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Research Papers

# Tactical Operating Room Planning Based on System Transient Performance ControlOPEN ACCESS

[+] Author and Article Information
Zhigang Zeng

Department of Mechanical and
Manufacturing Engineering,
University of Calgary,
Calgary, AB T2N 1N4, Canada
e-mail: zenz@ucalgary.ca

Robert W. Brennan

Department of Mechanical and
Manufacturing Engineering,
University of Calgary,
Calgary, AB T2N 1N4, Canada
e-mail: rbrennan@ucalgary.ca

Theodor Freiheit

Department of Mechanical and
Manufacturing Engineering,
University of Calgary,
Calgary, AB T2N 1N4, Canada
e-mail: tfreihei@ucalgary.ca

Manuscript received February 24, 2018; final manuscript received April 15, 2018; published online May 9, 2018. Assoc. Editor: Shijia Zhao.

ASME J of Medical Diagnostics 1(3), 031004 (May 09, 2018) (14 pages) Paper No: JESMDT-18-1011; doi: 10.1115/1.4040055 History: Received February 24, 2018; Revised April 15, 2018

## Abstract

Efficient management of operating room (OR) schedules is important as the OR is the largest cost and revenue center in a hospital and can substantially impact its staffing and finances. A major problem associated with developing OR schedules for elective surgeries is the schedule disruption from uncertainty inherent in the duration of surgical services. Another problem is the cascaded impact on overall system performance of facilities and resources upstream and downstream to the OR. Using a manufacturing system analytical approach, the peri-operative process is modeled as a transfer line with three machines and two buffers by a discrete time Markov chain. Uncertain surgical and recovery duration is quantified probabilistically and incorporated in the Markov chain model with multistate geometrical machines. Model predictive control (MPC) to pace patient release into the ORs is then applied to control system transient performance. With this model and empirical studies of surgery and recovery duration, guidance can be given to OR managers on how to dynamically schedule and reschedule patients throughout an OR's day that minimizes cost for a given workload. The proposed predictive control model can also control other transient performance metrics such as OR and recovery room (RR) utilization, patient flow, and cost.

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## Introduction

Surgical service is most amenable to cost control when a systematic process of planning is used [1]. As expense centers, operating rooms (ORs) must be scheduled and run efficiently because they impact on the financial health of the institution as a whole [2]. Efficiency improvement has been the objective for many years in the published research. However, most of the results so far are still far from being practical in their application [3].

Most literature in OR scheduling has focused on specific, isolated problems in the OR and propose methods to address those problems. For instance, some works [47] focus on the capacity of hospital based on the operating room. Other works [812] focus on trying to reduce the waiting time of patients. Still other works are related to trying to address a specific or partial problem of the OR system (e.g., the needs of OR manager versus the needs of OR [13]; OR planning versus intensive care unit admissions [14], etc.). In fact, the OR must be examined as a system that has shared upstream and downstream resources [3]. For example, if the postanalgesic care unit (i.e., the recovery room (RR)) is full, for safety, a patient is not allowed to be transferred out of the surgical suite (is blocked), preventing its preparation and use for a new operation. Therefore, the study of the peri-operative process as a system is needed, a system that integrates in its model the interaction between upstream and downstream resources and can be used to compensate for their systemic effect on operational management goals.

The following methodology is proposed to address the problem of short-term OR scheduling of elective surgeries through a multidisciplinary synthesis combining manufacturing systems engineering and modern control theory. Short-term decision-making questions such as “How should patients be dynamically scheduled throughout an operating room's day to minimize operating room cost based on a fixed workload?” and “How should the OR plan be rescheduled when an emergency surgery arises and disrupts the current plan, yet will still provide a minimum cost at a given level of workload?” could be answered. Consequently, a general model platform is built by considering the OR system as a manufacturing transfer line enables the probabilistic quantification of the uncertainty in the OR system. And then, optimal results of the model could practically instruct the behavior of OR manager to reach a desired target.

The model of the OR system proposed in this research is based on a serial production line, where dynamic scheduling of the OR is determined based on analysis and control of system's transient properties. At present, transient analysis of serial production lines is in the forefront of research in the field of manufacturing systems [15]. Recent literature includes Meerkov and Zhang [16] and Li and Meerkov [17], who analyzed the transients of a manufacturing system in which machines are modeled by Bernoulli random variables using mathematical analysis for a simple two machine system and discrete event simulation for larger systems. Similar works were completed by Meerkov et al. [18] for a two-machine system with the geometric reliability model using a Markov chain and by Chen et al. [19] for a serial line of multiple geometric machines, where recursive aggregation is used to approximate transient behavior with high accuracy. However, most of above literature is still in the stage of theoretical research. In this research, a system of three-machines with a multistate geometric, unreliable Markov chain model is developed for an application to operating room systems.

The outline of this paper is as follows: Sec. 2 presents the research methodology. A Markov chain model based on OR system is developed in Sec. 3. Then a predictive control model based on the Markov model is developed in Sec. 4. Section 5 is the model validation and numerical examples. The conclusion and future work are presented in Sec. 6.

## Methodology

An operating room system could be considered as a three-stage queuing system as shown in Fig. 1. A workday will be separated into serial time slots. In each time slot, the OR manager will make a decision whether transfer a patient from waiting pool to surgery room. According to the queuing theory, utilization = λ/μ; λ is average patient arrival rate, μ is average service rate of OR (or capacity of OR), and w = 1/(μλ) [20] (Little's law, usually appears in $L=λw,L$ is number of patients in OR); w is average delay time of patients. All these metrics could be calculated by transient performances of the transfer line system. Therefore, the decision-making question becomes that for given μ, to find an optimal sequence of $λ′$ (arrival rate at different time slot) to meet a fixed $utilization$ at minimal w (i.e., time related cost). This research aims at providing the “optimal” admission of prescheduled patients from the waiting room to the surgery room.

###### Transient Performance of Operating Room System.

This methodology focuses on dynamic operating room scheduling, where surgery duration is uncertain. Therefore, it adapts a mathematical model that is useful for the stochastic analysis of the operating room system. A peri-operative process can be considered to behave as a Markov chain, a stochastic process in which its future behavior depends only on its present state. Therefore, surgery (recovery) duration was modeled as a multistate Markov chain similar to a simpler serial production line [20], where events like surgery completion or patient release can be associated with its states. The use of multiple states facilitates the approximation of different statistical distributions for the duration of different surgery types or hospital processes. An extended operating room system, including the OR manager (the controller), the operating rooms, and the recovery rooms, was modeled as a three-machine and two-buffer transfer line using a Markov chain, as shown in Fig. 2. The Markov transition matrix is decomposed into a matrix of the two control variables ($u1$, the probability of a patient entering a wait state, and $u2$, the probability of releasing a patient to the operating room, and the detailed description of parameters in the model, see Sec. 3, part 1), plus a matrix of constants. From the decomposed transition matrix, state-space equations for the expected number of visits to each state are derived from recursion theory.

