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Invited Review

The hybrid ow shop scheduling problem

Rubén Ruiz a, * , José Antonio Vázquez-Rodríguez ba Grupo de Sistemas de Optimización Aplicada, Instituto Tecnológico de Informática, Universidad Politécnica de Valencia, Valencia, Spainb Automated Scheduling, Optimization and Planning Research Group, School of Computer Science, University of Nottingham, Jubilee Campus Wollaton Road, Nottingham NG8 1BB, UK

a r t i c l e i n f o

Article history:

Received 11 March 2009Accepted 17 September 2009Available online 13 October 2009

Keywords:SchedulingHybrid ow shopReview

a b s t r a c t

The scheduling of ow shops with multiple parallel machines per stage, usually referred to as the hybrid

owshop (HFS), is a complex combinatorial problemencountered in many real world applications. Givenits importance and complexity, the HFS problem has been intensively studied. This paper presents a lit-erature review on exact, heuristic and metaheuristic methods that have been proposed for its solution.The paper briey discusses and reviews several variants of the HFS problem, each in turn considering dif-ferent assumptions, constraints and objectivefunctions. Research opportunities in HFS are also discussed.

2009 Elsevier B.V. All rights reserved.

1. Introduction

Hybrid ow shops (HFS) are common manufacturing environ-ments in which a set of n jobs are to be processed in a series of m stages optimizing a given objective function. There are a numberof variants, all of which have most of the following characteristicsin common:

1. The number of processing stages m is at least 2.2. Each stage k has M ðkÞ P 1 machines in parallel and in at least

one of the stages M ðkÞ > 1.3. All jobs are processed following the same production ow:

stage 1, stage 2, . . . , stage m . A job might skip any number of stages provided it is processed in at least one of them.

4. Each job j requires a processing time p jk in stage k. We shallrefer to the processing of job j in stage k as operation o jk.

In the ‘‘standard” form of the HFS problem all jobs and ma-chines are available at time zero, machines at a given stage areidentical, any machine can process only one operation at a timeand any job can be processed by only one machine at a time; setuptimes are negligible, preemption is not allowed, the capacity of buffers between stages is unlimited and problemdata is determin-istic and known in advance. Although most of the problems de-scribed in this review do not fully comply with theseassumptions, they mostly differ in two or three aspects only; thestandard problem will serve as a ‘‘template” to which assumptions

and constraints will be added or removed to describe different HFSvariants.

The HFS problem is, in most cases, NP -hard. For instance, HFSrestricted to two processing stages, even in the case when onestage contains two machines and the other one a single machine,is NP -hard, after the results of Gupta [58] . Similarly, the HFSwhen machines are allowed to stop processing operations beforetheir completion and to resume them on different time slots(something referred to as preemption) results also in stronglyNP -hard problems even with m ¼ 2, according to Hoogeveenet al. [74] . Moreover, the special case where there is a single ma-chine per stage, known as the ow shop, and the case where thereis a single stage with several machines, known as the parallel ma-chines environment, are also NP -hard, [49] . However, with somespecial properties and precedence relationships, the problemmight be polynomially solvable [43] .

HFS is found in all kinds of real world scenarios including theelectronics [209,210,114,83] , paper, [170] and textile, [52] , indus-tries. Examples are also found in the production of concrete,[137] , the manufacturing of photographic lm, [189,4] , and others,[2,40,3,112,18,216,152] . We also nd examples in non-manufac-turing areas like civil engineering [44] , internet service architec-tures [8] and container handling systems [36,35] .

The HFS problem has attracted a lot of attention given its com-plexity and practical relevance. This paper describes the HFS prob-lem and reviews many of the solution approaches that have beenproposed for its solution. These include exact methods, heuristics,and metaheuristics. The present review lls in some of the gapsidentied in previous reviews, like those of, [196,113,205] or morerecently [145] , and briey describes some of the most recent ap-proaches. It also identies research opportunities and proposessome interesting research lines. It has to be mentioned that our

0377-2217/$ - see front matter 2009 Elsevier B.V. All rights reserved.doi: 10.1016/j.ejor.2009.09.024

* Corresponding author.E-mail addresses: [email protected] (R. Ruiz), [email protected] (J.A. Vázquez-

Rodríguez).

European Journal of Operational Research 205 (2010) 1–18

Contents lists available at ScienceDirect

European Journal of Operational Research

j ou rna l home page : www. e l s ev i e r. com/ loca t e / e jo r

http://dx.doi.org/10.1016/j.ejor.2009.09.024mailto:[email protected]:[email protected]://www.sciencedirect.com/science/journal/03772217http://www.elsevier.com/locate/ejorhttp://www.elsevier.com/locate/ejorhttp://www.sciencedirect.com/science/journal/03772217mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.ejor.2009.09.024


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eral cases that are the result of the addition or removal of a limitednumber of assumptions and/or constraints.

The rest of this section describes approaches to the differentvariants of the problem paying special attention, in Section 3.4 ,to the standard HSF problem. Given the large number of ap-proaches and problem variants we opted for a simple classicationwith the three very broad classes: exact algorithms, deterministicheuristics and metaheuristics, as we believe it to be more appropri-ate than other more sophisticated classications which after allcould not capture the wide variety of the HFS literature.

3.1. Exact algorithms

Without doubt, branch and bound (B&B) is the preferred tech-nique when solving to optimality the HFS problem. Most researchso far, however, has concentrated on simplied versions of theproblem. The simplest scenario, for example, considers only twostages with a single machine at the rst stage and two identicalmachines in the second stage ðm ¼ 2 ; M ð1Þ ¼ 1 ; M ð2Þ ¼ 2Þ. For thisspecic case, the earliest known B&B algorithm was proposed byRao [148] . Much later, Bolat et al. [19] studied the same problemand approached it with B&B, heuristics, and genetic algorithms.Another exact method for this problem, but without waiting al-lowed between the two stages is given in [57] . The opposite caseðm ¼ 2 ; M ð1Þ ¼ 2 ; M ð2Þ ¼ 1Þ was studied by Arthanary and Ramasw-amy [14] and also by Mittal and Bagga [123] . Problems with twostages and any number of identical parallel machines at the secondstage have been recently studied as well. Lee and Kim [108] pro-posed a B&B method with the minimization of total tardiness.Problem instances of up to 15 jobs were shown to be solvable inreasonable times. The case where stage one may have any numberof machines and stage two only one is studied in [59] . The authorsproposed a B&B that is able to obtain good solutions in a reason-able time. In Ref. [66] , the two-stage regular HFS (unconstrainednumber of machines in stages 1 and 2) with makespan criterionis solved with a very effective B&B method that produces optimalsolutions for problems up to 1000 jobs in size. However, the pro-posed algorithm could not solve many medium instances (20–50 jobs) and in some cases the observed average gap reached morethan 4%. Dessouky et al. [41] are the rst to approach two andthree-stage HFSs with uniform parallel machines at each stagewith B&B methods. Salvador [154] studied the no-wait HFS prob-lem variant. The proposed B&B explores only permutation se-quences and jobs are assigned to the earliest available machineat each stage. The author employed dynamic programming for in-stances of a small size. Recently, Choi and Lee [38] have studied atwo-stage problem with multiple identical parallel machines ateach stage for the minimization of tardy jobs. The authors proposea B&B method as well as some ad-hoc heuristics.

The earliest known B&B method for the general HFS problem,

with any number of stages and any number of parallel machinesper stage, is due to Refs. [24,22] . The tree structure that they pro-posed is an adaptation of that rst presented in [27] for the singlestage parallel machines problem, and has been the most widelyused when dealing with an indenite number of stages. Despiteproposing sophisticated lower bounds, at the time of Brah andHunsucker [24] , instances of a very limited size could be solvedto optimality. More specically, problems with up to eight jobsand two stages with three parallel machines each could be solvedwithin several hours of CPU time. Rajendran and Chaudhuri [147]also studied the same problemand although it did not compare re-sults directly with those of Brah and Hunsucker [24] , the perfor-mance of the later algorithm seems better. In a similar paper,Rajendran and Chaudhuri [146] and Vignier at al. [199] work with

the same methodology but in this case for the minimization of theowtime. An interesting aspect from Refs. [147] and [146] is that

only the sequencing problem is solved via B&B and jobs are as-signed to the rst available machine in each stage. An m-stageHFS with some production planning decisions was considered in[54] . The authors proposed a B&B method that was able to solveonly some small instances to optimality. The original lower boundin [24] was further improved in [198] by better exploiting theavailable information of the partial schedule at a given node. More

advanced lower bounds to that in [24] have also been presented in[156] or more recently, in [194] .In most B&B algorithms the construction of a solution starts in

stage 1, then it moves to stage 2 and so on until stage m . In Ref.[29] , a different strategy was adopted. At every decision pointthe critical stage, i.e., the bottleneck, and a job are selected. Theproblem is then stated as to decide if there is a schedule with amakespan value smaller than a certain upper bound, UB. Anothersuccessful strategy is to incorporate heuristic non-exact methodsinto the functioning of B&B algorithms. For example, in Refs.[144] and [124] heuristics were used to generate upper bound val-ues. Dispatching rules were used to generate the initial UB and ge-netic algorithm at the beginning of each stage. In Ref. [125] ,heuristic lower and upper bounds are also extensively used and,in [15] heuristic rules were used for initializing the root node forthe owtime criterion.

