Annals of Data Science

, Volume 6, Issue 1, pp 61–81 | Cite as

Multi-objective Inventory Model with Both Stock-Dependent Demand Rate and Holding Cost Rate Under Fuzzy Random Environment

  • Totan GaraiEmail author
  • Dipankar Chakraborty
  • Tapan Kumar Roy


In this paper, we investigated a multi-objective inventory model under both stock-dependent demand rate and holding cost rate with fuzzy random coefficients. Chance constrained fuzzy random multi-objective model and a traditional solution procedure based on an interactive fuzzy satisfying method are discussed. In addition, the technique of fuzzy random simulation is applied to deal with general fuzzy random objective functions and fuzzy random constraints which are usually difficult to converted into their crisp equivalents. The purposed of this study is to determine optimal order quantity and inventory level such that the total profit and wastage cost are maximized and minimize for the retailer respectively. Finally, illustrate example is given in order to show the application of the proposed model.


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



Possibility measure


Necessity measure


Credibility measure


Probability measure


Chance measure


Chance constrained multi-objective problem


Interactive fuzzy satisfied method

\({\mathop {a}\limits ^{\simeq }}\)

Fuzzy random variable


Compliance with Ethical Standards

Competing interests

We declare that ours has neither financial nor non-financial competing.

Availability of Sata and Materials

Not applicable.


