Journal of Intelligent Manufacturing

, Volume 29, Issue 8, pp 1753–1771 | Cite as

The (pqrl) model for stochastic demand under Intuitionistic fuzzy aggregation with Bonferroni mean

  • Sujit Kumar De
  • Shib Sankar SanaEmail author


This paper investigates a hill type economic production-inventory quantity (EPIQ) model with variable lead-time, order size and reorder point for uncertain demand. The average expected cost function is formulated by trading off costs of lead-time, inventory, lost sale and partial backordering. Due to the nature of the demand function, the frequent peak (maximum) and valley (minimum) of the expected cost function occur within a specific range of lead time. The aim of this paper is to search the lowest valley of all the valley points (minimum objective values) under fuzzy stochastic demand rate. We consider Intuitionistic fuzzy sets for the parameters and used Intuitionistic Fuzzy Aggregation Bonferroni mean for the defuzzification of the hill type EPIQ model. Finally, numerical examples and graphical illustrations are made to justify the model.


Manufacturing Lead time Fuzzy system Bonferroni mean Optimization 


  1. Allahviranloo, T., & Saneifard, R. (2012). Defuzzification method for ranking fuzzy numbers based on center of gravity. Iranian Journal of Fuzzy Systems, 9, 57–67.Google Scholar
  2. Atanassov, K., & Gargov, G. (1989). Interval valued intuitionistic fuzzy sets. Fuzzy Sets and Systems, 31, 343–349.CrossRefGoogle Scholar
  3. Atanassov, K. (1999). Intuitionistic fuzzy sets: Theory and applications. Berlin: Physica Verlag.CrossRefGoogle Scholar
  4. Atanassov, K. (1986). Intuitionistic fuzzy sets and system. Fuzzy Sets and Systems, 20, 87–96.CrossRefGoogle Scholar
  5. Atanassov, K. (1994). New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets and Systems, 61, 137–142.CrossRefGoogle Scholar
  6. Angelov, P. P. (1997). Optimization in an intuitionistic fuzzy environment. Fuzzy Sets and Systems, 86, 299–306.CrossRefGoogle Scholar
  7. Ayag, Z., Samanlioglu, F., & Bykzkan, G. (2013). A fuzzy QFD approach to determine supply chain management strategies in the dairy industry. Journal of Intelligent Manufacturing, 24, 1111–1122.CrossRefGoogle Scholar
  8. Bandyopadhyay, S., & Bhattacharya, R. (2015). Finding optimum neighbour for routing based on multi-criteria, multi-agent and fuzzy approach. Journal of Intelligent Manufacturing, 26, 25–42.CrossRefGoogle Scholar
  9. Banerjee, S., & Roy, T. K. (2010). Probabilistic inventory model with fuzzy cost components and fuzzy random variable. International Journal of Computational and Applied Mathematics, 5, 501–514.Google Scholar
  10. Beg, I., & Rashid, T. (2014). Multi-criteria trapezoidal valued intuitionistic fuzzy decision making with choquet, integral based TOPSIS. Opsearch, 51, 98–129.CrossRefGoogle Scholar
  11. Bellman, R. E., & Zadeh, L. A. (1970). Decision making in a fuzzy environment. Management Science, 17, B141–B164.CrossRefGoogle Scholar
  12. Beliakov, G., Pradera, A., & Calvo, T. (2007). Aggregation functions: A guide for practitioners. Heidelberg: Springer.Google Scholar
  13. Beliakov, G., & Jems, S. (2013). On extending generalized Bonferroni means to Attanasov orthopairs. Fuzzy Sets and Systems, 211, 84–98.CrossRefGoogle Scholar
  14. Bonferroni, C. (1950). Sulle medie multiple di potenze. Bolletino Mathematical Italiana, 5, 267–270.Google Scholar
  15. Ben-Daya, M., & Raouf, A. (1994). Inventory models involving lead time as decision variable. Journal of Operational Research Society, 45, 579–582.CrossRefGoogle Scholar
  16. Cardenas-Barron, L. E., Smith, N. R., Martinez-Flores, J. L., & Rodriguez-Salvador, M. (2010). Modelling lead time effects on joint inventory and price optimization. International Journal Logistics Economics and Globalisation, 2, 270–291.CrossRefGoogle Scholar
  17. Cardenas-Barron, L. E., Chung, K. J., & Trevino-Garza, G. (2014). Celebrating a century of the economic order quantity model in honor of Ford Whitman Harris. International Journal of Production Economics, 155, 1–7.CrossRefGoogle Scholar
  18. Chen, S. M., & Tan, J. M. (1994). Handling multi-criteria fuzzy decision making problems based on vague set theory. Fuzzy Sets and Systems, 67, 163–172.CrossRefGoogle Scholar
  19. Chuang, B. R., Ouyang, L. Y., & Lin, Y. J. (2004). Impact of defective items on (Q, r, L) inventory model involving controllable setup cost. Yugoslav Journal of Operations Research, 14, 247–258.CrossRefGoogle Scholar
