Soft Computing

pp 1–21

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

• Manoranjan De
• Barun Das
• Manoranjan Maiti
Methodologies and Application

## Abstract

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.

## Keywords

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

## Abbreviations

ACEC

Average carbon emission cost

ACER

Average carbon emission reward

ATP

Average total profit

CE

Carbon emission

CEC

Carbon emission cost

CER

Carbon emission reward

EOQ

Economic order quantity

EPL

Economic production lot size

EPQ

Economic production quantity

FDE

Fuzzy differential equation

FRI

Fuzzy Riemann integration

IFN

Intuitionistic fuzzy number

IFOT

Intuitionistic fuzzy optimization technique

IFS

Intuitionistic fuzzy set

MOOP

Multi-objective optimization problem

MOP

Multi-objective problem

TFN

Triangular fuzzy number

UPC

Unit production cost

## Notes

### Acknowledgements

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

### 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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