Makespan Minimization for a Two-Machine Scheduling Problem with a Single Server

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Makespan Minimization for a Two-MachineScheduling Problem with a Single Server

Keramat Hasani Svetlana A. KravchenkoFrank Werner

Islamic Azad University, Malayer Branch, Malayer, IranUnited Institute of Informatics Problems, Minsk, BelarusFakultät für Mathematik, Otto-von-Guericke-Universität

Magdeburg

1,2 2

3

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2

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Outline of the Talk

Introduction Setup sequence model Block models Computational results Conclusion References

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Introduction n jobs ,have to be scheduled on two

machines with single server. For each job , there are given: - processing time , - setup time All setups have to be done by a single server which

can handle at most one job at a time. Notation: P2, S1 || .This problem is strongly NP-

hard since the problem P2, S1 | = s | is strongly NP-hard, see Hall et al.

(2000).

jsjp

js

maxc

maxc

nJJJ ,...,, 21

jJ

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Introduction

The problem P2, S1 || Cmax was considered e.g. in:

Hall et al. (2000) Brucker et al. (2002) Werner and Kravchenko (2010) Gan et al. (2012)

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Setup sequence model (M0)

In the following model, the loading order of the jobs is used as in Gan et al. (2012).

.0,

,1,

= otherwise

setupbetojobiththeisJif

xj

ij

1=,1=

ji

n

j

x 1=,1=

ji

n

i

x

jij

n

ji xsss ,

1=

=

Let be the loading time of the ith loading job and be the processing time of the ith loading job, i.e., we have

.= ,1=

jij

n

ji xppp

iss ipp

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Setup sequence model (M0)(cont’d)

Now for the first and the second loading jobs, we can introduce the inequality

211,2 ssssF

211,2 ssppL

If the processing part of the first loading job is large enough, then one can introduce the inequality

and to denote the time interval when only one machine is busy, one can introduce with the inequalities

,21,22 ppLL .1,222 LppL 2L

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.0,

,,...,,1,

=

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otherwise

JJJjobstheamonglastfinishedisJif

x

jj

jlet

To estimate the overlapping part for the first two jobs, we introduce the inequalities:

),(1 21,22 xMLOF 222 MxppOF

where }{max= jj pM

To know the earliest time when one of the machines is available, we introduce the inequality

.21,22 OFFF

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For 1,2,= nj

,11, jjjj ssFF

,11, jjjj ssLL

,11,1 jjjj ppLL

,1,11 jjjj LppL

),(1 11,1 jjjj xMLOF

,111 jjj MxppOF

.11,1 jjjj OFFF

nnmax LFsC =)(

we have the following inequalities:

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Block Models (M1,M2) The problem can be considered as

a unit of blocks where .

Each block can be completely defined by the first level job and a set of second level jobs

where inequality

holds.

maxCSP ||12,,,,1 zBB nz

kB

aJ ,},,{ 1 aka JJ akaakaa ppssp 11

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Block Models (cont’d)

=f

1, if the level is the first one,

2, if the level is the second one.

jJ

jfkB ,,

1, if job is scheduled in level f in the k-th block,

0, otherwise.

jfkB ,,The variable is used for a block, ,,1,= nk

:,1,= nj

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1=,,

2

1=1=jyk

y

n

k

BEach job belongs to some block, i.e., for nj ,1,=

There is only one job of the first level for each block:

1,1,1=

jk

n

j

B

jkj

n

jk BsST ,1,

1=

The loading part of the block Bk has the length 0,kST

The objective part of the block Bk has the length

.)( ,2,1=

jkjj

n

j

Bps

, we have

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Block Models (cont’d)The processing part of the block has the length 0,kPT

jkjj

n

jjkj

n

jk BpsBpPT ,2,

1=,1,

1=

)(

1 jjj stSTst2 jjjj stPTSTst

We denote by F the total length of the modified schedule:

nnn PTSTstF 111 nnn PTSTstF

][ jch denotes the maximal number of second level jobs for the same block (only in model M1):

nxxnxx BnchBchBBa ,1,,1,1,2,,2,1 ][[1])(

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M1 contains all constraints.

