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Switched Max-Plus Linear-Dual Inequalities: Application in Scheduling of Multi-Product Processing Networks

“…It is always possible to transform a P-TEG into an equivalent one whose places have at most 1 initial token each [11]. Therefore, in the following we will only focus on P-TEGs in which the initial marking m(p) is either 0 or 1 for each place p ∈ P. Under this assumption, a consistent trajectory for a given P-TEG must satisfy LDIs as in (2), where matrices A 0 , A 1 ∈ R n×n max , B 0 , B 1 ∈ R n×n min take the name of characteristic matrices of the P-TEG, and are defined as follows. If there exists a place p ij with initial marking µ ∈ {0, 1}, upstream transition t j and downstream transition t i , then A µ ij = τ − pij and B µ ij = τ + pij ; otherwise, A µ ij = −∞ and B µ ij = ∞.…”

“…Time-window constraints arise in different settings. Typical examples from manufacturing are the bakery and the semi-conductor industries, where some processes (e.g., yeast fermentation in the former case, metal deposition in the latter) must be executed under strict temporal constraints in order to obtain the desired quality of the final product [1,2]. In transportation, time windows can be used to specify admissible pickup and delivery times for customers [3].…”

“…Given their ability to model time-window constraints, they have been applied to solve scheduling and control problems in various types of systems including bakeries, electroplating lines, and cluster tools [5][6][7][8]. The signal describing firing times of transitions in P-TEGs evolves -non-deterministically -according to max-plus linear-dual inequalities (LDIs), i.e., dynamical inequalities that are linear in the primal operations and in the dual operations of the max-plus algebra (see (2) in Section 2.4). The simple nature of the dynamics of P-TEGs has led to a number of theoretical results regarding the analysis of their cycle time [9][10][11][12], which is defined as the temporal difference between the occurrence of two repetitions of the same event in the system, assuming events occur in a periodic manner (for a formal definition, see Section 3.2).…”

“…• the number of transitions in the resulting P-TEGs increases (linearly) with V , which leads to a rather high computational complexity for the cycle time analysis where large values of V are considered; • for a different entrance order of jobs, a new P-TEG must be built. Moreover, when the number of jobs to be processed is infinite 2 and the entrance order is not periodic, no P-TEG can model the manufacturing system.…”

Model-based analysis, control and dosing of electroplating electrolytes

“…The RMH agent transports the carriers with products through the plating process chain. Solvers for the hoist scheduling problems (HSP) from operations research optimize the utilization of the electroplating line and schedule the operations of the hoist in the MES [34]. To import the activities of the hoist, an interface to the MES has been implemented.…”