  This study examines a resource-sharing problem involving multiple parties
that agree to use a set of capacities together. We start with modeling the
whole problem as a mathematical program, where all parties are required to
exchange information to obtain the optimal objective function value. This
information bears private data from each party in terms of coefficients used in
the mathematical program. Moreover, the parties also consider the individual
optimal solutions as private. In this setting, the concern for the parties is
the privacy of their data and their optimal allocations. We propose a two-step
approach to meet the privacy requirements of the parties. In the first step, we
obtain a reformulated model that is amenable to a decomposition scheme.
Although this scheme eliminates almost all data exchanges, it does not provide
a formal privacy guarantee. In the second step, we provide this guarantee with
a locally differentially private algorithm, which does not need a trusted
aggregator, at the expense of deviating slightly from the optimality. We
provide bounds on this deviation and discuss the consequences of these
theoretical results. We also propose a novel modification to increase the
efficiency of the algorithm in terms of reducing the theoretical optimality
gap. The study ends with a numerical experiment on a planning problem that
demonstrates an application of the proposed approach. As we work with a general
linear optimization model, our analysis and discussion can be used in different
application areas including production planning, logistics, and revenue
management.
