A modification to the CGPS algorithm for three-machine flow shop scheduling

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A modification to the CGPS algorithm for three-machine flow shop schedulingHui-Chin Tang aa Department of Industrial Engineering and Management Cheng Shiu Institute ofTechnology Kaohsiung Taiwan 83305 Republic of ChinaPublished online 18 Jun 2013

To cite this article Hui-Chin Tang (2001) A modification to the CGPS algorithm for three-machine flow shop schedulingJournal of Information and Optimization Sciences 222 321-331 DOI 10108002522667200110699494

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A modification to the CGPS algorithm for three-machine flow shop scheduling

Hui-Chin Tang

Department of Industrial Engineering and Management

Cheng ShilL Institute of TechnologY

Kaohsiung 83305

Taiwa1~

Republic of China

ABSTRACT

In this paper the three-machine flow shop schedtlling (3FSS) with the objective of minimizing the makespan is considered The algorithm developed by Chen Glass Potts and Strusevich (CGPS) is by far the best one in terms of the worst-case pelshyformance ratio for the 3F88 This paper analyzes a particular condition in a step of the CGP8 algorithm a11lt1 proposes a modification to the CGP8 algorithm by changing this condition It is shown that this modified algorithm is more general one for the 3F88

1 INTRODUCTION

Most of the scheduling problems are the combinatorial optimizashytion problems Among which the flow shop scheduling (FSS) [2 3 7] has been considered by many scholars as a significant role in developing general techniques for the combinatorial optimization problems The FSS is the problem of scheduling n jobs on In sequential machines Each job has to be processed on each machine At any time each machine processes at most one job and each job is processed on at most one machine Preemption is not allowed All jobs are available for processing at time zero The uninterrupted processing time of each job on each machine is given The ohjcctive is to minimize the compleshytion time of the last job the so-called makespan

An interesting special case is when the number of machines is three referred to as A Band C In that case there exists an optimal schedule with the same processing order on machines A and B and the same processing order on machines Band C [2] Then we only minishymize over all permutation schedules for which all machines process

Journal of Information amp Optimizatlon Sciences Vol 22 (200l) No2 pp 321-331 copy Analytic Publishing Co 0252-266701 $200+25

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322 H CTANG

the jobs according to the same job sequence for the three-machine FSS (3FSS)

It is recognized that the 3FSS is a NP-hard problem in the strong sense [4] Therefore several scholars have taken interest in proposing heuristic algorithms to get near-optimal solutions in a short time Two methods are distinguished constructive methods and improvement methods For more details see Lai [6] Smutnicki [9] and the references cited there As the worst-case performance ratio is concerned the algorithm developed by Chen Glass Potts and Strusevich (referred to as CGPS algorithm) [1] has a better worst-case performance ratio of 53 for the 3FSS This is very inspiring If the CGPS algorithm can be implemented more generally then its applicability will be even wider The primary purpose of this paper is to modify the CGPS algorithm by changing a particular condition that a particular step of the CGPS algorithm serves for branching subsequent actions In the following sections we first recall the CGPS algorithm for the 3FSS Next a modification to the CGPS algorithm is presented Finally some conshycluding remarks are given

2 THE CGPS ALGORITHM

In this section we will provide a concise review of the CGPS algoshyrithm For notational convenience we adopt the same notations as Chen et aL [1] Let ai bi and Ci be the uninterrupted processing times of job i EN = I 2 n on machines A Band C respectively Because we consider the problem of sequencing n jobs in a 3FSS with the obmiddot jective of minimizing the makespan this implies that the objective of the 3FSS is to minimize the function

r II v I 1

Cmax(n) max tL ail + L bi + L ciIl s Ii s Y s It ~ (1) IJ h= h=v J

over all permutations 1t (iI i2 ill) of jobs Let 1t be an optimal permutation that minimizes the function in (I) Also let indices ~t and Y be critical jobs in 1t which is the maximum attained in (1)

The CGPS algorithm is an O(nlogn) time heuristic that is based on Johnsons algorithm [5] There are two main stages for the CGPS algorithm Firstly the initial schedule 1t1 was obtained by using the machine aggregation method of Rock and Schmidt [8] Secondly apshyplying a transformation they attempted to improve 1t1 with respect to the worst-case performance ratio The CGPS algorithm is shown to generate a schedule with makespan at most 53 times that of an optimal schedule

