Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 5
Integer Programming
Chapter 5 - Integer Programming
1
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
2
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.
Mixed Integer Model: Some of the decision variables (but
not all) required to have integer values.
Chapter 5 - Integer Programming
3
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:
Required Floor
Space
(sq. ft.)
Purchase
Price
Press
15
$8,000
Lathe
30
4,000
Machine
Chapter 5 - Integer Programming
4
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 5 - Integer Programming
5
A 0 - 1 Integer Model (1 of 2)
Recreation facilities selection to maximize daily usage by
residents.
Resource constraints: $120,000 budget; 12 acres of land.
Selection constraint: either swimming pool or tennis center
(not both).
Data:
Recreation
Facility
Swimming pool
Tennis Center
Athletic field
Gymnasium
Chapter 5 - Integer Programming
Expected Usage
(people/day)
Cost ($)
Land
Requirement
(acres)
300
90
400
150
35,000
10,000
25,000
90,000
4
2
7
3
6
A 0 - 1 Integer Model (2 of 2)
Integer Programming Model:
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
x1 = construction of a swimming pool
x2 = construction of a tennis center
x3 = construction of an athletic field
x4 = construction of a gymnasium
Chapter 5 - Integer Programming
7
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 5 - Integer Programming
8
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 5 - Integer Programming
9
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 noninteger solution values but may result in a less than optimal
(sub-optimal) solution.
Chapter 5 - Integer Programming
10
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
11
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
12
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
Chapter 5 - Integer Programming
13
Computer Solution of IP Problems
0 – 1 Model with Excel (2 of 5)
Exhibit 5.2
Chapter 5 - Integer Programming
14
Computer Solution of IP Problems
0 – 1 Model with Excel (3 of 5)
Exhibit 5.3
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 (5 of 5)
Exhibit 5.5
Chapter 5 - Integer Programming
17
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
Chapter 5 - Integer Programming
18
Computer Solution of IP Problems
0 – 1 Model with QM for Windows (2 of 3)
Exhibit 5.6
Chapter 5 - Integer Programming
19
Computer Solution of IP Problems
0 – 1 Model with QM for Windows (3 of 3)
Exhibit 5.7
Chapter 5 - Integer Programming
20
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 5 - Integer Programming
21
Computer Solution of IP Problems
Total Integer Model with Excel (2 of 5)
Exhibit 5.8
Chapter 5 - Integer Programming
22
Computer Solution of IP Problems
Total Integer Model with Excel (3 of 5)
Exhibit 5.9
Chapter 5 - Integer Programming
23
Computer Solution of IP Problems
Total Integer Model with Excel (4 of 5)
Exhibit 5.10
Chapter 5 - Integer Programming
24
Computer Solution of IP Problems
Total Integer Model with Excel (5 of 5)
Exhibit 5.11
Chapter 5 - Integer Programming
25
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 5 - Integer Programming
26
Computer Solution of IP Problems
Total Integer Model with Excel (2 of 3)
Exhibit 5.12
Chapter 5 - Integer Programming
27
Computer Solution of IP Problems
Solution of Total Integer Model with Excel (3 of 3)
Exhibit 5.13
Chapter 5 - Integer Programming
28
Computer Solution of IP Problems
Mixed Integer Model with QM for Windows (1 of 2)
Exhibit 5.14
Chapter 5 - Integer Programming
29
