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Flowshop scheduling research after five decades
European Journal of Operational Research 169 (2006) 699–711 www.elsevier.com/locate/ejor
Editorial
Flowshop scheduling research after five decades Jatinder N.D. Gupta *, Edward F. Stafford Jr. College of Administrative Science, The University of Alabama in Huntsville, Huntsville, AL 35899, USA Available online 2 April 2005
Abstract Since JohnsonÕs seminal paper in 1954, flowshop scheduling problems have received considerable research attention over the last fifty years. As a result, several optimization and heuristic solution procedures are available to solve a variety of flowshop scheduling problems. This paper provides a brief glimpse into the evolution of flowshop scheduling problems and possible approaches for their solution over the last fifty years. It briefly introduces the current flowshop problems being solved and the approaches being taken to solve (optimally or approximately) them. The paper concludes with some fruitful directions for future research. 2005 Elsevier B.V. All rights reserved. Keywords: Flowshop scheduling; Historical developments; Problem assumptions; Current research status; Future research directions
1. Introduction A frequently occurring operational problem is one of processing a given number of jobs (commodities) on a specified number of machines (facilities)—referred to by various investigators as scheduling, dispatching, sequencing, or combinations thereof. The first of these terms will be used here since this encompasses dispatching and sequencing. The desire to process the jobs in a special order to achieve some objective function is what creates a problem that remains largely un*
Corresponding author. Tel.: +1 256 824 6593; fax: +1 256 824 2929. E-mail address: [email protected] (J.N.D. Gupta).
solved. The actual situations that give rise to scheduling problems are wide and varied. Thus, for example, we have single machine scheduling problem, multiple machine scheduling problem, manpower scheduling problem, to name a few. In this paper and in this feature issue, we will consider a specific class of multiple machine scheduling problems, called the flowshop scheduling problems. The first paper on flowshop problem was published fifty years ago. In the 50 years since Johnson (1954) published his seminal paper on flowshop scheduling, more than 1200 papers on various aspects of this problem have been published in the operational research literature. We are, therefore, quite pleased to be serving as guest editors
0377-2217/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.02.001
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of this special issue of European Journal of Operational Research on flowshop scheduling celebrating a golden jubilee of JohnsonÕs paper. This special issue also honors JohnsonÕs contributions by providing a glimpse of the research that evolved since his original paper. In his paper, Johnson credited Richard Bellman with giving him the problem for analysis and research. Our research revealed that shortly after JohnsonÕs work, Bellman published at least two papers that deal with two-machine flowshop problem (Bellman and Gross, 1954; Bellman, 1956; Bellman et al., 1982). A study of BellmanÕs work indicates that the two-machine flowshop problem in JohnsonÕs paper was inspired by a real-life situation. This settles the issue raised by Dudek et al. (1992) about the relevance of the flowshop problems. JohnsonÕs rule or algorithm, as his method is known today, is a simple technique for optimally sequencing a set of n jobs to be processed in a two-machine manufacturing system. Johnson also offered an optimizing technique for the three-machine system wherein the processing times for the jobs on the three machines had certain restrictive relative relationships. It is noteworthy that Johnson did not use the term flowshop to identify the production system represented by his model. As a part of our research for this paper, we attempted to identify the published paper wherein the term flowshop, flow shop, or flow-shop was first used. In our first pass, we found that Ignall and Schrage (1965) used the term in the title of their early paper on branch-and-bound for the regular flowshop. Subsequently, we have traced the term to an earlier paper by Heller (1960). In one of his previous working papers, Heller (1959) used the term conservative assembly line to describe a flowshop.1
This paper provides a glimpse of the developments in flowshop scheduling over the last fifty years.2 The problem assumptions, problem formulation, and solution approaches used in solving the flowshop scheduling problem are described and current state of research is introduced. Subsequently, several plausible and fruitful directions for future research are outlined. 2. The flowshop scheduling problem A flowshop is characterized by more or less continuous and uninterrupted flow of jobs through multiple machines in series. In such a shop, the flow of work is unidirectional since all jobs follow the same technological routing through the machines. Although this description of flowshop resembles an assembly-line operation, there are several differences. First: a flowshop is equipped to handle a variety of jobs as opposed to a standard product manufactured by an assembly-line. Second: the jobs in a flowshop do not have to be processed on all machines; that is, a job may skip some operations according to its technological requirements. However, in an assembly-line, all jobs have to move from one station to another without skipping any work-station. Third: in a flowshop, each machine is independent of other machines and can be loaded independently; whereas in assembly-line operations, each work station depends on the preceding one. And finally, each job has its own processing time at each machine in a flowshop; however, all units of a product have a standard time at each work station in an assembly-line (Ashour, 1972). Because of these differences, Heller (1959) characterized a flowshop as a conservative assembly line. 2.1. Flowshop problem definition
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To settle this issue once and for all, we make the following offer to all in the flowshop research community. We will take to dinner the first person to correctly identify the earliest published paper in which the term flowshop, or one of its variant spellings, was used. Simply send us a copy of that paper along with the correct citation. Deadline is December 31, 2005, and the dinner will be at a future operational research meeting, or in Huntsville, Alabama, USA. In Huntsville, we will also take you for a tour of the Space and Rocket Center.
As stated earlier, the work-flow in a flowshop is unidirectional. This means that the order in which 2
Since the purpose of this paper is to provide a glimpse into flowshop scheduling research and to introduce this special issue, references cited are neither exhaustive nor complete. They are cited for convenience of the reader to locate some additional readings if desired.
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jobs are processed on various machines is the same for all n jobs and is specified. Since the nomenclature of the machines is arbitrary, the machines may be numbered such that each job is processed on machine 1 first, machine 2 second,. . ., and machine m last. With this numbering of machines and the abstract terminology of jobs and machines, the traditional flowshop scheduling problem considered by Johnson may be defined as follows (see Tanaev et al., 1994) for a comprehensive treatment of flowshop problems): • Given n jobs to be processed on m machines in the same technological order; the processing time of job i on machine j being pij (i = 1,2, . . ., n; j = 1,2, . . ., m); it is desired to find the order (schedule) in which these n jobs should be processed on each of the m machines to minimize a well defined measure of production cost. The above problem, when considered in the general case, gives (n!)m possible schedules. Even for problems as small as n = m = 5, the number of possible schedules is so large that a direct enumeration is economically impossible. A simplified version of the above problem, applicable to many situations in computer centers, production shops, and other similar situations exists in which it is assumed that the order (sequence) in which these jobs are processed on each machine is the same. For this simplified version, the number of feasible schedules reduces to n!. This simplified version of the flowshop scheduling problem can be solved to seek an optimal schedule for the general flowshop problem described earlier if certain conditions regarding the number of machines (m) and the measure of production cost are satisfied. Conway et al. (1967) state the following two theorems to describe these conditions. Theorem 1. For the m-machine flowshop scheduling problem with all jobs simultaneously available and the objective of minimizing a non-decreasing function of completion times of jobs, it is sufficient to consider only those schedules in which the same job sequence occurs on the first two machines.
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Theorem 2. For the m-machine flowshop scheduling problem with the objective of minimizing make-span, it is sufficient to consider only those schedules in which the same job sequence occurs on the last two machines. Now, if the number of machines, m = 2, the simplified version will in fact find global optimal solution. Similarly, if the objective is minimization of make-span (defined as the total throughput time in which all jobs complete processing on all machines) and the number of machines m 6 3, the simplified version will lead to global optimal solution. The flowshop problems considered by Johnson (1954) satisfied these conditions. Even when it is sufficient to solve the simplified version to obtain global optima, it may be difficult to identify the optimal schedule. For this reason, the assumption that the same job sequence is followed at each machine is active in practically every research effort in flowshop scheduling. 2.2. Flowshop assumptions The foregoing assumptions are only a few of several assumptions imposed on the flowshop scheduling problem. Scheduling problems in general and flowshop cases in particular are structured rather restrictively from a practical viewpoint. Implicitly, and without specific statement, the formulations of these problems overlook several important factors that are encountered in operational situations. For example, Ashour (1972) cites several factors that may interrupt the flow of work in a shop and hence cause unexpected delays. Some of the delay causing factors are: (1) unpredictable breakdown of machines; (2) the variability in the performance and absenteeism of operators; (3) delay in supply of materials, fixtures, or tools; (4) the variability of processing times; (5) changes in job specifications; (6) rejection and changes in job identity due to defective production; and (7) customer pressures that may cause rush orders for early completion of jobs. The restrictive assumption of unidirectional flow pattern of jobs in a flowshop, though more general than that of an assembly-line, limits the variety of output of the shop. Indeed, the flowshop is very restrictive. Nevertheless, progress in scheduling
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theory is not appreciable even to solve this very restrictive class of scheduling problems. As a result, the already restrictive case of flowshop must have several additional simplifying assumptions imposed on it. Though these assumptions increase the generality of models thus developed and the possibility of finding an optimal schedule; they do, however, divorce these models from realistic situations. While not originally stated in JohnsonÕs paper, the additional restrictive flowshop assumptions include the characteristics of jobs, machines, and operating policies of the shop. Gupta et al. (1979) describes the following 21 assumptions for the traditional flowshop scheduling problem originally conceived in Johnson (1954) paper.
