mixed integer prog_excel solver practice example

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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

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$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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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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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 = 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


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


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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)


Maximize Z = $100x1 + $150x2

subject to:

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

15x1 + 30x2 200 ft2

0 d i t

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20Exhibit 5.8

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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 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 --

60

x1 = selection of web site projectx2 = selection of warehouse projectx = selection clothing department project


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 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 +


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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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


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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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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mixed integer prog_excel solver practice example

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