Panek Marseille

8/11/2019 Panek Marseille

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NIVERSITY OFU DORTMUND

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 problem

characteristics Additional restrictions for pairs of tasks:

start-start restrictions

end-start restrictions

end-end restrictions

Parallel allocation of mixing vessels

Machine allocation

allowed

interval


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Problem simplifications

Labs are non-bottleneck resources, no exclusive

resource allocation is needed (provided by Axxom)

Individual colors for lacquers => 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

Task 1

Task 2

State A

State B

State C1

1

1

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 tothe time horizon


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MILP formulation of the

continuous time model Real variables for starting and ending times of tasks

Binary variables for the machine allocation

task iis processed on machine k:

Binary variables for the sequencing of tasks task iis processed before task hon machine k:

ii es,

1ik

0,1 hikihk


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Starting and ending times for

allocated machines Starting and ending dates for tasks ion machines k

Extra linear equations are needed to express

nonlinear products of binary and real variables

ikiik

ikiik

ee

ss

:

:


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Restrictions on binary variables

Each task must be processed on 1 machine

If both tasks iand hare processed on machine kthen

either iis scheduled before hor vice versa

1k

ik

hkikhikihk


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Sequencing restrictions

Tasks i, hprocessed on the same machine kmust

not overlap each other

Set iff task iis finished before task h

))(1(

))(1()1(

hkikihkhkik

hkikihkhkik

Mmse

MMse

1ihk


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Objective function

Minimize too late and too early job completions

,0,max

0,maxmin

i

ii

i

ii

edeadline

deadlineeJ


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Additional heuristics

1. Non-overtaking of non-overlapping jobs

2. Non-overtaking of equal-typed jobs (M. Bozga)

3. Earliest Due Date (EDD)

hkikhi sereleasedeadline thenif

hkikhihi setypetypedeadlinedeadline thenandif

hkikhi sedeadlinedeadline thenif


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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 1+2

4. Solve the problem again reusing previous solutionas initial integer solution


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How is the model influenced

by the heuristics?

N: #Tasks, M: #Machines

Most binary variables are variables.

Worst case: # variables = O(N2

M) (!!!)(i,h=1...N, k=1...M)

# real variables = ~2NM

But:

When using heuristics, many binary variables are fixedand disappear from the model.

We want to reduce O(N2M) to O(NM)! How that?

ihk

ihk


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A little example:

1 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

ihk* **

* **

* * *

* * *

Matrix of

variablesi=1

2

3

4h=1 2 3

4


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Heuristics #1

non-overlapping jobs

Job# 1 2 3 4

Release 0 1 1 3

Deadline 2 3 4 5

Type 1 1 2 2

ihk* 1*

* 1*

* * *

0 0 *

Matrix of

variablesi=1

2

3

4h=1 2 3

4


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Heuristics #2

equal-typed jobs

Job# 1 2 3 4

Release 0 1 1 3

Deadline 2 3 4 5

Type 1 1 2 2

ihk1 1*

0 1*

* * 1

0 0 0

Matrix of

variablesi=1

2

3

4h=1 2 3

4


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Heuristics #2

EDD

Job# 1 2 3 4

Release 0 1 1 3

Deadline 2 3 4 5

Type 1 1 2 2

ihk1 11

0 11

0 0 1

0 0 0

Matrix of

variablesi=1

2

3

4h=1 2 3

4


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Empirical studies on the Axxom

Case Study

model scaled from 4up to 29jobs

Jobs in job table sorted according to deadlines

2-stage solution procedure (heuristics 3, 1+2) CPU usage limited to 20+20minutes

Measurement of solution time,

equations, real and binary variables,

objective values and bounds

Software: GAMS+Cplex

Hardware: 1.5 GHz Athlon, 1 GB Ram


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Objective values

lower

bounds

integer

solutions

gap


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Solution times

20 min.limit was

active for

>10 jobs


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Variables and Equations

equations

total

variables

binary

variables

~50% of all

Variables!

U D


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Gantt chart: 29 jobs

2h of computation time, first integer solution after few min.(node 173)

U D


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22 jobs, moving horizon procedure

Horizon: 7 jobs, 16 steps a 25 minutes, 300 MHz machine

U D


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Conclusions from empirical

studies

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 fromO(N2M)to O(NM)without restricting the problem too much

for

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