Ordinal Semi On Line Scheduling for Jobs with Arbitrary Release Times on Identical Parallel Machines

http://www.scirp.org/journal/iim ISSN Online: 2160-5920 ISSN Print: 2160-5912

DOI: 10.4236/iim.2017.96014 Nov. 15, 2017 245 Intelligent Information Management

Ordinal Semi On-Line Scheduling for Jobs with

Arbitrary Release Times on Identical Parallel

Machines

Sai Ji

1

, Rongheng Li

1*

, Yunxia Zhou

2

1_{Key Laboratory of High Performance Computing and Stochastic Information Processing, Department of Mathematics,} Hunan Normal University, Changsha, China

2_{Department of Computer, Hunan Normal University, Changsha, China}

Abstract

In this paper, we investigate the problem of semi-on-line scheduling n jobs on

m identical parallel machines under the assumption that the ordering of the

jobs by processing time is known and the jobs have arbitrary release times. Our aim is to minimize the maximum completion time. An ordinal algorithm is investigated and its worst case ratio is analyzed.

Keywords

Schedule, Algorithm, Worst Case Ratio, Parallel Machines

1. Introduction

The problem of minimizing the maximum completion time for scheduling n

jobs on m identical parallel machines (which is denoted by Pm/ /⋅ Cmax) have attracted the interests of many researchers since it was proposed by Graham in 1969 [1]. The problem is defined as follows: Given a job set L=

{

J J1, 2,,Jn

}

of n jobs and an identical parallel machine set

{

M M1, 2,,Mm

}

, where job Jj

has non-negative processing time pj, assign the jobs onto the machines so as to

minimize the maximum completion of the m machines.

A scheduling problem is called off-line if we have complete information about the job data before constructing a schedule. In contrast, the scheduling problem is called online if the jobs appear one by one and it requires scheduling the ar-riving job irrevocably on a machine without knowledge of the future jobs. The processing time of next job becomes available only after the current job is

sche-duled. Graham [1] proposed the List Scheduling (LS) algorithm to minimize the

How to cite this paper: Ji, S., Li, R.H. and Zhou, Y.X. (2017) Ordinal Semi On-Line Scheduling for Jobs with Arbitrary Release Times on Identical Parallel Machines. Intel-ligent Information Management, 9, 245-254. https://doi.org/10.4236/iim.2017.96014

Received: October 4, 2017 Accepted: November 12, 2017 Published: November 15, 2017

Copyright © 2017 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

DOI: 10.4236/iim.2017.96014 246 Intelligent Information Management

maximum completion time for online scheduling n jobs on m identical parallel

machines.

Li and Huang [2] generalized Graham’s classical on-line scheduling problem

to m identical machines. They describe the requests of all jobs in terms of order. For an order of the job Jj, the scheduler is informed of a 2-tuple

(

r pj, j

)

,

where rj and pj represent the release time and the processing time of the job

j

J , respectively. The orders of request have no release time but appear online

one by one at the very beginning time of the system. In this online situation, the jobs’ release times are assumed to be arbitrary. If all jobs’ release times are zero,

then the problem in Li and Huang [2] becomes the same as the Graham’s

clas-sical on-line scheduling problem.

For the classical online problem on the identical parallel machine system, we know that no algorithm can be better than algorithm LS when the number of machines is less than 4. In many applications, partial information about jobs can be made available in advance. This motivates us to study semi-online scheduling

problems when different types of partial information become available [3]. He

and Zhang [4], and He and Dosa [5] consider the system when the lengths of all jobs are known in

[ ]

1,r with

r

≥

1

. See Cheng et al. [6]; Li and Huang [7];

Seiden et al.[8]; Li et al.[9] for more recent results on semi-online scheduling. Liu et al. [10] firstly considered ordinal semi-online problem in which it is

assumed that the values of the processing times pj are unknown, but that the

order of the jobs by non-increasing processing time is known, i.e.,

1 2 n

p ≥p ≥≥p . The problem can be denoted as Pm/ ordinal /Cmax. They

proposed an algorithm with worst case performance ratio not greater than

1 1

2 m

m m

− +

 

+  _{ }. Later 2 max

/ ordinal /

Q C is considered by Tan and He [11] and

3/ ordinal / max

P C is considered by He and Tan [12].

