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27
Fuzzy Constraints in JobShop Scheduling
 Journal of Intelligent Manufacturing
, 1995
"... : This paper proposes an extension of the constraintbased approach to jobshop scheduling, that accounts for the flexibility of temporal constraints and the uncertainty of operation durations. The set of solutions to a problem is viewed as a fuzzy set whose membership function reflects preference. ..."
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Cited by 56 (9 self)
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: This paper proposes an extension of the constraintbased approach to jobshop scheduling, that accounts for the flexibility of temporal constraints and the uncertainty of operation durations. The set of solutions to a problem is viewed as a fuzzy set whose membership function reflects preference. This membership function is obtained by an egalitarist aggregation of local constraintsatisfaction levels. Uncertainty is qualitatively described is terms of possibility distributions. The paper formulates a simple mathematical model of jobshop scheduling under preference and uncertainty, relating it to the formal framework of constraintsatisfaction problems in Artificial Intelligence. A combinatorial search method that solves the problem is outlined, including fuzzy extensions of wellknown lookahead schemes. 1. Introduction There are traditionally three kinds of approaches to jobshop scheduling problems: priority rules, combinatorial optimization and constraint analysis. The first kind ...
A Flow Shop with Compatibility Constraints in a Steelmaking Plant
 IN ZWEBEN AND FOX (EDS) INTELLIGENT SCHEDULING
, 1994
"... We present a scheduling methodology for applications where the generation of schedules is constrained by antagonistic and vague knowledge. Besides temporal and capacity constraints, compatibility constraints between consecutive jobs are managed. We model the vague constraints and uncertain data b ..."
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Cited by 15 (12 self)
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We present a scheduling methodology for applications where the generation of schedules is constrained by antagonistic and vague knowledge. Besides temporal and capacity constraints, compatibility constraints between consecutive jobs are managed. We model the vague constraints and uncertain data by fuzzy set theory. The importance of single jobs and and the difficulty to schedule them is defined on the different constraints and is used to control the generation of schedules. A preliminary schedule is generated by considering the important jobs and those that are difficult to schedule first. Easy or not so important jobs are scheduled later. Finally, the achieved schedule is "repaired" until a schedule is found that achieves a given level of satisfaction. Since the goodness of solutions is rated by fuzzy sets, robust schedules achieve better evaluations than weak schedules. However, if no robust solution is found constraints will be relaxed. This methodology is appropriate for applications in process engineering where uncertain knowledge is dominant. We explain the methodology with a case study from a steelmaking plant for highgrade steel.
Gradual Numbers and their Application to Fuzzy Interval Analysis
, 2008
"... We introduce a new way of looking at fuzzy intervals. Instead of considering them as fuzzy sets, we see them as crisp sets of entities we call gradual (real) numbers. They are a gradual extension of real numbers, not of intervals. Such a concept is apparently missing in fuzzy set theory. Gradual num ..."
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Cited by 15 (4 self)
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We introduce a new way of looking at fuzzy intervals. Instead of considering them as fuzzy sets, we see them as crisp sets of entities we call gradual (real) numbers. They are a gradual extension of real numbers, not of intervals. Such a concept is apparently missing in fuzzy set theory. Gradual numbers basically have the same algebraic properties as real numbers, but they are functions. A fuzzy interval is then viewed as a pair of fuzzy thresholds, which are monotonic gradual real numbers. This view enable interval analysis to be directly extended to fuzzy intervals, without resorting tocuts, in agreement with Zadeh’s extension principle. Several results show that interval analysis methods can be directly adapted to fuzzy interval computation where end points of intervals are changed into left and right fuzzy bounds. Our approach is illustrated on two known problems: computing fuzzy weighted averages, and determining fuzzy floats and latest starting times in activity network scheduling.
Fuzzy set theory applications in production management research: a literature survey
 Journal of Intelligent Manufacturing
, 1998
"... ..."
On the sure criticality of tasks in activity networks with imprecise durations
 IEEE Transactions on Systsems, Man, and Cybernetics PartB 2002
"... Abstract—The notion of the necessary criticality (both with respect to path and to activity) of a network with imprecisely defined (by means of intervals or fuzzy intervals) activity duration times is introduced and analyzed. It is shown, in the interval case, that both the problem of asserting whet ..."