Some authors, including Sherali et al. [170] , have implicitlyusedB&B through mathematical programming, i.e., they represent theirproblemas an MIP model and use a regular solver to obtain a solu-tion. Gooding et al. [50] also employed mathematical programmingto model an HFS with a particular production cost minimizationobjective. It has to be noted that the literature on chemical engi-neering has been neglected in the scheduling literature, althoughit includes some notable papers. A good example is Ref. [116] ,where many mathematical models are given for the m -stage HFSwith no-wait and/or limited storage, batching, identical as wellas unrelated parallel machines and several optimization criteria.Sawik [162] modeled a exible ow line with blocking and reentry.This model was later improved by Sawik [163] . Related papers bythe same author appear in [164] and [166] . Pearn et al. [139] pro-posed a mathematical model and heuristics for a complex circuit-packaging problem with sequence dependent setup times. Sawik[165] proposes yet more mathematical models for a exible owline for tardiness related criteria. A regular HFS, this time withthe sum of weighted completion times criterion, is approachedwith mathematical programming and lagrangian relaxation in[182] . The same problem and objective, but with the addition of limited buffers, is studied in [181] . A rescheduling problem thatconsiders inventory constraints is dealt with in [159] . Some math-ematical models are proposed. The HFS is modeled as a resourceconstrained multi-project scheduling problem with setup timesin [201] . Apart from a mathematical formulation, some heuristicsare displayed. He et al. [71] also provides a mathematical formula-

tion and heuristics, this time for a special m-stage HFS with the no -idle constraint from the glass container manufacturing industry.

Despite the relative success of exact algorithms, they are stillincapable of solving medium and large instances and are too com-plex for real world problems. It is necessary to study non-exact butefcient heuristics. The reader is referred to Refs. [196] and [92] formore detailed reviews of B&B algorithms.

3.2. Heuristics

The simplest type of heuristics are dispatching rules, alsoknown as scheduling policies or list scheduling algorithms. Theseare simple rules of thumb for the ranking and assignment of jobsonto machines. A number of papers have been exclusively dedi-

cated to their study and comparison on a variety of problems.For instance [23] compares 10 dispatching rules for the m-stage

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problem with the maximumtardiness criterion. Later, in [26] , sim-ulation studies are carried out to further analyze the performanceof dispatching rules for the same problem with the makespan andthe maximum tardiness objectives. Simulation was also employedto analyze the effectiveness of dispatching rules in [53] . Anothercomparison study of dispatching rules both in static as well as indynamic HFSs is given in [88] . Further study of dynamic HFSsand dispatching rules has been given recently in [107] . Bragliaand Petroni [121] used data envelopment analysis from the resultsof [88] as a means to obtain accurate information about the perfor-mance of dispatching rules.

Dispatching rules have been used as solution tools for manyproblems of theoretical and practical relevance. In Ref. [55] , for in-stance, dispatching rules and tailored heuristics were used to min-imize makespan and maximum tardiness on the m-stage problem.For the same problem, but with the possibility of stage skipping,Kochhar and Morris [93] and Kochhar et al. [94] propose alterna-tive dispatching rules and a local search algorithm, respectively.The on-line version of the same problem, with tardiness criterion,and no buffers, was briey studied in [179] with dispatching rules.Gupta and Tunc [62] and Sriskandarajah and Sethi [175] proposeda set of dispatching rules based heuristics for the two-stage prob-lem investigated in [58] . Tsubone et al. [190] also studied a two-stage HFS with a single batching machine at the rst stage and ma-chine eligibility in the second stage. Hunsucker and Shah [77,78]studied the constrained HFS where there is a limit on the maxi-mum number of concurrent jobs. They evaluated a series of dis-patching rules under different due date criteria, makespan andowtime. The m-stage problem with the weighted tardiness objec-tive was approached by Kyparisis and Koulamas [102] using dis-patching rules. Verma and Dessouky [195] study an m-stageproblem with uniform parallel machines and identical jobs. Theyinvestigate the performance of dispatching rules, some tailoredheuristics, and derive lower bounds. Jayamohan and Rajendran[80] explore the idea of using different dispatching rules at differ-ent stages of the shop.

Dispatching rules are particularly suitable to deal with complex,dynamic, and unpredictable environments and hence their popu-larity in practice. As early as in [138] a simplied two-stage HFSarising in the glass container industry was studied and the authorproposed several ad-hoc dispatching rules. In Ref. [129] , theauthors applied dispatching rules to a simple two-stage problemfrom cable manufacturing with only one machine in the rst stage.The authors tested problems with up to 70 jobs and up to four ma-chines in the second stage. Extensions to these dispatching ruleswere later presented in [128] and [89] . Real problems are alsostudied in Refs. [189] and [2] where a photographic lm and apaper bag factories, respectively, were investigated. In both cases,scheduling systems based on dispatching rules were designed.A real rubber production problem was studied by Yanney and

Kuo [219] . Another real problem, this time based on textile manu-facturing, was approached by Guinet [137] with several ad-hocheuristics. A SMT circuit board exible ow line was analyzed in[142] . Further results in this line were shown in [143] . The authorsconsider family setup times and study several dispatching rules.Agnetis [5] studied a complex HFS problemarising in car manufac-turing where different performance criteria were studied.

A number of sophisticated heuristics use a divide-and-conquerstrategy in which the original problem is divided into smaller sub-problems that are solved one at a time and their solutions are inte-grated into a whole solution to the original problem. In Ref. [193] ,for example, a two-stage problemis simplied into a series of mul-tiple ow shops in order to reduce routing exibility and to reducemanufacturing costs. Suresh [178] studies another two-stage prob-

lem and divides it into two single parallel machines problems, oneper stage, where the release times in the second stage are the com-

pletion times of the jobs at the rst stage. Particularly effective di-vide-and-conquer strategies for the m-stage problem are thevariants of the shifting bottleneck procedure (SBP). These work un-der the principle of giving full priority to the bottleneck stage,maximizing in this way its productivity, and consequently the pro-ductivity of the entire shop. In Ref. [37] an SBP approach is pro-posed for the m-stage problem with the makespan criterion.Yang [218] proposes an SBP heuristic for the minimization of thetotal weighted tardiness, also for an m-stage problem. While mostof bottleneck exploiting methods divide the problem into stages,i.e., they schedule one stage at a time, in [140] this is done by jobs.Every time that a job has to be scheduled, the bottleneck is recal-culated, a job selected, and all its operations scheduled. This heu-ristic, named the progressive bottleneck improvement, competedwell with the SBP variant presented in [37] . This algorithm, how-ever, was only tested on HFS problems with 2 and 3 stages. Otherauthors exploiting the SBP idea are Acero-Domfnguez and Pater-mina-Arboleda [1] for the makespan objective, Lee et al. [109] forthe total tardiness objective and Chen and Chen [32] for the num-ber of tardy jobs. Recently [33] , the same authors have proposedsimilar heuristics but for the total tardiness objective. The idea of separating the sequencing and machine assignment has beenexploited in many studies. For example, Guinet at al. [56] workedwith the two-stage HFS and proposed heuristics that in a rst stepdene a job ordering and then assign jobs at every stage followingthis ordering. A similar method was devised in [157] , where spe-cic regular ow shop heuristics were applied to obtain initial job sequences, and in [185] , where the m-stage HFS problem withblocking is studied. Brah and Loo [25] also separated both decisionproblems, sequencing and machine assignment, to minimizemakespan and owtime in an m-stage problem.

The vast majority of work in non-exact approaches are tailoredheuristics to specic cases of the problem, mainly with 2 and 3stages. Gupta [58] , for instance, proposed a very simple heuristicfor an HFS with only two-stages, parallel machines in the rst stageand only one machine at the second stage. The same problem wasstudied later by Gupta et al. [59] who also proposed heuristicmethods and a B&B algorithm. A similar problemwas studied laterwith the total number of tardy jobs criterion in [64] . In Ref. [197]some mathematical programming formulations and heuristics areprovided for a simple two-stage HFS with only one machine atthe second stage. Kim et al. [91] have recently proposed heuristicsfor a similar two-stage problem with release dates and a product-mix ratio constraint. A similar problem with lot streaming and thetotal owtime criterion was studied by Zhang et al. [225] and laterby Liu [115] for the makespan criterion. Tseng et al. [188] tackledthe issue of missing operations at the rst stage and proposedsome simple heuristics. Another simplied setting is that studiedin [134] , where a problem with two stages with identical parallelmachines and unit processing times is approached with heuristic

methods. Riane et al. [150] study the case where there are onlytwo uniform machines at the second stage. Gupta and Tunc [63]studied the problem with a single machine in the rst stage, anynumber of identical parallel machines in the second stage, and sep-arable and sequence independent setup and removal times withthe makespan criterion. Chen [31] proposed heuristics and pro-vided worst-case analyses for the same problem and Uetakeet al. [192] study the case when machines at the second stageareunrelated. The two-stage case with any number of uniform par-allel machines is studied in [174] and later revisited by Kyparisisand Koulamas [104] . Caricato et al. [28] considers a problem witha set of initial batches, and jobs have a predened sequence insideeach batch. The authors proposed a number of approaches basedon heuristics originally designed to tackle the TSP problem. A sec-

ond article that investigates batching is [17] where two stages withthe second one containing batching machines is solved using tai-

