{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/109559"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/109559","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"M♮-convexity, S-convexity, and their applications in operations","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Li, Menglong"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Industrial Engineering","degree_department":null,"school":null,"contributors":["Chen, Xin","Garg, Jugal","Seshadri, Sridhar","Wang, Qiong"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-03-05T21:45:09Z","date_published":"2021-03-05T21:45:09Z","updated_at":"2026-07-22T22:24:50Z","subjects":["discrete convex analysis","S-convexity","decreasing optimal solution","inventory and production","supply chain management","gross substitutability"],"languages":["en"],"rights":["Copyright 2020 Menglong Li"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/109559","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Chen, Xin","Garg, Jugal","Seshadri, Sridhar","Wang, Qiong"]},{"key":"dc:creator","label":"Author","values":["Li, Menglong"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2021-03-05T21:45:09Z","2023-03-05T21:47:41Z","2020-09-04","2020-12"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Industrial Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["discrete convex analysis","S-convexity","decreasing optimal solution","inventory and production","supply chain management","gross substitutability"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2020 Menglong Li"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/109559"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2022-12-01","The student, Menglong Li, accepted the attached license on 2020-09-01 at 11:22.","The student, Menglong Li, submitted this Dissertation for approval on 2020-09-01 at 13:26.","This Dissertation was approved for publication on 2020-09-04 at 08:52.","DSpace SAF Submission Ingestion Package generated from Vireo submission #15793 on 2021-03-04 at 16:30:11","Made available in DSpace on 2021-03-05T21:45:09Z (GMT). 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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.","Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2022-12-01","The student, Menglong Li, accepted the attached license on 2020-09-01 at 11:22.","The student, Menglong Li, submitted this Dissertation for approval on 2020-09-01 at 13:26.","This Dissertation was approved for publication on 2020-09-04 at 08:52.","DSpace SAF Submission Ingestion Package generated from Vireo submission #15793 on 2021-03-04 at 16:30:11","Made available in DSpace on 2021-03-05T21:45:09Z (GMT). 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