The Cyclic Job Shop Problem with uncertain processing times
Résumé
Most models for scheduling problems assume deterministic parameters. In contrast, real
world scheduling problems are often subject to many sources of uncertainty, for example
activities duration can decrease or increase, machines can break down, new activities can be
incorporated, etc. In this paper, we focus on scheduling problems that are cyclic and where
activity durations are affected by uncertainty. Indeed, the best solution for a deterministic
problem can quickly become the worst one in the presence of uncertainties.
In this paper, we consider the Cyclic Job Shop Problem (CJSP) where processing times
are affected by uncertainty. Several studies were conducted on the deterministic CJSP.
The CJSP with identical parts is studied in (Roundy, R. 1992). The author shows that the
problem is NP-hard and designs a branch and bound algorithm to solve the problem. Hanen
(1994) investigates the general CJSP and presents a branch and bound procedure to tackle
the problem. A general framework for modeling and solving cyclic scheduling problems is
presented in (Brucker, P. and Kampmeyer, T. 2008). The authors present different models
for cyclic versions of the job shop problem. However, a few works consider cyclic scheduling
problems under uncertainty. Che, A. et. al. (2015) investigate the cyclic hoist scheduling
problem with processing time window constraints where the hoist transportation times
are uncertain. The authors define a robustness measure for cyclic hoist schedule and a
bi-objective mixed integer linear program to optimize the cycle time and the robustness.
In order to deal with uncertainty, we use a robust optimization approach. We model
the uncertain parameters by using the idea of uncertainty set proposed by Bertsimas and
Sim (2004). Each task duration belongs to an interval, and the number of parameters that
can deviate from their nominal values is bounded by a positive number called budget of
uncertainty. This parameter allows us to control the degree of conservatism of the resulting
schedule. Finally, we propose a branch and bound procedure that computes the minimum
cycle time for the robust CJSP such that, for each scenario in the uncertainty set, there
exists a feasible cyclic schedule.
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