, 44:177 | Cite as

An EPQ model for three-layer supply chain with partial backordering and disruption: Triangular dense fuzzy lock set approach



This article deals with three-echelon supply chain (SC) network involving flow of raw materials with imperfect quality, the manufacturer and multiple retailers under the effect of learning experiences in fuzzy decision-making process. Existing literature explores the SC model under full backordering and disruption. Thus, in this study we first develop a production-inventory control problem accompanied with partial backlogging and random disruptions. Any batch received from the supplier is inspected by the manufacturer and if any of them are found to be flawed then all the goods in the inspected batch are rejected. However, we present a case study for problem definition and to comprehend the model into practical applicability. To minimize the aggregate cost of the SC we have utilized the Triangular dense fuzzy lock set for controlling the cost vector of the proposed objective function of the model. Utilizing new defuzzification method and applying the proper keys, chosen by the decision maker, it is possible to minimize the average system cost exclusively. Finally, graphical illustrations and sensitivity analysis are made to justify the model.


Production inventory partial backlogging disruption supply chain triangular dense fuzzy lock set optimization 



The authors are thankful to the anonymous reviewers for their valuable comments and suggestions to improve the quality of the presentation of this article.


  1. 1.
    Harris F 1913 How many parts to make at once. Fact. Mag. Manag. 10: 135–136Google Scholar
  2. 2.
    Taft E W 1918 The most economical production lot. Iron Age 101: 1410–1412Google Scholar
  3. 3.
    Li J, Wang S and Cheng T C E 2008 Analysis of postponement strategy by EPQ-based models with planned backorders. Omega 36: 777–88Google Scholar
  4. 4.
    Zhang R Q 2009 A note on the deterministic EPQ with partial backordering. Omega 37 (5): 1036–1038Google Scholar
  5. 5.
    Chiu Y S P and Ting C K 2010 Determining the optimal run time for EPQ model with scrap, rework, and stochastic breakdowns. Eur. J. Oper. Res. 201 (2): 641–643zbMATHGoogle Scholar
  6. 6.
    Sana S S 2010 An economic production lot size model in an imperfect production system. Eur. J. Oper. Res. 201(1): 158–170zbMATHGoogle Scholar
  7. 7.
    Montgomery D C, Bazaraa M S and Keswani A K 1973 Inventory models with a mixture of backorders and lost sales. Nav. Res. Logist. Q. 20(2): 255-263zbMATHGoogle Scholar
  8. 8.
    Hsieh T P and Dye C Y 2012 A note on The EPQ with partial backordering and phase-dependent backordering rate. Omega 40(1): 131–133Google Scholar
  9. 9.
    Taleizadeh A A, Pentico D W, Aryanezhad M and Ghoreyshi S M 2012 An economic order quantity model with partial backordering and a special sale price. Eur. J. Oper. Res. 221(3): 571–583MathSciNetzbMATHGoogle Scholar
  10. 10.
    Pentico D W, Drake M J and Toews C 2011 The EPQ with partial backordering and phase-dependent backordering rate. Omega 39 (5): 574–577zbMATHGoogle Scholar
  11. 11.
    Zhang R Q, Kaku I and Xiao Y Y 2011 Deterministic EOQ with partial backordering and correlated demand caused by cross-selling. Eur. J. Oper. Res. 210 (3): 537–551MathSciNetzbMATHGoogle Scholar
  12. 12.
    Sicilia J, San-José L A and García-Laguna J 2012 An inventory model where backordered demand ratio is exponentially decreasing with the waiting time. Ann.Oper. Res. 199: 137–155MathSciNetzbMATHGoogle Scholar
  13. 13.
    San-José L A, García-Laguna J and Sicilia J 2009 An economic order quantity model with partial backlogging under general backorder cost function. TOP 17: 366–384MathSciNetzbMATHGoogle Scholar
  14. 14.
