Integer Programming

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Integer Linear Programming
Professor Ahmadi

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Integer 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

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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.

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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

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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
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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 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

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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.

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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
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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.

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Example All-Integer LPRounded down solution
x2
5
-x1 x2 lt 1
3x1 x2 lt 9
4
Max 3x1 2x2
3
Rounded down solution (2, 1)
2
x1 3x2 lt 7
1
x1
1 2 3 4
5 6 7
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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
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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

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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

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Example Chattanooga Electronics

• Chattanooga Electronics, 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
• Initial
  Floor Space Exp. Rate
• Product Line Investment
  (Sq.Ft.) of Return
• 1. Digital TVs 60,000 125
  8.1
• 2. Color 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

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Example Chattanooga Electronics

• Chattanooga has decided that they should not
  stock large screen TVs unless they stock either
  digital or color TVs. Also, they will not stock
  both types of DVDs, 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 Chattanooga to
  maximize its overall expected rate of return.

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Example Chattanooga Electronics

• Define the Decision Variables
• xj 1 if product line j is introduced
• 0 otherwise.
• Define the Objective Function
• Maximize total overall expected return
• Max .081(60000)x1 .09(12000)x2
  .11(20000)x3
• .102(14000)x4 .105(15000)x5
  .141(2000)x6
• .132(32000)x7 or
• Max 4860x1 1080x2 2200x3 1428x4
  1575x5
• 282x6 4224x7

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Example Chattanooga Electronics

• Define the Constraints
• 1) Money
• 2) Space
• 3) Stock large screen TVs only if stock
  digital or color

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Example Chattanooga Electronics

• Define the Constraints (continued)
• 4) Do not stock both types of DVDs
• 5) Stock video games if they stock color
  TV's
• 6) At least 3 new lines
• 7) Variables are 0 or 1
• xj 0 or 1 for j 1, , , 7

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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

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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).

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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

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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

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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

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The End of Chapter