Unformatted text preview: followed by city j on the tour
⎪
xij = ⎨
⎪ 0 otherwise.
⎩
The formulation is as follows:
n Minimize n ∑∑c x
i =1 j =1
n subject to ∑x
j =1 ij n ∑x
i =1 ∑ ij i∈S , j ∈N \ S ij ij =1 for all i = 1 to n =1 for all j = 1 to n xij ≥ 1 for all S ⊂ {1,2,..., n} with 1 ≤ S ≤ n − 1 xij ∈{0,1}, for all i, j = 1 to n. The first set of constraints ensures that there is some city that follows city i in the solution. The
second set of constraints ensures that there is some city that precedes city j in the solution. Unfortunately, there are not enough constraints to ensure that the solution is a tour. For example, a
feasible solution to the first two sets of constraints for the six city problem consists of x12 = x23 = x31 =
1, and x45 = x56 = x64 = 1. This "solution" to the first two sets of constraints corresponds to the cycles
1231 and 4564. A tour should be one cycle only.
The third set of constraints guarantees the following. For any nonempty subset...
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This note was uploaded on 03/18/2014 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.
 Spring '07
 JamesOrli
 Management

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