    San-José L A, Sicilia J and García-Laguna J 2009A General model for EOQ inventory systems with partial backlogging and linear shortage costs. Int. J. Syst. Sci. 40(1): 59–71MathSciNetzbMATHGoogle Scholar
  15. 15.
    Karimi-Nasab M and Wee H M 2015. An inventory model with truncated exponential replenishment intervals and special sale offer. J. Manuf. Syst. Google Scholar
  16. 16.
    San-José L A, Sicilia J and García-Laguna J 2014 Optimal lot size for a production–inventory system with partial backlogging and mixture of dispatching policies. Int. J. Prod. Econ. 155: 194–203Google Scholar
  17. 17.
    Mak K L 1987 Determining optimal production–inventory control policies for an inventory system with partial backlogging. Comput. Oper. Res. 14(4): 299–304zbMATHGoogle Scholar
  18. 18.
    Pentico D W and Drake M J 2009 The deterministic EOQ with partial backordering: a new approach. Eur. J. Oper. Res. 194(1): 102–113MathSciNetzbMATHGoogle Scholar
  19. 19.
    Parlar M and Berkin D 1991 Future supply uncertainty in EOQ models. Nav. Res. Logist. 38: 107–121MathSciNetzbMATHGoogle Scholar
  20. 20.
    Wee H M, Yu J and Chen M C 2007 Optimal inventory model for items with imperfect quality and shortage backordering. Omega 35: 7–11Google Scholar
  21. 21.
    Chang H C and Ho C H 2009 Exact closed-form solutions for optimal inventory model for items with imperfect quality and shortage backordering. Omega 38 (3–4): 233–237Google Scholar
  22. 22.
    Salehi H, Taleizadeh A A and Tavakkoli-Moghaddam R 2016 An EOQ model with random disruption and partial backordering. Int. J. Prod. Res. 54(9): 2600–2609Google Scholar
  23. 23.
    Chiu S W, Chou C L and Wu W K 2013 Optimizing replenishment policy in an EPQ-based inventory model with nonconforming items and breakdown. Econ. Model. 35: 330–337Google Scholar
  24. 24.
    Paul S K, Sarker R and Essam D 2015 Managing disruption in an imperfect production–inventory system. Comput. Ind. Eng. 84: 101–112Google Scholar
  25. 25.
    Hu F, Lim C C and Lu Z 2014 Optimal production and procurement decisions in a supply chain with an option contract and partial backordering under uncertainties. Appl. Math. Comput. 232 (1): 1225–1234MathSciNetzbMATHGoogle Scholar
  26. 26.
    Skouri K, Konstantaras I, Lagodimos A G and Papachristos S 2014 An EOQ model with backorders and rejection of defective supply batches. Int. J. Prod. Econ. 155: 148–154Google Scholar
  27. 27.
    Konstantaras I, Skouri K and Lagodimos A G 2019 EOQ with independent endogenous supply disruptions. Omega 83: 96–106Google Scholar
  28. 28.
    Ritha W and Francina Nishandhi I 2015 Single vendor multi buyer’s integrated inventory model with rejection of defective supply batches. Int. J. Math. Comput. Res. 3(10): 1182–1187Google Scholar
  29. 29.
    Heimann D and Waage F 2007 A closed-form approximation solution for an inventory model with supply disruption and non-ZIO reorder policy. J. Syst. Cybern. Inform. 5(4): 1–12Google Scholar
  30. 30.
    Synder L V 2014 A tight approximation for an EOQ model with supply disruptions. Int. J. Prod. Econ. 155: 91–108Google Scholar
  31. 31.
    Synder L V, Atan Z, Peng P, Rong Y, Schmitt A J and Sinsoysal B 2016 OR/MS models for supply chain disruptions: A review. IIE Trans. 48(2): 89–109Google Scholar
  32. 32.
    Berk E and Arreola-Risa A 1994 Note on “Future supply uncertainty in EOQ models”. Nav. Res. Logist. 41(1): 129–132zbMATHGoogle Scholar
  33. 33.
    Atan Z and Synder L V 2012 Inventory strategies to manage supply disruptions. In: Gurnani H, Mehrotra A, Ray S (Eds), Managing supply disruption. Berlin: SpringerGoogle Scholar
  34. 34.
    Kumar R S and Goswami A 2015 A fuzzy random EPQ model for imperfect quality items with possibility and necessity constraints. Appl. Soft Comput. 34: 838–850Google Scholar
  35. 35.
