An inventory model for partial backlogging items with memory effect

Inventory control is considered one of the most widely documented topics in the reality. Fractional derivatives and integration is the part of fractional calculus. Fractional calculus is the generalized part of ordinary calculus. The memory of physical phenomena is a highly concerning topic but it is neglected with describing in terms of integer-order differential equation. To discuss the memory of the inventory model, fractional derivative tools are considered. A fractional derivative at any point gives the previous marginal output and current point output for any current point input. In this model, a shortage is considered, and during the shortage period, demand is partially backlogged. Depending on the low partial backlogging rate and high partial backlogging rate, the result of the memory effect varies on the total average cost and optimal ordering interval. Moreover, in order to show the relationship between fractional models and ordinary classical model, two types of memory indices have been considered: (i) differential memory index and (ii) integral memory index. For a certain memory effect, minimized total average cost is the same for low partial backlogging rate and high partial backlogging rate.

ton suggested the current definition of derivative in 1666. The first-order, second-order, and more large-order derivatives of displacement are called velocity, acceleration, jerk, and jounce, respectively. The idea of fractional-order derivative was discovered by the suggestion of Gottfried Wilhelm Leibniz in 1695 (Miller and Ross 1993;Podlubny 1999;Kilbas et al. 2006). Fractional-order derivative and integration have a physical significance (Saeedian et al. 2017). The order of the fractional-order derivative (Du et al. 2013) and fractional-order integration is suggested as an index of memory (Pakhira et al. 2020). Because fractional-order derivative and fractional-order integration have the power to remember previous effects of input to determine the current value of output, such types of systems are called memory-affected systems. An inventory system is one of the perfect examples of memory-affected system. Fractional calculus has been applied to a different areas of science (Saeedian et al. 2017;Du et al. 2013). Different models using fractional derivative was formulated to study some real-life phenomena (Tarasova and Tarasov 2016;Tarasov and Tarasova 2016). In the literature, several authors have addressed the issues related to the memory effect in an EOQ model where partial backlogging is allowed during a shortage. Some of the important studies related to this topics are summarized in Table 1.  Saeedian et al. (2017) Memory effects on epidemic evolution Du et al. (2013) Measuring memory with the order of fractional derivative using experimental data Pakhira et al. (2020) Memory effects in an EOQ model using Mittag-Leffler function-type demand rate Tarasova and Tarasov (2016) Memory effects in hereditary Keynesian model Tarasov and Tarasova (2016) well-known square root formulae for the constant demand rate EOQ model. In the inventory system, the prominent factors are demand and replenishment of items. Donaldson (1977) extended to the linear time-dependent demand model analytically with a finite time horizon. Authors such as Datta and Pal (1992), Dave (1989), and Goswami and Chaudhuri (1991) developed deterministic lot size inventory models with shortages and a linear trend in demand. These models are developed using ordinary calculus. But, we intend to develop an inventory model using fractional calculus. Apart from this, the theoretical development of fractional calculus was done by different researchers like Miller and Ross (1993), Podlubny (1999), Caputo (1967), and Ghosh et al. (2015). Application of memory effect using fractional calculus was done by Saeedian et al. (2017) in the biological model. Tarasova and Tarasov (2016) and Tarasov and Tarasova (2016) has developed many research articles using the concept of the memory effect. Memory-dependent inventory models have been developed by Pakhira et al. (2021a), Pakhira et al. (2021b), and Pakhira et al. (2022). The literature related to the fractional-order inventory model is listed in Table 2.
To modify the inventory model, (Choudhury et al. 2022) and (Taleizadeh et al. 2020) gave their attention using ordinary calculus and get important results on an integrated inventory model. Partial backwarding inventory model with limited storage capacity under ordersize-dependent trade credit has been consider in Jiang et al. (2022). Production inventory models and more generalized version inventory models using ordinary calculus have been developed by the authors (San-Jose et al. 2022;De and Mahata 2020;Khan et al. 2021;Canyakmaz et al. 2019;Taleizadeh et al. 2022). The stochastic inventory model is considered by Taleizadeh et al. (2021), Taleizadeh et al. (2020. Multi-product single-machine economic production has been discussed in Taleizadeh et al. (2020). Replenishment of imperfect items are considered in Taleizadeh et al. (2020). Lashgari et al. (2018), Taleizadeh (2018), Diabat et al. (2017), Aslani et al. (2017), Salehi et al. (2016), andTaleizadeh et al. (2015) improved different models with modified EOQ technique.

