Chapter Topics Integer Programming

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5-1Copyright © 2016 Pearson Education, Inc.

Integer Programming

Chapter 5


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


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.


Machine

Required Floor Space (ft.2)

Purchase Price

Press Lathe

15

30

$8,000

4,000

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:


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


■ 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

Facility

Expected Usage (people/day)

Cost ($)

Land Requirement (acres)

Swimming pool Tennis Center Athletic field Gymnasium

300 90 400 150

35,000 10,000 25,000 90,000

4 2 7 3

A 0 - 1 Integer Model (1 of 2)

2



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

A 0 - 1 Integer Model (2 of 2)


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.



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 = condominiums purchased

x2 = acres of land purchased

x3 = bonds purchased

A Mixed Integer Model (2 of 2)


■ 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 Graphical Solution


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


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.

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 web site Module C – “Integer Programming: the Branch and Bound Method” for detailed description of this

method.

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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

Computer Solution of IP Problems

0 – 1 Model with Excel (1 of 5)


Exhibit 5.2



=C7*C12+D7*C13

+E7*C14+F7*C15Decision variables—

C12:C15


Exhibit 5.3



Restricts variables, C12:C15,

to integer and 0-1 values


Exhibit 5.4



Click on “bin” for 0-1.


Exhibit 5.5



Deactivate

Return to solver

window



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

4


Exhibit 5.6



5-20Copyright © 2016 Pearson Education, Inc.Exhibit 5.7



Click to solve.

Variable type Click on “0/1” to make variables 0 or 1.



Total Integer Model with Excel (1 of 6)

Integer Programming Model of Machine Shop:

Maximize Z = $100x1 + $150x2

subject to:

8,000x1 + 4,000x2 $40,000

15x1 + 30x2 200 ft2



Exhibit 5.8



Solution:

X1=1 swimming pool

X2=0 tennis center

X3=1 athletic field

X4=0 gymnasium

Z=700 people per day usage


Exhibit 5.9



Objective function

Slack, =G6-E6

=C6*B10+D6*B11Decision variables—

B10:B11


Exhibit 5.10



Integer variables

5


Exhibit 5.11



Click on “int”


Exhibit 5.12



Integer Solution


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



Mixed Integer Model with Excel (1 of 3)


Exhibit 5.13



Available to invest

=C4*B8+D4*B9+E4*B10


Exhibit 5.14


Solution of Total Integer Model with Excel (3 of 3)

Integer requirement for

condos (x1) and bonds (x3)

Constraints for acres,

condos, and bonds

5-30Copyright © 2016 Pearson Education, Inc.Exhibit 5.15


Mixed Integer Model with QM for Windows (1 of 2)

Click on “Real”

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Exhibit 5.16


Mixed Integer Model with QM for Windows (2 of 2)


■ University bookstore expansion project.

■ Not enough space available for both a computer department and a

clothing department.

Project NPV Return

($1,000s) Project Costs per Year ($1000)

1 2 3

1. Web site 2. Warehouse 3. Clothing department 4. Computer department 5. ATMs Available funds per year

$120 85

105 140

75

$55 45 60 50 30

150

$40 35 25 35 30

110

$25 20 --

30 --

60

0 – 1 Integer Programming Modeling Examples

Capital Budgeting Example (1 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




Exhibit 5.17



=SUMPRODUCT(C7:C11,E7:E11)

=C9+C10


Exhibit 5.18



0-1 integer

restriction

Mutually exclusive

constraint


Plant

Available Capacity

(tons,1000s)

A B C

12 10 14

Farms Annual Fixed Costs

($1000)

Projected Annual Harvest (tons, 1000s)

1 2 3 4 5 6

405 390 450 368 520 465

11.2 10.5 12.8 9.3 10.8 9.6

Farm

Plant ($/ton shipped)

A B C

1 2 3 4 5 6

18 15 12 13 10 17 16 14 18 19 15 16 17 19 12 14 16 12


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?

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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




Exhibit 5.19



Objective function

=SUM(C5:C10) =C10+D10+E10

=G10-C22*F10


Exhibit 5.20



0-1 integer

restriction

Plant capacity

constraints

Harvest constraints


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

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:


Set Covering Example (1 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: x2 + x10 1

Charlotte: x1 + x3 + x11 1

Cincinnati: x4 + x5 + x6 + 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





Set Covering Example in Excel (3 of 4)

Exhibit 5.21

Objective function

Decision variables

in row 20

=SUMPRODUCT(B18:M18,B20:M20)

8


Exhibit 5.22



City constraints

set > 1


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


■ Total annual expense budget is $72,000.

■ Sales generated each year:



■ How many salespeople should be transferred into each region in order to

maximize increased sales?


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


Model Formulation (2 of 3)



Solution with QM for Windows (3 of 3)


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Chapter Topics Integer Programming