CN116777063A - Two-dimensional boxing method based on one-tool cutting constraint and branch pricing algorithm - Google Patents

Two-dimensional boxing method based on one-tool cutting constraint and branch pricing algorithm Download PDF

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CN116777063A
CN116777063A CN202310738906.2A CN202310738906A CN116777063A CN 116777063 A CN116777063 A CN 116777063A CN 202310738906 A CN202310738906 A CN 202310738906A CN 116777063 A CN116777063 A CN 116777063A
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魏丽军
王孙康宏
姚绍文
刘强
舒文兰
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Guangdong University of Technology
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Abstract

The invention belongs to the technical field of boxing problem optimization, and particularly relates to a two-dimensional boxing method based on a one-tool constraint and branch pricing algorithm. The invention provides an accurate solving algorithm of branch pricing aiming at the two-dimensional rectangular boxing problem with one-cut constraint, the algorithm can accurately and rapidly solve the two-dimensional boxing optimal solution, and the preprocessing technology aiming at the container and box data is adopted in the calculation process to reduce the problem scale, greatly reduce the searching space of the algorithm and improve the convergence rate of the algorithm.

Description

Two-dimensional boxing method based on one-tool cutting constraint and branch pricing algorithm
Technical Field
The invention belongs to the technical field of boxing problem optimization, and particularly relates to a two-dimensional boxing method based on a one-tool constraint and branch pricing algorithm.
Background
The Two-Dimensional Bin-Packing Problems (2D-BPP) is equivalent to the Two-Dimensional rectangular layout problem, is a classical combination optimization problem, and early research on the combination optimization problem can be traced to the early 20 th century, and the research range comprises one-Dimensional layout and Two-Dimensional layout, wherein the Two-Dimensional layout is divided into Two-Dimensional rectangular layout, two-Dimensional special-shaped layout and the like. Research on this problem is closely related to actual production and logistics, and its application fields include manufacturing industry, logistics, computer aided design, etc.
In the manufacturing industry, the two-dimensional layout problem is an important link in the production process, such as the wood industry, the glass industry, the paper industry and the like, given a plurality of larger rectangular raw materials, manufacturers cut small rectangular finished products with different given sizes from the larger rectangular raw materials according to the requirements of customers, and the aim is to cut the raw materials with minimum cost. In addition, during the manufacturing process, many parts need to be assembled on the production line, and two-dimensional layout problems can be used to determine how to optimally place these parts on the production line to improve production efficiency and product quality. The two-dimensional layout problem can also be used for optimizing the package design so as to reduce the waste of packaging materials and improve the protection performance and the aesthetic property of the package; in the computer industry, the multi-CPU task scheduling problem of the computer system can be completely converted into an open dimension problem (Open Dimension Problem, ODP) of two-dimensional rectangular layout, and each task consumes a specified amount of resources and occupies a fixed period of time. Each task can be seen as a small box with width being the occupied resource and height being the occupied time, and the whole computer system can be seen as a plurality of two-dimensional open containers waiting to be filled with fixed width (total resource) and fixed height (time) limitlessly.
Therefore, the two-dimensional boxing problem has important theoretical research significance and practical application value.
The main solutions to the two-dimensional packing problem can be classified into heuristic algorithms and intelligent optimization algorithms. The accurate algorithm can obtain an optimal layout scheme, and for some important application scenarios, the optimal solution is often critical. The accurate algorithm can provide an exact optimal solution, so that the reliability and stability of the algorithm are improved; the intelligent optimization algorithm is an optimization algorithm based on intelligent technologies such as biological evolution and natural selection, and is an algorithm for solving complex problems by simulating a biological evolution process. Common intelligent optimization algorithms include: particle swarm optimization algorithm, genetic algorithm, ant colony algorithm and the like, which are all solved based on a random search mode, and whether the obtained feasible solution is the optimal solution cannot be judged even if the feasible solution obtained each time is unstable.
With the development of technology, artificial intelligence methods have also been proposed for solving two-dimensional boxing problems such as machine learning methods, deep learning methods, and reinforcement learning methods, but these methods have disadvantages in that they require strong computational power and a large amount of data set support, and that the interpretability of such methods is poor.
