A Decomposition Algorithm for the Two-Stage Chance-Constrained Operating Room Scheduling Problem

The required time for surgical interventions in operating rooms (OR) may vary significantly from the predicted values depending on the type of operations being performed, the surgical team, and the patient. These deviations diminish the efficient utilization of OR resources and result in the disruption of projected surgery start times. This paper proposes a two-stage chance-constrained model to solve the OR scheduling problem under uncertainty. The goal is to minimize the costs associated with OR opening and overtime as well as reduce patient waiting times. The risk of OR overtime is controlled using chance constraints. Numerical experiments show that the proposed model provides a better trade-off between minimizing costs and reducing solution variability when compared to two existing models in the literature. It is also shown that the three models converge as the overtime probability threshold approaches one. Moreover, it is observed that the individual chance constraints result in the opening of fewer rooms, lower waiting times, and shorter solution times when compared to that of joint chance constraints. A decomposition algorithm is applied that solves large test instances of the OR scheduling problem, that of which is known to be NP-hard. Strong valid inequalities are derived in order to accelerate the convergence speed. The proposed approach outperformed both a commercial solver and a basic decomposition algorithm after solving all test instances with up to 89 surgeries and 20 ORs in less than 48 minutes.


I. INTRODUCTION
Health care expenditures are expected to constitute 25% of the US gross domestic product (GDP) in 2025, an increase from 15.9% in 2005 [21]. Surgical expenses contribute to 30% of health care expenditures and are expected to grow from $572 billion in 2005 to $912 billion (2005-valuated dollars) in the year 2025. Surgical procedures are complex tasks requiring a variety of specialized and expensive resources. In 2011, hospitalizations involving surgical procedures constituted 29% of total hospital stays while contributing to 48% of total hospital costs in the US [33]. In light of these reports, surgeries are recognized as the most crucial activities performed in hospitals from a social, medical and economic point of view.
The associate editor coordinating the review of this manuscript and approving it for publication was Shih-Wei Lin .
Several survey articles have recognized the surgery duration uncertainty as a major obstacle to developing practical and cost-effective OR schedules [18]. This paper proposes a chance-constrained programming model that: 1) provides cost-effective OR schedules by considering both deterministic and stochastic costs, 2) maintains a low OR overtime probability and compares individual and joint chance constraints, 3) results in a better cost-variability trade-off compared to two existing models in the literature and 4) solves such problems at a faster rate than the two aforementioned existing models before applying any solution algorithms. Moreover, a computationally efficient solution method and strong valid inequalities are provided to facilitate timely decision-making in the case of disruptions in the schedule.
The OR scheduling literature has been reviewed in several survey articles [2], [18], [28]. The published literature has been classified using several categories, including uncertainty. Variable surgery duration is one of the most commonly studied sources of uncertainty by the Operations Research community. It is shown that mitigating the impact of disruptions in the schedule due to uncertainty can lead to higher capacity utilization and lower costs [19], [20]. Hence, it is crucial to ensure that the provided schedule works reliably in the presence of large variability in surgery durations.
