CN115455341B - Solving method for raw material blanking layout - Google Patents

Solving method for raw material blanking layout Download PDF

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CN115455341B
CN115455341B CN202211346050.6A CN202211346050A CN115455341B CN 115455341 B CN115455341 B CN 115455341B CN 202211346050 A CN202211346050 A CN 202211346050A CN 115455341 B CN115455341 B CN 115455341B
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梁桥康
肖海华
秦海
邹坤霖
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Abstract

A method for solving blanking and stock layout of raw materials comprises the following steps: s1, determining the length of a raw material, the number of each blanking workpiece and the size of each blanking workpiece; s2, establishing an original mathematical model, and performing matrixing on the original mathematical model to obtain a simplified mathematical model; s3, establishing an initial identity matrix, introducing constant parameters, and then performing initial row change on the simplified mathematical model to obtain a new mathematical model; s4, solving the new mathematical model by means of an optimizer to obtain an initial feasible solution; and S5, based on the initial feasible solution, solving the optimal relaxation solution by utilizing a CG algorithm, if the relaxation solution is not a non-negative integer, rounding the optimal relaxation solution to obtain an integer solution, and selecting the best solution in the S4 and the S5 as a raw material blanking layout scheme. The invention provides a CG algorithm improved based on a new mathematical model to solve the blanking problem with low demand, and the solving quality of the CG algorithm can be well improved.

Description

Solving method for raw material blanking layout
Technical Field
The invention relates to the technical field of raw material blanking, in particular to a solving method for raw material blanking stock layout.
Background
The method for reducing the raw material consumption is one of the necessary means for realizing green manufacturing of enterprises, is concerned with the problems of improving economic benefit and ecological environment of the enterprises, is the most direct and effective method for saving energy and reducing emission, and is an important means for helping the nation to realize the aim of carbon neutralization. Cutting problems directly related to raw material consumption are ubiquitous in practical engineering problems, such as cutting problems are indispensable in production plans of various industries such as paper making, steel, plastics, aluminum and wood. The cutting problem is also called as the blanking problem, and the solving effect of the cutting problem is directly related to the raw material consumption and the production efficiency of manufacturing enterprises. Specifically, a mathematical model of the classical one-dimensional cutting inventory problem (CSP) refers to the cutting of a set of available stock material into various different size specifications of products required by a customer order by optimizing a given objective function (raw material cost minimization). From a mathematical theory level, the cutting problem is a typical NP combination optimization problem. At present, researchers divide the algorithms for solving such problems into two types, one is to minimize the objective function (minimize material cost) by constructing a good (aiming at minimizing material waste) cutting layout mode and repeatedly using the cutting mode as much as possible, so as to meet the demands of customer orders for products with different specifications, and the method is generally called a constructive heuristic optimization algorithm; the other method is to firstly use a column generation precision method to obtain a relaxation solution of the original problem, then apply a rounding technology to the relaxation solution by combining a heuristic algorithm, and simultaneously update a cutting mode to obtain an optimal integer solution of the cutting optimization problem, and the method is generally called a residual heuristic algorithm.
For a method for solving a general problem of one-dimensional cutting blanking, the existing research technology basically tends to be mature, but for solving the problem of low-demand blanking, the existing approximate solution algorithm is almost difficult to obtain a satisfactory optimal solution. Although the optimal relaxation solution can be obtained by the column generation precise algorithm (CG algorithm), the difference between the integer solution after rounding and the ideal optimal integer solution is often large, and the number of different cutting modes (different blanking layout schemes) is large, which causes a lot of raw material waste in enterprise manufacturing and low production efficiency of enterprises.
Disclosure of Invention
The invention provides a method for solving a raw material blanking layout, which aims to solve the technical problem that raw materials are easy to waste due to an optimal solution obtained by the existing algorithm in the prior art.
In order to achieve the purpose, the technical scheme of the invention is realized as follows:
the invention provides a solving method for raw material blanking layout, which comprises the following steps:
s1, determining the length of raw materials, the number of each blanking workpiece and the size of each blanking workpiece;
s2, establishing an original mathematical model according to the length of the raw materials, the number of each blanking workpiece and the size of each blanking workpiece, and performing matrixing on the original mathematical model to obtain a simplified mathematical model;
s3, establishing an initial identity matrix, introducing constant parameters, and performing initial row change on the simplified mathematical model by using the initial identity matrix and the constant parameters to obtain a new mathematical model;
s4, solving the new mathematical model (called VTC model) by means of a Gurobi optimizer to obtain an initial feasible solution of the new mathematical model, wherein the initial feasible solution comprises a blanking layout scheme and decision variables, and the decision variables refer to variables corresponding to different layout schemes, namely the number of raw materials used corresponding to each layout scheme;
s5, continuously solving on the basis of an initial feasible solution by utilizing a CG algorithm to obtain an optimal relaxation solution, wherein the optimal relaxation solution refers to an optimal decision variable, if the optimal relaxation solution is all non-negative integers, the sum value of the optimal decision variable is compared with the sum value of the decision variables in the step S4, the decision variable generated by the party with the minimum sum is selected as a final decision variable, the corresponding blanking layout scheme is a final raw material blanking layout scheme, and if the optimal relaxation solution is not all non-negative integers, the step S6 is entered;
and S6, rounding the optimal relaxation solution obtained in the step S5 by means of a heuristic algorithm to obtain a non-negative integer solution, comparing the sum value of the optimal decision variables with the sum value of the decision variables in the step S4, selecting the decision variable generated by the party with the minimum sum as a final decision variable, and taking the corresponding blanking layout scheme as the final raw material blanking layout scheme.
In this embodiment, the original mathematical model in step S2 is:
a first objective function:
Figure 895870DEST_PATH_IMAGE001
the first constraint condition is:
Figure 157219DEST_PATH_IMAGE002
Figure 405797DEST_PATH_IMAGE003
Figure 394482DEST_PATH_IMAGE004
Figure 943275DEST_PATH_IMAGE005
，
Figure 234579DEST_PATH_IMAGE006
，
Figure 344355DEST_PATH_IMAGE007
，
Figure 12097DEST_PATH_IMAGE008
，
Figure 274451DEST_PATH_IMAGE009
（1）
wherein,
Figure 736656DEST_PATH_IMAGE010
representing a decision variable, in particular representing the quantity of raw material used in the j cutting mode;
Figure 835193DEST_PATH_IMAGE011
the number of ith products to be blanked in the jth cutting mode is represented;
Figure 41047DEST_PATH_IMAGE012
the total number of workpieces to be blanked is represented;
Figure 157907DEST_PATH_IMAGE013
the dimension of the required ith workpiece to be blanked is represented; l represents the length of the raw material;
Figure 322173DEST_PATH_IMAGE014
representing a non-negative integer.
Preferably, the simplified mathematical model in step S2 is specifically:
a second objective function:
Figure 409471DEST_PATH_IMAGE015
the second constraint condition is as follows:
Figure 153436DEST_PATH_IMAGE016
Figure 531327DEST_PATH_IMAGE017
Figure 459969DEST_PATH_IMAGE018
，
Figure 657732DEST_PATH_IMAGE019
，
Figure 80755DEST_PATH_IMAGE020
，
Figure 47574DEST_PATH_IMAGE021
Figure 412696DEST_PATH_IMAGE022
（2）
wherein,
Figure 832176DEST_PATH_IMAGE023
a matrix with 1 row and n columns is represented; solution vector matrix
Figure 557424DEST_PATH_IMAGE024
The representation is a matrix of n rows and 1 column, the elements of which are decision variables whose values are constrained to be greater than or equal to zero and to be integers; a represents a matrix of the layout scheme (cutting pattern), and is a matrix of m rows and n columns (
Figure 378750DEST_PATH_IMAGE025
) When all the cutting modes are solved, all the columns in the matrix A are formed by the generated cutting modes, each column represents one cutting mode, and before iteration starts, the cutting modes are an identity matrix with m rows and m columns;
Figure 914773DEST_PATH_IMAGE026
a column vector demand matrix representing m rows and 1 column, wherein the element values of the matrix represent the required number of workpieces with different sizes, and m represents that the number of the required workpiece types is equal to m;
Figure 821549DEST_PATH_IMAGE027
a matrix representing m rows and 1 columns, which is a column variable vector matrix, in which the values of the elements are non-negative integers,
Figure 851953DEST_PATH_IMAGE028
representing a cutting pattern currently to be solved by
Figure 793364DEST_PATH_IMAGE028
As
Figure 500289DEST_PATH_IMAGE029
The index of (a) is determined,
Figure 894362DEST_PATH_IMAGE028
represents all possible cutting pattern sets, such as:
Figure 964342DEST_PATH_IMAGE029
representing a cutting pattern that the current iteration needs to be solved for
Figure 494681DEST_PATH_IMAGE028
In which the elements
Figure 372507DEST_PATH_IMAGE030
Is shown in cutting mode
Figure 253875DEST_PATH_IMAGE028
The number of the i-th workpieces,
Figure 626082DEST_PATH_IMAGE031
；
Figure 542085DEST_PATH_IMAGE032
represents a 1 row and m columnsThe elements of the matrix represent the sizes of m different workpieces; such as
Figure 59654DEST_PATH_IMAGE033
Size of the ith workpiece
Figure 693898DEST_PATH_IMAGE031
；
Figure 728850DEST_PATH_IMAGE034
Is shown in cutting mode
Figure 873261DEST_PATH_IMAGE028
Wherein the sum of the product of the size of the different workpieces multiplied by the number of the respective corresponding different workpieces is less than or equal to the length of the raw material
Figure 702677DEST_PATH_IMAGE035
；
Figure 683271DEST_PATH_IMAGE036
In (1)0A column vector matrix representing m rows and 1 columns of all 0 elements, wherein the matrix
Figure 521914DEST_PATH_IMAGE029
All the variable values are constrained to be greater than or equal to zero and must be less than or equal to the required number corresponding to each type of workpiece, the values of which
Figure 756718DEST_PATH_IMAGE037
，
Figure 288193DEST_PATH_IMAGE014
Representing a non-negative integer.
Preferably, the step S3 specifically includes the following steps:
step S310, establishing an initial identity matrix, and introducing a constant parameter
Figure 490504DEST_PATH_IMAGE038
In each iteration, column k (c)Unknown column) is fixed as a constant parameter, and the constant parameter is equal to the value of the decision variable corresponding to the kth column obtained by the previous iteration solution; at the beginning of the iteration, take
Figure 132838DEST_PATH_IMAGE039
，
Figure 877940DEST_PATH_IMAGE040
Representing a column vector demand matrixd The value of the 1 st element in (c), corresponding to the required number of first workpieces,
Figure 691569DEST_PATH_IMAGE041
represents the decision variables corresponding to the 1 st column in the layout scheme matrix A and also represents the decision variables corresponding to the first cutting mode
Figure 256543DEST_PATH_IMAGE028
A decision variable of (c);
and step S320, carrying out primary variation on the simplified mathematical model by using the constant parameters to obtain a new mathematical model.
