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Title:M♮-convexity, S-convexity, and their applications in operations
Author(s):Li, Menglong
Director of Research:Chen, Xin
Doctoral Committee Chair(s):Chen, Xin
Doctoral Committee Member(s):Garg, Jugal; Seshadri, Sridhar; Wang, Qiong
Department / Program:Industrial&Enterprise Sys Eng
Discipline:Industrial Engineering
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):discrete convex analysis
decreasing optimal solution
inventory and production
supply chain management
gross substitutability
Abstract: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.
Issue Date:2020-09-04
Rights Information:Copyright 2020 Menglong Li
Date Available in IDEALS:2021-03-05
Date Deposited:2020-12

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