AN ESTIMATION OF EFFECTS OF MEMORY AND LEARNING EXPERIENCE ON THE EOQ MODEL WITH PRICE DEPENDENT DEMAND

. In this article, an economic order quantity model has been studied in view of joint impacts of the memory and learning due to experiences on the decision-making process where demand is considered as price dependant function. The senses of memory and experience-based learning are accounted by the fractional calculus and dense fuzzy lock set respectively. Here, the physical scenario is mathematically captured and presented in terms of fuzzy fractional differential equation. The 𝛼 -cut defuzzification technique is used for dealing with the crisp representative of the objective function. The main credit of this article is the introduction of a smart decision-making technique incorporating some advanced components like memory, self-learning and scopes for alternative decisions to be accessed simultaneously. Besides the dynamics of the EOQ model under uncertainty is described in terms of fuzzy fractional differential equation which directs toward a novel approach for dealing with the lot-sizing problem. From the comparison of the numerical results of different scenarios (as particular cases of the proposed model), it is perceived that strong memory and learning experiences with appropriate keys in the hand of the decision maker can boost up the profitability of the retailing process.

and constant demand with no shortage. Gradually, the literature of the lot sizing modelling was improved and matured incorporating more reliable assumptions serving different purposes [24,46]. In reality, the demand and pricing are a very crucial factor in any retailing business model. The demand rate may vary with respect to time [19,61], stock level [21], selling price [37], various dealing policy and so many other components in different supply situations. In a developing country, generally, the customers pay attentions towards a low-priced product even sacrificing over the quality in terms of durability and reliability [34][35][36]. So, the demand rate of product in these circumstances can be viewed as a negatively proportional to the selling price. In general, the demand pattern is imprecise in nature. So, the optimal business planning forecasting the demand is matter of great challenge for the manager. If an organization or an individual decision-maker is enriched with past memory of dealing with the similar type of circumstance, the overall present dealing management can be more effective in the fulfilment of its objectivity. Again, the learning through the experiences of the repeated tasks can make clear perceptions of the decision maker on the demand pattern and the measure of vagueness over the demand pattern can be reduced.
On the basis of above-mentioned intuitions, the following research questionnaires motivate the authors for introducing the theory in this present article: (i) What will be the optimal dealing strategy for an economic order quantity (EOQ) model motivated retailing scenario with price dependent imprecise demand pattern? (ii) How does the vagueness regarding the demand pattern create hindrances on the decision-making policy? (iii) How much the experience gained through doing repeated tasks can help the decision-maker for installing a better policy to attain the goal? (iv) How much the memory of the system (retailing organization) can create impacts in making the optimal strategy in favour of the profit maximization objectivity? (v) What will be the simultaneous effects of the memory and experience-based learning on the EOQ model? (vi) If the decision maker has keys in his/her hand, how much the keys can regulate the retailing process towards the profit maximization objectivity?
Focusing on the mentioned research questions, the above-mentioned intuitions on the decision-making phenomena are mathematically modelled and established in this present paper through the study of an EOQ model with price sensitive demand rate under the fuzzy fractional differential equation of Caputo type and triangular dense fuzzy lock set (TDFLS) decision making setup. The next section provides the details on the theoretical background of the present article.

Recent advancement on inventory modelling
As it is mentioned earlier that the selling price has very significant impacts on the demand pattern in a business scenario, following the footstep of Kim et al. [37], Polatoglu and Sahin [51] also considered price dependent demand in one of their studies. Mukhopadhyay et al. [45] include the deterioration of items beside the price dependency of the demand rate in the list of the assumptions. In another major finding on the consideration of theprice dependent demand, Pal et al. [47] discussed the problem with no shortage. The investigation for an EOQ model with price depended demand and fully backlogged shortage was credited by Sana [58]. In this context, the time varying holding cost and discount policy in inventory model with price dependent demand had been implemented by Alfares and Ghaithan [6]. Mishra et al. [42] considered an inventory model with stock and price dependent demand rate, controllable rate of deterioration under preservation technology. Mashud et al. [40] contributed a worthy investigation on an inventory control problem with stock and price dependent demand rate, deterioration and partial backlogged shortage. An inventory control management strategy for the perishable items was developed by Feng et al. [20] assuming the dependency of the demand rate on price, displayed stock, freshness of items simultaneously. On the other hand, Hendalianpour [26] developed an inventory control problem with price dependent demand in game theoretic approach using the double interval grey number for tracing the consumer behavior more accurately. The study of the distribution management of blood products (one kind of perishable items) was mathematically modeled under transshipment and uncertain demand consideration and optimized through heuristic algorithm by Liu et al. [38]. The potential of different contracts (like wholesale-price, revenue-share, quantity discount etc.) under the competitive structures of two manufactures and two retailers with price sensitive stochastic grey demand was examined by Hendalianpour et al. [27]. Also, Dey et al. [17] considered the discrete set up cost for an integrated inventory model with price dependent demand.