In this paper, we assume that we are given m identical parallel machines

{

M M1, 2,,Mm

}

and an ordinal job list L=

{

J J1, 2,,Jn

}

with arbitrary

re-lease times, i.e., each Jj has a release time rj and a processing size of pj

sa-tisfying p1≥p2≥≥pn. Our aim is to minimize the maximum completion

time. We denote this problem as Pm/ ordinal,

(

rj

)

/Cmax.

The rest of the paper is organized as follows. In Section 2, some definitions and the algorithm P are given. In Section 3, we analyze the algorithm P and show its the upper bound of the worst case ratio. In Section 4, we give some con-cluding remarks.

2. Some Definitions and the Algorithm

P

In this section we will give some definitions and an algorithm P.

Definition 1. Let algorithm

A

_{be a heuristic algorithm of scheduling job list}

L

. _maxA

( )

C L and _maxOPT

( )

C L

denote the makespan of algorithm

A

and an

DOI: 10.4236/iim.2017.96014 247 Intelligent Information Management

(

)

max

( )

( )

max

, sup

A OPT L

C L

R m A

C L

=

as the worst case performance ratio of algorithm

A

.

Definition 2. Suppose that Jj is the current job with information

(

r pj, j

)

, i.e., release time rj and size of pj, to be scheduled on machine Mi. We say

that machine Mi has an idle time interval for job Jj, if there exists a time

in-terval

[

T T1, 2

]

satisfying the following two conditions:

1) Machine Mi is idle in interval

[

T T1, 2

]

and a job with release time T2 has been assigned to machine Mi to start at time T2.

2) T2−maxT r1, j≥ pj.

It is obvious that if machine Mi has an idle time interval for job Jj then

we can assign Jj to machine Mi in the idle interval.

The algorithm P:

Let L=

{

J J1, 2,,Jn

}

be a job list of problem Pm/ ordinal,

(

ri

)

/Cmax. We assign jobs of

L

one by one on machine M i_i

(

=1, 2,,m

)

according to the

following rules: If 1

2

m

i  

≤ ≤  _{ } holds, we assign the following jobs on machine Mi:

{ }

2 1 2

| 0

i m

m i k m

J J _{  } k

+ − +_ + _{ }_

 

 _≥ 

 

 

 



.

If 1 2

m

i m

 

+_{ }≤ ≤

  holds, we assign the following jobs on machine Mi:

{ }

2 1 3 1

2 2

| 0 | 0 .

i m m

m i k m m i k m

J J _{  } k J _{  } k

+ − +_ + _{ }_ + − +_ + _{ }_

   

 _≥   _≥ 

   

   

   

 

Assignment example for m=5:

1 1 10 18

2 2 9 17

3 3 8 13 16 21

4 4 7 12 15 20

5 5 6 11 14 19

: : : : :

M J J J

M J J J

M J J J J J

M J J J J J

M J J J J J

    

Note: The above assignment just means the order of the job’s assigning, not mean the order of job’s processing because of the release times of the jobs. For example on machine M1, if r1=6,p1=8,r10=6,p10=6, then J10 begin to be processed at time zero and J1 begin to be processed at 6 even though J1 is assigned before J10 because J1 appears before J10.

The following symbols will be used in the analysis of this paper later on:

1) 1 1 , .

n n

n n

j k j

j j k

P =

∑

₌ p P =

∑

₌ p

2) Ui: the total sum of the idle time on machine Mi in the schedule P.

DOI: 10.4236/iim.2017.96014 248 Intelligent Information Management

the total sum of the processing time of the jobs assigned on machine Mi and

the idle time of machine Mi in the schedule P. It is easy to see that

( )

{

}

max max 1, 2, ,

P

m

C L = C C  C

holds.

1)

[ ]

hi: the index of the h-th job assigned on machine Mi in the schedule P.

2)  _{ }x : It represents the smallest integer not less than x.

3)  _{ }x : It represents the largest integer not bigger than x.