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Cited by 9 (5 self)
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Abstract—The notion of the necessary criticality (both with respect to path and to activity) of a network with imprecisely defined (by means of intervals or fuzzy intervals) activity duration times is introduced and analyzed. It is shown, in the interval case, that both the problem of asserting whether a given path is necessarily critical and the problem of determining an arbitrary necessarily critical path (more exactly, a subnetwork covering all the necessarily critical paths) are easy. The corresponding solution algorithms are proposed. However, the problem of evaluating whether a given activity is necessarily critical does not seem to be such. Certain conditions are formulated which in some situations (but not in all possible) allow evaluating the necessary criticality of activities. The results obtained for networks with interval activity duration times are generalized to the case of networks with fuzzy activity duration times. Two effective algorithms of calculating the degree of necessary criticality of a fixed path, as well as an algorithm of determining the paths that are necessarily critical to the maximum degree, are proposed. Index Terms—Fuzzy CPM, possibility and necessity, project management and scheduling. I.
Criticality in the Network with Imprecise Activity Times
 In Proceedings of 8th International Conference IPMU
"... Abstract. A review of the results obtained in the area of fuzzy network analysis is presented. The main approaches to the concept of criticality in a network with fuzzy activity times are described and classified. Against the background of this review some new results, obtained by the authors recent ..."
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Cited by 3 (1 self)
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Abstract. A review of the results obtained in the area of fuzzy network analysis is presented. The main approaches to the concept of criticality in a network with fuzzy activity times are described and classified. Against the background of this review some new results, obtained by the authors recently, are presented. The paper is an extended version of the work presented in IPMU’2000 (see [8]). 1
A Fuzzy Scheduling Problem with Dynamic Job Priorities and an Extension to Multiple Criteria 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 th ..."
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Cited by 2 (1 self)
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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.
Fuzzy critical path method in intervalvalued activity networks
 Int. J. Pure Appl. Sci. Technol
"... Abstract: Critical Path method (CPM) is widely used in project scheduling and controlling. In conventional project scheduling problem, the crisp numbers are used for the activity times. But in reality, in an imprecise and uncertain environment, it is an unrealistic assumption. To represent the uncer ..."
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Cited by 1 (0 self)
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Abstract: Critical Path method (CPM) is widely used in project scheduling and controlling. In conventional project scheduling problem, the crisp numbers are used for the activity times. But in reality, in an imprecise and uncertain environment, it is an unrealistic assumption. To represent the uncertainty involved we have considered the intervalvalued numbers to represent the activity times. In this paper, we compute project characteristics such as earliest times, latest times and float times in terms of intervals. In this method, we introduce a new approach to find latest times by removing negative intervals times which can be generated by other methods. Also a new fuzzy critical path method based on the fuzzy theory is developed to solve the project scheduling problem under the fuzzy environment. Through a numerical example, calculation steps in this method and the results are presented.
Research Issues and Challenges in Fuzzy Scheduling
, 1994
"... Scheduling is an activity fraught with fuzziness. The prevalence of Murphy's Law in real life situations, noise infected measurements, vague objectives of different importance, where compromises between antagonistic optimization criteria are generally allowed, and the difficulty of estimating p ..."
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Cited by 1 (0 self)
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Scheduling is an activity fraught with fuzziness. The prevalence of Murphy's Law in real life situations, noise infected measurements, vague objectives of different importance, where compromises between antagonistic optimization criteria are generally allowed, and the difficulty of estimating process times all contribute to make the construction of predictive schedules a complex task. Both authors have gained indepth practical experience in the field of fuzzy scheduling. Application domains of implemented systems ranged from production scheduling of mechanical and hydraulic presses [18, 19] to finegrained scheduling of highgrade steel production [25]. In addition to fundamental and advanced theoretic research on fuzzy scheduling, Slany's Ph.D. thesis [25] also contains an annotated bibliography with more than 400 references on fuzzy scheduling. The present paper takes a step back from actual implementations to introduce the main aspects of fuzzy scheduling to a broader audience...