1 1 Slide © 2005 Thomson/South-Western Chapter 8 Integer Linear Programming n Types of Integer Linear Programming Models n Graphical and Computer Solutions.

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© 2005 Thomson/South-Western© 2005 Thomson/South-Western

Chapter 8Chapter 8Integer Linear ProgrammingInteger Linear Programming

Types of Integer Linear Programming ModelsTypes of Integer Linear Programming Models Graphical and Computer Solutions for an All-Graphical and Computer Solutions for an All-

Integer Linear ProgramInteger Linear Program Applications Involving 0-1 VariablesApplications Involving 0-1 Variables Modeling Flexibility Provided by 0-1 Modeling Flexibility Provided by 0-1

VariablesVariables

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Types of Integer Programming ModelsTypes of Integer Programming Models

An LP in which all the variables are restricted An LP in which all the variables are restricted to be integers is called an to be integers is called an all-integer linear all-integer linear program program (ILP).(ILP).

The LP that results from dropping the integer The LP that results from dropping the integer requirements is called the requirements is called the LP RelaxationLP Relaxation of the of the ILP.ILP.

If only a subset of the variables are restricted If only a subset of the variables are restricted to be integers, the problem is called a to be integers, the problem is called a mixed-mixed-integer linear programinteger linear program (MILP). (MILP).

Binary variables are variables whose values Binary variables are variables whose values are restricted to be 0 or 1. If all variables are are restricted to be 0 or 1. If all variables are restricted to be 0 or 1, the problem is called a restricted to be 0 or 1, the problem is called a 0-1 or binary integer linear program0-1 or binary integer linear program. .

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Example: All-Integer LPExample: All-Integer LP

Consider the following all-integer linear Consider the following all-integer linear program:program:

Max 3Max 3xx11 + 2 + 2xx22

s.t. 3s.t. 3xx11 + + xx22 << 9 9

xx11 + 3 + 3xx22 << 7 7

--xx11 + + xx22 << 1 1

xx11, , xx22 >> 0 and integer 0 and integer

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LP RelaxationLP Relaxation

Solving the problem as a linear program Solving the problem as a linear program ignoring the integer constraints, the optimal ignoring the integer constraints, the optimal solution to the linear program gives fractional solution to the linear program gives fractional values for both values for both xx11 and and xx22. From the graph on . From the graph on the next slide, we see that the optimal solution the next slide, we see that the optimal solution to the linear program is:to the linear program is:

xx11 = 2.5, = 2.5, xx22 = 1.5, = 1.5, zz = 10.5 = 10.5

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LP RelaxationLP Relaxation

LP Optimal (2.5, 1.5)LP Optimal (2.5, 1.5)

Max 3Max 3xx11 + 2 + 2xx22

--xx11 + + xx22 << 1 1

xx22

xx11

33xx11 + + xx22 << 9 9

11

33

22

55

44

1 2 3 4 5 6 71 2 3 4 5 6 7

xx11 + 3 + 3xx22 << 7 7

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Rounding UpRounding Up

If we round up the fractional solution If we round up the fractional solution ((xx11 = 2.5, = 2.5, xx22 = 1.5) to the LP relaxation = 1.5) to the LP relaxation problem, we get problem, we get xx11 = 3 and = 3 and xx22 = 2. From the = 2. From the graph on the next slide, we see that this graph on the next slide, we see that this point lies outside the feasible region, making point lies outside the feasible region, making this solution infeasible. this solution infeasible.

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Rounded Up SolutionRounded Up Solution

LP Optimal (2.5, 1.5)LP Optimal (2.5, 1.5)

Max 3Max 3xx11 + 2 + 2xx22

--xx11 + + xx22 << 1 1

xx22

xx11

33xx11 + + xx22 << 9 9

ILP Infeasible (3, 2)ILP Infeasible (3, 2)

xx11 + 3 + 3xx22 << 7 7

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11

33

22

55

44

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Rounding DownRounding Down

By rounding the optimal solution down to By rounding the optimal solution down to xx11 = 2, = 2, xx22 = 1, we see that this solution indeed = 1, we see that this solution indeed is an integer solution within the feasible is an integer solution within the feasible region, and substituting in the objective region, and substituting in the objective function, it gives function, it gives zz = 8. = 8.

We have found a feasible all-integer We have found a feasible all-integer solution, but have we found the OPTIMAL all-solution, but have we found the OPTIMAL all-integer solution?integer solution?

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The answer is NO! The optimal solution The answer is NO! The optimal solution is is xx11 = 3 and = 3 and xx22 = 0 giving = 0 giving zz = 9, as evidenced = 9, as evidenced in the next two slides. in the next two slides.

