4.1.
THE LINEAR PROGRAMMING PROBLEM 31
4.1.2 Problem formulation.
A linear programming problem or LP in brief is any decision problem of the form 4.7. Maximize
c
1
x
1
+ c
2
x
2
+ . . . + c
n
x
n
subject to a
il
x
1
+ a
i2
x
2
+ . . . + a
in
x
n
≤ b
i
, l ≤ i ≤ k , a
il
x
1
+ . . . . . . . . . + a
in
x
n
≥ b
i
, k + 1 ≤ i ≤ ℓ , a
il
x
1
+ . . . . . . . . . + a
in
x
n
= b
i
, ℓ + 1 ≤ i ≤ m , and
x
j
≥ 0 , 1 ≤ j ≤ p ,
x
j
≥ 0 , p + 1 ≤ j ≤ q;
x
j
arbitary , q + 1 ≤ j ≤ n ,
4.7 where the
c
j
, a
ij
, b
i
are fixed real numbers. There are two important special cases:
Case I: 4.7 is of the form 4.8:
Maximize
n
X
j=1
c
j
x
j
subject to
n
X
j=1
a
ij
x
j
≤ b
i
, x
j
≥ 0 ,
1 ≤ i ≤ m , 1 ≤ j ≤ n
4.8
Case II: 4.7 is of the form 4.9:
Maximize
n
X
j=1
c
j
x
j
subject to
n
X
j=1
a
ij
x
j
= b
i
, x
j
≥ 0 ,
1 ≤ i ≤ m , 1 ≤ j ≤ n .
4.9
Although 4.7 appears to be more general than 4.8 and 4.9, such is not the case. Proposition:
Every LP of the form 4.7 can be transformed into an equivalent LP of the form 4.8. Proof.
Step 1: Replace each inequality constraint
P a
ij
x
j
≥ b
i
by P−a
ij
x
j
≤ −b
i
. Step 2:
Replace each equality constraint P a
ij
x
j
= b
i
by two inequality constraints: P a
ij
x
j
≤ b
i
, P−a
ij
x
j
≤ −b
i
. Step 3:
Replace each variable x
j
which is constrained x
j
≤ 0 by a variable y
j
= −x
j
constrained y
j
≥ 0 and then replace a
ij
x
j
by −a
ij
y
j
for every i and c
j
x
j
by −c
j
y
j
.
32
CHAPTER 4. LINEAR PROGRAMMING
Step 4: Replace each variable
x
j
which is not constrained in sign by a pair of variables y
j
−z
j
= x
j
constrained y
j
≥ 0, z
j
≥ 0 and then replace a
ij
x
j
by a
ij
y
j
+ −a
ij
z
j
for every i and
c
j
x
j
by c
j
y
j
+ −c
j
z
j
. Evidently the resulting LP has the form 4.8 and is equivalent to the original one.
♦ Proposition:
Every LP of the form 4.7 can be transformed into an equivalent LP of the from 4.9 Proof.
Step 1: Replace each inequality constraint
P a
ij
x
j
≤ b
i
by the equality constraint P a
ij
x
j
+ y
i
= b
i
where y
i
is an additional variable constrained y
i
≥ 0. Step 2:
Replace each inequality constraint P a
ij
x
j
≥ b
i
by the equality constraint P a
ij
x
j
− y
i
= b
i
where y
i
is an additional variable constrained by y
i
≥ 0. The new variables added in these steps are called slack variables.
Step 3, Step 4: Repeat these steps from the previous proposition. Evidently the new LP has the
form 4.9 and is equivalent to the original one. ♦
4.2 Qualitative Theory of Linear Programming
Kategori