A Model of Optimal Production Planning Based on the Hysteretic Demand Curve

Abstract

The article considers a hysteretic model of consumer behaviour in mono-product markets. Demand generation with regard to an individual consumer is modeled using a non-ideal relay with inverted thresholds. Therefore, the sales rate is defined as an analogue of the Preisach converter. The article considers the problem of the optimal production, storage, and distribution of goods, taking into account the hysteretic nature of the demand curve. The problem is reduced to a non-classical optimal control problem with hysteretic non-linearities. The latter is solved using Pontryagin’s maximum principle. The adopted economic model is based on the binary relationship of consumers to the product: the product is bought or the product is not bought. Transitions between these states are determined within the framework of our model only by the price of the goods; therefore, only the operator of a non-ideal relay can accurately describe such a dependence. The article presents the results of computational experiments illustrating the theoretical assumptions.

Keywords

• hysteresis
• non-ideal relay
• Preisach operator
• sales rate
• price function
• consumer behaviour