introduction-lp-duality1

2 put under the standard form p4 minimize 8x1 9x2 2x3

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Unformatted text preview: . Finally, c j denotes the benefit (the price) of a unit of product j. Hence, by checking the units of measure in the constraints ∑ ai j yi ≥ c j , the variable yi must represent a benefit per unit of resource i. Somehow, the variable yi measures the unitary value of the resource i. This is illustrated by the following theorem the proof of which is omitted. Theorem 9.13. If Problem 9.5 admits a non degenerated optimal solution with value z∗ , then there is ε > 0 such that, for any |ti | ≤ ε (i = 1, . . . , m), the problem Maximize ∑n= c j x j j Subject to ∑n=1 ai j x j ≤ bi + ti (i = 1, . . . , m) j xj ≥ 0 ( j = 1, . . . , n) admits an optimal solution with value z∗ + ∑m 1 y∗ti , where (y∗ , . . . , y∗ ) is the optimal solution m i= i 1 of the dual of Problem 9.5. Theorem 9.13 shows how small variations in the amount of available resources can affect the benefit of the company. For any unit of extra resource i, the benefit increases by y∗ . Sometimes, i y∗ is called the marginal cost of the resource i. i In many networks design problems, a clever interpretation of dual variables may help to achieve more efficient linear programme or to understand the problem better. 146 9.4 9.4.1 CHAPTER 9. LINEAR PROGRAMMING Exercices General modelling Exercise 9.1. Which problem(s) among P1, P2 and P3 are under the standard form? P1 : Maximize 3x1 Subject to: 4x1 6x1 x1 − + − + 5x2 5x2 6x2 8x2 x1 , x2 ≥3 =7 ≤ 20 ≥0 P2 : Minimize 3x1 + x2 + 4x3 + x4 + 5x5 Subject to: 9x1 + 2x2 + 6x3 + 5x4 + 3x5 ≤ 5 8x1 + 9x2 + 7x3 + 9x4 + 3x5 ≤ 2 x1 , x2 , x3 , x4 ≥ 0 P3 : Maximize Subject to: 8x1 − 4x2 3x1 + x2 ≤ 7 9x1 + 5x2 ≤ −2 x1 , x2 ≥ 0 Exercise 9.2. Put under the standard form: P4 : Minimize −8x1 + 9x2 + 2x3 − 6x4 − 5x5 Subject to: 6x1 + 6x2 − 10x3 + 2x4 − 8x5 ≥ 3 x1 , x2 , x3 , x4 , x5 ≥ 0 Exercise 9.3. Consider the following two problems corresponding to Problems 9.2 and 9.3 of the course. Prove that the first one is unfeasible and that the second one is unbounded. Maximize Subject to: 3x1 − x2 x1 + x2 ≤ 2 −2x1 − 2x2 ≤ −10 x1 , x2 ≥ 0 Maximize x1 − x2 Subject to: −2x1 + x2 ≤ −1 −x1 − 2x2 ≤ −2 x1 , x2 ≥ 0 Exercise 9.4. Find necessary and sufficient conditions on the numbers s and t for the problem P5 : Maximize Subject to: x1 + x2 sx1 + tx2 ≤ 1 x1 , x2 ≥ 0 9.4. EXERCICES 147 a) to admit an optimal solution; b) to be unfeasible; c) to be unbounded. Exercise 9.5. Prove or disprove: if the problem (9.1) is unbounded, then there exists an index k such that the problem: Maximize xk n Subject to: ∑ j=1 ai j x j ≤ bi xj ≥ 0 for 1 ≤ i ≤ m for 1 ≤ j ≤ n is unbounded. Exercise 9.6. The factory RadioIn builds to types of radios A and B. Every radio is produced by the work of three spets Pierre, Paul and Jacques. Pierre works at most 24 hours per week. Paul works at most 45 hours per week. Jacques works at most 30 hours per week. The resources necessary to build each type of radio and their selling prices as we...
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This note was uploaded on 11/20/2013 for the course CS 101 taught by Professor Smith during the Fall '13 term at Mitchell Technical Institute.

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