Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Study of Memory Effect in an EOQ Model with Fractional Polynomial Demand Rate Under Fuzzy Environment

Discontinuity, Nonlinearity, and Complexity 11(4) (2022) 583--598 | DOI:10.5890/DNC.2022.12.002

Rituparna Pakhira$^1$, Uttam Ghosh$^1$, Susmita Sarkar$^1$, Lakshmi Narayan Mishra$^2$

Download Full Text PDF



In current years, fractional calculus has received much attention because it can include memory effect to the inventory model. In the paper, fractional order derivative and integration have been applied to include the memory dependency to the polynomial type demand rate inventory model under the fuzzy environment. The advantage of fractional order derivative and integration have been applied to the EOQ model taking fuzzy environment. Inventory cost parameters, for example holding cost, purchasing cost and ordering cost are thought to be triangular fuzzy numbers. Two type memory indices (i) differential memory index, (ii) integral memory index have been established where differential memory index is associated with the fractional order derivative and integral memory index is associated with the fractional order integration. Defuzzification has been considered using graded mean integration method and signed distance method. Numerical examples and graphical presentations are considered to explain importance of the work.


The authors would like to thank the reviewers and the editor for the valuable comments and suggestions for the improvement of this article.


  1. [1]  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, E95, 022409.
  2. [2]  Tarasova, V.V and Tarasov, V.E. (2016), Memory effects in hereditary Keynesian model Problems of Modern Science and Education, 38(80), 38-44.
  3. [3]  Tarasova, V.V. and Tarasov, V.E. (2016), A generalization of the concepts of the accelerator and multiplier to take into account of memory effects in macroeconomics//Ekonomika I Predprinmatelstvo, Journal of Economy and Entrepreneurship, 10, No.10-3.
  4. [4]  Tarasova, V.V. and Tarasov, V.E. (2016), Marginal utility for economic processes with memory, Almanah Sovremennoj Nauki I Obrazovaniya [Almanac of Modern Science and Education], 7(109), 108-113.
  5. [5]  Tarasova, V.V. and Tarasov, V.E. (2016), Fractional dynamics of natural growth and memory effect in economics, European Research, 2016, 12(23), 30-37.
  6. [6]  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.
  7. [7]  Tarasov, V.E. and Tarasova, V.V. (2017), Economic interpretation of fractional derivatives, Progress in Fractional Differential and Applications, 3(1), 1-6.
  8. [8]  Das, T., Ghosh, U., Sarkar, S., and Das, S. (2018), Time independent fractional Schrodinger equation for generalized Mie-type potential in higher dimension framed with Jumarie type fractional derivative, Journal of Mathematical Physics, 59, 022111.
  9. [9]  Pakhira, R., Ghosh, U., and Sarkar, S. (2020), Study of memory effect in an inventory model for deteriorating items with partial backlogging, Computers $\&$ Industrial Engineering, 148.
  10. [10]  Pakhira, R., Ghosh, U., and Sarkar, S. (2018), Study of memory effects in an inventory model using fractional calculus, Applied Mathematical Sciences, 12, 797-824.
  11. [11]  Pakhira, R., Ghosh, U., and Sarkar, S. (2020), Study of memory effect between two memory dependent inventory models, Journal of Fractional Calculus and Applications, 11(1), 138-155.
  12. [12]  Miller, K.S. and Ross, B. (1993), An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley \& Sons, New York, NY, USA.
  13. [13]  Podubly, I. (1999), Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA. 198.
  14. [14]  Caputo, M. (1967), Linear models of dissipation whose frequency independent, Geophysical Journal of the Royal Astronomical Society, 13(5), 529-539.
  15. [15]  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.
  16. [16]  Majumder, P., Bera, U.K., and Maiti, M. (2015), An EPQ model of deteriorating items under partial trade credit financing and demand declining market in crisp and fuzzy environment, Procedia Computer Science, 45, 780-789.
  17. [17]  Mandal, N.K. (2012), Fuzzy economic order quantity model with ranking fuzzy number cost parameters, Yugoslav Journal of Operations Research, 22, 247-264.
  18. [18]  Sharmita, D. and Uthayakumar, R. (2015), Inventory model for deteriorating items involving fuzzy with shortages and exponential demand, International Journal of Supply and operations management, 2(3), 888-904.
  19. [19]  Dutta, D. and Kumar, P. (2013), Fuzzy inventory model for deteriorating items with shortage under fully backlogged condition, International Journal of Soft Computing and Engineering, 3(2), 393-398.
  20. [20]  Rotundo, G. (2005), in Logistic Function in Large Financial Crashes, The Logistic Map and the Route to Chaos: From the Beginning to Modern Applications, edited by M. Ausloos and M. Dirickx(Springer-Verlag, Berlin/Heidelberg, 239-258.