Solution of an economic production quantity model using the generalized Hukuhara derivative approach

Document Type : Article

Authors

1 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, India

2 Department of Mathematics, Raja Narendra Lal Khan Women's College (Autonomous), Midnapore-721102, W.B., India

3 Department of Mathematics, Midnapore College (Autonomous), Midnapore-721101, W.B, India

4 Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata, Nadia-741249, W.B, India

Abstract

In this study, an economic production quantity (EPQ) model with deterioration is developed where the production rate is stock dependent and the demand rate is unit selling price and stock dependent. The low unit selling price and more stocks correspond high demand but more stock corresponds to slow production because of the avoidance of unnecessary stocks. First of all, we develop the production model by solving some ordinary differential equations having deterministic profit function under some specific assumptions. Later, we develop the fuzzy model by solving the fuzzy differential equations using Generalized Hukuhara derivative. In fact, the differential equation of the model has been split into two parts namely gH(L-R) and gH(R-L) on the basis of left(L) and right(R) α- cuts of fuzzy numbers for which the problem itself is transformed into multi-objective EPQ problem. A new formula of aggregation of several objective values obtained at different aspiration levels has been discussed to defuzzify the fuzzy multi-objective problems. We solve the crisp and fuzzy models using LINGO software. Numerical and graphical illustrations confirm that the model under Generalized Hukuhara derivative of (R-L) type contributes more profit which is one of the basic novelties of the proposed approach.

