INVENTORY MODEL WITH DEMAND AS TYPE-2 FUZZY NUMBER: A FUZZY DIFFERENTIAL EQUATION APPROACH

Document Type : Research Paper

Authors

1 Department of Mathematics, National Institute of Technology, Agartala, 799046, India

2 Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, 721102, India

Abstract

An inventory model is formulated with type-2 fuzzy parameters under trade credit policy and solved by using Generalized Hukuhara derivative approach. Representing demand parameter of each expert's opinion is a membership function of type-1 and thus, this membership function again becomes fuzzy. The final opinion of all experts is expressed by a type-2 fuzzy variable. For this present problem, to get corresponding defuzzified values of the triangular type-2 fuzzy demand parameters, first critical value (CV)-based reduction methods are applied to reduce corresponding type-1 fuzzy variables which becomes pentagonal in form. After that $\alpha$- cut of a pentagonal fuzzy number is used to construct the upper $\alpha$- cut and lower $\alpha$- cut of the fuzzy differential equation. Different cases are considered for fuzzy differential equation: gH-(i) differentiable and gH-(ii) differentiable systems. The objective of this paper is to find out the optimal time so as to minimize the total inventory cost. The considered problem ultimately reduces to a multi-objective problem which is solved by weighted sum method and global criteria method. Finally the model is solved by generalised reduced gradient method using LINGO (13.0) software. The proposed model and technique are lastly illustrated by providing numerical examples. Results from two methods are compared and  some sensitivity analyses both in tabular and graphical forms are presented and discussed. The effects of total cost with respect to the change of demand related parameter ($\beta$),  holding cost parameter ($r$), unit purchasing cost parameter ($p$), interest earned $(i_e)$ and interest payable $(i_p)$ are discussed. We also find the solutions for type-1 and crisp demand as particular cases of type-2 fuzzy variable. This present study can be applicable in many aspects in many real life situations where type-1 fuzzy set is not sufficient to formulate the mathematical model. From the numerical studies, it is observed that under both gH-(i) and gH-(ii) cases, total cost of the system gradually reduces for the sub-cases - 1.1, 1.2 and 1.3 depending upon the positions of N(trade credit for customer) and M (trade credit for retailer) with respect to T (time period).

