Shipment Selection Problem (SSP)

• Shipment Selection Problem (SSP) is a combinatorial optimization challenge that assigns shipments to carriers, time slots, or package types under various capacity, cost, and service constraints.
• The topic integrates advanced formulations including MIQP, MIP, and heuristic algorithms while leveraging quadratic compatibility terms and quantum-classical hybrid methods.
• Empirical results demonstrate significant gains in schedule compatibility, cost reduction, and damage control, highlighting SSP’s impact on modern logistics efficiency.

The Shipment Selection Problem (SSP) refers to a class of combinatorial optimization problems arising in logistics, freight operations, and e-commerce fulfillment, where the objective is to assign shipments to options such as time slots, carrier types, vehicle schedule gaps, or package types under operational, cost, and service constraints. SSP formulations integrate quadratic compatibility structures, capacity limits, exclusivity conditions, and trade-offs between cost, service reliability, and shipment quality, with prominent instances handled via mixed-integer programming, quantum-classical hybrid methods, and scalable heuristic algorithms (Lopez-Ruiz et al., 13 Apr 2026, Yanchuk et al., 2020, Gurumoorthy et al., 2020).

1. Mathematical Formulations of SSP

At its core, the SSP requires selecting an optimal assignment of shipment units (or shipment-sequences) to a set of available logistical options, while respecting constraints on capacity, exclusivity, cost, and business rules. The formal structure is problem-dependent. Three principal instantiations include:

• Assignment to gaps in vehicle schedules (Lopez-Ruiz et al., 13 Apr 2026):
  • Sets: $G$ (schedule gaps), $Q_g$ (shipment-sequences for $g$), $S$ (shipments), binary $x_{gq}$ for sequence assignments.
  • Objective (MIQP):

  $$\max_{\{x_{gq}\in\{0,1\}\}} \sum_{g\in G}\sum_{q\in Q_g} v_{gq}x_{gq} +\sum_{(q_1,q_2)} w_{q_1q_2}x_{g_1q_1}x_{g_2q_2}$$ - Constraints: per-gap capacity, shipment exclusivity, sequence exclusivity per gap.

• Shipment–carrier option selection (Yanchuk et al., 2020):

  • Sets: Shipments $I$, carrier options $J$, binary decision $x_{i,j}$.
  • Objective (MIP with time penalty):

  $$\min_x\; \sum_{i\in I}\sum_{j\in J}[f_j+v_j w_i ] x_{i,j} +\lambda\sum_{i\in I}\sum_{j\in J}(\tau_{i,j}-T^{\max})_+ x_{i,j}$$ - Constraints: exactly one carrier per shipment, carrier capacity, option availability, rural-zone coverage.

• Optimal package-type assignment (Gurumoorthy et al., 2020):

  • Sets: Products $Q_g$0, package-types $Q_g$1, binary $Q_g$2 per product.
  • Objective (Tikhonov Lagrangian relaxation):

  $Q_g$3 - Constraints: exactly one package per product, business infeasibility mask.

The structural diversity of these models reflects the operational variety of the SSP family. Most variants are NP-(hard), especially when quadratic or coupling constraints are present.

2. Quadratic and Hybrid Optimization Approaches

The inclusion of quadratic “compatibility” terms in the objective is a distinctive feature in advanced SSP instances, especially freight schedule optimization (Lopez-Ruiz et al., 13 Apr 2026). Here, selecting pairs of shipment-sequences influences the solution’s quality due to spatial, temporal, or operational dependencies. These are modeled via quadratic weights $Q_g$4, producing a Mixed-Integer Quadratic Program (MIQP).

Quantum annealing and quantum approximate optimization algorithms (QAOA) have been deployed on these QUBO/Ising reductions, with the Iterative-QAOA procedure adapting non-variational linear-ramp schedules and iterative biasing to target low-energy solution regions efficiently. The Ising Hamiltonian encoding aggregates both assignment values and penalty-enforced constraints:

$Q_g$5

where $Q_g$6 and $Q_g$7 aggregate problem data and penalties.

