A heuristic method for optimizing the packing and stowage of cargo

Abstract

The application provides a material container loading optimization method with simple framework, high calculation efficiency and good stability. Aiming at the problem of how to load various and large amount of materials into containers or train and automobile compartments, the utilization rate and loading efficiency of the containers or compartments are effectively improved by optimizing the material container loading sequence and the layout and loading of the materials in the containers or compartments. In the method, the same kind of materials are packed into cuboid materials (blocks), so that the number of materials to be loaded is greatly reduced; the Pareto thought in the multi-objective optimization problem is introduced to determine the container loading batch and sequence of the materials; the material end face database and the available space database are established to improve the matching efficiency of the materials and the available space; and the available space is divided in different ways to reduce the probability of generating narrow and small waste space, which is helpful to improve the container loading efficiency and the utilization rate of the containers or compartments.

Description

Technical Field

[0001] This invention provides a heuristic method for optimizing the packing and loading of materials, applicable to the generation and optimization of packing and loading schemes in the material storage and transportation process, belonging to the field of logistics. Background Technology

[0002] Material loading and unloading, including the use of shipping containers and the loading of goods using trains, trucks, and other transport vehicles, is a crucial aspect of military logistics and modern logistics. Improved utilization of container / carriage space and load capacity translates to lower transportation costs, shorter transit times, and higher resource efficiency. During material loading and unloading, materials of different specifications and quantities are sequentially loaded into containers / carriages with specific volume and load capacity limitations. The loading sequence, placement within the container space, and arrangement of the materials significantly impact the space and load capacity utilization of the containers / carriages.

[0003] Currently, cargo loading and unloading operations in my country mainly rely on manual labor or manual operation of machinery. The loading sequence, layout, and placement of goods within containers / cargo compartments heavily depend on the experience of the operators, often resulting in low container / cargo compartment load-bearing capacity and space utilization. Statistics show that the utilization rate of a standard land and sea transport container is generally 60%-80%. The total mass and volume of goods inside a container are far less than the container's rated load capacity and volume, leading to low container utilization and reduced cargo transportation efficiency. Optimizing the loading sequence, placement, and arrangement of goods within containers / cargo compartments based on their geometric dimensions, mass, quantity, and loading constraints before actual loading, thereby increasing the utilization rate of individual containers / cargo compartments and reducing the number of containers / cargo compartments required to load a certain quantity of goods, is a crucial issue for reducing logistics costs, shortening cargo transportation cycles, and improving overall cargo transportation efficiency. This has significant theoretical and practical value.

[0004] The process of packing materials into containers is an NP-hard problem in combinatorial optimization, requiring solutions to various constraints such as geometric constraints, weight constraints, and rotational constraints. The 3D packing problem can be divided into two main categories: input minimization packing problems and output maximization packing problems. The former refers to packing all materials into containers (such as shipping containers) while minimizing the number of containers required; the latter refers to packing a portion of the materials into containers with a fixed size and quantity, maximizing the value (or mass and volume) of the packed materials.

[0005] Research on 3D container loading algorithms began with the earliest purely mathematical exact algorithms. However, the increase in the number of containers and materials leads to a combinatorial space explosion, making it extremely difficult to solve exact algorithms. In recent years, heuristic algorithms, intelligent algorithms, and combinatorial algorithms have become research hotspots both domestically and internationally. Heuristic material loading planning algorithms consist of a series of rules based on manual loading experience, including material selection rules, material stacking rules, space management rules, and center-of-gravity adjustment rules. The calculation results of this algorithm are often approximate optimal solutions to the problem. Artificial intelligence methods consist of models and their solution algorithms. The models of artificial intelligence methods include both commonly used linear optimization models such as linear programming models and nonlinear optimization models such as nonlinear mixed integer programming models. Therefore, optimization algorithms for artificial intelligence methods need to be able to handle nonlinear optimization models. Commonly used artificial intelligence algorithms include genetic algorithms, simulated annealing algorithms, and tabu search algorithms. While this type of algorithm possesses strong global optimization capabilities when solving the 3D packing problem, its convergence and search speed are easily affected by initial conditions. When the problem is highly complex, the length of a single instance can reach thousands of digits, undoubtedly increasing the difficulty of solving the problem. The definition of constraints needs to be reflected in the objective function or fitness function through penalty functions, which directly affects the algorithm's convergence and the quality of the computational results. Combinatorial algorithms combine two or more algorithms to leverage their respective advantages in solving the packing problem. These algorithms include combinations of similar algorithms as well as interactive combinations of dissimilar algorithms. Compared to mathematically exact algorithms, the above optimization algorithms have higher computational efficiency, but the optimization effect is random; the reliability of these methods decreases significantly with the increase in the number of containers / carriages and materials. Since large-scale packing problems are common in practice, the engineering application value and guiding significance of the existing research results of the above methods are limited.

