Drawing on the Stackelberg game approach to solving the pricing problem in a supply chain, this paper develops a bi-level model whereby a domestic company and a foreign manufacturer compete to gain more profit from the market of a retailer. The domestic company acts as the leader and the retailer as the follower. The domestic company has two manufacturers each of whom produces and sells a different quality of the product. The retailer decides to purchase products based on the prices offered by the low-quality manufacturer, the high-quality manufacturer, and the foreign manufacturer, known as an exogenous factor. In fact, the first level seeks to maximize its profits and the second level seeks to reduce the cost of purchasing. In this paper, the price of the products of each manufacturer is considered a contributing factor to the retailer's tendency to buy from each manufacturer. This assumption is designed by the multinomial logit model. As the proposed model has binary variables in its follower segment, a novel hybrid exact method based on explicit enumeration method and Lambert-W function is applied to solve it. In other words, to calculate the optimal selling price of domestic products and their profit first by using the explicit enumeration method, the bi-level model is transformed into a single-level problem. The problem is, then, solved precisely by applying the Lambert-W function. The efficiency of the proposed model is proven by the results obtained from solving the model and the sensitivity analysis of the main parameters of the model. Moreover, to have a detailed managerial analysis of each manufacturer's profit on the competitive market environment, the market is studied in view of three different scenarios: (1) when there is a sense of patriotism regarding domestic manufacturers; (2) when customers have low incomes; and (3) when customers have high incomes. Finally, the study results conclude that if the domestic company has two manufacturers that produce a different type of quality can lead to an increase in the profit of the domestic company. Indeed, the proposed model can increase the competitive power of the domestic company against imported products by providing appropriate pricing on its products.
