Integer Linear Programming
Professor Ahmadi
Slide 1
Chapter 11
Integer Linear Programming
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Types of Integer Linear Programming Models
Graphical Solution for an All-Integer LP
Spreadsheet Solution for an All-Integer LP
Application Involving 0-l Variables
Special 0-1 Constraints
Slide 2
Types of Integer Programming Models
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A linear program in which all the variables are
restricted to be integers is called an integer linear
program (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
program.
Slide 3
Example: All-Integer LP
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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
Slide 4
Example: All-Integer LP
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LP Relaxation
x2
-x1 + x2 < 1
5
3x1 + x2 < 9
4
Max 3x1 + 2x2
3
LP Optimal (2.5, 1.5)
2
x1 + 3x2 < 7
1
1
2
3
4
5
6
7
x1
Slide 5
Example: All-Integer LP
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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 previous slide, we
see that the optimal solution to the linear program is:
x1 = 2.5, x2 = 1.5, z = 10.5
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 page, we see
that this point lies outside the feasible region, making
this solution infeasible.
Slide 7
Example: All-Integer LP
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Rounded Up Solution
x2
5
-x1 + x2 < 1
3x1 + x2 < 9
4
Max 3x1 + 2x2
3
ILP Infeasible (3, 2)
2
LP Optimal (2.5, 1.5)
x1 + 3x2 < 7
1
1
2
3
4
5
6
7
x1
Slide 8
Example: All-Integer LP
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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 z = 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 z = 9, as evidenced in the next two
slides.
Slide 9
Example: All-Integer LP
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Complete Enumeration of Feasible ILP Solutions
There are eight feasible integer solutions to this
problem:
x1
x2 z
1. 0
0
0
2. 1
0
3
3. 2
0
6
4. 3
0
9
optimal solution
5. 0
1
2
6. 1
1
5
7. 2
1
8
8. 1
2
7
Slide 10
Example: All-Integer LP
x2
5
-x1 + x2 < 1
3x1 + x2 < 9
4
3
Max 3x1 + 2x2
2
ILP Optimal (3, 0)
x1 + 3x2 < 7
1
1
2
3
4
5
6
7
x1
Slide 11
Special 0-1 Constraints
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When xi and 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:
Sxj < k
• Project j is conditional on project i:
xj - xi < 0
• Project i is a co-requisite for project j:
xj - xi = 0
• Projects i and j are mutually exclusive:
xi + xj < 1
Slide 12
Example: Chattanooga Electronics
Chattanooga Electronics, Inc. is planning to expand
its operations into other electronic equipment. The
company has identified seven new product lines it can
carry. Relevant information about each line follows:
Initial Floor Space Exp. Rate
Product Line
Investment (Sq.Ft.)
of Return
1. Digital TVs
$6,000
125
8.1%
2. HD TVs
12,000
150
9.0
3. Large Screen TVs
20,000
200
11.0
4. DVDs
14,000
40
10.2
5. DVD/RWs
15,000
40
10.5
6. Video Games
2,000
20
14.1
7. PC Computers
32,000
100
13.2
Slide 13
Chattanooga Electronics - Continued
Define the Decision Variables
xj = 1 if product line j is introduced;
= 0 otherwise.
Where the Product lines are defined as:
1. (X1) = Digital TVs
2. (X2) = HD TVs
3. (X3) = Large Screen TVs
4. (X4) = DVDs
5. (X5) = DVD/RWs
6. (X6) = Video Games
7. (X7) = Computers
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Slide 14
Example: Chattanooga Electronics
Chattanooga Electronics has decided that:
1. they should not stock large screen TVs (X3) unless
they stock either digital (X1) or HD TVs (X2).
2. also, they will not stock both types of DVDs (X4 & X5).
3. they will stock video games (X6) only if they stock HD
TVs (X2).
4. the company wishes to introduce at least three new
product lines.
5. If the company has $45,000 to invest and 420 sq. ft. of
floor space available, formulate an integer linear
program for Chattanooga Electronics to maximize its
overall expected rate of return.
Slide 15
Example: Chattanooga Electronics
Define the Objective Function
Maximize total overall expected return:
Max .081(6000)x1 + .09(12000)x2 + .11(20000)x3
+ .102(14000)x4 + .105(15000)x5 + .141(2000)x6
+ .132(32000)x7
or
Max 486x1 + 1080x2 + 2200x3 + 1428x4 + 1575x5
+ 282x6 + 4224x7
Slide 16
Example: Chattanooga Electronics
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Define the Constraints
1) Money:
60x1 + 12x2 + 20x3 + 14x4 + 15x5 + 2x6 + 32x7 < 45
2) Space:
125x1 +150x2 +200x3 +40x4 +40x5 +20x6 +100x7 < 420
Slide 17
Example: Chattanooga Electronics
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Define the Constraints (continued)
3) Stock large screen TVs (X3) only if stock digital (X2) or
HD (X2):
4) Do not stock both types of DVDs (X4 & X5):
5) Stock video games (X6) only if they stock HD TV's
(X2):
6) At least 3 new lines:
7) Variables are 0 or 1:
xj = 0 or 1 for j = 1, , , 7
Slide 18
Example: Mo’s Programming
Mo's Programming has five idle Programmers and
four custom Programs to develop. The estimated time
(in hours) it would take each Programmer to write each
Program is listed below. (An 'X' in the table indicates
an unacceptable Programmer-Program assignment.)
Programmer
Program
1 2 3 4 5
Java
19 23 20 21 18
C++
11 14 X 12 10
Assembler
12 8 11 X 9
Pascal
X 20 20 18 21
Slide 19
Example: Mo’s Programming
Formulate an integer program for determining the
Programmer-Program assignments that minimize the
total estimated time spent writing the four Programs.
No Programmer is to be assigned more than one
Program and each Program is to be worked on by only
one Programmer.
-------------------This problem can be formulated as a 0-1 integer
program. The LP solution to this problem will
automatically be integer (0-1).
Slide 20
Example: Mo’s Programming
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Define the decision variables
xij = 1 if Program i is assigned to Programmer j
= 0 otherwise.
Number of decision variables =
[(number of Programs)(number of Programmers)]
- (number of unacceptable assignments)
= [4(5)] - 3 = 17
Define the objective function
Minimize total time spent writing Programs:
Min 19x11 + 23x12 + 20x13 + 21x14 + 18x15 + 11x21
+ 14x22 + 12x24 + 10x25 + 12x31 + 8x32 + 11x33
+ 9x35 + 20x42 + 20x43 + 18x44 + 21x45
Slide 21
Example: Mo’s Programming
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Define the Constraints
Exactly one Programmer per Program:
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
Slide 22
Example: Mo’s Programming
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Define the Constraints (continued)
No more than one Program per Programmer:
5) x11 + x21 + x31 < 1
6) x21 + x22 + x23 + x24 < 1
7) x31 + x33 + x34 < 1
8) x41 + x42 + x44 < 1
9) x51 + x52 + x53 + x54 < 1
Non-negativity: xij > 0 for i = 1, . . ,4 and j = 1, . . ,5
Slide 23
The End of Chapter 6
Slide 24