Question 1175634
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A manufacturer makes two items, A and B. Item A requires 3 minutes of labor to assemble and B requires 4 minutes of assembly time. 
Item A costs $2 in raw materials and B costs $1. There is a maximum of 3,000 labor minutes available for assembly and a budget 
of $1,000 in raw material costs per day. Assuming they sell all the produce and that the profit is $5 per item A and $4 per item B,
how many of each item must be produced in order to maximize profit?
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<pre>
Let X = # items A;  Y = # items B.


From the condition, we have this formulation of maximization problem:


    (1)  the objective function to maximize is the profit  P = 5X + 4Y  dollars.


Restrictions


    (2)  3X + 4Y <= 3000  minutes   (assembly time)

    (3)  2X +  Y <= 1000  dollars   (material cost)

    (4)   X >= 0,  Y >= 0.


You can make a plot of the feasibility domain.


It is a quadrilateral in QI with the vertices  (X,Y) = (0,0), (500,0), (200,600), (0,750).


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


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


    P(0,0)                     =    0

    P(500,0)   = 5*500 + 4*0   = 2500

    P(200,600) = 5*200 + 4*600 = 3400

    P(0,750)   = 5*0   + 4*750 = 3000.


Comparing these values, you find the optimal point.

It is  (X,Y) = (200,600),  200 items A and 600 items B, providing maximum profit of 3400 dollars.
</pre>

Solved.


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In this site, there is a 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> 

which explains, for beginners, metodology of solving such problems in more details.