Numerous works have used stochastic programming to model the uncertain surgery durations in the OR scheduling problems [1], [7], [25], [32]. The majority of these models consider optimizing the expectation of costs/revenues during the planning horizon [10]. For problems with moderate variability, using the expected value (EV) can result in desirable outputs. However, the obtained solutions may show poor performance for problems displaying frequent changes in a less predictable manner [24]. A number of articles considered using the Conditional Value-at-Risk (CVaR) [27] to account for undesirable realizations of the uncertain parameters [15], [22], [29]. The CVaR function minimizes the expected tail of costs.
Another array of articles have used the chance-constrained programming (CCP) models [3] to address the uncertainty. This approach mitigates the risk of disadvantageous events (e.g., OR overtime, patient waiting time) exceeding the specified thresholds, rather than merely minimizing their expected value [4], [13], [35]. Shylo et al. [30] applied chance-constraints to control the OR block overtime in the OR surgery planning problem. Zhang et al. [36] studied a chance-constrained OR surgery allocation problem. Deng and Shen [4] developed a two-stage stochastic model for the multi-server appointment scheduling problem with a joint chance constraint on server overtime. They applied the proposed model and solution approach to solve OR scheduling problem test instances. Jebali and Diabat [11] studied the surgery planning problem under uncertain surgery duration, length of stay in the intensive care unit (ICU), and emergency patient arrival. They employed chance constraints to control the violation of ICU capacity. Wang et al. [31] proposed a distributionally robust chance-constrained model for the surgery planning problem with stochastic surgery durations. Noorizadegan and Seifi [23] proposed a CCP model for the surgery planning problem with uncertain surgery durations. Kamran et al. [12] proposed a two-stage stochastic model with chance constraints on OR overtime for the advance scheduling problem. Deng et al. [5] developed a distributionally robust chance-constrained model for the OR scheduling problem. They control the risk of OR overtime and surgery waiting using joint chance constraints. Table 1 summarizes the selected published literature and identifies the research gaps that are addressed in this paper. First, it can be observed that very few articles proposed a CCP model for the OR scheduling problem under uncertainty [4], [5]. Surgery scheduling problems often have a more complex structure resulting from a variety of decisions, such as OR opening, patient-to-OR assignment, surgery sequencing, and projected and actual start times before and after the realization of random surgery durations, respectively. CCP-based models have the potential to effectively handle such large variabilities in daily surgery scheduling problems [18].
Second, a majority of the models have neglected the importance of minimizing the stochastic second-stage costs. Their primary focus has been on providing schedules within the specified risk tolerances while also aiming to minimize deterministic performance measures, such as fixed OR opening costs [5], [23]. Unlike existing approaches, this paper proposes a chance-constrained model that aims to minimize both deterministic and stochastic costs for the OR scheduling problem. The significance of considering both classes of costs is highlighted using numerical experiments.
Third, we provide insightful observations about the performance of three different models (CCP, CVaR and EV) in solving the stochastic OR scheduling problem under various risk thresholds. The proposed model is compared alongside EV and CVaR models using several metrics such as total costs, OR utilization and solution time. Moreover, the performances of both individual and joint chance constraints are compared in terms of OR opening decisions, minimizing the second-stage costs and computational efficiency.
Finally a computationally efficient decomposition algorithm is applied to provide high-quality solutions for the large-scale test instances within reasonable time frames. VOLUME 8, 2020 We proposed an algorithm to derive feasibility cuts using the first-stage solutions that accelerates finding feasible solutions and the convergence speed.

