Question

A city council voted to conduct a study on inner city community problems. A nearby university was contacted to provide sociologists and research assistants. Allocation of time and costs per week are given in the table. How many sociologists and how many research assistants should be hired to minimize the cost and meet the weekly labour-hour requirements? What is the weekly cost?

	LABOUR HOURS Research	LABOUR HOURS Research	MINIMUM LABOUR HOURS NEEDED PER WEEK
	Sociologist	Assistant
FIELD WORK	$10$	$30$	$180$
RESEARCH CENTRE	$30$	$10$	$140$
COSTS PER WEEK (E)	$500$	$300$

Answer 1

Given:

	Labour Hour Research	Labour Hour Research	Minimum Labour Hours Needed Per week
	Sociologist	Assistant
Field Work	10	30	180
Research centre	30	10	140
Lists Per Week	500	300

Solution:Let $x$ be the number of Sociologists

Let $y$ be the number of Research assistance write all the constraints implied by the table

i)$10x+30y\ge 180$

ii)$30x+10y\ge 140$

iii)$x\ge 0,y\ge 0$

To minimize the objective $c=500+300y$ subject to the above constraints

i)$10x+30y\ge 180$

$x$	0	18
$y$	6	0

ii)$30x+10y\ge 140$

$x$	0	$14/3$
$y$	14	0

Points A(0,6),B(18,0) points C(0,14) D$\left(14/3\right)$

iii)$10x+30y=180$

$\underset{–}{30x+10y=140}$

$x=3$ and $y=5$

E(3,5)

The Feasible region CEB

Corner Point	$x$	$y$	c=$500x+300y$
A	0	14	500(0)+300(14)=4200
B	3	5	500(3)+300(5)=3000
C	18	0	500(18)+300(0)=9000

The weekly cost we be at a minimum of $3000$ of $3$ Sociologist and $5$ Research Assistant are hired.

The labour hours are hired each week in field work

$10\left(3\right)+30\left(5\right)=180$hrs

The labour hour are used for each work in research

$30\left(3\right)+10\left(5\right)=140$hrs