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Optimal Pricing With Impatient Customers

Jieqi Di, Sigrún Andradóttir, Hayriye Ayhan

TL;DR

This paper addresses pricing in a congested service system where arriving customers decide to join based on a quoted price and may abandon during waiting or service. The authors formulate the problem as a Markov decision process using arrival-rate controls and prove that the optimal policy is always uni-modal in queue length and monotone decreasing under a cost-condition, contrasting with traditional queueing results. They introduce two practical heuristics—the cutoff static policy and the two price policy—that are simpler to implement and shown to achieve near-optimal performance, with the two-price policy exhibiting greater robustness. Numerical experiments across diverse parameter settings demonstrate the near-optimality and robustness of the two-price heuristic, suggesting actionable pricing rules for managers facing impatient customers and abandonment dynamics.

Abstract

We investigate the optimal pricing strategy in a service-providing framework, where customers can become impatient and leave the system prior to service completion. In this setting, a price is quoted to an incoming customer based on the current number of customers in the system. When the quoted price is lower than the price the incoming customer is willing to pay (which follows a fixed probability distribution), then the customer joins the system and a reward equal to the quoted price is earned. A cost is incurred upon abandonment and a holding cost is incurred for customers waiting to be served. Our goal is to determine the pricing policy that maximizes the long-run average profit. Unlike traditional queueing systems without abandonments, we show that the optimal quoted prices do not always increase with the queue length in this setting. In particular, we prove that the optimal pricing policy is always uni-modal and provide conditions guaranteeing that the optimal policy is increasing in the number of customers in the system. Moreover, we introduce two heuristics that simplify the optimal dynamic pricing policy. Both heuristics admit customers until the number of customers in the system reaches a certain threshold. The cutoff static policy charges all admitted customers a fixed price while the two price policy charges one price when the arriving customer can enter service immediately and another price if the customer needs to wait. By selecting the price(s) and threshold that maximize the long-run average profit, both heuristics achieve near optimality and the two price policy provides more robustness compared to the cutoff static policy.

Optimal Pricing With Impatient Customers

TL;DR

This paper addresses pricing in a congested service system where arriving customers decide to join based on a quoted price and may abandon during waiting or service. The authors formulate the problem as a Markov decision process using arrival-rate controls and prove that the optimal policy is always uni-modal in queue length and monotone decreasing under a cost-condition, contrasting with traditional queueing results. They introduce two practical heuristics—the cutoff static policy and the two price policy—that are simpler to implement and shown to achieve near-optimal performance, with the two-price policy exhibiting greater robustness. Numerical experiments across diverse parameter settings demonstrate the near-optimality and robustness of the two-price heuristic, suggesting actionable pricing rules for managers facing impatient customers and abandonment dynamics.

Abstract

We investigate the optimal pricing strategy in a service-providing framework, where customers can become impatient and leave the system prior to service completion. In this setting, a price is quoted to an incoming customer based on the current number of customers in the system. When the quoted price is lower than the price the incoming customer is willing to pay (which follows a fixed probability distribution), then the customer joins the system and a reward equal to the quoted price is earned. A cost is incurred upon abandonment and a holding cost is incurred for customers waiting to be served. Our goal is to determine the pricing policy that maximizes the long-run average profit. Unlike traditional queueing systems without abandonments, we show that the optimal quoted prices do not always increase with the queue length in this setting. In particular, we prove that the optimal pricing policy is always uni-modal and provide conditions guaranteeing that the optimal policy is increasing in the number of customers in the system. Moreover, we introduce two heuristics that simplify the optimal dynamic pricing policy. Both heuristics admit customers until the number of customers in the system reaches a certain threshold. The cutoff static policy charges all admitted customers a fixed price while the two price policy charges one price when the arriving customer can enter service immediately and another price if the customer needs to wait. By selecting the price(s) and threshold that maximize the long-run average profit, both heuristics achieve near optimality and the two price policy provides more robustness compared to the cutoff static policy.
Paper Structure (14 sections, 15 theorems, 97 equations, 13 figures, 3 tables, 1 algorithm)

This paper contains 14 sections, 15 theorems, 97 equations, 13 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Model Description
  4. Structure of the Optimal Dynamic Policy
  5. Major Results
  6. Proofs of Major Results
  7. Discussion: Is $\frac{c_h+c_s\theta_s}{\mu+\theta_s}\leq \frac{c_h}{\theta_q}+c_q$ Necessary for Monotonicity?
  8. Monotone Policy Iteration
  9. Heuristics
  10. Best Cutoff Static Policy
  11. Performance of the Best Cutoff Static Policy
  12. Best Two Price Policy
  13. Numerical Experiments
  14. Conclusion

Key Result

Theorem 1

The optimal arrival rate policy has a uni-modal structure.

Figures (13)

  • Figure 1: The plot of an optimal arrival rate policy for each state
  • Figure 2:
  • Figure 3:
  • Figure 4: Structure of the optimal policy with $m=1$ and evaluation distribution $U(20,50)$
  • Figure 5: Structure of the optimal policy with $m=1$ and evaluation distribution $Exp(1/35)$
  • ...and 8 more figures

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 16 more