Introduction to optimization and operations research (2019–2020) Lecturer: Michel Bierlaire, Teaching assistants: Nourelhouda Dougui, Stefano Bortolomiol Simplex (18 October 2019) Question 1: Consider the following linear optimization problem: min −2x1 + 2x2 subject to: x1 + x2 −x1 + 2x2 x1 x2 ≤5 ≤3 ≥0 ≥ 0. (1) (2) (3) (4) 1. Represent the feasible region graphically. 2. Identify the set of feasible solutions when constraints (1) and (2) are active. 3. Identify graphically all the basic solutions. 4. Identify graphically all the feasible basic solutions. 5. Identify the optimal solution using the graphical method. Question 2: Consider an optimization problem in standard form with the following data −2 −1 1 −1 1 0 2 A= , b= , c= 0 1 1 0 1 6 0 and the following simplex tableau representing a basic solution of the problem: x1 1 0 0 x2 -1 2 -3 x3 1 -1 2 1 x4 0 1 0 2 4 4 Introduction to optimization and operations research (2019–2020) Lecturer: Michel Bierlaire, Teaching assistants: Nourelhouda Dougui, Stefano Bortolomiol Simplex (18 October 2019) 1. What are the basic variables associated with this tableau? 2. What is the vertex x corresponding to this tableau? 3. What is the value of the objective function at that vertex? 4. What are the reduced costs of the non basic variables? 5. Calculate all basic directions associated with this basic solution, determine if they are feasible or not, and represent them on a graph. If the direction is feasible, what is the longest step than can be performed along the direction? Question 3: 1. Which of the following statements about a feasible basic solution is true for a minimization problem? (a) If c̄ < 0, that is, all reduced costs are negative, the solution is optimal. (b) If the values of the basic variables are negative, the solution is optimal. (c) If c̄N > 0, that is, the reduced costs of the non basic variables are positive, the solution is optimal. 2. Consider a linear minimization problem with four variables (x = (x1 , x2 , x3 , x4 )T ). In the current iteration of the simplex algorithm, we obtain the reduced costs vector c̄ = (−1, 0, 0, 0)T and the upper-right column of the simplex tableau is given by B −1 b = (2, 4, 3)T . Which of the following propositions is certainly false? (a) x1 enters the basis in the following iteration of the simplex. (b) x3 is a basic variable. (c) The current solution is not feasible. 3. Which is the feasible basic solution associated with the following simplex tableau? 2 Introduction to optimization and operations research (2019–2020) Lecturer: Michel Bierlaire, Teaching assistants: Nourelhouda Dougui, Stefano Bortolomiol Simplex (18 October 2019) x1 1 0 0 x2 4 1 2 x3 1 3 3 s1 0 1 0 s2 5 2 5 5 6 -11 (a) x1 = 5, x2 = 6, x3 = s1 = s2 = 0 (b) x3 = 5, s1 = 6, x1 = x2 = s2 = 0 (c) x3 = 5, x2 = 6, x1 = s1 = s2 = 0 (d) x1 = 5, s1 = 6, x2 = x3 = s2 = 0 Is this tableau optimal? What is the objective function value? 4. The following tableau represents a basic solution for phase I of the simplex algorithm. a1 and a2 are the auxiliary variables associated with the two constraints. x1 x2 2/3 1 -1/3 0 0 0 x3 0 1 0 x4 1/3 1/3 0 a1 0 -1 1 a2 1/3 1/3 1 6 2 0 Which of the following propositions is true? (a) There is a feasible solution for the original problem and there is no auxiliary variable in the basis. (b) There is a feasible solution for the original problem, but there are still auxiliary variables in the basis. (c) The original problem does not admit any feasible solution. Question 4: Consider the following linear optimization problem: min −3x1 + 4x2 3 Introduction to optimization and operations research (2019–2020) Lecturer: Michel Bierlaire, Teaching assistants: Nourelhouda Dougui, Stefano Bortolomiol Simplex (18 October 2019) subject to: x1 + x2 2x1 + 3x2 x1 x2 ≥4 ≤ 18 ≥0 ≥ 0. (1) (2) (3) (4) 1. Draw the feasible domain and solve the problem using the graphical method. 2. Solve the problem using the simplex algorithm in two phases. 4
