Abstract
“Time is life” is popular now, considering of some optimization models with time windows is necessary. In this paper, we propose a new type of time windows which is left-side time windows, and then consider the shortest path problems with left-side time windows as well as the shortest path problems with multi-left-side time windows. The objectives of these problems are not simply the sum of their weights on such paths, but their travelling cycles which make one day as their travelling time unit. The optimal algorithms are presented for both the shortest path problem with single left-side time windows and the shortest path problem with multi-left-side time windows.
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