix of rotation angle : 。 6. The method for optimizing regional area utilization according to claim 5, characterized in that, S2 includes S2.4, defining the objective function. : ； when At its maximum, the target area has the highest area occupancy rate.

7. The method for optimizing regional area utilization according to claim 6, characterized in that, S3 includes, S3.1, constructing mutually exclusive constraints for rotational states, where the sum of the position grid variables corresponding to all rotation angles of an item is 1: ； S3 includes S3.2, constructing the central grid position constraints, including setting... For the first The item When all projected shapes with various rotation angles are placed on the target area raster map, the maximum position value of the corresponding feature identifier raster center cell in the row direction is... For the first The item When all the projected shapes of various rotation angles are placed on the target area grid map, the maximum position value of the corresponding feature identifier grid center cell in the column direction; In the target area raster map, the location raster variable corresponding to a raster that cannot be used as a center cell has a value of 0, and the sum of the location raster variables corresponding to all raster cells that can be occupied by a center cell does not exceed 1. ； ； 。 8. The method for optimizing regional area utilization according to claim 7, characterized in that, S3 includes S3.3, constructing non-overlapping constraints, including setting the matrix. The sum of the occupied grid matrices corresponding to all rotation angles of all items: ； In the corresponding target area raster map, the raster value does not exceed 1: ； In the formula, for The corresponding target area grid map grid The value; S3 includes S3.4, constructing binary constraints for raster variables. The value can be 0 or 1: 。 9. The method for optimizing regional area utilization according to claim 8, characterized in that, S4 includes, S4.1, and will The elements in The element representation in the code, combined with the central grid position constraint, is rewritten. : ； In the formula, For the rewritten objective function, For the first The number of grid cells occupied in the feature identifier grid corresponding to each item; Establish The optimization problem under constraints includes mutual exclusion constraints for rotational states, central grid position constraints, non-overlapping constraints, and binary constraints for grid variables.

10. The method for optimizing regional area utilization according to claim 9, characterized in that, S4 includes S4.2, and the solution. The optimization problem under constraints yields the following... Maximum value under constraints The value of the maximum value corresponds to the value of the independent variable, which includes the position of the center cell of the feature identifier grid and the rotation angle of the item, thus obtaining the optimal layout of the items to be stored.

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