Disclosure of Invention
The invention provides a two-dimensional boxing method based on a one-cutter constraint and branch pricing algorithm, and aims to solve the problems of large calculation scale, large data size and low convergence speed in solving the two-dimensional boxing problem in the prior art.
Specifically, the invention provides a two-dimensional boxing method based on a one-knife constraint and branch pricing algorithm, which comprises the following steps:
s1, determining a box body of a rectangular block for boxing and a container for bearing the box body, constructing a two-dimensional boxing set coverage model related to the box body and the container, taking the box body and the container as initial data, and preprocessing the initial data;
s2, taking the initial data as an initial column of a branch pricing algorithm, and constructing pricing sub-problems related to the two-dimensional boxing set coverage model based on a cutter constraint;
s3, solving a branch pricing algorithm by taking the solution of the linear relaxation problem initially listed in the two-dimensional boxing set coverage model as a root node of a branch pricing tree, wherein a branch strategy based on box pairs is set, and the pricing sub-problem is solved according to a preset dynamic programming and packaging algorithm to obtain an optimal solution of the two-dimensional boxing set coverage model;
and S4, loading the box body into the container based on the optimal solution.
Still further, a set of said containers b= {1,2, …, m }, each of said containers having the same width W and height H, a set of said boxes n= {1,2, …, N }, each of said boxes having the following properties:
the two-dimensional boxing set coverage model meets the following conditions:
min λ r ；
wherein lambda is r For decision variables, indicating whether the packing scheme r is selected, if so, lambda r Taking 1, otherwise taking 0; omega is a complete set of feasible packaging schemes; omega' is the packing scheme part set; n is a box body set; alpha jr To specify whether the bin j is allocated in the packing scheme r, and if so, the value is 1, otherwise, 0.
Further, the step of preprocessing the initial data by using the box and the container as the initial data includes:
the container is reduced in size to define the width combination of any container body as the maximum value W which can be generated * W is less than or equal to W, and:
thereafter, let w=w * And reducing the height H of the container according to the relation;
amplifying the size of the box body, and defining the maximum width of any box bodyAnd:
thereafter, let w j ＝w j * And according to the relation, the height h of the box body j * And amplifying.
Still further, the two-dimensional packing set overlay model further includes a viable location constraint for the bins, each of the bins being in contact with a boundary of the other bin or the container on either a left side or a bottom of the bin when loaded into the container.
Still further, in the pricing sub-questions regarding the two-dimensional packing set coverage model constructed based on a cut constraint, let beta be j As a binary variable, beta if and only if the bin j is packed in the optimal packing scheme j Taking 1, otherwise taking 0, m j ，n j ，q j ，e j Are binary variables, M is a very large constant, and the j coordinate of the box body is (x j ，y j ) The shadow price of the box j in the two-dimensional box assembly covering model is mu j The pricing sub-problem satisfies:
still further, the solving of the branch pricing algorithm includes the steps of:
s31, constructing an initial column for a branch pricing algorithm, solving a linear relaxation problem that the two-dimensional boxing set coverage model is assembled in the initial column and does not contain integer constraint, taking a solution of the linear relaxation problem as a root node of a branch pricing tree, putting the root node into a node list of a current branch, defining the solution of the linear relaxation problem corresponding to the root node as an optimal solution, wherein the value of the solution of the two-dimensional boxing set coverage model corresponding to the optimal solution is an optimal function value, and the iteration number of the branch pricing algorithm is initialized to 0;
s32, judging whether the node list of the current branch is empty, if not, selecting a node to be optimized according to a preset strategy, and entering step S33; if yes, outputting the optimal solution;
s33, solving a linear relaxation problem of the node sub-problem of the node to be optimized to obtain a solution to be optimized;
s34, if the solution to be optimized meets 0-1 integer constraint and the function value to be optimized of the solution to be optimized corresponding to the two-dimensional boxing set coverage model is better than the optimal function value, updating the solution to be optimized and the function value to be optimized into the optimal solution and the optimal function value;
s35, substituting the optimal solution obtained in the step S33 into the pricing sub-problem to solve, adding part or all of the columns with the negative check numbers into a main problem scheme set, and if a new column meeting the condition exists, adding 1 to the iteration times, and returning to the step S33; otherwise, outputting the optimal solution;
s36, judging whether the optimal solution meets the integer constraint of the pricing sub-problem, if not, ending the branch, adding 1 to the iteration number, and returning to the step S32;
s37, judging whether the optimal solution meets the upper and lower boundary constraint of the target value of the pricing sub-problem, if not, ending the branch, adding 1 to the iteration number, and returning to the step S32;
s38, judging whether the optimal solution meets the 0-1 integer constraint of the pricing sub-problem, if not, ending the branch, adding 1 to the iteration number, and returning to the step S32;
and S39, branching according to the optimal solution to obtain two child nodes of the current branch, adding the child nodes into a node list, adding 1 to the iteration times, and returning to the step S32.
