Theoretical framework for formulating MIP scheduling models with multiple and non-uniform discrete-time grids

Abstract We present a framework for the formulation of MIP scheduling models based on multiple and nonuniform discrete time grids. In a previous work we showed that it is possible to use different (possibly non-uniform) time grids for each task, unit, and material. Here, we generalize these ideas to account for general resources, and a range of processing characteristics such as limited intermediate storage and changeovers. Each resource has its own grid based on resource consumption and availability allowing resource constraints to be modeled more accurately without increasing the number of binary variables. We develop algorithms to define the unit-, task-, material-, and resource-specific grids directly from problem data. Importantly, we prove that the multi-grid formulation is able to find a schedule with the same optimal objective as the discrete-time single-grid model with an arbitrarily fine grid. The proposed framework leads to the formulation of models with reduced number of binary variables and constraints, which are able to find good solutions faster than existing models.

[1]  Christos T Maravelias,et al.  Advances in mixed-integer programming methods for chemical production scheduling. , 2014, Annual review of chemical and biomolecular engineering.

[2]  Richard Bellman,et al.  ON A ROUTING PROBLEM , 1958 .

[3]  Gabriela P. Henning,et al.  A novel network-based continuous-time representation for process scheduling: Part I. Main concepts and mathematical formulation , 2009, Comput. Chem. Eng..

[4]  J. M. Pinto,et al.  A Continuous Time Mixed Integer Linear Programming Model for Short Term Scheduling of Multistage Batch Plants , 1995 .

[5]  Christos T. Maravelias,et al.  Mixed-Time Representation for State-Task Network Models , 2005 .

[6]  Dan Wu,et al.  Decomposition approaches for the efficient solution of short-term scheduling problems , 2003, Comput. Chem. Eng..

[7]  R. Sargent,et al.  A general algorithm for short-term scheduling of batch operations—II. Computational issues , 1993 .

[8]  Christodoulos A. Floudas,et al.  Effective Continuous-Time Formulation for Short-Term Scheduling. 2. Continuous and Semicontinuous Processes , 1998 .

[9]  Nilay Shah,et al.  Improving the efficiency of discrete time scheduling formulation , 1998 .

[10]  Christodoulos A. Floudas,et al.  Unit-specific event-based continuous-time approach for short-term scheduling of batch plants using RTN framework , 2008, Comput. Chem. Eng..

[11]  Christos T. Maravelias,et al.  Valid Inequalities Based on Demand Propagation for Chemical Production Scheduling MIP Models , 2013 .

[12]  Jie Li,et al.  A novel approach to scheduling multipurpose batch plants using unit-slots , 2009 .

[13]  Joseph F. Pekny,et al.  Application of distributed computing to batch plant design and scheduling , 1996 .

[14]  Christodoulos A. Floudas,et al.  Improving unit-specific event based continuous-time approaches for batch processes: Integrality gap and task splitting , 2008, Comput. Chem. Eng..

[15]  Iftekhar A. Karimi,et al.  A simpler better slot-based continuous-time formulation for short-term scheduling in multipurpose batch plants , 2005 .

[16]  Christos T. Maravelias,et al.  Polyhedral results for discrete-time production planning MIP formulations for continuous processes , 2009, Comput. Chem. Eng..

[17]  L. R. Ford,et al.  NETWORK FLOW THEORY , 1956 .

[18]  A. Barbosa‐Póvoa,et al.  An Improved RTN Continuous-Time Formulation for the Short-term Scheduling of Multipurpose Batch Plants , 2001 .

[19]  Ignacio E. Grossmann,et al.  Time representations and mathematical models for process scheduling problems , 2011, Comput. Chem. Eng..

[20]  Pedro M. Castro,et al.  Scheduling of a continuous plant with recycling of byproducts: A case study from a tissue paper mill , 2009, Comput. Chem. Eng..

[21]  Gabriela P. Henning,et al.  A novel network-based continuous-time representation for process scheduling: Part II. General framework , 2009, Comput. Chem. Eng..

[22]  Christos T. Maravelias,et al.  Simultaneous Batching and Scheduling Using Dynamic Decomposition on a Grid , 2009, INFORMS J. Comput..

[23]  C. Maravelias,et al.  Computational Study of Network-Based Mixed-Integer Programming Approaches for Chemical Production Scheduling , 2011 .

