This paper addresses an open challenge in the field of surgery scheduling: stochastic surgery
scheduling in multiple ORs with PACU constraints. The objective is to minimize the
expected cost of patient waiting time, surgeon idle time, OR blocking time, OR overtime and PACU overtime. With the surgery sequence predetermined in each OR, this problem is formulated as a stochastic optimization model based on a discrete event dynamic system (DEDS). With sample gradients derived by perturbation analysis, a sample-based gradient descent algorithm with random restarts (SAA-GDR) is proposed to solve the sample average approximation of our stochastic optimization model. Numerical experiments are conducted to select sample size and the number of random restarts. It is demonstrated that SAA-
GDR with 1000 scenarios and 50 random restarts could identify a near-optimal schedule for more than 20 patients within 1 minute. SAA-GDR is shown to outperform other methods, including the time-indexed model and stochastic approximation method, in terms of solution quality and running time. Lastly, we present the change in the schedule after including PACU constraints and demonstrate that considerable cost savings (11.8% on average) are possible in the many hospitals where PACU beds are a constraint.
Although the First-Come-First-Served (FCFS) rule is applied in the PACU admission process in our study, problems with other priority-based rules, as discussed in §3.1, could also be solved by SAA-GDR with modifications in the sample-gradient derivation.
Chapter 4
Surgery Sequencing and
Scheduling in multiple ORs with
PACU capacity constraints
4.1
Problem Description
In this study, we consider the surgery sequencing and scheduling problems after surgeons have determined which surgeries will be performed in their assigned blocks. We consider only the planned surgeries without considering emergency cases in this paper. An additional set of assumptions are made similar to those in [27, 68, 57].
• Each surgeon performs all his or her operations in a single designated OR.
• Surgeries in each OR are carried out in the same sequence as determined by our sequencing method.
• Surgeons arrive at the beginning of the day (time 0) and the first surgery in each OR is scheduled at time 0. All surgeries are scheduled to start within regular work hours. For example surgeries can only be scheduled to start within an 8-hour period, i.e. [0, 8] (although actual surgery start times may occur outside the 8-hour window).
• Patients arrive at their scheduled start times punctually and thus a surgery cannot be started before its scheduled start time.
• Patients’ surgical durations and length of stay (LOS) in the PACU follow known truncated random distributions. Surgical durations and LOS in the PACU of patients are independent from the surgery sequence and SST of surgeries. Surgical durations and LOS in the PACU of different patients are assumed mutually independent and the distributions differ by patient.
• Turnover times of pre/post-surgical operations are included in the surgical durations. • The time to transport a patient from the OR to the PACU and release a patient from
the PACU is included in the LOS in the PACU.
• The non-anticipative rule is enforced in our study which requires decisions not to depend on information from a later time. It has two implications in our study:
1. A surgery is started instantly when the OR, the surgeon and the patient are ready. This also means the first surgery in each OR starts at time 0.
2. A patient enters the PACU immediately after his or her surgery if a PACU bed is available.
• If two or more patients are waiting for a PACU bed, the one with the earliest surgery finish time will first enter the PACU, i.e. the First-Come-First-Served (FCFS) PACU admission rule. If there is a tie in surgery finish time, the patient from the smallest- indexed OR is arbitrarily admitted to the PACU first. The proposed method can easily accommodate alternative PACU admission rules with minimal modifications to the heuristic to construct feasible solutions to SURSAA in §4.2.3. FCFS is used in this study to demonstrate the development of our solution approach. We include more discussions about different PACU admission rules in the Conclusions.
Given these assumptions, the patient flow on the day of surgery is as follows: The first surgery in each OR is started at time 0 and other surgeries are carried out in the sequence
determined by our method. After surgery, a patient is admitted into the PACU if a PACU bed is available; otherwise the patient is blocked in the OR waiting for a PACU bed. If two or more patients are blocked and competing for a PACU bed, they are admitted following the FCFS PACU admission rule. When a patient is admitted into the PACU, the following surgery (if any) will be conducted as soon as possible at the maximum of its scheduled start time and previous surgery’s completion time. A patient spends LOS in the PACU before he or she can be discharged.
The surgery schedule is assessed by the expected cost of patient waiting time, surgeon idle time, OR blocking time, OR overtime and PACU overtime. Patient waiting time is the time when a patient waits before his or her surgery starts. Surgeon idle time is the time when a surgeon waits between the finish of a surgery and the start of the following surgery. Note that a surgeon leaves after his or her last surgery in the OR even if the last patient is held waiting for a PACU bed. OR blocking time is the time when a patient is blocked in an OR after surgery before his or her entry into the PACU. Note that the surgeon is idle while the patient is blocked in the OR and hence the OR blocking time is “double”-penalized. OR overtime is counted for each OR and it indicates the extra amount of time an OR that is used beyond regular work hours. PACU overtime is the time by which the last PACU release exceeds regular hours. Cost parameters are determined in §4.3.1.