The Miers Company produces small engines for several manufacturers. The company receives orders from two assembly plants for their Top-flight engine. Plant 1 needs at least 45 engines, and plant 2 needs at least 32 engines. The company can send at most 140 engines to these two assembly plants. It costs $35 per engine to ship to plant 1 and $50 per engine to ship to plant 2. Plant 1 gives Miers $20 in rebates toward its products for each engine they buy, while plant 2 gives similar
$10 rebates. Miers estimates that they need $1500 in rebates to cover products they plan to buy from the two plants. How many engines should be shipped to each plant to minimize shipping costs? What is the minimum cost?
Can anyone give me a step by step guide to graphing and solving this problem? Thank you.
Sure, don't ship anything and the shipping cost is zero
Actually, since there are 140 engines and a minimum of 77 is needed, both minimums can be met. So
- Let x be the number of engines shipped to plant 1 and our solution should have x \(\displaystyle \ge\) 45.
- Let y be the number of engines shipped to plant 1 and our solution should have y \(\displaystyle \ge\) 32.
- Let S be the total shipping cost.
- Let R be the total of the rebates and R \(\displaystyle \ge\) 1500.
- Let Q be the given number of engines shipped with Q \(\displaystyle \le\) 140. Note that generally it is assumed that Q is equal to the maximum but that might not be the case here.
Thus
x \(\displaystyle \ge\) 45
y \(\displaystyle \ge\) 32
77 \(\displaystyle \le\) Q = x + y \(\displaystyle \le\) 140
S = 35 x + 50 y
R = 20 x + 10 y \(\displaystyle \ge\) 1500
First, considering x and y, Q must be at least 77 and x can be no greater than Q-32. So x lies in the interval [45, Q-32]. Since we have assumed that Q is given we can find R in terms of Q and x. Using the bounds on Q and R, you can get another bound on x. We can also find S in terms of Q and x. Minimizing S (where S'=0 or at the end points). Since S is linear in x, the minimum will be at one end or the other and thus we want x as close to that end and still satisfy the conditions.
Putting this all together we start with R = 10(x+Q). Since it costs less to ship to plant 1 than plant 2 [the derivative wrt x of Q is smaller than the derivative wrt y of Q], if we increase Q, it would be best to increase x by the same amount to minimize shipping costs. So x = x
0 + \(\displaystyle \delta\)Q and Q = Q
0 + \(\displaystyle \delta\)Q. Thus
R = 10 (x
0 + Q
0 + 2 \(\displaystyle \delta\)Q) \(\displaystyle \ge\) 1500
Picking x
0 and Q
0 to be 45 and 77 gives
\(\displaystyle \delta\)Q \(\displaystyle \ge\) 14
and
x \(\displaystyle \ge\) 45 + 14 = 59
Now
S = 35 x + 50 y = 35 x + 50 (Q - x) = 50 Q - 15 x
What is the minimum of S in the restricted interval for x? What if \(\displaystyle \delta\)Q changed?
Edit: Fix typos - hopefully all of them
