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Top1. Introduction
The Transportation Problem (TP) plays an important role in many industrial real-life applications with the aim of ensuring the shipment of supplies from planned sources to specific destinations via minimal transportation costs.
The TP is widely studied in the literature. It was firstly designed by Hitchcock (1941). Dantzig (1951) solved the TP using the Simplex method. Dinagar and Palanivel (2009) studied the TP in trapezoidal fuzzy domain. Pandian and Natarajan (2010) designed the Fuzzy Zero Point method to optimize the fuzzy TP. Kaur and Kumar (2012) developed a new method to find an initial basic solution for TP where the transportation costs are denoted by generalized fuzzy numbers. Shanmugasundari and Ganesan (2013) introduced a fuzzy version of Vogel’s and MODI methods in order to generate basic initial solution for TP with imprecise variables described by triangular fuzzy numbers. Beaulaa and Priyadharsini (2015) introduced an iterative method to find the optimal solution for the TP in trapezoidal intuitionistic fuzzy environment using the Stepping Stone method with indices based ranking methods. Ahmed et al. (2016) designed a new method entitled Allocation Table Method (ATM) to find an initial basic feasible solution for the TP and its efficiency is tested by solving several TP. Muruganandam and Srinivasan (2016) investigated a new heuristic with the graded means ranking function to optimize the fully trapezoidal fuzzy TP. Hunwisai and Kumam (2016) solved the trapezoidal fuzzy TP using the ATM and the Modified Distribution methods to find the optimal crisp solution. Moreover, Kumar (2016) developed the PSK technique for the TP under triangular and trapezoidal fuzzy numbers.
Uthra et al. (2017) proposed a new ranking measure to transform the costs in TP from trapezoidal intuitionistic fuzzy numbers to crisp ones. Furthermore, Kumar (2018) presented two techniques: the first one is based on Linear Programming technique and the second is based on the Distribution Method to optimize the balanced and unbalanced intuitionistic fuzzy TP. Kumar (2019) presented a mixed and fully triangular intuitionistic fuzzy solid TP and solved it using the PSK method with graphical representation of the found triangular intuitionistic fuzzy optimal solution. Ngastiti et al (2020) solved a fully trapezoidal fuzzy TP using the Zero Point and the Zero Suffix methods; then, the two methods are compared based on the value of the basic feasible solution and the number of iterations.
Moreover, Gargouri and Bouamama (2020) presented a rich literature review for the TP and its variants (multi-objective optimization, multimodal, optimization methods, case study, etc.). Mhaske and Bondar (2020) optimized triangular, pentagonal and heptagonal fuzzy transportation problems using the North West Corner method, the Matrix Minima method and the Vogel’s approximation method; furthermore, they introduced a new function for nonagon and hendecagon fuzzy transportation cost numbers. Li et al. (2021) introduced a case study in a bike sharing system as a smart transportation system and then optimized it using the Deep Reinforcement Learning technique. Sikkannan (2021) solved the triangular fuzzy TP using an original heuristic based on the standard deviation metric and the magnitude ranking function.