Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 418 https://internationalpubls.com Inventory Optimization with Ramp-Type Demand, Time-Dependent Holding Costs, and Discounted Backorders under Inflation Ekta Chauhan1*, R. K. Shrivastav1 and Sangeeta Gupta2 1Department of Mathematics, Agra College, Agra, Dr. Bhimrao Ambedkar University, Agra, U.P., India 2The A.H. Siddiqi Centre for Advanced Research in Applied Mathematics and Physics, Sharda University, Greater Noida, U.P., India ekta.chauhan0562@gmail.com, dr.srivastavark@gmail.com, sangeeta147@gmail.com Article History: Received: 21-10-2024 Revised: 05-12-2024 Accepted: 12-12-2024 Abstract: This study presents an inventory model with a ramp-type demand pattern, where holding costs increase over time, and inflation is taken into account. The model is particularly relevant for inventories like seasonal produce and newly launched fashion items. When stock runs out, the inventory manager offers a discount to customers willing to backorder their demand. The goal is to determine the optimal ordering policy and backorder discount by minimizing the total cost across a replenishment cycle. Finally, numerical results are provided to assess how the optimal policies respond to changes in key system parameters. Graphs are used to visualize the impact of these parameters on the total inventory cost, enhancing the understanding of the model. Keywords: price discount on backorder, ramp-type demand, shortage, inflation, time dependent holding cost. 1. Introduction The demand rates are considered to be constant in classical inventory models, but in actuality, time is a significant factor in the inventory system. When the demand rate is ramp type, it rises gradually until it reaches a particular limit, at which point it stabilizes and reaches a constant level. This type of demand commonly occurs with the launch of new consumer goods like mobile phones, clothing, automobiles, fashion items and cosmetics. Hill [7] was the first to investigate inventory models for scenarios with an initial increase in demand, followed by a constant demand rate. Mandal and Pal [10] expanded on Hill's inventory model to include deteriorating items and allowed shortages. Panda et al. [18] presented optimal replenishment policy for perishable seasonal products in a season with ramp- type time dependent demand. Mandal [8] developed an EOQ inventory model for Weibull distributed deteriorating items under ramp type demand and shortages. Chandra [4] introduced an inventory model with ramp type demand, time varying holding cost and price discount on backorders. Saha et al. [19] proposed inventory model with ramp-type demand and price discount on back order for deteriorating items under partial backlogging. Chandra studied [3] two warehouse inventory model for deteriorating Items with ramp type demand and price discount on backorders. Mondal et al. [11] discussed an EOQ model for seasonal product with ramp-type time and stock dependent demand, shortage and partial backorder. In most models, holding cost is assumed to be known and constant. However, this is not always the case. For example, in the case of seasonal fruits and vegetables, extended storage time requires more mailto:ekta.chauhan0562@gmail.com mailto:dr.srivastavark@gmail.com mailto:sangeeta147@gmail.com%20n https://www.inderscienceonline.com/doi/abs/10.1504/IJMOR.2024.138057 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 419 https://internationalpubls.com advanced storage facilities and services which leads to higher holding costs. Muhlemann and Valtis- Spanopoulous [13] were the first to introduce variable holding costs, developing an EOQ model with constant demand in which the holding cost is a percentage of the average inventory value. Goh [6] explored two types of holding cost variations: (a) a nonlinear function of storage time and (b) a nonlinear function of storage level. Alfares [1] introduced an inventory system with stock-dependent demand, where the holding cost is a step function of storage time, considering two types of holding cost variation based on