Further, the preset dynamic programming packaging algorithm comprises the following steps:
constructing an initial upper bound u 0 Let temporary upper bound u++u 0 ；
Acquiring a feasible packing scheme set F (W, H, u) according to the width W, the height H and the temporary upper bound u of the container;
if the feasible packaging scheme set F (W, H, u) is an empty set, updating the temporary upper bound u, and recalculating the feasible packaging scheme set F (W, H, u); otherwise, outputting the feasible packing scheme set F (W, H, u).
Further, the solving process of the branch pricing algorithm follows a branch strategy based on a box pair:
definition of variable delta ij To assist the 0-1 variable, we use to indicate whether the container i and the bin j are selected in the same branch, select delta in the current branch ij Pairing (i, j) of said container and said box having a value closest to 0.5, said container and said box in said pairing (i, j) being assigned such that δ ij =1 or δ ij ＝1。
The invention has the beneficial effects that an accurate solving algorithm of branch pricing is provided for the two-dimensional rectangular boxing problem with one-cut constraint, the algorithm can accurately and rapidly solve the two-dimensional boxing optimal solution, and the preprocessing technology for the container and box data is adopted in the calculation process to reduce the problem scale, greatly reduce the searching space of the algorithm and improve the convergence rate of the algorithm.
Drawings
FIG. 1 is a flow chart of steps of a two-dimensional boxing method based on a cut constraint and branch pricing algorithm provided by an embodiment of the invention;
FIG. 2 is a schematic diagram showing the constraint distinction between a primary cut and a non-primary cut according to an embodiment of the present invention;
FIG. 3 is a schematic illustration of a process for downsizing a container provided in an embodiment of the present invention;
FIG. 4 is a schematic diagram of a process for enlarging the size of a box according to an embodiment of the present invention;
FIG. 5 is an iterative logic diagram of a branch pricing algorithm provided by an embodiment of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.
Referring to fig. 1, fig. 1 is a step flowchart of a two-dimensional boxing method based on a cutter constraint and a branch pricing algorithm according to an embodiment of the present invention, where the two-dimensional boxing method includes the following steps:
s1, determining a box body of a rectangular block for boxing and a container for bearing the box body, constructing a two-dimensional boxing set coverage model related to the box body and the container, taking the box body and the container as initial data, and preprocessing the initial data.
The one-cut constraint refers to a constraint that a cutting operation must cut from one side of a rectangular block to the opposite side thereof and divide it into two independent small rectangular blocks in cutting a plate, and in the case of boxing, refers to a constraint that boxes are boxed along edges, and the one-cut constraint is distinguished from the non-one-cut constraint as shown in fig. 2.
Still further, a set of said containers b= {1,2, …, m }, each of said containers having the same width W and height H, a set of said boxes n= {1,2, …, N }, each of said boxes having the following properties:
the two-dimensional boxing set coverage model meets the following conditions:
min λ r (2)；
wherein lambda is r For decision variables, indicating whether the packing scheme r is selected, if soLambda is the middle r Taking 1, otherwise taking 0; omega is a complete set of feasible packaging schemes; omega' is the packing scheme part set; n is a box body set; alpha jr To specify whether the bin j is allocated in the packing scheme r, and if so, the value is 1, otherwise, 0.
