Nested Bisection Algorithm for Scalable Dynamic Pricing
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Solution Overview
Problem
Current dynamic pricing systems face challenges in efficiently pricing a large set of items to maintain an aggregate gross margin above a specified level, especially under uncertain demand conditions, and are not scalable for online retailers with large catalogs.
Innovation Solution
A computationally efficient nested-bisection algorithm is developed, which is amenable to parallelization, allowing for the pricing of numerous items by formulating the revenue maximization under margin constraints as a tractable convex problem, leveraging a separable linear demand model and incorporating interval uncertainty to hedge profit losses with gains from higher-margin items.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If traditional dynamic pricing systems are used to price a large set of items, then revenue maximization can be achieved, but the system becomes computationally inefficient and not scalable for large product catalogs
Solution Approach 1:
The patent segments the pricing problem into individual item-level optimizations with a shared Lagrange multiplier for the aggregate margin constraint. Each item's price can be optimized independently given the multiplier value, allowing parallel computation across thousands of items while maintaining global margin constraints through iterative adjustment of the multiplier.
Solution Approach 2:
The patent transforms the constrained optimization problem into an unconstrained one by introducing a Lagrange multiplier parameter that enforces the aggregate margin constraint. By iteratively adjusting this parameter, the system achieves computational efficiency while maintaining constraint satisfaction, enabling scalable pricing for large catalogs.
2Reliability
If dynamic pricing is applied to maintain aggregate gross margin above a specified level, then margin constraints are satisfied, but the system cannot handle uncertain demand conditions effectively
Solution Approach 1:
The patent implements a feedback mechanism where the Lagrange multiplier is iteratively adjusted based on whether the aggregate margin constraint is satisfied. The system continuously monitors the relationship between optimized prices and margin requirements, adjusting the multiplier to ensure constraints are met while adapting to demand conditions.
Solution Approach 2:
The patent performs preliminary optimization of individual item prices based on expected demand and margin requirements before finalizing the pricing policy. By pre-computing price recommendations that account for uncertainty ranges and margin constraints, the system prepares robust pricing strategies that can handle realized demand variations.
Data Source
AI summary
A dynamic pricing algorithm is used to price a large set of items so that their aggregate margin is above a pre-specified level even under uncertain demand. The algorithm automatically hedges the profit loss associated to low-margin items with profit gains associated to higher-margin items, and does so even when the realized demand is different than the expected one. The algorithm leverages the separability of a re-formulation of the robust counterpart of the nominal revenue maximization problem. This separability results into a nested bisection algorithm where each iteration in the procedure requires only computation of a number of independent, one-dimensional optimization problems, one for each product to price. The algorithm is easily implemented in a parallel architecture such a multi-core computer or a cluster of computers, where each core handles an independent one-dimensional problem corresponding to an item and its data is stored locally.