Optimization of cutting sheet material on rectangles of various sizes. Associativity between the initial geometry and detail contours in TFlex Study

Economical linear cutting materials (cutting holder) is relevant for many industries and construction. It is cutting logs and boards in woodworking, cutting rods, reinforcement rods, corners, channels, pipes, billets on the workpiece ...

In the production of metal structures and mechanical engineering, transverse cutting rolls with paper and tissue in cellulose and light industry.

Despite the seeming simplicity, the solution of the problem of linear cutting is very not easy, but worth it. The introduction of a scientific approach to the cutting of jogging materials makes it possible to reduce the costs of them sometimes by more than 10%! Take the article to the end and make sure that these words are right.

The topic under consideration refers to the tasks of linear programming. To solve such tasks, scientists in the last 70 years have come up with several different methods.

Method of indexes L.V. Kantorovich and V.A. Zhalkaller with a certain skill allows "manually" without the use of computing technology to effectively perform linear cutting. I recommend curious readers with this method to get acquainted by reading the book of the above authors of the "rational cutting of industrial materials".

Simplex-method based on the ideas of L.V. Cantorovich, was described and developed in detail by a number of scientists from the United States in the middle of the 20th century. An add-in MS Excel "Solution Search" (Solver) uses this algorithm. It is with the help of this method andExcel We will solve the problem of linear cutting in this article.

Later, genetic, greedy and ant algorithms were developed and obtained. However, we restrict themselves with their transfer and proceed to the case, without climbing the debs of theories (although there, "in the wilds", very interesting).

Include Excel and on simple example Cutting metal rods to the details will get acquainted with one of the ways to solve the practical tasks of linear cutting. Often mathematics this task is called the "task of cutting".

Initial data for example, I did not invent, but took from the article Pokrovsky MA "Minimization of inevitable losses of materials in industrial production when they are accountable for piece billets" published in No. 5 (May 2015) of the electronic scientific and technical journal "Engineering Bulletin" published by the FGBOU VPO "MSTU them. AD Bauman "(link:engul. bmstu.. ru/ dOC/775784. hTML).

The goal I pursued is to compare the results of the problem of solving the problem.

An example of solving a linear cutting task in MS Excel.

Agree that:

1. Billets are the source material in the form of rods, strips, rods, etc. equal length.

2. Details are the items that need to be obtained by cutting the initial blanks on the part.

3. The width of the saw, cut, the rumble is accepted equal to zero.

The task:

To configure one of the orders, the blank area must be chopped on the combined scissors from the same rods-blanks with a length of 1500 mm Three part size:

151 pieces of 330 mm long

206 pieces of 270 mm long

163 pieces of 190 mm long

It is required to find the optimal cutting plan using the minimum amount of material and giving, respectively, the minimum amount of waste.

Initial data:

1. The length of the initial blanks L.z. In millimeters, write to the combined cell

D3E3F3: 1500

2. We assign numbers i. All sizes of parts, ranging from the longest and ending with the shortest in cells.

D4; E4; F4: 1; 2; 3

3. Length details L.d. I. in millimeters we write in

D5; E5; F5: 330; 270; 190

4. Number of details N.d. I. in pieces we enter in

D6; E6; F6: 151; 206; 163

5. We proceed to a very important stage - filling the variants of camp.

Need to remember and understand 2 principles of doing this work.

1. Waste lengths should be less than the smallest detail ( 0< Lo J. < L.d. MIN. ).

2. "Stacking" parts in the workpiece start with the biggest details and from the largest amount of them, consistently moving towards the decrease.

If there is no partial size in the formation option, then the cell is left blank, you will not write zero to facilitate the visual perception of the table.

Option of cutting №1:

Attempting to carve out 5 parts No. 1 from one workpiece 5, so we write to the cell

Add in the cutting detail number 2 or item number 3 is also impossible, so we leave empty cells

Option of cutting number 2:

We decrease by 1 from the previous version number 1 and write down in

We try to add 2 details No. 2 - it does not work, so we complement in

It remains the opportunity to comply with the detail number 3. We enters B.

Adhering to the voiced principles, fill in the analogy of all the options in this case, 18 variants of camp.

Having made a couple of tables of clarification options on your own, you understand the logic of actions and spend a few minutes to this work.

If the first principle is not fulfilled when the first principle is not fulfilled, the cell with a length of waste is automatically painted in red. Conditional formatting applied to G7 ... G24 cells will clearly help you in this work.

In the H7 ... H24 cells do not write anything! They are used to display the result of the solution!

