[Paper Review] Exact cost minimization of a series-parallel system

F. Castro, J. Gago+3 more|arXiv (Cornell University)|Mar 15, 2012

Reliability and Maintenance Optimization3 references3 citations

TL;DR

This paper presents an exact algebraic method using Gröbner bases to solve the cost-minimization problem in series-parallel reliable systems with multiple component choices. By deriving a closed-form expression for the test set, the approach avoids computationally expensive Gröbner basis computations and achieves significant speedups over prior methods, especially for problems with correlated component cost and reliability.

ABSTRACT

The redundancy allocation problem is formulated minimizing the design cost for a series-parallel system with multiple component choices whereas ensuring a given system reliability level. The obtained model is a nonlinear integer programming problem with a non linear, non separable constraint. We propose an algebraic method, based on Gröbner bases, to obtain the exact solution of the problem. In addition, we provide a closed form for the required Gröbner bases, avoiding the bottleneck associated with the computation, and promising computational results.

Motivation & Objective

• To address the NP-hard redundancy allocation problem in series-parallel systems with multiple component choices, minimizing total cost while satisfying a required system reliability level.
• To overcome the computational bottleneck of traditional methods like branch-and-bound and Lagrangian relaxation by using algebraic techniques.
• To develop an exact solution method that avoids iterative Gröbner basis computation through a closed-form formula for the test set.
• To improve computational efficiency and reduce CPU time compared to existing exact algorithms, particularly in high-dimensional cases.
• To validate the method on realistic test cases with correlated component cost and reliability, reflecting real-world engineering design constraints.

Proposed method

• Formulate the reliability optimization problem as a nonlinear integer programming problem with a non-separable, nonlinear reliability constraint.
• Construct a relaxed linear integer program by replacing the nonlinear reliability constraint with a linear one derived via a greedy algorithm to find a feasible starting point.
• Compute the Gröbner basis of the toric ideal associated with the linear constraints using a closed-form expression, eliminating the need for time-consuming reduction algorithms.
• Use the resulting test set (derived from the Gröbner basis) to perform a descent algorithm that increases the cost function at each step until an optimal solution is found.
• Implement the algorithm in MATLAB using a walk-back procedure to explore the solution space efficiently, guided by the test set directions.
• Apply the method to benchmark instances with varying system sizes and component choices, comparing performance against existing methods like those based on Lagrangian relaxation and branch-and-bound.

Experimental results

Research questions

• RQ1 Can an algebraic method based on Gröbner bases provide an exact solution to the cost-minimization problem in series-parallel systems with multiple component choices?
• RQ2 Can a closed-form expression for the test set be derived to avoid the computational cost of Gröbner basis computation in this context?
• RQ3 How does the proposed method compare in computational efficiency and scalability to existing exact methods such as branch-and-bound and Lagrangian relaxation?
• RQ4 What is the impact of component cost-reliability correlation on the performance and solution quality of the algorithm?
• RQ5 Can the method be effectively combined with heuristic strategies to achieve fast, near-optimal solutions while preserving optimality guarantees?

Key findings

• The proposed method achieves exact solutions with significantly reduced CPU time compared to prior exact algorithms, such as the cut-and-partition scheme in [11], especially for larger systems.
• For systems with 20 subsystems and 2 component types, the algorithm required only 276 seconds on average, while processing 294,747 nodes, outperforming the 85,103 iterations reported in [11] for a similar problem.
• In the case of correlated component cost and reliability, the method solved all test instances exactly, with average CPU times under 10 seconds for small systems (n=10, k≤5), and under 15 minutes for n=17, k=2.
• The algorithm’s performance degrades with less reliable components (r_ij ∈ [0.98, 0.99]), but still found optimal solutions, indicating robustness under challenging conditions.
• Stopping the algorithm at the first feasible solution found during the walk-back procedure yields high-quality approximations in seconds, suggesting strong potential for hybridization with heuristics.
• The closed-form Gröbner basis construction eliminates the need for costly reduction steps, making the method scalable and practical for real-world engineering design problems.