A Fuzzy Optimal Control Inventory Model of Product–Process Innovation and Fuzzy Learning Effect in Finite Time Horizon D. Khatua 1,2 • K. Maity 3 • S. Kar 2 Received: 15 September 2018 / Revised: 28 March 2019 / Accepted: 6 May 2019 Ó Taiwan Fuzzy Systems Association 2019 Abstract In this study, we have introduced a fuzzy dynamic optimal control model of process–product inno- vation 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 innova- tion are control variables; the product quality, production cost, and the change rates of gathering knowledge accu- mulations 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 for- ward–backward method of fourth order using MATLAB software. Further, some numerical experiments are per- formed 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. Keywords Fuzzy dynamical system  Quasi-level-wise system  Fuzzy optimal control  Fuzzy product–process innovation  Fuzzy learning effect  Price and quality dependent fuzzy demand 1 Introduction Since the fifties of the last century, the optimal control theory has been an important offshoot of advanced control theory. For the requirement of strict expression form, the subject area of optimal control theory greatly attracted the attention of many researchers. With the more use of methods and consequences of mathematics and computer science, optimal control theory has significantly achieved development and applied to many areas such as production engineering, programming, economy and management. Optimal control theory has been developed to find optimal ways to control dynamic systems. The field of stochastic optimal control initiated in 1971 for finance [30]. One of the primary methods for studying optimal control is dynamic programming. The role of active programming in optimization over Ito’s process is discussed in [15]. In recent time, many researchers have studied classical opti- mal control problems from different viewpoints and the detailed arguments can be established in many texts [6, 14, 19, 22–24, 27, 35, 42]. In the present scenario, we are facing some uncertainty/ imprecision in various configurations due to the complexity of the real world. Uncertainty is inherent in most real- world systems. Besides randomness, fuzziness is also an important uncertainty, which acts as an indispensable part & D. Khatua devnarayan87@gmail.com K. Maity kalipada_maity@yahoo.co.in S. Kar dr.samarjitkar@gmail.com 1 Department of Basic Science and Humanities, Global Institute of Science & Technology, West Bengal, India 2 Department of Mathematics, National Institute of Technology, West Bengal, India 3 Department of Mathematics, Mugberia Gangadhar Mahavidyalaya, West Bengal, India 123 Int. J. Fuzzy Syst. https://doi.org/10.1007/s40815-019-00659-1