Two-machine flowshop group scheduling problem

Computers & Operations Research 27 (2000) 975}985

Two-machine #owshop group scheduling problem

Dar-Li Yang!,*, Maw-Sheng Chern"

!Department of Industrial Engineering and Management, Nankai College of Technology and Commerce, Nantou 54210,Taiwan, ROC

"Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu 30043,Taiwan, ROC

Received 1 July 1998; received in revised form 1 April 1999

Abstract

This paper considers a two-machine #owshop group scheduling problem. The jobs are classi"ed intogroups and the jobs in the same group must be processed in succession. Each group requires a set up time andremoval time on both machines. A transportation time is required for moving the jobs from machine 1 tomachine 2. The objective is to minimize the maximum completion time (makespan). A polynomial timealgorithm is proposed to solve the problem. This generalizes the algorithms proposed by Baker and others.

Scope and purpose

Recently, an important class of scheduling problem is characterized by a group scheduling constraintwhere the jobs are classi"ed into groups by the same operation requirements or characters. Each grouprequires a setup time and removal time on both machines. That is, each machine needs a time to set up or toremove the tools, jigs and "xtures when the group starts processing or completes processing. A transporta-tion time is required to move the jobs between the machines. The objective is to "nd a sequence of groups andjobs in each group such that the maximum completion time (makespan) is minimized. Baker provided anoptimal algorithm for this problem in the case of two-machine #owshop group scheduling without consider-ing the removal and transportation times. But, in some manufacturing environments, it is required toconsider the group removal time and job transportation time. The main contribution of this paper is todevelop a polynomial time algorithm, which generalizes the algorithms proposed by Baker and others.( 2000 Elsevier Science Ltd. All rights reserved

Keywords: Two-machine #owshop; Group scheduling; Maximum completion time

*Corresponding author. Fax: #886-49-554412.E-mail address: [email protected] (D.-L. Yang)

0305-0548/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 3 0 5 - 0 5 4 8 ( 9 9 ) 0 0 0 7 0 - 2

1. Introduction

Johnson [1] "rst proposed an e$cient algorithm for the two-machine #owshop makespanscheduling problem. Yoshida and Hitomi [2] extended this to the problem with setup times, whichare separated from the processing times. Allahverdi [3] extended the problem further by consider-ing separated setup times from processing times and machines su!er random breakdowns. Sule andHuang [4] and Allahverdi [5] separately extended Yoshida and Hitomi [2] and Allahverdi [3]results to a similar problem with separated removal times. Maggu and Das [6] considered thetwo-machine permutation #owshop scheduling problem with transportation times of jobs betweenmachines. Nabeshima and Maruyama [7] extended Maggu and Das [6] result to a similar problemwith separated setup and removal times, start and stop lags and transportation times. Theimportance of above additional times is described in detail in papers [8,9].

Recently, a number of basic scheduling problems is also extended to the group schedulingproblems [10}14]. Baker [10] and Sekiguchi [11] considered a two-machine #owshop groupscheduling problem where each group required a setup time. Vickson and Alfredsson [12] assumedthat the jobs are identical in the same group and showed that such a two-machine #owshop groupscheduling problem can be solved by Johnson's rule. Cetinkaya and Kayaligil [13] extended this tothe problem with setup times, which are separated from the processing times. Cetinkaya [14] extendedCetinkaya and Kayaligil [13] result to the problem with separated removal times and transfer batches.

In this paper, we consider a two-machine #owshop group scheduling problem. The jobs areclassi"ed into groups and the jobs in the same group must be processed in succession. These jobsare processed on two machines and each group requires a setup time and removal time on bothmachines. For example, the setup time s