M3. Each machine in the shop operates independently of other machines and thus is capable of operating at its own maximum output rate. M4. Each machine can process at most one job at a time. This eliminates those machines that are designed to process several jobs simultaneously like multi-spindle drill. M5. Each machine is continuously available for processing jobs throughout the scheduling period and there are no interruptions due to breakdowns, maintenance or other such causes.
2.2.3. Assumptions concerning operating policies 2.2.1. Assumptions concerning job J1. Each job is released to the shop at the beginning of the scheduling period. J2. Each job may have its own due date which is fixed and is not subject to change. J3. Each job is independent of each other. J4. Each job consists of specified operations, each of which is performed by only one machine. J5. Each job has a prescribed technological order which is the same for all jobs and is fixed. J6. Each job (operation) requires a known and finite processing time to be processed by various machines. This processing time includes transportation and setup times, if any, and is independent of preceding and succeeding jobs. J7. Each job is processed no more than once on any machine. J8. Each job may have to wait between machines and thus in-process inventory is allowed.
2.2.2. Assumptions concerning machines M1. Each machine center consists of only one machine; that is, the shop has only one machine of each type. M2. Each machine is initially idle at the beginning of the scheduling period.
P1. Each job is processed as early as possible. Thus, there is no intentional job waiting or machine idle time. P2. Each job is considered an indivisible entity even though it may be composed of a number of individual units. P3. Each job, once accepted, is processed to completion; that is, no cancellation of jobs is permitted. P4. Each job (operation), once started on a machine, is completed to its completion before another job can start on that machine, that is, no preemptive priorities are assigned. P5. Each job is processed on no more than one machine at a time. (This is a result of assumptions J5 and P2.) P6. Each machine is provided with adequate waiting space for allowing jobs to wait before starting their processing. P7. Each machine is fully allocated to the jobs under consideration for the entire scheduling period; that is, machines are not used for any other purpose throughout the scheduling period. P8. Each machine processes jobs in the same sequence. That is, no passing or overtaking of jobs is permitted. The above list of assumptions indicates how explicitly practical situations have to be analyzed
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before conceptual scheduling models can be used. Relaxation of one or more of these assumptions yields various scheduling models, both for the flowshop and other more general shops. Over the years, researchers, in dropping or modifying one or more of the above assumptions, have introduced variants of the original Johnson model which more closely approximate real-life production systems. Terms like no intermediate queues (NIQ), sequence-dependent setup times (SDST), and hybrid flowshops show the degree to which JohnsonÕs original (classical or regular) flowshop problem has evolved to represent a myriad of related production systems.
3. Five decades of flowshop scheduling research The early group of flowshop researchers was quite small, and these people were concentrated in a few US academic and research institutions such as UCLA, Cornell, RAND Corporation, and Texas Tech University. TodayÕs flowshop research community, as evidenced by the authors appearing in this issue, is global. We received responses to our call for papers from every continent and every geographical region, Antarctica excepted. It is not unusual for a recently published flowshop paper to be authored by researchers from three different countries on two or more continents. This speaks well for the future of flowshop research. In this section, we provide a glimpse of the problems addressed, solution approaches adapted, and related research developments in flowshop scheduling by each decade. 3.1. First decade (1955–1964) After the publication of JohnsonÕs classical paper on the two- machine flowshop scheduling problem, developments in the first decade include the consideration of the m-machine flowshop problem to minimize makespan and the inclusion of start and stop time-lags in the two-machine flowshop scheduling problems. Early research on the flowshop problem was highly theoretical, and it tended toward optimization techniques such as
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mathematical programming (see, for example, Wagner, 1959; Manne, 1960). Thus, this decade saw consideration of mathematical programming approaches to flowshop problems on one hand and the use of Monte-Carlo simulation techniques on the other (for a review of these developments, see Sisson (1959) and Muth and Thompson (1963)). While only a few solution techniques were developed during this decade, several alternate processing configurations and scenarios were identified for scheduling problems. The problem sizes solvable in these early research efforts were quite small for three reasons: (1) lack of computer power; (2) lack of efficient computer programs; and, more importantly, (3) most variants of the two-machine flowshop problem are NP-hard. By the early 1960s, researchers were attempting to extend JohnsonÕs analysis to the m-machine case. Thus, for example, the combinatorial approach that Johnson used for the two-machine problem was extended to the m-machine flowshop problem by Dudek and Teuton (1964). Even though it was later shown to be incorrect, it began the analytical framework for the development of dominance conditions for flowshop scheduling.
3.2. Second decade (1965–1974) The second decade witnessed a rather wide range of solution techniques on one hand and the consideration of objective functions other than makespan on the other. The first paper of this decade is difficult to identify as there are several papers that appeared around the same time. The combinatorial approach started by Dudek and Teuton (1964) was corrected and improved by Smith and Dudek (1967), McMahon (1969), Gupta (1971), and Szwarc (1971, 1973). The branch and bound solution approaches were also developed during this time by Lomnicki (1965), Brown and Lomnicki (1966), McMahon and Burton (1967), Gupta (1969b, 1970), and others. These branch and bound approaches differ in terms of the lower bounds used and the branching strategies. A comprehensive framework for these branch and bound algorithms is provided by Lageweg et al. (1978). Except for the papers by Ignall and
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Schrage (1965) and Gupta (1972), all these papers consider makespan as the optimality criterion. Ignall and Schrage (1965) considered the twoand-three machine flowshop problems with total flowtime criterion as well. Gupta (1972) considered the existence of start and stop time-lags on one hand and several objective functions on the other. The difficulty in developing optimization algorithms for flowshop scheduling problem resulted in developing and using heuristics for finding good, near-optimal solutions for larger and larger flowshop problems (see Framinan et al. (2004) for a review of these and subsequent heuristic approaches to flowshop problems). 3.3. Third decade (1975–1984) The emergence of the theory of NP-completeness (which is now called NP-hardness) had a profound impact on the direction of developments in flowshop scheduling (see Garey and Johnson (1979) for a detailed study of NP-hardness). On one hand, there are developments to resolve the complexity status of various flowshop scheduling problems (see Brucker (1998), Chen et al. (1998), Lawler et al. (1993) and Pinedo (1995) for reviews of complexity of scheduling problems); on the other hand, many more heuristic approaches were developed during this time (see the book by Morton and Pentico (1993) for various heuristic approaches). In addition, this decade witnessed the proliferation of a variety of flowshop problems and consideration of flowshop problems that violate several restrictive assumptions listed earlier in this paper. Thus, for example, this decade witnessed an interest in identifying the worst-case performance of a heuristic algorithm on one hand (see Smutnicki (1998) for a review), and the consideration of flowshop scheduling problems with separable setup times on the other. Further, this decade also witnessed the consideration of due date related objective functions in flowshop scheduling (see Morton and Pentico (1993) for these developments). The third decade also witnessed the consideration of stochastic processing times in analyzing various flowshop problems. While the progress in the development of stochastic models for solving
flowshop problems has been slow, significant results are available and described by Dempster et al. (1982) and Pinedo (1995). 3.4. Fourth decade (1985–1994) This decade saw the emergence of hybrid flowshops where each stage of the flowshop could contain multiple parallel machines and the development of metaheuritics like Tabu Search, Genetic Algorithms, and Simulated Annealing (see Aarts and Lenstra (1997), Osman and Kelly (1996) and Rayward-Smith et al. (1996) for illustrations of such approaches). As a result of the consideration of several objective functions and various heuristic approaches, we witnessed the expansion of efforts to solve flowshop problems with separable setups which could be either sequence independent or sequence dependent (see the reviews by Allahverdi et al. (1999) and Cheng et al. (2000)). While such problems were identified as early as 1969 and some solution techniques were developed around that time (see Corwin (1969), Corwin and Esogbue (1974), Gupta (1969a, 1975) and Uskup and Smith (1975) for such early efforts), the developments in the fourth decade are significant. This decade also witnessed the use of artificial intelligence based techniques to solve the flowshop scheduling problems. Such attempts include the development of decision support systems and expert systems (see Brown et al. (1995), Byrd and Moore (1983) and Zweben and Fox (1994) as examples of such attempts). 3.5. Fifth decade (1995–2004) The proliferation of the variety of flowshop scheduling problems, objective functions, and solution approaches continued in this decade. This decade had seen several developments and improvements to the already identified and evolved flowshop problems. While it is difficult to identify any one single problem as being representative of developments during this decade that was not considered earlier, it is interesting to note that consideration and solution of multi-criteria
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flowshop scheduling became quite popular in this decade (see TÕKindt and Billaut (1993) for a review of multi-criteria scheduling problems). This decade did witness expanded attention on the simultaneous consideration of lot sizing and scheduling on one hand and the batch processing machines on the other (see Potts and Van Wassenhove (1992), Potts and Kovalyov (2000) and Trietsch and Baker (1993) as examples of these developments). Ruiz et al. (2005) investigate the effectiveness and efficiency of various metaheuristics in minimizing makespan in flowshops with separated sequence dependent setup times (SDST). These heuristics are adaption of existing metaheuritics for the regular flowshop to SDST flowshop. In addition to testing existing algorithms, these authors also propose and compare two advanced genetic algorithms for solving the SDST flowshop problem. Utilizing the existing benchmark problems, they show that their proposed algorithms provide better solutions than existing algorithms. This decade also saw significant progress in developing solution procedures for robotic flowshops and flowshops equipped with automatic guided vehicles (AGVs) as is evident from the recent review by Geismar et al. (2004). In addition, because of the recent advances in computational efficiencies, there is a renewed interest in the use of mathematical programming approaches to flowshop problems (see the recent paper by Tseng et al. (2004) as an example of such efforts). While the evolution of flowshop problems has been categorized by five decades since the publication of JohnsonÕs seminal paper, it should be mentioned that most flowshop problems were envisioned to some extent during the first two decades. Further, once a problem is identified and some progress in its solution is reported, research on further developments related to this problem continues through subsequent decades.
