Theoretical developments and application of variational principle in a production inventory problem with interval uncertainty

Abstract

The goal of this work is two folded: (i) theoretical development of optimality condition of a variational problem with interval uncertainty; (ii) application of the established results in a production inventory model with interval uncertainty. To serve the purpose of first fold, the definitions of optimizers of an interval valued variational problem (IVVP) are proposed by using interval order relations. Then both the necessary and sufficient optimality conditions of an IVVP with single as well as several variables are derived. In this derivation, the obtained necessary conditions are named as c-r Euler equations and sufficient conditions are named c-r as Legendre conditions. To achieve the purpose of second fold, a production inventory model is formulated by considering interval valued time dependent production and demand rates. Then the corresponding IVVP of the proposed model is obtained and its optimal policy is studied with the help of established c-r Euler’s equations and c-r Legendre conditions. Finally, to illustrate the optimal policy of the proposed model, a numerical example is considered and optimality of its obtained results are verified graphically.

KEYWORDS:

• IVVP
• optimizer of IVVP
• c-r Euler equations
• c-r Legendre conditions
• inventory under interval uncertainty
• interval valued production rate

Acknowledgements

All the authors of this manuscript would like to convey their sincere gratitude to the Editor and anonymous Reviewers for their constructive comments and suggestions to enrich the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The authors declared that they did not use any publish data inside this paper.

Additional information

Funding

This work was supported by Department of Science and Technology, Government of India [Grant Number (SR/FST/MSII/2017/10 (C))]; University Grants Commission [Grant Number File no. 16-6(DEC.2018)/2019(NET/CSIR)].

Notes on contributors

Subhajit Das

Subhajit Das received MSc degree in Mathematics from The University of Burdwan in India. He joined the Department of Mathematics at The University of Burdwan in 2019 as a Junior Research Fellow. His research interests include optimal control theory, industrial engineering, and interval mathematics. He has published six research articles in different reputed journals including Journal of the Franklin Institute, Soft Computing, Alexandria Engineering Journal etc.

Md Sadikur Rahman

Md Sadikur Rahman is working as a junior research fellow at the department of Mathematics, The University of Burdwan, India. He completed his MSc degree in Mathematics in the year of 2017 from the University of Kalyani, India. He has twenty research articles published in different reputed journals including Applied Mathematical Modelling, Computers and Industrial Engineering, International Journal of System Science: Operations & Logistic, Knowledge based System etc. Presently, he is working on various branches of mathematics including Inventory control, Interval Analysis, Optimal Control, Numerical Analysis, Theory of Optimization etc.

Ali Akbar Shaikh

Ali Akbar Shaikh is an Assistant Professor of Mathematics at The University of Burdwan, West Bengal, India. Earlier, he was a Postdoctoral Fellow at the School of Engineering and Sciences of Tecnológico de Monterrey, México. He has obtained the award SNI of level 1 (out of 0–3) presented by the National System of Researchers of México from Government of México, in 2017. He obtained his PhD and MPhil in Mathematics from The University of Burdwan, and MSc in Applied Mathematics from University of Kalyani, India. He has published more than 90 research papers in different international journals of repute. His research interests include inventory control, interval optimisation, and particle swarm optimisation.

Asoke Kumar Bhunia

Asoke Kumar Bhunia is a Professor at the Department of Mathematics, The University of Burdwan, West Bengal, India. He obtained his PhD in Mathematics and MSc in Applied Mathematics from Vidyasagar University, India. His research interests include computational optimisation, soft computing, interval mathematics, and interval ranking. He has published over 140 research papers in various national and international journals of repute. He is a reviewer of several SCI journals. He has guided 13 PhD and two MPhil students. He is an author of four research monographs and six book chapters. He has written two text/reference level books in the area of optimization and Operations Research. He is an INSA Visiting Fellow and former Associate Editor of the Springer’s journal, OPSEARCH.

Ali Ahmadian

Ali Ahmadian is an adjunct professor at the Kean University- Wenzhou Campus, Affiliated Researcher at Kean University, USA and Associate Researcher at the Mediterranea University of Reggio Calabria, Italy and former a Fellow Researcher at the Institute of IR 4.0, The National University of Malaysia. As a young researcher, he is dedicated to research in applied mathematics. He received his PhD in early of 2014 as the best postgraduate student from Universiti Putra Malaysia (UPM). After his PhD he took a postdoctoral fellowship at same university as part of the numerical analysis research group and at the same time joint to the University of Malaya as an Associate Researcher. He was promoted as a Fellow Researcher on December 2017 in UPM. In 2019 he joined the National University of Malaysia as a senior lecturer where he currently works in the Institute of IR 4.0. In general, his primary mathematical focus is the development of computational methods and models for problems arising in computer science, biology, physics, and engineering under fuzzy and fractional calculus (FC); in this context, he has worked on projects related to nano-communication networks, drug delivery systems, acid hydrolysis in palm oil frond, and carbon nanotubes dynamics, nanofluids, viscosity, AI and etc. He could successfully receive 17 national and international research grants (Worth: $ 800,000). He is a member of editorial board in Journal of Control and Decision, Progress in Fractional Differentiation and Applications (Natural Sciences Publishing) and Mathematical Problems in Engineering (Hindawi) and lead guest editor of special issues in more than 50 international journals from Elsevier, Springer, Wiley and so on. He is an author of more than 190 research papers published in as Nature, IEEE, Elsevier, Springer, Wiley and etc. He also received Premio Anassilaos Giovani Award for his achievement in art, culture, economics and science from Italy on November 20221.