An Exact Method for a Problem of Time Slot Pricing

TL;DR

This paper presents an exact algorithm for optimizing time slot prices to maximize revenue while respecting capacity constraints, applicable to convex distance functions and allowing for asymptotically optimal solutions with real prices.

Contribution

It introduces a novel exact algorithm for a time slot pricing problem with capacity constraints, applicable to convex distance functions, and extends to approximate solutions for real prices.

Findings

The algorithm runs in polynomial time, specifically O(n^3|P|^3).

It can find optimal or near-optimal prices under capacity constraints.

Applicable to convex distance functions and real-valued prices.

Abstract

A company provides a service at different time slots, each slot being endowed with a capacity. A non-atomic population of users is willing to purchase this service. The population is modeled as a continuous measure over the preferred times. Every user looks at the time slot that minimizes the sum of the price assigned by the company to this time slot and the distance to their preferred time. If this sum is non-negative, then the user chooses this time slot for getting the service. If this sum is positive, then the user rejects the service. We address the problem of finding prices that ensure that the volume of users choosing each time slot is below capacity, while maximizing the revenue of the company. For the case where the distance function is convex, we propose an exact algorithm for solving this problem in time $O(n^3|P|^3)$, where $P$ is the set of possible prices and $n$ is the number of time slots. For the case where the prices can be any real numbers, this algorithm can also be used to find asymptotically optimal solutions in polynomial time under mild extra assumptions on the distance function and the measure modeling the population.