Optimizing a multi-item economic order quantity problem with imperfect items, inspection errors, and backorders

  • Soheyl Khalilpourazari
  • Seyed Hamid Reza Pasandideh
  • Seyed Taghi Akhavan Niaki
Methodologies and Application


This paper proposes a multi-item economic order quantity model with imperfect items in supply deliveries. The inspection process to classify the items is not perfect and involves two types of error: Type-I and Type-II. To cope with the uncertainty involved in real-world applications and to bring the problem closer to reality, operational constraints are assumed stochastic. The aim is to determine the optimal order and back order sizes of the items in order to achieve maximum total profit. As the proposed mathematical model is a constrained nonlinear programming, three different solution methods including an exact method named the interior-point and two novel meta-heuristics named grey wolf optimizer (GWO) and moth-flame optimization (MFO) algorithms are utilized to solve the problem. In order to demonstrate the most efficient solution method, the performance of the three solution methods is evaluated when they solve some test problems of different sizes. Various comparison measures including percentage relative error, relative percentage deviation, and computation time are used to compare the solution methods. Based on the results, MFO performs better in small and medium instances in terms of percentage relative error; meanwhile, GWO shows a better performance in terms of relative percentage deviation in large-size test problems. In the end, sensitivity analyses are carried out to investigate how any parameter change affects the objective function value of the mathematical model in order to determine the most critical parameter.


Economic order quantity Imperfect items Inspection errors Stochastic constraints Grey wolf optimizer Moth-flame optimization 


Compliance with ethical standards

Conflict of interest

All authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Industrial Engineering, Faculty of EngineeringKharazmi UniversityTehranIran
  2. 2.Department of Industrial EngineeringSharif University of TechnologyTehranIran

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