A. PROBLEM DESCRIPTION
Let I be the set of elective surgeries and R be the set of operating rooms. The problem is to schedule surgeries over a daily planning horizon. We assume that operating rooms use the block booking policy [6]. Each OR is allocated to a surgical specialty according to the master surgery schedule. The incidence matrix E = {e ir } for all i ∈ I , r ∈ R allows specific surgery-to-OR assignments. The surgery duration is random, denoted by a random vector δ = (δ 1 , . . . , δ |I | ) T ∈ R |I | + where δ i shows the random duration for surgery i ∈ I . We assume that the random surgery duration has finite and discrete support S for δ. The probability density of each scenario s is denoted by p s , where s∈S p s = 1. Each realization of δ in scenario s is shown by δ s = (δ s 1 , . . . , δ s |I | ) T . Every surgery on the daily booking list must be operated. No interruption is allowed once an operation has started. It is desired to decrease the probability of working OR overtime. This restriction is enforced using chance constraints. Each surgery must be assigned an OR and a projected start time. Our goal is to minimize the sum of fixed OR opening costs and expected costs corresponding to OR overtime and patient waiting times.

B. CHANCE-CONSTRAINED OR SCHEDULING PROBLEM
We propose a two-stage stochastic model for the daily OR scheduling problem with individual chance constraints enforcing overtime restrictions on every OR. Table 2 introduces the notation used in our model.
The first-stage problem involves deterministic decision making (i.e., OR opening and surgery case assignment) prior to the realization of uncertain surgery durations. After the uncertain parameters are revealed, the second-stage problem determines recourse actions (e.g., adjusting start times and adding OR overtime) that incur additional costs to provide meaningful schedules based on first-stage decisions. The goal is to minimize total costs as well as satisfy the chance constraints on OR overtime.
The first-stage problem (M 1 ) can be formulated as follows: where x = (u, y) is the vector of first-stage variables. Set X is the resulting set from the deterministic constraints (3)-(7) formulated as follows: u r ∈ B |R| , y ikr ∈ B |I |×|K |×|R| , ∀i ∈ I , k ∈ K , r ∈ R (7) Objective function (1) minimizes the total cost of opening operating rooms. Constraints (3) and (4) ensure that every surgery will be assigned to one and only one spot in an open 80162 VOLUME 8, 2020 OR during the day. Constraint (5) enforces eligible surgeryto-OR assignments. Constraint (6) determines the order of operating surgical cases in each OR. Constraint (7) enforces binary values for the first-stage decision variables. The chance-constrained second-stage problem (M 2 ) is formulated as follows: tp kr , tr krs , w rs ≥ 0, ∀k ∈ K , r ∈ R, s ∈ S (13) Constraint (8) determines the projected start time for each surgery according to the sequencing decisions. Constraint (9) ensures that each surgery starts after its projected start time. Constraint (10) is similar to (8) in that the actual start times must follow the sequencing decisions. Constraint (11) calculates the amount of waiting time in every OR per scenario. The chance constraints (12) state that the surgeries assigned to an OR must be finished during the regular hours (i.e., no overtime) with high probability. Constraint (13) enforces the non-negativity of the second-stage decision variables. The objective function of the second-stage problem is formulated in the remainder of this section. The set P(s) of the first-stage solutions that are made to satisfy the chance-constrained second-stage problem is derived as follows: Proposition 1: Let α r |S| be an integer for every r. Then, chance constraints (12) are equivalent to: where binary variable z rs = 1 when the time capacity of room r is violated. Proof: See Appendix A-A. For each scenario s ∈ S, an operation may be completed during regular hours (z rs = 0) or may run into overtime (z rs = 1). Therefore, the second-stage cost, g(x, s), will be calculated differently in each case: where o rs is a non-negative variable representing overtime. Therefore, the objective function of the second-stage problem can be formulated as follows: Objective function (20) minimizes the expected costs corresponding to OR overtime and patient waiting times. The deterministic equivalent formulation for the two-stage chance-constrained OR scheduling model (M DEF ) can be modeled as follows: Using a big M value can lead to weak LP relaxations. Assigning a smaller value for M can help tighten the feasible region for the LP relaxation of M DEF . Therefore, instead of setting a single large value for M , constraint-specific formulae used to calculate the big M values are developed for constraints (22) and (23) as shown below: The values in (26) are valid because the total operation time in each OR does not exceed the duration of all surgical cases that can be allocated to the specific operating room.

III. SOLUTION APPROACH
This section describes a decomposition algorithm that solves the proposed model in Section II. The proposed algorithm can solve the model to optimality if the following assumptions are satisfied [16], [17]: 1) The random vector S has discrete and finite support. Specifically, p s = 1 |S| for s ∈ S. We have stated this assumption in the problem description in Section 2.1. 2) Set X and P(s), s ∈ S are non-empty compact sets.
Without loss of generality, we can assume that for every s ∈ S, there exists a feasible first-stage solution that satisfies the chance constraints. Therefore, sets X and P(s) are finite sets of points that qualifies them as compact sets. 3) Set conv (P(s)), s ∈ S have the same recession cone, i.e., there exists C ⊆ R N such that Proof: See Appendix A-B. 4) There does not exist an extreme rayθ of conv(X) with f Tθ < 0, i.e., the two-stage problem has a bounded optimal solution. Proof: See Appendix A-C. Proposition 2: Model M DEF has an optimal solution (x * , z * ) in which s z * rs = α r |S| for all s ∈ S [17].
We begin the decomposition algorithm by defining feasibility (F) and optimality (O) sets as follows: These sets will be approximated using feasibility and optimality cuts in the following master problem (MP): The setsF andÕ are the outer approximations of the feasibility (F) and optimality (O) sets, respectively. In the remainder of this section, we will derive strong valid inequalities to defineF andÕ.

