Consider the following resource allocation problem and the accompanying optimal tableau (14, 15, and re are slack variables). Z X maximize z = 4x + 6x + 5x3 subject to Z X1 0 3x + x + 2x3 s 64 x + x + x = 20 x + 2x + 3x3 = 30 X1, X2, X30 X2 0 1 0 0 0 0 0 1 0 1 0 XJ X 3 0 3 1 2 0 0 (Profit S) XS (Resource 1) (Resource 2) (Resource 3) 2 -S 2 2 2 I -1 RHS 100 24 10 10 (a) (8) Determine the range on the unit profit of Product 1 (c) for which the current basis remains optimal. (b) (8) What would be the new optimal solution if c was 8? Comment on the new solution. (c) (12) A new product is proposed. It is estimated that each unit of this new product consumes 4 units of Resource 1, 2 units of Resource 2, and 1 unit of Resource 3. What should be the unit profit of this new product for it to be profitable to manufacture? (d) (10) Suppose that the production requirements for Product 3 are modified such that each unit of product 3 now requires 2 units of resource 1, 1 unit of Resource 2, and 1 unit of resource 3. Find the new optimal solution.

2.2. (38 points, 40 minutes) Consider the following resource allocation problem and the accomparying optimal tableau (x4,x5, and x6 are slack variables). maximizez=4x1+6x2+5x1subjectto3x1+x2+2x364x1+x2+x320x1+2x2+3x330x1,x2,x10 (Profit \$) (Resource I) (Resource 2) (Resource 3) (a) (8) Determine the range on the unit profit of Product 1(c1) for which the current basis remains optimal. (b) (8) What would be the new optimal solution if c1 was 8 ? Comment on the new solution. (c) (12) A new product is proposed. It is estimated that each unit of this new product consumes 4 units of Resource 1,2 units of Resource 2, and 1 unit of Resource 3. What should be the unit profit of this new product for it to be profitable to manufacture? (d) (10) Suppose that the production requirements for Product 3 are modified such that each unit of product 3 now requires 2 units of resource 1,1 unit of Resource 2, and 1 unit of resource 3. Find the new optimal solution