The rank pricing problem: Models and branch-and-cut algorithms

Abstract

One of the main concerns in management and economic planning is to sell the right product to the right customer for the right price. Companies in retail and manufacturing employ pricing strategies to maximize their revenues. The Rank Pricing Problem considers a unit-demand model with unlimited supply and uniform budgets in which customers have a rank-buying behavior. Under these assumptions, the problem is first analyzed from the perspective of bilevel pricing models and formulated as a non linear bilevel program with multiple independent followers. We also present a direct non linear single level formulation bearing in mind the aim of the problem. Two different linearizations of the models are carried out and two families of valid inequalities are obtained which, embedded in the formulations by implementing a branch-and-cut algorithm, allow us to tighten the upper bound given by the linear relaxation of the models. We also study the polyhedral structure of the models, taking advantage of the fact that a subset of their constraints constitutes a special case of the Set Packing Problem, and characterize all the clique facets. Besides, we develop a preprocessing procedure to reduce the size of the instances. Finally, we show the efficiency of the formulations, the branch-and-cut algorithms and the preprocessing through extensive computational experiments.

The rank pricing problem with ties

“…Aggarwal et al [1] and Briest and Krysta [7] take up the work in [27] and present complexity results and approximation algorithm schemes for RPP and variants of it. To the best of our knowledge, Calvete et al [8] proposed the first formulations for the RPP, and thus [8] is a good starting point when tackling RPPT.…”

“…Note that, even though the results are stated for models (RM) and (3IM), they also apply to subproblems (BR SUBD k ) during the resolution of model (BM). This preprocessing is based on the one described in Calvete et al [8] for RPP problem.…”

“…These instances were proposed in [8] for RPP, so we modified them by adding ties in the ranked lists of preferences of the customers. Thus, for each size we generated three instances varying the number of ties in the list of preferences (denoted Ties in Tables 4 Figure 3 is a performance profile that shows the percentage of instances having an integrality gap less than or equal to the value on the x-axis.…”

“…Considering ties in the preference lists of the customers leads to a different bilevel structure of the problem. As detailed in Calvete et al [8], in the RPP (without ties), the second level problem associated to each customer has a unique optimal solution for a given vector of prices of the products. However, in this extension, the indifference results in second level problems that may have multiple optimal solutions.…”

Models and algorithms for the product pricing with single-minded customers requesting bundles

“…To the best of our knowledge, the SMBPP has not yet been tackled using exact approaches based on mathematical optimization. Mathematical formulations for pricing problems have attracted the attention of the literature in the recent years, see for instance recent work by Calvete et al 2019. This work contributes to the literature by proposing linear and nonlinear formulations for the SMBPP.…”