[0006] Considering the high complexity of packing large quantities of various types of goods, this invention utilizes the concept of "material blocks" to package similar materials into blocks with larger mass, volume, and geometric dimensions that closely match a specific end face of the container / carriage. This reduces the total number of materials to be packed and improves the space utilization of the container / carriage. Furthermore, thresholds for the total mass and volume of material blocks are set to prevent excessively high or low block density, which could lead to low space or load-bearing capacity utilization in the container / carriage. To improve the matching speed between materials and available space, a database of material blocks and available space is proposed. The material database includes end face data of the materials, reflecting placement constraints, while the available space database includes multiple partitioning methods for the same available space, avoiding the drawback of a single partitioning method potentially resulting in excessively small, unused subspaces. The length, width, and height of the materials are considered as solutions to a multi-objective optimization problem. The Pareto dominance concept from multi-objective optimization is introduced to group the materials to be packed. Packing batches are determined according to group numbers, and the packing order within each group is determined by comprehensively considering the volume, mass, and end face area of ​​the materials, thus obtaining the packing order for all materials. During the loading of materials into containers / carriages, a two-way matching process between materials and available space, along with database updates, is performed until all materials are loaded. This method reduces the complexity of loading large quantities of various types of materials, has high computational efficiency, and maintains high utilization rates of container / carriage space and load capacity. Summary of the Invention

[0007] The purpose of this invention is to address the shortcomings of existing container packing methods in terms of computational efficiency and stability, and to provide a heuristic optimization method for loading large quantities of various types of materials into rectangular containers / carriages. This method has a simple framework, high computational efficiency, good packing effect, and good stability. It can obtain a reasonable packing scheme in a short time, which helps to improve packing efficiency and container utilization.

[0008] First, define the container packing problem: Given a certain number of containers / carriages (sufficient quantity), their internal dimensions are... L × W × H (Unit: m), Rated load T (Unit: kg), overloading is not allowed; M 0 types of supplies are waiting to be delivered to a certain location. m ( m =1,…, M The quantity of type 0) materials is N 0,m Specifications are L 0,m × W 0,m × H 0,m(Unit: m), mass is T 0,m The permitted placement direction inside the container / carriage is S 0,m ( S 0,m (This refers to the set of orientations for transporting goods); the total cost of transporting goods is measured by the number of containers / carriages required, with the lowest transport cost corresponding to the fewest number of containers / carriages. The present invention adopts the following technical solution: (1) Import data and pre-packing processing: The imported data includes the internal dimensions of the container / carriage, rated load capacity, quantity, geometric dimensions, mass, and permissible placement direction of various materials; based on the total mass and volume of the materials, estimate the number of containers / carriages required. N 0 and the average density of containers / carriages after loading. q 0, respectively Preprocessing of material data before packing includes the following sub-steps: (1.1) Packing the same type of materials into several rectangular material "blocks": to maximize the container / carriage size. L × W The surface utilization rate is the objective function, and the first m The materials were packaged into several rectangular "blocks"; similarly, they were obtained from... L × H Face to face W × H The corresponding packaging schemes are determined; the utilization rates of the container / carriage end faces of the three schemes are compared to determine the first one. m Packaging scheme for this type of material. When the first m The density of this type of material is greater than 2 q At time 0, set the maximum mass constraint for the material block (≤0.5). T ); when the first m The density of this type of material is less than 0.5. q When the value is 0, set the maximum volume constraint for the material block (≤0.5). LWH For ease of description, the packaged materials are defined as follows: M species, first m ( m =1,…, M The quantity of these materials is N m Specifications are L m × W m × H m (Unit: m), mass is T m The permitted placement orientations within the container / carriage are: S m.Will L × W noodle, L × H Face to face W × H The face is collectively referred to as the container / carriage end face. L m × W m noodle, L m × H m Face to face W m × H m The surfaces are collectively referred to as the material end faces.