4. Current flowshop scheduling research When we first envisioned this special issue, we anticipated a set of survey papers covering the past, present, and future of flowshop research.
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Instead, we received submissions that were as eclectic as those appearing in any randomly chosen issue of this journal. Thus, the papers that survived the rigorous review process and included in this special issue represent the current state of the art in flowshop scheduling. Four papers deal with different aspects of the two-machine flowshop problem; and one paper examines the two job, m-machine problem. Two papers concentrate on the flowshop variants including separable setup times. Two papers involve different solution technique approaches: adaptive learning and traveling salesman modelling of flowshop problems. The final two papers involve analyses of the difficulty of solving flowshop problems. 4.1. Two-machine problems Briand, La and Erschler introduce a conceptual technique for associating jobs with intervals rather than sequence positions. They then use AllenÕs algebra to establish sufficient optimality conditions that characterize a large set of optimal sequences. This set includes the Johnson sequences among others. While not useful in minimizing makespan in a two-machine flowshop, their suggested approach may be useful to solve problems involving secondary criteria where the primary criterion is makespan. Mukherjee and Chatterjee examine the use of problem decomposition for solving the twomachine flowshop problem. Since the shifting bottleneck heuristic has proven successful in finding good solutions for the more complex job shop problem, they examine its applicability to flowshop problems. For the two-machine flowshop problem, their proposed decomposition based shifting bottleneck approach does generate an optimal makespan schedule even though the computational complexity is much higher than that of JohnsonÕs algorithm. Their main contribution is showing how the job shop decomposition techniques may be modified for finding good solutions to the permutation flowshop problem. In his original paper, Johnson assumed that job processing times were known and deterministic. Kalczynski and Kamburowski describe a heuristic for minimizing the expected makespan with
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consistent coefficients of variation in the job processing times. They assume that the processing times follow a Weibull distribution, that the known solution techniques like JohnsonÕs and TalwarÕs rules are special cases of their work. Using simulation, they examine the effectiveness of their heuristic to minimize expected makespan. Portougal and Trietsch consider the two-machine flowshop problem with stochastic processing time to minimize expected makespan and to find optimal service level. Since the identification of optimal service level requires the computation of the entire makespan distribution, they develop and empirically test two heuristics (JohnsonÕs rule applied to means of the service time distributions; and a pair-switching approach) for the twomachine flowshop problem with stochastic job processing times. They also investigate the impact of sequencing on the variance of the makespan and show that JohnsonÕs rule may also be effective to solve stochastic flowshop problems. 4.2. Two-job problems Averbakh revisits the two-job, m-machine permutation flowshop problem with stochastic processing times. For each operation, the processing time can take on any value with a given uncertainty time interval. Based on a geometric formulation of the problem, Averbakh proposes a linear time algorithm (with O(m) time complexity) algorithm for the minmax regret solution. 4.3. Flowshops with setup times While JohnsonÕs original paper assumed that setup times are included in the processing times, later studies have shown that separating the setup times may lead to better schedules. In addition, these times may be sequence dependent. Allahverdi and Al-Anzi use the three-machine flowshop problem with separated sequence independent setup times to model the three-site distributed database scheduling problem. Utilizing specifically developed dominance conditions and a lower bound, they use a branch-and-bound approach for finding a schedule that minimizes the sum of job completion times. To reduce the computa-
tional effort of their proposed branch and bound algorithm, they use a three-phase hybrid heuristic algorithm to generate an upper bound. This heuristic is also shown to be effective in finding approximately optimal schedule for problems containing a large number of jobs. In many practical applications, the two or more machines are available at each stage in a flowshop. These problems are called hybrid flowshops and may involve some machine eligibility constraints. Ruiz and Maroto consider a hybrid flowshop problem with machine eligibility requirements, sequence dependent setup times and unrelated parallel machines. They adapt the best genetic algorithm for the SDST flowshop developed by Ruiz et al. (2005) and show that their proposed algorithm is quite effective in minimizing makespan. 4.4. Solution approaches As the nature of flowshop problems becomes complex, improved and new solution approaches are needed to solve these problems. Many of these solution approaches are adaption of approaches used for successfully solving other combinatorial optimization problems. Agarwal, Colak, and Eryarsoy describe an improvement heuristic approach to solving the regular flowshop problem. Their technique starts with a one-pass starting solution which is then improved on by using a weighted parameter to perturb the original data. The learning done in their heuristic is similar to the approach used in neural networks and promises good results and possible use to solve other scheduling problems. Bagchi, Gupta, and Sriskandarajah provide a review of the traveling salesman problem (TSP) based techniques found in the literature for solving several variants of the flowshop problem. They show that the use of TSP based approaches is quite useful in optimally solving some rather complex flowshop problems found in robotic shops. In addition, they also review the computational complexity of a variety of flowshop models and show that TSP based heuristic approaches can be used to develop effective heuristic solution procedures for various scheduling problems. Based on their
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review of existing literature, they also provide some useful directions for future research including the application of Group TSP modelling approaches for flowshop problems. 4.5. Complexity and performance guarantees No special issue involving the flowshop problem would be complete without addressing the issues of computational complexity and performance guarantees for flowshop heuristics. Martinez, Dauze`re-Pe´re`s, Gue´ret, Mati and Sauer examine the complexity of flowshop problems under the condition of no intermediate storage between machines subject to the constraint that a machine is blocked until the job at the next stage is completed. Claiming this to be a new type of blocking constraint, they examine the complexity of systems with as many as five machines. Their work is based on systems they have observed in the real world. Therefore, their research extends the possible practical application of flowshop scheduling theory. In the final paper of this special issue, Gupta, Koulamas, and Kyparisis provide a survey of performance guarantees for flowshop heuristics of two types: (1) worst-case scenarios; and (2) absolute performance guarantees. Probabilistic analysis is used to show that best absolute performance heuristics outperform the tightest worst-case heuristics with a 99% probability when a certain number of jobs is present. Based on these results, the authors suggest that the use of probabilistic analysis may provide a bridge between the absolute and worst-case performance guarantees of heuristics for combinatorial problems.