ikof group G

iis the time needed to set up the tools, jigs,

"xtures on machine k, and the removal time rik

is the time needed to remove the tools, jigs, and"xtures once the group G

icompletes processing on machine k. We also assume that if the current

machine is idle, the setup can be started before the "rst job in the group arrives. More detaileddescription of the setup and removal times is provided in [2,4,8,10]. A transportation time isrequired for moving the job from machine 1 to machine 2. We assume that each machine canprocess only one job at a time and di!erent operations of the same job cannot be processedsimultaneously. All jobs are available simultaneously at time zero and preemption of jobs is notallowed. The jobs in the same group must be processed in succession and the processing order ofjobs is the same on each machine. That is, only the permutation schedules are considered. Ourobjective is to "nd a sequence of groups and sequence of jobs in each group such that the maximumcompletion time (makespan) on machine 2 is minimized. There are two di!erent de"nitions ofcompletion time, in the literature, while scheduling with separated removal times [8]. If the jobcompletion time does not include its removal time, it is called job-based (j-based) completion time.When it includes removal time, it is called machine-based (m-based) completion time. In this paper,we adopt the machine-based completion time. Our proposed problem extends the problems in [1,2, 4, 6] by introducing the group scheduling and extends the problems in [10}13] by introducingremoval times of group and transportation times of jobs.

The following notations are required:

n "number of groups,ni"number of jobs in group G

i,

976 D.-L. Yang, M.-S. Chern / Computers & Operations Research 27 (2000) 975}985

aij"processing time of the job J

ijin group G

ion machine 1, j"1,2, n

i.

bij"processing time of the job J

ijin group G

ion machine 2, j"1,2, n

i.

tij"transportation time for moving a job J

ijin group G

ifrom machine 1 to machine 2,

j"1,2, ni.

sik"setup time of group G

ion machine k, k"1, 2.

rik"removal time of group G

ion machine k, k"1, 2.

In the next section, we propose a polynomial time algorithm to solve the problem. Thisgeneralizes the algorithms proposed by Johnson [1] and others [2,4,6,10}14].

2. Maggu and Das's algorithm and the proposed algorithm

In order to solve the proposed problem, we have to determine the sequence of jobs in each groupand determine the sequence of groups. First, the optimal job sequence in each group can bedetermined by using Maggu and Das's algorithm [6].

Maggu and Das's algorithmStep 1: Let L"MJ

i1, J

i2,2, J

iniN.

Step 2: Determine the job processing order in the following way:2.1 Decompose set L into the following two sets:

U"MJijDa

ij#t

ij)b

ij#t

ijN and V"MJ

ijDa

ij#t

ij'b

ij#t

jN.

2.2 Arrange the members of set U in nondecreasing order of aij#t

ij, and arrange the

members of set V in nonincreasing order of bij#t

ij.

2.3 An optimal sequences p is the ordered set U followed by the ordered set V.

Without loss of generality, we assume that the sequence of jobs p"(Ji1, J

i2,2, J

ini) is obtained by

Maggu and Das's algorithm and the processing of each job is started as early as possible. Let Fj1

bethe completion time of job J

ijon machine M

1for j"1,2, n

i, and F

j2be its completion time on

machine M2. The completion time of sequences can then be determined by the following recurrence

relations:

Fj1"F

j~1,1#a

ijfor all j"1,2, n

i,

Fj2"maxMF

j1#t

ij, F

j~1,2N#b

ijfor all j"1,2, n

i, (1)

where the ready times on both machines are 0, F01"F

02"0. The following theorem immediately

holds.

Theorem 1 (Maggu and Das [6]). For a given group Gi, let p be the sequence determined by Maggu

and Das's algorithm and p@ be any sequence which is diwerent from p. We have

F1(p))F

1(p@) and F

2(p))F

2(p@),

where Fk(p), F

k(p@) are the completion times of sequences p and p@ on machine k, k"1, 2, respectively.

D.-L. Yang, M.-S. Chern / Computers & Operations Research 27 (2000) 975}985 977

Fig. 1.

By Theorem 1 and recurrence relations (1), it follows that Theorem 1 also holds for any given readytimes, F

01*0 and F

02*0. For each group G

i, we may then obtain an optimal job sequence by

using Maggu and Das's algorithm. Thus, it is su$cient to "nd an optimal group sequence. In thefollowing, we de"ne each group as a composition job, which is similar to the one presented in [11].The composition job is de"ned such that there is no intermediate idle time among these operationson each machine. Let ¹

ibe the minimum time length for processing group G

idetermined by

Maggu and Das's algorithm and Eq. (1) with F01"F

02"0 (Fig. 1). Without increasing the

maximum completion time, in the following discussion, we assume that the right-shifting has beendone such that there is no idle time among the operations on machine 2 (Fig. 1(b)). Thecomposition job corresponding to the optimal job sequence of group G

iis de"ned as follows.