[0009] (1.2) Based on the geometric dimensions and permissible placement orientation of the materials, establish a material end face database. This database contains three sub-databases, which respectively store materials that can be placed with containers / carriages. L × W noodle, L × H Face to face W × H The data of the parallel end faces of the materials are respectively called L × W Face database L × H Face database and W × H Face database. For example: if a certain material can be placed in any orientation, then the three sub-databases must each contain the three face surfaces of that material; if a certain material's... W m × H m The surface can only be used with containers / carriages W × H If the surfaces are parallel, then the material... W m × H m The face can only be saved to W × H Face database, L × W Face database does not contain W m × H m But contains L m × W m Face to face L m × H m noodle,L × W The same applies to face databases.

[0010] Initialize the available space database. The available space database can only contain one empty container / carriage space (i.e., L × W × H (A rectangular space); when a certain item is placed into this space, the database automatically adds a new item. L × W × H Space, i.e., sufficient number of containers / carriages. (2) Determine the packing order of the materials based on their geometric dimensions (length, width, height), weight, and other parameters.

[0011] (2.1) Based on the length, width, and height of each material, group the materials to be packed and the material blocks, and determine the packing batches of different materials according to the group number from smallest to largest. The first group of materials is used collectively. G 1 indicates, G 1. Determine as follows: If no materials exist. m '( m =1,…, M and m '≠ m Make the following system of inequalities true. Then supplies m Belongs to the material collection G 1. Elimination G After grouping all the materials in step 1, apply the above method to the remaining ungrouped materials to determine the second group of materials. G 2; and so on, until all supplies are grouped and obtained. N Group supplies.

[0012] (2.2) The packing order within each material group is as follows: materials with larger volume are packed first; if the volumes are equal, materials with the largest end face area are packed first; if the volume and the largest end face area are equal, materials with the longest side are packed first; for materials with the same geometric dimensions, materials with greater mass are packed first; for materials with the same geometric dimensions and mass, the packing order is randomly arranged.

[0013] (3) Load the materials one by one into the container / carriage. The specific loading method is as follows: (3.1) Matching available space for materials: Based on the material end face database, materials whose end face size is completely matched with the end face size of a certain available space are loaded into the available space; after the new materials are loaded into the container / carriage, the space for loading materials is divided into the order of "front-right (or left)-up" and "right (or left)-front-up" respectively, and the available spaces are merged. The available space database saves both division results at the same time; update the material end face database.

[0014] (3.2) Repeat step (3.1) until there are no materials that perfectly match the dimensions of the available space end face.

[0015] (3.3) Matching materials with available space: Select the material to be loaded that is ranked first and call it the material to be loaded; load the material to be loaded into the “optimal” available space that can accommodate the material; update the available space database by dividing and merging the space; update the material end face database.

[0016] "Optimal" available space refers to available space that simultaneously meets the following conditions: it can accommodate the materials to be loaded; it can maximize the utilization rate of the end face area; when the end face area utilization rate is the same, select the available space with the larger volume utilization rate. (4) Repeat step (3) until all materials are loaded into the container / carriage.

[0017] The beneficial effects of this invention are as follows: In the problem of packing large quantities of various types of materials, similar materials are packaged into larger rectangular blocks, significantly reducing the number of materials to be packed; by introducing the concept of Pareto dominance from multi-objective optimization problems, the packing order of materials is determined by comprehensively considering factors such as the length, width, height, and mass of the materials to be packed, avoiding packing "small" materials first and improving the utilization rate of containers / carriages; at the same time, the available space is divided using the "front-right (or left)-top" and "right (or left)-front-top" methods, reducing the probability of generating excessively small waste subspaces and improving the utilization rate of containers / carriages; by establishing a material end-face database and an available space database, the matching speed between materials and available space and the computational efficiency of the packing process optimization algorithm are improved. Attached Figure Description

[0018] Figure 1 Flowchart of the material packing and loading process; Figure 2 The optimal packing scheme for Category 1 materials for container W×H, L×H, and L×W sides; Figure 3 A schematic diagram of available space division. Detailed Implementation

[0019] To make the technical solution and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. This example uses the loading of goods into a container as a relevant illustration; the same principle applies to the loading process into train carriages, automobiles, and other vehicles.