This formulation allows for the transient performance of the system to be observed and calculated by the control variables over discrete time steps. This enables the control of transient performance (daily metrics) of the OR system to a desired level. The optimal control variable ($u1,u2)$ values derived at each time-step correspond to a dynamic patient arrival rate that is used to provide guidance to the OR manager on which time-step patients should be released to the OR during the workday.

###### Control of Transient Performance.

Model predictive control (MPC) is used to attain the desired system performance through the control variables. Control parameter values are determined at each time-step that will provide, for example, an overall minimum expected daily operational cost subject to the constraint of meeting a given utilization. Based on the acquired control parameter values, rules are developed to advise the OR manager in their decision for when to release a patient into the OR. If the original OR schedule is disrupted, e.g., by the need to include an emergency surgery, a new schedule could be calculated for the balance of the day that will be optimized for the performance objectives.

Model predictive control [21,22] is an advanced control concept that originated in the control of industrial processes. It relies on a state-space model of the system and generates control action based on the historic information of the system and a cost function that is minimized over a prediction horizon. Generalized predictive control [21] popularized the use of transfer functions or difference equations for the model in the predictive control, and is a widely used approach. The original formulation of predictive control, such as dynamic matrix control, used step or pulse response models instead of state-space models. The idea behind step response models is that a step input can be applied to each input of the plant, and the open-loop response of each output variable can be recorded until all the output variables have settled to constant values. This allows one to deduce the response to inputs. However, this approach has some drawbacks compared to model predictive control [21]. First, it is frequently impractical to apply step inputs as they can be too disruptive to normal operations. Second, step response models are only adequate if all the controlled variables are measured outputs. However, an OR system requires a model that can predict the status of unmeasured outputs.

## Markov Chain Model of the Operating Room System

A three-stage (three-machine) Markov chain model of the OR system was developed as shown in Fig. 2. The model has finite buffers that here denote the number of surgery rooms (or recovery rooms), where multiple surgeries are simultaneously performed in these rooms. Accordingly, the multistate Markov chain model represents an aggregated duration for all surgery rooms rather than a single, specific patient's surgery in a specific OR suite. Therefore, a buffer content transition event represents a nonpatient specific completion of a surgery in one OR suite and their transfer into the RR buffer. The model for the recovery room is similar, except a recovery represents a transfer out of the system. Between every two machines, only one patient is allowed to be transferred at a time unit.

This model is a modification of a simple manufacturing system model using a geometric distribution, where a production machine has two states: an operational (up or working) state, and under repair (down or failed) state. In a discrete-time model, a machine operation is completed in a fixed time unit, working at a production rate of one part per time unit. While the machine is operational, it may fail with a finite probability during a discrete time unit. When the machine is under repair, it will likewise be repaired with a finite probability during any time unit. This simplified model can be extended to a multistate Markov chain, where a machine in a repair state can have several states representing the progression of discrete time units, allowing for the approximation of statistical distributions for production duration beyond the geometric distribution. This modified model was adapted to represent the state transitions of surgery duration. The state transition probabilities in this multistate Markov chain model are determined from historic data on surgery and postsurgery recovery duration.

The overall model reflects the inherent uncertainty of surgical service delivery. The discrete event Markov model represents events that may cause buffer content change, e.g., release of patients into the system and/or the completion of a surgery or a recovery. As a discrete time model, there is a fixed time interval between state transitions (steps).

###### Model Description and Assumptions.

Proceeding from left to right in Fig. 2, machine 1 is the OR manager (a controller), who in this model decides whether to release a patient to an OR room or to keep them waiting. In machine 1, state 1 represents the event of releasing a patient into the OR, while state 2 represents the event of delaying (waiting) to release the patient during that time-step. The capacity of buffer 1 denotes the total numbers of OR suites in use, while the content of buffer 1 indicates the number of surgeries simultaneously being performed.

The states of machine 2 represent the progression of time (steps) between the dispatch of a patient into an OR suite and a surgery completion in any one of the suites. In this machine, states d2, d3, …, dj represent an abstraction of the aggregated (variable) duration for the completion of the next, single surgery from all surgeries currently occupying buffer 1. A transition from state 1 to state d2 denotes that none of the surgeries currently occupying the OR suites (the buffer) have been completed in a given step and therefore the buffer content cannot be decreased in that time-step. Similarly, a transition d2d3 denotes no surgery is completed for the following time-step, etc. In other words, when a state transits from dj to its sequential state dj+1, the patients currently in surgery continue without a surgical completion. Returning to state 1 from state dj represents the event that a single surgery currently in the OR suites has been completed. That is, if machine 2 enters state 1, one patient is thereby ready to transit to the RR, which triggers a buffer state change if the system can accommodate that buffer change. The probability that a surgery is completed during a given time-step is quantified by $pi1$.

Machine 3 represents the state transitions for the progression of time between the dispatch of patients into the recovery room and the transfer out of the system of a single postsurgery patient currently in buffer 2, and has a similar state transition structure as machine 2. The capacity of buffer 2 denotes the total number of recovery beds available.

The system state is represented as $s=(n1,n2,α1,α2,α3)$ following the notation of Gershwin for manufacturing system engineering [20], where $N1andN2$ denote the total number of operating rooms and recovery beds and $0≤n1≤N1$, and $0≤n2≤N2$, respectively. Parameters $α1,α2,α3$ denote the state of the controller, the state for the duration between surgery completions, and the state for the duration between recovery completions, respectively. The total number of system states is then $m=(N1+1)×(N2+1)×2×s×t$.

Even if a machine is operational, it cannot process patients if none are available to it or if there is no room in which to put processed patient. In the former condition, the machine is said to be starved; in the latter, it is blocked. It is assumed here that the first machine is never starved (i.e., the patients are prescheduled and are always available to surgery room) and the third one is never blocked (i.e., the recovery room will never be blocked). Blocked or starved machines, because they are not operating, are not vulnerable to waiting or continuing operation (processing). Once machine transitions take place, the new buffer level can be obtained and its value is determined based on the adjacent machines' new states. When a patient has been processed and delivered from the upstream machine to the downstream machine, the patient is added to the buffer; if the downstream machine has completed processing a patient, the patient is removed from the buffer. The new buffer level also depends on the buffer levels immediately upstream and downstream at the end of the previous cycle.

###### The Internal and Boundary Transition Equations.

In this three-machine transfer-line system, the nontransient states are grouped into internal states $2≤n1≤N1−2$; $2≤n2≤N2−2$, lower boundary states $n1≤1$; $n2≤1$, and upper boundary states $n1≥N1−1$, $n2≥N2−1$. According to the state-transition assumptions noted earlier, a buffer state change is dependent on adjacent machine state changes. Figure 3 represents different state change scenarios, where the three columns represent upstream-and-downstream machines. A “High” state denotes that a machine is in state 1, while a “Low” denotes that a machine is in state 2, $di$, or $uj$.

