Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Florentin
Operations Research OperationsOlariu
Research E. Operations
Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research OperationsNovember
Research Operations
Research Operations Research
15, 2016
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research - Course 7
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
1 / 35
Table of contents
1
2
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research
Operations Research Operations Research Operations Research
Branching
and Cutting
Operations Research Operations Research Operations Research Operations Research
LinearResearch
Programming
Operations
OperationsRelaxation
Research Operations Research Operations Research
Operations
Research
Operations
Research Operations Research Operations Research
Branch-and-Bound
Method
Operations Research Operations Research Operations Research Operations Research
Cutting
OperationsPlane
ResearchMethod
Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Bibliography
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
2 / 35
Introduction
Operations Research
Operations Research
Operations Research
Operations Research
Research again
Operations
Operations
Operations Research
Operations
LetOperations
us consider
the Research
ILP, MILP,
and Research
BILP problems
Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations
Research Operations
Operations Research
maximize
z = cTResearch
x;
Operations Research Operations Research Operations Research Operations Research
subject
to Operations
Ax Research
b;
Operations Research Operations
Research
Operations Research (1)
n : Research
Operations Research Operations Research x Operations
Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations
Operations Research
T x; Research
z = cOperations
Operations Research maximize
Operations Research
Research Operations Research
Operations Research Operations
Operations
subjectResearch
to Ax
b; Research Operations Research (2)
Operations Research Operations Research Operations Research Operations Research
; Research
i
:
Operations Research Operations Research xi Operations
Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
where
=
1; 2; : :Operations
:;n .
Operations
Research
Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations
Researchz =Operations
maximize
cT x; Research Operations Research
Operations Research Operations Research Operations Research Operations Research
(3)
subject Research
to AxOperations
b; Research Operations Research
Operations Research Operations
Operations Research Operations Research Operations
Operations Research
x
0; Research
1 n:
6
2 Z+
6
2 Z+ 8 2 I
?6 I(f
g
6
2f g
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
3 / 35
Linear Programming Relaxation
Operations Research
Operations Research
Operations Research
Operations Research
Operations
Research Operations
Research that
Operations
Operations
Any
ILP problem
admits Research
a naturalOperations
relaxation,
is theResearch
original
Research Operations Research Operations Research Operations Research
polyhedron
is replaced
with
another,
larger,
polyhedron
obtained
Operations Research
Operations
Research
Operations
Research
Operations Research
Operations Research Operations Research Operations Research Operations Research
byOperations
discarding
the
integrality
requirements
on
x.
Research Operations Research Operations Research Operations Research
Operations
Research
Operations
Research
Research Operations Research
Problems
(1) and
(2) have
the Operations
same relaxation:
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations
Research Operations
Operations Research
maximize
z = cT Research
x;
Operations Research Operations Research Operations Research Operations Research
(4)
subject
to Operations
Ax bResearch
;
Operations Research Operations
Research
Operations Research
n
Operations Research Operations Research Operations
Research
Operations
Research
x
:
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Problem
(3) has a specific relaxation:
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
T x;
Operations Research Operationsmaximize
Research Operations
Operations Research
z = cResearch
Operations Research Operations Research Operations Research Operations Research
subject
Ax Research
b;
Operations Research Operations
Research toOperations
Operations Research (5)
Operations Research Operations Research x
Operations
Operations Research
[0; 1Research
]n :
Operations Research Operations Research Operations Research Operations Research
6
2 R+
2
Olariu E. Florentin
6
Operations Research - Course 7
November 15, 2016
4 / 35
Linear Programming Relaxation
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Linear
relaxation of ILP problems usually gives greater objective
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations Research
Operationsenlarge
Research the
Operations
Research
function
value because
we eventually
feasible
region.
Operations Research Operations Research Operations Research Operations Research
If for solving
relaxation
alreadyResearch
have theoretical
tools such
Operations
Research the
Operations
ResearchweOperations
Operations Research
Operations
Research
Operations
Research
Operations
Research
Operations
as Simplex algorithm, ILP problems are harder to solve. Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations
Research
Operations
Research
Operations
Research
The
optimal
solutions
of the
relaxation
might
not be
a feasible
soluOperations Research Operations Research Operations Research Operations Research
tion
to theResearch
ILP original
problem,
to the
integrality
restrictions.
Operations
Operations
Research due
Operations
Research
Operations
Research
Operations Research Operations Research Operations Research Operations Research
There
are
situations
when
rounding
an
optimal
solution
for the
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations
Operations Researchsolution
Operations
relaxation
can give
an Research
optimal/sub-optimal
toResearch
the original
Operations Research Operations Research Operations Research Operations Research
ILP problem.
Operations
Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
5 / 35
Linear Programming Relaxation
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Researchwe Operations
Researchonly
For theResearch
sake of Operations
simplicity
of ourOperations
exposition
will analyze
Operations Research Operations Research Operations Research Operations Research
the
MILP
problem
in
the
following
form
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
T x + hT y;
z = cOperations
Operations Research maximize
Operations Research
Research Operations Research
Operations Research Operations Research Operations Research Operations Research
(6)
subject to Ax + Gy b;
Operations Research Operations Research Operations Research
Operations Research
p
n;y
Operations Research Operations Research x Operations
Research: Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Research Operations Research Operations Research Operations Research
Or,Operations
equivalently,
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
T
Operations Research(MILP
Operations
Research
Research (7)
) max
cT xOperations
+ h yResearch
: (x; y)Operations
;
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
where Research Operations Research Operations Research Operations Research
Operations
p
n
Operations Research= Operations
Research
(x; y) Research Operations
: AxResearch
+ Gy Operations
b
Operations Research Operations Research Operations Research Operations Research
6
2 Z+ 2 R+
f
P f
Olariu E. Florentin
2 Z+ R+
Operations Research - Course 7
2 Pg
6 g
November 15, 2016
6 / 35
Linear Programming Relaxation
Operations Research
Operations Research
Operations Research
Operations Research
Operations
Research
Operations
Operations Research
Operations
The
natural
relaxation
ofResearch
the polyhedron
P is Operations Research
Research Operations Research Operations Research Operations Research
Operations Research Operations Research
Research
n +p Operations Research
= f(x Research
y) 2 R+Operations
: Ax
+ Gy Operations
6 bOperations
g Research
Operations Research R
Operations
Research
Operations Research Operations Research Operations Research Operations Research
Research relaxation
Operations Research
Operations
Operations Research
Operations
The natural
of problem
(7)Research
is
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research
Operations
Research Operations Research
T
(LPOperations
) max fResearch
cT x + hOperations
y : (xResearch
y) 2 Rg
Operations Research
Operations Research (8)
Operations Research Operations Research Operations Research Operations Research
Research throughout
Operations Research
Operations
Research
Operations
Research
Operations
WeOperations
will assume
this section
that
problem
(7) has
finite
Research Operations Research Operations Research Operations Research
;
:
;
;
optimum.
