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We have so far developed a method to find the optimal tour between the above points and also tried to establish that this route is indeed the shortest, but one query arises that this route may be a local optimal tour and not a general optimal tour!
The concern arises only due to the fact that the tour may not be the optimal tour and a better tour may exist and our tour may be optimal locally not universally. So this section attempts to prove that our tour is a universal optimal tour and not a local optimal tour. So let us draw a comparison from no other than the Mr. optimal tour itself. Fig. 8 presents two hypothetical tours, Fig. 8(a) is a hypothetical universal optimal tour and Fig. 8(b) may be thought as a local optimal tour which may be got by many of the known methods or by our method, say!
So let us take a singularity and see whether in it arises in our method or not.
Space for Fig. 8(a) and 8(b)
Now as obvious from the above Fig.8 (a) & 8(b), Fig.8 (a) is the hypothetical local tour and Fig.8 (b) is the hypothetical optimal tour which we got by our method. We will see that whether these two are same or there may be a condition which may remain unturned by our method. So we try to list the differences which may remain when our method is completed. As you will recall that we got an optimal tour by series of steps. But it may still happen that a point may be joined to a side which is not in the mesh i.e. it is not present till now, for example, P1P2 is a diagonal which may exist in the optimal tour but not present in local tour. No other singularity may be counted here after as if a point