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In concern with the construction of initial permutation 1t1 Chen et a1 [1] applied Johnsons algorithm [5] to the artificial two-machine FSS created by the machine aggregation method of Rock and Schmidt [8] The steps of this initial permutation algorithm are as follows

Algorithm 1

Step 1 Obtain an schedule 1tl via Johnsons algorithm [5] for the artificial two-machine FSS with processing times (li = aj + bi and l3i =b + Ci for i E N Rearrange the jobs so that 1tl =(1 2 n)

Step 2 Evaluate Cmax(ltl) with respect to the original 3FSS D

In the following we present some notations and results that are useful in the CGPS algorithm

Let the critical jobs in 1tl be ~l and v where 1 S Il S v S n and

Nl =1 2 Jl-lN2 = )t+ 1 v I Ns=v+ 1 n (2)

Also let a(N)= 2 aj for N ~ N and beN) c(N) be defined simishyiEN

larly Then the makespan of 1t 1 is rewritten as

a(Nl) + aft + bp + CII + c(Ns) if ~l = v Cmax(1tl) = (3)

a(N1) + a + bl1 -I- beNz) + bv + Cv + c(N3) if Il v

Firstly Chen et a1 [1] showed some lower bounds on the optimal n1akespan for the 3FSS From the machine workloads the trivial lower bound of the optimal makespan is

Cmax(r) maxa(N) b(N) c(N) (4)

Moreover Chen et a1 [1] used the idea of Johnsons algorithm [5] to obtain a sharper lower bound on the optimal makespan The result is stated formally below

v

LEMMA 1 Cma-lt1tmiddot) 2 ak + b + L Clp for v = 1 2 n (5) k=1 k=v

PROOF See p 893 in Chen et a1 [1] C

Next Chen et a1 [1] established conditions under which the schedule 1tl generated by algorithm 1 has a worst-case performance ratio of 53

LEMMA 2 If 11 and v are critical jobs for the schedule 1tl generated by algorithm 1 Then

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324 HCTANG

(a) Ifl = v then 1t1 is an optimal permutation (b) Ifl lt v and

beNz) + minb bJ $ 23 beN) (6)

then we have

PROOF See p 893 in Chen et a1 [1] 0

If fl t= V and (6) is violated Chen et a1 [1] attempted to construct a permutation 1tz from 1tl So that at least one of these permutations has a worst-case performance ratio of 53 The construction of 1tz inshyvolves partitioning the jobs of N z and then rescheduling the jobs The verification of the partitioning of Nz is described in the following

LEMMA 3 If (6) is violated then there are jobs w E N z and w E Nz such that

w-l w

and v v

PROOF See p 893 in Chen et a1 [1]

From Lemma 3 the existence of job w w or w w is justified So that Chen et a1 [1] can partition the jobs of Nz into

N~ hl + 1 w - I w and N =w + 1 v - I (10)

and then improve 1tl by removing jobs of N~ and N frommiddot the middle to the beginning and end of the schedule respectively The resulting permutation is 1tz =(N~ N I ~l W v Na N~) To summarize the steps of the COPS algorithm are as follows

CGPS Algorithm

Step 1 Apply Algorithm 1 to obtain 1tl and Cmax(1tl) Rearrange the jobs so that 1tt =(12 n) Let fl and v be critical jobs in 1tl

where 1 $ l $ v $ n and N l Nz and lV be defined as (2)

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THE CGPS ALGORITHM 325

Step 2 If 11 v or (6) holds then rru = nl and Cmax(rrH) =Cmairrl) Otherwise go to step 3

Step 3 If

(11)

then there is a job w E N z such that (8) holds and sets w = w Othshyerwise there is a job w E N2 such that (9) holds and sets w = wl

Step 4 Let N and N be defined as (10) Form a new permutashytion n2 (N N 11 W v N3 N)

Step 5 Choose 1tH E 1t J n2 such that Cmax(1tH) minCmaxCnl) C (1t2) max

The following additional lower bounds are useful in the subshysequent analysis for the case that ~l lt v and (6) is violated

LEMMA 4 If 11 lt v and (6) is violated then

1(3 Cmairc) ~ b(N~) + bv + b(N3) (13)

113 Cmax(re) ~ beN) + b(N3) + maxbft bJ (14)


LEMMA 5 If 11 lt v (6) is violated and Cmax(rel)Cmax(rr) ~ 53 then

13 Cmai1t) ~ b(Nl) + a(N~) + aw + bl + CO + c(N) + b(Ng) (15)

PROOF From (3) and (4) it follows that

Cmax(n) - 113 Crnain)O a(Nl) + aft + Cv + c(N3) - b(Nl) b(N3)

