A Fuzzy Scheduling Problem with Dynamic Job Priorities and an Extension to Multiple Criteria Sanja Petrovic* Martin Josef Geiger** Automated Scheduling, Optimisation and Planning Group (ASAP) School of Computer Science & IT University of Nottingham Nottingham, Great Britain *Email:
[email protected] **Email:
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Abstract In real world scheduling problems, priorities of jobs may change over time. A less important job today may be of high importance tomorrow and vice versa. Another important aspect of decision making in manufacturing environments is often the impreciseness of the problem definition, comprising both the available data and the knowledge about the preference structure of the decision maker. The paper presents a study of neighbourhood search heuristics for fuzzy scheduling. We especially address the problem of changing job priorities over time as studied at the Sherwood Press Corporation, a Nottingham based printing company. It can be shown, that the use of multiple criteria within the search process may improve the effectiveness of local search operators. Keywords Fuzzy scheduling, dynamic scheduling, multi criteria optimisation, local search, preference structure.
1. INTRODUCTION Scheduling in manufacturing environments belongs to one of the most active fields of research within various related disciplines including operations research and artificial intelligence. The general problem can be defined to be the problem of assigning tasks to machines with respect to a set of side constraints in order to complete all tasks (Błażewicz et al. 2001). Typical general constraints are that each task can be processed by one machine at a time and each machine is able to process at most one task at a time. Other side constraints are often precedence relations due to technical requirements of the tasks, release dates and due dates. Numerous models and algorithmic solution approaches to different scheduling problems have been developed, ranging from exact methods like branch-and-bound (Carlier, Pinson 1989, Brucker et al. 1994) to heuristic and metaheuristic techniques (Bruns 1997). Although is has been noticed quite early, that the quality of a schedule in practice involves multiple criteria (Rinnooy Kan 1976, French 1982), research considering the problem as a multi criteria problem is a rather new though fast developing (Bagchi 1999). Most existing models assume the absence of uncertainty within the scheduling environment, and the problem data is described using crisp numbers (Brucker 2001). In many real world scheduling problems however, various kinds of uncertainty and vagueness often do exist which make the models more complex. Besides the development of stochastic approaches (Pinedo 2002), fuzzy set theory and fuzzy logic (Zadeh 1965) has attracted research interest of the community (Prade 1979). Different aspects of the scheduling problem were treated by applying this paradigm (Ishii 2000): • Fuzzy processing times: Imprecise processing times of the task have been described using triangular (Tsujimura et al. 1995) or trapezoidal fuzzy numbers (McCahon, Lee 1992). • Fuzzy due dates: The due dates for the tasks are defined using membership functions with decreasing satisfaction while increasing tardiness (Ishii et al. 1992, Sakawa, Kubota 2000). • Fuzzy precedence relations: Existing precedence relations among the tasks may be regarded as fuzzy, expressing uncertainty or preferences for certain task sequences (Ishii, Tada 1995, Fortemps 2000). • Fuzzy optimality criteria: The definition of optimality criteria often involves uncertain and vague scheduler’s preferences while fuzzy sets and fuzzy logic can accommodate graduality of the
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A Fuzzy Scheduling Problem with Dynamic Job Priorities and an Extension to Multiple Criteria satisfaction of the imposed constraints (Ishibuchi, Murata 2000). In some cases not a single but a set of optimality criteria is considered, leading to a multi objective formulation of the problem (Slany 1996, Lee 2002). In the current paper a study on scheduling in manufacturing environments within the Sherwood Press, a printing company based in Nottingham is presented. The company faces a problem of scheduling a large number of jobs (cards, brochures, etc.) on a set of machines of different purpose. Due dates of jobs are seldom known precisely. In addition priorities of jobs may change over time. The paper is organized as follows: A fuzzy scheduling model with dynamically changing job priorities is given in section 2. Initial experiments of local search neighbourhoods on benchmark problems are reported in section 3. An extension to a multi criteria problem is proposed in section 4. The extended multicriteria formulation of the problem is investigated and results and conclusions are given in sections 5 and 6 respectively.
