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Integer Programming Integer Programming ––
ModelingModeling
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ode gode g
Integer Programming (IP) Models
Integer Programming Graphical Solution
Computer Solution of Integer Programming Problems With Excel and QM for Windows
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Integer Programming ModelsTypes of Models
Total Integer Model: All decision variables required to have integer solution values.g
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
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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
Machine Required Floor
SpacePurchase
Price
$150/day.
Resource constraints: $40,000, 200 sq. ft. floor space.
Machine purchase prices and space requirements:
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p
(sq. ft.) Price
Press Lathe
15
30
$8,000
4,000
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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
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x1, x2 0 and integer
x1 = number of pressesx2 = number of lathes
Recreation facilities selection to maximize daily usage by residents.
Resource constraints: $120,000 budget; 12 acres of land.
S l ti t i t ith i i l t i t
A 0 - 1 Integer Model (1 of 2)
Selection constraint: either swimming pool or tennis center (not both).
Data:
Recreation Facility
Expected Usage (people/day)
Cost ($) Land
Requirement (acres)
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Swimming pool Tennis Center Athletic field Gymnasium
300 90 400 150
35,000 10,000 25,000 90,000
4 2 7 3
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Integer Programming Model:
Maximize Z = 300x1 + 90x2 + 400x3 + 150x4
A 0 - 1 Integer Model (2 of 2)
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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x1 = construction of a swimming poolx2 = construction of a tennis centerx3 = construction of an athletic fieldx4 = construction of a gymnasium
A Mixed Integer Model (1 of 2)
$250,000 available for investments providing greatest return after one year.
Data: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
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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
A Mixed Integer Model (2 of 2)
subject to:
50,000x1 + 12,000x2 + 8,000x3 $250,000x1 4 condominiumsx2 15 acresx3 20 bonds x2 0
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x1, x3 0 and integer
x1 = condominiums purchasedx2 = acres of land purchasedx3 = bonds purchased
Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution
Integer Programming Graphical Solution
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 ExampleGraphical Solution of Maximization Model
Maximize Z = $100x1 + $150x2
subject to:subject to:8,000x1 + 4,000x2 $40,000
15x1 + 30x2 200 ft2
x1, x2 0 and integer
Optimal Solution:Z = $1,055.56x = 2 22 presses
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x1 = 2.22 pressesx2 = 5.55 lathes
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 beBased 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.
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Recreational Facilities Example:
Computer Solution of IP Problems0 – 1 Model with Excel (1 of 5)
Maximize Z = 300x1 + 90x2 + 400x3 + 150x4
subject to:
$35,000x1 + 10,000x2 + 25,000x3 + 90,000x4 $120,000
4x1 + 2x2 + 7x3 + 3x4 12 acres
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x1 + x2 1 facility
x1, x2, x3, x4 = 0 or 1
Computer Solution of IP Problems0 – 1 Model with Excel (2 of 5)
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Computer Solution of IP Problems0 – 1 Model with Excel (3 of 5)
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Computer Solution of IP Problems0 – 1 Model with Excel (4 of 5)
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Computer Solution of IP Problems0 – 1 Model with Excel (5 of 5)
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Computer Solution of IP Problems0 – 1 Model with QM for Windows (1 of 3)
Recreational Facilities Example:
M i i Z 300 90 400 150Maximize 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
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x1, x2, x3, x4 = 0 or 1
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Computer Solution of IP ProblemsTotal 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
0 d i t
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x1, x2 0 and integer
Computer Solution of IP ProblemsTotal Integer Model with Excel (2 of 5)
20Exhibit 5.8
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Computer Solution of IP ProblemsTotal Integer Model with Excel (4 of 5)
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Computer Solution of IP ProblemsTotal Integer Model with Excel (3 of 5)
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Computer Solution of IP ProblemsTotal Integer Model with Excel (5 of 5)
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Integer Programming Model:
Maximize Z = $9 000x1 + 1 500x2 + 1 000x3
Computer Solution of IP ProblemsMixed Integer Model with Excel (1 of 3)
Maximize Z $9,000x1 + 1,500x2 + 1,000x3
subject to:
50,000x1 + 12,000x2 + 8,000x3 $250,000x1 4 condominiumsx2 15 acresx3 20 bonds
0
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x2 0x1, x3 0 and integer
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Computer Solution of IP ProblemsTotal Integer Model with Excel (2 of 3)
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Computer Solution of IP ProblemsSolution of Total Integer Model with Excel (3 of 3)
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Computer Solution of IP ProblemsMixed Integer Model with QM for Windows (1 of 2)
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Computer Solution of IP ProblemsMixed 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
0 – 1 Integer Programming Modeling ExamplesCapital Budgeting Example (1 of 4)
and a clothing department.
