# Hybrid Harmony Search for Stochastic Scheduling of Chemotherapy Outpatient Appointments
## 1 Introduction
Since this optimization method requires a prohibitive computational time to be solved, the authors introduced a heuristic algorithm to find approximate solutions in a reasonable time. [PERSON] and [PERSON] [22] used a mean-risk stochastic programming model powered by a specific heuristic to schedule patient appointments and resources for reducing patient waiting time and nurse overtime. [PERSON] et al. [24] exploited the principles of queueing theory for the offline appointment sequencing problem by engaging a large number of servers (chairs) and customers (patients) in a stochastic system wherein service duration and punctuality are subject to significant uncertainty. They proposed a data-driven approach based on the infinite-server queues whose effectiveness was proved by testing that approach against near-optimal algorithms. Both in deterministic and in stochastic mathematical programming models, the time to converge drastically depends on the number of patients and the number of resources as well. Hence, heuristic or metaheuristic algorithms represent a valid alternative to achieve a perfect compromise between the quality of solutions and computational times in solving COS problems. To this end, [PERSON] et al. [3] proposed a two-phase approach for the next-day COS problem. They used a specifically devised heuristic algorithm for handling the appointment scheduling for the laboratory tests, and two heuristics based on the multiple knapsack problem for the second phase in which patients have to be online assigned to the infusion seats. The work of [PERSON] et al. [25] represents a valuable contribution since a metaheuristic algorithm was adopted in the field of the stochastic COS problem. They developed a GRASP algorithm to generate sub-optimal lists of patients for consultation and injection phases in a same-day chemotherapy treatment scenario. Finally, [PERSON] et al. [26] developed a heuristic method based on a progressive hedging algorithm, since they experienced high computational times to solve instances with a two-stage stochastic programming model focused only on the treatment stage. The present paper, inspired by the health services provided by an oncology unit of a hospital located in Southern Italy, aims to solve the same-day offline stochastic COS problem, motivated by the activities of a real-life oncology department in which the medical consultations are performed on the same day as the chemotherapy treatments. Different from the literature, this research is the first to consider that the pharmacy collects the therapies in batches and a courier service delivers the batches to the oncology department. Our objective is to make sure that the time each patient spends in the system, i.e., the total flow time, is minimized. The aim of the proposed scheduling method is to enhance the daily experience of each patient by reducing their waiting time. Furthermore, the proposed research aims at reducing resource idle time and maximizing the number of patients scheduled in a day. The main contributions of the present research are the following:  
1. Adopting a stochastic scheduling strategy that is able to reduce the total flow time (i.e., idle times of patients among the stages) in a same-day COS problem;
2. Introducing a novel multi-stage simulation framework (based on discrete-time recursive equations [27; 28]) in which the pharmacy is detached from the oncology department and drugs are delivered in batches;
3. Developing a new hybrid metaheuristic algorithm, namely Harmony Search (HS), to solve the COS problem for the minimization of patient waiting time. The proposed algorithm is 'hybrid' since heuristic solutions in the initial population, the local search and the reinitialization procedure are embedded in the metaheuristic algorithm;
4. Differently from the existing scheduling approaches (e.g., templated scheduling), the proposed method allows the healthcare manager to identify the optimal schedule in the space of the solution with reasonable computational time.