In today’s era when a substitute for almost every product is readily available, acceptance and adoption of a new product in a market requires substantial amount of promotion. Here we formulate and analyze policies for promoting sales of a product in a market through optimal control theory problems. The market is partitioned into various segments depending upon multifarious demands of customers and promotion of the product is done segment-wise. The aim is to maximize the profits keeping in mind the demand requirements and the available budget for promotion. In order to provide a realistic model, the total available budget is taken to be imprecise. The optimal control model with fuzzy parameter is converted into crisp form using necessity and possibility constraints, and thereafter solved by using Pontryagin Maximum principle. To illustrate this technique, a numerical example is also considered by discretizing the model. The analysis also gives a deep insight of how the promotional effort should be planned by the decision makers keeping in mind the financial constrains without hindering the promotional effort at the end of the planning period. This paper mirrors the real time situation that could be faced by any industry, including that of software development, where budgets may have variable components and promotion of products may vary according to different regions and markets. The experimental data reveals that profitability can still be maximized if real-life constraints are applied in promotional planning by any industry.