Fuzzy lot-sizing models
In the real-world decision-making system, a decision maker must have to tackle the ambiguities involved in the parameters which have great impacts on the decision. To define, describe and measure the real-world uncertainty/ambiguity, fuzzy concept is regarded as one of the finest tools. Zadeh [63] was the first to introduce the concept of fuzzy uncertainty. This novel concept was incorporated for decision making problem by Bellman and Zadeh [9]. Park [49] was the pioneer to manifest the economic order quantity (EOQ) model under fuzzy uncertainty. Consequently, some worthy findings on the fuzzy inventory models were contributed by the researchers like Vujosevic et al. [62] and Hojati [29]. Chen et al. [13] was the first to build a fuzzy economic order quantity model with back order. Following their contribution, the advancement and enrichment of the literature of the fuzzy inventory model has been done in the recent decades. A very detailed and versatile review on the inventory model under fuzzy uncertainty was contributed by Shekarian et al. [60] in this regard. On the other hand, fuzzy arithmetic was quite different from the crisp counterpart. So, the calculus of fuzzy variables and fuzzy valued functions were developed separately. Chang and Zadeh [12] gave an insight of fuzzy mapping. Later, Kaleva [30] introduced the concept of fuzzy differential equation. Then, lots of improvements and advancements [4,7,8,11,43] were done in this direction. While discussing the inventory problems under fuzzy uncertainty, it is expected to use the fuzzy differential equation in order to describe the uncertain dynamics of the model. Few papers [14,22,23,39,44] used fuzzy differential equation approach to interpret the inventory control problems under fuzzy uncertainty.

Learning and memory-based decision making of inventory model
Experiences and past memory are two major effective components for self-learning and advancement of human knowledge. So, memory from the past experience may help the decision maker to accurate the dealing strategy. Also, earning the experiences through executing the repeated practice of similar kind of job may make the decision maker more mature in his/her field. So, these two features can be implemented in the study of the inventory control problems to develop more optimal business planning in favor of its objectivity. In this context, one of the closest physical interpretations of the fractional calculus is assumed that it represents the memory of a system through its iterative kernels. The theory of fractional calculus [1,18,41,50] gained attention of the scholar engaging on the exploration of truth in different research directions of science and technology. Nature and emotion-based dynamic cannot be described aptly by the classical integer order calculus. Thus, the theory of fractional differential equation has been implemented to estimate the effect of memory on inventory control problems [48,52,53]. In addition, if the function is fuzzy valued, then it called the fuzzy fractional differential equation. The fractional differential equation was first discussed in uncertain phenomena by Agarwal et al. [2]. Then, the theory of fuzzy Laplace transformation and fuzzy fractional differential equation in the sense of Riemann-Liouville and Caputo had been developed gradually by several novel investigations [3,5,28,56,57]. Very recently, the production phase of an EPQ model has been analysed by the fuzzy fractional differential equation under the Riemann-Liouville derivative by Rahaman et al. [54]. The theories of learning were implemented on the study of inventory control problems in different ways. In this context, Kazemi et al. [33] incorporated the sense of human learning to study an economic order quantity (EOQ) model with back order under fuzzy uncertainty. Also, Shekarian et al. [59] considered the fuzzy uncertainty and learning to discuss an EOQ model for the items of imperfect quality. Bousdekis et al. [10] developed sensor driven learning approach for the perspective analysis of time dependent parameters that leads to self-optimization through learning and feedback. An alternative intuition for the fuzzy learning-based decision making was established through the introduction of the triangular dense fuzzy set (TDFS) by De and Beg [16]. The concept of experience-based learning was further enriched with the key and lock facility in terms of the triangular lock fuzzy dense set (TLFDS) by De [15]. Consequently, the concept of the TDFS and TLFDS were aptly utilized to explore the environments favoured sustainable decision making of inventory control problems [31,32] Based on the existing literature related to our above-mentioned research questionaries, the following research gaps have been identified: (i) The past memory very frequently affects the present decision-making process of an organization in present time. Optimal strategy for the present dealing situation can be adapted from previous experiences. Thus, the study on the memory concerned decision making of the inventory management problems [48,52,53] is very new trend and many things can be explored in this arena. For example, memory effect with uncertainty can turn the decision-making process more interesting and reliable. (ii) The mathematical quantification of experience-based learning doing repeated tasks in terms of dense and lock fuzzy sets [15,16,31,32] provided a very effective alternative to analyse the uncertain decision-making situation under self-optimization. The combined effects of the memory and experiences-based learning may be more reliable scenario in reality. In the existing literature, we have not noticed much work (except the article [55]) where the memory and learning has been considered simultaneously to describe an EOQ model. (iii) Fuzzy numbers and variables are being run according to their own arithmetic rule and calculus that are different from its crisp counterpart. Thus, it is very much desirable to describe the dynamics of a fuzzy inventory model through the analysis of fuzzy differential equations. The literature of fuzzy inventory models expressed by the fuzzy differential equations [14,22,23,39,44] is a little bit inferior in the whole literature of fuzzy inventory control problems. Memory effected inventory models are seen to be described by the fractional differential equation. Thus, fuzzy fractional differential equations are expected to be used for dealing the memory concerned model under fuzzy uncertainty. In this context, we identified an article concerning the application of the differential equation under Riemann-Liouville derivative on an EPQ model [54]. However, the definition of initial conditions associated with the Riemann-Liouville fractional differential equations is little abstract in terms of the physical significance. Thus, the fuzzy fractional differential equation can be better alternative to fill the gap up. (iv) The fuzzy fractional differential equations have an enriched literature in the theoretical point of view.
The applications of the notions of fuzzy fractional differential equations are rarely discussed (at least in the operation research domain). So, there is a gap to explore the meaning of fuzzy fractional differential equation with respect to the decision-making scenario of lot sizing problems.

