Lagrangian relaxation of the generic materials and operations planning model

The supply chain management requires increasingly proposals for the production programming planning that brings together its special singularities. Solving coexisting products and alternative processes or by-products must be allowed by the mathematical programming models. The generic materials and operations planning (GMOP) formulation allows operating with different materials and process lists. The paper presents a procedure to solve the versatile GMOP model by the Lagrange Relaxation. The subgradient update method of the lagrangian multiplier is compared with a linear update method. Obtaining lower bound faster compared to the linear method is allowed by the subgradient method, but the linear method provides better solutions after certain iterations.

[1]  José Pedro García Sabater,et al.  Heuristic for Material and Operations Planning in Supply Chains with Alternative Product Structure , 2017 .

[2]  P. Wolfe.  Note on a method of conjugate subgradients for minimizing nondifferentiable functions , 1974 .

[3]  Jean-Louis Goffin,et al.  On convergence rates of subgradient optimization methods , 1977, Math. Program..

[4]  Shuzhi Sam Ge,et al.  Allocating Resources in Multiagent Flowshops With Adaptive Auctions , 2011, IEEE Transactions on Automation Science and Engineering.

[5]  Julien Maheut.  Modelos y Algoritmos Basados en el Concepto Stroke para la Planificación y Programación de Operaciones con Alternativas en Redes de Suministro , 2013 .

[6]  Bart De Schutter,et al.  Accelerated gradient methods and dual decomposition in distributed model predictive control , 2013, Autom..

[7]  Tianyou Chai,et al.  A novel Lagrangian relaxation approach for a hybrid flowshop scheduling problem in the steelmaking-continuous casting process , 2014, Eur. J. Oper. Res..

[8]  Bjarne A. Foss,et al.  Oil production optimization - A piecewise linear model, solved with two decomposition strategies , 2010, Comput. Chem. Eng..

[9]  D. Bertsekas.  Nondifferentiable optimization via approximation , 1975 .

[10]  Francisco Barahona,et al.  The volume algorithm: producing primal solutions with a subgradient method , 2000, Math. Program..

[11]  Richard M. Karp,et al.  The traveling-salesman problem and minimum spanning trees: Part II , 1971, Math. Program..

[12]  Iftekhar A. Karimi,et al.  Planning and scheduling of parallel semicontinuous processes. 1. Production planning , 1997 .

[13]  Danielle Zyngier,et al.  Hierarchical decomposition heuristic for scheduling: Coordinated reasoning for decentralized and distributed decision-making problems , 2008, Comput. Chem. Eng..

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

[15]  Rita Streblow,et al.  Decentralized scheduling strategy of heating systems for balancing the residual load , 2015 .

[16]  Arthur M. Geoffrion,et al.  Lagrangian Relaxation for Integer Programming , 2010, 50 Years of Integer Programming.

[17]  Luiz Antonio Nogueira Lorena,et al.  Lagrangean/surrogate relaxation for generalized assignment problems , 1999, Eur. J. Oper. Res..

[18]  Philip Wolfe,et al.  Validation of subgradient optimization , 1974, Math. Program..

[19]  Masahiro Inuiguchi,et al.  Lagrangian relaxation with cut generation for hybrid flowshop scheduling problems to minimize the total weighted tardiness , 2010, Comput. Oper. Res..

[20]  Zukui Li,et al.  Capacity expansion planning through augmented Lagrangian optimization and scenario decomposition , 2012 .

[21]  L. J. Savage,et al.  Three Problems in Rationing Capital , 1955 .

[22]  Jiehong Kong,et al.  Coordination between strategic forest management and tactical logistic and production planning in the forestry supply chain , 2014, Int. Trans. Oper. Res..

[23]  Pilar I. Vidal-Carreras,et al.  Economic lot scheduling with deliberated and controlled coproduction , 2012, Eur. J. Oper. Res..

[24]  Hanif D. Sherali,et al.  Recovery of primal solutions when using subgradient optimization methods to solve Lagrangian duals of linear programs , 1996, Oper. Res. Lett..

[25]  V. Jeet,et al.  Lagrangian relaxation guided problem space search heuristics for generalized assignment problems , 2007, Eur. J. Oper. Res..

[26]  Niels Kjølstad Poulsen,et al.  A Dantzig-Wolfe Decomposition Algorithm for Linear Economic Model Predictive Control of Dynamically Decoupled Subsystems , 2014 .

[27]  Monique Guignard-Spielberg,et al.  Lagrangean decomposition: A model yielding stronger lagrangean bounds , 1987, Math. Program..

[28]  B. J. Lageweg,et al.  Surrogate duality relaxation for job shop scheduling , 1983, Discret. Appl. Math..

[29]  Olga Battaïa,et al.  An improved Lagrangian relaxation-based heuristic for a joint location-inventory problem , 2015, Comput. Oper. Res..

[30]  C. Beltran,et al.  Unit Commitment by Augmented Lagrangian Relaxation: Testing Two Decomposition Approaches , 2002 .

[31]  T. Chang.  Comments on “Surrogate Gradient Algorithm for Lagrangian Relaxation” , 2008 .

[32]  Mohammad Marufuzzaman,et al.  Sustainable network design for multi-purpose pellet processing depots under biomass supply uncertainty , 2017, Comput. Ind. Eng..