The objective function (2) represents minimizing the number of containers used to pack all the bins; constraint (3) indicates that the number of times each time period is covered is greater than or equal to the number of staff needed for the time period; constraint (4) is decision variable lambda r Is defined as a constraint. From a computational standpoint, it is impractical to construct equations (2) - (4) by enumerating all shifts in Ω. Thus, embodiments of the present invention employ a column generation method to calculate the bin packing problem of the build.
Further, the step of preprocessing the initial data by using the box and the container as the initial data includes:
the container is reduced in size to define the width combination of any container body as the maximum value W which can be generated * W is less than or equal to W, and:
thereafter, let w=w * And the height H of the container is reduced according to the relation. W can be as follows j Replaced by h j W is replaced by H, thereby calculating H * Let h=h then * 。
FIG. 3 is in W * The calculation of (1) shows a process of downsizing the container.
Amplifying the size of the boxes to solve a subset and problem for each box j e N to determine a maximumSo that N\ { j } is stored inIn a group the total width is equal to +.>Is arranged in the box body. If->Then the width of the box j can be increased +.>Defining the maximum value of the width of any of the boxes +.>And:
thereafter, let w j ＝w j * And according to the relation, the height h of the box body j * And amplifying. Similarly, the maximum height h can be calculated j * Thereby increasing the height of the case j:
FIG. 4 toThe calculation of (2) shows the process of enlarging the size of the case. The aim of using two pretreatment techniques, shrinking the size of the container and enlarging the size of the tank, is to reduce the complexity of the example, thus obtaining a problem that is easier to solve in practice.
Still further, the two-dimensional packing set overlay model further includes a viable location constraint for the bins, each of the bins being in contact with a boundary of the other bin or the container on either a left side or a bottom of the bin when loaded into the container.
The feasible positions of the box body adopt a plurality ofThe well-known Normal Patterns principle is reduced. This principle indicates that there is an optimal solution in which each box is moved as downwards and leftwards as possible, and therefore, it is necessary to touch the other box or border on either the left side or the bottom of the box. More specifically, vertical and horizontal Normal Patterns in a container can be N from the collection v (W) and N h (H) Extracting, wherein N v (w) and N h (h) The definition is as follows:
in the embodiment of the invention, the feasible positions of the box bodies are reduced by taking the Raster Point principle into consideration. Raster Point can be further reduced on the basis of Normal Patterns. First, toThe definition is as follows:
for a given vertical position z ε N v (w) ifThen the placement of position z results in at least a loss +.>If there are other positions z' > z, make +.>The position z can be skipped because an equivalent or better solution can be obtained by placing the box at the position z'. Vertical and horizontal Raster Point, as follows:
s2, taking the initial data as an initial column of a branch pricing algorithm, and constructing pricing sub-questions about the two-dimensional boxing set coverage model based on a cutter constraint.
Defining the shadow price of the j-th row of constraint (3) as μ j In general, assume that RMP already has a base feasible solution:
B＝(α j1 ，a j2 ，…，α jr ) (15)；
and has been found:
according to the modified simplex method, if the condition (the number of tests is less than 0) is satisfied:
b is the optimal base, B -1 b is the solution.
Pricing sub-problems are equivalent to 2D-KP, where the value of each bin is determined by the corresponding shadow price μ j And (5) determining.
Still further, in the pricing sub-questions regarding the two-dimensional packing set coverage model constructed based on a cut constraint, let beta be j Is a binary variable, when andbeta only when the bin j is packed in the optimal packing scheme j Taking 1, otherwise taking 0, m j ，n j ，q j ，e j Are binary variables, M is a very large constant, and the j coordinate of the box body is (x j ，y j ) The shadow price of the box j in the two-dimensional box assembly covering model is mu j The pricing sub-problem satisfies:
the objective function (18) requires minimizing the Reduced Cost of the solution, which also amounts to maximizing the value of the packaged bin in the solution; constraint (19) - (20) requires that the tank in the solution cannot exceed the container; constraints (21) - (26) require that the cases in the solution cannot overlap; constraints (27) - (29) give a domain of decision variables.
