Column Generation Routing With Detour Inequalities and Graph Management
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Existing column generation methods for solving large-scale logistical optimization problems, such as vehicle routing and scheduling, face inefficiencies due to the need to consider all variables explicitly, leading to increased computational complexity and time in finding optimal solutions.
Innovation Solution
The implementation of dual optimal inequalities, graph generation, principled graph management, and local area routes within the column generation framework allows for the generation of optimized routes by splitting problems into master and subproblems, iteratively solving them using column generation methods, and adding columns to the master problem based on subproblem solutions, thereby reducing computational complexity and improving convergence speed.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If all variables are considered explicitly in linear programming models for routing and scheduling, then the solution completeness is improved, but the computational complexity and solving time increase significantly
Solution Approach 1:
The patent divides the large-scale linear programming problem into multiple smaller subproblems using column generation methodology. The master problem coordinates multiple subproblems, each handling a subset of variables. This segmentation allows the system to solve complex routing and scheduling problems by breaking them into manageable pieces that can be solved iteratively, maintaining solution completeness while reducing computational complexity at each step.
Solution Approach 2:
The patent transforms the problem from considering all variables in a single large linear program to a two-dimensional approach using column generation. The outer dimension represents iterations of the master problem, while the inner dimension represents subproblems generating candidate columns. This dimensional transformation enables the system to handle large-scale problems by exploring the solution space systematically across multiple levels rather than exhaustively in a single layer.
2Productivity
If column generation methods are used to reduce variables considered, then the computational speed is improved, but the convergence time may increase due to iterative solving
Solution Approach 1:
The patent applies preliminary action by pre-processing the problem data and initializing the column generation algorithm with a carefully selected initial set of columns. The system performs preliminary calculations to set up the master problem and subproblems in an optimized state before iterative solving begins. This preliminary preparation reduces the number of iterations needed for convergence, thereby reducing overall convergence time while maintaining the computational speed benefits of column generation.
3Productivity
If the problem is split into master and subproblems for column generation, then the computational efficiency is improved, but the algorithm complexity increases
Solution Approach 1:
The patent introduces an intermediary coordination mechanism between the master problem and subproblems. The master problem acts as an intermediary that receives candidate columns from subproblems, evaluates them, and directs subsequent subproblem generation. This intermediary structure simplifies the overall algorithm complexity by providing a clear coordination protocol, even though multiple components are involved. The master problem mediates the interaction between subproblems, making the complex system manageable through structured communication and control.
Data Source
AI summary
Disclosed are devices, systems, and methods for routing and scheduling using column generation (CG)-based techniques. The described embodiments include dual optimal inequalities that incorporate detours (referred to as detour-DOI), graph generation (GG), principled graph management (PGM), and column generation with local area (LA) route relaxations. These CG-based techniques can be applied to a variety of logistical optimization problems that include, but are not limited to, the facility location problem, the capacitated vehicle routing problem (CVRP), the CVRP with time windows, multi-robot routing (both with and without time windows), the bus driver scheduling problem, the supply chain scheduling problem, the shift/workforce scheduling problem, and the grocery picking for micro-fulfillment problem.