MILP Approach to the Axxom Case Study Sebastian Panek

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MILP Approach to theAxxom 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)• What‘s 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

• Additional restrictions for pairs of tasks:– start-start restrictions

– end-start restrictions

– end-end restrictions

• Parallel allocation of mixing vessels• Machine allocation

allowedinterval

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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 BState C

1

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 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 :

ii es ,

1ik

0,1 hikihk

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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

ikiik

ikiik

ee

ss

:

:

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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

1k

ik

hkikhikihk

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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

))(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

iii

iii

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 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,h=1...N, k=1...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?

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 ofvariables

i=1

2

3

4h=1 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

ihk* 1*

* 1*

* * *

0 0 *

Matrix ofvariables

i=1

2

3

4h=1 2 3

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

ihk1 1*

0 1*

* * 1

0 0 0

Matrix ofvariables

i=1

2

3

4h=1 2 3

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

ihk1 11

0 11

0 0 1

0 0 0

Matrix ofvariables

i=1

2

3

4h=1 2 3

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, 1+2)• CPU usage limited to 20+20 minutes• 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

lowerbounds

integersolutions

gap

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Solution times

20 min.limit wasactive for>10 jobs

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Variables and Equations

equations

totalvariables

binaryvariables

~50% of allVariables!

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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 <20 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