Planning and scheduling of semi-urgent surgeries
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Abstract
This paper investigates the trade-off between cancellations of elective surgeries due to semi-urgent surgeries, and unused operating room (OR) time due to excessive reservation of OR time for semi-urgent surgeries.Semi-urgent surgeries, to be performed soon but not necessarily today, pose an uncertain demand on available hospital resources, and interfere with the planning of elective patients. For a highly utilized OR, reservation of OR time for semi-urgent surgeries avoids excessive cancellations of elective surgeries, but may also result in unused OR time, since arrivals of semi-urgent patients are unpredictable. First, using a queuing theory framework, we evaluate the OR capacity needed to accommodate every incoming semi-urgent surgery. Second, we introduce another queuing model that enables a trade-off between the cancelation rate of elective surgeries and unused OR time. Third, based on Markov decision theory, we develop a decision support tool that assists the scheduling process of elective and semi-urgent surgeries. We demonstrate our results with actual data obtained from a department of neurosurgery.
Keywords
Surgical scheduling Operating rooms Emergency patient flow Queuing theory Markov decision processes1 Introduction
We consider a surgical department where elective, urgent and semi-urgent (synonym: semi-elective) patients are treated. An example of a department with such characteristics is a neurosurgery department. Urgent treatment is, among others, required for ruptured aneurysms, epidural or subdural hematomas, cauda equina syndrome, and (instable) spine fractures compromising the myelum or cauda equina. Semi-urgent pathologies include, among others, intracranial oncology, spine fractures with no or minimal neurological symptoms, drain dysfunctionalities, and disc herniations with unbearable pain or severe neurological deficits. Apart from these pathologies, the majority of neurosurgery patients do not require surgery within one or 2 weeks, and these are regarded as elective.
There is a definite trade-off between two major intertwined issues with respect to available surgical capacity: allocation of capacity to surgical departments and optimization of the surgical schedule within departments. On the one hand, when the target is minimal use of surgical resources, a more efficient surgical schedule may reduce the slack in the schedule, and therefore reduce the required capacity while keeping the societal costs due to patient cancellation and waiting constant. On the other hand, when the target is minimal societal costs due to patient cancellation and waiting, a more efficient surgical schedule may reduce these while keeping the allocated surgical resources constant. The trade-off is thus between societal costs and required surgical capacity. Allocating capacity to a surgical department usually is subject to additional constraints such as the restriction in the total available time, the time allocated to other departments, labor regulations (e.g., opening hours of the operating theater), staff restrictions (e.g., available number of surgeons), and the possibility to handle exceptions (e.g., in over-time).
In this paper we take the capacity allocated to a surgical department as a starting point. We aim for robust patient scheduling schemes. We focus on the setting of a neurosurgery department treating urgent, semi-urgent and elective patients. Urgent patients are usually treated in a separate operating room (OR), but semi-urgent patients need to be fitted in the regular OR schedule. When a semi-urgent patient arrives, an elective patient is canceled to accommodate this (prioritized) patient. The cancellation of a surgery negatively affects the patient [1]. Medical professionals tend to feel sorry for the canceled patient and aim to reschedule the surgery as soon as possible. Thus, a canceled elective patient receives a semi-urgent status, and rescheduling this surgery possibly causes the cancellation of another elective patient. This knock-on effect results in a clear dependency between semi-urgent patient arrivals and cancellation of elective patients in subsequent weeks.
Several strategies are known from literature to cope with non-elective patients. One strategy is to reserve a small amount of time for emergency patients for whom surgery is required on the day of arrival in each elective patient OR [2], instead of dedicating one or several ORs to emergent cases [3]. Another possibility is to determine the elective patient schedule given the expected number of emergencies [4]. In all papers reviewed in [5], acute cases have to be performed at least on the day of arrival, as opposed to the semi-urgent surgeries that are studied in this paper. In both [3] and [6] the authors distinguish between emergency surgeries (which have to be performed now) and urgent surgeries (which have to be performed within a day). In [4] and [7] stochastic programming is applied to support the scheduling of add-on cases, but in both papers these cases have to be completed on the day of arrival.
