On the Optimization of Benefit to Cost Ratios for Public Sector Decision Making

Abstract

Decision making in the public sector centers on delivering resources and services for the common good, emphasizing an expansive set of objectives such as equity and efficiency, beyond immediate short term returns to reflect the broader cares of society and public beneficiaries. Cost-benefit analysis is a prevailing decision-making framework in the public sector that often uses the benefit to cost ratio (BCR) to compare viable alternatives, yet no systematic framework exists for evaluating many alternatives beyond the status quo of doing nothing. We propose a new framework to maximize the BCR for public sector decisions, seeking the largest improvement per marginal deployment of capacity. Requiring a status quo representable through (constrained) decision variables, the framework is generally applicable and useful to a broad set of decision contexts that involve maximizing the BCR for marginal deployments of resources. We demonstrate the applicability of our framework on a compelling case study for the New York City runaway and homeless youth shelter system, an area of high societal need. We represent this problem as a mixed integer linear fractional program (MILFP) and employ Dinkelbach's algorithm that converts the MILFP to a series of linearized mixed-integer optimization problems, making our approach tractable for fairly large problem instances. Our optimization-based algorithmic framework yields data-informed recommendations for making New York City shelter expansion decisions to better serve runaway and homeless youth, and generalizes to reveal managerial insights for optimizing the BCR. More broadly, our algorithmic decision making framework allows for iteration and comparison across multiple potential constraints ensuring action away from the status quo, thereby empowering effective assessment of marginal deployment of additional resources