Granular Computing

, Volume 4, Issue 1, pp 71–88 | Cite as

A fuzzy rough multi-objective multi-item inventory model with both stock-dependent demand and holding cost rate

  • Totan GaraiEmail author
  • Dipankar Chakraborty
  • Tapan Kumar Roy
Original Paper


In this paper, we developed a multi-objective multi-item inventory model with fuzzy rough coefficients. Here, we have considered both demand and holding cost which is a non-linear function of the instantaneous stock level. Chance-constrained fuzzy rough multi-objective model and a traditional solution procedure based on an interactive fuzzy satisfying method are discussed. By examining the various definitions and theoretical results of fuzzy rough variables, we have designed a Tr-Pos chance constrained technique to determine the optimal solutions of a fuzzy rough multi-objective inventory problem. Finally, a numerical example is provided to illustrate the present model, and a sensitivity analysis of the optimal solution with respect to the major parameters is carried out.


Multi-objective Multi-item inventory Stock-dependent demand Stock-dependent holding cost Fuzzy rough variable Chance measure 


  1. Avinadav T, Herbon A, Spiegel U (2013) Optimal inventory policy for a perishable item with demand function sensitive to price and time. Int J Prod Econ 144:497–506CrossRefGoogle Scholar
  2. Alfares KH, Ghaithan MA (2013) Inventory and pricing model with price-dependent demand, time-varying holding cost and quantity discounts. Comput Ind Eng 94:170–177CrossRefGoogle Scholar
  3. Balkhi ZT, Foul A (2009) A multi-item production lot size inventory model with cycle dependent parameters. Int J Math Model Methods Appl Sci 3:94–104Google Scholar
  4. Chen SM, Lee SH, Lee CH (2001) A new method for generating fuzzy rules from numerical data for handling classification problems. Appl Artif Intell 15:645–664CrossRefGoogle Scholar
  5. Chen SM, Chien CY (2011a) Parallelized genetic colony systems for solving the traveling salesman problem. Expert Syst Appl 38:3873–3883CrossRefGoogle Scholar
  6. Chen SM, Chien CY (2011b) Solving the travelling salesman problem based on the genetic simulated annealing ant colony system with particle swarm optimization techniques. Expert Syst Appl 38:14439–14450CrossRefGoogle Scholar
  7. Chen SM, Chung NY (2006) Forecasting enrolments of students using fuzzy time series and genetic algorithms. Int J Inf Manage Sci 17:1–17Google Scholar
  8. Chen SM, Kao PY (2013) TAIEX forecasting based on fuzzy time series, particle swarm optimization techniques and support vector machines. Inf Sci 247:62–71MathSciNetCrossRefGoogle Scholar
  9. Chen SM (1996) A fuzzy reasoning approach for rule-based systems based on fuzzy logics. IEEE Trans Syst Man Cybern Part B Cybern 26:769–778CrossRefGoogle Scholar
  10. Chen SM, Chang TM (2001) Finding multiple possible critical paths using fuzzy PERT. IEEE Trans Syst Man Cybern Part B Cybern 31:930–937CrossRefGoogle Scholar
  11. Chakraborty D, Jana DK, Roy TK (2014) Multi-objective multi-item solid transportation problem with fuzzy inequality constraints. J Inequ Appl 1:1–22Google Scholar
  12. Dubois D, Prade H (1988) Possibility theory: an approach to computerized processing of uncertainty. Plenum, New YorkCrossRefzbMATHGoogle Scholar
  13. Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–208CrossRefzbMATHGoogle Scholar
  14. Gupta R, Vrat P (1986) Inventory model with multi-items under constraint systems for stock dependent consumption rate. Oper Res 24:41–42CrossRefGoogle Scholar
  15. Garai T, Chakraborty D, Roy TK (2017a) Expected Value of exponential fuzzy number and its application to multi-item deterministic inventory model for deteriorating items. J Uncert Anal Appl.
  16. Garai T, Chakraborty D, Roy TK (2017b) Possibility-necessity-credibility measures on generalized intuitionistic fuzzy number and their applications to multi-product manufacturing system. Granul Comput 1:1–15Google Scholar
  17. Horng NY, Chen SM, Chang YC, Lee CH (2005) A new method for fuzzy information retrieval based on fuzzy hierarchical clustering and fuzzy inference techniques. IEEE Trans Fuzzy Syst 13:216–228CrossRefGoogle Scholar
  18. Hartley R (1978) An existence and uniqueness theorem for an optimal inventory problem with forecasting. J Math Anal Appl 66:346–353MathSciNetCrossRefzbMATHGoogle Scholar
  19. Ishii H, Konno T (1998) A stochastic inventory problem with fuzzy shortage cost. Eur J Oper Res 106:90–94CrossRefGoogle Scholar
  20. Khouja M (1995) The economic production lot size model under volume flexibility. Comp Oper Res 22:515–525CrossRefzbMATHGoogle Scholar
  21. Liu B (2002) Theory and practice of uncertain programming. Physica-Verlag, HeidelbergCrossRefzbMATHGoogle Scholar
  22. Lushu S, Nair KPK (2002) Fuzzy models for single-period inventory model. Fuzzy Sets Syst 132:273–289CrossRefzbMATHGoogle Scholar
