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Unformatted text preview: ll are given in the
following table:
Radio A
Pierre
1h
Paul
2h
Jacques
1h
Selling prices 15 euros Radio B
2h
1h
3h
10 euros We assume that the company has no problem to sell its production, whichever it is.
a) Model the problem of ﬁnding a weekly production plan maximizing the revenue of RadioIn as a linear programme. Write precisely what are the decision variables, the objective
function and the constraints.
b) Solve the linear programme using the geometric method and give the optimal production
plan.
Exercise 9.7. The following table shows the different possible schedule times for the drivers
of a bus company. The company wants that at least one driver is present at every hour of the
working day (from 9 to 17). The problem is to determine the schedule satisfying this condition
with minimum cost.
Time 9 – 11h
Cost
18 9 – 13h
30 11 – 16h 12 – 15h 13 – 16h 14– 17h
38
14
22
16 16 – 17h
9 Formulate an integer linear programme that solves the company decision problem.
Exercise 9.8 (Chebyshev approximation). Data : m measures of points (xi , yi ) ∈ Rn+1 , i =
1, ..., m.
Objective: Determine a linear approximation y = ax + b minimizing the largest error of approximation. The decision variables of this problem are a ∈ Rn and b ∈ R. The problem may be 148 CHAPTER 9. LINEAR PROGRAMMING formulated as:
min z = max {yi − axi − b}.
i=1,...,m It is unfortunately not under the form of a linear program. Let us try to do some transformations.
Questions:
1. We call MinMax problem the problem of minimizing the maximum of a set of numbers:
min z = max{c1 x, ..., ck x}.
How to write a MinMax problem as an LP?
2. Can we express the following constraints
x ≤ b
or
x ≥ b in a LP (that is without absolute values)? If yes, how?
3. Rewrite the problem of ﬁnding a Chebyshev linear approximation as an LP. 9.4.2 Simplex Exercise 9.9. Solve with the Simplex Method the following problems:
a. Maximize 3x1 + 3x2
Subject to:
x1 + x2
2x1
2x1 + x2 + 4x3
+ 2x3
+ 3x3
+ 3x3
x1 , x2 , x3 ≤
≤
≤
≥ 4
5
7
0 9.4. EXERCICES 149 b.
Maximize 5x1 + 6x2 + 9x3 + 8x4
Subject to:
x1 + 2x2 + 3x3 + x4 ≤ 5
x1 + x2 + 2x3 + 3x4 ≤ 3
x1 , x2 , x3 , x4 ≥ 0
c.
Maximize 2x1
Subject to:
2x1
x1
2x1
4x1 + x2 + 3x2
+ 5x2
+ x2
+ x2
x1 , x2 ≤
≤
≤
≤
≥ 3
1
4
5
0 Exercise 9.10. Use the Simplex Method to describe all the optimal solutions of the following
linear programme:
Maximize 2x1 + 3x2 + 5x3 + 4x4
Subject to:
x1 + 2x2 + 3x3 + x4 ≤ 5
x1 + x2 + 2x3 + 3x4 ≤ 3
x1 , x2 , x3 , x4 ≥ 0
Exercise 9.11. Solve the following problems using the simplex method with two phases.
a.
Maximise
Subject to: 3x1 + x2
x1 − x2 ≤ −1
−x1 − x2 ≤ −3
2x1 + x2 ≤
4
x1 , x2 ≥ 0 b.
Maximise
Subject to: 3x1 + x2
x1 − x2 ≤ −1
−x1 − x2 ≤ −3
2x1 + x2 ≤
2
x1 , x2 ≥ 0 150 CHAPTER 9. LINEAR PROGRAMMING c. 3x1 + x2 Maximise
Subject to: x1 − x2 ≤ −1
−x1 − x2 ≤ −3
2x1 − x2 ≤
2
x1 , x2 ≥ 0 9.4.3 Duality Exercise 9.12. Write the dual of the following linear programme.
Maximize
Subject to: 7x1 + x2 4...
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 Fall '13
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