CS 600 Test II

November 9, 1998

1. Find an initial basic feasible solution for the following linear programming problem, and give the initial dictionary for the problem. [20 points]

Max z = x₁ + 3x₂
Subject to
$-x_{1} - x_{2} \leq -2$
$x_{1} - x_{2} \leq -1$
$2x_{1} + x_{2} \leq 4$
$x_{1} \geq 0$, $x_{2} \geq 0$.

2 (a) Find the dual of the following linear programming problem:

Min z = 8x₁ + 3x₂ + 9x₃ + 3x₄
Subject to
$x_{1} - x_{2} + 3x_{3} + 2x_{4} \geq 5$
$2x_{1} + x_{2} + x_{3} - 2x_{4} \geq 8$
$x_{i} \geq 0$ for i = 1, ..., 4. [20 points]

(b) Graphically solve the dual of (a) [20 points]

3. Use the separable programming technique to formulate an approximate linear programming model for the following problem. Use x₁,x₂ = 0, 1, 2, 3as the breakpoints of the piecewise linear functions.

Max z = 4x₁ + 4x₂ - x₁² - x₂³
Subject to
$2x_{1} + 5x_{2} \leq 14$
$3x_{1} + x_{2} \leq 8$
$x_{1} \geq 0$, $x_{2} \geq 0$. [20 points]

(b) Find an initial feasible solution for the approximation problem.
[20 points]