Robust Machine Scheduling Based on Group of Permutable Jobs

Abstract

This chapter presents the "group of permutable jobs" structure to represent set of solutions to disjunctive scheduling problems. Traditionally, solutions to disjunctive scheduling problems are represented by assigning sequence of jobs to each machine. The group of permutable jobs structure assigns an ordered partition of jobs to each machine, i.e. a group sequence. The permutation of jobs inside a group must be all feasible with respect to the problem constraints. Such a structure provides more flexibility to the end user and, in particular, allows a better reaction to unexpected events. The chapter considers the robust scheduling framework where uncertainty is modeled via a discrete set of scenarios, each scenario specifying the problem parameters values. The chapter reviews the models and algorithms that have been proposed in the literature for evaluating a group sequence with respect to scheduling objectives for a fixed scenario as well as the recoverable robust optimization methods that have been proposed for generating robust group sequence5) M 2 (4) 7 0 2 4 Fig. 9.9: Gantt representation of a group sequence for the single machine problem Example 2: Job Shop EnvironmentIn Fig. 9.10, two groups of permutable operations are proposed. The first one is composed by the operations of J 1 and J 3 performed on the first machine. The second is composed of the operations of the same jobs on the third machine. One can see that whatever the order of the operations inside each group, the sequence remains feasible. Of course, this flexibility has a price since the makespan is now equal to 32.

Evaluation of a new decision-aid parameter for job shop scheduling under uncertainties

“…To compute this lower bound and to avoid the exhaustive enumeration of all possible schedules, we can use a symmetrical calculation of the worst-case evaluation [2,3]. This new evaluation can be measured by computing the best-case-earlieststarting-time of operations and groups.…”

“…• worst-case(O 1 ) = 11 and worst-case(O 7 ) = 12 (using the polynomial algorithm of [2] described also in [3]), therefore O 1 is sequenced before O 7 using RA worst-case-2.…”

“…For each instance, we generate a GoPO schedule with a maximum possible flexibility and such that the best-case is not only the optimal schedule of GoPO but also the optimal schedule of the instance 3 . To do so, we use an adaptation of the algorithm of [22] mentioned in section 2.…”

“…Some of these methods can be found in [8,25,32,37,40,48]. Groups of Permutable Operations (GoPO) is one of the most studied proactive-reactive methods [2,3,9,13,22]. This method was created by LAAS-CNRS laboratory of Toulouse, France [21] and is implemented in an industrial software [41].…”

Tardiness minimisation heuristic for job shop scheduling under uncertainties using group sequences

“…One of the most studied approaches which consider these two scheduling phases is the group sequences method (Erschler and Roubellat, 1989;Billaut et al, 2008;Artigues et al, 2016). The goal of this approach is to produce during the offline phase a flexible reference schedule resilient to the effects of uncertain events.…”