ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Study of Memory Effect in an Inventory Model with Constant Deterioration Rate

Journal of Applied Nonlinear Dynamics 10(2) (2021) 229--243 | DOI:10.5890/JAND.2021.06.004

Rituparna Pakhira$^{1}$, Uttam Ghosh$^{1}$, Susmita Sarkar$^{1}$, Vishnu Narayan Mishra$^{2}$

$^{1}$ Department of Applied Mathematics, University of Calcutta, Kolkata

$^{2}$ Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak 484 887, Anuppur, Madhya Pradesh, India

Abstract

In any business memory effect has great importance in inventory management as memory or past history cannot be ignored from the practical life of the inventory system. Many exogenous inventory parameters and factors at a given time depends not only their current values but also on the past histories. Thus an inventory management is a non Markovian process which can be tackled with memory dependent kernel of fractional derivatives. Another important factor of inventory management is deterioration of items which is directly related with memory effect as long memory of high rate of deterioration leads to poor impact on the business. Effect of long memory on deterioration of items have been studied in this paper using memory kernel of fractional derivatives. Fractional integration has been used to derive fractional order holding cost, deterioration cost, shortage cost and opportunity cost. Our analysis shows that for gradually increasing memory effect profit gradually increases. Our model is developed as a memory dependent inventory model.

Acknowledgments

The authors would like to thank Editor and Reviewers for his valuable suggestions to improve the papers. The first author would also like to thank the Department of science and Technology, Government of India, New Delhi, for the financial assistance under AORC, Inspire fellowship Scheme towards her PhD research work.

