1 1 slide © 2005 thomson/south-western chapter 8 integer linear programming n types of integer...
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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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Example: All-Integer LPExample: All-Integer LP
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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Example: All-Integer LPExample: All-Integer LP
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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Example: All-Integer LPExample: All-Integer LP
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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Example: All-Integer LPExample: All-Integer LP
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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Example: All-Integer LPExample: All-Integer LP
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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Example: All-Integer LPExample: All-Integer LP
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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Example: All-Integer LPExample: All-Integer LP
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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Example: All-Integer LPExample: All-Integer LP
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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Example: All-Integer LPExample: All-Integer LP
Partial Spreadsheet Showing FormulasPartial Spreadsheet Showing Formulas
A B C D
8 X1 X2
9
10 =B6*C9+C6*D9
11
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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Example: All-Integer LPExample: All-Integer LP
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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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves
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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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves
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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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves
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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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves
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.
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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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves
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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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves
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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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves
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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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves
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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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves
Solver Parameters Dialog BoxSolver Parameters Dialog Box
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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves
Solver Options Dialog BoxSolver Options Dialog Box
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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves
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
Example: Metropolitan MicrowavesExample: Metropolitan Microwaves
Optimal SolutionOptimal Solution
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Example: Metropolitan MicrowavesExample: Metropolitan Microwaves
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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Example: Tina’s TailoringExample: Tina’s Tailoring
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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Example: Tina’s TailoringExample: Tina’s Tailoring
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
= 0 otherwise.= 0 otherwise.
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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Example: Tina’s TailoringExample: Tina’s Tailoring
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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Example: Tina’s TailoringExample: Tina’s Tailoring
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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Example: Tina’s TailoringExample: Tina’s Tailoring
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.