In this paper we address a novel extension of the well-known multiple strip packing problem. Given a set of rectangular items that must be packed into a set of strips of predefined widths. In contrast to the conventional problem, both height and width of each rectangle is dependent on the strip it can be packed into. We do no impose any additional conditions on the size of the rectangles or widths of the strips, except for there is at least one strip that can accommodate each rectangle. This problem naturally arises in several applications, e.g. task scheduling in highly heterogeneous cloud or grid environments; or harbor logistics where the problem can be considered as an extension of the berth assignment problem with multiple quays. In this paper we propose several integer programming formulations of the problem and its particular case where the rectangles can be placed only on the so-called levels. To find quality solutions of the problem, we propose solution algorithms of two types. The first one is a skyline best-fit heuristic combined with a fast parallel repeated randomized local search. The second type is based on the two-stage nature of the problem: it first distributes the rectangles among strips by solving an auxiliary integer program and then packs them with a best-fit heuristic. We demonstrate the effectiveness of proposed algorithms in extensive computational experiments on test instances adapted from the literature and large-scale synthetically generated problems related to task scheduling in SPARK systems.