A company produces three different products; x,y, and z. For the production of 1 unit of product x, 1 unit of A and 1 unit of B are used as input. 1 unit of A and 2 units of B are used to produce 1 unit of product y. To produce 1 unit of product z, only 1 unit of A is used. The company holds 40 units of A and 20 units of B periodically in total. By the way, the company can not produce product y more than the twice of product z.

Conversely, the sale price of 1 unit of product x is $10 , product y is $15 and product z is $12. Furthermore, the cost of 1 unit of product z is $8, product y is $9 and product z is $7.

A) Solve the above problem by using Simplex Method.

B)According to above information, what should be the company's optimal product mix to maximize its profit? Construct the problem as a LP model.