Max-Plus Generalization of a Flowshop Problem. Lower and Upper Bounds

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MAX-PLUS GENERALIZATION OF A FLOWSHOP PROBLEM. LOWER AND UPPER BOUNDS. Christophe LENTE· Jean-Charles BILLAUT· Jean-Louis BOUQUARD·

• Laboratoire d'Informatique Ecole d'Ingenieurs en Informatique pour l'Industrie 64 avenue Jean Portalis F-37200 Tours , France Tel: 0033 247 361 427 Fax: 00 33 247361 422 email: [email protected]

Abstract: A three machine permutation flowshop problem is studied using MaxPlus algebra. There exist constraints like removal and setup times no sequence dependant, time lags or predifined groups of jobs. The makespan is the criterion to minimize, its expression in Max-Plus algebra allows to generalize some results known for the unsconstrained flowshop problem and to derive upper and lower bounds. Computational experiments are proposed in order to compare these approximate methods. Copyright © 2001 IFAC Keywords: Scheduling algorithms, Algebraic approaches, Sequences.

sponding Max-Plus model based on matrices and the fundamental case of two machines. In section 3, one proposes, mainly for the three-machine case, some lower bounds that can be easily derived from the model, and a dominance condition. In section 4, five heuristic methods are developped, based on similar principles, and computational results of section 5 prove the efficiency of the algorithms. Some of these bounds correspond to already existing lower bounds but generalize them.

1. INTRODUCTION

A flowshop consists of a set of different. machines that perform tasks (or operations) of jobs. The machines are numbered from AIl to Mm. A job is composed of an ordered list of tasks where the ith task has to be executed on machine M i . Each job can be performed only on one machine at a time. It is assumed that tasks cannot be interrupted (non-preemption) and each machine can handle only one job at a time. The study is limited to the particular case of permutation flowshops, in which each machine processes the jobs in the same order. The problem is to find the job sequences on the machines which minimize the makespan, i.e. the maximum of the completion times of all tasks. It is well known that the problem is NP-hard in the strong sense for In ~ 3 (Garey et al. 1976) .

2. ST{;DIED PROBLEM A three-machine permutation flows hop problem is considered. One notes Px .k the processing time of the kth task of job :r (on machine Ah). Each operation needs setup S.r.k and removal times Tx.k that are non sequence dependant: they depend on machine Ah and job x but not on jobs executed before or after the job x. During these times, the

One considers in this paper some additional constraints that complicate the basic problem. In section 2 one presents these constraints, the corre-

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Proposition 1.

concerned machine is considered as busy but the presence of the job is not needed, so the ith task of a job can be performed while the (i + 1)th machine is beeing prepared for the next task of the same job. It is also assumed that there exists time lags a Xk between the completion time of ajob on machine Ah and its starting time on A/HI - for instance to model transportation delay or possible parallelisation. The purpose is to minimize the completion time of last removal time. Such a problem is denoted by F3/prmn, aij, Tnsd, Snsd/Cmax.

where Ck.t (a) is the completion time of a on machine M( in the flows hop limited to machines from Ah to M l , and Ct(a) stands for Cl.l(a) (Lente and Billaut 1999) .

Proposition 2. The makespan of a sequence a is given by

The model still holds if there are predefined groups of jobs and is adaptable to constraints of precedence.

C max (a)

~ T(a) (1I]I 1I)

3

3

= ~~Ck.t(a)

2.2 Generalization Let M = (M(k») (l
(())

(k)

(()

(k)

