An algorithm for a Parallel Machine Problem with Eligibility and Release and Delivery times, considering setup times Manuel Mateo [email protected] Management.

An algorithm for a Parallel Machine Problem with Eligibility and Release and Delivery

times, considering setup times

Manuel Mateo [email protected]

Management Department,

Universitat Politècnica Catalunya.

Barcelona (Spain)

HAROSA, Barcelona (10/07/13)

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2

Summary

Introduction: the times, the eligibility and the setup times

Notation and definition of the problem Pm/rj,qj,sj,Mj/cmax

Proposed algorithms

Initial Solution

Heuristic algorithm

Genetic Algorithm (crossover, mutation, local search)

Computational experiments and results

Conclusions

Remark: this work is based on the Master Thesis in Engineering done by David Miquel.

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Introduction: the times

3

The manufacturing of products is usually divided in operations or phases of transformation.

Usually one of them becomes the bottleneck of the process. In the presented problem, we suppose this bottleneck is an

intermediate phase. Therefore, some operations are done before (the total time

to work them out leads to a release time) and some others are done after (their total time is called delivery or queue time).

j j1 j2 j3 j4 j5

p1,j 2 4 0 3 5

p2,j 8 5 4 4 7

p3,j 3 2 1 0 3

bottleneck

j

rj

pj

qj

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Line 4

Line 3

Line 2

Line 1

Introduction: eligibility

Manufacturing plants usually have several machines or assembly lines (i.e., parallel machine).

There are several products to be manufactured. A usual situation is a product that is assigned to a machine

(line) and will be only manufactured in that machine.

4

A B C

INITIAL SITUATION

D

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Low-level machines

Medium-level machines

High-level machines

Introduction: eligibility (II)

But more product-machine assignments are possible:

5

Line 1

Line 2

Line 4 1 l DC

Line 3

CO

NSI

DER

ED

SITU

ATIO

N

B DC

B DC

B DC

2 l

A

2 l

2 l

Medium-level

Jobs Machines

High-level

Medium-level

Low-level

A

B

D

C

High-level

Low-level

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There are no setup times if the jobs classified in one level are done in the machines of the same level:

If the jobs in one level are assigned in the machines of different level, setup times appear (between the schedule of jobs belonging to different levels):

Introduction: the setup times

6

Medium-level

Jobs Machines

High-level

Medium-level

Low-level

High-level

Low-level

Medium-level

Jobs Machines

High-level

Medium-level

Low-level

High-level

Low-level

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Notation of the problem

The machines are distributed among p groups or levels (k=1,…,p). Particularly, we propose an algorithm for p=3 (high-level, medium-level and low-level).

A set of n jobs (j=1,…,n) to be scheduled on m parallel machines (i=1,…,m).

Given a job j, it is known: the processing time pj for the operation,

the release time rj (also called head times),

the delivery or queue time qj (also tail times),

the associated level lj.

Any machine i and job j is classified into one of the levels.

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Notation of the problem (II)

The n jobs to be manufactured are also divided in three groups: Ji is the subset of jobs of level i, with |Ji|=ni.

  n=n1+n2+n3

Eligibility restrictions:

A machine in the level k can manufacture jobs of its own level k and also of levels k+1,…,p.

The processing time of a job is the same for any machine.

8

Medium, k=2

Jobs Machines

High-level (k=1)

Medium-level (k=2)

Low-level (k=3)

High, k=1

Low, k=3

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The problem Pm | rj,qj,sj,Mj | cmax

The number of machines and jobs is initially known. A schedule is feasible if the next conditions are accomplished:

Each machine processes at most one job at a time. A job is only processed in a single machine. Pre-emption is not allowed. Starting time is not lower than the release time: A job of level k is processed in a machine of level k or higher.

Setup times are required when a job classified in a level is going to be manufactured after another of a different level.

Machines are initially prepared for the jobs of their own level. It is not necessary another setup at the end of the last job

scheduled if it is from a different level.

9

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The problem Pm | rj,qj,sj,Mj | cmax (II)

The processing in all machines for each job is the same, i.e., we consider identical parallel machines.

Given tj the starting time for any job j, the completion time of the job is obtained:

The makespan can be determined:

cmax= max{cj} For any feasible schedule, the objective is:

Min {cmax}

10

j jt rj j j jc t p q

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Proposed algorithms

11

INITIAL SOLUTION

HEURISTIC

Preprocessing

Insertion improvement

Flexible improvement

Genetic Algorithm (GA)

METAHEURISTIC

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At each level (3 times), it is necessary to solve a problem of a single machine (mk=1) or parallel machines (mk>1).

mk?

mk = 1 NEH-like methodMinimum {rj} & {qj}

Maximum {rj} & {qj}

mk > 1Parallel-machine

method Preprocessing + Algorithm

Gharbi & Houari (2002)

Medium-level

Jobs Machines

High-level

Medium-level

Low-level

High-level

Low-level

Initial solution (no setup times)

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Description of the Heuristic

Sequences of jobs: SPT (shortest processing time) SRT (shortest release time) SQT (shortest queue time) LPT (longest processing time) LRT (longest release time) LQT (longest queue time) 6 random sequences

13

Determine the sequence of jobs

j=1

For each position pos of j in the sequence, cmax(pos)

Assign the job in pos such that cmax is minimum; j=j+1

j=n?

