## 3 Problem Description  
This research aims to develop an optimization methodology and design an H&S transportation system to regulate passenger flows within a cluster of quite sparsely located islands (or any other similar kinds of transport terminals) with a small number of carriages. In such types of problems, the passenger flows typically initiate from central ferry terminals, placed on the mainland, and distributed to the cluster islands. Such clusters involve mostly small-sized islands or tourist destinations; thus, the problem emphasizes passenger flows rather than vehicle and cargo transport. In these cases, it is not efficient to interconnect all islands directly to central ports. Instead, it is preferable to transfer passengers to a central node of the cluster (hub island) and then split the flows by routing additional lines (usually lower capacity and speed vessels) to different destinations in order to reduce carrier costs and passenger travel times. This problem structure differentiates from the typical H&S problem (Figure 1), as islands are not directly connected to hub nodes. However, the algorithmic structure and computational approach fall within the same class of problems. Short Sea Shipping transportation in a scatterly distributed islands constitutes a multi-parameter optimization problem in which decisions should be made regarding the number of vessels deployed, the hub selection, the route development, and the island service priority, subject to certain constraints and preferences. SSS is mainly adapted to the passenger movement particularities of scatter-distributed island regions, such as the Caribbean, Indonesia, the Baltic Sea, etc. Such an SSS network involves short irregular-shaped routes with several intermediate stops prioritizing popular destinations. This network development is similar in principle but different in application from the corresponding one for Artic waterways, which refers mainly to liner shipping networks involving long-distance and rather linear routes within limited navigable waters due to ice. In this problem formulation, one or more main lines originate from the central ports and reach hub node islands where passengers may either continue their journey or transfer to other vessels toward their destination. The optimization model that is developed aims to provide proposals referring to the following:  
1. the islands that are assigned as hub nodes within several potential alternatives;  
Figure 1: (**a**) Conventional H&S structure, (**b**) Short Sea Shipping transportation H&S structure. 2. the islands that are to be served by each route (vessel) originating either at a central port or at a hub node;
3. the island service priority within each route. The problem formulation contains elements of the Hub-and-Spoke and Travelling Salesman structures, making it more demanding than the typical H&S one, as the hub selection within nodes and the shortest routes among islands are internal optimization goals. Considering, in addition, that the passenger flows change along route nodes, the use of formal optimization seems rather imperative to provide efficient routing solutions. A 3-D tensor is employed to represent the solution structure for the proposed H&S model (Figure 2a). In this formulation, the number of blocks along the X-axis indicates the islands to be served, while the number of blocks along the Y and Z axes reflects the number of lines (vessels) and the ports--potential origins (central ports or hub nodes), respectively. If the number of ports involved is equal to the number of lines (vessels), the problem is downgraded to a two-dimensional problem (island-to-line allocation and prioritization), and each solution can be represented by a 2-D permutation matrix, as shown in Figure 2b. A number of tensor blocks (as many as the islands) are numbered with a sequence of integers indicating the order of island service, and all others have no value.