![](bg8.png)
Proposition 6. If any of the previous cases holds
then the no-idle schedule is unique; it is
consequently the optimal solution for J2|n=2,no-
idle|C
max
problem. Otherwise, it is impossible to get
a no-idle schedule.
Proof. A feasible schedule for J2|n=2|C
max
problem
takes the format presented in figure 5 above.
Clearly, having this format, it is not possible to
transform any schedule to a no-idle schedule. There
are at most max(n
1
, n
2
) – 2 idle intervals.
However, we deduce that if any of the previous
cases holds then the no-idle schedule is unique. It is
consequently the optimal solution for J2|n=2,no-
idle|C
max
problem. Otherwise, it is impossible to get
a no-idle schedule.
6 CONCLUSIONS
In this paper, we have studied the impact of adding
the no-idle constraint to the problem of minimizing
the makespan in a two-machine job shop. We have
studied separately the case where the number of
operations per job isn’t greater than two and the case
where all operations are of unit time. In the first case
we have showed that there exists usually an optimal
schedule which we can calculate using Jackson’s
rule and then fixing the last operation scheduled on
the machine which contains an idle time and then
scheduling the other operations consecutively
without idle times. However, in the second case, we
showed that it is not usually possible to build a
feasible no-idle schedule. Then, we have proposed
the IT algorithm which minimizes first the C
max
then
the number of idle times (Ā). We have shown that it
is impossible to build a schedule which contains a
number of idle times smaller than that of the
schedule obtained by applying IT algorithm.
Consequently, if this schedule is no-idle then it is
also optimal for the corresponding problem with
adding the no-idle constraint. Moreover, in the
general case, where the number of operations per job
can be greater than two and all operations do not
have the same processing time, we have shown that
where the number of jobs is equal to two there are
only few cases numbered from 1 to 9 which are
efficiently solvable and where the set of feasible no-
idle schedules contains a unique schedule. In
conclusion, we deduce that it is not usually possible
to construct a feasible no-idle schedule for the two-
machine job shop problem and that in the majority
of cases, this set is empty.
REFERENCES
Adiri, I., Pohoryles, D., 1982. Flow-shop/no-idle or no-
wait scheduling to minimise the sum of completion
times.
Naval Research Logistics Quarterly, Vol. 29,
pp. 495-504.
Baptiste, P., Lee, K.H., 1997. A branch and bound
algorithm for the F|no-idle|C
max
. Proceedings of the
International Conference on Industrial Engineering
and Production Management (IEPM’1997)
, Lyon, vol.
1, pp. 429-438.
Brucker, P., 1994. A polynomial algorithm for the two
machine job-shop scheduling problem with fixed
number of jobs.
Operations Research Spektrum, Vol.
16, pp. 5-7.
Brucker, P., 1995. Scheduling Algorithms,
Springer,
ISBN: 3-540-60087-6.
Chrétienne, P., 2008. On single-machine scheduling
without intermediate delays.
Discrete Applied
Mathematics, 156, p. 2543-2550.
Garey, M.R.D., Johnson, D.S., Sethi, R., 1976. The
complexity of flowshop and jobshop scheduling.
Mathematics of Operations Research, Vol. 1, pp. 117-129.
Hefetz, N. , Adiri, I., 1982. An Efficient Optimal
Algorithm for the Two-Machines Unit-Time Jobshop
Schedule-Length Problem.
Mathematics of Operations
Research, Vol. 7, No. 3., pp. 354-360.
Jackson, J.R., 1956. An extension of Johnson’s results on
job lot scheduling.
Naval Research Logistic Quarterly,
Vol. 3, pp. 201-203.
Johnson, S.M., 1954. Optimal two- and three-stage
production schedules with setup times included,
Naval
Research Logistics Quarterly, Vol. 1, pp. 61-68.
Kalczynski, P. J., Kamburowski, J., 2007. On no-wait and
no-idle flow shops with makespan criterion.
European
Journal of Operational Research, Vol. 178, pp. 677–685.
Lenstra, J. K., Rinnooy Kan, A.H.G., Brucker, P., 1977.
Complexity machine scheduling problems.
Annals of
Discrete Mathematics
, Vol. 1, pp. 343-362.
Saadani, H., Guinet, A., Moalla, M., 2001. A travelling
salesman approach to solve the
F|no-idle|C
max
problem.
Proceedings of the International Conference on
Industrial Engineering and Production Management
(IEPM’ 2001)
, Quebec, vol. 2, 880-888.
Saadani, H., Guinet, A., Moalla, M., 2003. Three stage no-
idle flow-shops.
Computers and Industrial
Engineering
, 44, 425-434.
Valente, J. M. S., Alves, R. A. F. S., 2005. Improved
Heuristics for the Early/Tardy scheduling problem with
no idle time.
Computers & Operations Research, 32: p.
557-569.
Valente, J. M. S., 2006. Heuristics for the single machine
scheduling problem with early and quadratic tardy
penalities. Working paper 234, Faculdade de Economia
do Porto, Portugal, December.
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