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An exact method for a problem of time slot pricing

Olivier Bilenne, Frédéric Meunier

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.

An exact method for a problem of time slot pricing

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 , where is the set of possible prices and 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.

Paper Structure

This paper contains 15 sections, 18 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Problem formulation
  4. Contributions
  5. Preliminary results
  6. Algorithm when $P$ is finite
  7. Description of the algorithm and proof of \ref{['thm:semi-disc']}
  8. Proofs of the lemmas
  9. Algorithm when $P$ is $\mathbb{R}$
  10. Proof of \ref{['thm:cont']}
  11. Proofs of \ref{['prop:upper_bound', 'prop:bounded-delta']}
  12. Preliminaries
  13. Proof of \ref{['prop:bounded-delta']}
  14. Proof of \ref{['prop:upper_bound']}
  15. Concluding remarks

Key Result

Theorem 1.1

When $P$ is finite, the problem can be solved in time $O(n^3|P|^3)$.

Figures (2)

  • Figure 1: The black dots form an example of a feasible price profile when $n=8$ and $P$ has five elements. The lower envelope of the functions $d(s-t_{j})+p_{j}$ determines the intervals $J_{j}({p})$, which are ordered as in \ref{['prop:intervals']}. Its nonpositivity further delimits the intervals $I_{j}({p})$. All time slots with prices above the threshold $-d(0)$ are therefore avoided. The solution is feasible as the service loads $\mu(I_{j}({p}))$ are less than the capacities $\nu_{j}$.
  • Figure 2: Example of a source-to-sink path in $D$ with its associated path in $L(D)$ (dashed lines) when $n=8$ and $P$ has five elements. The corresponding price profile derived in \ref{['lem:path-price']} is given by the sequence of black dots.

Theorems & Definitions (36)

  • Remark 1
  • Theorem 1.1
  • Proposition 1.2
  • proof : Proof of \ref{['prop:bounded']}
  • Theorem 1.3
  • Lemma 2.2: Monotonicity
  • proof
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • ...and 26 more