2. A SCHEDULING PROBLEM WITH FUZZY DUE DATES The investigated scheduling problem of the Sherwood Press Corporation may be modelled as a job shop problem, in which a set of machines M = {M1 ,..., M m } exists that perform operations on a set of jobs J = {J1 ,..., J n } (Błażewicz et al. 1996). Each job J i , i = 1,..., n consists of an ordered set of tasks Ti = {Ti1 ,..., Tit } , and each task Tij , i = 1,..., n, j = 1,..., t may be processed on a certain machine with a nonnegative processing time pij . It is assumed that each machine may process at most one task and each task may be processed by at most on machine at a certain time. Due to the technical nature of the jobs, jobs can have different machine routings as presented in figure 1.
Printing Embossing
Cutting
Cutting
Embossing Folding Packaging
Figure 1: Example of different machine routings for different jobs Although the machine sequences are not necessarily identical for all jobs, it is possible to identify a typical material flow scheme. The jobs are always printed first as several products may be placed on a single sheet of paper before the paper is cut into pieces which are then processed further. Possible operations include printing of the products, cutting, embossing, folding, cards insertion, gathering/stitching/trimming, and finally packaging. Various kinds of uncertainty and vagueness within the described production environment exist. In the current study, we particularly address two aspects as they are of high importance in the regarded real world system. The practical example of the Sherwood Press shows that due dates for jobs sometimes are not properly representable using crisp numbers. In the actual case, so called ‘promised delivery dates’ exist, based upon an individual agreement with the customer. The goods are supposed to be delivered within a time interval, not necessarily on a certain day. This leads to a fuzzy interpretation of the linguistic notion of ‘tardiness’, as gradual degrees of tardiness within the promised delivery date are possible. As described in figure 2, fuzzy due ~ ~ dates Di for each job J i are introduced (Ishibuchi, Murata 2000), and a corresponding fuzzy tardiness Ti with the linear membership function can be derived. 1
~ Di
~ Ti
0 d i1
di2
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Decision Support in an Uncertain and Complex World: The IFIP TC8/WG8.3 International Conference 2004 ~ ~ Figure 2: Fuzzy due date Di and fuzzy tardiness Ti definition ~ A fuzzy due date Di consists of a doublet (di1 , di2 ) , defining an interval within which the satisfaction degree of completing the job J i is monotonically decreasing, starting from a maximum satisfaction value of one for a
completion time lower or equal to di1 to a minimum satisfaction value of zero for a completion time greater or ~ equal to di2 . Accordingly, the fuzzy tardiness value Ti can be defined as monotonically increasing from zero to one within (di1 , di2 ) . Consequently for each job J i , ‘tardiness’ is expressed with a fuzzy membership function ~ Ti . In practical situations, some jobs turn out to be more important than others. If information about relative preferences of the decision maker is available, scheduling systems should be able to integrate the notion of relative importance into the construction of the schedules. In order to do so, weights wi expressing the relative importance of jobs J i are introduced. It should be noticed that the weight assignments may not be stable over time but allow changes. A solution technique must be able to reschedule jobs, taking into consideration the new preferences of the decision maker. The overall performance measure integrating the relative importance of the jobs and considering the fuzzy ~ definition of the due dates can then for each solution x be expressed by the function c1 ( x) = ∑ wi Ti , and the overall objective is to find a schedule x that minimises c1 ( x) . We refer to this optimality criterion with c1 as another criterion c2 will be additionally introduced in section 4.
