Addressing a fixed charge transportation problem with multi-route and different capacities by novel hybrid meta-heuristics

Document Type : Research Paper


Industrial Engineering Department, Yazd University, Yazd, Iran


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.


Main Subjects

Baidya, A., Bera, U. K., & Maiti, M. (2017). Models for solid transportation problems in logistics using particle swarm optimisation algorithm and genetic algorithm. International Journal of Logistics Systems and Management27(4), 487-526.
Baidya, A., Bera, U. K., & Maiti, M. (2016). The grey linear programming approach and its application to multi-objective multi-stage solid transportation problem. OPSEARCH53(3), 500-522.
Chopra S. Meindl P., Supply Chain Management, Pearson, 4th edition, (2010), 978-0-130608040-4.
Chopra, S., Meindel, P. (2007). Supply chain management: Strategy, planning and operation, Pearson Prentice Hall.
Engin, O., Ceran, G., & Yilmaz, M. K. (2011). An efficient genetic algorithm for hybrid flow shop scheduling with multiprocessor task problems. Applied Soft Computing11(3), 3056-3065.
Fakhrzad, M.B., A. Sadri Esfahani, (2013a). Modeling the time windows vehicle routing problem in cross-docking strategy using two meta-heuristic algorithms. International Journal of Engineering-Transactions A: Basics 27 (7), 1113.
Fakhrzad, M. B., Sadeghieh, A., & Emami, L. (2013b). A new multi-objective job shop scheduling with setup times using a hybrid genetic algorithm.
Fakhrzad, M.B., Heydari, M., (2008). A Heuristic Algorithm for Hybrid Flow-shop Production Scheduling to Minimize the Sum of The Earliness ANDF Tardiness Costs. Journal of the Chinese Institute of Industrial Engineers. 25 (2), 105-115.
Fard, A. M. F., & Hajiaghaei-Keshteli, M. (2018). A bi-objective partial interdiction problem considering different defensive systems with capacity expansion of facilities under imminent attacks. Applied Soft Computing68, 343-359.
Fard, A. F., & Hajiaghaei-Keshteli, M. (2016). Red deer algorithm (RDA); a new optimization algorithm inspired by red deer's mating. In International Conference on Industrial Engineering, IEEE., (2016 e) (pp. 33-34).
Fathollahi-Fard, A. M., Hajiaghaei-Keshteli, M., & Tavakkoli-Moghaddam, R. (2018a). The Social Engineering Optimizer (SEO). Engineering Applications of Artificial Intelligence72, 267-293.
Fathollahi-Fard, A. M., & Hajiaghaei-Keshteli, M. (2018b). A stochastic multi-objective model for a closed-loop supply chain with environmental considerations. Applied Soft Computing69, 232-249.
Fathollahi-Fard, A. M., Hajiaghaei-Keshteli, M., & Mirjalili, S. (2018c). Hybrid optimizers to solve a tri-level programming model for a tire closed-loop supply chain network design problem. Applied Soft Computing.
Fathollahi-Fard, A. M., Hajiaghaei-Keshteli, M., & Mirjalili, S. (2018d). Multi-objective stochastic closed-loop supply chain network design with social considerations. Applied Soft Computing71, 505-525.
Fathollahi-Fard, A. M., Hajiaghaei-Keshteli, M., & Tavakkoli-Moghaddam, R. (2018e). A bi-objective green home health care routing problem. Journal of Cleaner Production200, 423-443.
Glover, F. (1977). Heuristics for integer programming using surrogate constraints. Decision Sciences8(1), 156-166.
Goldberg, D. E., & Holland, J. H. (1988). Genetic algorithms and machine learning. Machine learning3(2), 95-99.
Golmohamadi, S., Tavakkoli-Moghaddam, R., & Hajiaghaei-Keshteli, M. (2017). Solving a fuzzy fixed charge solid transportation problem using batch transferring by new approaches in meta-heuristic. Electronic Notes in Discrete Mathematics58, 143-150.
Hajiaghaei-Keshteli, M., & Fard, A. M. F. (2018a). Sustainable closed-loop supply chain network design with discount supposition. Neural Computing and Applications, 1-35.
Hajiaghaei-Keshteli, M., & Fathollahi-Fard, A. M. (2018b). A set of efficient heuristics and metaheuristics to solve a two-stage stochastic bi-level decision-making model for the distribution network problem. Computers & Industrial Engineering123, 378-395.