S3, solving the branch pricing algorithm by taking the solution of the linear relaxation problem initially listed in the two-dimensional boxing set coverage model as a root node of a branch pricing tree, wherein a branch strategy based on box pairs is set, and solving the pricing sub-problem according to a preset dynamic programming and packaging algorithm to obtain the optimal solution of the two-dimensional boxing set coverage model.
Still further, referring to the iterative logic diagram of the branch pricing algorithm shown in fig. 5, the solving of the branch pricing algorithm includes the following steps:
s31, constructing an initial column for a branch pricing algorithm, solving a linear relaxation problem that the two-dimensional boxing set coverage model is assembled in the initial column and does not contain integer constraint, taking a solution of the linear relaxation problem as a root node of a branch pricing tree, putting the root node into a node list of a current branch, defining the solution of the linear relaxation problem corresponding to the root node as an optimal solution, wherein the value of the solution of the two-dimensional boxing set coverage model corresponding to the optimal solution is an optimal function value, and the iteration number of the branch pricing algorithm is initialized to 0;
s32, judging whether the node list of the current branch is empty, if not, selecting a node to be optimized according to a preset strategy, and entering step S33; if yes, outputting the optimal solution;
s33, solving a linear relaxation problem of the node sub-problem of the node to be optimized to obtain a solution to be optimized;
s34, if the solution to be optimized meets 0-1 integer constraint and the function value to be optimized of the solution to be optimized corresponding to the two-dimensional boxing set coverage model is better than the optimal function value, updating the solution to be optimized and the function value to be optimized into the optimal solution and the optimal function value;
s35, substituting the optimal solution obtained in the step S33 into the pricing sub-problem to solve, adding part or all of the columns with the negative check numbers into a main problem scheme set, and if a new column meeting the condition exists, adding 1 to the iteration times, and returning to the step S33; otherwise, outputting the optimal solution;
s36, judging whether the optimal solution meets the integer constraint of the pricing sub-problem, if not, ending the branch, adding 1 to the iteration number, and returning to the step S32;
s37, judging whether the optimal solution meets the upper and lower boundary constraint of the target value of the pricing sub-problem, if not, ending the branch, adding 1 to the iteration number, and returning to the step S32;
s38, judging whether the optimal solution meets the 0-1 integer constraint of the pricing sub-problem, if not, ending the branch, adding 1 to the iteration number, and returning to the step S32;
and S39, branching according to the optimal solution to obtain two child nodes of the current branch, adding the child nodes into a node list, adding 1 to the iteration times, and returning to the step S32.
Embodiments of the present invention describe a method for solving a pricing sub-problem (SP) associated with a two-dimensional bin packing problem (2D-BPP) with a one-cut constraint g ) Dynamic programming packing algorithm (Dynamic Programming Packaging A, gorithm, DPPA). The algorithm can easily deal with prohibiting or requiring a request to be madeBranching rules for the case package in the same container.
Before introducing a dynamic programming algorithm, the embodiment of the invention basically defines the algorithm:
definition 1: a sub-container. The sub-container is one of the input parameters of the dynamic programming algorithm. With vector s= (x) S ，y S ，W S ，H S ) Representing sub-containers, where (x S ，y S ) Is the left lower corner coordinate of the sub-container, W S ，H S The width and height of the sub-containers, respectively.
Definition 2: a packing scheme is possible. For a given sub-container S, p is used S Representing the packing scheme on S. A packing scheme is said to be viable if it meets all the constraints of 2D-KP o|g. Any feasible packing scheme can be used with vectorsAnd (3) representing. Wherein for->There is->And->Is p s The number of middle boxes j.
According to definition 2, a packing scheme p can be obtained S The total profit of (2) is as follows:
definition 3: dominant relationship. If a possible scheme p S Is put into another possible scheme p S′ Dominant then are:
definition 4: maximum feasible packing scheme. If a packing scheme p is available S Not by any other possible packing scheme p S′ Is governed by, then called p S Is one of the largest possible packing schemes.
