We consider three problems on inventory, transportation and location in a supply chain. In Chapter 2, we study Multilevel Rationing (MR) and Strict Priority (SP) stock allocation policies for a centralized single product multi-class M/G/1 make-to-stock queueing systems. To obtain the total cost of the system under these policies, we introduce a new method called “customer composition”. Using this method, we focus on the proportion of customers of each class out of the total number of customers in the queue since the number of customers in M/G/1 queues is invariant for any non-idling and non-anticipating policy. We consider a series of two-priority M/G/1 queues with an exceptional service time in each busy period to characterize the customer composition. We derive closed form expressions for the costs of SP and MR policies using these results.

In Chapter 3, we consider a two-echelon inventory system with a congested centralized production facility and several Distribution Centers (DCs). We assume that the production and transportation times are stochastic that are generally distributed, and customers arrive to each DC according to an independent Poisson process. Inventory at DCs is managed using the one-for-one replenishment policy. We use the customer composition approach to characterize the total inventory carrying and backlog costs of the system under the FCFS, SP and MR allocation policies at the warehouse. For the special case of exponentially distributed production and transportation times, we use the unit-flow method and derive closed form expressions for the optimal cost and base-stock level of the DCs. We numerically demonstrate that prioritization using either the SP or the MR policy could be very beneficial in comparison with the FCFS policy.

In Chapter 4, we study a two-echelon supply chain with a set of suppliers, a set of retailers and a set of capacitated cross-docks which are to be established. The demand of the retailers could be satisfied from the suppliers through the cross-docks. The objective is to determine the number and location of cross-docks, the assignment of retailers to suppliers so that the total cost of pipeline and retailers’ inventory, transportation, and facility location is minimized. We formulate the problem as a non-linear mixed integer programming and derive several structural results for special cases of the problem. To solve the general problem, we show that it can be written as a cutting stock problem and develop a column generation algorithm to solve it. We investigate the efficiency of the proposed algorithm numerically.