chapter 9 - integer programming 1 chapter 9 integer programming introduction to management science...

Chapter 9 - Integer Programming 1

Chapter 9

Integer Programming

Introduction to Management Science

8th Edition

by

Bernard W. Taylor III


Chapter Topics

Integer Programming (IP) Models

Still “linear programming models”

Variables must be integers, and so forma subset of the original, real-number models

Integer Programming Graphical Solution

Computer Solution of Integer Programming Problems With Excel and QM for Windows


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.

Also referred to as Boolean and True/False

Mixed Integer Model: Some of the decision variables (but not all) required to have integer values.


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:


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


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)



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 (either do or don’t)

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)


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.



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)


Rounding non-integer solution values up (down) to the nearest integer value can result in an infeasible solution

A feasible solution may be found by rounding down (up)non-integer solution values, but may result in a less than optimal (sub-optimal) solution.

Whether a variable is to be rounded up or down depends on where the corresponding point is with respect to the constraint boundaries.

Integer Programming Graphical Solution


Integer Programming ExampleGraphical Solution of Maximization Model

Maximize Z = $100x1 + $150x2

subject to: 8,000x1 + 4,000x2 $40,000 15x1 + 30x2 200 ft2


Optimal Solution:Z = $1,055.56x1 = 2.22 pressesx2 = 5.55 lathes

Figure 9.1Feasible Solution Space with Integer Solution Points

Both of these may be found by judiciousrounding, but the value of the objective function needs to be checked for each.


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 and QM for Windows used in this book.

See CD-ROM Module C – “Integer Programming: the Branch and Bound Method” for detailed description of method.


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 Problems0–1 Model with Excel (1 of 5)


Exhibit 9.2



Exhibit 9.3


Instead, one could justspecify $C$12:$C$15as “binary” (= ‘0’ or ‘1’).


Exhibit 9.4


To constrain a range of variables to be integers, enter:

… and note that it doesn’t matter what you put in the right-hand side field, but it must not be empty.

Instead, one could just specify the “bin” option (= ‘0’ or ‘1’),thus avoiding the additional, “≥ 0” and “≤ 1” constraints.


Exhibit 9.5



Computer Solution of IP Problems0–1 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


Exhibit 9.6



Exhibit 9.7



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



Exhibit 9.8



Exhibit 9.9


… and, again, just to note that this field will automatically

fill, but Excel wants you to put something in it anyway.


Exhibit 9.10



Exhibit 9.11




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 ProblemsMixed Integer Model with Excel (1 of 3)


Exhibit 9.12



Exhibit 9.13

Computer Solution of IP ProblemsSolution of Total Integer Model with Excel (3 of 3)


Exhibit 9.14

Computer Solution of IP ProblemsMixed Integer Model with QM for Windows (1 of 2)


Exhibit 9.15

Computer Solution of IP ProblemsMixed Integer Model with QM for Windows (2 of 2)


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)


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 + $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



Exhibit 9.16



Exhibit 9.17


Could have chosen ‘bin’ (= ‘0’ or ‘1’) instead!


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 the six farms should be purchased that will meet current production capacity at minimum total cost, including annual fixed costs and shipping costs?

Data:


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 = 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 + x1C - 11.2y1 ≤ 0 x2A + x2B + x2C -10.5y2 ≤ 0 x3A + x3B + 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 + x3A + x4B + x5B + x6B = 10 x1C + x2C + x3C+ x4C + x5C + x6C = 14 xij ≥ 0 yi = 0 or 1


The 6 farms’ capacities.�

The 3 plants’ capacities.


Exhibit 9.18



Exhibit 9.19



Cities Cities within 300 miles 1. Atlanta Atlanta, Charlotte, Nashville 2. Boston Boston, New York 3. Charlotte Atlanta, Charlotte, Richmond 4. Cincinnati Cincinnati, Detroit, 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, Richmond10. 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)


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 + 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



Exhibit 9.20


the decision variables


Exhibit 9.21


For “N7”, enter “= sumproduct(B7:M7,B$20:M$20)” and the down;the last constraint requires that these (hubs within 300 mi) equal at least one.


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/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

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 ExampleModel Formulation (2 of 3)


Total Integer Programming Modeling ExampleSolution with QM for Windows (3 of 3)

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