Integer Programming

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Introduction to Management Science 8th
Edition by Bernard W. Taylor III
Chapter 5 Integer Programming
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Chapter Topics

• Integer Programming (IP) Models
• Integer Programming Graphical Solution
• Computer Solution of Integer Programming Problems
  With Excel and QM for Windows

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Integer Programming Models Types of Models

• Maximize Z 10x1 15x2
• subject to 8x1 4x2 ? 40
• 15x1 30x2 ? 200
• Total Integer Model All decision variables
  required to have integer solution values (e.g.,
  x1, x2 ? 0 and integer).
• 0-1 Integer Model All decision variables
  required to have integer values of zero or one
  (e.g., x1, x2 0 or 1).
• Mixed Integer Model Some of the decision
  variables (but not all) required to have integer
  values (e.g., x1? 0 and integer, x2 ? 0 ).

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Total Integer Model (1 of 2)

• Machine shop obtaining new presses and lathes.
• Marginal profitability each press 100/day
  each lathe 150/day.
• Resource constraints 40,000, 200 sq. ft. floor
  space.
• Machine purchase prices and space requirements

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Total Integer Model (2 of 2)
Integer Programming Model Define x1 number
of presses x2 number of
lathes Maximize Z 100x1 150x2
subject to 8,000x1 4,000x2 ?
40,000 15x1 30x2 ?
200 ft2 x1, x2 ? 0
and integer
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0 - 1 Integer Model (1 of 2)

• Recreation facilities selection to maximize daily
  usage by residents.
• Resource constraints 120,000 budget 12 acres
  of land.
• Selection constraint either swimming pool or
  tennis center (not both).
• Data

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0 - 1 Integer Model (2 of 2)
Integer Programming Model Define x1
construction of a swimming pool
x2 construction of a tennis center
x3 construction of an athletic field
x4 construction of a
gymnasium Maximize Z 300x1 90x2 400x3
150x subject to 35,000x1 10,000x2
25,000x3 90,000x4 ? 120,000
4x1 2x2 7x3 3x3 ? 12 acres
x1 x2 ? 1 facility x1,
x2, x3, x4 0 or 1
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Mixed Integer Model (1 of 2)

• 250,000 available for investments providing
  greatest return after one year.
• Data
• Condominium cost 50,000/unit, 9,000 profit if
  sold after one year.
• Land cost 12,000/ acre, 1,500 profit if sold
  after one year.
• Municipal bond cost 8,000/bond, 1,000 profit if
  sold after one year.
• Only 4 condominiums, 15 acres of land, and 20
  municipal bonds available.

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Mixed Integer Model (2 of 2)
Integer Programming Model Maximize Z
9,000x1 1,500x2 1,000x3 subject
to 50,000x1 12,000x2 8,000x3 ?
250,000 x1 ? 4 condominiums
x2 ? 15 acres x3 ? 20 bonds
x2 ? 0 x1, x3 ? 0 and
integer x1 condominiums
purchased x2 acres of land
purchased x3 bonds purchased
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Integer Programming Graphical Solution

• Rounding non-integer solution values up to the
  nearest integer value can result in an infeasible
  solution
• A feasible solution is ensured by rounding down
  non-integer solution values but may result in a
  less than optimal (sub-optimal) solution.

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Integer Programming Example Graphical Solution of
Maximization Model
Maximize Z 100x1 150x2 subject to
8,000x1 4,000x2 ? 40,000 15x1 30x2
? 200 ft2 x1, x2 ? 0 and integer Optimal
Solution Z 1,055.56 x1 2.22 presses x2
5.55 lathes
Figure 5.1 Feasible Solution Space with Integer
Solution Points
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Branch and Bound Method

• Traditional approach to solving integer
  programming problems.
• Based on principle that total set of feasible
  solutions can be partitioned into smaller
  subsets of solutions.
• Smaller subsets evaluated until best solution is
  found.
• Method is a tedious and complex mathematical
  process.
• Excel and QM for Windows used in this book.
• See CD-ROM Module C Integer Programming the
  Branch and Bound Method for detailed description
  of method.

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Computer Solution of IP Problems 0 1 Model with
Excel (1 of 5)
Recreational Facilities Example Maximize Z
300x1 90x2 400x3 150x4 subject to
35,000x1 10,000x2 25,000x3 90,000x4 ?
120,000 4x1 2x2 7x3 3x3 ?
12 acres x1 x2 ? 1 facility
x1, x2, x3, x4 0 or 1
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Computer Solution of IP Problems 0 1 Model with
Excel (2 of 5)
Exhibit 5.2
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Computer Solution of IP Problems 0 1 Model with
Excel (4 of 5)
Exhibit 5.4
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Computer Solution of IP Problems 0 1 Model with
Excel (3 of 5)
Exhibit 5.3
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Computer Solution of IP Problems 0 1 Model with
Excel (5 of 5)
Exhibit 5.5
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Computer Solution of IP Problems 0 1 Model with
QM for Windows (1 of 3)
Recreational Facilities Example Maximize Z
300x1 90x2 400x3 150x4 subject to
35,000x1 10,000x2 25,000x3 90,000x4 ?
120,000 4x1 2x2 7x3 3x3 ?
12 acres x1 x2 ? 1 facility
x1, x2, x3, x4 0 or 1
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Computer Solution of IP Problems 0 1 Model with
QM for Windows (2 of 3)
Exhibit 5.6
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Computer Solution of IP Problems 0 1 Model with
QM for Windows (3 of 3)


Exhibit 5.7
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0 1 Integer Programming Modeling
Examples Capital Budgeting Example (1 of 3)

• University bookstore expansion project.
• Not enough space available for both a computer
  department and a clothing department.
• Data

