Skip to main content

Advertisement

SpringerLink
Log in

A Fuzzy Optimal Control Inventory Model of Product–Process Innovation and Fuzzy Learning Effect in Finite Time Horizon

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

In this study, we have introduced a fuzzy dynamic optimal control model of process–product innovation with learning by doing. The firm’s cost functions of product and process innovation depend on investment in both the innovations and the gathering of knowledge of product and process innovation. In this model, the product price, and the investments of product and process innovation are control variables; the product quality, production cost, and the change rates of gathering knowledge accumulations of product and process innovation are state variables. To represent the model more realistically, we considered all the variables are fuzzy in nature. The main goal of this report is to probe the relationships between these variables and investigate the optimality criteria for the model. The model is formulated as a single objective profit maximization problem in the single period finite time horizon and solved numerically using Runge–Kutta forward–backward method of fourth order using MATLAB software. Further, some numerical experiments are performed and the graphical representation of the results are also depicted to illustrate the model. Also a sensitivity analysis is conducted to study the issue of varying the parameters and coefficients on the target function value.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Abbasbandy, S., Nieto, J.J., Alavi, M.: Tuning of reachable set in one dimensional fuzzy differential inclusions. Chaos Solitons Fractals 26(5), 1337–1341 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahmad, M.Z., Hasan, M.K., De Baets, B.: Analytical and numerical solutions of fuzzy differential equations. Inf. Sci. 236, 156–167 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arrow, K.J.: The economic implications of learning by doing. Rev. Econ. Stud. 29(3), 155–173 (1962)

    Article  Google Scholar 

  4. Avagyan, V., Esteban-Bravo, M., Vidal-Sanz, J.M.: Licensing radical product innovations to speed up the diffusion. Eur. J. Oper. Res. 239(2), 542–555 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bede, B., Gal, S.G.: Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst. 151(3), 581–599 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bhowmick, J.: Optimal inventory policies for imperfect inventory with price dependent stochastic demand and partially backlogged shortages. Yugosl. J. Oper. Res. 22(2), 199–223 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Buckley, J.J., Feuring, T.: Fuzzy initial value problem for nth-order linear differential equations. Fuzzy Sets Syst. 121(2), 247–255 (2001)

    Article  MATH  Google Scholar 

  8. Chalco-Cano, Y., Roman-Flores, H.: On new solutions of fuzzy differential equations. Chaos Solitons Fractals 38(1), 112–119 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, B., Liu, X.: Reliable control design of fuzzy dynamic systems with time-varying delay. Fuzzy Sets Syst. 146(3), 349–374 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chenavaz, R.: Dynamic pricing, product and process innovation. Eur. J. Oper. Res. 222(3), 553–557 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chenavaz, R., et al.: Dynamic pricing rule and R&D. Econ. Bull. 31(3), 2229–2236 (2011)

    Google Scholar 

  12. Cheng, S., Zadeh, L.: On fuzzy mapping and control. IEEE Trans. Syst. Man Cybern. 2, 30–34 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  13. De, A., Maity, K., Maiti, M.: Stability analysis of combined project of fish, broiler and ducks: dynamical system in imprecise environment. Int. J. Biomath. 8(05), 1550067 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dimitrov, S., Ceryan, O.: Optimal inventory decisions when offering layaway. Int. J. Prod. Res. 57, 1–15 (2018)

    Google Scholar 

  15. Dixit, A.K., Dixit, R.K., Pindyck, R.S., Pindyck, R.: Investment Under Uncertainty. Princeton University Press, Princeton (1994)

    Book  Google Scholar 

  16. Dubois, D., Prade, H.: Towards fuzzy differential calculus part 3: differentiation. Fuzzy Sets Syst. 8(3), 225–233 (1982)

    Article  MATH  Google Scholar 

  17. Goetschel Jr., R., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets Syst. 18(1), 31–43 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaleva, O.: Fuzzy differential equations. Fuzzy Sets Syst. 24(3), 301–317 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  19. Katsifou, A., Seifert, R.W., Tancrez, J.S.: Joint product assortment, inventory and price optimization to attract loyal and non-loyal customers. Omega 46, 36–50 (2014)

    Article  Google Scholar 

  20. Khatua, D., De, A., Maity, K., Kar, S.: Use of e and g operators to a fuzzy production inventory control model for substitute items. RAIRO-Oper. Res. 53, 473–486 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Khatua, D., Maity, K.: Stability of fuzzy dynamical systems based on quasi-level-wise system. J. Intell. Fuzzy Syst. 33(6), 3515–3528 (2017)

    Article  Google Scholar 

  22. Khatua, D., Maity, K., Kar, S.: Determination of advertisement control policy for complementary and substitute items for a class inventory problem. Int. J. Bus. Forecast. Mark. Intell. 3(3), 223–247 (2017)

