Optimization of nesting and order batching based on features of square parts and pearson correlation coefficient

Abstract

The application relates to a layout optimization and order batch method based on square piece features and a Pearson correlation coefficient, and relates to a layout optimization and order batch method applied to the field of intelligent manufacturing. The application solves the optimal cutting problem of square pieces in current individualized industrial products. The steps of the application are as follows: 1. determining a similar condition, establishing a one-dimensional array of required materials for each order; 2. applying a Pearson correlation coefficient to determine the similarity of each order, and combining similar orders into a batch; 3. in the same batch, cutting by material, and preprocessing the square piece data of the same material; 4. applying a large product item cutting method with the original sheet width as the resolution reference to start cutting; and 5. applying a small product item dense paving method to arrange the remaining small product items. The application fully utilizes order information and product information, combines with the actual production, and proposes a two-stage cutting method, effectively improves the plate utilization, and is suitable for batch cutting of large-quantity and multi-type individualized customized square pieces.

Description

Technical Field

[0001] This invention relates to methods in the field of intelligent manufacturing, specifically to a nesting optimization and order batching method based on the characteristics of square parts and the Pearson correlation coefficient, for the optimal cutting of square parts in personalized industrial products. Background Technology

[0002] Given the large variety of personalized customization options and the sheer volume of orders, current production organization often employs a "batch ordering + mass production + order sorting" model. Under this model, order batching and sample layout optimization are crucial.

[0003] Order batching is the process of combining different orders into a certain number of batches under the constraints of actual production capacity. When batching, it is necessary to resolve the contradiction between personalization and production efficiency. Layout optimization is essentially a material cutting problem. It optimizes the layout of square parts on the original sheet material to reduce material waste during the cutting process and simplify the cutting process.

[0004] When batching orders, similar orders are usually grouped into several batches based on their similarity. This facilitates batch processing, improves production efficiency, and shortens delivery cycles.

[0005] When cutting, depending on the cutting process, there are two methods: flush cutting and non-flush cutting. Flush cutting involves making a straight cut perpendicular to one side of the square part, dividing it into two pieces with each cut. Non-flush cutting, on the other hand, does not require dividing the square part into two pieces with each cut. Comparing the two methods, flush cutting is simpler, while non-flush cutting offers more diverse material preparation options.

[0006] Cutting can be further divided into precise and non-precise methods. Precise cutting yields all products after each specific cutting stage, while non-precise cutting requires an additional cutting stage for some products. Once the cutting stages are determined, precise cutting yields all products after completing all stages, while non-precise cutting increases the workload beyond completing all stages.

[0007] The cutting stages mentioned above arise because the cutting direction varies each time during the cutting process; the cutting direction is the same within the same stage. Too few stages will not yield the desired product, while too many stages will increase the workload. Therefore, it is necessary to select appropriate cutting stages to complete the cutting task with maximum efficiency. Common cutting stages typically consist of 3-4 stages. Taking a 3-stage cutting as an example, the modules generated by the first stage are called stripes in this invention, such as Stripe1 and Stripe2; the modules generated by the second stage are called stacks in this invention, such as Stripe1 being further cut into Stack1, Stack2, etc.; the modules generated by the third stage are called items in this invention, such as Stack1 being further cut into Item1, Item2, etc.

[0008] The primary goal of order batching and layout optimization is to maximize the utilization rate of the board material, that is, to satisfy:

[0009]

[0010] Where γ is the board utilization rate, S i n is the area of ​​each product item, and n is the total number of product items. 原 This refers to the number of original films, S 原 It is the area of ​​the original film.

[0011] Current research on order batching mostly employs clustering methods, selecting appropriate targets based on the characteristics of different orders and establishing clustering models under specific constraints. However, batching product orders from different production fields presents different constraints, and the optimization objectives also differ. For square parts, there is a coupling issue between order batching and layout optimization, thus requiring coordinated optimization of both. Current research on layout optimization is largely theoretical, failing to consider the impact of actual conditions such as flush cutting or non-flush cutting methods and cutting stages on production efficiency. Optimization is often tailored to square parts with special shapes, such as blocks of equal height. Faced with diverse product orders and a large number and variety of square parts, many methods are not universally applicable. Summary of the Invention

