Integer Programming
Mixed Integer Linear
Programming
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D Nagesh Kumar, IISc
Optimization Methods: M7L2
Objectives
2
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To discuss about the Mixed Integer Programming (MIP)

To discuss about the Mixed Integer Linear Programming (MILP)

To discuss the generation of Gomory constraints

To describe the procedure for solving MILP
D Nagesh Kumar, IISc
Optimization Methods: M7L2
Introduction
Mixed Integer Programming:


Mixed Integer Linear Programming:



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Some of the decision variables are real valued and some
are integer valued
MIP with linear objective function and constraints
The procedure for solving an MIP is similar to that
of All Integer LP with some exceptions in the
generation of Gomory constraints.
D Nagesh Kumar, IISc
Optimization Methods: M7L2
Generation of Gomory Constraints
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
Consider the final tableau of an LP problem
consisting of n basic variables (original variables)
and m non basic variables (slack variables)

Let xi be the basic variable which has integer
restrictions
D Nagesh Kumar, IISc
Optimization Methods: M7L2
Generation of Gomory Constraints …contd.


From the ith equation,
m
xi  bi   cij y j
j 1

Expressing bj as an integer part plus a fractional part,
bi  bi   i

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Expressing cij as cij  cij  cij
D Nagesh Kumar, IISc
where
cij if cij  0

0 if cij  0
0 if cij  0
 
cij  
cij if cij  0
cij
Optimization Methods: M7L2
Generation of Gomory Constraints …contd.

Thus,
 c
m
j 1


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
ij

 cij y j   i  bi  xi 
Since xi and bi are restricted to take integer values and also 0  i  1
the value of  i  bi  xi can be  0 or <0
Thus we have to consider two cases.


D Nagesh Kumar, IISc
Optimization Methods: M7L2
Generation of Gomory Constraints …contd.


Case I:  i  bi  xi  0

For xi to be an integer,
i  bi  xi   i or i  1or i  2,....

Therefore,
 c
m

ij
j 1


 cij y j  i
Finally it takes the form,
m
 cij y j  i
j 1
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D Nagesh Kumar, IISc
Optimization Methods: M7L2
Generation of Gomory Constraints …contd.
Case II:

 i  bi  xi   0
For xi to be an integer,
 i  bi  xi   1  i or  2   i or  3   i ,....

Therefore,
 cij  cij y j  i  1
m
j 1

Finally it takes the form,
m

c
 ij y j  i  1
j 1
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D Nagesh Kumar, IISc
Optimization Methods: M7L2
Generation of Gomory Constraints …contd.

Dividing this inequality by i  1 and multiplying with  i , we have
i
i
m
cij y j   i

1
j 1
Now considering both cases I and II, the final form of the Gomory
constraint after adding one slack variable si is,

m
si  
j 1
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cij y j

i
i
m
cij y j    i

1
D Nagesh Kumar, IISc
j 1
Optimization Methods: M7L2
Procedure for solving Mixed-Integer LP
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
Solve the problem as an ordinary LP problem neglecting the
integrality constraints.
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Generate Gomory constraint for the fractional valued
variable that has integer restrictions.
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Insert a new row with the coefficients of this constraint, to
the final tableau of the ordinary LP problem.
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Solve this by applying the dual simplex method

The process is continued for all variables that have
integrality constraints
D Nagesh Kumar, IISc
Optimization Methods: M7L2
Thank You
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D Nagesh Kumar, IISc
Optimization Methods: M7L2