integer programming - 0702112

8/2/2019 Integer Programming - 0702112

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11SlideSlideRajendraRajendra Desai, XIME, 2012Desai, XIME, 2012

Integer Linear ProgrammingInteger Linear Programming


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Integer Linear ProgrammingInteger Linear Programming

Types of Integer Linear Programming ModelsTypes of Integer Linear Programming Models

Graphical and Computer Solutions for an AllGraphical and Computer Solutions for an All--Integer Linear ProgramInteger Linear Program

Applications Involving 0Applications Involving 0--1 Variables1 Variables

Modeling Flexibility Provided by 0Modeling Flexibility Provided by 0--1 Variables1 Variables


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Types of Integer Programming ModelsTypes of Integer Programming Models

An LP in which all the variables are restricted to beAn LP in which all the variables are restricted to beintegers is called anintegers is called an allall--integer linear programinteger linear program (ILP).(ILP).

The LP that results from dropping the integerThe LP that results from dropping the integerrequirements is called therequirements is called the LP RelaxationLP Relaxation of the ILP.of the ILP.

If only a subset of the variables are restricted to beIf only a subset of the variables are restricted to beintegers, the problem is called aintegers, the problem is called a mixedmixed--integer linearinteger linearprogramprogram (MILP).(MILP).

Binary variables are variables whose values areBinary variables are variables whose values arerestricted to be 0 or 1. If all variables are restricted torestricted to be 0 or 1. If all variables are restricted to

be 0 or 1, the problem is called abe 0 or 1, the problem is called a 00--1 or binary integer1 or binary integerlinear programlinear program..


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Example: AllExample: All--Integer LPInteger LP

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

Max 3Max 3xx11 + 2+ 2xx22

s.t. 3s.t. 3xx11 ++ xx22


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LP RelaxationLP RelaxationSolving the problem as a linear program ignoringSolving the problem as a linear program ignoring

the integer constraints, the optimal solution to thethe integer constraints, the optimal solution to thelinear program gives fractional values for bothlinear program gives fractional values for both xx11 andandxx22. From the graph on the next slide, we see that the. From the graph on the next slide, we see that the

optimal solution to the linear program is:optimal solution to the linear program is:

xx11 = 2.5,= 2.5, xx22 = 1.5,= 1.5,

Max 3Max 3xx11 + 2+ 2xx22 = 10.5= 10.5


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LP RelaxationLP Relaxation

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


xx11 ++ xx22


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Rounding UpRounding UpIf 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 problem, we get= 1.5) to the LP relaxation problem, we get xx11 = 3= 3andand xx22 = 2. From the graph on the next slide, we= 2. From the graph on the next slide, wesee that this point lies outside the feasible region,see that this point lies outside the feasible region,making this solution infeasible.making this solution infeasible.


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Rounded Up SolutionRounded Up Solution

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

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


xx11 ++ xx22


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Complete Enumeration of Feasible ILP SolutionsComplete Enumeration of Feasible ILP SolutionsThere are eight feasible integer solutions to thisThere are eight feasible integer solutions to this

problem:problem:

xx11

xx22

33xx11 + 2+ 2xx22

1.1. 0 00 0 002.2. 1 01 0 333.3. 2 02 0 664.4. 3 03 0 9 optimal solution9 optimal solution5.5. 0 10 1 226.6. 1 11 1 557.7. 2 12 1 888.8. 1 21 2 77


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


xx11 ++ xx22


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Example:Example: ShyamsShyams TailoringTailoring

ShyamsShyams Tailoring has five idle tailors and fourTailoring has five idle tailors and fourcustom garments to make. The estimated time (incustom garments to make. The estimated time (inhours) it would take each tailor to make each garmenthours) it would take each tailor to make each garmentis shown in the next slide. (An 'X' in the table indicatesis shown in the next slide. (An 'X' in the table indicatesan unacceptable tailoran 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 10Admiral'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 determiningthe tailorthe tailor--garment assignments that minimizegarment 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 onlygarment and each garment is to be worked on by onlyone tailor.one tailor.

----------------------------------------

This problem can be formulated as a 0This problem can be formulated as a 0--1 integer1 integer

program. The LP solution to this problem willprogram. The LP solution to this problem willautomatically be integer (0automatically be integer (0--1).1).


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Define the decision variablesDefine the decision variablesxxijij = 1 if garment i is assigned to tailor= 1 if garment i is assigned to tailorjj

= 0 otherwise.= 0 otherwise.

