Fixed charge 4-dimensional transportation problem for breakable incompatible items with type-2 fuzzy random parameters under volume constraint

Abstract

Real-life problems are full of uncertainties combined with impreciseness and randomness. Transportation problems become difficult when its parameters are type-2 fuzzy random type. This paper presents a 4-dimensional fixed charge transportation problem for incompatible and compatible breakable items with type-2 fuzzy random parameters under the availability, demand, conveyance’s weight and volume capacity constraint. The model is developed for usual transportation cost minimization objective, where transportation costs, fixed charges, availabilities, demands, conveyances’ weights and volume capacities are type-2 fuzzy random. A type-2 fuzzy random parameter is converted to type-1 fuzzy random one by critical value-based reduction method and then converted to a random parameter using two different methods-generalized credibility measure and centroid method. The random parameter is converted to crisp parameter using probability chance constraint method for constraints and expectation for objective. A detailed conversion procedure from type-2 fuzzy random to crisp is also presented. The reduced deterministic problem is solved by the Generalized Reduced Gradient method using Lingo 11.0 and numerically illustrated. A real-life example is introduced. Particular models are deduced for non-breakable items or different pairwise incompatible items. The sensitivity analyses against different generalized credibility levels of different parameters are presented for total transportation cost. The numerical results show the efficiency of the proposed critical value reduction-generalized credibility-chance constraint, and expectation method. The importance of uncertainty, multi-path and vehicles’ volume capacity in transportation problems is demonstrated.