Keywords

References

References:
1. Harris, F.W. “How many parts to make at once. Factory”, The Magazine of Management, 10(2), pp. 135–136 (1915). 
2. Taft, E.W. “The most economical production lot”, Iron Age, 101, pp. 1410–1412 (1918). 
3. Taleizadeh, A.A. “Lot sizing model with advance payment pricing and disruption in supply under planned partial back ordering”, International Transactions in Operational Research, 24(4), pp.783–
800 (2017). https://doi.org/10.1111/itor.12297.
4. Lashgari, M., Taleizadeh, A.A., and Sana, S.S. “An inventory control problem for deteriorating items with back ordering and financial considerations under two level of trade credit linked to order quantity”, Journal of
Industrial and Management Optimization, 12(3), pp. 1091–1119 (2016). http://dx.doi.org/10.3934/jimo.2016.12.1091.
5. Taleizadeh, A.A. and Pentico, D.W. “An economic order quantity model with a known price increase and partial backordering”, European Journal of Operational Research, 228(3), pp. 516–525 (2013).
https://doi.org/10.1016/j.ejor.2013.02.014.
6. Zhang, C., You, M., and Han, G. “Integrated supply chain decisions with credit-linked demand: A Stackelberg approach”, Scientia Iranica, 28(2), pp. 927–949 (2021). DOI: 10.24200/sci.2019.50342.1646.
7. Al-Amin Khan, M., Shaikh, A.A., Konstantaras, I., et al. “Inventory models for perishable items with advanced payment, linearly time-dependent holding cost and demand dependent on advertisement and selling price”, International Journal of Production
Economics, 230, 107804 (2020). https://doi.org/10.1016/j.ijpe.2020.107804.
8. Al-Amin Khan, M., Ahmed, S., Babu, M.S., et al. “Optimal lot-size decision for deteriorating items with price-sensitive demand, linearly time dependent holding cost under all-units discount environment”,
International Journal of Systems Science: Operations & Logistics, 9(1), pp. 61–74 (2020). https://doi.org/10.1080/23302674.2020.1815892.
9. Al-Amin Khan, M., Shaikh, A.A., Panda, G., et al. “Inventory system with expiration date: Pricing and replenishment decisions”, Computers & Industrial Engineering, 132, pp. 232–247 (2019). https://doi.org/10.1016/j.cie.2019.04.002
10. Kim, J., Huang, H., and Shinn, S. “An optimal policy to increase supplier’s profit wit price dependent demand functions”, Production Planning and Control, 6(1), pp. 45–50 (1995). https://doi.org/10.1080/09537289508930252
11. Mukhopadhyay, S., Mukherjee, R.N., and Chaudhuri, K.S. “Joint pricing and ordering policy for a deteriorating inventory”, Computers & Industrial Engineering, 47(4), pp. 339–349 (2004). https://doi.org/10.1016/j.cie.2004.06.007
12. Sana, S.S. “Price sensitive demand and random sales price- a news boy problem”, International Journal of Systems Science, 43(3), pp. 491–498 (2012). https://doi.org/10.1080/00207721.2010.517856
13. Pal, B., Sana, S.S., and Chaudhuri, K. “Two echelon manufactures retailer supply chain strategies with price, quality and promotional effort sensitive demand”, International Transaction of Operational
Research, 22(6), pp. 1071–1095 (2015). https://doi.org/10.1111/itor.12131.
14. Alfares, H.K. and Ghaithan, A.M. “Inventory and pricing model with price-dependent demand, time varying holding cost and quantity discounts”, Computers & Industrial Engineering, 94, pp. 170–177
(2016). https://doi.org/10.1016/j.cie.2016.02.009.
15. Giri, B.C., Pal, S., Goswami, A., et al. “An inventory model for deteriorating items with stock dependent demand rate”, European Journal of Operational Research, 95(3), pp. 604–610 (1996).
https://doi.org/10.1016/0377-2217(95)00309-6.
16. Mandal, M., Roy, T.K., and Maity, M. “A fuzzy inventory model of deteriorating items with stock dependent demand under limited storage space”, OPSEARCH, 35(4), pp. 323–337 (1998). https://doi.org/10.1007/BF03398552
17. Mandal, M. and Maity, M. “Inventory of damageable items with variable replenishment rate, stock dependent demand and some units in hand”, Applied Mathematical Modelling, 23(10), 799–807 (1999).
https://doi.org/10.1016/S0307-904X(99)00018-9.
18. Dye, C.Y. “A deteriorating inventory model with stock dependent demand and partial backlogging under conditions of permissible delay in payments”, OPSEARCH, 39(3–4), pp. 189–201 (2002).
https://doi.org/10.1007/BF03398680.
19. Alfares, H.K. “Inventory model with stock level dependent demand rate and variable holding cost”, International Journal of Production Economics, 108(1–2), pp. 259–265 (2007). https://doi.org/10.1016/j.ijpe.2006.12.013
20. Datta, T.K. and Paul, K. “An inventory system with stock dependent, price sensitive demand rate”, Production Planning and Control, 12(1), pp. 13–20 (2001). https://doi.org/10.1080/09537280150203933
21. Teng, J.T. and Chang, C.T. “Economic production quantity models for deteriorating items with pricestock dependent demand”, Computers and Operations Research, 32(2), pp. 297–308 (2005).
