Image-Based Decomposition for Linear Programming Solve
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Solution Overview
Problem
Monolithic linear programming (LP) problems in supply chain planning are not amenable to standard decomposition techniques, leading to prolonged solve times and resource-intensive processes during supply chain planning, as existing methods require prior knowledge of the problem structure and functional insights which may not be available or are too complex.
Innovation Solution
The method involves image-based decomposition, where the LP problem is converted into images to identify clusters and contours, allowing for the decomposition of the problem into smaller subproblems, which can be solved sequentially or in parallel, using an image-based partitioning method and masterless iteration to generate a globally-optimal solution.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If monolithic LP problems are solved using standard decomposition techniques, then problem structure knowledge is required, but this approach is not amenable to standard decomposition and requires prior knowledge of problem structure and functional insights which may not be available
Solution Approach 1:
The patent creates a visual representation (image) of the LP problem structure where rows and columns are mapped to spatial coordinates. This visual copy allows decomposition based on spatial proximity rather than requiring prior knowledge of problem structure, enabling automatic identification of subproblems through image clustering and contour detection
Solution Approach 2:
The patent segments the LP problem into smaller subproblems by dividing the visual representation into distinct spatial regions. Connected components in the image are identified as separate subproblems, each with its own constraints and variables, allowing independent solving while maintaining global optimality through masterless iteration
2Reliability
If monolithic LP problems are solved using standard methods, then optimal solutions can be generated, but solve times are overly time consuming and resource intensive
Solution Approach 1:
By segmenting the LP problem into smaller subproblems based on visual clustering, the patent enables parallel processing and reduces computational complexity. Each subproblem can be solved independently and simultaneously, dramatically reducing total solve time while maintaining solution optimality through coordinated masterless iteration
Solution Approach 2:
The patent transforms the LP problem from a mathematical abstraction into a visual dimension where spatial proximity indicates problem connectivity. This dimensional transformation enables intuitive decomposition and parallel solving, converting a time-consuming sequential process into an efficient parallel computation
3Reliability
If monolithic LP problems are solved using standard methods, then optimal solutions are generated, but the process requires simplifying constraints or objectives to finish within pre-specified batch solve windows
Solution Approach 1:
The patent divides the LP problem into smaller subproblems that can be solved in parallel within the batch window. This segmentation allows the system to process multiple subproblems simultaneously, increasing throughput without requiring simplification of constraints or objectives, thereby maintaining both optimality and productivity
4Speed
If standard decomposition techniques are used, then solve speed improves, but prior knowledge of problem structure and functional insights are required which are not always available
Solution Approach 1:
The patent creates a visual copy of the LP problem structure where mathematical relationships are represented as spatial arrangements. This visual representation eliminates the need for prior knowledge of problem structure, as the image itself reveals connectivity patterns through clustering and contour analysis, enabling automatic decomposition and fast solving
Solution Approach 2:
The patent replaces traditional mechanical decomposition methods (which require explicit problem structure knowledge) with a visual-based approach. Image processing techniques automatically identify problem components and relationships, substituting human expertise with automated visual analysis that works regardless of problem structure complexity
Data Source
AI summary
A system and method are disclosed for solving a supply chain planning problem modeled as a linear programming (LP) problem. Embodiments include receiving a matrix formulation of at least a portion of the LP problem representing a supply chain planning problem for a supply chain network, generating an image based on the matrix formulation to identify connected components, partitioning the matrix formulation based, at least in part, on the connected components constraint into at least two partitions, formulating an LP subproblem from each of the at least two partitions, and solving the LP subproblems to generate a global solution to the supply chain planning problem.