  20. Dabois, D., Gottwald, S., Hajek, P., Kacprzyk, J., & Prade, H. (2005). Terminological difficulties in fuzzy set theory, the case of intuitionistic fuzzy sets. Fuzzy Sets and Systems, 156, 485–491.CrossRefGoogle Scholar
  21. Deep, K., Singh, K. P., & Kansal, M. L. (2011). Genetic algorithm based fuzzy weighted average for multi-criteria decision making problems. Opsearch, 48, 96–108.CrossRefGoogle Scholar
  22. De, S. K. (2013). EOQ model with natural idle time and wrongly measured demand rate. International Journal of Inventory Control and Management, 3, 329–354.Google Scholar
  23. De, S. K., Biswas, R., & Roy, A. R. (2000). Some operations on intuitionistic fuzzy sets. Fuzzy Sets and Systems, 114, 477–484.CrossRefGoogle Scholar
  24. De, S. K., & Goswami, A. (2006). An EOQ model with fuzzy inflation rate and fuzzy deterioration rate when a delay in payment is permissible. International Journal of Systems Science, 37, 323–335.CrossRefGoogle Scholar
  25. De, S. K., Goswami, A., & Sana, S. S. (2014). An interpolating by pass to Pareto optimality in intuitionistic fuzzy technique for an EOQ model with time sensitive backlogging. Applied Mathematics and Computation, 230, 664–674.CrossRefGoogle Scholar
  26. De, S. K., & Sana, S. S. (2013). Fuzzy order quantity inventory model with fuzzy shortage quantity and fuzzy promotional index. Economic Modelling, 31, 351–358.CrossRefGoogle Scholar
  27. De, S. K., & Sana, S. S. (2015). Backlogging EOQ model for promotional effort and selling price sensitive demand—an intuitionistic fuzzy approach. Annals of Operations Research, 233, 57–76.CrossRefGoogle Scholar
  28. Dymova, L., & Sevastjanov, P. (2011). Operations on intuitionistic fuzzy values in multiple criteria decision making. Scientific Research of the Institute of Mathematics and Computer Science, 1, 41–48.Google Scholar
  29. Grzegorzewski, P. (2002). Nearest interval approximation of a fuzzy number. Fuzzy Sets and Systems, 130, 321–330.CrossRefGoogle Scholar
  30. He, Y. D., Chen, H. Y., Zhou, L. G., Liu, J. P., & Tao, Z. F. (2014a). Intuitionistic fuzzy geometric interaction averaging operators and their application to multi-criteria decision making. Information Sciences, 259, 142–159.CrossRefGoogle Scholar
  31. He, Y. D., Chen, H. Y., Zhou, L. G., Han, B., & Zhao, Q. Y. (2014b). Generalised Intuitionistic fuzzy geometric interaction operators and their application to decision making. Expert Systems with Applications, 41, 2484–2495.CrossRefGoogle Scholar
  32. Hong, D. H., & Choi, C. H. (2000). Multicriteria decision making problems based on vague set theory. Fuzzy Sets and Systems, 114, 103–113.CrossRefGoogle Scholar
  33. Hsu, S. L., & Lee, C. C. (2009). Replenishment and lead time decisions in manufacturer–retailer chains. Transportation Research Part E, 45, 398–408.CrossRefGoogle Scholar
  34. Jaggi, C. K., & Sharma, A. (2014). Fuzzification of EOQ model with allowable shortage under the condition permissible delay in payments. In Mathematical modeling and application (pp. 239–258).Google Scholar
  35. Jakovljevic, Z., Petrovic, P. B., Mikovic, V. D., & Pajic, M. (2014). Fuzzy inference mechanism for recognition of contact states in intelligent robotic assembly. Journal of Intelligent Manufacturing, 25(3), 571–587.CrossRefGoogle Scholar
  36. Kashif, M., & Shahzad, K. H. H. (2013). Integrated supply chain and product family architecture under highly customized demand. Journal of Intelligent Manufacturing., 24, 1005–1018.CrossRefGoogle Scholar
  37. Liao, C. J., & Shyu, C. H. (1991). An analytical determination of lead time with normal demand. International Journal of Operation and Production Management, 11, 72–78.CrossRefGoogle Scholar
  38. Olsson, F. (2014). Analysis of inventory policies for perishable items with fixed lead times and lifetimes. Annals of Operations Research, 217, 399–423.CrossRefGoogle Scholar
  39. Ouyang, L. Y., Yeh, N. C., & Wu, K. S. (1996). Mixture inventory model with backorders and lost sales for variable lead time. Journal of the Operational Research Society, 47, 829–832.CrossRefGoogle Scholar
  40. Ouyang, L. Y., & Wu, K. S. (1997). Mixture Inventory model involving variable lead time with a service level constraint. Computers and Operations Research, 24, 875–882.CrossRefGoogle Scholar
