Consider the following linear programming problem P

Minimize

z=x_{1}+x_{2}

subject to

x_{1}+2x_{2}=10

2x_{1}+x_{2}\ge2

x_{2}\ge0

xunconstrained x_{1} in sign

Let the surplus of constraint (2) be x). Answer the following independent questions:

1. Solve the problem graphically. Identify the feasible region by its comer points (coordinatesx and x) and edges, and shade it. Find the optimal point on the graph and write the optimalvalues of the variables and below.

{x_{l}}^{*}=

x^{*}=

xy^{*}=

z^{*}=

Answer the following independent questions:

(a) Determine the optimal solution if instead of minimization the objective was maximization.

(b) Determine the largest possible value of the RHS of constraint (2) for which problem is still feasible. Note that the current RHS value of constraint (2) is 2.

(e) Determine the optimal solution if the direction of the inequality of constraint (2) is opposite (5)?

(d) Write an objective function that has multiple optima on the feasible region of problem P

3. (a) Construct 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 7

(b) Set up Big M method iteration (0) tableau.

(c) Indicate the entering and leaving variable and perform a single iteration. Write the resulting basic solution (all variables with values) of iteration (1).

(d) Identify and mark the points associated with the iteration (0) and iteration (1) tableaux on the graph you constructed in Question 1 above. Indicate whether these two points are feasible or infeasible to problem and the constraints (if any) that are violated? Are these points corner points of the feasible region?