3. INVESTIGATION OF LOCAL SEARCH NEIGHBOURHOOD STRUCTURES A solution x of the problem can be represented by a single job permutation π = {π 1 ,..., π n } which is decoded into an active schedule, following the decoding proposed by Giffler, Thompson (1960). In an active schedule it is not possible to execute a task earlier, for example by changing the sequence on a certain machine, without delaying another task. As it can be shown that for performance measures like c1 ( x) at least one of the optimal schedules is an active schedule (Brucker 2001), we restrict the search to the set of active schedules. The decoding procedure of the job permutation π successively schedules all tasks, starting with the tasks that can possibly be scheduled first with respect to their release dates and predecessors. It may occur that tasks compete for execution during an overlapping time period on the same machine. In this case, conflict resolution is done with respect to the position of the corresponding jobs in the sequence π , and jobs with lower position indices are preferred over jobs with higher ones. There are a number of papers which investigate different representation and decoding for job shop problems (Bierwirth 1995, Mattfeld 1996, Bruns 1997). Due to technical requirements the jobs share similar machine sequences, and therefore a single permutation has been chosen in the studied scheduling problem. Three neighbourhood definitions (Reeves 1999) have been investigated within the hillclimbing framework described in figure 3. Starting with a randomly generated initial solution x, the neighbours of x, denoted by nh( x) , are generated and the best alternative among them is selected for further investigation, following accordingly a best-improvement strategy. It may be worth to mention that a first-improvement strategy which takes the first found solution that is better than the current one has been tested, too, however leading to inferior results. This strategy has therefore not been further investigated. Initialize Generate random initial solution x Repeat Generate nh( x) If ∃x'∈ nh( x ) | c1 ( x' ) < c1 ( x ) then Set x ← x ' Otherwise stop Figure 3: Single criterion hillclimbing framework The neighbourhood operators in detail are defined as follows:
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A Fuzzy Scheduling Problem with Dynamic Job Priorities and an Extension to Multiple Criteria •
nhBSH (x) Backward shift (BSH). Removes a job π i from the job sequence and inserts it before a preceding job π j ( j < i ).
•
nhFSH (x ) Forward shift (FSH). Removes a job π i from the job sequence and inserts it after a succeeding job π j ( j > i ).
•
nhEX (x) Exchange (EX). Exchanges the position of two jobs with each other.
The well known neighbourhood operator ‘Adjacent Pairwise Exchange’ (APEX), which allows only exchanges of adjacent jobs π i , π i +1 , has not been included in the analysis as previous investigations report comparably inferior results (Reeves 1999). Furthermore, it is valid to state that the mapping nhAPEX (x ) defines a subset of any of the described neighbourhood search operators. All neighbourhoods lead to a neighbourhood size of identical cardinality. For each alternative x, for n jobs a n(n − 1) alternatives are associated in the ‘neighbourhood’. It has to be pointed out that the number of 2 neighbourhoods do not result into entirely disjunct sets, as nhBSH ( x) ∩ nhFSH ( x ) ∩ nhEX ( x ) ≠ ∅ . The number of elements included in the intersection of the operators is however rather small (n − 1) , and we therefore conclude that a comparative analysis is possible and justified. 3.1 Definition of benchmark instances and test setup A variety of benchmark instances for job scheduling problems exist (Beasley 1990, Taillard 1993, Beasley 1996). The particular case of scheduling problems with fuzzy due dates addressed here is however comparably less often considered. In order to obtain quantitative results, we created 50 benchmark instances of shop scheduling problems with fuzzy due dates based on the proposal of Demirkol et al. (1998), as several advantages arise from using benchmark instances, instead of testing algorithms on a number of real world data sets. It gave us the possibility of deriving more stable conclusions based on results from several instances with different characteristics. Also it is possible to control the problem instance characteristics which are derived from the observed real world case within defined boundaries. The described methodology for generating job scheduling problems with due dates has been extended to fuzzy ~ due dates. The fuzzy due date Di = (d i1 , d i2 ) of each job i is computed such that d i1 lies within the interval
µR R 2 1 1 µ − 2 , µ + 2 and d i is within 1.05di ,1.1di . Here, µ is µ = (1 − T )nO with T being the expected proportion of tardy jobs, R being a due date range parameter, n the number of jobs, and O being the expected processing time of the tasks. The processing times of the tasks have been derived from a discrete uniform distribution, with lower and upper values based on the observed real world data from Sherwood Press. Parameters T = 0.5 and R = 0.5 have been set, leading to test instances with tight fuzzy due dates and an expected percentage of late jobs of 50%. The expected operation time O=960 minutes of the jobs was based on the analyzed real world data. The instances comprise 20 jobs that have to be processed on 10 machines. A nonnegative weight wi has been assigned to each job such that
∑ wi = 1 , modelling the relative importance
of the jobs given by the decision maker. Each problem instance has been solved with each neighbourhood structure in 100 test runs and average objective values of the weighted fuzzy tardiness have been computed. To investigate the behaviour of the search strategies in dynamic environments, the weights wi of the jobs have been altered within each test run three times after each 5,000 iterations, leading to a total of 20,000 evaluated solutions. The search had accordingly to continue with the changed priorities and continue with the current best found alternative. It should be noticed, that the fitness landscape of the problem instances with altering weight settings is partially linked. Good solutions given a certain importance of the jobs may also be of high quality with different weights. However, this does not necessarily have to be true for all alternatives, as tardy jobs in the past may be prioritized comparably higher in the future and therefore diminish the quality of the schedule. In practice, priorities do change, but in most cases they do not lead to an entirely new configuration of the search space.
