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Note 5: Duality and Dual Simplex Method
If the primal problem (P) is
max
Z
=
∑
n
j
=1
c
j
x
j
s.t.
∑
n
j
=1
a
ij
x
j
≤
b
i
∀
i
= 1
,
2
,...,m
x
j
≥
0
∀
j
= 1
,
2
,...,n,
(1)
then the dual problem (D) is
min
W
=
∑
m
i
=1
b
i
y
i
s.t.
∑
m
i
=1
a
ij
y
i
≥
c
j
∀
j
= 1
,
2
,...,n
y
i
≥
0
∀
i
= 1
,
2
,...,m.
(2)
There are
m
food types and
n
nutritional ingredients. Each unit of food type
i
contains
a
ij
units of nutritional ingredient
j
. Food type
i
sells for
b
i
, while a
person’s daily requirement of nutritional ingredient
j
is
c
j
. Let
x
j
be the fair worth
of nutritional ingredient
j
to the person and
y
i
be the number of units of food type
i
to be bought daily by the person. Then (P) maximizes the total required fair worth
of all nutritional ingredient on the basis that the total fair worth of all nutritional
ingredient of any food type does not exceed its selling price, while (D) minimizes the
total cost of satisfying the daily nutritional needs.
In essence,
(P)’s variable corresponds to (D)’s constraint, and (D)’s variable corresponds to
(P)’s constraint;
If (P) is a maximization (minimization) problem, then (D) is a minimization
(maximization) problem;
(P)’s free variable corresponds to (D)’s = constraint and its nonnegative variable
corresponds to (D)’s max
/
≤
constraint (meaning that (D) is a maximization problem
and the constraint is of the “
≤
” type) or min
/
≥
constraint; on the other hand, (D)’s
free variable corresponds to (P)’s = constraint, its nonnegative variable corresponds
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This document was uploaded on 11/03/2009.
 Spring '09

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