CH_05_INTEGER_PROGRAMMING

Integer Programming

Chapter 5 5-1

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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
0-1 Integer Programming Modeling Examples

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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.

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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 budget, 200 sq. ft. floor space.
■ Machine purchase prices and space requirements:

Required
Machine Floor Space (ft.2) Purchase Price

Press 15 $8,000

Lathe 30 4,000

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A Total Integer Model (2 of 2) 5-5

Integer Programming Model:

Maximize Z = $100x1 + $150x2
subject to:
$8,000x1 + 4,000x2 ≤ $40,000

15x1 + 30x2 ≤ 200 ft2
x1, x2 ≥ 0 and integer
x1 = number of presses
x2 = number of lathes

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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).

Recreation Expected Usage Land Requirement
Facility (people/day)
Cost ($) (acres)
Swimming pool 300
Tennis Center 90 35,000 4
Athletic field 400 10,000 2
Gymnasium 150 25,000 7
90,000 3

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A 0 - 1 Integer Model (2 of 2) 5-7

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

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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.

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A Mixed Integer Model (2 of 2) 5-9

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 Machine Shop 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.
Feasible solutions can be partitioned into smaller subsets
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 book’s companion website – “Integer Programming: the
Branch and Bound Method” for detailed description of this

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 + 3x4 ≤ 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 5-14

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Computer Solution of IP Problems
0 – 1 Model with Excel (3 of 5)

Exhibit 5.3 5-15

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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 (5 of 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 + 3x4 ≤ 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)

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Computer Solution of IP Problems
Total Integer Model with Excel (1 of 5)

Integer Programming Model of Machine Shop:

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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Computer Solution of IP Problems
Total Integer Model with Excel (2 of 5)

Exhibit 5.8Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 5-22

Computer Solution of IP Problems
Total Integer Model with Excel (4 of 5)

Exhibit 5.9

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Computer Solution of IP Problems
Total Integer Model with Excel (3 of 5)

Exhibit 5.10

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Computer Solution of IP Problems
Total Integer Model with Excel (5 of 5)

Exhibit 5.11 5-25

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Computer Solution of IP Problems
Mixed Integer Model with Excel (1 of 3)

Integer Programming Model for Investments Problem:

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

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Computer Solution of IP Problems
Total Integer Model with Excel (2 of 3)

Exhibit 5.12

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Computer Solution of IP Problems
Solution of Total Integer Model with Excel (3 of 3)

Exhibit 5.13

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Computer Solution of IP Problems
Mixed Integer Model with QM for Windows (1 of 2)

Exhibit 5.14 5-29

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Computer Solution of IP Problems
Mixed Integer Model with QM for Windows (2 of 2)

Exhibit 5.15

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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.

Project NPV Return Project Costs per Year ($1000)
($1,000s) 123
1. Web site
2. Warehouse $120 $55 $40 $25
3. Clothing department 85
4. Computer department 45 35 20
5. ATMs 105
140 60 25 --
Available funds per year
75 50 35 30

30 30 --

150 110 60

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0 – 1 Integer Programming Modeling Examples
Capital Budgeting Example (2 of 4)

x1 = selection of web site project 5-32
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 (3 of 4)

Exhibit 5.16

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0 – 1 Integer Programming Modeling Examples
Capital Budgeting Example (4 of 4)

Exhibit 5.17

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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?

Plant Available Plant ($/ton shipped)
Capacity
(tons,1000s) Farm A B C

A 12 1 18 15 12
B 10 2 13 10 17
C 14

Farms Annual Fixed Projected Annual 3 16 14 18
Costs Harvest (tons, 1000s) 4 19 15 16
1 ($1000) 5 17 19 12
2 11.2 6 14 16 12
3 405 10.5
4 390 12.8
5 450
6 368 9.3
520 10.8
465
9.6

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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 (1000 tons) shipped from farm I to plant j; j = A,B,C.

Minimize Z = 18x1A+ 15x1B+ 12x1C+ 13x2A+ 10x2B+ 17x2C+ 16x3+ 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

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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 these 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, 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, 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 = 1 if it is.

Minimize Z = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12

subject to: Atlanta: x1 + x3 + x8 ≥ 1
Boston:
Charlotte: x2 + x10 ≥ 1
Cincinnati: x1 + x3 + x11 ≥ 1
Detroit: x4 + x5 + x6 + x8 + x10 ≥ 1
Indianapolis: x4 + x5 + x6 + x7 + x10 ≥ 1
Milwaukee: x4 + x5 + x6 + x7 + x8 + x12 ≥ 1
Nashville: x5 + x6 + x7 ≥ 1
New York: x1 + x4 + x6+ x8 + x12 ≥ 1
Pittsburgh: x2 + x9+ x11 ≥ 1
Richmond: x4 + x5 + x10 + x11 ≥ 1
St Louis: x3 + x9 + x10 + x11 ≥ 1
x6 + x8 + x12 ≥ 1
xij = 0 or 1

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0 – 1 Integer Programming Modeling Examples
Set Covering Example (3 of 4)

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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)

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