Preparation for the decision:

* In the cells G7 ... G24, waste lengths (trimming) are calculated, which remain as a result of the execution of the deflection, by the formula

Lo J. = L. s -Σ (L.d. I. * N.d. IJ. )

6. The number of parts of each size, manufactured by all applied cutting options, will be counted in the D26, E26 and F26 cells by the formula

N.d. I. calculation = Σ (N.d. IJ. * N.z. J. )

The number of parts in the foundation found at the end of the solution should fully comply with the specified number of parts!

7. The required number of blanks to perform the optimal formation plan will be determined in the combined D27E27F27 cell by the formula

N. z set \u003dΣN. z. J.

8. The total length of all the billets necessary to perform a linear cutting of all parts will be counted in the combined D28E28F28 cell by the formula

L.z. Σ = L. s *N.z set

9. The total length of all waste received when performing the foundation plan found will be considered in the combined D29E29F29 cell by the formula

L.about Σ = Σ (L.about J. * N.z. J. )

10. The proportion of waste obtained when performing the optimal linear cutting plan on the total amount of the material used will be calculated in the combined D30E30F30 cell by the formula

ΩO. = Lo Σ / Lz Σ

Decision:

Preparation Completed, 18 options are determined by the most optimal splits of one workpiece into parts and all necessary formulas are inscribed. Now to solve the main task: to determine the optimal planning plan is how many blanks, and for what options to cutTo ultimately get all the necessary parts in the right amount at a minimum of waste.

1. Choose in the main menu "Service" - "Search solutions ...".

2. In the "Solution Search" window that appears in the same name, we make settings.

2.1. Appoint a target function with the total length of waste Lo Σ. And enter the link to the target cell window.

2.2. Install the switch "Equal:" to the "minimum value" position.

2.3. Indicate cells with variables NZ J. In the "Changing Cell" window.

2.4. Enter the restrictions on the window of the same name. As conditions, we indicate the need for the equality of the specified ND I. and calculated ND IRISCH the number of parts, as well as the variables NZ J. - The calculated number of blanks according to the variants of the formation - impose a limit: it must be integers.

3. Press the "Parameters" button and in the "Solution Saying Settings" dropped by the settings as shown in the following screenshot. Close the window with the "OK" button.

4. In the Solution Search window, click the "Run" button and wait until Excel finds the solution. This can last a few minutes.

5. After saving the solution found by the "OK" button, the results are displayed in the H7 ... H24 cells on the Excel sheet.

The following picture shows the found optimal linear cutting plan.

What is the result?

The linear cutting in Excel blanks for the tasks of such considered in this article is performed as described above for 10-15 minutes! "Manual", not knowing the method of Cantorovich indexes, you will not find solutions for this time.

By running "Solution Search" several times with different search options, you managed to find 5 different plans for cutting blanks. All 5 plans require the same number of blanks - 93 and give waste to just 2.21% !!! These plans are almost 6% better than a plan calculated by Pokrovsky and more than 10% more important than the "traditional" plan (see the reference to the original source in the first part of the article). A very worthy result will be achieved quickly and without the use of expensive programs.

It should be noted that the Excel Solver superstructure ("Solution Search") using a simplex method when solving a linear programming tasks can not work with no more than 200 variables. In the annex to the linear cutting problem considered by us, this means that the number of codes cannot exceed 200 options. For simple tasks, this is enough. For more complex tasks, you should try to apply the "mix" "greedy" algorithm and the Solver simplex method, which full list Recalls are not more than 200 most economical. Next, we reserve patience and achieve results. You can try to break the challenging task for a few simple, but the "optimality level" of the found solution will be at the same time, is likely to be lower.

Perhaps the considered version of the solution of the linear cutting issues and not the "highest pilot", but definitely step forward compared to the "traditional" approach in many industries.

Using the add-on MS Excel "Solution Search" (Solver) was on the blog already once reviewed in the article. I think that this wonderful tool is worthy of close attention and will no longer help elegantly and quickly solve a number of new non-trivial tasks.

P.S. Links to the best free linear cutting programs found by me online:

http://stroymaterial-buy.ru/raschet/70-raskroynih-izdeliy.html.

http://forum-okna.ru/index.php?pp\u003dcore&module\u003dattach§ion\u003dattach & attach_id \u003d 7508

http://forum.dwg.ru/attachment.php?attachmentid\u003d114501&d\u003d13823277 74.

http://www.planetcalc.ru/917/

Programs according to the last two links are implementing greedy heuristics and perform a linear cutting in a task from the article using already 103 blanks. The use of greedy algorithms is justified in cases of having to reduce the total time of cutting operation at too large quantities Options for revealing in more optimal plans.