[0012] The purpose of this invention is to propose a layout optimization and order batching method based on the characteristics of square parts and the Pearson correlation coefficient. This method overcomes the main problems of current research on order batching and layout optimization, which fails to consider the coupling between the two and the inability of traditional layout optimization methods to be universally applicable to "small batch, multi-variety" square part orders, thus making them unsuitable for actual production. By using the sheet material required by different orders as similar targets and applying the Pearson correlation coefficient to batch the orders, the same sheet material method is used as much as possible within the same batch, solving the problem of collaborative optimization between batching and layout. Under the constraints of end-cutting, three-stage cutting, and precise layout, the method preprocesses a group of square part data requiring the same sheet material by considering the characteristics of the original sheet material and the characteristics of different square parts, distinguishing between large and small product items, and then arranging the square parts. This layout method has higher cutting efficiency when cutting square parts and is universally applicable to various square parts, adapting to the actual production of square parts and better guiding enterprise production work.

[0013] The objective of this invention is achieved through the following technical solution: A one-dimensional array of required materials is created for each order. This array is normalized, and the similarity between orders is obtained using the Pearson correlation coefficient. Orders with high similarity are grouped into a batch according to production constraints. Within the same batch, all square parts requiring the same raw material are cut uniformly. Before cutting, the product item data is preprocessed to separate large and small product items. Large product items are cut first, and finally, small product items are densely laid out to optimize the layout.

[0014] The flowchart of this invention is as follows Figure 1 As shown, the specific steps are as follows:

[0015] Step 1: Determine the similarity conditions.

[0016] This invention first counts all types of materials in all orders and creates a one-dimensional array of required materials for each order. If an order contains a product item that requires a certain material, the corresponding position in the array is filled with the number of product items that require that material; otherwise, the corresponding position is filled with zero. Finally, the array is normalized.

[0017] Step 2: Apply the Pearson correlation coefficient to group similar orders into a batch.

[0018] The material array for each order reflects the material requirements of that order. Therefore, this invention can determine whether the material requirements of each order are similar simply by comparing the similarity of their material arrays. If the similarity is 1, the two orders require the same materials. If the similarity is not greater than 0, the two orders do not require the same materials. This invention calculates the correlation coefficient between each material array by applying the Pearson correlation coefficient formula, thereby obtaining the correlation coefficient matrix between each order. The Pearson correlation coefficient formula is as follows:

[0019]

[0020] Where r is the correlation coefficient, and x and y are two variables.

[0021] Correlation coefficient matrix as follows Figure 2 As shown, after obtaining the correlation coefficient matrix between each order, different orders are selected to form the same batch according to the order of correlation coefficient from largest to smallest.

[0022] Step 3: Within the same batch, cut materials separately and preprocess the data of square parts made of the same material.

[0023] First, take the length of the longer side of each product item as its length, and the length of the shorter side as its width. Taking a sheet material with dimensions of 2440*1220 (mm) as an example, extract the data from each product item whose length falls between 1220mm and 2440mm to form set A.

[0024] Step 4: Begin cutting using the large product item cutting method with the original film width as the resolution benchmark.

[0025] The maximum length L of product items in A a1 The first piece of the original image is cut, and the arrangement begins in the left space. The length L is subtracted from the longer side of the original image. a1 The obtained length L b1 As the first original image data classification standard, all images with a length less than 1220mm and a width less than or equal to L... b1 The data forms the corresponding data group B1 for the first original piece. If the corresponding data group is empty, the short boards obtained after the first cut are all scrap. If the corresponding data group is not empty, after the first stage of cutting, the long boards from L... a1 Begin cutting the sheet material in set A in descending order of length, constrained by width, until the remaining width of the original sheet is insufficient to accommodate the next product item. At this point, the remaining sheet material is temporarily added to set C (waste). This is the first stage of cutting on the left side. Figure 3 As shown.

[0026] During the second stage of cutting on the left, unarranged product items with a width smaller than the remaining width of the current original sheet are searched. If the length of such a product item is less than La1, it can be placed in the temporary waste set C, and the waste is reused. This method helps reduce the waste output rate. However, within the remaining width of the current original sheet, only one product can be arranged in the length direction; otherwise, a fourth stage of cutting will occur. A schematic diagram of the second stage of cutting on the left is shown below. Figure 4 As shown.