[0020] This invention presents a heuristic-based optimization algorithm for cargo loading and distribution, applicable to scenarios involving the loading of various types of large quantities of goods into containers / carriages, such as cargo transportation and distribution. It exhibits high computational efficiency and robustness in practical applications.

[0021] like Figure 1 As shown in the figure, the specific process of loading various types of large quantities of materials into containers / carriages in this embodiment of the invention is as follows: (1) Import data and preprocess it before packing.

[0022] (1.1) The imported data includes the internal dimensions of the container, rated load capacity, quantity, dimensions, and weight of various materials. The imported container and material data are shown in Table 1 and Table 2.

[0023] Table 1

[0024] Table 2 (1.2) Estimated number of containers required N 0 and the average density of the container after loading. q 0. Based on the dimensions, weight, and quantity of all materials, calculate the total mass and volume of all materials as 1.70 × 10⁻⁶. 5 kg, 221.8m 3 Therefore, the required number of containers can be estimated based on formulas (1) and (2). N 0=9 and the average density of the container after loading materials q 0≈670kg / m 3 .

[0025] (1.3) Pack all kinds of materials into several rectangular material "blocks": Calculate the volume and density of each material according to the size and weight of the materials to be packed, as shown in Table 3. To maximize container capacity W × H The objective function is the utilization rate of the material blocks. The volume constraint is that the packaged material blocks can fit into the container, and the mass constraint is that the mass of the material blocks is less than or equal to the rated load capacity of the container. An enumeration method is used to determine the first type of material pairs. W × H The optimal packaging solution for the surface, such as Figure 2 As shown in Table 4; repeating this process, the specific requirements for Category 1 materials can be determined. L × H Face to face L × W The optimal packaging schemes for the three schemes are shown in Table 4. Comparing the three schemes, [the following is a summary of the optimal packaging schemes]. L × H The above solution is the optimal packaging solution for Category 1 materials.

[0026] Table 4

[0027] Similarly, packaging schemes for other types of materials can be determined, as shown in Table 5.

[0028] It should be noted that the density of materials in categories 2 and 4 is less than 0.5. q 0. When optimizing the packaging scheme, set its maximum size to 0.5. LWH The density of materials in categories 9 and 10 is greater than 2. q 0. When determining its packaging plan, the maximum weight is set to 0.5. T .

[0029] (1.4) Establish a materials end-face database. The materials end-face database contains three sub-databases, which store each material separately and can be linked to containers. L × W noodle, L × H Face to face W × H The data of the parallel end faces of the materials are respectively called L × W Face database L × H Face database and W × H Face database.

[0030] (1.5) Initialize the available space database. The available space database is used to store the coordinates of the left-bottom-right corners of each available space and the dimensions of the available space in the length, width, and height directions. The initial available space database contains only one empty container space, and it is stipulated that after an empty container is loaded with goods, the database will automatically add a new empty container space.

[0031] (2) Determine the packing order of the materials based on their geometric dimensions (length, width, height), weight, etc. The data for the materials to be packed after packaging are shown in Table 6. Although the variety of materials increased after packaging, the quantity of materials to be packed decreased significantly. Table 6 To be packed M Among these supplies, when m When taking the values ​​14, 15, 16, 17, 18, 19, 20, 22, 24, they do not exist. m '≠ mFormula (3) holds true, therefore, materials of types 14, 15, 16, 17, 18, 19, 20, 22, and 24 are loaded into the container in the first batch. Then, based on volume, maximum end face area, longest side, and weight priority, the loading order of the first batch of materials is determined as 17, 18, 15, 24, 14, 20, 16, 19, and 22. After removing the first batch of materials, the above steps are repeated to determine the materials loaded into the container in the second, third, fourth, and subsequent batches, and the loading order of each batch, as shown in Table 7.