(k)

m 1 ,3 m 2 ,3 m3 One defines for each permutation a of {I, ... , n}

~~~ is also

1

the matrix A1(a)

k=1

= Q9M(O"(k» k=n

see

(()

mi~J m~k)

= (

n

= EB [AL,k EEl [Bh ,j'

For more informations, 1992, Gunawardena 1998).

1I) (

\:Ial , a2 subsequences: T(ala2) = T(a2)T(al)

One notes additively EB the maximum and mutiplicatively 0 the addition. The first law, EEl , is idem potent, commutative, associative and has a neutral element (-00) noted ({]). The second law ,0 , is commutative, associative, distributive on d'l , has a neutral element (0) noted II and (() is an absorbing element for 0 . One can summarize that by saying that IRmax = (JR U -00, EEl , 0 ) is a dioid. It is possible to extend these two laws to n x n matrices of elements of IRmax, in that case d'l and 0 are defined by [A EEl Bkj = [Al;,j EEl [Bkj

a dioid.

=

Proposition 3. (l'vlatrix of a con catenation of two sequences)

2.1 Max-Plus modeling

and [A 0 BL,j

Cl(a) (() (()) C.1,2(a) C 2(a) (() ( C1.3(a) C 2 •3 (a) C 3 (a)

T(a) =

(Gaubert

minimi,. c
(D

and one wants to

M(a) (TI TI

!)

For more convenience, in the following, all matrices will be denoted by capital letters and their componants by small letters with column

The calculus of the makespan can be modelled by a product of m x m matrices in Max-Plus algebra (Lente and Billaut 1999),(Lente et al. 1999). The case of three machines is presented below but the model is easy to generalize. One associates to each job x a triangular MaxPlus matrix noted Tx or T(x) and defined by

an(~, :a~ ingc)e~. For instance, A is the matri.x a1.3 a2 ,3 a3

2.3 The special case of the two machines fiowshop problems About the basic problem, Johnson has proved that when two jobs a and b are adjacents, it does not increase the makespan to schedule a before b when min(Pa.l, Ph.2) ::; min(pa ,2, Ph,1 ) (Johnson 1954). Unfortunately, this rule is not transitive. In fact Johnson's order corresponds to the rule:

Then one associates to each sequence the triangular matrix T(a): 1

T(a)

= Q9T(a(i)) d%!

1

a precedes b iff

Q9Ti (a) i=IO"I

where a(i) is the job in ith position in sequence

OR { OR

a.

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Pa.l ::; Pa.2 & Ph.l ::; Ph.2 & Pa.l ::; Ph.l Pa.l ::; Pa.2 AND Ph.l > Pb.2 Pa.l > Pa.2 & Ph.l > Ph.2 & Pa .2 ~ Ph,2

flowshop problems, componant m13 is dominant and f(a) is always equal to ml3(a), so one does not really want to minimize m12(a). In that case, a more accurate calculus can be made. In the following 1\ stands for min. Let

This problem is generalized b(Y ~A~rodUC)t of 2 x 2 . • (k ml 0 tnangular matnces AI ) = (k) (k) m I ,2 m 2

Proposition 4. Given two 2 x 2 triangular matrices A and B: B 0 A < A <& B {::} aI ~ ~ C < -bI e -a2 {::} min(aI2 - a2 , bI2 b12 - bI2 aI2 aI2 lr2) :S min(al2 - aI , bl 2 - b2) (division in MaxPlus algebra corresponds to substraction in usual algebra, if the denominator is not 0) . The last part of the equivalence has to be understood in the usual algebra.

Q(k)

=

AI(k) IAII(k)

m1~) ( = m~A' )

:

(k)

1

M(a)=

® M(u(k»

1

2

=

1

®

M(u(n» <&

MiU(k»

A'=71-1

> Q(O'(Il»

1I1(k)

m 23

)

oc m I mu m2 be the right residuation of M(k) by AI: k ), that is the greatest matrix X such as X ® Mi k ) :S AI(k) (Blyth and Janowitz 1972). One has the following minorations:

1

1I[(O'(n » 0

-

3. LOWER BOUNDS

EB

(k )

-- --1\-(k ) (k) (k)

