This study presents an algorithm for deriving the long-term polices of quality level, price and advertisement for a product. The diVusion models and cost functions are combined to formulate pro® t functions capable of determining future pro® t trends. The algorithm ® rst implements the optimal control theory to derive the optimal conditions of the pro® t function. Then the genetic algorithm is employed to search for the approximate solutions of quality level, price and advertising expenditure at each period on the planning horizon (life cycle). Examples of diVerent scenarios of the model parameters are presented to describe the results obtained herein. Sensitivity analysis for the major parameters is performed to specify their eVects on pro® ts. Results in this study allow us, ® rstly, to obtain explicit solutions simultaneously with respect to quality, price and advertising policies, secondly, to propose an appropriate algorithm for solving the diVerent scenarios of the dynamic pro® t function, which consists of the diVusion function and cost function, and thirdly, to enhance the long-term pro® t performance via the polices proposed herein, that is the approximation of the best solution. NomenclatureA…t † rate of advertising expenditure at time t C…x…t †; q…t † † total cost per unit at time t for a cumulative sales volume x…t † and quality level q…t † d price parameter h quality parameter M total number of potential purchasers over the life cycle of the product P…t † unit price at time t q…t † quality level at time t, 0 < q…t † 4 1 r discount rate T period of the life cycle x…t † cumulative sales volume by time t x 0 …t †ˆg…x…t †; p…t †; q…t †; A…t † †, sales rate at time t for a cumulative sales volume x…t †, price p…t †, quality level q…t † and advertising expenditure A…t † ¬ innovation coe cient reaction coe cient for advertising ® imitation coe cient
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