EPL models with fuzzy imperfect production system including carbon emission: a fuzzy differential equation approach

  • Manoranjan DeEmail author
  • Barun Das
  • Manoranjan Maiti
Methodologies and Application


The paper outlines the production policies for maximum profit of a firm producing imperfect economic lot size with time-dependent fuzzy defective rate under the respective country’s carbon emission rules. Generally in economic production lot-size models, defective production starts after the passage of some time from production commencement. So the starting time of producing defective units is normally uncertain and imprecise. Thus, produced defective units are fuzzy, partially reworked instantly and sold as fresh units. As a result, the inventory level at any time becomes fuzzy and the relation between the production, demand and inventory level becomes a fuzzy differential equation (FDE). Nowadays, different governments have made environmental regulations following the United Nations Framework Convention on Climate Change to reduce carbon emission. Some governments use cape and trade policy on emission. Due to this, firms are in fix how to optimize the production. If the firms produce more, the profit increases along with more emission and corresponding tax. Here, models are formulated as profit maximization problems using FDE, and the corresponding inventory and environmental costs are calculated using fuzzy Riemann integration. An \(\alpha \)-cut of average profits is obtained and the reduced multi-objective crisp problems are solved using intuitionistic fuzzy optimization technique. The models are illustrated numerically and results are presented graphically. Considering different carbon regulations, an algorithm for a firm management is presented to achieve the maximum profit. Real-life production problems for the firms in Annex I and developing countries are solved.


Fuzzy imperfect production Carbon emission Fuzzy differential equation Intuitionistic fuzzy optimization technique 



Average carbon emission cost


Average carbon emission reward


Average total profit


Carbon emission


Carbon emission cost


Carbon emission reward


Economic order quantity


Economic production lot size


Economic production quantity


Fuzzy differential equation


Fuzzy Riemann integration


Intuitionistic fuzzy number


Intuitionistic fuzzy optimization technique


Intuitionistic fuzzy set


Multi-objective optimization problem


Multi-objective problem


Triangular fuzzy number


Unit production cost



The authors are greatly indebted to the referees for their valuable observations and suggestions for improving the presentation of the paper.

Compliance with ethical standards

Conflict of interest

The 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 Applied Mathematics with Oceanology and Computer ProgrammingVidyasagar UniversityMidnaporeIndia
  2. 2.Department of MathematicsSidho-Kanho-Birsha UniversityPuruliaIndia

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