ESI 4313 Operations Research 2

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ESI 4313Operations Research 2

• Introduction to Integer Programming
• Lecture 27 March 09, 2006

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Set Covering Problems

• Western Airlines has decided to have hubs in USA.
  
• Western runs flights between the following
  cities Atlanta, Boston, Chicago, Denver,
  Houston, Los Angeles, New Orleans, New York,
  Pittsburgh, Salt Lake City, San Francisco, and
  Seattle.
• Western needs to have a hub within 1000 miles of
  each of these cities.
• Determine the minimum number of hubs

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Set Covering Problems-contd

• Decision Variables
• xi 1 if a hub is located in city i
• xi 0 if a hub is not located in city i
• Formulation
• Minimize xAT xBO xCH xDE xHO xLA xNO
  xNY xPI xSL xSF xSE
• subject to

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Either-Or Constraints

• Dorian Auto is considering manufacturing three
  types of auto compact, midsize, large.
• Resources required and profits obtained from
  these cars are given below.
• We have 6,000 tons of steel and 60,000 hours of
  labor available.
• If any car is produced, we must produce at least
  1,000 units of that car.
• Find a production plan to maximize the profit.

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Either-Or Constraints-contd

• Decision Variables
• x1, x2, x3 number of compact, midsize and large
  cars produced
• y1, y2, y3 1 if compact , midsize and large
  cars are produced or not
• Formulation
• Maximize z 2x1 3x2 4x3
• subject to
• x1 My1 x2 My2 x3 My3
• 1000 - x1 M(1-y1)
• 1000 - x2 M(1-y2)
• 1000 - x3 M(1-y3)
• 1.5 x1 3x2 5x3 6000
• 30 x1 25x2 40 x3 60000
• x1, x2, x3 ³ 0 and integer y1, y2, y3 0 or 1

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Additional Applications

• Location of fire stations needed to cover all
  cities
• Location of fire stations to cover all regions
• Truck despatching problem
• Political redistricting
• Capital investments

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Techniques for solving IP

• Enumeration Techniques
• Complete Enumeration
• list all solutions and choose the best
• Branch and Bound
• Implicitly search all solutions, but cleverly
  eliminate the vast majority before they are even
  searched
• Implicit Enumeration
• Branch and Bound applied to binary variables
• Cutting Plane Techniques

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Capital Budgeting problem
maximize 16x1 22x2 12x3 8x4 11x5
19x6 subject to 5x1 7x2 4x3 3x4 4x5
6x6 ? 14 xj binary for j 1
to 6
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Complete Enumeration

• Systematically considers all possible values of
  the decision variables.
• If there are n binary variables, there are 2n
  different ways.
• Usual idea iteratively break the problem in
  two. At the first iteration, we consider
  separately the case that x1 0 and x1 1.

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An Enumeration Tree
Original problem
x1 0
x1 1
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On complete enumeration

• Suppose that we could evaluate 1 billion
  solutions per second.
• Let n number of binary variables
• Solutions times
• n 30, 1 second
• n 40, 17 minutes
• n 50 11.6 days
• n 60 31 years

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Branch and Bound

• The basic idea search the enumeration tree, but
  at each node
• Solve the linear program at the node
• Eliminate the subtree (fathom it) if
• The solution is integer (there is no need to go
  further) or
• The best solution in the subtree cannot be as
  good as the best available solution (the
  incumbent) or
• There is no feasible solution