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lored heuristics. Batching is also studied in [191] , where a two-stage cyclic HFS with product-family batches is considered. Li[111] also considers batch production allowing split on the singlemachine at stage 1 and proposed several heuristic methods. Thesame problem with two stages and identical machines in the rststage was studied by Lee and Vairaktarakis [105] . The authors pro-posed a fast heuristic running in O ðn log nÞ, derived error boundsand proposed some lower bounds. He and Babayan [69] considera two-stage HFS with the makespan criterion where there is a sin-gle machine in the rst stage and multiple assembly machines atthe second stage. Mathematical models and heuristics are pro-posed. Similarly, Sung and Kim [177] tackle a two-stage assemblyshop with two independent machines in the rst stage and twoassembly identical machines in the second stage. Dispatchingrules, heuristics, and B&B methods are proposed. Assembly stagesare also considered in [106] which studied problems with one ma-chine either at the rst or at the second stage of the shop and pro-posed a number of heuristic procedures. A mathematicalformulation for the same two-stage assembly problem and make-span was given in [70] along with some heuristic methods. A twostages problem with unrelated parallel machines in the rst stageis considered in [120] . Figielska [46] considered a variation of thesame problemwhere preemption of jobs is allowed, and some lim-ited resources (besides machines) are needed at the rst stage. Re-cently, Figielska [47] , the same author has proposed moreheuristics, together with metaheuristics including simulatedannealing and genetic algorithms for the same problem. A problemwhich considers unrelated parallel machines and proportionateprocessing times is studied by Sundararaghavan et al. [176] . Simi-larly, Dessouky et al. [41] study twoand three-stage problems withuniform parallel machines. They propose lower bounds, polyno-mial algorithms and metaheuristics. Huang and Li [75] study theuniform parallel machines case, but only at the second stage. Theypropose several heuristics. Two and three-stage problems are ap-proached by Koulamas and Kyparisis [95] with the aid of simplemethods with known worst-case bounds. The symmetric problemwith unavailability constraints was approached in [12] with differ-ent heuristics and a B&B algorithm. The no-wait constraint wasconsidered in a two-stage problem and approached with tailoredheuristics by Xioe et al. [214] .

Some highly peculiar two-stage problem variants approachedheuristically are for example, the transfer batch two-stage HFSscheduling problem with both separable and inseparable setuptimes studied in [90] . Kouvelis and Vairaktarakis [97] study theproblemwhere jobs can either be fully processed in one stage (sin-gle machine problems) or at both stages and provide some heuris-tic methods. Another interesting paper is that of Schuurman andWoeginger [167] where a minimum makespan two-stage HFS withidentical parallel machines is studied. However, in this case, thenumber of machines at each stage is also part of the problem data.

Some heuristics and approximation schemes are provided. Otherspecial HFSs are multiprocessor problems where operations re-quire to be processed by more than one machine simultaneously.Oguz et al. [135] proposed heuristics for a two-stage multiproces-sor HFS. In order to deal with uncertainty, in [72] a two-stage prob-lem with makespan criterion is modeled using fuzzy processingtimes and approached using fuzzy heuristics. A similar problembut with unavailability periods for the machines is approachedby Xie and Wang [213] .

A relatively small number of heuristics have been proposed forvariants of the m -stage problem. A number of such heuristics, forinstance, are proposed in [42] and [160] for the m-stage problemwith any number of machines per stage. Later, Santos et al. [158]proposed an improvement heuristic for the m-stage HFS with the

makespan criterion. Sawik [161] proposes a simple heuristic forthe minimization of makespan in a similar shop but with limited

buffers. Heuristics for the HFS with m-stages, uniform parallel ma-chines and makespan criterion are due to [169] and [103] . [215]study the case where the last of m stages is a batching stage. Theresults over a large set of small instances indicated that the pro-posed heuristic obtains close to optimal solutions for very smallproblem instances. An m-stage with unrelated parallel machinesproblem is studied in [172] . On-line methods are given in [68]for the m-stage HFSwith the makespan criterion. Group schedulingfor an m-stage HFS was approached by [117] . [60] consider a real-istic problem where due date assignment is part of the problem.The authors also consider limited production resources besidesthe machines. [20] considered precedence relationships between jobs and proposed several heuristics. [21] studied a fairly complexproblem with lags, setup and removal times and precedence con-straints with the maximum lateness criterion. Sawik [161] An m -stage problem, with job recirculation, common job due dates,and the weighted tardiness criterion was analyzed and solvedusing a number of heuristics, including dispatching rules, in [39] .Hong et al. [73] extends the work in [72] on fuzzy heuristics forHFS with uncertain data by extending the application of the pro-posed fuzzy heuristics to the m stages case.

Most real world applications of tailored heuristics are for rela-tively simple problems with two or three stages only. In Ref.[209] a real electronics production system is studied. The authorconsiders three stages, identical parallel machines per stage, andthe feature that jobs can skip one or two of the processing stages.The author proposes a pseudo-dynamic programming heuristic. Tothe best of our knowledge, [209] is the rst study in the HFS liter-ature where more than one objective is considered. More precisely,the author minimizes C max and among the minimum C max se-quences, work in progress (WIP) is considered (lexicographicaloptimization). In Ref. [210] , blocking is added to the same applica-tionof Wittrock [209] . Another interesting application of heuristicsappeared in [40] , where a two-stage petrochemical productionproblem with a limited buffer in between the stages was studied.In Ref. [44] , civil engineering projects are modeled as three stageHFSs with preemption and precedence constraints among jobs.The optimization criterion considered is total tardiness and somesimple heuristic methods are proposed. Another reality-inspiredproblem is approached by Riane [149] . The problem has threestages with one or two machines per stage. The authors presentedboth exact as well as heuristic algorithms. In a related and laterwork, the same problem, this time with any number of identicalparallel machines per stage, is studied by Soewandi and Elmagh-raby [173] . Later, Koulamas and Kyparisis [96] studied the sameproblem and proposed improved heuristics with best known per-formances. Lin and Liao [112] studied a real label sticker manufac-turing problem modeled as a two-stage HFS with sequencedependent setup times and machine eligibility. Ruiz et al. [153]have studied a complex problem from the ceramic tile sector with

unrelated parallel machines, sequence dependent setup times, ma-chine eligibility, machine release dates, positive and negative lags,stage skipping, non-anticipatory setups and job precedence con-straints. Some mathematical models and heuristics are proposed.Group technology for setup family grouping in the ceramic tile sec-tor was studied by Andres [13] for the case of three stages.

3.3. Metaheuristics

For the last 20 years, researchers in the combinatorial optimiza-tion community have developed successful generic strategies toimprove on the performance of simple deterministic heuristics.Most of these methods, known as metaheuristics, add an elementof randomness into deterministic heuristics with the idea that its

repeated usage may lead to better solutions than the ones gener-ated deterministically. In simulated annealing (SA), for example,



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a local search heuristic is called many times. Randomness is inte-grated into the acceptance criterion of non-improving solutionswith the idea of preventing the stagnationon a local optima, whichis the weakness of common steepest descent local search proce-dures. Tabu search (TS) achieves the same purpose by allowingnon-improving movements and avoiding certain improving move-ments that may take the search to an already visited point. Geneticalgorithms (GA) maintain a ‘‘population” of solutions and carry outa simultaneous exploration of different parts of the search space.These three metaheuristics are, by far, the most widely used inHFS scheduling, and probably in scheduling in general. Other meta-heuristics that have also been used in HFS are ant colony optimiza-tion (ACO), articial immune systems (AIS), neural networks (NN)and others.

Most metaheuristics proposed for the HFS exploit a simplestrategy: to restrict the search to the space of job permutations.The idea is to nd a permutation of the n jobs and build a scheduleby assigning jobs onto the machines according to this ordering.One of the earliest papers to exploit this idea is Voss [200] , wherea dispatching rule was used to seed a TS algorithm to solve a two-stage HFS with a single machine in the second stage and stage-based sequence dependent setup times. Haouari and M’Hallah[67] also studied the two-stage case but with identical parallel ma-chines at each stage. The authors presented a lower bound, a TSheuristic, and an SA algorithm. Both methods work over the per-mutation of jobs and a simple heuristic is used for job assignmentat each stage. Chen et al. [34] presented a TS heuristic for a two-stage exible ow line with makespan criterion. Another TS wasproposed by Grabowski and Pempera [52] in a no-wait HFS in-spired by a real manufacturing problem. In Ref. [208] , the capacitybetween stages is limited ( block). A TS that explores the permuta-tion space was developed and compared with the approaches pro-posed in [210] and in [160] . TS outperformed both heuristics. InRefs. [36] and [35] a similar TS was also developed and used tosolve real instances of a container handling system. The multipro-cessor task problem with precedence relationships is approachedby Oguz et al. [136] . A TS metaheuristic is proposed, some specialcases of the problem are proved to be polynomially solvable,whereas others are proved to be NP -hard. A simplication of the HFS in which machine assignments are known in advance,was also approached with TS by Finke et al. [48] . The authors min-imize the sum of earliness and tardiness. An m-stage problemwithgroup scheduling is approached in [118] with TS. Yet anotherapplication of TS is for the regular HFS with the sum of theweighted completion times criterion and limited buffers, [207] .The same objective, but with all jobs having the same processingtimes on all machines, is solved in [76] using a column generationstrategy and constructive heuristics. No much later, the sameauthors improved on their results with a constructive GA [171] .There are few studies that consider unrelated parallel machines

at each stage. An example is [119] , that proposed a SA methodfor this problem with total owtime minimization. He also consid-ered sequence dependent removal times ( Rsd ) and sequence inde-pendent setup times. Allahverdi and Al-Anzi [8] presentedseveral heuristics and a SA method for an m-stage HFS that modelsclient–server requests. Another SA method with the same solutionrepresentation was given in [84] together with a lower bound toevaluate algorithm performance. This bound was proved wrongand corrected later by Haouari and Hidri [65] . Simulated annealinghas been used to solve a HFS with sequence dependent setup timesand transportation times between stages of an automated guidedvehicle in [126] . In a similar paper [127] , SA is applied to the sameproblem but without transportation constraints.

An interesting randomized local search heuristic is due to [110] .

