show all work A manufacturing company receives orders for en

show all work

A manufacturing company receives orders for engines from two assembly plants. Plant l eeds at least 45 engines, and plant ll needs at least 32 engines. The company can send at most 120 engines to these assembly plants. It costs $30 per engine to ship to plant land $50 per engine to ship to plant ll. Plant l gives the manufacturing company $20 in rebates toward its products for each engine they buy, while plant ll gives similar $15 rebates. The manufacturer estimates that they need at least $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? How many engines should be shipped to each plant to minimize shipping costs, subject to the given constraints? The number of engines to send to plant l is The number of engines to send to plant ll is What is the minimum shipping cost, subject to the given constraints?

Solution

Let the number of engines shipped by the manufacturing company to the Plant I and the Plant II be x and y respectively. Then x 45 and y 32. The rebate received by the manufacturing company is 20x + 15 y. If the rebates are to be atleast $ 1500, then 20x + 15y 1500 or, 4x + 3y 300. Since the total number of engines that can be shipped to the assembly plants is 120, we have x + y 120. The manufacturing

Also, the shipping cost is 30x + 50y which will be lowest when y is lowest possible number ( as the coefficient of y is larger than that of x). which is 32. Then the shipping cost is 30x + 50*32 = 30x + 1600. However, we also must have 4x + 3y 300. When y = 32, we have 4x +3*32 300 or, 4x +96 300 or, 4x 300 -96 or, 4x 204 so that x 51. When x = 51, the shipping cost is 30*51 + 1600 = 1530 + 1600 = $ 3130. Also, then x + y = 51 + 32 = 83 ( and 83 < 120).Thus, when 51 engines are shipped to the Plant I and 32 engines are shipped to the Plant II, all the given conditions are satisfied and the shipping cost is minimum ( $ 3130).