# Hybrid Harmony Search for Stochastic Scheduling of Chemotherapy Outpatient Appointments
## 1 Introduction
[9] studied the same-day appointment scheduling problem with two major resources, namely oncologists for consultation and beds for injection, and proposed a combination of heuristics based on the Lagrangian relaxation to minimize the patient waiting times and the makespan. [PERSON] et al. [10] used mathematical programming models to hierarchically tackle both the planning and scheduling of chemotherapy patients to minimize the patient waiting time while considering the limited availability of nurses. Later, [PERSON] et al. [11] improved the model proposed by [PERSON] et al. [10] so that it could be solved in a smaller computational time and also for larger-sized instances. [PERSON] and [PERSON] [12] exploited mathematical programming to generate a multi-level template, which aims to minimize the patient waiting time and nurse workload in an outpatient clinic. Discrete event simulation and mathematical programming were used by [PERSON] et al. [13] to compare the proposed appointment scheduling tool with the current practice under a series of operational measures such as patient waiting time, clinic total working time and resource utilization. Notably, they solved a linear programming model to schedule patient appointments according to a same-day offline strategy. [PERSON] and [PERSON] [14] distinguished between functional and primary care delivery models to provide chemotherapy treatments to cancer patients, depending on the availability of nurses. They consider a single-stage system where a set of patients have to be scheduled offline on the same day only for the infusion phase and proposed two multi-objective optimization models based on mathematical programming that ignored both oncology consultation and lab tests. [PERSON] et al. [15] introduced a mixed integer programming (MIP) model for reducing the patient waiting time of a same-day offline COS problem in a real-life oncology unit. They modeled the whole set of system stages, namely oncology consultation, drug preparation and injection, also considering the limited number of resources at the different stages. To reduce the number of binary variables and constraints, thus enhancing the computational efficiency of their COS approach, [PERSON] et al. [16] devised a two-stage COS method,properly inspired by cellular manufacturing systems, which involved a clustering phase and a mathematical programming phase for the minimization of the total completion time referred solely to the injection stage of an oncology unit. Another valuable contribution to the COS scenario is attributable to [PERSON] et al. [2], who focused on the infusion stage to generate an online scheduling method subject to the nursing constraint. They used integer programming to design a template of vacant appointment slots that follows a bicriteria objective based on the combination of weighted flow time and makespan. A different perspective emerges from the research by [PERSON] and [PERSON] [17]. They presented a mathematical model for the next-day COS problem that managed the chemotherapy patient appointments while taking into consideration the workload of nurses and pharmacists as a constraint of the optimization problem in a cancer clinic. Constraint programming was applied by [PERSON] et al. [18] to the next-day outpatient scheduling problem considering the online approach. A similar approach for the same-day case can be attributed to [PERSON] et al. [19], who developed a chemotherapy outpatient scheduling template by reducing the violation between resource assignment and treatment requirements. Recently, a research stream emphasized the need for managing uncertainty in healthcare scheduling problems [20; 21]. Several sources of uncertainty may have a notable impact on the chemotherapy path of oncology outpatients, therefore deterministic models could represent a strong simplification [22]. [PERSON] et al. [23] presented a two-stage stochastic programming model for the next-day COS problem of an oncology unit located in the USA.