integer programming - jeryfarel | ahmad jaeri berbagi ilmu · pdf fileinteger programming...

Introduction to Management Science

(8th Edition, Bernard W. Taylor III)

Chapter 9

Chapter 9 - Integer Programming 1

Integer Programming

Chapter Topics

Integer Programming (IP) ModelsStill “linear programming models”

Variables must be integers, and so formVariables 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 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.


A Total Integer Model (1 of 2)

Machine shop obtaining new presses and lathes.

Marginal profitability: each press $100/day; each lathe $150/day.

MachineRequired Floor

SpacePurchase

Price

$150/day.

Resource constraints: $40,000, 200 sq. ft. floor space.

Machine purchase prices and space requirements:


Space(sq. ft.)

Price

Press

Lathe

15

30

$8,000

4,000

A Total Integer Model (2 of 2)

Integer Programming Model:

Maximize Z = $100x1 + $150x2

subject to:subject to:

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

15x1 + 30x2 200 ft2

x1, x2 0 and integer

x1 = number of presses


1

x2 = number of lathes

Recreation facilities selection to maximize daily usage by residents.

Resource constraints: $120,000 budget; 12 acres of land.

A 0–1 Integer Model (1 of 2)

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)


Facility (people/day)(acres)

Swimming poolTennis CenterAthletic fieldGymnasium

30090

400150

35,00010,00025,00090,000

4273


Maximize Z = 300x1 + 90x2 + 400x3 + 150x4

subject to:

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 (either do or don’t)


1 2 3 4

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


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:

A Mixed Integer Model (2 of 2)

subject to:

50,000x1 + 12,000x2 + 8,000x3 $250,000x1 4 condominiumsx2 15 acresx3 20 bonds x2 0x , x 0 and integer


2


x1 = condominiums purchasedx2 = acres of land purchasedx3 = bonds purchased

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)

Integer Programming Graphical 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 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 = $1,055.56


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 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


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

Computer Solution of IP Problems0–1 Model with Excel (1 of 5)

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


x1, x2, x3, x4 = 0 or 1



Exhibit 9.2



Exhibit 9.3

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


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

Exhibit 9.4

integers, enter:


Exhibit 9.4

… 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


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 + $150x2Maximize 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

Computer Solution of IP ProblemsMixed Integer Model with Excel (1 of 3)

subject to:

50,000x1 + 12,000x2 + 8,000x3 $250,000x1 4 condominiumsx2 15 acresx3 20 bonds x2 0


x2 0x1, x3 0 and integer



Exhibit 9.12

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


Exhibit 9.13

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


Exhibit 9.14

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


Exhibit 9.15

University bookstore expansion project.

Not enough space available for both a computer department and a clothing department.

0–1 Integer Programming Modeling ExamplesCapital Budgeting Example (1 of 4)

and a clothing department.

Data:

ProjectNPV Return

($1000)Project Costs per Year ($1000)

1 2 3

1. Website2. Warehouse

120 85

5545

4035

2520


2. Warehouse3. Clothing department4. Computer department5. ATMs

Available funds per year

85105140 75

45605030

150

35253530

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: 55x + 45x + 60x + 50x + 30x 150


55x1 + 45x2 + 60x3 + 50x4 + 30x5 15040x1 + 35x2 + 25x3 + 35x4 + 30x5 11025x1 + 20x2 + 30x4 60

x3 + x4 1xi = 0 or 1



Exhibit 9.16



Exhibit 9.17

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

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?

PlantAvailable Capacity

(tons,1000s)ABC

121014

Farms Annual Fixed Costs

($1000)

Projected Annual Harvest (tons, 1000s)

12

405390

11.210.5 Plant

annual fixed costs and shipping costs?

Data:


23456

390450368520465

10.512.8 9.310.8 9.6

FarmPlant

A B C

123456

18 15 1213 10 1716 14 1819 15 1617 19 1214 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.


= 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 ≤ 0x3A + x3B + x3C - 12.8y3 ≤ 0 x4A + x4B + x4C - 9.3y4 ≤ 0


x3A + x3B + x3C - 12.8y3 ≤ 0 x4A + x4B + x4C - 9.3y4 ≤ 0x5A + x5B + x5B - 10.8y5 ≤ 0 x6A + x6B + X6C - 9.6y6 ≤ 0x1A + x2A + x3A + x4A + x5A + x6A =12x1B + x2B + x3A + x4B + x5B + x6B = 10x1C + x2C + x3C+ x4C + x5C + x6C = 14xij ≥ 0 yi = 0 or 1

The 6 farms’ capacities� .

The 3 plants’ capacities.



Exhibit 9.18



Exhibit 9.19

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)

Cities Cities within 300 miles1. Atlanta Atlanta, Charlotte, Nashville2. Boston Boston, New York3. Charlotte Atlanta, Charlotte, Richmond4. Cincinnati Cincinnati, Detroit, Nashville, Pittsburgh5. Detroit Cincinnati, Detroit, Indianapolis, Milwaukee, Pittsburgh6. Indianapolis Cincinnati, Detroit, Indianapolis, Milwaukee, Nashville, St.

Louis

miles of every city:


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

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:


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: x + x + x 1


Milwaukee: 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/salesperson


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


x1, x2 0 or integer

Step 2:

Solve the Model using QM for Windows

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


integer programming - jeryfarel | ahmad jaeri berbagi ilmu · pdf fileinteger programming...

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