Integer%20Programming

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Chapter 9
Binary Integer
Programming
Dr. Ardavan Asef-Vaziri Industrial Engineering
April 2002
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Computational Complexity
Computationally speaking, we can partition
problems into two categories. Easy Problems and
Hard Problems We can say that easy problem ( or
in some languages polynomial problems) are those
problems with their solution time proportional
to nk. Where n is the number of variables in the
problem, and k is a constant, say 2, 3, 4 Lets
assume that k 2. Therefore, easy problems are
those problems with their solution time
proportional to n2. Where n is the number of
variables in the problem. Difficult problems are
those problems with their solution time
proportional to kn, or in our case 2n.
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Computational Complexity
Now supposed we have an easy problem with n
variables. Therefore, the solution time is
proportional to n2 . We have solved this problem
in one hour using a computer with 2 GHz
CPU. Suppose we have a new computer with its
processing capabilities 10000 times of a 2 GHz
computer, and we have one century time. What is
the size of the largest problem that we can solve
with this revolutionary computer in one
century. 1 n2 (10000) (100)(10000) (nx)2
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Computational Complexity
1 n2 (10)10 (nx)2 1 (nx)2 (10)10
n2 (nx)2 / n2 (10)10 (nx / n ) 2
(10)10 nx / n (10)5 100000 nx
100,000n The number of variables in the new
problem is 100,000 times greater that the number
of variables in the old problem.
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Computational Complexity
Now supposed we have a hard problem with n
variables. Therefore, the solution time is
proportional to 2n . We have solved this problem
in one hour using a computer with 900 MHz
CPU. Suppose we have a new computer with 900,000
MHz CPU, and we have one century time. What is
the size of the largest problem that we can solve
in one century using a 900, 000 MHz CPU. 1
2n (10)10 2(nx)
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Computational Complexity
1 2n (10)10 2(nx) (1) 2(nx)
(10)102n 2(nx) / 2n (10)10 2x
(10)10 log 2x log (10)10 x log 2 10 log
(10) x ( .301) 10(1) x 33 The number of
variables in the new problem is 33 variables
greater that the number of variables in the old
problem.
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Working with Binary Variables
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Exactly one, at least one, at most one, none
1- Exactly one of the two projects is
selected 2- At least one of the two projects
is selected 3-At most one of the two projects
is selected 4- None of the projects should be
selected
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Both, at most 2
1- Both projects must be selected 2- none, or
one or both of projects are selected 3- If
project 1 is selected then project 2 must be
selected 4- If project 1 is selected then
project 2 could not be selected
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More Fun
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More Practice
Either project 12 or projects 345 are selected
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More on Binary Variables
Suppose y1 is our production of product i . Due
to technical considerations, we want one of the
following two constraints to be satisfied. If
the other one is also satisfied it does not
create any problem. If the other one is violated
it does not create any problem.
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One constraint out of two
We can produce 3 products but due to managerial
difficulties, we want to produce only two of
them. We can produce these products either in
plant one or plant two but not in both. In other
words, we should also decide whether we produce
them in plant one or plant two. Other
information are given below Required
hrs Available hrs Product 1 Product
2 Product 3 Plant 1 3 4 2
30 Plant 2 4
6 2 40 Unit profit 5
7 3 Sales potential 7 5
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Meeting a Subset of Constraints
Consider a linear program with the following set
of constraints 12x1 24x2 18x3 2400 15x1
32x2 12x3 1800 20x1 15x2 20x3
2000 18x1 21x2 15x3 1600 Suppose that
meeting 3 out of 4 of these constraints is good
enough.
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Still Fun
We want two of the following 4 constraints to be
satisfied. The other 2 are free, they may be
automatically satisfied, but if they are not
satisfied there is no problem y1 y2 ?
100 y1 2 y3 ? 160 y2 y3 ? 50 y1 y2 y3
? 170 In general we want to satisfy k
constraints out of n constraints
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A Constraint With k Possible Values
Mastering Formulation of Binary Variables
y1 y2 10 or 20 or 100
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A Location Allocation Problem
We have 5 demand centers, referred to as 1, 2, 3,
4,5. We plan to open one or more Distribution
Centers (DC) to serve these markets. There are 5
candidate locations for these DCs, referred to as
A, B, C, D, and E. Annual cost of meeting all
demand of each market from a DC located in each
candidate location is given below
DC1 DC2 DC3 DC4 DC5 Market
1 1 5 18 13 17 Market 2 15 2 26 14 14 Market
3 28 18 7 8 8 Market 4 100 120 8 8 9 Market
5 30 20 20 30 40 Each DC can satisfy the demand
of one, two, three, four, or five market
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A Location Allocation Problem
Depreciated initial investment and operating cost
of a DC in location A, B, C, D, and E is 199,
177, 96, 148, 111. The objective function is to
minimize total cost (Investment Operating and
Distribution) of the system. Suppose we want to
open only one DC. Where is the optimal
location. Suppose we don't impose any constraint
on the number of DCs. What is the optimal number
of DCs.
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Example 1 (Capital Budgeting)
Mercer Development is considering the potential
of four different development projects. Each
project would be completed in at most three
years. The required cash outflow for each project
is given in the table below, along with the net
present value of each project to Mercer, and the
cash that is available (from previous projects)
each year.
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Maximum Flow Problem D.V. and OF
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Maximum Flow Problem Arc Capacity Constraints
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Maximum Flow Problem Flow Balance Constraint
? tij ? tji ? i ? N \ O and D
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Maximum Flow Problem Flow Balance Constraint
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Maximum Flow Problem with Restricted Number of
Arcs
xij The decision variable for the directed arc
from node i to nod j. xij 1 if arc ij is on
the flow path xij 0 if arc ij is not on the
flow path ? xij ? 4
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Maximum Flow Problem with Restricted Number of
Arcs
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The relationship between flow and arc variables

