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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.
 Fall '13
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