    Google Scholar 

  23. Kogan, K., Chernonog, T.: Competition under industry-stock-driven prevailing market price: environmental consequences and the effect of uncertainty. Eur. J. Oper. Res. 276(3), 929–946 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kutzner, S.C., Kiesmüller, G.P.: Optimal control of an inventory-production system with state-dependent random yield. Eur. J. Oper. Res. 227(3), 444–452 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Levitt, S.D., List, J.A., Syverson, C.: Toward an understanding of learning by doing: Evidence from an automobile assembly plant. J. Polit. Econ. 121(4), 643–681 (2013)

    Article  Google Scholar 

  26. Ma, M., Friedman, M., Kandel, A.: A new fuzzy arithmetic. Fuzzy Sets Syst. 108(1), 83–90 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Malekitabar, M., Yaghoubi, S., Gholamian, M.: A novel mathematical inventory model for growing-mortal items (case study: rainbow trout). Appl. Math. Model. 71, 96–117 (2019)

    Article  MathSciNet  Google Scholar 

  28. Mavrikios, D., Papakostas, N., Mourtzis, D., Chryssolouris, G.: On industrial learning and training for the factories of the future: a conceptual, cognitive and technology framework. J. Intell. Manuf. 24(3), 473–485 (2013)

    Article  Google Scholar 

  29. Mazandarani, M., Najariyan, M.: A note on a class of linear differential dynamical systems with fuzzy initial condition. Fuzzy Sets Syst. 265, 121–126 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Merton, R.C.: Optimum consumption and portfolio rules in a continuous-time model. In: Stochastic Optimization Models in Finance, Academic Press, pp. 621–661. Elsevier (1975)

  31. Mosleh, M., Otadi, M.: Approximate solution of fuzzy differential equations under generalized differentiability. Appl. Math. Model. 39(10–11), 3003–3015 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Najariyan, M., Farahi, M.H.: Optimal control of fuzzy linear controlled system with fuzzy initial conditions. Iran. J. Fuzzy Syst. 10(3), 21–35 (2013)

    MathSciNet  MATH  Google Scholar 

  33. OSullivan, D., Rolstadås, A., Filos, E.: Global education in manufacturing strategy. J. Intell. Manuf. 22(5), 663–674 (2011)

    Article  Google Scholar 

  34. Pal, D., Mahaptra, G., Samanta, G.: Optimal harvesting of prey-predator system with interval biological parameters: a bioeconomic model. Math. Biosci. 241(2), 181–187 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pan, X., Li, S.: Optimal control of a stochastic production-inventory system under deteriorating items and environmental constraints. Int. J. Prod. Res. 53(2), 607–628 (2015)

    Article  Google Scholar 

  36. Pan, X., Li, S.: Dynamic optimal control of process-product innovation with learning by doing. Eur. J. Oper. Res. 248(1), 136–145 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pearson, D.: A property of linear fuzzy differential equations. Appl. Math. Lett. 10(3), 99–104 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  38. Puri, M.: Ralescu, DA: fuzzy random variables. Math. Anal. Appl. 114, 409–422 (1986)

    Article  MathSciNet  Google Scholar 

  39. Puri, M.L., Ralescu, D.A.: Differentials of fuzzy functions. J. Math. Anal. Appl. 91(2), 552–558 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  40. Reimann, M., Xiong, Y., Yu, Z.: Managing a closed-loop supply chain with process innovation for remanufacturing. Eur. J. Oper. Res. 276, 510–518 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  41. Seikkala, S.: On the fuzzy initial value problem. Fuzzy Sets Syst. 24(3), 319–330 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  42. Stecca, G., Baffo, I., Kaihara, T.: Design and operation of strategic inventory control system for drug delivery in healthcare industry. IFAC-PapersOnLine 49(12), 904–909 (2016)

    Article  Google Scholar 

  43. Thompson, P.: Learning by doing. In: Handbook of the economics of innovation, vol. 1, pp. 429–476 (2010)

  44. Xu, J., Liao, Z., Nieto, J.J.: A class of linear differential dynamical systems with fuzzy matrices. J. Math. Anal. Appl. 368(1), 54–68 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  Google Scholar 

Download references

Acknowledgement

Dr. Kalipada Maity thanks to Department of Science & Technology and Biotechnology [475(Sanc.)/ST/P/S&T/16G-31/2018 dated 15.03.2019] for financial help.

Author information

Authors and Affiliations

  1. Department of Basic Science and Humanities, Global Institute of Science & Technology, West Bengal, India

    D. Khatua

  2. Department of Mathematics, National Institute of Technology, West Bengal, India

    D. Khatua & S. Kar

  3. Department of Mathematics, Mugberia Gangadhar Mahavidyalaya, West Bengal, India

    K. Maity

Authors
  1. D. Khatua
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Maity
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Kar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Khatua.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khatua, D., Maity, K. & Kar, S. A Fuzzy Optimal Control Inventory Model of Product–Process Innovation and Fuzzy Learning Effect in Finite Time Horizon. Int. J. Fuzzy Syst. 21, 1560–1570 (2019). https://doi.org/10.1007/s40815-019-00659-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-019-00659-1

Keywords

Search

Navigation