MILP Approach to the

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MILP Approach to the
Axxom Case Study
Sebastian Panek
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Introduction

• What is this talk about?
• MILP formulation for the scheduling problem
  provided by Axxom (lacquer production)
• Whats new since our meeting in Sept. 02?
• Improved model and solution procedure, new
  results
• What about modeling TA as MILP?
• This work is still in progress...

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Overview

• Short problem description
• MILP formulation
• Solution procedure
• Emprical studies
• Conclusions

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Short problem description
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Additional problemcharacteristics
allowed interval

• Additional restrictions for pairs of tasks
• start-start restrictions
• end-start restrictions
• end-end restrictions
• Parallel allocation of mixing vessels
• Machine allocation

6
Problem simplifications

• Labs are non-bottleneck resources, no exclusive
  resource allocation is needed (provided by Axxom)
• Individual colors for lacquers gt many different
  products
• No batch merging is possible
• Only few jobs exceeding max. batch capacity
• Batch splitting is not considered

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General approaches

• For short-term scheduling problems in the
  processing industry Kondili,Floudas, Pantelides,
  Grossmann,...
• State Task Networks (STN)
• Resource Task Networks (RTN)
• Early formulations discrete time
• Recent work continuous time

1
1
Task 1
State A
State C
1
Task 2
State B
1
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General approaches (2)

• Advantages
• Batch splitting/merging
• Mass balances
• Individual modeling of products
• Restrictions on storages
• Disadvantages
• Continuous and discrete time models tend to
  require many points of time, number difficult to
  estimate
• Very detailed view of the problem not always
  necessary
• Problem large models, difficult to solve

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Our approach sequencing based continuous time
model

• Continuous time
• Individual representation of time for machines
• Focused on tasks and machines
• Products (states) are not considered explicitly
• Fixed batch sizes (no merging and splitting of
  batches)
• Grows according to the number of tasks and not to
  the time horizon

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MILP formulation of thecontinuous time model

• Real variables for starting and ending times of
  tasks
• Binary variables for the machine allocation
• task i is processed on machine k
• Binary variables for the sequencing of tasks
• task i is processed before task h on machine k

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Starting and ending times forallocated machines

• Starting and ending dates for tasks i on machines
  k
• Extra linear 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 scheduled before h or vice versa

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Sequencing restrictions

• Tasks i, h processed on the same machine k must
  not overlap each other
• Set iff task i is finished before
  task h

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Objective function

• Minimize too late and too early job completions

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Additional heuristics

• Non-overtaking of non-overlapping jobs
• Non-overtaking of equal-typed jobs (M. Bozga)
• Earliest Due Date (EDD)

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2-step solution procedure

1. Apply heuristics 3 (EDD) by fixing some
   variables
2. Solve the problem
3. Relax and fix some variables according to
   heuristics 12
4. Solve the problem again reusing previous solution
   as initial integer solution

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How is the model influencedby the heuristics?

• N Tasks, M Machines
• Most binary variables are variables.
• Worst case variables O(N2M) (!!!)
• (i,h1...N, k1...M)
• real variables 2NM
• But
• When using heuristics, many binary variables are
  fixed and disappear from the model.
• We want to reduce O(N2M) to O(NM)! How that?

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A little example1 machine, 4 jobs, 1 task/job

• Job 1 2 3 4
• Release 0 1 1 3
• Deadline 2 3 4 5
• Type 1 1 2 2

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h1
2
3
Matrix of variables



i1



2



3



4
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Heuristics 1non-overlapping jobs

• Job 1 2 3 4
• Release 0 1 1 3
• Deadline 2 3 4 5
• Type 1 1 2 2

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h1
2
3
Matrix of variables

1

i1

1

2



3
0
0

4
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Heuristics 2equal-typed jobs

• Job 1 2 3 4
• Release 0 1 1 3
• Deadline 2 3 4 5
• Type 1 1 2 2

4
h1
2
3
Matrix of variables
1
1

i1
0
1

2


1
3
0
0
0
4
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Heuristics 2EDD

• Job 1 2 3 4
• Release 0 1 1 3
• Deadline 2 3 4 5
• Type 1 1 2 2

4
h1
2
3
Matrix of variables
1
1
1
i1
0
1
1
2
0
0
1
3
0
0
0
4
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Empirical studies on the AxxomCase Study

• model scaled from 4 up to 29 jobs
• Jobs in job table sorted according to deadlines
• 2-stage solution procedure (heuristics 3, 12)
• CPU usage limited to 2020 minutes
• Measurement of
• solution time,
• equations, real and binary variables,
• objective values and bounds
• Software GAMSCplex
• Hardware 1.5 GHz Athlon, 1 GB Ram

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Objective values
integer solutions
gap
lower bounds
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Solution times
20 min. limit was active for gt10 jobs
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Variables and Equations
equations
50 of all Variables!
total variables
binary variables
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Gantt chart 29 jobs
2h of computation time, first integer solution
after few min.(node 173)
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22 jobs, moving horizon procedure
Horizon 7 jobs, 16 steps a 25 minutes, 300 MHz
machine
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Conclusions from empiricalstudies

• EDD heuristics at 1. stage helps finding integer
  solutions quickly (even for large instances!)
• 2. stage usually cannot find better solutions (in
  short time)...
• but the number of binary variables is
  significantly reduced from O(N2M) to O(NM)
  without restricting the problem too much
• for lt20 jobs very good gaps can be expected in
  short time
• first integer solutions within few minutes for
  the 29 jobs instance
• efficiency comparable to TA model from M. Bozga
  (VERIMAG)...
• but quantitative infos about integer solutions
  from the gaps
• A decomposition strategy helps improving the
  efficiency and the quality of results