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Product process innovation model of fuzzy optimal control of nonlinear system with finite time horizon under granular differentiability concept

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Abstract

This article explores developed a fuzzy production inventory control model of product-process innovation effecting with learning by doing. It is also a single objective profit maximization problem with the single period finite time horizon under uncertain environment.In real world, most of the companies are going to face some uncertainty for inventory cost, prices, demand, stock, etc. To make the proposed model more realistic, we consider all variables (control variable, state variable, co-state variable) are fuzzy in nature. In fuzzy optimal control system, to convert all fuzzy variables into crisp variables, here used granular differentiability, a new concept of fuzzy dynamical system. In this study, the demand of the product depends on quality, selling price and stock. Here the stock and quality have positive effect to the demand but the selling price has negative effect to it. Moreover the quality/cost of the product is increased / deceased with increase the gathering knowledge accumulation of product and process innovation. Here selling price, production rate, instantaneous investment rate on process innovation and product innovation are control variables and quality, cost, stock, knowledge gathering on process innovation and product innovation are state variables. Finally, this optimal control problem is solved by using Pontryagin Maximum principle and for numerical result and graphical representation we use Runge–Kutta forward-backward method of fourth order in MATLAB software. Subsequently, the numerical results are presented both in tabular form and graphically.

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References

  1. Utterback, J.M.: Adynamic model of process and product innovation. Omega 3(6), 639–656 (1975)

    Article  Google Scholar 

  2. Reinhard, A.: Product innovation a tool for competitive advantage. Eur. J. Oper. Res. 47, 182–189 (1990)

    Article  Google Scholar 

  3. Piney, B.J., Victor, B., Boyton, A.C.: Making mass customization work. Havard Bus. Rev. 71, 108–119 (1993)

    Google Scholar 

  4. Lambertini, L., Mantovani, A.: Process and product innovation by a multiproduct monopolist. A dynamic approach. Int. J. Ind. Org. 27(4), 508–518 (2009)

    Article  Google Scholar 

  5. Kamine, M.I.: Optimal licensing of cost reducing innovation. J. Math. Econ. 5, 483–508 (1992)

    Article  Google Scholar 

  6. Athey, S., Schmutzle, A.: Product and process flexibility in an innovation environment. Rand J. Econ. 26(4), 557–574 (1995)

    Article  Google Scholar 

  7. Klepper, S.: Entry exit and innovation over the product life cycle. Am. Rev. 86, 562–583 (1996)

    Google Scholar 

  8. Saha, S.: Consumer preferences and process R and D. RAND J. Oper. Res. 47, 182–189 (2007)

    Google Scholar 

  9. Lara, K., Divakaran, L.: Valuing the flexibility of investing in security process innovations. Eur. J. Oper. Res. 192, 216–235 (2009)

    Article  Google Scholar 

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

    Article  Google Scholar 

  11. 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 53, 473–486 (2017)

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. Khatua, D., Maity, K., Kar, S.: Product-process innovation and Fuzzy learning effect in finite time horizon. Int. J. Fuzzy System 21(5), 1560–1570 (2019)

    Article  Google Scholar 

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

    Article  Google Scholar 

  15. Chang, S.L., Zadeh, L.A.: On fuzzy map** and control. IEEE Trans. Syst. Man Cybern. 2, 30–34 (1972)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  18. Najariyan, M., Farahi, M.H.: A new approach for solving a class of Fuzzy optimal control system under generalised Hukuhara differentiability. J. Frankl. Inst. 352, 1836–49 (2015)

    Article  Google Scholar 

  19. 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  Google Scholar 

  20. Mohammed, M., Zengtai, G., Maura, O.: The solution of fuzzy variational problem and fuzzy optimal control problem under granular differentiability concept. Int. J. Comput. Math. 98, 1495–1520 (2020)

    Google Scholar 

  21. De, A., Khatua, D., Kar, S.: Control the preservation cost of a fuzzy production inventory model of assortment items by using the granular differentiability approach. Comput. Appl. Math. 39, 285 (2020)

