Farmer Leary grows wheat and corn on his 45-acre farm. He can sell at most 140 b
ID: 426936 • Letter: F
Question
Farmer Leary grows wheat and corn on his 45-acre farm. He can sell at most 140 bushels of wheat and 120 bushels of corm. Each acre planted with wheat yields 5 bushels of wheat, and each planted with corn yields 4 bushels of corn. Wheat sells for $30 per bushels, and corn sells for $50 per bushel. To harvest an acre of wheat requires 6 hours of labor; 10 hours are needed to harvest an acre of corn. Up to 350 hours of labor can be purchased at $10 per hour. Let AI = acres planted with wheat, A2- acres planted with corn; and L hours of labor that are purchased. To maximize the profits. Leary should solve the following LP: 4. Max Z-150A1 + 200A2-10L s.t. Al+A2 S45 6A1+ 10A2-L 0Explanation / Answer
a) If everything remains constant and the area of land available changes from 45 to 40, then the area for wheat and corn will change. This means that the value of A1 and A2 will change too. The reduction in constraint is within the allowable limit and hence the basis will not change but the value of the optimal solution will change. Now we can see that the dual prices for constraint number 2 is 75. This means for every tightening of constraint by 1 unit, our optimal solution will drop by 75 units. Currently the optimal solution is 4250 but it will drop by 5*75 = 375 and become 3875.
Now there may be more impact on this due to the changing value of A1, A2 and L. However, those information are not presented. That can only be obtained by running the solution with new constraints.
b) The objective coefficient is now a product of number of bushels and price of wheat. If the price changes to $26, then the coefficient will change from 150 to 130. Now the coefficient’s allowable decrease is up to 30 and the basis of the LP will not change. However, the optimal function will change by 20 times the A1 value 20*25 = 500. The optimal solution will become 4250-500 = 3750
c) Using the slack portion we can see that there is a surplus of 15 for wheat (constraint 5). This means we could change the value of the allowable decrease up to 15 and not impact the optimal solution. So, if we change the value from 140 to 130, it will not impact the optimal solution.
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