This chapter explores the possible application of fractional calculus in the field of operation research, more specifically inventory control problem. The sense of memory can be implemented in a dynamical system with the mathematical manipulation through fractional calculus. In this chapter, some recently published papers on generalized lot-sizing models described by fractional differential equation in crisp as well as uncertain environments are reviewed. The intuitional applicability, obstacles, and challenges for studying the inventory management problems under fractional differential equation (in Riemann-Liouville and Caputo approaches) are discussed in this chapter.
Top1. Introduction
The journey of fractional calculus started with the famous conversation between L’Hospital and Liebnitz. Liebnitz’s response to the question of L’Hospital on the existence of derivative of 
order was “an apparent paradox from which one day useful consequences will be drawn”. The theory of fractional calculus grew very slowly through the last the centuries. However, the worldwide fervourfor fractional calculus (FC) as well as fractional order system (FOS) has been apparently exponential in the most recent decades because of its exactness on portraying the dynamical nature associated with different physical procedures as a general rule. As of late, this idea has been prominently utilized for the demonstrating in the broad area of AppliedMathematics, Physical Science, Technology and Management replacing the Newtonian calculus (Agila et al., 2016; Agrawal et al., 2004; Kilbas et al., 2006; Machado & Mata, 2015; Miller & Ross, 1993; Podlubny, 1999). Differential equation is one of the frequently used mathematical tools to describe the variability of a dynamic state over time. If the integer order derivative is replaced by the fractional counterpart, the differential equation is called the fractional differential equation (FDE). In reality has replaced the differential equation of integer order proving its smartness for describing dynamical system in a complicated situation. Several Studies and findings (Abbasbandy, 2007; Arikoglu & Ozkol, 2009; Bhrawy et al., 2013; Duan et al., 2013; Hajipour et al., 2019; Mainardi et al., 2007) are carried out to establishing that fact which ultimately enriched the world of technology and innovation it’s applications. The physical meaning of fractional calculus seems to be little abstract. However, several studies proved the exactness of fractional calculus against the conventional integer order calculus on the fitting of real data from memory involved systems. Thus, interpretation of memory is regarded as one of the physical meaning of the fractional calculus. The Riemann-Liouville fractional integration and derivative and Caputo fractional derivative gained the popularity in this context. Here, in this present article, we fix our focuses on the exploration of the possible implementation of fractional differential equations on the inventory control problems in both crisp and uncertain environments. Thus, the literature review in the next section will be limited into some specific keywords, namely theory of fractional differential equations in uncertainty, crisp fractional inventory models and fuzzy fractional inventory models.
The rest of this article is decorated as the following: Brief literature reviewing about some particular keywords has been carried out in the section 2. Some basic theory of the fractional calculus is presented in the section 3. The section 4 is involved in the comparison of fractional and integer order system to describe the inventory models. The section 5 presents the comparisons between Riemann-Liouville and Caputo derivative to describe the inventory models with different assumptions. Several other obstacles and challenges are described in section 6. Finally, concluding remarks are made in the section 7.