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Introduction to Management Science 8th

Edition by Bernard W. Taylor III

Chapter 5 Integer Programming

Chapter Topics

- Integer Programming (IP) Models
- Integer Programming Graphical Solution
- Computer Solution of Integer Programming Problems

With Excel and QM for Windows

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.

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

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

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

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

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.

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

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.

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

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.

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

Computer Solution of IP Problems 0 1 Model with

Excel (2 of 5)

Exhibit 5.2

Computer Solution of IP Problems 0 1 Model with

Excel (3 of 5)

Exhibit 5.3

Computer Solution of IP Problems 0 1 Model with

Excel (4 of 5)

Exhibit 5.4

Computer Solution of IP Problems 0 1 Model with

Excel (5 of 5)

Exhibit 5.5

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

Computer Solution of IP Problems 0 1 Model with

QM for Windows (2 of 3)

Exhibit 5.6

Computer Solution of IP Problems 0 1 Model with

QM for Windows (3 of 3)

Exhibit 5.7

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

Computer Solution of IP Problems Total Integer

Model with Excel (2 of 5)

Exhibit 5.8

Computer Solution of IP Problems Total Integer

Model with Excel (3 of 5)

Exhibit 5.9

Computer Solution of IP Problems Total Integer

Model with Excel (4 of 5)

Exhibit 5.10

Computer Solution of IP Problems Total Integer

Model with Excel (5 of 5)

Exhibit 5.11

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

Computer Solution of IP Problems Total Integer

Model with Excel (2 of 3)

Exhibit 5.12

Computer Solution of IP Problems Solution of

Total Integer Model with Excel (3 of 3)

Exhibit 5.13

Computer Solution of IP Problems Mixed Integer

Model with QM for Windows (1 of 2)

Exhibit 5.14

Computer Solution of IP Problems Mixed Integer

Model with QM for Windows (2 of 2)

Exhibit 5.15

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

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

0 1 Integer Programming Modeling

Examples Capital Budgeting Example (3 of 4)

Exhibit 5.16

0 1 Integer Programming Modeling

Examples Capital Budgeting Example (4 of 4)

Exhibit 5.17

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

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 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 x3A x4b x5B x6B 10

x1B x2C x3C x4C x5B x6C 14 xij

0 yi 0 or 1

0 1 Integer Programming Modeling Examples Fixed

Charge and Facility Example (3 of 4)

Exhibit 5.18

0 1 Integer Programming Modeling Examples Fixed

Charge and Facility Example (4 of 4)

Exhibit 5.19

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

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 ? 1 xij 0 or 1

0 1 Integer Programming Modeling Examples Set

Covering Example (3 of 4)

Exhibit 5.20

0 1 Integer Programming Modeling Examples Set

Covering Example (4 of 4)

Exhibit 5.21

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?

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

Total Integer Programming Modeling

Example Solution with QM for Windows (3 of 3)

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