Problema Solution
A car part manufacturing plant makes steering wheels for Ford. Machine A can produce 30 steering wheels per hour at a cost of $16 per hour. Machine B can produce 40 steering wheels per hour at a cost of $22 per hour. At least 360 steering wheels must be made in each 8-hour shift to satisfy the Ford CEO. What is the least possible cost of making steering wheels in one shift? How many steering wheels will each machine produce to equal this cost?
Answer provided by our tutors
Let
a = the time in hours that Machine A works
b = the time in the hours that Machine B works
a >= 0
b >= 0
the working day is 8 hours thus we have
a <= 8
b <= 8
At least 360 steering wheels must be made in each 8-hour shift to satisfy the Ford CEO means:
30a + 40b >= 360 divide both sides by 10
3a + 4b >= 36
the least possible cost of making steering wheels in one shift is described with the objective function:
F(a, b) = 16a + 22b
We have the following constrains:
0<= a <=8
0 <= b <=8
3a + 4b >= 36
click here to see the feasible region:
The corner points are:
for a = 8 and 3a + 4b = 36 we find b = (36-3a)/4 that is b = 3
one corner point is (8, 3)
for b = 8 and 3a + 4b = 36 we find a = (36-4b)/3 that is a = 4/3
second corner point is (4/3, 8)
third corner point is (8, 8)
Evaluating the function at the corner points, we find:
F(a, b) = 16a + 22b
F(8, 3) = 16*8 + 22*3 = $194
F(4/3, 8) = 16*(4/3) + 22*8 = $197.33
F(8, 8) = 16*8 + 22*8 = $304
The objective function, which represents the possible cost , is minimized when a = 8 and b = 3.
The least possible cost of making steering wheels in one shift is $194.
Machine A will produce: 8*30 = 240 wheels
Machine B will produce: 3*40 = 120 wheels.