On the sensitivity of linear resource sharing problems to the arrival of new agents

TL;DR

This paper analyzes how the arrival of new agents impacts resource sharing in a linear programming model, providing a probabilistic sensitivity index crucial for managing multi-agent systems effectively.

Contribution

It introduces a probabilistic sensitivity index for resource sharing problems, linking agent arrivals to potential changes in allocations using dual reformulations and scenario approaches.

Findings

Derived a formula for the probability of resource share impact.

Applied the sensitivity index to a cargo loading example.

Demonstrated the approach's effectiveness through numerical analysis.

Abstract

We consider a multi-agent optimal resource sharing problem that is represented by a linear program. The amount of resource to be shared is fixed, and agents belong to a population that is characterized probabilistically so as to allow heterogeneity among the agents. In this paper, we provide a characterization of the probability that the arrival of a new agent affects the resource share of other agents, which means that accommodating the new agent request at the detriment of the other agents allocation provides some payoff. This probability represents a sensitivity index for the optimal solution of a linear programming resource sharing problem when a new agent shows up, and it is of fundamental importance for a correct and profitable operation of the multi-agent system. Our developments build on the equivalence between the resource sharing problem and certain dual reformulations which can be interpreted as scenario programs with the number of scenarios corresponding to the number of agents in the primal problem. The recent "wait-and-judge" scenario approach is then used to obtain the sought sensitivity index. Our theoretical findings are demonstrated through a numerical example on optimal cargo aircraft loading.