Let (x ;Operations
y ) be an
optimal
solution
and zOperations
optimal
Operations Research
Research
Operations
Research
be theResearch
Operations Research Operations Research Operations Research Operations Research
value
of this problem.
Operations Research Operations Research Operations Research Operations Research
Research Operations
Researchimplies
Operations
Operations
Research
Operations
The assumption
from above
thatResearch
problem
(8) has
an optimal
Operations Research Operations Research Operations Research Operations Research
0 ) with optimal value z .
rational
solution
(x0 ; yResearch
Operations
Research
Operations
Operations Research 0 Operations Research
Operations Research Operations Research Operations Research Operations Research
This result comes without a proof and is based on the rationality
Operations Research Operations Research Operations Research Operations Research
Research
Operations Research
Research
Operations
of Operations
the input
data (why?).
Always Operations
an optimal
solution
is aResearch
rational
Operations Research Operations Research Operations Research Operations Research
one (why?).
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
7 / 35
Linear Programming Relaxation
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations
Research Operations Research Operations Research
Suppose
that (x0 ; y0 ) and
z0 can be
obtained by using an LP solver
Operations Research Operations Research
Operations Research Operations Research
Operations
Research
Operations
Research Operations Research Operations Research
(like the
simplex
algorithm).
Operations Research Operations Research Operations Research Operations Research
n , then (x0 ; y0 )
Since Research , we
mustResearch
have z Operations
z0 . IfResearch
x0
Operations
Operations
Operations Research
Operations
Research
Research
and
therefore
z =Operations
z0 - in Research
this caseOperations
problem
(6) is Operations
solved. Research
Operations Research Operations Research Operations Research Operations Research
n , that
Operations
Research when
Operations
Operations
Research of
What
happens
x0 Research
is, Research
at least Operations
a component
Operations
Research
Operations
Research
Operations Research Operations Research
0
x Operations
is fractional?
possible
answers
gives
different
strategies
Research The
Operations
Research
Operations
Research
Operations
Research for
Operations
Research
Operations
Research
Operations
Research
Operations
Research
solving problem (6).
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations
Operationsstrategies:
Research Operations
Research
We will
describe
two Research
such distinct
the Branch-andOperations Research Operations Research Operations Research Operations Research
BoundResearch
MethodOperations
and the
Cutting
PlaneResearch
Method.
Operations
Research
Operations
Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
PR
Olariu E. Florentin
6
2Z
2P
62 Z
Operations Research - Course 7
November 15, 2016
8 / 35
Branch-and-Bound Method
Operations Research
Operations Research
Operations Research
Operations Research
Research this
Operations
Research
OperationsSuppose
Research that
Operations
Research
Operations
WeOperations
first present
method
informally.
xj0 62
Z, then
Research Operations Research Operations Research Operations Research
weOperations
define Research
two newOperations
polyhedra
Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
p Research
n Operations
0
Operations Research= Operations
Research
(x;Research
y)
: x
xOperations
;
j
Operations Research 1 Operations Research Operations Research j Operations
Research
Operations Research Operations Research Operations Research Operations Research
p
n
Operations Research 2 Operations
Research
=
(x; y) Operations Research
: xj Operations
xj0 :Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research =Operations
Research
Operations Research
Obviously,
.
1
2 and 1Operations
2 =Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations
Research
Operations Research Operations Research
Consider
two new
MILP
problems
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research T Operations
Research Operations Research
(MILP
c x +Operations
hT y : Research
(x; y) Operations
1 ) maxResearch
1 ;
Operations Research
Operations
Research (9)
Operations Research Operations Research Operations Research Operations Research
T
T
Operations Research
Operations
Research
Operations
Research Operations
Research
(MILP
(10)
2 ) max c x + h y : (x; y)
2 :
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
The
optimal solution of (6) is the best among the optimal solutions
Operations Research Operations Research Operations Research Operations Research
Operations
Operations
Research
Operations
Researchthe
of Operations
(9) andResearch
(10). The
idea Research
of branching
comes
here:
by solving
Operations Research Operations Research Operations Research Operations Research
P P \f
P P \f
P P [P
2 Z+ R+
2 Z+ R+
P \P ?
f
f
6 b cg
> d eg
2P g
2P g
two new sub-problems we will solve the original problem.
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
9 / 35
Branch-and-Bound Method
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Research linear
Operations
Research of
Operations
Research
LetOperations
the natural
relaxations
1 and
2 be Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations
Research 0Operations Research
n p
(x; y) Operations Research
: xj
xj ; Research
Operations Research Operations
Research
Operations
1 =
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations
Operations
Research
n pResearch
x; y) Operations
: Research
xj
xj0Operations
:
Operations Research2 =Operations(Research
Research
Operations Research Operations Research Operations Research Operations Research
The
natural
relaxation
corresponding
problems
Operations
Research
Operations
Research Operations
Research areOperations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research
Operations
Research Operations Research
T
T
(LP1 ) max c x + h y : (x; y)
1 ; Research (11)
Operations Research Operations
Research Operations Research Operations
Operations Research Operations Research Operations Research Operations Research
Operations Research (LP
Operations
Research
Operations
cT x +
hT y Research
: (x; y) Operations
2 ) max
2 : Research (12)
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Research
Operations Research weOperations
Research
Regarding
eachOperations
of these
two sub-problems
have two
possible
Operations Research Operations Research Operations Research Operations Research
situations.
Operations
Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
P
R
R\f
2 R++
P
6 b cg
R
R\f
2 R++
> d eg
f
f
Olariu E. Florentin
Operations Research - Course 7
2R g
2R g
November 15, 2016
10 / 35
Branch-and-Bound Method
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Research
Research
Research
Research then
1. Operations
Obviously,
if Operations
(that is,Operations
problem
(LPi )Operations
is infeasible),
i =
Operations Research Operations Research Operations Research Operations Research
and (MILP
also infeasible.
We say Operations
that this
problem is
i = Research
i ) is
Operations
Operations
Research
Operations Research
Research
Operations Research
Operations
Research Operations Research Operations Research
fathomed
or pruned
by infeasibility.
Operations Research Operations Research Operations Research Operations Research
Operations Research
Research
Operations
Research
Operations
2. Otherwise,
let (xi ;Operations
yi ) be an
optimal
solution
of (LP
zi be its
i ) andResearch
Operations Research Operations Research Operations Research Operations Research
optimal
value.
Operations Research Operations Research Operations Research Operations Research
Operations Research
Operations Research
Operations Research Operations Research
2.1
If xi Research
2 n , then
(xi ; yi ) is an optimal solution of (MILPi ) and
Operations
Operations
Research Operations Research Operations Research
zi Research
z . WeOperations
say thatResearch
(MILPi Operations
) is fathomed
integrality.
