Slides by
JOHN
LOUCKS
St. Edward’s
University
© 2008 Thomson South-Western. All Rights Reserved
Slide 1
Chapter 7
Integer Linear Programming
Types of Integer Linear Programming Models
Graphical and Computer Solutions for an AllInteger Linear Program
Applications Involving 0-1 Variables
Modeling Flexibility Provided by 0-1 Variables
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Slide 2
Types of Integer Programming Models
An LP in which all the variables are restricted to be
integers is called an all-integer linear program (ILP).
The LP that results from dropping the integer
requirements is called the LP Relaxation of the ILP.
If only a subset of the variables are restricted to be
integers, the problem is called a mixed-integer linear
program (MILP).
Binary variables are variables whose values are
restricted to be 0 or 1. If all variables are restricted to
be 0 or 1, the problem is called a 0-1 or binary integer
linear program.
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Slide 3
Example: All-Integer LP
Consider the following all-integer linear program:
Max 3x1 + 2x2
s.t.
3x1 + x2 < 9
x1 + 3x2 < 7
-x1 + x2 < 1
x1, x2 > 0 and integer
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Slide 4
Example: All-Integer LP
LP Relaxation
Solving the problem as a linear program ignoring
the integer constraints, the optimal solution to the
linear program gives fractional values for both x1 and
x2. From the graph on the next slide, we see that the
optimal solution to the linear program is:
x1 = 2.5, x2 = 1.5,
Max 3x1 + 2x2 = 10.5
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Slide 5
Example: All-Integer LP
LP Relaxation
x2
5
- x1 + x2 < 1
4
3x1 + x2 < 9
3
Max 3x1 + 2x2
LP Optimal (2.5, 1.5)
2
x1 + 3x2 < 7
1
1
2
3
4
5
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6
7
x1
Slide 6
Example: All-Integer LP
Rounding Up
If we round up the fractional solution (x1 = 2.5,
x2 = 1.5) to the LP relaxation problem, we get x1 = 3
and x2 = 2. From the graph on the next slide, we
see that this point lies outside the feasible region,
making this solution infeasible.
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Slide 7
Example: All-Integer LP
Rounded Up Solution
x2
- x1 + x2 < 1
5
3x1 + x2 < 9
4
Max 3x1 + 2x2
3
ILP Infeasible (3, 2)
LP Optimal (2.5, 1.5)
2
x1 + 3x2 < 7
1
1
2
3
4
5
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6
7
x1
Slide 8
Example: All-Integer LP
Rounding Down
By rounding the optimal solution down to x1 = 2,
x2 = 1, we see that this solution indeed is an integer
solution within the feasible region, and substituting
in the objective function, it gives 3x1 + 2x2 = 8.
We have found a feasible all-integer solution, but
have we found the OPTIMAL all-integer solution?
--------------------The answer is NO! The optimal solution is x1 = 3
and x2 = 0 giving 3x1 + 2x2 = 9, as evidenced in the
next two slides.
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Slide 9
Example: All-Integer LP
Complete Enumeration of Feasible ILP Solutions
There are eight feasible integer solutions to this
problem:
x1
x2 3x1 + 2x2
1.
2.
3.
4.
5.
6.
7.
8.
0
1
2
3
0
1
2
1
0
0
0
0
1
1
1
2
0
3
6
9
2
5
8
7
optimal solution
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Slide 10
Example: All-Integer LP
Optimal All-Integer Solution
x2
- x1 + x2 < 1
5
3x1 + x2 < 9
4
Max 3x1 + 2x2
3
ILP Optimal (3, 0)
2
x1 + 3x2 < 7
1
1
2
3
4
5
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6
7
x1
Slide 11
Example: Tina’s Tailoring
Tina's Tailoring has five idle tailors and four
custom garments to make. The estimated time (in
hours) it would take each tailor to make each garment
is shown in the next slide. (An 'X' in the table indicates
an unacceptable tailor-garment assignment.)