In [8] the authors start from a different viewpoint and determine, using a simulation model, how many elective cases can be performed in a dedicated orthopedic trauma OR. They state that when elective patients are willing to accept that their surgery might be canceled because of an incoming trauma patient, a higher throughput can be achieved. In [9] a trade-off is made between overtime and unused OR time. The paper has an operational viewpoint, by scheduling patients on an individual level. This is similar to the methodology presented in [10], where mathematical algorithms are used to schedule individual cases in available OR blocks.
The problem setting described here shows a similarity with the news vendor problem, where at the start of each decision period for that period the available capacity is matched with the required resources, and unmatched requests are discarded at the end of the period (see e.g. [9, 11, 12, 13] for news vendor problems applied to OR problems). The news vendor problem does not incorporate scheduling of discarded requests in subsequent periods, which is precisely the problem when elective surgeries are canceled and re-scheduled in subsequent periods. Modeling this knock-on effect is the natural domain of queuing theory. In this paper, we therefore invoke the powerful theory of queues to analyze the cancellation rate of elective patients given a pre-specified surgical capacity, and the influence of canceling patients on the cancellation rate in the future.
For a surgical department with given capacity handling elective, urgent and semi-urgent patients, this paper investigates reservation schemes of OR time for semi-urgent surgeries. As the arrival pattern of semi-urgent patients is unpredictable, the reserved OR may remain unused since elective patients cannot be scheduled so shortly before their surgery. We study the trade-off between cancellations of elective surgeries due to semi-urgent surgeries, and unused OR time due to excessive reservation of OR time for semi-urgent surgeries.
In the next section we first evaluate, using a queuing theory framework, the long run OR capacity needed to accommodate every incoming semi-urgent surgery. Second, we introduce another queuing model that enables a trade-off between the cancellation rate of elective surgeries and unused OR time. In Section 3 we develop a decision support tool, based on Markov decision theory, that assists the scheduling process of elective and semi-urgent surgeries. We demonstrate our results in Section 4 with actual data obtained from a department of neurosurgery, followed by the discussion and conclusion in Section 5.
2 Model and long term behavior
The goal of the strategic model presented in this section is to provide an estimate for the amount of OR time that should be reserved for all semi-urgent surgeries in the long run. Therefore, we do not distinguish between the 1- and 2-week streams or take overtime into account. These components of the problem are discussed in the tactical model presented in Section 3. Obviously, dynamically adjusting the amount of reserved OR time according to the effectuated inflow of semi-urgent surgeries would result in little unused OR time. However, given hospital policy that dictates that elective patients should be planned weeks in advance, such an adaptive policy would impose canceling the elective patients that were planned in the claimed slots. In order to make the trade-off between cancellation of surgeries and unused OR capacity, a constant amount of OR time is reserved for semi-urgent surgeries.
Notation introduced in Section 2
Symbol | Description |
---|---|
K | Number of slots available per OR day |
m | Total number of slots assigned to department |
s | Number of slots reserved for semi-urgent surgeries |
W_{n} | Number of semi-urgent slots waiting for surgery at the start of week n |
W | Number of semi-urgent slots waiting for surgery at the start of a week in a stationary regime |
\(\textbf{q}\) | Equilibrium distribution of W |
P_{W}(z) | Generating function of W |
λ | Arrival rate of semi-urgent surgeries |
p_{k} | ℙ(Surgery is of length k slots), k = 1,2,..K |
R_{n} | Number of semi-urgent slots that arrive during week n |
P_{R}(z) | Generating function of the number of arrivals per week |
N_{e} | Number of unused reserved semi-urgent slots per week |
N_{c} | Number of canceled elective slots per week |
C_{e} | Cost of one unused reserved semi-urgent slot |
C_{c} | Cost of one canceled elective slot |
C_{t} | Total Costs |
2.1 Assumptions and model parameters
The time available per OR day is divided into K slots of equal length. Surgeries can have a duration of 1,2,..,K slots (K < ∞), and are categorized according to this duration.