3. Main Results

The following simple inequality will be referred to later on:

( )

1

max max , , 1, 2, , , , 1, 2, ,

n j j OPT

j j i i

p

C L r p j n U p i m

m

=

 

 

≥ _ + = + = _

 

 

∑

 

Furthermore it is easy to get

( )

max , 1, 2, ,

OPT

i

C L ≥U i=  m

Lemma 1: For any job list L=

{

J J1, 2,,Jn

}

from problem

(

)

max

/ ordinal, /

m i

P r C , suppose that h jobs are assigned on machine M_i by

algorithm P. If h≥2 and

[ ] [ ] [ ]

h_i 1_i h_i i h 1 x −

− = − ≥ hold, then we have

1 .

n

i i i i

C ≤xP₊ +U +p

Proof: It is easy to see that 2,

[ ]

1

i

h

h x

h i

−

∀ ≥ ≥

− holds. Firstly we prove the

following statement by induction for h:

[ ]

[ ] [ ]

(

[ ] [ ]

)

(

)

[ ]

1 2 1 1

i

i i i

h

i h i i h

xP₊ ≥ p + + p +_x h − − h− _p (1)

For h=2 we have

[ ]

[ ] [ ]

(

)

(

[ ] [ ]

)

[ ] [ ]

(

[ ] [ ]

)

(

)

[ ] 2

1 1 1 2 2

2 2

2 1

2 1 2 1

i

i i i

i i

i i i

i i

xP x p p x p

p x p

+ = + + + ≥ _ − _

 

= +_ − − − _



That means the statement is true for h = 2. Now suppose (1) holds for h = k,

i.e.,

[ ]

[ ] [ ] [ ]

(

[ ] [ ]

)

(

)

[ ]

1 2 3 1 1 .

i

i i i i

k

i k i i k

xP₊ ≥p +p + + p +_x k − − k− _p

Then for h= +k 1 we have

[ ]

[ ] [ ] [ ]

(

[ ] [ ]

)

(

)

[ ]

[ ] [ ] [ ]

(

)

[ ] [ ] [ ]

(

[ ] [ ]

)

(

)

[ ]

[

] [ ]

(

)

[ ]

[ ] [ ] [ ]

(

[ ] [ ]

)

(

)

[ ] 1

1 2 3

1 2 1

2 3

1

2 3 1

1 1

1 1

1

1 1

i

i i i i

i i i

i i i i

i

i i i i

k

i k i i k

k k k

k i i k

k

i i

k i i k

xP p p p x k k p

x p p p

p p p x k k p

x k k p

p p p x k k p

+ +

+ + +

+

+

 

≥ + + + +_ − − − _

+ + + +

 

≥ + + + +_ − − − _

 

+_ + − _

 

≥ + + + +_ − − − _







DOI: 10.4236/iim.2017.96014 249 Intelligent Information Management

[

] [ ]

(

)

[ ]

[ ] [ ] [ ]

(

[

] [ ]

)

(

)

[ ] 1

2 3 1 1

1

1 1 1 1

i

i i i i

k

i i

k i i k

x k k p

p p p x k k p

+

+ +

 

+_ + − _

 

= + + + +_ + − − + − _

By (1) we have

[ ]

[ ] [ ] [ ]

[ ] [ ]

(

)

(

)

_{[ ]}

[ ] [ ] [ ]

1 1 2 3

2 3

1 1

i

i i i

i

i i i

h n

i i i i h

i k

i i

i i i

h

xP U xP U p p p

x h h p U

p p p U C p

+ + ≥ + + ≥ + + +

 

+_ − − − _ +

≥ + + + + = −





where the last equality results from

[ ]

1_i=i by the rules of algorithm P. The claim is proved.

Lemma 2: For any job list L=

{

J J1, 2,,Jn

}

from problem

(

)

max

/ ordinal, /

m i

P r C , we have

(

)

(

)

1

1

1

2

1 ; is even

3 1

2 1

1 ; is odd

3 1 1 2

4 1

; is odd

3 1 2

n i

i i

n i

i i i

n i

i i p

U p i m m

m i

p m

C U p i m

m i

p m

U p i m m

m +

+

+ 

+ + ≤ ≤

 _{− +}



 ₋



≤_ + + ≤ ≤

− + + 

 ₊

 + + ≤ ≤

 +



Proof: Case 1: m is even and 1≤ ≤i m. Case 1.1: 1

2

m i

≤ ≤ _and h= +k 2.