Garment
Wedding gown
Clown costume
Admiral's uniform
Bullfighter's outfit
1
19
11
12
X
Tailor
2 3 4 5
23 20 21 18
14 X 12 10
8 11 X 9
20 20 18 21
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Slide 12
Example: Tina’s Tailoring
Formulate an integer program for determining
the tailor-garment assignments that minimize
the total estimated time spent making the four
garments. No tailor is to be assigned more than one
garment and each garment is to be worked on by only
one tailor.
-------------------This problem can be formulated as a 0-1 integer
program. The LP solution to this problem will
automatically be integer (0-1).
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Slide 13
Example: Tina’s Tailoring
Define the decision variables
xij = 1 if garment i is assigned to tailor j
= 0 otherwise.
Number of decision variables =
[(number of garments)(number of tailors)]
- (number of unacceptable assignments)
= [4(5)] - 3 = 17
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Slide 14
Example: Tina’s Tailoring
Define the objective function
Minimize total time spent making garments:
Min 19x11 + 23x12 + 20x13 + 21x14 + 18x15 + 11x21
+ 14x22 + 12x24 + 10x25 + 12x31 + 8x32 + 11x33
+ 9x35 + 20x42 + 20x43 + 18x44 + 21x45
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Slide 15
Example: Tina’s Tailoring
Define the Constraints
Exactly one tailor per garment:
1) x11 + x12 + x13 + x14 + x15 = 1
2) x21 + x22 + x24 + x25 = 1
3) x31 + x32 + x33 + x35 = 1
4) x42 + x43 + x44 + x45 = 1
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Slide 16
Example: Tina’s Tailoring
Define the Constraints (continued)
No more than one garment per tailor:
5) x11 + x21 + x31 < 1
6) x12 + x22 + x32 + x42 < 1
7) x13 + x33 + x43 < 1
8) x14 + x24 + x44 < 1
9) x15 + x25 + x35 + x45 < 1
Nonnegativity: xij > 0 for i = 1, . . ,4 and j = 1, . . ,5
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Slide 17
Modeling Flexibility Provided by 0-1 Variables
When xi and xj represent binary variables designating
whether projects i and j have been completed, the
following special constraints may be formulated:
•
•
•
•
At most k out of n projects will be completed:
xj < k
j
Project j is conditional on project i:
xj - xi < 0
Project i is a corequisite for project j:
xj - xi = 0
Projects i and j are mutually exclusive:
xi + xj < 1
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Slide 18
Example: Metropolitan Microwaves
Metropolitan Microwaves, Inc. is planning to
expand its sales operation by offering other electronic
appliances. The company has identified
seven new product lines it can carry.
Relevant information about each line
follows on the next slide.
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Slide 19
Example: Metropolitan Microwaves
Product Line
1.
2.
3.
4.
5.
6.
7.
TV/VCRs
TVs
Projection TVs
VCRs
DVD Players
Video Games
Home Computers
Initial Floor Space Exp. Rate
Invest.
(Sq.Ft.)
of Return
$ 6,000
12,000
20,000
14,000
15,000
2,000
32,000
125
150
200
40
40
20
100
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8.1%
9.0
11.0
10.2
10.5
14.1
13.2
Slide 20
Example: Metropolitan Microwaves
Metropolitan has decided that they should not
stock projection TVs unless they stock either
TV/VCRs or TVs. Also, they will not stock both
VCRs and DVD players, and they will stock video
games if they stock TVs. Finally, the company
wishes to introduce at least three new product lines.
If the company has $45,000 to invest and 420 sq.
ft. of floor space available, formulate an integer linear
program for Metropolitan to maximize its overall
expected return.
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Slide 21
Example: Metropolitan Microwaves
Define the Decision Variables
xj = 1 if product line j is introduced;
= 0 otherwise.
where:
Product line 1 = TV/VCRs
Product line 2 = TVs
Product line 3 = Projection TVs
Product line 4 = VCRs
Product line 5 = DVD Players
Product line 6 = Video Games
Product line 7 = Home Computers
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Slide 22
Example: Metropolitan Microwaves
Define the Decision Variables
xj = 1 if product line j is introduced;
= 0 otherwise.
