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Pricing, bundling, and driver behavior in crowdsourced delivery

Alim Buğra Çınar, Claudia Archetti, Wout Dullaert, Markus Leitner, Stefan Waldherr

Abstract

Challenges in last-mile delivery have encouraged innovative solutions like crowdsourced delivery, where online platforms leverage the services of drivers who occasionally perform delivery tasks for compensation. A key challenge is that occasional drivers' acceptance behavior towards offered tasks is uncertain and influenced by task properties and compensation. The current literature lacks formulations that fully address this challenge. Hence, we formulate an integrated problem that maximizes total expected cost savings by offering task bundles to occasional drivers. To this end, we simultaneously determine the optimal bundle set, their assignment to occasional drivers, and compensations for each pair while considering acceptance probabilities, which are captured via generic logistic functions. The vast number of potential bundles, combined with incorporating acceptance probabilities leads to a mixed-integer nonlinear program (MINLP) with exponentially many variables. Using mild assumptions, we address these complexities by exploiting properties of the problem, leading to an exact linearization of the MINLP which we solve via a tailored exact column generation algorithm. Our algorithm uses a variant of the elementary shortest path problem with resource constraints (ESPPRC) that features a non-linear and non-additive objective function as its subproblem, for which we develop tailored dominance and pruning strategies. We introduce several heuristic and exact variants and perform an extensive set of experiments evaluating the algorithm performances and solution structures. The results demonstrate the efficiency of the algorithms for instances with up to 120 tasks and 60 drivers and highlight the advantages of integrated decision-making over sequential approaches. The sensitivity analysis indicates that compensation is the most influential factor in shaping the bundle structure.

Pricing, bundling, and driver behavior in crowdsourced delivery

Abstract

Challenges in last-mile delivery have encouraged innovative solutions like crowdsourced delivery, where online platforms leverage the services of drivers who occasionally perform delivery tasks for compensation. A key challenge is that occasional drivers' acceptance behavior towards offered tasks is uncertain and influenced by task properties and compensation. The current literature lacks formulations that fully address this challenge. Hence, we formulate an integrated problem that maximizes total expected cost savings by offering task bundles to occasional drivers. To this end, we simultaneously determine the optimal bundle set, their assignment to occasional drivers, and compensations for each pair while considering acceptance probabilities, which are captured via generic logistic functions. The vast number of potential bundles, combined with incorporating acceptance probabilities leads to a mixed-integer nonlinear program (MINLP) with exponentially many variables. Using mild assumptions, we address these complexities by exploiting properties of the problem, leading to an exact linearization of the MINLP which we solve via a tailored exact column generation algorithm. Our algorithm uses a variant of the elementary shortest path problem with resource constraints (ESPPRC) that features a non-linear and non-additive objective function as its subproblem, for which we develop tailored dominance and pruning strategies. We introduce several heuristic and exact variants and perform an extensive set of experiments evaluating the algorithm performances and solution structures. The results demonstrate the efficiency of the algorithms for instances with up to 120 tasks and 60 drivers and highlight the advantages of integrated decision-making over sequential approaches. The sensitivity analysis indicates that compensation is the most influential factor in shaping the bundle structure.

Paper Structure

This paper contains 41 sections, 10 theorems, 19 equations, 11 figures, 17 tables.

Table of Contents

  1. Introduction
  2. Literature review
  3. Problem definition and mathematical formulation
  4. Assumptions:
  5. Example problem instance
  6. Mixed-integer nonlinear programming formulation
  7. Modeling and fitting acceptance probability functions
  8. Methodology
  9. Linearization of formulation (2)
  10. Column generation
  11. Solving the pricing subproblem
  12. Initialization
  13. Extension
  14. Example
  15. Dominance and label pruning
  16. ...and 26 more sections

Key Result

Proposition 1

There exists an optimal solution to formulation eq:minlp in which each (optimal) compensation value $C^*_{wb}$ for every bundle $b$ offered to occasional driver $w$ is calculated as

Figures (11)

  • Figure 1: An instance with $D=2, M=5, N=2$, and two offers (indicated in red and blue) as well as the driver interface of the (red) offer to driver one.
  • Figure 2: Overview of the exact solution algorithm.
  • Figure 3: Example corridor space defined around $\vec{v}_1=e_1-s_1$ for occasional driver 1 containing $d_1$, $t_2$, and $t_4$.
  • Figure 4: Acceptance probabilities of different occasional driver classes towards different offers
  • Figure 5: Performance of exact algorithms.
  • ...and 6 more figures

Theorems & Definitions (15)

  • Definition 1: Logistic acceptance probability
  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Corollary 1
  • Corollary 2: Reduced cost-based label pruning
  • Corollary 3: Route-based dominance
  • Corollary 4: Detour upper bound
  • ...and 5 more