Significance of the fractional calculus
Fractional-order derivative for its memory kernel function can incorporate memory effect. Order of the fractional derivative and fractional integration is index of memory. It can collect the past experience. The key merit of fractional calculus is that it differs from integer-order calculus from the view of memory effect. The derivative and integration of the non-integer order is the suitable tool for the description of memory and hereditary properties. To the best of our knowledge, it is known that the fractional derivative could give nonlinear (power law) approximation of the local behavior of non-differentiable functions (Du et al. 2013).To describe the memory and hereditary properties in financial systems, an excellent tool is the fractional calculus (Pakhira et al. 2020;Tarasova and Tarasov 2016;Tarasov and Tarasova 2016).

Contribution of the proposed model and importance of the proposed model in real EOQ model
In classical inventory models, the differential equation is considered the first-order differential equation, and all associated costs are evaluated by the integer-order integration. But, here, we want to consider the fractional-order differential equation, and all associated costs are evaluated by the fractional-order integration. Two types of memory indices are established here (i) differential memory index (α) (ii) integral memory index (β). Different memory is considered for different range of memory index. Long memory is considered for α, β ∈ [0, 0.5] because α, β are considered order of the fractional derivative and integration. If α, β = 1, then we get particular output corresponding particular input. If α, β < 1, then small previous output includes to the current output. If α, β << 1, then many long output includes to the current output. Depending on this concept, we consider long memory, short memory, strong memory, and poor memory. Figure 3 shows the nature of changing of the minimized total average cost against differential memory index. From this figure, it is clear that the minimized total average cost of the two cases (a) (δ = 0.01), (b)(δ = 0.2) coincides at α = 0.4786695534. Thus, profit is equal for both low and high partial backlogging rate at α = 0.4786695534.
In this paper, we have made a memory-dependent inventory model where a time-dependent linear-type demand rate is assumed during the positive stock period. During the shortage period, all demand is either completely backlogged or partially backlogged. In reality, it is observed that most time, demand is partially backlogged. Here, shortages are allowed and partially backlogged at a constant rate. For bad experiences, profit loss, and for good experiences maximum time profit gains so the past experience has a good impact on the real market system. From the numerical example, it is clear that the system behaves qualitatively different in the presence of low as well as high partial backlogging rate. It is also observed from the numerical examples that in case of high partial backlogging rate, the business takes less time to reach the minimum value of the total average cost compared to the case of low partial backlogging rate. This is a real situation of the market. Critical value of the differential memory index (α) has been found for which the minimized total average cost coincides for the two different partial backlogging rate situations. Thus, for two different situations of the inventory system, inventory system gives the same output. Using sensitivity analysis, sensitive parameters are identified for both memory affected system as well as memoryless system, lost sales cost per unit time per unit item (C 4 ), ordering cost per order (C 3 ), and demand rate (γ ). Thus, to maintain the business properly, the business person should use the previous experience but the experience should be of short time scale but not for highly long time scale for this model.

Organization of the paper
The rest part of the paper has been arranged as follows: The model formulation is discussed in Sect. 2, the classical model is given in Sect. 2.3, the fractional-order inventory model is presented in Sect. 2.4, and the fractional-order model analysis is given in Sect. 2.5. Section 3 presents a numerical example to describe the applicability of the proposed work. Section 4 presents the graphical representation of the results. Finally, Sect. 5 concludes the study.

