d nagesh kumar, iiscoptimization methods: m7l2 1 integer programming mixed integer linear...
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D Nagesh Kumar, IISc Optimization Methods: M7L21
Integer Programming
Mixed Integer Linear Programming
D Nagesh Kumar, IISc Optimization Methods: M7L22
Objectives
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: M7L23
Introduction
Mixed Integer Programming: Some of the decision variables are real valued and some
are integer valued
Mixed Integer Linear Programming: 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: M7L24
Generation of Gomory Constraints
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: M7L25
Generation of Gomory Constraints …contd.
From the ith equation,
Expressing bj as an integer part plus a fractional part,
Expressing cij as where
m
jjijii ycbx
1
iii bb
ijijij ccc
0
00
00
0
ijij
ijij
ij
ijijij
cifc
cifc
cif
cifcc
D Nagesh Kumar, IISc Optimization Methods: M7L26
Generation of Gomory Constraints …contd.
Thus,
Since xi and are restricted to take integer values and also
the value of can be or <0
Thus we have to consider two cases.
iiij
m
jijij xbycc
1
ib 10 i iii xb 0
D Nagesh Kumar, IISc Optimization Methods: M7L27
Generation of Gomory Constraints …contd.
Case I:
For xi to be an integer,
Therefore,
Finally it takes the form,
0 iii xb
,....21 iiiiii ororxb
ij
m
jijij ycc
1
ij
m
jij yc
1
D Nagesh Kumar, IISc Optimization Methods: M7L28
Generation of Gomory Constraints …contd.
Case II:
For xi to be an integer,
Therefore,
Finally it takes the form,
0 iii xb
,....321 iiiiii ororxb
11
ij
m
jijij ycc
11
ij
m
jij yc
D Nagesh Kumar, IISc Optimization Methods: M7L29
Generation of Gomory Constraints …contd.
Dividing this inequality by and multiplying with , we have
Now considering both cases I and II, the final form of the Gomory constraint after adding one slack variable si is,
1i i
ij
m
jij
i
i yc
11
ij
m
jij
i
ij
m
jiji ycycs
11 1
D Nagesh Kumar, IISc Optimization Methods: M7L210
Procedure for solving Mixed-Integer LP
Solve the problem as an ordinary LP problem neglecting the
integrality constraints.
Generate Gomory constraint for the fractional valued
variable that has integer restrictions.
Insert a new row with the coefficients of this constraint, to
the final tableau of the ordinary LP problem.
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: M7L211
Thank You