Preferably, the new mathematical model in step S320 is specifically:
assuming that the number of decision variables is m and the number of different workpiece types is m, each iteration generates a new column, which also represents a new cutting pattern
Figure 561622DEST_PATH_IMAGE028
When iterative solution is carried out, the kth cutting mode is generated
Figure 895651DEST_PATH_IMAGE028
Its objective function and constraint conditions can be written as follows:
a third objective function:
Figure 378716DEST_PATH_IMAGE042
the third constraint condition is as follows:
Figure 430986DEST_PATH_IMAGE043
Figure 274177DEST_PATH_IMAGE044
Figure 728292DEST_PATH_IMAGE045
Figure 880794DEST_PATH_IMAGE046
, and
Figure 685939DEST_PATH_IMAGE047
，
Figure 332821DEST_PATH_IMAGE048
，
Figure 641442DEST_PATH_IMAGE049
Figure 466310DEST_PATH_IMAGE050
，
Figure 227593DEST_PATH_IMAGE051
，
Figure 678165DEST_PATH_IMAGE052
Figure 841294DEST_PATH_IMAGE053
，
Figure 326809DEST_PATH_IMAGE054
（3）
in the formula (3), the compound represented by the formula (I),
Figure 840967DEST_PATH_IMAGE028
representing a cutting pattern currently to be solved by
Figure 829651DEST_PATH_IMAGE028
As
Figure 847286DEST_PATH_IMAGE055
The index of (a) is determined,
Figure 279535DEST_PATH_IMAGE028
a set of all possible cutting patterns is represented,
Figure 15410DEST_PATH_IMAGE027
a matrix representing m rows and 1 columns, which is a column variable vector matrix, such as:
Figure 807786DEST_PATH_IMAGE055
representing the cutting pattern currently required to be solved
Figure 679927DEST_PATH_IMAGE028
In which the elements
Figure 781613DEST_PATH_IMAGE056
Is shown in cutting mode
Figure 4783DEST_PATH_IMAGE055
The number of the i-th workpieces,
Figure 476216DEST_PATH_IMAGE031
(ii) a Wherein,
Figure 593077DEST_PATH_IMAGE038
indicating that in the current iteration, the cutting mode is to be used
Figure 491763DEST_PATH_IMAGE028
Fixing a decision variable corresponding to a kth column (in a corresponding matrix A) as a constant parameter, wherein the constant parameter is equal to a value of the decision variable corresponding to the kth column obtained by previous iteration solution;
Figure 812017DEST_PATH_IMAGE057
representing solution to cutting patterns
Figure 87140DEST_PATH_IMAGE028
(corresponding to the k-th column in matrix A), by matrix-encoding the column vector
Figure 324086DEST_PATH_IMAGE058
The kth element of (a) is forced to be 0; in the same way, the method for preparing the composite material,
Figure 128094DEST_PATH_IMAGE059
representing solution to cutting patterns
Figure 702689DEST_PATH_IMAGE028
By fitting a matrix
Figure 250345DEST_PATH_IMAGE060
The k-th row of the matrix A is obtained by forcing the elements of the k-th row to be 0 elements, the function of the matrix A is to replace the matrix A, repeated and complicated matrix change of the matrix A can be avoided, and only the previous iteration solution needs to be provided
Figure 341797DEST_PATH_IMAGE061
And
Figure 316707DEST_PATH_IMAGE062
then the process is carried out;
Figure 877132DEST_PATH_IMAGE063
representing a column vector variable matrix
Figure 228479DEST_PATH_IMAGE055
Participating in a matrix expression after the initial change of the matrix; to restrain
Figure 174438DEST_PATH_IMAGE064
The decision variable corresponding to the k column in the decision variable matrix is represented as a fixed value
Figure 851407DEST_PATH_IMAGE065
Besides, all the other decision variables are constrained to be greater than or equal to zero; in the constraint
Figure 758183DEST_PATH_IMAGE034
In (1),
Figure 21543DEST_PATH_IMAGE066
representing a 1-row-m-column matrix of row vectors, the elements of which represent the dimensions of m different workpieces, e.g.
Figure 228534DEST_PATH_IMAGE067
Size of the ith workpiece
Figure 669879DEST_PATH_IMAGE031
；
Figure 798372DEST_PATH_IMAGE034
Is shown in cutting mode
Figure 632467DEST_PATH_IMAGE028
Wherein the sum of the product of the size of the different workpieces multiplied by the number of the respective corresponding different workpieces is less than or equal to the length of the raw material
Figure 162806DEST_PATH_IMAGE068
；
Figure 40632DEST_PATH_IMAGE062
Variable matrix representing unknown column vectors
Figure 922000DEST_PATH_IMAGE029
After the participation matrix A is subjected to the initial row change, the expression of the k row and k column elements in the matrix A has the value equal to 1, wherein
Figure 530092DEST_PATH_IMAGE069
Is a matrix with 1 row and m columns,
Figure 180517DEST_PATH_IMAGE069
is obtained by combining a matrix
Figure 229244DEST_PATH_IMAGE060
Line vector constructed by the k-th line of (1)A matrix; 0 is less than or equal toa p ≤d Representing a column vector variable matrix
Figure 597909DEST_PATH_IMAGE029
All the variable values are constrained to be greater than or equal to zero and must be less than or equal to the required number corresponding to each type of workpiece, the values of which
Figure 508227DEST_PATH_IMAGE070
，
Figure 13158DEST_PATH_IMAGE071
Represents a non-negative integer; objective function
Figure 232786DEST_PATH_IMAGE072
In
Figure 88747DEST_PATH_IMAGE073
Is a 1-row m-column matrix with all 1 elements;
note that: when the iteration number k is more than or equal to m +1, each cutting mode is generated
Figure 35712DEST_PATH_IMAGE074
Replace an old column, the specific replacement rule: m +1 cutting mode
Figure 660728DEST_PATH_IMAGE074
Substitution of the 1 st column, m +2 th cutting pattern in matrix A
Figure 316838DEST_PATH_IMAGE074
Substitute column 2 in matrix A, and so on, 2m cutting patterns
Figure 128936DEST_PATH_IMAGE074
Replace the mth column in matrix a; note also that after matrix a is subjected to the change of the initial rows of the matrix, each column in a is not a cutting pattern, and only after the solution of all cutting patterns is completed, the cutting patterns are combined into each column in matrix a, where each column in a represents one cutting pattern.
Preferably, the step S4 specifically includes the following steps:
step S410, solving the new mathematical model by using a Gurobi optimizer;
step S420, passing
Figure 646636DEST_PATH_IMAGE075
，
Figure 126159DEST_PATH_IMAGE076
And
Figure 953169DEST_PATH_IMAGE062
updating a third objective function and a third constraint in the new mathematical model, wherein
Figure 518143DEST_PATH_IMAGE077
，
Figure 809840DEST_PATH_IMAGE078
And
Figure 675028DEST_PATH_IMAGE069
can pass through the initial identity matrix
Figure 141782DEST_PATH_IMAGE079
Obtaining; solved by means of Gurobi optimizer to obtain
Figure 194051DEST_PATH_IMAGE075
And
Figure 178188DEST_PATH_IMAGE062
at this time, the first cutting mode
Figure 507669DEST_PATH_IMAGE080
Is solved and used
Figure 286269DEST_PATH_IMAGE075
And
Figure 950469DEST_PATH_IMAGE062
updating the first column of the matrix A, performing the initial row change on the updated matrix A, and simultaneously performing the initial unit matrix
Figure 738296DEST_PATH_IMAGE081
Sum matrix
Figure 420819DEST_PATH_IMAGE082
Making the same initial row change as the matrix A to obtain an updated matrix
Figure 370321DEST_PATH_IMAGE081
The latter matrix
Figure 256237DEST_PATH_IMAGE083
Then according to the matrix
Figure 847755DEST_PATH_IMAGE083
To obtain a matrix
Figure 10884DEST_PATH_IMAGE084
While obtaining the updated matrix
Figure 6652DEST_PATH_IMAGE082
The latter matrix
Figure 520810DEST_PATH_IMAGE085
According to a matrix
Figure 775074DEST_PATH_IMAGE085
To obtain a matrix
Figure 792709DEST_PATH_IMAGE086
(ii) a In the same way, use
Figure 738142DEST_PATH_IMAGE087
，
Figure 474017DEST_PATH_IMAGE086
And
Figure 141758DEST_PATH_IMAGE062
updatingAfter 1 iteration, the third objective function and the third constraint condition corresponding to the new mathematical model are determined, and at this time,
Figure 404113DEST_PATH_IMAGE080
unknown, needs to be solved, wherein
Figure 131897DEST_PATH_IMAGE069
By passing
Figure 230434DEST_PATH_IMAGE083
Obtaining;
step S430, performing k iterations, k =2.. K, on the new mathematical model by means of a Gurobi optimizer, calculating
Figure 701867DEST_PATH_IMAGE063
And
Figure 818727DEST_PATH_IMAGE062
: will be provided with
Figure 451834DEST_PATH_IMAGE063
And
Figure 427880DEST_PATH_IMAGE062
as the kth column of matrix a, where,
Figure 811326DEST_PATH_IMAGE069
can pass through the matrix
Figure 658059DEST_PATH_IMAGE060
Obtaining, and then performing an initial row change on the matrix A, and simultaneously performing the matrix change on the matrix A
Figure 117859DEST_PATH_IMAGE060
Sum matrix
Figure 784464DEST_PATH_IMAGE058
Making the same initial row change as the matrix A to obtain the matrix
Figure 738645DEST_PATH_IMAGE088
Then according to the matrix
Figure 705464DEST_PATH_IMAGE088
Obtaining a matrix
Figure 805007DEST_PATH_IMAGE089
While obtaining the updated matrix
Figure 490066DEST_PATH_IMAGE058
The latter matrix
Figure 218244DEST_PATH_IMAGE090
Then according to the matrix
Figure 39570DEST_PATH_IMAGE090
Obtaining a matrix
Figure 450959DEST_PATH_IMAGE091
In combination with each other
Figure 482369DEST_PATH_IMAGE092
，
Figure 637407DEST_PATH_IMAGE091
And
Figure 188605DEST_PATH_IMAGE062
and updating a third objective function and a third constraint condition corresponding to the new mathematical model after the iteration is performed for k times, wherein at the moment,
Figure 36476DEST_PATH_IMAGE080
unknown, needs to be solved, wherein
Figure 555182DEST_PATH_IMAGE069
Is obtained by combining a matrix
Figure 248331DEST_PATH_IMAGE088
A row vector matrix constructed by the elements of the (k + 1) th row;
step S440, the step S430 is circulated until the iteration times k = beta m, the value of m is equal to the number of different workpiece types to be blanked, beta represents the coefficient of iteration, and beta =1or2; and after the iteration is carried out for k times, a feasible solution of a new mathematical model formed by a third constraint condition and a third objective function is obtained.