When a surgery has an expected duration of more than K slots, it is also included into the category of surgeries with length K slots. The total number of OR slots assigned to the department per week (m) equals the number of OR days per week multiplied by K. In order to accommodate semi-urgent patients, every week a fixed number of slots (s) is reserved (0 ≤ s ≤ m). Given the impact of the surgery on the patient and the undesirability of performing semi-urgent surgeries in overtime, we assume, in line with medical practice (see Section 1), that canceled elective patients become semi-urgent patients the following week. These patients need to undergo surgery within 1 week of their canceled surgery.
2.1.1 Progression of the number of semi-urgent slots
2.1.2 The arrival process
2.2 Stability of the system
2.3 Stationary distribution of the number of semi-urgent slots waiting
2.4 Performance measures
2.5 Cost structure
3 Optimal allocation of surgery slots
Additional notation introduced in Section 3
Symbol | Description |
---|---|
W1_{n} | Number of 1-week semi-urgent slots waiting for surgery at the start of week n |
W2_{n} | Number of 2-week semi-urgent slots waiting for surgery at the start of week n |
w_{n} = (w1_{n}, w2_{n}) | System state at start of week n |
a_{n} | Action chosen in week n |
R1_{n} | Number of 1-week semi-urgent slot arrivals during week n |
R2_{n} | Number of 2-week semi-urgent slot arrivals during week n |
λ_{1} | Arrival rate of 1-week semi-urgent surgeries |
λ_{2} | Arrival rate of 2-week semi-urgent surgeries |
p_{1k} | ℙ(1-week semi-urgent surgery is of length k slots), k = 1,2,..,K |
p_{2k} | ℙ(2-week semi-urgent surgery is of length k slots), k = 1,2,..,K |
N_{e,n} | Number of unused reserved semi-urgent slots during week n |
N_{c,n} | Number of canceled elective slots during week n |
N_{o,n} | Number of slots performed in overtime during week n |
C_{o} | Cost of performing one slot in overtime |
C_{t,n} | Total costs incurred in week n |
α | Discount factor |
δ^{*} | Optimal policy |
δ_{M} | Monotone policy |
3.1 Assumptions
In this model we employ a more detailed view of the process, and consider the inflow of the two types of semi-urgent surgeries separately: the first type of semi-urgent surgeries need to be performed within one week, the second type of semi-urgent surgeries need to be performed within two weeks. Given the system status at the beginning of week n, we decide how many 1- and 2-week semi-urgent slots should be performed this week. Since 1-week semi-urgent surgeries have to be performed this week, all incoming surgeries of this type are scheduled for this week. First the reserved slots (1,2,..,s^{*}) are used, and if additional 1-week semi-urgent demand remains, elective slots are canceled.
All 1-week semi-urgent slots are planned this week.
Two week semi-urgent slots not planned this week become 1-week semi-urgent slots next week.
Elective slots canceled this week become 2-week semi-urgent slots next week.
3.2 The Markov decision model
We use a Markov decision model with infinite planning horizon to support the department in deciding how many 2-week slots should be planned in a certain week (action a_{n}). The system state at the start of week n, (n = 0,1,..., ∞ ), is given by w_{n}= (w1_{n}, w2_{n}), where w1_{n} and w2_{n} are the number of 1- and 2-week semi-urgent slots waiting at that moment. The action chosen depends on the number of 2-week slots waiting and on the part of capacity that is already allocated to 1-week slots. Summarizing, the range for action a_{n} is determined by \((0,1,..,\mbox{min}(w2_n,(m-w1_n)^+))\).
3.2.1 Transition probabilities
Let the random variables R1_{n} and R2_{n} denote the number of 1- and 2-week semi-urgent slot arrivals during week n, where R1_{n} + R2_{n} = R_{n}. Similarly to the queuing model presented in Section 2, R1 and R2 follow a compound Poisson distribution, with arrival rates λ_{1} and λ_{2}, and p_{1k} and p_{2k} the probability that a 1- and 2-week surgery is of length k slots.
