# Hybrid Harmony Search for Stochastic Scheduling of Chemotherapy Outpatient Appointments
## 2 Problem Statement  
The oncology department under consideration can be considered as the counterpart of a three-stage hybrid flow shop manufacturing system in which the first stage is related to the medical consultation, the second stage consists of the pharmacy laboratory and, finally, the third stage involves a set of chairs in parallel for the chemotherapy treatment [15; 18]. The oncology unit disposes of \(O\) oncologists (\(o=1,\,\ldots,\,O\)), \(N\) nurses (\(n=1,\,\ldots,\,N\)) and \(C\) identical chairs (\(c=1,\,\ldots,\,C\)) for the treatment. The oncology unit collaborates with the pharmacy laboratory, whose capacity depends on \(D\) pharmacist technicians (\(d=1,\,\ldots,\,D\)) in charge of the drug preparation. Every day, a number of \(P\) patients (\(p=1,\,\ldots,\,P\)) need to be scheduled to receive oncology therapy. A referee oncologist assists the patient during the whole therapeutic path, in accordance with the care protocol [29]. Hence, the long/medium-term appointment-planning phase is managed by the referee oncologist, who decides upon the days any patient undergoes the therapy. For that reason, the set of patients to be treated every day is known a priori. The sequence of steps that a patient encounters during the treatment day are the following. Once the patient arrives at the oncology department, he/she is registered in a welcome room. The patient waits until the referee oncologist becomes available. If the oncologist is available, the medical consultation starts and a decision on the chemotherapy is taken based on the health status of the patient and the blood tests executed the day before by the patient in the same hospital or an external laboratory. Therefore, there exists a deferral probability [25; 30] that the patient \(p\) is not ready to receive the chemotherapy on the same day. In case the medical consultation is successful, the referee oncologist sends a request (i.e., a prescription) to the pharmacy, which includes the type and dose of the drug [31]. Once the prescription for the therapy arrives at the pharmacy, the pharmacist technicians can start the drug preparation. The pharmacy gathers the ready therapies in batches and makes use of a courier service to send the batches to the oncology department. Finally, once the drug arrives at the oncology center and both a nurse and a chair are available, the patient starts the treatment. After the therapy infusion is completed, the patient is discharged from the oncology ward. Additionally, to thoroughly describe the problem under investigation regarding the ward we observed, the following assumptions, which are common to several literary contributions [2; 26; 32; 25], can be summarized as follows:  
* The number of patients to be treated on a given day is known in advance as it arises from the planning phase, conforming to the patient care protocol decided by the oncologists;
* Each patient is assigned to a referee oncologist for the medical consultation;
* Cancellations and no-shows at the consultation session are disregarded;
* The desk registration time is negligible;
* The results of blood exams are immediately available for medical consultation since the exams are executed by the patient in the previous days;
* Drug preparation devices never break down;
* Each pharmacist technician can prepare a therapy at a time;
* Pre-emption on the different activities is not allowed;
* Nurses have identical skills, i.e., each patient can be treated by any available nurse;* Each nurse can prepare only one patient at a time;
* A nurse can monitor four patients simultaneously at most;
* The time any patient needs to leave the chair is negligible. To model the uncertainties of the problem, the Sample Average Approximation framework was adopted [33]. This consists of generating a finite number of scenarios \(\Omega\) in which stochastic parameters are independently sampled from the corresponding stochastic distributions, while deterministic parameters are kept unchanged for each scenario \(\omega\) (\(\omega\) = 1,..., \(\Omega\)).