Year: 2025 | Month: December | Volume: 12 | Issue: 12 | Pages: 93-103
DOI: https://doi.org/10.52403/ijrr.20251211
Delay-Induced Oscillations and Instability in a Nonlinear Supply Chain Inventory System
Hafidh Khoerul Fata1, Handika Lintang Saputra1
1Department of Mathematics, Faculty of Sciences and Mathematics, Universitas Diponegoro, Semarang, Indonesia.
Corresponding Author: Hafidh Khoerul Fata
ABSTRACT
This study examines delay-induced oscillations and stability loss in a continuous-time supply chain inventory system. The focus is on a single-echelon inventory model characterized by constant customer demand and an order-up-to policy, subject to a fixed replenishment lead time. The ordering rate is influenced by the deviation of on-hand inventory from the desired target level through linear and cubic feedback terms, resulting in a scalar delay differential equation with nonlinear delayed feedback. By analyzing deviations from the target inventory level, we derive the equilibrium point and assess its local stability by examining the associated characteristic equation. For positive feedback gain, we derive an explicit expression for the critical lead time at which a Hopf bifurcation occurs. Below this threshold, the equilibrium is locally asymptotically stable, and inventory deviation diminishes to zero; when the delay surpasses the critical value, a pair of complex conjugate eigenvalues crosses the imaginary axis, leading to persistent oscillatory adjustments in the system. The linear–cubic feedback structure underscores how stronger corrective reactions to inventory discrepancies, combined with significant lead times, can amplify rather than mitigate fluctuations. These findings elucidate a clear delay–margin relationship for a simple order-up-to policy and provide a dynamic explanation for structurally induced bullwhip behavior: even with constant demand, the interaction between feedback gains and lead time can generate sustained oscillations in inventory levels. Managerial implications regarding the joint selection of feedback parameters and permissible lead times are discussed, and extensions to more realistic multi-echelon settings and stochastic demand models are proposed.
Keywords: Delay differential equations, inventory control, Hopf bifurcation, bullwhip effect, delay-induced oscillations.
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