Question

ohn Powell, chief of airport security at a large Canadian airport, is responsible for scheduling security agents around the clock. Powell has divided the day into six 4-hour periods: midnight to 4:00 A.M., 4:00 A.M. to 8:00 A.M., etc. Security agents work 8-hour shifts. Passenger traffic in the airport is determined by domestic and international flight schedules, and Powell has decided that the minimum number of security agents that he needs on duty during the different 4-hour shifts is as shown in the table:

Shift	Time	Min. #of agents required
1	Midnight to 4:00	6
2	4:00 to 8:00	10
3	8:00 to 12:00	24
4	12:00 to 16:00	24
5	16:00 to 20:00	16
6	20:00 to 24:00	12

Since Powell has been requested to keep the security budget at a minimum, he needs to minimize the number of security agents who work in the airport in a day. In making up a daily roster of security agents, Powell has to specify the starting time for each agent, therefore he has to decide how many agents should start their 8-hour shift at the beginning of each 4-hour time block. Formulate the security agent scheduling problem as a linear program.

Powel should schedule  agents in shift 1,  agents in shift 2,  agents in shift 3,  agents in shift 4,  agents in shift 5, and  agents in shift 6, for a total of  agents.

Answer 1

Let the decision variables be:

	Decision variables
Midnight - 4	x1
4 - 8	x2
8-12	x3
12-16	x4
16-20	x5
20-24	x6

Objective function = x1+x2+x3+x4+x5+x6 and this will have to be minimized.

Constraints:

(1): x6+x1 >= 6

(2): x1+x2 >= 10

(3): x2+x3 >= 24

(4): x3+x4 >= 24

(5): x4+x5 >= 16

(6): x5+x6 >= 12

Lastly all the variables >=0 and should be integers.

Solving in excel, using the solver function, we get the following solution:

	Decision variables
Midnight - 4	3
4 - 8	7
8-12	17
12-16	7
16-20	9
20-24	3
Objective function	46
Constraints:
6	>=	6
10	>=	10
24	>=	24
24	>=	24
16	>=	16
12	>=	12

Thus Powel should schedule 3 agents in shift 1, 7 agents in shift 2, 17 agents in shift 3, 7 agents in shift 4, 9 agents in shift 5 and 3 agents in shift 6 for a total of 46 agents.

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