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Same-day or next-day? Transparent time-dependent shipment pricing for e-fulfillment

Uta Mohring, Melvin Drent, Ivo Adan, Willem van Jaarsveld

TL;DR

The paper addresses balancing same-day and next-day e-fulfillment demand through transparent, time-dependent shipment pricing. It develops a discrete-time periodic Markov-chain model, and a problem-domain transform that maps policies to cumulative express-demand profiles, enabling tractable analysis. Key results show the optimal policy has a cutoff point and monotonically increasing express-fees, with a simple two-level fee structure achieving near-optimal performance and practical interpretability. Numerical experiments demonstrate substantial profit gains from time-dependent pricing, validating the approach as a transparent mechanism to align demand with fulfillment capacity while preserving customer fairness. The work provides actionable guidelines and a path toward polynomial-time optimization for complex time-dependent pricing in deadline-driven fulfillment settings.

Abstract

We develop a parsimonious model of an e-commerce fulfillment center that offers time-dependent shipment options and corresponding fees to utility-maximizing customers arriving according to a Poisson process. For any such policy, we provide an exact steady-state analysis using the underlying periodic Markov chain to characterize system performance. Because shipment fees shape both the volume and timing of same-day demand, direct optimization over the price domain is analytically intractable. To enable structural and computational insights, we introduce a transformation that maps each shipment-fee policy to its induced cumulative demand profile. This reformulation reveals that the optimal policy features a cutoff time and monotonically increasing fees, and it yields a supermodular profit function that can be optimized in polynomial time. We also propose a simple two-level time-dependent fee structure that is intuitive for customers and achieves near-optimal performance. Numerical experiments show that introducing a cutoff time substantially improves profits under static fees, and that using time-dependent fees produces further significant gains. Overall, transparent time-dependent shipment policies help firms align same-day demand with fulfillment capacity while maintaining transparency and fairness for customers.

Same-day or next-day? Transparent time-dependent shipment pricing for e-fulfillment

TL;DR

The paper addresses balancing same-day and next-day e-fulfillment demand through transparent, time-dependent shipment pricing. It develops a discrete-time periodic Markov-chain model, and a problem-domain transform that maps policies to cumulative express-demand profiles, enabling tractable analysis. Key results show the optimal policy has a cutoff point and monotonically increasing express-fees, with a simple two-level fee structure achieving near-optimal performance and practical interpretability. Numerical experiments demonstrate substantial profit gains from time-dependent pricing, validating the approach as a transparent mechanism to align demand with fulfillment capacity while preserving customer fairness. The work provides actionable guidelines and a path toward polynomial-time optimization for complex time-dependent pricing in deadline-driven fulfillment settings.

Abstract

We develop a parsimonious model of an e-commerce fulfillment center that offers time-dependent shipment options and corresponding fees to utility-maximizing customers arriving according to a Poisson process. For any such policy, we provide an exact steady-state analysis using the underlying periodic Markov chain to characterize system performance. Because shipment fees shape both the volume and timing of same-day demand, direct optimization over the price domain is analytically intractable. To enable structural and computational insights, we introduce a transformation that maps each shipment-fee policy to its induced cumulative demand profile. This reformulation reveals that the optimal policy features a cutoff time and monotonically increasing fees, and it yields a supermodular profit function that can be optimized in polynomial time. We also propose a simple two-level time-dependent fee structure that is intuitive for customers and achieves near-optimal performance. Numerical experiments show that introducing a cutoff time substantially improves profits under static fees, and that using time-dependent fees produces further significant gains. Overall, transparent time-dependent shipment policies help firms align same-day demand with fulfillment capacity while maintaining transparency and fairness for customers.
Paper Structure (28 sections, 12 theorems, 84 equations, 6 figures, 2 tables)

This paper contains 28 sections, 12 theorems, 84 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature review
  3. Pricing in queueing systems
  4. Pricing as part of order fulfillment operations
  5. Deadline-oriented order fulfillment operations
  6. Model
  7. Problem description
  8. Model
  9. Discrete-time Markov chain
  10. Performance measures
  11. Structural properties of the shipment policy
  12. Constant shipment policy
  13. Constant cutoff point shipment policy
  14. Two-level time-dependent shipment policy
  15. General time-dependent shipment policy
  16. ...and 13 more sections

Key Result

Lemma 1

Let $\pi=(\bar{\tau},f_0, f_1, \dots, f_{\bar{\tau}})\in\Pi$. The expected revenue $\mathbb{E}[R^{\pi}]$ is separable and concave in the shipment fees $f_\tau$, $\tau = 0,1,\dots, \bar{\tau}$. The unique revenue-maximizing shipment policy is $\pi^r=(T-1,f^r,\dots, f^r)$ with $f^r = \max\{U_{\textrm{

Figures (6)

  • Figure 1: Representation of the time line and corresponding notation.
  • Figure 2: Example express shipment demand profile (dashed line for guidance only).
  • Figure 3: Express shipment demand profile of constant cutoff point shipment policy (dashed line for guidance only).
  • Figure 4: Illustration of the two-level time-dependent shipment policy.
  • Figure 5: Express shipment demand profiles of two-level time-dependent shipment policies (dashed line for guidance only).
  • ...and 1 more figures

Theorems & Definitions (18)

  • Definition 1: Set of shipment policies
  • Lemma 1: Revenue-maximizing shipment policy
  • Definition 2: Express shipment demand profile
  • Definition 3: Constant shipment policy
  • Lemma 2: Constant shipment policy
  • Definition 4: Constant cutoff point shipment policy
  • Lemma 3: Effect of cutoff point
  • Definition 5: Two-level time-dependent shipment policy
  • Lemma 4: Type of two-level fee
  • Lemma 5: Effect of two-level time-dependent fee
  • ...and 8 more