A. FEASIBILITY CUTS
Two sets of subproblems are required to formulate the strong feasibility cuts: single-scenario optimization and single-scenario separation [16]. The optimization subproblem for the OR scheduling problem is formulated as follows: where γ ∈ R N andX ⊇ X , such that P(r, s) ∩X = .
Proposition 3: Problem (30) is feasible and has a finite optimal value if γ ∈ R N .
Proof: See Appendix A-E. The separation subproblem can be formulated as follows: Solving this subproblem to optimality returns a separating hyperplane of the form γ x ≥ β for all x ∈ P(r, s).
Letx be a solution to the master problem (MP). If rs (x) > 0 andπ is the optimal solution, the separating hyperplane − kπ 4 k id i y ikr ≥ − kπ 4 k cap r cuts offx fromF. Therefore, we define the valid feasibility cuts as follows: Theorem 1: The following sets of inequalities are valid for F: The remainder of this section presents another class of feasibility cuts derived from the solutions to the first-stage problem (M 1 ) or master problem (MP) that result in the violation of chance constraints (12).
Theorem 2: The following set of inequalities are valid for F: where TB is the subset of surgeries that lead to the violation of chance constraints when they are assigned to the same OR. Letŷ ikr be the set of surgery-to-OR assignments obtained from solving (M 1 ) or (MP). Algorithm 1 generates inequalities of type (39). Algorithm 1 Feasibility Cut Generation 1: input: Sets and parameters in Table 2,ŷ ikr . 2: initialize: v rs ← 0, Cnt r ← 0, TB ← . 3: Calculate v rs = k,i δ isŷikr − cap r , ∀r ∈ R, s ∈ S. 4: if v rs > 0, ∀r ∈ R, s ∈ S then 5: • Cnt r ← Cnt r + 1. 6: if Cnt r > α r |S|, ∀r ∈ R then 7: • TB ← {i|ŷ ikr = 1}.

B. OPTIMALITY CUTS
In this section, we derive optimality cuts to add toÕ. First, we formulate the dual problems for regular and overtime modes of the second-stage problem. For every r ∈ R and s ∈ S where z rs = 0, the regular mode dual problem is formulated as follows: π 2 (k−1)rs −π 2 krs +π 3 krs −π 4 krs − π 5 rs ≤ 0, ∀k ∈ K (42) π 5 rs ≤ cw r (43) k π 1 krs + π 2 krs + π 3 krs + π 4 krs + π 5 rs = 1 (44) π 1 krs , π 2 krs , π 3 krs , π 4 krs , π 5 rs ≥ 0, ∀k ∈ K Since the only difference between the regular and overtime mode is the introduction of overtime variables (when z rs = 1), the overtime mode dual problem can be derived by replacing π withπ and adding the dual constraints corresponding to overtime variables: The set of dual optimal solutions for the regular and overtime modes are shown by rs and¯ rs , respectively. Then, we formulate optimality subproblems for each mode. For a given τ ∈ R N , r ∈ R and s ∈ S, we formulate the optimality subproblem for the regular mode as follows: Similarly, we formulate the optimality subproblem for the overtime mode as follows:

C. DECOMPOSITION ALGORITHM
A decomposition algorithm is proposed to solve the two-stage chance-constrained OR scheduling problem. This algorithm has a similar structure to the Benders decomposition algorithm [26]. Rather than traditional Benders cuts, we use strong valid inequalities derived in Section III-A and Section III-B. Parameter in Algorithm 2 represents the upper bound on the relative optimality gap, calculated as UB−LB UB . Table 2 Solve master problem (29). 5: if (29) is infeasible then 6: Stop. Original problem is infeasible. Let x,ẑ,ρ be an optimal solution to (29). 9:

Theorem 3: Algorithm 2 converges to an optimal solution in finite iterations.
Proof: See Appendix A-G.

