Question 1202803
.
Sally Sethness assembles stereo equipment for resale in her shop. 
She offers two products, turntables and cassette players. 
She makes a profit of $10 on each turntable and $6 on each cassette. 
Both must go through two steps in her shop—assembly and bench checking. 
A turntable takes 12 hours to assemble and 4 hours to bench check. 
A cassette player takes 4 hours to assemble but 8 hours to bench check. 
Looking at this month's schedule, Sally sees that she has 60 assembly hours uncommitted 
and 40 hours of bench-checking time available. 
Use graphic linear programming to find her best combination of turntables and cassette players. 
What is the total profit on the combination you found?
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<pre>
Let X be the number of turntable  and Y be the number of cassette players.


We want maximize the profit function

    P(X,Y) = 10X + 6Y


under the following restrictions

    12X + 4Y <= 60      (assembly time)

     4X + 8Y <= 40      (bench check time)

     X >= 0,  Y >= 0


The plot of the feasibility domain is shown in the Figure below.


    {{{graph( 400, 400, -2, 22, -2, 22,
          (60-12x)/4,  (40-4x)/8
)}}}


    Plots y = {{{(60-12x)/4}}}  (red) and y = {{{(40-4x)/8}}} (green)



It is a quadrilateral in QI with the vertices  at points P1 = (0,0), P2 = (0,5), P3 = (4,3), P4 = (5,0).


We apply the standard Linear Programming method in its geometric interpretation.


The solution is one of these 4 points, where the objective function (profit) has a maximum.


You calculate the values of the objective function  P(X,Y)  (profit)  at listed points


    P1:  P(0,0) = 10*0 + 6*0 =    0,

    P2:  P(0,4) = 10*0 + 6*5 =   30,

    P3:  P(4,3) = 10*4 + 6*3 =   58,

    P4:  P(6,0) = 10*5 + 6*0 =   50.


Comparing these values, you find the optimal point.

It is  P3:  (X,Y) = (4,3),  which means  4 turntables and 3 cassette players, providing maximum PROFIT of 58 dollars.
</pre>

Solved.


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If you want to see many other similar and different solved problems, &nbsp;look into the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-minimax-problems-by--the-Linear-Programming-method.lesson>Solving minimax problems by the Linear Programming method</A> 

in this site.



/////////////////////



In opposite to Calculus, which was invented about three hundred years ago, 
the Linear Programming method for solving linear min-max problems
was invented about only 80 years ago, in 1940'.


It quickly became widely accepted and widely extended method 
of solving many important problems in Economics.


The size of solved problems quickly increased from tens to hundreds,
thousands and more variables. The best mathematical minds took part
in creating and developing this method.


So, now this method is studied in high schools, colleges, and universities.


There are several levels of learning/teaching this method.


First, the lowest/starting level is its geometric version, as I presented it
in my post. At this level, students learn it in high school. 
Usually, only two unknown variables are considered, and the purpose of 
this learning is to make first steps: to get a first acquaintance 
with the method, to get its idea, and to get familiarity on how it works.


<pre>
    The major idea of the LP-method is that linear function over a polyhedron
    in multi-dimensional linear space gets its maximum-minimum values
    at the corners/vertices of the polyhedron.
    The idea is simple, but enormously productive.
</pre>

The second level is advanced consideration of the 2D case with two unknowns
and 3D case. At this level, it is appropriate to consider version of the method
with slopes of the constraint boundary lines and the slope of the objective function.
It is the level of teaching this method in colleges, where Math is not a major specialty.


Third level is teaching students to use existing standard calculators
from the Internet, like Theo does it in his post.
This third level can sometimes be combined with the first or the second level.
At this level, the major goal is to teach students to setup problems.


Fourth level is studying this method in the matrix version, oriented
for work with thousands unknowns. It is intended for those students who are 
going/targeted to be the upper level users and/or professional DEVELOPERS 
in this area. It is the level of renowned Math colleges and Universities,
and the level of graduate students, working on their PhD degrees in the area.


In the pedagogic literature, these levels are strictly separated.
There are, probably, tens of textbooks for the first level;
there are textbooks for the second level, although not so many.
On my bookshelf, I have three college Math textbooks teaching at this level,
and I am sure that every student, who really needs and wants to know
the subject at this level, has an ability to learn from such textbooks.
Always there is a possibility to buy affordable used textbook for low price,
since at this level nothing is changed in decades. 
(And, empirically, than older is a textbook, than it is better).


For the highest, fourth level, there are standard textbooks and monographs
on simplex method, as well as Proceedings of numerous conferences.
They are used by PhD students in their Computer Science courses or in 
their Computational Math courses. And at the level 4, the simplex method technically 
has nothing in common with the levels 1, 2 and 3, except of general idea.


So, regarding the LP-method or simplex method, there are enough educational sources 
for each level, and they are strictly/accurately separated by levels of complexity.


Next, when students come to this forum asking to enlighten them about the LP-method,
such requests are 100% from beginner students, who need to get basic knowledge at level number one.


As soon as they know the subject at this level one, they will not come to this forum 
asking to enlighten them at level two: they will go further on their own legs, 
without spikes/toggles.


It is why I always present them the LP-method at the level one, as I did it in my post above.


Not because I am a bad or a limited person, but because the life is arranged this way, and not otherwise.