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0 1 Integer Programming Modeling
Examples Capital Budgeting Example (2 of 3)
x1 selection of web site project x2 selection
of warehouse project x3 selection clothing
department project x4 selection of computer
department project x5 selection of ATM
project xi 1 if project i is selected, 0 if
project i is not selected Maximize Z 120x1
85x2 105x3 140x4 70x5 subject to
55x1 45x2 60x3 50x4 30x5 ? 150
40x1 35x2 25x3 35x4 30x5 ? 110
25x1 20x2 30x4 ? 60 x3 x4 ? 1
xi 0 or 1
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0 1 Integer Programming Modeling
Examples Capital Budgeting Example (2 of 3)
What additional constraint/s are needed for the
following a) If the website is chosen, then the
ATMs must be selected? b) The clothing
department can be selected only if the
computer department is chosen. c) If both the
clothing and computer departments are chosen,
then the warehouse must be built?
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0 1 Integer Programming Modeling Examples Fixed
Charge and Facility Example (1 of 4)

• Which of six farms should be purchased that will
  meet current production capacity at minimum total
  cost, including annual fixed costs and shipping
  costs?
• Data

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0 1 Integer Programming Modeling Examples Fixed
Charge and Facility Example (2 of 4)
yi 0 if farm i is not selected, and 1 if farm i
is selected, i 1,2,3,4,5,6 xij potatoes
(tons, 1000s) shipped from farm i, i
1,2,3,4,5,6 to plant j, j A,B,C. Minimize Z
18x1A 15x1B 12x1C 13x2A 10x2B 17x2C
16x3A 14x3B 18x3C
19x4A 15x4b 16x4C 17x5A 19x5B
12x5C 14x6A 16x6B 12x6C 405y1
390y2 450y3 368y4 520y5
465y6 subject to x1A x1B x1B -
11.2y1 lt 0 x2A x2B x2C -10.5y2 lt 0
x3A x3A x3C - 12.8y3 lt 0 x4A
x4b x4C - 9.3y4 lt 0 x5A x5B x5B -
10.8y5 lt 0 x6A x6B X6C - 9.6y6 lt 0
x1A x2A x3A x4A x5A x6A 12
x1B x2B x3A x4b x5B x6B 10
x1B x2C x3C x4C x5B x6C 14 xij
0 yi 0 or 1
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0 1 Integer Programming Modeling Examples Fixed
Charge and Facility Example (3 of 4)
Exhibit 5.18
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0 1 Integer Programming Modeling Examples Fixed
Charge and Facility Example (4 of 4)
Exhibit 5.19
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0 1 Integer Programming Modeling Examples Set
Covering Example (1 of 4)

• APS wants to construct the minimum set of new
  hubs in the following twelve cities such that
  there is a hub within 300 miles of every city

Cities Cities
within 300 miles 1. Atlanta Atlanta, Charlotte,
Nashville 2. Boston Boston, New York 3.
Charlotte Atlanta, Charlotte, Richmond 4.
Cincinnati Cincinnati, Detroit, Nashville,
Pittsburgh 5. Detroit Cincinnati, Detroit,
Indianapolis, Milwaukee, Pittsburgh 6.
Indianapolis Cincinnati, Detroit, Indianapolis,
Milwaukee, Nashville, St. Louis 7.
Milwaukee Detroit, Indianapolis, Milwaukee 8.
Nashville Atlanta, Cincinnati, Indianapolis,
Nashville, St. Louis 9. New York Boston, New
York, Richmond 10. Pittsburgh Cincinnati,
Detroit, Pittsburgh, Richmond 11.
Richmond Charlotte, New York, Pittsburgh,
Richmond 12. St. Louis Indianapolis, Nashville,
St. Louis
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0 1 Integer Programming Modeling Examples Set
Covering Example (2 of 4)
xi city i, i 1 to 12, xi 0 if city is not
selected as a hub and xi 1if it is. Minimize Z
x1 x2 x3 x4 x5 x6 x7 x8 x9
x10 x11 x12 subject to Atlanta x1 x3
x8 ? 1 Boston x2 x10 ? 1 Charlotte x1
x3 x11 ? 1 Cincinnati x4 x5 x8 x10 ?
1 Detroit x4 x5 x6 x7 x10 ?
1 Indianapolis x4 x5 x6 x7 x8 x12 ?
1 Milwaukee x5 x6 x7 ? 1 Nashville x1
x4 x6 x8 x12 ? 1 New York x2 x9 x11 ?
1 Pittsburgh x4 x5 x10 x11 ? 1 Richmond
x3 x9 x10 x11 ? 1 St Louis x6 x8
x12 ? 1 xij 0 or 1
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0 1 Integer Programming Modeling Examples Set
Covering Example (3 of 4)
Exhibit 5.20
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0 1 Integer Programming Modeling Examples Set
Covering Example (4 of 4)
Exhibit 5.21
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Total Integer Programming Modeling
Example Problem Statement (1 of 3)

• Textbook company developing two new regions.
• Planning to transfer some of its 10 salespeople
  into new regions.
• Average annual expenses for sales person
• Region 1 - 10,000/salesperson
• Region 2 - 7,500/salesperson
• Total annual expense budget is 72,000.
• Sales generated each year
• Region 1 - 85,000/salesperson
• Region 2 - 60,000/salesperson
• How many salespeople should be transferred into
  each region in order to maximize increased sales?

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Total Integer Programming Modeling Example Model
Formulation (2 of 3)
Step 1 Formulate the Integer Programming
Model Maximize Z 85,000x1
60,000x2 subject to x1 x2 ? 10
salespeople 10,000x1
7,000x2 ? 72,000 expense budget
x1, x2 ? 0 or integer Step 2 Solve the
Model using QM for Windows
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Total Integer Programming Modeling
Example Solution with QM for Windows (3 of 3)