introduction-lp-duality1

942 simplex exercise 99 solve with the simplex method

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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 Min-Max problem the problem of minimizing the maximum of a set of numbers: min z = max{c1 x, ..., ck x}. How to write a Min-Max 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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