https://doi.org/10.1016/S0305-0548(03)00237-5.
22. Sana, S.S. “An EOQ model for salesman initiatives, stock and price dependent demand of similar products –a dynamical system”, Applied Mathematics and Computation, 218(7), pp. 3277–3288 (2011). https://doi.org/10.1016/j.amc.2011.08.067
23. Al-Amin Khan, M., Shaikh, A.A., Panda, G.C., et al. “The effect of advance payment with discount facility on supply decisions of deteriorating products whose demand is both price and stock dependent”,
International Transactions in Operational Research, 27(3), pp. 1343–1367 (2020). https://doi.org/10.1111/itor.12733.
24. Sarkar, B., Mandal, B., and Sarkar, S. “Quality improvement and backorder price discount under controllable lead time in an inventory model”, Journal of Manufacturing Systems,35, pp. 26–36 (2015).
https://doi.org/10.1016/j.jmsy.2014.11.012.
25. Sarkar, B., Tayyab, M., Kim, N., et al. “Optimal production delivery policies for supplier and manufacturer in a constrained closed-loop supply chain for returnable transport packaging through
metaheuristic approach”, Computers & Industrial Engineering, 135, pp. 987–1003 (2019). https://doi.org/10.1016/j.cie.2019.05.035.
26. Sebatjane, M. and Adetunji, O. “Optimal lot-sizing and shipment decisions in a three-echelon supply chain for growing items with inventory level- and expiration date-dependent demand”, Applied Mathematical
Modelling, 90, pp. 1204–1225 (2021). https://doi.org/10.1016/j.apm.2020.10.021.
27. Sarkar, B., Sarkar, M., Ganguly, B., et al. “Combined effects of carbon emission and production quality improvement for fixed lifetime products in a sustainable supply chain management”, International
Journal of Production Economics, 231, 107867 (2021). https://doi.org/10.1016/j.ijpe.2020.107867.
28. Zadeh, L.A. “Fuzzy sets”, Information and Control, 8(3), pp. 338-353 (1965). 
29. Bellman, R.E. and Zadeh, L.A. “Decision making in a fuzzy environment”, Management Science, 17, pp. 141–164 (1970). https://doi.org/10.1287/mnsc.17.4.B141.
30. Park, K.S. “Fuzzy set theoretic interpretation of economic order quantity”, IEEE Transactions on Systems, Man and Cybernetics, 17(6), pp. 1082–1084 (1987). DOI: 10.1109/TSMC.1987.6499320.
31. De, S.K. and Beg, I. “Triangular dense fuzzy sets and new defuzzification methods”, Journal of Intelligent & Fuzzy Systems, 31, pp. 469–477 (2016). DOI:  10.3233/IFS-162160.
32. De, S.K. and Beg, I. “Triangular dense fuzzy Neutrosophic sets”, Neutrosophic Sets & Systems, 13, pp. 24–37 (2016). DOI: 10.3233/IFS-162160.
33. De, S.K. “Triangular dense fuzzy lock set”, Soft Computing, 22(21), pp. 7243–7254 (2018). https://doi.org/10.1007/s00500-017-2726-0.
34. De, S.K., and Mahata, G.C. “A comprehensive study of an economic order quantity model under fuzzy monsoon demand”, Sadhana, 44(89), pp. 1–12 (2019). https://doi.org/10.1007/s12046-019-1059-3.
35. Maity, S., Chakraborty, A., De, S.K., et al. “A comprehensive study of a backlogging EOQ model with nonlinear heptagonal dense fuzzy environment”, RAIRO-Operations Research, 56, pp. 267–286 (2018).
36. Maity, S., De, S.K., and Mondal, S.P. “A study of an EOQ model under lock fuzzy environment”, Mathematics, 7(75), pp. 1–23 (2019). https://doi.org/10.3390/math7010075.
37. Maity, S., De, S.K., and Pal, M. “Two decision makers’ single decision over a back order EOQ model with dense fuzzy demand rate”, Finance and Market, 3(1), pp.1–11 (2018).
38. Karmakar, S., De, S.K., and Goswami, A. “A pollution sensitive dense fuzzy economic production quantity model with cycle time dependent production rate”, Journal of Cleaner Production, 154, pp. 139–150
(2017). https://doi.org/10.1016/j.jclepro.2017.03.080.
39. Karmakar, S., De, S.K., and Goswami, A. “A pollution sensitive remanufacturing model with waste items: Triangular dense fuzzy lock set approach”, Journal of Cleaner Production, 187, pp. 789–803 (2018).
https://doi.org/10.1016/j.jclepro.2018.03.161.
40. Rahaman, M., Mondal, S.P., Alam, S., et al. “Synergetic study of inventory management problem in uncertain environment based on memory and learning effects”, Sādhanā, 46(39), pp. 1–20 (2021).
41. De, S.K. and Mahata, G.C. “A cloudy fuzzy economic order quantity model for imperfect-quality items with allowable proportionate discounts”, Journal of Industrial Engineering International, 15, pp. 571–583
(2019). https://doi.org/10.1007/s40092-019-0310-1.
42. De, S.K. and Mahata, G.C. “Decision of a fuzzy inventory with fuzzy backorder model under cloudy fuzzy demand rate”, International Journal of Applied and Computational Mathematics, 3, pp. 2593–2609
(2017). https://doi.org/10.1007/s40819-016-0258-4.
43. Aghamohagheghi, M., Hashemi, S., and Tavakkoli- Moghaddam, R. “A new decision approach to the sustainable transport investment selection based on the generalized entropy and knowledge measure under an interval-valued Pythagorean fuzzy
