Optimization/Linear Programming Question: Please answer parts D,E,F

5. While solving some linear programming problem, we encounter the following dictionary after some iterations (altogether 7 variables and 3 constraints). xi = 9-0.5T2-1.25a4-T5-0.5r7 Zg = 15-2.522-2.25x4-x5-0.5r7 x3-4-0.5r2-0.25r4-0.5r7 z 22 + 1.5x2 + 1.25x4-225-1.517 = (a) Write down (a representation of) the linear programming problem that is being op- timized. Also, write the linear programming problem in matrix form with the usual notation for the c, b, x vectors and the A matrix. (b) What is the basic solution described by the dictionary? Give the values of all variables and the obiective function (c) Without performing an iteration, write down the solution that would be obtained if r were to enter the basis, and the usual rule is used to determine the exiting variable. Note: The simplex method will not make this choice because increasing rs will lower the objective value.) (d) What variable should enter the basis to obtain the greatest total increase in the objec- tive at the next iteration? e) Define the largest coefficient rule as follows: among all non-basic variables that are candidates to enter the basis, pick one with the largest coefficient in absolute value. Suppose this rule is used to determine the entering variable in the simplex method what variables will enter and exit the basis at the next iteration? (f) If a7 enters the basis, which variable must leave the basis? Perform an iteration and show the next dictionary. 5. While solving some linear programming problem, we encounter the following dictionary after some iterations (altogether 7 variables and 3 constraints). xi = 9-0.5T2-1.25a4-T5-0.5r7 Zg = 15-2.522-2.25x4-x5-0.5r7 x3-4-0.5r2-0.25r4-0.5r7 z 22 + 1.5x2 + 1.25x4-225-1.517 = (a) Write down (a representation of) the linear programming problem that is being op- timized. Also, write the linear programming problem in matrix form with the usual notation for the c, b, x vectors and the A matrix. (b) What is the basic solution described by the dictionary? Give the values of all variables and the obiective function (c) Without performing an iteration, write down the solution that would be obtained if r were to enter the basis, and the usual rule is used to determine the exiting variable. Note: The simplex method will not make this choice because increasing rs will lower the objective value.) (d) What variable should enter the basis to obtain the greatest total increase in the objec- tive at the next iteration? e) Define the largest coefficient rule as follows: among all non-basic variables that are candidates to enter the basis, pick one with the largest coefficient in absolute value. Suppose this rule is used to determine the entering variable in the simplex method what variables will enter and exit the basis at the next iteration? (f) If a7 enters the basis, which variable must leave the basis? Perform an iteration and show the next dictionary