[0027] There are two scenarios for the third-stage cutting on the left side, one of which is, for example... Figure 5 Item 2, as shown, is produced directly through the third-stage cutting process. (Second example) Figure 5 As shown in item4, since the width is not full, the remaining width needs to be cut off in the second stage before the third stage of cutting can be used to complete the production of the product item.

[0028] When there is no space on the left, begin arranging on the right. For example... Figure 6 As shown, the product items on the right are arranged vertically. First, select a product item whose width is less than the current maximum remaining length and whose length is less than 1220mm, and place it in the lower left corner of the original piece on the right. Select a product item whose width is less than the previous product item and whose length is less than the remaining width of the original piece after placing the previous product item, and place it above the previous product item, left-aligned. When the remaining width is insufficient to place the next product item, start a new column and rearrange from the bottom. When the right side is filled, place the next piece of the original piece.

[0029] At this point, the first stage of cutting on the right side will generate three temporary scrap items: item10, item11, and item12. Product items with the same dimensions from the unarranged product items will be placed into these scrap items. This approach further increases material utilization. The vertical arrangement on the right side has certain advantages, as the first stage of cutting can first cut the original sheet into vertical strips, such as... Figure 7 As shown.

[0030] The vertical strip is then cut in the second stage to obtain... Figure 8 The sheet material shown.

[0031] The third stage of cutting on the right side is as follows Figure 9 As shown, this completes the process of cutting large product items using the original width as the resolution standard. The remaining product items will be arranged using the dense laying method for small product items.

[0032] Step 5: Arrange the remaining small product items using the dense placement method for small product items.

[0033] Sort all remaining product items by width from largest to smallest, and arrange them sequentially from the bottom left corner of the original sheet. When the total length exceeds 2440mm, the next sheet begins arranging upwards, starting the second row. When the total width exceeds 1220mm, the next product item begins arranging from the bottom left corner of the next original sheet. Continue arranging product items on the original sheet using this method until all product items are arranged. This method allows product items with similar widths to be arranged in a row, reducing the scrap rate. This method generates the same temporary scrap in length and width as the method using the original sheet width as the dividing line for large product items. Using the same method, prioritizing the product item with the largest area among the scrap sizes and including it in the scrap can increase material utilization.

[0034] Compared with the prior art, the present invention has the following advantages:

[0035] 1) This invention, combined with actual production practice, overcomes the shortcomings of current research that ignores practical limitations such as cutting methods. Under the constraints faced in actual cutting such as flush cutting and three-stage cutting, it proposes a two-stage method using the length and width information of product items and original sheets, namely, a large product item cutting method and a small product item dense laying method, which use the width of the original sheet as the distinguishing benchmark. While meeting order requirements and related constraints, it greatly improves the utilization rate of the board material and reduces the generation of waste, and can be used to guide actual production.

[0036] 2) In solving the problem, this invention makes full use of order and product information. When analyzing order correlation, a correlation coefficient matrix was established based on the Pearson coefficient; during layout optimization, the length and width information of product items and original pieces were utilized. The use of this information greatly ensures the rationality and operability of product item layout and order batching. Attached Figure Description

[0037] Figure 1 This is a flowchart of a layout optimization and order batching method based on square part features and Pearson correlation coefficient.

[0038] Figure 2 This is the correlation matrix.

[0039] Figure 3 This is a schematic diagram of the first stage of cutting on the left.

[0040] Figure 4 This is a schematic diagram of the second stage of cutting on the left.

[0041] Figure 5 This is a schematic diagram of the third stage of cutting on the left.

[0042] Figure 6 This is a schematic diagram of the cut on the right.

[0043] Figure 7This is a schematic diagram of the first stage of cutting on the right.

[0044] Figure 8 This is a schematic diagram of the second stage of cutting on the right.

[0045] Figure 9 This is a schematic diagram of the third stage of cutting on the right.

[0046] Figure 10 This is a schematic diagram of the cutting result of a large product item cutting method using the original film width as the resolution benchmark during specific implementation.

