A common technical challenge encountered in many operations management models is that decision variables are truncated by some random variables and the decisions are made before the values of these random variables are realized, leading to non-convex minimization problems. To address this challenge, we develop a powerful transformation technique which converts a non-convex minimization problem to an equivalent convex minimization problem. We show that such a transformation enables us to prove the preservation of some desired structural properties, such as convexity, submodularity, and L-convexity, under optimization operations, that are critical for identifying the structures of optimal policies and developing efficient algorithms. We then demonstrate the applications of our approach to several important models in inventory control and revenue management: dual sourcing with random supply capacity, assemble-to-order systems with random supply capacity, and capacity allocation in network revenue management.

In many operations management problems, the decisions are truncated by random variables. Take a dual sourcing inventory management problem as an example: the suppliers may have random capacities, and the actual received quantity from ordering is truncated by this random capacity. Often the random capacities of different suppliers may be dependent. An interesting challenge is that due to the truncation, the optimization problem may not be convex. In “Stochastic Optimization with Decisions Truncated by Positively Dependent Random Variables”, X. Chen and X. Gao propose a transformation technique to convert the original nonconvex minimization problem to an equivalent convex one. They demonstrate the application of their method using an inventory substitution problem with dependent random supply capacities and a two-part fee cost structure. In addition, their method can also incorporate the decision maker’s risk attitude.

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In many real-life situations, the inventory record may not match the actual stock perfectly. This can happen due to distortion of inventory data, such as transaction errors, misplaced inventories, and spoilage. In these cases, because the decision maker only has incomplete information about the inventory levels, many well-known inventory policies are not even admissible, and our understanding of the optimal policies, even their existence, is very limited. In “Average Cost Optimality in Partially Observable Lost-Sales Inventory Systems,” Bai et al. consider the classical lost-sales inventory model, in which the inventory level is only observed when it becomes zero. They formulate the cost-minimization problem as a partially observable Markov decision process. By exploiting the vanishing discount factor approach, they provide a way to verify the existence of optimal policies under the average cost criterion. The key step in their analysis is the construction of a valid policy, which, in a certain sense, copies the actions of another policy for the process starting from another initial state.

The concept of supermodularity has received considerable attention in economics and operations research. It is closely related to the concept of complementarity in economics and has also proved to be an important tool for deriving monotonic comparative statics in parametric optimization problems and game theory models. However, only certain sufficient conditions (e.g., lattice structure) are identified in the literature to preserve the supermodularity. In this article, new concepts of mostly sublattice and additive mostly sublattice are introduced. With these new concepts, necessary and sufficient conditions for the constraint structures are established so that supermodularity can be preserved under various assumptions about the objective functions. Furthermore, some classes of polyhedral sets that satisfy these concepts are identified, and the results are applied to assemble-to-order systems.