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  We consider criticality in project networks having imprecise activity duration times. It is well known that finding all possibly critical paths of an imprecise project network is an NP-hard problem. Here, based on a method for finding critical paths of crisp networks by using only the forward recursion of critical path method, for the first time an algorithm is proposed which can find all possibly critical paths of interval-valued project networks. The proposed algorithm considers interactivity among paths which has not been yet considered in the fuzzy project scheduling literature. The extension of the proposed algorithm to the fuzzy network calculates criticality degrees of activities and paths of projects without any need to enumerate all project paths. Although algorithms for calculating criticality degrees in fuzzy networks have been previously proposed, despite the fact that they mostly consider a specific type of fuzzy numbers as activity duration times, the exiting algorithms do not discriminate possibly critical paths before calculating the criticality degrees. The computational experience on a series of well-known project samples confirms the algorithm to be remarkably more efficient than similar algorithms for fuzzy networks.

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Batch scheduling is among the important problems in industrial engineering and has been widely attendant in practical applications. Clustering is the set of observation assignment into some subsets so that the observations in the same cluster are similar in some sense and the similarity of generated clusters is very low. Clustering is considered as one of the approaches in unsupervised learning and a common technique for statistical data analysis which has been applied in many fields, including machine learning, data mining and etc. This paper studies a case study in Iran Puya company (as a home appliance maker company in Iran). In the production line of refrigerator of the current company, a cutting machine is identified as a bottleneck that can process several iron plates simultaneously. In this regard a good scheduling on this cutting machine improves the effectiveness of production line in terms of cost and time. The objective is to minimize the total tardiness and maximizing the job values when the deteriorated jobs are delivered to each customer in various size batches. Based on these assumptions a mathematical model is proposed and two hybrid algorithms based on simulation annealing and clustering methods are offered for solving it and the results are compared with the global optimum values generated by Lingo 10 software. Based on the effective factors of the problem, a number of sensitivity analyses are also implemented including number of jobs and rate of deterioration. Accordingly, the running time grows exponentially when the number of jobs increases. However the rate of deterioration could not affect the running time. Computational study demonstrates that using clustering methods leads an specified improvements in total costs of company between 15 to 41 percent.