Reformulation, linearization, and decomposition techniques for balanced distributed operating room scheduling

Abstract We study the balanced distributed operating room (OR) scheduling (BDORS) problem as a location-allocation model, encompassing two levels of balancing decisions: (i) daily macro imbalance among collaborating hospitals in terms of the number of allocated ORs and (ii) daily micro imbalance among open ORs in each hospital in terms of the total caseload assigned. BDORS is formulated as a novel mixed-integer nonlinear programming (MINLP) in which the macro and micro imbalance are penalized using absolute value and quadratic functions. We develop various reformulation-linearization techniques (RLTs) for the MINLP models, leading to three mathematical modelling variants: (i) a mixed-integer quadratically constrained program (MIQCP) and (ii) two mixed-integer programs (MIPs) for the absolute value penalty function and an MIQCP for the quadratic penalty function. Two novel exact techniques based on reformulation-decomposition techniques (RDTs) are developed to solve these models: a uni- and a bi-level logic-based Benders decomposition (LBBD). We motivate the LBBD methods with an application to BDORS in the University Health Network (UHN), consisting of three collaborating hospitals: Toronto General Hospital, Toronto Western Hospital, and Princess Margaret Cancer Centre in Toronto, Ontario, Canada. The uni-level LBBD method decomposes the model into a surgical suite location, OR allocation, and macro balancing master problem (MP) and micro OR balancing sub-problems (SPs) for each hospital-day. The bi-level approach uses a relaxed MP, consisting of a surgical suite location and relaxed allocation/macro balancing MP and two optimization SPs. The primary SP is formulated as a bin-packing problem to allocate patients to open operating rooms to minimize the number of ORs, while the secondary SP is the uni-level micro balancing SP. Using UHN datasets consisting of two datasets, hard MP/easy SPs and easy MP/hard SPs, we show that both LBBD approaches and both MIP models solved via Gurobi converge to  ≈ 2% and  ≈ 1–2% optimality gaps, on average, respectively, within 30 minutes runtime, whereas the MIQCP solved via Gurobi could not solve any instance of the UHN datasets given the same runtime. The uni- and bi-level LBBD approaches solved all instances of hard MP/easy SPs dataset to  ≈ 11% and  ≈ 2% optimality gaps, on average, respectively, within 30 minutes runtime, whereas MIQCP solved via Gurobi could not solve any of these instances. Additionally, we show that convergence of each LBBD varies depending on where in the decomposition the actual computational complexity lies.