k=1l

M?)

I

(k)

m I3 m I 3

\

So one can generalize Johnson 's order on the matrices and optimally minimize the product, and not only r.

Let 's decompose each matrix

:\

1

fV\

~

A1(0'(k» 1

k=11-1

fl,f(k )

in a sum

A(1~:k~ 1I~'(A~)Where tk )

(k ) ""

m 1 •2 m 2 I[}

and M,(k)

=

I[}

u

,

1I1(k)

3

=

I[}

m ( k) I[} I[} ) I[} I[} (k) (k) ( m1.3 I[} m3

0

It is obvious that M(a)

k=n

2

k=n

k=n

d!l Ml(a) EB M2(a) EB M'(a). These products of matrices are equivalent to products of second order square matrices, so they can be minimized in O(n.ln(n)) using adequat Johnson's order (see 2.3).

3.3 J23 algorithm

Similarly, there exists a sequence J23 that minimizes the product of matrices fl,f3 . One obtains three minorations: (m3)23(J23)

:S

(m)23(a) ,

(k)

(k)

A m l3 m I2 1\ ((k) 1\ (k)) 0 (m 3)23(J23) I ::'O k ::'0 11 m 23 m2

3.1 J ' algorithm

:S

m13(a)

(k)

Such a technique applied to the product of matrices M' gives a schedule J' and coefficient m~.3(J') is a minorant of r. This result is closed to those presented in (Szwarc 1983, Sule and Huang 1983), but it can be extended with no more calculus to the problem of scheduling groups of jobs or subsequences.

and

A m 13 1\ ((k)) ® (m3)(J23)

:S

mI3(a)

l ::'O k ::'O ll m3

(m3 does not depend on the schedule).

It is eas~' to adapt the previous methods to the m-machine flowshop problem, and so to generalize results presented in (Lageweg et al. 1978).

3.2 J12 algorithm

3.4 Adjacent pairwise interchange

Let J12 be the Johnson 's order relatiw to k matrices ). Coefficient (md I2 (J12) is a minoration of all coefficients mI2(a) , so it is a lower bound of criterion r. But in a lot of

M:

Let A and B be two triangular matrices and a and b the corresponding jobs. It is easy to check

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that BA = BIAI EB B3A3 EB B' A' EB B3AI and in B3AI' the only coefficient that is not dominated is the bottom left one which is equal to aI2~3 . So if a precedes b in the three Johnson 's order JI2,

J23 and J' and if a12 :::; a23

~~2

4 .3 Heuristic H2

A sufficient condition for inequality (1) is bl1 al3 ti' bl2 a 23 bl3 a 33 > IT W h·IC h· . - -
then BA :::; AB and

"'23

it is better to schedule a before b when they are adjacents. Of course, there exists a better condition for the basic three-machine flowshop problem (Burns and Rooker 1976) but it doesn 't hold if constraints are added.

al1 al2 23 b33 to Johnson's one can be derived.

OR {

OR

4. UPPER BOUNDS

To obtain an upper bound it is sufficient to construct a feasible sequence through any heuristic algorithm.

4.4 Heuristic H3

In that approach, the members of inequation (1) are raised to power 3 and one extracts the following condition by keeping only one specific term by member of the new inequation:

4.1 Johnson's type upper bound

al1 b13a12~3a13b33 :::; bl1 al3bl2a23b13a33 Following that idea, once the previous lower bounds have been computed, it is easy to compute the corresponding upper bounds r(J12) , r(J23) and r(J/) .

That leads to the rule:

Concerning the unconstrained flows hop problem, there is an other approach that consists in computing the Johnson 's order JI3 of the extracted two machines flowshop problem composed of the first and the last machine. In particular, JI3 is used in CDS heuristic (Campbell et al. 1970). A generalization is to define JI3 as the John-