[0032] (3) Load the blocks of materials to be packed into the container one by one. The loading process is as follows: (3.1) Arrange the available spaces in ascending order of volume, let i =1; (3.2) If the first i The available space has been matched with the material end-face database. i = i +1, repeat step (3.2); otherwise, L × W Search within the Face database and the first i Available space L × W The end faces that are perfectly matched are identified, and the corresponding materials are determined; if one material that can be loaded is found, it is loaded into the first material. i If there is available space, proceed to step (3.5); if multiple materials that can be loaded are matched, the type and quantity of materials to be loaded are determined by enumeration with the goal of maximizing the total volume of loaded materials, and then proceed to step (3.5); otherwise, proceed to step (3.3). (3.3) will L × H Search within the Face database and the first i Available space L × H The end faces that are perfectly matched are identified, and the corresponding materials are determined; if one material that can be loaded is found, it is loaded into the first material. i If there is available space, proceed to step (3.5); if multiple materials that can be loaded are matched, the type and quantity of materials to be loaded are determined by enumeration with the goal of maximizing the total volume of loaded materials, and then proceed to step (3.5); otherwise, proceed to step (3.4). (3.4) will W × H Search within the Face database and the first i Available space W × HThe end faces that are perfectly matched are identified, and the corresponding materials are determined; if one material that can be loaded is found, it is loaded into the first material. i If there is available space, proceed to step (3.5); if multiple items can be loaded, then with the goal of maximizing the total volume of loaded items, use enumeration to determine the type and quantity of loaded items, and proceed to step (3.5); otherwise, i = i +1 and proceed to step (3.6). (3.5) Update the material end-face database: Delete the data corresponding to the newly packed materials in the material end-face database; Update the available space database: according to... Figure 3 The diagram shows the deletion of available space, merging, and partitioning of available space.

[0033] (3.6) If i If the number of available spaces is greater than the number of available spaces in the available space database, proceed to step (3.7); otherwise, proceed to step (3.1). (3.7) Select the material to be loaded at the top of the sort and load it into the available space that meets the following conditions: it can accommodate the material to be loaded; it can maximize the utilization rate of the end face area; if the end face area utilization rate is the same, select the available space with the larger volume utilization rate.

[0034] (3.8) Update the material end face database and available space database according to step (3.5).

[0035] (4) Repeat step (3) until all materials are loaded into the container.

[0036] Finally, it should be noted that the purpose of the above embodiments is only to make the technical solutions of the present invention clearer, and not to limit the present invention. 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 of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A heuristic method for optimizing the packing and loading of supplies, characterized in that, Define the container packing problem as follows: Given a sufficient quantity of one type of container or wagon, whose internal dimensions are... L × W × H , L For the length of the supplies, W For the width of the materials, H For the height of the goods, the rated load T ;have M 0 types of supplies are waiting to be delivered to a certain location. m The quantity of these supplies is N 0,m Specifications are L 0,m × W 0,m × H 0,m The quality is T 0,m ,in m =1,…, M 0; The permissible placement direction inside the container or carriage is S 0,m , S 0,m The set of orientations for operation; the total cost of transporting goods is measured by the number of containers or wagons required, with the lowest transportation cost corresponding to the fewest number of containers / wagons; The specific packing process is as follows: Step 1: Import Data and Pre-packaging: Import data includes the internal dimensions of the container / carriage, rated load capacity, quantity, geometric dimensions, weight, and permissible placement orientation of various materials; based on the total weight and volume of the materials, estimate the required number of containers or carriages. N 0 and the average density of containers or wagons after loading goods q 0, respectively: Preprocessing of material data before packing includes the following sub-steps: a) Pack the same type of materials into several rectangular material blocks: To maximize the use of containers or carriages L × W The surface utilization rate is the objective function, and the first m The materials are packaged into several rectangular blocks; similarly, they are obtained from... L × H Face to face W×H Corresponding packaging solutions; Compare the container / carriage end-face utilization rates of the three schemes to determine the first one. m Packaging scheme for this type of material; when the first m The density of this type of material is greater than 2 q At time 0, set a maximum mass constraint on the material block; the maximum mass of the material block ≤ 0.

5. T When the first m The density of this type of material is less than 0.

5. q When the value is 0, set a maximum volume constraint for the material block; the maximum volume of the material block ≤ 0.