storage time: retroactive increase and incremental increase. Palanivel and Suganya [19] suggested an inventory model with partial backlogging, demand depend on price and stock level, time varying holding cost and quantity discounts. Sharma and Sharma [21] discussed on the effect of inflation on a deteriorating inventory model with non-linear holding cost and non-linear demand. Unmet demand is usually considered either completely lost or fully backlogged in inventory models with shortages. Therefore, in practice, some customers may choose to wait for their orders to be fulfilled while others might abandon their purchase. In such cases, inventory managers may implement strategies like offering discounts on backorders or reducing waiting times to encourage customers to wait patiently for their desired items. A continuous review inventory model with negotiable backorders and order quantity as decision variables was presented by Pan and Hsiao [17]. In order to account for more realistic elements of the actual inventory systems, Ouyang et al. [14] created a periodic review inventory model with backorder discounts. Uthayakumar and Parvathi [23] have studied an inventory model with mixture of backorders involving reducible lead time and setup cost. Pal and Chandra [15] investigated a deterministic inventory model with permissible delay in payment and price discount on backorders. Pal and Chandra [16] developed a periodic review inventory model with stock dependent demand, permissible delay in payment and price discount on backorders. Chauhan and Tayal [5] suggested an order quantity scheme for ramp type demand and backlogging during stock out with discount strategy. Mrudul et al. [12] introduced Optimal pricing policy for deteriorating items with continuous compounding under price-sensitive demand and shortages. Kumar P et al. [8] studied a two-warehouse inventory system with time-dependent demand and preservation technology. In today's business environment, inflation is a significant concern. It represents a persistent increase in the overall price levels of goods and services over time, which leads to reduction in the purchasing power of money. High inflation rates can significantly decrease the purchasing power of businesses. Consequently, when constructing any inventory models, it is crucial to consider for the impact of inflation. Buzacott [2] was among the first to incorporate inflation rates into inventory modelling alongside different pricing strategies. Under the impact of inflation. Economic ordering policy with price and advertisement dependent demand and allowable payment delay under inflation was described by Udayakumar et al. [22] for non-instantaneously deteriorating items. This paper introduces an inventory model where holding cost increases as linear increase function of time, and the demand rate is represented as a ramp type demand. If the consumer is willing to backorder his demand during a stock-out, the manager offers him a discount. Additionally, the effects of inflation are also incorporated into the model. Numerical results are provided to examine the impact of change in various system parameters on the sensitivity of the optimal policies. Graphical representations https://www.inderscienceonline.com/doi/abs/10.1504/IJSOI.2021.114113 https://www.inderscienceonline.com/doi/abs/10.1504/IJSOI.2021.114113 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 420 https://internationalpubls.com clarify the relationships between various parameters and the total inventory cost, thereby enhancing the understanding of the inventory models. 2. Assumptions and Notations: The following assumptions and notations are used in the development of the model. 