Operations
Researchor pruned
Operationsby
Research
n
Operations
Research
Operations
Research
Operations
2.2
If xi 62Research
, andOperations
zi is smaller
(or equal)
than
the best
lowerResearch
bound of
Operations Research Operations Research Operations Research Operations Research
z
,
then
(
LP
)
doesn’t
have
a
better
solution
than
the
best
current
Operations
Research i Operations Research Operations Research Operations Research
Operations
Research
Operations
Research
Operations
Research
Operations
Research
solution and the problem is fathomed or pruned by bound.
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
P
R
?
6
?
Z
Z
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
11 / 35
Branch-and-Bound Method
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
n
Operations
Research
Researchthan
Operations
Research
2.3
If xi 62
, andOperations
zi is greater
the best
lower Operations
bound ofResearch
z , then
Operations Research Operations Research Operations Research Operations Research
(MILP
optimalOperations
solutionResearch
of (MILP
). Let Research
xhi a noni ) can contain
Operations
Research
Operationsan
Research
Operations
i
Operations
Researchcomponent
Operations Research
Research
Operations Research
integral
of x . WeOperations
define the
polyhedra
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research OperationsnResearch
Pi 1 = Pi \ f(x; y) 2 +p : xh Operations
bxhi cg; Research
Operations Research Operations
Research Operations Research Operations
Research
Operations Research Operations Research Operations Research Operations Research
n +p
i
f(x; y) Operations
2
: xh
dxOperations
Operations Research POperations
Research
Research
Research
i 2 = Pi \
h eg:
Operations Research Operations Research Operations Research Operations Research
We construct
new sub-problems
(MILP
(MILP
Operations
Research two
Operations
Research Operations
Research
Operations
Research
i 1 ) and
i 2 ); after
Operations
Research
Operations
Research Operations Research Operations Research
that
we iterate
the process.
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Z
R
R
Olariu E. Florentin
Operations Research - Course 7
6
>
November 15, 2016
12 / 35
Branch-and-Bound Method - Example
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations the
Research
Operations
Research
Operations
Research
Operations Research
Consider
following
ILP
problem
and its
LP relaxation
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
max
zOperations
= 17x1Research
+ 12x2 Operations Research Operations Research
Operations Research
max
z = 17x1 + 12x2
Operations
Research
Operations
s. t.
10x1 +
7x2 Research
40 Operations Research Operations Research
Operations Research Operations Research Operations Research
Operations
Research
s. t. 10x
40
1 + 7x
2
(POperations
)
R)
2x2 Research5 (Operations
Researchx1 +
Operations
Research Operations Research
x1 + 2x
5
2
Operations Research Operations
Research Operations
Research
x1 ; xResearch
0Operations
2 Research
Operations Research Operations
Operations Research Operations
x1 ; x2 Research 0
x1 ; x2Research Operations Research Operations Research
Operations Research Operations
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
AnOperations
optimal
solution to the natural LP relaxation, ( ), is x =
Research Operations Research Operations Research Operations Research
Operations
Research
(x1 ; x2 )=
(1:67;Operations
3:33)T ,Research
and the Operations
optimalResearch
value isOperations
z = 205Research
=3
68:33.
Operations Research Operations Research Operations Research Operations Research
By branching
variable
x1 weOperations
createResearch
two newOperations
ILP sub-problems
Operations
Research on
Operations
Research
Research
Operations
Research
Operations
Research
Operations
Research
Operations
Research
(P1 ) and (P2 ), those relaxations are (R1 ) and (R2 ), respectively.
Operations
Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
8
>>
><
>>
>:
6
6
>
2Z
8
>
>
<
>
>
:
6
6
>
R
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
13 / 35
Branch-and-Bound Method - Example
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
max Research
z = 17xOperations
max
17x1 +Research
12x2
1 + 12xResearch
2
Operations
Operations
Researchz =
Operations
Operations
Research
Operations
Research
Operations
Research
Operations
Research
s. t. 10x1 + 7x2
40
s. t. 10x1 + 7x2
40
Operations Research Operations Research Operations Research Operations Research
(R1Operations
)
(
R
)
x
+
2x
5
x
+
2x
5
2
2 Research
1
2
Research 1 Operations
Operations
Research Operations
Research
Operations Research Operations
Research
Operations
Research
Operations
Research
x1
1
x1
2
Operations Research Operations Research Operations Research Operations Research
x2
0
x1 ; x2 Research 0
Operations Researchx1 ; Operations
Research
Operations Research Operations
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations
Research
The problem
the optimal
solution
= (1; 3Research
)T , and the
1 ) hasResearch
Operations
Research (R
Operations
Operations
Research x Operations
Operations
Research
Research
Operations Research
optimal
value
z = Operations
65; thisResearch
problemOperations
is pruned
by integrality.
Operations Research Operations Research Operations Research Operations Research
z Operations
= 65 becomes
newResearch
best lower
bound
for the
optimal
value
Research the
Operations
Operations
Research
Operations
Research
Operations
Research
Operations
Research
Operations
Research
Operations
Research
of original problem (P ) and the current best solution becomes x =
Operations Research Operations Research Operations Research Operations Research
(1; 3)T Research
.
Operations
Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
The problem
R2 ) hasResearch
the optimal
solution
= (2; Research
2:86)T , and
Operations
Research (Operations
Operations
Research xOperations
Operationsvalue
Research
Operations
Operations
optimal
z =Operations
68:29; Research
this solution
isResearch
not feasible
forResearch
the ILP
Operations Research Operations Research Operations Research Operations Research
problem (P2 ).