Hybrid workflows combine quantum solvers for assignment generation with classical metaheuristics (such as large-neighborhood search) to refine solutions and enforce operational feasibility (Lopez-Ruiz et al., 13 Apr 2026).

3. Heuristics and Scalable Assignment Algorithms

Not all SSP variants require complex MIQPs or quantum methods. When the assignment constraints decouple across products (e.g., package-type selection with per-item independence), the problem admits scalable $Q_g$8 algorithms. Each product $Q_g$9 chooses package $g$0, and a bisection search over $g$1 tunes the shipment–damage trade-off to meet exogenous business constraints (Gurumoorthy et al., 2020). This approach is tractable even at the scale of $g$2 assignments and is deployed for Amazon’s e-commerce package-type selection.

4. Empirical Results and Operational Impact

Table: Operational KPIs and Impact Metrics Reported Across SSP Deployments

SSP Variant	Primary KPIs	Observed Effects (Best Case)
Gap-filling (Quantum Hybrid)	SD, SCS, TDD	+12% SD, –9.4% TDD, ΔCost ≈ 0 (Lopez-Ruiz et al., 13 Apr 2026)
Carrier selection	Cost, Timeliness	14.6% cost reduction (vs. all-WW), >95% on-time (Yanchuk et al., 2020)
Package-type assignment	Cost, Damage	–24% damage, –5% transport cost (Gurumoorthy et al., 2020)

Quantum hybrid workflows yield compatibility-aware assignments with SCS (Schedule Compatibility Score) increase (mean +0.02, up to +0.14), occasionally enabling up to +12.1% delivered shipments. Warm-starts from quantum assignments further improve shipment delivery rates and reduce per-shipment drive distance. In shipment–carrier selection, MIP optimization reduces baseline cost by up to 14.6% and achieves high on-time fulfillment. Package-type optimization reduces both damage rates and shipping cost, with a –24% in-transit damage rate in live systems and measurable logistics cost savings.

5. Solution Methodologies and Workflow Integration

SSP solution workflows are highly context-dependent, typically comprising:

1. Data ingestion from ERP, schedule, and routing systems.

2. SSP instance construction as a MIP/MIQP/QUBO/IP.

3. Application of assignment algorithm:

   • Quantum-classical hybrid (Iterative-QAOA, warm start for classical VRP solvers) (Lopez-Ruiz et al., 13 Apr 2026).
   • Classical MIP solver (CPLEX, SCIP) (Yanchuk et al., 2020).
   • Heuristic linear-time decoupling (package assignment) (Gurumoorthy et al., 2020).
4. Post-processing for operational feasibility, KPI evaluation, and cost assessment.
5. Bisection or grid search in trade-off parameter space (e.g., $g$3) to enforce business constraints.
6. Integration of user-behavior modules and scenario simulation for variance and sensitivity analysis.

In advanced deployments, quantum solution modules supply compatibility-prioritized assignments which downstream classical metaheuristics can further refine within limited neighborhoods, leveraging the global structure provided by quantum routines (Lopez-Ruiz et al., 13 Apr 2026).

6. Practical Considerations, Limitations, and Outlook

All published SSP solutions require careful tuning of penalty weights, compatibility functions, and solver parameters to ensure feasibility and practical utility. The computational tractability varies; MIQP and QUBO/Ising approaches can become severe bottlenecks as problem size and quadratic weights increase, rendering traditional solvers (e.g., SCIP) slow for large $g$4 (Lopez-Ruiz et al., 13 Apr 2026). Quantum-classical hybrids bypass some of these barriers by quickly generating warm-start solutions that are further refined classically.

A plausible implication is that SSPs with complex quadratic or higher-order dependencies stand to benefit from near-term quantum optimization technologies, whereas problems with independent per-shipment assignments remain well-served by classical scalable heuristics. Future research directions include higher-order compatibility modeling, automated penalty tuning, hardware quantum optimization, and further integration of user-behavior modeling for demand-side optimization (Lopez-Ruiz et al., 13 Apr 2026, Yanchuk et al., 2020, Gurumoorthy et al., 2020).