For each system of linear equations and/or linear inequalities, graph the system and identify the solution set if there is one:
(a)
8x1 + 16x2 = 0
2x1 + 4x2 = 0
2x1 + 3x2 = 1
(b)
x1 - x2 = -6
3x1 - 3x2 = 9
(c)
-2x1 + x2 ≤ 30
x1 + 2x2 ≥ 100
(d)
5x1 + 3x2 ≥ 25
-3x1 + 7x2 ≤ 15
(2) Consider the linear program:
Maximize: Z = -5x1 - 5x2
Subject to :
x1 + 3x2 ≤ 8
x1 + x2 ≤ 4
x1 ≥ 0
x2 ≥ 0
(a) Graph the linear program.
(b) Identify all corner point solution(s), if they exist, on your graph.
(c) If there are multiple solutions, indicate that on your graph, or indicate that there are no solutions or one solution.
(d) Identify the optimal solution, if it exists, on your graph.
(e) Determine the optimal value for Z, if an optimal solution exists.
(3) Consider the linear program:
Maximize: Z = 5x1 + 7x2
Subject to :
2x1 + 3x2 ≥ 42
3x1 + 4x2 ≥ 60
x1 + x2 ≥ 18
x1 ≥ 0
x2 ≥ 0
(a) Use the graphical method to solve this problem. Identify all the corner points on the graph.
(b) For each CPF solution, identify the pair of constraint boundary equations it satisfies.
(c) For each CPF solution, identify its adjacent CPF solutions.
(d) Calculate Z for each CPF solution. Use this information to identify an optimal solution.
(e) Describe graphically what the simplex method does step by step to solve the problem.
(4) Suppose that the following constraints have been provided for a linear programming model with decision variables x1 and x2:
-x1 + 3x2 ≤ 30
-3x1 + x2 ≤ 30
x1 ≥ 0
x2 ≥ 0
(a) Demonstrate graphically that the feasible region is unbounded.
(b) If the objective is to maximize Z = -x1 + x2, does the model have an optimal solution? If so, find it. If not, explain why not.
(c) Repeat part (b) when the objective is to maximize Z = x1 - x2.
(d) For objective functions where this model has no optimal solution, does this mean that there are no good solutions according to the model? Explain. What probably went wrong when formulating the model?
(5) Consider the following problem:
Maximize: Z = 2x1 + 3x2
Subject to :
-3x1 + x2 ≤ 1
4x1 + 2x2 ≤ 20
4x1 - x2 ≤ 10
-x1 + 2x2 ≤ 5
x1 ≥ 0
x2 ≥ 0
(a) Develop a table giving each of the CPF solutions and the corresponding defining equations, BF (basic feasible) solution, and nonbasic variables. Calculate Z for each of these solutions, and use just this information to identify the optimal solution.
(b) Develop the corresponding table for the corner-point infeasible solutions, etc. Also identify the sets of defining equations and nonbasic variables that do not yield a solution.