In there, randomness is introducedby modifying the problem data.Each variant of the original problem is solved with a deterministic

heuristic and the solution is evaluated using the original data. Theidea is, of course, that the solution to one of the modied problemsimproves on that generated with the data of the original problem.The authors concluded that the method is easy to adapt to copewith different objective functions. In the same paper, the authorsproposed a lower bound for the makespan criterion which was la-ter improved by Kurz and Askin [99] . Other authors have used arepresentation where a permutation for each machine at eachstage is maintained. This is sometimes referred to as operation pro-cessing order or exact representation. Janiak et al. [79] employ thisscheme and propose several TS and SA methods. In order to copewith such large search space, the schedule is constructed, at eachstage, after the solution of a linear program. Similarly, in [51] aSA algorithm that does not separate the job sequencing and ma-chine assignment decisions is applied to an m-stage unrelated par-allel machine problem.

Genetic algorithms have also been widely used. In Ref. [212] , GAwas used to searchthe permutation space for the solutionof the m-stage problem with makespan as objective. The same problem,with the addition of sequence dependent setup times was ap-proached in [101] usinga GA with the random keys representation.Theproposed RKGA outperformed several other specialized heuris-tics including those of [100] . In Ref. [133] , a multiprocessor prob-lem was considered. A GA, similar to the one presented in [212] ,was comparedwith the sophisticated TS of [136] and obtained bet-ter results. A realistic problem from a check-processing companythat considered recirculation and the sum of weighted tardinessobjective was approached using GA by Bertel and Billaut [18] . InRef. [152] , a GA was employed to minimize makespan on an m-stage problem with unrelated parallel machines, sequence depen-dent setup times and machine eligibility. The proposed GA wassuperior to a wide range of heuristics and other metaheuristics,among them, ACO based heuristics, TS procedures, other GAs, SAand deterministic procedures. The proposed GA also obtained bet-ter schedules than the ones generated manually by the personnelof a real world ceramic tiles production shop. A similar GA was re-cently proposed for the same problem but with the additional con-sideration of limited buffers by Yaurima et al. [220] . A similarproblem, with unrelated parallel machines at each stage, and setuptimes, was approached in [85] with GAs and later in [86] with sev-eral heuristics including dispatching rules, tailored heuristics, GA,TS and SA. In these two papers, the authors study a linear combi-nation of the makespan and the number of tardy jobs as an objec-tive. Another real world application is given in [83] , where a realprinted circuit board manufacturing system modeled as a three-stage HFS is approached with GA. The HFS with multiprocessortasks is studied for the makespan criterion by Serifoglu and Ulusoy[168] and by Oguz and Ercan [133] and approached with GA. A GAand a SA were proposed for a fairly complex cyclic scheduling ex-ible ow line with lot sizing in [81] . Another complex problem

with production time windows in exible ow lines is exposedin [145] . In Ref. [202] , the authors use GA to explore the permuta-tion space for the rst stage of the shop and dispatching rules oth-ers than the rst in rst out for the rest. They carried outexperiments on different objective functions and obtained good re-sults, among them, improved on the performance of the SBP withlocal search presented in [218] for the sum of weighted tardinessobjective. Later, the same authors use a permutation for the rststage of the shop and combinations of dispatching rules for therest. Both the permutation and the combination of heuristics arefound by GA. A similar GA is shown in [203] , where several criteriaare jointly considered and in [204] , where a number of compositeobjective functions are investigated. A simplistic GA approach wasproposed in [211] for an HFS with machine eligibility constraints

and makespan criterion. The same SMT circuit board exible owline studied in [142,143] has been studied more recently in [87] ,



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where a mathematical programming based solution approach, dis-patching rules and a TS algorithm are proposed. Very recently, amemetic algorithm (GA with embedded local search) was pro-posed for a similar problem [184] .

Other metaheuristics have been used less frequently. Articialimmune systems (AIS), for instance, have been used in [45] andin [224] . In the later case, the proposed AIS outperformed the GAproposed in [101] . Neural Networks were used in [61] as a mech-anism to decide among several heuristics, for the one to be usedon a particular instance. Another application of NN is given in[206] where it was used for the m -stage HFS with the makespancriterion. However, the proposed approach is complex and the re-sults are relatively poor. A similar work with NN is due to [180] . Inboth works, unfortunately, comparisons are carried out onlyagainst simple heuristics. The addition of sequence dependent set-up times is shown in [183] . Their results, together with those in[206] and [180] suggest that NN, at least as proposed by theauthors, are not as successful as the other reviewed metaheuristics.Ying and Lin [222] proposed an ant colony optimization (ACO)metaheuristic for the multiprocessor task problem with prece-dence relationships and showed superior results than [136] . A par-ticle swarm optimization (PSO) metaheuristic, which allegedlyimproved upon the results in [222] , was proposed by Tseng andLiao [187] . The same problem is approached with tailored heuris-tics in [223] and with an iterated greedy metaheuristic in [221] .Alaykyran et al. [6] studied the regular m-stage HFS with makespancriterion. They proposed an ACO method which produced betterresults than a curtailed B&B algorithm. A radically different ap-proach, namely an agent system method, was used in [16] for athree stage identical parallel machine HFS.

Simulationtools,whichareadequate tomodel theinherentcom-plexity of real world problems, have beenused in combinationwithcommon metaheuristics. For example, TS has been combined withsimulationin [216] tosolve a realmulti-layerceramiccapacitorpro-duction problem. The same problemwas approached using GA andsimulation tools in [217] . A similar problem is approached by Kuoet al. [98] also with the aid of simulation tools. Recently, Aleri [7]alsomix simulationtoolswithTStosolve a very complexschedulingproblem. Riane et al. [151] consideran m-stage HFS with the make-spancriterion. Theauthorsusesimulationtoolsfor theplanningandbatching stages and SA for the scheduling phase. Another realisticproblemapproach with SA and simulation tools is due to [11] .

3.4. The m-stage hybrid ow shop with identical parallel machines andmakespan minimization

The standard HFS problem, with m stages, identical parallel ma-chines and makespan minimization is the natural starting point forthose interested on the HFS problem. Whether researchers belongto the academia or to the industry, it is a good idea to study the HFS

in this standard form rst, to learn techniques that are successfulon this variant of the problem, and to try to extrapolate this knowl-edge to other particular forms of the problem. In the following wediscuss to a greater detail, and when possible compare, those fewtechniques that have had the most inuence in the standard HFSproblem, either because themselves have lead to powerful algo-rithms or because they have inspired the work of otherresearchers.

A number of publications, starting with Brah and Hunsucker[22,24] have proposed B&B algorithms to solve the standard HFSproblem. Most algorithms differ from each other on the lowerbounds that are used to prune the search tree. We explain nextthe operation of these lower bounds starting with the work in Refs.[22,24] which exposes the ideas that are the basis of more elabo-

rated approaches proposed in more recent years. The bounds pro-posed in [22,24] have also served as the basis for lower bounds that

have been used to evaluate heuristic methods for the standard HFSproblem [156,99,194] and some of its variants [101] . In what fol-lows, any non-terminal node of a search tree represents a partialschedule. Suppose, then, that one is analyzing the partial scheduleat a particular non-terminal node. Let J k be the set of job indiceswith an operation in stage k, i.e., j 2 J k if o jk 2 O

k, and let b J k J kbe the subset of already scheduled operations. This means that

operation o jk, such that j 2 J k n b J k, has not been scheduled yet. Let b S k be the schedule of the elements in b J k represented by the branchof the tree corresponding to the non-terminal node being analyzed

and b S kl be the schedule formachine l. Let b S klðiÞ denote the job indexof the ith operation assigned to machine l in b S kl and let C ð b S klðiÞÞ ¼start b S klðiÞ;k þ p b S klðiÞ;k, where start b S klðiÞ;k is the starting time of operationo b S klðiÞ;k, be its completion time. C ð b S

klÞ ¼ max iC ð b S klðiÞÞ is the comple-tion time of the partial schedule b S kl. The average completion time( ACT ), i.e., minimum processing requirement for J k given b S k, canbe expressed as:

ACT ð bS k

Þ ¼

1

M ðkÞ XM ðkÞ

l¼1 C ð bS kl

Þ þ X j2 J k n b J k p jk8>: 9>=>;

:

Because b S k is only a partial schedule, it is clear that any completeschedule S k, obtained by adding operations to b S k must have a com-pletion time, C ðS kÞ ¼ max lC ðS klÞ, that is P ACT ð b S kÞ. Therefore ACT ð b S kÞ is a lower bound on the makespan ðC max Þ of a partial orincomplete schedule b S k . Note that the maximum completion time(MCT ) for a workload, MCT ð b S kÞ ¼ max lC ð b S klÞ, is also a lower bound.

When considering a node that is in a stage other than the lastone, the previous bounds can be improved by adding the time thatthe last job b | to leave stage k requires to nish its processing in therest of the stages (> k). It is not possible to be certain about such a job. However, it is possible to determine the set of jobs that containit and among them to choose the one with the less work remaining.

If ACT ð b S kÞ P MCT ð b S

kÞ, ̂| belongs to the not yet scheduled jobs,

J k n b J k, otherwise ̂| may be in J k . Considering this, the following ma-chine based lower bound on the maximum completion time wasproposed in [24] .