    Article  Google Scholar 

  22. Khatua, D., Maity, K., Kar, S.: A fuzzy production inventory control model using granular differentiability approach. Soft. Comput. 25, 2687–2701 (2021)

    Article  Google Scholar 

  23. Hati, S., Maity, K.: Pollution control multi-objective inventory model of product-process innovation with stock dependent demand and gathering knowledge accumulation in finite time horizon. Contemp. Issues Bus. Econ. 1, 83–100 (2022)

    Google Scholar 

  24. Hati, S., Maity, K.: Reliability dependent imperfectproduction inventory optimal control fractional order model for uncertain environment under granular differentiability. Fuzzy Inf. Eng. 14, 1–28 (2022)

    Article  Google Scholar 

  25. Najariyan, M., Zhao, Y.: Granular fuzzy PID controller. Expert Syst. Appl. 167, 114182 (2021)

    Article  Google Scholar 

  26. Najariyan, M., Qiu, L.: Interval type-2 fuzzy differential equation and stability. IEEE Trans. Fuzzy Syst. 30(8), 2915–2929 (2021)

    Article  Google Scholar 

  27. Grcek, A., Pedrycz, W.: Clustering granular data and their characterization with information granules of higher type. IEEE Trans Fuzzy Syst. 23, 850–860 (2015)

    Article  Google Scholar 

  28. Qian, Y., Li, Y., Linga, J., Lin, G., Dang, C.: Fuzzy granular structure distance. IEEE Trans. Fuzzy Syst. 23, 2245–59 (2015)

    Article  Google Scholar 

  29. Piegat, A., Landowski, M.: Horizontal membership function and examples of its applications. Int. J. Fuzzy Syst. 17, 22–30 (2015)

    Article  Google Scholar 

  30. Piegat, A., Landowski, M.: Is fuzzy number the right result of arithmetic operations on fuzzy number? Adv. Fuzzy Logic Technol. 3, 181–194 (2017)

    Google Scholar 

  31. Mazandarani, M., Pariz, N., Kamyad, A.V.: Granular differentiability of Fuzzy-number-valued function. IEEE Trans. Fuzzy Syst. 26(1), 310–323 (2018)

    Article  Google Scholar 

  32. Mazandarani, M., Pariz, N.: Sub-optimal control of fuzzy linear dynamical system under granular differentiability concept. ISA Trans. 76, 1–17 (2018)

    Article  Google Scholar 

  33. Zimmermann, H.J.: Fuzzy Set Theory and Its Appl. S**er Business Economics, Berlin (2001)

    Book  Google Scholar 

  34. Bede, B., Stefanini, L.: Generalized differentiability of fuzzy valued function. Fuzzy Set Syst. 230, 119–141 (2013)

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank to Department of Science and Technology and Biotechnology, Govt of West Bengal [475(Sanc.)/ST/P/S & T/16 G-31/2018 dated 15.03.2019] for financial help.

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Authors and Affiliations

  1. Department of Mathematics, Mugberia Gangadhar Mahavidyalaya, Bhupatinagar, Purba Medinipur, West Bengal, 721425, India

    S. Hati & K. Maity

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  1. S. Hati
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  2. K. Maity
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Contributions

The main achievements of the paper are as follows: \(\bullet \,\) Presenting a dynamical optimal control production inventory model of process and product innovation with learning by doing in fuzzy environment. \(\bullet \,\) Investigating the relationships among price, stock of product, quality and product -process innovation investments in fuzzy environment. \(\bullet \,\) Solve this optimal control fuzzy dynamical model by using granular differentiability concept and the optimal control law. \(\bullet \,\) Fuzzy nonlinear demand function depends on price of product, stock level of product and quality of product.

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Correspondence to S. Hati.

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Hati, S., Maity, K. Product process innovation model of fuzzy optimal control of nonlinear system with finite time horizon under granular differentiability concept. OPSEARCH 60, 753–775 (2023). https://doi.org/10.1007/s12597-023-00630-7

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