De5nition 1. For a group Gi, the associated composition job is de"ned as a processing vector

(ai, d

i, b

i) where

ai"¹

i!

ni+j/1

bij#s

i1!s

i2,

bi"¹

i!

ni+j/1

aij#r

i2!r

i1, d

i"¹

i#s

i1#r

i2!maxMa

i, 0N!maxMb

i, 0N. (2)

We note that aiand b

imay be negative. There are four cases illustrated in Fig. 2, where Da

iD and

DbiD are the length of time period in which exactly one of the machines is busy and d

iis the length of

time period in which both machines are busy. For a group sequence S, let Cij

be the completiontime of group G

ion machine j. We will regard a composition job as a single job and assume,

without loss of generality, that the processing of each composition job is started as early as possible.Let group G

hbe processed immediately before G

i. Then the completion time of group G

ican be

obtained as follows:

If ai*0 and b

i*0, then C

i1"C

h1#a

i#d

iand C

i2"maxMC

h1#a

i, C

h2N#d

i#b

i. (3)

If ai*0 and b

i)0, then C

i1"C

h1#a

i#d

i!b

iand C

i2"maxMC

h1#a

i, C

h2N#d

i. (4)


Fig. 2.

If ai)0 and b

i*0, then C

i1"C

h1#d

iand C

i2"maxMC

h1, C

h2!a

iN#d

i#b

i. (5)

If ai)0 and b

i)0, then C

i1"C

h1#d

i!b

iand C

i2"maxMC

h1, C

h2!a

iN#d

i. (6)

By (3)}(6), a general expression of Ci1

and Ci2

can be written as

Ci1"C

h1#maxMa

i, 0N#d

i!minM0, b

iN,

Ci2"maxMC

h1#maxMa

i, 0N, C

h2!minM0, a

iNN#d

i#maxMb

i, 0N. (7)

We will show that an optimal group sequence can be characterized by the properties proposed inthe following theorem.


Theorem 2. (i) If ak"minMa

1, a

2,2, a

n, b

1, b

2,2, b

nN, then there is an optimal sequence in which

Gk

is the xrst processed group. (ii) If bk"minMa

1, a

2,2, a

n, b

1, b

2,2, b

nN, then there is an optimal

sequence in which Gk

is the last processed group.

Proof. We now prove (i). Let S"(%, Gh, G

l, G

k, %

2) be a group sequence, where %

1and %

2are

subsequences and %1

or %2

may be empty. We will show that interchanging the order of Gland

Gk

in S does not increase the maximum completion time. Thus, we deduce that (i) holds byrepeatedly interchanging G

kwith the groups before it until it is the "rst processed group. Let

S@"(%1, G

h, G

k, G

l, %

2) and denote C

ijand C@

ijas the completion times of group G

ion machine

j in sequences S and S@, respectively.By (7), we have

Cl1"C

h1#maxMa

l, 0N#d

l!minM0, b

lN, (8)

Cl2"maxMC

h1#maxMa

l, 0N, C

h2!minM0, a

lNN#d

l#maxMb

l, 0N. (9)

Using Eqs. (8) and (9), we obtain

Ck1"C

h1#maxMa

l, 0N#d

l!minM0, b

lN#maxMa

k, 0N#d

k!minM0, b

kN, (10)

Ck2"maxMC

l1#maxMa

k, 0N#d

k, C

l2!minM0, a

kN#d

kN#maxMb

k, 0N

"max GC

h1#maxMa

l, 0N#d

l!minM0, b

lN#maxMa

k, 0N#d

k,

H#maxMbk, 0NmaxG

Ch1#maxMa

l, 0N,

Ch2!minM0, a

lN H#d

l#maxMb

l, 0N!minM0, a

kN#d

k

"maxGC

h1#maxMa

l, 0N!minM0, b

lN#maxMa

k, 0N#maxMb

k, 0N,

Ch1#maxMa

l, 0N#maxMb

l, 0N!minM0, a

kN#maxMb

k, 0N,

Ch2!minM0, a

lN#maxMb

l, 0N!minM0, a

kN#maxMb

k, 0N H#d

l#d

k. (11)

