MILP Approach for the

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

• Problem description (Dagmar Ludewig)
• Problem characteristics
• Discrete time model
• Continuous time model
• Tests and Results
• Conclusions

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Problem characteristics

• 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

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Problem characteristics

• Additional restrictions for tasks
• start-start restrictions
• end-start restrictions
• end-end restrictions

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Discrete time model

• Based on the model of Kondili, Pantelides and
  Sargent (1993)
• Suitable for batch operations in chemical
  processing
• Uses the state-task network concept

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1
Task 1
State A
State C
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Task 2
State B
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The principle of the state tasknetwork model

• Discrete time points starting times and
  durations of processing steps

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2
A
B
C
t
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3
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MILP model for the state tasknetwork

• 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

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MILP model for the state-tasknetwork

• Product flow balance
• Processing starts when external demand is created

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Difficulties and problems

• 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!)

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Continuous time model

• 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

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MILP formulation of the continuoustime model

• 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

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MILP formulation of the continuoustime model

• 2 tasks on 2 different machines
• 2 tasks on the same machine

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2
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Start and end times for allocatedmachines

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

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Restrictions on binary variables

• 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

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Order restrictions

• 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)

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Objective function

• Minimize the sum of latenesses of all tasks
• Alternatively other objectives like makespan or
  cost minimization available

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Tests

• 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

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Gantt diagramm for 20 lacqeurs
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Solution with GAMS/Cplex

• Integer solutions for a problem with 10 lacqeurs
• Node Objective Time
• 185 10607 lt20s
• 1031 2469 36s
• 2015 669 56s
• 3962 185 92s
• Poor solution can be obtained quickly, good
  solutions need much time and memory.

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Conclusions

• 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