Neural Computing and Applications

, Volume 31, Issue 12, pp 9073–9093 | Cite as

Mathematical programming and three metaheuristic algorithms for a bi-objective supply chain scheduling problem

  • Hamid Zarei
  • Morteza Rasti-BarzokiEmail author
Original Article


In this study, a bi-objective optimization problem for a supply chain with different transportation modes is addressed. The first objective function is minimizing costs imposed by production, batching, due date assignment and transportation. The second one is minimizing inventory and tardiness costs. Also, a heuristic rule is developed to choose non-dominated transportation modes. Three metaheuristic algorithms including multi-objective particle swarm optimization (MOPSO), non-dominated sorting genetic algorithm (NSGA-II) and hybrid NSGA-II (HNSGA-II) are customized to solve the problem. In addition, a theoretical improvement in the non-dominated sorting procedure called improved efficient non-dominated sorting (IENS) is proposed. Computational tests are used for comparing and evaluating the proposed methods and algorithms. The results show that IENS reduces running time compared to the modern method of non-dominated sorting, the efficient non-dominated sorting method, and this reduction is statistically significant. Also, the HNSGA-II has an average but robust performance compared to the other two algorithms.


Supply chain scheduling Non-dominated sorting genetic algorithm Multi-objective particle swarm optimization Transportation modes 



Particle swarm optimization


Multi-objective particle swarm optimization


Non-dominated sorting genetic algorithm


Hybrid non-dominated sorting genetic algorithm


Efficient non-dominated sorting


Improved efficient non-dominated sorting


Random key encoding



The authors acknowledge the editorial team, four anonymous reviewers and the Associate Editor Prof. Haider Abbas, for their helpful comments to improve the paper. We also thank Mr. Hossein Khosroshahi (Ph.D. Candidate, Isfahan University of Technology) for his useful feedback and suggestions.

Compliance with ethical standards

Conflict of interest

Hamid Zarei and Morteza Rasti-Barzoki certify that they have no affiliations with or involvement in any organization or entity with any financial interest (such as honoraria, educational grants, participation in speakers’ bureaus, membership, employment, consultancies, stock ownership, or other equity interest, and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.

Supplementary material

521_2018_3898_MOESM1_ESM.docx (33 kb)
Supplementary material 1 (DOCX 33 kb)


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© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringIsfahan University of TechnologyIsfahanIran

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