A lower bound on the expected optimal value of certain random linear programs and application to shortest paths and reliability

TL;DR

This paper derives a computable lower bound on the expected optimal value of certain random linear programs, with applications to shortest path problems and system reliability analysis, especially when direct computation is infeasible.

Contribution

It introduces a general lower bound for the expected optimal value of linear programs with random nonnegative costs, inspired by prior upper bounds, applicable to complex systems.

Findings

Provides a practical lower bound for expected shortest path length

Applicable to large systems where Monte Carlo methods are computationally expensive

Extends theoretical understanding of random linear program expectations

Abstract

The paper studies the expectation of the inspection time in complex aging systems. Under reasonable assumptions, this problem is reduced to studying the expectation of the length of the shortest path in the directed degradation graph of the systems where the parameters are given by a pool of experts. The expectation itself being sometimes out of reach, in closed form or even through Monte Carlo simulations in the case of large systems, we propose an easily computable lower bound. The proposed bound applies to a rather general class of linear programs with random nonnegative costs and is directly inspired from the upper bound of Dyer, Frieze and McDiarmid [Math.Programming {\bf 35} (1986), no.1,3--16].