# A Mathematical Model for Scheduling and Assignment of Customers in Hospital Waste Collection Routes
## 1 Introduction  
Because of difficulties or limitations in assigning resources, including machines, facilities, staff, or inputs, one of the leading problems facing any company's operations management, whether in manufacturing or services, is customer-proposed scheduling work activities. Such problems are difficult to solve and have been the subject of several research studies. Many experts have proposed solutions, ranging from heuristics and meta-heuristics approaches to the application of precisely defined rules. Both manual techniques and specialized software programs have been employed by [PERSON] and [PERSON] [1].  
The problem of Hospital Waste Collection (HWC) includes dividing customers into specific groups (clusters), scheduling visits within a particular time frame, and searching for efficient routes for visiting customers. From a theoretical standpoint, the HWC problem involves interdependent vehicle clustering, planning, and routing sub-problems with specific complexities imposed by constraints. The classical models in the literature overlook such issues as described by [PERSON] et al. [2].  
The HWC problem involves a set of customers to whom a waste collection service is provided. The problem consists of a subset of Hospital Waste (corresponding to special waste), which requires scheduling of customers and vehicle assignment, with a predefined frequency of visit. In particular, for the HWC problem, not visiting customers at the given frequency and time implies economic losses for the service provider. The suppliers haspublicly awarded contracts through tenders, and non-compliance is punishable by waste management laws. As a variant of the vehicle routing problem (VRP), the HWC problem is considered a non-deterministic polynomial-time hard (NP-Hard) problem. No exact algorithms with short solution times that can solve real-life instances with a large number of customers are known.  
In this study, a mixed-integer linear programming (MILP) model is proposed that solves the scheduling of customers and collection vehicle assignment for HWC. The model was programmed in A Mathematical Programming Language (AMPL) and solved using the GUROBI 9.0 and CPLEX 12.10 solvers. To validate the effectiveness of the method, data from a Chilean company that provides these services and a subsidiary of an international holding company was used. The results show the proposed method's efficiency in support of decision making. The company's business collects and temporarily stores pollution and disease-causing Hospital Waste generated by hospitals, family health centers, dialysis centers, first-aid rooms, and private healthcare providers in Chile. Most of its customers are public bodies. Thus, their Hospital Waste removal and final disposal operations are regulated by the Chilean Ministry of Health.  
The main contribution of this paper is the mathematical structure of the proposed model. This structure allows balancing the number of customers and the workload during each day. The model has been tested in a real case of a waste collection company. Under this context, this strategy is expected to bring the theory closer to the current reality by analyzing a productive system sector capturing this environment's aspects. The proposed model is generic and could be adapted to other companies with similar requirements.  
State of the art concerning problems related to collecting hazardous materials and solid waste is presented in Section 2. The method proposed to address the problem is performed in Section 3, whereas the computational results are shown in Section 4. Lastly, Section 5 includes conclusions and future perspectives.