Derivation of the analytical expression for the complexity of container retrieval from a stack

A container terminal warehouse complex is a sophisticated system that performs the functions of servicing container flows passing through it. At large terminals, tens or even hundreds of thousands of containers are stored at any given time in multi-tiered stacks to minimize the extremely limited area of the port territory. This necessary storage approach increases the labour intensity of container retrieval operations for loading onto ships or adjacent transport vehicles. Increasing stack height complicates the procedure by blocking target containers with other containers, requiring these to be moved to gain access. This leads to an increase in the number of additional movements per commercial (i. e., customer-paid) movement to retrieve the target container. Consequently, productivity decreases and operational costs increase. In particular, an increase in auxiliary movements reduces container selectivity — defined as the ratio of commercial movements to the total number of movements — and increases labour intensity, defined as its inverse. The values of selectivity and labour intensity largely depend on the type of technological equipment used and storage organisation schemes. To assess the operational characteristics of designed port warehouse complexes and determine the necessary fleet size of technological equipment, it is necessary to develop quantitative metrics that allow estimation of the labour intensity of warehousing operations under different transport and technological schemes. Within this study, the authors have derived analytical expressions for all classes of warehousing equipment in the form of combinatorial formulas, which can serve as objective and easily computable metrics. The use of these metrics enables more accurate calculations at the technological design, planning, and management stages of container terminals. Thus, more accurate forecasting of operational indicators is thereby achieved, taking into account the specifics of the technologies and organisational solutions used.