Reliability Dependent Imperfect Production Inventory Optimal Control Fractional Order Model for Uncertain Environment Under Granular Differentiability

ABSTRACT

In this paper, we present a general formulation for the production inventory optimal control problem to a class of fractional fuzzy differential systems with reliability-dependent imperfect production. we investigate this model under the uncertain environment of fuzzy numbers with the rate of change of stock level expressed by granular Caputo fractional fuzzy derivative of order $\text{β}\in [0,1]$. The demands depend on the selling price, product quality, and stock level of the product displayed in the store. The unit production cost depends on development cost due to reliability, raw material cost, and wear tear cost. Here state, co-state variables and parameters are taken as uncertain variables with triangular fuzzy numbers and using granular dierentiability the existence and uniqueness of solution to the fractional dierential system with fuzzy variables and parameters. The production inventory optimal control problem for the fractional system is proposed and a necessary condition for optimality is obtained. Finally, this optimal control problem was solved by using the Pontryagin Maximum principle and for numerical results and graphical representation.

KEYWORDS:

• Reliability
• fuzzy dynamical system
• fuzzy granular differentiability
• fractional dynamic system
• production inventory

Ethical approval

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

Informed consent

Informed consent was obtained from all individual participants included in the study.

Disclosure statement

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

Additional information

Funding

The authors thank to Department of Science & Technology and Biotechnology, Govt. of West Bengal [475(Sanc.)/ST/P/S & T/16G-31/2018 dated 15.03.2019] for financial help.

Notes on contributors

S. Hati

S. Hati is a Teacher in Department of Mathematics in Mugberia Gangadhar Mahavidyalaya, West Bengal, India. He qualified GATE in the year 2019. His research interests are in Inventory and optimal control of production systems in fuzzy, uncertain environments, fuzzy differential equations and fuzzy dynamical system.

K. Maity

K. Maity is an Associate Professor in Department of Mathematics in Mugberia Gangadhar Mahavidyalaya, West Bengal, India. He has received Prof. M.N.Gopalan Award for best doctoral thesis in Operational Research organized by ORSI, India, 2006. Dr. Maity has published at least 60 papers in International journals like IJOR, EJOR, AMC, AMM, MCM, Inf. Science, FODM, Opt. and Eng., Int. J. of System Science and other journal. His research led is inventory management, operation research, uncertain theory, fuzzy-rough optimization.