1 Chapter 5 Integer Programming 2 Chapter Topics Integer Programming (IP) Models Integer Programming Graphical Solution Computer Solution of Integer Programming Problems With Excel
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Chapter 5Integer Programming
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Chapter Topics
Integer Programming (IP) Models
Integer Programming Graphical Solution
Computer Solution of Integer Programming Problems With Excel
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Integer Programming ModelsTypes 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 (sq. ft.)
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, 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 integerx1 = number of pressesx2 = number of lathes
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Recreation facilities selection to maximize daily usage by residents.Resource constraints: $120,000 budget; 12 acres of land.Selection constraint: either swimming pool or tennis center (not both).Data:
Recreation Facility
Expected Usage (people/day) Cost ($)
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 acresx1 + x2 ≤ 1 facilityx1, x2, x3, x4 = 0 or 1x1 = construction of a swimming poolx2 = construction of a tennis centerx3 = construction of an athletic fieldx4 = 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 condominiumsx2 ≤ 15 acresx3 ≤ 20 bonds x2 ≥ 0x1, x3 ≥ 0 and integerx1 = condominiums purchasedx2 = acres of land purchasedx3 = 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 ExampleGraphical Solution of Maximization Model
Maximize Z = $100x1 + $150x2subject to:
8,000x1 + 4,000x2 ≤ $40,000 15x1 + 30x2 ≤ 200 ft2
x1, x2 ≥ 0 and integer
Optimal Solution:Z = $1,055.56x1 = 2.22 pressesx2 = 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.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 used in this book.
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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 acresx1 + x2 ≤ 1 facilityx1, x2, x3, x4 = 0 or 1
Computer Solution of IP Problems0 – 1 Model with Excel (1 of 5)
14Exhibit 5.2
Computer Solution of IP Problems0 – 1 Model with Excel (2 of 5)
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15Exhibit 5.3
Computer Solution of IP Problems0 – 1 Model with Excel (3 of 5)
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Exhibit 5.4
Computer Solution of IP Problems0 – 1 Model with Excel (4 of 5)
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17Exhibit 5.5
Computer Solution of IP Problems0 – 1 Model with Excel (5 of 5)
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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
x1, x2 ≥ 0 and integer
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19Exhibit 5.8
Computer Solution of IP ProblemsTotal Integer Model with Excel (2 of 5)
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Exhibit 5.10
Computer Solution of IP ProblemsTotal Integer Model with Excel (3 of 5)
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Exhibit 5.9
Computer Solution of IP ProblemsTotal Integer Model with Excel (4 of 5)
22Exhibit 5.11
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
subject to:50,000x1 + 12,000x2 + 8,000x3 ≤ $250,000
x1 ≤ 4 condominiumsx2 ≤ 15 acresx3 ≤ 20 bonds x2 ≥ 0x1, x3 ≥ 0 and integer
Computer Solution of IP ProblemsMixed Integer Model with Excel (1 of 3)
24Exhibit 5.12
Computer Solution of IP ProblemsTotal Integer Model with Excel (2 of 3)
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Exhibit 5.13
Computer Solution of IP ProblemsSolution of Total Integer Model with Excel (3 of 3)
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University bookstore expansion project.Not enough space available for both a computer department and a clothing department.Data:
Project NPV Return ($1000)
Project Costs per Year ($1000) 1 2 3
1. Website 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 ExamplesCapital Budgeting Example (1 of 4)
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x1 = selection of web site projectx2 = selection of warehouse projectx3 = 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 + $70x5subject to:
55x1 + 45x2 + 60x3 + 50x4 + 30x5 ≤ 15040x1 + 35x2 + 25x3 + 35x4 + 30x5 ≤ 11025x1 + 20x2 + 30x4 ≤ 60
x3 + x4 ≤ 1xi = 0 or 1
0 – 1 Integer Programming Modeling ExamplesCapital Budgeting Example (2 of 4)
28Exhibit 5.16
0 – 1 Integer Programming Modeling ExamplesCapital Budgeting Example (3 of 4)
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Exhibit 5.17
0 – 1 Integer Programming Modeling ExamplesCapital Budgeting Example (4 of 4)
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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 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 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?Data:
Shipping Costs
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yi = 0 if farm i is not selected, and 1 if farm i is selected, i = 1,2,3,4,5,6xij = 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 ≤ 0x3A + x3A + x3C - 12.8y3 ≤ 0 x4A + x4b + x4C - 9.3y4 ≤ 0x5A + x5B + x5B - 10.8y5 ≤ 0 x6A + x6B + X6C - 9.6y6 ≤ 0
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
0 – 1 Integer Programming Modeling ExamplesFixed Charge and Facility Example (2 of 4)
32Exhibit 5.18
0 – 1 Integer Programming Modeling ExamplesFixed Charge and Facility Example (3 of 4)
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Exhibit 5.19
0 – 1 Integer Programming Modeling ExamplesFixed Charge and Facility Example (4 of 4)
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Cities Cities within 300 miles1. 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. 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
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)
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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 ≥ 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 ≥ 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
0 – 1 Integer Programming Modeling ExamplesSet Covering Example (2 of 4)
36Exhibit 5.20
0 – 1 Integer Programming Modeling ExamplesSet Covering Example (3 of 4)
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Exhibit 5.21
0 – 1 Integer Programming Modeling ExamplesSet Covering Example (4 of 4)