Auction, Pickup & Delivery Optimization

• Auction, Pickup, and Delivery Problem is a complex optimization problem that combines auction mechanisms with multi-agent routing and capacity constraints.
• It employs mixed-integer programming and integrated heuristic methods to balance task assignments and minimize total travel delay in MAPD and CSD settings.
• The framework uses a hierarchical fluid–particle approach with VCG auctions and convex optimization to achieve near-optimal performance and incentive compatibility.

The auction, pickup, and delivery problem encapsulates a broad class of optimization and market design models that address the allocation of delivery tasks to agents (robots or human drivers) who must pick up and deliver items across a network, with incentives or market mechanisms (such as auctions) to handle strategic information and scale. This domain includes both industrial “Multi-Agent Pickup and Delivery” (MAPD) with robotic fleets and large-scale crowdsourced delivery (CSD) with self-interested human agents. Recent developments combine exact mixed-integer programs, fluid-relaxed master formulations, Vickrey–Clarke–Groves (VCG) auction truthfulness, and network-based convex optimization to achieve computational tractability, incentive compatibility, and near-optimality at massive scales (Chen et al., 2021, Akamatsu et al., 2023, Oyama et al., 29 Dec 2024).

1. Formal Problem Definitions and Capacity Constraints

The prototypical MAPD instance consists of a set of packages $P = \{1, ..., n\}$, agents/robots $R = \{1,...,m\}$, task locations $I$, earliest pickup times, and a travel time matrix $t(u,v)$ over $I$. Each task $i$ has a pickup $s_i$ and dropoff $g_i$, robots have a start $s_k$ and capacity $C$, and the combined objective is to minimize total travel delay (TTD) over theoretical minimums: $f = \sum_{i\in P} [ a(g_i) - (r_i + t(s_i,g_i)) ].$

Assignment and routing are formulated as a mixed-integer program with variables denoting agent-task matches ($\mu_{i,k}$), route arc selections, precise operation timing, and capacity tracking. For $C>1$ (robots/drivers carry multiple packages), constraints enforce that agent load never exceeds $C$ at any time (Chen et al., 2021). In two-sided market CSD, shippers and drivers are indexed with heterogeneous preferences, demand/supply elasticities, and potential task-bundling, leading to a social cost minimization: $$\min_{\{n^b_t, m^a_r\}} \sum_{j,b,t} c^b_t n^b_t + \sum_{t,w,a,r} c^a_r m^a_r$$ subject to assignment and supply constraints (Oyama et al., 29 Dec 2024).

2. Hierarchical and Fluid-Particle Formulations

Large-scale auction-based delivery systems are decomposed using a hierarchical “fluid–particle” approach. The original integer matching is split into:

• Master Problem (“fluid”): Determines aggregate flows—how many tasks of each type are assigned to each agent group—using continuous relaxations and random utility (ARUM/logit) theory.
• Sub-Problems (“particle”): Each agent group (e.g., drivers with common OD, time window) runs a local combinatorial auction to assign micro-tasks, revealing private costs/values.

For example, with $y_{j,t}$ denoting shippers choosing window $t$ for task $j$, and $f_{(t,w),r}$ drivers of OD $w$ in $t$ executing bundle $r$, the master-level fluid program becomes (Akamatsu et al., 2023, Oyama et al., 29 Dec 2024): $$\min_{y,f} \sum_j \mathbf{C}^S_j \cdot y_j - \hat{\mathcal{H}}^S_j(y_j) + \sum_{t,w} \mathbf{C}^D_{(t,w)} \cdot f_{(t,w)} - \hat{\mathcal{H}}^D_{(t,w)}(f_{(t,w)})$$ with entropic regularizations $\hat{\mathcal{H}}$ representing information from random utility (entropy) models.

This relaxation reduces the dimensionality and allows solution via scalable convex optimization (e.g., Sinkhorn–Knopp for regularized Optimal Transport), with negligible loss in optimum ($<1\%$ error for $10^5$ participants) (Akamatsu et al., 2023, Oyama et al., 29 Dec 2024).

3. Auction Mechanisms and Truthful Cost Revelation

The matching sub-problems within each agent group are solved as VCG auctions. For drivers, each agent submits a bid $b^a_r$ for bundle $r$ representing their opportunity cost (often unobservable directly). The allocation maximizes surplus: $\max_{m^a_r} \sum_{a,r} b^a_r m^a_r$ subject to each participant executing exactly one bundle, and total task assignment matching the master’s allocation.

