CEE
LSA_7_TSP_VRP_asymptotic_results.pdf

Combining a and b we have 2 6 n f n m m f m optimal

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Combining (a) and (b) we have 2 ( ) 6 ( ) N f N m m f m + optimal tour length 6 (individual optimal tour lengths) m A + 2 2 2 ( ) 6 ( ) A N A f N m A m f m m + This is true for every realization (throw of our darts) and therefore also true for the average: We can write this because the squares are identical and the expected number of hits in each is N / m 2 . Thus… + lengths) tour optimal individual ( 4 length tour optimal A m One can use similar logic to establish that 2 2 4 ( ) ( ) 6 ( ) N N m m f f N m m f m m + + Note that as N , f ( N ). Then the above expression implies that the –4 m and 4 m terms become negligible and that 1 ) ( ) ( lim , given any for 2 = N f m N f m m N This can be proven by considering an optimal grand tour and its portions within each little square. The individual optimal tour is shorter than the portion of the grand tour plus the perimeter of the little square.
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Logistics Systems Analysis TSP on a Surface (Planar Problem) Since the asymptotic result is true for every m , we see that for large N f ( N ) ~ N 1/2 . (why?) (Hint: Let n 2 = N / m 2 , then for all n and large N , f ( N )= f ( n 2 )/ n * N 1/2 holds.) This completes the proof for a square-shaped area. Thus… N k N A kN AN k 2 / 1 * L = = = δ Expected Distance per point Where δ is the expected density of points As N the standard deviation of the number of points is a vanishingly small part of the expectation and one can then show that the above formula holds if N is fixed across realization. = = s s s s S s s s s s N A k N A k L L for , ) ( 1 * * δ If all the squares are identical, then, i.e. L * for an arbitrary arrangement of squares is the same as for a uniform square area. Equation (*) shows that the formula Effect of Shape and variable density : The logic of the previous proof shows that if we have s = 1,…, S contiguous squares with N s points in each and areas A s , then we can add the lengths of individual tours; i.e., (*) , * N A k N A A A k N A k L s s s s s s = = = = s N N k N A k L for , 2 / 1 * δ is independent of shape.
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  • Fall '16
  • Distance, cij, Ñ È

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