SOLUTION: 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. T

Question 1175634: 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?
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
x = number of items of A.
y = number of items of B.

labor = 3x + 4y <= 3000 minutes per day.
raw materials = 2x + y <= 1000 dollars per day.

profit is equal to 5x + 4y dollars.
this is your objective function.
it's what you want to maximize.

you can solve this graphically using the desmos.com calculator.

using that calculator, you graph the opposite of the inequalities.
the feasible region is the area of the graph that is not shaded.

your constraint inequalities are:

3x + 4y <= 3000
2x + y <= 1000
x >= 0
y >= 0

you are graphing the opposite of these inequalities.
you are evaluating the objective functio0n at the corner points of the feasible region.
the feasible region is the area on the graph that is not shaded.

evaluation of the corner points yields the maximum profit at (200,600).
the maximum profit is 3400 dollars.
all the constraint inequalities are satisfied at this point.

the graph looks like this.

i also used a simplex method tool that provide the same answer.

the results of using that tool are shown below.

the calculator can be found at https://www.desmos.com/calculator

the simplex method tool can be found at https://www.zweigmedia.com/RealWorld/simplex.html

Answer by ikleyn(52858)   (Show Source): You can put this solution on YOUR website!
.
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?
~~~~~~~~~~~~~~~

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.

Solved.

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In this site, there is a lesson
    - Solving minimax problems by the Linear Programming method
which explains, for beginners, metodology of solving such problems in more details.

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