    Kumar R S and Goswami A 2015 EPQ model with learning consideration, imperfect production and partial backlogging in fuzzy random environment. Int. J. Syst. Sci. 46: 1486–1497zbMATHGoogle Scholar
  36. 36.
    Mahata G C 2017 A production-inventory model with imperfect production process and partial backlogging under learning considerations in fuzzy random environments. J. Intell. Manuf. 28(4): 883–897Google Scholar
  37. 37.
    Shekarian E, Olugu E U, Abdul-Rashid S H and Kazemi N 2016 An economic order quantity model considering different holding costs for imperfect quality items subject to fuzziness and learning. J. Intell. Fuzzy Syst. 30(5): 2985–2997zbMATHGoogle Scholar
  38. 38.
    De S K and Beg I 2016 Triangular dense fuzzy sets and new defuzzification methods. Int. J. Intell. Fuzzy Syst. 31(1): 469–477zbMATHGoogle Scholar
  39. 39.
    [39]De S K and Beg I 2016 Triangular dense fuzzy Neutrosophic sets. Neutrosophic Sets Syst. 13: 1–12Google Scholar
  40. 40.
    De S K and Mahata G C 2017 Decision of a fuzzy inventory with fuzzy backorder model under cloudy fuzzy demand rate. Int. J. Appl. Comput. Math. 3(3): 2593–2609MathSciNetzbMATHGoogle Scholar
  41. 41.
    Karmakar S, De S K and Goswami A 2017 A pollution sensitive dense fuzzy economic production quantity model with cycle time dependent production rate. J. Clean. Prod. 154: 139–150Google Scholar
  42. 42.
    De S K and Sana S S 2016 An EOQ model with backlogging. Int. J. Manag. Sci. Eng. Manag. 11: 143–154Google Scholar
  43. 43.
    De S K and Sana S S 2015 Backlogging EOQ model for promotional effort and selling price sensitive demand- an intuitionistic fuzzy approach. Ann. Oper. Res. 233(1): 57–76MathSciNetzbMATHGoogle Scholar
  44. 44.
    De S K and Sana S S 2013 Fuzzy order quantity inventory model with fuzzy shortage quantity and fuzzy promotional index. Econ. Model. 31: 351–358Google Scholar
  45. 45.
    Karmakar S, De S K and Goswami A 2017 A deteriorating EOQ model for natural idle time and imprecised demand: Hesitant fuzzy approach. Int. J. Syst. Sci. Oper. Logist. 4(4): 297–310Google Scholar
  46. 46.
    Chakraborty D, Jana D K and Roy T K 2015 Multi-item integrated supply chain model for deteriorating items with stock dependent demand under fuzzy random and bifuzzy environments. Comput. Ind. Eng. 88: 166–180Google Scholar
  47. 47.
    Mahata G C and Goswami A 2013 Fuzzy inventory models for items with imperfect quality and shortage backordering under crisp and fuzzy decision variables. Comput. Ind. Eng. 64: 190–199Google Scholar
  48. 48.
    Mahata G C and Goswami A 2007 An EOQ model for deteriorating items under trade credit financing in the fuzzy sense. Prod. Plann. Control 18: 681–692Google Scholar
  49. 49.
    Mahata G C and Mahata P 2011 Analysis of a fuzzy economic order quantity model for deteriorating items under retailer partial trade credit financing in a supply chain. Math. Comput. Modell. 53: 1621–1636MathSciNetzbMATHGoogle Scholar
  50. 50.
    Xu Z S and Zhou W 2017 Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment. Fuzzy Optim. Decis. Mak. 16(4): 481–503MathSciNetzbMATHGoogle Scholar
  51. 51.
    Wang H and Xu Z S 2016 Multi groups decision making using intuitionistic-valued hesitant fuzzy information. Int. J. Comput. Intell. Syst. 9: 468–482Google Scholar
  52. 52.
    Ding J, Xu Z S and Zhao Z 2017 An interactive approach to probabilistic hesitant fuzzy multi-attribute group decision making with incomplete weight information. J. Intell. Fuzzy Syst. 32: 2523–2536zbMATHGoogle Scholar
  53. 53.
    De S K 2018 Triangular dense fuzzy lock sets. Soft Comput. 22(21): 7243–7254zbMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of Mathematics (UG & PG Units)Midnapore College (Autonomous)Midnapore, Paschim MedinipurIndia
  2. 2.Department of MathematicsSidho-Kanho-Birsha University, Purulia Sainik SchoolPuruliaIndia

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