3.2.2 Performance measures
3.2.3 Cost structure
3.3 Determination of optimal policy
Note that it is never optimal to perform 2-week slots in overtime, since even if they are postponed and then cannot be treated in regular time, they can be treated in overtime next week as well.
4 Capacity planning and scheduling at a Department of Neurosurgery
In this section we illustrate our modeling and optimization approach by considering a department of neurosurgery situated in an academic hospital in the Netherlands. Department staff feared that dedicating scarce OR time to the uncertain stream of semi-urgent patients would lead to an excessive amount of unused OR capacity, and therefore decided to plan almost only elective patients in the available OR time. As a consequence, in daily operation, a large portion of elective surgeries was canceled in order to accommodate semi-urgent surgeries. Furthermore, many ad hoc decisions were needed to ensure that all patients would receive the care they needed. Supported by our models, we show possibilities for improvement.
All surgeries performed by the department can be characterized by the estimated OR time as follows: a) one third of an OR day, b) two thirds of an OR day, c) one OR day, and d) more than one OR day. With this in mind the OR day is divided into three slots of equal length (K = 3). Type 1 surgeries have an estimated duration of one slot, type 2 of two slots, and type 3 surgeries an estimated duration of three or more slots. Therefore, it is either possible to perform in one OR day i) three type 1 surgeries, ii) one type 1 and one type 2 surgery, or iii) one type 3 surgery. The department is assigned eight OR days each week. With each day consisting of three slots, the department has 24 slots per week at its disposal (i.e. m = 24).
4.1 Data
Parameter values for queuing model (Section 2)
Parameter | Value |
---|---|
λ | 11/2 |
p_{1} | 29/55 |
p_{2} | 11/55 |
p_{3} | 15/55 |
Parameter values for Markov decision model (Section 3)
Parameter | Value |
---|---|
λ_{1} | 31/10 |
λ_{2} | 12/5 |
p_{11} | 20/31 |
p_{12} | 5/31 |
p_{13} | 6/31 |
p_{21} | 9/24 |
p_{22} | 6/24 |
p_{23} | 9/24 |
Cost combinations
Name | C_{e} | C_{c} | C_{o} |
---|---|---|---|
CC_{1} | 1 | 1 | 100 |
CC_{2} | 10 | 1 | 100 |
CC_{3} | 1 | 10 | 100 |
4.2 Determining the required number of semi-urgent slots
Queuing model outcomes
s | \(\mathbb{E}[N_{e}]\) | \(\mathbb{E}[N_{c}]\) | \(\mathbb{E}[C_t(CC_1)]\) | \(\mathbb{E}[C_t(CC_2)]\) | \(\mathbb{E}[C_t(CC_3)]\) |
---|---|---|---|---|---|
10 | 0.40 | 23.81 | 24.21 | 27.81 | 238.54 |
11 | 1.40 | 5.42 | 6.82 | 19.42 | 55.64 |
12 | 2.40 | 2.50 | 4.90 | 26.50 | 27.36 |
13 | 3.40 | 1.37 | 4.77 | 35.37 | 17.14 |
14 | 4.40 | 0.82 | 5.22 | 44.82 | 12.58 |
15 | 5.40 | 0.51 | 5.91 | 54.51 | 10.47 |
16 | 6.40 | 0.32 | 6.72 | 64.32 | 9.61 |
17 | 7.40 | 0.21 | 7.61 | 74.21 | 9.45 |
18 | 8.40 | 0.13 | 8.53 | 84.13 | 9.72 |
19 | 9.40 | 0.08 | 9.48 | 94.08 | 10.25 |
20 | 10.40 | 0.05 | 10.45 | 104.05 | 10.94 |
21 | 11.40 | 0.03 | 11.43 | 114.03 | 11.74 |
22 | 12.40 | 0.02 | 12.42 | 124.02 | 12.62 |
23 | 13.40 | 0.01 | 13.41 | 134.01 | 13.54 |
24 | 14.40 | 0.01 | 14.41 | 144.01 | 14.48 |
4.3 Allocation of 2-week semi-urgent slots
We now use the Markov decision model to schedule the 1- and 2-week semi-urgent slots. Our goal is to find an optimal policy that prescribes the number of 2-week semi-urgent slots to plan, given any possible system state.