  41. Ouyang, L. Y., & Wu, K. S. (1998). A minimax distribution free procedure for mixed inventory model with variable lead time. International Journal of Production Economics, 56–57, 551–516.Google Scholar
  42. Ouyang, L. Y., & Chuang, B. R. (1999). (Q, R, L) inventory model involving quantity discounts and a stochastic backorder rate. Production Planning and Control, 10, 426–433.CrossRefGoogle Scholar
  43. Ouyang, L. Y., Chuang, B. R., & Wu, K. S. (1999). Optimal inventory policies involving variable lead time with defective items. Journal of the Operational Research Society of India, 36, 374–389.Google Scholar
  44. Ouyang, L. Y., Chuang, B. R., & Lin, Y. J. (2003). Impact of backorder discounts on periodic review inventory model. Information and Management Sciences, 14, 1–13.Google Scholar
  45. Ramli, N., & Mohamad, D. (2009). A comparative analysis of centroid methods in ranking fuzzy numbers. European Journal of Scientific Research, 28, 492–501.Google Scholar
  46. Shin, S. J., Kim, D. B., Shao, G., Brodsky, A., & Lechevalier, D. (2015). Developing a decision support system for improving sustainability performance of manufacturing processes. Journal of Intelligent Manufacturing. doi: 10.1007/s10845-015-1059-z.CrossRefGoogle Scholar
  47. Singh, A., Datta, S., Mahapatra, S. S., Singha, T., & Majumdar, G. (2013). Optimization of bead geometry of submerged arc weld using fuzzy based desirability function approach. Journal of Intelligent Manufacturing, 24, 35–44.CrossRefGoogle Scholar
  48. Takeuti, G., & Tinani, S. (1984). Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. Journal of Symbolic Logic, 49, 851–866.CrossRefGoogle Scholar
  49. Voskoglou, M. G. (2013). Application of the centroid technique for measuring learning skills. Journal of Mathematical Sciences and Mathematics Education, 8, 34–45.Google Scholar
  50. Wang, Z. X., Liu, Y. J., Fan, Z. P., & Feng, B. (2009). Ranking L–R fuzzy number based on deviation degree. Information Sciences, 179, 2070–2077.CrossRefGoogle Scholar
  51. Wei, G. W., Wang, H. J., & Lin, R. (2011). Application of correlation to interval valued intuitionistic fuzzy multiple attribute decision making with incomplete weight information. Knowledge and Information Systems, 26, 337–349.CrossRefGoogle Scholar
  52. Wei, G. W. (2010). Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making. Applied Soft Computing, 10, 423–431.CrossRefGoogle Scholar
  53. Wu, K. S. (2000). (Q, r) inventory model with variable lead time when the amount received is uncertain. Information and Management Sciences, 11, 81–94.Google Scholar
  54. Xia, M. M., Xu, Z. S., & Zhu, B. (2011). Generalized intuitionistic fuzzy Bonferroni means. International Journal of General Systems, 27, 23–47.CrossRefGoogle Scholar
  55. Xia, M. M., Xu, Z. S., & Zhu, B. (2013). Geometric Bonferroni means with their application in multi-criteria decision making. Knowledge Based Systems, 40, 88–100.CrossRefGoogle Scholar
  56. Xu, Z. S. (2007). Intuitionistic fuzzy aggregation operations. IEEE Transactions on Fuzzy Systems, 15, 1179–1187.CrossRefGoogle Scholar
  57. Xu, Z. S., & Hu, H. (2010). Projection models for intuitionistic fuzzy multiple attribute decision making. International Journal of Information Technology and Decision Making, 9, 267–280.CrossRefGoogle Scholar
  58. Xu, Z. S., & Yager, R. R. (2006). Some geometric aggregation operators based on intuitionistic fuzzy sets. International Journal of General Systems, 35, 417–433.CrossRefGoogle Scholar
  59. Xu, Z. S., & Yager, R. R. (2011). Intuitionistic fuzzy Bonferroni means. IEEE Transactions on Fuzzy Cybernetics, Man and Cybernetics-Part-B, 41, 568–578.CrossRefGoogle Scholar
  60. Yager, R. R. (1981). A procedure for ordering fuzzy subsets of the unit interval. Information Sciences, 24, 143–161.CrossRefGoogle Scholar
  61. Yuan, B., Zhang, C., & Shao, X. (2015). A late acceptance hill-climbing algorithm for balancing two-sided assembly lines with multiple constraints. Journal of Intelligent Manufacturing, 26, 159–168.CrossRefGoogle Scholar
  62. Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8, 338–356.CrossRefGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsMidnapore College (Autonomous)Paschim MedinipurIndia
  2. 2.Department of MathematicsBhangar MahavidyalayaSouth 24 ParganasIndia

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