Similarly, we have

C@l1"C@

h1#maxMa

k, 0N#d

k!minM0, b

kN#maxMa

l, 0N#d

l!minM0, b

lN, (12)

C@l2"maxG

C@h1#maxMa

k, 0N!minM0, b

kN#maxMa

l, 0N#maxMb

l, 0N,

C@h1#maxMa

k, 0N#maxMb

k, 0N!minM0, a

lN#maxMb

l, 0N,

C@h2!minM0, a

kN#maxMb

k, 0N!minM0, a

lN#maxMb

l, 0N H#d

k#d

l. (13)

Now by comparing S and S@, we have Ch1"C@

h1and . C

h2"C@

h2. Thus, C@

l1"C

k1holds.


By (11) and (13), in order to have C@l2)C

k2, it is su$cient to have

maxGmaxMa

k, 0N!minM0, b

kN#maxMa

l, 0N#maxMb

l, 0N,

maxMak, 0N#maxMb

k, 0N!minM0, a

lN#maxMb

l, 0NH

)maxGmaxMa

l, 0N!maxM0, b

lN#maxMa

k, 0N#maxMb

k, 0N

maxMal, 0N#maxMb

l, 0N!minM0, a

kN#maxMb

k, 0N H. (14)

Subtracting maxMak, 0N#maxMa

l, 0N#maxMb

k, 0N#maxMb

l, 0N from both sides of (14), we

have an equivalent inequality

maxM!(minM0, bkN#maxMb

k, 0N),!(minM0, a

lN#maxMa

l, 0N)N

)maxM!(minM0, blN#maxMb

l, 0N),!(minM0, a

kN#maxMa

k, 0N)N. (15)

Since minM0, xN#maxMx, 0N"x, (15) is equivalent to the inequality

maxM!bk,!a

lN)maxM!b

l, !a

kN or minMa

k, b

lN)minMa

l, b

kN.

Therefore, if minMak, b

lN)minMa

l, b

kN holds, then C@

l2)C

k2holds. By the assumption that

ak"minMa

1, a

2,2, a

n, b

1, b

2,2, b

nN, we have minMa

k,b

lN)minMa

l, b

kN. Hence C@

l2)C

k2.

The proof of (ii) is similar to that of (i). Therefore, the theorem holds. h

With an adaptation of Theorem 2, an optimal group sequence for the proposed problem isdirectly constructed by an algorithm which is similar to the Johnson's rule. A stepwise descriptionof the algorithm is given as follows:

Algorithm 1.Step 1: (Determining the optimal job sequences and composition jobs) For each group

Gi, i"1, 2,2, n, "nd an optimal job sequence and composition job (a

i, d

i, b

i) by using

Maggu and Das's algorithm and Eq. (2). Set k"1, l"n and )"M1, 2,2, nN and go toStep 2.

Step 2: (Determining the optimal group sequence) Find the minimum value m"min6i|6Ma

i, b

iN.

Ties may be broken arbitrarily. If m"ajfor some j3), then go to Step 2.1. Otherwise, go

to Step 2.2.2.1 Place G

jin the kth position of the processing sequence and set )")!MjN. If )O/,

set k"k#1 and go to Step 2. Otherwise, go to Step 3.2.2 Place G

jin the lth position of the processing sequence and set )")!MjN. If )O/,

set l"l!1 and go to Step 2. Otherwise, go to Step 3.Step 3: (Determining the completion time) Determine the completion time by the recurrence

equations (7) where the processing of each composition job is started as early as possible.


