Stochastic Network Path Optimization via Mean-Variance Projection
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Solution Overview
Problem
Existing methods for finding stochastic shortest paths in networks with randomly distributed edge lengths are inefficient, often relying on approximate heuristic processes that fail to maximize the probability of reaching a destination within a given constraint, especially when edge lengths follow non-linear distributions.
Innovation Solution
A method that models stochastic networks with normally distributed edge lengths using quasi-convex maximization, reducing the problem to a parametric shortest path problem, allowing for exact computation of optimal paths by projecting the network onto a mean-variance plane and identifying vertices that maximize the probability of reaching the destination within a constraint.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If adaptive heuristic processes are used to find stochastic shortest paths, then the method can handle non-linear distributions, but the solution accuracy is approximate and lacks error bounds
Solution Approach 1:
The patent replaces adaptive heuristic search processes with a systematic transformation approach. By transforming the stochastic network into an equivalent deterministic network using mean-variance analysis, the method eliminates the need for iterative heuristics while providing exact error bounds through the mathematical properties of the transformation.
Solution Approach 2:
The patent changes the parameter space by projecting stochastic edge parameters (random variables with non-linear distributions) onto a deterministic mean-variance plane. This parameter transformation allows the use of exact deterministic algorithms while preserving the statistical properties of the original stochastic problem, achieving both accuracy and adaptability.
2Measurement precision
If exact methods are used to solve stochastic shortest path problems, then solution accuracy is improved, but the computational time complexity becomes exponential or worse
Solution Approach 1:
The patent extracts the stochasticity from the network by separating the mean and variance components of edge distributions. This extraction transforms the complex stochastic optimization problem into a simpler deterministic problem on a projected network, reducing computational complexity from exponential to polynomial time while maintaining exact solution accuracy.
Solution Approach 2:
The patent adds a new dimension to the problem by introducing the mean-variance plane as a projection space. By mapping stochastic edges to deterministic edges in this two-dimensional parameter space, the method enables efficient computation through standard deterministic algorithms while preserving the ability to compute exact error bounds.
3Productivity
If the network is projected onto a mean-variance plane, then the number of deterministic problems to solve is reduced, but the transformation requires computing mean and variance for each edge
Solution Approach 1:
The patent performs preliminary computation of mean and variance for each edge before the main path-finding algorithm. This pre-processing step transforms all stochastic edges into deterministic mean-variance edges in advance, allowing the subsequent deterministic shortest path algorithm to run efficiently without repeated stochastic simulations or computations during the search process.
Data Source
AI summary
A method finds an optimal path from a source to a destination. The possible paths from the source to the destination are represented as a stochastic graph of nodes connected by edges. Each edge has an independent probability distribution over a cost of the edge. A constraint for reaching the destination is defined. The graph is reduced to a relatively small set of deterministic minimum cost problems, which can be solved to determine an optimal path that maximizes a probability of reaching the destination within the constraint.