Multi-Compartment VRP with Multiple Time Windows

• The paper introduces an MCVRPMTW formulation that rigorously integrates compartment flexibility, multiple time windows, and driver operational regulations.
• It employs a branch-and-price algorithm with resource-constrained shortest path labeling and rolling-space clustering for efficient large-scale solution generation.
• Computational results reveal cost reductions, a ~20% decrease in vehicles, and significant performance improvements compared to manual planning.

The Multi-Compartment Vehicle Routing Problem with Multiple Time Windows (MCVRPMTW) generalizes the classical vehicle routing problem with time windows (VRPTW) by introducing vehicles equipped with several independent compartments and customers requesting service within multiple possible time windows. The formulation incorporates compartment flexibility, rigorous item-to-compartment and item-to-item compatibility constraints, and driver operational considerations such as mandatory breaks and daily regulations. Recent advances include exact algorithms based on branch-and-price frameworks equipped with resource-constrained shortest path labeling, dominance rules, and large-scale decomposition via rolling-space clustering (Raqabi et al., 22 Jan 2026).

1. Mathematical Formulation

In MCVRPMTW, a fleet $\mathcal{K}$ of vehicles, each with compartments $\mathcal{M}_k$, services a set of clients $\mathcal{N}$ over a planning horizon $\mathcal{D}$. Vehicles start and finish routes at depot nodes $\sigma, \delta$ and traverse arc sets $\mathcal{A}$. Customers may specify multiple delivery windows, denoted $\Theta_n$ for client $n$, with allowable intervals $[s_n^p,e_n^p]$.

Indices and Parameters

• $x_{ij}^{kd}$: vehicle $k$ traverses $(i,j)$ on day $d$
• $z_{mn}^k$: client $n$ assigned to compartment $m$ of $k$
• $y_{mn}^k$: quantity of item for client $n$ in compartment $m$ of $k$
• Compartment-related constraints include $b_{nm}^k \in \{0,1\}$ (client-to-compartment compatibility), $f_{ij}\in\{0,1\}$ (item-to-item compatibility), total and per-compartment capacity limits ($L_k, \overline L_{mk}$), mandatory driver breaks $B$, daily time and distance caps ($T$, $\mathit{Dist}$).

Set Partitioning and MILP Formulation

The canonical set partitioning model minimizes total cost:

$$\min_{y}\;\sum_{r\in\Omega}c_r y_r \quad\text{s.t.}\;\sum_{r\in\Omega}a_{nr}y_r=1\quad\forall n\in\mathcal{N}$$

where $y_r$ is a binary route variable and $a_{nr}$ indicates client inclusion. For MILP, explicit variable definitions and constraints encode assignment, compartment loads, compatibility, regulatory limits, and feasibility:

\begin{align} \min\;&\sum_{k,d}\sum_{(i,j)\in\mathcal{A}}c_{ij}\,x_{ij}^{kd} \ &z_{mn}^k \le b_{nm}^k \ &y_{mn}^k \le Q_n\,z_{mn}^k \ &z_{mi}^k+z_{mj}^k \le 1+f_{ij} \ &\sum_m y_{mn}^k = Q_n\sum_{i,d}x_{in}^{kd} \ &\sum_{n,m}y_{mn}^k \le L_k \ &\sum_{n}y_{mn}^k \le \overline L_{mk}\sum_{n}z_{mn}^k \end{align}

Time window, flow, and driver break constraints ensure operational feasibility; big-M terms enforce conditional constraints for arrivals, departures, and scheduling.

2. Branch-and-Price and Labeling Algorithm

The branch-and-price (B&P) approach decomposes MCVRPMTW into a restricted master problem (RMP) and pricing subproblem. The RMP selects routes minimizing adjusted cost, subject to customer coverage. Pricing involves solving an Elementary Shortest Path Problem with Resource Constraints (ESPPRC), reflecting compartmental, compatibility, and time window constraints.

