Use Simplex method to solve the following problems

1.  Min   3x + 7y

 s.t.

9x + 3y >= 36

4x + 5y >= 40

 x -  y <= 0

2x <= 6

       x, y   >= 0

2.  Max   3x + 7y

s.t.

     2x +  y <= 6

     4x + 5y <= 20

   2x      >= 7

          2y <= 7

       x, y   >= 0

3.  Max   2x + 3y

s.t.

     x + 2y >= 12

      8x + 7y >= 56

       x      <= 9

           2y >= 5               

         x, y >= 0

4.  

Urban Renewal Model:

The city of Kingsville is faced with a severe budget shortage. Seeking a long-term solution, the city council votes to improve the tax base by condemning an inner-city housing area and replacing it with a modern development.

The project involves two phases: (1) demolishing substandard houses to provide land for the new development, and (2) building the new development. The following is the situations.

A. As many as 300 substandard houses can be demolished. Each house occupies a .25-acre lot. The cost of demolishing a condemned house is $2000.

B. Lot size for new single, double, triple and quadruple-family homes (units) are .1 8, .28, .4 and .5 acre, respectively. Streets, open space, and utility easements account for 15% of available acreage.

C. In the new development the triple and quadruple units account for at least 25% of the total. Single units must be at least 20% of all units and double units at least 10%.

D. The tax per unit for single. double. triple, and quadruple units is $1.000. $1,900. $2.700, and $3,400, respectively.

E. The construction cost per unit for single, double. triple, and quadruple family homes is $50,000, $70,000, $130,000, and $160,000, respectively. Financing through a local bank can amount to a maximum of $15 million.

How many units of each type should be constructed to maximize tax collection?

Hint: use following decision variables:

s = number of single homes constructed
d = number of double homes constructed
t = number of triple homes constructed
q = number of quadruple homes constructed
de = number of substandard houses demolished.