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Lecture 9

# Lecture 9 - Integer Programming 1 Integer Programming...

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1 Integer Programming

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2 Integer Programming Integer (Linear) Programming: ILP or IP A linear programming problem with additional constraints that some or all variables must be integers Also called mixed integer programming (MIP) if some variables are continuous, and some are integers integers are , , 0 , 0 8 2 10 subject to 2 3 max y x y x y x y x y x z - + + =
3 Solving IP Is Difficult Rounding the solution of the corresponding LP to nearby integers often does not work Optimal solution for LP Optimal solution to IP

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4 Example: Site Selection Problem Planning to construct new buildings at 4 potential sites: 1, 2, 3, and 4 There are 3 possible designs: A, B, and C Problem: maximum profit subject to budget constraints Net profits (\$M) p ij 1 2 3 4 A 6 7 9 11 B 12 15 5 8 C 12 16 19 20 Investment (\$M) ≤100 a ij 1 2 3 4 A 13 20 24 30 B 39 45 12 20 C 30 44 48 55
5 IP Model Define variables: y ij for i in I ={A,B,C}, j in J ={1,2,3,4} such that y ij =1 if design i is used at site j , and y ij =0 for otherwise . , }, 1 , 0 { 100 subject to max J j I i y y a y p z ij I i J j ij ij I i J j ij ij = ∑∑ ∑∑

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6 Optimal Solution y A 1 = y A 3 = y B 3 = y B 4 = y C 1 =1, other y ij ’s are 0. Implying that Site 1 has two buildings: A and C Site 3 has two buildings: A and B Site 4 has one building: B Site 2 has no buildings More constraints may be needed to describe other requirements
7 Additional Constraints Site 2 must have exactly one building: y A2 + y B2 + y C2 =1 Site 2 must have at least one building: y A2 + y B2 + y C2 ≥1 Site 1 can only have at most one building: y A1 + y B1 + y C1 ≤1 Design A can be used at sites 1,2,3 only if it is also selected for site 4 y A1 + y A2 + y A3 ≤ 3 y A4

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8 Additional Constraints At most two designs can be used Define new binary variables w i : for i =A,B,C.
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