LBM ð bS kÞ ¼

ACT ð bS kÞ þ min

j2 J kn b J k Pk0¼kþ 1

p jk0 if ACT ð bS kÞP MCT ð bS

kÞ;

MCT ð bS kÞ þ min

j2 b J k Pk0¼kþ 1

p jk0 otherwise :

8>>>>>:A job based lower bound can be calculated as:LBJ ð bS

kÞ ¼ minl

C ð bS klÞ þ max

j2 J kn b J k Xk0¼kþ 1 p jk0 ;

which is the earliest time when a machine in k is free, plus the min-imum processing time required for a non-scheduled job to exit the

shop. Usually, LBM dominates LBJ , however, the importance of thelatter becomes apparent as the number of schedulable jobs at a gi-ven stage approximates the number of parallel machines on it. It isconvenient, then, to merge both lower bounds into a single one:

LBð bS kÞ ¼ max f LBM ð bS

kÞ; LBJ ð bS kÞg:

The largest problems tackled in [24] had six jobs and ve machines(30 operations). The progress since then has been slow; themost re-cent algorithms can solve, within reasonable time, problems of be-tween 50 and 80 operations only. In Ref. [144] , for example, theabove lower bounds were improved after observing that when (1) J n J 0– £ , and (2) ACT ð b S kÞ ¼ MCT ð b S kÞ, the following inequality mayhold:

min j2 J kn b J

k

Xm

k0¼kþ 1

p jk

0 < min j2 b J

k

Xm

k0¼kþ 1

p jk

0 ;



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and consequently, LBM may be underestimated. This problem wasalleviated by modifying LBM when conditions (1)and (2)hold [144] ,

LBM ¼ ACT ð bS kÞ þ max min

j2 J kn b J k Xm

k0¼kþ 1

p jk0 ; min j2 b J k X

m

k0¼kþ 1

p jk0( ):In the same work, the authors also proposed to use genetic algo-rithm (GA) to aid the calculation of theupperbound before and dur-ing the exhaustive search. The authors solved problems with up to15 jobs and ve machines, and although the authors did not com-pare their algorithms directly with those in [24] their results seemto be superior. A very similar but more recent work was presentedin [124] where GA was also used to improve the bound calculation.However, in [124] a different branching scheme, originally pro-posed in [15] for the minimization of total ow time, was used.Problems of no more than 18 operations (six jobs and three stages)were solved. It is important to mention that the branching schemeproposed in [15] can only enumerate ðn !Þm schedules, much fewerthan the totality of feasible solutions:

Ym

k¼1

n 1

M ðkÞ 1 n!M ðkÞ! ;as calculated in [24] . Therefore, and to the best of our knowledge,the known theory does not support the claim that the algorithmsin [15] and [124] guarantee optimality. [125] also uses a differentbranching scheme and ad-hoc lower bounds and managed to solveproblems of up to 20 jobs and four machines in relatively shorttimes. Unfortunately, theauthors did not compare their results withthose in [144] nor presented results using longer running times. In[132] two B&B variants were enhanced with the so called ‘‘energeticreasoning” and ‘‘global operations” strategies. The proposed algo-rithms solved instances of up to 15 jobs and ve stages. Althoughthe authors did not compare their results directly with other algo-rithms from the literature, they did compare with standard B&Bs,in line with those in Refs. [24] and [125] , and demonstrated thatthe proposed ideas do improve algorithm performance.

The HFS problem is remarkably complex, and as such, there is apoor understanding about its properties and the way to exploitthem in non-exact algorithms. Successful metaheuristics exploitin one way or another one of the following two broad ideas: touse the properties of the graph representationof the problemin or-der to reduce the search, or to limit the search to the space of per-mutations of the job indices.

The rst approach, which is particularly successful on themakespan objective, exploits the simple observation that in orderto improve a schedule it is enough to consider modications thatinvolve one or more jobs in the critical path. Moreover, any modi-cation that does not involve an operation in the critical path leadsto a schedule that is at best not worse than the current schedule.This idea alone can drastically reduce the size of the neighborhood

explored by local search algorithms. Other properties of the graphrepresentation of the problem, implied by the critical blocks the-ory, can be used to further speed up the search. Details can befound in [131] and in [130] . In [131] a tabu search (TS) algorithmthat obtained good solutions to problems with up to 3000 opera-tions was proposed. This TS is very similar to those designed bythe same authors to solve other classical scheduling problems,some of which are recognized as state-of-the-art. In the case of the HFS problem, however, no other authors have compared theiralgorithms with those in [131] and is therefore difcult to assessif this good performance also holds for the HFS. Other similar algo-rithms, including SA, variable-depth search and TS were proposedin [130] . There, the authors compared different neighborhoodfunctions and concluded that algorithms that use the function pro-

posed in [131] are the best performers. An important drawback of

this type of approaches is that, since they rely on a theory that isonly applicable to a reduced number of HFS cases, they are difcultto generalize to other more realistic shop scenarios.

The second type of metaheuristics implement the idea pre-sented in [155] in the commented owmult method. This algo-rithm enumerates just the n! permutations of the job indices.Each permutation dictates the priority of jobs in the rst stage of the shop. In the remainingstages, jobs are given preference accord-ing to the rst in rst out scheduling policy. In all stages jobs areassigned to the rst available machine or to the machine that re-sults in the earliest completion time. Enumerating n ! schedules isstill intractable and it is typical to use metaheuristics to searchfor good permutations. This idea is easy to conceptualize, to imple-ment, and to generalize to most variants of the problem, and istherefore very commonly used. For example, GA is used to searchfor good job permutations for the standard HFS problem in[212,202] and for other variants of the problem in[101,133,18,152,220,168,133] . SA is used in [84] for the standardproblem and in [67,8,84] for other variants; articial immune sys-tems are used in [45] . TS is used in [216] for the standard problemand in [200,67,52] for other variants. A drawback of exploring thepermutation space is, of course, that the reduced space may notcontain the optimum solution and there is no known upper boundon the deviation between the solutions of the ‘‘owmult optimum”and the ‘‘real optimum”. However, the experiments in [155] showthat owmult was capable of nding the optimum makespan on90% of the problems in which it was tested and on 98% of the casesit found solutions within 5% of the optimum.

Unfortunately, even if we restrict our attention to the standardHFS problem, it is still impossible to make a fair comparison of algorithms given that most authors test their algorithms on theirown randomly generated instances or real world instances. Indeed,there is not an agreed set of benchmarks for the standard HFSproblem. The only test set that has been referred to in few articlesis the test set presented in [210] . This only includes a handful of small, easy and very peculiar instances. Moreover, more often thannot, authors compare their algorithms with relatively simple heu-ristics, with their own variants of certain metaheuristics or withadaptations of methods originally designed to solve other prob-lems. The authors are not aware of any work comparing both of the above mentioned type of metaheuristics and therefore we donot know with any certainty under what circumstances one ap-proach shouldbe preferred over the other. What one should expectis for meta-heuristics that exploit any of the above ideas to outper-form those that explore the whole search space indiscriminately,such as the NN proposed in [180] .

4. Analysis of the literature

This review has examined more than 200 papers, mainly deal-ing with the HFS problem and its many variants. As with otherelds of study, the number of papers being published has beensteadily raising over the past few decades, as Fig. 1 shows.

As canbe seen, there is a clear increasing trend whichshows thegrowing interest in this eld. It is reasonable to expect that in thecoming years the HFS problem will receive an even larger amountof attention.

There are, however, some important remarks to be made. Table2 shows the percentage of papers that deal with 2, 3 or m-stageproblems and whether the machines at each stage are identical,uniform or unrelated.

As shown, a quarterof the reviewed literature deals with simpletwo-stage problems with identical parallel machines and almost athird only tackles two-stage problems. While these problems are of

theoretical interest, many times, the developed methods are not



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easily extendible to threeor more stages. Similarly, a large percent-age of the reviewed papers consider identical machines at eachstage (83.72%) and only a meager 6.97% of the literature tacklesm-stage problems with unrelated parallel machines at each stage.It is clear that the m-stage problem with unrelated parallel ma-chines is the most general case and therefore, the most likely tobe found in practice. As a matter of fact, from the reviewed litera-ture, most papers dealing with real problems do so with m stagesand unrelated parallel machines.

Similarly, in Fig. 2 we separate the reviewed literature amongthe different objective functions. Notice that ‘‘Other” includes costfunctions and/or problem or situation specic objective functions.

Clearly, the literature is heavily biased towards the C max crite-rion with a 60% of the references studying this single objective. To-tal/average completion time or owtime, both in their unweighted

and weighted forms, add up another 11%. It is striking to see thatfrom all surveyed papers, only a total of 1% deal with the earli-ness–tardiness criterion, which is so important for real problems.Another relevant observation is that only a handful of papers dealwith multiple objectives, and, to the best of our knowledge, the pa-pers dealing with more than one objective do so separately. We areonly aware of the recent papers of [85] and [86] that consider aweighted sum of objectives. Multi-objective scheduling is a veryrich eld of study as recent works show [186] . For regular owshop problems, the number of existing multi-objective approachesis large as reviewed by Minella et al. [122] . Therefore, multi-objec-tive scheduling for HFS is a necessary venue of research that hasnot been explored so far.

It is also interesting to study the different methodologies and

techniques that the authors apply in the reviewed literature.Fig. 3 shows a pie chart with this distribution. First of all, adding up B&B and mathematical programs andmodels we have a full 25%of reviewed papers. We have to consider

Fig. 1. Evolution of number of papers per year.

Table 2

Percentage of the reviewed papers according to number of stages and type of parallelmachines.

Number of stages Type of parallel machines Total

Identical Uniform Unrelated

2 25.12 1.86 4.65 31.633 4.19 1.4 0 5.59m 54.41 1.4 6.97 62.78

Total 83.72 4.66 11.62 100.00

C

60 %

F

1 %

L

1 %

T

2 %

C / F 9 %

T 6 %

U

4 %

C / F 2 %

T 2 %

Other10 %

Multi2 %

E / T 1 %

Fig. 2. Distribution of objective functions C /F = total/average completion or owtime, T = total/average tardiness, E /T = total/average sum of earliness and tardiness,multi = multiples objectives.