Data:
Project NPV Return
($1000) Project Costs per Year ($1000)
1 2 3
1. Website 120 55 40 25
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2. Warehouse 3. Clothing department 4. Computer department 5. ATMs Available funds per year
85 105 140 75
45 60 50 30
150
35 25 35 30
110
20 -- 30 --
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x1 = selection of web site projectx2 = selection of warehouse projectx = selection clothing department project
0 – 1 Integer Programming Modeling ExamplesCapital Budgeting Example (2 of 4)
x3 = selection clothing department projectx4 = selection of computer department projectx5 = selection of ATM projectxi = 1 if project “i” is selected, 0 if project “i” is not selected
Maximize Z = $120x1 + $85x2 + $105x3 + $140x4 + $70x5
subject to:
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55x1 + 45x2 + 60x3 + 50x4 + 30x5 15040x1 + 35x2 + 25x3 + 35x4 + 30x5 11025x1 + 20x2 + 30x4 60
x3 + x4 1xi = 0 or 1
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0 – 1 Integer Programming Modeling ExamplesCapital Budgeting Example (3 of 4)
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0 – 1 Integer Programming Modeling ExamplesCapital Budgeting Example (4 of 4)
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0 – 1 Integer Programming Modeling ExamplesFixed 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 Capacity
(tons,1000s) A B C
12 10 14
Farms Annual Fixed Costs
($1000)
Projected Annual Harvest (tons, 1000s)
1 405 11.2
Plant
Data:Shipping Costs
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2 3 4 5 6
390 450 368 520 465
10.5 12.8 9.3 10.8 9.6
Farm 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
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 = 18x + 15x + 12x + 13x + 10x + 17x + 16x +
0 – 1 Integer Programming Modeling ExamplesFixed Charge and Facility Example (2 of 4)
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 ≤ 0x3A + x3A + x3C - 12.8y3 ≤ 0 x4A + x4b + x4C - 9.3y4 ≤ 0x5A + x5B + x5B - 10.8y5 ≤ 0 x6A + x6B + X6C - 9.6y6 ≤ 0
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x1A + x2A + x3A + x4A + x5A + x6A = 12x1B + x2B + x3B + x4B + x5B + x6B = 10x1C + x2C + x3C + x4C + x5C + x6C = 14
xij ≥ 0 yi = 0 or 1
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0 – 1 Integer Programming Modeling ExamplesFixed Charge and Facility Example (3 of 4)
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0 – 1 Integer Programming Modeling ExamplesFixed Charge and Facility Example (4 of 4)
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Cities Cities within 300 miles
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:
0 – 1 Integer Programming Modeling ExamplesSet Covering Example (1 of 4)
1. Atlanta Atlanta, Charlotte, Nashville2. Boston Boston, New York3. Charlotte Atlanta, Charlotte, Richmond4. Cincinnati Cincinnati, Detroit, Indianapolis, Nashville, Pittsburgh5. Detroit Cincinnati, Detroit, Indianapolis, Milwaukee,
Pittsburgh6. Indianapolis Cincinnati, Detroit, Indianapolis, Milwaukee, Nashville,
St. Louis7 Mil k D t it I di li Mil k
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7. Milwaukee Detroit, Indianapolis, Milwaukee8. Nashville Atlanta, Cincinnati, Indianapolis, Nashville, St. Louis9. New York Boston, New York, Richmond
10. Pittsburgh Cincinnati, Detroit, Pittsburgh, Richmond11. Richmond Charlotte, New York, Pittsburgh, Richmond12. St. Louis Indianapolis, Nashville, St. Louis
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:
Atl t 1
0 – 1 Integer Programming Modeling ExamplesSet Covering Example (2 of 4)
Atlanta: x1 + x3 + x8 1Boston: x2 + x10 1Charlotte: x1 + x3 + x11 1Cincinnati: x4 + x5 + x6 + x8 + x10 1Detroit: x4 + x5 + x6 + x7 + x10 1Indianapolis: x4 + x5 + x6 + x7 + x8 + x12 1Milwaukee: x5 + x6 + x7 1Nashville: x1 + x4 + x6+ x8 + x12 1
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Nashville: x1 x4 x6 x8 x12 1New York: x2 + x9+ x11 1Pittsburgh: x4 + x5 + x10 + x11 1Richmond: x3 + x9 + x10 + x11 1St Louis: x6 + x8 + x12 1 xij = 0 or 1
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0 – 1 Integer Programming Modeling ExamplesSet Covering Example (3 of 4)
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0 – 1 Integer Programming Modeling ExamplesSet Covering Example (4 of 4)
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Total Integer Programming Modeling ExampleProblem 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/salespersonRegion 2 - $7,500/salesperson
Total annual expense budget is $72,000.
Sales generated each year:
Region 1 $85 000/salesperson
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Region 1 - $85,000/salespersonRegion 2 - $60,000/salesperson
How many salespeople should be transferred into each region in order to maximize increased sales?
Step 1:
Formulate the Integer Programming Model
Total Integer Programming Modeling ExampleModel Formulation (2 of 3)
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
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x1, x2 0 or integer
Step 2:
Solve the Model using QM/Excel for Windows
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Total Integer Programming Modeling ExampleSolution with QM for Windows (3 of 3)
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