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MILP Approach for the Axxom Case Study (Lacquer Production)

MILP Approach for the Axxom Case Study (Lacquer Production). Sebastian Panek. Problem description (Dagmar Ludewig) Problem characteristics Discrete time model Continuous time model Tests and Results Conclusions. Overview. 29 types of lacqeur to be produced

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MILP Approach for the Axxom Case Study (Lacquer Production)

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  1. MILP Approach for the Axxom Case Study (Lacquer Production) Sebastian Panek

  2. Problem description (Dagmar Ludewig) Problem characteristics Discrete time model Continuous time model Tests and Results Conclusions Overview

  3. 29 types of lacqeur to be produced Some batches are too large and must be splitted to be processed on the existing machines Variable processing times on mixing vessels start time and end time or start time and duration must be considered for each task Problem characteristics

  4. Additional restrictions for tasks: start-start restrictions end-start restrictions end-end restrictions Problem characteristics

  5. Based on the model of Kondili, Pantelides and Sargent (1993) Suitable for batch operations in chemical processing Uses the state-task network concept Discrete time model 1 1 State A Task 1 State C 1 State B Task 2 1

  6. Discrete time points: starting times and durations of processing steps The principle of the state tasknetwork model 1 2 A B C t 1 2 3 4 5 6 7

  7. Sets: i tasks, j products,k machines Processing indicator variabless the machine k starts the execution of task i at t Batch size variables the batch size for task i on machine k at t Stock limitations for states Batch limitations for machines MILP model for the state tasknetwork

  8. Product flow balance Processing starts when external demand is created MILP model for the state-tasknetwork

  9. The problem size depends essentially on the number of time points, tasks and machines A coarse time grid is needed to keep the model small A fine time grid (atmost 2h/unit) is needed to satisfy restrictions between particular tasks With this time grid the solution of the model failed due to the extremely high memory usage (on a 640 MB computer, 1 lacqeur!) Difficulties and problems

  10. Another approach: continuous time Easy formulation of restrictions on start and processing times Focused on tasks Products (states) are not considered explicitly Fixed batch sizes (no merging and splitting of batches) Capacity restrictions for states are difficult Continuous time model

  11. Real variables for start and end times Binary variables for the machine allocation task i is processed on machine k Binary variables for the ordering of the tasks task i is processed before task h on machine k MILP formulation of the continuoustime model

  12. 2 tasks on 2 different machines 2 tasks on the same machine MILP formulation of the continuoustime model 1 1 2 2 1 2 1 2

  13. Start and end times for tasks i on particular machines k Additional equations are needed to express nonlinear products of binary and real variables Start and end times for allocatedmachines

  14. Each task must be processed on 1 machine If both tasks i and h are processed on machine k then either i is before h or vice versa Restrictions on binary variables

  15. Tasks processed on the same machine must exclude each other Set iff task i ends before task h starts and set it 0 otherwise (M formulation) Order restrictions

  16. Minimize the sum of latenesses of all tasks Alternatively: other objectives like makespan or cost minimization available Objective function

  17. After manual batch splitting: 38 individual batches to be processed 14 machines (5 mixing vessels, 2 carussels) 22 tasks (8 for uni lacqeurs, 6 for metalic lacqeurs, 8 for special metalic lacqeurs) MILP models with different sizes up to 22 lacqeurs have been solved Larger models are very difficult to solve Tests

  18. Gantt diagramm for 20 lacqeurs

  19. Integer solutions for a problem with 10 lacqeurs Node Objective Time 185 10607 <20s 1031 2469 36s 2015 669 56s 3962 185 92s Poor solution can be obtained quickly, good solutions need much time and memory. Solution with GAMS/Cplex

  20. The continuous time model requires smaller models and therefore is easier to solve The discrete time model is more flexible and powerful (batch merging and splitting, external components, product restrictions) Both types are strongly limited with respect to the solution time of large models A kind of meta-strategy (i.e. moving horizon) is needed to solve large models Conclusions

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