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Complete Enumeration of Feasible ILP SolutionsComplete Enumeration of Feasible ILP Solutions

There are eight feasible integer solutions There are eight feasible integer solutions to this problem:to this problem:

xx11 xx22 zz

1. 0 0 01. 0 0 0 2. 1 0 32. 1 0 3 3. 2 0 63. 2 0 6 4. 3 0 9 optimal 4. 3 0 9 optimal

solutionsolution 5. 0 1 25. 0 1 2 6. 1 1 56. 1 1 5 7. 2 1 87. 2 1 8

8. 1 2 78. 1 2 7

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ILP Optimal (3, 0)ILP Optimal (3, 0)

Max 3Max 3xx11 + 2 + 2xx22

--xx11 + + xx22 << 1 1

xx22

xx11

33xx11 + + xx22 << 9 9

xx11 + 3 + 3xx22 << 7 7

11

33

22

55

44

1 2 3 4 5 6 71 2 3 4 5 6 7

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Partial Spreadsheet Showing Problem DataPartial Spreadsheet Showing Problem Data

A B C D12 Constraint X1 X2 RHS

3 #1 3 1 9

4 #2 1 3 7

5 #3 -1 1 1

6 Obj.Coefficients 3 2

LHS Coefficients

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Partial Spreadsheet Showing FormulasPartial Spreadsheet Showing Formulas

A B C D

8 X1 X2

9

10 =B6*C9+C6*D9

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12 Constraints LHS RHS

13 #1 =B3*$C$9+C3*$D$9 <= 9

14 #2 =B4*$C$9+C4*$D$9 <= 7

15 #3 =B5*$C$9+C5*$D$9 <= 1

Maximized Objective Function

Optimal Decision Variable Values

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Partial Spreadsheet Showing Optimal SolutionPartial Spreadsheet Showing Optimal Solution

A B C D

8 X1 X2

9 3.000 2.9419E-12

10 9.000

11

12 Constraints LHS RHS

13 #1 9 <= 9

14 #2 3 <= 7

15 #3 -3 <= 1

Maximized Objective Function

Optimal Decision Variable Values

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Special 0-1 ConstraintsSpecial 0-1 Constraints

When When xxii and and xxjj represent binary variables represent binary variables designating whether projects designating whether projects ii and and jj have been have been completed, the following special constraints may completed, the following special constraints may be formulated:be formulated:

• At most At most kk out of out of nn projects will be completed: projects will be completed: xxjj << kk jj

• Project Project jj is is conditionalconditional on project on project ii: :

xxjj - - xxii << 0 0

• Project Project ii is a is a corequisitecorequisite for project for project jj: :

xxjj - - xxii = 0 = 0

• Projects Projects ii and and jj are are mutually exclusivemutually exclusive: :

xxii + + xxjj << 1 1

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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves

Metropolitan Microwaves, Inc. is planning Metropolitan Microwaves, Inc. is planning toto

expand its operations into otherexpand its operations into other

electronic appliances. The companyelectronic appliances. The company

has identified seven new product lineshas identified seven new product lines

it can carry. Relevant informationit can carry. Relevant information

about each line follows on the next slide. about each line follows on the next slide.

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Initial Floor Space Exp. Rate Initial Floor Space Exp. Rate

Product Line Product Line Invest. (Sq.Ft.) of Invest. (Sq.Ft.) of ReturnReturn

1. TV/VCRs1. TV/VCRs $ 6,000 125$ 6,000 125 8.1% 8.1%2. Color TVs 2. Color TVs 12,000 150 12,000 150

9.0 9.0 3. Projection TVs 3. Projection TVs 20,000 200 20,000 200

11.0 11.0 4. VCRs4. VCRs 14,000 40 14,000 40

10.2 10.2 5. DVD Players5. DVD Players 15,000 40 15,000 40 10.5 10.5 6. Video Games 6. Video Games 2,000 20 2,000 20

14.1 14.1 7. Home Computers 7. Home Computers 32,000 100 32,000 100

13.2 13.2

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Metropolitan has decided that they Metropolitan has decided that they should not stock projection TVs unless they should not stock projection TVs unless they stock either TV/VCRs or color TVs. Also, they stock either TV/VCRs or color TVs. Also, they will not stock both VCRs and DVD players, and will not stock both VCRs and DVD players, and they will stock video games if they stock color they will stock video games if they stock color TVs. Finally, the company wishes to introduce TVs. Finally, the company wishes to introduce at least three new product lines. at least three new product lines.

If the company has $45,000 to invest If the company has $45,000 to invest and 420 sq. ft. of floor space available, and 420 sq. ft. of floor space available, formulate an integer linear program for formulate an integer linear program for Metropolitan to maximize its overall expected Metropolitan to maximize its overall expected return.return.

