Skip to main content
SpringerLink
Log in

Solving fixed-charge transportation problem using a modified particle swarm optimization algorithm

  • Original article
  • Published:
International Journal of System Assurance Engineering and Management Aims and scope Submit manuscript

Abstract

Particle Swarm Optimization (PSO) has emulated the social behaviour of some animals such as a flock of birds and a school of fish, searching for food. This communicative sociality when modelled as computational procedure has solved a wide range of complex problems. Over the years, PSO has undergone transformations and numerous variants have come up. In this paper, PSO has been hybridized with two new algorithms to solve the fixed charge transportation problem to minimize the transportation cost (variable and fixed) of delivering goods while satisfying supply/demand constraints. The method considers the reduction of objective function defined by Balinski, Adlakha et al., Yousefi et al. and is incorporated within the PSO. An independent approach of solving the problem on the basis of variable cost initially followed by addition of fixed cost has also been explored. It was observed that proposed PSO works best without reducing the objective function. The simulation results reveal a substantial gain of the proposed method in terms of its efficiency and effectiveness examined on different test problems. To validate the claims, the proposed PSO has also been compared with the solutions attained by other existing methods (either exact or heuristics).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Adlakha V, Kowalski K (2003) A simple heuristic for solving small fixed-charge transportation problems. Omega 31(3):205–211

    Article  Google Scholar 

  • Adlakha V, Kowalski K, Vemuganti R (2006) Heuristic algorithms for the fixed-charge transportation problem. Opsearch 43(2):132–151

    Article  Google Scholar 

  • Adlakha V, Kowalski K, Wang S, Lev B, Shen W (2014) On approximation of the fixed charge transportation problem. Omega 43:64–70

    Article  Google Scholar 

  • Akbari M, Molla-Alizadeh-Zavardehi S, Niroomand S (2020) Meta-heuristic approaches for fixed-charge solid transportation problem in two-stage supply chain network. Oper Res 20(1):447–471

    Google Scholar 

  • Balinski ML (1961) Fixed-cost transportation problems. Naval Res Logistics Q 8(1):41–54

    Article  Google Scholar 

  • Barr RS, Glover F, Klingman D (1981) A new optimization method for large scale fixed charge transportation problems. Oper Res 29(3):448–463

    Article  MathSciNet  Google Scholar 

  • Cabot AV, Erenguc SS (1984) Some branch-and-bound procedures for fixed-cost transportation problems. Naval Resh Logistics Q 31(1):145–154

    Article  MathSciNet  Google Scholar 

  • Chen J, Pan F, Cai T (2006) Acceleration factor harmonious particle swarm optimizer. Int J Automation Comput 3(1):41–46

    Article  Google Scholar 

  • Clerc M, Kennedy J (2002) The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Trans Evolut Comput 6(1):58–73

    Article  Google Scholar 

  • Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the Sixth International Symposium on Micro Machine and Human Science, pp. 39–43. Nagoya, Japan

  • El-Sherbiny MM, Alhamali RM (2013) A hybrid particle swarm algorithm with artificial immune learning for solving the fixed charge transportation problem. Comput Ind Eng 64(2):610–620

    Article  Google Scholar 

  • Fakhrzad MB, Goodarzian F, Golmohammadi A (2019) Addressing a fixed charge transportation problem with multi-route and different capacities by novel hybrid meta-heuristics. J Ind Syst Eng 12(1):167–184

    Google Scholar 

  • Fernandez-Martinez JL, Garcia-Gonzalo E (2011) Stochastic stability analysis of the linear continuous and discrete pso models. IEEE Trans Evolutionary Comput 15(3):405–423

    Article  Google Scholar 

  • Garg H (2016) A hybrid pso-ga algorithm for constrained optimization problems. Appl Math Comput 274:292–305

    MathSciNet  MATH  Google Scholar 

  • Garg H (2019) A hybrid gsa-ga algorithm for constrained optimization problems. Inf Sci 478:499–523

    Article  Google Scholar 

  • Gazi V, Passino KM (2004) Stability analysis of social foraging swarms. IEEE Trans Syst Man Cybernetics Part B (Cybernetics) 34(1):539–557

    Article  Google Scholar 

  • Gottlieb J, Paulmann L (1998) Genetic algorithms for the fixed charge transportation problem. In: Evolutionary Computation Proceedings, 1998. IEEE World Congress on Computational Intelligence., The 1998 IEEE International Conference on, pp. 330–335. IEEE

  • Gray P (1971) Exact solution of the fixed-charge transportation problem. Oper Res 19(6):1529–1538

    Article  Google Scholar 

  • Hirsch WM, Dantzig GB (1968) The fixed charge problem. Naval Res Logistics Q 15(3):413–424

    Article  MathSciNet  Google Scholar 

  • Hitchcock F (1941) The distribution of a product from several sources to numerous locations. J Math Phys 20:224–230