Definition 5: and merging packaging schemes. Given two non-overlapping sub-containers S 1 And S is 2 One possible packing schemeAnd->Let->And->The horizontal and vertical combinations of (2) are +.>
Assume thatIs composed of->And->The combination is that:
if S 3 Not exceeding the range of the container, the combination isIn a row. Finally, also beRemoving redundant items:
definition 6: and merging the packaging scheme sets. Let P be S Representing a set of possible packing schemes for the sub-containers S. Given two sets of possible packaging schemesAnd->Let->And->The horizontal and vertical combinations of (2) are +.>And:
the upper bound calculation method comprises the following steps: by u (p) S ) Representing packaging scheme p s Is the profit upper bound of (2). Let 0-1 variable z j Indicating whether the bin j is packed. For 2D-KP O|G, the value of the relaxed one-dimensional backpack problem is an effective upper bound. The embodiment of the invention adopts three upper bounds UK, UV and UW, which respectively represent the relaxed area dimension, the vertical dimension and the horizontal dimension.
The mathematical model for calculating the UK is as follows:
z j ∈{0，1}，j∈N (38)；
the mathematical model for calculating UV is as follows:
z j ∈{0，1}，j∈N (41)；
the mathematical model for calculating UW is as follows:
z j ∈{0，1}，j∈N (44)；
now consider a possible packing scheme p S Using UK (p S )，UV(p S )，UW(p S ) Respectively represent p S Is the upper boundary of three profit types. Then, give u (p S ) The calculation formula of (2) is as follows:
u(p S )＝v(p S )+min(UK(p S )，UV(p S )，UW(p S )) (45)；
the embodiment of the invention solves the problem of loosening the one-dimensional knapsack by adopting a classical dynamic programming method.
The dynamic programming packaging algorithm is based on a dynamic programming packaging process, letting F (W S ，H S U) is the maximum feasible packing scheme set for S. The pseudo code for the dynamic programming packaging process of the embodiments of the present invention is shown in table 1 below.
TABLE 1 pseudo code for dynamic planning of packing procedures
Based on the above dynamic programming process, further, the preset dynamic programming packaging algorithm includes the following steps:
constructing an initial upper bound u 0 Let temporary upper bound u++u 0 ；
Acquiring a feasible packing scheme set F (W, H, u) according to the width W, the height H and the temporary upper bound u of the container;
if the feasible packaging scheme set F (W, H, u) is an empty set, updating the temporary upper bound u, and recalculating the feasible packaging scheme set F (W, H, u); otherwise, outputting the feasible packing scheme set F (W, H, u)
In summary, the pseudo code of the dynamic programming packing algorithm provided by the present invention is shown in table 2 below.
Table 2 pseudo code for dynamic programming packing algorithm
The conventional branch policy of the branch pricing tree is variable-based, that is, a variable row r cannot appear in the left-side optimal solution of the branch tree, a variable row r must be included in the right-side optimal solution of the branch tree (left-side and right-side can be interchanged), so that the variable row r needs to be deleted from the main problem of the left-side branch tree, and in the following calculation of the sub-problem of the left-side branch tree, the deleted variable row r needs to be prevented from being added to the main problem again because the Reduced Cost is smaller than 0 as a base row, and the branch tree is extremely unbalanced, and the branch searching efficiency is Reduced, so that the conventional branch policy based on the variable is not suitable for the branch pricing algorithm.
The prior art indicates that when the optimal solution for RMP is fractional, there must be a pair of bins (i, j) such that 0 < δ ij ＝∑λ r (α ir α jr ) < 1, r.epsilon.OMEGA'. Variable delta ij May be interpreted as an auxiliary (0-1) variable to indicate whether item i and item j are selected in the same column.
Therefore, the solution process of the branch pricing algorithm in the embodiment of the present invention adopts a branch strategy based on the bin pairs:
definition of variable delta ij To assist the 0-1 variable, we use to indicate whether the container i and the bin j are selected in the same branch, select delta in the current branch ij Pairing (i, j) of said container and said box having a value closest to 0.5, said container and said box in said pairing (i, j) being assigned such that δ ij =1 or δ ij =1. The left child node requires that item i and item j must be assigned to the same column and that alpha be ir +α jr Column r=1 is deleted from RMP. The right child node requires that item i and item j cannot be assigned to the same column and that alpha be ir +α jr Column r > 1 is deleted from RMP.