Present study as a junction of different research directions
In this article, an EOQ model of price dependent demand rate is developed and analysed incorporating impacts of the past memory and learning due to experiences on the decision-making process. Intuitively, the optimal dealing strategy for price sensitive business scenario is subjected to the learning and training facility of the decision maker(s). Mathematically, the present study connects different research disciplines according to the intuitive meanings. The sense of memory is adapted in the study in terms of fractional calculus and uncertainty with associated parameters and variables is represented by the fuzzy set up. We, therefore, utilized Caputo type fuzzy fractional differential equations to trace the dynamical behaviour of the system in the present study. The decreasing natures of the vagueness regarding the demand pattern as the time forwards (through repetitions of similar kind of the decision-making phenomena) are accounted by the theory of TDFLS. Thus, the introduction of present study may be viewed as a theoretical amalgamation of distinct research directions.

Novelty of the present study
The main goal of this paper is to consider the demand to be selling price dependent in a TDFLS set up. Price is a very crucial factor in marketing-retailer scenario. Low price of product can attract the customer attention. So, the demand rate can be boosted up by lowering the selling price. Thus, the demands were regarded as the functions of selling price in the existing literature. But, in reality the demand pattern is not precise to the retailer and thus the selling price may not be crisps. In this circumstance, the decision maker has to find the optimal retailing policy anticipating the demand pattern more precisely. Suppose, the selling price is primarily traced with the association of impreciseness and is given loosely as {price is not suitable for the expected profit, price is suitable for the expected price, price is suitable for the profit above the expectation}. The decision maker(s) had to take an optimal decision to choose a selling price which can meet an expected level of the profit at least. The whole discussion of this paper is established on this novel intuition of managerial phenomena. Secondly, the present study chooses the fuzzy fractional differential equation to describe the proposed EOQ model under uncertainty and memory sensitivity. To describe the dynamics of the fuzzy parameters carrying memory, the theory of the fuzzy fractional differential equation under Caputo derivative is utilized for the first time (according to the authors' knowledge) in the discussion of an inventory control problem. Thirdly, the proposed theory is represented in this paper as a more generalized phenomenon of interpretation over the other phenomena on the different combinations of crisps, fuzzy, dense fuzzy, memory and so many components as the decision-making features. Also, the superiority of the memory motivated inventory model under TDFLS learning consideration over the models discussed as the particular cases is established here.

