Question 1131906
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<pre>
Let X = # of standard beds;
    Y = # of luxury   beds.


The objective (profit) function is F(X,Y) = 6X + 8Y, in thousand dollars.


The constraints are

2.3X + 3.7Y <= 851      (1)    (maximum weekly cost, in thousand dollars)

0.3X + 0.4Y <= 120      (2)    (maximum weekly shipping, in thousand dollars)

X + Y <= 300            (3)

X >= 0,  Y >= 0.        (4)    (non-negativity)


The problem is to maximize the objective function under the given restrictions.


The feasible domain is shown in the Figure below.

It is a quadrilateral in QI under the red, green and blue lines - factually, under the red and blue lines.



{{{graph( 330, 330, -100, 500, -100, 500,
          (851-2.3x)/3.7,  (120-0.3x)/0.4, 300-x
)}}}


Plot  2.3X + 3.7Y = 851 (red),  0.3X + 0.4Y = 120 (green)  and  X + Y <= 300 (blue)


The maximum (the solution to the problem) is achieved in one of the three corner points:

    P1 = (0,230)      (red line Y-intercept)

    P2 = (185,115)    (red line and blue line intersection point)

    P3 = (300,0)      (blue line X-intercept)


Now, calculate the value of the objective function at each of this three corner points

    at P1:  F(0,230)   = 6*0 + 8*230   = 1840 thousand dollars;

    at P2:  F(185,115) = 6*185 + 8*115 = 2030 thousand dollars;   and  

    at P3:  F(300,0)   = 6*300 + 8*0   = 1800 thousand dollars.


The maximum is achieved at P2, and this point gives the solution.


<U>ANSWER</U>.  The maximum profit is achieved when  185 standard beds and 115 luxury beds are produced per week.

         The maximum profit then is 2030 thousand dollars per week.
</pre>

Solved.


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To see other similar problems solved by the Linear Programming method, &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.


Also, &nbsp;look into the solutions on other similar problems in the archive to this forum under the links


https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1131906.html

https://www.algebra.com/algebra/homework/coordinate/word/Linear_Equations_And_Systems_Word_Problems.faq.question.1131043.html

https://www.algebra.com/algebra/homework/word/finance/Money_Word_Problems.faq.question.1129285.html

https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1128383.html

https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.1123217.html

https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1102103.html