1. Linear model

The PAR company is dedicated to the sale of rice in different areas of the country. To carry out the sale, it has a system of representatives, who are in charge of traveling to the different areas and promoting the product, with the aim of ensuring sales.

The sales management, in charge of assigning the representatives, has divided the country into N different zones in which a team of M representatives must distribute.

According to the characteristics of each zone and each representative, the sales management has estimated that the sales volume generated by representative i in zone j is vij, (i = 1, ..., M, j = 1, ..., N). Suppose this volume is independent of the number of representatives assigned to zone j, and the number of zones assigned to representative i.

Sales management must assign at least one representative in each zone and for workload reasons, one representative cannot deal with more than R zones, with R

The sales management wants to assign its sales team to the different areas in such a way as to maximize the company's sales volume.

(a) Formulate an integer linear optimization model that allows the manager to determine the zones to be assigned for each representative, clearly indicating the decision variables, the constraints (title and formula), and the objective function.

(b) Now consider that the sales volume does depend on the number of representatives assigned to each zone. Thus, for each additional representative in the area, above the mandatory, the sales volume of each representative is decreased by 10%. So if there are two representatives each will generate 90% of the original sales volume, if there are three, 80% and so on.

Consider that in no zone can more than 5 representatives be assigned. Indicate and make the necessary adjustments to the model developed in (a) to represent the new situation