13 Cma-ln) ~ CmaA1t) - [a(N j ) + a fl + Cv + c(N3) b(Nl) - b(N3)]middot

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326 rL c TANG

Using (5) with v = w it further follows that

lC I

13 Cmaln) I ail + b + I Ch - [a(Nl) + all + Cv + C(N3)] k=1 h=

Inequality (15) is now derived which completes the proof of the lemma 0

In Lemma 5 we recognized that the inequality

in Chen et aL [1] is replaced by the less restrictive condition (15) We also show that condition (11) is not necessary for inequality (15) This inequality (15) is useful in our algorithm

LEMMA 6 If 11 lt v (6) is violated (11) holds and Cmai1CI) I Cmai1C) 53 then

(16)

PROOF See p 894 in Chen et aL [1]

In the following lemma Chen et al [1] derived the worst-case performance ratio of the CGPS algorithm for the case that 11 lt v and (6) is violated

LEMMA 7 If ~ lt v (6) is violated (11) holds and Cmax(nj) I Cmax(n) 53 then the permutation TC2 generated by the COPS algoshyrithm satisfies

PROOF See p 895 in Chen et al [1] 0

From Lemma 2 and Lemma 7 there are three cases for the worst-case performance of the CGPS algorithm Firstly for the case that ~ = v the worst-case performance ratio is one from Lemma 2 Secondly for the case that ~l lt v and (6) holds the worst-case performshyance ratio is 53 from Lemma 2 Thirdly for the case that ~ lt v and (6) is violated the worst-case performance ratio is 53 from Lemma 7

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Therefore in each of these cases we have that the worst-case pershyformance of the CGPS algorithm is 53

3 THE REVISED CGPS ALGORITHM

Before presenting the revised CGPS algorithm we analyze the necessity of condition (11) and revise condition (11) to make the CGPS algorithm more general In the CGPS algorithm it is well known that condition (11) serves for branching subsequent actions In concern with the necessity of condition (11) we first present some additional notations and results that are used in our analysis

Assume that rt2 =(N~ N l l W v Ns N~) =01 i 2 i) and thatj andjs are critical jobs in rt2 where 1 S IS S S n Then we have

I S lL

Cmax(rt) = middot + bJ + CJ bull (18) aJ middotL-h~hL-h 11=1 11=1 It=s

Also the inequalities

G w-l

L ali + L bk S bll + b(N~) for 11 + 1 S ~ S w - I (19) k=I(+1 II=

and F It

L bk + L cil S Cv + c(N) for v + 1 S ~ S n (20) k=v+l k=

are valid since Jl and v are critical jobs in rt1 From the proof of Lemma 7 in Chen et aI [11 it is recognized that condition (11) is necesshysary for the case that j r E N~ and is E w v U N3 U N~ In fact condishytion (11) is necessary only for the case that jl E N~ and is E v U N3 In other words it is not necessary for the case that ir E N~ and is E w U N~ This result is stated formally below

LEMMA 8 For jr E N~ and js E w U N~ if ~l lt v (6) is violated and Cma(rtl)Cmax(rt) 53 then the permutation rt2 generated by the COPS algorithm satisfies (17)

PROOF For il E N~ we can rewrite (18) as

j S II

Cmai 1t2) L ak + L bj + L Cjll (21) 1I=ll+1 h= h=s

Two cases are distinguished Firstly in the case of is w using (5) with v =w and (12) we have

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328 H C TANG

i 11-1

= ak + L bk + b(Nl) + b + b + cII + c + c(Ns) + c(N~) k=l+1 k=j

i w-l

=L Ok + bw + C + Cv + c(Na) + c(N~) + b(Nl ) + b + I bl I=~t+l

(22)

Secondly in the case ofjs E N~ applying (19) with ~ j and (4) to (21) yields

From (14) and (15) it further follows that

(23)

The desired inequality (17) is now derived from (22) and (23)

However for the case that jr E N2 and js E v V Ns the lack of same result as Lemma 8 is due to a lack of condition (11) Then conshydition (11) is necessary for this case

Now we state our revised CGPS algorithm We first replace conshydition (11) in the CGPS algorithm by

(11)

