Answer:
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 :)