By the rules of P we have

[ ]

(

)

(

)(

)

(

)

(

)

(

)(

)

(

)

1 2 1 2 2

3 _{4 4 2 3}

2 1 2 2

2 1

3 1 1 3 1 1

2 1 2

3 1 1 3 1

i

h k k

km

h i _{m i} m i km

k

k m i k i m i

k

k m i m i

− + − +

= =

− _{+ − +} − + +

+ =

+ − + + − + − −

+

≤ =

+ − + − +

Case 1.2:

2

m

i m

< ≤ _and h=2k+1.

[ ]

(

)

(

)

(

)

1 2

3 1

2 1 2 2

2

3 1 3 1 2 1

2

2

3 1

i

h k

km h

m i

k m

k m i k i m i

m i −

=

− _{+ − +}

=

 

− + + _ − − _+ − +

 

≤

− +

Case 1.3:

2

m

i m

DOI: 10.4236/iim.2017.96014 250 Intelligent Information Management

[ ]

(

)

(

)

1 2 1

3 1

3 1 2

2

1 2

2

1 3 1

3 1 3 1

2 2 2

2

3 1

i

h k

km h

m i

k

m m i

k m i k i

m i

− +

=

− _{+ − +}

 ₊ 

 

 

=

− −

 ₊  _{− + +}  _{− −} ₊

   

   

≤

− +

Let

(

2

)

3 1

x

m i =

− + , by Lemma 1 when m is even and 1≤ ≤i m we have

(

2 1

)

3 1

n i

i i i

P

C U p

m i

+

≤ + +

− +

Case 2: m is odd. 1 1

2

m

i −

≤ ≤ _and h= +k 2 hold.

[ ]

(

)

(

)

(

)(

) (

)

(

)

(

)

(

)(

) (

)

(

)

(

)

(

)(

) (

)

(

)

1 2 1

3 1 1

2 1 2

2

2 1

3 1 1 1 3 1 2

2 1

3 1 1 1 3 1 1 2

2 1

3 1 1 1

2

3 1 1

i

h k

k m

h

m i

k

k m i k k i m i

k

k m i k k i i

k

k m i k

m i

− ₌ + −

+ −

+ − +

+ =

+ − + + + + − + − −

+ ≤

+ − + + + + − + + −

+ ≤

+ − + + +

≤

− + +

Let

(

2

)

3 1 1

x

m i =

− + + , by Lemma 1, when m is odd and

1 1

2

m

i −

≤ ≤ _we

have

(

2 i 1

)

3 1 1

n

i i i

P

C U p

m i

+

≤ + +

− + +

Case 3: m is odd and 1

2

m

i m

+ ≤ ≤

Case 3.1: h=2k+1

[ ]

1 2

(

3 1

)

3 4 1 2 1 2

2

i

h k

k m

h i m

m i

−

= ≤

+

− _{+ − +} +

Case 3.2: h=2k+2

(

)

1 2 1

3 2 3 1 2

2

i

h k

k m

h i

m i

− +

=

+ −

 

DOI: 10.4236/iim.2017.96014 251 Intelligent Information Management

(

)

1 4

2

1 3 9

3 1 4

2 2 2

4 3 1

k

m

k m i

m  ₊      =  ₊  _{+ + + −}     ≤ +

Let 4

3 1

x m =

+ , when m is odd and

1 2 m i m + _{≤ ≤} we have 1 4 3 1 n i

i i i

P

C U p

m

+

≤ + +

+ Hence the Lemma is proved.

Theorem 3 For any job list L=

{

J J1, 2,,Jn

}

from problem

(

)

max

/ ordinal, /

m i

P r C we have:

(

)

1

, 2

2 m R m P

m m − ≤ +   +  _{ }

Proof: Case 1: If m is even and 1≤ ≤i m holds, by Lemma 2 we have

(

)

(

)

(

)

(

)

(

)

(

)

1 1 1 1 1 2 3 1 2 2

3 1 3 1

3 1 2

2

3 1 2 3 1

n i

i i i

i n i i n i i i P

C p U

m i

P P

p U

m i m i

m i P

p P U

m i m i

+ ≤ + + − + = − + + − + − + − +   =  − + + − + _{ } − +

(

)

(

)

(

)

(

)

( )

( )