Different notations and assumptions
The following notations are used to denote the parameters, decision variables and objective functions to develop and optimize the model:︀ : Fuzzy demand (units)︀ : Fuzzy selling price ($/units)︀ ( ): Fuzzy inventory level at timẽ︀ : Fuzzy lot size (units) defuzz : Defuzzified lot size (units) (decision variable) : Total time cycle (month) (decision variable) : Differential memory index (decision variables) : Integral memory index (decision variables) ℎ: Holding cost (per unit per unit time) : Ordering cost (per complete cycle) TAP , : Total average profit (in fuzzy) TAP defuzz : Total average profit (after defuzzification) (objective function) The proposed EOQ model is developed based on the following assumptions:

Formulation of the model
The classical EOQ model under uncertainty can be described by the fuzzy differential equation with the terminal conditions as follows: The equation (4.1) represents a memory free uncertain economic order quantity model. The sense of memory can be added in this regard by reconstructing the mathematical model with the help of the following differ-integral equations: In the equation (4.2), ( , ) is an iterative kernel carrying the notion of memory. If the kernel ( , ) is chosen to be ( − ) −2 Γ( −1) , then the equation (4.2) can be rewritten as:︀ Thus, incorporating the sense of memory in the fuzzy EOQ model given by the equation (4.1), the memory motivated uncertain scenario can be depicted by the following fuzzy fractional differential equation under Caputo differentiability along with the terminal conditions: The fuzzy demand is taken as a function of the fuzzy selling price in the assumptions for developing the proposed model. So, putting̃︀ = −̃︀ (where the selling pricẽ︀ is a fuzzy number), the system represented by the equation (4.4) can be rewritten as: (4.5)

Existence, uniqueness of the solution
Now, one of the major concerns about the system given by the equation (4.5) is existence and uniqueness of its solution. Here, the theory provided by the Theorem 5.1 of [54] is imitated to discuss the existence and uniqueness criteria of the solution of the system given by the equation (4.5). The following notes about the equation (4.5) can be observed: Then, the Theorem 5.1 of [54] provides the guarantee of the existence and uniqueness of the solution of the fuzzy fractional differential equation (4.5). Thus, the system represented by the equation (4.5) can produce a unique solution, provided the demand to have a bounded value. Then, the components of the parametric representation of the demand ratẽ︀( ) are derived as:

Analytical solution
Taking fuzzy Laplace transformation [3,56,57] of the equation (4.5): Here, two different cases are considered based on the fuzzy fractional Caputo differentiability [5] of the inventory level functioñ︀( ).
Then, from the equation (4.7), the following equation is derived: The parametric representation of the equation (4.8) can be written as: Taking the inverse Laplace transformation of the system represented by the equation (4.9), the components of the parametric representation of the fuzzy inventory level are given by: . (4.10) Using the initial conditions in the equation (4.10), the components of the parametric representation of the fuzzy lot size are given by: . (4.11) Then, ultimately the equation (4.10) takes the form: . and Total earned revenue: and So, total average profit is given by: And therefore, the components of the parametric representation of TAP , are the follows: Then, proceeding in the similar path as of case I (details are given in Appendix A), the components of the parametric representation of the total average profit (TAP , ) and lot size and, . (4.20)

TDFLS decision making
In this subsection, the triangular dense fuzzy lock set approach is incorporated to make a proper decision in the uncertain environment using the learning experience and memory. Here, the unit selling price is assumed to be triangular dense fuzzy lock number [15] with two keys and the other involved parameters are assumed to be constants.
Let, the selling price, a triangular lock fuzzy dense number is given by:  .22) and For the case of [(1) − ] differentiability of̃︀( ), the defuzzified value of the total average profit is obtained using the -cut defuzzification approach as follows (details are given in Appendix B): Also, the defuzzified value of the lot size in the case of [(1) − ] differentiability of̃︀( ) will be: For the case of [(2) − ] differentiability of̃︀( ), the defuzzified value of the total average profit is obtained using the -cut defuzzification approach as follows (details are given in Appendix B): Also, defuzz1 = defuzz2 . Interestingly, there is no distinction between two cases regarding fuzzy fractional differentiability of̃︀( ) after the defuzzifications in the above mentioned technique. So, without the loss of generality, defuzz1 and TAP defuzz1 are replaced by the notations defuzz and TAP defuzz respectively.
So, the optimization problem will be

Special cases for optimization of the problems
In this section, some popular models are seen as particular cases of the proposed inventory model under memory and learning based study with double keys in the hand of the decision maker. The approach of deduction may be specified through the values of the memory index or the values of the involved parameters to present the degree of fuzziness. The categorization of the proposed problem is based on the memory index which corresponds to the main problem itself and its integer order version (stands for memory free situation). For both the fractional and integer order model, more categorizations are carried out on the basis of the values of keys and the variance of the parameters of dense lock fuzzy system.

