Discrete Optimization
An approach to optimize block surgical schedules

https://doi.org/10.1016/j.ejor.2013.10.040Get rights and content

Highlights

  • We propose a new approach to optimize a master surgical schedule (MSS).

  • We employ newsvendor models to prescribe optimal block duration and block sequence.

  • We deal with uncertainty, minimizing expected earliness and tardiness costs.

  • We give a closed-form solution for the case with normally distributed surgery times.

  • We give a closed-form solution for optimal block durations when no shows can occur.

Abstract

We provide an approach to optimize a block surgical schedule (BSS) that adheres to the block scheduling policy, using a new type of newsvendor-based model. We assume that strategic decisions assign a specialty to each Operating Room (OR) day and deal with BSS decisions that assign sub-specialties to time blocks, determining block duration as well as sequence in each OR each day with the objective of minimizing the sum of expected lateness and earliness costs. 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.

Introduction

A block surgical schedule (BSS) prescribes the duration of the time block reserved for each specified surgery sub-specialty and sequences time blocks within each operating room (OR) day to achieve the objective of minimizing the sum of expected earliness and tardiness costs. Tactical-level decisions compose a BSS for use over the intermediate-term (e.g., month or quarter), allowing flexibility within a long-term, strategic plan, for example, to accommodate seasonal demand changes. With the goal of synthesizing a methodology to prescribe a BSS, specific research objectives of this paper are 1 a method to optimize the planned duration of each block, minimizing the sum of expected earliness and lateness costs; 2 a method to optimize the sequence (i.e., permutation) of blocks in each OR day; and 3 an extension of our method to prescribe an optimal planned block duration when no-shows are considered.

Each hospital provides a unique capacity for performing surgery through the numbers of ORs and surgical skills it offers. A surgical suite typically comprises several ORs, each of which is equipped to support one (e.g., cardiology, neurological, or orthopedic) or several (e.g., general surgery, ENT) specialties.

The typical surgical specialty comprises a number of sub-specialties. For instance, orthopedics includes hip replacement, knee replacement, femur fixation, and shoulder repair sub-specialties. Surgeries that require the same sub-specialty are medically homogeneous and require the same medical expertise of the surgeon or surgeons involved (van Oostrum et al., 2008).

Allocation (or assignment) decisions are made for the longer term (e.g., six or 12 months); we assume that they assign one specialty to each OR day. Based on specialty-to-OR-day-assignment decisions, the current paper prescribes time blocks for sub-specialties within the specialty for the intermediate term (e.g., month or quarter). To the best of our knowledge, little research had dealt with determining block duration and sequence within an OR day.

We deal with the block scheduling policy in this study. A block is the amount of time during which a specific sub-specialty is assigned to an OR. A block may be planned with the duration of two hours, half of a day, or a day, for example, to permit a surgeon to perform a series of surgeries. An alternative, the open scheduling policy, under which each surgeon can schedule his/her surgeries at any time, was common in the 1960s and 1970s but is rarely used in practice today, because it does not utilize surgeons’ time as efficiently as block scheduling (Blake, Dexter, & Donald, 2002).

Once BSS determines a schedule of time blocks, including the duration and sequence of each, the day-by-day schedule for a week may be used cyclically, that is, for each week over the intermediate-term planning horizon. A cyclic schedule avoids the need to prescribe a new schedule every week and promotes coordination among surgeons, staff and other departments (e.g., post-anesthesia care unit (PACU), intensive care unit (ICU)), affording each surgeon the opportunity to promote his/her efficiency by performing surgeries consecutively and by establishing routine office hours that are compatible with the BSS.

A BSS, which is analogous to a master production schedule in a manufacturing environment, has a number of important uses. A BSS defines aggregate resource requirements of peri-operative activities and ancillary departments (e.g., PACU, ICU, nursing), not only of ORs and surgeons. Nurse managers must ensure that the set of ORs and PACUs run compatibly each day of the week (Blake & Donald, 2002) so that actual decisions adhere to the BSS as strictly as possible. Like Dexter & Hopwood, 1999, Rohleder et al., 2005, Samanlioglu et al., 2010, this paper focuses on ORs and does not deal with other departments. An appropriate BSS allows hospital managers to accommodate random events (e.g., a short-term shortage of surgeons or anesthetists), seasonal fluctuations in demand (e.g., summer or Christmas time), or strategic decisions that alter program emphasis (e.g., to respond to an increasing popularity of cosmetic surgery) (Blake & Donald, 2002). In particular, the operational-level uses the BSS to schedule individual patients; if actual demand levels were to deviate significantly from the those upon which BSS was based, a hospital manager should update the BSS to better accommodate them.

