Discrete OptimizationOn no-wait and no-idle flow shops with makespan criterion
Introduction
A set of n jobs available at time zero has to be processed in a shop with m machines M1, M2, … , Mm. Each job is processed first on M1, next on M2, and so on, and lastly on Mm. No machine can process more than one job at a time, no job preemption is allowed, all setup times are included into the job processing times, and there is unlimited storage between the machines. The problem, commonly referred to as Fm∥Cmax, is to determine a schedule that minimizes the completion time of the last job on Mm, also known as the makespan. When the schedule must have the same job sequence on every machine, the corresponding permutation problem is denoted by Fm∣prmu∣Cmax.
The paper deals with minimizing the makespan in permutation flow shops under no-wait and no-idle conditions. When each job must be processed from the start to finish without any interruption between the machines, the problem Fm∣no-wait∣Cmax is defined. The no-wait condition secures that any no-wait schedule must be a permutation schedule, and thus Fm∣no-wait∣Cmax and Fm∣prmu, no-wait∣Cmax are the same [9]. On the other hand, the no-idle permutation problem Fm∣prmu, no-idle∣Cmax is formulated when each machine must process the jobs without any idle time. As in the case of Fm∥Cmax, there are instances of Fm∣no-idle∣Cmax with m ⩾ 4 for which the restriction to permutation schedules can be costly [11].
Piehler [14] was the first to show that Fm∣no-wait∣Cmax reduces to an instance of the traveling salesman problem (TSP); see also [16], [21]. When m = 2, this instance becomes solvable by the algorithm of Gilmore–Gomory [7]; see [16]. Adiri and Pohoryles [3] observed that F2∣prmu, no-idle∣Cmax and F2∣prmu∣Cmax are equivalent, and thus both problems can be solved by Johnson’s [10] algorithm. Röck [17] and Baptiste and Lee [5] proved that F3∣no-wait∣Cmax and F3∣prmu, no-idle∣Cmax are strongly NP-hard. Numerous practical examples of job scheduling in no-wait environments are summarized in the review paper of Hall and Sriskandarajah [9]. Several applications under the no-idle condition are reported by Saadani et al. [18], and in some of their references.
It is well known that the makespan of a given job sequence in Fm∣prmu∣Cmax can be represented as the length of a critical (longest) path in a network; see e.g. [15, p. 131]. To the best of our knowledge, similar network representations are unknown in the case of Fm∣no-wait∣Cmax and Fm∣prmu, no-idle∣Cmax.
In this paper we present the two missed network representations. They allowed us to reveal a duality relationship that exists between Fm∣no-wait∣Cmax and Fm∣prmu, no-idle∣Cmax, and explain clearly an observed anomaly in flow shop scheduling under the no-wait and no-idle conditions. This virtual anomaly is manifested in the possible reduction in the makespan as a consequence of prolonging the processing of a job on an intermediate machine; see [1], [12]. Our network representations also lead to a natural reduction of Fm∣no-wait∣Cmax to TSP, some lower bounds on the shortest makespans, and new efficiently solvable special cases.
Section snippets
Two-machine flow shops
Suppose a set of jobs, {1, 2, … , n}, available at time zero has to be processed in a flow shop with two machines A and B in series. Let ak and bk be the processing times of job k on A and B, respectively, and let Cmax(π; A, B), Cmax(π; no-wait, A, B), and Cmax (π; no-idle, A, B) be the makespans of a job sequence π = (π(1), π(2), … , π(n)) in F2∣prmu∣Cmax, F2∣no-wait∣Cmax, and F2∣prmu, no-idle∣Cmax.
It was observed in [3], that for every job sequence π,
Network representations of the makespans
Let pij be the processing time of job j on machine Mi for i = 1, 2, … , m and j = 1, 2, … , n. Let Cmax(π; no-wait) and Cmax(π; no-idle) denote the makespans of a job sequence π = (π(1), π(2), … , π(n)) in Fm∣no-wait∣Cmax and Fm∣prmu, no-idle∣Cmax.
