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Can network analyst tool solve multiple vehicle routing problem, or is it only solved by using ArcLogistics Route?

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Network Analyst extension for ArcGIS has a solver Vehicle Routing Problem (VRP). This solver does have support for multiple vehicles.

Look for the Routes class inside the Help page.

This network analysis class stores the routes that are part of a given vehicle routing problem analysis layer. A route specifies the vehicle and driver characteristics, and it represents the traversal between depots and orders. In Network Analyst, vehicles, routes, and drivers are synonymous, and the term route is used to encompass all three.

To get started using VRP solver, go through these two tutorials:

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ArcLogistics Route has been retired since 1 Dec 2013, so so it is no longer available to solve your multiple vehicle routing problem.

Consequently, I think you should follow the advice in the answer from @AlexTereshenkov.

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