An important characteristic of the branching strategy based on the box pair is that the structure of the pricing sub-problem is reserved, and the branching searching efficiency is high.
And S4, loading the box body into the container based on the optimal solution.
The invention has the beneficial effects that an accurate solving algorithm of branch pricing is provided for the two-dimensional rectangular boxing problem with one-cut constraint, the algorithm can accurately and rapidly solve the two-dimensional boxing optimal solution, and the preprocessing technology for the container and box data is adopted in the calculation process to reduce the problem scale, greatly reduce the searching space of the algorithm and improve the convergence rate of the algorithm.
Those skilled in the art will appreciate that implementing all or part of the above-described methods in accordance with the embodiments may be accomplished by way of a computer program stored on a computer readable storage medium, which when executed may comprise the steps of the embodiments of the methods described above. The storage medium may be a magnetic disk, an optical disk, a Read-Only Memory (ROM), a random access Memory (Random Access Memory, RAM) or the like.
It should be noted that, in this document, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, the element defined by the phrase "comprising one … …" does not exclude the presence of other like elements in a process, method, apparatus or device comprising such element.
From the above description of the embodiments, it will be clear to those skilled in the art that the above-described embodiment method may be implemented by means of software plus a necessary general hardware platform, but of course may also be implemented by means of hardware, but in many cases the former is a preferred embodiment. Based on such understanding, the technical solution of the present invention may be embodied essentially or in a part contributing to the prior art in the form of a software product stored in a storage medium (e.g. ROM/RAM, magnetic disk, optical disk) comprising instructions for causing a terminal (which may be a mobile phone, a computer, a server, an air conditioner, or a network device, etc.) to perform the method according to the embodiments of the present invention.
While the embodiments of the present invention have been illustrated and described in connection with the drawings, what is presently considered to be the most practical and preferred embodiments of the invention, it is to be understood that the invention is not limited to the disclosed embodiments, but on the contrary, is intended to cover various equivalent modifications and equivalent arrangements included within the spirit and scope of the appended claims.

Claims (8)

1. A two-dimensional boxing method based on a cut constraint and branch pricing algorithm, characterized by comprising the following steps:
s1, determining a box body of a rectangular block for boxing and a container for bearing the box body, constructing a two-dimensional boxing set coverage model related to the box body and the container, taking the box body and the container as initial data, and preprocessing the initial data;
s2, taking the initial data as an initial column of a branch pricing algorithm, and constructing pricing sub-problems related to the two-dimensional boxing set coverage model based on a cutter constraint;
s3, solving a branch pricing algorithm by taking the solution of the linear relaxation problem initially listed in the two-dimensional boxing set coverage model as a root node of a branch pricing tree, wherein a branch strategy based on box pairs is set, and the pricing sub-problem is solved according to a preset dynamic programming and packaging algorithm to obtain an optimal solution of the two-dimensional boxing set coverage model;
and S4, loading the box body into the container based on the optimal solution.
2. A two-dimensional boxing method based on a cut constraint and branch pricing algorithm according to claim 1, wherein a set of said containers b= {1,2, …, m }, each of said containers having the same width W and height H, a set of said boxes n= {1,2, …, N }, each of said boxes having the following properties:
the two-dimensional boxing set coverage model meets the following conditions:
min λ r ；
wherein lambda is r For decision variables, indicating whether the packing scheme r is selected, if so, lambda r Taking 1, otherwise taking 0; omega is a complete set of feasible packaging schemes; omega' is the packing scheme part set; n is a box body set; alpha jr To specify whether the bin j is allocated in the packing scheme r, and if so, the value is 1, otherwise, 0.
3. A two-dimensional boxing method based on a one-cut constraint and branch pricing algorithm according to claim 2, wherein the step of preprocessing the initial data with the box and the container as initial data comprises:
the container is reduced in size to define the width combination of any container body as the maximum value W which can be generated * W is less than or equal to W, and:
thereafter, let w=w * And reducing the height H of the container according to the relation;
amplifying the size of the box body, and defining the maximum width of any box bodyAnd:
thereafter, let w j ＝w j And according to the relation, the height h of the box body j And amplifying.
