Question 1188897
(a)
constraint equations are:
x + y <= 40
10x + 30y >= 280
30x + 10y >= 360
x, y >= 0
objective function is:
minimize 5000x + 3000y


(b)
constraint equations are:
x + y <= 40
10x + 30y >= 280
30x + 10y >= 360
x, y >= 0
x - y <= 0
objective function is:
minimize 5000x + 3000y


(b) is the same as (a) but with the additional requirement that x <= y.
if you subtract y from both sides of this inequality, you get x - y <= 0.


the graphical analysis a and the simplex tool analysis both indicate that:
minimum cost for (a) is 68,000.
minimum cost for (b) is 72,000


for (a) at (10,6), the constraint are:
x + y <= 40 = 10 + 6 <= 40 which is true.
10x + 30y >= 280 = 10*10 + 30*6 >= 280 = 280 >= 280 which is true.
30x + 10y >= 360 = 10*30 + 6*10 >= 360 = 360 >= 360 which is true. 
x >= 0, y >= 0 which is true.


for (b) at (9,9), the constraints are:
x + y <= 40 = 9 + 9 <= 40 = 18 <= 40 which is true.
10x + 30y >= 280 = 10*9 + 30*9 >= 280 = 360 >= 280 which is true.
30x + 10y >= 360 = 30*9 + 10*9 >= 360 = 360 >= 360 which is true.
x >= 0, y >= 0 which is true.
x - y <= 0 = 9 - 9 <= 0 = 0 <= 0 which is true.


all the constraints are met for both (a) and (b) at the minimum cost point.
for (a), the minimum cost point is (10,6).
for (b), the minimum cost point is (9,9).


for (a), the results are shown below, first for the graphical analysis and next for the simplex tool analysis.


<img src = "http://theo.x10hosting.com/2021/121601.jpg" >


<img src = "http://theo.x10hosting.com/2021/121602.jpg" >


for (b), the results are shown below, first for the graphical analysis and next for the simplex tool analysis.


<img src = "http://theo.x10hosting.com/2021/121603.jpg" >


<img src = "http://theo.x10hosting.com/2021/121604.jpg" >


with the graphical analysis, using the desmos.com calculator, you are graphing the opposite of the constraint inequalities.
the area of the graph that is not shaded is the feasible region.
the minim cost will be at the corner points of the feasible region, as shown.
you evaluate the objective function at each of the corner points to find the corner point that has the minimum cost.


with the simplex tool, you need to make each inequality have the variables on the left and the constant on the right.
that is why x <= y had to be shown as x - y <= 0.


the problem statement is a little unrealistic because it allowed for 0 sociologist or 0 research assistants.
this would not be a possible combination in a realistic environment where you would need at least some sociologists and some research asistants.
despite that, the minimum cost solution did include some sociologists and some research assistants.