An efficient auction mechanism for regional logistics synchronization

Abstract

This paper is the first attempt to propose an efficient auction mechanism for the regional logistics synchronization (RLS) problem, which aims to capture both logistics punctuality and simultaneity in a regional logistics network. The main motivation of RLS is motivated by our industrial collaborator, i.e. a third-party logistics (3PL) company, that if the delay has already occurred or will occur, the customers tend to pursue the simultaneity. We develop the one-sided Vickrey–Clarke–Groves (O-VCG) auction to realize incentive compatibility (on the buy side), budget balance, and individual rationality. The vehicle routing problem faced by the 3PL company is formulated as the lane covering problem with RLS requirements. Given the complexity of the proposed model, three canonical swarm intelligence meta-heuristics are employed to address the auction-based RLS problem. Besides, a superior tracking artificial bee colony with novel information learning mechanism is further developed to explore better solutions. Comparison results reveal the effectiveness of the proposed optimizers in terms of realized social welfare. Experimental results show that the O-VCG auction can achieve high synchronization level, approximately allocative efficiency and (ex post) budget balance.

Appendix A

Proof of Theorem 1

It suffices to prove for the case of $u_{3PL} >0$. Suppose that agents in $I\backslash \{i\}$ bid truthfully. If agent i bids truthfully, then he makes a VCG-like payment, $p_i (t_i )=v_i (t_i )-(\mathrm{Z}(I)-\mathrm{Z}(I\backslash i))$ (note: $t_i$ can be empty), where

$$\begin{aligned} \mathrm{Z}(I)=V(\varphi )-\hat{{C}}(\varphi )=\sum _{i\in I(\varphi )} {v_i (t_i )} -\hat{{C}}(\varphi ). \end{aligned}$$

Similarly, if he bids $\hat{{v}}_i (t)$, then he makes the corresponding VCG-like payment, $\hat{{p}}_i (\hat{{t}}_i )=\hat{{v}}_i (\hat{{t}}_i )-(\hat{{\mathrm{Z}}}(I)-\mathrm{Z}(I\backslash i))$, where

$$\begin{aligned} \hat{{\mathrm{Z}}}(I)&=V({\varphi }')-\hat{{C}}({\varphi }')\\&=\left[ \sum _{j\in I({\varphi }')\backslash \{i\}} {v_j (\hat{{t}}_j )} +\hat{{v}}_i (\hat{{t}}_i )\right] -\hat{{C}}({\varphi }'). \end{aligned}$$

If truthful bidding is not a dominant strategy for agent i, then

$$\begin{aligned} u_i (t_i )<\hat{{u}}_i (\hat{{t}}_i )=v_i (\hat{{t}}_i )-\hat{{p}}_i (\hat{{t}}_i ). \end{aligned}$$

The above inequality can be rewritten as

$$\begin{aligned}&\mathrm{Z}(I)-\mathrm{Z}(I\backslash i)<v_i (\hat{{t}}_i )-[\hat{{v}}_i (\hat{{t}}_i )-(\hat{{\mathrm{Z}}}(I)-\mathrm{Z}(I\backslash i))] \\&\quad \Leftrightarrow \mathrm{Z}(I)-\mathrm{Z}(I\backslash i)\\&\quad<\left[ \sum _{j\in I({\varphi }')\backslash \{i\}} {v_j (\hat{{t}}_j )} +v_i (\hat{{t}}_i )\right] -\hat{{C}}({\varphi }')-\mathrm{Z}(I\backslash i) \\&\quad \Leftrightarrow \mathrm{Z}(I)<\left[ \sum _{j\in I({\varphi }')\backslash \{i\}} {v_j (\hat{{t}}_j )} +v_i (\hat{{t}}_i )\right] -\hat{{C}}({\varphi }'), \end{aligned}$$

which contradicts the fact that $\varphi$ is an efficient allocation achieving $\mathrm{Z}(I)$. $\square$

Proof of Theorem 2

It is clear that $u_i (t_i )=\mathrm{Z}(I)-\mathrm{Z}(I\backslash i)\ge 0$. Thus, O-VCG auction is individually rational for each agent $i\in I$. Since the 3PL company is the auctioneer, the trade will fail if $u_{3PL} \le 0$. Also, the O-VCG auction is budget-balanced for the auctioneer. Based on Theorem 1 and the assumption that the 3PL company tells the truth, O-VCG auction finds an allocatively efficient allocation that maximizes social welfare. $\square$