Multi-item 4D-TPs under budget constraint using rough interval

Multi-item four dimensional transportation problems (MI4D-TPs) are formulated with budget constraint under rough and fuzzy interval environments. Here unit selling prices, unit purchasing costs, unit transportation costs, fixed charges, sources, demands, capacities of conveyances and total budget are represented by rough intervals for rough model and fuzzy numbers for fuzzy model. In this paper, four dimensional transportation problems (4D-TPs) under rough interval environment are transformed to deterministic ones by three different methods - (i) upper and lower approximation intervals, (ii) Expected Value Technique and (iii) rough chance-constraint programming technique. The fuzzy 4D-TP is made deterministic using possibility measures. Reduced problems are solved by generalized reduced gradient (GRG) method using LINGO 12.0 software. The models are illustrated with numerical examples and natures of the solutions i.e. complete and rather satisfactory solutions are presented. As particular cases, the results of solid transportation problems (3D-TPs) and conventional transportation problems (2D-TPs) are obtained. Importance of consideration of routes in MI4D-TPs is pointed out. As a particular case, the results of an earlier investigation are derived. Sensitivity of profit in the rough model through Expected Value Technique is presented against the arithmetic combination parameter. Percentage of increase of profit against the expectation parameter is furnished.