Linear Algebra, Theory And Applications, 2012a

152 LINEAR PROGRAMMING
Example 6.4.3 Maximize x 1 − x 2 +2x 3 subject to the constraints, 2x 1 + x 2 − x 3 ≥ 3,x 1 +
x 2 + x 3 ≥ 2,x 1 + x 2 + x 3 ≤ 7 and x ≥ 0.
From (6.20) you can immediately assemble an initial simplex tableau. You begin with
the first 6 columns and top 3 rows in (6.20). Then add in the column and row for z. This
yields
⎛
⎞
⎜
⎝
2
1
3
0 − 1 3
− 1 5
3
0 0
3
1
1
0
3
1
3
− 2 1
3
0 0
3
0 0 0 0 1 1 0 5
−1 1 −2 0 0 0 1 0
and you first do row operations to make the first and third columns simple columns. Thus
the next simplex tableau is
⎛
⎞
2
1
3
0 − 1 3
− 1 5
3
0 0
3
1 1
⎜ 0
3
1
3
− 2 1
3
0 0 3 ⎟
⎝ 0 0 0 0 1 1 0 5 ⎠
7 1
0
3
0
3
− 5 7
3
0 1
3
You are trying to get rid of negative entries in the bottom left row. There is only one, the
−5/3. The pivot is the 1. The next simplex tableau is then
⎛
⎞
2
1
3
0 − 1 1 10
3
0
3
0
3
1 1 2 11
⎜ 0
3
1
3
0
3
0 3 ⎟
⎝ 0 0 0 0 1 1 0 5 ⎠
7 1 5 32
0
3
0
3
0
3
1
3
and so the maximum value of z is 32/3 and it occurs when x 1 =10/3,x 2 = 0 and x 3 =11/3.
6.5 Duality
You can solve minimization problems by solving maximization problems. You can also go
the other direction and solve maximization problems by minimization problems. Sometimes
this makes things much easier. To be more specific, the two problems to be considered are
A.) Minimize z = cx subject to x ≥ 0 and Ax ≥ b and
B.) Maximize w = yb such that y ≥ 0 and yA ≤ c,
(
equivalently A T y T ≥ c T and w = b T y T ) .
⎟
⎠
In these problems it is assumed A is an m × p matrix.
I will show how a solution of the first yields a solution of the second and then show how
a solution of the second yields a solution of the first. The problems, A.) andB.) are called
dual problems.
Lemma 6.5.1 Let x be a solution of the inequalities of A.) and let y be a solution of the
inequalities of B.). Then
cx ≥ yb.
and if equality holds in the above, then x is the solution to A.) and y is a solution to B.).
Proof: This follows immediately. Since c ≥ yA, cx ≥ yAx ≥ yb.
It follows from this lemma that if y satisfies the inequalities of B.) andx satisfies the
inequalities of A.) then if equality holds in the above lemma, it must be that x is a solution
of A.) andy is a solution of B.).