Planning and Scheduling Surgeries under Stochastic Environment

This dissertation presents an integrated approach to planning and scheduling surgeries in operating-rooms (ORs) at strategic, tactical and operational levels. We deal with uncertainties of surgery demand and durations to reflect a reality in OR management.

The strategic part of the dissertation studies capacity decisions that allocate surgical specialties to OR days with the objective of minimizing total expected costs due to penalties for any patients who are not accommodated and for under- (i.e., idleness) and over- (i.e., overtime) usage of OR capacity. It presents a prototypical non-linear, stochastic programming model to structure the problem and four adaptations, along with associated solution approaches, with the goal of facilitating solution by overcoming the computational disadvantages of the prototype. Each of these models offers advantages but is also attended by disadvantages. Computational tests compare the four models and solution approaches with respect to solution quality and run time.

The tactical part of the dissertation prescribes an approach to optimize a master surgical schedule (MSS), which adheres to the block scheduling policy, using a new type of newsvendor-based model. Our newsvendor approach prescribes the optimal duration of each block and the best permutation, obtained by solving the sequential newsvendor problem, determines the optimal block sequence. We obtain closed-form solutions for the case in which surgery durations follow the normal distribution. Furthermore, we give a closed-form solution for optimal block duration with no-shows. We conduct numerical tests for surgery durations that follow normal, lognormal and gamma distributions. Results show that the closed-form solutions associated with the normal distribution gives close approximations to solutions associated with log-normal and gamma distributions.

The operational part of the dissertation prescribes an optimal rule to sequence two or three surgeries in a block. The smallest-variance-first-rule (SV) is generally accepted as the optimal policy for sequencing two surgeries, although it has been proven formally only for several restricted cases. We extend prior work, studying three distributions as models of surgery duration (the lognormal, gamma, and normal) and including overtime in a total-cost objective function comprising surgeon-and-patient- waiting-, operating-room-idle-, and staff over-times. We specify expected waiting- and idle- time as functions of the parameters of surgery duration to identify the best rule to sequence two surgeries. We compare the relative values of expected waiting- and idle- times numerically with that of expected overtime. Results recommend that the SV rule be used to minimize total expected cost of waiting-, idle- and over-time. We find that gamma and normal distributions with the same mean and variance as the lognormal give nearly the same expected waiting- and idle- times, observing that the lognormal in combination with either the gamma or normal gives a similar result.

Lastly, the dissertation investigates an appointment system with deterministic arrival times (D) and non-identical exponential service times (M). For two customers, we show that both the smallest-mean-first-rule and the SV minimize the sum of expected waiting- and idle-times. We prove that neither is optimal for three customers, but verifies that the first customer in the sequence should be the one with the smallest variance (mean).