A state-space approach to study the bullwhip effect under the rationing game

Driven by the challenges posed by supply chain disruptions when demand surpasses supply, this study introduces a novel stochastic state-space approach to examine how the rationing game influences the reverse bullwhip effect and inventory fluctuations. The rationing mechanism is represented through a proportional controller, which regulates the allocation of goods to downstream supply chain participants. The analysis focuses on a three-node supply chain comprising a single distributor supplying two retailers. Each retailer manages its inventory using a base stock policy while responding to stochastic customer demand patterns. The dynamic behaviour of the supply chain is captured through a closed-form covariance matrix, which is formulated in terms of the proportional control parameters and the proportion of the distributor’s inventory allocated to retailers. By analysing the model under stationary conditions, the study explores how distributor inventory variability and correlated demand patterns impact upstream demand amplification and broader instability phenomena within supply chains.