Internal transition equations are those that involve only internal states. According to Gershwin's book [20], they are equations in which the final state and all the initial states are internal. The derivation process of one internal equation satisfied by the probability distribution $p(n1,n2,α1,α2,α3)$ of the system state is indicated as follows: Display Formula

(1)$pn1,n2,1,di,1=1−u1p12d1−p12upn1−1,n2+1,1,1,1+1−u1pi−1,id1−p12upn1−1,n2+1,1,di−1,1+1−u11−pi1d−pi,i+1d1−p12upn1−1,n2+1,1,di,1+∑j=2t1−u1p12dpj1u1−p12upn1−1,n2+1,1,di,uj+∑j=2t1−u1pi−1,idpj1u1−p12upn1−1,n2+1,1,di−1,uj+∑j=2t1−u11−pi1d−pi−1,idpj1u1−p12upn1−1,n2+1,1,di,uj+1−u2p12d1−p12upn1−1,n2+1,2,1,1+1−u2pi−1,id1−p12upn1−1,n2+1,2,di−1,1+1−u2(1−pi1d−pi,i+1d)1−p12upn1−1,n2+1,2,di,1+∑j=2t1−u2p12dpj1u1−p12upn1−1,n2+1,2,di,uj+∑j=2t1−u2pi−1,idpj1u1−p12upn1−1,n2+1,2,di−1,uj+∑j=2t1−u21−pi1d−pi−1,idpj1u1−p12upn1−1,n2+1,2,di,uj$

In the above Eq. (1), a final state in the left side indicates the total probability of entering the state $n1,n2,1,di,1$ from the different initial states (right side); and the right side represents the summation of probabilities that different states change into the state of the left side. In Eq. (1), the term $1−u1p12d1−p12upn1−1,n2+1,1,1,1$, which is the first term in the right side of Eq. (1), is the probability from state $n1−1,n2+1,1,1,1$ entering into state $n1,n2,1,di,1$, while $1−u1p12d1−p12u$ means the product of the transition probability for the three stages (controller, machine 2, and machine 3) in Fig. 2, and these three stages could be understood as three steps of an event. Correspondingly, $1−u1$ is the transition probability in the controller (state 1 to state 1); $p12d$ is the transition probability in the production duration of machine 2 (state 1 to state $d2$); $1−p12u$ is the transition probability in the production duration of machine 3 (state 1 to state 1). The probability of transition from state $(n1−1,n2+1,1,1,1)$ to state $(n1,n2,1,di,1)$ is $1−u1p12d1−p12u$ in the case that machine 1 stays up, machine 2 fails down from up, and machine 3 stays up. The derivation of the other terms of the equation is similar. In Eq. (1), the terms of right hand include all possible states, which could enter in the state of left hand.

According to the aforementioned assumptions, buffer content changes depend on the adjacent machine state variation. The corresponding buffer changes illustrated in Fig. 3 occur in both the upstream buffer $n1$ (occupancy of the OR suites) and the downstream buffer $n2$ (occupancy of the RR). In Fig. 3, the three columns represent upstream-and-downstream machines, “High” denotes Machine state 1. “Low” denotes the Machine state 2, $di$, or $uj$. “High” to “High,” “High” to “Low,” “Low” to “Low,” “Low” to “High,” represent the machine state transition. Each row of Fig. 3 represents one scenario of buffer changes versus the adjacent machine state variation. Take the first row for example, if the upstream machine transits from state “Low” to state “High” and the downstream machine transits from state “Low” to state “Low,” the buffer content between these two machines will increase 1.

There are total of 11 scenarios for internal state transitions and Eq. (1) is just one of them. Similarly, the boundary transition equations involving state transitions when the buffers are either empty or full have been derived.

###### The Transition Matrix and Transient Performance of the Operating Room System.

The transition matrix of this Markov chain model expresses the state-transition relationship between the system states that was expressed in the state transition equations. When the buffer capacity and the state transition probabilities of the OR and RR duration are assigned, the transition matrix is fully defined.

Deviating from a traditional MPC that uses a control vector, to accommodate the Markov transition matrix, the transition matrix T is decomposed into two constant matrices (B) and a diagonal control variable matrix (U), Eq. (2). In this case, the control vector variables one would see in a traditional MPC problem formulation are distributed along the diagonal of matrix U, whereby the decomposition can separate out the control variables yet maintain the matrix form necessary for a Markovian transition matrix. Display Formula

(2)$T=B0+B1U$

The duration of surgery or recovery is approximated by selecting appropriate state-transition probabilities for the transition matrix. These probabilities are calculated by temporarily adding an absorbing state “A,” which is added to the system to facilitate the determination of the probability a state is reached after a given number of state transitions. An absorbing state is a state that, once entered, cannot be exited. All state transitions entering state 1 are thus redirected into the absorbing state “A,” thereby terminating state transitions when a surgery is completed. The addition of the absorbing state and the redirection of the state transitions from the absorbing state into state 1 is illustrated in Fig. 4.

The probability for the cumulative number of time steps until the task duration is completed is found by selecting state transition probabilities that have the same probability of occurrence as the cumulative distribution function (CDF) probability at each time-step, for example, of the recovery time duration. The state-transition probabilities are found by minimizing the error between the probability of absorption at a given time-step and the CDF probability at that time-step. Since the system has more degrees-of-freedom than equations, the solution is not unique.

The error, $ϵi$, in the relationship between transition and the CDF probabilities is shown in the following equation: Display Formula

(3)$ϵi=Φi−Taiba$

where $Φi$ is the cumulative probability for task completion at step $i$ taken from the CDF, $Ta$ is the modified transition matrix of state transition probabilities p, including the new absorbing state, and $ba$ is a vector that extracts the transient probability of being in the absorbing state. By making a suitable initial guess for p, the error can be minimized to find a set of values for the original transition matrix

$minpϵ$
Display Formula
(4)$Subjectto:0≤p≤1$

For the purposes of this example, we can assume the buffer capacity (number of rooms) for both the OR and RR as 6 (the average number of surgery room in an American hospital is six [23]). In this paper, the number of states for machine 2 and machine 3 are assumed to be 7 and 4, respectively, which are both defined to give 1 h time steps when approximating the cumulative distribution functions for surgery or recovery completion time. The CDFs come from data provided by a healthcare organization and are given in Tables 1 and 2.

Correspondingly, Tables 3 and 4 are transition probabilities of the OR and RR as calculated from Eqs. (3) and (4).

