Question

A city police department has the following minimal daily requirement for policeman. Note, you are to consider period 1 as following immediately after period 6. Each policeman works eight consecutive hours. Let X denote the number of men starting work in period t everyday. The police department seeks a daily manpower schedule that employs the least number of policemen, provided that each of the above requirements is met. Formulate linear programming model to find an optimal schedule.

Time of Day

Period

Minimal number of police required during a period

2 – 6

1

20

6 – 10

2

50

10 – 14

3

80

14 – 18

4

100

18 – 22

5

40

22 - 2

6

30

Answer 1

Let X1, X2, X3, X4, X5, and X6 denote the number of policemen starting work in periods 1, 2, 3, 4, 5, and 6, respectively. The objective is to minimize the total number of policemen employed, which can be expressed as:

Minimize: X1 + X2 + X3 + X4 + X5 + X6

Subject to:

X1 + X6 >= 20 (at least 20 policemen required during period 1)

X1 + X2 >= 50 (at least 50 policemen required during period 2)

X1 + X2 + X3 >= 80 (at least 80 policemen required during period 3)

X1 + X2 + X3 + X4 >= 100 (at least 100 policemen required during period 4)

X4 + X5 >= 40 (at least 40 policemen required during period 5)

X5 + X6 >= 30 (at least 30 policemen required during period 6)

All variables are non-negative integers.

I hope this is what you are looking for :)