The generalization of scheduling with machine cost

We study a new online resource allocation problem. Usually in resource allocation problems the tasks are presented by rectangles, where the sides describe the time and the resource needed to perform the task. In the online problem the tasks/rectangles arrive one by one according to a list. If we are to perform the task during the shortest possible time, say H, with fixed amount of resource or by minimal resource keeping the completion time below a given bound W, then the problem can be described as online strip packing. Here we consider the problem where neither the resource nor the time is limited; we minimize the objective @[email protected]?H+W, where @c is a fixed positive parameter. In the special case @c=1 we have to pack the incoming rectangles into a container rectangle with perimeter as small as possible. This problem is a generalization of scheduling with machine cost. We present and analyse a shelf-based online algorithm for the solution of the problem. We also consider the semi online case where the list of rectangles is ordered by decreasing height. Finally we study the version where the sizes of the rectangles are not fixed but the algorithm can modify them keeping their area fixed.