Theorem 2 shows that the "rst cycle of Algorithm 1 positions a group optimally. By induction, thealgorithm also positions the remaining (n!1) groups optimally. We now consider the followingspecial cases:

(a) For each i, if ni"1, s

i1"0, s

i2"0, r

i1"0, r

i2"0, and t

i1"0, then the proposed algorithm

reduces to the algorithm presented by Johnson [1].(b) For each i, if n

i"1, s

i1"0, s

i2"0, r

i1"0, r

i2"0, then the proposed algorithm reduces to the

algorithm presented by Maggu and Das [6].(c) For each i, if n

i"1 and t

i1"0, then the proposed algorithm reduces to the algorithm presented

by Sule and Huang [4].(d) For each i, if s

i1"0, s

i2"0, r

i1"0, r

i2"0, t

ij"0, a

ij"a

iand b

ij"b

i, j"1,2, n

i, then the

proposed algorithm reduces to the algorithm presented by Vickson and Alfredsson [12].(e) For each i, if r

i1"0, r

i2"0, t

ij"0, a

ij"a

iand b

ij"b

i, j"1,2, n

i, then the proposed

algorithm reduces the algorithm presented by Cetinkaya and Kayaligil [13].(f) For each i, if t

ij"0, a

ij"a

i, b

ij"b

i, j"1,2, n

i, and unit sized transfer batch, then the

proposed algorithm reduces the algorithm presented by Cetinkaya [14].(g) For each i, if r

i1"0, r

i2"0 and t

ij"0 for each j, then the proposed algorithm reduces to the

algorithm presented in [2,10,11].

3. A numerical illustration

In the following, the proposed algorithm is implemented to solve a "ve-group problem which isgiven in Table 1.

Step 1: The optimal job sequences of "ve groups determined by Maggu and Das's algorithm are(J

11, J

13, J

12), (J

24, J

23, J

21, J

22), (J

33, J

31, J

32), (J

41, J

42) and (J

52, J

54, J

53, J

51). The

composition jobs (ai, d

i, b

i), i"1,2, 5 are given in Table 2. For example, the optimal

job sequence of group G3

is (J33

, J31

, J32

) and ¹3

determined by recurrence equations (1)is 16. Hence, a

3"¹

3!+3

j/1b3j#s

31!s

32"16!(3#2#5)#5!3"8, b

3"¹

3!

+3j/1

a3j#r

32!r

31"16!(5#4#3)#5!4"5 and d

3"¹

3#s

31#r

32!maxMa

3, 0N

!maxMb3, 0N"16#5#5!8!5"13. The Gantt diagram is given in

Fig. 3.Step 2: By Theorem 2, the group sequence is determined as follows:

Group G2

scheduled: G2

- - - -Group G

1scheduled: G

2- - - G

1Group G

4scheduled: G

2G

4- - G

1Group G

5scheduled: G

2G

4- G

5G

1Group G

3scheduled: G

2G

4G

3G

5G

1Thus, the optimal group sequence (G

2, G

4, G

3, G

5, G

1) is obtained.

Step 3: The maximum completion time determined by the recurrence equations (7) is 110.

The Gantt diagram is given in Fig. 4.


Table 1Data for "ve-group problem

i ni

si1

si2

ri1

ri2

aij

bij

tij

2 3 31 3 3 5 4 2 4 2 2

5 3 2

3 2 42 4 4 7 5 2 5 4 1

2 5 22 3 1

5 3 23 3 5 3 4 5 4 2 2

3 5 1

4 2 4 6 5 4 2 4 15 3 2

3 2 25 6 3

5 4 3 6 7 5 6 4 28 5 4

Table 2List of a

i, b

iand d

i

i 1 2 3 4 5

ai

5 0 8 3 8bi

2 2 5 4 4di

13 21 13 13 24

Fig. 3. The composition job of group G3.


Fig. 4.

4. Conclusions

We investigate a two-machine #owshop group scheduling problem with transportation times ofjobs. A polynomial algorithm is proposed for solving it. This generalizes the algorithm proposed byJohnson and others. In this paper and the previous research [6], it is assumed that onlypermutation schedules are considered in the two-machine #owshop problem with transportationtimes. However, for the general case where non-permutation schedules are also considered, thisproblem becomes NP-hard [15]. In the future research, some performance measure, such as totalcompletion time or maximum lateness, may be considered.

Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments andsuggestions on improving the presentation of this paper. This work was supported in part by theNational Science Council of the Republic of China under grant NSC-88-2213-E252-001.

References

[1] Johnson SM. Optimal two- and three-stage production schedules with setup times included. Naval ResearchLogistics Quarterly 1954;1:61}8.

[2] Yoshida T, Hitomi K. Optimal two-stage production scheduling with setup times separated. AIIE Transactions1979;11:261}3.

[3] Allahverdi A. Two-stage production scheduling with separated setup times and stochastic breakdowns. Journal ofthe Operational Research Society 1995;46:896}904.

[4] Sule DR, Huang KY. Sequency on two and three machines with setup, processing and removal times separated.International Journal of Production Research 1983;21:723}32.

[5] Allahverdi A. Scheduling in stochastic #owshops with independent setup, processing and removal times. Com-puters & Operations Research 1997;24:955}60.

[6] Maggu PL, Das G. On 2]n sequencing problem with transportation times of jobs. Pure and Applied MathematikaSciences 1980;12:1}6.

[7] Nabeshima I, Maruyama S. Two- and three-machine #ow-shop makespan scheduling problems with additionaltimes separated from processing times. Journal of the Operations Research Society of Japan 1984;27:348}66.

[8] Allahverdi A, Gupta JND, Aldowaisan T. A review of scheduling research involving setup considerations. OMEGAThe International Journal of Management Science 1999;27:219}39.

[9] Proust C, Gupta JND, Deschamps V. Flowshop scheduling with set-up, processing and removal times separated.International Journal of Production Research 1991;29:479}93.

[10] Baker KR. Scheduling groups of jobs in the two-machine #ow shop. Mathematical and Computer Modeling1990;13:29}36.


[11] Sekiguchi Y. Optimal schedule in a GT-type #ow-shop under series-parallel precedence constraints. Journal of theOperations Research Society of Japan 1983;26:226}51.

[12] Vickson RG, Alfredsson BE. Two- and three-machine #ow shop scheduling problems with equal sized transferbatches. International Journal of Production Research 1992;30:1551}74.

[13] Cetinkaya FC, Kayaligil MS. Unit sized transfer batch scheduling with setup times. Computers & IndustrialEngineering 1992;22:177}83.

[14] Cetinkaya FC. Lot streaming in a two-stage #ow shop with set-up, processing and removal times separated.Journal of the Operational Research Society 1994;45:1445}55.

[15] Chern M-S, Yang D-L. On the computational complexity of #owshop scheduling problem with transportationtimes of jobs. Working Paper, Department of Industrial Engineering, National Tsing Hua University, 1994.

Dar-Li Yang is an Associate Professor in the Department of Industrial Engineering and Management of NankaiCollege, Taiwan. He is also the Chairman in the Night School of Nankai College. He received his Ph.D. in IndustrialEngineering from National Tsing Hua University, Taiwan. His main research area is deterministic scheduling. Dr. Yang'spublications appeared in Computers & Operations Research, Computers and Industrial Engineering, International Journal ofOperations and Quantitative Management and others.

Maw-Sheng Chern received a M.S. degree in Mathematics from National Tsing Hua Universitiy, a M. Math inCombinatorics & Optimization, and a Ph.D. in Management Sciences from the University of Waterloo. He is currentlythe program director of Industrial Engineering, National Science Council, ROC. He also serves on the editorial board ofIIE Transactions on Scheduling and Logistics and Chiao-Ta Management Review. His research interests includecombinatorial optimization, production scheduling and network programming. His publications appeared in Mathemat-ics of Operations Research, Operations Research Letters, Transportation Science, Naval Research Logistics, European Journalof Operational Research, Computers & Operations Research, Computers and Industrial Engineering, Journal of the Opera-tions Research Society of Japan, Fuzzy Sets and Systems, Information Processing Letters, BIT, IEEE Transactions onRealiability and others.


Documents

Two-machine flowshop group scheduling problem