Below the articles in the "Reviews" block can write your comments, dear readers.

Exceptional simplicity is a special feature of the Astra S-Nesting program. It is unusual for professional programBut we specifically put efforts to solve all tasks from ordering to edit carding cards and the appointment of the clipping route was intuitive. As a result, this ensures rapid implementation and successful operation.

Download

Details about the program Astra S-Nesting

Import from CAD / CAM

Import details is executed from DXF files. For imported parts, properties are specified: the name of the material, the thickness, the number and number of the drawing. All properties of parts can be changed after import. The order may contain details of different thicknesses and materials - the program automatically sorts the part to the collaboration groups.

Optimization of cutting

Fast automatic cutting provides a high material use ratio. If required, cutting cards can be edited manually. For this, the program offers convenient tools: moving parts with a given step, up to the stop and free movement of the mouse; rotation of parts to a given and arbitrary angle, the edge of the edges of the parts; Scaling commands; The ability to cancel and repeat teams, etc.

Calculation of the clipping route

An excellent algorithm allows you to calculate the cutting route in seconds, providing an optimal processing sequence. Depending on the specified parameters, it is set to cutting methods into the outer and internal contour of the parts, the direction of bypass, the presence of bridges or jumpers, etc. When manually edited the clipping route, you can change all the route elements and the circuit processing sequence.

Printing report reports

For the order, a set of reports is formed, including specifications and sketches of cutting cards. Report templates in Astra S-Nesting customizable! This means that you can change them yourself under adopted at your enterprise forms. Any report can be exported to Excel.

Integration with CAD / CAM

One of key principlesSupported in the Astra S-Nesting program is integration with other CAD / CAM systems. Data that prepares and stores your CAD can be saved as an order for the Astra S-Nesting program and get back the results of the calculation. The program provides launch from command line, Automatic data import, ordering, and export results back to processing in the external system.

Our price list presents three products related to the overall theme of sorting and optimization:

• Linear cutting program and long material
• Program of two-dimensional cutting glass, sandwiches, chipboard and other sheet materials
• Route Optimization Program for Solving Logistics Tasks

The supply of cutting modules is possible both as part of a comprehensive solution. Okrufustoms: management of bypass production and in the form individual programs. When calling out 1C cuts, it does not require import - export data to intermediate files. The user works in the standard interface, and all the subtleties of interaction 1c with external optimizers perform filling processing table portions. For the purpose of taking into account warehouse residues, business cuts and materials in production, standard documents and registers of type configurations 1C are used.

Linear cutting (profile, pipe, log)

Provides inventory confirmed percentage of cropping<1%. Ряд клиентов приобрели наши алгоритмы для замены программ оптимизации, поставлявшихся производителями отрезных станков. В программе использован алгоритм плотной укладки и генетический алгоритм поиска решения. На вход поступают данные о количестве и размерах изделий и деловых отходов. На выходе формируются карты раскроя с указанием тележек и ячеек. При необходимости, формируются файлы для обрабатывающих центров, станков с ЧПУ и этикетки с подробной информацией об отрезаемой заготовке и примыкающих элементах.

Live demo on the site

The example below is not a static picture, but a working web application.
You can start the profile cutting button Start, Set your product size and blanks, change optimization settings and evaluate the solution.
Of course, the optimizer in the browser works more slowly than a native program, but allows for free to get suitable results without having to download and install and install on a computer.

20,000 p.

Cutting glass and leafy materials

Forms the cards of the highest quality. Provides a percentage of material savings close to theoretical limit. 10-12% surpasses popular Opty-Way programs, Maxcut, Perfectcut, Cutting, etc. by such indicators such as the area of \u200b\u200bnon-flying residues, the total area of \u200b\u200bthe declared material and the amount of used entire sheets

All two-dimensional cutting algorithms are developed in the "Cutting Programs", Novosibirsk, Developer: Schilyaev Vladimir Henrikhovich. "Okosoft" is the official developer dealer and under the sub-license agreement has the right to sell a program in the form of a separate product, and used as part of our developments.

40000 p.

Why optimize the cutting in the program?

Many customers say: "I have a good stuffer. It perfectly carins the glass and the profile in the head. Only triangles fall into the garbage."
Most often, it is true. At the same time, one of the tasks of the leader is the organization of a predictable process, the stability of which does not depend on the genius of the performers. Software optimization broke on the basis of the production plan is one of the measures close to this purpose.

