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Integer Programming (BRS)

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Professor Ahmadi Chapter 11 Integer Linear Programming Types of Integer Linear Programming Models Graphical Solution for an All-Integer LP Spreadsheet Solution for an ... – PowerPoint PPT presentation

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Title: Integer Programming (BRS)


1
Integer Linear Programming
Professor Ahmadi

2
Chapter 11Integer Linear Programming
  • 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

3
Types of Integer Programming Models
  • 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.

4
Example All-Integer LP
  • Consider the following all-integer linear
    program
  • Max 3x1 2x2
  • s.t. 3x1 x2 lt 9
  • x1 3x2 lt
    7
  • -x1 x2
    lt 1
  • x1, x2 gt 0
    and integer

5
Example All-Integer LP
  • LP Relaxation

x2
-x1 x2 lt 1
5
3x1 x2 lt 9
4
Max 3x1 2x2
3
LP Optimal (2.5, 1.5)
2
x1 3x2 lt 7
1
x1
1 2 3 4
5 6 7
6
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
    previous slide, we see that the optimal solution
    to the linear program is
  • x1 2.5, x2 1.5, z 10.5

7
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.

8
Example All-Integer LP
  • Rounded Up Solution

x2
-x1 x2 lt 1
5
3x1 x2 lt 9
4
Max 3x1 2x2
3
ILP Infeasible (3, 2)
2
LP Optimal (2.5, 1.5)
x1 3x2 lt 7
1
x1
1 2 3 4
5 6 7
9
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
    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.

10
Example All-Integer LP
  • 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

11
Example All-Integer LP
x2
5
-x1 x2 lt 1
3x1 x2 lt 9
4
Max 3x1 2x2
3
2
ILP Optimal (3, 0)
x1 3x2 lt 7
1
x1
1 2 3 4
5 6 7
12
Special 0-1 Constraints
  • 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 lt k
  • Project j is conditional on project i
  • xj - xi lt 0
  • Project i is a co-requisite for project j
  • xj - xi 0
  • Projects i and j are mutually exclusive
  • xi xj lt 1

13
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

14
Chattanooga Electronics - Continued
  • Define the Decision Variables
  • xj 1 if product line j is introduced
  • 0 otherwise.
  • Where the Product lines are defined as
  • (X1) Digital TVs
  • (X2) HD TVs
  • (X3) Large Screen TVs
  • (X4) DVDs
  • (X5) DVD/RWs
  • (X6) Video Games
  • (X7) Computers

15
Example Chattanooga Electronics
  • Chattanooga Electronics has decided that
  • they should not stock large screen TVs (X3)
    unless they stock either digital (X1) or HD TVs
    (X2).
  • also, they will not stock both types of DVDs (X4
    X5).
  • they will stock video games (X6) only if they
    stock HD TVs (X2).
  • 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 Chattanooga
    Electronics to maximize its overall expected rate
    of return.

16
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

17
Example Chattanooga Electronics
  • Define the Constraints
  • 1) Money
  • 60x1 12x2 20x3 14x4 15x5 2x6
    32x7 lt 45
  • 2) Space
  • 125x1 150x2 200x3 40x4 40x5
    20x6 100x7 lt 420

18
Example Chattanooga Electronics
  • 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

19
Example Mos 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

20
Example Mos 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).

21
Example Mos Programming
  • 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

22
Example Mos Programming
  • 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

23
Example Mos Programming
  • Define the Constraints (continued)
  • No more than one Program per Programmer
  • 5) x11 x21 x31 lt 1
  • 6) x21 x22 x23 x24 lt 1
  • 7) x31 x33 x34 lt 1
  • 8) x41 x42 x44 lt 1
  • 9) x51 x52 x53 x54 lt 1
  • Non-negativity xij gt 0 for i 1, . . ,4
    and j 1, . . ,5

24
The End of Chapter 6
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