1. Introduction to Management Science
8th Edition by
Bernard W. Taylor III
Chapter 9
Integer Programming
Chapter 9 - Integer Programming

2. Chapter Topics Integer Programming (IP) Models
Still “linear programming models”
Variables must be integers, and so form a subset of the original, real-number models
Integer Programming Graphical Solution
Computer Solution of Integer Programming Problems With Excel and QM for Windows
Chapter 9 - Integer Programming

3. Integer Programming Models Types 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.
Also referred to as Boolean and True/False
Mixed Integer Model: Some of the decision variables (but not all) required to have integer values.
Chapter 9 - Integer Programming

4. 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:
Chapter 9 - Integer Programming

5. 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
Chapter 9 - Integer Programming

6. A 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 9 - Integer Programming

7. A 0–1 Integer Model (2 of 2) Integer Programming Model:
Maximize Z = 300x1 + 90x x x4
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 (either do or don’t)
x1 = construction of a swimming pool
x2 = construction of a tennis center
x3 = construction of an athletic field
x4 = construction of a gymnasium
Chapter 9 - Integer Programming

8. 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.
Chapter 9 - Integer Programming

9. A 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 9 - Integer Programming

10. Integer Programming Graphical Solution
Rounding non-integer solution values up (down) to the nearest integer value can result in an infeasible solution
A feasible solution may be found by rounding down (up) non-integer solution values, but may result in a less than optimal (sub-optimal) solution.
Whether a variable is to be rounded up or down depends on where the corresponding point is with respect to the constraint boundaries.
Chapter 9 - 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
Both of these may be found by judicious rounding, but the value of the objective function needs to be checked for each.
Figure 9.1
Feasible Solution Space with Integer Solution Points
Chapter 9 - 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 9 - 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 + 3x4  12 acres
x1 + x2  1 facility
x1, x2, x3, x4 = 0 or 1
Chapter 9 - Integer Programming

14. Computer Solution of IP Problems 0–1 Model with Excel (2 of 5)
Exhibit 9.2
Chapter 9 - Integer Programming

15. Computer Solution of IP Problems 0–1 Model with Excel (3 of 5)
Instead, one could just specify $C$12:$C$15 as “binary” (= ‘0’ or ‘1’).
Exhibit 9.3
Chapter 9 - Integer Programming

16. To constrain a range of variables to be integers, enter:
Computer Solution of IP Problems
0–1 Model with Excel (4 of 5)
To constrain a range of variables to be integers, enter:
Instead, one could just specify the “bin” option (= ‘0’ or ‘1’), thus avoiding the additional, “≥ 0” and “≤ 1” constraints.
… and note that it doesn’t matter what you put in the right-hand side field, but it must not be empty.
Exhibit 9.4
Chapter 9 - Integer Programming

17. Computer Solution of IP Problems 0–1 Model with Excel (5 of 5)
Exhibit 9.5
Chapter 9 - 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 + 3x4  12 acres
x1 + x2  1 facility
x1, x2, x3, x4 = 0 or 1
Chapter 9 - Integer Programming

19. Computer Solution of IP Problems
0–1 Model with QM for Windows (2 of 3)
Exhibit 9.6
Chapter 9 - Integer Programming

20. Computer Solution of IP Problems
0–1 Model with QM for Windows (3 of 3)
Exhibit 9.7
Chapter 9 - Integer Programming

21. Computer Solution of IP Problems
Total 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
Chapter 9 - Integer Programming

22. Computer Solution of IP Problems
Total Integer Model with Excel (2 of 5)
Exhibit 9.8
Chapter 9 - Integer Programming

23. Computer Solution of IP Problems
Total Integer Model with Excel (3 of 5)
… and, again, just to note that this field will automatically
fill, but Excel wants you to put something in it anyway.
Exhibit 9.9
Chapter 9 - Integer Programming

24. Computer Solution of IP Problems
Total Integer Model with Excel (4 of 5)
Exhibit 9.10
Chapter 9 - Integer Programming

25. Computer Solution of IP Problems
Total Integer Model with Excel (5 of 5)
Exhibit 9.11
Chapter 9 - Integer Programming

26. Computer Solution of IP Problems
Mixed Integer Model with Excel (1 of 3)
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
Chapter 9 - Integer Programming

27. Computer Solution of IP Problems
Total Integer Model with Excel (2 of 3)
Exhibit 9.12
Chapter 9 - Integer Programming

28. Computer Solution of IP Problems
Solution of Total Integer Model with Excel (3 of 3)
Exhibit 9.13
Chapter 9 - Integer Programming

29. Computer Solution of IP Problems
Mixed Integer Model with QM for Windows (1 of 2)
Exhibit 9.14
Chapter 9 - Integer Programming

30. Computer Solution of IP Problems
Mixed Integer Model with QM for Windows (2 of 2)
Exhibit 9.15
Chapter 9 - Integer Programming

31. 0–1 Integer Programming Modeling Examples
Capital Budgeting Example (1 of 4)
University bookstore expansion project.
Not enough space available for both a computer department and a clothing department.
Data:
Chapter 9 - Integer Programming

32. 0–1 Integer Programming Modeling Examples
Capital Budgeting Example (2 of 4)
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 9 - Integer Programming

33. 0–1 Integer Programming Modeling Examples
Capital Budgeting Example (3 of 4)
Exhibit 9.16
Chapter 9 - Integer Programming

34. 0–1 Integer Programming Modeling Examples
Capital Budgeting Example (4 of 4)
Could have chosen ‘bin’ (= ‘0’ or ‘1’) instead!
Exhibit 9.17
Chapter 9 - Integer Programming

35. 0–1 Integer Programming Modeling Examples
Fixed Charge and Facility Example (1 of 4)
Which of the six farms should be purchased that will meet current production capacity at minimum total cost, including annual fixed costs and shipping costs?
Data:
Chapter 9 - Integer Programming

36. 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 + x1C y1 ≤ x2A + x2B + x2C -10.5y2 ≤ 0
x3A + x3B + 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
x1C + x2C + x3C+ x4C + x5C + x6C = 14
xij ≥ yi = 0 or 1
The 6 farms’ capacities.
The 3 plants’ capacities.
Chapter 9 - Integer Programming

37. 0–1 Integer Programming Modeling Examples
Fixed Charge and Facility Example (3 of 4)
Exhibit 9.18
Chapter 9 - Integer Programming

38. 0–1 Integer Programming Modeling Examples
Fixed Charge and Facility Example (4 of 4)
Exhibit 9.19
Chapter 9 - Integer Programming

39. 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 9 - Integer Programming

40. 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 9 - Integer Programming

41. 0–1 Integer Programming Modeling Examples
Set Covering Example (3 of 4)
Exhibit 9.20
the decision variables
Chapter 9 - Integer Programming

42. 0–1 Integer Programming Modeling Examples
Set Covering Example (4 of 4)
Exhibit 9.21
For “N7”, enter “= sumproduct(B7:M7,B$20:M$20)” and the down; the last constraint requires that these (hubs within 300 mi) equal at least one.
Chapter 9 - Integer Programming

43. 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 9 - Integer Programming

44. 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 9 - Integer Programming

45. Total Integer Programming Modeling Example
Solution with QM for Windows (3 of 3)
Chapter 9 - Integer Programming

46. Chapter 9 - Integer Programming