[33]  George B. Dantzig,et al.  Decomposition Principle for Linear Programs , 1960 .

[34]  Yoram Singer,et al.  Adaptive Subgradient Methods for Online Learning and Stochastic Optimization , 2011, J. Mach. Learn. Res..

[35]  S. Agmon.  The Relaxation Method for Linear Inequalities , 1954, Canadian Journal of Mathematics.

[36]  J. F. Benders.  Partitioning procedures for solving mixed-variables programming problems , 1962 .

[37]  C. Maranas,et al.  A Hierarchical Lagrangean Relaxation Procedure for Solving Midterm Planning Problems , 1999 .

[38]  Jairo Rafael,et al.  Análisis del efecto de algunos factores de complejidad e incertidumbre en el rendimiento de las Cadenas de Suministro. Propuesta de una herramienta de valoración basada en simulación. , 2016 .

[39]  Grit Walther,et al.  Negotiation-based coordination in product recovery networks , 2008 .

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

[41]  Hartmut Stadtler,et al.  Supply Chain Management and Advanced Planning: Concepts, Models, Software, and Case Studies , 2010 .

[42]  Hai Jiang,et al.  A Lagrangian relaxation based approach for the capacitated lot sizing problem in closed-loop supply chain , 2012 .

[43]  D. Bertsekas.  Convexification procedures and decomposition methods for nonconvex optimization problems , 1979 .

[44]  Ali Diabat,et al.  A Lagrangian relaxation approach for solving the integrated quay crane assignment and scheduling problem , 2015 .

[45]  Mikko Kurttila,et al.  An Application of a Reduced Cost Approach to Spatial Forest Planning , 2009 .

[46]  José P. García-Sabater,et al.  Modelo estocástico para planificar cadenas de suministro con productos de ciclos de vida cortos , 2010 .

[47]  Steven Y. P. Lu,et al.  A hybrid solution to collaborative decision-making in a decentralized supply-chain , 2012 .

[48]  Julio Juan García Sabater,et al.  Modelo de optimización estocástica para la planificación de cadenas de suministro para productos con ciclo de vida cortos. , 2010 .

[49]  J. E. Rooda,et al.  An augmented Lagrangian relaxation for analytical target cascading using the alternating direction method of multipliers , 2006 .

[50]  Vladimir Marianov,et al.  Lagrangean Relaxation-Based Techniques for Solving Facility Location Problems , 2011 .

[51]  In-Jae Jeong,et al.  A job shop distributed scheduling based on Lagrangian relaxation to minimise total completion time , 2009 .

[52]  G. Bitran,et al.  Computational Complexity of the Capacitated Lot Size Problem , 1982 .

[53]  Richard M. Karp,et al.  The Traveling-Salesman Problem and Minimum Spanning Trees , 1970, Oper. Res..

[54]  Giovanna Miglionico,et al.  On solving the Lagrangian dual of integer programs via an incremental approach , 2009, Comput. Optim. Appl..

[55]  Josefa Mula,et al.  The Generic Materials and Operations Planning (GMOP) Problem Solved Iteratively: A Case Study in Multi-site Context , 2011, APMS.

[56]  Jesús Sáez,et al.  Solving linear programming relaxations associated with Lagrangean relaxations by Fenchel cutting planes , 2000, Eur. J. Oper. Res..

[57]  Mikael Rönnqvist,et al.  Use of Lagrangian decomposition in supply chain planning , 2011, Math. Comput. Model..

[58]  D. Bertsekas,et al.  Convergen e Rate of In remental Subgradient Algorithms , 2000 .

[59]  Boris Polyak.  Minimization of unsmooth functionals , 1969 .

[60]  Ignacio E. Grossmann,et al.  A Lagrangean based branch-and-cut algorithm for global optimization of nonconvex mixed-integer nonlinear programs with decomposable structures , 2008, J. Glob. Optim..

[61]  José Alberto Araúzo Araúzo,et al.  Programación y Control de Sistemas de Fabricación Flexibles: un Enfoque Holónico , 2015 .

[62]  Claude Lemaréchal,et al.  Lagrangian Relaxation , 2000, Computational Combinatorial Optimization.

[63]  Martin J. Wainwright,et al.  Randomized smoothing for (parallel) stochastic optimization , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[64]  José P. García-Sabater,et al.  A new formulation technique to model materials and operations planning: the generic materials and operations planning (GMOP) problem , 2013 .

[65]  George Q. Huang,et al.  Optimal configuration of cluster supply chains with augmented Lagrange coordination , 2015, Comput. Ind. Eng..

[66]  Lucio Grandinetti,et al.  Auction algorithms for decentralized parallel machine scheduling , 2006, Parallel Comput..

[67]  Hartmut Stadtler,et al.  Supply Chain Management and Advanced Planning , 2000 .

[68]  P. Camerini,et al.  On improving relaxation methods by modified gradient techniques , 1975 .

[69]  Takahito Kuno,et al.  A Lagrangian Based Branch-and-Bound Algorithm for Production-transportation Problems , 2000, J. Glob. Optim..

[70]  X. Zhao,et al.  Surrogate Gradient Algorithm for Lagrangian Relaxation , 1999 .