Fractional order sub problems
This is the case of fractional model in the lock fuzzy dense environment with single key.

When
This is the case of fractional model in the fuzzy dense environment.

When
This is the case of fractional in the general fuzzy environment.

When
This is the case of fractional in the crisp environment.

When
This is the case of integer order model in the lock fuzzy dense environment with single key.

When
This is the case of integer order model in the fuzzy dense environment.

When
This is the case of integer order model in the general fuzzy environment.

When
This is the case of integer order model in the crisp environment.

Graphical depiction of the sense of generalization
The senses of generalization discussed in the Sections 5.1 and 5.2 are presented graphically by Figures 1  and 2. Figure 1 represents the intuitions for the generalization of a crisp model (may be integer or fractional order) to complicated model under dense fuzzy lock system. On the other hand, Figure 2 explores the fact that every integer order model is a particular case of the corresponding fractional order model (better to say the arbitrary order model).

Numerical exploration
In this section, some numerical problems are analysed to study the effect of memory and learning experiencebased decision making in an inventory control problem with the selling price dependent demand rate. The following algorithm developed in the first subsection of the present section is used to measure the impact of memory and learning experiences.

Solution algorithm
Step 1. Input the value of the parameters , ℎ, and constant , .
Step 13. Do fuzzification of the selected crisp model. Input and .
Step 14. Calculate   Step 15. Apply dense fuzzy environment on the selected crisp model. Input and .
Step 17. Apply dense fuzzy lock (single key) environment on the selected crisp model. Input , and .
Calculate   Then, the optimum values of the objective function and the decision variables in the classical EOQ model are obtained as TAP defuzz = 4106.55; * defuzz = 387.298 and * = 2.066. Now, the impacts of memory on the classical EOQ model are examined letting the order of differentiation and integration to be constants. The Tables 1 and 2 represent the memory sensitivity individually with respect to the differential and integral memory index respectively. One memory index is fixed to the value 1 while measuring the impact of another memory index. The Table 3 shows the memory sensitivity of the result varying both memory indexes simultaneously.

Memory sensitivity in crisp system
The Figures 3, 4 and 5 present graphically the variance of the total average profit, lot size and total cycle time, respectively with respect to the memory indexes.
From the Figure 1, it is seen that as the memory index decreases, the graph of the optimal values of total average profit maintains a strictly increasing pattern. However, the influence of the integral memory index on the total average profit is far superior in comparison to the differential memory index. From the Figure 2, it is seen that as the differential memory index decreases, the graph of the optimal lot size increases initially and then ultimately decreases with a bell-shaped curve. However, for the case of integral memory index as well as the case of combined effects, the graphs maintain steady decreasing patterns. Also, in the Figure 3, it is noticed that the graph of the total cycle time follows an increasing pattern with respect to the decreasing trends of the differential memory index. Again, the scenarios for the case of the integral memory index and the combined case are totally reversed.

Decision making under uncertainty
Apart from the memory less case (corresponding to = = 1), we select three different fractional cases on the basis of maximization objective of total average profit subject to the condition of feasible values of total time cycle and lot size from the analysis in the Section 6.2. The performance of the selected pairs of memory index in crisp and different fuzzy situations are displayed in the Table 4. Together with the non-fuzzy parameters, described in the Section 6.2, the below mentioned values of the fuzzy parameters are considered here: The bar diagram with fifty-six bars in the Figure 6 is regarded as a graphical correspondence of the results displayed by the Table 4.
The most preferable solution in the sense of profit maximization is therefore given by TAP * defuzz = 4836.45,

Sensitivity analysis
The sensitivity of the most preferable solution with respect to the non-fuzzy parameters is discussed in this subsection. Changing the values of ordering cost ( ), the unit holding cost (ℎ), unit selling price ( ) from −50% to +50%, the sensitivity of the optimal solution is structured in the tabular and graphical forms by the Table 5 and the Figure 7 respectively. The graph of total average profit preserves the expected facts that it is an increasing function with respect to and is a decreasing function of and ℎ. Moreover, the total average profit is highly sensitive with respect to the parameter compared to the parameters and ℎ. The sensitivity patterns are just reversed for the cases of the lot size and the total cycle time.

