[SOLVED] CS Problem 2 (75 points)

Problem 2 (75 points)
Consider the following linear programming problem P:

Maximize z =x1+2 x2

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Subject tox1 +x2 29(1)
x1 +x2 19(2)
x1 +42 16(3)
____________________________________
x2 0
x1 unconstrained in sign

Let the slack of constraint (1) and (2) be x3 and x4, respectively, and the surplus of constraint (3) be x5. Answer the following independent questions:
1Solve the problem graphically:
Identify the feasible region by its corner points (coordinates x1 and x2 ) and shade it. Find the optimal point on the graph and write the optimal values of the variables and z, below.

2 Determine the optimal solution, if instead of maximization the objective was minimization.

3 Write a maximization objective function that has multiple optima on the feasible region of Problem P.

4Determine the range of values of b3 (the rhs of constraint (3), whose current value is 16) that renders Problem P infeasible.

5Consider decreasing b3 from its current value of 16. Find the critical value of b3 beyond which constraint (3) becomes redundant.

6 What is the optimal solution of Problem Pif constraint (3) is removed from the formulation? What about the optimal solution if constraint (1) is removed from the formulation?

7For each one of the three points given by their coordinates (x1, x2) below, determine if the point is feasible, infeasible, and if a constraint is binding at that point (if infeasible, indicate the violated constraint(s) and if binding indicate which constraint is binding). Justify your answer by checking each constraint.

8Construct the initial basic solution by adding artificial variables and making the necessary variable transformations so that you can apply the Big-M method to Problem P. Fill-in the (0) iteration tableau. Indicate the entering and leaving variable and perform a single Big-M iteration (1). Write the resulting basic solution (all variables with values) of iteration (1) and indicate whether it is feasible or infeasible to Problem P. Indicate on the graph the point this basic solution corresponds to and the constraints (if any) that are violated.

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