Concerning the problem F3jprmnjCmax , heuristics HI , H2 and H3 are the same.

al1-a 12 bl1 bl2 a preced es b I·ff
5. RESULTS This section presents the results of some computational experiments on the F3jprmn, aij, rnsd, snsdjCmax. The studied criterion is the completion time of the last removal time on the third machine. Processing times are randomly generated in {I, ... , lOO} , time lags, setup times and removal times in {O, ..., 25}.

:~~~~ :ch("!1i: ,e~~;)ve to the product of matrices m l3 m 23

4.2 Heuristic Hl

For all generated problems, one computes three minorations LB of the optimal makespan using method JI , JI2 and J23 then heuristics J/ , JI3. H3 and Random are run in order to obtain a majoration of the makespan. For each set of problems and each heuristic, one determines Pm, the proportion of problems for which the heuristic gives the best value (the greatest for the lower bounds and the smallest for the upper bounds) , Pum , the proportion of problems for which the heuristic was the only one to give the best value, Po the proportion of problems for which the optimality is proven and m the mean difference between the best upper bound and the studied lower bound or between studied upper bound and the best lower bound. Finally one have indicated the percentage of problems for which the best lower and upper

The adjacent pairwise interchange rule suggests an other heuristic consisting in sequencing ma(k)

trices by non-decreasing ratio

m:;). In a basic m 23

flowshop , jobs are scheduled by non decreasing difference between processing times on first and third machine. There is an other way to derive this rule. If one focus only on left bottom componants of the matrices, product BA is not worse than product AB if

By neglecting the two middle terms, one retrieves J' algorithm, by neglecting the four exterior terms one obtains HI.

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bounds are equal, so corresponding sequences are optimaL In SI (resp. S2 and S3) the number of jobs is equal to 50 (resp. 75 and 100), in S4 (resp. S5 and S6) 50 subsequences (or groups of jobs) have to be scheduled each one composed of 5 jobs (resp. 10 and 15) and in S7 (resp. S8 and S9), times lag, setup and removal times are ignored, one deal only with basic flows hop problems, the number of j~bs is 50 (resp. 75 and 100). 50000 instances of each problem are generated. SI Pm LB J' 71% LB J12 28% LB J23 28% VB J' 61% VB J13 21% VB H3 26% tuB Rand. 0% Solved 61.5%

Pllm

Po

70% 1% 1% 55% 18% 20% 0%

48% 13% 13% 47% 15% 6% 0%

S2 Pm LB J' 70% LB J12 29% LB J23 29% VB J' 62% VB J13 20% VB H3 24% tuB Rand. 0% Solved 64.5%

Pllm

Po

69% 1% 1% 57% 18% 19% 0%

50% 14% 14% 50% 15% 5% 0%

S3 Pm LB J' 70% LB J12 30% LB J23 30% VB J' 62% VB J13 20% VBH3 23% 0% tuB Rand Solved 67.1%

Pllm

Po

69% 1% 1% 58 % 19% 19% 0%

52% 15% 15% 51% 16% 4% 0%

S6 Pm LB J' 69% LB J12 23% LB J23 23o/c VB J' 49% VB J13 20% VB H3 39% tuB Rand 0% Solved 50.2%

Pllm

Po

m

68% 8% 8% 42% 18% 32% 0%

44% 6% 6% 42% 8o/c 8% 0%

276.5 337.1 340.6 238.4 339.5 120.7 896.6

m 63.1 96.7 96.2 42.0 87.2 45.5 284.9

S7 Pm LB J' 69% LB J12 32% LB J23 32% VB J' 60% VB J13 23% UBH3 26% luB Rand 0% Solved 67.7%

Pllm

Po

m

67 % lo/c 1% 53 % 20% 19% 0%

51% 17o/c 17% 50% 19% 7o/c 0%

60.8 74.6 75.1 35.7 69.9 34.4 259.0

m 75.6 109.2 108.5 44.5 103.4 46.6 342.9

