MPM = Maximum Permitted Mileage
In fare construction, the permitted total of ticket point mileages allowed between two fare break points

or, by paying the airfare between two points you are allowed to fly up to this mileage.

TPM = Ticketed Point Mileage
An officially stated distance between two cities, used in airline fare calculations regardless of the airports used
or, Actual mileage distance between a city pair

Every city pair has a MPM and a TPM.

For example:
Miami - Frankfurt - Miami
Airline: ??
Fare code: ABCDEFGH
Fare: $ 395.00 USD plus Q's and taxes
TPM = 4824 miles
MPM = 5788 miles
Real miles distance is 4824

You are allowed to fly up to 5788 miles without any additional charge. That is why you have the same fare for:

Miami - Frankfurt - Berlin - Frankfurt - Miami
Airlines: ??
Fare code: IJKLMNOPQ
Fare: $ 395.00 USD plus Q's and taxes
Frankfurt - Berlin
TPM = 269

So if you add:
Miami - Frankfurt
TPM = 4824 miles

You have a total of 5093 miles, which is less than the MPM of the Miami - Frankfurt fare: 5788 miles

Hope that it became now more easier for understanding.

Wish you all the best.

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