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A single season production and advertising control problem with bounded final goodwill. (English) Zbl 1001.91057

The authors consider the problem of a firm which seeks the maximum profit from offering a seasonal product on the market over a single season and has to read a fixed goodwill value at the end of the season. Production and sale take place in two disjoint and consecutive subintervals of each seasonal period. While in the first interval the firm produces the good and keeps it in store, the second interval is used to sell the product. It is assumed that the firm can advertise the product at every time of each season. The advertising effect is modelled by using the goodwill state function. In contrast to earlier papers a lower bound constraint for the goodwill level at the end of the season is considered. The authors formulate the original optimal control problem. In the presented solution approach, three subproblems are identified: the optimal production problem, the optimal advertising problem in the production interval and the optimal advertising problem in the sales interval. After solving these (simpler) optimal control problems a nonlinear program is solved where the inventory level and the goodwill level are decision variables. Finally, the authors analyze the characteristics of the solutions of the nonlinear programming problem.

MSC:

91B38 Production theory, theory of the firm
49N90 Applications of optimal control and differential games
90C30 Nonlinear programming
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References:

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