Preferably, the initial identity matrix in step S420
Figure 418150DEST_PATH_IMAGE079
The matrix a before the iteration starts is specifically:
Figure 436922DEST_PATH_IMAGE093
（4）
Figure 52711DEST_PATH_IMAGE094
（5）
preferably, the k iterations of the calculation in step S430 are solved by using a Gurobi optimizer.
Preferably, the step S5 specifically includes the following steps:
step S510, an initial feasible solution is that each column in a matrix A is formed by m cutting modes generated by the solution, a decision variable corresponding to each column (each cutting mode) in the matrix A is established according to the matrix A, and an initial iteration base matrix B of a CG algorithm is established;
step S520, a new mathematical model (VTC model) is provided, iterative training is carried out on the VTC model by means of a Gurobi optimizer until the iterative times reach a set value, an initial feasible solution is obtained, and an initial iteration base matrix B is constructed;
and S530, based on the initial iteration base matrix B, carrying out iteration training on the initial iteration base matrix B by utilizing a CG algorithm until the value of the second objective function is not reduced any more, obtaining an optimal relaxation solution, comparing the value of the sum of the optimal decision variables with the value of the sum of the decision variables in the step S4, selecting the decision variable generated by the party with the minimum sum as a final decision variable, wherein the corresponding blanking layout scheme is the final raw material blanking layout scheme, and if the optimal relaxation solution is not completely a non-negative integer, entering the step S6.
The invention has the beneficial effects that:
the method comprises two different stages which are respectively a first stage and a second stage, wherein the first stage is a VTC algorithm solving stage in a step S4, the second stage is a CG algorithm solving stage in a step S5, a VTCCG algorithm is formed by a CG algorithm improved based on a VTC new model, the number of cutting and layout schemes obtained through the VTCCG algorithm is small, the cutting and layout schemes are superior to the existing similar solving algorithm, and the solving quality of a column generation accurate algorithm can be well improved when the blanking problem with small specification and low requirement is solved; meanwhile, the solving result of the method has the advantages of saving more materials, saving cost and reducing the number of cutting modes.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a layout plan for solving example 1 according to the present invention;
figure 3 is a layout plan for the creative door and window blanking software solution example 1.
Detailed Description
The invention is further described in detail below with reference to the drawings and specific embodiments.
Referring to fig. 1, an embodiment of the present application provides a method for solving a raw material blanking layout, including the following steps:
s1, determining the length of raw materials, the number of each blanking workpiece and the size of each blanking workpiece;
s2, establishing an original mathematical model according to the length of the raw materials, the number of each blanking workpiece and the size of each blanking workpiece, and performing matrixing on the original mathematical model to obtain a simplified mathematical model;
s3, establishing an initial identity matrix, introducing constant parameters, and performing initial row change on the simplified mathematical model by using the initial identity matrix and the constant parameters to obtain a new mathematical model;
s4, solving the new mathematical model (called VTC model) by means of a Gurobi optimizer to obtain an initial feasible solution of the new mathematical model, wherein the initial feasible solution comprises a blanking layout scheme and decision variables, and the decision variables refer to variables corresponding to different layout schemes, namely the number of raw materials used corresponding to each layout scheme;
s5, continuously solving on the basis of an initial feasible solution by utilizing a CG algorithm to obtain an optimal relaxation solution, wherein the optimal relaxation solution refers to an optimal decision variable, if the optimal relaxation solution is all non-negative integers, the sum value of the optimal decision variable is compared with the sum value of the decision variables in the step S4, the decision variable generated by the party with the minimum sum is selected as a final decision variable, the corresponding blanking layout scheme is a final raw material blanking layout scheme, and if the optimal relaxation solution is not all non-negative integers, the step S6 is entered;
and S6, rounding the optimal relaxation solution obtained in the step S5 by means of a heuristic algorithm to obtain a non-negative integer solution, comparing the sum value of the optimal decision variables with the sum value of the decision variables in the step S4, selecting the decision variable generated by the party with the minimum sum as a final decision variable, and taking the corresponding blanking layout scheme as the final raw material blanking layout scheme.
In this embodiment, the original mathematical model in step S2 is:
a first objective function:
Figure 674185DEST_PATH_IMAGE001
the first constraint condition is:
Figure 324609DEST_PATH_IMAGE002
Figure 389648DEST_PATH_IMAGE003
Figure 758313DEST_PATH_IMAGE004
Figure 917899DEST_PATH_IMAGE005
，
Figure 688409DEST_PATH_IMAGE006
，
Figure 783404DEST_PATH_IMAGE007
，
Figure 750616DEST_PATH_IMAGE008
，
Figure 589259DEST_PATH_IMAGE009
（1）
wherein,
Figure 338909DEST_PATH_IMAGE095
representing a decision variable, in particular representing the quantity of raw material used in the j cutting mode;
Figure 604805DEST_PATH_IMAGE096
the number of ith products to be blanked in the jth cutting mode is represented;
Figure 557849DEST_PATH_IMAGE097
the total quantity of the i type workpieces to be blanked is represented;
Figure 934604DEST_PATH_IMAGE098
the dimension of the required ith workpiece to be blanked is represented; l represents the length of the raw material;
Figure 538760DEST_PATH_IMAGE014
representing a non-negative integer.
In this embodiment, the simplified mathematical model in step S2 is specifically:
a second objective function:
Figure 975558DEST_PATH_IMAGE099
the second constraint condition is as follows:
Figure 648854DEST_PATH_IMAGE100
Figure 94878DEST_PATH_IMAGE101
Figure 819121DEST_PATH_IMAGE102
，
Figure 426820DEST_PATH_IMAGE103
，
Figure 88876DEST_PATH_IMAGE020
，
Figure 73013DEST_PATH_IMAGE104
Figure 917341DEST_PATH_IMAGE022
（2）
wherein,
Figure 695941DEST_PATH_IMAGE023
a matrix with 1 row and n columns is represented; solution vector matrix
Figure 346759DEST_PATH_IMAGE024
The representation is a matrix of n rows and 1 column, the elements of which are decision variables whose values are constrained to be greater than or equal to zero and to be integers; a represents a matrix of the layout scheme (cutting pattern), and is a matrix of m rows and n columns (
Figure 134586DEST_PATH_IMAGE105
) When all the cutting modes are solved, all the columns in the matrix A are formed by the generated cutting modes, each column represents one cutting mode, and before iteration starts, the cutting modes are an identity matrix with m rows and m columns;
Figure 443208DEST_PATH_IMAGE026
a column vector demand matrix representing m rows and 1 column, the elements of the matrix representing the required number of workpieces of different sizes, mThe number of workpiece types required is equal to m;
Figure 517343DEST_PATH_IMAGE027
a matrix representing m rows and 1 columns, which is a column variable vector matrix, in which the values of the elements are non-negative integers,
Figure 544205DEST_PATH_IMAGE028
representing a cutting pattern currently to be solved by
Figure 745510DEST_PATH_IMAGE028
As
Figure 174217DEST_PATH_IMAGE029
The index of (a) is determined,
Figure 419254DEST_PATH_IMAGE028
represents all possible cutting pattern sets, such as:
Figure 667833DEST_PATH_IMAGE029
a cutting pattern representing the current iteration needs to be solved
Figure 797463DEST_PATH_IMAGE028
In which the elements
Figure 454578DEST_PATH_IMAGE030
Is shown in cutting mode
Figure 745882DEST_PATH_IMAGE028
The number of the i-th workpieces,
Figure 871970DEST_PATH_IMAGE031
；
Figure 539711DEST_PATH_IMAGE032
representing a row vector matrix of 1 row and m columns, the elements of which represent the dimensions of m different workpieces, e.g.
Figure 552798DEST_PATH_IMAGE106
Size of the ith workpiece
Figure 15003DEST_PATH_IMAGE031
；
Figure 503753DEST_PATH_IMAGE034
Is shown in cutting mode
Figure 99820DEST_PATH_IMAGE028
Wherein the sum of the product of the size of the different workpieces multiplied by the number of the respective corresponding different workpieces is less than or equal to the length of the raw material
Figure 92047DEST_PATH_IMAGE107
；
Figure 90266DEST_PATH_IMAGE036
In (1)0A column vector matrix representing m rows and 1 columns of all 0 elements, wherein the matrix
Figure 66312DEST_PATH_IMAGE029
All the variable values are constrained to be greater than or equal to zero and must be less than or equal to the required quantity and value corresponding to each type of workpiece
Figure 200490DEST_PATH_IMAGE037
，
Figure 47223DEST_PATH_IMAGE108
Representing a non-negative integer.
In this embodiment, the step S3 specifically includes the following steps:
step S310, establishing an initial identity matrix, and introducing a constant parameter
Figure 382390DEST_PATH_IMAGE038
In each iteration process, fixing a decision variable corresponding to a kth column (unknown column) as a constant parameter, wherein the constant parameter is equal to a value of the decision variable corresponding to the kth column obtained by the previous iteration solution; at iterationAt the beginning, get
Figure 189940DEST_PATH_IMAGE039
，
Figure 3175DEST_PATH_IMAGE040
Representing a column vector demand matrixd The value of the 1 st element in (c), corresponding to the required number of first workpieces,
Figure 829049DEST_PATH_IMAGE041
the decision variables corresponding to the 1 st column in the layout scheme matrix A are also represented corresponding to the first cutting mode
Figure 69537DEST_PATH_IMAGE028
A decision variable of (c);
and step S320, carrying out primary variation on the simplified mathematical model by using the constant parameters to obtain a new mathematical model.
