Chapter 3.5, Problem 2P

Explanation of Solution

Formulation of LP:

Let,

${\text{x}}_{\text{i}}$ be the number of police officer’s works on shift $\text{i}$.

$x_1$= Police officer work on shift 1.

$x_2$= Police officer work on shift 2.

$x_3$= Police officer work on shift 3.

$x_4$= Police officer work on shift 4.

$x_5$= Police officer work on shift 5.

$x_6$= Police officer work on shift 6.

The objective is to minimize the number of police officer’s needed to meet daily requirements.

That is,

$\left(\text{Number of police officers on shift 1}\right)\text{+}\left(\text{ Number of police officers on shift 2}\right)\text{+}\mathrm{.....}\text{+}\left(\text{ Number of police officers on shift 6}\right)$$=x_1+x_2+x_3+x_4+x_5$

Objective function:

Minimize,

$z=x_1+x_2+x_3+x_4+x_5$

Now, the requirement of number of on-duty police officers during midnights 12 to 4 am is 8 and each police officer work two consecutive 4-hours shift.

Therefore, the constraints will be,

${\text{x}}_{\text{1}}{\text{+x}}_{\text{6}}\ge \text{8}$

Similarly the requirement of number of on-duty police officers during 4 am to 8 am is 7 second constraint is,

${\text{x}}_{\text{1}}{\text{+x}}_2\ge 7$

Requirement of number of on-duty police officer’s during 8 am to 12 noon is 6,

${\text{x}}_2{\text{+x}}_3\ge 6$

Requirement of number of on-duty police officers during 12 noon to 4 pm is 6.

${\text{x}}_3{\text{+x}}_4\ge 6$

Requirement of number of on-duty police officer’s during 4 pm to 8 pm is 5.

${\text{x}}_4{\text{+x}}_5\ge 5$

Requirement of number of on-duty police officer’s during 8 pm to 12 midnight is 4.

$\begin{array}{l}{\text{x}}_{\text{1}}{\text{+x}}_{\text{4}}{\text{+x}}_{\text{5}}{\text{+x}}_{\text{6}}{\text{+x}}_{\text{7}}{\text{+y}}_{\text{1}}{\text{+y}}_{\text{3}}{\text{+y}}_{\text{4}}{\text{+y}}_{\text{5}}{\text{+y}}_{\text{6}}{\text{+y}}_{\text{7}}\ge \text{11 }\left(\text{Sunday}\right)\\ {\text{x}}_{\text{1}}{\text{+x}}_{\text{2}}{\text{+x}}_{\text{5}}{\text{+x}}_{\text{6}}{\text{+x}}_{\text{7}}{\text{+y}}_{\text{1}}{\text{+y}}_{\text{2}}{\text{+y}}_{\text{4}}{\text{+y}}_{\text{5}}{\text{+y}}_{\text{6}}{\text{+y}}_{\text{7}}\ge \text{17 }\left(\text{Monday}\right)\\ {\text{x}}_{\text{1}}{\text{+x}}_{\text{2}}{\text{+x}}_{\text{3}}{\text{+x}}_{\text{6}}{\text{+x}}_{\text{7}}{\text{+y}}_{\text{1}}{\text{+y}}_{\text{2}}{\text{+y}}_{\text{3}}{\text{+y}}_{\text{5}}{\text{+y}}_{\text{6}}\end{array}$