![](bg4.png)
T
i
=
S
i1
p
i1
R
i1
S
i1
p
i1
R
i2
p
i2
α
i2
S
i1
p
i1
R
i3
p
i2
α
i2
p
i3
α
i3
S
i2
R
i1
β
i2
S
i2
p
i2
R
i2
S
i2
p
i2
R
i3
p
i3
α
i3
S
i3
R
i1
β
i3
1
p
i2
β
i2
S
i3
R
i2
β
i3
S
i3
p
i3
R
i3
(14)
It must be noted that this model can be generalized
to additional constraints such as release dates (r
i
), la-
tency duration (q
i
) or some batch constraints.
5 CENTRAL PROBLEM
In (Augusto et al., 2006) or (Fondrevelle et al., 2006),
authors proposed a way to calculate lower bounds
and optimal schedules for problem Fm—min − max
delays; perm—C
max
. This section represents now
how to transform our studied problem into flowshop
with only minimal and maximal delays. To do that,
terms relating to setup and removal times must be in-
tegrated to the definition of new processing times and
minimal and maximal delays. It will define a new
flowshop problem whose characteristics are described
below. In this new formulation, a job J
i
is only defined
by operations’ processing times p
ik
and minimal and
maximal delays α
ik
and β
ik
. But its associated matrix
T
i
remains unchanged.
p
ik
= S
ik
p
ik
R
ik
(1 ≤ k ≤ m)
β
ik
=
β
ik
S
ik
R
i(k−1)
(2 ≤ k ≤ m)
α
ik
=
α
ik
S
ik
R
i(k−1)
(2 ≤ k ≤ m)
(15)
For example, in case of three machines, the matrix
associated to job J
i
becomes
T
i
=
p
i1
p
i1
α
i2
p
i2
p
i1
α
i2
p
i2
α
i3
p
i3
1
β
i2
p
i2
p
i2
α
i3
p
i3
1
β
i2
p
i2
β
i3
1
β
i3
p
i3
(16)
6 CONCLUSIONS
Thanks to MaxPlus approach, it is possible to trans-
form a flowshop scheduling problem into a matrix
problem. Some manipulations over these matrices al-
low us to exhibit a sort of central problem. Permu-
tation flowshop with several classical constraints are
equivalent to some permutation flowshop with min-
imal and maximal delays. Calculus have been pre-
sented with non sequence dependent setup and re-
moval times, but the equivalence can also be stated for
release dates or groups of jobs for example. This re-
sult means that one can focus our efforts to solve this
central problem. It also means that it must be possible
to adapt automatically a method developed for a type
of constraints to another type of constraints.
Further research will concern modeling a greater
set of constraints, like limited stocks between ma-
chines or blocking constraints and, if possible and
if necessary, the definition of an other central prob-
lem. They will also concern improvements of solving
methods presented in (Augusto et al., 2006).
REFERENCES
Augusto, V., Lente, C., and Bouquard, J. L. (2006).
R
´
esolution d’un flowshop avec delais minimaux et
maximaux. In MOSIM.
Baccelli, F., Cohen, G., Olsder, G., and Quadrat, J. (1992).
Synchronisation and Linearity: An Algebra for Dis-
crete Event Systems. John Wiley and Sons, New York.
Bouquard, J. and Lent
´
e, C. (2006). Two-machine flow shop
scheduling problems with minimal and maximal de-
lays. 4’OR, 4(1):15–28.
Bouquard, J., Lente, C., and Billaut, J. (2006). Applica-
tion of an optimization problem in Max-Plus algebra
to scheduling problems. Discrete Applied Mathemat-
ics, 154(15):2064–2079.
Carpaneto, G., Dell’amico, M., and Toth, P. (1995).
Carpaneto, Dell’amico, Toth - 1995 - Exact solu-
tion of large asymmetric traveling salesman prob-
lems.pdf. ACM Transactions on Methematical Soft-
ware, 21(4):394–409.
Cohen, G., Dubois, D., Quadrat, J., and Viot, M. (1985).
A linear system-theoretic view of discret-event pro-
cesses and its use for performance evaluation in man-
ufacturing. IEEE Trans. Automatic Control, 30:210–
220.
Fondrevelle, J., Oulamara, A., and Portmann, M.-C. (2006).
Permutation flowshop scheduling problems with max-
imal and minimal time lags. Computers & Operations
Research, 33:1540–1556.
Gaubert, S. (1992). Th
´
eorie des syst
`
emes lin
´
eaires dans les
dio
¨
ıdes. PhD thesis.
Gaubert, S. and Mairesse, J. (1999). Modeling and analysis
of timed petri nets using heaps of pieces. IEEE Trans.
Automatic Control, 44(4):683–698.
Giffler, B. (1963). Schedule algebras and their use in
formulating general systems simulations,. in ”Indus-
trial scheduling”, Muth and Thompson (eds.),Prentice
Hall, New Jersey (U.S.A.).
Graham, R. L., Lawler, E. L., Lenstra, J. K., and Rinnooy
Kan, A. H. (1979). Optimization and approximation
in deterministic sequencing and scheduling: A survey.
Annals of Discrete Mathematics, 5(2):287–326.
Gunawardena, J. (1998). Idempotency. Publications of the
Newton Institute, Cambridge University Press.
Hanen, C. and Munier, A. (1995). Cyclic scheduling on
parallel processors: an overview. In Chretienne, P.,
Coffman, E., Lenstra, J., and Liu, Z., editors, Schedul-
ing Theory and its Applications, pages 193–226. John
Wiley.
Lent
´
e, C. (Nov. 2001). Analyse Max-Plus de probl
`
emes
d’ordonnancement de type Flowshop. Th
`
ese de doc-
torat, Universit
´
e Frano¸is Rabelais de Tours.
EquivalencebetweenTwoFlowshopProblems-MaxPlusApproach
177