This chapter presents a case study to organize the sales territories for a company with 11 sales managers to be assigned to 111 sales coverage units in Mexico. The assignment problem is modeled as a mathematical program with two objective functions. One objective minimizes the maximum distance traveled by the manager, and the other objective minimizes the variation of the sales growth goals with respect to the national average. To solve the bi-objective non-linear mixed-integer program, a weights method is selected. Some instances are solved using commercial software with long computational times. Also, a heuristic and a metaheuristic based on simulated annealing were developed. The design of the heuristic generates good solutions for the distance objective. The metaheuristic produces better results than the heuristic, with a better balance between the objectives. The heuristic and the metaheuristic are capable of providing good results with short computational times.
TopIntroduction
The problem addressed in this chapter is based on the case of a direct sales company. This is an assignment problem that results in a configuration of territories for the sales managers of the company. The problem is modeled as an optimization problem and is described below.
In the highest level of the sales structure, 11 managers must serve a market consisting of 111 sales coverage units. A sales coverage unit (SCU) is a geographic area with dispersed sellers and a supervisor with an office located in a specific city or town into the area. Managers have regional offices with fixed locations, from where they visit the different supervisors of the sales coverage units. These visits will depend on the sales coverage units assigned to each manager. In the most basic structure, this is an assignment problem, where sales coverage units are assigned to managers. Figure 1 shows the structure of the sales organization.
Figure 1. Structure of the sales organization
The visits of managers to the cities of the supervisors are scheduled at different moments during the year, with a long separation in time between visits. The visits are done to check the results of sales of the SCU, to define strategies for the short-term future, and to make training of the supervisors and the sellers. Because of the policies of the company, the manager must spend some time in the regional office between visits for administrative activities. Therefore, there is no opportunity to sequence visits in single or multiple routes. The visits must consider the movement from the city of origin of the manager to the city of the supervisor, and back to the city of origin. Transportation costs have a positive correlation with the distance between cities.
Currently, the company has an inherited structure that was determined with unknown objectives. The basic structure was of SCUs already assigned to managers. The basic structure of assignment is updated each year when new SCUs are added or some SCUs are deleted because of lack of activity. The new SCUs added are assigned to the closest manager. At this point, the company wanted to reorganize its sales structure based on two objectives: to minimize the distance each manager traveled and to balance the territories with respect to the expected average sales growth. The motivation of the study was to solve this problem for the company while some operational constraints were observed.
To avoid spending too much time on transportation, one objective of the problem is to minimize the maximum distance that any manager must travel. This also helps to balance the size of each territory, regarding the number of customers. In the other side, one of the constraints of the problem is that the sum of all the transportation costs should not exceed an annual budget.
For each one of the sales coverage units, a goal of sales growth is determined for the year. Together, the growth goals of each SCU result in a national growth average. An individual growth average can be calculated for each one of the managers according to the sales coverage units assigned to them. One of the objectives of the problem is to make the assignment such that the differences of the individual growth averages with respect to the national growth average are minimized. In this way, the managers have a feeling of equity in the distribution of responsibilities.
The problem must consider that each one of the managers must have assigned at least one sales coverage unit and that each sales coverage unit is assigned to a single manager. The idea is to keep the current number and location of managers. In a feasible solution without this consideration, some managers might be fired, but the company wants to keep the salesforce because of social responsibility concerns. Another issue to be observed is to avoid duplication of assignments, to elude administrative confusion of the supervisors reporting to multiple managers, and to keep order in the computation of the individual growth averages.