Minimum Cost Network Design with k-Coverage Approximation
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Solution Overview
Problem
Existing methods fail to efficiently design minimum-cost networks with large numbers of network elements that provide k-coverage, requiring excessive computational resources and time to find accurate solutions to the k-set covering problem.
Innovation Solution
The development of hybrid heuristics such as GRASP with path-relinking, Lagrangian heuristic, and Greedy Lagrangian Heuristic (GLH) and GRASP Lagrangian Heuristic (LAGRASP) to approximate solutions to the k-set covering problem, allowing for the selection of the least-cost network elements that provide k-coverage.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If exact methods are used to solve the k-set covering problem, then solution accuracy is improved, but computational resources and time increase excessively
Solution Approach 1:
The patent transforms the original k-set covering problem into a Lagrangian relaxation problem by introducing Lagrange multipliers and modifying the objective function parameters. This parameter transformation allows the problem to be solved more efficiently while maintaining solution quality through iterative refinement of the Lagrange multipliers
Solution Approach 2:
The patent introduces Lagrange multipliers as intermediary variables that mediate between the conflicting requirements of solution accuracy and computational efficiency. These multipliers act as a bridge, allowing the algorithm to converge to accurate solutions without requiring exhaustive search of all possible network configurations
2Adaptability or versatility
If the number of network elements increases, then network coverage capability is improved, but problem complexity and computational difficulty increase
Solution Approach 1:
The patent segments the large-scale k-set covering problem into smaller subproblems by utilizing the Lagrangian relaxation structure. This segmentation allows the computational burden to be distributed and managed more effectively, enabling the handling of networks with large numbers of elements without exponential increase in complexity
Solution Approach 2:
The patent moves the problem from the original combinatorial optimization dimension into the Lagrangian dual space, adding a new dimensional perspective. This dimensionality change transforms the intractable combinatorial problem into a more manageable form that can be solved efficiently even as network size increases
3Reliability
If redundant coverage (k-coverage) is required, then network reliability is improved, but network cost increases
Solution Approach 1:
The patent implements feedback through iterative updates of Lagrange multipliers based on the coverage constraints. This feedback mechanism allows the algorithm to learn from constraint violations and progressively refine the solution to achieve the desired k-coverage level at minimum cost, rather than requiring manual trial and error
Solution Approach 2:
The patent applies partial action by selectively activating only those network elements necessary to achieve the k-coverage requirement. Rather than deploying all potential elements, the Lagrangian relaxation approach identifies and activates only the minimal subset needed, reducing overall network cost while maintaining reliability
Data Source
AI summary
Methods, systems, and products determine minimum cost networks. A k-fold coverage formulation is applied to potential network elements having cost and coverage parameters. Iterative heuristics are used to find an optimal solution to the k-set coverage formulation. The approximate solutions to the k-set coverage formulation are then used to select at least some of the potential network elements for use in a minimum cost network.