# A Mathematical Model for Scheduling and Assignment of Customers in Hospital Waste Collection Routes
## 4 Results  
The transitional cost of the proposed approach is related to the training of the decision-makers in its use. The company's data must be ported over and validated to complete the manual planning transition to use the proposed approach. If it is significant enough, the company may have some downtime during the transition as well. However, free commercial solvers to solve the problem make it so that the costs of transitioning are trivial compared to the expensive commercial licenses commonplace for solutions. Indeed, it is essential to mention that the implementation and transitional costs must be compared to using a manual solution of the problem which often uses suboptimal considerations. Implementation and transitional costs are significant to calculate because they require upfront capital. Training and software need to be purchased, which means the company needs to have the liquidity to afford the project. Implementation and transitional costs are also a one-time expenditure, however, which means that long-term, they are not as important as the ongoing and total cost of obtained solutions.  
To measure the angle \(\theta_{i}\) of each customer about the depot, each of them was georeferenced using the Google Maps tool. Subsequently, the Coticchia-Surace equations were used to transform UTM coordinates into Euclidian coordinates (x,y). Lastly, the angle was calculated using the function atan2. Thus, \(\theta_{i}\) is the angle in degrees between the positive x-axis of a plane and the point given by the coordinates (x,y). In this case, \(\theta_{i}\) ranges from 0 to \(2*\pi\), thus enabling us to divide the region into sectors to cluster customers based on their polar coordinate in the plane. The values of \(\theta max\) and \(CusMax\) are obtained from the company; the suggested value was near the minimal value to obtain a feasible solution and have a solution with a good work balance.  
In particular, we considered a month with four weeks and each week with five days, with the following waste collection frequencies:  
* Daily: Customers visited every day of the month.
* Thrice-weekly: Customers visited on Monday, Wednesday, and Friday, every week.
* Semi-weekly: Customers visited two times, every week of the month.
* Weekly: Customers visited the same days of the week, every week.
* Biweekly: Customers visited 2 times, two weeks apart.
* Monthly: Customers only visited once a month.  
Table 1 outlines the number of customers distributed by customer frequency.  
Demand was estimated based on the mean monthly kilograms of waste removed by the trucks during 2019. The capacity of the vehicles was determined based on the maximal number of containers per truck. Medium trucks have a capacity of 12 containers with a capacity of 40 kg each. Large trucks have a capacity of 16 containers.  
The customers to whom the particular waste transport service is provided were selected for the computational experiments because they account for more than 80 percent of all the company customers (Table 1). The categories that correspond to the mean monthly kilograms generated by the customers to whom the special waste collection service is provided are shown in Table 2. We use the customers with ten or more kilograms per month of demand, 85 customers from the Maule region, and 214 from the Bio-bio region.