Time-varying capacity planning for designing large-scale homeless care systems

Many people in communities around the world are facing homelessness due to housing shortages. The San Francisco Bay Area has struggled to provide housing for thousands of people who are unsheltered. Permanent housing is the ideal solution for most people entering the system, but temporary shelter is also critical. Investment in housing and shelter is paramount to providing a long-term solution to serve the current and future homeless population. We construct a queueing model for tracking the flow of single adults through shelter and housing based on Alameda County’s coordinated entry system. In contrast to routing or allocation policies, we optimize the system through increasing shelter and housing server capacities. We formulate optimization problems to reduce the size of the unsheltered population given cost constraints by varying investment in housing and shelter over time. Additionally, we impose policy-based shape constraints to reflect the time-dependence and feasibility constraints associated with planning decisions. We thus show how resources can be allocated between housing and shelter over time. While this joint optimization approach can be used to analyze homeless populations outside of Alameda County, it also can be broadly applied to capacity investment decisions for tandem queueing systems, for example, in healthcare settings.