If you assume that a person can go through the head of combinations more than a computer for the same period of time obtained by automatic cutting coefficient of ~ 1%, it looks more attractive than uncontrollable and uncontrollable 1%, which can provide genius. Crossing the optimization tasks for the program can free up additional time (1-2 hours a day), which it will spend with benefit for business.

In fact, the situation with cutting on most enterprises is worse. The specification lay the coefficients of about 4-7% and if the workshop will work with a rough cut, this is considered a good result. The decrease in the real coefficient of cutting by 3-5% is 30-50 thousand rubles saved on every million, spent on materials. And also, it will make it possible not to lay extra rugs in the planned cost price and offer to the buyer more favorable prices.

The task of optimal consumption of materials consists of several parts

Warehouse accounting of measuring materials

Depending on business features, customers use several accounting schemes:

• Based on the production plan, requirements are formed - invoices indicating products. The issuance of additional materials (not enough) is affected by individual documents. Specifying products in these documents - preferably, but is not a prerequisite. In this case, only those materials that are in the specifications produced today are issued in the workshop and only in the desired quantity. The disadvantage of this approach can be called the need to make up more documents and the lack of stock of materials on the sites (maybe this is a dignity?)
• Requirements - Overhead are formed asynchronously, without binding to the production plan on the basis of applications of Masters. Such an approach allows you to get a "live warehouse" with the minimum operator cost of the storekeeper, but does not protect against the overpowering of materials. All responsibility for matching the consumption of specifications lies in this case on masters and workers. A factory cost analysis will show deviations, but it may be late

Accounting for business cutting

It is possible in the script that the requirements are formed on the basis of the production plan. The remnants of the business cutting at the beginning of the cutting are taken from a special register and can be adjusted by the operator in accordance with the real residues. When conducting a job on production, data on the number of materials that must be obtained from the warehouse is placed in the requirements - invoices, and the data on the resulting business cutting is folded back to the register.

The interaction of performers

Taking a decision on the use of the opening optimizer, it is necessary to consider:

• With automatic depth, it is impossible to organize the welding (assembly) of products "from - under saws", since the segments related to one product will be "scattered" throughout the entire optimization map
• The production cycle is elongated, it is required to organize a pool for storing blanks. Compromise is a cutting part of 30 - 50 products. At the same time, high cropping rates are achieved and the uniform loading of welding and fittings
• The efficiency is reduced with which the workshop can respond to changes in the plan. If the manager needs to inject a new product into today's plan, it will worsen optimization results

"BCUT" - a program for automating the cutting of sheet materials intended, mainly for the production of cabinet furniture. It allows you to quickly perform a layout of rectangular blanks on the sheets of rectangular shape, taking into account the technological parameters of equipment and edge bandage, printing cards and labels for details, create programs for cutting and printing labels on Altendorf ® machines.

The BCUT program is a continuation of the development of the "cutting" module, which is part of the BCAD Furniture CAD. She not only absorbed all the best achievements of the predecessor, but also received a number of new opportunities.

If you are interested in the features of the BCUT package, you can download. With prices for "BCUT" you can read in the section of our site.

Fast creation of order

"BCUT" allows you to quickly. Key features are optimized for quick input. For example, parts sizes, and incoming can be entered by one hand using only an additional digital keyboard. Immediately you can use the built-in calculator to recalculate numeric values.

Knowledge of materials used and equipment features are stored in a data bank. They are filled in advance, not when client. Nevertheless, if the client has a special material, then you can fill in information about it quickly and simply without affecting the bank.

Imports of blanks

There is an import (read data) of the workpieces of parts along with the description of the materials and. Excel 2003 and Excel 2007 are supported. In this way, you can quickly receive orders scored by the customer manually or made in other programs.

Easy work with the program

In appearance and methods of controlling "BCUT" no different from any modern program. Built at work with the usual visual elements for Windows, for which they click on the mouse or pen, and also introduce text or numbers from the keyboard. Pictograms are clear and clear. The buttons are equipped with understandable inscriptions and pop-up tips. Therefore, you can easily master the "BCUT".

On the screen, everything relating to the opening assignment is simultaneously visible. This is especially convenient on modern displays. Quite comfortable work is possible even on the laptop monitor, with a resolution of 1024 × 600. There is a color illumination of the status of parts in the table.

Adjustment accounting

Accounting for the edge material is carried out on all reveal details, taking into account the parameters of the machine on which the edgeless is produced.
When replacing the machine, the edge consumption is automatically recalculated taking into account the parameters of this machine.

Accounting for equipment features

All features of cutting or edging are entered into a data bank. "BCUT" takes into account the maximum lengths of the cuts, the thickness of the saw and the cuts of the edge. You can take into account the differences in the saw along and across. In advance to specify the values \u200b\u200bfor different thicknesses of the saw, which will be taken into account when automatically calculating points. The thickness of the saw and allowances are set up to 0.1 mm.

