1chapter 5 - integer programming chapter 5 integer programming introduction to management science 9...
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1Chapter 5 - Integer Programming
Chapter 5
Integer Programming
Introduction to Management Science
9th Edition
by Bernard W. Taylor III
© 2007 Pearson Education
2Chapter 5 - Integer Programming
Chapter TopicsChapter Topics
Integer Programming (IP) ModelsInteger Programming (IP) Models
Integer Programming Graphical SolutionInteger Programming Graphical Solution
Computer Solution of Integer Programming Problems Computer Solution of Integer Programming Problems With Excel and QM for WindowsWith Excel and QM for Windows
3Chapter 5 - Integer Programming
Integer Programming ModelsTypes of Models
Total Integer Model: All decision variables required to have integer solution values.
0-1 Integer Model: All decision variables required to have integer values of zero or one.
Mixed Integer Model: Some of the decision variables (but not all) required to have integer values.
4Chapter 5 - Integer Programming
Machine
Required Floor Space (sq. ft.)
Purchase Price
Press Lathe
15
30
$8,000
4,000
A 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:
5Chapter 5 - Integer Programming
A Total Integer Model (2 of 2)
Integer Programming Model:
Maximize Z = $100x1 + $150x2
subject to:
8,000x1 + 4,000x2 $40,000
15x1 + 30x2 200 ft2
x1, x2 0 and integer
x1 = number of presses x2 = number of lathes
6Chapter 5 - Integer Programming
Recreation facilities selection to maximize daily usage by residents.
Resource constraints: $120,000 budget; 12 acres of land.
Selection constraint: either swimming pool or tennis center (not both).
Data:
Recreation Facility
Expected Usage (people/day)
Cost ($) Land
Requirement (acres)
Swimming pool Tennis Center Athletic field Gymnasium
300 90 400 150
35,000 10,000 25,000 90,000
4 2 7 3
A 0 - 1 Integer Model (1 of 2)
7Chapter 5 - Integer Programming
Integer Programming Model:
Maximize Z = 300x1 + 90x2 + 400x3 + 150x4
subject to:
$35,000x1 + 10,000x2 + 25,000x3 + 90,000x4 $120,000
4x1 + 2x2 + 7x3 + 3x4 12 acres
x1 + x2 1 facility
x1, x2, x3, x4 = 0 or 1
x1 = construction of a swimming pool x2 = construction of a tennis center x3 = construction of an athletic field x4 = construction of a gymnasium
A 0 - 1 Integer Model (2 of 2)
8Chapter 5 - Integer Programming
A 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.
9Chapter 5 - Integer Programming
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
A Mixed Integer Model (2 of 2)
10Chapter 5 - Integer Programming
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.
Integer Programming Graphical Solution
11Chapter 5 - Integer Programming
Integer Programming ExampleGraphical 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 = $1,055.56x1 = 2.22 pressesx2 = 5.55 lathes
Figure 5.1 Feasible Solution Space with Integer Solution Points
12Chapter 5 - Integer Programming
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.
13Chapter 5 - Integer Programming
Recreational Facilities Example:
Maximize Z = 300x1 + 90x2 + 400x3 + 150x4
subject to:
$35,000x1 + 10,000x2 + 25,000x3 + 90,000x4 $120,000
4x1 + 2x2 + 7x3 + 3x4 12 acres
x1 + x2 1 facility
x1, x2, x3, x4 = 0 or 1
Computer Solution of IP Problems0 – 1 Model with Excel (1 of 5)
14Chapter 5 - Integer Programming
Exhibit 5.2
Computer Solution of IP Problems0 – 1 Model with Excel (2 of 5)
15Chapter 5 - Integer ProgrammingExhibit 5.3
Computer Solution of IP Problems0 – 1 Model with Excel (3 of 5)
16Chapter 5 - Integer Programming
Exhibit 5.4
Computer Solution of IP Problems0 – 1 Model with Excel (4 of 5)
17Chapter 5 - Integer ProgrammingExhibit 5.5
Computer Solution of IP Problems0 – 1 Model with Excel (5 of 5)
18Chapter 5 - Integer Programming
Computer Solution of IP Problems0 – 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 + 3x4 12 acres
x1 + x2 1 facility
x1, x2, x3, x4 = 0 or 1
19Chapter 5 - Integer Programming
Computer Solution of IP ProblemsTotal Integer Model with Excel (1 of 5)
Integer Programming Model:
Maximize Z = $100x1 + $150x2
subject to:
8,000x1 + 4,000x2 $40,000
15x1 + 30x2 200 ft2
x1, x2 0 and integer
20Chapter 5 - Integer ProgrammingExhibit 5.8
Computer Solution of IP ProblemsTotal Integer Model with Excel (2 of 5)
21Chapter 5 - Integer Programming
Exhibit 5.9
Computer Solution of IP ProblemsTotal Integer Model with Excel (4 of 5)
22Chapter 5 - Integer Programming
Exhibit 5.10
Computer Solution of IP ProblemsTotal Integer Model with Excel (3 of 5)
23Chapter 5 - Integer Programming
Exhibit 5.11
Computer Solution of IP ProblemsTotal Integer Model with Excel (5 of 5)
24Chapter 5 - Integer Programming
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
Computer Solution of IP ProblemsMixed Integer Model with Excel (1 of 3)
25Chapter 5 - Integer ProgrammingExhibit 5.12
Computer Solution of IP ProblemsTotal Integer Model with Excel (2 of 3)
26Chapter 5 - Integer Programming
Exhibit 5.13
Computer Solution of IP ProblemsSolution of Total Integer Model with Excel (3 of 3)
27Chapter 5 - Integer Programming
Exhibit 5.14
Computer Solution of IP ProblemsMixed Integer Model with QM for Windows (1 of 2)
28Chapter 5 - Integer Programming
Exhibit 5.15
Computer Solution of IP ProblemsMixed Integer Model with QM for Windows (2 of 2)
29Chapter 5 - Integer Programming
University bookstore expansion project.