Test problem instances are obtained from Leeftink and
Hans [14]. The instances consist of different surgical specialties such as orthopedic, otorhinolaryngology, and oncology. The surgery durations follow a three-parameter lognormal distribution [9]. We used the Monte Carlo sampling method to generate a finite set of scenarios for random surgery durations. The overhead and variable costs for operating rooms are determined using the cost settings in [7]. The OR opening cost is calculated by multiplying the overhead cost by the OR available time. Each OR operates an 8-hour workday and has one block that is assigned to a surgical specialty. Optimization models are implemented in Python using IBM CPLEX on a workstation with 24 cores, 3 GHz processors, and 384 GB of memory. A time limit of one hour is imposed for all instances. The valid cuts are implemented using the CPLEX lazy constraint callback function. VOLUME 8, 2020

A. COMPARING CCP WITH OTHER STOCHASTIC MODELS
We compare the performance of the proposed chanceconstrained model with the two models proposed in [22]: SDORS-EV and SDORS-CVaR. SDORS-EV is a stochastic programming model that attempts to optimize the expected value of OR overtime and patient waiting costs. SDORS-CVaR is a risk-based model that minimizes the expected tail of overtime and waiting costs by using the CVaR function [22]. For simplicity, the following terminology is used in our experiments: CCP (chance-constrained), CVaR (SDORS-CVaR) and EV (SDORS-EV). The performance of these models are evaluated using several criteria. Table 3 compares the performance of the three models after solving eight OR scheduling instances within the specified time limit. A finite set of 100 scenarios is generated for each surgery, and the parameter α r is set to 0. 10   In Figure 1, the trade-off between minimizing costs and controlling the variability of costs is compared for each model. Several values are used for the confidence level parameter α in order to mimic the behavior of these models under different risk attitudes. A high α resembles an aggressive approach to minimize expected costs while accepting a substantial risk of OR overtime. On the contrary, a low α depicts conservative decision making (i.e., accepting higher costs given that the chance of OR overtime is low). As observed in Figure 1, CVaR places emphasis on minimizing variability while EV focuses on providing the minimum average costs. However, CCP provides a more moderate trade-off between minimizing average costs and reducing variability. Assuming a given tolerance for the OR schedule variability, CCP outperforms CVaR by providing more cost-effective solutions. Similarly, CCP outperforms EV by providing OR schedules with lower variability, assuming a fixed budget.
In Figure  For small α values, CCP and CVaR suggest opening more ORs to reduce the risk of overtime and reduce the expected tail costs, respectively. Therefore, they incur higher average total costs and lower OR utilization than EV. EV displays better OR utilization at the risk of experiencing increased overtime. CPP is the superior method in terms of reducing overtime and patient waiting times. Moreover, CCP performs best in reducing the number of scenarios where overtime occurs. Overall, using CCP results in fewer occurrences of overtime and better OR utilization than CVaR when α is not very restrictive (i.e., α > 0.1 in our numerical experiments). It is also observed that the three models converge in all metrics as α increases.
Proposition 6: CCP, CVaR and EV provide the same optimal solution when α = 1.
Proof: See Appendix A-H.

B. SOLVING LARGE-SCALE TEST INSTANCES
It is observed in Table 3 that the solution times increase exponentially as the problem size grows. Therefore, we apply the valid inequalities and the decomposition algorithm presented in Section III to solve larger test instances in shorter time periods. We used the Monte Carlo sampling method to generate a set of 100 scenarios, and the parameter α r was set to 0.10. Table 4 compares the performance of feasibility cuts (38) and (39) when used separately and combined to solve the test problem instances shown in Table 3. It can be observed that both valid inequalities are effective in reducing the solution time when compared to the cuts generated by the CPLEX solver. It is also observed that using both types of cuts leads to longer solution times for small test instances due to the time spent for generating inequalities. However, as problem size grows, adding both types of feasibility cuts toF leads to significantly faster convergence than applying them separately. Therefore, we use valid inequalities (38) and (39) to generate feasibility cuts in the following numerical experiments. Larger test problem instances are solved and reported in Table 5 to evaluate the performance of the proposed decomposition algorithm. We compare the performance of our algorithm with that of the IBM CPLEX MIP Solver 12.9 [38] and a decomposition algorithm that uses the big-M optimality cuts introduced in [17]. The solution time and the optimality gap are reported for each algorithm. The column Basic Decomposition illustrates the results from the decomposition algorithm using feasibility cuts (38) and big-M optimality cuts. The last column highlights the results of the proposed decomposition algorithm in this paper. As shown in Table 5, we observe that the CPLEX solver is the least desirable option for solving M DEF , as expected. For the largest problem instance, the CPLEX solver does not find any feasible solutions within the time limit. User-defined feasibility and optimality cuts can improve solution speed. It can also be observed that using stronger optimality cuts rather than the big-M cuts can reduce solution time significantly. Neither the CPLEX solver nor the basic decomposition algorithm can solve any of the instances to global optimality within the time limit. Nevertheless, the proposed decomposition algorithm outperforms other methods by solving all test instances to optimality within 48 minutes.