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Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
14 / 35
Branch-and-Bound Method - Example
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations xResearch
Operations Research Operations Research
1 = 1.67, x2 = 3.33
Operations Research Operations Research Operations Research Operations Research
z = 68.33
Operations Research Operations(R)
Research Operations Research Operations Research
x ≤1
x ≥2
Operations Research Operations1 Research Operations1 Research Operations Research
Operations Research x1Operations
Operations
Operations Research
= 1, x2 = 4 Research
x1 = 2, x2Research
= 2.86
Operations Research (R
Operations
Research
Operations
Research
Operations
Research
z
=
65
z
=
68.29
(R2 )
1)
Operations Research Operations Research Operations Research Operations Research
Operations Research
Research Operations
Research
Operations
Figure:Operations
The enumeration
tree after
solving
(R2 ) Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Since
the Research
optimal Operations
value ofResearch
problemOperations
(R2 ) is
better Operations
than theResearch
current
Operations
Research
Operations Research Operations Research Operations Research Operations Research
best
optimal value of an integer solution, we create two new subOperations Research Operations Research Operations Research Operations Research
problems
(P3 ) and
(P4Research
), by branching
on the variable
Operations
Research
Operations
Operations Research
Operations x
Research
2 : one with
Operations Research Operations Research Operations Research Operations Research
the additional constraint x2 2,
the other with x2 3.Research
Operations Research Operations Research
Operations Research Operations
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
6
Olariu E. Florentin
Operations Research - Course 7
>
November 15, 2016
15 / 35
Branch-and-Bound Method - Example
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
max
z =Operations
17x1 +Research
12x2
max Operations
z = 17xResearch
Operations
Research
Operations Research
1 + 12x2
Operations
Research
Operations
Research
Operations
Research
Operations
s. t. 10x1 + 7x2
40
s. t. 10x1 + 7x2 Research40
Operations Research Operations Research Operations Research Operations Research
x1 + 2x
5
+ 2x2 Research 5
Operations Research
Operations
Research
Operations Research x1Operations
2
(R3Operations
)
(R4 ) Research Operations Research
Research Operations
Research
Operations
x1
2
x1
2
Operations Research Operations
Research Operations Research Operations
Research
x2 Research2 Operations Research Operations Research
x2
3
Operations Research Operations
Operations Research Operations Research Operations Research Operations Research
x1 ; x2 Research0 Operations Research Operations
x1 ; Research
x2
0
Operations Research Operations
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
T and the
The
problem
(R3 )Operations
has theResearch
optimalOperations
solution
x = (2Operations
:6; 2) ,Research
Operations
Research
Research
Operations
Operations Research
Operations Research
optimalResearch
value zOperations
= 68:Research
2; this solution
is not feasible
for the ILP
Operations Research Operations Research Operations Research Operations Research
problem
(P3 ). Operations Research Operations Research Operations Research
Operations
Research
Operations Research Operations Research Operations Research Operations Research
The problem
not feasible
solutions
pruned by
4 ) has Research
Operations
Research (R
Operations
Operations
Research therefore
Operations is
Research
Operations Research Operations Research Operations Research Operations Research
infeasibility.
Operations Research Operations Research Operations Research Operations Research
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Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
16 / 35
Branch-and-Bound Method - Example
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
x1 =
1.67, x2 = 3.33Operations Research
Operations Research Operations
Research
Operations Research
Operations Research Operations
Research Operations Research Operations Research
z
= 68.33
(R)
Operations Research Operations
Research Operations
Research Operations Research
x1 ≤ 1
x1 ≥ 2
Operations Research Operations Research Operations Research Operations Research
x2 = 4
x1 = 2, x2 = 2.86
Operations Researchx1 = 1,Operations
Research Operations
Research Operations Research
z = 65 Operations Research
= 68.29
(R2 ) zOperations
Operations (R
Research
Research Operations Research
1)
x2 ≤ 2 Operations Research
x2 ≥ 3 Operations Research
Operations Research Operations Research
Operations Research Operations
Research
Operations
Research
Operations Research
x1 = 2.6, x2 = 2
inf easible
Operations Research Operations Research Operations Research
Operations Research
(R3 ) z = 68.2
(R4 )
Operations Research Operations
Research Operations
Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Figure: The enumeration tree after solving (R4 )
Operations Research Operations Research Operations Research Operations
Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Since
the Research
optimal Operations
value ofResearch
problemOperations
(R3 ) is
better Operations
than theResearch
current
Operations Research Operations Research Operations Research Operations Research
best
optimal
valueOperations
of an integer
we create
two Research
new subOperations
Research
Research solution,
Operations Research
Operations
Operations
Research
Operations
Research
Operations
Research
Operations
Research
problems (P5 ) and (P6 ), by branching on the variable x1 : one with
Operations Research Operations Research Operations Research Operations Research
the additional
x1 2,
the other
withOperations
x1 3.Research
Operations
Research constraint
Operations Research
Operations
Research
6
Olariu E. Florentin
Operations Research - Course 7
>
November 15, 2016
17 / 35
Branch-and-Bound Method - Example
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
max Research
z = 17xOperations
max
17x1 +Research
12x2
1 + 12xResearch
2
Operations
Operations
Researchz =
Operations
Operations
Research
Operations
Research
Operations
Research
Operations
Research
s. t. 10x1 + 7x2
40
s. t. 10x1 + 7x2
40
Operations Research Operations Research Operations Research Operations Research
x
+
2x
5
x
+
2x
5
2 Research
1
2
Operations Research 1 Operations
Operations Research Operations
Research
(R5 )
(R6 )
Operations Research Operations
Research
Operations
Research
Operations
Research
x1 = 2
x2
2
Operations Research Operations Research Operations Research Operations Research
x2
2
x1 Research 3
Operations Research Operations
Research
Operations Research Operations
Operations Research Operations
x1 ; x2 Research0 Operations Research Operations
x1 ; Research
x2
0
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Research
Operations
Research
The
problem
(R5 )Operations
has the
optimal
solution
x = Operations
(2; 2)T ,Research
and the
Operations Research Operations Research Operations Research Operations Research
optimal
z =Operations
58; this
problem
is pruned
integrality
Operationsvalue
Research
Research
Operations
Research by
Operations
Researchbut
Operations
Research
Operations
Research
Operations
Research
Operations
Research
its optimal value is no better than the current best optimal value
Operations Research Operations Research Operations Research Operations Research
which Research
is 65. Operations Research Operations Research Operations Research
Operations
Operations Research Operations Research Operations Research Operations
Research
The problem
the optimal
solution
(3; 1:43Research
)T , and the
6 ) has Research
Operations
Research (R
Operations
Operations
Researchx =Operations
Operationsvalue
Research
Operations
Operations
optimal
z =Operations
68:14; Research
this solution
isResearch
not feasible
forResearch
the ILP
Operations Research Operations Research Operations Research Operations Research
problem (P6 ).