  23. Li FD (2005) An approach to fuzzy multi-attribute decision-making under uncertainty. Inf Sci 169:107–112CrossRefGoogle Scholar
  24. Lee H, Yao JS (1998) Economic production quantity for fuzzy demand quantity and fuzzy production quantity. Eur J Oper Res 109:203–211CrossRefzbMATHGoogle Scholar
  25. Morsi NN, Yakout MM (1998) Axiomatics for fuzzy rough sets. Fuzzy Sets Syst 100:327–342MathSciNetCrossRefzbMATHGoogle Scholar
  26. Mondal M, Maity KA, Maiti KM, Maiti M (2013a) A production-repairing inventory model with fuzzy rough coefficients under inflation and time value of many. Appl Math Model 37:3200–3215MathSciNetCrossRefzbMATHGoogle Scholar
  27. Mondal M, Maity KA, Maiti KM, Maiti M (2013b) A production-repairing inventory model with fuzzy rough coefficients under inflation and time value of money 37:3200–3215Google Scholar
  28. Maiti MK, Maiti M (2005) Production policy for damageable items with variable cost function in an imperfect production process via genetic algorithm. Math Comput Modell 42:977–990MathSciNetCrossRefzbMATHGoogle Scholar
  29. Maity KA (2011) One machine multiple-product problem with production-inventory system under fuzzy inequality constraint. Appl Soft Comput 11:1549–1555CrossRefGoogle Scholar
  30. Min J, Zhou YW, Liu GQ, Wang SD (2012) An EPQ model for deteriorating items with inventory level dependent demand and permissible delay in payments. Int Syst Sci 43:1039–1053MathSciNetCrossRefzbMATHGoogle Scholar
  31. Pedrycz W, Chen SM (2011) Granular computing and intelligent systems: design with information granules of high order and high type. Springer, Heidelberg, GermanyCrossRefGoogle Scholar
  32. Pedrycz W, Chen SM (2015a) Information granularity, big data, and computational intelligence. Springer, Heidelberg, GermanyCrossRefGoogle Scholar
  33. Pedrycz W, Chen SM (2015b) Granular computing and decision-making: interactive and iterative approaches. Springer, Heidelberg, GermanyCrossRefGoogle Scholar
  34. Pando V, Garcia-Lagunaa J, San-Jose LA, Sicilia J (2012) Maximizing profits in an inventory model with both demand rate and holding cost per unit time dependent on the stock level. Comput Ind Eng 62:599–608CrossRefGoogle Scholar
  35. Pando V, San-jose LA, Garcia-Laguna J, Sicilia J (2013) An economic lot-size model with non-linear holding cost hinging on time quantity. Int J Prod Econ 145:294–303CrossRefGoogle Scholar
  36. Radzikowska MA, Kerre EE (2002) A comparative study of rough sets. Fuzzy Sets Syst 126:137–155MathSciNetCrossRefzbMATHGoogle Scholar
  37. Roy A (2008) An inventory model for deteriorating items with price dependent demand and time varying holding cost. Advanced modelling and optimization 10:25–37MathSciNetzbMATHGoogle Scholar
  38. Sakawa K (1993) Fuzzy sets an interactive multi-objective optimization. Plenum, New YorkCrossRefzbMATHGoogle Scholar
  39. Tsai P W, Pan J S, Chen S M, Liao B Y, Hao S P (2008) Parallel cat swarm optimization. In: Proceedings of the 2008 International Conference on Machine Learning and Cybernetics, Kunming, China, vol 6, pp 3328–3333Google Scholar
  40. Tsai SM, Pan JS, Chen SM, Liao BY (2012) Enhanced parallel cat swarm optimization based on the Taguchi method. Expert Syst Appl 39:6309–6319CrossRefGoogle Scholar
  41. Taleizadeh AA, Sadjadi SJ, Niaki STA (2011) Multi-product EPQ model with single machine, back-ordering and immediate rework process. Eur J Ind Eng 5:388–411CrossRefGoogle Scholar
  42. Taleizadeh AA, Wee MH, Jolai F (2013) Revisiting a fuzzy rough economic order quantity model for deteriorating items considering quantity discount and prepayment. Math Comput Modell 57:1466–1479MathSciNetCrossRefGoogle Scholar
  43. Tripathi EP (2013) Inventory model with different demand rate and different holding cost. Int J Ind Eng Comput 4:437–446Google Scholar
  44. Wu KS, Ouyang LY, Yang CT (2006) An optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partial backlogging. Int J Prod Econ 101:369–384CrossRefGoogle Scholar
  45. Yang TC (2014) An inventory model with both stock-dependent demand rate and stock-dependent holding cost rate. Int J Prod Econ 155:214–221CrossRefGoogle Scholar
  46. Xu J, Zhao L (2010) A multi-objective decision-making model with fuzzy rough coefficients and its application to the inventory problem. Inf Sci 180:679–696MathSciNetCrossRefzbMATHGoogle Scholar
  47. Xu J, Zaho L (2008) A class of fuzzy rough expected value multi-objective decision making model and its application to inventory problems. Comput Math Appl 56:2107–2119MathSciNetCrossRefzbMATHGoogle Scholar
  48. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefzbMATHGoogle Scholar
  49. Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1:3–28MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Totan Garai
    • 1
    Email author
  • Dipankar Chakraborty
    • 1
  • Tapan Kumar Roy
    • 1
  1. 1.Department of MathematicsIndian Institute of Engineering Science and Technology, ShibpurHowrahIndia

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