如何處理旅行商問題，以期在一定天數內使每個訪問的城鎮收益最大化（城鎮總數大於在給定時間範圍內可以訪問的城鎮）？對銷售產品的需求因鎮而異。有什麼我可以參考的出版文獻來介紹這種TSP變體嗎？

謝謝

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如何處理旅行商問題，以期在一定天數內使每個訪問的城鎮收益最大化（城鎮總數大於在給定時間範圍內可以訪問的城鎮）？對銷售產品的需求因鎮而異。有什麼我可以參考的出版文獻來介紹這種TSP變體嗎？

謝謝

optimizationlinear-programmingtraveling-salesman

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Search for *orienteering problem* or *prize-collecting TSP*.

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There are a few problems in this category. They are variants of the Travelling Salesman Problem in which you don't need to visit all customers, but can choose which ones to visit. They fall under the general name of Travelling Salesman Problems with Profits.

The main variants are:

- The
**Orienteering Problem**(OP) in which you want to maximise the profit collected at visited customers, under an upper bound on the trip length. - The
**Prize-collecting TSP**in which you want to minimise the trip length, under a lower bound on the amount of profit collected. - The
**Profitable Tour Problem**in which instead of considering travel time, you consider travel*costs*. In this case you can express profits collected and travel costs with the same unit of measure (say, €). The objective is to maximise the*revenue*(= profit collected, minus travel costs).

Most of these problems also appear in the literature with variants (multi-vehicle, time windows, etc.) which are common for other "classic" routing problems. For example, the multi-vehicle OP is called the Team Orienteering Problem (TOP). Adding time windows gives the OPTW and TOPTW. Adding vehicle capacities gives the Capacitated TOP. Some stochastic versions include the TSP with Profits and Stochastic Customers, the OP with stochastic travel times and the OP with stochastic profits.

As per the solution methods for the Orienteering Problem (which seems the one most suited for your question), as far as I know the state-of-the-art exact algorithm is still the one by Fischetti, Salazar-Gonzalez, and Toth. In the last couple of years, new heuristics have improved on previously best-known results: a genetic algorithm and an ALNS.