Model formulation
In this section, first, the classical inventory model will be formulated and then that will be fractionalized. Both classicaland fractional-order EOQ models are developed under the following assumptions.

Assumptions
The following are the assumptions taken in the formulation of the model.

Notations
The following notations are used in the entire paper.

Classical inventory model
The positive inventory level (I 1 (T )) reduces due to constant demand rate (γ ) during the time interval [0, t 1 ], whereas the negative inventory level (I 2 (T )) develops at the same demand rate (γ ) during the time interval [t 1 , T ]; partial backlogging in this time interval [t 1 , T ] is allowed; such a system can be governed by the two ordinary differential equations in the following form: where I 1 (t 1 ) = I 2 (t 1 ) = 0 and 0 < δ < 1 is the backlogging rate of the inventory system.

Fractional-order inventory model
To incorporate the influence of memory effect to the model (1) and (2), first, the fractional-order inventory model will be developed. Using the memory kernel function, the ordinary differential equations (1) and (2) become, Here, I 1 (t) is the positive inventory level at time t and I 2 (t) is the negative inventory level at time t where 0 < α ≤ 1 along with boundary conditions I 1 (t 1 ) = I 2 (t 1 ) = 0. These two fractional-order differential equations (3) and (4) along with boundary condition represents a fractional-order inventory model.

Fractional-order inventory model analysis
In this section, the solution of the proposed fractional-order inventory models (3) and (4) has been found in the form, where I 1 (t) is the memory-dependent positive inventory level at time t, and I 2 (t) is the memory-dependent negative inventory level at time t. The maximum positive inventory level M attained at t = 0 is and the maximum backorder units S attained at t = T is Using (7) and (8), the order size Q during [0, T ] becomes where Q is the total order quantity. M and S are getting from (7) and (8), respectively. Now, total average cost which is the average of the inventory holding cost, shortage cost, lost sales cost, will be evaluated to get the minimized total average cost of the business. Those individual costs for this memory-affected system are mentioned below: (i) In classical inventory model, the inventory holding cost is calculated by ordinary integration D −1 (I 1 (t)) for 0 ≤ inventory holding cost for the fractional inventory model denoted by HOC α,β (T ) for 0 ≤ t ≤ t 1 is where D −β for 0 ≤ t ≤ t 1 is the fractional integration of order β used in the Riemann-Liouville sense (β is considered as an integral memory index), where per unit inventory holding cost is assumed as C 1 t α . Here, I 1 (t) is the positive inventory level getting from Eq. (5). (ii) Shortage cost without memory effect using ordinary integration is whereas with memory effect, it is This cost is associated with the negative inventory level during the time interval t 1 ≤ t ≤ T .
where D −β for t 1 ≤ t ≤ T is the fractional integration of order β used in the Riemann-Liouville sense. Here, I 2 (t) is the negative inventory level getting from (6). To integrate the above, we can consider ( T and here C 2 is the shortage cost per unit per unit time. (iii) Lost sales cost without memory effect is whereas with memory effect, it is Lost sales cost with memory effect is denoted by LSC α,β and defined for t 1 ≤ t ≤ T as where D −β for t 1 ≤ t ≤ T is the fractional integration of order β used in the Riemann-Liouville sense where C 4 is the lost sales cost per unit time per unit item. Therefore, total average cost for the fractional-order inventory model is Here, C 3 is the ordering cost per order. HOC α,β (T ) is β th -order total inventory holding cost getting from (10), SC α,β (T ) is the total shortage cost with fractional effect getting from (11), LSC α,β is the total lost sales cost with fractional effect getting from (12) and + 1) , To analyze the impact of the parameters α and β on to the total average cost, we discuss their impact by considering (a) 0 < α ≤ 1.0, 0 < β ≤ 1.0 (b) β = 1.0, 0 < α ≤ 1.0 (c) α = 1.0, 0 < β ≤ 1.0 and cases. The impact for each case is discussed as below.
(a) Case 1 0 < α ≤ 1.0, 0 < β ≤ 1.0: In this case, the differential memory index α (rate of change of inventory level) and the integral memory index β are both fractional. To find the minimum value of the total average cost TOC av α,β (T ), the nonlinear programming is proposed. The proposed nonlinear programming problem is in the following form: + 1) ,