2.1 Notations I(t) = The inventory quantity at a given time t. b = The fraction of demand fulfilled through backorders when there is a stockout. P = The cost of purchasing each unit s1= The backorder cost per unit on backorder for each time period s2 = Cost associated with a lost sale T = duration of a cycle of replenishing T1= time it takes for the available goods to be sold out, 0 < T1< T 𝑆𝑟 = maximum height of stock during a replenishing cycle s = The inventory shortage at the end of the cycle of replenishment 2.2 Assumptions • A single inventory item is taken consideration by the model. • There is no lead time because inventory is replenished instantly upon ordering. • Shortages are permitted, with b fraction of unfulfilled demands during stockouts being backlogged. • The holding cost per item per unit time, denoted as h(t), is considered depend on time h(t) = h + αt where α > 0, b >0 • It is assumed that the demand rate R(t) is a ramp-type function of time t. R(t) = D0[t − (t − μ)H(t − μ)] Where D0 and µ are positive constants and H(t − μ) is the Heaviside’s function defined as follows: H(t − μ) = { 1 for t ≥ μ 0 for t < μ Figure (1): The ramp type demand rate • The time required for available stock to run out (T1) is more than µ. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 421 https://internationalpubls.com • The fraction b of backorder during the stock out period is directly proportional to the price discount provided by the inventory manager. 3. Mathematical model: The planning period consists of reorder intervals, each of length T units, with orders placed at T, 2T, 3T, and so on, with the order amount being only enough to raise the stock height to a maximum level 𝑆𝑟. Demand depletes inventory during the times T1 < T and (0, T1), and shortages arise during the interval (T1, T), with a fraction b being backlogged. Therefore, the change in inventory level over time is described by Figure (2): Graphical representation of Inventory model d dt I(t) = −D0t if 0 < t < μ d dt I(t) = −D0μ if μ < t < T1 d dt I(t) = −bD0μ if T1 < t < T With boundary conditions 𝐼 (0) = 𝑆𝑟 𝑎𝑛𝑑 𝐼(𝑇1) = 0, The solutions to the differential equation for the three different cases (I, II, III) are as follows: Case I: 0 < t < μ I(t) = D0t2 2 + 𝑆𝑟 … (1) Case II: μ < t < T1 I(t) = D0μ(T1 − t) … (2) Case III: T1 < t < T I(t) = bD0μ(T1 − t) … (3) Hence, 𝑆𝑟 = D0μ 2 (2T1 − μ) … (4) s = bD0μ(T − T1) … (5) Then, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 422 https://internationalpubls.com Ordering cost per cycle (OC) =A Holding cost for a cycle (HC) = ∫ h(t) I(t)dt T1 0 HC = ∫ h(t) I(t)dt μ 0 + ∫ h(t) I(t)dt T1 μ HC = D0μ r2 [e−rT1 + e−rμ(T1r − 1 − rμ)] − hD0 2r3 [2 − e−rμ(r2μ2 + 2rμ + 2) − μr2(e−rμ − 1)(μ + 2T1)] − aD0 2r4 [{6 − e−rμ(r3μ3 + 3r2μ2 + 6rμ + 6)} + r2μ2{1 − e−rμ(rμ + 1)} − 2rμ{e−rt(r2T1 2 + 2rT1 + 2) + e−rμr2μ2 + 2rμ + 2} + 2T1μr2{e−rT1(rT1 + 1) − 1}] … (6) The backorder cost, at the end of a cycle = −s1 ∫ I(t)dt T T1 BC = s1bD0μ r2 (T1re−rT − rTe−rT − e−rT + e−rT1) … (7) Lost sales cost for a cycle (LS) = s2D0μ(1 − b)(T − T1) … (8) Purchase cost for a cycle (PC) = P ( D0μ 2 (2T1 − μ) + bD0μ(T − T1)) … (9) Consequently, the cost per replenishment cycle unit is represented by TC = 1 T [OC + HC + BC + LC + PC] TC = A + D0μ r2 [e−rT1 + e−rμ(T1r − 1 − rμ)] − hD0 2r3 [2 − e−rμ(r2μ2 + 2rμ + 2) − μr2(e−rμ − 1)(μ + 2T1)] − aD0 2r4 [{6 − e−rμ(r3μ3 + 3r2μ2 + 6rμ + 6)} + r2μ2{1 − e−rμ(rμ + 1)} − 2rμ{e−rt(r2T1 2 + 2rT1 + 2) + e−rμr2μ2 + 2rμ + 2} + 2T1μr2{e−rT1(rT1 + 1) − 1}] + s1bD0μ r2 (T1re−rT − rTe−rT − e−rT + e−rT1) + s2D0μ(1 − b)(T − T1) + P ( D0μ 2 (2T1 − μ) + bD0μ(T − T1)) … (10) 4. Numerical Example: Using the statistical programme MATLAB, we numerically solve the equations for given sets of costs. A = 500, a = 0.6, P = 5, x = 0.25, s = 6, r = 0.05, D = 100, h = 3, s2 = 7 and b = 0.7 We obtained the following optimum values t =3.6354, T = 5.65794 and Total Cost= ₹ 300.04158 5. Sensitive Analysis: Sensitivity analysis is given by taking one parameter at a time, increasing or reducing it by 10% and 20% while maintaining the original values of the remaining parameters. The results can be shown in the table. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 423 https://internationalpubls.com Table (1): Variation in system parameters Parameters % Change Value 𝐓𝟏 T TC A -20% 400 3.3152 5.0480 281.3582 -10% 450 3.4810 5.3607 290.9658 +10% 550 3.7802 5.9422 308.6623 +20% 600 3.9171 6.2157 316.8876 α -20% 0.48 3.8605 5.8494 295.8814 -10% 0.54 3.7416 5.7478 298.0349 +10% 0.66 3.5395 5.7765 301.9204 +20% 0.72 3.4524 5.5053 303.6872 p -20% 4 3.6766 5.6434 278.2421 -10% 4.5 3.6561 5.6509 289.1500 +10% 5.5 3.6143 5.6644 310.9176 +20% 6 3.5930 5.6703 321.7754 µ -20% 0.2 4.0034 6.3584 255.9501 -10% 0.225 3.8067 5.9788 278.2303 +10% 0.275 3.4835 5.6579 321.4527 +20% 0.3 3.3469 5.3809 342.5141 s1 -20% 4.8 3.5519 6.0137 293.0917 -10% 5.4 3.5971 5.8171 296.8455 +10% 6.6 3.6683 5.5263 302.7976 +20% 7.2 3.6968 5.4156 305.2000 r -20% 0.04 3.5847 5.5099 302.1615 -10% 0.045 3.6096 5.5821 301.1090 +10% 0.055 3.6622 5.7378 298.9582 +20% 0.06 3.6902 5.8222 297.8578 D0 -20% 80 3.9828 6.3489 256.6936 -10% 90 3.7958 5.9731 278.6353 +10% 110 3.4955 5.3883 320.9927 +20% 120 3.3719 5.1542 341.5506 h -20% 2.4 3.6443 5.6456 297.7484 -10% 2.7 3.6443 5.6518 298.8970 +10% 3.3 3.6263 5.6639 301.1818 +20% 3.6 3.6172 5.6698 302.3180 s2 -20% 5.6 3.5866 5.9770 296.2199 -10% 6.3 3.6132 5.8007 298.1477 +10% 7.7 3.6540 5.5397 301.9011 +20% 8.4 3.6698 5.4402 303.7259 b -20% 0.56 3.5866 5.9770 295.9244 -10% 0.63 3.6132 5.8007 298.1733 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 424 https://internationalpubls.com +10% 0.77 3.6540 5.5397 301.6135 +20% 0.84 3.6698 5.4402 302.9501 Figure 3: Variation in system parameters table (1) Table (2): Effect of backorder rate (b) and Inflation (r) b r 0.56 0.63 0.7 0.77 0.84 0.04 T1 3.5348 3.5621 3.5847 3.6037 3.6199 T 5.8037 5.641 5.5099 5.4005 5.3081 TC 298.1047 300.3201 302.1615 303.7112 305.0305 0.045 T1 3.5602 3.5872 3.6096 3.6284 3.6444 T 5.8879 5.7191 5.5621 5.4685 5.3726 TC 297.0230 299.2546 301.1090 302.6697 303.9971 0.05 T1 3.5866 3.6132 3.6354 3.6540 3.6698 T 5.9770 5.8007 5.6579 5.5397 5.4402 TC 295.9244 298.1733 300.0415 301.6135 302.9501 0.055 T1 3.6042 3.6404 3.6622 3.6805 3.6961 T 6.0714 5.8869 5.7378 5.6146 5.5110 TC 294.8075 297.0751 298.9582 300.5421 301.8886 0.06 T1 3.6432 3.6688 3.6902 3.7082 3.7234 T 6.1718 5.9782 5.8222 5.6935 5.5855 TC 293.6711 295.9588 297.8578 299.4547 300.8175 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 425 https://internationalpubls.com Figure 4: Effect of backorder rate (b) and Inflation (r) 6. Conclusion: The paper examines an inventory model where demand is considered as ramp-type, inflation and allowing for shortages. The holding cost is assumed to increase linearly over time. This study is relevant to inventories such as seasonal vegetables and fruits, newly launched fashion items, and electronic goods. A portion of the demand is backlogged, and customers who are willing to wait are offered a discount. The goal is to determine the optimal ordering policy and backorder discount that minimizes the total cost over a replenishment interval. Numerical analysis indicates that offering substantial discounts on backorders is beneficial for the inventory manager when backorder costs are low. Sensitivity analysis and numerical examples are used to illustrate the model. When increase parameters A, a, p, b, h, µ, s1, s2 and D0 the total cost increase significantly. When decrease parameter r the total cost decrease marginally. References [1] Alfares, H. K. (2007). Inventory model with stock-level dependent demand rate and variable holding cost. International Journal of Production Economics, 108(1), 259-265. [2] Buzacott J. A. (1975). Economic order quantities with inflation, Operational Research Quarterly, 26(3), 553–558. [3] Chandra S., (2021) Two Warehouse Inventory Model for Deteriorating Items with Ramp Type Demand and Price Discount on Backorders, Journal of Scientific Research, 13 (2), 455-465. [4] Chandra S., (2017) An inventory model with ramp type demand, time varying holding cost and price discount on backorders, Uncertain Supply Chain Management, 5(1), 51-58. [5] Chauhan A. and Tayal S. (2021) , An order quantity scheme for ramp type demand and backlogging during stock out with discount strategy, International Journal of Service Operations and Informatics, 11(1), 27-40. [6] Goh, M. (1994). EOQ models with general demand and holding cost functions. European Journal of Operational Research, 73(1), 