In this embodiment, the new mathematical model in step S320 is specifically:
assuming that the number of decision variables is m and the number of different workpiece types is m, each iteration generates a new column, which also represents a new cutting pattern
Figure 128498DEST_PATH_IMAGE028
When iterative solution is carried out, the kth cutting mode is generated
Figure 479845DEST_PATH_IMAGE028
Its objective function and constraint conditions can be written as follows:
a third objective function:
Figure 425804DEST_PATH_IMAGE042
the third constraint condition is as follows:
Figure 102773DEST_PATH_IMAGE109
Figure 9549DEST_PATH_IMAGE110
Figure 39953DEST_PATH_IMAGE111
Figure 715785DEST_PATH_IMAGE046
, and
Figure 688289DEST_PATH_IMAGE047
，
Figure 82361DEST_PATH_IMAGE048
，
Figure 152342DEST_PATH_IMAGE049
Figure 948260DEST_PATH_IMAGE112
，
Figure 560506DEST_PATH_IMAGE051
，
Figure 441875DEST_PATH_IMAGE113
Figure 204294DEST_PATH_IMAGE114
，
Figure 730085DEST_PATH_IMAGE115
（3）
in the formula (3), the compound represented by the formula (I),
Figure 654179DEST_PATH_IMAGE028
representing a cutting pattern currently to be solved by
Figure 881898DEST_PATH_IMAGE028
As
Figure 182429DEST_PATH_IMAGE055
The index of (a) is determined,
Figure 326840DEST_PATH_IMAGE028
a set of all possible cutting patterns is represented,
Figure 421835DEST_PATH_IMAGE027
a matrix representing m rows and 1 columns, which is a column variable vector matrix, such as:
Figure 277796DEST_PATH_IMAGE055
representing cutting patterns currently required to be solved
Figure 241072DEST_PATH_IMAGE028
In which the elements
Figure 866089DEST_PATH_IMAGE030
Is shown in cutting mode
Figure 7351DEST_PATH_IMAGE055
The number of the (i) th workpieces,
Figure 350608DEST_PATH_IMAGE031
(ii) a Wherein,
Figure 851996DEST_PATH_IMAGE038
indicating that in the current iteration, the cutting mode is to be used
Figure 331519DEST_PATH_IMAGE028
Fixing a decision variable corresponding to a kth column (in a corresponding matrix A) as a constant parameter, wherein the constant parameter is equal to a value of the decision variable corresponding to the kth column obtained by previous iteration solution;
Figure 33896DEST_PATH_IMAGE057
representing solution to cutting patterns
Figure 710121DEST_PATH_IMAGE028
(corresponding to the k-th column in matrix A), byMatrix the column vector
Figure 156146DEST_PATH_IMAGE058
The kth element of (a) is forced to be 0; in the same way, the method for preparing the composite material,
Figure 614809DEST_PATH_IMAGE059
representing solution to cutting patterns
Figure 222508DEST_PATH_IMAGE116
By combining the matrices
Figure 150144DEST_PATH_IMAGE060
The k-th row of the matrix A is obtained by forcing the elements of the k-th row to be 0 elements, the function of the matrix A is to replace the matrix A, repeated and complicated matrix change of the matrix A can be avoided, and only the previous iteration solution needs to be provided
Figure 399860DEST_PATH_IMAGE117
And
Figure 713030DEST_PATH_IMAGE062
then the process is carried out;
Figure 491630DEST_PATH_IMAGE063
representing a column vector variable matrix
Figure 405097DEST_PATH_IMAGE080
Participating in a matrix expression after the initial change of the matrix; to restrain
Figure 192924DEST_PATH_IMAGE064
The decision variable corresponding to the k column in the decision variable matrix is represented as a fixed value
Figure 501546DEST_PATH_IMAGE038
Besides, all the other decision variables are constrained to be larger than or equal to zero; in the constraint
Figure 575681DEST_PATH_IMAGE034
In (1),
Figure 602543DEST_PATH_IMAGE118
representing a row vector matrix of 1 row and m columns, the elements of which represent the dimensions of m different workpieces, e.g.
Figure 803848DEST_PATH_IMAGE119
Size of the ith workpiece
Figure 966976DEST_PATH_IMAGE031
；
Figure 477592DEST_PATH_IMAGE034
Is shown in cutting mode
Figure 991750DEST_PATH_IMAGE028
Wherein the sum of the product of the size of the different workpieces multiplied by the number of the respective corresponding different workpieces is less than or equal to the length of the raw material
Figure 855801DEST_PATH_IMAGE120
；
Figure 515846DEST_PATH_IMAGE121
Variable matrix representing unknown column vectors
Figure 807150DEST_PATH_IMAGE080
After the participation matrix A is subjected to the initial row change, the expression of the k-th row and k-th column element in the matrix A has the value equal to 1, wherein
Figure 933238DEST_PATH_IMAGE069
Is a matrix with 1 row and m columns,
Figure 600979DEST_PATH_IMAGE069
is obtained by combining a matrix
Figure 614066DEST_PATH_IMAGE060
The row vector matrix constructed by the k line of (1), wherein 0 is less than or equal toa p ≤d Representing a column vector variable matrix
Figure 341850DEST_PATH_IMAGE080
All the variable values are constrained to be greater than or equal to zero and must be less than or equal to the required number corresponding to each type of workpiece, the values of which
Figure 565021DEST_PATH_IMAGE037
，
Figure 895509DEST_PATH_IMAGE122
Represents a non-negative integer; objective function
Figure 887735DEST_PATH_IMAGE072
In
Figure 425902DEST_PATH_IMAGE123
Is a 1-row m-column matrix with all 1 elements;
note that: when the iteration number k is more than or equal to m +1, each cutting mode is generated
Figure 136369DEST_PATH_IMAGE028
Replace an old column, the specific replacement rule: m +1 cutting mode
Figure 145913DEST_PATH_IMAGE074
Substitution of the 1 st column, m +2 th cutting pattern in matrix A
Figure 382859DEST_PATH_IMAGE028
Substitute column 2 in matrix A, and so on, 2m cutting patterns
Figure 186867DEST_PATH_IMAGE028
Replace the mth column in matrix a; note also that after the matrix a is subjected to the matrix elementary row change, each column in a cannot cut the pattern, and only after the solution of all the cutting patterns is completed, the cutting patterns are combined into each column in the matrix a, where each column in a represents one cutting pattern.
In this embodiment, the step S4 specifically includes the following steps:
step S410, solving the new mathematical model by using a Gurobi optimizer;
step S420, passing
Figure 259997DEST_PATH_IMAGE075
，
Figure 807653DEST_PATH_IMAGE124
And
Figure 164685DEST_PATH_IMAGE062
updating a third objective function and a third constraint in the new mathematical model, wherein
Figure 405173DEST_PATH_IMAGE077
，
Figure 213203DEST_PATH_IMAGE078
And
Figure 564550DEST_PATH_IMAGE069
can pass through the initial identity matrix
Figure 651455DEST_PATH_IMAGE081
Obtaining; solved by means of a Gurobi optimizer
Figure 187478DEST_PATH_IMAGE075
And
Figure 94254DEST_PATH_IMAGE062
at this time, the first cutting mode
Figure 859079DEST_PATH_IMAGE080
Is solved and used
Figure 800490DEST_PATH_IMAGE075
And
Figure 772994DEST_PATH_IMAGE062
updating the first column of the matrix A, performing the initial row change on the updated matrix A, and simultaneously performing the initial unit matrix
Figure 167067DEST_PATH_IMAGE081
Sum matrix
Figure 234118DEST_PATH_IMAGE082
Making the same initial row change as the matrix A to obtain an updated matrix
Figure 764456DEST_PATH_IMAGE081
The latter matrix
Figure 783228DEST_PATH_IMAGE083
Then according to the matrix
Figure 789230DEST_PATH_IMAGE083
To obtain a matrix
Figure 286070DEST_PATH_IMAGE084
While obtaining the updated matrix
Figure 811861DEST_PATH_IMAGE082
The latter matrix
Figure 735954DEST_PATH_IMAGE085
According to a matrix
Figure 229252DEST_PATH_IMAGE085
To obtain a matrix
Figure 264205DEST_PATH_IMAGE086
(ii) a In the same way, use
Figure 300294DEST_PATH_IMAGE087
，
Figure 506540DEST_PATH_IMAGE086
And
Figure 362501DEST_PATH_IMAGE062
and updating a third objective function and a third constraint condition corresponding to the new mathematical model after 1 iteration, wherein at the moment,
Figure 325778DEST_PATH_IMAGE080
unknown, needs to be solved, wherein
Figure 685215DEST_PATH_IMAGE069
By passing
Figure 92057DEST_PATH_IMAGE083
Obtaining;
step S430, performing k iterations, k =2.. K, on the new mathematical model by means of a Gurobi optimizer, calculating
Figure 169734DEST_PATH_IMAGE063
And
Figure 936702DEST_PATH_IMAGE062
: will be provided with
Figure 416225DEST_PATH_IMAGE063
And
Figure 118601DEST_PATH_IMAGE062
as the kth column of matrix a, where,
Figure 57476DEST_PATH_IMAGE069
can pass through the matrix
Figure 503501DEST_PATH_IMAGE060
Obtaining, and then performing an initial row change on the matrix A, and simultaneously performing the matrix change on the matrix A
Figure 962164DEST_PATH_IMAGE060
Sum matrix
Figure 569863DEST_PATH_IMAGE058
Making the same initial row change as the matrix A to obtain the matrix
Figure 497499DEST_PATH_IMAGE088
Then according to the matrix
Figure 216056DEST_PATH_IMAGE088
Obtaining a matrix
Figure 935751DEST_PATH_IMAGE089
While obtaining the updated matrix
Figure 838985DEST_PATH_IMAGE058
The latter matrix
Figure 378550DEST_PATH_IMAGE090
Then obtaining the matrix according to the matrix
Figure 277630DEST_PATH_IMAGE090
In combination with each other
Figure 586251DEST_PATH_IMAGE092
，
Figure 925966DEST_PATH_IMAGE091
And
Figure 687248DEST_PATH_IMAGE062
and updating a third objective function and a third constraint condition corresponding to the new mathematical model after the iteration is performed for k times, wherein at the moment,
Figure 888554DEST_PATH_IMAGE080
unknown, needs to be solved, wherein
Figure 51682DEST_PATH_IMAGE069
By passing
Figure 437664DEST_PATH_IMAGE088
Obtaining;
step 440, the step 430 is circulated until the iteration times k = beta m, the value of m is equal to the number of different workpiece types to be blanked, beta represents the iteration coefficient, and beta =1or2; and after the iteration is carried out for k times, a feasible solution of a new mathematical model formed by a third constraint condition and a third objective function is obtained.
In this embodiment, the initial identity matrix in step S420
Figure 76455DEST_PATH_IMAGE081
The matrix a before the iteration starts is specifically:
Figure 940506DEST_PATH_IMAGE125
（4）
Figure 332042DEST_PATH_IMAGE126
（5）
in this embodiment, the k iterations of the calculation in step S430 are solved by using a Gurobi optimizer.
The steps S410 to S440 are the solving phase of the VTC new model, which is illustrated by a simple example below.
Assuming that the order requires the blanking of 4 different types of workpieces, the original material lengthL =300，Column vector demand matrix
Figure 154505DEST_PATH_IMAGE127
(ii) a The size of 4 workpieces is required
Figure 15013DEST_PATH_IMAGE128
，
Figure 682755DEST_PATH_IMAGE129
Is a row vector matrix.
