ORIE 3300/5300
ASSIGNMENT 13
Fall 2008
Individual work.
Not to be graded.
1. Consider the following integer program:
maximize
x
1
+ 3
x
2
subject to 2
x
1
+ 5
x
2
≤
9
x
1
,
x
2
≥
0
,
integer
.
Solve this problem by branch and bound by starting from an optimal
tableau for the initial linear programming relaxation, and then using
the dual simplex method to solve the linear programming relaxation of
each subproblem obtained by adding the extra constraint deﬁning the
branch.
2. The diet problem can be written as min
c
T
x,Ax
≥
b,x
≥
0, where
x
j
is the amount of food
j
to be purchased at a unit cost of
c
j
>
0,
a
ij
is
the amount of nutrient
i
provided by each unit of food
j
, and
b
i
is the
daily requirement of nutrient
i
. After adding surplus variables
t
and
putting into standard equality form, this becomes max(

c
)
T
x,Ax

t
=
b,x
≥
0
,t
≥
0. Show that the initial basis consisting of all the surplus
variables is dualfeasible, so that this problem is ideal for solution by
the dual simplex method.
3. This exercise illustrates the connection between the dual simplex method
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 Fall '08
 TODD
 Linear Programming, Optimization, aij, dual simplex method

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