Supply Chain LP Decomposition by Variable Fixing and Bound Splitting
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Solution Overview
Problem
Monolithic linear programming (LP) formulations of multi-objective supply chain planning problems are inefficient and complex, making them unsuitable for standard decomposition techniques, which hinders solving speed and accuracy.
Innovation Solution
A variable-fixing decomposition method is applied to decompose LP supply chain planning problems by fixing variables to their upper or lower bounds, allowing the problem to be divided into subproblems, which are then solved independently and combined for a globally optimal solution.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If monolithic LP formulation is used for multi-objective supply chain planning problems, then the problem can be solved with a single unified model, but the solve time and computational complexity increase significantly
Solution Approach 1:
The patent applies segmentation by dividing the monolithic LP problem into multiple subproblems based on hierarchical objectives. Each subproblem corresponds to a specific objective level and can be solved independently. The decomposition is achieved by identifying variables that appear in multiple objectives and using them as decomposition points, creating smaller, more manageable subproblems that can be solved in parallel or sequentially, thereby reducing overall solve time while maintaining solution optimality.
2Reliability
If monolithic LP formulation is used for multi-objective supply chain planning problems, then the complete problem structure is preserved, but the computational complexity increases making standard decomposition techniques inapplicable
Solution Approach 1:
The patent segments the complex monolithic LP problem into hierarchical subproblems based on objective priorities. By identifying shared variables across objectives and using them as decomposition boundaries, the method creates a segmented structure that preserves the original problem's integrity while reducing computational complexity to a level where standard solution techniques become applicable.
Solution Approach 2:
The patent introduces a hierarchical dimension to the problem structure by organizing objectives into levels. This dimensional transformation allows the complex multi-objective problem to be viewed and solved as a series of simpler single-objective problems at different hierarchical levels, effectively reducing computational complexity while maintaining the complete problem structure through the hierarchical framework.
3Productivity
If standard decomposition techniques are applied to monolithic LP problems, then solving speed may be improved, but these techniques are generally not amenable to monolithic LP formulations
Solution Approach 1:
The patent makes monolithic LP problems amenable to decomposition techniques by systematically segmenting them into hierarchical subproblems. This segmentation is achieved by identifying variables that connect different objectives and using them as decomposition boundaries, thereby creating a structure that standard decomposition techniques can effectively process while improving solving speed.
Solution Approach 2:
The patent changes the structural parameters of the monolithic LP problem by transforming it from a single unified formulation into a hierarchical set of subproblems. This parameter change involves modifying how variables and constraints are organized across different objective levels, making the problem adaptable to standard decomposition techniques while maintaining the original problem's solution properties.
Data Source
AI summary
A system and method are disclosed including a computer that receives a formulation of a multi-objective linear programming planning problem, the formulation including at least one variable fixed at an upper bound or a lower bound. The computer also solves the formulation for a higher-order objective and fixes the upper bound or the lower bound of at least one variable to preserve a solution of the formulation for the higher-order objective. The computer also replaces at least one variable in the formulation with a value of the upper bound or the lower bound, when the upper bound of at least one variable is fixed at the lower bound or the lower bound of at least one variable is fixed at the upper bound, and checks whether replacing at least one variable with the value of the upper bound or the lower bound divides the formulation into two independent components.