Matakuliah : K0442-Metode Kuantitatif
Tahun : 2009

Integer Linear Programming
• Types of Integer Linear Programming Models
• Graphical and Computer Solutions for an All-Integer
Linear Program
• Applications Involving 0-1 Variables
• Modeling Flexibility Provided by 0-1 Variables
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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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Example: All-Integer LP
• Consider the following all-integer linear program:
Max 3 x
1
+ 2 x
2 s.t. 3 x
1
+ x
2 x
1
+ 3 x
2
x
1
+ x
2
< 9
< 7
< 1 x
1
, x
2
> 0 and integer
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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 x
1 and x
2
. From the graph on the next slide, we see that the optimal solution to the linear program is: x
1
= 2.5, x
2
= 1.5, z = 10.5
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4
3
2 x a
2
5
1
Example: All-Integer LP
• LP Relaxation
-x
1
+ x
2
< 1
3x
1
+ x
2
< 9
Max 3x
1
+ 2x
2
LP Optimal (2.5, 1.5) x
1
+ 3x
2
< 7
1 2 3 4 5 6 7 x
1
7

• Rounding Up
If we round up the fractional solution ( x
1
= 2.5, x
2
= 1.5) to the LP relaxation problem, we get x
1 x
2
= 3 and
= 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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Example: All-Integer LP
• Rounded Up Solution
5
4
3
-x
1
+ x
2
< 1
3x
1
+ x
2
< 9
Max 3x
1
+ 2x
2
ILP Infeasible (3, 2)
2
1
LP Optimal (2.5, 1.5) x
1
+ 3x
2
< 7
1 2 3 4 5 6 7 x
1
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Example: All-Integer LP
• Rounding Down
By rounding the optimal solution down to x
1
= 2, x
2
= 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?
--------------------and
The answer is NO! The optimal solution is x
1 x
2
= 3
= 0 giving z = 9, as evidenced in the next two slides.
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Example: All-Integer LP
• Complete Enumeration of Feasible ILP Solutions
There are eight feasible integer solutions to this problem: x
1 x
2 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
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5
4
3
2
1 x
2
Example: All-Integer LP
-x
1
+ x
2
< 1
3x
1
+ x
2
< 9
Max 3x
1
+ 2x
2
ILP Optimal (3, 0) x
1
+ 3x
2
< 7
1 2 3 4 5 6 7 x
1
12

Special 0-1 Constraints
• When x i and x j 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:
 x j
< k j
– Project j is conditional on project i : x j
x i
< 0
– Project i is a corequisite for project j : x j
x i
= 0
– Projects i and j are mutually exclusive: x i
+ x j
< 1
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Example: Metropolitan Microwaves
Metropolitan Microwaves, Inc. is planning to expand its operations into 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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Example: Metropolitan Microwaves
Initial Floor Space Exp. Rate
Product Line Invest. (Sq.Ft.) of Return
1. TV/VCRs $ 6,000 125
2. Color TVs 12,000 150
3. Projection TVs
4. VCRs
8.1%
9.0
20,000 200 11.0
14,000 40 10.2
5. DVD Players 15,000 40 10.5
6. Video Games 2,000 20 14.1
7. Home Computers 32,000 100 13.2
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Example: Metropolitan Microwaves
Metropolitan has decided that they should not stock projection TVs unless they stock either TV/VCRs or color TVs. Also, they will not stock both VCRs and DVD players, and they will stock video games if they stock color 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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Example: Metropolitan Microwaves
• Define the Decision Variables x j
= 1 if product line j is introduced;
= 0 otherwise.
Bina Nusantara University where:
Product line 1 = TV/VCRs
Product line 2 = Color 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
17


Define the Decision Variables x j
= 1 if product line j is introduced;
= 0 otherwise.

Define the Objective Function
Maximize total expected return:
Max .081(6000)x
1
+ .09(12000)x
+ .102(14000)x
4
2
+ .11(20000)x
3
+ .105(15000)x
5
+ .141(2000)x
6
+ .132(32000)x
7
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Example: Metropolitan Microwaves
• Define the Constraints
1) Money:
6 x
1
+ 12 x
2
+ 20 x
3
+ 14 x
4
+ 15 x
5
+ 2 x
6
+ 32 x
7
< 45
2) Space:
125 x
1
+150 x
2
+200 x
3
+40 x
4
+40 x
5
+20 x
6
+100 x
7
< 420
3) Stock projection TVs only if stock TV/VCRs or color TVs: x
1
+ x
2
> x
3 or x
1
+ x
2
x
3
> 0
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Example: Metropolitan Microwaves
• Define the Constraints (continued)
4) Do not stock both VCRs and DVD players: x
4
+ x
5
< 1
5) Stock video games if they stock color TV's: x
2
x
6
> 0
6) Introduce at least 3 new lines: x
1
+ x
2
+ x
3
+ x
4
+ x
5
+ x
6
+ x
7
> 3
7) Variables are 0 or 1: x j
= 0 or 1 for j = 1, , , 7
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• Optimal Solution
Introduce:
TV/VCRs, Projection TVs, and DVD Players
Do Not Introduce:
Color TVs, VCRs, Video Games, and Home Computers
Total Expected Return :
$4,261
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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.)
Tailor
Garment 1 2 3 4 5
Wedding gown 19 23 20 21 18
Clown costume 11 14 X 12 10
Admiral's uniform 12 8 11 X 9
22

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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Example: Tina’s Tailoring
• Define the decision variables x ij
= 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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Example: Tina’s Tailoring
• Define the objective function
Minimize total time spent making garments:
Min 19 x
11
+ 23 x
12
+ 20 x
13
+ 21 x
14
+ 18 x
15
+ 11 x
21
+ 14 x
22
+ 12 x
24
+ 10 x
25
+ 12 x
31
+ 8 x
32
+ 11 x
33
+ 9x
35
+ 20 x
42
+ 20 x
43
+ 18 x
44
+ 21 x
45
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Example: Tina’s Tailoring
• Define the Constraints
Exactly one tailor per garment:
1) x
11
+ x
12
+ x
13
+ x
14
+ x
15
= 1
2) x
21
+ x
22
+ x
24
+ x
25
= 1
3) x
31
+ x
32
+ x
33
+ x
35
= 1
4) x
42
+ x
43
+ x
44
+ x
45
= 1
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Example: Tina’s Tailoring
• Define the Constraints (continued)
No more than one garment per tailor:
5) x
11
+ x
21
+ x
31
< 1
6) x
12
+ x
22
+ x
32
+ x
42
< 1
7) x
13
+ x
33
+ x
43
< 1
8) x
14
+ x
24
+ x
44
< 1
9) x
15
+ x
25
+ x
35
+ x
45
< 1
Nonnegativity: x ij
> 0 for i = 1, . . ,4 and j = 1, . . ,5
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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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