Solve the following linear programming problems 20 Business

Solve the following linear programming problems.

20. Business: The Miers Company produces small engines for several manufacturers. The company receives orders from two assembly plants for its engine. Plant I needs at least 50 engines, and plant II needs at least 27 engines. The company can send at most 105 engines to these two assembly plants. It costs $ 20 per engine to ship to plant I and $ 35 per engine to ship to plant II. Plant I gives Miers $ 15 in rebates toward its products for each engine Miers buys, while plant II gives similar $ 10 rebates. Miers estimates that it needs at least $ 1200 in rebates to cover products it plans to buy from the two plants. How many engines should be shipped to each plant to minimize ship-ping costs? What is the minimum cost?

Solution

Let x = the number of engines shipped to plant 1
Let y = the number of engines shipped to plant 2

Plant 1 needs at least 45 engines.
x > 50
Plant 2 needs at least 32 engines.
y > 27

The company can send at most 105 engines to these two assembly plants
x + y <= 105


It cost $20 per engine to ship plant 1 and $35 per engine to ship plant 2
20x represents dollars to ship x engines to plant 1
35y represents dollars to ship y engines to plant 2
The sum of these two expressions represents the total shipping cost.
Total shipping cost=20x + 35y

Plant 1 gives Miers $15 in rebates.
15x represents dollars rebated from plant 1
0y represents dollars rebated from plant 2
Miers estimates that they need $1200 in rebates to cover products they plan to buy from the two plants.
15x + 0y >= 1200 (note the rebate line will be out of the solution area, the graph will show it)
now
Graph the four inequalities above, and then evaluate Total shipping cost at the vertex points of the solution area.
http://img341.imageshack.us/img341/6620/...
x > 50
y > 27
x + y <= 105
15x >= 1200
Total shipping cost=20x + 35y
The nearest solution to have 15x >= 1200
Now, you test each corner solution:
(50,55) Total shipping cost=20(50) + 35(55)=2925 with Rebate=15(50)=750
(50,27) Total shipping cost=20(50) + 35(27)=1945 with Rebate=15(50)=750
(78,27) Total shipping cost=20(78) + 35(27)=2505 with Rebate=15(78)=1170
non of the point satisfy the rebates condition so we will go with the less cost
(50,27)
Thus
50 engines should be shipped to plant 1
27 engines should be shipped to plant 2
Total shipping cost(minimum cost)= 1945
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just to make things right I will assume
Plant 2 gives Miers $10 in rebates.
10y represents dollars rebated from plant 2
Plant 1 gives Miers $15 in rebates.
15x represents dollars rebated from plant 1
15x + 10y >= 1200
now
Graph the four inequalities above, and then evaluate Total shipping cost at the vertex points of the solution area.
http://img825.imageshack.us/img825/7858/...
x > 50
y > 27
x + y <= 105
15x + 10y >= 1200
Total shipping cost=20x + 35y
Now, you test each corner solution:
(50,55) Total shipping cost=20(50) + 35(55)=2925 with Rebate= 15(50)+10(55)= 1300
(78,27) Total shipping cost=20(78) + 35(27)=2505 with Rebate= 15(78)+10(27)= 1440
(62,27) Total shipping cost=20(62) + 35(27)=2185 with Rebate= 15(62)+10(27)= 1200
(50,45) Total shipping cost=20(50) + 35(45)=2575 with Rebate= 15(50)+10(45)= 1200
Thus
the best solution
(62,27) Total shipping cost=2185 with Rebate=1200
62 engines should be shipped to plant 1
27 engines should be shipped to plant 2
Total shipping cost(minimum cost)= 2185 to have rebate of 1200