A simple Heuristic For m-Machine flow-shop & Its application in Routing-Scheduling problems

IGOR AVERBAKH, ODED BERMANThe theory and application of scheduling

Yang che kai. 楊哲愷

05/02/2023

The breakthrough of this paper• The special application of Heuristic

SYMM.• In Routing scheduling flow shop• Not only good at routing part, but

scheduling part.

• Focus on traveling’s servers, not traveling jobs.

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Outline• M-machine flow shop problem • The worst case performance• A Simple Heuristic performance

• Application to Routing-Scheduling problem• Case 1. Permutation Routing-Scheduling

flow shop• Case 2. General Routing scheduling flow-

shop• Further research in the future.

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Intro. m-M/Cs flow shop

※ Definition• is the Machines, i=1…m.• is the Jobs, j=1…n.• is processing time of in .• is the time interval when in .

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m-M/Cs flow shop performance of the worst case

M/C 1

M/C 2

Job1

Job2

Job3

… Jobn-2

Jobn-1

Job n

m=2

Job1

Job2

Job3

… Jobn-2

Jobn-1

Job n

Scheduling jobs by a sequence S, obtained by O(m*n).OPT-1(S)

Each machine processes the jobs according to sequence S without unnecessary idle times, starting a new job as soon as the previous job is processed on that machine and the new job is available after its execution on the previous machine.

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m-M/Cs flow shop performance of heuristic SYMM (1/2) • Minimize the makespan: Min max( for i=

1…n) ※Heuristic SYMM.1. Take an arbitrary sequence S.2. Compute and Compare OPT-1(S) & OPT-

1().3. If OPT-1(S)OPT-1(), take the permutation

schedule as an approximate solution.4. Else, take permutation schedule as an

approximate solution.

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• Minimize the makespan: Min max( for i= 1…n)

※Although there have already known Heuristics, SYMM is still useful because1. O(mn), better than O(mn) & O(mn+n).2. Simplicity.3. Allow to obtain many approximate

solution.4. Do not reorder the jobs.

※Ref: Gonzalez and Sahni(1978) Rock and Schmidt(1983)

m-M/Cs flow shop performance of heuristic SYMM (2/2)

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Intro. Case 1Permutation Routing-Scheduling ※Definition• is jobs located at different nodes of

network G=(N,E).• N= {, for i=0,1..n}.• All M/Cs are initially located at .• d(i, k)= distance from to .

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• The makespan is time between the first M/C leaves to the last M/C returns.• Minimize the makespan: Min max( for i=

1…n) • The performance of sequence max{m,

1+}

Intro. Case 1. Permutation Routing-Scheduling

𝐽 1𝐽 2 𝐽 3

𝐽𝑛−2𝐽𝑛−1𝐽𝑛

…

M/C 1

M/C 2

M/C m

…

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Case 1. By applying Heuristic SYMM-2※Heuristic SYMM-2.1. Compute and Compare values OPT-2( &

OPT-2().2. If OPT-2( OPT-2(), take sequence as an

approximate solution.3. Else, take

※Makespan to the solution obtained by SYMM-2 is not greater than max{(m+1)/2, 1+} ．OPT-2.

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Case 2. General routing-scheduling• The sequences of processing the jobs may

be different for different M/C.• = the service sequences for i M/C.• Applying Heuristic SYMM-2, we still can

get good approximate solution! (by pass the proof)

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Further research• Routing open shop scheduling problem.• Much precisely(efficient) heuristic for

routing-scheduling problem• Aerial fleet refueling problem

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• thanks.

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Reference and link• Paper links: http://

pubsonline.informs.org/doi/pdf/10.1287/opre.47.1.165• Slide share links: • http://www.slideshare.net/chekaiyang/pr

esentation-schuduling

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