This research contributes from several perspectives. It provides a closed form for optimal block durations, which are given by newsvendor solutions, for the case in which surgery durations are independent and normally distributed. Hospital managers can make use of this closed form to balance the risk of a planned duration that is too short, which could force a late start of the next block (i.e., delay), should actual time exceed it; and the risk of a planned duration that is too long, which could result in expediting the next block, should actual time be less. Furthermore, we deal with a new type of newsvendor problem, which is, in fact, a series of time-based newsvendor problems that we call the sequential newsvendor (SNV) problem. The classic newsvendor model deals with a single time period with random demand, which is a known distribution. It prescribes the optimal order quantity to minimize the sum of costs related to expected demand over and under the order quantity. Our model prescribes the duration of each surgery time block to minimize the sum of costs related to expected early and late completion (i.e., before and after the end of the time block, respectively). Optimal block durations can be obtained via a newsvendor problem that prescribes the optimal planned ending time. We prove that the smallest-variance-first-rule (SV) optimally sequences blocks if each surgery follows a normal distribution. This research also suggests an approach to find the optimal block duration when subject to no-shows, a new and emerging topic in the healthcare setting. However, because surgery typically deals with serious health issues, no-shows are not likely to occur with the high frequency they do, for example, at primary care clinics. If no-shows occur frequently, the risk of idleness increases and is often hedged by overbooking. This research analyzes the effect of no-shows using both the ratio of earliness cost to lateness cost and the probability that a patient will be a no-show to hedge by managing planned block duration.

The remainder of this paper is organized as follows. Section 2 reviews the intermediate-term surgical scheduling literature. Section 3 presents preliminaries and Section 4 describes our solution approach. Section 5 describes the optimal block duration with no-shows. Section 6 provides insights for hospital management. Finally, Section 7 concludes and offers suggestions for future research.

Section snippets

Literature review

Few studies have addressed the tactical level of decision making that prescribes a block surgical schedule for the intermediate term. Complicating matters, there is no commonly accepted definition of intermediate-term surgical scheduling (Testi et al., 2007, van Oostrum et al., 2008). Blake & Donald, 2002, Blake & Donald, 2002, Fei et al., 2008, and Fei et al. (2008) described the intermediate-term surgical scheduling process in detail, comparing it with master production scheduling in

Preliminaries

This section introduces notation and assumptions used in the subsequent presentation. We also discuss both decision and associated random variables. Finally, we formulate the objective function, which minimizes the sum of expected earliness and lateness costs.

Solution approach

In this section, we describe solution approaches to prescribe optimal block durations and optimal sequence. We show that the newsvendor solution, NV[k]δy[k]δ,kK, gives the optimal planned end times for a given sequence δΔ. We derive the closed form of the objective value and prove that the SV rule is optimal to sequence blocks for the case in which surgeries are independent and normally distributed.

Section 4.1 devises optimal block durations for the unconstrained version of SNV(y) and Section

Extensions: no-shows

Patient no-shows play a major role in deteriorating schedule performance (Lin, Muthuraman, & Lawley, 2011) because the no-show rate can be significant; for example, they have been reported to be from 22% to more than 50% (Guse, Richardson, Carle, & Schmidt, 2003) in health-care clinics. Surgery-patient no-shows may result from immediate cancellations before scheduled surgery, due, for example, to failure of patients to prepare for surgery as instructed. Hospital managers can overbook patients

Managerial insights

This paper provides managerial insights into BSS, based on the assumptions that forecasts provide the expect number of surgeries to be performed by each surgical sub-specialty, that a representative surgery-duration distribution that is normally distributed can be derived for each sub-specialty based on historical data, that all surgery durations are mutually independent, and that each surgery begins when the previous one ends. Our analysis results in an easy way to compute the optimal planned

Conclusion

This paper presents new methods to prescribe optimal planned duration and sequence of time blocks, each of which reserves OR resources for a particular surgical sub-specialty. Further, rather than using an overbooking policy, it gives a closed form to prescribe optimal planned block duration to hedge no shows. Results lend considerable insights for managing OR resources.

The methods we propose can be implemented easily and, we expect, would result in improved performance through managing the BSS

Acknowledgements

This material is based in part on work supported by the National Science Foundation on Grant No. CMMI 1129693. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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