Integrated multi‐echelon supply chain design with inventories under uncertainty: MINLP models, computational strategies

We address in this article a problem that is of significance to the chemical industry, namely, the optimal design of a multi-echelon supply chain and the associated inventory systems in the presence of uncertain customer demands. By using the guaranteed service approach to model the multi-echelon stochastic inventory system, we develop an optimization model to simultaneously determine the transportation, inventory, and network structure of a multi-echelon supply chain. The model is an MINLP with a nonconvex objective function including bilinear, trilinear, and square root terms. By exploiting the properties of the basic model, we reformulate this problem as a separable concave minimization program. A spatial decomposition algorithm based on the integration of Lagrangean relaxation and piecewise linear approximation is proposed to obtain near global optimal solutions with reasonable computational expense. Examples for specialty chemicals and industrial gas supply chains with up to 15 plants, 100 potential distribution centers, and 200 markets are presented. © 2009 American Institute of Chemical Engineers AIChE J, 2010

[1]  Kenneth F. Simpson.  In-Process Inventories , 1958 .

[2]  J. Little.  A Proof for the Queuing Formula: L = λW , 1961 .

[3]  L. Goddard,et al.  Operations Research (OR) , 2007 .

[4]  J. E. Falk,et al.  An Algorithm for Separable Nonconvex Programming Problems , 1969 .

[5]  F. Glover.  IMPROVED LINEAR INTEGER PROGRAMMING FORMULATIONS OF NONLINEAR INTEGER PROBLEMS , 1975 .

[6]  G. D. Eppen.  Note---Effects of Centralization on Expected Costs in a Multi-Location Newsboy Problem , 1979 .

[7]  Awi Federgruen,et al.  Computational Issues in an Infinite-Horizon, Multiechelon Inventory Model , 1984, Oper. Res..

[8]  Marshall L. Fisher,et al.  An Applications Oriented Guide to Lagrangian Relaxation , 1985 .

[9]  E. Balas.  Disjunctive programming and a hierarchy of relaxations for discrete optimization problems , 1985 .

[10]  Stephen C. Graves,et al.  Safety stocks in manufacturing systems , 1987 .

[11]  Miao-Sheng Chen,et al.  Effects of Centralization on Expected Costs in a Multi-location Newsboy Problem , 1989 .

[12]  M. D. Wilkinson,et al.  Management science , 1989, British Dental Journal.

[13]  K. Inderfurth.  Safety stock optimization in multi-stage inventory systems , 1991 .

[14]  Hau L. Lee,et al.  Material Management in Decentralized Supply Chains , 1993, Oper. Res..

[15]  Karl Inderfurth.  Valuation of Leadtime Reduction in Multi-Stage Production Systems , 1993 .

[16]  J. Beasley.  Lagrangean heuristics for location problems , 1993 .

[17]  R. Raman,et al.  Modelling and computational techniques for logic based integer programming , 1994 .

[18]  Paul Glasserman,et al.  Sensitivity Analysis for Base-Stock Levels in Multiechelon Production-Inventory Systems , 1995 .

[19]  Nikolaos V. Sahinidis,et al.  BARON: A general purpose global optimization software package , 1996, J. Glob. Optim..

[20]  S. Minner.  Strategic safety stocks in reverse logistics supply chains , 2001 .

[21]  Stefan Minner,et al.  Safety stocks in multi-stage inventory systems under different service measures , 1998, Eur. J. Oper. Res..

[22]  Alan Scheller-Wolf,et al.  Development of a Rapid-Response Supply Chain at Caterpillar , 2000, Oper. Res..

[23]  Costas D. Maranas,et al.  Mid-term supply chain planning under demand uncertainty: customer demand satisfaction and inventory management , 2000 .

[24]  Russell D. Meller,et al.  The interaction of location and inventory in designing distribution systems , 2000 .

[25]  Paul H. Zipkin,et al.  Foundations of Inventory Management , 2000 .

[26]  Manfred W. Padberg,et al.  Approximating Separable Nonlinear Functions Via Mixed Zero-One Programs , 1998, Oper. Res. Lett..

[27]  Sunwon Park,et al.  Supply chain optimization in continuous flexible process networks , 2000 .

[28]  Sean P. Willems,et al.  Optimizing Strategic Safety Stock Placement in Supply Chains , 2000, Manuf. Serv. Oper. Manag..

[29]  David D. Yao,et al.  A Supply Network Model with Base-Stock Control and Service Requirements , 2000, Oper. Res..

[30]  C. Pantelides,et al.  Design of Multi-echelon Supply Chain Networks under Demand Uncertainty , 2001 .

[31]  Shang Zhi,et al.  A proof of the queueing formula: L=λW , 2001 .