[0047] Figure 11 This is a schematic diagram showing the cutting results of the dense laying method for small product items during specific implementation. Detailed Implementation

[0048] The specific implementation of this invention is illustrated below using data from Group B2 of Problem B in the 2022 Postgraduate Mathematical Modeling Contest:

[0049] Group B2 data contains information such as product item length, width, required materials, and associated orders, involving a total of 146 types of raw material and 403 orders.

[0050] Step 1: Determine similarity conditions and create a one-dimensional array based on the total number of materials required.

[0051] We counted all material types in all orders, resulting in 146 types. For each order, we created a one-dimensional array of required materials with a length of 146. If an order contains a product item that requires a certain material, we filled the corresponding position in the array with the number of product items that require that material; otherwise, we filled the corresponding position with zero. Finally, we normalized the array.

[0052] Step 2: Apply the Pearson correlation coefficient to group similar orders into a batch.

[0053] The material array for each order reflects the material requirements of that order. Therefore, this invention can determine whether the material requirements of each order are similar simply by comparing the similarity of their material arrays. If the similarity is 1, the two orders require the same materials. If the similarity is not greater than 0, the two orders do not require the same materials. This invention calculates the correlation coefficient between each material array by applying the Pearson correlation coefficient formula, thereby obtaining the correlation coefficient matrix between each order. The Pearson correlation coefficient formula is as follows:

[0054]

[0055] After obtaining the correlation coefficient matrix between each order, different orders are selected to form the same batch according to their correlation coefficients from largest to smallest. During batching, certain batching constraints will apply based on actual production limitations. In this example, due to capacity constraints, the total number of product items in a single batch cannot exceed 1000, and the total area of ​​product items in a single batch cannot exceed 250m². 2 Ultimately, all orders were divided into 26 batches.

[0056] Step 3: Process the 26 batches separately. Within the same batch, cut by material, and preprocess the data of square parts made of the same material.

[0057] All orders are processed batch by batch. Within each batch, square parts requiring the same material are grouped together for cutting. The longer side of each product item requiring the same material is used as the product item's length, and the shorter side is used as its width. The original sheet size is 2440*1220 (mm). Data from product items requiring the same material with lengths between 1220mm and 2440mm are extracted and grouped into sets, with different sets created based on different materials.

[0058] Step 4: Apply the large product item cutting method, which uses the original film width as the resolution benchmark, to different batches and sets to begin cutting.

[0059] Cutting is performed batch by batch, with each batch cut separately according to the different materials. For example, the first batch requires 28 different materials, with each material's product items forming a separate set, resulting in 28 sets. For each set, a large product item cutting method based on the original sheet width is applied to cut out the large product items and some smaller product items within each set, such as... Figure 10 As shown.

[0060] Step 5: Apply the dense placement method of small product items to different batches and sets to arrange the remaining small product items.

[0061] For each set in each batch, the remaining small product items are cut into smaller pieces, and then arranged using a dense placement method. For example... Figure 11 As shown. After the arrangement is completed, all product items in each set have been cut. Processing is still done batch by batch; after all sets in one batch are cut, the next batch is cut, until all batches are completed, thus completing all order tasks.

[0062] For the two-stage cutting method proposed in this invention, which uses the original sheet width as the resolution criterion for large product item cutting and small product item dense laying, this invention was tested with four sets of data, and obtained a very good material utilization rate, as shown in Table 1. Applying this invention to the B2 set of data in Problem B of the 2022 Postgraduate Mathematical Modeling Contest, the material utilization rate of the first four batches is shown in Table 2. Comparing Tables 1 and 2 shows that in practical problems, considering order batching, the material utilization rate will decrease. This invention still has a good material utilization rate under actual production constraints and can be well applied in production practice.

