M♮-convexity, S-convexity, and their applications in operations

Many problems in operations management are embedded with substitute structures which often result in parametric optimization models maximizing submodular objective functions, and it is desirable to derive structural properties including monotone comparative statics of the optimal solutions or preservation of submodularity under the optimization operations. Yet, this task is challenging because the classical and commonly used results in lattice programming, applicable to optimization models with supermodular objective function maximization, does not apply. In this thesis, by employing a key concept in discrete convex analysis, M♮-convexity, we establish conditions under which the optimal solutions are nonincreasing in the parameters and the preservation property holds for parametric maximization models with submodular objectives, together with the development of several new fundamental properties of M♮-convexity. Furthermore, we propose a new concept of S-convexity (and its variant SSQS- convexity) which includes M♮-convexity as a subclass, and extend those results established for M♮-convexity to continuous S-convexity. In addition, we show that S-convex functions form a subclass of supermodular functions which have a one-to-one correspondence with jointly submodular and convex functions through the conjugate operator under mild conditions. A new preservation property which is not enjoyed by M♮-convexity is presented. Our theoretical results are applied to several notable operations models: a classical multi-product dynamic stochastic inventory model, an assemble-to-order inventory model, a production control problem with two products or facilities, a portfolio contract model, a discrete choice model, and a random yield inventory model. We illustrate that looking from the lens of M♮-convexity and S-convexity allows to facilitate the analysis of monotone comparative statics, simplify or unify the complicated proofs in the literature, and extend the results to more general settings.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Industrial Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2021

Author dc:creator

• Li, Menglong

Contributors dc:contributor

• Chen, Xin
• Garg, Jugal
• Seshadri, Sridhar
• Wang, Qiong

• discrete convex analysis
• S-convexity
• decreasing optimal solution
• inventory and production
• supply chain management
• gross substitutability

Statement dc:rights

• Copyright 2020 Menglong Li

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/109559

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/109559