Problem description
The optimization problem (14) will then be solved by the following primal geometric programming method (Bhrawy et al. 2013). Let w i , i = 1, ...5, w i ≥ 0 be the dual variables, then the system (14) in terms of dual variable can be written as where normalized condition is, and the orthogonal condition is, The primal-dual relations can then be written as Now, from these primal-dual relations (18), the following relations have been obtained as: and Thus, we have a set of five nonlinear equations (16), (17), with five non-negative unknowns w 1 , w 2 , w 3 , w 4 , w 5 . Solution of (16), (17) and (19) gives the optimal values w * 1 , w * 2 , w * 3 , w * 4 , w * 5 of the dual variables w 1 , w 2 , w 3 , w 4 , w 5 , and finally, the optimum value T * α,β of T α,β can be obtained from equation (20). Next, the minimized total average cost TOC * α,β can be obtained by substituting T * α,β in (14). (b) Case 2: β = 1.0, 0 < α ≤ 1.0: In this case, the rate of change of the inventory level (differential memory index α) is fractional but the integral memory index β is unity. In such case, the nonlinear programming problem for finding the minimization of the total average cost TOC av α,1 (T ) is where A = C 2 δγ (α + 1)Γ (α + 1) , Using the parallel analogy as earlier, the minimized total average cost TOC * α,1 (T ) and the optimal ordering interval T * α,1 can be calculated from (21). (c) Case 3: α = 1.0, 0 < β ≤ 1.0: Here, the differential memory index is unity but the integral memory index β is fractional. In this case, the nonlinear programming problem (14) becomes as where Proceeding in the same way as in case 1 and case 2, the minimized total average cost TOC * 1,β (T ) and the optimal ordering interval T * 1,β can be evaluated from (22). The bold mark indicating the Maximum value of T * α,β or TOC * α,β

Numerical example
To illustrate numerical results of the proposed fractionalorder inventory model, empirical values of the various parameters have been considered in proper units as C 3 = 70, C 1 = 1.5, C 2 = 2.2, C 4 = 2.5, γ = 12, δ = 0.01, t 1 = 2.3456. The required results of the minimized total average cost and the optimal ordering interval have been illustrated using MATLAB minimization method, and the results are placed in Tables 3, 4. From Table 3, it is found that as the differential memory index α run from 0.1 to 1.0, the minimized total average cost(TOC * α,β ) increases, and the optimal ordering interval (T * α,β ) decreases which implies for gradually increasing memory effect (α from 1.0 to 0.1), the minimized total average cost is gradually decreasing, whereas the optimal ordering interval is gradually increasing. Thus to reach at maximum profit, longer time interval is needed.
In Table 4 (α = 1; 0 < β ≤ 1.0) for integral memory index β < 0.5, the optimal ordering interval T * 1,β is very high  The bold mark indicating the Maximum value of T * α,β or TOC * α,β which means very long time is required to get the maximum profit. In the next numerical example, the partial backlogging rate is increasing keeping all other parameters same.
From Table 5, it is observed that for high partial backlogging rate ((δ = 0.2), 0 < α ≤ 1, β = 1.0), the business takes less time to reach the minimum value of the total average cost compared to the case of low partial backlogging rate here (δ = 0.01, 0 < α ≤ 1, β = 1.0), (see Table  3). Moreover, in long memory effect, minimum value of the total average cost is less when partial backlogging rate is high for β = 1.0 and 0 < α ≤ 1.0. Partial backlogging of the inventory is profitable in long run.