[0063] Table 1. Material utilization rate of 4 sets of data

[0064]

[0065] Table 2B2 Data Material Utilization Rate

[0066]

Claims

1. A method for layout optimization and order batching based on square component features and Pearson correlation coefficient, characterized in that... It includes the following steps: Step 1: First, count all types of materials in all orders, and create a one-dimensional array of required materials for each order; Step 2: The material array for each order can reflect the material requirements of each order. Pearson correlation coefficient is applied to group similar orders into a batch; Step 3: Use the length of the longer side of each product item as the length of the product item, and the length of the shorter side of each product item as the width of the product item. Combine this with the original sheet specifications to filter out the largest product items. Step 4: Begin cutting using the large product item cutting method with the original sheet width as the distinguishing criterion. First, cut the large product items. The remaining portion of the original sheet after cutting the large product items is used to cut suitable small product items, as follows: 1) Cut the first original sheet using the maximum length La1 of product items in A. First, arrange the left space. Use the length Lb1 obtained by subtracting La1 from the long side of the original sheet as the data classification criterion for the first original sheet. All data with a length less than 1220mm and a width less than or equal to Lb1 form the corresponding data group B1 for the first original sheet. If the corresponding data group is empty, the short boards obtained after the first cut are all waste. If the corresponding data group is not empty, after the first stage of cutting, the boards in set A are cut from La1 on the long board in descending order of length, with width as the constraint, until the remaining width of the original sheet cannot accommodate the next product item. At this time, the remaining boards are temporarily included in the waste set C. This is the first stage of cutting on the left. 2) When performing the second stage of cutting on the left, look for unarranged product items whose width is smaller than the current remaining width of the original sheet. If the length of the product item is less than La1, it can be placed in the temporary waste set C. The waste is reused. This method helps to reduce the waste output rate, but within the current remaining width of the original sheet, only one product can be arranged in the length direction. Otherwise, a fourth stage of cutting will occur; 3) There are two situations for the third stage cutting on the left: one is to directly complete the production of the product item by using the third stage cutting, and the other is that since the width is not fully occupied, the remaining width needs to be cut off by the second stage cutting before the third stage cutting can be used to complete the production of the product item; 4) When there is no space for arrangement on the left, start arranging on the right. The product items on the right are arranged vertically. First, select the product item whose width is less than the current maximum remaining length and whose length is less than 1220mm, and place it in the lower left corner of the original piece on the right. Select the product item whose width is less than the previous product item and whose length is less than the remaining width of the original piece after placing the previous product item. Arrange the product items on the top of the previous product items, left-aligned. When the remaining width is insufficient to arrange the next product item, start a new column and rearrange from the bottom. When the right side is filled, arrange the next piece of the original sheet; 5) The first stage of cutting on the right side will generate some temporary waste. Find product items of the same size from the unarranged product items and place them into these waste items. This scheme can further increase the material utilization rate. The vertical arrangement on the right side has certain advantages, as it can use the first stage of cutting to cut the original sheet into vertical strips first; 6) After completing the entire process of the large product item cutting method based on the width of the original sheet, the remaining product items will continue to be arranged by the small product item dense laying method; Step 5: Apply the dense placement method for small product items to arrange the remaining small product items, and finally complete the cutting of all products. The specific steps are as follows: 1) Sort all remaining product items from widest to narrowest width, and arrange them sequentially from the bottom left corner of the original sheet. When the length exceeds 2440mm, the next sheet starts to be arranged upwards, opening the second row. When the width exceeds 1220mm, the next product item starts to be arranged from the bottom left corner of the next original sheet. Arrange the product items on the original sheet according to the above arrangement method until all product items are arranged. This method can arrange product items with similar widths in one row, thus reducing the waste rate; 2) This method will generate the same temporary waste in length and width as the method of cutting large product items based on the width of the original sheet. Using the same method, prioritize finding the product item with the largest area that meets the waste size and place it in the waste, which can increase the material utilization rate.

2. The nesting optimization and order batching method based on square component features and Pearson correlation coefficient according to claim 1, characterized in that... The first step is as follows: If an order contains a product item that requires a certain material, the corresponding position in the array is filled with the number of product items that require that material. If not, the corresponding position is filled with zero. Finally, the array is normalized.

3. The nesting optimization and order batching method based on square component features and Pearson correlation coefficient as described in claim 1, characterized in that... The second step is as follows: 1) The correlation coefficient between each material array is calculated by applying the Pearson correlation coefficient formula, thereby obtaining the correlation coefficient matrix between each order. The Pearson correlation coefficient formula is as follows: Where r is the correlation coefficient, and x and y are two variables; 2) After obtaining the correlation coefficient matrix between each order, select different orders to form the same batch according to the order of correlation coefficient from largest to smallest.

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