INEN 622: Group Quiz 1

 

 

1.     (10pts) A farmer, George W. Carver, with 350 acres of farmland and a budget of $25,000 for wages of laborers for the next season is trying to determine what he should plant.  He pays laborers $8 per hour.  Other relevant information is provided in the following table.

 

Crop

Hours of labor/acre

Yield bushels/acre

Profit/bushel

Corn

2

80

50 cents

Oats

2

60

30 cents

Flax

5

50

60 cents

Wheat

7

60

70 cents

Soybeans

9

55

90 cents

  

     To feed his own livestock, the farmer must plant 45 acres of corn and oats (in any combination).  He has also made arrangements to sell the produce of 20 acres devoted to flax and 30 acres devoted to wheat.  His farm equipment can handle at most 100 acres of soybean.  Formulate the farmer’s land allocation problem as a linear program to maximize the net profit.

 

 

2.     (10 pts) Give the auxillary problem for the following LP (standard form) and complete one iteration of the simplex method on initial dictionary for the auxillary problem:

 

Min    4x1+ 4x2 + x3

s.t.     x1 + x2  + x3 ≤ 2

          2x1 + x2 ≤ 3

2x1 + x2 + 3x3  ³ 3

          x1, x2, x3 ³ 0

3.     (10 points)  Use the optimal dictionary for the auxillary problem in #2, given below, to solve the original problem of #2.

 

 

 

x1 = 1.5 - 0.5 x2 - 1.5 x3 + 0.5 x6 – 0.5 x0

x4 = 0.5 - 0.5 x2 + 0.5 x3 - 0.5 x6 + 1.5 x0

x5 = 0.0 +                 3x3       - x6 +      2x0

z  = 0.0 – x0

 

 

 

4.     (10pts) Show that setting x1, x2 and x4 as basic variables and the other variables as nonbasic corresponds to a basic feasible solution for the LP below.  Find the value of the corresponding BFS and perform one iteration of the revised simplex method using the given BFS.

 

 

Max   2x1- x2 + x3

s.t.     3x1 + x2  + x3 £ 60

          x1 - x2  + 2x3 £ 10

x1 +  x2 – x3 £ 20

          x1, x2 , x3 ³ 0

 

 

5.     (10 pts) Solve the following LP by the graphical method

 

Max   5x1 - x2

s.t.     2x1 + x2 = 6

          x1 + x2 £ 4

x1 + 2x2 £ 5

                           x1, x2, ³ 0

 

 

 

 

 

6.  (10 pts) The Rosa Parks Bus Company believes that it will need the following number of bus drivers during each of the next five years:  year 1, 60 drivers; year 2, 70 drivers; year 3, 50 drivers; year 4, 65 drivers; year 5, 75 drivers.  On January 2nd of each year the company must decide how many drivers should be hired or fired.  A driver is only hired or fired on January 2nd.  Thus, all drivers hired for the year will receive their full salaries.  It costs $4,000 to hire a driver and $2,000 to fire a driver.  A driver’s yearly salary is $25,000 per year.  At the beginning of year 1 the company has 50 drivers.  Fractional drivers can exist.  Formulate an LP for this problem to minimize cost.