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Define the Decision VariablesDefine the Decision Variables

xxjj = 1 if product line = 1 if product line jj is introduced; is introduced;

= 0 otherwise.= 0 otherwise.

where:where:Product line 1 = TV/VCRsProduct line 1 = TV/VCRsProduct line 2 = Color TVsProduct line 2 = Color TVsProduct line 3 = Projection TVsProduct line 3 = Projection TVsProduct line 4 = VCRsProduct line 4 = VCRsProduct line 5 = DVD PlayersProduct line 5 = DVD PlayersProduct line 6 = Video GamesProduct line 6 = Video GamesProduct line 7 = Home ComputersProduct line 7 = Home Computers

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Define the Decision VariablesDefine the Decision Variables

xxjj = 1 if product line = 1 if product line jj is introduced; is introduced;


Define the Objective FunctionDefine the Objective Function

Maximize total expected return:Maximize total expected return:

Max .081(6000)Max .081(6000)xx11 + .09(12000) + .09(12000)xx22 + .11(20000)+ .11(20000)xx33

+ .102(14000)+ .102(14000)xx4 4 + .105(15000)+ .105(15000)xx55 + + .141(2000).141(2000)xx66

+ .132(32000)+ .132(32000)xx77

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Define the ConstraintsDefine the Constraints

1) Money: 1) Money:

66xx11 + 12 + 12xx22 + 20 + 20xx33 + 14 + 14xx44 + 15 + 15xx55 + 2 + 2xx66 + + 3232xx77 << 45 45

2) Space: 2) Space:

125125xx11 +150 +150xx22 +200 +200xx33 +40 +40xx44 +40 +40xx55 +20 +20xx66 +100+100xx77 << 420 420

3) Stock projection TVs only if 3) Stock projection TVs only if

stock TV/VCRs or color TVs:stock TV/VCRs or color TVs:

xx11 + + xx22 > > xx33 or or xx11 + + xx22 - - xx33 >> 0 0

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Define the Constraints (continued)Define the Constraints (continued)

4) Do not stock both VCRs and DVD players: 4) Do not stock both VCRs and DVD players:

xx44 + + xx55 << 1 1

5) Stock video games if they stock color TV's: 5) Stock video games if they stock color TV's:

xx22 - - xx66 >> 0 0

6) Introduce at least 3 new lines:6) Introduce at least 3 new lines:

xx11 + + xx22 + + xx33 + + xx44 + + xx55 + + xx66 + + xx77 >> 3 3

7) Variables are 0 or 1: 7) Variables are 0 or 1:

xxjj = 0 or 1 for = 0 or 1 for jj = 1, , , 7 = 1, , , 7

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Partial Spreadsheet Showing Problem DataPartial Spreadsheet Showing Problem Data

A B C D E F G H I

1

2 Constraints X1 X2 X3 X4 X5 X6 X7 RHS

3 #1 6 12 20 14 15 2 32 45

4 #2 125 150 200 40 40 20 100 420

5 #3 1 1 -1 0 0 0 0 0

6 #4 0 0 0 1 1 0 0 1

7 #5 0 1 0 0 0 -1 0 0

8 #6 1 1 1 1 1 1 1 3

9 Obj.Func.Coeff. 486 1080 2200 1428 1575 282 4224

LHS Coefficients

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Partial Spreadsheet Showing Example Partial Spreadsheet Showing Example FormulasFormulasA B C D E F G H

12 X1 X2 X3 X4 X5 X6 X7

13 Dec.Values 0 0 0 0 0 0 0

14 Maximized Total Expected Return 0

15

16 LHS RHS

17 0 <= 45

18 0 <= 420

19 0 >= 0

20 0 <= 1

21 0 >= 0

22 0 >= 3

Video

Lines

VCRs

TVs

Constraints

Money

Space

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Solver Parameters Dialog BoxSolver Parameters Dialog Box

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Solver Options Dialog BoxSolver Options Dialog Box

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Integer Options Dialog BoxInteger Options Dialog Box

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A B C D E F G H

12 X1 X2 X3 X4 X5 X6 X7

13 Dec.Values 1 0 1 0 1 0 0

14 Maximized Total Expected Return 4261

15

16 LHS RHS

17 41 <= 45

18 365 <= 420

19 0 >= 0

20 1 <= 1

21 0 >= 0

22 3 >= 3

Video

Lines

VCRs

TVs

Constraints

Money

Space


Optimal SolutionOptimal Solution

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Optimal SolutionOptimal Solution

Introduce:Introduce: TV/VCRs, Projection TVs, and DVD PlayersTV/VCRs, Projection TVs, and DVD Players

Do Not Introduce:Do Not Introduce: Color TVs, VCRs, Video Games, and Home Color TVs, VCRs, Video Games, and Home ComputersComputers

Total Expected ReturnTotal Expected Return:: $4,261$4,261

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Example: Tina’s TailoringExample: Tina’s Tailoring

Tina's Tailoring has five idle tailors and Tina's Tailoring has five idle tailors and four custom garments to make. The estimated four custom garments to make. The estimated time (in hours) it would take each tailor to make time (in hours) it would take each tailor to make each garment is shown in the next slide. (An 'X' each garment is shown in the next slide. (An 'X' in the table indicates an unacceptable tailor-in the table indicates an unacceptable tailor-garment assignment.)garment assignment.)