5. Conclusions and directions for future research This paper has provided a brief excursion into the developments in flowshop scheduling theory during the past fifty years. During these fifty years, flowshop scheduling research has seen an evolution from theoretical and abstract modeling to the development of models that more nearly match the conditions found in real-life production systems. Recent advances in heuristic approaches,
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the availability of reasonably inexpensive highspeed microcomputers for nearly every researcher, and ongoing developments in integer programming software have resulted in optimal or nearoptimal solutions for larger and larger problems. In most of these developments, JohnsonÕs rule and analysis methods play a significant role. Thus, contrary to the viewpoint expressed by Dudek et al. (1992), our research reveals that JohnsonÕs early work on flowshop scheduling had a profound effect and utility in subsequent research over the past fifty years. In fact, the influence of JohnsonÕs work extends to several other scheduling problems and is not limited to flowshops only. In spite of several developments in the last fifty years, even the theoretical flowshop scheduling problem remains largely unsolved. The reported progress in flowshop scheduling theory, while advancing at a rapid pace, is not appreciable to solve practical flowshop scheduling problems optimally and efficiently. Further, there is a lack of integrative and interactive decision making in the field of scheduling as many aspects of business practices are not included in the development of scheduling algorithms. The future research directions suggested here are intended to bridge the gap between the development of theory and practical applications of theory. Three areas of research are identified: theoretical, computational, and empirical research. 5.1. Theoretical research The development of flowshop scheduling techniques thus far is essentially curtailed enumeration schemes. The dominance conditions developed (in combinatorial and branch and bound procedures) depend on partial schedules that precede a job candidate. Theoretical research in flowshop scheduling should attempt to develop dominance conditions that are either independent of partial schedules that precede a job candidate or are such that a large number of partial schedules containing a lesser number of jobs are rejected quickly. For the general m-machine problems, it is unlikely that dominance conditions independent of preceding and proceeding partial schedules can be developed since this would imply the existence of a
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polynomial bounded algorithm—a result which contradicts the well established results on the complexity of scheduling and related combinatorial problems. The combinatorial analysis approach, however, can be specialized to develop polynomial-bounded algorithms for several special structure flowshop problems if there is enough justification for special cases. The unidirectional nature of workload and restrictive assumptions outlined earlier do imply some structural relationships among the processing times of various jobs on different machines. It is reasonable to deduce that the restrictive flowshop scheduling problem will have well established structural relationships which are situation dependent. Therefore, theoretical research should consider many more special cases of flowshop scheduling that have been considered before and develop efficient optimization techniques for their solution. Simultaneously, more quick, perhaps dirty but reliable heuristic procedures should be developed. Consideration of hybrid heuristic approaches for these problems provides another fruitful area for future theoretical research. The measures of performance used in scheduling depend on such considerations as the interactions between scheduling, inventory, and plant layout. The theoretical research in scheduling theory should quantify these interactions and develop general purpose models of scheduling problems and general purpose measures of performance. These general purpose models could then provide an insight and perhaps the directions, for the development of solution techniques for only a narrow range of measures of performance. In addition, they may provide a motivation for the practical use of the theoretical developments. 5.2. Computational research The past fifty years of flow shop scheduling research has provided a panorama of techniques that claim to find acceptable solutions. A practical scheduler has difficulty in selecting an algorithm to solve a given flowshop scheduling problem. The computational research should consider such aspects as comparative efficiency of various algorithms for a specified problem with given data
set. Thus, new measures of computational effort required should be developed. The existing measures, average numbers of schedules and average computational time on one hand and the worst-case performance bounds on the other may be inappropriate measures for algorithm selection. In addition, artificial intelligence techniques, such as neural networks should be further exploited to select specific heuristics to be used for a given problem (see Gupta et al. (2000) for one such effort). The mathematical programming approach to flowshop scheduling has been rejected because of excessive computational burden and heuristic solution procedures are being developed instead. Recent advances in solving large-scale mathematical programming problems and the availability of approximate solution procedures for the integer programming problem may show that the mathematical approach can, in fact, be used to find realistic schedules with less computational effort. A comparative computational investigation of the mathematical programming techniques and other flowshop scheduling procedures (exact and approximate) would be of immense value in indicating future research trends. In this regard, it may well turn out that heuristic solution procedures based on mathematical programming approaches provide better solutions that other heuristics. 5.3. Empirical research The mathematical theory of flowshop scheduling suffers from too much abstraction and too little application. Research in flowshop scheduling seem to be motivated by what the researchers can achieve rather that what is important. The practical use of flowshop scheduling techniques, therefore, is rare. This questions their suitability. In spite of fifty years of research, we know very little about the practical flowshop scheduling problem except that it is an often occurring problem. Perhaps we have over-emphasized the rigor in mathematical development at the cost of underemphasizing the realism of problem formulation. Future research in flowshop scheduling should be inspired more by real life problems rather than problems encountered in mathematical abstractions.
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For a realistic problem formulation, empirical research is necessary to understand the practical situations. The flowshop scheduling is only one of a few areas where no case histories are available. Empirical research should answer such questions as: What is the maximum problem size encountered in practice? What specific situations give rise to flowshop scheduling problems? What are the desired objectives of scheduling? What is the nature of processing times? How rigid (or flexible) are the operating policies? Empirical research, therefore, needs to include a survey of industrial scheduling practices and situations. Without such a survey, we may in fact spend another twenty-five years in solving a problem that perhaps needs no solution, since it may be the wrong problem (from practical consideration). The above suggested research directions are not meant to replace the existing efforts in solving flowshop scheduling problems. Rather, they are designed to provide a framework for viewing several problems and the contributions of various research efforts in solving practical scheduling problems. The recent developments in supply chain management, internet, and e-commerce have created new and complex scheduling and coordination problems that we have just begun to understand. Therefore, we need to diversify our research efforts in scheduling to include these new and emerging problems. Along with these research efforts, a survey of industrial scheduling problems and practices would aid in the identification and clear definition of scheduling problems encountered in practice and the need to solve such problems. This alone can provide the needed impetus to bridge the gap between theoretical developments and industrial practices in production scheduling and control.
Acknowledgments The completion of this special issue has involved hard work and contributions of several people in addition to the authors of the papers. Professor Roman Slowinski, one of the three editors of the European Journal of Operational Research, has worked closely with us to get this
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issue to publication. His experience, dedication, and work has improved the quality of this issue. The large number of submissions for this issue resulted in our calling on the services of many referees. We therefore thank these anonymous individuals for helping with the review process. They each spent many hours in reviewing, critiquing, and re-reviewing the many papers considered for this issue. Without their efforts, this issue could not have been completed. We are also grateful to those authors whose papers could not be included in this issue for a variety of reasons. We trust that their work will eventually appear elsewhere in the flowshop research literature. We appreciate the informal review of this paper and several discussions about various editorial issues provided by our colleague, Fan T. Tseng throughout the editing of this special issue. Finally, we are thankful to Dr. C. David Billings, Dean of our College of Administrative Science at the University of Alabama in Huntsville for allowing us to work on this special issue.
References Aarts, E., Lenstra, J.K., 1997. Local Search in Combinatorial Optimization. Wiley Interscience, Chichester, England. Allahverdi, A., Gupta, J.N.D., Aldowaisan, T., 1999. A survey of scheduling research involving setup considerations. OMEGA, International Journal of Management Science 27, 219–239. Ashour, S., 1972. Sequencing Theory. Springer-Verlag, Berlin. Bellman, R., 1956. Mathematical aspects of scheduling theory. Journal of Society of Industrial and Applied Mathematics 4, 168–205. Bellman, R., Gross, O., 1954. Some combinatorial problems arising in the theory of multi-stage processes. Journal of Society of Industrial and Applied Mathematics 2, 175– 183. Bellman, R., Esogbue, A.O., Nabeshima, I., 1982. Mathematical Aspects of Scheduling and Applications. Pergamon Press, New York. Brown, A.P.G., Lomnicki, Z.A., 1966. Some applications of the branch and bound algorithm to the machine scheduling problem. Operational Research Quarterly 17, 173–186. Brown, D.E., Scherer, W.T. (Eds.), 1995. Intelligent Scheduling Systems. Kluwer, Boston. Brucker, P., 1998. Scheduling Algorithms. Springer-Verlag, Berlin. Byrd, J., Moore, T., 1983. Decision Models for Managers. McGraw Hill, New York.
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Gupta, J.N.D., Sexton, R.S., Tunc, E.A., 2000. Selecting a scheduling heuristic through neural networks. INFORMS Journal of Computing 12, 150–162. Heller, J., 1959. Combinatorial, probabilistic, and statistical aspects of an M · J scheduling problem. AEC Research and Development Report NYO-2540, New York University. Heller, J., 1960. Some numerical experiments for an M · J flowshop and its decision-theoretical aspects. Operations Research 8, 178–184. Ignall, E., Schrage, L., 1965. Application of branch-and-bound technique to some flow shop problems. Operations Research 13, 400–412. Johnson, S.M., 1954. Optimal two- and three-stage production schedules with setup times included. Naval Research logistics Quarterly 1, 61–68. Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B., 1993. Sequencing and scheduling: Algorithms and complexity. In: Graves, S.C. (Ed.), Handbooks in Operations Research and Management Science, Vol. 4. Elsevier Science Publishers, Amsterdam, pp. 445–552. Lageweg, B.J., Lenstra, J.K., Rinnooy Kan, A.H.G., 1978. A general bounding scheme for the permutation flow-shop problem. Operations Research 26, 53–67. Lomnicki, Z.A., 1965. A branch and bound algorithm for the exact solution of the three machine scheduling problem. Operational Research Quarterly 16, 89–100. McMahon, G.B., 1969. Optimal production schedules for flowshops. Canadian Operational Research Society Journal 7, 141–151. McMahon, G.B., Burton, P.G., 1967. Flowshop scheduling with the branch and bound method. Operations Research 15, 473–481. Manne, A., 1960. On the job-shop scheduling problem. Operations Research 8, 219–223. Morton, T.E., Pentico, D.W., 1993. Heuristic Scheduling Systems. Wiley, New York. Muth, J., Thompson, G.L., (Eds.), 1963. Industrial Scheduling. Prentice-Hall, Englewood Cliffs, N.J. Osman, I.H., Kelly, J.P., 1996. Meta-heuristics: Theory and Applications. Kluwer Academic Publishers, Boston. Pinedo, M., 1995. Scheduling: Theory, Algorithms, and Systems. Prentice, Englewood Cliffs. Potts, C.N., Van Wassenhove, L.N., 1992. Integrating scheduling with batching and lot sizing: A review of algorithms and complexity. Journal of the Operational Research Society 43, 395–406. Potts, C.N., Kovalyov, M.Y., 2000. Scheduling with batching: A review. European Journal of Operational Research 120, 228–249. Rayward-Smith, V.J., Osman, I.H., Reeves, C.R., Smith, G.D., 1996. Modern Heuristic Search Methods. Wiley, Chichester, England. Ruiz, R., Maroto, C., Alcaraz, J., 2005. Solving the flowshop scheduling problem with sequence dependent setup times using advanced metaheuristics. European Journal of Operational Research 165, 34–54.