Number of decision variables =Number of decision variables =

[(number of garments)(number of tailors)][(number of garments)(number of tailors)]-- (number of unacceptable assignments)(number of unacceptable assignments)

= [4(5)]= [4(5)] -- 3 = 173 = 17


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Define the objective functionDefine the objective functionMinimize total time spent making garments:Minimize total time spent making garments:

Min 19Min 19xx1111 + 23+ 23xx1212 + 20+ 20xx1313 + 21+ 21xx1414 + 18+ 18xx1515 + 11+ 11xx2121+ 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 ConstraintsExactly 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= 14)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


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Modeling Flexibility Provided by 0Modeling Flexibility Provided by 0--1 Variables1 Variables

WhenWhen xxii andand xxjj represent binary variables designatingrepresent binary variables designatingwhether projectswhether projects ii andandjj have been completed, thehave been completed, thefollowing special constraints may be formulated:following special constraints may be formulated:

At mostAt most kk out ofout of nn projects will be completed:projects will be completed:77

xxjj


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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves

Metropolitan Microwaves is planning to expand itsMetropolitan Microwaves is planning to expand itssales operation by offering other electronic appliances.sales operation by offering other electronic appliances.

The company has identified seven new product lines itThe company has identified seven new product lines itcan carry. Relevant information about each linecan carry. Relevant information about each line

follows on the next slide.follows on the next slide.


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Initial Floor Space Exp. RateInitial Floor Space Exp. Rate

Product LineProduct Line Invest. (Invest. (Sq.FtSq.Ft.) of Return.) of Return

1. TV/VCRs1. TV/VCRs 6,000 1256,000 125 8.1%8.1%2. TVs2. TVs 12,000 15012,000 150 9.09.03. Projection TVs3. Projection TVs 20,000 20020,000 200 11.011.04. VCRs4. VCRs 14,000 4014,000 40 10.210.25. DVD Players5. DVD Players 15,000 4015,000 40 10.510.56. Video Games6. Video Games 2,000 202,000 20 14.114.1

7. Home Computers7. Home Computers3

2,000 1003

2,000 100 13

.213

.2


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Metropolitan has decided that they should not stockMetropolitan has decided that they should not stockprojection TVs unless they stock either TV/VCRs orprojection TVs unless they stock either TV/VCRs orTVs. Also, they will not stock both VCRs and DVDTVs. Also, they will not stock both VCRs and DVDplayers, and they will stock video games if they stockplayers, and they will stock video games if they stockTVs. Finally, the company wishes to introduce atTVs. Finally, the company wishes to introduce at

least three new product lines.least three new product lines.

If the company has 45,000 to invest and 420 sq. ft. ofIf the company has 45,000 to invest and 420 sq. ft. offloor space available, formulate an integer linearfloor space available, formulate an integer linear

program for Metropolitan to maximize its overallprogram for Metropolitan to maximize its overallexpected return.expected return.


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Define the Decision VariablesDefine the Decision Variablesxxjj = 1 if product line= 1 if product linejj is introduced;is introduced;


where:where:

Product line 1 = TV/VCRsProduct line 1 = TV/VCRsProduct line 2 = TVsProduct line 2 = TVs

Product line 3 = Projection TVsProduct line 3 = Projection TVs

Product line 4 = VCRsProduct line 4 = VCRs

Product line5

= DVD PlayersProduct line5

= DVD PlayersProduct line 6 = Video GamesProduct line 6 = Video Games

Product line 7 = Home ComputersProduct line 7 = Home Computers


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Define the Decision VariablesDefine the Decision Variablesxxjj = 1 if product line= 1 if product linejj 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)xx44+ .105(15000)+ .105(15000)xx55 + .141(2000)+ .141(2000)xx66+ .132(32000)+ .132(32000)xx77


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Define the ConstraintsDefine the Constraints1) Money:1) Money:

66xx11 + 12+ 12xx22 + 20+ 20xx33 + 14+ 14xx44 + 15+ 15xx55 + 2+ 2xx66 + 32+ 32xx77 xx33 oror xx11 ++ xx22 -- xx33 >> 00


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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 > 006) Introduce at least 3 new lines:6) Introduce at least 3 new lines:

xx11 ++ xx22 ++ xx33 ++ xx44 ++ xx55 ++ xx66 ++ xx77 >> 33

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

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


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Partial Spreadsheet Showing Problem DataPartial Spreadsheet Showing Problem DataA 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 FormulasPartial Spreadsheet Showing Example FormulasA 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

1516 LHS RHS

17 0 = 3

Constraints

Money

Space

Video

Lines

VCRs

TVs


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Optimal SolutionOptimal SolutionA 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

1516 LHS RHS

17 41 = 3

Video

Lines

VCRs

TVs

Constraints

Money

Space


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Optimal SolutionOptimal SolutionIntroduce:Introduce:

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

Do Not Introduce:Do Not Introduce:

TVs, VCRs, Video Games, and Home ComputersTVs, VCRs, Video Games, and Home Computers

Total Expected ReturnTotal Expected Return::

$4,261$4,261


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Cautionary Note About Sensitivity AnalysisCautionary Note About Sensitivity Analysis

Sensitivity analysis often is more crucial for ILPSensitivity analysis often is more crucial for ILPproblems than for LP problems.problems than for LP problems.

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

Recommendation: Resolve the ILP problem severalRecommendation: Resolve the ILP problem severaltimes with slight variations in the coefficients beforetimes with slight variations in the coefficients beforechoosing the best solution for implementation.choosing the best solution for implementation.

integer programming - 0702112

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