Not enough space available for both a computer department and a clothing department.
Data:
Project NPV Return
($1000) Project Costs per Year ($1000)
1 2 3
1. Website 2. Warehouse 3. Clothing department 4. Computer department 5. ATMs Available funds per year
120 85 105 140 75
55 45 60 50 30
150
40 35 25 35 30
110
25 20 -- 30 --
60
0 – 1 Integer Programming Modeling ExamplesCapital Budgeting Example (1 of 4)
30Chapter 5 - Integer Programming
x1 = selection of web site projectx2 = selection of warehouse projectx3 = selection clothing department projectx4 = selection of computer department projectx5 = selection of ATM projectxi = 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
0 – 1 Integer Programming Modeling ExamplesCapital Budgeting Example (2 of 4)
31Chapter 5 - Integer Programming
Exhibit 5.16
0 – 1 Integer Programming Modeling ExamplesCapital Budgeting Example (3 of 4)
32Chapter 5 - Integer Programming
Exhibit 5.17
0 – 1 Integer Programming Modeling ExamplesCapital Budgeting Example (4 of 4)
33Chapter 5 - Integer Programming
Plant
Available Capacity
(tons,1000s) A B C
12 10 14
Farms Annual Fixed Costs
($1000)
Projected Annual Harvest (tons, 1000s)
1 2 3 4 5 6
405 390 450 368 520 465
11.2 10.5 12.8 9.3 10.8 9.6
Farm
Plant A B C
1 2 3 4 5 6
18 15 12 13 10 17 16 14 18 19 15 16 17 19 12 14 16 12
0 – 1 Integer Programming Modeling ExamplesFixed 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:Shipping Costs
34Chapter 5 - Integer Programming
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 ≤ 0 x2A + x2B + x2C -10.5y2 ≤ 0 x3A + x3A + x3C - 12.8y3 ≤ 0 x4A + x4b + x4C - 9.3y4 ≤ 0 x5A + x5B + x5B - 10.8y5 ≤ 0 x6A + x6B + X6C - 9.6y6 ≤ 0
x1A + x2A + x3A + x4A + x5A + x6A = 12 x1B + x2B + x3B + x4B + x5B + x6B = 10 x1C + x2C + x3C + x4C + x5C + x6C = 14
xij ≥ 0 yi = 0 or 1
0 – 1 Integer Programming Modeling ExamplesFixed Charge and Facility Example (2 of 4)
35Chapter 5 - Integer ProgrammingExhibit 5.18
0 – 1 Integer Programming Modeling ExamplesFixed Charge and Facility Example (3 of 4)
36Chapter 5 - Integer Programming
Exhibit 5.19
0 – 1 Integer Programming Modeling ExamplesFixed Charge and Facility Example (4 of 4)
37Chapter 5 - Integer Programming
Cities Cities within 300 miles 1. Atlanta Atlanta, Charlotte, Nashville 2. Boston Boston, New York 3. Charlotte Atlanta, Charlotte, Richmond 4. Cincinnati Cincinnati, Detroit, Indianapolis, 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, Richmond10. Pittsburgh Cincinnati, Detroit, Pittsburgh, Richmond11. Richmond Charlotte, New York, Pittsburgh, Richmond12. St. Louis Indianapolis, Nashville, St. Louis
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:
0 – 1 Integer Programming Modeling ExamplesSet Covering Example (1 of 4)
38Chapter 5 - Integer Programming
xi = city i, i = 1 to 12, xi = 0 if city is not selected as a hub and x i = 1if it is.
Minimize Z = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12
subject to:
Atlanta: x1 + x3 + x8 1Boston: x2 + x10 1Charlotte: x1 + x3 + x11 1Cincinnati: x4 + x5 + x6 + x8 + x10 1Detroit: x4 + x5 + x6 + x7 + x10 1Indianapolis: x4 + x5 + x6 + x7 + x8 + x12 1Milwaukee: x5 + x6 + x7 1Nashville: x1 + x4 + x6+ x8 + x12 1New York: x2 + x9+ x11 1Pittsburgh: x4 + x5 + x10 + x11 1Richmond: x3 + x9 + x10 + x11 1St Louis: x6 + x8 + x12 1 xij = 0 or 1
0 – 1 Integer Programming Modeling ExamplesSet Covering Example (2 of 4)
39Chapter 5 - Integer ProgrammingExhibit 5.20
0 – 1 Integer Programming Modeling ExamplesSet Covering Example (3 of 4)
40Chapter 5 - Integer Programming
Exhibit 5.21
0 – 1 Integer Programming Modeling ExamplesSet Covering Example (4 of 4)
41Chapter 5 - Integer Programming
Total Integer Programming Modeling ExampleProblem 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/salespersonRegion 2 - $7,500/salesperson
Total annual expense budget is $72,000.
Sales generated each year:
Region 1 - $85,000/salespersonRegion 2 - $60,000/salesperson
How many salespeople should be transferred into each region in order to maximize increased sales?
42Chapter 5 - Integer Programming
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
Total Integer Programming Modeling ExampleModel Formulation (2 of 3)
43Chapter 5 - Integer Programming
Total Integer Programming Modeling ExampleSolution with QM for Windows (3 of 3)