# A Mathematical Model for Scheduling and Assignment of Customers in Hospital Waste Collection Routes
## 3 Materials and Methods  
A mixed-integer linear programming model is proposed to support decision-making in a Hospital Waste Collection company where it must be decided: (a) which day each customer should be visited and (b) which vehicle should be used to visit each customer. These customers are characterized according to three aspects. The first is the frequency of visit, which can be daily, weekly, monthly, or other. The second is the quantity and type of waste to be collected. Finally, there is the geographic location, expressed in polar coordinates, to obtain the angle \(\theta\), associated with its sector in the city. In addition, a four-week month, with a Monday through Friday workday, and a fleet of vehicles with capacity must be considered. A block chart of the proposed solution method is presented in Figure 1.  
The proposed mathematical model schedules visits satisfying the frequency associated with each customer, without exceeding the capacity of the vehicle associated with each route and the maximum number of customers that can be visited in a day. The customers of each route belong to a limited sector of the city through polar coordinates. Among all the possible combinations, the one that minimizes the maximum demand of a route should be selected as a key performance indicator (KPI) to balance each day's work. Input datasets  
\(D\): Customers with daily frequency without considering non-business (civil or religious) days fixed by the country's authorities. \(TW\): Customers with thrice-weekly frequency (visited on Monday, Wednesday and Friday). \(SW\): Customers with semi-weekly frequency (visited on Monday and Wednesday, or on Tuesday and Thursday). Figure 1: Block chart of the proposed solution method. Source: By the authors. \(W\): Customers with weekly frequency (visited once a week, always on the same day of the week). \(B\): Customers with biweekly frequency (visited every two weeks). \(M\): Customers with monthly frequency (visited once a month). \(I\): All Customers \((D\cup TW\cup SW\cup W\cup B\cup M)\). \(J\): Days of the month. \(V\): Trucks available daily for Hospital Waste Collection. Model parameters:  
\(F_{i}\): Number of visits in a month to customer \(i\). \(K_{i}\): Demand of customer \(i\) (containers). \(C_{k}\): Capacity of truck \(k\) (containers per day). \(\theta_{i}\): Angle in radian associated with customer \(i\) in relation to the depot. \(\theta max\): Maximum angle of each sector of customer. \(CusMax\): Maximum number of customers assigned to a truck daily. \(M\): Angle in radian of a full circle, \(2*\pi\), used as Big-M.  
Decision variables:  
\(x_{ij}=1\) if the customers \(i\) is assigned to day \(j\), and \(0\) otherwise. \(y_{ijk}=1\) if the customers \(i\) is assigned to day \(j\) and to truck \(k\), and \(0\) otherwise. \(capmax=\) Maximum demand collected on one day (containers). \(\theta m_{jk}=\) Minimum angle of the customers on day \(j\) to truck \(k\). \(\theta p_{jk}=\) Maximum angle of the customers on day \(j\) to truck \(k\). The proposed mathematical model of integer linear programming seeks to solve customers' scheduling and assignment in a monthly calendar, scheduled according to their visit frequency (daily, weekly, three-weekly, weekly, biweekly, and monthly). In addition, customers' assignment to the different collection trucks is performed according to their location concerning the located warehouse. The main goal is to balance loads of the trucks daily and cluster by geographical areas. The model only considers customers having a particular infectious waste product for transportation. The proposed model uses homogeneous resources for the hazardous waste collection service that have a daily truck assigned, and also, the visit to these customers is coordinated in advance and at their request.