Figure 820475DEST_PATH_IMAGE130
Is a column vector variable matrix that represents a cutting pattern being solved for. Is provided witht= (1,1,1,1) is a matrix with row 1 and column 4. Before the iteration starts, the following two matrices are initialized:
Figure 423626DEST_PATH_IMAGE131
(6)
Figure 646797DEST_PATH_IMAGE132
(7)
establishing a matrix type new mathematical model:
an objective function:
Figure 977284DEST_PATH_IMAGE133
(8)
constraint conditions are as follows:
Figure 969511DEST_PATH_IMAGE134
(9)
at this point, we can obtain:
Figure 245028DEST_PATH_IMAGE135
(10)
Figure 689916DEST_PATH_IMAGE136
(11)
start of 1 st iteration (1 st cutting pattern generation):
according to equation (3), we obtain the objective optimization problem as follows:
an objective function:
Figure 824094DEST_PATH_IMAGE137
(12)
constraint conditions are as follows:
Figure 936406DEST_PATH_IMAGE138
(13)
Figure 881360DEST_PATH_IMAGE139
(14)
Figure 79123DEST_PATH_IMAGE140
，
Figure 751413DEST_PATH_IMAGE141
，
Figure 718232DEST_PATH_IMAGE142
，
Figure 958720DEST_PATH_IMAGE143
(15)
Figure 17681DEST_PATH_IMAGE144
，
Figure 369028DEST_PATH_IMAGE145
，
Figure 314987DEST_PATH_IMAGE146
，
Figure 991956DEST_PATH_IMAGE147
，
Figure 774098DEST_PATH_IMAGE148
(16)
to simplify the repetitive tedious first row change of the matrix, we will change the matrix
Figure 663557DEST_PATH_IMAGE149
The first column in (a) is expressed as follows:
Figure 729602DEST_PATH_IMAGE150
(17)
will decide variables
Figure 46314DEST_PATH_IMAGE151
Fixed as constant parameters
Figure 705965DEST_PATH_IMAGE152
So we can get:
Figure 764227DEST_PATH_IMAGE153
(18)
Figure 560145DEST_PATH_IMAGE154
(19)
Figure 437971DEST_PATH_IMAGE155
(20)
Figure 584918DEST_PATH_IMAGE156
(21)
by the formula (7)
Figure 691546DEST_PATH_IMAGE157
We can obtain
Figure 341970DEST_PATH_IMAGE158
By passing
Figure 390697DEST_PATH_IMAGE159
To obtain
Figure 24941DEST_PATH_IMAGE160
Thus, equations (18) through (20) can be converted into:
Figure 59893DEST_PATH_IMAGE161
(22)
we now substitute equations (18) through (20) into equation (12), so that the objective optimization problem can be written as:
an objective function:
Figure 938725DEST_PATH_IMAGE162
(23)
constraint conditions are as follows:
Figure 299299DEST_PATH_IMAGE163
(24)
Figure 279894DEST_PATH_IMAGE164
(25)
Figure 852957DEST_PATH_IMAGE165
(26)
Figure 353340DEST_PATH_IMAGE166
(27)
Figure 619236DEST_PATH_IMAGE167
，
Figure 962493DEST_PATH_IMAGE168
，
Figure 729461DEST_PATH_IMAGE169
，
Figure 208984DEST_PATH_IMAGE170
(28)
Figure 22612DEST_PATH_IMAGE171
we can now convert equations (24) to (27) into the following matrix form:
Figure 587586DEST_PATH_IMAGE172
(29)
Figure 892665DEST_PATH_IMAGE173
(30)
solving the above optimization problem with the Gurobi optimizer we can get:
Figure 492274DEST_PATH_IMAGE174
，
Figure 975339DEST_PATH_IMAGE175
，
Figure 293188DEST_PATH_IMAGE176
，
Figure 11745DEST_PATH_IMAGE177
；
Figure 590494DEST_PATH_IMAGE178
，
Figure 369094DEST_PATH_IMAGE179
，
Figure 548140DEST_PATH_IMAGE180
，
Figure 70389DEST_PATH_IMAGE181
；
Figure 503644DEST_PATH_IMAGE182
therefore, we can obtain the 1 st column in the cutting mode update matrix A obtained by solving
Figure 453145DEST_PATH_IMAGE183
The following were used:
Figure 480007DEST_PATH_IMAGE184
(31)
start of iteration 2 (generation of cutting pattern 2):
also, according to equation (31), we obtain the objective optimization problem as follows:
an objective function:
Figure 681313DEST_PATH_IMAGE185
(32)
constraint conditions are as follows:
Figure 844441DEST_PATH_IMAGE186
(33)
Figure 355056DEST_PATH_IMAGE187
(34)
Figure 869214DEST_PATH_IMAGE188
，
Figure 110096DEST_PATH_IMAGE189
，
Figure 127731DEST_PATH_IMAGE190
，
Figure 950193DEST_PATH_IMAGE191
(35)
Figure 810702DEST_PATH_IMAGE192
，
Figure 212864DEST_PATH_IMAGE193
，
Figure 491530DEST_PATH_IMAGE194
，
Figure 219315DEST_PATH_IMAGE195
.
Figure 301540DEST_PATH_IMAGE196
(36)
now, we perform the matrix elementary row change on equation (33), and at the same time, the matrix in equation (7)
Figure 772973DEST_PATH_IMAGE081
Making the same matrix elementary line change as formula (33) to obtain
Figure 139101DEST_PATH_IMAGE197
The following expressions (37) and (38), respectively:
Figure 37787DEST_PATH_IMAGE198
(37)
Figure 748254DEST_PATH_IMAGE199
(38)
at this time, the decision variables are
Figure 882432DEST_PATH_IMAGE200
Is fixed to
Figure 729165DEST_PATH_IMAGE201
So we can get:
Figure 939698DEST_PATH_IMAGE202
(39)
Figure 871882DEST_PATH_IMAGE203
(40)
Figure 809751DEST_PATH_IMAGE204
(41)
Figure 42149DEST_PATH_IMAGE205
(42)
in the same way, by
Figure 17058DEST_PATH_IMAGE206
Order to
Figure 78949DEST_PATH_IMAGE207
By the formula (37), let
Figure 430295DEST_PATH_IMAGE208
Thus, the equations (39) to (41) can be written in the form of a matrix as follows:
Figure 641834DEST_PATH_IMAGE209
(43)
therefore, the above objective optimization problem can be updated to the form:
an objective function:
Figure 787645DEST_PATH_IMAGE210
(44)
constraint conditions are as follows:
Figure 569787DEST_PATH_IMAGE211
(45)
Figure 724825DEST_PATH_IMAGE212
(46)
Figure 525290DEST_PATH_IMAGE213
(47)
Figure 373161DEST_PATH_IMAGE214
(48)
Figure 767233DEST_PATH_IMAGE215
(49)
Figure 834284DEST_PATH_IMAGE216
，
Figure 630202DEST_PATH_IMAGE217
，
Figure 773607DEST_PATH_IMAGE218
，
Figure 389396DEST_PATH_IMAGE219
(50)
Figure 761603DEST_PATH_IMAGE220
the constraints (45) to (48) can be written in the form of a matrix as follows:
Figure 677606DEST_PATH_IMAGE221
(51)
Figure 601700DEST_PATH_IMAGE222
(52)
will matrix
Figure 94998DEST_PATH_IMAGE083
Row 2 of (a) constructs a row vector matrix
Figure 129950DEST_PATH_IMAGE223
Thus, the equation constraint (49) can be written as follows:
Figure 23431DEST_PATH_IMAGE224
(53)
solving the above optimization problem with the Gurobi optimizer we can get:
Figure 384005DEST_PATH_IMAGE225
，
Figure 99020DEST_PATH_IMAGE226
，
Figure 937663DEST_PATH_IMAGE227
，
Figure 438045DEST_PATH_IMAGE228
；
Figure 969521DEST_PATH_IMAGE229
，
Figure 47198DEST_PATH_IMAGE230
，
Figure 548587DEST_PATH_IMAGE231
，
Figure 28110DEST_PATH_IMAGE232
；
Figure 838809DEST_PATH_IMAGE233
therefore, the 2 nd cutting pattern is solved and the 2 nd column in the matrix A is updated with the cutting pattern to obtain
Figure 669361DEST_PATH_IMAGE234
As follows:
Figure 974441DEST_PATH_IMAGE235
(54)
start of 3 rd iteration (3 rd cutting pattern generation):
at iteration 3, the objective optimization problem is updated as follows:
an objective function:
Figure 574049DEST_PATH_IMAGE236
(55)
constraint conditions are as follows:
Figure 447327DEST_PATH_IMAGE237
(56)
Figure 109384DEST_PATH_IMAGE238
(57)
Figure 93521DEST_PATH_IMAGE216
，
Figure 937849DEST_PATH_IMAGE239
，
Figure 450870DEST_PATH_IMAGE240
，
Figure 101687DEST_PATH_IMAGE241
(58)
Figure 623935DEST_PATH_IMAGE242
，
Figure 57191DEST_PATH_IMAGE243
，
Figure 616479DEST_PATH_IMAGE244
，
Figure 908920DEST_PATH_IMAGE245
.
