Loss Ratio as a Risk Constraint: Analysis and Optimization

42 Pages Posted: 25 Apr 2023 Last revised: 14 Mar 2024

See all articles by Opher Baron

Opher Baron

University of Toronto - Rotman School of Management

Oded Berman

University of Toronto - Rotman School of Management

Andre Augusto Cire

University of Toronto at Scarborough - Division of Management; University of Toronto - Operations Management

Vahid Roshanaei

University of Toronto - Rotman School of Management

Date Written: April 20, 2023

Abstract

Operational models often balance assignment and capacity decisions under data uncertainty. For example, in operating room scheduling the decision maker must define appropriate limits for personnel overtime while accounting for random surgery durations. Typical approaches to hedge against uncertainty are pervasive and include probabilistic constraints, robust optimization, and risk measures such as conditional value-at-risk (CVaR) inequalities. In this work, we propose to safeguard linear capacity constraints by limiting the relative expected shortage with respect to the capacity, denoted here by the loss ratio. Our motivation is to investigate a simple-to-interpret metric as a function of allocated capacity while also exploiting tractable structure. Specifically, we characterize the analytical minimum capacities for a given assignment and show its connection to existing techniques such as chance constraints and CVaR, and provide probabilistic guarantees on constraint violations. We also show a sensitivity analysis to assess the impact of a controllable safety level on costs, as well as evaluate the loss ratio for nominal deterministic models based on expectation. We then discuss optimization aspects that incorporate the proposed risk constraints under scenario-based uncertainty, presenting a polynomial-time algorithm on the number of samples to find minimum capacities as well as marginals to be used in dual-based decomposition techniques. We illustrate our models on a home care network design problem inspired by practice.

Keywords: Stochastic Programming, Inventory/Production:Uncertainty:Stochastic, Integer Programming:Algorithms:Benders/decomposition, Healthcare

Suggested Citation

Baron, Opher and Berman, Oded and Cire, Andre Augusto and Roshanaei, Vahid, Loss Ratio as a Risk Constraint: Analysis and Optimization (April 20, 2023). Rotman School of Management Working Paper No. 4425015, Available at SSRN: https://ssrn.com/abstract=4425015 or http://dx.doi.org/10.2139/ssrn.4425015

Opher Baron

University of Toronto - Rotman School of Management ( email )

Oded Berman

University of Toronto - Rotman School of Management ( email )

105 St. George Street
Toronto, Ontario M5S 3E6 M5S1S4
Canada

Andre Augusto Cire

University of Toronto at Scarborough - Division of Management ( email )

1265 Military Trial
Scarborough, Ontario M1C 1A4
Canada

University of Toronto - Operations Management ( email )

105 St. George st
Toronto, ON M5S 3E6
Canada

Vahid Roshanaei (Contact Author)

University of Toronto - Rotman School of Management ( email )

105 St. George Street
Toronto, Ontario M5S 3E6 M5S1S4
Canada

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