Label Structure

A label for a partial path encodes:

• Last node $i(L)$
• Accumulated time $t(L)$
• Reduced cost $\bar c(L)$
• Current day $d(L)$
• Set of served clients $\mathcal{S}(L)$
• Load in each compartment $l_m^k(L)$
• Items in each compartment $O_m^k(L)$
• Working time $g(L)$ and driving distance $o(L)$

Resource extension applies to moves $(i,j)$, window choice $p$, compartment $m^h$, and optional break $d_{ij}$, updating label attributes according to feasibility checks: window adherence, capacity, compatibility, and regulatory limits.

Dominance and Pricing

Dominance rules prune inferior labels:

• Same-day: $L_1$ dominates $L_2$ if it matches node, achieves lower time/cost/load, contains a superset of clients/compatibles, and does not exceed time/distance.
• Inter-day: also checks for possible advancement via breaks.
• Aggressive early dominance uses relaxed tolerances for feasibility.

Reduced arc cost updates as $\bar c_{ij} = c_{ij} - \pi_i$, with $\pi_i$ the dual for each customer.

3. Acceleration and Stabilization Strategies

Large-scale MCVRPMTW instances require acceleration beyond standard B&P:

• Symmetry-limiting: Label extensions prioritize earliest feasible window and smallest compartment index, minimizing redundant path exploration.
• Dual stabilization: RMPs warm-start from a global pool, with penalty or stabilization intervals to damp oscillation in column selection.
• Column selection: Only the top $K$ columns by reduced cost and select high-quality diverse columns are used per iteration.
• Branching heuristics: Branch on arc-inclusion variables; primal-diving heuristics prioritize near-integral variables along a depth-first trajectory, expediting feasible solution construction and primal bound improvement.

4. Rolling-Space Decomposition and Large-Scale Solution

For instances with up to 400 customers, decomposition by rolling-space clustering enables tractable exact solution:

• Clustering: A sweep window around the depot creates spatially overlapping clusters of 20–25 customers. Each customer belongs to several clusters, enhancing coverage.
• Parallel pricing: Each cluster independently solves the labeling ESPPRC, initially disregarding multi-compartment dominance rules for efficiency.
• Multi-compartment feasibility: Candidate routes exceeding load threshold $\kappa$ or involving potentially incompatible items are validated via a 0–1 feasibility MILP.
• Integration: Feasible columns from clusters merge into the global RMP; column generation continues until all clusters yield no negative cost routes. Branching constraints propagate to all clusters as needed.

5. Computational Results and Managerial Insights

Instances are based on operational data from a US weekly distribution scenario:

Instance Class	Customers	Vehicle Compartments	Solution Algorithm	CPU (sec)	Optimality Gap
M1–M5	10–30	6	B&P Exact	~14,892	0.00%
M1–M5	10–30	6	RS-B&P	~187	0.07–3.7%
M6–M7	35–40	6	RS-B&P	389–557	2.5–5.2%
L1–L7	50–400	6	RS-B&P	~3,110	5–6% fewer vehicles

Key findings include:

• Increasing compartments (2 to 6): cost decreases, fewer vehicles, but higher CPU.
• Multiple time windows: cost decreases by 5–7%, CPU increases by ~30%.
• Optimal cluster sizing (20–25): best trade-off in solution quality and computing time.
• Compared to manual planning, vehicle count reduced ~20%, total distance by ~7,000 km/week ($\sim\$250,000$/year) at one plant.</li> <li>Compartment flexibility and automated break scheduling outperform manual rules in consolidation and legal compliance.</li> </ul> <h2 class='paper-heading' id='limitations-and-further-research'>6. Limitations and Further Research</h2> <p>Current rolling-space B&amp;P enables exact solutions for up to 400-customer instances, but high density or larger scales reveal computational bottlenecks. The route feasibility filter threshold $\kappa$ must be tuned carefully to balance solution quality and search space.

  Future research includes joint integration of inventory and routing decisions, extension to heterogeneous fleets, dynamic or stochastic travel times and demands, advanced cutting-plane strategies for compartment compatibility, and machine learning for column generation and branching prediction (Raqabi et al., 22 Jan 2026).