Defective Sheet Packing Using Branch-and-Cut Under Cutting Constraints
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Solution Overview
Problem
Current packing methods for defective sheets, particularly using heuristic and intelligent optimization algorithms, face inefficiencies and instability in finding optimal solutions for two-dimensional rectangular packing under non-guillotine cutting constraints, leading to suboptimal material utilization and high production costs.
Innovation Solution
A branch-and-cut algorithm is employed to acquire and process information about defective sheets, establishing primal and relaxed problem models, using a third-party solver and x-check method to iteratively add constraints and find optimal cutting plans that maximize material utilization and adhere to cutting requirements.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Loss of time
If heuristic algorithms are used for packing defective sheets, then a feasible solution can be obtained within a certain period of time, but the solution quality cannot be guaranteed and the results are unstable
Solution Approach 1:
The patent segments the complex two-dimensional packing problem into multiple one-dimensional strip cutting problems by horizontally dividing the defective sheet into several strips. This segmentation allows the use of efficient one-dimensional algorithms while maintaining overall solution quality, resolving the contradiction between solving speed and solution optimality.
Solution Approach 2:
The patent introduces an x-check method as an intermediary verification mechanism. This method checks whether the solution obtained from the relaxed problem model satisfies the constraints of the original primal problem model, ensuring solution quality without requiring complex optimization algorithms.
2Measurement precision
If simulated annealing algorithm is used, then the global optimal solution may be obtained, but the solving process is complicated and solving efficiency is low
Solution Approach 1:
The patent divides the two-dimensional packing problem into multiple one-dimensional strip cutting problems. This segmentation simplifies the optimization process, allowing efficient solving of each strip independently while maintaining global optimality, thus resolving the contradiction between solution optimality and solving efficiency.
Solution Approach 2:
The patent extracts and removes the complex two-dimensional optimization constraints by transforming the problem into a relaxed one-dimensional model. This extraction eliminates the need for complicated iterative optimization processes while preserving the essential packing requirements.
3Reliability
If genetic algorithm is used, then satisfactory global searching can be enabled, but local searching ability is poor, time consumption is large, and convergence speed is slow
Solution Approach 1:
The patent segments the global optimization problem into multiple local one-dimensional optimization problems for each strip. This segmentation enables efficient local optimization within each strip while the overall arrangement ensures global optimality, resolving the contradiction between global searching capability and convergence speed.
4Measurement precision
If complex optimization algorithms are used for defective sheet packing, then solution accuracy may be improved, but computing time and resource consumption increase significantly
Solution Approach 1:
The patent segments the complex two-dimensional optimization into multiple simple one-dimensional strip cutting problems. This segmentation dramatically reduces computational complexity while maintaining packing optimization accuracy, as each strip can be solved independently and efficiently.
Solution Approach 2:
The patent extracts the essential packing requirements from the complex optimization model and formulates a relaxed one-dimensional problem model. This extraction removes unnecessary computational complexity while preserving the key constraints and objectives, achieving accurate solutions with minimal computing time.
Data Source
AI summary
A packing method for a defective sheet based on branch-and-cut algorithm, including: acquiring sheet information, where the sheet information includes size information of the defect sheet, size information of a target block and position information of a defect; based on actual cutting requirements, establishing constraints; and establishing a primal problem model according to the constraints; converting the primal problem model into a relaxed problem model; checking and verifying the obtained solution by x-check method to determine whether the obtained solution meets the constraints of the primal problem model; outputting the optimal solution to obtain an optimal cutting plan that meets the actual cutting requirements; and cutting the defective sheet according to the optimal cutting plan. A system for implement the packing method is also provided.