Sensitivity analysis
The sensitivity analysis has been presented by changing each of the parameters by +50%, +10%, −10%, −50% to observe the effects on the optimal ordering interval T * α,1 and minimized total average cost TOC α,1 , taking one parameter at a time and keeping the remaining parameters unchanged. Sensitivity analysis can help to take a decision about most sensitive and least sensitive parameters of that inventory system. Sensitivity analysis on the minimized total average cost TOC * α,1 and optimal ordering interval T * α,1 is placed in Tables  7, 8.
From this table, we conclude that if we increase demand rate (γ ), optimal ordering interval changes from 28.8383 unit to 11.1613 unit and minimized total average cost changes from 18.3470 to 48.0410 unit. However, if we increase carrying cost per order (C 1 ), then the optimal ordering interval changes from 12.4376 unit to 21.1844 unit and minimized total average cost changes from 32.3643 to 34.6685 unit. Further, if we increase shortage cost per order (C 2 ), optimal ordering interval changes from 24.4376 unit to 14.2383 unit and minimized total average cost changes from 32.6161 to 34.4095 unit. The impact of the cost C 3 and C 4 are also analyzed from this table. For instance, if we increase ordering cost per order (C 3 ), optimal ordering interval changes from 6.0159 unit to 23.8011 unit and minimized total average cost changes from 30.6690 to 35.3643 unit. Lastly, if we increase lost sales cost per unit time per unit item (C 4 ), optimal ordering interval changes from 23.7744 unit to 6.1207 unit and minimized total average cost changes from 20.5072 to 45.5466 unit. Table 7 also suggest that if we increase partial backlogging rate (δ), optimal ordering interval changes from 24.3280 unit to 14.3006 unit and minimized total average cost changes from 32.7517 to 34.2842 unit; while by increasing the shortage starting time (t 1 ), optimal ordering interval changes from 17.4488 unit to 27.2199 unit and minimized total average cost changes from 33.9969 to 35.9572 unit. Table 8 suggests that by increasing the demand rate (γ ), optimal ordering interval changes from 41.1063 unit to 9.9668 unit, and minimized total average cost changes from 17.3403 to 46.3846 unit. Also, if we increase carrying cost per order (C 1 ), optimal ordering interval changes Table 7 Sensitivity analysis on the optimal ordering interval T * α,1 and cost TOC * α,1 at the differential memory index α = 1.0, β = 1.0

