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≤5(1)
−x1+ 2x2≤3(2)
x1≥0(3)
x2≥0.(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
A=1−1 1 0
1 1 0 1, b =2
6, c =
−2
−1
0
0
and the following simplex tableau representing a basic solution of the prob-
lem:
x1x2x3x4
1 -1 1 0 2
0 2 -1 1 4
0 -3 2 0 4
1
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 xcorresponding 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, deter-
mine 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 ¯c < 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 ¯cN>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 ¯c= (−1,0,0,0)Tand the upper-right column of the sim-
plex tableau is given by B−1b= (2,4,3)T. Which of the following
propositions is certainly false?
(a) x1enters the basis in the following iteration of the simplex.
(b) x3is a basic variable.
(c) The current solution is not feasible.
3. Which is the feasible basic solution associated with the following sim-
plex tableau?
2
Introduction to optimization
and operations research (2019–2020)
Lecturer: Michel Bierlaire, Teaching assistants: Nourelhouda Dougui, Stefano Bortolomiol
Simplex (18 October 2019)
x1x2x3s1s2
1 4 1 0 5 5
0 1 3 1 2 6
0 2 3 0 5 -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. a1and a2are the auxiliary variables associated
with the two constraints.
x1x2x3x4a1a2
2/3 1 0 1/3 0 1/3 6
-1/3 0 1 1/3 -1 1/3 2
0 0 0 0 1 1 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≥4(1)
2x1+ 3x2≤18 (2)
x1≥0(3)
x2≥0.(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.
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