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Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
18 / 35
Branch-and-Bound Method - Example
Operations Research Operations Research Operations Research Operations Research
x1 = 1.67,Research
x2 = 3.33
Operations Research Operations
Operations Research Operations Research
Operations Research Operations
Operations Research Operations Research
z = 68.33
(R) Research
Operations Research x1 ≤Operations
Research x1 ≥Operations
Research Operations Research
1
2
Operations Research Operations Research Operations Research Operations Research
x1 = 1, x2 = 4
2, x2 = 2.86
Operations Research
Operations Researchx1 =Operations
Research Operations Research
z=
65
z = 68.29Research
(R1 )
Operations Research
Operations
Research (ROperations
Operations Research
2)
x2 ≤ 2
x2 ≥ 3
Operations Research Operations Research
Operations Research
Operations Research
Operations Research Operationsx1Research
Operations Research
= 2.6, x2 = 2 Operations Research
inf easible Operations Research
Operations Research Operations Research Operations Research
(R3 ) z = 68.2
(R4 )
Operations Research Operations
Research Operations Research
Operations Research
x1 ≤ 2
x1 ≥ 3
Operations Research Operations Research Operations Research Operations Research
x2 = 2
x1 = 3, x2 = 1.43
Operations Researchx1 = 2,Operations
Research Operations
Research Operations Research
Operations (R
Research
Research Operations Research
z = 58 Operations Research
= 68.14
(R6 ) z Operations
5)
Operations Research Operations Research Operations Research Operations Research
OperationsFigure:
Research The
Operations
Research tree
Operations
Operations
Research
enumeration
after Research
solving (R
6)
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations
Research
Operations
Research
Operations Research
Since the
optimal
value
of problem
(R
6 ) is better than the value
Operations Research Operations Research Operations Research Operations Research
of the best-so-far
integer
solution,
we create
sub-problems
Operations
Research Operations
Research
Operations
ResearchtheOperations
Research (P7 )
Operations
Research Operations
Research
Operations
Research
and
(P8 ): Research
one withOperations
the additional
constraint
x2 1,
the other
with
Operations Research Operations Research Operations Research Operations Research
x2 2.
6
>
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
19 / 35
Branch-and-Bound Method - Example
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
max
z =Operations
17x1 +Research
12x2
max Operations
z = 17xResearch
Operations
Research
Operations Research
1 + 12x2
Operations
Research
Operations
Research
Operations
Research
Operations
s. t. 10x1 + 7x2
40
s. t. 10x1 + 7x2 Research40
Operations Research Operations Research Operations Research Operations Research
x1 + 2x
5
+ 2x2 Research 5
Operations Research
Operations
Research
Operations Research x1Operations
2
(R7Operations
)
(R8 ) Research Operations Research
Research Operations
Research
Operations
x2
1
x2 = 2
Operations Research Operations
Research Operations Research Operations
Research
x1 Research3 Operations Research Operations Research
x1
3
Operations Research Operations
Operations Research Operations Research Operations Research Operations Research
x1 ; x2 Research0 Operations Research Operations
x1 ; Research
x2
0
Operations Research Operations
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
T and the
The
problem
(R7 )Operations
has theResearch
optimalOperations
solution
x = (3Operations
:3; 1) ,Research
Operations
Research
Research
Operations
Operations Research
Operations Research
optimalResearch
value zOperations
= 68:Research
1; this solution
is not feasible
for the ILP
Operations Research Operations Research Operations Research Operations Research
problem
(P7 ). Operations Research Operations Research Operations Research
Operations
Research
Operations Research Operations Research Operations Research Operations Research
The problem
not feasible
solutions
pruned by
8 ) has Research
Operations
Research (R
Operations
Operations
Research therefore
Operations is
Research
Operations Research Operations Research Operations Research Operations Research
infeasibility.
Operations Research Operations Research Operations Research Operations Research
8
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Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
20 / 35
Branch-and-Bound Method - Example
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations
Research Operations Research Operations Research
x1 = 1.67, x2 = 3.33
Operations Research Operations Research Operations Research Operations Research
(R) z = 68.33
Operations Research Operations Research Operations Research Operations Research
x ≤1
x ≥2
Operations Research 1 Operations Research 1 Operations Research Operations Research
Operations Researchx1 = 1,Operations
Research Operations
Operations Research
x2 = 4
x1 = 2, x2 = Research
2.86
Operations (R
Research
Operations Research Operations Research
z = 65 Operations Research
(R2 ) z = 68.29
1)
Operations Research Operations Research
x2 ≤ 2 Operations Research
x2 ≥ 3 Operations Research
Operations Research Operations Research Operations Research Operations Research
= 2.6, x2 = 2
Operations Research Operationsx1Research
Operations Research
Operations Research
inf easible
(R3 ) z = 68.2
(R4Research
)
Operations Research Operations
Research Operations
Operations Research
x1 ≤ 2
x1 ≥ 3
Operations Research Operations
Research Operations
Research Operations Research
Operations Research
Operations Researchx1 = 3,Operations
Research Operations Research
x1 = 2, x2 = 2
x2 = 1.43
Operations Research Operations Research Operations Research Operations Research
z
=
58
z
=
68.14
(R6 )
5)
Operations (R
Research
Operations Research
Operations Research Operations Research
x2 ≥ 2
x2 ≤ 1
Operations Research Operations Research
Operations Research
Operations Research
Operations Research Operations
Research
Operations Research Operations Research
x1 = 3.3,
x2 = 1
inf easible
Operations Research Operations
Research Operations Research
Operations Research
(R7 ) z = 68.1
(R8 )
Operations Research Operations Research Operations Research Operations Research
Operations Research
Research Operations
Research
Operations
Figure:Operations
The enumeration
tree after
solving
(R8 ) Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
21 / 35
Branch-and-Bound Method - Example
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations
Operations
Operations
Since the
optimal
valueResearch
of problem
(R7Research
) is better
thanResearch
the current
Operations Research Operations Research Operations Research Operations Research
best optimal
of an
integer
solution,
and Operations
its solution
Operations
Research value
Operations
Research
Operations
Research
Researchis fracOperations
Operations
Operations
Operations
Research
tional,
we Research
create two
new Research
sub-problems
(PResearch
)
and
(
P
)
branch9
10 , by
Operations Research Operations Research Operations Research Operations Research
ingOperations
on theResearch
variableOperations
x1 : one
with the
additional
x1
3,
Research
Operations
Research constraint
Operations Research
Operations
Research
the other
with Operations
x1 4. Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
max Research
z = 17xOperations
max
17x1 +
12x2
1 + 12xResearch
2
Operations
Operations
Research z =
Operations
Research
Operations
Research
Operations
Research
Operations
Research
Operations
Research
s. t. 10x1 + 7x2
40
s. t. 10x1 + 7x2
40
Operations Research Operations Research Operations Research Operations Research
x
+
2x
5
x
+
2x
5
Operations Research 1 Operations
Operations Research Operations
Research
2 Research
1
2
(R9 ) Operations Research Operations Research(ROperations
10 )
Research
Operations
Research
x2
1
x2
1
Operations Research Operations Research Operations Research Operations Research
x
=
3
x
Operations Research Operations
Research Operations Research Operations
1
1 Research 4
Operations Research x
Operations
Research
Operations
Research
Operations
Research
; x2
0
x1 ; x2
0
Operations Research 1 Operations
Research Operations Research Operations
Research
Operations Research Operations Research Operations Research Operations Research
6
>
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:
Olariu E. Florentin
6
6
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>
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>
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>
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Operations Research - Course 7
6
6
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November 15, 2016
22 / 35
Branch-and-Bound Method - Example
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Research
Operations
Research
The
problem
(R9 )Operations
has the
optimal
solution
x = Operations
(3; 1)T ,Research
and the
Operations Research Operations
Research Operations Research Operations Research
optimal
value
z
=
63;
this
problem
is
pruned
by
integrality
Operations Research Operations Research Operations Research Operations Researchbut
Operations Research Operations Research Operations Research Operations Research
itsOperations
optimal
value is no better than the current best optimal value
Research Operations Research Operations Research Operations Research
which Research
is 65. Operations Research Operations Research Operations Research
Operations
Operations Research Operations Research Operations Research Operations
Research
The problem
the optimal
solution
= (4; 0Research
)T , and the
10 ) has
Operations
Research (R
Operations
Research
Operations
Research xOperations
Operations
Research
Research
Operations Research
optimal
value
z = Operations
68; thisResearch
problemOperations
is pruned
by integrality
and its
Operations Research Operations Research Operations Research Operations Research
optimal
value
is
better
than
the
current
best
optimal
value
which
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations
Research
Operations
Research
Operations
Research
is 65.