  1. 1.
    Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefGoogle Scholar
  2. 2.
    Ishii H, Konno T (1998) A stochastic inventory problem with fuzzy shortage cost. Eur J Oper Res 106:90–94CrossRefGoogle Scholar
  3. 3.
    Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1:3–28CrossRefGoogle Scholar
  4. 4.
    Dubois D, Prade H (1988) Possibility theory: an approach to computerized processing of uncertainty. Plenum, New YorkCrossRefGoogle Scholar
  5. 5.
    Kwakernaak H (1978) Fuzzy random variables, definitions and theorems. Inf Sci 15:1–29CrossRefGoogle Scholar
  6. 6.
    Liu B (2001) Fuzzy random chance-constrained programming. IEEE Trans Fuzzy Syst 9:713–720CrossRefGoogle Scholar
  7. 7.
    Liu B (2001) Fuzzy random dependent-chance programming. IEEE Trans Fuzzy Syst 9:721–726CrossRefGoogle Scholar
  8. 8.
    Luhandjula MK (2004) Optimisation under hybrid uncertainty. Fuzzy Sets Syst 146:187–203CrossRefGoogle Scholar
  9. 9.
    Qiao Z, Wang G (1993) On solutions and distributions problems of the linear programming with fuzzy random variable coefficients. Fuzzy Sets Syst 58:155–170CrossRefGoogle Scholar
  10. 10.
    Majumder S, Kar S, Pal T (2018) Mean-entropy model of uncertain portfolio selection problem. Springer, SingaporeCrossRefGoogle Scholar
  11. 11.
    Kar MB, Kar S, Guo S, Li X, Majumder S (2018) A new bi-objective fuzzy portfolio selection model and its solution through evolutionary algorithms. Soft Comput.
  12. 12.
    Garai T, Chakraborty D, Roy TK (2018) Possibility mean, variance and covariance of generalized intuitionistic fuzzy numbers and its application to multi-item inventory model with inventory level dependent demand. J Intell Fuzzy Syst 35:1021–1036CrossRefGoogle Scholar
  13. 13.
    Mondal SP (2018) Interval valued intuitionistic fuzzy number and its application in differential equation. J Intell Fuzzy Syst 34:677–687CrossRefGoogle Scholar
  14. 14.
    Salahshour S, Mahata A, Mondal SP, Alam S (2018) The behavior of logistic equation with alley effect in fuzzy environment: fuzzy differential equation approach. Int J Appl Comput Math 4:1–20CrossRefGoogle Scholar
  15. 15.
    Dutta P, Chakraborty D, Roy AR (2005) A single-period inventory model with fuzzy random variable demand. Math Comput Model 41:915–922CrossRefGoogle Scholar
  16. 16.
    Dey O, Chakraborty D (2011) A fuzzy random continuous review inventory system. Int J Prod Econ 132:101–106CrossRefGoogle Scholar
  17. 17.
    Wang X (2011) Continuous review inventory model with variable lead time in a fuzzy random environment. Expert Syst Appl 38:11715–11721CrossRefGoogle Scholar
  18. 18.
    Kumar RS, Goswami A (2015) A continuous review production-inventory system in fuzzy random environment: min–max distribution free procedure. Comput Ind Eng 79:65–75CrossRefGoogle Scholar
  19. 19.
    Iltaf Hussain AS, Mandal UK, Mondal SP (2018) Decision maker priority index and degree of vagueness coupled decision making method: a synergistic approach. Int J Fuzzy Syst 20:1551–1566CrossRefGoogle Scholar
  20. 20.
    Mondal SP (2016) Differential equation with interval valued fuzzy number and its applications. Int J Syst Assur Eng Manag 7:370–386CrossRefGoogle Scholar
  21. 21.
    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
  22. 22.
    Kar MB, Kundu P, Kar S, Pal T (2018) A multi-objective multi-item solid transportation problem with vehicle cost, volume and weight capacity under fuzzy environment. J Intell Fuzzy Syst 35:1991–1995CrossRefGoogle Scholar
  23. 23.
    Majumder S, Kundu P, Kar S, Pal T (2018) Uncertain multi-objective multi-item fixed charge solid transportation problem with budget constraint. Soft Comput.
  24. 24.
    Kundu P, Kar S, Maiti M (2014) Multi-objective solid transportation problems with budget constraint in uncertain environment. Int J Syst Sci 45:1668–1682CrossRefGoogle Scholar
  25. 25.
    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
  26. 26.
    Garai T, Chakraborty D, Roy TK (2018) A multi-objective multi-item inventory model with both stock-dependent demand rate and holding cost rate under fuzzy rough environment. J Granul Comput 3:1–18CrossRefGoogle Scholar
  27. 27.
    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
  28. 28.
    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
  29. 29.
    Garai T, Chakraborty D, Roy TK (2018) Expected value of exponential fuzzy number and its application to multi-item deterministic inventory model for deteriorating items. J Uncertain Anal Appl.
  30. 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 43:1039–1053CrossRefGoogle Scholar
  31. 31.
    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 Model 57:1466–1479CrossRefGoogle Scholar
  32. 32.
    Jana DK, Das B, Maiti M (2014) Multi-item partial backlogging inventory models over random planning horizon in random fuzzy environment. Appl Soft Comput 21:12–27CrossRefGoogle Scholar
  33. 33.
    Chakraborty D, Jana D k, Roy TK (2015) Multi-item integrated supply chain model for deteriorating items with stock dependent demand under fuzzy random and bi-fuzzy environments. Comput Ind Eng 88:166–180CrossRefGoogle Scholar
  34. 34.
    Chakraborty D, Jana KD, Roy KT (2017) A new approach to solve intuitionistic fuzzy optimization problem using possibility, necessity and credibility measures. Int J Eng Math 1:1–12Google Scholar
  35. 35.
    Xu J, Zaho L (2008) A class of fuzzy rough expected value multi-objective decision making model and its application to inventory problems. Computers Math Appl 56:2107–2119CrossRefGoogle Scholar
  36. 36.
    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 62:599–608Google Scholar
  37. 37.
    Tripaathi EP (2013) Inventory model with different demand rate and different holding cost. Int J Ind Eng Comput 4:437–446Google Scholar
  38. 38.
    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
  39. 39.
    Roy A (2008) An inventory model for deteriorating items with price dependent demand and time varying holding cost. Adv Model Optim 10:25–37Google Scholar
  40. 40.
    Liu YK, Liu B (2003) Fuzzy random variables: a scalar expected value operator. Fuzzy Optim Decis Mak 2:143–160CrossRefGoogle Scholar
  41. 41.
    Li J, Xu J, Gen MA (2006) class of multi-objective linear programming model with fuzzy random coefficients. Math Comput Model 44:1097–1113CrossRefGoogle Scholar
  42. 42.
    Xu J, Yao L (2009) A class of multi-objective linear programming models with random rough coefficients. Math Comput model 49:189–206CrossRefGoogle Scholar
  43. 43.
    Liu B (2002) Theory and practice of uncertain programming. Physica-Verlag, HeidelbergCrossRefGoogle Scholar
  44. 44.
    Sakawa K (1993) Fuzzy sets an interactive multi-objective optimization. Plenum, New YorkCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Totan Garai
    • 1
    Email author
  • Dipankar Chakraborty
    • 2
  • Tapan Kumar Roy
    • 1
  1. 1.Department of MathematicsSilda Chandra Sekhar College, SildaJhargramIndia
  2. 2.Department of MathematicsHeritage Institute of TechnologyAnandapur, KolkataIndia

Personalised recommendations