Urban logistics systems often face disruptions such as warehouse shutdowns, vehicle shortages, and sudden demand spikes, making it difficult to meet all customer needs. This paper presents a scenario-based optimization framework that supports resilient fulfillment planning when resources are limited. We combine two solver approaches—Linear Programming (LP) and Unbalanced Optimal Transport (UOT)—to allow partial demand fulfillment, with penalties for unmet demand. LP focuses on minimizing total cost, while UOT offers more flexible, fairness-aware delivery by relaxing strict fulfillment constraints. The framework uses a detailed cost model that includes distance, fuel usage, emissions, delays, and zone-based charges to reflect real operational conditions. Using case studies from Chennai, we compare solver performance under different disruption scenarios. Results show that LP provides low-cost solutions but may neglect high-cost zones, while UOT ensures broader coverage with more balanced allocations. The scientific novelty of this work lies in integrating soft-constrained optimization with UOT to explicitly capture trade-offs between cost efficiency and service equity under disruption—an aspect often overlooked in existing logistics models. This unified framework contributes not only a practical decision-support tool for urban planners but also a generalizable methodological advance for resilience-oriented logistics optimization. This approach helps logistics planners explore trade-offs between cost and service equity and design more robust delivery strategies under uncertain conditions.