On the dynamics of stochastic diffusion of manufacturing technology

Manufacturing technology diffusion (MTD), a close relative of demand diffusion in marketing, refers to the transition of technology's economic value during the transfer and operation phases of a technology life cycle. Since manufacturing technology is rooted in manufacturing activities (e.g. production planning and control) as opposed to marketing alternatives (e.g. pricing and advertising), the modeling of MTD must inevitably address two aspects: regularity (drift) and uncertainty (disturbance). The MTD model proposed herein addresses the problem of how to regulate MTD in the face of uncertainty in order to maximize expected total profit. The MTD model adopts stochastic differential equations (SDEs), to overcome the limitations of invariance (e.g. a fixed market size and the absence of disturbance) as suffered by a typical product life cycle (PLC) model. First we derive a drift function in the context of MTD and address the drift-only MTD model (i.e. with zero disturbance). With reference to a specific application of flexible manufacturing, we find an optimal control for the regulation of MTD and in addition we prove the optimality of early technology phase-out, which interestingly coincides with the pervasive phenomena of life-cycle-shortening in manufacturing. Then, by variational calculus in combination with applications of Ito's formula, we obtain an augmented Hamilton–Jacobi variational equation for the solution of the MTD model. An early phase-out policy is also proved to be optimal for the flexible manufacturing case when disturbance is present.