The number of machines in the data bank is not limited. You can combine in one job cutting on different machines.

High speed cutting

BCUT provides high cutting speed. This is achieved at the expense of fast algorithms and the use of multiprocessing of modern computers.

The layout algorithm is optimized for work on multi-core processors, such as Intel Core2DUO. When optimizing, all cores of the processor are used.

This allows you to set several materials in the reservation and cover them simultaneously by pressing one button. With very large tasks, it is possible by entering parts for one material, start cutting, and during optimization to set parts for another material.

Optimization

"BCUT" has simple and clear. There is an opportunity and individual cards by changing the optimization methods and / or add additional details to task.

Automatic calculation of billets

It is possible to automatically calculate the allowances and the size of the workpieces, taking into account the modes of the cut, the allowance for incoming, type edge, including taking into account the edge. Calculation is possible. The allowances are automatically recalculated when settings change, for example, when selecting another mode of cutting and source data.

However, it remains the opportunity to specify the allowance of the allowance manually. Just entering its value, including using the built-in calculator.

More clearly, the operation of the automatic allowance is shown in the video. Size ~ 2 MB.

Save order

Weekend documents

Comfortable, clear weekend documents: cards, summits, labels for details and business residues. On the cards of the disclock uses bar coding. It is possible to configure the tags "for yourself". You can simply remember the image of the map in the buffer and insert as a picture, for example, to the Word document.

The report and task on cutting and edging is formed for all materials. The number and area of \u200b\u200bsheets, blanks, parts, residues are indicated.

purpose of work: Consolidation of knowledge in the field of economic and mathematical modeling, familiarity with the method of solving the problem of rational cutting materials based on solving the optimization linear programming task.

Source positions. The manufacture of many species of modern industrial products begins with the cutting of materials, which is one of the important production problems for the preparatory production and the logistics organs.

The tasks of the optimal cutting of materials are one of the first tasks, to solve which linear programming methods were used. They consist in determining the best method of cutting the incoming material, in which the largest number of finished products in a given assortment will be made or the smallest amount of waste will be obtained.

The first work dedicated to solving the tasks, subsequently called linear programming tasks, appeared in 1939. It was the book of L.V. Kantorovich "Mathematical Methods of Organization and Planning Production". The task set up to the Institute of Mathematics and Mechanics of the Leningrad State University of the Laboratory of Plywood Trust was the impetus for her appearance. In other industries, economic and mathematical methods for optimizing the cutting of materials were also successfully used. So, back in 1948 - 1949. Mathematical methods of cutting were successfully applied on the car-building factory. Egorova in Leningrad, which made it possible to reduce several times with waste with different materials.

Mathematical model of the problem.

The materials entering the enterprise are subject to cutting on the workpiece. From the correctness of the cutting depends the cost of production (used, for example, on automobiles and in other).

In most cases, cutting materials on the workpiece is made in a certain proportion, which ensures the receipt of a kit of blanks (i.e., a multiple kit).

The task of optimizing the formation of materials is to develop such cutting options at which a certain number of blanks are obtained in this assortment (different types) with minimal waste.

To compile a mathematical model of the optimal cutting problem, we introduce the following notation:

L. - the length of the material; S. - surface area of \u200b\u200bleaf or rolled material; N. - Number of source units.

Need to get m. various types of blanks or long L I. or square S I. where i. - type of workpiece ( i \u003d 1, 2, ..., m ).

The number of procurements is known i. -Ho type of product, i.e. The number of billets that is necessary for the production of one product - b I. . The number of sets of products manufactured by the enterprise will be denoted by k. .

Cutting material can be produced n. ways. Known and ij. - number of workpieces i. -to type obtained j. -M way ( j \u003d 1, 2, ..., n ).

The amount of waste obtained with a starting unit of the source material j. -H way - With j. .

It is required to make such a plan for cutting to ensure complete sets of blanks with minimal waste.

Denote by x J. Number of units of the source material, cut out j. -H way. Find such x j ³ 0 who satisfy the following restrictions:

(limit on the number of source material)

(limit on the production plan)

So much turns out the blanks of the i-th view with all the options for cutting. Based on the condition of the package, we obtain the following limitations on the production plan:

The total amount of waste must be minimal, then the function of the target takes the form:

An example of calculations in the problem of optimal cutting materials.

Metal rods 6 m long each available in the amount of 100 pcs. It is necessary to make the design shown in Fig. 1.