50-54. [7] Hill, R. M. (1995). Inventory models for increasing demand followed by level demand. Journal of the Operational Research Society, 46(10), 1250-1259. [8] Kumar P., Saxena A., Kumar K., (2024). A two-warehouse inventory system with time-dependent demand and preservation technology. Communications on Applied Nonlinear Analysis, 31(2), 240-247. [9] Mandal, B. (2010). An EOQ inventory model for Weibull distributed deteriorating items under ramp type demand and shortages. Opsearch,47(2), 158-165. [10] Mandal, B., & Pal, A. K. (1998). Order level inventory system with ramp type demand rate for deteriorating items. Journal of interdisciplinary Mathematics, 1(1), 49-66. [11] Mondal P., Das P. and Khanra S.,(2024), An EOQ model for seasonal product with ramp-type time and stock dependent demand, shortage and partial backorder, International Journal of Mathematics in Operational Research, 27( 3) , 393-413. [12] Mrudul Y. J., Bhavisha H. K. and Manish R. B., (2023), Optimal pricing policy for deteriorating items with continuous compounding under price-sensitive demand and shortages, International Journal of Procurement Management, 18(1), 20-43. https://www.inderscienceonline.com/doi/abs/10.1504/IJSOI.2021.114113 https://www.inderscienceonline.com/doi/abs/10.1504/IJSOI.2021.114113 https://www.inderscienceonline.com/doi/abs/10.1504/IJMOR.2024.138057 https://www.inderscienceonline.com/doi/abs/10.1504/IJMOR.2024.138057 https://www.inderscienceonline.com/doi/abs/10.1504/IJMOR.2024.138057 https://www.inderscienceonline.com/journal/ijmor https://www.inderscienceonline.com/toc/ijmor/27/3 https://www.inderscienceonline.com/journal/ijpm https://www.inderscienceonline.com/toc/ijpm/18/1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 426 https://internationalpubls.com [13] Muhlemann, A. P., & Valtis-Spanopoulos, N. P. (1980). A variable holding cost rate EOQ model. European Journal of Operational Research, 4(2), 132-135. [14] Ouyang, L. Y., Chuang, B. R., & Lin, Y. J. (2003). Impact of backorder discounts on periodic review inventory model. International Journal of Information and Management Sciences, 14(3), 1-14. [15] Pal, M., & Chandra, S. (2012). A deterministic inventory model with permissible delay in payment and price discount on backorders. Opsearch,49(3), 271-279. [16] Pal, M., & Chandra, S. (2014). A periodic review inventory model with stock dependent demand, permissible delay in payment and price discount on backorders. Yugoslav Journal of Operations Research, 24(1), 99-110. [17] Pan, J. C. H., & Hsiao, Y. C. (2001). Inventory models with back-order discounts and variable lead time. International Journal of Systems Science,32(7), 925-929. [18] Panda, S., Senapati, S., & Basu, M. (2008). Optimal replenishment policy for perishable seasonal products in a season with ramp-type time dependent demand. Computers & Industrial Engineering, 54(2), 301-314. [19] Palanivel M. and Suganya M. (2021), Partial backlogging inventory model with price and stock level dependent demand, time varying holding cost and quantity discounts, Journal of Management Analytics, 9(1), 32-59. [20] Saha S., Sen N., and Nath B.K. (2018). Inventory Model with Ramp-type Demand and Price Discount on Back Order for Deteriorating Items under Partial Backlogging, Applications and Applied Mathematics: An International Journal (AAM), 13(1), 472 – 483. [21] Sharma G. and Sharma A., (2022). Effect of Inflation on a Deteriorating Inventory Model with Non-linear Holding Cost and Non-linear Demand, International Journal of Contemporary Mathematical Sciences, 17(2), 47 – 60. [22] Udayakumar R., Geetha K.V. and Sana S.S., (2020). Economic ordering policy for non-instantaneous deteriorating items with price and advertisement dependent demand and permissible delay in payment under inflation, Mathematical Methods in the Applied Sciences, 44(9), 1-25. [23] Uthayakumar, R., & Parvathi, P. (2008). Inventory models with mixture of backorders involving reducible lead time and setup cost. Opsearch,45(1), 12-33. https://www.tandfonline.com/author/Palanivel%2C+M https://www.tandfonline.com/author/Suganya%2C+M