TailorTailor

GarmentGarment 11 22 33 44 55 Wedding gown 19 23 20 21 18Wedding gown 19 23 20 21 18

Clown costume 11 14 X 12 10Clown costume 11 14 X 12 10 Admiral's uniform 12 8 11 X 9Admiral's uniform 12 8 11 X 9 Bullfighter's outfit X 20 20 18 21Bullfighter's outfit X 20 20 18 21

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Formulate an integer program for determiningFormulate an integer program for determining

the tailor-garment assignments that minimizethe tailor-garment assignments that minimize

the total estimated time spent making the fourthe total estimated time spent making the four

garments. No tailor is to be assigned more than onegarments. No tailor is to be assigned more than one

garment and each garment is to be worked on by garment and each garment is to be worked on by onlyonly

one tailor.one tailor.

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This problem can be formulated as a 0-1 This problem can be formulated as a 0-1 integerinteger

program. The LP solution to this problem willprogram. The LP solution to this problem will

automatically be integer (0-1).automatically be integer (0-1).

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Define the decision variablesDefine the decision variables

xxijij = 1 if garment i is assigned to tailor = 1 if garment i is assigned to tailor jj


Number of decision variables = Number of decision variables =

[(number of garments)(number of [(number of garments)(number of tailors)] tailors)]

- (number of unacceptable assignments) - (number of unacceptable assignments)

= [4(5)] - 3 = 17= [4(5)] - 3 = 17

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Define the objective functionDefine the objective function

Minimize total time spent making garments:Minimize total time spent making garments:

Min 19Min 19xx1111 + 23 + 23xx1212 + 20 + 20xx1313 + 21 + 21xx1414 + 18 + 18xx1515 + + 1111xx2121

+ 14+ 14xx2222 + 12 + 12xx2424 + 10 + 10xx2525 + 12 + 12xx3131 + 8 + 8xx3232 + 11+ 11xx3333

+ 9x+ 9x3535 + 20 + 20xx4242 + 20 + 20xx4343 + 18 + 18xx4444 + 21 + 21xx4545

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Define the ConstraintsDefine the Constraints

Exactly one tailor per garment:Exactly one tailor per garment:

1) 1) xx1111 + + xx1212 + + xx1313 + + xx1414 + + xx1515 = 1 = 1

2) 2) xx2121 + + xx2222 + + xx2424 + + xx2525 = 1 = 1

3) 3) xx3131 + + xx3232 + + xx3333 + + xx3535 = 1 = 1

4) 4) xx4242 + + xx4343 + + xx4444 + + xx4545 = 1 = 1

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Define the Constraints (continued)Define the Constraints (continued)

No more than one garment per tailor:No more than one garment per tailor:

5) 5) xx1111 + + xx2121 + + xx3131 << 1 1

6) 6) xx1212 + + xx2222 + + xx3232 + + xx4242 << 1 1

7) 7) xx1313 + + xx3333 + + xx4343 << 1 1

8) 8) xx1414 + + xx2424 + + xx4444 << 1 1

9) 9) xx1515 + + xx2525 + + xx3535 + + xx4545 << 1 1

Nonnegativity: Nonnegativity: xxijij >> 0 for 0 for ii = 1, . . ,4 and = 1, . . ,4 and jj = = 1, . . ,5 1, . . ,5

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Cautionary Note About Sensitivity Cautionary Note About Sensitivity AnalysisAnalysis

Sensitivity analysis often is more crucial for ILP Sensitivity analysis often is more crucial for ILP problems than for LP problems.problems than for LP problems.

A small change in a constraint coefficient can A small change in a constraint coefficient can cause a relatively large change in the optimal cause a relatively large change in the optimal solution.solution.

Recommendation: Resolve the ILP problem Recommendation: Resolve the ILP problem several times with slight variations in the several times with slight variations in the coefficients before choosing the “best” coefficients before choosing the “best” solution for implementation.solution for implementation.

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End of Chapter 8End of Chapter 8

1 1 Slide © 2005 Thomson/South-Western Chapter 8 Integer Linear Programming n Types of Integer Linear Programming Models n Graphical and Computer Solutions.

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