Based on the parameters given in Tables 3 and 4, Figs. 58 show the transient performance of various system measures of the OR as the control variables ($u1$,$u2$) change. The initial system state is $(0,0,2,4,4)$, where all three machines are initially down and the buffers are empty. Blockage is the probability that the OR buffer will be full at a given time-step, while starvation is the probability that the OR buffer will be empty at a given time-step. Expected utilization is the expected occupancy of the OR buffer at a given time-step divided by the OR buffer capacity. Since the OR buffer is initially empty, the blockage probability of the OR rises from 0. The starvation probability of the OR is reducing when $u1$ is smaller and $u2$ is larger, which indicates an increasing possibility that a patient is released to the OR. The transient performance of the recovery room (machine 3) is shown in Figs. 911, but since it is the last “machine” in the line, it cannot be blocked.

It is worthwhile noting that the utilization of the OR and RR are mostly diverged during the transient process, but as the number of time steps $n$ increases, they converge to steady-state values. Additional experiments indicate that the tendency in variation in the blockage or starvation probability is similar with increased buffer size. These plots ultimately indicate that there is a relationship between the control variables $u1$,$u2$ and the system transient performance. In Sec. 4, a predictive control model will be developed to control the system transient performance to a given set point.

## Predictive Control Model of the Operating Room System

Model predictive control is an advanced control concept, which originated in the control of industrial processes. It relies on a state-space model of the system and generates controls based on the history information of the system and a cost function that is minimized over a prediction horizon.

###### Recursion Model for the Expected Number of State Visits.

The state-space model representation used in model predictive control typically assumes the following vector-matrix form: Display Formula

(5a)$xk+1=Axk+Bu(k)$
Display Formula
(5b)$yk=Cxk+Du(k)$

A recursive relationship between the Markov model states must therefore be developed. To this end, the cumulative expected number of visits to the various states of the system's Markov chain after k time steps is denoted by Y(k). This can be expressed as the summation of powers up to k of the transition matrix Display Formula

(6)$Y1=TY2=T+T2⋮Yk=T+T2+⋯+Tk$

A recursive equation for the expected probability of being in a system state after k time steps can then be derived from Eq. (6) as follows:

$X1=T$
Display Formula
(7a)$X2=12X(1)+T2⋮$
Display Formula
(7b)$Xk=k−1kX(k−1)+1kTk$

In Eq. (7), the square matrix X(k) denotes the system of state spaces at time-step k. Each column vector in matrix X(k) represents the expected state probability vector when the system originates from each respective state. This recursive relationship can be adapted to system state x(k) in the right-hand side of Eq. (5). Thus, from Eq. (5a) with Eq. (7b) and control variables from the Markov chain's transition matrix decomposition of Eq. (2), all future steps can be predicted from an estimate of the future Markov chain transition matrices and the current state (shown in Eq. (8) only for the first step for reasons of space). Display Formula

(8)$X̂(k|k−1)=k−1kX(k−1)+1kT̂(k|k−1)∏j=k−10T(j)$

The future Markov chain transition matrices in Eq. (8) can be written a predictive sequence of steps up to Hp steps into the future (the control horizon) as follows: Display Formula

(9)$T̂(k|k−1)=B0+B1Û(k|k−1)T̂k+1k−1=B0+B1Ûk+1k−1⋮T̂(k+Hp−1|k−1)=B0+B1Û(k+Hp−1|k−1)$

where the control variable matrix consists of an initial value and incremental changes for the control variables over the control horizon Display Formula

(10)$Ûk+j=Uk−1+∑i=kk+jΔÛ(i|k−1)$

System output is extracted from the cumulative expected number of visits to the various states of the system's Markov chain by a diagonal matrix, $Czz$, of coefficient matrices, $Cz$, whose elements are weighting constants that calculate, for example, the expected buffer content or the expected operational cost from the estimated system state probabilities $X̂$. Thus, Eq. (5b) becomes Eq. (11), where the output is now a matrix rather than a vector and the control variables are embedded within $X̂$. Display Formula

(11)$Ẑ(k)=CzzX̂(k|k−1)X̂(k+1|k−1)⋮X̂(k+Hp−1|k−1)$

An appropriate vector, selected based on the system's starting state, can be extracted from the output matrix. This vector's elements represent the expected performance of the system at each discrete time-step.

The following is a simple example to demonstrate how $Cz$ factors out the desired states. Assume a system of two machines that has a buffer of capacity one between them, where the system states are denoted $(n,α1,α2)$, where the buffer capacity is $n=0,1$ and the machine states $αi=0,1$. The total number of states in this system is eight, and the states in this system are

$0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,1,0,1,1,1,0,1,1,1$

A general transition matrix, T, for this system has a state transition probability from state i to state j of $pij$. For a $Cz$ to factor out the states where the first machine cannot transfer a part into the buffer, i.e., the blocked state of first machine, the states that would need to be selected are where the first machine is operational and the buffer is full, i.e., states $1,1,∀α2$, i.e., states $1,1,0$ and $1,1,1$. The corresponding $Cz$ is $(0,0,0,0,0,0,0,1,1)$, where the probability of blockage is calculated by

$Cz*T=(p71+p81,p72+p82,p73+p83,p74+p84,p75+p85,p76+p86,p77+p87,p78+p88)$

Each element in this vector corresponds to the probability of blockage when starting from a different initial state i.

Equations (8) through (10) are extensively manipulated to isolate the incremental change of the control values over the control horizon, and thus a sequence of control variables values can be found by minimizing a cost function [21,24]. The general form of the output in terms of the incremental change in the control variables can be further simplified as [21] Display Formula

(12a)$Ẑk=ΨXk−1+Υ+∑i=1HpΘiΔÛ(k)Ai−k$
Display Formula
(12b)$Ψ=CzzIk−1kk−1k+1k−1k+2…k−1k+Hp−1$
Display Formula
(12c)$Υ=CzzM0∏j=k−10T(j)$
Display Formula
(12d)$Θi=CzzMi$

where A and M are matrices with complex expressions involving T, B0, and B1.

The cost function maps one or more variables representing an event to a “cost” associated with that event. Minimization is used to determine the values for $ΔÛ$, where a quadratic cost function is employed to penalize the deviation of the predicted outputs $Ẑ$ from a reference trajectory R over its prediction horizon [21]. The cost function, which also includes a term to penalize for excessive control changes $ΔÛ$, is Display Formula

(13)$Vk=∑i=1HpΘiΔÛ(k)Ai−k−ε(k)2+ΔÛ(k)2$
$εk=Rk−ΨXk−1−Υ$

###### Expected Operating Room System Operational Cost and Utilization.

The expected operational cost of the OR system requires a coefficient vector $Cz$ tailored to the costs associated with different system states. The cost is based on the occupancy or use time of the surgery suites. The general cost equation is Display Formula

(14)$Cost=∑k=1TmaxCzrX̂kk−1+∑k=Tmax+1HpCzoX̂kk−1$

where $Tmax$ is the final time-step associated with regular work hours, and $Tmax+1⋯Hp$ represents overtime hours. Overtime is only included if $Hp>Tmax$. The coefficients of vector $Cz$ define the system state costs, where the expected regular time cost is $Czri=n1(i)μ1+(N1−n1(i))μ2$ and the expected overtime cost is $Czoi=n1(i)μ3$, where $μ$ are cost weighting factors. An evaluation of the system's Hessian equation will determine if the function provides a minimum.

