Document Type : Research Paper

**Authors**

Industrial Engineering Department, Yazd University, Yazd, Iran

**Abstract**

In most real world application and problems, a homogeneous product is carried from an origin to a destination by using different transportation modes (e.g., road, air, rail and water). This paper investigates a fixed charge transportation problem (FCTP), in which there are different routes with different capacities between suppliers and customers. To solve such a NP-hard problem, four meta-heuristic algorithms include Red Deer Algorithm (RDA), Stochastic Fractal Search(SFS), Genetic Algorithm (GA), and Simulated Annealing (SA) and two new hybrid meta-heuristics include hybrid RDA & GA (HRDGA) algorithm and Hybrid SFS & SA (HSFSA) algorithm are utilized. Regarding the literature, this is the first attempt to employ such optimizers to solve a FCTP. To tune up their parameters of algorithms, various problem sizes are generated at random and then a robust calibration is applied by using the Taguchi method. The final output shows that Simulated Annealing (SA) algorithm is the better than other algorithms for small-scale, medium-scale, and large-scale problems. As such, based on the Gap value of algorithms, the results of LINGO software shows that it reveals a better outputs in comparison with meta-heuristic algorithms in small-scale and simulated annealing algorithm is better than other algorithms in large-scale and medium-scale problems. Finally, a set of computational results and conclusions are presented and analyzed.

**Keywords**

- Fixed-charge transportation problem
- SA algorithm
- GA algorithm
- SFS algorithm
- RDA algorithm
- Taguchi method

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