Please help me solve this Linear Programming problem by introducing additional variables
Solving a linear programming problem by introducing additional variables. 2. Consider the LP problem: Maximize z = 3x: + 4X2 Subject to: -3X: + 5x2 S 32 2x: + X2 S 22 5x + 9x2 2 68 X1, X220 1. Consider the LP problem: Maximize z = 5x1 + 7x2 Subject to: X1 + 6x2 5 48 3X: + 2x2 5 32 2x: + X2 S 20 X1, X220 (a) Rewrite the LP problem by introducing slack or surplus variables. (a) Rewrite the LP problem by introducing slack or surplus variables. (b) Fill in the following table. In the top row put the names of all the variables; in the remaining rows, fill in all possible ways of assigning two zeros to the set of variables, then solve for the values of the remaining variables (it is convenient to write the revised system as a matrix, drop out the variables that are assigned zero, then use Gauss-Jordan (rref) to solve for the remaining variables). (b) Fill in the following table. In the top row put the names of all the variables; in the remaining rows, fill in all possible ways of assigning two zeros to the set of variables, then solve for the values of the remaining variables (it is convenient to write the revised system as a matrix, drop out the variables that are assigned zero, then use Gauss-Jordan (rref) to solve for the remaining variables). (c) Determine, by examining the values, which rows represent feasible solutions (which will be all rows with all non-negative values). Indicate those any way you like on the grid above. (c) Determine, by examining the values, which rows represent feasible solutions (which will be all rows with all non-negative values). Indicate those any way you like on the grid above. Page 1 of 4 Page 3 of 4 (d) Now that you have identified all (X1, Xa) pairs that are feasible solutions, substitute those into the objective function and determine which gives the maximum, and where that occurs. (d) Now that you have identified all (X1, X2) pairs that are feasible solutions, substitute those into the objective function and determine which gives the maximum, and where that occurs. Page 4 of 4 Page 2 of 4 Solving a linear programming problem by introducing additional variables. 2. Consider the LP problem: Maximize z = 3x: + 4X2 Subject to: -3X: + 5x2 S 32 2x: + X2 S 22 5x + 9x2 2 68 X1, X220 1. Consider the LP problem: Maximize z = 5x1 + 7x2 Subject to: X1 + 6x2 5 48 3X: + 2x2 5 32 2x: + X2 S 20 X1, X220 (a) Rewrite the LP problem by introducing slack or surplus variables. (a) Rewrite the LP problem by introducing slack or surplus variables. (b) Fill in the following table. In the top row put the names of all the variables; in the remaining rows, fill in all possible ways of assigning two zeros to the set of variables, then solve for the values of the remaining variables (it is convenient to write the revised system as a matrix, drop out the variables that are assigned zero, then use Gauss-Jordan (rref) to solve for the remaining variables). (b) Fill in the following table. In the top row put the names of all the variables; in the remaining rows, fill in all possible ways of assigning two zeros to the set of variables, then solve for the values of the remaining variables (it is convenient to write the revised system as a matrix, drop out the variables that are assigned zero, then use Gauss-Jordan (rref) to solve for the remaining variables). (c) Determine, by examining the values, which rows represent feasible solutions (which will be all rows with all non-negative values). Indicate those any way you like on the grid above. (c) Determine, by examining the values, which rows represent feasible solutions (which will be all rows with all non-negative values). Indicate those any way you like on the grid above. Page 1 of 4 Page 3 of 4 (d) Now that you have identified all (X1, Xa) pairs that are feasible solutions, substitute those into the objective function and determine which gives the maximum, and where that occurs. (d) Now that you have identified all (X1, X2) pairs that are feasible solutions, substitute those into the objective function and determine which gives the maximum, and where that occurs. Page 4 of 4 Page 2 of 4
