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Unformatted text preview: we obtain
m m ∑ biyi − ∑ i=1 n n ∑ ai j x j yi + ∑ i=1 j=1 m n m n j=1 i=1 j=1 ∑ ai j yix j − ∑ c j x j = ∑ biyi − ∑ c j x j = 0. j=1 i=1 144 CHAPTER 9. LINEAR PROGRAMMING Therefore, Inequalities 9.24 and 9.25 must be equalities. As the variables are positive, we
further get that
for all i, and for all j, n bi − ∑ ai j x j yi = 0 j=1 m ∑ ai j yi − c j i=1 x j = 0. A product is equal to zero if one of its two members is null and we obtain the desired result.
Theorem 9.12. A feasible solution (x1 , . . . , xn ) of Problem 9.5 is optimal if and only if there is
a feasible solution (y1 , . . . , ym ) of Problem 9.6 such that:
xj > 0
∑m ai j yi = c j i f
i=
yi = 0 i f ∑m 1 ai j x j < bi
j= (9.26) Note that, if Problem 9.5 admits a nondegenerated solution (x1 , . . . , xn ), i.e., xi > 0 for any
i ≤ n, then the system of equations in Theorem 9.12 admits a unique solution.
Optimality certiﬁcates  Examples. Let see how to apply this theorem on two examples.
Let us ﬁrst examine the statement that
∗
∗
∗
∗
∗
∗
x1 = 2, x2 = 4, x3 = 0, x4 = 0, x5 = 7, x6 = 0 is an optimal solution of the problem
Maximize
Subject to: 18x1
2x1
−3x1
8x1
4x1
5x1 − 7x2
− 6x2
− x2
− 3x2
+ 2x2 + 12x3
+ 2x3
+ 4x3
+ 5x3
+ 8x3
− 3x3 +
+
−
−
+
+ 5x4
7x4
3x4
2x4
7x4
6x4 + 8x6
+ 3x5 + 8x6
+ x5 + 2x6
+ 2x6
− x5 + 3x6
− 2x5 − x6
x1 , x2 , · · · , x6 ≤
≤
≤
≤
≤
≥ 1
−2
4
1
5
0 In this case, (9.26) says:
2y∗ − 3y∗ + 8y∗ + 4y∗ + 5y∗
1
2
3
4
5
−6y∗ − y∗ − 3y∗
+ 2y∗
1
2
3
5
3y∗ + y∗
− y∗ − 2y∗
1
2
4
5
y∗
2
y∗
5 = 18
= −7
=
0
=
0
=
0 As the solution ( 1 , 0, 5 , 1, 0) is a feasible solution of the dual problem (Problem 9.6), the pro3
3
posed solution is optimal. 9.3. DUALITY OF LINEAR PROGRAMMING
Secondly, is 145 ∗
∗
∗
∗
∗
x1 = 0, x2 = 2, x3 = 0, x4 = 7, x5 = 0 an optimal solution of the following problem?
Maximize 8x1
Subject to: 2x1
x1
5x1 −
−
+
+ 9x2
3x2
7x2
4x2 + 12x3
+ 4x3
+ 3x3
− 6x3 Here (9.26) translates into: + 4x4 + 11x5
+ x4 + 3x5
− 2x4 +
x5
+ 2x4 + 3x5
x1 , x2 , · · · , x5 ≤
≤
≤
≥ 1
1
22
0 −3y∗ + 7y∗ + 4y∗ = −9
1
2
3
y∗ − 2y∗ + 2y∗ =
4
1
2
3
∗
y2
=
0
As the unique solution of the system (3.4, 0, 0.3) is not a feasible solution of Problem 9.6, the
proposed solution is not optimal. 9.3.5 Interpretation of dual variables As said in the introduction of this section, one of the major interests of the dual programme is
that, in some problems, the variables of the dual problem have an interpretation.
A classical example is the economical interpretation of the dual variables of the following
problem. Consider the problem that consits in maximizing the beneﬁt of a company building
some products. Each variable x j of the primal problem measures the amount of product j that is
built, and bi the amount of resource i (needed to build the products) that is available. Note that,
for any i ≤ n, j ≤ m, ai, j represents the number of units of resource i needed per unit of product
j...
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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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