Even if NP-hard, the TPP can be solved efficiently in practice by many different exact methods (as the TSP). Dynamic programming, Mathematical Programming (branch-and-cut, branch-and-bound), and Constraint Programming -based algorithms exist and have been proposed for the TPP in the Operation Research/Computer Science literature. I suggest you to read a very recent survey on the topic: "Manerba et al., 2017. The Traveling Purchaser Problem and its Variants, European Journal of operational Research 259, 1-18".
Of course, the practical complexity of solving exactly the TPP grows with the number of markets and products (especially the market). However, by using the proper methods, exact solutions can be found in a reasonable amount of time on modern computers for instances with 150/200 markets and 200/300 products.
Some known solvers (Cplex, Gurobi, ..) are also able to solve exactly some small/medium instances of the TPP if its model is given as input. This will be very useful if you need those solutions just for berchmarking your heuristics. In that case, the mathematical formulation of the problem is a critical issue to address.
PS: For the folklore, if you find an exact algorithm having a polynomial time complexity that solves a NP-hard problem, then you also solve the
"P vs NP" open problem proposed by the Clay Institute as one of the Millenium Problems and win 1 million of dollars.