Pricing for Multi-modal Pickup and Delivery Problems with Heterogeneous Users

TL;DR

The result shows that one can simplify the Linear Program formulations used to solve for edge prices by first finding the path prices combinatorially, and this is connected with prior works that consider non-atomic congestion games.

Abstract

In this paper, we study the pickup and delivery problem with multiple transportation modalities, and address the challenge of efficiently allocating transportation resources while price matching users with their desired delivery modes. More precisely, we consider that orders are demanded by a heterogeneous population of users with varying trade-offs between price and latency. To capture how prices affect the behavior of heterogeneous selfish users choosing between multiple delivery modes, we construct a congestion game taking place over a form of star network, where each source-sink pair is composed of parallel links connecting users with their preferred delivery method. Using the unique geometry of this network, we prove that one can set prices explicitly to induce any desired network flow, i.e, given a desired allocation strategy, we have a closed-form solution for the delivery prices. We conclude by performing a case study on a meal delivery problem with multiple courier modalities using data from real world instances.

Figures (3)

• Figure 1: We represent the pickup and delivery problem as a congestion game played over a star network. Each source-sink pair is denoted by $i\in\mathcal{I}$, which can be viewed as a population of users at some location demanding a particular order at a certain rate. Each source-sink pair is connected by a set of parallel edges $j\in\mathcal{J}$, which can be viewed as the set of delivery modes the users choose from. Note that we are not concerned with how the couriers are routed to the pickup or delivery location, and instead focus on how we allocate the different delivery modes for each order. Specifically, our goal is to induce an optimal allocation of transportation modalities by appropriately setting prices for each order-modality pair.
• Figure 2: A sketch depicting how a canonical Nash flow splits the population $a\in[a_0,a_{J}]$ into subintervals $[a_{j-1},a_j): x(a)=j$, where $a_0=0$, $a_{J}=1$, and $j\in\mathcal{J}$. Note that order $i$ is left out from notation.
• Figure 3: The above schematic is an example of the user preference functions $\alpha_{i,j}$ for three modes $\mathcal{J}=\{1,2,3\}$, where the index corresponding to order $i\in\mathcal{I}$ has been left out for convenience. The $x$-axis represents users $a\in[0,1]$ and the $y$-axis represents hours per dollar, the inverse of the value of time. We can see that the three lines all intersect at $a=1$, signifying that the most frugal users do not care about the modality and prefer the cheapest option. All three lines have varying slopes, and we sort their indices so that for $a<a'$, $\frac{\alpha_{i,j+1}(a)}{\alpha_{i,j}(a)}\geq \frac{\alpha_{i,j+1}(a')}{\alpha_{i,j}(a')}$ for all $j\in\{1,\ldots,J-1\}$. The dashed gray lines help visualize this property: as we increase $a$ to $1$, the ratio $\frac{\alpha_{3}(a)}{\alpha_{2}(a)}$ decreases due to $\alpha_2$ having a larger slope than $\alpha_3$, reaching a minimum value of $1$ when $a=1$. Intuitively, this orders modalities by their slopes in decreasing order, representing a decrease in relative luxury. For our setting, this order is eVTOL aircrafts, luxury vehicles, and standard vehicles.

Theorems & Definitions (11)

• Definition 1
• Proposition 1
• Definition 2
• Definition 3
• Proposition 2
• Theorem 1
• proof
• Corollary 1
• Corollary 2
• Remark 1