Keywords

References

[1] T. Allahviranloo and M. Afshar Kermani, Numerical methods for fuzzy linear partial dif-
ferential equations under new defi nition for derivative, Iranian Journal Fuzzy Systems, 7(3)
(2010), 33{50.
[2] S. Arshed, On existence and uniqueness of solution of fuzzy fractional differential equations,
Iranian Journal Fuzzy Systems, 10(6) (2013), 137{151.
[3] B. Bede and S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued func-
tions with applications to fuzzy differential equations, Fuzzy Sets Syst., 151(4) (2005), 581{
599.
[4] B. Bede and L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets
and Systems, 230(5) (2013), 119{141.
[5] K. J. Chung and L. E. Cardenas-Barron, The simpli ed solution procedure for deteriorating
items under stock-dependent demand and two-level trade credit in the supply chain manage-
ment, Appl. Math. Model., 37(7) (2013), 4653{4660.
[6] K. J. Chung, L. E. Cardenas-Barron and P. S. Ting, An inventory model with non-instan-
taneous receipt and exponentially deteriorating items for an integrated three layer supply
chain system under two levels of trade credit, Int. J. Prod. Eco., 155(5) (2014), 310{317.
[7] S. C. Chen, L. E. Cardenas-Barron and J. T. Teng, Retailer's economic order quantity when
the supplier offers conditionally permissible delay in payments link to order quantity, Int. J.
Prod. Econ., 155(3) (2014), 284{291.
[8] L. E. Cardenas-Barron, K. J. Chung and G. Trevio-Garza, Celebrating a century of the
economic order quantity model in honor of For Whitman Harris, Int. J. Prod. Econ., 155(7)
(2014), 1{7.
[9] B. Das, N. K. Mahapatra and M. Maiti, Initial-valued fi rst order fuzzy differential equation
in Bi-level inventory model with fuzzy demand, Math. Model. Anal., 13(4) (2008), 493{512.
[10] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press,
New York, 1980.
[11] D. Dubois and H. Prade, Operations on fuzzy numbers, Int. J. Syst. Sci., 9(6) (1978), 613{
626.
[12] R. Ezzati, K. Maleknejad, S. Khezerloo and M. Khezerloo Convergence, Consistency and
stability in fuzzy differential equations, Iranian Journal Fuzzy Systems, 12(3) (2015), 95{
112.
[13] S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, J.
Oper. Res. Soc., 36(4) (1985), 335{338.
[14] P. Guchhait, M. K. Maiti and M. Maiti, A production inventory model with fuzzy produc-
tion and demand using fuzzy differential equation: An interval compared genetic algorithm
approach, Eng. Appl. Artif. Intel., 26(7) (2013), 766{778.
[15] Y. F. Huang, Supply chain model for the Retailer's ordering policy under two levels of delay
payments derived algebraically, Opsearch, 44(8) (2007), 366{377.
[16] N. N. Karnik and J. M. Mendel, Centroid of a type-2 fuzzy set, Information Sciences, 132(6)
(2001), 195{220.
[17] A. Kandel and W. J. Byatt, Fuzzy differential equations. In Proceedings of the International
Conference on Cybernetics and Society, Tokyo, November 1978, 1213{1216.
[18] F. Liu, An efficient centroid type-reduction strategy for general type-2 fuzzy logic system,
Information Sciences, 178(7) (2008), 2224{2236.
[19] P. Majumder, U. K. Bera and M. Maiti, An EPQ model for two-warehouse in unremitting
release pattern with two level trade credit period concerning both supplier and retailer, Appl.
Math. Comput., 274(6) (2016), 430{458.
[20] M. Mizumoto and K. Tanaka, Fuzzy sets of type-2 under algebraic product and algebraic sum,
Fuzzy Sets and Systems, 5(3) (1981), 277{280.
[21] J. S. Martnez, R. I. John, D. Hissel and M. C. Pera, A survey-based type-2 fuzzy logic system
for energy management in hybrid electrical vehicles, Information Sciences, 190(9) (2012),
192{207.
[22] J. M. Mendel and R. I. John, Type-2 fuzzy sets made simple, IEEE Transactions on Fuzzy
Systems, 10(2) (2002), 307{315.
[23] M. K. Maiti and M. Maiti, Fuzzy inventory model with two warehouses under possibility
constraints, Fuzzy Sets Syst, 157(8) (2006), 52{73.
[24] S. M. Mousavi, S. Hajipour and N. N. Aalikar, A multi-product multi-period inventory
control problem under inflation and discount: a parameter-tuned particle swarm optimization
algorithm, Int. J. Adv. Manuf. Tech., 33(4) (2013), 1{18.
[25] S. M. Mousavi, J. Sadeghi, S. T. A. Niaki, N. Alikar, A. Bahreininejad and H. Metselaar,
Two parameter-tuned meta-heuristics for a discounted inventory control problem in a fuzzy
environment, Information Sciences, 276(8) (2014), 42{62.
[26] S. M. Mousavi, J. Sadeghi, S. T. A. Niaki and M. Tavana, A bi-objective inventory optimiza-
tion model under inflation and discount using tuned Pareto-based algorithms: NSGA-II,
NRGA, and MOPSO, Applied Soft Computing, 43(6) (2016), 57{72.
[27] S. M. Mousavi, A. Bahreininejad, N. Musa and F. Yusof, A modi fied particle swarm opti-
mization for solving the integrated location and inventory control problems in a two-echelon
supply chain network, J. intell. Manuf., 23(4) (2014), 1{16.
[28] L. Y. Ouyang, C. H. Hob and C. H. Su, An optimization approach for joint pricing and
ordering problem in an integrated inventory system with order-size dependent trade credit,
Comput. Indust. Eng., 57(7) (2009), 920{930.
[29] S. Pal, M. K. Maiti and M. Maiti, An EPQ model with price discounted promotional demand
in an imprecise planning horizon via Genetic Algorithm, Comput. Indust. Eng., 57(6) (2009),
181{187.
[30] S. H. R. Pasandideh, S. T. A. Niaki and S. M. Mousavi, Two metaheuristics to solve a multi-
item multiperiod inventory control problem under storage constraint and discounts, Int. J.
Adv. Manuf. Technol., 69(7) (2013), 1{14.
[31] T. Pathinathan and K. Ponnivalavan, Pentagonal fuzzy numbers, Int. J. Comput. Algm.,
3(4) (2014), 1003{1005.
[32] R. Qin, Y. K. Liu and Z. Q. Liu, Methods of critical value reduction for type-2 fuzzy variables
and their applications, J. Comput. Appl. Math., 235(7) (2011), 1454{1481.
[33] S. Sharan, S. P. Tiwary and V. K. Yadav, Interval type-2 fuzzy rough sets and interval type-2
fuzzy closure spaces, Iranian Journal of Fuzzy Systems, 12(3) (2015), 113{125.
[34] N. H. Shah and L. E. Cardenas-Barron, Retailer's decision for ordering and credit policies
for deteriorating items when a supplier offers order-linked credit period or cash discount,
Appl. Math. Comp., 259(5) (2015), 569{578.
[35] L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions
and interval differential equations, Nonlinear Analysis, 71(4) (2009), 1311{1328.
[36] B. Sarkar, S. Saren and L. E. Cardenas-Barron, An inventory model with trade-credit policy
and variable deterioration for fi xed lifetime products, Ann. Oper. Res., 229(1) (2015), 677{
702.
[37] S. Tiwari, L. E. Cardenas-Barron, A. Khanna and C. K. Jaggi, Impact of trade credit and
inflation on retailer's ordering policies for non-instantaneous deteriorating items in a two-
warehouse environment, Int. J. Prod. Econ., 176(3) (2016), 154{169.
[38] J. Wu, F. B. Al-khateeb, J. T. Teng and L. E. Cardenas-Barron, Inventory models for dete-
riorating items with maximum lifetime under downstream partial trade credits to credit-risk
customers by discounted cash-flow analysis, Int. J. Prod. Eco., 171(1) (2016), 105{115.
[39] J. Wu, L. Y. Ouyang, L. E. Cardenas-Barron and S. K. Goyal, Optimal credit period and lot
size for deteriorating items with expiration dates under two-level trade credit fi nancing, Eur.
J. Oper. Res., 237(3) (2014), 898{908.
[40] P. S. You, S. Ikuta and Y. C. Hsieh, Optimal ordering and pricing policy for an inventory
system with trial periods, Appl. Math. Model., 34(4) (2010), 3179{3188.
[41] L. A. Zadeh, The concept of a linguistic variable and its application to approximate resoning
I, Information Sciences, 8(2) (1975), 199{249.
[42] L. A. Zadeh, The concept of a linguistic variable and its application to approximate resoning
II, Information Sciences, 8(2) (1975), 301{357.
Iranian Journal of Fuzzy Systems
Volume 15, Issue 1
January and February 2018
Pages 1-24

Files

Share

How to cite

Statistics