##
###
The hybrid flow shop scheduling problem

Rubén Ruiz, José Antonio Vázquez-Rodríguez

2010
*
European Journal of Operational Research
*

The scheduling of flow shops with multiple parallel machines per stage, usually referred to as the Hybrid Flow Shop (HFS), is a complex combinatorial problem encountered in many real world applications. Given its importance and complexity, the HFS problem has been intensively studied. This paper presents a literature review on exact, heuristic and metaheuristic methods that have been proposed for its solution. The paper discusses several variants of the HFS problem, each in turn considering
## more »

... erent assumptions, constraints and objective functions. Research opportunities in HFS are also discussed. Z is restricted to be greater or equal to the completion time of the last operation to finish its processing, i.e., the makespan. Several other criteria will be discussed in Section 3. The set of constraints (3) , guarantees that all operations are assigned strictly to one machine at each stage. Constraint set (4), restrict the starting time of operation o jk to be greater or equal to its release time from the previous stage. Constraint sets (5) and (6) prevent any two operations from overlapping in a common machine. Constraint sets (7) , (8) and (9) define the domains of the decision variables. The HFS problem can also be represented as a graph G (N, A) , where N is a set of nodes corresponding to each operation, and A is a set of disjunctive arcs describing the set of possible paths in the graph. A solution is a graph G(N, S), where S is a subset of the arcs in A but with a fixed direction, i.e., S represents an assignment and ordering of the job operations. Several heuristics have been devised using these representation, these will be discussed in Section 4. Naming hybrid flowshop variants The modification, removal, or addition of assumptions and/or constraints to the standard problem described above leads to different HFS variants. To refer to them, the nomenclature presented in [56] and further extended for the HFS case in [204] , is adopted. Scheduling problems are described with a triplet α|β|γ, where α describes a shop config-Without doubt, Branch and Bound (B&B) is the preferred technique when solving to optimality the HFS problem. Most research so far, however, has concentrated on simplified versions of the problem. The simplest scenario, for example, considers only two stages with a single machine at the first stage and two identical machines in the second stage (m = 2, M (1) = 1, M (2) = 2). For this specific case, the earliest known B&B algorithm was proposed by [155] . Much later, [21] studied the same problem and approached it with B&B, heuristics, and genetic algorithms. Another exact method for this problem, but without waiting allowed between the two stages is given in [61] . The opposite case (m = 2, M (1) = 2, M (2) = 1) was studied by [14] and also by [128] . Problems with two stages and

doi:10.1016/j.ejor.2009.09.024
fatcat:g2rd6vfrsjdyxasfiwduf3hile