Figure 828335DEST_PATH_IMAGE246
(59)
in this case, although equation (56) is changed to equation (60) by performing an initial row change, we can choose to perform an initial row change directly on column 2 in equation (37) in order to simplify the repetitive complicated matrix change and reduce the calculation time of the matrix change, and this column can be passed through
Figure 257042DEST_PATH_IMAGE087
And
Figure 751346DEST_PATH_IMAGE062
obtained because of the 2 nd cutting mode
Figure 999925DEST_PATH_IMAGE080
Has been solved by iteration 2. Thus, in particular, we are directed to
Figure 988610DEST_PATH_IMAGE087
And
Figure 271823DEST_PATH_IMAGE062
carrying out the primary variation to obtain
Figure 438494DEST_PATH_IMAGE083
Is updated to
Figure 174368DEST_PATH_IMAGE247
The following expressions (60) and (61) are used:
Figure 107689DEST_PATH_IMAGE248
(60)
Figure 370043DEST_PATH_IMAGE249
(61)
at this point, we will decide on the variables
Figure 832249DEST_PATH_IMAGE250
Fixed as a constant
Figure 432251DEST_PATH_IMAGE251
In this way we can get:
Figure 169263DEST_PATH_IMAGE252
(62)
Figure 286123DEST_PATH_IMAGE253
(63)
Figure 919230DEST_PATH_IMAGE254
(64)
Figure 895276DEST_PATH_IMAGE255
(65)
in the same way, by
Figure 780187DEST_PATH_IMAGE256
Can obtain
Figure 626920DEST_PATH_IMAGE257
，
Figure 821141DEST_PATH_IMAGE258
Can be obtained by the formula (60), and then by
Figure 18904DEST_PATH_IMAGE259
Can obtain
Figure 206041DEST_PATH_IMAGE260
Therefore, equations (62) to (64) can be written as:
Figure 907280DEST_PATH_IMAGE262
(66)
for this iteration, the objective function can be written as follows:
an objective function:
Figure 272403DEST_PATH_IMAGE263
(67)
constraint conditions are as follows:
Figure 691883DEST_PATH_IMAGE264
(68)
Figure 308809DEST_PATH_IMAGE265
(69)
Figure 5500DEST_PATH_IMAGE266
(70)
Figure 416890DEST_PATH_IMAGE267
(71)
Figure 448300DEST_PATH_IMAGE268
(72)
Figure 868917DEST_PATH_IMAGE269
，
Figure 921580DEST_PATH_IMAGE270
，
Figure 503871DEST_PATH_IMAGE240
，
Figure 22577DEST_PATH_IMAGE271
(73)
Figure 715727DEST_PATH_IMAGE246
constraints (68) through (71) can be written as follows:
Figure 777224DEST_PATH_IMAGE272
(74)
Figure 140203DEST_PATH_IMAGE273
(75)
will matrix
Figure 21571DEST_PATH_IMAGE274
Line 3 in (a) is converted into a row vector matrix
Figure 643046DEST_PATH_IMAGE275
Thus, the equation constraint (72) can be written as:
Figure 293470DEST_PATH_IMAGE276
(76)
the above optimization problem is solved by means of a Gurobi optimizer, so that we obtain the result of the 3 rd iteration as follows:
Figure 591465DEST_PATH_IMAGE277
，
Figure 694550DEST_PATH_IMAGE278
，
Figure 854136DEST_PATH_IMAGE279
，
Figure 624646DEST_PATH_IMAGE280
；
Figure 595007DEST_PATH_IMAGE281
，
Figure 185388DEST_PATH_IMAGE282
，
Figure 148665DEST_PATH_IMAGE283
，
Figure 773682DEST_PATH_IMAGE284
；
Figure 404690DEST_PATH_IMAGE285
note that: result of iteration 2
Figure 482368DEST_PATH_IMAGE286
With the result of iteration 3
Figure 983756DEST_PATH_IMAGE287
Are identical to each other, and
Figure 463279DEST_PATH_IMAGE288
and
Figure 509863DEST_PATH_IMAGE289
are identical to each otherTherefore, the temperature of the molten steel is controlled,
Figure 199471DEST_PATH_IMAGE290
as follows:
Figure 379916DEST_PATH_IMAGE291
(77)
the 4 th iteration starts (4 th cutting pattern generation):
the objective optimization problem for the 4 th iteration is updated as follows:
an objective function:
Figure 822268DEST_PATH_IMAGE292
(78)
constraint conditions are as follows:
Figure 429967DEST_PATH_IMAGE293
(79)
Figure 606870DEST_PATH_IMAGE294
(80)
Figure 591007DEST_PATH_IMAGE295
，
Figure 654909DEST_PATH_IMAGE296
，
Figure 433509DEST_PATH_IMAGE297
，
Figure 832129DEST_PATH_IMAGE298
(81)
Figure 731209DEST_PATH_IMAGE242
，
Figure 39830DEST_PATH_IMAGE243
，
Figure 113965DEST_PATH_IMAGE244
，
Figure 140827DEST_PATH_IMAGE245
.
Figure 466766DEST_PATH_IMAGE246
(82)
likewise, as in iteration 3, directly on
Figure 505261DEST_PATH_IMAGE299
And
Figure 891243DEST_PATH_IMAGE300
making an initial change while obtaining
Figure 530034DEST_PATH_IMAGE301
Is updated to
Figure 394085DEST_PATH_IMAGE302
It is made as follows:
Figure 51200DEST_PATH_IMAGE303
(83)
Figure 342504DEST_PATH_IMAGE304
(84)
observation of (83) formula
Figure 343958DEST_PATH_IMAGE305
And in (84) formula
Figure 136334DEST_PATH_IMAGE302
The following equation holds true for the relationship of (1):
Figure 274054DEST_PATH_IMAGE306
(85)
therefore, variables will be decided
Figure 611626DEST_PATH_IMAGE307
Is fixed to
Figure 100376DEST_PATH_IMAGE308
Thereafter, the remaining decision variables and column variables
Figure 430863DEST_PATH_IMAGE080
The relationship between them is as follows:
Figure 423090DEST_PATH_IMAGE309
(86)
Figure 587355DEST_PATH_IMAGE310
(87)
Figure 674653DEST_PATH_IMAGE311
(88)
Figure 418618DEST_PATH_IMAGE312
(89)
by means of a matrix
Figure 655564DEST_PATH_IMAGE313
Can obtain
Figure 459572DEST_PATH_IMAGE314
By the same token
Figure 798281DEST_PATH_IMAGE315
Can obtain
Figure 345937DEST_PATH_IMAGE316
Thus, equations (86) through (88) are written as:
Figure 437390DEST_PATH_IMAGE317
(90)
thus, the objective optimization problem can be written as:
an objective function:
Figure 677878DEST_PATH_IMAGE318
(91)
constraint conditions are as follows:
Figure 362938DEST_PATH_IMAGE319
(92)
Figure 88186DEST_PATH_IMAGE320
(93)
Figure 909511DEST_PATH_IMAGE321
(94)
Figure 711114DEST_PATH_IMAGE322
(95)
Figure 352311DEST_PATH_IMAGE323
(96)
Figure 382715DEST_PATH_IMAGE324
，
Figure 324126DEST_PATH_IMAGE325
，
Figure 31051DEST_PATH_IMAGE326
，
Figure 690702DEST_PATH_IMAGE327
.
Figure 383852DEST_PATH_IMAGE328
(97)
at this time, the constraints (92) to (95) can be written in the form of a matrix as follows:
Figure 291022DEST_PATH_IMAGE329
(98)
Figure 44214DEST_PATH_IMAGE330
(99)
will matrix
Figure 315795DEST_PATH_IMAGE313
Line 4 in (a) is converted into a row vector matrix
Figure 812636DEST_PATH_IMAGE331
Thus, the equation constraint (96) can be written as follows:
Figure 338426DEST_PATH_IMAGE332
(100)
solving the above optimization problem with the Gurobi optimizer, the result of the 4 th iteration (4 th cutting mode) can be obtained as follows:
Figure 262520DEST_PATH_IMAGE333
，
Figure 631184DEST_PATH_IMAGE334
，
Figure 790770DEST_PATH_IMAGE335
，
Figure 826859DEST_PATH_IMAGE336
；
Figure 764597DEST_PATH_IMAGE337
，
Figure 886137DEST_PATH_IMAGE338
，
Figure 849414DEST_PATH_IMAGE339
，
Figure 208851DEST_PATH_IMAGE340
；
Figure 350113DEST_PATH_IMAGE341
finally, we write the solution result in the form of a matrix as follows:
Figure 427791DEST_PATH_IMAGE342
；
Figure 335704DEST_PATH_IMAGE343
(ii) a Value of objective function
Figure 939861DEST_PATH_IMAGE344
This case shows the specific steps of solving the new mathematical model established by the present invention, since the 4 th iteration has reached the optimal integer solution (the optimal relaxed solution can be obtained by the column generation exact algorithm to be 5.9, so the lower bound of the optimal integer solution is 6), and the decision variable value has no decimal, so there is no need to round the decision variable. The above case only shows the solving step of the iteration number k = β m (β = 1), and when the iteration number k = β m (β = 1), the solving step is the same, and only the cutting pattern generated by the m +1 th iteration is required to replace the 1 st column in the matrix a, and so on until the iteration number k =2 m.
In each iteration process, the decision variable corresponding to the current solution column (cutting mode) is set as a constant, and the value of the constant is equal to the value of the decision variable corresponding to the column after the previous iteration. Therefore, all the other decision variables and the unknown column vector variable matrix needing to be solved currently
Figure 376658DEST_PATH_IMAGE080
(cutting mode)
Figure 408261DEST_PATH_IMAGE080
) A distinct linear relationship is established.
In this embodiment, the step S5 specifically includes the following steps:
step S510, an initial feasible solution is that each column in a matrix A is formed by m cutting modes generated by the solution, a decision variable corresponding to each column (each cutting mode) in the matrix A is established according to the matrix A, and an initial iteration base matrix B of a CG algorithm is established;
step S520, a new mathematical model (VTC model) is provided, iterative training is carried out on the VTC model by means of a Gurobi optimizer until the iterative times reach a set value, an initial feasible solution is obtained, and an initial iteration base matrix B is constructed;
and S530, performing iterative training on the initial iteration base matrix B by using a CG algorithm based on the initial iteration base matrix B until the value of the second objective function is not reduced any more to obtain an optimal relaxation solution, comparing the value of the sum of the optimal decision variables with the value of the sum of the decision variables in the step S4, selecting the decision variable generated by the party with the minimum sum as a final decision variable, wherein the corresponding blanking layout scheme is the final raw material blanking layout scheme, and if the optimal relaxation solution is not a non-negative integer, entering the step S6.
The algorithm provided by the invention is tested and applied in a certain company in Hunan, aiming at the test of a certain batch of zero single engineering data (the length L =6000cm of the aluminum material as a raw material, and the rest data are shown in Table 4), different algorithms and professional commercial software are adopted for solving, and the results are shown in the following table 1:
introduction of related solution algorithms and commercial software:
creating a door and window blanking software: the creative door and window blanking software is developed by a certain limited company in China, which is a well-known professional door and window design and management software developer in China.
Gurobi software: gurobi is a new generation of large-scale optimizer developed by the United states. The method is widely applied to multiple fields of finance, logistics, manufacturing, aviation, petroleum and petrochemical industry, commercial service and the like, provides a solid foundation for intelligent decision making, and becomes a core optimization engine of thousands of mature application systems.
CG: classical column generation exact algorithms, proposed by Gilmore P, gorory RE in 1961, which until now have been used as core algorithms by almost all relevant optimization optimizers or professional commercial software.
FFD and Greedy are two mainstream heuristic approximate solution algorithms at present, particularly the Greedy algorithm is the most advanced approximate solution algorithm for calculating the cutting optimization problem at present.
According to literature investigations: most of other related solving algorithms round the relaxation solution generated based on CG to obtain an integer solution of an optimization problem, and belong to research on a strategy level. Therefore, the invention provides the VTCCG algorithm for improving CG, which has theoretical significance and practical application value.
