## 6 Case of Study  
As mentioned previously, this problem is inspired by the situation faced by a company that produces and distributes oxygen. The company is located in the northeast region of Mexico and has two different types of vehicles: (i) a big truck with 22 tons of capcity and (ii) a mid-size one with a capacity of 11 tons approximately.  
This company has a team of planners who are responsible for creating the distribution routes following specific rules. The primary rule is to visit the first hospitals and medical facilities to guarantee a minimum quality level of the product. The secondary rule is to maintain a minimum stock level at the customer tank. Every day, the team checks the oxygen inventory level at the customer's facility, and, according to their consumption rate, they decide if the customer will be visited the next day or not. Once they have selected a group of customers, they look over their location. After that,  
Figure 4: Example without a cost increase on the routing inventory problem.  
Figure 5: Influence of priorities on the routing inventory problem.  
the planner defines which type of vehicle (truck) should be used and designs the distribution routes considering the priority constraints.  
This case of study considers an instance with 28 customers (7 hospitals and 21 industrial clients) distributed in the metropolitan area of the city of interest. Three trucks of different capacities (22, 22 and 11 tons) are available at the depot. In Figure 6, the location of customers is presented. Hospitals are colored in red, industrial customers in blue, and the depot is marked in black. It is important to notice that the location of the depot as well as the cost units have been modified due to confidentiality issues. The software used by the company is different from the one used here. Although the model is the same.  
In the first period, the model suggests three routes, the first one (blue) only visits industrial customers, while routes two and three (black and red respectively) visit first hospitals and then industrial customers. Then, in period two, every route includes a hospital that is visited first. Finally, in period three, the model creates two routes (see Figure 7). The total cost of this solution is $635.59 and the time used to obtain the solution was 7200 s. Comparing the routes designed by the model with the routes created by the planner, the model reduces the total distribution cost in more than 30%. The inventory routing problem is a combinatorial optimization problem of type NP-Hard. Therefore, finding reasonable quality solutions is a complicated process when done without the aid of computing tools. Consider that, for a planner, designing a route implies a considerable effort since he/she needs to consider every assigned customer, verify its inventory level, and, according to its consumption rate, decide whether to include it or not in the next day delivery program. The task is complicated for the planners because, if a customer does not receive a sufficient quantity in an adequate time, it could run out of the product and be forced to stop operations. With this tool, the company could save money, while guaranteeing that every customer has a minimum stock level of product and will help the planners in their work. There are still some characteristics to be considered in order to make this solution closer to the real operation of the company. Notice that the location of the depot and the cost units have been modified due to confidentiality issues.  
Figure 6: Location of customers.