J.N.D. Gupta, E.F. Stafford Jr. / European Journal of Operational Research 169 (2006) 699–711 Sisson, R.L., 1959. Methods of sequencing in job shops—a review. Operations Research 7, 10–29. Smith, R.D., Dudek, R.A., 1967. A General algorithm for the solution of the n job, m machine sequencing problem of the flowshop. Operations Research 15, 71–82. Also, see their Errata Operations Research 17, 756. Smutnicki, C., 1998. Some results of the worst-case analysis for flow shop scheduling. European Journal of Operational Research 109, 66–87. Szwarc, W., 1971. Elimination methods in the m · n sequencing problem. Naval Research Logistics Quarterly, 295–305. Szwarc, W., 1973. Optimal elimination methods in the m · n flowshop scheduling problem. Operations Research 21, 1250–1259. Tanaev, V.S., Sotskov, Y.N., Strusevich, V.A., 1994. Scheduling Theory: Multi-stage Systems. Kluwer Academic Publishers, Dordrecht.
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TÕKindt, V., Billaut, J-C., 1993. Multi-criteria Scheduling. Springer-Verlag, Berlin. Trietsch, D., Baker, K.R., 1993. Basic techniques of lotstreaming. Operations Research 41, 1065–1076. Tseng, F.T., Stafford, E.F., Gupta, J.N.D., 2004. An empirical analysis of the integer programming formulations for the permutation flowshop. OMEGA: The International Journal of Management Science 32, 285– 293. Uskup, G., Smith, S.B., 1975. A branch and bound for twostage production sequencing. Operations Research 23, 118– 136. Wagner, H.M., 1959. An integer linear-programming model for machine scheduling. Naval Research Logistics Quarterly 6, 131–140. Zweben, M., Fox, M.S., 1994. Intelligent Scheduling. Morgan Kauffman Publishers, San Francisco.
Editorial
Flowshop scheduling research after five decades Jatinder N.D. Gupta *, Edward F. Stafford Jr. College of Administrative Science, The University of Alabama in Huntsville, Huntsville, AL 35899, USA Available online 2 April 2005
Abstract Since JohnsonÕs seminal paper in 1954, flowshop scheduling problems have received considerable research attention over the last fifty years. As a result, several optimization and heuristic solution procedures are available to solve a variety of flowshop scheduling problems. This paper provides a brief glimpse into the evolution of flowshop scheduling problems and possible approaches for their solution over the last fifty years. It briefly introduces the current flowshop problems being solved and the approaches being taken to solve (optimally or approximately) them. The paper concludes with some fruitful directions for future research. 2005 Elsevier B.V. All rights reserved. Keywords: Flowshop scheduling; Historical developments; Problem assumptions; Current research status; Future research directions
1. Introduction A frequently occurring operational problem is one of processing a given number of jobs (commodities) on a specified number of machines (facilities)—referred to by various investigators as scheduling, dispatching, sequencing, or combinations thereof. The first of these terms will be used here since this encompasses dispatching and sequencing. The desire to process the jobs in a special order to achieve some objective function is what creates a problem that remains largely un*
Corresponding author. Tel.: +1 256 824 6593; fax: +1 256 824 2929. E-mail address: [email protected] (J.N.D. Gupta).
solved. The actual situations that give rise to scheduling problems are wide and varied. Thus, for example, we have single machine scheduling problem, multiple machine scheduling problem, manpower scheduling problem, to name a few. In this paper and in this feature issue, we will consider a specific class of multiple machine scheduling problems, called the flowshop scheduling problems. The first paper on flowshop problem was published fifty years ago. In the 50 years since Johnson (1954) published his seminal paper on flowshop scheduling, more than 1200 papers on various aspects of this problem have been published in the operational research literature. We are, therefore, quite pleased to be serving as guest editors
0377-2217/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.02.001
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of this special issue of European Journal of Operational Research on flowshop scheduling celebrating a golden jubilee of JohnsonÕs paper. This special issue also honors JohnsonÕs contributions by providing a glimpse of the research that evolved since his original paper. In his paper, Johnson credited Richard Bellman with giving him the problem for analysis and research. Our research revealed that shortly after JohnsonÕs work, Bellman published at least two papers that deal with two-machine flowshop problem (Bellman and Gross, 1954; Bellman, 1956; Bellman et al., 1982). A study of BellmanÕs work indicates that the two-machine flowshop problem in JohnsonÕs paper was inspired by a real-life situation. This settles the issue raised by Dudek et al. (1992) about the relevance of the flowshop problems. JohnsonÕs rule or algorithm, as his method is known today, is a simple technique for optimally sequencing a set of n jobs to be processed in a two-machine manufacturing system. Johnson also offered an optimizing technique for the three-machine system wherein the processing times for the jobs on the three machines had certain restrictive relative relationships. It is noteworthy that Johnson did not use the term flowshop to identify the production system represented by his model. As a part of our research for this paper, we attempted to identify the published paper wherein the term flowshop, flow shop, or flow-shop was first used. In our first pass, we found that Ignall and Schrage (1965) used the term in the title of their early paper on branch-and-bound for the regular flowshop. Subsequently, we have traced the term to an earlier paper by Heller (1960). In one of his previous working papers, Heller (1959) used the term conservative assembly line to describe a flowshop.1
This paper provides a glimpse of the developments in flowshop scheduling over the last fifty years.2 The problem assumptions, problem formulation, and solution approaches used in solving the flowshop scheduling problem are described and current state of research is introduced. Subsequently, several plausible and fruitful directions for future research are outlined. 2. The flowshop scheduling problem A flowshop is characterized by more or less continuous and uninterrupted flow of jobs through multiple machines in series. In such a shop, the flow of work is unidirectional since all jobs follow the same technological routing through the machines. Although this description of flowshop resembles an assembly-line operation, there are several differences. First: a flowshop is equipped to handle a variety of jobs as opposed to a standard product manufactured by an assembly-line. Second: the jobs in a flowshop do not have to be processed on all machines; that is, a job may skip some operations according to its technological requirements. However, in an assembly-line, all jobs have to move from one station to another without skipping any work-station. Third: in a flowshop, each machine is independent of other machines and can be loaded independently; whereas in assembly-line operations, each work station depends on the preceding one. And finally, each job has its own processing time at each machine in a flowshop; however, all units of a product have a standard time at each work station in an assembly-line (Ashour, 1972). Because of these differences, Heller (1959) characterized a flowshop as a conservative assembly line. 2.1. Flowshop problem definition
1
To settle this issue once and for all, we make the following offer to all in the flowshop research community. We will take to dinner the first person to correctly identify the earliest published paper in which the term flowshop, or one of its variant spellings, was used. Simply send us a copy of that paper along with the correct citation. Deadline is December 31, 2005, and the dinner will be at a future operational research meeting, or in Huntsville, Alabama, USA. In Huntsville, we will also take you for a tour of the Space and Rocket Center.
As stated earlier, the work-flow in a flowshop is unidirectional. This means that the order in which 2
Since the purpose of this paper is to provide a glimpse into flowshop scheduling research and to introduce this special issue, references cited are neither exhaustive nor complete. They are cited for convenience of the reader to locate some additional readings if desired.
