A Dynamic Programming Approach for a Travelling Purchaser Problem With Additional Constraints

The traveling purchaser problem (TPP) is the problem of determining a tour of a purchaser that needs to buy several items in different shops such that the total amount of travel and purchase costs is minimized. We study a version of the problem with additional constraints, namely, a limit on the maximum number of markets to be visited, a limit on the number of items bought per market and where only one copy per item needs to be bought. We present a method based on a dynamic programming formulation for the problem incorporating market and item information in the state definition. Due to the expected exponentially sized state space of the proposed dynamic program, we develop a state space relaxation (SSR) that provides a lower bound on the cost of the optimal solution. The limited number of markets and purchases per market permit us to incorporate, in a practical way, this relaxation within a subgradient optimization procedure for improving the original SSR lower bound. We also propose a heuristic that is used together with the subgradient optimization procedure, in order to attempt to transform relaxed solutions into feasible solutions for the problem. We present computational results taken from instances with up to 300 markets. The results show that the dynamic programming based method appears to be a good method for giving good intervals on the optimal value for this restricted version of the TPP.