An algorithm finds all best trade-offs between speed and cost in two-phase transportation.

Moving goods efficiently can be tricky when shipments must happen in steps. The authors studied a shipping problem where travel routes are divided into Phase-I and Phase-II. Routes in Phase-I run at the same time in parallel. Routes in Phase-II only start after all Phase-I shipments finish. The problem tries to minimize two competing goals at once: the total shipping time and the total shipping cost across both phases. Because saving money often takes more time, the researchers designed a step-by-step method called the BTPTP-Algorithm. At each step, it solves cost problems with specific limits based on the travel times found in the previous step. This process removes all weaker choices and keeps every non-dominated solution, which are the best trade-offs where cost cannot drop without increasing time. The authors proved the method finds all these best solutions and tested its run times on problems of various sizes.

Why it matters

Companies shipping goods in stages, like from factories to warehouses to stores, often struggle to balance time and cost. This work gives them a systematic way to see all optimal choices, so they can decide based on their priorities without guesswork.

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