[32]  Zuo-Jun Max Shen,et al.  An Inventory-Location Model: Formulation, Solution Algorithm and Computational Results , 2002, Ann. Oper. Res..

[33]  Martin W. P. Savelsbergh,et al.  Inventory Routing in Practice , 2002, The Vehicle Routing Problem.

[34]  Thomas L. Magnanti,et al.  A Comparison of Mixed - Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems , 2003, Manag. Sci..

[35]  J. Winch,et al.  Supply Chain Management: Strategy, Planning, and Operation , 2003 .

[36]  Iftekhar A. Karimi,et al.  Resource-constrained scheduling of parallel production lines using asynchronous slots , 2003 .

[37]  I. Grossmann,et al.  Temporal Decomposition Scheme for Nonlinear Multisite Production Planning and Distribution Models , 2003 .

[38]  Zuo-Jun Max Shen,et al.  A Joint Location - Inventory Model , 2003, Transp. Sci..

[39]  Sean P. Willems,et al.  Supply Chain Design: Safety Stock Placement and Supply Chain Configuration , 2003, Supply Chain Management.

[40]  Herbert E. Scarf,et al.  Optimal Policies for a Multi-Echelon Inventory Problem , 1960, Manag. Sci..

[41]  Cheng-Liang Chen,et al.  Multi-objective optimization of multi-echelon supply chain networks with uncertain product demands and prices , 2004, Comput. Chem. Eng..

[42]  Marshall L. Fisher,et al.  The Lagrangian Relaxation Method for Solving Integer Programming Problems , 2004, Manag. Sci..

[43]  J. A. Bandoni,et al.  Supply chain optimization of large-scale continuous processes , 2005, Comput. Chem. Eng..

[44]  Christodoulos A. Floudas,et al.  Global Solution Approach for a Nonconvex MINLP Problem in Product Portfolio Optimization , 2005, J. Glob. Optim..

[45]  Sean P. Willems,et al.  Optimizing the Supply Chain Configuration for New Products , 2005, Manag. Sci..

[46]  Ignacio E. Grossmann,et al.  Enterprise‐wide optimization: A new frontier in process systems engineering , 2005 .

[47]  Johannes Weber.  Please Scroll down for Article , 1992 .

[48]  A. Barbosa‐Póvoa,et al.  Pipeline Scheduling and Inventory Management of a Multiproduct Distribution Oil System , 2006 .

[49]  Sean P. Willems,et al.  Optimizing Strategic Safety Stock Placement in Supply Chains with Clusters of Commonality , 2006, Oper. Res..

[50]  David Simchi-Levi,et al.  Inventory placement in acyclic supply chain networks , 2006, Oper. Res. Lett..

[51]  Navneet Vidyarthi,et al.  Integrated Production-Inventory-Distribution System Design with Risk Pooling: Model Formulation and Heuristic Solution , 2007, Transp. Sci..

[52]  Reinhard Hübner,et al.  Strategic Supply Chain Management in Process Industries: An Application to Specialty Chemicals Production Network Design , 2007 .

[53]  Gintaras V. Reklaitis,et al.  Enterprise-wide modeling & optimization - An overview of emerging research challenges and opportunities , 2007, Comput. Chem. Eng..

[54]  Martin W. P. Savelsbergh,et al.  Inventory routing with continuous moves , 2007, Comput. Oper. Res..

[55]  Reha Uzsoy,et al.  A single-product network design model with lead time and safety stock considerations , 2007 .

[56]  S. Chopra,et al.  Supply Chain Management: Strategy, Planning & Operation , 2007 .

[57]  Sean P. Willems,et al.  A Periodic-Review Modeling Approach for Guaranteed Service Supply Chains , 2007, Interfaces.

[58]  Gintaras V. Reklaitis,et al.  Integrated safety stock management for multi-stage supply chains under production capacity constraints , 2008, Comput. Chem. Eng..

[59]  Mark S. Daskin,et al.  Capacitated warehouse location model with risk pooling , 2008 .

[60]  Raul Rivas Perez,et al.  System identification of steam pressure in a fire-tube boiler , 2008, Comput. Chem. Eng..

[61]  Peter M. Verderame,et al.  Integrated Operational Planning and Medium-Term Scheduling for Large-Scale Industrial Batch Plants , 2008 .

[62]  Ignacio E. Grossmann,et al.  An improved piecewise outer-approximation algorithm for the global optimization of MINLP models involving concave and bilinear terms , 2008, Comput. Chem. Eng..

[63]  I. Karimi,et al.  Piecewise MILP under‐ and overestimators for global optimization of bilinear programs , 2008 .

[64]  Manufacturing Trade Inventories Sales.  US Census Bureau MTIS How the Data are Collected page , 2009 .