    Article  Google Scholar 

  • Huang H, Zhifang H (2009) Particle swarm optimization algorithm for transportation problems. Particle Swarm Optimization, Intech, Shanghai, pp 275–290

  • Ide A, Yasuda K (2005) A basic study of adaptive particle swarm optimization. Electr Eng Japan 151(3):41–49

    Article  Google Scholar 

  • Kadirkamanathan V, Selvarajah K, Fleming PJ (2006) Stability analysis of the particle dynamics in particle swarm optimizer. IEEE Trans Evolutionary Comput 10(3):245–255

    Article  Google Scholar 

  • Koopmans T (1947) Optimum utilization of the transport system. In: Proceedings of International Statistical Conferences : Econometric, pp. 136–145

  • Lotfi M, Tavakkoli-Moghaddam R (2013) A genetic algorithm using priority-based encoding with new operators for fixed charge transportation problems. Appl Soft Comput 13(5):2711–2726

    Article  Google Scholar 

  • Madenoğlu, FS (2021) Managing transportation in supply chain: Metaheuristics for solving a capacitated fixed-charge transportation problem. In: Handbook of Research on Recent Perspectives on Management, International Trade, and Logistics, pp. 198–218. IGI Global

  • Murty KG (1968) Solving the fixed charge problem by ranking the extreme points. Oper. Res. 16(2):268–279

    Article  Google Scholar 

  • Palekar US, Karwan MH, Zionts S (1990) A branch-and-bound method for the fixed charge transportation problem. Manag Sci 36(9):1092–1105

    Article  MathSciNet  Google Scholar 

  • Pop P, Matei O, Sitar CP, Zelina I (2016) A hybrid based genetic algorithm for solving a capacitated fixed-charge transportation problem. Carpath J Math 32:225–232

    Article  MathSciNet  Google Scholar 

  • Qin Z, Yu F, Shi Z, Wang Y (2006) Adaptive inertia weight particle swarm optimization. In International conference on Artificial Intelligence and Soft Computing, pp. 450–459. Springer

  • Raj KAAD, Rajendran C (2011) A hybrid genetic algorithm for solving single-stage fixed-charge transportation problems. Technol Oper Manag 2(1):1–15

    Article  Google Scholar 

  • Ratnaweera A, Halgamuge SK, Watson HC (2004) Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients. IEEE Trans Evolut Comput 8(3):240–255

    Article  Google Scholar 

  • Robers P, Cooper L (1976) A study of the fixed charge transportation problem. Comput Math Appl 2(2):125–135

    Article  MathSciNet  Google Scholar 

  • Sadagopan S, Ravindran A (1982) A vertex ranking algorithm for the fixed-charge transportation problem. J Optim Theory Appl 37(2):221–230

    Article  MathSciNet  Google Scholar 

  • Sagratella S, Schmidt M, Sudermann-Merx N (2020) The noncooperative fixed charge transportation problem. Euro J Oper Res 284(1):373–382

    Article  MathSciNet  Google Scholar 

  • Shi Y, Eberhart R (1998) A modified particle swarm optimizer. In: 1998 IEEE international conference on evolutionary computation proceedings. IEEE world congress on computational intelligence (Cat. No. 98TH8360), pp. 69–73. IEEE

  • Steinberg DI (1970) The fixed charge problem. Naval Res Logistics Q 17(2):217–235

    Article  MathSciNet  Google Scholar 

  • Sun M, Aronson JE, McKeown PG, Drinka D (1998) A tabu search heuristic procedure for the fixed charge transportation problem. Euro J Oper Res 106(2–3):441–456

    Article  Google Scholar 

  • Tari FG, Hashemi Z (2018) Prioritized k-mean clustering hybrid ga for discounted fixed charge transportation problems. Comput Ind Eng 126:63–74

    Article  Google Scholar 

  • Van den Bergh F, Engelbrecht AP (2006) A study of particle swarm optimization particle trajectories. Inf Sci 176(8):937–971

    Article  MathSciNet  Google Scholar 

  • Yousefi K, J Afshari A, Hajiaghaei-Keshteli M (2019) Solving the fixed charge transportation problem by new heuristic approach. J Optim Ind Eng 12(1):41–52

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IKG Punjab Technical University Jalandhar, Punjab, India

    Gurwinder Singh

  2. Department of Applied Sciences, BBSBEC, Fatehgarh Sahib, Punjab, India

    Amarinder Singh

Authors
  1. Gurwinder Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amarinder Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gurwinder Singh.

Ethics declarations

Conflict of interest

The authors do not have any conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Trace of Gbest and corresponding optimal solution matrix

Trace of Gbest and corresponding optimal solution matrix

Fig. 3
figure 3

Plot of Optimal value(Gbest) & corresponding optimal solution

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singh, G., Singh, A. Solving fixed-charge transportation problem using a modified particle swarm optimization algorithm. Int J Syst Assur Eng Manag 12, 1073–1086 (2021). https://doi.org/10.1007/s13198-021-01171-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13198-021-01171-2

Keywords

Search

Navigation