The predicted average utilization of the OR during regular work time is calculated per the following equation: Display Formula

(15)$Utilization=∑j=1TmaxCzu*X̂kk−1Tmax$

where coefficient vector $Czui=n1(i)/N1$. In summary, based on Eqs. (14) and (15), minimizing the daily cost under the constraint of utilization is Display Formula

(16a)$minΔû1,Δû2∑k=1TmaxCzrX̂kk−1+∑k=TmaxHpCzoX̂kk−1$
subject to Display Formula
(16b)$∑j=1TmaxCzuX̂jk−1Tmax=UtilizationTarget$
Display Formula
(16c)$0≤u10+∑j=1kΔû1j≤1,∀k∈1,…,Hp$
Display Formula
(16d)$0≤u20+∑j=1kΔû2j≤1,∀k∈1,…,Hp$

It should be noted that the predictive control model simultaneously calculates an optimal value for the control parameters $Δû1$ and $Δû2$ over all time steps.

## Model Validation and Numerical Examples

Discrete-event simulation (DES) has been used by others to help operating room system decision-makers and support their efforts in achieving desired objectives. DES is based on an operations research modeling and analysis methodology that permits end users (such as hospital administrators or OR managers) to evaluate the efficiency of existing health care delivery systems, and to ask “what if?” questions [25]. Such information allows OR managers to identify management alternatives. A discrete event simulation process illustrated in Fig. 12 was used to describe the movement of patients through the OR and RR. This DES will be used to validate the estimated expected utilization on the optimal control parameter values, ($u1$,$u2$), acquired for different time steps during a work day. A set of real data obtained from a health center was used for this validation. In this section, some experiments are used to investigate if the outputs from the DES (Fig. 12) model and the proposed control model are close to each other based on the same inputs. Numerical examples are also discussed in this section.

###### Discrete Events Simulation for Validation.

To validate the estimated expected utilization, data on surgery and recovery duration were obtained from a health center as shown in the CDF of Tables 1 and 2. Based on the Eqs. (14) and (15) and the cost weighting factors: regular busy time $μ1=0.5$; regular idle time $μ2=0.1$; over time busy time $μ2=0.8$, Fig. 13 is the instantaneous cost at each time-slot at different utilization levels from the MPC model, where the initial number of patients in the OR at the beginning of the day is four. The total work time is 24 h, where a performance metric of interest is the utilization during regular work time, time-steps 0–8. After time-step 8, no new patients are released to the OR. As is expected and can be seen in the plot, the cost increases as utilization increases. As utilization is increased, the time at which the maximum instantaneous cost occurs later as a result of greater overtime costs from late released patients. However, since the optimization objective is to minimize total cost, the instantaneous cost decreases as utilization increases during time-steps 16–24.

Eight time steps (Hp = 8) of $u1$ and $u2$ are acquired from the MPC model, representing the standard working hours of the operating room, subjected to the constraint that a target OR utilization must be met. Using these release/wait probabilities, the release of patients into the system is simulated as follows (Fig. 12):

Step 1: if initial state of controller is state 1, compare with the first group's $u1$ with a uniform random number U[0,1], if $u1$ > U[0,1], do not release, controller state will be shifted to state 2. If $u1$U[0,1], release a patient and controller state will still stay in state 1, and Buffer1 will increase by 1. If initial state of controller is state 2, if $u2$U[0,1], release a patient and controller state will be shifted to state 1, and buffer1 will increase by 1. If $u1$ < U[0,1], do not release, and stay in state 2.

Step 2: select $ps1$ from Table 1 and make a comparison with a uniform random number U[0,1], if 1 − $ps1$U[0,1], release a patient to the recovery room, and buffer 1 decrease by 1 and buffer 2 increase by 1 (the premise is buffer 1 is not empty, buffer 2 is not full; or don't release). If 1 − $ps1$ > U[0,1], do not release.

Step 3: select $pr1$ from Table 2 and make a comparison with a uniform random number U[0,1], if 1−$pr1$U[0,1], release a patient out of recovery room, and buffer 2 decrease by 1 (the premise is buffer 2 is not empty). If 1 − $pr1$ > U[0,1], do not release.

Figures 1417 are a comparison of MPC model that results at different initial patient number of OR, where the dashed line is the result of predictive model and the dash-dot line denotes the simulation results. The x-axis in Figs. 1417 denotes different control targets for utilization (constraints) and the y-axis denotes the utilization value achieved. The simulation was repeated a sufficient number of times to provide a good estimate of the 95% confidence interval, shown as the red error bars in Figs. 1417. Six DES scenarios for a target expected utilization ranging from 45% to 95% were compared to the MPC model. For the DES, the maximum capacity of the surgery and recovery rooms are both six, while the initial control parameter values are set at $u1=0.5$ and $u2=0.5$.

The system utilization achieved with the MPC model and an initial content of the OR of zero (buffer 1) is (0.45, 0.55, 0.65, 0.75, –, –) as shown in Fig. 14. In this example, a high level of utilization (0.85 and 0.95) cannot be reached. This is because when the OR rooms start the day empty, it is not possible to reach high levels of utilizations, as the release of patients is paced over the first few hours of the day. In Fig. 15, the initial OR buffer content is now one, and the system utilization achieved is the same at (0.45, 0.55, 0.65, 0.75, –, –), and again unable to reach the higher utilization levels.

Increasing the initial OR buffer content again to four, Fig. 16, now results in reaching a utilization level of 0.85 (0.45, 0.55, 0.65, 0.75, 0.85, –), but an initial buffer content of six, Fig. 17, loses the lower utilization level of 0.45 (–, 0.55, 0.65, 0.75, 0.85, –). Hospitals concerned with maintaining higher OR utilization, rather than minimizing operational costs, usually start the day at full capacity, but a controller set to achieve lower levels of utilization is not possible with these initial conditions. It is is also worth noting that the maximum (steady-state) utilization of the system is 0.9125, so the system cannot be controlled to reach a utilization greater than this limit even if the OR buffer starts the day full.

The three steps of the simulation are repeated using values for the control variables determined from the MPC model over the eight steps of the control horizon. Each sample observed the number of patients in the OR buffer over the course of a day, and averaging these observations gives a measure of utilization. From Figs. 1417, as one can see, the solutions are valid (the optimization constraints have been met), the MPC results fall within the confidence interval of the simulation. This indicates that the control variable values determined through the MPC will, on average over a sufficient number of days, provide a level of OR utilization consistent with the target utilization.

