4.2 Two-Stage Solution Method
4.2.1 Surrogate Model
A surrogate model with a different (but closely related) objective is built in the first stage. The Lagrangian relaxation of the surrogate model is decomposed by patients and the sub- problems are solved by dynamic programming. After iterations of the subgradient method, the best sequence is passed to the second stage.
In the multiple-OR SSSP with PACU constraints, the time of the last surgery comple- tion, the last PACU admission and the last PACU release are required to determine surgeon idle time, OR overtime and PACU overtime, respectively. Consequently, extra constraints are built among patients to determine the “last one”, which significantly complicates the problem. Without these “last-one” constraints, the problem can be decomposed by patients after relaxation.
We propose a surrogate problem with a new objective to minimize the costs of patient waiting time, OR blocking time, patients’ completion time, patients’ LOS in the OR after regular hours and patients’ LOS in the PACU after regular hours. The components in the surrogate objective are defined with correspondence to those in the original objective. Patient waiting time and OR blocking time are defined in the same manner as in our original problem. Patient’s completion time is the time of a patient’s discharge from the PACU. By penalizing patients’ completion time, we hope to force every surgery to be finished as soon as possible and hence reduce surgeons’ idle time. If a patient stays in the OR or the PACU after regular hours, his or her LOS in the overtime period is penalized, i.e., we penalize patients’ LOS in the OR or in the PACU after regular hours. Patients’ LOS in the OR or in the PACU after regular hours, though not exactly the same, serve as the surrogate
objective of OR overtime and PACU overtime in the original objective, respectively. This new objective facilitates relaxation and decomposition of the problem. More importantly, the surrogate objective function with selected cost parameters is shown experimentally to be strongly correlated with the original objective in §4.3.2.
We also note that the surrogate problem and the following surrogate SAA model are inspired by Augusto et al [5], who study a deterministic surgery planning problem in pooled ORs to minimize the cost of patients’ completion times. In spite of significant differences between models, we were inspired by the decomposition approach in their model after Lagrangian relaxation.
We formulate the surrogate problem as a time-indexed SAA model; i.e. SURSAA. First we will define the notation used in SURSAA. In total, orr ORs are under study and the OR index set is J = {0, 1, . . . , orr − 1}. In OR j ∈ J , SRj patients or surgeries (these two terms are interchangeable in this study) are scheduled and the corresponding index set is Kj = {0, 1, . . . , SRj− 1}. yjk indicates patient or surgery k in OR j where j ∈ J, k ∈ Kj and Y is the set of all patients. Scenarios are indexed by ω and the index set of all N scenarios is Ω. We use dωjk and pωjk to indicate the surgery duration and LOS in the PACU of patient yjk in scenario ω, respectively. The total number of PACU beds is pcap.
We discretize the time horizon under study into HT time buckets and the length of each time bucket is referred to as a time unit. The length of regular work time is M T time units. Time buckets are indexed by t and their index set is T = {0, 1, . . . , HT − 1}. Parameters CP W, CI, CB, CO and CP O represent the unit cost of patient waiting time, surgeron idle time, OR blocking time, OR overtime and PACU overtime in the original objective, respectively. CsP W, CsCP, CsB, CsO and CsP O are the cost per time unit of patient waiting time, patient completion time, OR blocking time, and patients’ stay in the OR and in the PACU after regular hours in the surrogate objective, respectively. For simplicity, we assume identical CP W and CB for all patients, identical CI and CO for all surgeons and ORs and the same work hours for ORs and the PACU. Our method can, however, be easily modified to handle distinct unit costs and different work hours for ORs
and the PACU. In our test problems, CP W, CI, CB, CO and CP O are randomly generated within reasonable ranges. Since patient waiting time and OR blocking time are defined in the same way in the original and surrogate objective functions, CsP W = CP W and CsB = CB are used in the surrogate model. Other cost parameters CsCP, CsO and CsP O are selected as shown in §4.3.1 to match the surrogate objective with the original objective.