Operations Research Operations Research Operations Research Operations Research
Operations
Operations
Research
Research for
Operations
Research value
z = 68Research
becomes
the new
best Operations
lower bound
the optimal
Operations Research Operations Research Operations Research Operations Research
of original
problem
(P )Research
and theOperations
currentResearch
best solution
Operations
Research
Operations
Operationsbecomes
Research x =
(4;Operations
0)T . Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
23 / 35
Branch-and-Bound Method - Example
Operations Research Operations Research Operations Research Operations Research
x1 = 1.67,Research
x2 = 3.33
Operations Research Operations
Operations Research Operations Research
z = 68.33
Operations Research Operations
Operations Research Operations Research
(R) Research
1
2
Operations Research x1 ≤Operations
Research x1 ≥Operations
Research Operations Research
Operations Researchx = 1,Operations
Research Operations
Research Operations Research
x2 = 4
x1 = 2, x2 = 2.86
1
Operations Research Operations Research Operations Research Operations Research
z = 68.29
z = 65
(R1 )
2)
Operations Research
Operations Research (ROperations
Research Operations Research
x2 ≤ 2
x2 ≥ 3
Operations Research Operations Research
Operations Research
Operations Research
Operations Research Operationsx1Research
Operations Research
= 2.6, x2 = 2 Operations Research
inf easible
Operations Research Operations
Research Operations
Research
Operations Research
(R3 ) z = 68.2
(R4 )
Operations Research Operations
Research
Operations
Research
Operations Research
x1 ≤ 2
x1 ≥ 3
Operations Research Operations Research Operations Research Operations Research
x1 = 3, x2 = 1.43
x2 = 2
Operations Researchx1 = 2,Operations
Research Operations
Research Operations Research
z = 58 Operations Research
= 68.14
Operations (R
Research
Research Operations Research
(R6 ) z Operations
5)
x2 ≤ 1
x2 ≥ 2 Operations Research
Operations Research Operations Research
Operations Research
Operations Research Operations
Research Operations Research Operations Research
x1 = 3.3, x2 = 1
inf easible
Operations Research Operations Research Operations Research
Operations Research
68.1
(R7 ) z = Research
(R8Research
)
Operations Research Operations
Operations
Operations Research
x1 ≤ 3
x1 ≥ 4
Operations Research Operations Research Operations Research Operations Research
Operations Research
Research x1 =Operations
Research Operations Research
x1 = 3, x2 =Operations
1
4, x2 = 0
Operations Research
Research (ROperations
Operations Research
z Operations
= 63
z = 68 Research
(R9 )
10 )
Operations Research Operations Research Operations Research Operations Research
Operations Research Figure:
OperationsThe
Research
Research
final Operations
enumeration
tree.Operations Research
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
24 / 35
Notes Regarding the Algorithm
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Research from
Operations
Research example
Operationsthat
Research
Operations
Research a
WeOperations
can observe
the above
we have
to traverse
Operations Research Operations Research Operations Research Operations Research
binary
tree,
havingOperations
on its Research
nodes LP
problems.
Operations
Research
Operations
Research Operations Research
Operations Research Operations Research Operations Research Operations Research
After
we solve such a problem, the corresponding node becomes a
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations
Research
Operations
Research
leaf (when
the Operations
problemResearch
is pruned)
or it
expands
creating
two new
Operations Research Operations Research Operations Research Operations Research
children.
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations
Research
Research Operations Research
Usually
the
traversal
strategy
usedOperations
is depth-first-search.
Operations Research Operations Research Operations Research Operations Research
Researchdiscuss
Operations
Research
Operations
WeOperations
will shortly
theResearch
reasons Operations
for using
dfs strategy
inResearch
building
Operations Research Operations Research Operations Research Operations Research
and
traversing
the
enumeration
tree.
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
25 / 35
Notes Regarding the Algorithm
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operationsshows
Researchthat
Operations
Research solutions
Operations Research
Empirical
knowledge
most integer
of the origiOperations Research Operations Research Operations Research Operations Research
nal problem
deep inResearch
the tree.Operations
The advantages
for finding
Operations
Research lie Operations
Research Operations
Researchinteger
Operations early
Researchare:Operations Research Operations Research Operations Research
solutions
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations
Research
Operations
Research
Operations
Research
It is better
to have
at least
a feasible
solution
for the
original
problem
Operations Research Operations Research Operations Research Operations Research
(in case that we want to abort the tree building),
Operations Research Operations Research Operations Research Operations Research
Secondly,
such a feasible
solution
result Research
in subsequent
Operations
Research identifying
Operations Research
Operations
Research may
Operations
Operations
Operations
Research of Operations
Operations Research
nodesResearch
become leaves
(because
pruningResearch
by bound).
Operations Research Operations Research Operations Research Operations Research
Operations
Operations
Research based
Operations
It Operations
is very Research
simple to
codeResearch
a recursive
algorithm
onResearch
the dfs
Operations Research Operations Research Operations Research Operations Research
strategy.
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
While
building
tree, the
subsequent
problems
are Research
obtained
Operations
ResearchtheOperations
Research
OperationsLP
Research
Operations
Operations
Research
Operations
Research
Operations
Research
Operations
Research
by adding or refining a bound on one specific variable. This leads us
Operations Research Operations Research Operations Research Operations Research
to the Research
observation
thatResearch
one canOperations
reuse Research
the optimal
simplex
tableau
Operations
Operations
Operations
Research
Research
Research
Operationsproblems.