B&B10 %

MPF15 %

DR13 %

Heuristics37 %

TS6 %

SA5 %

GA8 %

Other6 %

Fig. 3. Distribution of employed techniques B&B = branch and bound, MPF = math-ematical programming and formulation, DR = dispatching rules, TS = tabu search,SA = simulated annealing, GA = genetic algorithms.



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Table 3

Summary of reviewed papers.

Year References Problem Comments

1970 [148] FH 2 ; ðð1 ð1Þ; P 2ð2ÞÞÞkC max B&B, small instances1971 [14] FH 2 ; ððP 2ð1Þ; 1ð2ÞÞÞkC max B&B, small instances

1973 [154] FHm; ððPM ðkÞÞmk¼1 Þjno wait jC max B&B, small instances[123] FH 2 ; ððP 2ð1Þ; 1ð2ÞÞÞkC max B&B, small instances

1979 [138] FH 2 ; ððPM ðkÞÞ2k¼1 Þjno idleð1Þjf T ; U g dispatching rules

1984 [129] FH 2 ; ðð1 ð1Þ; R2ð2ÞÞÞkwaiting and idleness Dispatching rules1985 [209] FH 3 ; ððPM ðkÞÞ3k¼1 ÞjskipjC max MPF and heuristics, ow lines

1987 [128] FH 2 ; ðð1 ð1Þ; R2ð2ÞÞÞkwaiting and idleness Dispatching rules[93] FHm; ððPM ðkÞÞmk¼1 Þjblock ; brkdwn ; S nsd ; skipjF Dispatching rules, ow lines

1988 [210] FH 3 ; ððPM ðkÞÞ3k¼1 Þjblock; skip jf C max ; WIP g Pseudo DP, circuit board manufacturing, ow lines[22] FHm; ððPM ðkÞÞmk¼1 ÞkC max B&B, MPF[58] FH 2 ; ððP 2ð1Þ; 1ð2ÞÞÞkC max Heuristics, NP-hard proof [94] FHm; ððPM ðkÞÞmk¼1 Þjblock ; brkdwn ; S nsd jF Dispatching rules, local search

1989 [175] FH 2 ; ððPM ðkÞÞ2k¼1 ÞkC max Dispatching rules based heuristics[219] FH 3 ; ððP 2ð1Þ; 1ð2Þ; R5ð3ÞÞÞjskip ; re v isit jC max Dispatching rules

1990 [170] FH 2 ; ððP 10 ð1Þ; P 12 ð2ÞÞÞkad-hoc MPF, paper industry

1991 [24] FHm; ððPM ðkÞÞmk¼1 ÞkC max B&B[62] FH 2 ; ðð1 ð1Þ; PM ð2ÞÞÞkC max Dispatching rules based heuristics[137] FHm; ððPM ðkÞÞmk¼1 ÞjS sd jf C max ; T g Ad-hoc heuristics, textile industry

1992 [77] FHm; ððPM ðkÞÞmk¼1 ÞÞkfT ; U g Restricted jobs. dispatching rules

[146] FHm; ððPM ðkÞÞmk¼1 ÞÞj prmu jF B&B

FH 2 ; ððPM ðkÞÞ2k¼1 ÞÞkF Heuristics[147] FHm; ððPM ðkÞÞmk¼1 ÞÞj prmu jC max B&B

1993 [2] FHm; ððRM ðkÞÞmk¼1 ÞjS sd jT w Dispatching rules, paper bags factory

[160] FHm; ððPM ðkÞÞmk¼1 ÞÞjbuffer ; skipjC max Heuristics, ow lines[189] FH 2 ; ððPM ðkÞÞ2k¼1 ÞÞkad-hoc Dispatching rules, photographic lm production[200] FH 2 ; ððPM ð1Þ; 1ð2ÞÞÞjS sd jC max TS and heuristics

1994 [40] FH 2 ; ððPM ð1Þ; PM ð2ÞÞÞjbuffer jC max Heuristics, petrochemical production[30] FHm; ððPM ðkÞÞmk¼1 ÞjblockjE

w þ T w MPF based heuristic[78]

FHm; ððPM ðkÞ

Þmk¼1 ÞkfC max ; F ; F max g

Restricted jobs. dispatching rules

[42] FHm; ððPM ðkÞÞmk¼1 ÞjskipjC max Heuristics, ow lines[50] FHm; ððPM ðkÞÞmk¼1 ÞkCost MPF, specic problem[63] FH 2 ; ðð1 ð1Þ; PM ð2ÞÞÞjS nsd ; Rnsd jC max Heuristics[105] FH 2 ; ððPM ðkÞÞ2k¼1 ÞkC max Heuristics, bounds[142] FHm; ððPM ðkÞÞmk¼1 ÞkF Dispatching rules

1995 [156] FHm; ððPM ðkÞÞmk¼1 ÞkC max Lower bounds[155] FHm; ððPM ðkÞÞmk¼1 ÞkC max MPR heuristic[31] FH 2 ; ððPM ð1Þ; 1ð2ÞÞÞkC max Heuristics, worst-case performance

FH 2 ; ðð1 ð1Þ; PM ð2ÞÞÞkC max Heuristics, worst-case performance[3] FHm; ððRM ðkÞÞmk¼1 ÞjS sd jf C max ; F max ; F g MPF, heuristics, carpet manufacturing[161] FHm; ððPM ðkÞÞmk¼1 ÞÞjblocking ; skipjC max Heuristics, ow lines[192] FH 2 ; ðð1 ð1Þ; RM ð2ÞÞÞkfC max ; F max g Heuristics

1996 [23] FHm; ððPM ðkÞÞmk¼1 ÞÞkT max Dispatching rules

[44] FH 3 ; ððPM ðkÞÞmk¼1 ÞÞj pmtn ; prec jT Heuristics, civil engineering[56] FH 2 ; ððPM ðkÞÞmk¼1 ÞÞkC max Heuristics, lower bounds[55] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max ; T max Heuristics, dispatching rules[71] FHm; ððPM ðkÞÞmk¼1 Þjno idlejC max MPF, heuristics[74] FH 2 ; ððPM ðkÞÞ2k¼1 ÞÞj pmtn jC max NP-hard proof [157] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Heuristics[190] FH 2 ; ðð1 ð1Þ; RM ð2ÞÞÞjM ð2Þ j jC max ; WIP ; CU Dispatching rules, lot streaming

[197] FH 2 ; ððPM ð1Þ; 1ð2ÞÞÞj pmtn ð1ÞjT max MPF, heuristics[199] FHm; ððPM ðkÞÞmk¼1 ÞÞkC B&B

1997 [5] FHm; ððPM ðkÞÞmk¼1 ÞÞjr j; other jseveral Real car production problem, dispatching rules

[20] FHm; ððPM ðkÞÞmk¼1 ÞÞj prec jC max Heuristics[54] FHm; ððPM ðkÞÞmk¼1 ÞÞjskipjC max Production planning, MPF, B&B[59] FH 2 ; ððPM ð1Þ; 1ð2ÞÞÞkC max B&B, heuristics[67] FH 2 ; ððPM ðkÞÞ2k¼1 ÞÞkC max Lower bound, SA, TS

[89] FH 2 ; ðð1 ð1Þ; R2ð2ÞÞÞkwaiting and idleness Dispatching rules. Note for [128]



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Table 3 (continued )


[88] FHm; ððPM ðkÞÞmk¼1 ÞÞkseveral Dispatching rules[90] FH 2 ; ððPM ðkÞÞ2k¼1 ÞÞjS nsd jC max Transfer batch problem, heuristics[110] FHm; ððPM ðkÞÞmk¼1 ÞÞjskipjf C max ; T

w g Problem space based local search

[111] FH 2 ; ðð1ð1Þ; PM ð2ÞÞÞjbatch ; S sd ; split jC max Heuristics[134] FH 2 ; ððPM ðkÞÞ2k¼1 ÞÞjsize jk; p j ¼ 1 jC max Heuristics, lower bounds

[176] FH 2 ; ððRM ðkÞÞ2k¼1 ÞÞj proportional jC max MPF, heuristics[178] FH 2 ; ððRM ðkÞÞ2k¼1 ÞÞkC max Heuristics[198] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max B&B, new lower bound

1998 [26] FHm; ððPM ðkÞÞmk¼1 ÞÞkfC max ; T g Dispatching rules[34] FH 2 ; ððPM ðkÞÞ2k¼1 Þjskip jC max TS[41] FH ðm 6 3Þ; ððQM ðkÞÞmk¼1 ÞÞkC max Polynomial algorithm when m ¼ 2 ;

B&B and heuristics when m ¼ 3[64] FH 2 ; ðð1ð1Þ; PM ð2ÞÞÞkU Heuristics[68] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Heuristics[75] FH 2 ; ðð1ð1Þ; Q 2ð2ÞÞÞkC max Heuristics[97] FH 2 ; ððPM ðkÞÞ2k¼1 ÞÞkC max Complex routings, heuristics[106] FH 2 ; ððP 1ð1Þ; 1ð2ÞÞÞjassembly ð2ÞjC max Heuristics, also symmetric problem[131] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max TS[143] FHm; ððPM ðkÞÞmk¼1 ÞkF Dispatching rules

[144] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Hybrid B&B-GA[149] FH 3 ; ð1ð1Þ; R2ð2Þ; 1ð3ÞÞjM ð2Þ j jC max B&B and heuristics

