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Exam I and Key Feb 16, 2005 MAP 5326 S Hudson
Name
Show all your work. Use the space provided, or leave a note. Don’t use a calculator or your own extra paper. Problems 1-4 refer to this final tableaux arising from some linear programming problem with non-negative decision variables x 1 , x 2 , x 3 and three ”≤” constraints.
Basic Eqn Z x 1 x 2 x 3 x 4 x 5 x 6 RHS
Z (0) 1 0 0 0 1 0 1 7
x 1 (1) 0 1 0 0 -4 -5 5 3 x 3 (2) 0 0 0 1 3 1 -1 2 x 2 (3) 0 0 1 0 6 5 -5 6
Explain why this problem has multiple solutions.
Find at least two optimal solutions. For extra-credit, give a formula including all optimal solutions.
Choose one of your solutions, and find its complementary basic solution (for the dual problem).
What is the value of W ∗^ for the dual problem?
For questions 5-7), consider this problem: Maximize Z = 3x 1 + 2x 2 subject to
2 x 1 + x 2 ≤ 6 x 1 + 2x 2 ≤ 6 and x 1 ≥ 0 and x 2 ≥ 0.
Graph the feasible region and find all the CPF solutions. Solve the problem (any method is OK, but show/explain your work).
Find the allowable increase in b 2 = 6.
State the dual problem.
Answer True or False and explain briefly. Assume “the problem” is a linear programming one, but do not assume it has a unique optimal solution.
Suppose there are three decision variables and the constraints include 0 ≤ x 1 ≤ 100 and 0 ≤ x 2 ≤ 100 and 0 ≤ x 3 ≤ 100. Then the problem must have an optimal solution.
The best CPF solution is always an optimal solution.
Only CPF solutions can be optimal.
In the Wyndor example, since the second constraint, 2x 2 ≤ 12, has a positive shadow price, it is a binding constraint.
In the R.S.M., a formula for part of row zero is cBB−^1 A − c.
- Consider this problem: Minimize Z = 8x 1 + 4x 2 + 6x 3 + 3x 4 + 9x 5 subject to
x 1 + 2x 2 + 3x 3 + 3x 4 ≤ 180 4 x 1 + 3x 2 + 2x 3 + x 4 + x 5 ≤ 270 x 1 + 3x 2 + x 4 + 3x 5 ≤ 180 xi ≥ 0.
You are told that x 3 , x 1 , and x 5 are basic in the optimal solution and that
− 1
=
Find the optimal solution.
- Consider this problem: Minimize Z = 3x 1 + 2x 2 + 4x 3 subject to
2 x 1 + x 2 + 3x 3 = 60 3 x 1 + 3x 2 + 5x 3 ≥ 120 xi ≥ 0.
Start to solve this using the Big-M method in tableaux form. You can stop in the very first iteration, after you have found (and stated) which variable enters and which leaves.
- The World Light Co. makes two light fixtures (products 1 and 2) that require both metal frame parts and electrical components. Management wants to determine how many units of each product to produce so as to maximize profit. For each unit of product 1, 1 unit of frame parts and 2 units of electrical components are required. For each unit of product 2, 3 units of frame parts and 2 units of electrical components are required.The company has 200 units of frame parts and 300 units of electrical components. Each unit of product 1 gives a profit of 1 dollar, and each unit of product 2, up to 60 units, gives a profit of 2 dollars. Any excess over 60 units brings no profit, so this has been ruled out.
a) Formulate a linear programming model for this problem.
b) Solve it graphically. What is the maximal profit?