Table 1 test results of field data
Figure 588706DEST_PATH_IMAGE345
From the results shown in table 1, it can be seen that: compared with two mainstream heuristic algorithms and professional well-known commercial software-creation, the number of different stock layout schemes generated by the algorithm (VTCCG) provided by the invention is reduced obviously on the premise of ensuring the optimal objective function value Obj (the total amount of raw material consumption is minimum). From the aspect of practical engineering application, the number of different stock layout schemes is directly related to the cutting efficiency. Since the position coordinates of the workpiece to be cut, i.e. the position of the cutting tool, must be readjusted each time a different cutting pattern is changed, the number of different patterns is directly related to the time cost of the cutting process of the enterprise product.
To further verify the validity of the algorithm proposed by the present invention, a common data set generated by an authoritative random data generator was used for testing, and the parameters were consistent except for the different length parameters of the raw material (raw material parameter L =1500 in tables 2 and 3), i.e. the data characteristics were consistent with the common data set (same as the parameters shown in the literature). For space reasons, only the data of the 1 st and 13 th categories in the public data set are tested (see the literature: cerqueira, G., aguiar, S.S., marques, M. (2021). Modified greedy theoretical for the one-dimensional viewing storage protocol, journal of composite Optimization, 1-18.), and the solution results of the first 13 cases are shown in Table 2 and Table 3:
table 2 partial simulation test results for common data set
Figure 47369DEST_PATH_IMAGE346
TABLE 3 partial simulation test results for common data set
Figure 655068DEST_PATH_IMAGE348
As can be seen from the results shown in tables 2 and 3: the algorithm provided by the invention obtains a relatively ideal integer solution, and the generated different layout schemes are least in quantity and are greatly less than the column generation algorithm and the other two famous heuristic algorithms, thereby being beneficial to simplifying the cutting process and improving the production efficiency of enterprises.
The invention is explained by selecting a common casement window in the market at present, the specific blanking data is shown in table 4 in detail, table 4 comprises 20 examples, the blanking data of the examples in table 1 is from table 4, and the blanking data of the examples in tables 2 and 3 is from a common data set-simulation test data (see literature).
Table 4, blanking data:
Figure 972917DEST_PATH_IMAGE349
Figure 566841DEST_PATH_IMAGE350
in order to complete the processing of door and window products, enterprises need to cut parts with different sizes and different quantities from standard lengths of 6000cm of raw materials so as to complete the manufacture of door and window products. However, the number of different cutting patterns is directly related to the station coordinate setting of the workpiece to be cut, and the more the number of patterns, the more the number of times the station coordinate setting. The study of the literature reveals that: on the mathematical theory level, for solving an integer solution of the one-dimensional cutting problem, the minimization of the total amount of raw material consumption and the number of times of setting the station coordinates (the number of different stock layout schemes) are a pair of conflicting indexes, that is, when the primary target and the auxiliary target are simultaneously used as the target optimization (the dual-target optimization problem), the smaller the number of different stock layout schemes is, the more the total amount of raw material consumption is. For the research of a single-target optimization problem (the optimization problem researched by the invention), most of the existing solving algorithms are difficult to obtain satisfactory stock layout schemes (the smaller the number of different stock layout schemes is, the better the number is) under the condition of ensuring the optimal target. Compared with other existing classical algorithms, the algorithm provided by the invention can better solve the problem.
Case comparison effect display: in table 1, the layout of example 1 was solved using the algorithm of the present invention and the creative commercial door and window blanking software as shown in figures 2 and 3, respectively, below:
in fig. 2, for scenario 1: the total cutting length is 5828cm, and 4 aluminum tubes need to be cut; for scheme 2: the total cutting length is 5949cm, and 8 aluminum tubes need to be cut; for scheme 3: the total cutting length is 5896cm, and 4 aluminum tubes need to be cut; for scheme 4: the total cutting length is 5976cm, and 4 aluminum tubes need to be cut; the cutting is carried out according to the 4 different stock layout schemes, and the required quantity of different parts in example 1 can be met.
In fig. 3, for scenario 1: the total cutting length is 5965cm, and 4 aluminum pipes need to be cut; for scheme 2: the total cutting length is 5658cm, and 1 aluminum tube needs to be cut; for scheme 3: the total cutting length is 5975cm, and 6 aluminum tubes need to be cut; for scheme 4: the total cutting length is 5965cm, and 3 aluminum pipes need to be cut; for scheme 5: the total cutting length is 5973cm, and 3 aluminum tubes need to be cut; for scheme 6: the total cutting length is 5343cm, and 1 aluminum tube needs to be cut; for scheme 7: the total cutting length is 5964cm, and 1 aluminum pipe needs to be cut; for scheme 8: the total cutting length is 5949cm, and 1 aluminum pipe needs to be cut; the cutting is carried out according to the above 8 different stock layout schemes, so that the required quantity of different parts in example 1 can be met.
According to the display effect of fig. 2 and 3, it can be seen that: compared with domestic famous professional door and window blanking software (creative), the algorithm (VTCCG) provided by the invention has the advantages that the types of generated stock layout schemes are fewer, namely the number of different blanking stock layout schemes is reduced by 50%, and the total number of raw materials consumed by the two is 20. This is equivalent to cutting station coordinate setting number of times reduces half, can save the adjustment time of station coordinate greatly, is favorable to promoting the production efficiency of enterprise.
The above description is only an embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can easily conceive of changes or substitutions within the technical scope of the present invention, and all such changes or substitutions are included in the scope of the present invention. Moreover, the technical solutions in the embodiments of the present invention may be combined with each other, but it is necessary to be able to be realized by a person skilled in the art, and when the technical solutions are contradictory or cannot be realized, the combination of the technical solutions should be considered to be absent, and is not within the protection scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the appended claims.

Claims (6)

1. A method for solving blanking and stock layout of raw materials is characterized by comprising the following steps: the method comprises the following steps:
s1, determining the length of raw materials, the number of each blanking workpiece and the size of each blanking workpiece;
s2, establishing an original mathematical model according to the length of the raw materials, the number of each blanking workpiece and the size of each blanking workpiece, and performing matrixing on the original mathematical model to obtain a simplified mathematical model;
s3, establishing an initial identity matrix, introducing constant parameters, and performing initial row change on the simplified mathematical model by using the initial identity matrix and the constant parameters to obtain a new mathematical model;
s4, solving the new mathematical model by means of a Gurobi optimizer to obtain an initial feasible solution of the new mathematical model, wherein the initial feasible solution comprises blanking layout schemes and decision variables, and the decision variables refer to variables corresponding to different layout schemes, namely the number of raw materials used corresponding to each layout scheme;
s5, continuously solving on the basis of an initial feasible solution by utilizing a CG algorithm to obtain an optimal relaxation solution, wherein the optimal relaxation solution refers to an optimal decision variable, if the optimal relaxation solution is all non-negative integers, the sum value of the optimal decision variable is compared with the sum value of the decision variables in the step S4, the decision variable generated by the party with the minimum sum is selected as a final decision variable, the corresponding blanking layout scheme is a final raw material blanking layout scheme, and if the optimal relaxation solution is not all non-negative integers, the step S6 is entered;
s6, rounding the optimal relaxation solution obtained in the step S5 by means of a heuristic algorithm to obtain a non-negative integer solution, comparing the sum value of the optimal decision variables with the sum value of the decision variables in the step S4, selecting the decision variable generated by the party with the minimum sum as a final decision variable, and setting the corresponding blanking layout scheme as a final raw material blanking layout scheme;
the original mathematical model in step S2 is:
a first objective function:
Figure 447815DEST_PATH_IMAGE001
the first constraint condition is:
Figure 688304DEST_PATH_IMAGE002
Figure 701259DEST_PATH_IMAGE003
Figure 114923DEST_PATH_IMAGE004
Figure 201828DEST_PATH_IMAGE005
，
Figure 941114DEST_PATH_IMAGE006
，
Figure 113469DEST_PATH_IMAGE007
，
Figure 330824DEST_PATH_IMAGE008
，
Figure 334552DEST_PATH_IMAGE009
（1）
wherein,
Figure 916843DEST_PATH_IMAGE010
representing a decision variable, in particular representing the quantity of raw material used in the j cutting mode;
Figure 373232DEST_PATH_IMAGE011
the number of ith products to be blanked in the jth cutting mode is represented;
Figure 394278DEST_PATH_IMAGE012
the total number of workpieces to be blanked is represented;
Figure 455774DEST_PATH_IMAGE013
the size of the required ith workpiece to be blanked is largeSmall; l represents the length of the raw material;
Figure 536863DEST_PATH_IMAGE014
represents a non-negative integer;
the simplified mathematical model in the step S2 is specifically:
a second objective function:
Figure 683811DEST_PATH_IMAGE015
the second constraint condition is as follows:
Figure 242968DEST_PATH_IMAGE016
Figure 955709DEST_PATH_IMAGE017
Figure 145382DEST_PATH_IMAGE018
，
Figure 576363DEST_PATH_IMAGE019
，
Figure 939211DEST_PATH_IMAGE020
，
Figure 709721DEST_PATH_IMAGE021
Figure 132612DEST_PATH_IMAGE022
（2）
wherein,
Figure 254152DEST_PATH_IMAGE023
a matrix with 1 row and n columns is represented; solution vector matrix
Figure 155112DEST_PATH_IMAGE024
The representation is a matrix of n rows and 1 column, the elements of which are decision variables whose values are constrained to be greater than or equal to zero and to be integers; a represents a matrix of the layout scheme, is a matrix of m rows and n columns,
Figure 45708DEST_PATH_IMAGE025
when all the cutting modes are solved, all the columns in the matrix A are formed by the generated cutting modes, each column represents one cutting mode, and before iteration starts, the cutting modes are an identity matrix with m rows and m columns;
