- Stoch. Syst.
- Volume 3, Number 2 (2013), 500-573.
Brownian inventory models with convex holding cost, Part 2: Discount-optimal controls
We consider an inventory system in which inventory level fluctuates as a Brownian motion in the absence of control. The inventory continuously accumulates cost at a rate that is a general convex function of the inventory level, which can be negative when there is a backlog. At any time, the inventory level can be adjusted by a positive or negative amount, which incurs a fixed positive cost and a proportional cost. The challenge is to find an adjustment policy that balances the inventory cost and adjustment cost to minimize the expected total discounted cost. We provide a tutorial on using a three-step lower-bound approach to solving the optimal control problem under a discounted cost criterion. In addition, we prove that a four-parameter control band policy is optimal among all feasible policies. A key step is the constructive proof of the existence of a unique solution to the free boundary problem. The proof leads naturally to an algorithm to compute the four parameters of the optimal control band policy.
Stoch. Syst., Volume 3, Number 2 (2013), 500-573.
First available in Project Euclid: 11 February 2014
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 90B05: Inventory, storage, reservoirs 93E20: Optimal stochastic control
Dai, J. G.; Yao, Dacheng. Brownian inventory models with convex holding cost, Part 2: Discount-optimal controls. Stoch. Syst. 3 (2013), no. 2, 500--573. doi:10.1214/11-SSY046. https://projecteuclid.org/euclid.ssy/1392131423