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

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.