Hajiaghaei-Keshteli, M., & Aminnayeri, M. (2014). Solving the integrated scheduling of production and rail transportation problem by Keshtel algorithm. Applied Soft Computing25, 184-203.
Hajiaghaei-Keshteli, M. (2011). The allocation of customers to potential distribution centers in supply chain networks: GA and AIA approaches. Applied Soft Computing11(2), 2069-2078.
Hajiaghaei-Keshteli, M., Molla-Alizadeh-Zavardehi, S., & Tavakkoli-Moghaddam, R. (2010). Addressing a nonlinear fixed-charge transportation problem using a spanning tree-based genetic algorithm. Computers & Industrial Engineering59(2), 259-271.
Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. P. (1983). Optimization by simulated annealing. science220(4598), 671-680.
Lotfi, M. M., & Tavakkoli-Moghaddam, R. (2013). A genetic algorithm using priority-based encoding with new operators for fixed charge transportation problems. Applied Soft Computing13(5), 2711-2726. (14)
Midya, S., & Roy, S. K. (2017). Analysis of interval programming in different environments and its application to fixed-charge transportation problem. Discrete Mathematics, Algorithms and Applications. (7)
Molla-Alizadeh-Zavardehi, S., Nezhad, S. S., Tavakkoli-Moghaddam, R., & Yazdani, M. (2013). Solving a fuzzy fixed charge solid transportation problem by metaheuristics. Mathematical and Computer Modelling57(5), 1543-1558. (16)
Mingozzi, A., & Roberti, R. (2017). An Exact Algorithm for the Fixed Charge Transportation Problem Based on Matching Source and Sink Patterns. Transportation Science. (8)
Pop, P., Matei, O., Sitar, C. P., & Zelina, I. (2016). A hybrid based genetic algorithm for solving a capacitated fixed-charge transportation problem. Carpathian Journal of Mathematics, 225-232. (10)
Pramanik, S., Jana, D. K., Mondal, S. K., & Maiti, M. (2015). A fixed-charge transportation problem in two-stage supply chain network in Gaussian type-2 fuzzy environments. Information Sciences325, 190-214. (12)
Salimi, H. (2015). Stochastic fractal search: a powerful metaheuristic algorithm. Knowledge-Based Systems75, 1-18. (19)
Samadi, A., Mehranfar, N., Fathollahi Fard, A. M., & Hajiaghaei-Keshteli, M. (2018). Heuristic-based metaheuristics to address a sustainable supply chain network design problem. Journal of Industrial and Production Engineering35(2), 102-117. (37)
Sadeghi-Moghaddam, S., Hajiaghaei-Keshteli, M., & Mahmoodjanloo, M. (2017). New approaches in metaheuristics to solve the fixed charge transportation problem in a fuzzy environment. Neural Computing and Applications, 1-21. (4)
Sahebjamnia, N., Fard, A. M. F., & Hajiaghaei-Keshteli, M. (2018). Sustainable tire closed-loop supply chain network design: Hybrid metaheuristic algorithms for large-scale networks. Journal of Cleaner Production. (34)
Saxena, R. R., & Upmanyu, M. (2016). A compromise method for solving fuzzy multi objective fixed charge transportation problem. Lecture Notes in Management Science8, 9. (11)
Shoushtary, M.A., Hoseini Nasab, H., Fakhrzad, M.B.(2014), Team robot motion planning in dynamics environments using a new hybrid algorithm (honey bee mating optimization-tabu list).Chinese Journal of Engineering, 1-8. (38)
Sanei, M., Mahmoodirad, A., & MOLLA, A. Z. S. (2013). An electromagnetism-like algorithm for fixed charge solid transportation problem. (15)
Storn, R., & Price, K. (1997). Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. Journal of global optimization11(4), 341-359. (23)
Zhang, B., Peng, J., Li, S., & Chen, L. (2016). Fixed charge solid transportation problem in uncertain environment and its algorithm. Computers & Industrial Engineering102, 186-197. (9)
Volume 12, Issue 1 - Serial Number 1
January 2019
Pages 167-184
  • Receive Date: 03 April 2018
  • Revise Date: 02 October 2018
  • Accept Date: 21 November 2018
  • First Publish Date: 06 January 2019