Mathematical Modelling of Inventory Systems for Optimal Production Planning under Cost– Service Trade-offs

Abstract: This study develops a mathematical framework for inventory-based production planning under explicit cost– service trade-offs by formulating a continuous-time nonlinear dynamic system in which inventory level, backlog, service responsiveness, and production rate evolve simultaneously. The proposed model captures service-sensitive demand, backlog recovery, linear and nonlinear inventory loss, and the effect of production responsiveness on service performance, thereby providing a more realistic representation than classical linear inventory models. A discounted cost functional is constructed to balance production, holding, shortage, service-support, and deterioration costs subject to service-level requirements. To obtain analytical insight, the full model is reduced under practically meaningful assumptions to a Riccati-type nonlinear differential equation, from which a closed-form inventory trajectory, equilibrium solution, and stability condition are derived. A numerical case study demonstrates monotone convergence of inventory to a stable equilibrium and reveals a clear cost–service frontier: higher service targets improve fill rate and customer fulfillment, but they also require larger equilibrium inventory, greater production effort, and higher total cost. The results show that nonlinear deterioration plays a critical role in limiting feasible inventory expansion and highlight the importance of balancing customer service ambitions with operational cost efficiency in production planning.

Keywords: Inventory systems, optimal production planning, nonlinear differential equations, cost–service trade-offs, service-sensitive demand, Riccati equation, equilibrium analysis, fill rate, nonlinear deterioration, production control.