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jobs are processed on various machines is the same for all n jobs and is specified. Since the nomenclature of the machines is arbitrary, the machines may be numbered such that each job is processed on machine 1 first, machine 2 second,. . ., and machine m last. With this numbering of machines and the abstract terminology of jobs and machines, the traditional flowshop scheduling problem considered by Johnson may be defined as follows (see Tanaev et al., 1994) for a comprehensive treatment of flowshop problems): • Given n jobs to be processed on m machines in the same technological order; the processing time of job i on machine j being pij (i = 1,2, . . ., n; j = 1,2, . . ., m); it is desired to find the order (schedule) in which these n jobs should be processed on each of the m machines to minimize a well defined measure of production cost. The above problem, when considered in the general case, gives (n!)m possible schedules. Even for problems as small as n = m = 5, the number of possible schedules is so large that a direct enumeration is economically impossible. A simplified version of the above problem, applicable to many situations in computer centers, production shops, and other similar situations exists in which it is assumed that the order (sequence) in which these jobs are processed on each machine is the same. For this simplified version, the number of feasible schedules reduces to n!. This simplified version of the flowshop scheduling problem can be solved to seek an optimal schedule for the general flowshop problem described earlier if certain conditions regarding the number of machines (m) and the measure of production cost are satisfied. Conway et al. (1967) state the following two theorems to describe these conditions. Theorem 1. For the m-machine flowshop scheduling problem with all jobs simultaneously available and the objective of minimizing a non-decreasing function of completion times of jobs, it is sufficient to consider only those schedules in which the same job sequence occurs on the first two machines.
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Theorem 2. For the m-machine flowshop scheduling problem with the objective of minimizing make-span, it is sufficient to consider only those schedules in which the same job sequence occurs on the last two machines. Now, if the number of machines, m = 2, the simplified version will in fact find global optimal solution. Similarly, if the objective is minimization of make-span (defined as the total throughput time in which all jobs complete processing on all machines) and the number of machines m 6 3, the simplified version will lead to global optimal solution. The flowshop problems considered by Johnson (1954) satisfied these conditions. Even when it is sufficient to solve the simplified version to obtain global optima, it may be difficult to identify the optimal schedule. For this reason, the assumption that the same job sequence is followed at each machine is active in practically every research effort in flowshop scheduling. 2.2. Flowshop assumptions The foregoing assumptions are only a few of several assumptions imposed on the flowshop scheduling problem. Scheduling problems in general and flowshop cases in particular are structured rather restrictively from a practical viewpoint. Implicitly, and without specific statement, the formulations of these problems overlook several important factors that are encountered in operational situations. For example, Ashour (1972) cites several factors that may interrupt the flow of work in a shop and hence cause unexpected delays. Some of the delay causing factors are: (1) unpredictable breakdown of machines; (2) the variability in the performance and absenteeism of operators; (3) delay in supply of materials, fixtures, or tools; (4) the variability of processing times; (5) changes in job specifications; (6) rejection and changes in job identity due to defective production; and (7) customer pressures that may cause rush orders for early completion of jobs. The restrictive assumption of unidirectional flow pattern of jobs in a flowshop, though more general than that of an assembly-line, limits the variety of output of the shop. Indeed, the flowshop is very restrictive. Nevertheless, progress in scheduling
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theory is not appreciable even to solve this very restrictive class of scheduling problems. As a result, the already restrictive case of flowshop must have several additional simplifying assumptions imposed on it. Though these assumptions increase the generality of models thus developed and the possibility of finding an optimal schedule; they do, however, divorce these models from realistic situations. While not originally stated in JohnsonÕs paper, the additional restrictive flowshop assumptions include the characteristics of jobs, machines, and operating policies of the shop. Gupta et al. (1979) describes the following 21 assumptions for the traditional flowshop scheduling problem originally conceived in Johnson (1954) paper.
M3. Each machine in the shop operates independently of other machines and thus is capable of operating at its own maximum output rate. M4. Each machine can process at most one job at a time. This eliminates those machines that are designed to process several jobs simultaneously like multi-spindle drill. M5. Each machine is continuously available for processing jobs throughout the scheduling period and there are no interruptions due to breakdowns, maintenance or other such causes.
2.2.3. Assumptions concerning operating policies 2.2.1. Assumptions concerning job J1. Each job is released to the shop at the beginning of the scheduling period. J2. Each job may have its own due date which is fixed and is not subject to change. J3. Each job is independent of each other. J4. Each job consists of specified operations, each of which is performed by only one machine. J5. Each job has a prescribed technological order which is the same for all jobs and is fixed. J6. Each job (operation) requires a known and finite processing time to be processed by various machines. This processing time includes transportation and setup times, if any, and is independent of preceding and succeeding jobs. J7. Each job is processed no more than once on any machine. J8. Each job may have to wait between machines and thus in-process inventory is allowed.
2.2.2. Assumptions concerning machines M1. Each machine center consists of only one machine; that is, the shop has only one machine of each type. M2. Each machine is initially idle at the beginning of the scheduling period.
P1. Each job is processed as early as possible. Thus, there is no intentional job waiting or machine idle time. P2. Each job is considered an indivisible entity even though it may be composed of a number of individual units. P3. Each job, once accepted, is processed to completion; that is, no cancellation of jobs is permitted. P4. Each job (operation), once started on a machine, is completed to its completion before another job can start on that machine, that is, no preemptive priorities are assigned. P5. Each job is processed on no more than one machine at a time. (This is a result of assumptions J5 and P2.) P6. Each machine is provided with adequate waiting space for allowing jobs to wait before starting their processing. P7. Each machine is fully allocated to the jobs under consideration for the entire scheduling period; that is, machines are not used for any other purpose throughout the scheduling period. P8. Each machine processes jobs in the same sequence. That is, no passing or overtaking of jobs is permitted. The above list of assumptions indicates how explicitly practical situations have to be analyzed
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before conceptual scheduling models can be used. Relaxation of one or more of these assumptions yields various scheduling models, both for the flowshop and other more general shops. Over the years, researchers, in dropping or modifying one or more of the above assumptions, have introduced variants of the original Johnson model which more closely approximate real-life production systems. Terms like no intermediate queues (NIQ), sequence-dependent setup times (SDST), and hybrid flowshops show the degree to which JohnsonÕs original (classical or regular) flowshop problem has evolved to represent a myriad of related production systems.
3. Five decades of flowshop scheduling research The early group of flowshop researchers was quite small, and these people were concentrated in a few US academic and research institutions such as UCLA, Cornell, RAND Corporation, and Texas Tech University. TodayÕs flowshop research community, as evidenced by the authors appearing in this issue, is global. We received responses to our call for papers from every continent and every geographical region, Antarctica excepted. It is not unusual for a recently published flowshop paper to be authored by researchers from three different countries on two or more continents. This speaks well for the future of flowshop research. In this section, we provide a glimpse of the problems addressed, solution approaches adapted, and related research developments in flowshop scheduling by each decade. 3.1. First decade (1955–1964) After the publication of JohnsonÕs classical paper on the two- machine flowshop scheduling problem, developments in the first decade include the consideration of the m-machine flowshop problem to minimize makespan and the inclusion of start and stop time-lags in the two-machine flowshop scheduling problems. Early research on the flowshop problem was highly theoretical, and it tended toward optimization techniques such as
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mathematical programming (see, for example, Wagner, 1959; Manne, 1960). Thus, this decade saw consideration of mathematical programming approaches to flowshop problems on one hand and the use of Monte-Carlo simulation techniques on the other (for a review of these developments, see Sisson (1959) and Muth and Thompson (1963)). While only a few solution techniques were developed during this decade, several alternate processing configurations and scenarios were identified for scheduling problems. The problem sizes solvable in these early research efforts were quite small for three reasons: (1) lack of computer power; (2) lack of efficient computer programs; and, more importantly, (3) most variants of the two-machine flowshop problem are NP-hard. By the early 1960s, researchers were attempting to extend JohnsonÕs analysis to the m-machine case. Thus, for example, the combinatorial approach that Johnson used for the two-machine problem was extended to the m-machine flowshop problem by Dudek and Teuton (1964). Even though it was later shown to be incorrect, it began the analytical framework for the development of dominance conditions for flowshop scheduling.