###### Decision Support Rules and Example.

Operating room managers need a simple yes/no signal for when to release a patient into the OR system. Therefore, the optimum values of $u1$ and $u2$ derived by the MPC model must be converted into a practical rule so that a clear recommendation can be given. Yes/no guidance on patient release can be provided by comparing $u1$ and $u2$ to predetermined threshold values, $ϕ1$ and $ϕ2$, that provide the same long-term, expected system utilization. The threshold values, $ϕ1$ and $ϕ2$, can be determined in a manner similar to the DES process of Sec. 5.1; however, $u1$ and $u2$ are compared to $ϕ1$ and $ϕ2$ in the step 1 instead of a random value. The best values for $ϕ1$ and $ϕ2$ are obtained by minimizing the squared error between the target utilization obtained, $ΩM$, and the expected utilization determined from a simulation, $ΩS$. Figures 18 and 19 compare the utilization obtained from the simulation that have the optimum values for $ϕ1$ and $ϕ2$ at different initial patients Display Formula

(17)$minϕ1,ϕ2ΩM−ΩS2$
Subject to:
$0≤ϕ1≤1;0≤ϕ2≤1$

Table 5 illustrates how the OR manager is provided guidance on the expected patient release schedule over the course of the day given the optimally calculated values for $u1$ and $u2$ and their respective release threshold criteria to $ϕ1$ and $ϕ2$. In this example, the expected utilization is 70%, and the optimal threshold criteria are $ϕ1=0.82$ and $ϕ2=0.41$. In this example, the system starts with two patients in the OR and the initial controller state is 2 (the wait state). The initial control parameter values are $u1=0.5$ and $u2=0.5$.

Since the initial state of the controller at step 1 is state 2 (the wait state), the OR manager would be given the advice to release a patient, as the computer compares the $u2$ value (0.85) with the $ϕ2$ value (0.41), and since $u2$ > $ϕ2$, the recommendation is to release a patient to the OR. At step 2, the controller is in state 1 (because the last step had released a patient, changing the state), and the computer compares the value of $u1$(0.0) with $ϕ1$(0.82). Since $u1$ < $ϕ1$, it is again recommended that a patient be released to the OR. Patient releases are similarly determined for the rest of the steps until the end of the work day. Figure 20 shows a simulation results based on the above guidance, for this example when a patient is predicted to be released into the system, and the OR occupancy during the regular course of a work day. The green bar denotes a newly released patient, while the blue bar denotes patients who are still being operated on in the OR, and the yellow bar are the patients whose operations continue into overtime.

The MPC model also supplies for reference a probability for transferring a patient into the OR at each step of the day. In the 70% utilization example of Fig. 20, the patient transfer probabilities are shown in Fig. 21. The expected (average) occupancy of OR at each time-step is shown in Fig. 22. This information can be supplied in advance so that the OR manager can brief his/her staff on the expected workload over the course of the day.

The expected time that an emergency surgery must wait before an OR suite will become available at each time-step can also be calculated by the expected occupancy of OR as followed:

Step 1: calculate the expected OR spare suites ($Ns$) at each time-step

$Nsk=(1−Ek)×N1$

$Ek$ is the expected OR occupancy at time $k$, $N1$ the total number of OR suites. If $Nsk≥1$, it denotes the waiting time for an OR suite is available is 0 at time $k$. For a given utilization, the waiting time will reduce by the increasing of $N1$.

Step 2: if $Nsk<1$, it denotes an emergency surgery must wait at time $k$. The general equation for predicting the waiting time for an available OR resource is Display Formula

(18)$wk=∑j=kt−1∏i=kj(1−Nsi)×(j−k+1)$

where $w(k)$ is predicted wait at time $k$, and $t$ is time-step where $Nst≥1;t>k$.

Based on the expected occupancy of OR (Fig. 22), the expected time that an emergency surgery must wait before an OR suite will become available at each time-step is shown in Fig. 23.

This metric can be supplied to the OR manager so that they can assess the risk of fitting an emergency surgery into the current schedule. For example, if an emergency surgery arises during step 4, the expected wait is about 0.392 h before an operating room will become available. And the expected waiting time will increase by the increasing of desired utilization target. The expected wait is calculated under the assumption that, for patient safety, no elective surgery would be halted once a patient has been released to the OR. This allows for the potential of needing fewer rooms dedicated to emergency surgeries, or that the expected emergency wait time could also be a minimization objective or a constraint to the cost function.

###### Rescheduling When Emergency Surgeries Occur.

When an emergency surgery must be included in the elective OR schedule, a patient may be released when the next OR room becomes available when surgery is deemed critical. However, the original elective plan would be disrupted. This model can be used to revise the schedule for the rest of the workday, optimizing to the desired target metrics (e.g., flow versus cost), by changing its initial state and recalculating optimal values for $Δû1$ and $Δû2$. Alternatively, overtime costs can be predicted with the current desired patient load before the emergency surgery to determine if the budget for completing all elective surgeries for that day is acceptable.

For example, an original schedule was determined for a desired utilization of 85% based on a starting OR content of six, the simulation result is shown in Fig. 24. At time-step 3, an emergency surgery is released, as shown by yellow bar in Fig. 24. In order for a utilization of 85% at a minimal operational cost to be achieved for the rest of the workday, the model sets the state of time-step 3 as an initial state, and recalculates $ϕ1,ϕ2$ and $Δû1,Δû2$ for the new schedule that is shown in Fig. 25. As one can see, from Figs. 24 and 25, the new schedule releases one less elective patient for the rest of the workday and maintain the utilization of 85.4% as the original schedule for all regular work time. And both schedules have the same overtime patient hours.

Figures 26 and 27 compare the patient release probabilities and the expected OR occupancy of the original schedule and the revised schedule calculated by the MPC model. As can be seen, the new schedule now has less probability that a patient will be released into the OR, and a correspondingly higher expected OR occupancy earlier in the day. After regular work time, limited by the setting of the model, no new patient will be allowed to release for both schedules.

## Conclusion

This research proposes a three-stage multistate Markov chain model to describe an operating room system. An efficient algorithm and closed-form expressions based on aggregation are used to approximate the transient performance measures of the OR system. A recursion model of the transition matrix is derived based on a Markov model. Considering this recursion model as a state space, a predictive control model of an OR system is developed. A Markov chain model is used to provide for variance in surgery duration, while model predictive control is used to determine optimal control parameters at each time-step over the course of the day so that the OR system can attain a predefined expected utilization at minimal cost. A practical procedure for determining predetermined thresholds to compare these control variable values gives the OR manager clear guidance for patient release decisions. Further, the model can help mitigate the disruptive effect of an emergency surgery on an elective surgery schedule by providing a revised schedule for the rest of a work day that is optimized to the desired performance metrics.

There are different subtleties (e.g., different costs for different surgeries; different costs for different stages; a surgeon might have a number of scheduled surgeries in a day, etc.) in specific surgery systems, which pose unique modeling challenges. However, the proposed general model platform has the potential to address such subtleties through setting parameters and constraints.

