Deterministic models / linear programming. Please help asap
1. Consider the following parametric problem:
z(θ) = min - 10x1 + 16x2 - x3
x1 - 2x2 + x3 ≤ 2 + 2θ
x1 - x2 ≤ 4 + θ
x1, x2, x3 ≥ 0,
where θ is a parameter.
(a) Solve the LP for θ = 0 and compute the optimal shadow prices.
(b) For what values of θ does the basis computed in part a) remain optimal?
(c) Solve the LP for all values of θ.2.
2. Consider the following optimization problem:
z(θ) = max (-3 + 3θ)x1 + (1 - 2θ)x2
- 2x1 + x2 ≤ 2
x1 - 2x2 ≤ 2x1 - x2 ≤ 4
x1, x2 ≥ 0.
(a) Use the parametric programming algorithm to find the optimal solution for all values of θ. Are there values of θ such that the objective is unbounded?
(b) Plot the optimal objective value as a function of θ.
(c) Graph the feasible region for the above problem (in terms of x1 and x2) and interpret the parametric algorithm on the graph.