Parameter
Percentage of change (%) T * from 14.7896 unit to 26.6463 unit, and minimized total average cost changes from 31.5853 to 33.0567 unit. To analyze the impact of the shortage cost per order (C 2 ), if we increase them, then the optimal ordering interval changes from 31.274 unit to 17.1636 unit and minimized total average cost changes from 31.6459 to 2.9919 unit. The table also suggest that by increasing the ordering cost per order (C 3 ), optimal ordering interval changes from 6.3759 × 10 −5 unit to 32.4751 unit and minimized total average cost changes from 20.5436 to 33.7283 unit. The impact of the lost sales cost per unit time per unit item (C 4 ) is clearly seen from the table that by increasing C 4 cost, we have observer that the optimal ordering interval changes from 32.4294 unit to 6.3759×10 −5 unit and minimized total average cost changes from 18.8732 to 5.6382 unit. Lastly, if we increase partial backlogging rate (δ), optimal ordering interval changes from 2.5217 unit to 17.2697 unit and minimized total average cost changes from 34.2932 to 32.8624 unit, and by increasing shortage starting time (t 1 ), optimal ordering interval changes from 24.6540 unit to 28.9032 unit and minimized total average cost changes from 33.0592 to 33.1053 unit.
In the proposed memory-affected system when α = 0.8, β = 1.0, the most sensitive parameters (w.r.t. sensitive changes of the minimized total average cost) are demand rate (γ ), ordering cost per order (C 3 ), and lost sales cost per unit time per unit item (C 4 ). In memoryless system (α = 1.0, β = 1.0), the most sensitive parameters are identified as demand rate (γ ), ordering cost per order (C 3 ), and lost sales cost per unit time per unit item (C 4 ). Hence, demand rate (γ ), ordering cost per order (C 3 ), and lost sales cost per unit time per unit item (C 4 ) are identified sensitive for both memory-affected and memoryless system.  . 1 Length of the optimal ordering interval against differential memory index (α) for δ = 0.01 (black color) and δ = 0.2 (red color) From Fig. 1, it is observed that the length of the optimal ordering interval is less in case of high partial backlogging rate compared to the case of low partial backlogging rate. On 0.01) is low, the optimal ordering interval is much longer compared to the high partial backlogging rate system (δ = 0.2). Figure 3 shows the nature of changing of the minimized total average cost against differential memory index α. From this figure, it is clear that the minimized total average cost of the two cases (a) (δ = 0.01), (b) (δ = 0.2) coincides at α = 0.4786695534. Thus, profit is equal for both low and high partial backlogging rate at α = 0.4786695534. From Figs. 1, 2, and 3, we conclude that in the proposed inventory system, length of the optimal ordering interval corresponding high partial backlogging rate (δ = 0.2) is less compared to the low partial backlogging rate (δ = 0.01). At δ = 0.4786695534, the minimized total average cost coincides for both cases of high partial backlogging rate and low partial backlogging rate. In this paper, two situations of the inventory system, i.e., (i) high partial backlogging rate and (ii) low partial backlogging rate are studied, and results have been compared. In Figs. 4,5,6,7,8,9, and 10, we have plotted the positive inventory level (I 1 (t)) against time t for different values of the differential memory index (α). Figure 4 shows that in strong memory effect (α → 0), inventory level suddenly falls down and then gradually decreases smoothly but in poor memory effect (α → 1.0) or memory less system (α = 1.0), inventory level decreases smoothly.
Next, we have plotted inventory holding cost against shortage starting time-t 1 for different values of the differential memory index α at the integral memory index β = 0.1 and β = 1.0. We also plotted shortage cost against ordering interval for different values of the differential memory index α at the integral memory index β = 0.1, 0.5, 0.9, 1.0.. From Figs. 7,8,9,and 10, it is observed that shortage cost with respect to ordering interval is gradually increasing with gradually decreasing memory effect (w.r.t memory index β), i.e., β → 1.0. Mainly, it is also observed that memory affected shortage cost (SC (α,β) , 0 < α < 1.0, 0 < β < 1.0) almost closes to the memoryless shortage cost, i.e., (SC 11 ).
From Figs. 11, 12, and 13, it is clear that when total average cost is minimum, length of the ordering interval is long. When integral memory index (β) is unity but differential memory index is fractional, cost function becomes convex type. Total average cost TOC av α,β is minimum for long length shortage starting time t 1 , i.e., for long stock period.

Conclusions
In this paper, a memory-dependent EOQ model has been proposed to establish memory dependency in the EOQ system where partial backlogging is taken during the shortage. Memory effect and constant partial backlogging have been incorporated which are the important content of the model. The fractional-order model has some limitations because the calculation of the fractional-order cost is difficult. Here, using the mathematical formulation, it is found that the system behaves qualitatively differently in the presence of low as well as high partial backlogging rates. It is also seen from the numerical examples that in the case of a high partial backlogging rate, the business takes less time to reach the minimum value of the total average cost compared to the case of a low partial backlogging rate of the inventory system. This is a realistic situation. We get a critical value of the differential memory index for which minimized total cost same for the two cases (i) high partial backlogging rate and (ii) low partial backlogging rate. Thus, for two different situations of the inventory system, the inventory system gives the same output. Using sensitivity analysis, sensitive parameters are identified for both memory-affected systems as well as memoryless systems, lost sales cost per unit time per unit item, ordering cost per order, and demand rate. Sensitivity analysis is aware that sensitive parameters should be taken carefully. Thus, to maintain the business properly, the business person should use the previous experience but the experience should be of a short time scale but not for a highly long time scale for this model. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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