I am not sure if you have a separate location table. If you have one, build a dimension from it, otherwise use the Location column from the LocationProductsMapping table to build the dimension.
Then create a new measure group from your LocationProductsMapping table. As Analysis Services cannot have measure groups without measures, use the count which the wizard normally suggests. Make this measure invisible, as it is not useful for users. Then on the "Dimension Usage" tab of cube designer, make sure your mapping measure group is related to the Product and Location dimensions, and set the relationship from the main measure group to location to "Many-to-many", selecting the mapping measure group.
And you are done. Analysis Services handles the rest for you.