C. IMPORTANCE OF MINIMIZING EXPECTED COSTS
Our numerical experiments show that a significant portion of total costs comes from the expected overtime and waiting time costs. Neglecting these measures in OR scheduling models can result in surpassing the predicted overtime budget by 200%, disheartening staff from longer-than-expected shifts, and causing dissatisfaction to patients [8]. We solved 10 replications of all test instances in Table 3 using two different objective functions: objective (21); and objective (21) minus the expected second-stage costs. Then, we calculated the sum of costs pertaining to OR opening, OR overtime, and patient waiting time for each optimal solution of each case. The percentage of savings obtained from optimizing both  The chance constraints in Section II-B are enforced on each OR independently. However, a decision-maker might be interested in controlling the chance of OR overtime on an aggregate level. In such cases, the chance constraints (12) Table 5. Figure 4 illustrates that M Joint tends to open more ORs to satisfy the tighter limit on OR overtime. The joint chance constraints restrict the occurrence of overtime to α|S| scenarios while the individual chance constraints allow up to min{α r |S||R|, |S|} scenarios with OR overtime. As the probability threshold α loosens, the gap between the optimal number of open ORs obtained by the two models shrinks due to the converging feasible regions. Similar to Proposition 6, it can be shown that the two models achieve equivalent optimal solutions when α = 1. Figure  scenario, respectively. It can be observed that M Joint achieves greater success at controlling OR overtime by opening more ORs and setting the projected start times earlier to satisfy the stricter chance constraints. However, these measures lead to lower OR utilization and higher waiting times when compared to that of M DEF . Table 6 compares the run times of the two models after solving the test instances presented in Table 3 and Table 5. Column First Feasible shows the run time to find the first feasible solution. Column 1% Gap details the run time to reach 1% optimality gap. It is observed that M DEF can find feasible solutions and attain near-optimal solutions in shorter time lengths compared to M Joint . For the largest test instance, M DEF finds a feasible solution within half the time required by M Joint and solves the problem to optimality within 48 minutes. On the other hand, M Joint fails to reach 1% optimality gap within the one-hour time limit.

V. CONCLUSIONS
In this paper, a chance-constrained mixed-integer programming model was proposed for the OR scheduling problem with stochastic surgery durations. The individual chance constraints controlled the risk of OR overtime. The goal was to minimize the sum of OR opening, OR overtime and patient waiting costs. Our model was compared with two other stochastic models in the literature: an expected value model and a CVaR-based model. We demonstrated that minimizing the expected costs when solving the chance-constrained OR scheduling model results in significant savings compared to the case where only the deterministic costs are minimized. Moreover, we compared the individual and joint chance constraints in terms of allocated ORs, second-stage stochastic costs and solution times. A decomposition algorithm with strong feasibility and optimality cuts was applied to effectively solve large-scale test instances. We proposed an algorithm that generated feasibility cuts using the first stage solutions, and as a result, significantly reduced the time required to find feasible solutions. Numerical experiments demonstrated that the decomposition algorithm outperformed both the IBM CPLEX solver and a basic decomposition algorithm by solving the largest test instances to optimality within the one-hour time limit. Moreover, it is shown that the individual chance constraints lead to higher OR utilization, reduced patient waiting times and shorter solution times. This work has the following limitations. First, it is assumed that the surgery duration follows a known probability distribution. Second, we considered the upstream and downstream resources (e.g. nurses, beds) to be sufficiently available. However, such assumptions do not hold true for all real-life cases. It is also demonstrated that finding strong cuts can increase the convergence speed significantly. Therefore, relaxing the above assumptions and discovering stronger feasibility and optimality cuts in order to solve more complex problems in shorter time frames can be a promising topic for future research.

