Preservation of Structural Properties in Optimization with Decisions Truncated by Random Variables and Its Applications

Abstract

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.

L♮-convexity and its applications in operations

“…The above result first appears in Chen et al (2017) and is a slight generalization of a corresponding one in Zipkin (2008) which deals with m ¼ 1.…”

“…In an attempt to deal with inventory models with random capacity or revenue management problems using booking limit control in which decisions are truncated by random variables, Chen et al (2017) derive useful properties of convexity, submodularity and L \ -convexity for the following problem.…”

“…The set A ¼ ðx,z,uÞ f jAu£b,u 1 ³u 1 , :::,u k ³u k ,u kþ1 £u kþ1 ,:::,u n £u n g, where b, u 1 , :::,u k , u kþ1 ,:::,u n are parameters that may depend on x and z, A ¼ ða ij Þ with entries a ij > 0 for any i and j ¼ 1, :::,k, and a ij £0 for any i and j ¼ k þ 1,:::,n. In addition, X j is contained in u j -z j , þ 1 Â Á for j ¼ 1, :::,k, and X j is contained in -1,u j -z j À Ã for j ¼ k þ 1,:::,n. THEOREM 9. (Chen et al 2017) Consider the optimization problem (6). Under Assumptions 1-3, problem (6) and the following optimization problem have the same optimal objective value.…”

“…Indeed, it was used by Zipkin (2008) to derive the optimal structural policy of lost-sales inventory models with positive lead times, which shed new lights on a classical result of Karlin and Scarf (1958) and Morton (1969). Since then, L \ -convexity was found to be powerful to establish the structures of optimal policies in various other operations models: serial inventory systems (Huh and Janakiraman, 2010); inventory-pricing models with positive lead times (Pang et al, 2012); capacitated inventory systems with remanufacturing (Gong and Chao, 2013); perishable inventory models ; dualsourcing models with random capacities, assemble-toorder systems with random capacities, and revenue management using booking limits (Chen et al, 2017); a joint inventory and transshipment control model with random capacities (Chen et al, 2015); an instantaneous control model of Brownian motion with positive lead time (Xu et al, 2016); etc.…”

“…In addition, the definition can be extended to functions defined on ℜ n . For some applications, we need the extension of L \ -convexity to more general spaces (see for example Chen et al 2017 andXu et al 2016). For simplicity, we restrict to finite dimensional spaces.…”