• Divisibility
• 1.5, 500.3, 111.11
• Certainty
• cj, aij, bi
• Linearity
• No x1 x2, x12, 1/x1, sqrt (x1)
• aijxj
• aijxj aikxk
• Nonnegativity

t23 could be greater than 0 while x23 is 0,
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Relationship between Flow and Arcs
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On a Path
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Maximum Flow on a Path
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The Shortest Route Problem
The shortest route between two points l ij The
length of the directed arc ij. l ij is a
parameter, not a decision variable. It could be
the length in term of distance or in terms of
time or cost ( the same as c ij ) For
those nodes which we are sure that we go from i
to j we only have one directed arc from i to j.
For those node which we are not sure that we go
from i to j or from j to i, we have two
directed arcs, one from i to j, the other from j
to i. We may have symmetric or asymmetric
network. In a symmetric network lij lji ?
ij In a asymmetric network this condition does
not hold
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Example
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Decision Variables and Formulation
xij The decision variable for the directed arc
from node i to nod j. xij 1 if arc ij is on
the shortest route xij 0 if arc ij is not
on the shortest route ? xij - ? xji 0
? i ? N \ O and D ? xoj 1 ? xiD 1
Min Z ? lij xij
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Example
x13 x14 x12 1 - x57 - x67 -1 x34 x35
- x43 - x13 0 x42 x43 x45 x46 -
x14 - x24 - x34 0 . .. Min Z
5x12 4x13 3x14 2x24 6x26 2x34 3x35
2x43 2x42 5x45 4x46 3x56 2x57 3x65
2x67
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The ShR Problem Binary Decision Variables
Find the shortest route of these two networks.
But for red bi-directional edges you are not
allowed to define two decision variables. Only
one. Solve the small problem. Only using 5
variables
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The ShR Problem Binary Decision Variables
2
1
4
3
Do not worry about the length of the arcs, we do
not need to write the objective functions. Note
that we do not know whether we may go from node
2 to 3 or from 3 to 2. Now we want to
formulate this problem as a shortest route.
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The ShR Problem Binary Decision Variables
To formulate the problem as a shortest route, you
probably want to define a pair of decision
variables for arcs 2-3 and 3-2, and then for
example write the constraint on node 2 as
follows X23X24 X12 X32 But you are
not allowed to define two variables for 2-3 and
3-2. You should formulated the problem only
using the following variables X23 is a binary
variable corresponding to the NON-DIRECTED edge
between nodes 2 and 3. It is equal to 1 if
arc 2-3 or 3-2 is on the shortest route and it
is 0 otherwise.
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The ShR Problem Smaller Number of Binary
Variables
As usual we can have the following variables
X12 X13 X24 X34 each corresponding to a
directed arc The solution of (X12 1 , X23
1, X34 1 all other variables are 0) means
that the shortest route is 1-2-3-4. The
solution of (X13 1 , X32 1, X24 1 all
other variables are 0) ) means that the shortest
route is 1-3-2-4. The solution of (X13 1 ,
X34 1 all other variables are 0) ) means that
the shortest route is 1-3-4.
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The ShR Problem Smaller Number of Binary
Variables
Now you should formulate the shortest route
problem using defining only one variable for
each edge. Your formulation should be general,
not only for this example. How many variables
do you need to formulate this problem?
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How Many Binary Variables for this ShR problem
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Integer Programming

• When are non-integer solutions okay?
• Solution is naturally divisible
• Solution represents a rate
• Solution only for planning purposes
• When is rounding okay?

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The Challenge of Rounding

• Rounded solution may not be feasible
• Rounded solution may not be close to optimal
• There can be many rounded solutions

Example Consider a problem with 30 variables
that are non-integer in the LP solution. How many
possible rounded solutions are there?
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How Integer Programs are solved
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Spreadsheet Solution to Example 1
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Example 2 (Set Covering Problem)
Suppose the Washington State legislature is
trying to decide on locations at which to base
search-and-rescue teams. The teams are expensive,
and hence they would like as few as possible.
However, since response time is critical, they
would like every county to either have a team
located in that county, or in an adjacent county.
Where should the teams be located?
Counties
1. Clallum 2. Jefferson 3. Grays Harbor
4. Pacific 5. Wahkiakum 6. Kitsap 7.
Mason 8. Thurston 9. Whatcom 10.
Skagit 11. Snohomish 12. King 13. Pierce 14.
Lewis 15. Cowlitz 16. Clark 17. Skamania 18.
Okanogan 19. Chelan
20. Douglas 21. Kittitas 22. Grant 23.
Yakima 24. Klickitat 25. Benton 26. Ferry 27.
Stevens 28. Pend Oreille 29. Lincoln 30.
Spokane 31. Adams 32. Whitman 33. Franklin 34.
Walla Walla 35. Columbia 36. Garfield 37.
Asotin
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Spreadsheet Solution to Example 2
Counties
1. Clallum 2. Jefferson 3. Grays Harbor
4. Pacific 5. Wahkiakum 6. Kitsap 7.
Mason 8. Thurston 9. Whatcom 10.
Skagit 11. Snohomish 12. King 13. Pierce 14.
Lewis 15. Cowlitz 16. Clark 17. Skamania 18.
Okanogan 19. Chelan
20. Douglas 21. Kittitas 22. Grant 23.
Yakima 24. Klickitat 25. Benton 26. Ferry 27.
Stevens 28. Pend Oreille 29. Lincoln 30.
Spokane 31. Adams 32. Whitman 33. Franklin 34.
Walla Walla 35. Columbia 36. Garfield 37.
Asotin

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Spreadsheet Solution to Example 2
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Spreadsheet Solution to Example 2 (Formulas)