Bipolar neutrosophic multi-item four-dimensional transportation problem with variable routes for breakable items

Generally, in case of transportation problem, mathematical frameworks predominantly focus on positive beliefs to manage uncertain and vague information, often overlooking the negative perspectives in decision-making. To address this gap, we introduce a four-dimensional transportation problem (4D-TP) model within a bipolar neutrosophic environment, which incorporates both positive and negative aspects of human cognition. The proposed model is formulated as a multi-item 4D-TP with breakability, aimed at maximizing profit by considering key parameters such as costs, accessibility, demands, and transportation capacities, all represented using single-valued triangular bipolar neutrosophic numbers. Unlike traditional approaches that assume uniformity in the number of routes for each origin-destination pair, our model accommodates varying numbers of routes, reflecting more realistic transportation scenarios. Furthermore, this is the first attempt to solve the 4D-TP using possibility measure associated with single-valued triangular bipolar neutrosophic number, which includes truth, indeterminacy, and falsity membership functions to capture both positive and negative dimensions. To validate the model, we provide a numerical example and solve it using the Generalized Reduced Gradient method, implemented in the LINGO-14.0 solver. Some managerial implications are also included at the end.