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Decision Support in an Uncertain and Complex World: The IFIP TC8/WG8.3 International Conference 2004 3.2 Results of the neighbourhood search The results show, that the EX-neighbourhood is able to produce superior schedules to the BSH one in most instances. On the other hand it is observed that the BSH is superior to FSH. It converges faster and to local optima of lower objective value. To illustrate this result in one example, figure 4 plots the average weighted fuzzy tardiness for one problem instance. 0.9 0.8
c1
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Figure 4: Average fuzzy tardiness results for one problem instance It can be observed, that the changes of the weights after 5,000, 10,000 and 15,000 evaluations distract the search as expected. Improvements are however again obtained within a few iterations afterwards, and again EX leads to comparably better results.
Outperformance in % of all problem instances
The question remains however, if a general conclusion is possible upon the gathered data, as the computed average weighted fuzzy tardiness varies over the problem instances, and the differences between the neighbourhood structures are in some cases not as illustrative as shown in figure 4. We therefore determined for each problem instance and each evaluation step if the difference between the average fuzzy weighted tardiness for different neighbourhoods is significant given a significance level of 0.99, taking into consideration the variance of the results over the test runs for each instance. Figure 5 shows the aggregation of significance tests over all 50 problem instances. 100% 90% 80% 70% 60%
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Figure 5: Aggregated significance test results over all problem instances After 1,000 evaluations, the EX operator shows a significantly better performance in around 40% of the problem instances. Similar to the results shown in figure 4, changes of the preference structure reduce the quality of the so far best found schedule and accordingly the advantage of EX over the other neighbourhood
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A Fuzzy Scheduling Problem with Dynamic Job Priorities and an Extension to Multiple Criteria definitions. However, the relative advantage of EX is afterwards increasing even more, leading to significantly best results in around 90-100% of the problem instances. Although the hillclimbing approach was able to improve the quality of the schedules within short time, it is possible to show that the search quickly gets stucked into local optima. We therefore kept found solutions with identical objective values c1 ( x) during the search and computed the average number of these solutions for the 50 problem instances. Figure 6 shows the increasing number of solutions with identical quality. 2000 1750 Size of plateau
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750 500 250 0 0
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Figure 6: Average size of the fitness plateaus We conclude, that due to the definition of the objective function c1 ( x) , the value of the membership function is equal to 1 for all tardy jobs. Therefore fitness plateaus exist in the search space which increase the difficulty for the neighbourhood search algorithms in identifying good schedules for further improvements.
4. AN EXTENSION TO MULTIPLE CRITERIA 4.1 Reformulation of the problem To overcome the difficulties observed in section 3.2, we reformulate the problem by introducing an additional criterion, following a similar approach of Knowles et al. (2001). In addition to criterion c1 ( x) a second criterion c2 ( x) has been proposed, based on a tardiness penalty Ti ' for job J i (Vlach 2000). As shown in figure 7 the tardiness definition now includes values greater than 1 for completion times greater than d i2 , and the overall tardiness penalty c2 ( x) is measured using a weighted sum c2 ( x ) =
1
∑wT ' . i i
Ti '
~ Di
0 di1
d i2
Figure 7: Definition of the tardiness penalty Ti ' as a second criterion It is obvious, that the two criteria c1 ( x) and c2 ( x) lead to similar but not identical quality assignments for the schedules. The overall objective can now be formulated as the vector optimisation problem ‘minimise’ C ( x) = (c1 ( x ), c 2 ( x )) . The minimisation is understood in the sense of Pareto-optimality, and the goal is thus to find all efficient solutions according to the definitions 1 and 2 given below.