[218] FHm; ððPM ðkÞÞmk¼1 ÞÞkT w B&B, SBP, hybrid SBP-LS

1999 [121] FHm; ððPM ðkÞÞmk¼1 ÞÞksev eral DEA analysis over [88][25] FHm; ððPM ðkÞÞmk¼1 ÞÞkfC max ; F g Flow shop based heuristics[51] FHm; ððRM ðkÞÞmk¼1 ÞÞkC max SA[53] FHm; ððPM ðkÞÞmk¼1 ÞÞkfC max ; F ; WIP ; CU } Simulation model, dispatching rules[113] FH j j Review paper[195] FHm; ððQM ðkÞÞmk¼1 ÞÞjr jjC max Dispatching rules, heuristics, NP-hard proof

[196] FH j j Review paper

2000 [21] FHm; ððPM ðkÞÞmk¼1 ÞÞj prec ; lags ; S nsd ; Rnsd jLmax Heuristics[29] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max B&B[52] FHm; ððPM ðkÞÞmk¼1 ÞÞjno wait jC max TS, concrete blocks production[61] FH 2 ; ðð1ð1Þ; PM ð2ÞÞÞkC max Neural networks for choosing heuristics[72] FH 2 ; ððPM ðkÞÞ2k¼1 ÞÞkC max Fuzzy heuristics, fuzzy processing times[80] FHm; ððPM ðkÞÞmk¼1 ÞÞkseveral Dispatching rules[95] FH f m ¼ 2 ; 3g; ððPM ðkÞÞmk¼1 ÞÞkC max Heuristics[114] FHm; ððPM ðkÞÞmk¼1 ÞÞjS sd ; blockjE

w þ T w MPF based heuristics[125] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max B&B[162] FHm; ððPM ðkÞÞmk¼1 ÞÞjskip ; block ; reentry jC max MPF, ow lines[167] FH 2 ; ððPM ðkÞÞ2k¼1 ÞÞkC max Heuristics, machine number part of input[191] FH 2 ; ðð1ð1Þ; PM ð2ÞÞÞkC max Heuristics[193] FH 2 ; ððPM ðkÞÞ2k¼1 ÞÞkC max Heuristics, problem simplication[212] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max MPR-GA

2001 [15] FHm; ððPM ðkÞÞmk¼1 ÞkbarT B&B[37] FHm; ððPM ðkÞÞmk¼1 ÞÞkfC max ; Lmax g SB heuristic[70] FH 2 ; ððP 1ð1Þ; 1ð2ÞÞÞjassembly ð2ÞjC max Heuristics, also MPF[73] FHm; ððPM ðkÞÞm

k¼1ÞÞkC max Fuzzy heuristics, fuzzy processing times

[99] FHm; ððPM ðkÞÞmk¼1 ÞÞjskipjC max Lower bound[102] FHm; ððPM ðkÞÞmk¼1 ÞÞkT

w Dispatching rules

[130] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max LS, SA, TS[132] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max B&B[151] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max SA, planning and simulation[158] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Heuristics[163] FHm; ððPM ðkÞÞmk¼1 ÞÞjskip ; block ; reentry jC max MPF, ow lines[173] FH 3 ; ððPM ðkÞÞ3k¼1 ÞÞkC max Heuristics

2002 [4] FH 2 ; ððPM ðkÞÞ2k¼1 ÞÞknon-regular MPF, heuristics, photographic lm production[43] FHm; ððPM ðkÞÞmk¼1 ÞÞj pmtn ; prec jC max Polynomial algorithm for special case of prec ([74] )[60] FHm; ððPM ðkÞÞmk¼1 ÞÞkE

u þ T v þ C w þ d z Heuristic, assignable due dates,Controllable processing times

[69] FH 2 ; ðð1ð1Þ; Pmð2ÞÞÞjassembly ð2ÞjC max MPF, heuristics

(continued on next page )



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[83] FH 3 ; ððPM ðkÞÞ3k¼1 ÞÞkC max GA, printed circuit boards[150] FH 2 ; ðð1 ð1Þ; Q 2ð2ÞÞÞkC max MPF, DP, heuristics, NP-complete proof [164] FHm; ððPM ðkÞÞmk¼1 ÞÞjskip ; block ; reentry jC max MPF, ow lines[166] FHm; ððPM ðkÞÞmk¼1 ÞÞjskip ; block ; reentry jC max MPF, ow lines[169] FHm; ððQM ðkÞÞmk¼1 ÞÞkC max Heuristics

2003 [100] FHm; ððPM ðkÞÞmk¼1 ÞÞjS sd jC max Heuristics[112] FH 2 ; ððRM ðkÞÞ2k¼1 ÞÞjS

ð1Þsd ; M

ð2Þ j jwT max Heuristic, label sticker manufacturing

[135] FH 2 ; ððPM ðkÞÞmk¼1 ÞÞjsize jkjwT max Heuristics, lower bounds

[140] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Bottleneck exploiting heuristic[174] FH 2 ; ððQM ðkÞÞ2k¼1 ÞÞkC max Heuristics, lower bounds[206] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Neural Networks[211] FHm; ððRM ðkÞÞmk¼1 ÞÞjM jjC max GA

2004 [1] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Bottleneck exploiting heuristic[11] FHm; ððPM ðkÞÞmk¼1 ÞÞjav ailjsev eral Simulation, heuristics, SA[16] FH 3 ; ððPM ðkÞÞ3k¼1 ÞÞkC max Agent-based approach[18] FHm; ððPM ðkÞÞmk¼1 ÞÞjrecrc jU

w MPF, GA, lower bounds, checks processing

[45] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Articial immune systems[87] FHm; ððPM ðkÞÞmk¼1 ÞÞjblocking ; skipjC max Flow lines, MPF, TS, huristics

[101] FHm; ððPM ðkÞÞmk¼1 ÞÞjS sd jC max MPF, MPR-GA[108] FH 2 ; ðð1 ð1Þ; PM ð2ÞÞÞkT B&B[109] FHm; ððPM ðkÞÞmk¼1 ÞÞkT Bottleneck exploiting heuristic[136] FHm; ððPM ðkÞÞmk¼1 ÞÞjsize jk; prec jC max TS, polynomial and NP-hard cases proofs

[168] FHm; ððPM ðkÞÞmk¼1 ÞÞjsize jkjC max MPR-GA

[185] FHm; ððPM ðkÞÞmk¼1 ÞÞjno wait jC max MPR heuristics[208] FHm; ððPM ðkÞÞmk¼1 ÞÞjblockjC max MPR-TS[214] FH 2 ; ððPM ðkÞÞ2k¼1 ÞÞjno wait jC max Heuristics[216] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max TS-simulation, ceramic capacitor manufacturing

2005 [13] FH 3 ; ððPM ðkÞÞ3k¼1 ÞjS sd jother MPF, group technology[19] FH 2 ; ðð1 ð1Þ; P 2ð2ÞÞÞkC max B&B, GA, heuristics[39] FHm; ððPM ðkÞÞmk¼1 ÞÞjrecrc jT

w Dispatching rules, heuristics

[57] FH 2 ; ðð1 ð1Þ; P 2ð2ÞÞÞjno wait ; ð p j ¼ 1Þ1 jC max Exact method

[92] FHm; ððPM ðkÞÞmk¼1 ÞÞkfC max ; C g Review on exact solution methods

[117] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Group scheduling, setups, heuristics[119] FHm; ððRM ðkÞÞmk¼1 ÞÞjS nsd ; Rsd jC SA[124] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Hybrid B&B-GA[133] FHm; ððPM ðkÞÞmk¼1 ÞÞjsize jkjC max MPR-GA

[139] FHm; ððPM ðkÞÞmk¼1 ÞÞjS sd jT MPF, heuristics[165] FHm; ððPM ðkÞÞmk¼1 ÞÞjskipjf barT ; T max g MPF, ow lines[179] FHm; ððPM ðkÞÞmk¼1 ÞjskipjT Dispatching rules, ow lines[180] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Neural networks[183] FHm; ððPM ðkÞÞmk¼1 ÞÞjS sd jC max Neural networks[194] FHm; ððPM ðkÞÞmk¼1 ÞkC max Lower bounds[202] FHm; ððPM ðkÞÞmk¼1 Þjr jjf C max ; T max ; T

w ; C g Several variants of MPR-GA

[205] FH jskip ; j Review paper[213] FH 2 ; ððPM ðkÞÞ2k¼1 Þjav ailjC max Approximation algorithms[225] FH 2 ; ððPM ð1Þ; 1ð2ÞÞÞkF Heuristics, bounds, lot streaming

2006 [8] FHm; ððPM ðkÞÞmk¼1 ÞÞkC MPR-SA[12] FH 2 ; ðð1 ð1Þ; PM ð2ÞÞÞjav ailjC max B&B, heuristics, complexity[36] FH 3 ; ððRM ðkÞÞ3k¼1 ÞÞj prec ; block; S nsd jC max MPR-TS[66] FH 2 ; ððPM ðkÞÞ2k¼1 ÞkC max B&B[84] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max MPR-SA, lower bounds[103] FHm; ððQM ðkÞÞmk¼1 ÞÞkC max FS based heuristics[104] FH 2 ; ððQM ðkÞÞ2k¼1 ÞÞkC max Heuristics, lower bounds. Note for [174][118] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Group scheduling, TS[152] FHm; ððRM ðkÞÞmk¼1 ÞÞjS sd ; M jjC max MPR-GA

[172] FHm; ððRM ðkÞÞmk¼1 ÞÞjbuffer jsev eral Heuristics[182] FHm; ððPM ðkÞÞmk¼1 ÞkF

w MPF, lagrangian relaxation

[181] FHm; ððPM ðkÞÞmk¼1 Þjbuffer jF w MPF, lagrangian relaxation

[203] FHm; ððPM ðkÞÞmk¼1 ÞÞk composite functions Hybrid GA + dispatching rules[222] FHm; ððPM ðkÞÞmk¼1 ÞÞjsize jkjC max ACO

[224] FHm; ððPM ðkÞ

Þmk¼1 ÞÞjS sd jC max Articial immune system



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that these techniques have proven so far useful either for simpli-ed problems, specic settings and/or small problems so a largerfocus is needed on approaches able to solve more general and lar-ger problems. Under heuristics we have classied many differentalgorithms and ad-hoc methods that are specic and do not con-tain a well known metaheuristic template. Most times these meth-ods are single-pass heuristics able to solve large problems but witha limited performance. Together with dispatching rules, thesemethods add up a 50% of the reviewed papers. This means thatonly the remaining 25% is left for SA, GA, TS and other metaheuris-tic techniques like ACO, PSO and other recent methodologies.Clearly, there is a large opportunity for research here. Metaheuris-tics have long ago established themselves as state-of-the-art meth-

odologies for the vast majority of scheduling problems andtherefore, HFS should be no exception.