4. A two-dimensional boxing method based on a one-cut constraint and branch pricing algorithm according to claim 3, wherein the two-dimensional boxing set cover model further comprises a viable location constraint for the boxes, each box being in contact with the boundary of the other box or container on the left or bottom of the box when loaded into the container.
5. The two-dimensional packing method based on a one-cut constraint and branch pricing algorithm according to claim 4, wherein β is set among the pricing sub-questions regarding the two-dimensional packing set coverage model constructed based on a one-cut constraint j As a binary variable, beta if and only if the bin j is packed in the optimal packing scheme j Taking 1, otherwise taking 0, m j ，n j ，q j ，e j Are binary variables, M is a very large constant, and the j coordinate of the box body is (x j ，y j ) The shadow price of the box j in the two-dimensional box assembly covering model is mu j The pricing sub-problem satisfies:
6. the two-dimensional boxing method based on a cut constraint and branch pricing algorithm according to claim 5, wherein the solving of the branch pricing algorithm comprises the steps of:
s31, constructing an initial column for a branch pricing algorithm, solving a linear relaxation problem that the two-dimensional boxing set coverage model is assembled in the initial column and does not contain integer constraint, taking a solution of the linear relaxation problem as a root node of a branch pricing tree, putting the root node into a node list of a current branch, defining the solution of the linear relaxation problem corresponding to the root node as an optimal solution, wherein the value of the solution of the two-dimensional boxing set coverage model corresponding to the optimal solution is an optimal function value, and the iteration number of the branch pricing algorithm is initialized to 0;
s32, judging whether the node list of the current branch is empty, if not, selecting a node to be optimized according to a preset strategy, and entering step S33; if yes, outputting the optimal solution;
s33, solving a linear relaxation problem of the node sub-problem of the node to be optimized to obtain a solution to be optimized;
s34, if the solution to be optimized meets 0-1 integer constraint and the function value to be optimized of the solution to be optimized corresponding to the two-dimensional boxing set coverage model is better than the optimal function value, updating the solution to be optimized and the function value to be optimized into the optimal solution and the optimal function value;
s35, substituting the optimal solution obtained in the step S33 into the pricing sub-problem to solve, adding part or all of the columns with the negative check numbers into a main problem scheme set, and if a new column meeting the condition exists, adding 1 to the iteration times, and returning to the step S33; otherwise, outputting the optimal solution;
s36, judging whether the optimal solution meets the integer constraint of the pricing sub-problem, if not, ending the branch, adding 1 to the iteration number, and returning to the step S32;
s37, judging whether the optimal solution meets the upper and lower boundary constraint of the target value of the pricing sub-problem, if not, ending the branch, adding 1 to the iteration number, and returning to the step S32;
s38, judging whether the optimal solution meets the 0-1 integer constraint of the pricing sub-problem, if not, ending the branch, adding 1 to the iteration number, and returning to the step S32;
and S39, branching according to the optimal solution to obtain two child nodes of the current branch, adding the child nodes into a node list, adding 1 to the iteration times, and returning to the step S32.
7. The two-dimensional packing method based on a one-tool constraint and branch pricing algorithm according to claim 6, wherein the preset dynamic programming packing algorithm comprises the steps of:
constructing an initial upper bound u 0 Let temporary upper bound u++u 0 ；
Acquiring a feasible packing scheme set F (W, H, u) according to the width W, the height H and the temporary upper bound u of the container;
if the feasible packaging scheme set F (W, H, u) is an empty set, updating the temporary upper bound u, and recalculating the feasible packaging scheme set F (W, H, u); otherwise, outputting the feasible packing scheme set F (W, H, u).
8. The two-dimensional boxing method based on a one-tool constraint and branch pricing algorithm according to claim 7, wherein the solving process of the branch pricing algorithm follows a case pair-based branch strategy:
definition of variable delta ij To assist the 0-1 variable, we use to indicate whether the container i and the bin j are selected in the same branch, select delta in the current branch ij Pairing (i, j) of said container and said box having a value closest to 0.5, said container and said box in said pairing (i, j) being assigned such that δ ij =1 or δ ij ＝1。
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