5. LWH ; For ease of description, the packaged materials are defined as follows: M species, first m The quantity of these supplies is N m Specifications are L m × W m × H m The quality is T m The permitted placement orientations within the container / carriage are: S m ， in m =1,…, M ; Will L × W noodle, L × H Face to face W × H The face is collectively referred to as the end face of a container or carriage. L m × W m noodle, L m × H m Face to face W m × H m The surfaces are collectively referred to as the material end faces; b) Establish a database of material end faces based on the geometric dimensions and permissible placement orientation of the materials: The database contains three sub-databases, each storing information related to a container / carriage. L × W noodle, L × H Face to face W × H The data of the parallel end faces of the materials are respectively called L × W Face database L × H Face database and W × H Face database; initialize the available space database, which can only contain one empty container space, i.e. L × W × H A rectangular space; when this space is filled with a certain item, the database automatically adds a new item. L × W × H Space, meaning a sufficient number of containers or carriages; Step 2: Determine the packing order of the materials based on their length, width, height, and weight parameters; Step 3: Load the supplies one by one into the container or truck. The specific loading method is as follows: a) Available space for resource matching: Based on the material end face database, materials whose end face dimensions perfectly match the end face dimensions of a certain available space are loaded into that available space; after new materials are loaded into the container / carriage, the space for loading materials is divided into front, right, top or front, left, top and right, front, top or left, front, top, and available spaces are merged. The available space database stores both division results simultaneously; the material end face database is updated. b) Repeat step a) until there are no materials that perfectly match the dimensions of the available space end face; c) Available space for matching supplies: Select the material that appears first in the sorted list and call it the material to be loaded; load the material into the optimal available space that can accommodate it; update the available space database by dividing and merging the space; update the material end face database. Step 4: Repeat step 3 until all supplies are loaded into containers or trucks.

2. The heuristic method for optimizing material packing and loading as described in claim 1, characterized in that: During the pre-packing stage, materials with identical geometric dimensions are packaged into blocks based on the internal dimensions and rated load of the container or carriage; during optimized packing, each block is treated as a whole.

3. The heuristic method for optimizing material packing and loading as described in claim 1, characterized in that: Introducing the Pareto dominance concept from multi-objective optimization problems, the materials to be packed and the material blocks are grouped according to the length, width, and height of each material. The packing batches of different materials are determined based on the group number from smallest to largest. The first group of materials is used collectively. G 1 indicates, G 1. Determine as follows: If no materials exist. m ' , m =1,…, M and m '≠ m This makes the following system of inequalities hold true. Then supplies m Belongs to the material collection G 1; Remove G After grouping all the materials in step 1, apply the above method to the remaining ungrouped materials to determine the second group of materials. G 2; and so on, until all supplies are grouped and obtained. N Grouped supplies; the packing order within each supplies group is as follows: Larger items are packed first; if the volumes are equal, items with the largest end face area are packed first; if both the volume and the largest end face area are equal, items with the longest side are packed first; for items with the same geometric dimensions, items with the greater mass are packed first; for items with identical geometric dimensions and mass, the packing order is randomly assigned.

4. The heuristic method for optimizing material packing and loading as described in claim 1, characterized in that: Before starting the packing algorithm, an end-face database containing three sub-databases is established to store information related to containers / carriages. L × W noodle, L × H Face to face W × H Data on parallel material end faces; updating the end face dimension database during packing and loading.

5. The heuristic method for optimizing the packing and loading of materials as described in claim 1, characterized in that, Establish and update the available space database online: After new materials are loaded into containers or carriages, the space for loading materials is divided into front, right, top or front, left, top and right, front, top or left, front, top, and available spaces are merged. The available space database saves both division results at the same time.

6. The heuristic method for optimizing material packing and loading as described in claim 1, characterized in that: After the packing algorithm is activated, the materials are loaded into containers or carriages one by one using two methods: matching available space with materials and matching materials with available space.

7. The heuristic method for optimizing cargo packing and loading as described in claim 1, characterized in that: In step 3, the optimal available space refers to the available space that simultaneously satisfies the following conditions: It can accommodate the materials to be loaded; it can maximize the utilization rate of the end face area; when the end face area utilization rate is the same, the available space with the larger volume utilization rate is selected.

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