Research
forOperations
an LP Research
problemOperations
for solving
one Operations
of its two
subsequent
Operations Research Operations Research Operations Research Operations Research
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
26 / 35
Branch-and-Bound Algorithm
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
The
algorithm builds a tree having nodes N ; N1 ; : : : Research
Operations Research Operations Research Operations Research0 Operations
Researchan Operations
Research Operations
Research
ForOperations
every Research
node Ni Operations
we associate
MILP problem
(Pi ) and
its natOperations Research Operations Research Operations Research Operations Research
ural
linearResearch
relaxation
(Ri )Research
.
Operations
Operations
Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
ToOperations
the root
N0 we Operations
associate
the original
problem
) and Research
its relaxResearch
Research
Operations
Research (P
Operations
Operations
Research
Operations
Research
Operations
Research
Operations
Research
ation (R ).
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations
Operations
Research Operations Research
zi is the
optimal
value Research
of (Ri ) (if
feasible).
Operations Research Operations Research Operations Research Operations Research
z is theResearch
best-so-far
lower
boundOperations
for theResearch
optimalOperations
value of
(P ), z .
Operations
Operations
Research
Research
Operations Research Operations Research Operations Research Operations Research
(x ; y Research
) is the best-so-far
feasible
solution
for (P
).
Operations
Operations Research
Operations
Research
Operations
Research
Operations Research Operations Research Operations Research Operations Research
is
a
list
containing
the
nodes
that
must
be
still
be
solved.
Usually
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations Research Operations Research
would be
a stackOperations
(for a Research
dfs strategy).
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
L
L
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
27 / 35
Branch-and-Bound Algorithm
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research =Operations
Operations Research.
Operations Research
=
; (x ; y ) =
0. Initialize.
N0 ; zResearch
Operations Research Operations
Research Operations Research Operations Research
Research If Operations
Operations
Research
1. Operations
Terminate?
= ,Research
then (x Operations
; y ) is Research
an optimal
solution.
Operations Research Operations Research Operations Research Operations Research
OperationsaResearch
Research
2. Operations
Select Research
node. Choose
node NiOperations
in and
deleteOperations
it fromResearch
.
Operations Research Operations Research Operations Research Operations Research
3. Operations
Bound.Research
If (Ri )Operations
is infeasible,
to stepResearch
1. Otherwise
let
(xi ; yi ) be
Research goOperations
Operations
Research
Operations Research Operations Research Operations Research Operations Research
an optimal solution to (Ri ), and
zi its Research
optimal Operations
value. Research
Operations Research Operations Research
Operations
Operations
Research
Operations Research
Research to
4. Prune.
IfResearch
zi
z , Operations
go to step
1. Otherwise,
if (xi ; Operations
yi ) is feasible
Operations Researchi Operations
Research Operations Research Operationsi Research
n
i
(POperations
) (i. e.,Research
x
), then Research
set z = zOperations
x ; y ) =Operations
(x ; y Research
), and go
Operations
i and (Research
Operations
Research
Operations
Research
Operations
Research
Operations
Research
to the step 1.
Operations Research Operations Research Operations Research Operations Research
Research
Operations
Operations
Research
Operations Research
5. Operations
Branch.
Choose
xji Research
. Create
two new
sub-problems
(Pi1 ) and
Operations Research Operations Research Operations Research Operations Research
branching
on Research
the variable
xj : Research
one withOperations
the additional
(Pi2 ), by
Operations
Research
Operations
Operations
Research conOperations
Operations
Research
Operations
Research
straint
xj Research
xji , the
other
with xjOperations
xji Research
. Add the
corresponding
Operations Research Operations Research Operations Research Operations Research
nodes
Ni1 Research
and Ni2 Operations
to , and
go toOperations
the step
1.
Operations
Research
Research
Operations Research
Operations Research Operations Research Operations Research Operations Research
L f g
L ?
1
?
L
6
2Z
L
62 Z
6b c
Olariu E. Florentin
L
>d e
Operations Research - Course 7
November 15, 2016
28 / 35
Cutting Plane Method
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations Research
Operations Research
The
cutting
plane Operations
methodResearch
was introduced
by R. Gomory
(1958).
Operations Research Operations Research Operations Research Operations Research
Operations
Operations
Research
Operations Research
The
basicResearch
idea is Operations
to solveResearch
the natural
linear
LP relaxation
of the
Operations Research Operations Research Operations Research Operations Research
(pure)
ILP
problem.
If
the
resulting
solution
is
integer
(that
is, all
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations values),
Research then
Operations
Operations Research
variables
have integer
weResearch
are finished.
Operations Research Operations Research Operations Research Operations Research
Otherwise
we add
a constraint
cutsResearch
(eliminates)
some
infeasible
Operations
Research
Operations
Research that
Operations
Operations
Research
Operations
Research
Operations
Research
Operations
Research
Operations
Research
solutions (including the one just obtained).
Operations Research Operations Research Operations Research Operations Research
Operations
Research Operations
Research in
Operations
Research
The
new constraint
is constructed
such aResearch
way thatOperations
it not eliminate
Operations Research Operations Research Operations Research Operations Research
any
feasible
integerOperations
solution
of theOperations
originalResearch
problem.
Operations
Research
Research
Operations Research
Operations Research Operations Research Operations Research Operations Research
The
new problem
solvedResearch
and theOperations
processResearch
is iterated.
After
adding
Operations
Research is
Operations
Operations
Research
Operations
Research
Operations
Research
Operations
Research
Operations
Research
a number of constraints, we eventually find an optimal solution to
Operations Research Operations Research Operations Research Operations Research
the ILPResearch
problem.
Operations
Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
29 / 35
Cutting Plane Method
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Research
Operations
Research
Operations
Research
WeOperations
consider
again Operations
a (pure)Research
ILP problem,
this
time in
standard
form,
Operations Research Operations Research Operations Research Operations Research
and
its
natural
linear
relaxation:
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
maximize
z = cTResearch
x;
Operations Research Operations
Research Operations
Operations Research
Operations Research Operations
Research
Operations
subject to Ax = b; Research Operations Research
Operations Research Operations Research Operations Research Operations Research(13)
0
Operations Research Operations Research x Operations
Research Operations Research
Operations Research Operations Research OperationsnResearch Operations Research
x
:
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations
Research Operations
Operations Research
maximize
z = cTResearch
x;
Operations Research Operations Research Operations Research Operations Research
subject
to Operations
Ax = Research
b;
Operations Research Operations
Research
Operations Research(14)
Operations Research Operations Research x Operations
0: Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
>
2Z
>
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
30 / 35
A generic cutting plane algorithm
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Operations Research
Operations
Operations
ResearchLet
1. Solve.
Solve
the appropriate
natural
LP Research
relaxation
problem.
Operations Research Operations Research Operations Research Operations Research
x Operations
be an optimal
to this problem.
