Product line design under multinomial logit model

Abstract

Product line design has significant effects on the level of profitability and the market share of a firm. Firms attempt to make their product lines more diverse in order to satisfy the increasingly heterogenous demand of their customers. However, the production and operational costs increase as the product line becomes more diverse. Hence, designing a product line that balances the potential increase in profit due to the high variety of the products against the costs is a crucial decision. We study the capacitated product line design problem of a firm wishing to introduce a new product line. Given a set of attributes, the firm decides on how many products to offer and which attributes to include in each product. Customer choice is modeled by a multinomial logit (MNL) model and the average utilities are assumed to be linear in product attributes. We study the scenarios in which the sales prices of the products are exogenous and endogenous, with a greater emphasis on the former. For the first scenario, we study the case in which the firm has two attributes in consideration and different capacity levels. We characterize the necessary inequalities to choose one assortment over another for each capacity level. For the 1-capacitated problem, we show that the optimal product can be characterized with two inequalities. We later extend this result to the case where the firm has a finite number of attributes in consideration. We also elaborate on how changes in the parameters of the model affect the choice of the optimal product for the 1-capacitated problem. We propose two rules to find assortments that are never optimal for the case where the firm has a capacity greater than one. With these rules, we reduce the number of assortments that needs to be checked for optimality. Furthermore, we introduce a procedure to find the optimal assortment in the uncapacitated problem. For the endogenous price scenario, we assume the firm has a finite set of attributes. We find the closed form solutions for the optimal prices of the products. For the 1-capacitated problem, we show that it is optimal to include all attributes for which the additional average utility of including the attribute is larger than the additional cost. Lastly, we extend this result to the case in which the firm can offer more than one product: The firm always fills up the capacity and the products having the largest utility markup are offered.