3.2. Second decade (1965–1974) The second decade witnessed a rather wide range of solution techniques on one hand and the consideration of objective functions other than makespan on the other. The first paper of this decade is difficult to identify as there are several papers that appeared around the same time. The combinatorial approach started by Dudek and Teuton (1964) was corrected and improved by Smith and Dudek (1967), McMahon (1969), Gupta (1971), and Szwarc (1971, 1973). The branch and bound solution approaches were also developed during this time by Lomnicki (1965), Brown and Lomnicki (1966), McMahon and Burton (1967), Gupta (1969b, 1970), and others. These branch and bound approaches differ in terms of the lower bounds used and the branching strategies. A comprehensive framework for these branch and bound algorithms is provided by Lageweg et al. (1978). Except for the papers by Ignall and
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Schrage (1965) and Gupta (1972), all these papers consider makespan as the optimality criterion. Ignall and Schrage (1965) considered the twoand-three machine flowshop problems with total flowtime criterion as well. Gupta (1972) considered the existence of start and stop time-lags on one hand and several objective functions on the other. The difficulty in developing optimization algorithms for flowshop scheduling problem resulted in developing and using heuristics for finding good, near-optimal solutions for larger and larger flowshop problems (see Framinan et al. (2004) for a review of these and subsequent heuristic approaches to flowshop problems). 3.3. Third decade (1975–1984) The emergence of the theory of NP-completeness (which is now called NP-hardness) had a profound impact on the direction of developments in flowshop scheduling (see Garey and Johnson (1979) for a detailed study of NP-hardness). On one hand, there are developments to resolve the complexity status of various flowshop scheduling problems (see Brucker (1998), Chen et al. (1998), Lawler et al. (1993) and Pinedo (1995) for reviews of complexity of scheduling problems); on the other hand, many more heuristic approaches were developed during this time (see the book by Morton and Pentico (1993) for various heuristic approaches). In addition, this decade witnessed the proliferation of a variety of flowshop problems and consideration of flowshop problems that violate several restrictive assumptions listed earlier in this paper. Thus, for example, this decade witnessed an interest in identifying the worst-case performance of a heuristic algorithm on one hand (see Smutnicki (1998) for a review), and the consideration of flowshop scheduling problems with separable setup times on the other. Further, this decade also witnessed the consideration of due date related objective functions in flowshop scheduling (see Morton and Pentico (1993) for these developments). The third decade also witnessed the consideration of stochastic processing times in analyzing various flowshop problems. While the progress in the development of stochastic models for solving
flowshop problems has been slow, significant results are available and described by Dempster et al. (1982) and Pinedo (1995). 3.4. Fourth decade (1985–1994) This decade saw the emergence of hybrid flowshops where each stage of the flowshop could contain multiple parallel machines and the development of metaheuritics like Tabu Search, Genetic Algorithms, and Simulated Annealing (see Aarts and Lenstra (1997), Osman and Kelly (1996) and Rayward-Smith et al. (1996) for illustrations of such approaches). As a result of the consideration of several objective functions and various heuristic approaches, we witnessed the expansion of efforts to solve flowshop problems with separable setups which could be either sequence independent or sequence dependent (see the reviews by Allahverdi et al. (1999) and Cheng et al. (2000)). While such problems were identified as early as 1969 and some solution techniques were developed around that time (see Corwin (1969), Corwin and Esogbue (1974), Gupta (1969a, 1975) and Uskup and Smith (1975) for such early efforts), the developments in the fourth decade are significant. This decade also witnessed the use of artificial intelligence based techniques to solve the flowshop scheduling problems. Such attempts include the development of decision support systems and expert systems (see Brown et al. (1995), Byrd and Moore (1983) and Zweben and Fox (1994) as examples of such attempts). 3.5. Fifth decade (1995–2004) The proliferation of the variety of flowshop scheduling problems, objective functions, and solution approaches continued in this decade. This decade had seen several developments and improvements to the already identified and evolved flowshop problems. While it is difficult to identify any one single problem as being representative of developments during this decade that was not considered earlier, it is interesting to note that consideration and solution of multi-criteria
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flowshop scheduling became quite popular in this decade (see TÕKindt and Billaut (1993) for a review of multi-criteria scheduling problems). This decade did witness expanded attention on the simultaneous consideration of lot sizing and scheduling on one hand and the batch processing machines on the other (see Potts and Van Wassenhove (1992), Potts and Kovalyov (2000) and Trietsch and Baker (1993) as examples of these developments). Ruiz et al. (2005) investigate the effectiveness and efficiency of various metaheuristics in minimizing makespan in flowshops with separated sequence dependent setup times (SDST). These heuristics are adaption of existing metaheuritics for the regular flowshop to SDST flowshop. In addition to testing existing algorithms, these authors also propose and compare two advanced genetic algorithms for solving the SDST flowshop problem. Utilizing the existing benchmark problems, they show that their proposed algorithms provide better solutions than existing algorithms. This decade also saw significant progress in developing solution procedures for robotic flowshops and flowshops equipped with automatic guided vehicles (AGVs) as is evident from the recent review by Geismar et al. (2004). In addition, because of the recent advances in computational efficiencies, there is a renewed interest in the use of mathematical programming approaches to flowshop problems (see the recent paper by Tseng et al. (2004) as an example of such efforts). While the evolution of flowshop problems has been categorized by five decades since the publication of JohnsonÕs seminal paper, it should be mentioned that most flowshop problems were envisioned to some extent during the first two decades. Further, once a problem is identified and some progress in its solution is reported, research on further developments related to this problem continues through subsequent decades.
4. Current flowshop scheduling research When we first envisioned this special issue, we anticipated a set of survey papers covering the past, present, and future of flowshop research.
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Instead, we received submissions that were as eclectic as those appearing in any randomly chosen issue of this journal. Thus, the papers that survived the rigorous review process and included in this special issue represent the current state of the art in flowshop scheduling. Four papers deal with different aspects of the two-machine flowshop problem; and one paper examines the two job, m-machine problem. Two papers concentrate on the flowshop variants including separable setup times. Two papers involve different solution technique approaches: adaptive learning and traveling salesman modelling of flowshop problems. The final two papers involve analyses of the difficulty of solving flowshop problems. 4.1. Two-machine problems Briand, La and Erschler introduce a conceptual technique for associating jobs with intervals rather than sequence positions. They then use AllenÕs algebra to establish sufficient optimality conditions that characterize a large set of optimal sequences. This set includes the Johnson sequences among others. While not useful in minimizing makespan in a two-machine flowshop, their suggested approach may be useful to solve problems involving secondary criteria where the primary criterion is makespan. Mukherjee and Chatterjee examine the use of problem decomposition for solving the twomachine flowshop problem. Since the shifting bottleneck heuristic has proven successful in finding good solutions for the more complex job shop problem, they examine its applicability to flowshop problems. For the two-machine flowshop problem, their proposed decomposition based shifting bottleneck approach does generate an optimal makespan schedule even though the computational complexity is much higher than that of JohnsonÕs algorithm. Their main contribution is showing how the job shop decomposition techniques may be modified for finding good solutions to the permutation flowshop problem. In his original paper, Johnson assumed that job processing times were known and deterministic. Kalczynski and Kamburowski describe a heuristic for minimizing the expected makespan with
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consistent coefficients of variation in the job processing times. They assume that the processing times follow a Weibull distribution, that the known solution techniques like JohnsonÕs and TalwarÕs rules are special cases of their work. Using simulation, they examine the effectiveness of their heuristic to minimize expected makespan. Portougal and Trietsch consider the two-machine flowshop problem with stochastic processing time to minimize expected makespan and to find optimal service level. Since the identification of optimal service level requires the computation of the entire makespan distribution, they develop and empirically test two heuristics (JohnsonÕs rule applied to means of the service time distributions; and a pair-switching approach) for the twomachine flowshop problem with stochastic job processing times. They also investigate the impact of sequencing on the variance of the makespan and show that JohnsonÕs rule may also be effective to solve stochastic flowshop problems. 4.2. Two-job problems Averbakh revisits the two-job, m-machine permutation flowshop problem with stochastic processing times. For each operation, the processing time can take on any value with a given uncertainty time interval. Based on a geometric formulation of the problem, Averbakh proposes a linear time algorithm (with O(m) time complexity) algorithm for the minmax regret solution. 4.3. Flowshops with setup times While JohnsonÕs original paper assumed that setup times are included in the processing times, later studies have shown that separating the setup times may lead to better schedules. In addition, these times may be sequence dependent. Allahverdi and Al-Anzi use the three-machine flowshop problem with separated sequence independent setup times to model the three-site distributed database scheduling problem. Utilizing specifically developed dominance conditions and a lower bound, they use a branch-and-bound approach for finding a schedule that minimizes the sum of job completion times. To reduce the computa-
tional effort of their proposed branch and bound algorithm, they use a three-phase hybrid heuristic algorithm to generate an upper bound. This heuristic is also shown to be effective in finding approximately optimal schedule for problems containing a large number of jobs. In many practical applications, the two or more machines are available at each stage in a flowshop. These problems are called hybrid flowshops and may involve some machine eligibility constraints. Ruiz and Maroto consider a hybrid flowshop problem with machine eligibility requirements, sequence dependent setup times and unrelated parallel machines. They adapt the best genetic algorithm for the SDST flowshop developed by Ruiz et al. (2005) and show that their proposed algorithm is quite effective in minimizing makespan. 4.4. Solution approaches As the nature of flowshop problems becomes complex, improved and new solution approaches are needed to solve these problems. Many of these solution approaches are adaption of approaches used for successfully solving other combinatorial optimization problems. Agarwal, Colak, and Eryarsoy describe an improvement heuristic approach to solving the regular flowshop problem. Their technique starts with a one-pass starting solution which is then improved on by using a weighted parameter to perturb the original data. The learning done in their heuristic is similar to the approach used in neural networks and promises good results and possible use to solve other scheduling problems. Bagchi, Gupta, and Sriskandarajah provide a review of the traveling salesman problem (TSP) based techniques found in the literature for solving several variants of the flowshop problem. They show that the use of TSP based approaches is quite useful in optimally solving some rather complex flowshop problems found in robotic shops. In addition, they also review the computational complexity of a variety of flowshop models and show that TSP based heuristic approaches can be used to develop effective heuristic solution procedures for various scheduling problems. Based on their
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review of existing literature, they also provide some useful directions for future research including the application of Group TSP modelling approaches for flowshop problems. 4.5. Complexity and performance guarantees No special issue involving the flowshop problem would be complete without addressing the issues of computational complexity and performance guarantees for flowshop heuristics. Martinez, Dauze`re-Pe´re`s, Gue´ret, Mati and Sauer examine the complexity of flowshop problems under the condition of no intermediate storage between machines subject to the constraint that a machine is blocked until the job at the next stage is completed. Claiming this to be a new type of blocking constraint, they examine the complexity of systems with as many as five machines. Their work is based on systems they have observed in the real world. Therefore, their research extends the possible practical application of flowshop scheduling theory. In the final paper of this special issue, Gupta, Koulamas, and Kyparisis provide a survey of performance guarantees for flowshop heuristics of two types: (1) worst-case scenarios; and (2) absolute performance guarantees. Probabilistic analysis is used to show that best absolute performance heuristics outperform the tightest worst-case heuristics with a 99% probability when a certain number of jobs is present. Based on these results, the authors suggest that the use of probabilistic analysis may provide a bridge between the absolute and worst-case performance guarantees of heuristics for combinatorial problems.