The set of scheduled start times of all surgeries is [SST00, . . . , SSTjk, . . . ] where SSTjk is the scheduled start time of surgery yjk ∈ Y . The actual surgery start time, the PACU admission time and the completion time of yjk in scenario ω ∈ Ω are represented by ASTjkω, AP Tjkω and sCPjkω, respectively. We assume all SSTjk, ASTjkω and AP Tjkω are integer multiples of the time unit. P Wjkω and Bωjk are patient waiting time and OR blocking time of yjk in scenario ω, respectively. sOωjk and sP Oωjk are after-hour stay of yjk in the OR and PACU in scenario ω, respectively. αωjkt and βjktω are binary variables such that αωjkt = 1 if yjk is in the OR in time bucket t in scenario ω and βjktω = 1 if yjk is in the PACU in time bucket t in scenario ω.
In the SURSAA model, we leave out the non-anticipativity rule and the FCFS PACU admission rule to produce a relaxed model that can be solved by Lagrangian relaxation very efficiently. Later these two rules will be re-applied when we use a heuristic to construct feasible solutions to the surrogate problem in the subgradient method. In solving SURSAA by Lagrangian relaxation, the key is to place the surgery into, roughly, the right time slots and then reimpose all constraints to obtain a feasible solution. After OR and PACU resource constraints (4.2.1.9) and (4.2.1.10) are relaxed in Lagrangian relaxation, the non- anticipativity rule and the FCFS PACU admission rule are not key factors in determining the schedule since they are closely related to the relaxed resource constraints. In addition, leaving the non-anticipativity rule and the FCFS PACU admission rule out is equivalent to relaxing them with zero-valued multipliers in Lagrangian relaxation.
SURSAA Model: min1 N X j∈J X k∈Kj X ω∈Ω (CsP WP Wjkω + CsBBjkω + CsCPsCPjkω + CsOsOωjk+ CsP OsP Ojkω) (4.2.1.1) s.t. SSTjk ≤ M T ∀j ∈ J, k ∈ Kj (4.2.1.2) ASTjkω = SSTjk+ P Wjkω (4.2.1.3) AP Tjkω = ASTjkω + dωjk+ Bjkω (4.2.1.4) sCPjkω = AP Tjkω + pωjk (4.2.1.5) sCPjkω ≤ HT (4.2.1.6) sOjω≥ AP Tω jk− M T (4.2.1.7) sP Ojω≥ sCPjkω − M T (4.2.1.8) ∀j ∈ J, k ∈ Kj, ω ∈ Ω αωjkt= 1ASTjkω ≤ t ≤ AP Tjkω − 1 ∀j ∈ J, k ∈ Kj, ω ∈ Ω (4.2.1.9) βjktω = 1AP Tω jk ≤ t ≤ AP Tjkω + pωjk − 1 ∀j ∈ J, k ∈ Kj, ω ∈ Ω (4.2.1.10) X k∈Kj αωjkt ≤ 1 ∀j ∈ J, t ∈ T, ω ∈ Ω (4.2.1.11) X j∈J X k∈Kj βjktω ≤ pcap ∀t ∈ T, ω ∈ Ω (4.2.1.12)
(4.2.1.1) The objective function is the expected cost of patient waiting time, OR blocking time, patient completion time, and after-hour stay in ORs and in the PACU.
(4.2.1.2) No surgery can be scheduled to start after regular work hours.
(4.2.1.3) to (4.2.1.8) Constraint (4.2.1.3) requires a surgery to start no earlier than its scheduled start time and also calculates the associated patient waiting time. Con- straint (4.2.1.4) makes sure that a patient is admitted into the PACU after his or her surgery and the corresponding OR blocking time is also determined. Patients’ completion times are calculated in Constraint (4.2.1.5). Constraint (4.2.1.6) ensures that all patients are released from the PACU within the time horizon. Constraint (4.2.1.7) and Constraint (4.2.1.8) penalize patients’ stay in the OR and the PACU after regular hours, respectively.
(4.2.1.9) to (4.2.1.10) In equation (4.2.1.9), OR j is occupied by patient yjk from his or her actual surgery start time to the time when he or she leaves for the PACU. Note that
αωjkt = 0 when t = AP Tjkω, since yjk leaves the OR at time AP Tjkω and time bucket AP Tjkω is not occupied. Similarly in Constraint (4.2.1.10), the patients’ stay in the PACU is defined by βjktω .
(4.2.1.11) OR resource constraints ensure that at most one patient can occupy the OR in any time bucket.
(4.2.1.12) PACU resource constraints ensure that the number of patients in the PACU is always within the PACU capacity.