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Decision Support in an Uncertain and Complex World: The IFIP TC8/WG8.3 International Conference 2004 Definiton 1 [Pareto-dominance]. An objective vector C ( x ) is said to dominate C ( x ') if and only if ∀i ci ( x) ≤ ci ( x ') ∧ ∃i ci ( x) < ci ( x ') . The domination of a vector C ( x ) over a vector C ( x ') is here denoted with C ( x) p C ( x' ) . Definiton 2 [Efficiency of solutions, Pareto set]. A solution x is said to be efficient in the sense of Paretooptimality if and only if ¬∃x' | C ( x ' ) p C ( x) . The set of all efficient solutions is called the Pareto set P. Using the proposed reformulation of the initial problem, we aim to add additional information to the search. As the optimal solution for the single-criterion version of the problem is always element of P, by solving the multi criteria extension we are implicitly solving the single criterion problem which is still our primary objective. 4.2 An extension of the hillclimbing framework and extended tests The hillclimbing framework presented in figure 3 has been extended to handle the existence of multiple nondominated solutions by integrating a set P approx containing best found solutions in the sense of definition 1 and 2. As shown in figure 8, the local search now performs additional steps, namely maintaining P approx and randomly selecting an element x ∈ P approx for generating further neighbouring solutions. Within the maintenance of P approx , dominated alternatives are simply removed and new found nondominated alternatives are added instead. Intialize Generate random intial solution x Set P approx = {x} Repeat Randomly select x ∈ P approx Generate nh( x) Update P approx with nh( x) Until stopping condition is met Figure 8: Multi criteria hillclimbing framework The same 50 problem instances proposed in section 3.1 have been solved with the multi criteria extension MOEX, using the observed best neighbourhood structure EX, and average values of sum of the weighted fuzzy tardiness c1 ( x) have been computed for 100 test runs. The performance of the multi criteria extension is judged only upon the quality of the c1 ( x) criterion, although the whole set of nondominated schedules is kept and improved in each iteration. Clearly, this creates a computational overhead compared with the single criterion approach that however might be justified by superior results.
5. RESULTS The results of the MO-EX approach have been directly compared with the best results obtained in section 3. Again, average values of c1 have been compared with statistical significance at a level of 0.99, and the number of problem instances in which one approach significantly outperforms the other have been calculated for each evaluation step. As figure 9 shows, the multi criteria reformulation is able to outperform the EX hillclimber after around 2,000 evaluations. The opposite result is true prior to 2,000 evaluations are reached. It can be seen, that the performance of the approaches are reversed after the weight settings are changed. In all tested cases, the single criterion EX neighbourhood was able to improve comparably faster, while MO-EX starts to outperform EX after around 2,000 evaluations in more test instances. This behaviour appears to be quite stable, unlike the growing advantage of EX over the other neighbourhood operators as explained in section 3.2. The main advantage of the multi criterion extension lies in the improved capability of distinguishing between locally optimal solutions with respect to c1 . While the single criterion hillclimbing technique tends to remain on a fitness plateau, the additional information introduced by criterion c2 helps to guide the search towards superior regions over time by the expense of additional computations and a slower convergence.
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A Fuzzy Scheduling Problem with Dynamic Job Priorities and an Extension to Multiple Criteria
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Figure 9: Comparison between multi criteria and single criterion approach, aggregated significance test results over all problem instances The guiding effect of the multi criteria extension can be demonstrated by analysing the cardinality of P approx during the search as shown in figure 10. While the number of nondominated solutions kept in P approx does grow over time, the increase is, compared to the number of reached solutions in the plateaus for the single criterion search presented in figure 6, significantly smaller.
Number of nondominated solutions
12 10 8 MO-EX
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Figure 10: Average number of elements in P approx We conclude that in real time scheduling systems where the available computation time is limited, a very fast improving hillclimbing method may be preferable over a later outperforming multi criterion reformulation. However given the necessary time, the presented bicriteria extension leads to better results even in terms of only one criterion, c1 .
6. CONCLUSIONS Experiments on the artificially generated data sets with characteristics similar to observed data sets from a realworld scheduling environment were performed with two aims. A study of a scheduling problem with fuzzy due dates and dynamically changing preferences has been presented, based upon an investigation of a real world problem. A fuzzy membership function has been used in order to describe the imprecise due dates and tardiness.
