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Published in Harald Paganetti, Proton Therapy Physics, 2018
A probability density function (PDF) expresses the relative likelihood that a variable will have a certain value, as determined by a random number. In mathematical terms, PDF(x) represents the probability of finding the random variable x′ within dx of a given value x. Considering a PDF(x) defined for a certain interval [a,b], the goal is to sample randomly between a and b according to the PDF(x) (a and b being finite and b > a). In case of a continuous uniform distribution, one obtains PDF(x) = 1/(b − a) for a ≤ x ≤ b and 0 for x < a and x > b. Assuming that random numbers Ri generated by a random number generator are uniformly distributed in [0,1], one can obtain random events via PDF(x) dx = dCDF. Thus, CDF(x)=∫axPDF(x′)dx′
Batching and scheduling for a single-machine flexible machining cell with multi-fixturing pallets and controllable processing times
Published in International Journal of Production Research, 2020
Jeong-Hoon Shin, Jae-Min Yu, Hyoung-Ho Doh, Hyung-Won Kim, Dong-Ho Lee
For the test, we generated 60 instances for 300 parts with different loading/unloading times, arrival times, processing times and due-dates. Since the FMC is being developed, we could not obtain the detailed operation data such as part loading (unloading) times, part arrival times, part processing times, part processing costs and due-dates. Hence, we generated them randomly with interval data provided by project partners. The detailed data were generated as follows. The part loading (unloading) times were set to 5 for the first 20 instances, 10 for the next 20 instances and 15 for the last 20 instances. The arrival times of a half of the parts were set to 0 randomly and those for the others were generated from , where denotes the discrete uniform distribution with range . The available part processing times and costs were generated from and , respectively. Also, the due-dates were generated from where denotes the minimum available processing time of part . Also, RDD (range of due-dates) and TF (tardiness factor) were randomly generated from and , respectively. Here, denotes the continuous uniform distribution with range . Note that the due date of a part was set to 0 if it was generated as a negative value.