The proposed model could also be applied for the planning of nondeterministic processing time production lines, as its approach is universal to the efficient operation and management of other processes. In future work, other problems of the OR such as alternating the release of surgeries from different duration categories (e.g., surgeries that have had their duration distributions categorized into short, medium and long average durations) into the system, examining the effect of different control strategies on different kinds of disturbances, how much operational cost can we save if we include one emergent OR room in a reschedulable operating room system? etc., will be explored.

## References

Wickizer, T. M. , 1991, “Effect of Hospital Utilization Review on Medical Expenditures in Selected Diagnostic Areas—An Exploratory Study,” Am. J. Public Health, 81(4), pp. 482–484. [PubMed]
May, J. H. , Spangler, W. E. , Strum, D. P. , and Vargas, L. G. , 2011, “The Surgical Scheduling Problem: Current Research and Future Opportunities,” Prod. Oper. Manage., 20(3), pp. 392–405.
Cardoen, B. , Demeulemeester, E. , and Belien, J. , 2010, “Operating Room Planning and Scheduling: A Literature Review,” Eur. J. Oper. Res., 201(3), pp. 921–932.
Lovejoy, W. S. , and Li, Y. , 2002, “Hospital Operating Room Capacity Expansion,” Manage. Sci., 48(11), pp. 1369–1387.
Ballard, S. M. , and Kuhl, M. E. , 2006, “The Use of Simulation to Determine Maximum Capacity in the Surgical Suite Operating Room,” Winter Simulation Conference (WSC), Monterey, CA, Dec. 3–6, pp. 433–438.
Modigliani, F. , and Hohn, F. E. , 1955, “Production Planning Over Time and the Nature of the Expectation and Planning Horizon,” Econometrica, 23(1), pp. 46–66.
Belien, J. , Demeulemeester, E. , and Cardoen, B. , 2009, “A Decision Support System for Cyclic Master Surgery Scheduling With Multiple Objectives,” J. Scheduling, 12(2), pp. 147–161.
Gladish, B. P. , Parra, M. A. , Terol, A. B. , and Uria, M. V. R. , 2005, “Management of Surgical Waiting Lists Through a Possibilistic Linear Multiobjective Programming Problem,” Appl. Math. Comput., 167(1), pp. 477–495.
VanBerkel, P. T. , and Blake, J. T. , 2007, “A Comprehensive Simulation for Wait Time Reduction and Capacity Planning Applied in General Surgery,” Health Care Manage. Sci., 10(4), pp. 373–385.
Persson, M. , and Persson, J. A. , 2007, “Optimization Modelling of Hospital Operating Room Planning: Analyzing Strategies and Problem Settings,” Operational Research for Health Policy: Making Better Decisions, 31st Annual Conference of the European Working Group on Operational Research Applied to Health Services, Southampton, UK, p. 137.
Cardoen, B. , and Demeulemeester, E. , 2008, “Capacity of Clinical Pathways—A Strategic Multi-Level Evaluation Tool,” J. Med. Syst., 32(6), pp. 443–452. [PubMed]
Addis, B. , Carello, G. , Grosso, A. , and Tanfani, E. , 2016, “Operating Room Scheduling and Rescheduling: A Rolling Horizon Approach,” Flexible Serv. Manuf. J., 28(1–2), pp. 206–232.
Dexter, F. , Macario, A. , and O'Neill, L. , 2000, “Scheduling Surgical Cases Into Overflow Block Time—Computer Simulation of the Effects of Scheduling Strategies on Operating Room Labor Costs,” Anesth. Analg., 90(4), pp. 980–988. [PubMed]
Blake, J. T. , Dexter, F. , and Donald, J. , 2002, “Operating Room Managers' Use of Integer Programming for Assigning Block Time to Surgical Groups: A Case Study,” Anesth. Analg., 94(1), pp. 143–148. [PubMed]
Ju, F. , Li, J. , and Horst, J. A. , 2017, “Transient Analysis of Serial Production Lines With Perishable Products: Bernoulli Reliability Model,” IEEE Trans. Autom. Control, 62(2), pp. 694–707.
Meerkov, S. M. , and Zhang, L. , 2008, “Transient Behavior of Serial Production Lines With Bernoulli Machines,” IIE Trans., 40(3), pp. 297–312.
Li, J. , and Meerkov, S. M. , 2008, Production Systems Engineering, Springer Science & Business Media, Berlin.
Meerkov, S. M. , Shimkin, N. , and Zhang, L. , 2010, “Transient Behavior of Two-Machine Geometric Production Lines,” IEEE Trans. Autom. Control, 55(2), pp. 453–458.
Chen, G. , Wang, C. , Zhang, L. , Arinez, J. , and Xiao, G. , 2016, “Transient Performance Analysis of Serial Production Lines With Geometric Machines,” IEEE Trans. Autom. Control, 61(4), pp. 877–891.
Gershwin, S. B. , 1994, Manufacturing Systems Engineering, Prentice Hall, Upper Saddle River, NJ.
Maciejowski, J. M. , 2002, Predictive Control: With Constraints, Pearson Education, London.
Mayne, D. Q. , 2014, “Model Predictive Control: Recent Developments and Future Promise,” Automatica, 50(12), pp. 2967–2986.
Hall, M. J. , Schwartzman, A. , Zhang, J. , and Liu, X. , 2017, “Ambulatory Surgery Data From Hospitals and Ambulatory Surgery Centers: United States, 2010,” U.S. National Center for Health Statistics, Atlanta, GA, National Health Statistics Reports No. 102.
Fazlirad, A. , and Freiheit, T. , 2016, “Application of Model Predictive Control to Control Transient Behavior in Stochastic Manufacturing System Models,” ASME J. Manuf. Sci. Eng., 138(8), p. 081007.
Jacobson, S. H. , Hall, S. N. , and Swisher, J. R. , 2006, “Discrete-Event Simulation of Health Care Systems,” Patient Flow: Reducing Delay in Healthcare Delivery, Springer, Berlin, pp. 211–252.
Copyright © 2018 by ASME
View article in PDF format.