Discussion and managerial insight
In this subsection, we have enlisted our observations that are obtained from the numerical analysis and enlighten the managerial strategies.   Figure 3 of Section 6.2, we may conclude that stronger system's memory always favours towards the profit maximization goal. Thirdly, we get the following results on the optimality of the total average profit incorporating different kinds of uncertainties, learning experience and key facilities of dense fuzzy lock situations as described in Table 4: (i) Ordering with respect to memory index: ( = = 1) < ( = 1, = 0.9) < ( = 0.9, = 0.9) < ( = 0.4, = 1). (ii) Ordering with respect to degree of uncertainty: Crisp< Dense fuzzy < General fuzzy < Lock fuzzy dense (single key) < Lock fuzzy dense (double keys). (iii) Ordering with respect to degree of experiences: (iv) Ordering with respect to number of keys: Single key < Double keys.
From the above results, it appears that the stronger memory and TDFLS fuzzy set up with double keys facilities is the most desirable and generalized decision-making methodology among the different discussed phenomena.

Managerial insights
It may be noted that in general, the demand rate of any product in the retailing process is not deterministic at all; rather lots of uncertainties are involved with it. Moreover, pricing is also a very sensitive decision to control demand pattern. In a developing country like India, common people pay concerns for the low-priced product even sacrificing the durability and quality of a product. In these circumstances, the demand can be viewed as a linear function of selling price. Now, to trace the demand pattern which is not precise to the retailer, the adjustment of the pricing will be a vague phenomenon. So, fuzzy decision-making technique is more desirable in this context. From the numerical simulation, it is perceived that the fuzzy decision making can favour the goal of the retailer. However, the retailer may have memory from the past dealing procedures. If there is any memory experience of the retailer, it can help to make a more accurate dealing policy to gain customer's faith which leads towards the ultimate gain of the organization. It is one of the major outcomes from the discussion of this article that stronger memory always helps to attain the highest gain of the retailer. However, the stronger memory sometimes may boost up the profit through lowering the values of the lot size and total cycle time in a very alarming measure which may not be cases of feasible phenomena in the practical sense. A smart decision maker must consider the system with moderate sense of memory for the feasibility of his/her decision in the real context. Another way to go for the accuracy in the optimal retailing policy is that the decision makers have to go through the self-learning procedure doing repeated deals in a particular retailing cycle. The demand pattern can be more precise to the decision-maker due to regular repetitions of the tasks.
The numerical simulation in this study do explore the fact that the dense fuzzy frame carrying the sense of learning can give better result compared to the crisp and general fuzzy sense. The decision-making methodology can be further improved incorporating the sense of key and lock in the dense fuzzy decision-making process. The decision makers have better control on the system in terms of execution of desirable conditions when different key features are available in the system. In this study, we have established that smart decisions (through the selection of suitable keys) of the experts can increase the profitability of the organization and several key options may contribute for the more accuracy on the decisions which also helps to the profit maximization. Actually, the mathematical concept of double keys represents two available alternatives of business strategies to be executed. Suppose, there are two experts engaged to make the decisions of their own. That is, two keys are available in the hand of the retailing organization to solve the puzzle due to impreciseness about the decision making for the optimal pricing to reach the maximum profit. Thus, optimal choice of the pricing through memory, learning experiences and key-lock technique can fulfil the retailer's objective of the profit maximization.

Conclusion
In this article, an EOQ model with price dependent demand is developed in a memory and learning motivated uncertain situation. Here, the fuzzy fractional differential equations of Caputo types are used to describe the proposed physical scenario. Also, the TDFLS decision making setup is utilized here to incorporate the experiences-based learning concept. The TDFLS decision making mechanism is designed for the optimal pricing which favours for increasing the total profit crystallizing the demand pattern. From the very discussion, it is perceived that memory has a very positive impact on the profit maximization goal of the retailer. The second observation is that through more learning experience, the decision will be matured and thus it will lead the system towards more profits. The keys facility in the TDFLS decision making mechanism also boosts up the total profit regulating the uncertain parameters through smart decisions. Besides the mentioned intuitions and managerial insights, this present article can be viewed as an initiative for finding the applications of fuzzy fractional calculus in the domain of the operation research. Though the present article is engaging in the exploration of a very interesting and effective managerial insight, we also acknowledge the limitation of the present study. The modelling and the decision-making components in this article are very well connected with the real business scenarios. However, the numerical simulation is carried out on the hypothetical data. In future, this drawback can be overcome by collecting raw data from real world business sector to validate the proposed methodology of this present article. Searching for the analytical solution approach replacing the numerical simulation will be future challenges in this research direction.
Appendix A.