Winning agents’ payments or rewards are derived from standard VCG “externality” formulas, ensuring dominant-strategy truthfulness and allocative efficiency (Akamatsu et al., 2023, Oyama et al., 29 Dec 2024). This incentivizes the revelation of true private costs, enabling accurate estimation of cost distributions required by the master problem.

For MAPD with robots, auction-based protocols such as TPTS assign tasks using distance-only bids, but are empirically dominated by integrated marginal-cost assignment heuristics that account for dynamic routing conflicts (Chen et al., 2021).

4. Integrated Assignment and Path Planning

MAPD and CSD problems historically used a sequential process: (1) assign tasks using lower-bound estimates, (2) solve path-planning/route optimization given the assignment. This sequential approach can lead to large inefficiencies due to ignoring emergent path conflicts and capacity dynamics.

A fully integrated algorithmic paradigm maintains, for each unassigned task, a set of (agent, marginal cost) pairs, continuously updated with real collision-free routing costs (PBS, A*). Task assignments are performed greedily or by regret-minimization across the true planning costs, incrementally updating only those assignments/path plans affected by new commitments. This “informed assignment” penalizes agent-task matches that would induce routing congestion or collision, minimizing realized TTD (Chen et al., 2021). For CSD, integrated approaches tightly couple the stochastic, private-value-driven assignment with the global partition, ensuring load balancing and resilience to preference heterogeneity (Oyama et al., 29 Dec 2024).

5. Task-Bundling, Network Transformations, and Convexified TAP

For agents with $C>1$ (multi-task), full enumeration of possible bundles is intractable. A task-chain network is constructed as a layered graph, with state transitions representing execution of a sequence of tasks or a return/dummy transition (allowing for chains shorter than $C$). The optimal assignment becomes equivalent to a multi-commodity traffic assignment problem (TAP) over this network, combining capacity, reward-adjusted detour costs, and entropy-based randomness (Oyama et al., 29 Dec 2024). This network representation admits convex Markovian flow optimization: $$\max_{y,x} \left[\text{shipper cost/entropy}\right] + \left[\text{driver network flow/entropy}\right]$$ subject to flow conservation and supply constraints, enabling efficient solution by accelerated first-order methods (FISTA, AGD).

The dual formulation has a low-dimensional price/reward vector $p_{(t,j)}$, corresponding to per-task incentives at equilibrium. The master solution produces assignment flows, and each sub-problem auction resolves the micro-allocation.

6. Performance and Empirical Findings

Both MAPD and CSD auction–pickup–delivery frameworks achieve significant computational and operational improvements:

Setting	Method	Opt. Gap	Solve Time	Key Findings
Warehouse MAPD ($C=1$)	Integrated MCA/PBS	0%	0.1–1 s/step	Min. TTD, load balanced (Chen et al., 2021)
Warehouse MAPD ($C>1$)	RMCA(r), LNS	<10%	<1 s/step	30–60% TTD reduction (Chen et al., 2021)
CSD, 10,000+ agents	Fluid-Particle + VCG	<1%	<1 s	$>100\times$ faster (Akamatsu et al., 2023)
CSD, 100,000 agents	TAP Dual AGD FPD	0.38%	<8 s	Bundle + truthfulness (Oyama et al., 29 Dec 2024)

Integrated approaches referencing actual delivery costs outperform both decoupled “Hungarian + MAPF” methods and auction approaches using only lower-bound distances (Chen et al., 2021). The fluid–particle with randomized utilities and auction integration achieves near-optimal surplus with sub-second response even at scale, and enables day-to-day updating of agent heterogeneity from revealed bids (Akamatsu et al., 2023, Oyama et al., 29 Dec 2024).

7. Limitations, Implications, and Future Research

Current formulations operate under static, deterministic network travel times. Extensions to dynamic, congestion-dependent times, real-time recomputation, or online learning of utility distributions remain open. VCG mechanisms, while truthful, may cause excessive subsidies in certain regimes; budget-balanced mechanisms are an area for future work (Oyama et al., 29 Dec 2024).

The combination of task-bundling, demand/supply elasticities, and truthful information elicitation is structurally robust for large-scale implementations in both industrial and crowdsourced settings. Iterative auction-based feedback enables market platforms to adapt allocations to evolving participant preferences and load patterns. These advances bridge the gap between computational tractability, market-theoretic optimality, and scalable real-world implementation for pickup and delivery problems (Chen et al., 2021, Akamatsu et al., 2023, Oyama et al., 29 Dec 2024).