The following result shows network representations of Cmax(π; no-wait) and Cmax(π; no-idle). Theorem 1 The makespans Cmax(π; no-wait) and Cmax(π; no-idle), where for simplicity π = (1, 2, … , n), are the lengths of critical paths in the networks shown in Figs. 1a and b, and 2a and b,
Final remarks
The technique we adapted to model the makespans in no-wait and no-idle flow shops can also be implemented for some hybrid flow shops. To illustrate, consider the problem Fm∣block(1, 2), no-wait∣Cmax, that is, there is no storage between M1 and M2, and the no-wait condition must be respected by the remaining machines. When the arcs with negative weights −p2j are deleted from Fig. 1a, the critical path length in the resulting network is the corresponding makespan. In particular, this shows that
Acknowledgements
The authors are grateful to three anonymous referees for their valuable suggestions and comments.
References (21)
- et al.
No-wait flowshops with bicriteria of makespan and maximum lateness
European Journal of Operational Research
(2004) - et al.
Some local search algorithms for no-wait flow-shop problem with makespan criterion
Computers and Operations Research
(2005) - et al.
A heuristic for minimizing the makespan in no-idle flow shops
Computers and Industrial Engineering
(2005) More on three-machine no-idle flow shops
Computers and Industrial Engineering
(2004)- et al.
Minimizing cycle time in a blocking flowshop
Operations Research
(2000) A special case of the (n/m/F/F max) problem
Opsearch
(1977)- et al.
Flow-shop/no-idle or no-wait scheduling to minimise the sum of completion times
Naval Research Logistics Quarterly
(1982) - P. Baptiste, K.H. Lee, A branch and bound algorithm for the F∣no-idle∣Cmax, in: International Conference on Industrial...
- et al.
The analysis of activity networks under generalized precedence relations (GPRs)
Management Science
(1992) - et al.
Sequencing a one state variable machine: A solvable case of the traveling salesman problem
Operations Research
(1964)
Cited by (59)
Metaheuristics with restart and learning mechanisms for the no-idle flowshop scheduling problem with makespan criterion
2022, Computers and Operations ResearchNo-idle parallel-machine scheduling of unit-time jobs with a small number of distinct release dates and deadlines
2021, Computers and Operations ResearchCitation Excerpt :An essential loss of profit or efficiency can happen in workforce timetabling if the idle times between successive working periods are useless for the business and for the workers. The bibliography of the scheduling research considering no-idling machines include Kovalyov and Shafransky (1998), Narain and Bagga (2005), El Houda Saadani et al. (2005), Kalczynski and Kamburowski (2007), Kovalyov and Werner (2007), Valente and Alves (2008), Chrétienne (2008, 2014, 2016), Carlier et al. (2010), Jouglet (2012), Quilliot et al. (2013), Kacem and Kellerer (2014), Yazdania and Naderi (2017), Pempera (2017), Wang et al. (2018), Cheng et al. (2019), Nagano et al. (2019), Shao et al. (2018), Billaut et al. (2019), and Antoniadis et al. (2019). Preliminary results of this study are presented at the conference CoDIT’2019 (Brauner et al., 2019).
Heuristics and iterated greedy algorithms for the distributed mixed no-idle flowshop with sequence-dependent setup times
2021, Computers and Industrial EngineeringCitation Excerpt :As each of the factories has to work under different labour costs, local laws and trading policies, managing the distributed production is more intricate than single site manufacturing systems (Komaki & Malakooti, 2017). Aside from distributed flowshops, another important variation is the no-idle PFSP (Kamburowski, 2004; Narain & Bagga, 2005b; Narain & Bagga, 2005a; Kalczynski & Kamburowski, 2005; Kalczynski & Kamburowski, 2007; Pan & Wang, 2008b; Pan & Wang, 2008a; Ruiz, Vallada, & Fernández-Martínez, 2009; Sun, Sun, Cui, & Wang, 2010; Tasgetiren, Pan, Suganthan, & Buyukdagli, 2013a; Tasgetiren, Pan, Suganthan, & Oner, 2013b; Shen, Wang, & Wang, 2015). In this environment, machines are not permitted to idle and require an uninterrupted processing of the production sequence.
Relationship between common objective functions, idle time and waiting time in permutation flow shop scheduling
2020, Computers and Operations ResearchAnalysis of flow shop scheduling anomalies
2020, European Journal of Operational ResearchCitation Excerpt :□ Kalczynski and Kamburowski (2007) showed that a no-wait job in the no-wait flow shop can be transposed to a no-idle machine in the no-idle time flow shop. We have used similar transpositions to demonstrate few other anomalies presented in the previous section.
Heuristics for the mixed no-idle flowshop with sequence-dependent setup times and total flowtime criterion
2019, Expert Systems with Applications