environment”, Scientia Iranica, 28(2), pp. 892–911 (2021). DOI: 10.24200/sci.2019.50131.1529.
44. Guleria, A. and Bajaj, R. “T-spherical fuzzy soft sets and its aggregation operators with application in decision-making”, Scientia Iranica, 28(2), pp. 1014– 1029 (2021). DOI: 10.24200/sci.2019.53027.3018
45. Chang, S. L. and Zadeh, L.A. “On fuzzy mapping and control”, a IEEE Transaction on Systems, Man and Cybernetics, 2(1), pp. 30–34 (1972). DOI: 10.1109/TSMC.1972.5408553.
46. Kaleva, O. “Fuzzy differential equations”, Fuzzy Sets and Systems, 24(3), pp. 301–317 (1987). https://doi.org/10.1016/0165-0114(87)90029-7.
47. Bede, B. “A note on two-point boundary value problems associated with nonlinear fuzzy differential equations”, Fuzzy Sets and Systems, 157(7), pp. 986– 989 (2016). https://doi.org/10.1016/j.fss.2005.09.006
48. Bede, B., Rudas, I.J, and Bencsik, A.L. “First order linear fuzzy differential equations under generalized differentiability”, Information Science, 177(7), 1648– 1662 (2007). https://doi.org/10.1016/j.ins.2006.08.021
49. Bede, B. and Gal, S.G. “Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations”, Fuzzy Sets and Systems, 151(3), pp. 581–599 (2005).
https://doi.org/10.1016/j.fss.2004.08.001.
50. Allahviranloo, T., Abbasbandy, S., Salahsour, S., et al. “A new method for solving fuzzy linear differential equations”, Computing, 92(2), pp. 181–197 (2011). https://doi.org/10.1007/s00607-010-0136-6
51. Buckley, J.J. and Feuring, T. “Fuzzy differential equations”, Fuzzy Sets and Systems, 110(1), pp. 43–54 (2000). https://doi.org/10.1016/S0165-0114(98)00141-9.
52. Buckley, J.J. and Feuring, T. “Fuzzy initial value problem for n-th order linear differential equations”, Fuzzy Sets and Systems, 121(2), pp. 247–255 (2001). https://doi.org/10.1016/S0165-0114(00)00028-2
53. Allahviranloo, T. and Ahmadi, M.B. “Fuzzy Lapalace transforms”, Soft Computing, 14(3), pp. 235–243 (2010). https://doi.org/10.1007/s00500-008-0397-6.
54. Mondal, S.P. and Roy, T.K. “First order linear homogeneous fuzzy ordinary differential equation based on the Lagrange multiplier method”, Journal of Soft Computing and Applications, pp. 1–17 (2013).
DOI: 10.5899/2013/jsca-00032.
55. Rahaman, M., Mondal, S.P., Algehyne, E.A., et al. “A method for solving linear difference equation in Gaussian fuzzy environments”, Granular Computing, 7, pp. 63-76 (2021). https://doi.org/10.1007/s41066-
020-00251-1.
56. Das, B., Mahapatra, N.K., and Maity, M. “Initialvalued first order fuzzy differential equation in bi-level inventory model with fuzzy demand”, Mathematical Modelling and Analysis, 13(4), pp. 493–512 (2008).
https://doi.org/10.3846/1392-6292.2008.13.493-512.
57. Guchhait, P., Maity, M.K., and Maity, M. “A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm approach”, Engineering Applications of Artificial
Intelligence, 26(2), pp. 766–778 (2013). https://doi.org/10.1016/j.engappai.2012.10.017.
58. Mondal, M., Maity, M.K., and Maity, M. “A productionrecycling model with variable demand, demand dependent fuzzy return rate: A fuzzy differential approach”, Computer & Industrial Engineering, 64(1), pp. 318–332 (2013).
https://doi.org/10.1016/j.cie.2012.10.014.
59. Majumder, P., Mondal, S.P., Bera, U.K., et al. “Application of Generalized Hukuhara derivative approach in an economic production quantity model with partial trade credit policy under fuzzy environment”, Operations Researches Perspectives, 3,
pp. 77–91 (2016). https://doi.org/10.1016/j.orp.2016.08.002.
60. Mondal, S.P. “Solution of basic inventory model in fuzzy and interval environment: Fuzzy and interval differential equation approach”, Fuzzy and Rough Set Theory in Organizational Decision Making, 4, (2017).
DOI: 10.4018/978-1-5225-1008-6.ch004.
61. Debnath, B.K., Majumder, P., and Bera, U.K. “Multiobjective Sustainable Fuzzy Economic Production Quantity (SFEPQ) model with demand as type-2 fuzzy number: A fuzzy differential equation approach”, Hacettepe Journal of Mathematics & Statistics, 48(1), pp. 112–139 (2019). 
62. Rahaman, M., Mondal, S.P., Alam, S., et al. “Interpretation of exact solution for fuzzy fractional nonhomogeneous differential equation under the Riemann– Liouville sense and its application on the inventory management control problem”, Granular Computing, 6, pp. 953-976 (2020). https://doi.org/10.1007/s41066-020-00241-3.
63. Rahaman, M., Mondal, S.P., Shaikh, A.A., et al. “Artificial bee colony optimization-inspired synergetic study of fractional-order economic production quantity model”, Soft Computing, 24, pp. 15341–15359 (2020). https://doi.org/10.1007/s00500-020-04867-y.
Scientia Iranica
Volume 31, Issue 22
Transactions on Industrial Engineering (E)
November and December 2024
Pages 2096-2111

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  • Receive Date: 13 May 2020
  • Revise Date: 18 July 2021
  • Accept Date: 22 November 2021

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