APPENDIX A PROOFS
A. PROPOSITION 1 Proof: According to constraints (12) and (13), an overtime occurs if: Therefore, the value of z rs in (16) captures scenarios where an OR runs overtime. Given that the random surgery duration has a discrete and finite support, constraint (17) limits the number of scenarios where each OR can run overtime.

B. ASSUMPTION 3
Proof: Since all of the first-stage decision variables are binary, P(s) is bounded by a hypercube of dimension N . Therefore, θ = 0 is the only solution that satisfies the condition in the definition of C. In other words, C = {0} for all s ∈ S.
C. ASSUMPTION 4 Proof: We need to show that both first-stage and second-stage problems have bounded optimal solutions. We know from Assumption 2 that both problems are feasible. The highest objective function value for the first-stage problem is when all operating rooms are open, i.e., r f r , which is bounded. Given the first-stage solution, every open operating room will run overtime in up to α r |S| scenarios. The amount of overtime is bounded by max k∈K {tr krs + i δ is y ikr − cap r }, which is also a finite value given tr krs ≥ 0 and the minimization objective function in the second-stage problem.

D. PROPOSITION 2
Proof: This holds for the individual chance constraints in our model without loss of generality. Assume that in the optimal solution to our model, there exists r ∈ R where s z * rs = < α r |S|. The optimal solution will allow (α r |S| − ) scenarios to run in overtime mode (z rs = 1) with the corresponding overtime variables o rs set to zero.

E. PROPOSITION 3
Proof: First, we know from Assumption 2 that P(s) ∩ X = . Given thatX ⊇ X , we have P(s) ∩X = that ensures the feasibility of (30). Second, the single-scenario optimization subproblem is bounded if γ is chosen to be a vector in the dual cone of C (recession cone). The dual cone of C is defined as C * = γ ∈ R N | γ θ ≥ 0, ∀θ ∈ C . From Assumption 3, we know that C = {0} for model (M 1 ). Therefore, any γ ∈ R N is a vector in C * .

G. THEOREM 2
Proof: The feasibility cuts are added to the master problem (29) to remove the first-stage decisions that result in infeasible M 2 . It is known from Assumption 2 that the set of feasible solutions to M 1 is finite. Therefore, a finite number of inequalities of type (38) and (39) can be added to (29). Moreover, the sets rs and¯ rs of the optimal solutions to the dual problems in Section III-B are finite since there is no constraint parallel to the objective function (40). Therefore, a finite number of optimality cuts (50) will be generated. Given that there are finite numbers of feasibility and optimality cuts, Algorithm 2 converges in a finite time following the convergence of the Benders decomposition algorithm [26].
H. PROPOSITION 6 Proof: When α = 1, the chance constraints (12) can be written as: Pr{tr krs + i δ is y ikr ≤ cap r : ∀k, s} ≥ 0, ∀r ∈ R (53) which holds for all feasible solutions to the first-stage problem. Therefore, the chance constraints are redundant and we have CCP ≡ EV when α = 1. Now, it suffices to show that CVaR ≡ EV . CVaR is defined as the expectation of those outcomes where total costs exceed a threshold value, called Value-at-Risk (VaR) [27]. For α = 1, the VaR of second-stage costs is defined as: where CDF represents the cumulative density function. Given that CDF ≥ 0 for every random variable, we conclude: g(x, s) ≥ VaR 1 , ∀x ∈ X , s ∈ S (55) Therefore, from Assumption 2 and inequality (55), the CVaR model minimizes the total costs over all scenarios, thus indicating equivalence to using the EV model.