5. Conclusions and directions for future research This paper has provided a brief excursion into the developments in flowshop scheduling theory during the past fifty years. During these fifty years, flowshop scheduling research has seen an evolution from theoretical and abstract modeling to the development of models that more nearly match the conditions found in real-life production systems. Recent advances in heuristic approaches,
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the availability of reasonably inexpensive highspeed microcomputers for nearly every researcher, and ongoing developments in integer programming software have resulted in optimal or nearoptimal solutions for larger and larger problems. In most of these developments, JohnsonÕs rule and analysis methods play a significant role. Thus, contrary to the viewpoint expressed by Dudek et al. (1992), our research reveals that JohnsonÕs early work on flowshop scheduling had a profound effect and utility in subsequent research over the past fifty years. In fact, the influence of JohnsonÕs work extends to several other scheduling problems and is not limited to flowshops only. In spite of several developments in the last fifty years, even the theoretical flowshop scheduling problem remains largely unsolved. The reported progress in flowshop scheduling theory, while advancing at a rapid pace, is not appreciable to solve practical flowshop scheduling problems optimally and efficiently. Further, there is a lack of integrative and interactive decision making in the field of scheduling as many aspects of business practices are not included in the development of scheduling algorithms. The future research directions suggested here are intended to bridge the gap between the development of theory and practical applications of theory. Three areas of research are identified: theoretical, computational, and empirical research. 5.1. Theoretical research The development of flowshop scheduling techniques thus far is essentially curtailed enumeration schemes. The dominance conditions developed (in combinatorial and branch and bound procedures) depend on partial schedules that precede a job candidate. Theoretical research in flowshop scheduling should attempt to develop dominance conditions that are either independent of partial schedules that precede a job candidate or are such that a large number of partial schedules containing a lesser number of jobs are rejected quickly. For the general m-machine problems, it is unlikely that dominance conditions independent of preceding and proceeding partial schedules can be developed since this would imply the existence of a
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polynomial bounded algorithm—a result which contradicts the well established results on the complexity of scheduling and related combinatorial problems. The combinatorial analysis approach, however, can be specialized to develop polynomial-bounded algorithms for several special structure flowshop problems if there is enough justification for special cases. The unidirectional nature of workload and restrictive assumptions outlined earlier do imply some structural relationships among the processing times of various jobs on different machines. It is reasonable to deduce that the restrictive flowshop scheduling problem will have well established structural relationships which are situation dependent. Therefore, theoretical research should consider many more special cases of flowshop scheduling that have been considered before and develop efficient optimization techniques for their solution. Simultaneously, more quick, perhaps dirty but reliable heuristic procedures should be developed. Consideration of hybrid heuristic approaches for these problems provides another fruitful area for future theoretical research. The measures of performance used in scheduling depend on such considerations as the interactions between scheduling, inventory, and plant layout. The theoretical research in scheduling theory should quantify these interactions and develop general purpose models of scheduling problems and general purpose measures of performance. These general purpose models could then provide an insight and perhaps the directions, for the development of solution techniques for only a narrow range of measures of performance. In addition, they may provide a motivation for the practical use of the theoretical developments. 5.2. Computational research The past fifty years of flow shop scheduling research has provided a panorama of techniques that claim to find acceptable solutions. A practical scheduler has difficulty in selecting an algorithm to solve a given flowshop scheduling problem. The computational research should consider such aspects as comparative efficiency of various algorithms for a specified problem with given data
set. Thus, new measures of computational effort required should be developed. The existing measures, average numbers of schedules and average computational time on one hand and the worst-case performance bounds on the other may be inappropriate measures for algorithm selection. In addition, artificial intelligence techniques, such as neural networks should be further exploited to select specific heuristics to be used for a given problem (see Gupta et al. (2000) for one such effort). The mathematical programming approach to flowshop scheduling has been rejected because of excessive computational burden and heuristic solution procedures are being developed instead. Recent advances in solving large-scale mathematical programming problems and the availability of approximate solution procedures for the integer programming problem may show that the mathematical approach can, in fact, be used to find realistic schedules with less computational effort. A comparative computational investigation of the mathematical programming techniques and other flowshop scheduling procedures (exact and approximate) would be of immense value in indicating future research trends. In this regard, it may well turn out that heuristic solution procedures based on mathematical programming approaches provide better solutions that other heuristics. 5.3. Empirical research The mathematical theory of flowshop scheduling suffers from too much abstraction and too little application. Research in flowshop scheduling seem to be motivated by what the researchers can achieve rather that what is important. The practical use of flowshop scheduling techniques, therefore, is rare. This questions their suitability. In spite of fifty years of research, we know very little about the practical flowshop scheduling problem except that it is an often occurring problem. Perhaps we have over-emphasized the rigor in mathematical development at the cost of underemphasizing the realism of problem formulation. Future research in flowshop scheduling should be inspired more by real life problems rather than problems encountered in mathematical abstractions.
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For a realistic problem formulation, empirical research is necessary to understand the practical situations. The flowshop scheduling is only one of a few areas where no case histories are available. Empirical research should answer such questions as: What is the maximum problem size encountered in practice? What specific situations give rise to flowshop scheduling problems? What are the desired objectives of scheduling? What is the nature of processing times? How rigid (or flexible) are the operating policies? Empirical research, therefore, needs to include a survey of industrial scheduling practices and situations. Without such a survey, we may in fact spend another twenty-five years in solving a problem that perhaps needs no solution, since it may be the wrong problem (from practical consideration). The above suggested research directions are not meant to replace the existing efforts in solving flowshop scheduling problems. Rather, they are designed to provide a framework for viewing several problems and the contributions of various research efforts in solving practical scheduling problems. The recent developments in supply chain management, internet, and e-commerce have created new and complex scheduling and coordination problems that we have just begun to understand. Therefore, we need to diversify our research efforts in scheduling to include these new and emerging problems. Along with these research efforts, a survey of industrial scheduling problems and practices would aid in the identification and clear definition of scheduling problems encountered in practice and the need to solve such problems. This alone can provide the needed impetus to bridge the gap between theoretical developments and industrial practices in production scheduling and control.
Acknowledgments The completion of this special issue has involved hard work and contributions of several people in addition to the authors of the papers. Professor Roman Slowinski, one of the three editors of the European Journal of Operational Research, has worked closely with us to get this
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issue to publication. His experience, dedication, and work has improved the quality of this issue. The large number of submissions for this issue resulted in our calling on the services of many referees. We therefore thank these anonymous individuals for helping with the review process. They each spent many hours in reviewing, critiquing, and re-reviewing the many papers considered for this issue. Without their efforts, this issue could not have been completed. We are also grateful to those authors whose papers could not be included in this issue for a variety of reasons. We trust that their work will eventually appear elsewhere in the flowshop research literature. We appreciate the informal review of this paper and several discussions about various editorial issues provided by our colleague, Fan T. Tseng throughout the editing of this special issue. Finally, we are thankful to Dr. C. David Billings, Dean of our College of Administrative Science at the University of Alabama in Huntsville for allowing us to work on this special issue.
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