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

2. Chapter Topics Integer Programming (IP) Models
Integer Programming Graphical Solution
Computer Solution of Integer Programming Problems With Excel and QM for Windows
Chapter 5 - Integer Programming

3. 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 ).
Chapter 5 - Integer Programming

4. 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:
Chapter 5 - Integer Programming

5. 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
Chapter 5 - Integer Programming

6. 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:
Chapter 5 - Integer Programming

7. 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 + 90x x x
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
Chapter 5 - Integer Programming

8. 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.
Chapter 5 - Integer Programming

9. 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
Chapter 5 - Integer Programming

10. 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.
Chapter 5 - Integer Programming

11. Feasible Solution Space with Integer Solution Points
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
Chapter 5 - Integer Programming

12. 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.
Chapter 5 - Integer Programming

13. Computer Solution of IP Problems 0 – 1 Model with Excel (1 of 5)
Recreational Facilities Example:
Maximize Z = 300x1 + 90x x x4
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
Chapter 5 - Integer Programming

14. Computer Solution of IP Problems 0 – 1 Model with Excel (2 of 5)
Exhibit 5.2
Chapter 5 - Integer Programming

15. Computer Solution of IP Problems 0 – 1 Model with Excel (4 of 5)
Exhibit 5.4
Chapter 5 - Integer Programming

16. Computer Solution of IP Problems 0 – 1 Model with Excel (3 of 5)
Exhibit 5.3
Chapter 5 - Integer Programming

17. Computer Solution of IP Problems 0 – 1 Model with Excel (5 of 5)
Exhibit 5.5
Chapter 5 - Integer Programming

18. Computer Solution of IP Problems
0 – 1 Model with QM for Windows (1 of 3)
Recreational Facilities Example:
Maximize Z = 300x1 + 90x x x4
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
Chapter 5 - Integer Programming

19. Computer Solution of IP Problems
0 – 1 Model with QM for Windows (2 of 3)
Exhibit 5.6
Chapter 5 - Integer Programming

20. Computer Solution of IP Problems
0 – 1 Model with QM for Windows (3 of 3)
Exhibit 5.7
Chapter 5 - Integer Programming

21. 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:
Chapter 5 - Integer Programming

22. 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
Chapter 5 - Integer Programming

23. 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?
Chapter 5 - Integer Programming

24. 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:
Chapter 5 - Integer Programming

25. 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 x5C + 14x6A + 16x6B + 12x6C + 405y y y y y y6
subject to:
x1A + x1B + x1B y1 < x2A + x2B + x2C -10.5y2 < 0
x3A + x3A + x3C y3 < x4A + x4b + x4C - 9.3y4 < 0
x5A + x5B + x5B y5 < x6A + x6B + X6C - 9.6y6 < 0
x1A + x2A + x3A + x4A + x5A + x6A =12
x1B + x2B + x3A + x4b + x5B + x6B = 10
x1B + x2C + x3C+ x4C + x5B + x6C = 14
xij = yi = 0 or 1
Chapter 5 - Integer Programming

26. 0 – 1 Integer Programming Modeling Examples
Fixed Charge and Facility Example (3 of 4)
Exhibit 5.18
Chapter 5 - Integer Programming

27. 0 – 1 Integer Programming Modeling Examples
Fixed Charge and Facility Example (4 of 4)
Exhibit 5.19
Chapter 5 - Integer Programming

28. 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
Chapter 5 - Integer Programming

29. 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  xij = 0 or 1
Chapter 5 - Integer Programming

30. 0 – 1 Integer Programming Modeling Examples
Set Covering Example (3 of 4)
Exhibit 5.20
Chapter 5 - Integer Programming

31. 0 – 1 Integer Programming Modeling Examples
Set Covering Example (4 of 4)
Exhibit 5.21
Chapter 5 - Integer Programming

32. 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?
Chapter 5 - Integer Programming

33. 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
Chapter 5 - Integer Programming

34. Total Integer Programming Modeling Example
Solution with QM for Windows (3 of 3)
Chapter 5 - Integer Programming