Bridging the Linear-Quadratic Gap: A Quantum-Classical Hybrid Approach to Robust Supply Chain Design

TL;DR

The paper tackles robust design of urban supply chain networks by addressing the Linear-Quadratic Gap created when linear greedy methods ignore quadratic overlap effects. It proposes a QUBO formulation for the CFLP with a geospatial overlap penalty $O_{ij}=e^{-d_{ij}/\lambda}$, validated on a digital twin of Delhi NCR with $N=30$ candidate hubs, and benchmarks Greedy, Exact MILP, and Quantum-Inspired Reverse Annealing. The results show the Greedy approach achieves $473$ demand but with overlap $5.08$, while the quantum-inspired method achieves $450$ demand with $3.26$ overlap, representing a $21.8\%$ reduction in risk and a $3.2\%$ demand loss relative to the exact optimum $465$, indicating a path to robust, polycentric network designs. The findings demonstrate that quantum-inspired optimization can close the Linear-Quadratic Gap and provide operationally resilient, risk-aware supply chain configurations with implications for planning in megacities.

Abstract

The design of supply chain networks in densely populated urban logistics systems faces a timely dilemma: the traditional optimisation approaches are effective to maximise the level of demand perfusion, but they are limited to embracing large expenses in overlapping the facilities and cannibalisation in the market. When tested on a high-fidelity digital twin of the Delhi NCR road network of thirty candidate sites, we establish that Classical Greedy algorithms using the theoretical maximum demand of (473 units) lack any theoretical overlap penalty, but incur a prohibitive overlap penalty (5.08). Here, in comparison, the Quantum-Inspired solution only losses 3.2% of demand (450 compared to 465 units relative to the optimal solution), but the solution preserves 21.8% less operational overlap risk (3.26 compared to 4.17), which can be viewed as a 35.8% improvement compared to the Greedy solution. Geospatial analysis shows that it can be attributed to a shift in strategies: This, in contrast to Classical approaches, which focus on locating facilities in the high-density central areas (North/Central Delhi), the quantum-inspired solver autonomously chooses the diversified topology of the North-south network, penetrating into the underserved periphery growth markets. This is a spatially balanced arrangement which is congruent to the polycentric structure of modern time megacities, and displays better stability to volatility in demand. We have shown that quantum-inspired optimisation methods can close the so-called Linear-Quadratic Gap phenomenon, i.e. the systematic inability of greedy methods to capture the actual quadratic interactions between facilities, and offer a way of computing the pathway to operationally robust and risk-optimised supply chain networks in dense urban conditions.

Figures (2)

• Figure 1: Pareto Frontier analysis showing the trade-off between Demand Capture and Overlap Risk, highlighting the Quantum solution (Green Star) as the robust optimum.
• Figure 2: Geospatial distribution of selected hubs in Delhi NCR, illustrating the Quantum solver's strategic expansion into the southern region (Green Markers) compared to the Greedy algorithm's central clustering.