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Decision Support in an Uncertain and Complex World: The IFIP TC8/WG8.3 International Conference 2004 Firstly, the effectiveness of different neighbourhood search operators within hill climbing has been investigated. In was observed that some neighbourhood structures have a significant advantage over others. In addition dynamic changes of the preference structure were studied and it was observed that these changes do have an effect on the search which has to continue from its current position (best solution found so far). The advantage of good search operators over others grows over time, independent from occurring changes. However, the simple local search quickly gets stuck in local optima, and especially the problem of the existence of a large number of solutions with identical objectives values makes the further search difficult. Further improvements were achieved by extending the single criterion problem to a multiple criteria problem, successfully overcoming local optimality by the expense of additional computations. The results show that multi objective extensions may help guiding the search within the search space, overcoming local optimality. The presented model will be further investigated on real-world data from Sherwood Press Corporation using the insights gained in the experiments on artificially generated data sets. The experiments showed that the performance of the different search algorithms depends on the allowed number of computations. This has to be taken into consideration when applying the search algorithms to the real world problem. Especially in dynamic environments, where relative importance of jobs may change over time, a fast reacting search algorithm is needed in order to adapt quickly to changes of the situation.
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A Fuzzy Scheduling Problem with Dynamic Job Priorities and an Extension to Multiple Criteria Ishii, H., Tada, M., & Masuda, T. (1992) Two scheduling problems with fuzzy duedates. Fuzzy Sets and Systems, 46, 339-347. Knowles, J.D., Watson, R.A., & Corne, D.W. (2001) Reducing local optima in single-objective problems by multi-objectivization, in: E. Zitzler, K. Deb, L. Thiele & C.A. Coello Coello, Evolutionary Multi-Criterion Optimization: Proceedings of the First International Conference EMO 2001 (pp. 269-283), Berlin: Springer. Lee, H.T., Chen, S.H., & Kang, H.Y. (2002) Multicriteria scheduling using fuzzy theory and tabu search. International Journal of Production Research, 40 (5), 1221-1234. Mattfeld, D.C. (1996) Evolutionary Search and the Job Shop: Investigations on Genetic Algorithms for Production Scheduling, Heidelberg: Physica. McCahon, C.S., Lee, E.S. (1992) Fuzzy job sequencing for flow shop. European Journal of Operational Research, 62, 294-301. Pinedo, M. (2002) Scheduling: Theory, Algorithms, and Systems, Upper Saddle River: Precentice Hall. Prade, H. (1979) Using fuzzy set theory in a scheduling problem: A case study. Fuzzy Sets and Systems, 2 (2), 153-165. Reeves, C.R. (1999) Landscapes, operators and heuristic search. Annals of Operations Research, 86, 473-490. Rinnooy Kan, A.H.G. (1976) Machine Scheduling Problems: Classification, complexity and computations, The Hague: Martinus Nijhoff. Sakawa, M., Kubota, R. (2000) Fuzzy programming for multiobjective job scheduling with fuzzy processing time and fuzzy duedate through genetic algorithms. European Journal of Operational Research, 120, 393407. Slany, W. (1996) Scheduling as a fuzzy multiple criteria optimization problem. Fuzzy Sets and Systems, 78, 197-222. Taillard, E. (1993) Benchmarks for basic scheduling problem. European Journal of Operational Research, 64, 278-285. Tsujimura, Y., Gen, M., & Kubota, E. (1995) Solving job-shop scheduling problem with fuzzy processing time using genetic algorithm. Journal of Japan Society for Fuzzy Systems, 7 (5), 1073-1083. Vlach, M. (2000) Single machine scheduling under fuzziness, in: R. Słowiński, M. Hapke, Scheduling Under Fuzziness (pp. 223-245), Heidelberg: Physica. Zadeh, L.A. (1965) Fuzzy sets. Information and Control, 8, 338-353.
ACKNOWLEDGEMENTS This research is supported by the Engineering and Physical Sciences Research Council (EPSRC), UK (Grant No. GR/R95319/01). We also want to thank our industrial collaborator the Sherwood Press Ltd, Nottingham, and two anonymous referees for their helpful comments.
COPYRIGHT Sanja Petrovic, Martin Josef Geiger © 2004. The authors grant a non-exclusive licence to publish this document in full in the DSS2004 Conference Proceedings. This document may be published on the World Wide Web, CD-ROM, in printed form, and on mirror sites on the World Wide Web. The authors assign to educational institutions a non-exclusive licence to use this document for personal use and in courses of instruction provided that the article is used in full and this copyright statement is reproduced. Any other usage is prohibited without the express permission of the authors.
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