SEQUENCE

YES

NO

j=j+1

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Preprocess of the GA

14

21

21

Initial Solution

Insertion improvemen

t

Flexible improvemen

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Genetic Algorithm

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General view of the Genetic Algorithm

Initial population: 1 (Initial solution + insertion

+ flexible improvements). 6 (SPT, SRT,SQT, LPT, LRT,

LQT) 73 random sequences

(generate 100 sequences + apply the heuristic)

Selection: 10% of individuals with best

fitness 90% roulette rule

15

Initial population

Fitness

Selection

Crossover

Mutation

Local search (IMI)

Regeneration

end?

SOLUTIONYES

NO

Fitness

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sGenetic Algorithm: crossover

16

Crossover Share many of the

characteristics in the

parents’ solutions Pc = 50%

Reference:

Vallada & Ruiz (2011)

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Genetic Algorithm: mutation & local search

17

Mutation Shift Pm = 50%

(Vallada & Ruiz, 2011)

Local search (IMI, Inter-Machine Insertion neighborhood)

Two machines Pls = 100% (Vallada & Ruiz, 2011) Acceptance criterion:

Makespans of both machines are reduced.

Makespan of one machine is reduced, although the other is increased.

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Computational experience

To check the efficiency of the algorithm, a set of instances similar to those used by Gharbi & Haouari (2002) are created: # jobs (n = {20,50,100,200}) 1000 instances Jobs of high level, 10-30% of the total number;

jobs of medium level, 10-50% of the total;

jobs of low level, the rest. # machines (m = {4,5,6,8,10}) Processing time: discrete uniform distribution . Release and delivery times: discrete distribution with K={3,5} Setup times: 3 sets with uniform distributions of [1, 3], [1, 9] and [1, 19].

1, ( / )K n m

1,10

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Computational experience (II)

Initial study on 220 instances to determine the computing time: balance between the results and the computational cost.

CPU time = n · (m/2) · (time) ms

Improvement 5% time = 240

Complete study on 4000 instances with different configurations of parallel machines to determine the improvement on the cmax of the initial solution.

Time 30 60 60 120 120 240 240 300

n = 20 2 0 0 0

n = 50 8 3 1 0

n = 100 22 17 13 1

n = 200 37 34 28 10

Total 69 / 220 54 / 220 42 / 220 11 / 220

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Some results

20

Number of jobs:

Proportion

of machines:

n IH IGA

20 27,7% 28,6%

50 32,4% 33,2%

100 34,1% 34,5%

200 33,6% 34,0%

Case IH IGA

1 mh > mm ; mh > ml 39,9% 40,0%

2 mm > mh ; mm > ml 38,1% 38,4%

3 ml > mh ; ml > mm 18,9% 20,0%

4 mh < mm ; mh < ml 20,7% 21,6%

5 mm < mh ; mm < ml 30,4% 30,7%

6 ml < mh ; ml < mm 48,6% 48,7%

7 mh = mm = ml 31,2% 31,5%

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Window for the setup times:

Influence of the parameter K ={3,5}.

Some results (II)

21

Setup time IH IGA

1 33,7% 34,1%

2 33,4% 33,8%

3 33,1% 33,6%

nK = 3 K = 5

IH IGA IH IGA

20 32,8% 34,3% 23,0% 23,4%

50 39,4% 40,8% 26,2% 26,3%

100 42,3% 43,1% 26,6% 26,8%

200 38,7% 39,1% 24,9% 25,1%

Average 39,6% 40,2% 25,5% 25,6%

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sConclusions from the computational experience

We have not shown the results, but the introduction of the heuristic on the initial population improves the quality of solutions. Once it is introduced: Some proportions of machines can lead to a improvements of nearly

50%; the best ones are: mh > mm ; mh > ml ml < mh ; ml < mm

On the other hand, improvements about 20% are obtained by:mb > mh ; ml > mm mh < mm ; mh < ml

The lower the parameter K is, the greater the improvement is (40% for K=3 and 25% for K=5).

Any number of jobs and any window for setup times give similar results (improvements around 30%).

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Conclusions

We studied the problem of parallel machines with eligibility and release and queue times.

The problem has the objective to minimize the makespan. The setup times are introduced when a job of a different level from

the one of the previous one is going to be manufactured. We proposed a heuristic procedure, based on some initial

sequences of jobs. A Genetic Algorithm is developed , which takes the advantages of the heuristic in the preprocess.

Improvement respect to initial solution varies from 20% to 50%. About the current and future research:

We hope to tune the parameters of the Genetic Algorithms.

Another metaheuristic (for instance ILS) could be developed.

An algorithm for a Parallel Machine Problem with Eligibility and Release and Delivery times, considering setup times

Thank you for your attention!