5. Research opportunities and conclusions

In this paper we have reviewed and analyzedmore than 200 pa-pers dealing with the hybrid ow shop (HFS) or related variants.This eld of study is attracting more research efforts due to themany applications that this realistic problem setting has in prac-tice. In the review, we have classied all the papers according tomany parameters, including problem variant studied, constraints,objective functions and employed methodologies. However, suchextensive analysis is necessarily shallow in the sense that we have



2007 [6] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Ant colony optimization[28] FH 2 ; ððP 2ð1Þ; 1ð2ÞÞÞjbatch jC max TSP-based heuristics[35] FH 3 ; ððRM ðkÞÞ3k¼1 ÞÞjS sd ; block ; prec jC max MPF, lower bounds, TS[48] FHm; ððPM ðkÞÞmk¼1 ÞÞjassign j

ET TS, special problem[79]

FHm; ððPM ðkÞ

Þm

k¼1 ÞÞjr jjCost TS, SA, heuristics

[81] FHm; ððRM ðkÞÞmk¼1 ÞÞjlot ; skipjCost GA, SA, ow lines[96] FH 3 ; ððPM ðkÞÞ3k¼1 ÞÞkC max Heuristics[145] FHm; ððPM ðkÞÞmk¼1 ÞÞjbatch ; skipjf F ; Cost g GA, SA, ow lines[145] FH j j Review paper[159] FHm; ððPM ðkÞÞmk¼1 ÞÞjr j; skipjU MPF, inventory constraints, rescheduling

[204] FHm; ððPM ðkÞÞmk¼1 ÞÞjr j; d jjseveral FS based heuristics

[201] FHm; ððPM ðkÞÞmk¼1 ÞÞjS sd jT w MPF, DR, heuristics. Multi-project RCPSP

[217] FHm; ððPM ðkÞÞmk¼1 ÞÞkseveral GA, simulation[215] FHm; ððPM ð1Þ; PM ð2ÞÞÞjbatch ðmÞjF MPF, lagrangian relaxation, heuristics

2008 [32] FHm; ððRM ðkÞÞmk¼1 ÞÞjskip jU Heuristics, ow lines[46] FH 2 ; ððRM ð1Þ; 1ð2ÞÞÞj pmtn ; resources ð1ÞjC max Heuristics[65] FHm; ððPM ðkÞÞmk¼1 ÞÞkC max Lower bounds, note over [84][76] FHm; ððPM ðkÞÞmk¼1 ÞÞj p jjF

w Proportionate shop, heuristics, column generation

[85] FHm; ððRM

ðkÞ

Þmk¼1 ÞÞjS sd ; r jjaC max þ ð 1 aÞU

MPF, heuristics, GA, SA, TS

[98] FHm; ððPM ðkÞÞmk¼1 ÞÞkseveral Heuristics, simulation[115] FH 2 ; ððPM ð1Þ; 1 ð2ÞÞÞkC max Heuristics, bounds, lot streaming[116] FHm; ððRM ðkÞÞmk¼1 ÞÞjno wait ; S nsd jsev eral MPF[120] FH 2 ; ððRM ð1Þ; 1ð2ÞÞÞjM jjC max Heuristics, eligibility on rst stage

[153] FHm; ððRM ðkÞÞmk¼1 ÞÞjskip ; rm ; lag ; S sd ; M j; prec jC max MPF, heuristics

[171] FHm; ððPM ðkÞÞmk¼1 ÞÞj p jjF w Proportionate shop, GA

[177] FH 2 ; ðð2ð1Þ; P 2ð2ÞÞÞjassembly ð2ÞjF Heuristics[187] FHm; ððPM ðkÞÞmk¼1 ÞÞjsize jkjC max Particle swarm optimization

[188] FH 2 ; ðð1ð1Þ; P 2ð2ÞÞÞjskipð1ÞjC max Heuristics

2009 [86] FHm; ððRM ðkÞÞmk¼1 ÞÞjS sd ; r jjaC max þ ð 1 aÞU MPF, heuristics, dispatching rules, GA[91] FH 2 ; ððPM ð1Þ; 1 ð2ÞÞÞkC max Heuristics, product-mix[184] FHm; ððPM ðkÞÞmk¼1 ÞÞjskip ; block ; reentry jC max GA mixed with LS[221] FHm; ððPM ðkÞÞmk¼1 ÞÞjsize jkjC max Iterated greedy (IG)

[17] FH 2 ; ððPM ð1Þ; PM ð2ÞÞÞjbatch ð2ÞjC max Heuristics[207] FHm; ððPM ðkÞÞmk¼1 Þjbuffer jF

w TS

[107] FHm; ððPM ðkÞÞmk¼1 ÞÞjr jjF Dispatching rules

[126] FHm; ððPM ðkÞÞmk¼1 ÞÞjS sd ; transport jf F ; T g SA[220] FHm; ððRM ðkÞÞmk¼1 ÞÞjS sd ; M j; buffer jC max MPR-GA

[33] FHm; ððRM ðkÞÞmk¼1 ÞÞjskip jT Heuristics, ow lines[47] FH 2 ; ððRM ð1Þ; 1ð2ÞÞÞj pmtn ; resources ð1ÞjC max Heuristics, GA, SA[38] FH 2 ; ððPM ðkÞÞmk¼1 ÞÞkU B&B, heuristics[127] FHm; ððPM ðkÞÞmk¼1 ÞÞjS sd jf F ; T g SA[7] FHm; ððPM ðkÞÞmk¼1 ÞÞjS sd ; reentry ; batch jsev eral Simulation, heuristics, TS[223] FHm; ððPM ðkÞÞmk¼1 ÞÞjsize jkjC max Heuristics

Mathematical programming formulation (MPF), dynamic programming (DP), tabu search (TS).Branch and bound (B&B), multiple permutation representation (MPR), work in progress (WIP).Local search (LS), shifting bottleneck procedure (SBP), simulated annealing (SA).



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not focused our attention in any particular HFS variant or solutiontechnique. In any case, we are certain that this review work will behelpful forother researchers in the area as well as for establishingareference starting point for new research efforts.

In practice, objectives vary and hence a variety of HFS modelsare possible. It is unrealistic for the minimization of C max to matchall cases. Nevertheless, 60% of the reviewed papers are exclusivelyconcerned with it. A very small percentage of the remaining pa-pers, on the other hand, are dedicated to the solution of problemswith real world motivated functions. This imbalance seems to beunjustied. Minimizing makespan may be relevant in several casessince it optimizes the use of limited resources. However, there areother objectives that in practice are sensible too. For instance, min-imizing holding costs (inventory costs) may be more relevant thanminimizing makespan, or to meet the clients demands on time, orboth of them at the same time. Unfortunately, it is not feasible tostudy all possible cost functions that could arise in practice. Thesame situation occurs with the constraints and assumptions, it isunlikely that a real world problem exactly matches any of themodels intensively studied in the literature. It seems to be a morepromising strategy to generate heuristics which show exibility ona wide range of HFS problems. It is also important to consider thatthe real world is unpredictable and dynamic. Algorithms must beable to nd solutions which remain robust under different scenar-ios. No results have been found on robust scheduling in HFS. More-over, the equally important problem of rescheduling has notreceived the attention it deserves. In both cases, technologies thathave been developed to address such problems in other schedulingscenarios, such as job shop [82] , should be adapted to HFS.

Production scheduling problems are multi-objective (MO) bynature, which means that several criteria, in conict with eachother, have to be considered at a time. Research in MO optimiza-tion is concerned with the generation of solutions in which noneof the objective functions can be improved without paying a costin other objective(s) (usually referred to as non-dominated solu-tions). In [210] , makespan and the inventory in process were con-sidered simultaneously, however, the heuristic developedconcentrated on minimizing makespan and the inventory in pro-cess was treated as a secondary objective. No attempt on ndingnon-dominated solutions in HFS has beenreported in the literatureto the best of our knowledge. Setup times in scheduling have re-cently attracted a lot of attention. The recent review paper of [9]and the study of setup importance from [10] are just some exam-ples. Apart from some isolated reviewed papers, sequence depen-dent setup times have been scarcely studied in HFS settings andmore research is needed in this regard. Finally, more effectivemetaheuristic templates are being proposed. The bulk of this re-search, however, only concentrates on relatively simple combina-torial optimization problems. A concentrated effort is required inorder to apply these recent methodologies to complex HFS

problems.

Acknowledgement

Rubén Ruiz is partially funded by the Spanish Ministry of Sci-ence and Innovation, under the Projects DPI2008-03511/DPI andIAP-020100-2008-11.

Appendix A

Table 3 .

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