Introduction to management science 10e by bernard taylor chapter 05

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Integer Programming

Chapter 5

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Chapter Topics

Programming Problems With Excel and

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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:

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■ Selection constraint: either swimming pool or

tennis center (not both).Recreation Facility Expected Usage (people/ day) Cost ($) Land Requirement (acres)

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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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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

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Branch and Bound Method

■ Traditional approach to solving integer

■ Excel and QM for Windows used in this book.

■ See book’s companion website – “Integer

Programming: the Branch and Bound

Method” for detailed description of this method.

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Computer Solution of IP Problems

0 – 1 Model with Excel (1 of 5)

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Exhibit 5.2

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Exhibit 5.3

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Exhibit 5.4

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Exhibit 5.5

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0 – 1 Model with QM for Windows (1

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Exhibit 5.6

of 3)

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Exhibit 5.7

of 3)

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Total Integer Model with Excel (1 of

5)

Integer Programming Model of

Machine Shop:

Maximize Z = $100x 1 + $150x 2 subject to:

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Exhibit 5.8

5)

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Exhibit 5.9

5)

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Exhibit 5.10

5)

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Exhibit 5.11

5)

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Mixed Integer Model with Excel (1 of 3)

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Exhibit 5.12

3)

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Exhibit 5.13

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

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Exhibit 5.14

Mixed Integer Model with QM for

Windows (1 of 2)

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Exhibit 5.15

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

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

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Exhibit 5.16

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Exhibit 5.17

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Plant

Available Capacity (tons,1000s)

Projected Annual Harvest (tons, 1000s)

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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Cities Cities within 300 miles

7 Milwaukee Detroit, Indianapolis, Milwaukee

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)

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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

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Exhibit 5.21

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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

■ 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?

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Step 1:

Formulate the Integer Programming Model

Maximize Z = $85,000x1 + 60,000x2subject 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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