Then the resulting algorithm is called the revised CGPS algorithm Because condition (11) is necessary for the case that j E N~ and

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js E v U N3 we will discuss the effectiveness and generality of (11) for this case

LEt1MA 9 For the case that jr E N~ and js E v U N3 if Jl lt v (6) is violated Cmai1tl)Cmai1t) 2 53 and (11) holds then the pellnutashytion 1tz generated by the revised COPS algorithm satisfies (17)

PROOF We prove the lemma by considering two cases Firstly in the case of jl E N~ and js =v after usmg (15) and (12) to obtain

i III-I

L ak + L bk + b(NI ) + bll + bw + bv + Cv + C(N3) + c(N~) k=~I+1 Il=j

s 13 Cmai7t) - b(N3) + bll + b(N~) + bv + Cv + c(N3)

s 13 Cmax(7t)-b(N3) +213 Cmail1) b(N1) + bv +cv +c(N3)

From (11) it further follows that

Secondly in the case of j E N~ and js E N3 applying (20) with ~ we have

i w-l j II

L ak + L bk + b(N1) + bll + bl( + bv + L bk + L cli + c(N~) k=v+I k=j

1 w-I

s L ak + L bk + b(N1) + bf + bll + bv + Cv + c(Ns) + C(N~) k=~+1

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330 H CTANG

The right side is identical to that of (18) withj E N~ andjs = v Thereshyfore inequality (17) is valid in each of these cases This completes the proof of the lemma

The verification of (11) for the case that j E N~ and js E v uNa is justified by Lemma 9 Note that such a replacement can not affect the worst-case performance ratio In the following we will show condition (11) is less restrictive than (11)

LEMMA 10 For the holds then (U) holds

case that j E N~ and js E v U N3 if (11)

PROOF Because j E N~ is critical job in fez we have

w-I w-l

(24)

Moreover from (12) and (14) we obtain

b(N~) s 13 beN) (25)

We now apply (11) (24) (14) and (25) to establish

by + Cv+ c(Na)

s by + a(Nl ) + a

w-l

S by + L bk + b(Nl )

s by + b(N~) + b(Nl)

s 113 beN) + 13 beN)

s 23 beN) + b(Nl) + b(N3)

This completes the proof of the lemma 0

From the result of Lemma 10 we can replace condition (11) by condition (11) that is less restrictive Therefore the revised CGPS alshygorithm is more general than the CGPS algorithm

4 CONCLUSION

We consider the problem of sequencing n jobs in a 3FSS with minimum makespan criterion Up to now the algorithm proposed by

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THE CGPS ALGORITHM a31

Chen et a1 [1] IS the best heuristic algorithm in terms of the worstshycase performance ratio for the 3FSS Two conclusions can be drawn from this paper Firstly we establish the case that are necessary for condition (11) in the CGPS algorithm In Lflmma 9 we can recognize that condition (11) is necessary only for the case that j E N~ and js E v U N3 Secondly we show the effectiveness and generality of condition (11) In other words we can replace condition (11) by conshydition (11) that is less restrictive Theoretically if condition (11) holds then condition (11) holds for the case that j E N~ and j E v U N3 Therefore the modification to the CGPS algorithm is more general one for the 3FSS

REFERENCES

1 n Cllen C A Glass C N Potts and V A Strusevich A new heuristic for three-machine flow shop scheduling Operatiolls Research Vol 44 (1996) pp 891-898

2 R W Conway W L Maxwell and L W Miller Theory of Schednlillg Addison-Wesley Reading ]VlA 1967

3 R A Dudek S S Panwalkal and M L Smith The lessons of flowshop scheduling research Operations Research Vol 40 (1992) pp 7-13

4 M R Garey D S Johnson and R Sethi The complexity of flowshop and jobshop scheduling Mathematics Operations Research Vol 1 (1976) pp 117-129

5 S M Johnson Optimal two- and three-stage production schedules with setup times included Naval Research Logistic Quarterly Vol 1 (1954) pp 61middot68

6 T C Lai A Note on heuristic of flow-shop scheduling Operations Resealch Vol 44 (1996) pp 648-652

7 E L Lawler J K Lenstra A H G Rinnooy Kan and D B Shmoys Sequencing and Scheduling Algorithms and Complexity in S C Graves A H G Rinnooy Kan and P H Zipkin (cds) Handbooks in Operations Research and Management Science Vol 4 Logistics 0 PlOdllctioll and [llventmy North Holland Amsterdam pp 445middot522