M2 contains all except (a).

The objective function is :

jxjj

n

j

n

xmax BpsFsC ,2,

1=1=

)(=)(

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Lower Bound

To evaluate the results obtained, we use the known lower bound:

},,,{max= 321 LBLBLBLB

,}{min)(2

1=1

i

Jiii

Ji

spsLB

},{min=2 iJi

iJi

psLB

}{max=3 ii

JipsLB

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Computational results (first experiment)

The performance of the models M0, M1 and MP was tested on the data generated in the same way as it is described in Abdekhodaee and Wirth (2002) and Gan et al. (2012).

For 10 instances were generated for ..8,2.0}.8,1,1.5,1{0.1,0.5,0L

,6,18,20}{8,10,14,1n

)(0,100=

(0,100)=

LUs

Upd

j

d

j

First experiment

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Computational results (first experiment)

For n = {8,10}, all models could obtain an optimal solution for all instances.

For n = 14, the models M0 and M1 were preferable to the model MP but for L = 1, the model MP appeared to be the best one in terms of average time. The time limit was 525 seconds.

For n = 16, the model M0 was the best for most instances. The models M1 and MP were comparable in terms of average time. Here, the time limit was 600 seconds.

For n = 18, the model M0 turned out to be the best with respect

to the quality of the obtained solutions. However, it is difficult to say which model turned out to be the fastest one. Here, the time limit was 675 seconds.

For n = 20, the models M0 and M1 were preferable in terms of the quality of the obtained solution. Here, the time limit was 750 seconds.

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Computational results (second experiment)

The models M1 and M2 were compared with the results of Gan et al. (2012)

The model M1 was used for

The model M2 was used for

.{8,20,50}n

50}.{100,200,2n

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For n = 8, M1 and Gan et al. (2012) could find an optimal solution, but M1 was faster.

For n = 20 and n = 50, M1 was always better than the models in Gan et al. (2012).

For n = 100, M2 was always better than the models in Gan et al. (2012).

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Load 0.1 0.5 0.8 1.0 1.5 1.8 2.0

Avg(Cmax/LB) 1.02 1.07 1.10 1.10 1.02 1.00 1.00 Max(Cmax/LB) 1.03 1.09 1.11 1.12 1.04 1.00 1.01

Load 0.1 0.5 0.8 1.0 1.5 1.8 2.0

Avg(Cmax/LB) 1.01 1.04 1.07 1.09 1.01 1.00 1.00 Max(Cmax/LB) 1.01 1.08 1.10 1.12 1.02 1.01 1.01

The average and the maximal gaps for 200=n

The average and the maximal gaps for 250=n

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Conclusion Three models were tested and the

performance was compared with that of the heuristics developed in Gan et al. (2012). The computational results show that the new models outperform all heuristics proposed in Gan et al. (2012) for most types of instances.

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References

A. Abdekhodaee and A. Wirth (2002). Scheduling parallel machines with a single server: some solvable cases and heuristics. Computers & Operations Research, 29:0 295–315, 2002.

P. Brucker, C. Dhaenens-Flipo, S. Knust, S.A. Kravchenko, F. Werner. Complexity results for parallel machine problems with a single server. Journal of Scheduling, 5:0 429–457, 2002.

H.S. Gan, A. Wirth, A. Abdekhodaee. A branch-and-price algorithm for the general case of scheduling parallel machines with a single server. Computers & Operations Research, 39:0 2242–2247, 2012.

N. Hall, C. Potts, C. Sriskandarajah. Parallel machine scheduling with a common server. Discrete Applied Mathematics, 102:0 223–243, 2000.

F. Werner, S.A. Kravchenko. Scheduling with multiple servers. Automation and Remote Control, 71:0 2109–2121, 2010.

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Makespan Minimization for a Two-Machine Scheduling Problem with a Single Server