Crime in the country is one of the critical aspects pointed out by the population. While it is true, this problem has a greater impact on people who suffer all kinds of actions that threaten not only private property but also the health and integrity of people. Companies are no strangers to these acts and for this reason the management of security and the management of the personnel that deals with the surveillance of people, goods and premises has taken on more importance in recent years. Roberto Ramirez, personnel manager of the Securitas company, must program the security force in such a way that it satisfies the personnel requirements shown in the following table 1.

Table 1. Security Personnel Requirements

Schedule	Minimum number of guards required
Midnight - 4 am 4 am - 8 am 8 am to 12 pm 12 pm to 4 pm 4 pm - 8 pm 8 pm - Midnight	5 7 15 7 12 9

The guards work 8-hour shifts. There are 6 such shifts each day. The start and end times of each shift are given in the following table 2.

Table 2. Shift schedule

Turns	Start time	End time
1 2 3 4 5 6	Midnight 4 am 8 am 12 pm 4 pm 8 pm	8 am 12 pm 4 pm 8 pm Midnight 4 am

What is the minimum number of security personnel to hire?

Would you say that the linear programming model makes it possible to totally optimize the assignment of security guards in the different shifts? Justify your answer (show model - Z ecuation and restrictions).

If the minimum number of guards required between 4 am and 8 am increases by 4 units, what is the impact on the total number of guards hired (objective function)?

An increase or decrease in the guard requirement, on which turn does it have the greatest impact on the value of the objective function? How much is that impact?

If the company to which Securitas provides surveillance services asks you to make a "gesture" in terms of increasing the number of security guards - at no additional cost - regardless of the requirements established in Table 2, what would you recommend? and because? Justify your answer.