Figure 394429DEST_PATH_IMAGE026
a column vector demand matrix representing m rows and 1 column, wherein the element values of the matrix represent the required quantity of workpieces with different sizes, and m represents the quantity of the required workpiece types;
Figure 800002DEST_PATH_IMAGE027
a matrix representing m rows and 1 columns, which is a column matrix of column vector variables, in which the values of the elements are non-negative integers,
Figure 707915DEST_PATH_IMAGE028
representing a cutting pattern currently to be solved by
Figure 249755DEST_PATH_IMAGE028
As
Figure 14449DEST_PATH_IMAGE029
The index of (a) is determined,
Figure 845002DEST_PATH_IMAGE028
a set of all possible cutting patterns is represented,
Figure 822185DEST_PATH_IMAGE029
a cut representing the current iteration needs to be solvedMode(s)
Figure 749690DEST_PATH_IMAGE028
In which the elements
Figure 622968DEST_PATH_IMAGE030
Is shown in cutting mode
Figure 3133DEST_PATH_IMAGE028
The number of the i-th workpieces,
Figure 784008DEST_PATH_IMAGE031
；
Figure 503702DEST_PATH_IMAGE032
a row vector matrix of 1 row and m columns is represented, and elements of the matrix represent the sizes of m different workpieces;
Figure 610198DEST_PATH_IMAGE033
is shown in cutting mode
Figure 415343DEST_PATH_IMAGE028
Wherein the sum of the product of the size of the different workpieces multiplied by the number of the respective corresponding different workpieces is less than or equal to the length of the raw material
Figure 999908DEST_PATH_IMAGE034
；
Figure 636426DEST_PATH_IMAGE035
In (1)0A column vector matrix representing m rows and 1 columns of all 0 elements, wherein the matrix
Figure 851507DEST_PATH_IMAGE029
All the variable values are constrained to be greater than or equal to zero and must be less than or equal to the required number corresponding to each type of workpiece, the values of which
Figure 940685DEST_PATH_IMAGE036
；
The new mathematical model in step S3 is specifically:
assuming that the number of decision variables is m and the number of different workpiece types is m, each iteration generates a new column, which also represents a new cutting pattern
Figure 532204DEST_PATH_IMAGE028
When iterative solution is carried out, the kth cutting mode is generated
Figure 757649DEST_PATH_IMAGE028
Then, the corresponding third objective function and third constraint may be written as follows:
a third objective function:
Figure 205948DEST_PATH_IMAGE037
the third constraint condition is as follows:
Figure 985685DEST_PATH_IMAGE038
，
Figure 177632DEST_PATH_IMAGE039
，
Figure 460845DEST_PATH_IMAGE040
，
Figure 80046DEST_PATH_IMAGE041
，and
Figure 146746DEST_PATH_IMAGE042
，
Figure 814488DEST_PATH_IMAGE043
，
Figure 280104DEST_PATH_IMAGE044
Figure 273468DEST_PATH_IMAGE045
，
Figure 293377DEST_PATH_IMAGE046
，
Figure 92705DEST_PATH_IMAGE047
Figure 350511DEST_PATH_IMAGE048
，
Figure 311514DEST_PATH_IMAGE049
（3）
in the formula (3), the compound represented by the formula (I),
Figure 287561DEST_PATH_IMAGE050
indicating that in the current iteration, the cutting mode is to be used
Figure 359422DEST_PATH_IMAGE028
Fixing the corresponding decision variable as a constant parameter, wherein the constant parameter is equal to the value of the corresponding decision variable of the kth column obtained by the previous iteration solution;
Figure 534051DEST_PATH_IMAGE051
representing solution to cutting patterns
Figure 665955DEST_PATH_IMAGE028
By matrixing the column vectors
Figure 863718DEST_PATH_IMAGE052
The kth element of (a) is forced to be 0; in the same way, the method for preparing the composite material,
Figure 4850DEST_PATH_IMAGE053
representing solution to cutting patterns
Figure 237248DEST_PATH_IMAGE028
By combining the matrices
Figure 540053DEST_PATH_IMAGE054
The element of the kth line of (1) is forced to be 0 element;
Figure 287429DEST_PATH_IMAGE055
representing a column vector variable matrix
Figure 904356DEST_PATH_IMAGE029
Participating in a matrix expression after the initial change of the matrix;
Figure 53577DEST_PATH_IMAGE056
the decision variable corresponding to the k column in the decision variable matrix is represented as a fixed value
Figure 464967DEST_PATH_IMAGE050
Besides, all the other decision variables are constrained to be larger than or equal to zero;
Figure 434060DEST_PATH_IMAGE057
variable matrix representing unknown column vectors
Figure 916994DEST_PATH_IMAGE029
After the participation matrix A is subjected to the initial row change, the expression of the k row and k column elements in the matrix A has the value equal to 1, wherein
Figure 858405DEST_PATH_IMAGE058
Is a matrix with 1 row and m columns,
Figure 768592DEST_PATH_IMAGE058
by combining matrices
Figure 428244DEST_PATH_IMAGE059
A row vector matrix constructed in the k-th row of (a); objective function
Figure 449289DEST_PATH_IMAGE060
In
Figure 304594DEST_PATH_IMAGE061
Is a 1 row m column matrix with all 1 elements.
2. The method according to claim 1, wherein the step S3 specifically includes the steps of:
step S310, establishing an initial identity matrix, and introducing a constant parameter
Figure 323366DEST_PATH_IMAGE050
In each iteration process, fixing the decision variable corresponding to the kth column as a constant parameter, wherein the constant parameter is equal to the value of the decision variable corresponding to the kth column obtained by the previous iteration solution; at the beginning of the iteration, take
Figure 267051DEST_PATH_IMAGE062
，
Figure 826208DEST_PATH_IMAGE063
Representing a column vector demand matrixd The value of the 1 st element in (c), corresponding to the required number of first workpieces,
Figure 742212DEST_PATH_IMAGE064
the decision variables corresponding to the 1 st column in the layout scheme matrix A are also represented corresponding to the first cutting mode
Figure 994202DEST_PATH_IMAGE028
A decision variable of (c);
and step S320, carrying out primary variation on the simplified mathematical model by using the constant parameters to obtain a new mathematical model.
3. The method according to claim 1, wherein the step S4 specifically includes the steps of:
step S410, solving the new mathematical model by using a Gurobi optimizer;
step S420, passing
Figure 628445DEST_PATH_IMAGE065
、
Figure 991293DEST_PATH_IMAGE066
And
Figure 558541DEST_PATH_IMAGE067
updating a third objective function and a third constraint in the new mathematical model, wherein
Figure 919115DEST_PATH_IMAGE068
，
Figure 837393DEST_PATH_IMAGE069
And
Figure 941615DEST_PATH_IMAGE070
can pass through the initial identity matrix
Figure 894527DEST_PATH_IMAGE071
Obtaining; solved by means of Gurobi optimizer to obtain
Figure 222741DEST_PATH_IMAGE065
And
Figure 565997DEST_PATH_IMAGE067
at this time, the first cutting mode
Figure 270648DEST_PATH_IMAGE072
Is solved and used
Figure 15750DEST_PATH_IMAGE065
And
Figure 780444DEST_PATH_IMAGE067
updating the first column of the matrix A, performing the initial row change on the updated matrix A, and simultaneously performing the initial unit matrix
Figure 407734DEST_PATH_IMAGE071
Sum matrix
Figure 853759DEST_PATH_IMAGE073
Making the same initial row change as the matrix A to obtain an updated matrix
Figure 781264DEST_PATH_IMAGE071
The latter matrix
Figure 654542DEST_PATH_IMAGE074
Then according to the matrix
Figure 503549DEST_PATH_IMAGE074
To obtain a matrix
Figure 815582DEST_PATH_IMAGE075
While obtaining the updated matrix
Figure 332014DEST_PATH_IMAGE073
The latter matrix
Figure 376193DEST_PATH_IMAGE076
According to a matrix
Figure 981005DEST_PATH_IMAGE076
To obtain a matrix
Figure 768833DEST_PATH_IMAGE077
(ii) a In the same way, use
Figure 405351DEST_PATH_IMAGE078
，
Figure 417169DEST_PATH_IMAGE079
And
Figure 709610DEST_PATH_IMAGE080
and updating a third objective function and a third constraint condition corresponding to the new mathematical model after 1 iteration, wherein at the moment,
Figure 97866DEST_PATH_IMAGE081
is unknown, wherein
Figure 792153DEST_PATH_IMAGE082
By passing
Figure 240451DEST_PATH_IMAGE074
Obtaining;
step S430, performing k iterations, k =2.. K, on the new mathematical model by means of a Gurobi optimizer, calculating
Figure 551347DEST_PATH_IMAGE083
And
Figure 680977DEST_PATH_IMAGE067
: will be provided with
Figure 292087DEST_PATH_IMAGE083
And
Figure 848970DEST_PATH_IMAGE067
as the kth column of matrix a, where,
Figure 647162DEST_PATH_IMAGE082
can pass through the matrix
Figure 642800DEST_PATH_IMAGE084
Obtaining, and then performing an initial row change on the matrix A, and simultaneously performing the matrix change on the matrix A
Figure 46099DEST_PATH_IMAGE084
Sum matrix
Figure 836201DEST_PATH_IMAGE085
Making the same initial row change as the matrix A to obtain the matrix
Figure 324951DEST_PATH_IMAGE086
Then according to the matrix
Figure 858701DEST_PATH_IMAGE086
Obtaining a matrix
Figure 178823DEST_PATH_IMAGE087
While obtaining the updated matrix
Figure 77509DEST_PATH_IMAGE085
The latter matrix
Figure 850293DEST_PATH_IMAGE088
Then according to the matrix
Figure 187734DEST_PATH_IMAGE088
Obtaining a matrix
Figure 300046DEST_PATH_IMAGE089
In combination with each other
Figure 697529DEST_PATH_IMAGE090
、
Figure 692030DEST_PATH_IMAGE091
And
Figure 770845DEST_PATH_IMAGE067
updating a third objective function and a third constraint condition corresponding to the new mathematical model after iteration for k times;
step S440, the step S430 is circulated until the iteration times k = beta m, the value of m is equal to the number of different workpiece types to be blanked, beta represents the coefficient of iteration, and beta =1or2; and after the iteration is carried out for k times, a feasible solution of a new mathematical model formed by a third constraint condition and a third objective function is obtained.
4. The solution method according to claim 3, wherein the initial identity matrix in step S420
Figure 531472DEST_PATH_IMAGE071
The matrix a before the iteration starts is specifically:
Figure 99856DEST_PATH_IMAGE092
（4）
Figure 784916DEST_PATH_IMAGE093
（5）。
5. the solution method according to claim 3, wherein the k iterations of the calculation in step S430 are solved by using a Gurobi optimizer.
6. The method according to claim 3, wherein the step S5 specifically comprises the steps of:
step S510, forming each row in a matrix A by the initial feasible solution, namely m cutting modes generated by the solution, establishing an initial iteration base matrix B of the CG algorithm according to the decision variables corresponding to each row in the matrix A;
step S520, a new mathematical model is provided, iterative training is carried out on the new mathematical model by means of a Gurobi optimizer until the number of iterations reaches a set value, an initial feasible solution is obtained, and an initial iteration base matrix B is constructed according to the initial feasible solution;
and S530, based on the initial iteration base matrix B, carrying out iteration training on the initial iteration base matrix B by utilizing a CG algorithm until the value of the second objective function is not reduced any more, obtaining an optimal relaxation solution, comparing the value of the sum of the optimal decision variables with the value of the sum of the decision variables in the step S4, selecting the decision variable generated by the party with the minimum sum as a final decision variable, wherein the corresponding blanking layout scheme is the final raw material blanking layout scheme, and if the optimal relaxation solution is not completely a non-negative integer, entering the step S6.
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