Define the Objective Function
Maximize total expected return:
Max .081(6000)x1 + .09(12000)x2 + .11(20000)x3
+ .102(14000)x4 + .105(15000)x5 + .141(2000)x6
+ .132(32000)x7
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Slide 23
Example: Metropolitan Microwaves
Define the Constraints
1) Money:
6x1 + 12x2 + 20x3 + 14x4 + 15x5 + 2x6 + 32x7 < 45
2) Space:
125x1 +150x2 +200x3 +40x4 +40x5 +20x6 +100x7 < 420
3) Stock projection TVs only if
stock TV/VCRs or TVs:
x1 + x2 > x3 or x1 + x2 - x3 > 0
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Slide 24
Example: Metropolitan Microwaves
Define the Constraints (continued)
4) Do not stock both VCRs and DVD players:
x4 + x5 < 1
5) Stock video games if they stock TV's:
x2 - x6 > 0
6) Introduce at least 3 new lines:
x1 + x2 + x3 + x4 + x5 + x6 + x7 > 3
7) Variables are 0 or 1:
xj = 0 or 1 for j = 1, , , 7
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Slide 25
Example: Metropolitan Microwaves
Partial Spreadsheet Showing Problem Data
A
B
C
G
H
I
2
3
Constraints
#1
X1
6
X2
12
X3
20
X4
14
X5
15
X6
2
X7
32
RHS
45
4
5
#2
#3
125
1
150
1
200
-1
40
0
40
0
20
0
100
0
420
0
6
7
#4
#5
0
0
0
1
0
0
1
0
1
0
0
-1
0
0
1
0
8
#6
1
1
1
1
1
1
1
3
1080
2200
1428
1575
282
4224
1
9 Obj.Func.Coeff. 486
D
E
F
LHS Coefficients
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Slide 26
Example: Metropolitan Microwaves
Partial Spreadsheet Showing Example Formulas
A
12
13
14
15
16
Dec.Values
B
X1
0
C
X2
0
D
X3
0
E
X4
0
Maximized Total Expected Return
F
X5
0
G
X6
0
H
X7
0
0
Constraints
LHS
17
18
19
Money
Space
TVs
0
0
0
<=
<=
>=
45
420
0
20
21
22
VCRs
Video
Lines
0
0
0
<=
>=
>=
1
0
3
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RHS
Slide 27
Example: Metropolitan Microwaves
Solver Parameters Dialog Box
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Slide 28
Example: Metropolitan Microwaves
Solver Options Dialog Box
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Slide 29
Example: Metropolitan Microwaves
Integer Options Dialog Box
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Slide 30
Example: Metropolitan Microwaves
Optimal Solution
A
12
13
14
15
16
Dec.Values
B
X1
1
C
X2
0
D
X3
1
Maximized Total Expected Return
F
X5
1
E
X4
0
G
X6
0
H
X7
0
4261
RHS
Constraints
LHS
17
18
19
Money
Space
TVs
41
365
0
<=
<=
>=
45
420
0
20
21
22
VCRs
Video
Lines
1
0
3
<=
>=
>=
1
0
3
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Slide 31
Example: Metropolitan Microwaves
Optimal Solution
Introduce:
TV/VCRs, Projection TVs, and DVD Players
Do Not Introduce:
TVs, VCRs, Video Games, and Home Computers
Total Expected Return:
$4,261
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Slide 32
Cautionary Note About Sensitivity Analysis
Sensitivity analysis often is more crucial for ILP
problems than for LP problems.
A small change in a constraint coefficient can cause a
relatively large change in the optimal solution.
Recommendation: Resolve the ILP problem several
times with slight variations in the coefficients before
choosing the “best” solution for implementation.
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Slide 33
End of Chapter 7
© 2008 Thomson South-Western. All Rights Reserved
Slide 34