## References

Wickizer, T. M. , 1991, “Effect of Hospital Utilization Review on Medical Expenditures in Selected Diagnostic Areas—An Exploratory Study,” Am. J. Public Health, 81(4), pp. 482–484. [PubMed]
May, J. H. , Spangler, W. E. , Strum, D. P. , and Vargas, L. G. , 2011, “The Surgical Scheduling Problem: Current Research and Future Opportunities,” Prod. Oper. Manage., 20(3), pp. 392–405.
Cardoen, B. , Demeulemeester, E. , and Belien, J. , 2010, “Operating Room Planning and Scheduling: A Literature Review,” Eur. J. Oper. Res., 201(3), pp. 921–932.
Lovejoy, W. S. , and Li, Y. , 2002, “Hospital Operating Room Capacity Expansion,” Manage. Sci., 48(11), pp. 1369–1387.
Ballard, S. M. , and Kuhl, M. E. , 2006, “The Use of Simulation to Determine Maximum Capacity in the Surgical Suite Operating Room,” Winter Simulation Conference (WSC), Monterey, CA, Dec. 3–6, pp. 433–438.
Modigliani, F. , and Hohn, F. E. , 1955, “Production Planning Over Time and the Nature of the Expectation and Planning Horizon,” Econometrica, 23(1), pp. 46–66.
Belien, J. , Demeulemeester, E. , and Cardoen, B. , 2009, “A Decision Support System for Cyclic Master Surgery Scheduling With Multiple Objectives,” J. Scheduling, 12(2), pp. 147–161.
Gladish, B. P. , Parra, M. A. , Terol, A. B. , and Uria, M. V. R. , 2005, “Management of Surgical Waiting Lists Through a Possibilistic Linear Multiobjective Programming Problem,” Appl. Math. Comput., 167(1), pp. 477–495.
VanBerkel, P. T. , and Blake, J. T. , 2007, “A Comprehensive Simulation for Wait Time Reduction and Capacity Planning Applied in General Surgery,” Health Care Manage. Sci., 10(4), pp. 373–385.
Persson, M. , and Persson, J. A. , 2007, “Optimization Modelling of Hospital Operating Room Planning: Analyzing Strategies and Problem Settings,” Operational Research for Health Policy: Making Better Decisions, 31st Annual Conference of the European Working Group on Operational Research Applied to Health Services, Southampton, UK, p. 137.
Cardoen, B. , and Demeulemeester, E. , 2008, “Capacity of Clinical Pathways—A Strategic Multi-Level Evaluation Tool,” J. Med. Syst., 32(6), pp. 443–452. [PubMed]
Addis, B. , Carello, G. , Grosso, A. , and Tanfani, E. , 2016, “Operating Room Scheduling and Rescheduling: A Rolling Horizon Approach,” Flexible Serv. Manuf. J., 28(1–2), pp. 206–232.
Dexter, F. , Macario, A. , and O'Neill, L. , 2000, “Scheduling Surgical Cases Into Overflow Block Time—Computer Simulation of the Effects of Scheduling Strategies on Operating Room Labor Costs,” Anesth. Analg., 90(4), pp. 980–988. [PubMed]
Blake, J. T. , Dexter, F. , and Donald, J. , 2002, “Operating Room Managers' Use of Integer Programming for Assigning Block Time to Surgical Groups: A Case Study,” Anesth. Analg., 94(1), pp. 143–148. [PubMed]
Ju, F. , Li, J. , and Horst, J. A. , 2017, “Transient Analysis of Serial Production Lines With Perishable Products: Bernoulli Reliability Model,” IEEE Trans. Autom. Control, 62(2), pp. 694–707.
Meerkov, S. M. , and Zhang, L. , 2008, “Transient Behavior of Serial Production Lines With Bernoulli Machines,” IIE Trans., 40(3), pp. 297–312.
Li, J. , and Meerkov, S. M. , 2008, Production Systems Engineering, Springer Science & Business Media, Berlin.
Meerkov, S. M. , Shimkin, N. , and Zhang, L. , 2010, “Transient Behavior of Two-Machine Geometric Production Lines,” IEEE Trans. Autom. Control, 55(2), pp. 453–458.
Chen, G. , Wang, C. , Zhang, L. , Arinez, J. , and Xiao, G. , 2016, “Transient Performance Analysis of Serial Production Lines With Geometric Machines,” IEEE Trans. Autom. Control, 61(4), pp. 877–891.
Gershwin, S. B. , 1994, Manufacturing Systems Engineering, Prentice Hall, Upper Saddle River, NJ.
Maciejowski, J. M. , 2002, Predictive Control: With Constraints, Pearson Education, London.
Mayne, D. Q. , 2014, “Model Predictive Control: Recent Developments and Future Promise,” Automatica, 50(12), pp. 2967–2986.
Hall, M. J. , Schwartzman, A. , Zhang, J. , and Liu, X. , 2017, “Ambulatory Surgery Data From Hospitals and Ambulatory Surgery Centers: United States, 2010,” U.S. National Center for Health Statistics, Atlanta, GA, National Health Statistics Reports No. 102.
Fazlirad, A. , and Freiheit, T. , 2016, “Application of Model Predictive Control to Control Transient Behavior in Stochastic Manufacturing System Models,” ASME J. Manuf. Sci. Eng., 138(8), p. 081007.
Jacobson, S. H. , Hall, S. N. , and Swisher, J. R. , 2006, “Discrete-Event Simulation of Health Care Systems,” Patient Flow: Reducing Delay in Healthcare Delivery, Springer, Berlin, pp. 211–252.

## Figures

Fig. 1

A three-stage operating room dynamical scheduling system

Fig. 2

Markov chain model of OR system

Fig. 3

Machine state transition and buffer changing

Fig. 4

Calculating transition probabilities using an absorbing state

Fig. 5

Expected blockage of OR versus (u1 and u2)

Fig. 6

Expected starvation of OR versus (u1 and u2)

Fig. 7

Expected occupancy of OR versus (u1 and u2)

Fig. 8

Expected utilization of the OR versus (u1 and u2)

Fig. 9

Expected starvation of RR versus (u1 and u2)

Fig. 10

Expected occupancy of RR versus (u1 and u2)

Fig. 11

Expected utilization of RR versus (u1 and u2)

Fig. 12

Steps of the discrete event simulation of the operating room system

Fig. 13

Instantaneous cost under the different utilization

Fig. 14

Estimated utilization versus simulation (n1=0)

Fig. 15

Estimated utilization versus simulation (n1=1)

Fig. 16

Estimated utilization versus simulation (n1=4)

Fig. 17

Estimated utilization versus simulation (n1=6)

Fig. 18

Estimated utilization versus simulation with ϕ1 and ϕ2 (n1=1)

Fig. 19

Estimated utilization versus simulation with ϕ1 and ϕ2 (n1=3)

Fig. 20

A simulation patient transfer to the OR

Fig. 21

Probability of releasing patients to OR

Fig. 22

Expected occupancy of OR as a percent of the total number of OR suites

Fig. 23

Expected waiting time for an OR suite availability

Fig. 24

Original schedule when emergency surgery is admitted at time-step 3

Fig. 25

Revised schedule for the rest work time

Fig. 26

Comparison of expected patient release probabilities

Fig. 27

Comparison of expected OR occupancy as a percent of the total number of OR suites

## Tables

Table 1 Surgery room cumulative probability
Table 2 Recovery room cumulative probability
Table 3 Transition probabilities of OR
Table 4 Transition probabilities of RR
Table 5 Guidance for patient release (regular work hours)

## Discussions

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