4.3.1 Monotone policy
It is possible that in the optimal policy action a is not monotone increasing in w2_{n}. Although this form of the optimal policy is not uncommon in literature [23], it may be hard for medical professionals to implement. Therefore we proceed as follows. We determine an optimal policy δ^{*}, as described in Section 3.3. We then check whether a is monotone increasing in w2_{n}. If this is the case, we maintain this optimal policy. Otherwise, we create a monotone policy, δ_{M}, based on the optimal policy, where the number of 2-week slots to plan (the chosen action) is not allowed to decrease. Such a monotone policy is not necessarily optimal, even in the class of monotone policies.
4.3.2 Obtained policies
While δ^{*} for CC_{1} and CC_{3} is straightforward—plan 2-week slots up to s^{*} and postpone the remaining 2-week slots until next week, the policy obtained for CC_{2} is quite different. In several states it occurs that even when the number of 1-week slots exceeds s^{*}, elective slots are canceled in order to accommodate 2-week slots. This action is chosen to avoid overtime, a result of s^{*} being close to s_{min}. Similar to the queuing model outcomes, this shows that maintaining a cost structure similar to CC_{2}, which results in choosing an s^{*} which is close to E[R], leads to the cancellation of elective slots.
5 Discussion and conclusion
In this paper we have developed a methodology to handle the semi-urgent patient flow at a surgical department. On a strategic level, we have determined the OR capacity needed to accommodate all semi-urgent patients on the long run, and we have described a queuing model that allows for a trade-off between the number of elective patients canceled and the amount of unused OR time. Given the amount of slots dedicated to semi-urgent patients, the distribution of the number of elective slots canceled, and the distribution of the number of unused semi-urgent slots can be derived with the queuing model, as is shown in Section 4. An insight that follows from these results is that focusing on only the average behavior of a system can result in undesired system outcomes, in this case the cancellation of many elective patients. Since semi-urgent patient arrivals and elective cancellations are dependent, even over consecutive periods, a natural modeling approach lies in the area of queuing theory.
On a tactical level, we have outlined a Markov decision model that supports the allocation of 1- and 2-week semi-urgent surgeries. This model provides a guideline for the weekly scheduling of semi-urgent patients. The policies obtained with the model can be transferred to a spreadsheet program and with little effort developed into a tool that is easy to use. The added value of the Markov decision model is that it simplifies the scheduling task substantially. Note that all models can be used for arbitrary parameter values.
In the methodology presented, both models involve the planning and scheduling of individual slots. It is not taken into account that when a surgery takes more than one slot, all slots must be scheduled adjacently in the same OR on the same day. To quantify this effect, we calculated the expected number of semi-urgent slots treated for the example in the case study where s^{*} = 13. When considering all possible states, consisting of the number of one-, two- and three-slot semi-urgent surgeries waiting, this expectation equals 8.86 when taking into account the adjacency requirement (i.e. in the situation where we have four full OR days of three slots and a single slot on another OR day). Note that in these calculations we assumed that a rational planner would aim to maximize the number of semi-urgent slots treated in the available time. Given that we consider an instance of the problem where s^{*} is relatively small, so there is little freedom to fill the OR days, the deviation of 7.7% from the value of 9.60 slots (calculated with the queuing model) will be smaller in most other (larger) instances of the problem. However, the adjency requirement results in a slightly higher demand for semi-urgent slots.
A topic for further research would be to extend the presented methodology with an operational model that schedules individual surgeries. We consider the total OR time allocated to a surgical department by OR management as given. Of course, it is possible to establish the optimal amount of allocated OR time, and doing so first could result in a better performance. One of our other aims is to carry out an extensive data analysis to support an implementation of our methodology at the neurosurgery department discussed in Section 4.
Notes
References
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