1 1 max max

3 5 3 2

3 1 3 1

8 8 6

max , ,

3 1

8 8 6

3 1 8 2 3 n i i n i i OPT OPT P

m i m

p U

m i m i m

P

m i

p U

m i m

m i C L m i m C L m − + ≤ + × + − + − +   − + ≤ _{ } − + _{ } − + ≤ − + − ≤

where the last inequality results from the fact that 8

(

8 6

)

3 1

m x

m x

− +

− + is a decreasing

function of x in interval x∈

[ ]

1,m

Case 2: m is odd and 1 1

2

m

i −

≤ ≤

By Lemma 2 we have

(

)

(

)

(

)

1

1 1

2

3 1 1

2 2

3 1 1 3 1 1

n i

i i i

i n

i i

P

C p U

m i

P P

p U

m i m i

+

≤ + +

− + +

= − + +

DOI: 10.4236/iim.2017.96014 252 Intelligent Information Management

(

)

(

)

(

)

(

)

(

)

(

)

( )

( )

1 1 1 1 max max

3 1 1 2

2

3 1 1 2 3 1 1

3 5 4 2

3 1 1 3( 1) 1

8 8 8

max , ,

3 1 1

8 8 8

3 1 1

8 3 1 n i i i n i i n i i OPT OPT

m i P

p P U

m i m i

P

m i m

p U

m i m i m

P

m i

p U

m i m

m i C L m i m C L m − + +   = _ − _+ + − + + _{ } − + + − + ≤ + × + − + + − + +   − + ≤   − + + _{ } − + ≤ − + + ≤ +

where the last inequality results from the fact that

(

8 8

)

6

3 1 1

m x

m x

− +

− + + is a

de-creasing function of x in interval x∈

[ ]

1,m .

Case 3: m is odd and 1

2

m

i m

+ ≤ ≤

By Lemma 2 we have

(

)

(

)

(

)

(

)

1

1 1

1 2 1

1 4 3 1 4 3 1 3 1 4 4

3 1 4 3 1

3 1

4 4

3 1 4 3 1

n

i i i i

n i i i i n i i i n i i

C p P U

m

p P P U

m

m p

p p p P U

m m

m p

ip P U

m m + ≤ + + + = + − + + +   = _ − + + + _+ + + _{ } + +   ≤ _ − _+ + + _{ } + 

( )

( )

1 1 max max

3 4 1 4

3 1 3 1

10 4 2

max , ,

3 1

10 4 2

3 1 8 3 1 n i i n i i OPT OPT m i

p P U

m m P m i p U m m m i C L m m C L m − + = + + + +   − + ≤ _{ } + _{ } − + ≤ + ≤ +

By the above conclusions, when m is even we have 8 2 _max

( )

3

OPT i

m

C C L

m −

≤

hold for i=1, 2,,m. Hence we get

( )

( )

( )

( )

max max max max 8 2

8 2 1

3 ₂ 3 2 OPT P OPT OPT m C L

C L _m m m

m m

C L C L

m − − − ≤ = = +   +  _{ }

When m is odd then max

( )

8 3 1

OPT i

m

C C L

m ≤

+ hold for i=1, 2,,m. Hence

DOI: 10.4236/iim.2017.96014 253 Intelligent Information Management

( )

( )

( )

( )

max max

max max

8

8 1

3 1 ₂

3 1

2

OPT P

OPT OPT

m

C L

C L _m m m

m m

C L C L

m − +

≤ = = +

+  

+  _{ }

Thus we get

(

)

max

( )

( )

max

1

, sup 2

2

P OPT L

C L m

R m P

m

C L

m −

= ≤ +

  +  _{ }

Hence the theorem is proved.

4. Concluding Remarks

In this paper, we consider the semi-online scheduling problem

(

)

max

/ ordinal, /

m i

P r C in which the job list has non-increasing processing times

and arbitrary release times. An algorithm is investigated and it is shown that its

worse case performance ratio is bounded by 2

(

1

)

2 m

m m  

+ − _ +_{ }

 

  for all

values of m. For this problem, to investigate better algorithms or give lower

bound and upper bound would be worth doing. There are many other schedul-ing problems where ordinal algorithms could be developed. The investigations to find good algorithms for these problems would be also of interest to the scheduling community.

Acknowledgements

This work was partly supported by the Chinese National Natural Science Foun-dation Grant (No.11471110) and the FounFoun-dation Grant of Education Depart-ment of Hunan (No. 16A126).

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