Optimal (r, Q) policy in a stochastic inventory system with limited resource under incremental quantity discount

- We model a practical problem in stochastic inventory systems which has not been investigated yet.
- We prove the essential properties of an optimal solution.
- Presenting an algorithm based on the optimal properties which is able to find the optimal solution of the problem.
- We examine our algorithm efficiency via generating and solving some numerical examples.

Resource constraint such as constraint on purchasing budget, and quantity discount are two common characteristics of inventory systems. While interacting these characteristics in an actual inventory system is important in practice, most of the existing (r, Q) models in the literature have considered them separately. This paper investigates a single-item (r, Q) model with a limited resource and incremental quantity discount under perpetual review where demand is stochastic and discrete. The lead time is constant and unsatisfied demands are backordered. Most actual inventory systems tend to offset resource shortages by renting the missing amount of the resource, instead of keeping a surplus resource in the system. Therefore, considering a soft resource constraint, beside the incremental quantity discount where the resource is price-dependent, makes the model more practical. An optimization problem is formulated to find an optimal (r, Q) policy which minimizes the expected system costs. Based on the mathematical properties of the cost function, the search region for the optimal solution of the problem is reduced to a one-dimensional search region which is proven to be a finite enumerable set of order quantities. A one-dimensional search algorithm is presented to find the optimal (r, Q) policy through this region. Finally, some numerical examples are provided to demonstrate the algorithm performance and its sensitivity to parameters variation.