S8 Pm LB J' 68% LB J12 32% 32% LB J23 VB J' 60% VB J13 23% VBH3 25% VB Rand 0% Solved 71.3%

PlJm

Po

m

67% 0% 0% 54% 20% 19% 0%

53% 19% 19% 52% 20% 6% 0%

72.6 84.8 86.4 39 80.4 34.3 309.9

m 88.0 120.4 120.5 45.7 115.7 47.2 388.6

S9 Pm LB J' 68% LB J12 33% LB J23 33% VB J' 60% VB J13 23% UBH3 23% luB Rand 0% Solved 73.6%

Pllm

Po

m

66 % 0% 0% 55 % 21% 18% 0%

54% 20% 20% 53% 21% 6% 0%

85.0 93.5 95.2 41.2 87.9 34.0 353.1

S4 Pm LB J' 69% LB J12 24% LB J23 24% VB J' 53% VB J13 20% VB H3 36% ICB Rand 0% Solved 52.6o/c

Pllm

Po

68% 7% 7% 45% 18% 28% 0%

45% 7% 7% 43% 9% 8% 0%

S5 Pm LB J' 69% LB J12 23% LB J23 23% "CB J' 51% DB J13 20% "CB H3 38o/c. I,"~B Rand 0% Solved 50.9%

Pum

Po

m

68% 87c 8o/c 43o/c 18% 31% 0%

44% 7% 6% 42% 8% 8% 0%

222.9 282.1 281.7 184.6 277.0 101.7 741.2

It appears that J' gives the best results and J13 and H3 are comparable. Howewer these three heuristics, J', J13 and H3 have fields of efficiency that don't overlap much and H3 is the more efficient in mean, especially when subsequences are scheduled rather than jobs - which corresponds to schf>duling jobs with negative time lags. Concerning the lower bounds, J12 and J23 seem to become useful only when predifined groups of jobs are involved.

m 153.7 204.7 205.9 121.5 196.4 77.0 539.4

6. CONCL"CSIO:K Applying Ma-x-Plus algebra to permutation flowshop scheduling allows to unify modeling and resolution. l\'umerous known results can be unified. uniformly presented and generalized. and new methods, like H3. established from the generalization are useful on basic problems. All bounds presented in this paper are computable in O(n.ln(n))

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time, they are efficient and not too redund ant, howewer it is necessa ry to improve the lower bounds .

7. REFER ENCES BIyth, T.S. and M.F. Janowi tz (1972). Residuation Theory. Pergam on Press. Burns, F. and J. Rooker (1976) . Johnso n's threemachin e flow-shop conject ure. Op. Research 24(3) , 578-580. Campb ell, H.G., RA . Dudek and l\LL. Smith (1970) . A heurist ic algorit hm for the n jobs, m machin e sequencing problem. Management Sci. 16B, 63CH>37. Garey, M.R, D.S. Johnso n and R Sethi (1976). The comple xity of flowshop and jobsho p scheduling. Math. Oper. Res. I , 117-12 9. Gaube rt, S. (1992) . Theorie des system es lineaires dans les dioi"des . These de doctora t. Ecole des Mines de Paris. Gunaw ardena , J. (1998). Idempotency. Publica tions of the Newton Institu te. Cambr idge Univer sity Press. Johnso n, S.M. (1954). Optima l two- and threestage produc tion schedules with setup times included. Naval Res Log Quart I, 61-68. Lageweg, B.J., J.K. Lenstr a and A.H.G. Rinnoo y Kan (1978). A general boundi ng scheme for the flow-shop problem . Operations Research 26(3), 53-67. Lente, C. and J.C. Billaut (1999). Une applicatio n des algebres tropica les aux problemes d 'ordonn anceme nt de type flowshop. In: MOSIM'99. Annecy (France ). pp. 177182. Lente, C., J.C. Billaut and S. Dufau (1999). Scheduling in max-pl us algebra . In: IEPM' 99. Vol. 1/ 2. Glasgow (Ecosse). pp. 22-30. Sule, D.R and K.Y. Huang (1983). Sequen cy on two and three machines with setup, processing and removal times separat ed. Int J Prod Res 21(5),7 23-732 . Szwarc , W. (1983). Flow shop problem s with time lags. Management Sc. 29(4) , 477-48 1.

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