# Optimized Risk-based CUSUM Control Charts Calculate Optimized Risk-based Univariate cumulative sum (CUSUM) Control Chart ```r rbcusumcc_opt(X, UC, C, n, T=5, se.shift=1, K_init= 0, LKL=0, UKL=6) ``` ## Arguments - `X`: vector of variable (numeric vector). Either can be simulated using data_gen or defined by using available data set. - `UC`: vector of measuerement error (numeric vector).Either can be simulated using data_gen or defined by using available previous information. - `C`: vector of decision costs (default value is vector of 1). - `n`: the sample size for grouping. For individual obervations use n=1). - `T`: A numeric value specifying the number of standard errors of the summary statistics at which the cumulative sum is out of control (The defualt value is 5). - `se.shift`: The amount of shift to detect in the process, measured in standard errors of the CUSUM statistics (default value is 1). - `K_init`: Set correction component to 0 by default (default value is 0) - `LKL`: Lower limit of K parameter (default value is 0) - `UKL`: Upper limit of K parameter (default value is 6) ## Returns - **cost0**: Total cost of a monitoring process - **cost1**: Total cost of correct acceptance related to a process monitoring - **cost2**: Total cost of decision error type 1 related to a process monitoring - **cost3**: Total cost of decision error type 2 related to a process monitoring - **cost4**: Total cost of correct reject related to a process monitoring - **LCLx**: Lower decision bound of CUSUM chart for a given data - **UCLx**: Upper decision bound of CUSUM control chart for a given data - **LCLy**: Lower decision bound of CUSUM chart for a given data with measurement uncertainity - **UCLy**: Upper decision bound of CUSUM chart for a given data with measurement uncertainity - **cusumx**: Real values of CUSUM statistic for a given data - **cusumy**: Observed values of CUSUM statistic for a given data with measurement errors - **reall**: Below target real values of CUSUM statistic for a given data - **realu**: Above target real values of CUSUM statistic for a given data - **obsl**: Below target observed values of CUSUM statistic for a given data with measurement errors - **obsu**: Below target observed values of CUSUM statistic of a given data with measurement errors - **Kopt**: Optimal 'K' paramater of a risk-based CUSUM control chart ## References Katona, A. I., Saghir, A., Hegedűs, C., & Kosztyán, Z. T. (2023). Design of Risk-Based Univariate Control Charts with Measurement Uncertainty. IEEE Access, 11, 97567-97573. Kosztyán, Z. T., & Katona, A. I. (2016). Risk-based multivariate control chart. Expert Systems with Applications, 62, 250-262. ## Author(s) Aamir Saghir, Attila I. Katona, Zsolt T. Kosztyan* e-mail: kzst@gtk.uni-pannon.hu ## See Also `data_gen`, `rbcc`, `rbcc_opt`,`rbcusumcc`, `rbewmacc`,`rbewmacc_opt`, `rbmacc`, `rbmacc_opt`, `rbmcc`, `rbmcc_opt`, `plot.rbcc`, `summary.rbcc`. ## Examples ```r # Data generation for vector X mu_X <- c(0) # Define data mean. va_X <- c(1) # Define data standard deviation. sk_X <- c(0) # Define data skewness. ku_X <- c(3) # Define data kurtosis. obs <- 200 # Total number of observations of a process. X <- data_gen (obs, mu_X, va_X, sk_X, ku_X) # Data generation for measurement error vector UC mu_UC <- c(0) # Define mean of measurement errors. va_UC <- c(1) # Define standard deviation of measurement errors. sk_UC <- c(0) # Define skewness of measurement errors. ku_UC <- c(3) # Define kurtosis of measurement errors. n <- 1 # For individual obervations use n=1 UC <- data_gen(obs,mu_UC, va_UC, sk_UC, ku_UC) C <- c(1,1,1,1) # Define a vector of decision costs. H <- rbcusumcc(X, UC, C, n, T=5, se.shift=1, K=5) # for subgroups of size 1 plot(H) # plot RBCC # optimal risk-based CUSUM control chart H_opt <- rbcusumcc_opt(X, UC, C, n, T=5, se.shift=1, K_init= 0, LKL=0, UKL=6) # with vector of proportional decision costs C <- c(1, 5, 60, 5) # vector of decision costs H <- rbcusumcc(X, UC, C, n, T=5, se.shift=1, K=5) H_opt <- rbcusumcc_opt(X, UC, C, n, T=5, se.shift=1, K_init= 0, LKL=0, UKL=6) # optimal risk-based CUSUM control chart summary(H_opt) # summarize the reults # Plot of traditional and optimal risk based cusum control charts plot(H_opt) ```

rbcusumcc_opt function