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Integer Programming
Chapter 5
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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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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:
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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
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■ 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)
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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
A 0 - 1 Integer Model (2 of 2)
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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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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
A Mixed Integer Model (2 of 2)
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■ 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
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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 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)
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Exhibit 5.2
Computer Solution of IP Problems
0 – 1 Model with Excel (2 of 5)
=C7*C12+D7*C13
+E7*C14+F7*C15Decision variables—
C12:C15
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Exhibit 5.3
Computer Solution of IP Problems
0 – 1 Model with Excel (3 of 5)
Restricts variables, C12:C15,
to integer and 0-1 values
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Exhibit 5.4
Computer Solution of IP Problems
0 – 1 Model with Excel (4 of 5)
Click on “bin” for 0-1.
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Exhibit 5.5
Computer Solution of IP Problems
0 – 1 Model with Excel (5 of 5)
Deactivate
Return to solver
window
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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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Exhibit 5.6
Computer Solution of IP Problems
0 – 1 Model with QM for Windows (2 of 3)
5-20Copyright © 2016 Pearson Education, Inc.Exhibit 5.7
Computer Solution of IP Problems
0 – 1 Model with QM for Windows (3 of 3)
Click to solve.
Variable type Click on “0/1” to make variables 0 or 1.
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Computer Solution of IP Problems
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
x1, x2 0 and integer
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Exhibit 5.8
Computer Solution of IP Problems
Total Integer Model with Excel (2 of 6)
Solution:
X1=1 swimming pool
X2=0 tennis center
X3=1 athletic field
X4=0 gymnasium
Z=700 people per day usage
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Exhibit 5.9
Computer Solution of IP Problems
Total Integer Model with Excel (3 of 6)
Objective function
Slack, =G6-E6
=C6*B10+D6*B11Decision variables—
B10:B11
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Exhibit 5.10
Computer Solution of IP Problems
Total Integer Model with Excel (4 of 6)
Integer variables
5
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Exhibit 5.11
Computer Solution of IP Problems
Total Integer Model with Excel (5 of 6)
Click on “int”
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Exhibit 5.12
Computer Solution of IP Problems
Total Integer Model with Excel (6 of 6)
Integer Solution
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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
Computer Solution of IP Problems
Mixed Integer Model with Excel (1 of 3)
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Exhibit 5.13
Computer Solution of IP Problems
Total Integer Model with Excel (2 of 3)
Available to invest
=C4*B8+D4*B9+E4*B10
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Exhibit 5.14
Computer Solution of IP Problems
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
Computer Solution of IP Problems
Mixed Integer Model with QM for Windows (1 of 2)
Click on “Real”
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Exhibit 5.16
Computer Solution of IP Problems
Mixed Integer Model with QM for Windows (2 of 2)
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■ 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)
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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 (2 of 4)
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Exhibit 5.17
0 – 1 Integer Programming Modeling Examples
Capital Budgeting Example (3 of 4)
=SUMPRODUCT(C7:C11,E7:E11)
=C9+C10
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Exhibit 5.18
0 – 1 Integer Programming Modeling Examples
Capital Budgeting Example (4 of 4)
0-1 integer
restriction
Mutually exclusive
constraint
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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
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?
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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
0 – 1 Integer Programming Modeling Examples
Fixed Charge and Facility Example (2 of 4)
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Exhibit 5.19
0 – 1 Integer Programming Modeling Examples
Fixed Charge and Facility Example (3 of 4)
Objective function
=SUM(C5:C10) =C10+D10+E10
=G10-C22*F10
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Exhibit 5.20
0 – 1 Integer Programming Modeling Examples
Fixed Charge and Facility Example (4 of 4)
0-1 integer
restriction
Plant capacity
constraints
Harvest constraints
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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:
0 – 1 Integer Programming Modeling Examples
Set Covering Example (1 of 4)
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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
0 – 1 Integer Programming Modeling Examples
Set Covering Example (2 of 4)
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0 – 1 Integer Programming Modeling Examples
Set Covering Example in Excel (3 of 4)
Exhibit 5.21
Objective function
Decision variables
in row 20
=SUMPRODUCT(B18:M18,B20:M20)
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Exhibit 5.22
0 – 1 Integer Programming Modeling Examples
Set Covering Example (4 of 4)
City constraints
set > 1
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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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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
Model Formulation (2 of 3)
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Total Integer Programming Modeling Example
Solution with QM for Windows (3 of 3)
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