We study a multiproduct pricing problem in which the prices of the products are restricted to discrete and finite sets. The demand for a product is a function of the prices of all the products. The prices of the products can be changed through time, subject to the aggregate consumption of each resource not exceeding its availability over the planning horizon. The focus of our work is the deterministic variant of this problem (wherein customer-arrival rates are deterministic), which is a key subproblem whose solution can be used to build effective policies for the stochastic variant (as in Gallego and Van Ryzin 1997). When the demand rate of each product is a concave function of the prices, we obtain an efficient and effective solution to the deterministic problem; the worst-case optimality gap of our solution depends on the curvature of the objective function. We obtain a similar performance guarantee for our solution under the linear attraction demand model and a special case of the MNL demand model. For a general demand function, the worst-case optimality gap of our solution depends on the curvature of both the objective function and the demand function. For the special case where the demand rate of a product depends only on its own price and not on the prices of the other products, we show that the deterministic problem can be efficiently solved via a linear program.

We consider a retailer offering an assortment of differentiated substitutable products to price-sensitive customers. Prices are chosen to maximize profit, subject to inventory/ capacity constraints, as well as more general constraints. The profit is not even a quasi-concave function of the prices under the basic multinomial logit (MNL) demand model. Linear constraints can induce a non-convex feasible region. Nevertheless, we show how to efficiently solve the pricing problem under three important, more general families of demand models. Generalized attraction (GA) models broaden the range of nonlinear responses to changes in price. We propose a reformulation of the pricing problem over demands (instead of prices) which is convex. We show that the constrained problem under MNL models can be solved in a polynomial number of Newton iterations. In experiments, our reformulation is solved in seconds rather than days by commercial software. For nested-logit (NL) demand models, we show that the profit is concave in the demands (market shares) when all the price-sensitivity parameters are sufficiently close. The closed-form expressions for the Hessian of the profit that we derive can be used with general-purpose nonlinear solvers. For the special (unconstrained) case already considered in the literature, we devise an algorithm that requires no assumptions on the problem parameters. The class of generalized extreme value (GEV) models includes the NL as well as the cross-nested logit (CNL) model. There is generally no closed form expression for the profit in terms of the demands. We nevertheless how the gradient and Hessian can be computed for use with general-purpose solvers. We show that the objective of a transformed problem is nearly concave when all the price sensitivities are close. For the unconstrained case, we develop a simple and surprisingly efficient first-order method. Our experiments suggest that it always finds a global optimum, for any model parameters. We apply the method to mixed logit (MMNL) models, by showing that they can be approximated with CNL models. With an appropriate sequence of parameter scalings, we conjecture that the solution found is also globally optimal.

In principle, a multiproduct firm can set separate prices for all possible bundled combinations of its products (i.e., "mixed bundling"). However, this is impractical for firms with more than a few products, because the number of prices increases exponentially with the number of products. In this study we show that simple pricing strategies are often nearly optimal -- i.e., with surprisingly few prices a firm can obtain 99% of the profit that would be earned by mixed bundling. Specifically, we show that bundle-size pricing -- setting prices that depend only on the size of bundle purchased -- tends to be more profitable than offering the individual products priced separately, and tends to closely approximate the profits from mixed bundling. These findings are based on an array of numerical experiments covering a broad range of demand and cost scenarios, as well as an empirical analysis of the pricing problem for an 8-product firm (a theater company).

In principle, a multiproduct firm can set separate prices for all possible bundled combinations of its products (i.e., quot;mixed bundlingquot;). However, this is impractical for firms with more than a few products, because the number of prices increases exponentially with the number of products. In this study we show that simple pricing strategies are often nearly optimal -- i.e., with surprisingly few prices a firm can obtain 99% of the profit that would be earned by mixed bundling. Specifically, we show that bundle-size pricing -- setting prices that depend only on the size of bundle purchased -- tends to be more profitable than offering the individual products priced separately, and tends to closely approximate the profits from mixed bundling. These findings are based on an array of numerical experiments covering a broad range of demand and cost scenarios, as well as an empirical analysis of the pricing problem for an 8-product firm (a theater company).

What do phone rates, frequent flyer programs, and railroad tariffs all have in common? They are all examples of nonlinear pricing. Pricing is nonlinear when it is not strictly proportional to the quantity purchased. The Electric Power Research Institute has commissioned Robert Wilson to review the various facets of nonlinear pricing. The work starts with a general non-mathematical discussion, followed by a more technical presentation intended for readers with a fairly advanced background. Thorough and detailed, this study has ample examples of case studies from a variety of industries.

We study multiproduct firms in the contexts of unregulated monopoly, regulated monopoly and Cournot oligopoly. Using the concept of consumer surplus as a function of quantities (rather than prices), we present simple formulas for optimal prices and show that Cournot equilibrium exists and corresponds to a Ramsey optimum. We then discuss a tractable class of preferences that involve a generalized form of homotheticity. Profit-maximizing quantities are proportional to efficient quantities. We discuss optimal monopoly regulation when the firm has private information about its cost vector, and find situations where optimal regulation leaves relative price decisions to the firm.

Michael Becher develops a concept for an integrated capacity and price control in revenue management. His concept is based on fuzzy expert controllers and complies with the defined business and application requirements.