随机参考价格影响下带成本约束的库存系统

Cost-Constrained Inventory System under the Stochastic Reference Price Effect

摘要: 本文深入探讨了制造商在考虑随机参考价格效应的情况下，对生产和定价做出最佳决策的问题。我们将其表述为随机控制问题，目的是在成本约束下最大化净利润函数。首先，为了解决约束优化问题，我们可以利用拉格朗日乘子方法将其转换为无约束问题。通过动态规划方法和粘性解方法，本文研究结果揭示了无约束问题的值函数是汉密尔顿–雅可比–贝尔曼(HJB)方程的唯一粘性解。然而，HJB方程难以得到显式解，我们利用有限差分理论和随机递归算法建立了合适的数值近似方案。我们证明了HJB方程的有限差分方案的收敛性，并证明了该方案的解收敛于HJB方程的解。此外，我们通过确定适当的拉格朗日乘子，在约束问题的最优解和无约束问题的最优解之间建立联系。最后，我们进行了数值实验并分析了关键参数的灵敏度。我们的实验结果可以为制造商提供库存和定价的参考。

Abstract: This paper delves into the manufacturer’s problem of making optimal decisions regarding production and pricing, taking into account the stochastic reference price effect. We formulate it as a problem of stochastic control, of which the aim is to maximum the net profit function subject to the cost constraint. First, to tackle a constrained optimization problem, we can utilize the Lagrange multiplier method to convert it into an unconstrained problem. Through the dynamic programming approach and the viscosity solutions method, our findings reveal that the value function of the unconstrained problem serves as the unique viscosity solutions to the Hamilton-Jacobi-Bellman (HJB) equation. However, the HJB equation is difficult to obtain an explicit solution, we use finite difference theory and stochastic recursive algorithm to establish a suitable numerical approximation scheme. We demonstrate the convergence of the finite difference scheme of the HJB equation and demonstrate that the solution of the scheme converges to the solution of HJB equation. Additionally, we establish a connection between the optimal solutions of the constrained problem and the unconstrained one by identifying an appropriate Lagrange multiplier. Finally, we carry out the experiments and analyze a sensitivity of the key parameters. The results of our experiments can provide manufacturers with references of inventory and pricing.