An Optimal Control Problem for An Inventory Model for Deteriorating Items Considering Advertising Dependent Demand

With an increase in market competition, the association between marketing and inventory management has become more important. The commercial activities are more rapid through social media, and advertising has played a crucial role in reaching the product to the consumers before it hits the market. It has thus become normal in an oligopolistic marketing system to increase sales through advertising effort and gain more profit from potential market. It is challenging, nevertheless, to calculate demand and costs related to advertising efforts. As a result, the purpose of this study is to identify the best advertising approach and its potential impact on demand in order to optimize the firm's overall profit. In this paper, we develop an inventory model for deteriorating items to obtain an optimal advertising and inventory strategy, where the consumer demand rate depends on advertising effort and inventory of the items displayed in the store. We have formulated two optimal control problems with the assumption that the replenishment cycle is longer than the fresh product time or not. It is assumed that products do not decay within the fresh product time interval, and inventory decreases due to consumer demand. Next, items will deteriorate and inventory level decreases because of the combined effects of customer demand and deterioration. The analytical solution for the optimal dynamic advertising effort strategies obtained by applying Pontryagin's maximum principle to maximize overall profit over the planning period. The efficiency of the proposed model is demonstrated by numerical examples. A parameter sensitivity analysis is also performed, providing suggestions for enhancing the firm's profitability when dealing with deteriorating products.