abstract

This paper develops and compares two direct zero-based procedures for fuzzy transportation problems: the Single Zero Method (SZM) and the Zero Counting Method (ZCM). Generalised trapezoidal fuzzy numbers are retained as the basic representation of uncertain transportation costs, while a declared ranking rule supplies scalar values for row reduction, column reduction and route selection. SZM gives priority to a zero that is unique in an active row or column. ZCM instead measures the scarcity of each zero through dynamic row and column zero counts and prefers the structurally most constrained zero. Both procedures allocate the maximum feasible amount, update the residual table, reconstruct the fuzzy objective from the original fuzzy costs and verify ranked optimality using the Modified Distribution Method. Two numerical examples are used for comparison. In a 3 by 4 problem, both methods produce the same ranked-optimal plan with cost 381 and fuzzy objective (306, 381, 381, 456; 1). In a 4 by 4 problem, the methods produce different shipment patterns but the same ranked-optimal cost of 381 and fuzzy objective (301, 381, 381, 461; 1), illustrating the presence of alternative optima. The results show that zero-based allocation is transparent and computationally accessible, but its claims should be supported by explicit ranking assumptions, deterministic tie rules and an independent optimality check.