If the solution,
x , is an
Research solution
Operations Research
Operations Research
Operations Research
Operations
Research Operations Research Operations Research
integerResearch
vector, Operations
then stop.
Operations Research Operations Research Operations Research Operations Research
Research
Research
Operations
Research plane
Operations
Research sep2. Operations
Cutting
Plane.Operations
Otherwise
generate
a cutting
constraint
Operations Research Operations Research Operations Research Operations Research
aratingResearch
x fromOperations
the feasible
(a Research
constraint
whichResearch
is satisfied
Operations
Research region
Operations
Operations
Research
Operations
Operations
Research
byOperations
all integer
solutions
of Research
the problem,
but
not byOperations
x ). Research
Operations Research Operations Research Operations Research Operations Research
Operations
Operations
Operations
Research
Operations
3. Refine
theResearch
Problem.
AddResearch
the cutting
plane
constraint
toResearch
the ILP
Operations Research Operations Research Operations Research Operations Research
problem
go toOperations
step 1.Research Operations Research Operations Research
Operationsand
Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
31 / 35
Gomory fractional cut
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Operations Research
Operationssolution
Research to
Operations
Research (13)
Let x0Research
be an optimal
basic feasible
the problem
Operations Research Operations Research Operations Research Operations Research
with optimal
z0 . Research
SupposeOperations
also that
the base
corresponding
to
Operations
Research value
Operations
Research
Operations
Research
Operations
Research
this
solution
is B . Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research corresponding
Operations Research to
From
the Research
optimal Operations
tableauResearch
we get Operations
the equations
Operations Research Operations Research Operations Research Operations Research
basic
variables
Operations
Research Operations Research Operations Research Operations Research
1Operations Research
1
Operations Research Operations
Operations Research
xResearch
B + B NxN = B b:
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations
Research Operations Research Operations~Research 1
1
Define
a~ij Research
= (B Aj )i , for all i
B and j N , and b = B B.
Operations
Operations Research Operations Research Operations Research
Operations
Operations Research
Operations
If everyResearch
basic variable
xi has anOperations
integerResearch
value (that
is, Research
xi0
; i
Operations Research Operations Research Operations Research Operations
Research
B ), then
we areOperations
finished.
Operations
Research
Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
2
Olariu E. Florentin
Operations Research - Course 7
2
2Z 8 2
November 15, 2016
32 / 35
Gomory fractional cut
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
0
Operations
Research
Operations
Otherwise,
there
exist Research
an i BOperations
with xResearch
. Operations Research
i
Operations Research Operations Research Operations Research Operations Research
The equation
xi isOperations Research Operations Research
Operations
Research labeled
Operationswith
Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations
Research
Operations Research
0
xi + Research
a~hj xj =Operations
b~h = xResearch
:
i
Operations Research Operations
Operations Research
j N
Operations Research Operations Research
Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Since xResearch
0; jOperations
N , and
a~hj Operations
a~hj ; Research
j N , Operations
we mustResearch
have
j
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research
Research
Research
xi +Operations
a~hjResearch
xj xiOperations
+
a~hj xj = b~Operations
h
Operations Research Operations
Research
Operations
Research
Operations Research
j N
j N
Operations Research Operations Research Operations Research Operations Research
Operations Research nOperations Research Operations Research Operations Research
Now,
as xResearch, we
get Research Operations Research Operations Research
Operations
Operations
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations
Operations Research
xi +Researcha~hj Operations
xj
b~Research
(15)
h :
Operations Research Operations Research Operations Research Operations Research
j
N
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
2
62 Z
X
62 Z
2
> 8 2
b c6
X
b c 6
2
2Z
X
2
Olariu E. Florentin
8 2
X
2
b c 6b c
Operations Research - Course 7
November 15, 2016
33 / 35
Gomory fractional cut
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research (15)
Operations
Operations
Operations
Research of
The
inequation
mustResearch
be satisfied
by Research
every integer
solution
Operations Research Operations Research Operations Research Operations Research
theOperations
problem
(13). Operations Research Operations Research Operations Research
Research
Operations0Research Operations Research Operations
Research
Research
0
~ and xOperations
But
x doesn’t
it because
0 for all
j N,
j =
i = bh , Research
Operations
Researchsatisfy
Operations
Research xOperations
Operations
Research
~
Operations
Research
Operations
Research
Operations
Research
Operations
Research
and bh is fractional.
Operations Research Operations Research Operations Research Operations Research
Inequation
(15)Operations
is a Gomory
cut. Operations Research
Operations
Research
Research fractional
Operations Research
Operations Research Operations Research Operations Research Operations Research
It was Research
proved (for
a pure
ILP) Operations
that byResearch
systematically
Operations
Operations
Research
Operations adding
Research these
Operations
Research
Operations
Research
Operations
Research
Operations
Research
cuts, and using the (dual) simplex algorithm, the finite convergence
Operations Research Operations Research Operations Research Operations Research
of Operations
the above
method
is assured.
Research
Operations
Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
2
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
34 / 35
Bibliography
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Bertsimas,
D., J. N.
Tsitsiklis,
to Linear
OptimizaOperations Research
Operations
ResearchIntroduction
Operations Research
Operations
Research
Operations
Research Scientific,
Operations Research
Operations
Research Operations
tion, Athena
Belmont,
Massachusetts,
1997. Research
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Conforti,
M., G. Cornuejols, G. Zambelli, Integer Programming,
Operations Research Operations Research Operations Research Operations Research
Graduate
TextsOperations
in Mathematics,
Springer,
2014.Operations Research
Operations
Research
Research Operations
Research
Operations Research Operations Research Operations Research Operations Research
Operations
Research
Research
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Kolman,
B., R.Operations
E. Deck,
Elementary
Linear Operations
Programming
Operations Research Operations Research Operations Research Operations Research
Applications, Elsevier Science and Technology Books, 1995.
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Schrijver,
A., AOperations
Course
in Combinatorial
Operations
Research
Research
Operations Research Optimization,
Operations Research ElecOperations
Research
Operations
Research
Operations
Research
Operations Research
tronic Edition: homepages.cwi.nl/~lex/files/dict.pdf,
2013.
Operations Research Operations Research Operations Research Operations Research
Operations Research
OperationsProgramming
Research Operations
Research Operations
Research
Vanderbei,
R., J., Linear
- Foundations
and
ExtenOperations Research Operations Research Operations Research Operations Research
sions,
International
Series
in
Operations
Research
&
Management
Operations Research Operations Research Operations Research Operations Research
Operations
Operations
Research
Operations 2014.
Research Operations Research
Science,Research
Springer
Science,
4th edition,
Operations Research Operations Research Operations Research Operations Research
Operations Research Operations Research Operations Research Operations Research
Olariu E. Florentin
Operations Research - Course 7
November 15, 2016
35 / 35