8 H Rock and G Schmidt Machine agglegation heuristics in shop scheduling Methods 0 Operatiolts Research Vol 45 (1983) pp 303middot314

9 C Smlltnicki Some results of the worst-case analysis for flow shop scheduling European Journal 0 Operational Research Vol 109 (1998) pp 66-87

Received August 2000

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A modification to the CGPS algorithm for three-machine flow shop scheduling

Hui-Chin Tang

Department of Industrial Engineering and Management

Cheng ShilL Institute of TechnologY

Kaohsiung 83305

Taiwa1~

Republic of China

ABSTRACT

In this paper the three-machine flow shop schedtlling (3FSS) with the objective of minimizing the makespan is considered The algorithm developed by Chen Glass Potts and Strusevich (CGPS) is by far the best one in terms of the worst-case pelshyformance ratio for the 3F88 This paper analyzes a particular condition in a step of the CGP8 algorithm a11lt1 proposes a modification to the CGP8 algorithm by changing this condition It is shown that this modified algorithm is more general one for the 3F88

1 INTRODUCTION

Most of the scheduling problems are the combinatorial optimizashytion problems Among which the flow shop scheduling (FSS) [2 3 7] has been considered by many scholars as a significant role in developing general techniques for the combinatorial optimization problems The FSS is the problem of scheduling n jobs on In sequential machines Each job has to be processed on each machine At any time each machine processes at most one job and each job is processed on at most one machine Preemption is not allowed All jobs are available for processing at time zero The uninterrupted processing time of each job on each machine is given The ohjcctive is to minimize the compleshytion time of the last job the so-called makespan

An interesting special case is when the number of machines is three referred to as A Band C In that case there exists an optimal schedule with the same processing order on machines A and B and the same processing order on machines Band C [2] Then we only minishymize over all permutation schedules for which all machines process

Journal of Information amp Optimizatlon Sciences Vol 22 (200l) No2 pp 321-331 copy Analytic Publishing Co 0252-266701 $200+25

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322 H CTANG





r II v I 1




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Algorithm 1





Nl =1 2 Jl-lN2 = )t+ 1 v I Ns=v+ 1 n (2)








v





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324 HCTANG



then we have




w-l w

and v v



N~ hl + 1 w - I w and N =w + 1 v - I (10)


CGPS Algorithm



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Step 3 If

(11)






1(3 Cmairc) ~ b(N~) + bv + b(N3) (13)








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326 rL c TANG


lC I






(16)






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I S lL



G w-l


and F It





j S II



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328 H C TANG

i 11-1


i w-l


(22)



(23)




(11)


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i III-I






i w-l j II


1 w-I


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330 H CTANG






w-I w-l

(24)


b(N~) s 13 beN) (25)


by + Cv+ c(Na)

s by + a(Nl ) + a

w-l



s 113 beN) + 13 beN)

s 23 beN) + b(Nl) + b(N3)



4 CONCLUSION


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REFERENCES











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322 H CTANG





r II v I 1




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Algorithm 1





Nl =1 2 Jl-lN2 = )t+ 1 v I Ns=v+ 1 n (2)








v





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324 HCTANG



then we have




w-l w

and v v



N~ hl + 1 w - I w and N =w + 1 v - I (10)


CGPS Algorithm



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Step 3 If

(11)






1(3 Cmairc) ~ b(N~) + bv + b(N3) (13)








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326 rL c TANG


lC I






(16)






Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014






I S lL



G w-l


and F It





j S II



Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

328 H C TANG

i 11-1


i w-l


(22)



(23)




(11)


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014





i III-I






i w-l j II


1 w-I


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

330 H CTANG






w-I w-l

(24)


b(N~) s 13 beN) (25)


by + Cv+ c(Na)

s by + a(Nl ) + a

w-l



s 113 beN) + 13 beN)

s 23 beN) + b(Nl) + b(N3)



4 CONCLUSION


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



REFERENCES











Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



Algorithm 1





Nl =1 2 Jl-lN2 = )t+ 1 v I Ns=v+ 1 n (2)








v





Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

324 HCTANG



then we have




w-l w

and v v



N~ hl + 1 w - I w and N =w + 1 v - I (10)


CGPS Algorithm



Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



Step 3 If

(11)






1(3 Cmairc) ~ b(N~) + bv + b(N3) (13)








Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

326 rL c TANG


lC I






(16)






Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014






I S lL



G w-l


and F It





j S II



Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

328 H C TANG

i 11-1


i w-l


(22)



(23)




(11)


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014





i III-I






i w-l j II


1 w-I


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

330 H CTANG






w-I w-l

(24)


b(N~) s 13 beN) (25)


by + Cv+ c(Na)

s by + a(Nl ) + a

w-l



s 113 beN) + 13 beN)

s 23 beN) + b(Nl) + b(N3)



4 CONCLUSION


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



REFERENCES











Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

324 HCTANG



then we have




w-l w

and v v



N~ hl + 1 w - I w and N =w + 1 v - I (10)


CGPS Algorithm



Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



Step 3 If

(11)






1(3 Cmairc) ~ b(N~) + bv + b(N3) (13)








Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

326 rL c TANG


lC I






(16)






Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014






I S lL



G w-l


and F It





j S II



Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

328 H C TANG

i 11-1


i w-l


(22)



(23)




(11)


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014





i III-I






i w-l j II


1 w-I


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

330 H CTANG






w-I w-l

(24)


b(N~) s 13 beN) (25)


by + Cv+ c(Na)

s by + a(Nl ) + a

w-l



s 113 beN) + 13 beN)

s 23 beN) + b(Nl) + b(N3)



4 CONCLUSION


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



REFERENCES











Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



Step 3 If

(11)






1(3 Cmairc) ~ b(N~) + bv + b(N3) (13)








Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

326 rL c TANG


lC I






(16)






Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014






I S lL



G w-l


and F It





j S II



Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

328 H C TANG

i 11-1


i w-l


(22)



(23)




(11)


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014





i III-I






i w-l j II


1 w-I


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

330 H CTANG






w-I w-l

(24)


b(N~) s 13 beN) (25)


by + Cv+ c(Na)

s by + a(Nl ) + a

w-l



s 113 beN) + 13 beN)

s 23 beN) + b(Nl) + b(N3)



4 CONCLUSION


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



REFERENCES











Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

326 rL c TANG


lC I






(16)






Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014






I S lL



G w-l


and F It





j S II



Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

328 H C TANG

i 11-1


i w-l


(22)



(23)




(11)


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014





i III-I






i w-l j II


1 w-I


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

330 H CTANG






w-I w-l

(24)


b(N~) s 13 beN) (25)


by + Cv+ c(Na)

s by + a(Nl ) + a

w-l



s 113 beN) + 13 beN)

s 23 beN) + b(Nl) + b(N3)



4 CONCLUSION


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



REFERENCES











Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014






I S lL



G w-l


and F It





j S II



Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

328 H C TANG

i 11-1


i w-l


(22)



(23)




(11)


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014





i III-I






i w-l j II


1 w-I


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

330 H CTANG






w-I w-l

(24)


b(N~) s 13 beN) (25)


by + Cv+ c(Na)

s by + a(Nl ) + a

w-l



s 113 beN) + 13 beN)

s 23 beN) + b(Nl) + b(N3)



4 CONCLUSION


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



REFERENCES











Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

328 H C TANG

i 11-1


i w-l


(22)



(23)




(11)


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014





i III-I






i w-l j II


1 w-I


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

330 H CTANG






w-I w-l

(24)


b(N~) s 13 beN) (25)


by + Cv+ c(Na)

s by + a(Nl ) + a

w-l



s 113 beN) + 13 beN)

s 23 beN) + b(Nl) + b(N3)



4 CONCLUSION


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



REFERENCES











Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014





i III-I






i w-l j II


1 w-I


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

330 H CTANG






w-I w-l

(24)


b(N~) s 13 beN) (25)


by + Cv+ c(Na)

s by + a(Nl ) + a

w-l



s 113 beN) + 13 beN)

s 23 beN) + b(Nl) + b(N3)



4 CONCLUSION


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



REFERENCES











Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

330 H CTANG






w-I w-l

(24)


b(N~) s 13 beN) (25)


by + Cv+ c(Na)

s by + a(Nl ) + a

w-l



s 113 beN) + 13 beN)

s 23 beN) + b(Nl) + b(N3)



4 CONCLUSION


Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



REFERENCES











Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014



REFERENCES











Dow

nloa

ded

by [

Tex

as A

ampM

Uni

vers

ity L

ibra

ries

] at

14

38 0

8 O

ctob

er 2

014

Documents

A modification to the CGPS algorithm for three-machine flow shop scheduling