References

1.  [1] Tarasova, V.V. and Tarasov, V.E. (2017), Logistic map with memory from economic model, Chaos, Solitons and Fractals, 95, 84-91. doi: 10.1016,j.chaos.2016.12.012.
2.  [2] Saeedian, M., Khalighi, M., Azimi-Tafreshi, N., Jafari, G.R., and Ausloos, M. (2017), Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model, Physical Review E, 95, 022409.
3.  [3] Pakhira, R., Ghosh, U., and Sarkar, S. (2018), Study of Memory Effects in an Inventory Model Using Fractional Calculus, Applied Mathematical Sciences, 12(17), 797-824.
4.  [4] Scalas, E., Gorenflo, R., and Mainardi, F. (2000), Fractional calculus and continuous-time finance, Physica A, 284(1-4), 376-384.
5.  [5] Wang, J.L. and Li, H.F. (2011), Surpassing the fractional derivative: Concept of the memory-dependent derivative, Computers and Mathematics with Applications, 62, 1562-1567.
6.  [6] Banerjee, J., Ghosh, U., Sarkar, S., and Das, S. (2017), A Study of Fractional Schrodinger Equation-composed via Jumarie fractional derivative Pramana, J. Phys., 88, 70.
7.  [7] Das, A.K. and Roy, T.K. (2014), Role of fractional calculus to the generalized inventory model, Journal of global Research in computer science, 5(2), 11-23.
8.  [8] Miller, K.S. and Ross, B. (1993), An introduction to the Fractional calculus and Fractional Differential Equations. John Wileys sons, New York, NY, USA.
9.  [9] Caputo, M. (1967), Linear models of dissipation whose frequency independent, Geophysical Journal of the Royal Astronomical Society, 13(5), 529-539.
10.  [10] Pakhira, R., Ghosh, U., Sarkar, S., and Mishra, V.N. (2019), Study of memory effect in an Economic Order Quantity model for completely backlogged demand during shortage, Progr. Fract. Differ. Appl. (accepted for publiaction).
11.  [11] Das, S. (2008), Functional Fractional Calculus for system Identification and Controls. Springer Berlin Heidelberg New York ISBN 978-3-540-72702.
12.  [12] Debnath, L. (2003), Fractional integral transform and Fractional equation in Fluid Mechanics, to appear in Fract. Cal. Anal.
13.  [13] Tarasova, V.V. and Tarasov, V.E. (2016), Memory effects in hereditary Keynesian model // Problems of Modern Science and Education. No. 38 (80). P. 38-44. DOI: 10.20861/2304-2338-2016-80-001 [in Russian].
14.  [14] Pakhira, R., Ghosh, U., and Sarkar, S. (2019), Study of Memory effect in an Inventory model with price dependent demand, Journal of Applied Economic Sciences, Volume XIV, Summer, 2(64), 360-367.
15.  [15] Pakhira.R., Ghosh.U., Sarkar.S.(2019).Application of memory effect in an inventory model with price dependent demand rate during shortage, I.J. Education and Management Engineering, DOI: 10.5815.
16.  [16] Pakhira, R., Ghosh, U., and Sarkar, S. (2019), Study of memory effect in an economic order quantity model with quadratic type demand rate,doi.10.12921/cmst.
17.  [17] Pakhira, R., Ghosh, U., and Sarkar, S. (2019), Study of Memory Effect in an Inventory Model with Quadratic Type Demand Rate and Salvage Value, Applied Mathematical Sciences, 13(5), 209-223.
18.  [18] Choi, K.U., Kang, B., and Koo, N. (2014), Stability for Caputo fractional differential systems, Abstract and Applied Analysis, Article ID 631419, 6 pages.
19.  [19] Tarasov, V.E. and Tarasova, V.V. (2017), Economic interpretation of fractional derivatives, Progress in Fractional Differential and Applications, 3(1), 1-6.
20.  [20] Silver, E.A. and Meal, H.C. (1969), A simple modification of the EOQ for the case a varying demand rate. Production of Inventory Management, 10, 52-65.
21.  [21] Goyal, S.K. and Giri, B.C. (2001), Recent trends in modeling of deteriorating inventory, European Journal of Operational Research, 36, 335-338.
22.  [22] Goswami, A. and Chaudhuri, K.S. (1991), An EOQ model for deteriorating items with shortages and time varying demand and costs, International Journal of system and Science, 28, 53-159.
23.  [23] Aggarwal, S.P. (1978), A Note an order Level Model for a system with Constant Rate of Deterioration, Opsearch, 15(4), 184-187.
24.  [24] Shah, Y.K. and Jaiswal, M.C. (1978), An order level inventory model for a system with constant rate of deterioration, Opsearch, 15, 184-187.
25.  [25] Podlubny, I. (1999), Fractional Differential Equations, Mathematics in science and Engineering, Academic press, SanDiego, Calif, USA, 198.
26.  [26] Abdulaziz, O., Hashim, I., and Momani, S. (2008), Application of Homotopy -perturbation method to fractional IVPs, J. Comput. Appl. Math, 216, 574-584.
27.  [27] Zheng, B. (2013), Exp-Function Method for Solving Fractional Partial Differential Equations. Hindawi Publishing Corporation. The Scientific-World Journal 1-8.
28.  [28] Zhang, S. and Zhang, H.Q. (2011), Fractional Sub-equation method and its applications to nonlinear fractional PDEs, phys. Lett A, 375.1069.
29.  [29] Erdelyi, A. (1954), On some functional transformation Univpotitec Torino.
30.  [30] Ghosh, U., Sengupta, S., Sarkar, S., and Das, S. (2015), Analytic Solution of linear fractional differential equation with Jumarie derivative in term of Mittag-Leffler function, American Journal of Mathematical Analysis, 3(2), 32-38.
31.  [31] Ojha, A.K. and Biswal, K.K. (2010), Posynomial Geometric Programming Problems with Multiple parameters, Journal of computing, 2(1), ISSN2151-9617.
32.  [32] Mittag-Leffler, G.M. (1903), Sur lanouvelle function E, C.R. Acad.sci. Paris, (Ser.II), 137, 554-558.
33.  [33] Manna, S.K. and Chaudhuri, K.S. (2014), An Order level Inventory Model for a deteriorating item with quadratic time varying demand, shortage and partial backlogging.
34.  [34] Erdelyi, A. (1954), On some functional transformation Univpotitec Torino.
35.  [35] Sing, T. and Pattnayak, H. (2013), An EOQ Model for Deteriorating Items with Linear trade demand, Variable Deterioration and partial backlogging, Journal of service science and Management, 6, 186-190.
36.  [36] Chang, H.J. and Lin, W.F. (2010), A partial backlogging inventory model for non-instantaneous deteriorating items with stock-dependent consumption rate under inflation, Yugoslav Journal of Operations Research, 20(1), 35-54, 10.2298/YJOR1001035C.
37.  [37] Tarasov, V.E. and Tarasova, V.V. (2016), Long and short memory in economics: fractional-order difference and differentiation, IRA-International Journal of Management and Social Sciences, 5(2), 327-334. DOI: 10.21013/jmss.v5.n2.p10.
38.  [38] Vandana (2017), A study of dynamic inventory involving economic ordering of commodity, PhD thesis, Pt. Ravishankar Shukla University Raipur, 492010, Chhattisgarh, India.
39.  [39] Vandana, D.R., Deepmala, M.L., and Mishra, V.N. (2018), Duality relations for a class of a multiobjective fractional programming problem involving support functions, American J. Operations Research, 8, 294-311. DOI: 10.4236/ajor.2018.84017.
40.  [40] Vandana and Sharma, B.K. (2016), An inventory model for Non-Instantaneous deteriorating items with quadratic demand rate and shortages under trade credit policy, Journal of Applied Analysis and Computation, 6(3), pp. 720-737, DOI: 10.11948/2016047.
41.  [41] Vandana and Sharma, B.K. (2016), An EOQ model for retailers partial permissible delay in payment linked to order quantity with shortages, Math. Comput. Simulation, Vol. 125, pp. 99-112. http://dx.doi.org/10.1016/j.matcom.2015.11.008.