In real-world inventory problems, the deterioration rate is usually treated as an uncontrolled factor. However, by adopting certain preservation techniques, such deterioration rate can be controlled up to a desired level. Moreover, in certain business affairs, the supplier also provides a permissible delay in payment to encourage the vendors to have further sales. In this paper, we develop an inventory model that considers the power pattern demand under trade credit, permissible completely backlogged shortages, and suitable investment in preservation technology. Additionally, to make the model more realistic, we apply the learning effect to the holding costs. We develop the mathematical model of the problem and its solving policy in both crisp and fuzzy environments. The main purpose of the model is to obtain the optimum preservation technology-based cost and cycle time that maximize the overall profit. Furthermore, to validate our findings, we consider numerical examples and subsequently demonstrate the concavity of the profit function via Mathematica 11.1.1 software. Finally, we study the sensitivity analysis and accordingly present several managerial insights for the benefit of inventory managers.

Recently, the notion of positive linear operators by means of basic (orq-) Lagrange polynomials and 𝒜 {\mathcal{A}} -statistical convergence was introduced and studied in [M. Mursaleen, A. Khan, H. M. Srivastava and K. S. Nisar, Operators constructed by means ofq-Lagrange polynomials andA-statistical approximation, Appl. Math. Comput. 219 2013, 12, 6911–6918]. In our present investigation, we introduce a certain deferred weighted 𝒜 {\mathcal{A}} -statistical convergence in order to establish some Korovkin-type approximation theorems associated with the functions 1,tand t 2 {t^{2}} defined on a Banach space C ⁢ [ 0 , 1 ] {C[0,1]} for a sequence of (presumably new) positive linear operators based upon ( p , q ) {(p,q)} -Lagrange polynomials. Furthermore, we investigate the deferred weighted 𝒜 {\mathcal{A}} -statistical rates for the same set of functions with the help of the modulus of continuity and the elements of the Lipschitz class. We also consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.

The concept of the deferred Nörlund equi-statistical convergence was introduced and studied by Srivastava et al. [Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM) 112 (2018), 1487–1501]. In the present paper, we have studied the notion of the deferred Nörlund statistical convergence and the statistical deferred Nörlund summability for sequences of real numbers defined over a Banach space. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of real numbers on a Banach space and demonstrated that our theorem effectively extends and improves most of the earlier existing results (in classical and statistical versions). Finally, we have presented an example involving the generalized Meyer–König and Zeller operators of a real sequence demonstrating that our theorem is a stronger approach than its classical and statistical versions.

The concept of statistically deferred-weighted summability was recently studied by Srivastava et al. (Math. Methods Appl. Sci. 41 (2018), 671–683). The present work is concerned with the deferred-weighted summability mean in various aspects defined over a modular space associated with a generalized double sequence of functions. In fact, herein we introduce the idea of relatively modular deferred-weighted statistical convergence and statistically as well as relatively modular deferred-weighted summability for a double sequence of functions. With these concepts and notions in view, we establish a theorem presenting a connection between them. Moreover, based upon our methods, we prove an approximation theorem of the Korovkin type for a double sequence of functions on a modular space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results. Finally, an illustrative example is provided here by the generalized bivariate Bernstein–Kantorovich operators of double sequences of functions in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.