Integer Programming

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Chapter 5

• Integer Programming

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What is an integer program (IP)?

• IP is a linear program in which all or some
  variables can only take integral values.
• A satisfiable solution of IP cannot be obtained
  by rounding the corresponding LP solution.
• Solution method of IP is different from, and more
  difficult than, solution method of LP.

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Example, p.185

• Machine Space Unit Profit
• required Price predicted
• Press 15 ft2/unit 8,000 100/day
• Lathe 30 ft2/unit 4,000 150/day
• There are 200 ft2 and 40,000 available.
• How many presses and lathes to purchase to
  maximize daily profit?

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Example

• Investment Cost Profit Available
• opportunity per unit per unit (units)
• Condominium 50,000 9,000 4
• Land 12,000 1,500 15
• Bond 8,000 1,000 20
• There are 250,000 available.
• Where to invest to maximize the profit?

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Types of IP models

• Total IP model
• All variables must be integral.
• 0-1 IP model
• Variables can be 0 or 1 only.
• Mixed IP model
• variables can be 0-1, integral or non-integral.

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Logical Representations by Using 0-1 variables

• 0-1 variables is used to represent various
  logical relationships in integer programming
  formulations.

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0-1 variables

• Let
• i1, 2, 3,

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Mutually exclusive relation.

• Either X1 or X2 (but not neither, not both)
• X1X21

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Contingency relation

• Either X1 or X2, or neither (but not both)
• X1X2lt1
• At least one of X1 and X2
• X1X2gt1

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Co-requisite Relation

• X1 and X2 must be on or off together
• X1X2

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Mutually exclusiveness on more than two variables

• Select exactly one of X1, X2, X3
  (Multiple-choice relation)
• X1X2X31

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Contingency relations on more than two variables

• Select no more than one from X1, X2, X3
• X1X2X3lt1
• Select no more than two from X1, X2, X3
• X1X2X3lt2
• Select at least one of X1, X2, X3
• X1X2X3gt1

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Conditional relation (if then )

• If X1 is on, then X2 must be on, (and if X1
  is off then X2 can be either on or off.)
• X1ltX2

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Example, p.186

• Facility Usage Cost Land (acres)
• considered people/day required
• Swimming pool 300 35,000 4
• Tennis center 90 10,000 2
• Athletic field 400 25,000 7
• Gymnasium 150 90,000 3
• There are 12 acres and 120,000 available.
• (continued on next page)

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Example, p.186 (cont.)

• Additional restriction on selection of the
  facilities
• (1) Between swimming pool and tennis center, only
  one can be constructed.
• (2) Between athletic field and tennis center, at
  least one must be built.
• (3) If athletic field is built, then swimming
  pool must be built.
• (4) Among the four, at least two must be built.
• Which facilities should be constructed to
  maximize the daily usage?

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Example p.205 A Set Covering Problem

• APS wants to build package distribution hubs to
  cover 12 cities. A hub can cover cities within
  300 miles, as shown on p.205.
• Which cities should be selected as hubs so that
  number of hubs to be built is minimized?

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Example p.205 Define Variables

• xi 0 if city i is not selected as a hub, and
• xi 1 if city i is selected as a hub
• where i 1, 2, 3, , 12 such that
• 1 for Atlanta, 2 for Boston 3 for Charlotte
• 4 for Cincinnati 5 for Detroit 6 for
  Indianapolis
• 7 for Milwaukee 8 for Nashville 9 for New York
• 10 for Pittsburgh 11 for Richmond 12 for St.
  Louis

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Example p.205 Set up Integer Program

• Minimize total number of hubs to build
• For each of the 12 cities
• It must be covered by at least a hub within 300
  miles i.e., there must be at least a hub within
  300 miles of it.
• The complete integer program is on p.206.

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Solution Methods of IP

• Solving IP is more complicated than solving LP.
  Two main solution methods of IP
• Branching and bound method
• Cutting plain method

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Integer Programming (IP), Discrete Optimization,
and NP Complete

• An integer program has finite number of feasible
  solutions.
• IP is a typical problem of discrete optimization
  that selects the best from a finite number of
  alternatives.
• IP is a computationally hard problem (NP complete
  problem). That is, no method has been found to
  solve IP efficiently.