[24]  Christos T. Maravelias,et al.  A General Framework for Process Scheduling , 2011 .

[25]  C. Floudas,et al.  Effective Continuous-Time Formulation for Short-Term Scheduling. 1. Multipurpose Batch Processes , 1998 .

[26]  Lazaros G. Papageorgiou,et al.  A hybrid MILP/CLP algorithm for multipurpose batch process scheduling , 2005, Comput. Chem. Eng..

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

[28]  Matthew H. Bassett,et al.  Decomposition techniques for the solution of large-scale scheduling problems , 1996 .

[29]  Pedro M. Castro,et al.  Scope for industrial applications of production scheduling models and solution methods , 2014, Comput. Chem. Eng..

[30]  Rainer E. Burkard,et al.  Review, extensions and computational comparison of MILP formulations for scheduling of batch processes , 2005, Comput. Chem. Eng..

[31]  Christos T. Maravelias,et al.  Multiple and nonuniform time grids in discrete-time MIP models for chemical production scheduling , 2013, Comput. Chem. Eng..

[32]  Christos T. Maravelias,et al.  A branch-and-bound algorithm for the solution of chemical production scheduling MIP models using parallel computing , 2013, Comput. Chem. Eng..

[33]  Ignacio E. Grossmann,et al.  New general continuous-time state-task network formulation for short-term scheduling of multipurpose batch plants , 2003 .

[34]  Christos T. Maravelias,et al.  Reformulations and Branching Methods for Mixed-Integer Programming Chemical Production Scheduling Models , 2013 .

[35]  Christos T. Maravelias,et al.  General framework and modeling approach classification for chemical production scheduling , 2012 .

[36]  Andres F. Merchan,et al.  Tightening methods for continuous‐time mixed‐integer programming models for chemical production scheduling , 2013 .

[37]  R. Sargent,et al.  A general algorithm for short-term scheduling of batch operations */I , 1993 .

[38]  Christos T. Maravelias.  On the combinatorial structure of discrete-time MIP formulations for chemical production scheduling , 2012, Comput. Chem. Eng..

[39]  Ignacio E. Grossmann,et al.  A hybrid MILP/CP decomposition approach for the continuous time scheduling of multipurpose batch plants , 2004, Comput. Chem. Eng..

[40]  Egon Balas,et al.  Machine Sequencing Via Disjunctive Graphs: An Implicit Enumeration Algorithm , 1969, Oper. Res..

[41]  Dan Wu,et al.  Cyclic short-term scheduling of multiproduct batch plants using continuous-time representation , 2004, Comput. Chem. Eng..

[42]  N. Giannelos,et al.  A novel event-driven formulation for short-term scheduling of multipurpose continuous processes , 2002 .

[43]  G. Reklaitis,et al.  Continuous Time Representation Approach to Batch and Continuous Process Scheduling. 1. MINLP Formulation , 1999 .

[44]  Jaime Cerdá,et al.  State-of-the-art review of optimization methods for short-term scheduling of batch processes , 2006, Comput. Chem. Eng..

[45]  Yves Pochet,et al.  A tighter continuous time formulation for the cyclic scheduling of a mixed plant , 2008, Comput. Chem. Eng..

[46]  C. Pantelides,et al.  A simple continuous-time process scheduling formulation and a novel solution algorithm , 1996 .

[47]  Ignacio E. Grossmann,et al.  Minimization of the Makespan with a Discrete-Time State−Task Network Formulation , 2003 .

[48]  C. Pantelides,et al.  Optimal Campaign Planning/Scheduling of Multipurpose Batch/Semicontinuous Plants. 2. A Mathematical Decomposition Approach , 1996 .

[49]  R. Sargent,et al.  The optimal operation of mixed production facilities—a general formulation and some approaches for the solution , 1996 .

[50]  I. Grossmann,et al.  MINLP model for cyclic multiproduct scheduling on continuous parallel lines , 1991 .

[51]  Christos T. Maravelias,et al.  Mixed-Integer Programming Model and Tightening Methods for Scheduling in General Chemical Production Environments , 2013 .

[52]  Jeffrey D. Kelly,et al.  Multi-Product Inventory Logistics Modeling in the Process Industries , 2009 .