Computer Solution of IP Problems
Mixed Integer Model with QM for Windows (2 of 2)
Exhibit 5.15
Chapter 5 - Integer Programming
30
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:
Project
1. Website
2. Warehouse
3. Clothing department
4. Computer department
5. ATMs
Available funds per year
Chapter 5 - Integer Programming
NPV Return
($1000)
120
85
105
140
75
Project Costs per Year ($1000)
1
2
3
55
45
60
50
30
40
35
25
35
30
25
20
-30
--
150
110
60
31
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 5 - Integer Programming
32
0 – 1 Integer Programming Modeling Examples
Capital Budgeting Example (3 of 4)
Exhibit 5.16
Chapter 5 - Integer Programming
33
0 – 1 Integer Programming Modeling Examples
Capital Budgeting Example (4 of 4)
Exhibit 5.17
Chapter 5 - Integer Programming
34
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:
Farms
1
2
3
4
5
6
Available
Capacity
(tons,1000s)
12
10
14
Plant
Annual Fixed
Costs
($1000)
405
390
450
368
520
465
Chapter 5 - Integer Programming
Projected Annual
Harvest (tons, 1000s)
11.2
10.5
12.8
9.3
10.8
9.6
A
B
C
Farm
A
Plant
B
1
2
3
4
5
6
18
13
16
19
17
14
15
10
14
15
19
16
C
12
17
18
16
12
12
35
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 + x1C - 11.2y1 < 0
x2A + x2B + x2C -10.5y2 < 0
x3A + x3B + x3C - 12.8y3 < 0
x4A + x4B + x4C - 9.3y4 < 0
x5A + x5B + x5C - 10.8y5 < 0
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 = 0
yi = 0 or 1
Chapter 5 - Integer Programming
36
0 – 1 Integer Programming Modeling Examples
Fixed Charge and Facility Example (3 of 4)
Exhibit 5.18
Chapter 5 - Integer Programming
37
0 – 1 Integer Programming Modeling Examples
Fixed Charge and Facility Example (4 of 4)
Exhibit 5.19
Chapter 5 - Integer Programming
38
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
1. Atlanta
2. Boston
3. Charlotte
4. Cincinnati
5. Detroit
6. Indianapolis
7. Milwaukee
8. Nashville
9. New York
10. Pittsburgh
11. Richmond
12. St. Louis
Cities within 300 miles
Atlanta, Charlotte, Nashville
Boston, New York
Atlanta, Charlotte, Richmond
Cincinnati, Detroit, Nashville, Pittsburgh
Cincinnati, Detroit, Indianapolis, Milwaukee, Pittsburgh
Cincinnati, Detroit, Indianapolis, Milwaukee, Nashville, St.
Louis
Detroit, Indianapolis, Milwaukee
Atlanta, Cincinnati, Indianapolis, Nashville, St. Louis
Boston, New York, Richmond
Cincinnati, Detroit, Pittsburgh, Richmond
Charlotte, New York, Pittsburgh, Richmond
Indianapolis, Nashville, St. Louis
Chapter 5 - Integer Programming
39
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:
Boston:
Charlotte:
Cincinnati:
Detroit:
Indianapolis:
Milwaukee:
Nashville:
New York:
Pittsburgh:
Richmond:
St Louis:
Chapter 5 - Integer Programming
x1 + x3 + x8  1
x2 + x9  1
x1 + x3 + x11  1
x4 + x5 + x8 + x10  1
x4 + x5 + x6 + x7 + x10  1
x4 + x5 + x6 + x7 + x8 + x12  1
x5 + x6 + x7  1
x1 + x4 + x6+ x8 + x12  1
x2 + x9+ x11  1
x4 + x5 + x10 + x11  1
x3 + x9 + x10 + x11  1
x6 + x8 + x12  1
xij = 0 or 1
40
0 – 1 Integer Programming Modeling Examples
Set Covering Example (3 of 4)
Exhibit 5.20
Chapter 5 - Integer Programming
41
0 – 1 Integer Programming Modeling Examples
Set Covering Example (4 of 4)
Exhibit 5.21
Chapter 5 - Integer Programming
42
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
43
Total Integer Programming Modeling Example
Model Formulation (2 of 3)
Step 1:
Formulate the Integer Programming Model
xi = number of sales person assigned to region i, i = 1, 2.
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
44
Total Integer Programming Modeling Example
Solution with QM for Windows (3 of 3)
Chapter 5 - Integer Programming
45