Study of Memory Effect in a Fuzzy EOQ Model with No Shortage

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Rituparna Pakhira 1,* Uttam Ghosh 1 Susmita Sarkar 1

1. Department of Applied Mathematics, University of Calcutta, Kolkata, 700009, West Bengal, India

* Corresponding author.


Received: 9 Apr. 2019 / Revised: 10 Jun. 2019 / Accepted: 12 Aug. 2019 / Published: 8 Nov. 2019

Index Terms

Fractional order derivative, Fractional order inventory model under fuzzy environment, Long memory effect, Short memory effect


The feature of the fractional order derivative and fractional order integration is one of the important tools to realize the beauty of the fractional calculus. Fractional order derivative and integration has a long history like classical calculus but its users are much less compared to the classical calculus. The purpose of this paper is to study an inventory model with linear type demand rate under the fuzzy environment. This paper also wants to introduce the memory effect property of fractional order derivative which can help to setup the model more authentic. Two advantages have been included to the model (i) memory effect,(ii) fuzzy environment. Here, the fractional order model is defuzzyfied using (i) signed distance method,(ii) graded mean integration method. Fuzzification can close to the reality with incorporating uncertainty behavior of some economic parameters of the inventory system and fractional order can explain the memory phenomena. For this problem due to illustrate defuzzification, set up cost, holding cost per unit, per unit cost are assumed as triangular fuzzy numbers. Fractional order derivative and integration are applied to develop the whole work. It is known that fractional calculus is a valuable tool to describe memory phenomena. Fractional order is established as the index of the memory. In this paper, depending on strength of memory, memory phenomena considered in two steps(i) long memory,(ii) short memory. The proposed fuzzy models and technique lastly have been illustrated. Results of two defuzzyfications are compared with graphical presentations. This present studies can help to moderate the classical fuzzy inventory model. From the numerical studied it is observed that in long memory effect, profit is good compared to the low memory effect or memory less system.

Cite This Paper

Rituparna Pakhira, Uttam Ghosh, Susmita Sarkar, "Study of Memory Effect in a Fuzzy EOQ Model with No Shortage", International Journal of Intelligent Systems and Applications(IJISA), Vol.11, No.11, pp.58-68, 2019. DOI:10.5815/ijisa.2019.11.06


[1]M.Saeedian, M. Khalighi, N. Azimi-Tafreshi, G. R. Jafari, M. Ausloos(2017), Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model, Physical Review E95,022409.
[2]V.V.Tarasova, V.E.Tarasov., (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].
[3]V.V, Tarasova, V.E.Tarasov. 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], 2016.vol.10.No.10-3.P.[12]-1129.[in Russian].
[4]V.V, Tarasova, V.E.Tarasov. Marginal utility for economic processes with memory//AlmanahSovremennojNauki I Obrazovaniya [Almanac of Modern Science and Education], 2016.No.7(109).P.108-113[in Russian].
[5]V.V, Tarasova, V.E.Tarasov. Fractional Dynamics of Natural Growth and Memory Effect in Economics,European Research.2016.No.12(23).P.30-37.
[6]V.E.Tarasov, V.V.Tarasova., (2016)." Long and short memory in economics: fractional-order difference and differentiation "IRA-International Journal of Management and Social Sciences. Vol. 5.No. 2. P. 327-334. DOI: 10.21013/jmss.v5.n2.p10.
[7]V.E.Tarasov, V.V.Tarasova., (2017).”Economic interpretation of fractional derivatives”. Progress in Fractional Differential and Applications.3.No.1, 1-6.
[8]T.Das, U.Ghosh, S.Sarkar and S.Das,(2018)” Time independent fractional Schrodinger equation for generalized Mie-type potential in higher dimension framed with Jumarie type fractional derivative”. Journal of Mathematical Phy sics,59, 022111; doi: 10.1063/1.4999262.
[9]R.Pakhira, U.Ghosh, S.Sarkar., (2018) Study of Memory Effects in an Inventory Model Using Fractional Calculus, Applied Mathematical Sciences, Vol. 12, no. 17, 797 - 824.
[10]R.Pakhira, U.Ghosh, S.Sarkar., ”Application of Memory effects In an Inventory Model with Linear Demand and No shortage”, International Journal of Research in Advent Technology, Vol.6, No.8, 2018.
[11]R.Pakhira., U.Ghosh., S.Sarkar.,(2018).Study of Memory Effect in an Inventory Model with Linear Demand and Salvage Value, International Journal of Applied Engineering ResearchISSN 0973-4562 Volume 13, Number 20 (2018) pp. 14741-14751.
[12]K.S.Miller, B.Ross., (1993).“An Introduction to the Fractional Calculus and Fractional Differential Equations”.JohnWiley&Sons, New York, NY, USA.
[13]I.Podubly(1999), “Fractional Differential Equations, Mathematics in Science and Engineering”, Academic Press, San Diego, Calif,USA.198.
[14]M.Caputo.,(1967).Linear models of dissipation whose frequency independent, “Geophysical Journal of the Royal Astronomical Society. 13(5), 529-539.
[15]U.Ghosh, S. Sengupta, S.Sarkar, S.Das., 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]P.Majumder,U.K.Bera, and M.Maiti., An EPQ modelof deteriorating items under partial trade credit financing and demand declining market in crisp and fuzzy environment, Procedia Computer Science, 45 (2015), 780-789.
[17]N. K. Mandal, Fuzzy economic order quantity modelwith ranking fuzzy number cost parameters, Yugoslav Journal of Operations Research, 22 (2012), 247-264.
[18]D. Sharmita and R.Uthayakumar,”Inventory model for deteriorating items involving fuzzy with shortages and exponential demand” International Journal of Supply and operations management, Nov 2015,volume 2,Issue 3, pp.888-904.
[19]G. Rotundo, 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, 2005), pp. 239–258.
[20]R.Pakhira, U.Ghosh, S.Sarkar.,(2019). Study of Memory Effect in an Inventory Model with Quadratic Type Demand Rate and Salvage Value, Applied Mathematical Sciences, Vol. 13, 2019, no. 5, 209 - 223 .
[21]Tarasov.V.E, Tarasova.V.V, Elasticity for economic processes with memory: fractional differential calculus approach, Fractional Differential Calculus, Vol-6, No-2(2016),219-232.
[22]S.Saha, T. Chakrabarti,A Fuzzy Inventory Model for Deteriorating Items with Linear Price Dependent Demand in a Supply Chain,Intern. J. Fuzzy Mathematical Archive, Vol. 13, No. 1, 2017, 59-67, ISSN: 2320 –3242 (P), 2320 –3250 (online).
[23]R.Pakhira, U.Ghosh, S.Sarkar., (2019). Application of memory effect in an inventory model with price dependent demand rate during shortage, I.J. Education and Management Engineering, 2019,DOI: 10.5815.
[24]R.Pakhira,U.Ghosh,S.Sarkar.,((2019).Study of Memory Effect In an Inventory Model with Linear Demand and Shortage, International Journal of Mathematical Sciences and Computing(IJMSC), ISSN: 2310-9025(Print), DOI:10.5815/ijmsc.2019.02.05.