A major research facility in Canada must maintain tight security 24 hours a day

ID: 353167 • Letter: A

Question

A major research facility in Canada must maintain tight security 24 hours a day and 7 days a week. Fulltime security personnel work 8 hour shifts that start every four hours throughout the day. The number of security personnel needed at different times throughout the day varies. The following table summarizes the minimum number of security personnel needed in each 4-hour period.

12am-

4am-

8am-

12pm-

4pm-

8pm-

The wage is $20 per hour from 12am to 8am, and $15 per hour from 8am to 12am. The Chief of Security wants to determine the number of security staff to schedule to meet the minimum security requirements with the minimum cost. Only formulate a linear programming model to determine the optimal security scheduling plan.

Time periods

12am-

4am

4am-

8am

8am-

12pm

12pm-

4pm

4pm-

8pm

8pm-

12am Min. # of security staff 14 22 18 24 19 10

Explanation / Answer

Answer:

Linear Programming Problem: The linear programming problem statement is the objective functions which maximize or minimize or optimize the function for the interested variable of the problem statement. The aim of objective function is to have the optimized solution with meeting the constraints of each of the variables. We can say that the linear programming problem objective optimize the function with compiling all the constraints equations.

Linear Objective Function: This is the main function of the problem, made up by including all relevant functional variables, which impacts the objective function significantly.

Constraints Conditions: These are the conditions which makes the boundary conditions for the problem statement. Thus in the given problem statement, this conditions are must meet requirements.

Time periods

12am-4am

4am-8am

8am-12pm

12pm-4pm

4pm-8pm

8pm- 12am

Min. # of security staff

Variable

Wages Rates

$ 20 / hour

$ 15 / hour

Objective Function: We need to minimize the cost of the security personals. Thus the function is to minimize the cost of securities.

Functional Variable: The X1, X2, X3, X4, X5 and X6 are the functional variable for the fgiven time period as mentioned above.

Objective Function: Min. 20X1 + 20X2 + 15X3 + 15X4 + 15X5 + 15X6

Constraints

X1 >,= 14

X2 >,= 22

X3 >,= 18

X4 >,= 24

X5 >,= 19

X6 >,= 10

So the cost of the objective function can be minimized by optimizing the objective function with meeting the constraints conditions for each variable.